Ann. Henri Poincar´e 11 (2010), 1–21 c 2010 Springer Basel AG 1424-0637/10/010001-21 published online May 11, 2010 DOI 10.1007/s00023-010-0033-8
Annales Henri Poincar´ e
On Hawking’s Local Rigidity Theorem for Charged Black Holes Pin Yu Abstract. We show the existence of the Hawking vector field in a full neighborhood of a local, regular, bifurcate, non-expanding horizon embedded in a smooth Einstein–Maxwell space–time without assuming the underlying space–time is analytic. This extends a result of Friedrich et al. (Commun Math Phys 204:691–707, 1971), which holds in the interior of the black hole region. Moreover, we also show, in the presence of an additional Killing vector field T which is tangent to the horizon and not vanishing on the bifurcate sphere, then space–time must be locally axially symmetric without the analyticity assumption. This axial symmetry plays a fundamental role in the classification theory of stationary black holes.
1. Introduction Let (M, g, F ) be a smooth and time oriented Einstein–Maxwell space–time of dimension 3 + 1 with electromagnetic field F . Let S be a smoothly embedded space-like 2-sphere in M and H+ , H− be the corresponding null boundaries of the causal future and the causal past of S. We also assume that both H+ and H− are regular, achronal, null hypersurfaces in a neighborhood O of S. The triplet (S, H+ , H− ) is called a local, regular bifurcate horizon in O. The main result of the paper asserts if (S, H+ , H− ) is non-expanding (see Definition 2.1), then it must be a Killing bifurcate horizon. More precisely, we have the following theorem: Theorem 1.1. Given a local, regular, bifurcate, non-expanding horizon (S, H+ , H− ) in a smooth and time oriented Einstein–Maxwell space–time (O, g, F ), there exists a neighborhood O ⊂ O of S and a non-trivial Killing vector field K in O , which is tangent to the null generators of H+ and H− . Moreover, the Maxwell field F is also invariant with respect to K, i.e., LK F = 0.
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The vector field K is called the Hawking vector field in the literature. Its existence is already known (see [6]) under the assumption that the space– time is real analytic. In the work of Friedrich et al. [5], the authors showed, by solving wave equations, the existence of the Hawking vector field K without the analyticity assumption, but K could only be constructed inside the domain of dependence of H+ ∪H− due to the fact that the corresponding wave equations are not well-posed outside this region. The new ingredient in this paper is to extend the Hawking vector field K to a full neighborhood of the bifurcate sphere S, without making any additional regularity assumptions on the underlying space–time (M, g). We shall achieve this goal following an idea of Alexakis et al., who proved a corresponding result for vacuum space-times in [2]. We also prove the following theorem: Theorem 1.2. Given a local, regular, bifurcate and non-expanding horizon (S, H+ , H− ) in a smooth and time oriented Einstein–Maxwell space–time (O, g, F ). Assume there is a Killing vector field T tangent to H+ ∪ H− which does not vanish identically on S. Then there is a neighborhood O ⊂ O of S, such that we can find a rotational Killing vector Z in O , i.e., the orbits of Z are closed. Moreover, [Z, T ] = 0. If in addition LT F = 0, then LZ F = 0. In reality, when one tries to prove global rigidity theorems for stationary space-times, one does not make the non-expanding assumption on the horizon H+ ∪ H− , since it is well-known that the non-expansion is a consequence of the fact that the Killing vector field T is tangent to H+ ∪ H− , see [6]. So the first theorem will produce a Hawking vector field K in a full neighborhood of S. The rotational vector field given by Theorem 1.2 then can be written as a linear combination of T and K, i.e., there exists a constant λ such that the one parameter group of diffeomorphism on M generated by the vector field Z = T + λK is a rotation with a period t0 . The part LZ F = 0 in the theorem follows immediately. In the proof, we will focus on the geometric construction of Z. We show that the period t0 is exactly the period of rotations generated by T on the bifurcate sphere S, while to determine λ, it suffices to know the action of T and K on a particular null geodesic on H+ ∪ H− , see the proof for more details. Once more, under the restrictive additional assumption of analyticity of the space–time (M, g), this second theorem is also known for Einstein vacuum space-times. It is usually called Hawking’s rigidity theorem, see [6], which asserts that under some global causality, asymptotic flatness and connectivity assumptions, a stationary analytic black hole whose horizon is non-degenerate must be axially symmetric. In the smooth category, one can find a proof in [2] based on the idea that, under a suitable conformal rescaling of null generators on the bifurcate sphere, the level sets of the affine parameters of the null generators on the horizon should represent the integrable surface ruled by the closed rotational orbits.
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These two theorems play an important role in the classification theory of stationary black holes, since they reduce the classifications to the cases which are covered by the well-known uniqueness theorems for electro-vac black holes in general relativity, see [3,4,6,9,11]. For more a historical account of this issue, we refer the reader to the paper [2]. We now describe the main ideas of the proofs. The first step towards the proof of Theorem 1.1 is to construct the Hawking vector field K. Since K is a Killing vector field, K satisfies the following covariant linear wave equations: g Kα = −Rα β Kβ
(1.1)
where Rαβ is the Ricci curvature tensor for the Lorentzian metric g. We hope to reconstruct K by solving this wave equation. This is precisely the strategy used in [5]. According to a famous result of Rendall (see [10]), the equation can be solved in the domain of dependence if initial data is prescribed on the characteristic hypersurfaces, see [10] for the proof. So one needs to find the correct initial data for (1.1) on H+ ∪ H− . The choice of the data can be rediscovered by the following heuristic argument: because K is Killing, the restriction of K to a geodesic should be a Jacobi field, so one may guess the initial data on H+ should be the non-trivial parallel Jacobi field uL where L is a null geodesic generator on H+ and u is the corresponding affine parameter, i.e., L(u) = 1; another way to guess the initial data is to compute the explicit formula for the exact Kerr or Kerr–Newman families of black holes. The second step is to extend the vector field K to the bad region, i.e., the region with is not covered by the domain of dependence. While the Cauchy problem for (1.1) is ill-posed on the bad region, solving (1.1) will no longer work. We have to rely on the new techniques used in [2]. A careful calculation shows K also solves an ordinary differential equation (ODE) which is well-posed in the ill-posed region for (1.1). So one can extend K into the bad region by solving this ODE. This is the way we construct K in a full neighborhood of S. Notice that although K is constructed, we still need to check that K is Killing since to derive Eq. (1.1) and the ODE we have already ignored a certain amount of information. One turns to proving the one parameter group φt generated by K acts isometrically. We need to show that, for each small t, the pull-back metric φ∗t g must coincide with g, in view of the fact that they are both solutions of Einstein–Maxwell equations and coincide on H+ ∪ H− . Now the Carleman type uniqueness techniques come into play, see also the results of Alexakis [1], Alexakis et al. [2] and Ionescu and Klainerman [7,8]. The paper is organized as follows. In Sect. 2, we construct a canonical null frame associated with the bifurcate horizon (S, H+ , H− ) and derive a set of partial differential equations (PDE) for various geometric quantities, as consequences of the non-expansion condition and the Einstein–Maxwell equations; in Sect. 3, we give a self-contained proof of Theorem 1.1 in the domain of dependence of H+ ∪ H− , which is Proposition B.1 in [5]; in Sect. 4, based on the Carleman type estimates proved in [7,8], we extend the Hawking vector field to a full neighborhood of S which completes the proof of Theorem 1.1; the last section is devoted to a geometric proof of Theorem 1.2.
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Notations The Greek indices α, β, γ, δ, ρ ranging from 1 to 4, and the Roman letters a, b, c from 1 to 2; one uses Dα to denote the covariant derivative Deα ; the curvature tensor is defined by Rαβγδ = g(Dα Dβ eγ −Dβ Dα eγ , eδ ), where Dα Dβ X = Dα (Dβ X)−DDα eβ X; repeated indices are always understood as subject to the Einstein summation convention; since during the proof of our main theorems, we will keep shrinking the open neighborhood O of S, we will keep denoting such neighborhoods by the same O for simplicity; we will often use the notation X Y whenever there exists some constant C so that X ≤ CY, C can depend on some given background metrics g and g and background fields F and F . To simplify the formulas, without losing information, we will use the ∗ notation. The expression A ∗ B is a linear combination of tensors, each formed by starting with A ⊗ B, using the metric to take any number of contractions. So A ∗ B should be treated as a quadratic expression in A and B, while the exact numerical coefficients are irrelevant.
2. Preliminaries We first set up the double null foliation in O. One can choose a smooth futuredirected null pair (L, L) along S with normalization g(L, L) = g( L, L) = 0,
g(L, L) = −1
+
such that L is tangent to H and L is tangent to H− . In a small neighborhood of S, we extend L along the null geodesic generators of H+ via parallel transport; we also extend L along the null geodesic generators of H− via parallel transport. So DL L = 0 and D L L = 0. We now define two optical functions u and u near S. The function u (respectively, u) is defined along H+ (respectively, H− ) by setting initial value u = 0 (respectively, u = 0) on S and solving L(u) = 1 (respectively, L(u) = 1). Let Su (respectively, Su ) be the level surfaces of u (respectively, u) along H+ (respectively, H− ). We define L (respectively, L) on each point of the hypersurface H+ (respectively, H− ) to be unique, future directed null vector orthogonal to the surface Su (respectively, Su ) passing though that point and such that g(L, L) = −1. The null hypersurface Hu− (respectively, Hu+ ) is defined to be the set of null geodesics initiating on Su ⊂ H+ (respectively, Su ⊂ H− ) in the direction of L (respectively, L). We require the null hypersurfaces Hu− (respectively, Hu+ ) to be the level sets of the function u (respectively, u). By this condition, u and u are extended into a neighborhood of S from the null hypersurface H+ ∪H− . Then we can extend both L and L into a neighborhood of S as gradients of the optical functions u and u: L = −gμν ∂μ u∂ν ,
L = −gμν ∂μ u∂ν .
Since u and u are null optical functions, we know g(L, L) = g( L, L) = 0
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while g(L, L) = −1 only holds on the null surface H+ ∪ H− . Moreover, we have L(u) = 1
on H+
and
L(u) = 1
on H− .
We define Suu = Hu+ ∩ Hu− . Using the null pair (L, L) one can choose a null frame {e1 .e2 , e3 = L, e4 = L} such that g(ea , eb ) = δab ,
g(ea , e3 ) = g(ea , e4 ) = 0,
a, b = 1, 2.
At each point p ∈ Suu ⊂ O, e1 , e2 form an orthonormal frame along the 2-surface Suu . It is easy to see that we have a lot of freedom to choose e1 and e2 . To make use of this point, we shall modify the frame by Fermi transport later. Recall the null second fundamental forms χ, χ and torsion ζ are defined on H+ ∪ H− via the given null pair (L, L): χab = g(Dea L, eb ),
χab = g(Dea L, eb ),
ζa = g(Dea L, L).
The traces of χ and χ are defined by trχ = δ ab χab and trχ = δ ab χab . Definition 2.1. We say that H+ is non-expanding if trχ = 0 on H+ ; similarly H− is non-expanding if trχ = 0 on H− . The bifurcate horizon (S, H+ , H− ) is called non-expanding if both H+ , H− are non-expanding. The non-expanding condition imposes a very strong restriction on the geometry of the Einstein–Maxwell space–time. We recall the Einstein–Maxwell equations: ⎧ R − 1 Rg = Tαβ ⎪ ⎪ ⎨ αβ 2 αβ D[α Fβγ] = 0 ⎪ ⎪ ⎩ Dα Fαβ = 0 where Tαβ = Fα μ Fβμ − 14 gαβ F μν Fμν is the energy–momentum tensor for the corresponding electromagnetic field. Since the dimension of the underlying manifold is four, the field theory is conformal, i.e., trT = 0. So by tracing the first equation in the system, we know the scalar curvature R = 0. One then rewrites the system as ⎧ Rαβ = Fα μ Fβμ − 14 gαβ F μν Fμν ⎪ ⎪ ⎨ D[α Fβγ] = 0 (2.1) ⎪ ⎪ ⎩ α D Fαβ = 0 We recall that the positive energy condition is valid for Einstein–Maxwell energy–momentum tensor, i.e., T (X, Y ) ≥ 0 where (X, Y ) is an arbitrary pair of future-directed causal vectors. Let 1 χ ˆab = χab − trχδab 2 be the traceless part of χ, so on H+ , according to the Raychaudhuri equation: 1 L(trχ) = −R44 − |χ| ˆ 2 − (trχ)2 , 2
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The non-expansion condition on H+ implies R44 + |χ| ˆ 2 = 0, Thanks to the positive energy condition, one knows R44 = T44 ≥ 0, so R44 = 0,
χ ˆ = 0 on H+ .
So χ = 0 on H+ . According to the untraced formulation of the Raychaudhuri equation: L(χ) + χ2 + R(−, L)L = 0, we know for all X ∈ T H+ , R(X, L)L = 0 In view of the first equation in (2.1), R44 = 0 implies F4a = 0, and the vanishing of these quantities implies R4a = 0. Combined with R(X, L)L = 0, we know R4aba = 0. To summarize, the non-expansion condition implies, on the null hypersurface H+ , that ⎧ χ=0 ⎪ ⎪ ⎪ ⎪ R ⎨ 4a = 0 R4aba = 0 (2.2) ⎪ ⎪ = 0 R ⎪ 344a ⎪ ⎩ F4a = 0 Similar identities hold on H− when one replaces the index 4 by 3 in this set of equations. It is precisely this set of geometric information that we shall use in the proof of our main theorems. Recall also our choice of the frame e1 , e2 is arbitrary on H+ . Since we know χ = 0, we can make this choice more rigid by using Fermi transport along L, i.e., we first pick a local orthonormal basis on the bifurcate sphere S, then use the Lie transport relation LL ea = 0 to get a basis on Su (which is not an orthonormal frame in general), the vanishing of χ on H+ guarantees {e1 , e2 } is still an orthonormal frame. On H+ , by virtue of the frame {e1 .e2 , e3 = L, e4 = L}, we have the following Ricci equations: ⎧ D4 L = 0, Da L = −ζa L, ⎪ ⎪ ⎨ D4 L = −ζa ea Da L = χab eb + ζa L (2.3) ⎪ ⎪ ⎩ D e = −ζ L D e = ∇ e + χ L 4 a
a
a b
a b
ab
where ∇a eb is the projection of Da eb onto the surface Su . A corresponding set of equations hold on H− . We shall use the following lemma: Lemma 2.2. On H+ , we have R(−, L, L, −) = −∇ζ − ∇4 χ + ζ ⊗ ζ i.e., Ra43b = −∇a ζb − ∇4 χab + ζa ζb . where ∇ denotes the projection of D onto Su ; a corresponding result holds on H− .
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Proof. For X, Y ∈ T Su , by definition, one has R(X, L, L, Y ) = g(DX D4 L, Y ) − g(D4 DX L, Y ) − g(DDX L L, Y ) + g(DD4 X L, Y ) We replace X and Y by ea and eb , so Ra43b = g(Da (ζ c ec ), eb ) − g(D4 (χa c ec + ζa L), eb ) + ζa g(D4 L, eb ) +g(D∇4 ea −ζa L L, eb ) = −(∇ζ)(ea , eb ) − g(D4 (χa c ec ), eb ) + g(D∇4 ea L, eb ) − ζa g(D4 L, eb ) = −∇a ζb − ∇4 χab + ζa ζb .
This completes the proof.
3. The Hawking vector Field Inside the Black Hole We define the following four regions I ++ , I −− , I +− and I −+ : I ++ = {p ∈ O|u(p) ≥ 0 and u(p) ≥ 0}, I −− = {p ∈ O|u(p) ≤ 0 and u(p) ≤ 0}, I +− = {p ∈ O|u(p) ≥ 0 and u(p) ≤ 0},
(3.1)
I −+ = {p ∈ O|u(p) ≤ 0 and u(p) ≥ 0}. In this section, we prove the following proposition: Proposition 3.1. Under the assumptions of Theorem 1.1, in a small neighborhood O of S, there exists a smooth Killing vector field K in O ∩ (I ++ ∪ I −− ) such that K = uL − u L
on (H+ ∪ H− ) ∩ O.
Moreover, LK F = 0 and [ L, K] = − L. The region O ∩ (I ++ ∪ I −− ) is the domain of dependence of H+ ∪ H− . As we mentioned in the introduction, by using the Newman–Penrose formalism, the first part of the proposition is shown by Friedrich et al. [5]. For the sake of completeness, we provide a direct proof without using Newman–Penrose formalism. As mentioned in the introduction, we consider the following characteristic initial value problem g Kα = −Rα β Kβ , (3.2) K = uL − u L on (H+ ∪ H− ) ∩ O. According to [10], this system of equation is well-posed in O ∩ (I ++ ∪ I −− ). A smooth vector field K is now constructed by solving (3.2) in the domain of dependence of H+ ∪ H− . To show K is indeed a Killing vector field, one has to show the deformation tensor of K παβ = LK g = Dα Kβ + Dβ Kα is zero in O ∩ (I ++ ∪ I −− ).
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Since K solves (3.2), by commuting derivatives, we know the deformation tensor παβ solves the following covariant wave equation: g παβ = −2Rρ αβ δ πρδ + Rαρ π ρ β + Rβρ π ρ α − 2LK Rαβ The geometric part of Einstein–Maxwell equations (2.1) provides LK Rαβ = LK Tαβ = Fα ρ LK Fβρ + Fβ ρ LK Fαρ − πρδ Fα ρ Fβ δ 1 1 1 − παβ Fμν F μν − gαβ F μν LK Fμν + gαβ πρδ F δ γ F ργ 4 2 2 Combined with the electromagnetic part of the Einstein–Maxwell equations (2.1), this formula allows one to derive a set of PDEs satisfied by LK Fαβ : D[α LK Fβγ] = 0 Dα LK Fαβ = παγ Dγ F α β + 12 (Dα πβγ + Dβ παγ − Dγ παβ ) Put all the equations together, one concludes that παβ and LK Fαβ solve the characteristic initial value problem for the following closed symmetric hyperbolic system: ⎧ g παβ = −2Rρ αβ δ πρδ + Rαρ π ρ β + Rβρ π ρ α ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −2(Fα ρ LK Fβρ + Fβ ρ LK Fαρ − πρδ Fα ρ Fβ δ ) ⎪ ⎪ ⎨ + 21 παβ Fμν F μν + gαβ F μν LK Fμν − gαβ πρδ F δ γ F ργ (3.3) ⎪ ⎪ ⎪ ⎪ D[α LK Fβγ] = 0 ⎪ ⎪ ⎪ ⎪ ⎩ Dα L F = π Dγ F α + 1 (D π + D π − D π ) β K αβ αγ α βγ β αγ γ αβ 2 So to show παβ = 0 and LK F = 0 in O, it suffices to show παβ = 0 +
LK F = 0
on H+ ∪ H− .
(3.4)
−
We only check (3.4) on H ; on H , the argument is exactly the same. In view of the expression of K = uL on H+ (since u = 0 on H+ ) and (2.3), it is easy to show ⎧ Da Kb = D4 Ka = Da K4 = D4 K4 = 0, D4 K3 = −1 ⎨ Dc Da Kb = D4 Da Kb = Db D4 Ka = D4 D4 Ka = Da Db K4 = 0, (3.5) ⎩ D4 Da K4 = Da D4 K4 = D4 D4 K4 = D4 D4 K3 = Da D4 K3 = 0. We see immediately that as long as a component of παβ does not involve the direction L, it is zero. More precisely, πab = π4a = π44 = 0 on H+
(3.6)
To prove the remaining components of π vanish, we need to make serious use of (3.2) to get derivatives in the L direction. Equation (3.2) gives D3 D4 Kβ + D4 D3 Kβ =
2
Da Da Kβ + Rβ ρ Kρ
a=1
Combined with the curvature identity D3 D4 Kβ − D4 D3 Kβ = −R34β ρ Kρ , we have
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2D4 D3 Kβ =
2
9
Da Da Kβ + Rβρ K ρ + R34βρ K ρ
(3.7)
a=1
Claim 3.2. We have D3 K4 = 1, Da D3 K4 = D4 D3 K4 = 0. Proof. We set β = 4 in (3.7), it is easy to check the left hand side of (3.7) is 2D4 D3 K4 = 2L(D3 K4 ) while the right-hand side is 0 by (2.2). So L(D3 K4 ) = 0 on H+ . It implies the value of D3 K4 on H+ is determined by its value on S which is 1. The other two identities can also be proved in a similar way. In particular, Claim 3.2 implies π34 = 0. Claim 3.3. We have Da K3 = uζa and D3 Ka = −uζa . Proof. The first identity in the claim is easy to check by direct computations; we now prove the second one. We first show that L(ζa ) = 0.
(3.8)
L(ζa ) = L(g(Da L, L)) = g(Da L, D4 L) + g(D4 Da L, L) = g(D4 Da L, L) = R4a43 = 0
(3.9) (3.10)
We use (2.2):
We now set β = b in (3.7), this implies D4 D3 Kb = 0, in view of D3 K4 = 1, we can show L(D3 Ka ) = −ζa Combined with (3.8), this shows D3 Ka = −uζa .
In particular, Claim 3.3 implies π3a = 0. Claim 3.4. We have π33 = 2D3 K3 = 0. Proof. To prove the claim, one needs help from the Einstein–Maxwell equations (2.1). Since F4a = 0, we have 2 L(R44 ) = L(F4a ) = 2F4a L(F4a ) = 0
which implies L(R4aa4 ) = 0
(3.11)
Recall the second Bianchi identity D4 R3aa4 + D3 Ra4a4 + Da R43a4 = 0
(3.12)
A simple computation with the help (2.2) and (3.11) shows the last two terms in (3.12) are zero. So we have L(R3aa4 ) = D4 R3aa4 = 0.
(3.13)
+
We compute L(trχ) along H : L(trχ) = L(g(Da L, ea )) = g(D4 Da L, ea ) + g(Da L, D4 ea ) = R4a3a + g(Da D4 L, ea ) + |ζ|2 = R4a3a + |ζ|2 − g(Da ζ , ea )
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where ζ is the dual vector field of ζ. In view of (3.13) and (3.8), we have L(L(trχ)) = −L(g(Da ζ , ea )) = −g(D4 Da ζ , ea ) = −R4aba ζ b − g(Da D4 ζ , ea ) = −g(Da D4 (ζ b eb ), ea ) = −ζ b g(Da (ζb L), ea ) This shows L(L(trχ)) = 0.
(3.14)
Now we are ready to prove the claim. We set β = 3 in (3.7), so 2D4 D3 K3 = Da Da K3 + R3ρ K ρ + R343ρ K ρ = Da Da K3 + uR34 + uR3434 = Da Da K3 + uR3aa4 By Lemma 2.2, we have R3aa4 = −∇a ζa − ∇4 χaa + ζa2 = −(∇ea ζ)(ea ) − ∇4 (trχ) + |ζ|2 = −divζ + ζ(∇a ea ) − L(trχ) + |ζ|2 We also compute Da Da K3 = u(divζ − ζ(∇a ea ) − |ζ|2 ) + trχ The previous computations showed 2D4 D3 K3 = trχ − uL(trχ) So in view of (3.14) L(D4 D3 K3 ) = −uL(L(trχ)) = 0
(3.15)
On S, it is easy to see D4 D3 K3 = 0, so D4 D3 K3 = 0 on H+ , which once again implies D3 K3 = 0 by solving transport equations along L. So we proved παβ = 0 on H+ . One still needs to show LK Fαβ = 0. Claim 3.5. We have the following identities: Da F4b = D4 F4b = D4 Fab = D4 F43 = 0
(3.16)
Proof. We will use (2.2) repeatedly: Da F4b = (Da F )(L ⊗ eb ) = ea (F4b ) − F (Da L ⊗ eb ) − F (L ⊗ Da eb ) = 0. Same argument shows D4 F4b = 0. We use Bianchi identity: D4 Fab = −Da Fb4 − Db F4a = 0. We now use the divergence free equation in Einstein–Maxwell equations (2.1): Dα Fα4 = 0 ⇒ Da Fa4 − D4 F34 = 0
so D4 F34 = Da Fa4 = 0. Claim 3.6. On H+ , we have LK F = 0
(3.17)
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Proof. Recall that LK Fαβ = DK Fαβ + g ρδ Dα Kδ Fρβ + g ρδ Dβ Kδ Fαρ . We show each component of LK F vanishes on H+ . The following three terms are relatively easy: LK Fab = DK Fab + g ρδ Da Kδ Fρb + g ρδ Db Kδ Faρ = uD4 Fab = 0. LK Fa4 = DK Fa4 + g ρδ Da Kδ Fρ4 + g ρδ D4 Kδ Faρ = uD4 Fa4 = 0. LK F43 = DK F43 + g ρδ D4 Kδ Fρ3 + g ρδ D3 Kδ F4ρ = DK F43 − D4 K3 F43 − D3 K4 F43 = uD4 F43 − π43 F43 = 0 We need some preparation to show that the most difficult term LK F3a vanishes. From the electromagnetic part of the Einstein–Maxwell equations (2.1), we have D4 F3b − D3 F4b + Db F43 = 0 and −(D4 F3b + D3 F4b ) + Da Fab = 0 So one has 2D4 F3b = Da Fab − Db F43
(3.18)
Apply the vector field L on (3.18), we have 2L(D4 F3b ) = L(Da Fab ) − L(Db F43 ) = D4 Da Fab − D4 Db F43 = (Da D4 Fab − R4aa ρ Fρb − R4ab ρ Faρ ) − (Db D4 F43 − R4b4 ρ Fρ3 −R4a3 ρ F4ρ ) = Da D4 Fab − Db D4 F43 = [ea (D4 Fab ) − D4 F (Da ea , eb ) − D4 F (ea , Da eb )] −[eb (D4 F43 ) − D4 F (Db L, L) − D4 F (L, Da L)]. In view of (2.3), one concludes that L(D4 F3b ) = 0
(3.19)
Now we compute LK F3b = DK F3b + g ρδ D3 Kρ Fδb + g ρδ Db Kρ F3δ = uD4 F3b + D3 Ka Fab − D3 K4 F3b − Db K3 F34 = uD4 F3b − uζa Fab − F3b − uζb F34 In particular, this shows LK F3b = 0 on S. Notice that L(F4b ) = L(Fab ) = L(F43 ) = 0, now apply L on LK F3b , so we have L(LK F3b ) = L(uD4 F3b ) − L(uζa Fab ) − L(F3b ) − L(uζb F34 ) (3.19)
= D4 F3b − ζa Fab − [D4 F3b + F (D4 L, Eb ) + F ( L, D4 eb )] − ζb F34 = 0
Now solving this ODE on H+ completes the proof.
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Remark 3.7. It follows from the previous computation that Dα πβγ = 0 on H+ . In fact, Da παβ = 0 and D4 παβ = 0 trivially comes from the fact that παβ = 0 on H+ ; to see D3 παβ = 0, we need to investigate the first equation in (3.3), in view of the facts that παβ = 0 and LK F = 0, this gives D4 D3 παβ + D3 D4 παβ = 0. Combined with the curvature identity D4 D3 παβ − D3 D4 παβ = −R34α ρ πρβ − R34β ρ παρ = 0, it gives L(D3 παβ )=0. Once more, by solving an ODE on H+ , D3 παβ = 0 follows from the fact that D3 παβ vanishes on S. We now prove the last statement of Proposition 3.1: [ L, K] = − L in the domain of dependence. We claim that D3 W = −DW L where W = [ L, K] + L, (3.20) W = 0 on H+ ∩ O. Since K is Killing vector field, for arbitrary vector fields X and Y , we have LK (DX Y ) = DX (LK Y ) + DLK X Y. Therefore, D3 W = D3 (−LK L + L) = −D3 (LK L) = −(LK (D3 L) − DLK L L) = DLK L L = −D[ L,K]+ L L = −DW L It remains to show W = 0 on H+ . W = D3 K − DK L + L = D3 K − uD4 L + L Since we have already computed the components D3 Kα , it is straightforward to show that W = 0 on H+ . By solving the ODE (3.20), we get W = 0 in the domain of dependence. This completes the proof.
4. The Hawking Vector Field Outside the Black Hole In the previous section we constructed the Hawking vector field K inside the black hole region. To extend K to a full neighborhood of the bifurcate sphere S, because the characteristic initial value problem is not well-posed on the full neighborhood as we explained in the introduction, we need to rely on a completely different strategy. The idea is, instead of solving a hyperbolic system, we now solve the ODE [ L, K] = − L for K. This ODE is well-posed in the complement of the domain of dependence since L is transversal to H+ . That is the construction of K. One has to show that K constructed in this way is a Killing vector field. Let φt be the one parameter family of diffeomorphisms generated by K. When t is small, both of (g, F ) and (φ∗t g, φ∗t F ) solve the Einstein–Maxwell equations and they coincide on H+ ∪ H− . To prove K is Killing, it suffices to show (g, F ) and (φ∗t g, φ∗t F ) coincide in a full neighborhood of S. We first define a vector field K by setting K = uL on H+ ∩ O and solving the ODE [ L, K ] = − L. The vector field K is defined and smooth in
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a full neighborhood O of S (since L = 0 on S). Moreover, in view of Proposition 3.1, K coincides with K in I ++ ∪ I −− . Thus K := K defines the desired extension. This construction is summarized in the following lemma: Lemma 4.1. There exists a smooth extension of the vector field K to a full neighborhood O of S such that [ L, K] = − L d dt
in O.
(4.1)
Let gt = φ∗t g and Lt = (φ−t )∗ L. In view of (4.1) of K, we know that Lt = − Lt . This implies that Lt = e−t L.
(4.2)
Let Dt be the Levi–Civita connection of gt , by the tensorial nature, we know that DtLt Lt = D L L = 0, (4.2) implies that 0 = DtLt Lt = e−2t DtL L. This proves the following lemma Lemma 4.2. Assume K is the smooth vector field constructed in (4.1) and Dt the Levi–Civita connection of the metric φ∗t g. We have DtL L = 0
in O.
(4.3)
To summarize, let Ft = φ∗t F , then we have a family of metrics and two forms (gt , Ft ) which solves the Einstein–Maxwell equations (2.1). Moreover, they coincide in the domain of dependence of H+ ∪ H− and for each t, one has DtL L = 0. So Theorem 1.1 is a consequence of the following uniqueness statement: Proposition 4.3. Assume in a full neighborhood O of S, g is a smooth Lorentzian metric and F is a smooth two form, such that (g , F ) solves Einstein– Maxwell equations (2.1). If g = g and F = F in (I ++ ∪ I −− ) ∩ O, in O, DL L = 0 where D denotes the Levi–Civita connection of the metric g . Then g = g and F = F in a full neighborhood O ⊂ O of S. The corresponding proposition for Einstein vacuum space-times was first proved in [1]. A simplified version can be found in [2]. In [7], the authors proved uniqueness results for covariant semi-linear wave equations of a fixed metric. But for the uniqueness at the level of metrics, since the corresponding PDEs are quasi-linear, one has to couple the system with a system of ODEs to recover the semi-linearity. In this section, we use this idea to prove uniqueness for the full curvature tensor and the electromagnetic field. Since the metric is uniquely determined by the curvature tensor, that will prove Proposition 4.3. Proof. We first derive a system of covariant wave equations for the full curvature tensor Rαβγδ of the metric g and Fαβ . Recall the second Bianchi identity: Dα Rβγρδ + Dβ Rγαρδ + Dγ Rαβρδ = 0
(4.4)
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Taking the divergence of (4.4) and commuting derivatives, we have Dα Dα Rβγρδ = −[Dα , Dβ ]Rγαρδ − [Dα , Dγ ]Rαβρδ − Dβ Dα Rγαρδ − Dγ Dα Rαβρδ = Rα βγμ Rμ αρδ + Rα βαμ Rγ μ ρδ + Rα βρμ Rγα μ δ + Rα βδμ Rγαρ μ +Rα γαμ Rμ βρδ + Rα γβμ Rα μ ρδ + Rα γρμ Rαβ μ δ + Rα γδμ Rαβρ μ +Dβ Dδ Rργ + Dγ Dρ Rδβ − Dβ Dρ Rδγ − Dγ Dδ Rρβ Schematically, we can write the previous formula as g Rαβγδ = (R ∗ R)αβγδ + Dγ Dδ Rαβ
(4.5)
where the last term denotes a linear combination of certain components of the Hessian of the Ricci curvature. In this expression, only the structure of the terms is important, the exact numerical coefficients are irrelevant. To have a closed system, we need to compute the Hessian of the Ricci tensor. By the gravitational part of (2.1), we have the following schematically expression: Dγ Dδ Rαβ = Fβμ Dγ Dδ Fα μ + Fα μ Dγ Dδ Fβμ + Dδ Fα μ Dγ Fβμ + Dγ Fα μ Dδ Fβμ 1 − gαβ (F μν Dγ Dδ Fμν + Dγ Fμν Dδ F μν ) 2 = (F ∗ D2 F )αβγδ + (DF ∗ DF )αβγδ Plugging this into (4.5), we have g Rαβγδ = (R ∗ R)αβγδ + (F ∗ D2 F )αβγδ + (DF ∗ DF )αβγδ
(4.6)
This equation involves two derivatives of F . In principle, the electromagnetic part of the Einstein–Maxwell equations (2.1) controls only one derivative of F through the second order system: Dα Dα Fβγ = −Dα Dβ Fγα − Dα Dγ Fαβ = −[Dα , Dβ ]Fγα − [Dα , Dγ ]Fαβ − Dβ Dα Fγα − Dγ Dα Fαβ = Rα βγμ F μ α + Rα βαμ Fγ μ which can be expressed in the following schematic form: g Fαβ = (R ∗ F )αβ
(4.7)
Since for the Einstein–Maxwell equations, the electromagnetic part of is almost decoupled from the gravitational part, we can actually control second derivatives of F by a cost of one derivative on the curvature tensor Rαβγδ . More precisely, we apply the covariant derivative Dρ on the second equation of (2.1): d(Dδ F )αβ = Dα Dδ Fβγ + Dβ Dδ Fγα + Dγ Dδ Fαβ = [Dα , Dδ ]Fβγ + [Dβ , Dδ ]Fγα + [Dγ , Dδ ]Fαβ + Dρ (D[α Fβγ] ) = −Rαδβμ F μ γ − Rαδγμ Fβ μ − Rβδγμ F μ α − Rβδαμ Fγ μ −Rγδαμ F μ β − Rγδβμ Fα μ
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where d stands for the exterior derivative on two forms. Schematically, this gives D[α (DF )βγ] = (R ∗ F )αβγ Similarly, we have Dα (DF )αβ = (R ∗ F )β Applying covariant derivative on these last two equations gives g (DF )αβ = (R ∗ DF )αβ + (DR ∗ F )αβ
(4.8)
We summarize (4.6), (4.7) and (4.8) in the following system of equations ⎧ g Rαβγδ = (R ∗ R)αβγδ + (F ∗ D2 F )αβγδ + (DF ∗ DF )αβγδ ⎪ ⎪ ⎨ g Fαβ = (R ∗ F )αβ (4.9) ⎪ ⎪ ⎩ g (DF )αβ = (R ∗ DF )αβ + (DR ∗ F )αβ and Fαβ . We have a corresponding system of equations for Rαβγδ We shall prove Proposition 4.3 in a neighborhood O(p) of a point p ∈ S. To do so, we first introduce a fixed coordinate system (xk ) for k = 1, 2, 3, 4. We emphasize that the coordinate (xk ) is chosen for both metrics g and g . In the proof we shall keep shrinking the neighborhoods of p; to simplify notations we keep denoting such neighborhoods by O(p). We now fix the null frame {e1 , e2 , e3 = L, e4 = L} on the null hypersurface H+ ∩ O(p). We use two different Levi-Civita connections to parallel transport the given null frame along L: D L vα = 0 with vα = eα on H+ ∩ O(p) (4.10) DL vα = 0 with vα = eα on H+ ∩ O(p)
The frames {vα } and {vα } are smoothly defined in O(p). We will express all = g (vα , vβ ). the geometric quantities in these frames. Let gαβ = g(vα , vβ ), gαβ Since D L vα = D L vα = 0, we know L(gαβ ) = L(gαβ ) = 0, so gαβ = gαβ . We define hαβ = gαβ = gαβ
L(hαβ ) = 0
in O(p).
(4.11)
We define the Christoffel symbols, curvature tensors and their differences Γγαβ = g(Dvα vβ , vγ ),
Γγ vβ , vγ ), αβ = g (Dvα
γ δΓγαβ = Γγ αβ − Γαβ
Rαβγδ = g(R(vα , vβ )vγ , vδ ), Rαβγδ = g (R (vα , vβ )vγ , vδ ), δRαβγδ = Rαβγδ − Rαβγδ
We also introduce δR and δF to denote the collections of all the δRαβγδ ’s and γ δΓγαβ ’s. One observes that Γγ3β = Γγ 3β = δΓ3β = 0. In view of D L vα = 0, we drive a system of ODEs for Γγαβ and Γγ αβ : L(Γγαβ ) = L(g(Dvα vβ , vγ )) = g(Dv3 Dvα vβ , vγ ) + g(Dvα vβ , Dv3 vγ ) = R3αβγ + g(D[v3 ,vα ] vβ , vγ ) + g(Dvα vβ , Dv3 vγ ) = R3αβγ + Γρ3α Γγρβ − Γρα3 Γγρβ + gρδ Γδαβ Γρ3γ
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Schematically, we have L(Γγαβ ) = R3αβγ + (Γ ∗ Γ)γαβ
(4.12)
γ L(Γγ αβ ) = R3αβγ + (Γ ∗ Γ )αβ
(4.13)
Similarly, we have Taking the difference of (4.12) and (4.13), one has L(δΓγαβ ) = δR3αβγ + (Γ ∗ Γ − Γ ∗ Γ)γαβ = δR3αβγ + (Γ ∗ δΓ)γαβ + (Γ ∗ δΓ)γαβ Since the smooth metrics g and g are given, so their corresponding Christoffel symbols are also fixed in the given coordinate (xk ). One treat these objects as given backgrounds data, in particular, one can bound these objects in L∞ norm, so we have: | L(δΓ)| |δΓ| + |δR|.
(4.14) {vα }
in terms of the Now we also need to express the frames {vα } and fixed coordinate vector fields ∂k relative to our local coordinates xk . We define vα = v α ∂k ,
(δv)kα = v α − vαk
k
vα = vαk ∂k ,
k
Consider [v3 , vα ] = −Dvα v3 = −Γβα3 vβ = −Γβα3 vβk ∂k , it implies v3j ∂j (vαk ) − vαj ∂j (v3k ) = −Γβα3 vβk so we have L(vαk ) = ∂j (v3k )vαj − Γβα3 vβk
(4.15)
Similarly, we have L(v α ) = ∂j (v 3 )v α − Γ α3 v β k
k
j
β
k
(4.16)
Notice that ∂j (v3k ) = ∂j (v 3 ) are fixed functions (since v3 = v3 = L), so by taking the difference, we have the following differential inequality k
L(δv) |δΓ| + |δv|.
(4.17) (δv)kα ’s.
where we use δv to denote the collection of all the We can also apply coordinate derivatives ∂k to (4.12), (4.13), (4.15) and (4.16), after taking the differences, one has the following estimates | L(∂δΓ)| |δΓ| + |∂δΓ| + |δR| + |∂δR|. | L(∂δv)| |δΓ| + |∂δΓ| + |δv| + |∂δv|.
(4.18) (4.19)
Now we derive a set covariant of wave equations for δR and δF, δDF (where δF is defined in an obvious way). In view of (4.9), when we take the differences, the following terms may cause trouble (g − g )R,
(g − g )F
and
(g − g )DF
The estimates are the same for all of them, to illustrate the idea, we now deal , it is easy to see it has the following form with the first one. Since gαβ = gαβ |(g − g )R| |δΓ| + |∂δΓ|.
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Similar relations hold for the other terms. Together with (4.14), (4.17), (4.18) and (4.19), we have the following system of differential inequalities: ⎧ | L(δΓ)| |δΓ| + |δR|, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ | L(∂δΓ)| |δΓ| + |∂δΓ| + |δR| + |∂δR|, ⎪ ⎪ ⎪ ⎪ ⎪ | L(δv)| |δΓ| + |δv|, ⎪ ⎪ ⎨ | L(∂δv)| |δΓ| + |∂δΓ| + |δv| + |∂δv|, (4.20) ⎪ ⎪ ⎪ ⎪ |g δR| |δR| + |δF | + |δDF | + |∂δDF | + |δΓ| + |∂δΓ|, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ |g δF | |δR| + |δF | + |δΓ| + |∂δΓ|, ⎪ ⎪ ⎪ ⎩ |g δDF | |δR| + |δDF | + |∂δR| + |δF | + |δΓ| + |∂δΓ|. Since in I ++ ∪ I −− , g = g and F = F , on the bifurcate horizon H+ ∪ H− , the following functions δΓ, ∂δΓ, δv, ∂δv, δR, δF and δDF vanish to infinite order. We now show that they vanish completely in a full neighborhood of S. It is an immediate consequence of the following uniqueness theorem, based on the Carleman estimates developed in [7], due to Alexakis [1], see also Lemma 4.4 of [2]. Proposition 4.4. Assume Gi , Hj : O(p) → R are smooth functions, i = 1, . . . , I, j = 1, . . . , J. Let G = (G1 , . . . , GI ), H = (H1 , . . . , HJ ), ∂G = (∂1 G1 , ∂2 G1 , ∂3 G1 , ∂4 G1 , . . . , ∂4 GI ) and assume that in O(p), |g G| |G| + |∂G| + |H|; | L(H)| |G| + |∂G| + |H|. Assume that G = 0 and H = 0 on (H+ ∪H− )∩O(p). Then, there exists a neighborhood O (p) ⊂ O(p) of x0 such that G = 0 and H = 0 in (I +− ∪I −+ )∩O (p). In particular, applied to (4.20), Proposition 4.4 implies δR = 0 and δF = 0 in a full neighborhood of S, which shows that the vector field K is Killing in a full neighborhood of S and LK F = 0. Remark 4.5. The vector field K is time-like in O ∩ (I +− ∪ I −+ ) which follows directly from the fact that L(g(K, K)) ≥ 0.
5. The Rotational Killing Vector Field We prove Theorem 1.2 in this section. In addition to the Hawking vector field K constructed in Theorem 1.1, we assume (O, g, F ) has another Killing vector field T which preserves F (i.e., LK F = 0 ) such that it is tangent to H+ ∪ H− and it is not identically zero on S. We will find a constant λ, such that Z = T + λK is a rotational Killing vector field, i.e., all the orbits of Z are closed circles. We study the action of T on the bifurcate sphere S. Because T is a smooth vector field tangent to the bifurcate horizon H+ ∪ H− , it must be tangent to S. Since T is not identically zero, thanks to Lemma A.1 in “Appendix”, the
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restriction of the metric g on S is rotational symmetric. In particular, the vector field X = T |S on S has a period t0 and has two zeroes. One fixes one of the zeroes and denotes it by p ∈ S. Let γ + be the null geodesic emanating from g(T, L) p on H+ ; similarly, we define γ − . On γ + , we define a function λ(u) = g(K, L) which measures the projection of T in the K direction. The key observation is Claim 5.1. On γ + , λ(u) is a constant. Proof. We show that [T, L] is parallel to L on H+ , i.e., there is a function f : H+ → R, such that [T, L] = f L. Since both vectors are tangent to H+ , so is [T, L]. It suffices to show g([T, L], ea ) = 0. g([T, L], ea ) = g(DT L, ea ) − g(DL T, ea )
Killing
=
g(DT L, ea ) + g(Da T, L)
= g(DT L, ea ) − g(T, Da L) = χ(T, ea ) − χ(ea , T ) = 0. We claim that L(f ) = 0 which follows from the 0 = LT (DL L) = DLT L L + DL (LT L) = Df L L + DL (f L) = L(f )L. When one restricts f to γ + , this implies that f (u) = f (p) is a constant. On the bifurcate sphere S, we can compute f = f L(u) = [T, L](u) = −L(T (u)) so on γ
+
T (u) = −f (p)u. We turn to L(λ(u)): L(λ(u)) = L
g(T, L) g(K, L)
= −L
T (u) u
= L(f ) = 0.
This shows λ(u) is a constant on γ + .
Remark 5.2. One can construct λ(u) on γ − and show that λ(u) = λ(u) = λ(p) is a constant. Now we can define the rotational vector field Z: Claim 5.3. Let λ = −f (p), then Z = T + λK is a rotational vector field with period t0 . Proof. Since K = 0 on S, Z|S = T |S has the same period t0 . We denote ψt the one parameter isometry group generated by Z on O. To show Z is rotational, it suffices to show ψt0 = id. We study the action of ψt on the null geodesic γ + . For all t, since p is a fixed point of ψt and ψt is an isometry, we know that ψt (γ) ⊂ γ is a reparametrization of γ + . In particular, it implies Z|γ + is proportional to K|γ + . In view of the definition of λ, we know that Z|γ + = 0 since we have subtracted the corresponding portion of K from T . So ψt |γ + = id. In particular, ψt0 |γ + = id.
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We turn to the action of ψt0 on the full tangent space of p. The previous argument shows (ψt0 )∗ L = L; the same considerations on γ − shows (ψt0 )∗ L = L. We also know that (ψt0 )∗ ea = ea because p is a zero on T on S and K vanishes on S. (ψt0 )∗ is the identity map on the tangent space of p. We can use Lemma A.2 in “Appendix” to conclude that ψt0 is the identity in a small neighborhood of p. Finally, we can use the compactness of S and the standard open-closed argument on S to conclude ψt0 is the identity map in a small neighborhood of S. To finish the proof of Theorem 1.2, we have: Claim 5.4. [Z, K] = 0. Proof. It suffices to show [T, K] = 0. Since both K and T are Killing, in view of the fact that all the Killing vector fields on a manifold form a Lie algebra under [−, −], we know that W = [T, K] is also Killing, so it solves the following equation: g Wα = −Rα β Wβ
(5.1)
We show that W =0
on H+ ∪ H− .
It is an easy consequence of the calculations in the proof of Claim 5.1: W = [T, K] = [T, uL] = u[T, L] + T (u)L = uf L − uf L = 0 By solving (5.1), we know that W = 0 in I ++ ∪ I −− . To show W = 0 in a full neighborhood of S, once again we have to use Proposition 4.4 in the straightforward way. This completes the proof.
Acknowledgements The author would like to thank Professor Sergiu Klainerman for suggesting the problem, and Willie Wai-Yeung Wong for valuable discussions.
Appendix A: Two Lemmas on Geometry Lemma A.1. Assume h is a Riemannian metric on the topological sphere S 2 which admits a non-trivial Killing vector field X, then (S 2 , h) is a Riemannian warped product ([0, 1], dr2 ) ×φ(r) (S 1 , dσ 2 ). In particular, each orbit of X is closed and has a common period t0 . Proof. First, we observe that, if X is non-trivial, then the set Z(X), which consists all zeroes of X, is discrete. It follows from the fact that, the zero locus of a Killing vector field is a disjoint union of totally geodesic sub-manifolds each of even dimension. Since we are on a surface, the zeroes must be discrete. In particular, since the S 2 is compact, X has only finite many zeroes.
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The second observations is that, for each zero p of X, indX (p) the index of X at p is either 1 or −1. It following from the fact that, X induces an isometry on Tp S 2 , which is a 2D rotation. So its index must be 1 or −1. Now we can apply the Poincar´e–Hopf index theorem: indX (p) = χ(S 2 ) = 2. p∈Z(X)
The previous observation imply that the cardinal number |Z(X)| ≥ 2. We can pick up two points p, q ∈ Z(X). Now let us fix a minimal geodesic γ(t) between p and q. Let φt be the flow generated by X. Since on Tp M, (φt )∗ is a rotation, it has a period t0 . Let x = p, q be a point on γ. We show that the orbit of x under φt is a closed non-degenerate circle, more precisely, it is the image {φt (x)|t ∈ [0, t0 )}. It trivially holds when x is close to either p or q, i.e., in the normal coordinate of p or q, since it will stay on the geodesic sphere which is a circle around either p or q. Since γ is minimal and X(q) = 0, so φt (γ) is also a minimal geodesic between p and q. When t varies, φt (γ) sweeps the whole S 2 , we know that all points except q is in the normal coordinate of p, so the orbit x is closed. This finishes the proof of the lemma. Lemma A.2. Assume (M, g) is a Lorentzian manifold, φ : M → M is an isometry and p ∈ M is a fixed point of φ. If φ∗p = id, then φ = id locally around p. Proof. In Riemannian geometry, this is easy since we have the concept of length; in our case, the difficulty comes from the fact that on the light-cone, the length is not well defined. But the proposition holds inside the light-cone since we can consider maximal time-like geodesics. Since locally the light-cone is the boundary of the future of the point p, the identity map can be continued to the boundary.
References [1] Alexakis, S.: Unique continuation for the vacuum Einstein equations. gr-qc0902. 1131 (2008, preprint) [2] Alexakis, S., Ionescu, A.D., Klainerman, S.: Hawking’s local rigidity theorem without analyticity. gr-qc0902.1173 (2009, preprint) [3] Bunting, G.L.: Proof of the Uniqueness Conjecture for Black Holes, Ph.D. Thesis, University of New England, Armidale (1983) [4] Carter, B.: An axi-symmetric black hole has only two degrees of freedom. Phys. Rev. Lett. 26, 331–333 (1971) [5] Friedrich, H., R´ acz, I., Wald, R.: On the rigidity theorem for space-times with a stationary event horizon or a compact Cauchy horizon. Commun. Math. Phys. 204, 691–707 (1999) [6] Hawking, S.W., Ellis, G.F.R.: The large scale structure of space–time. Cambridge University Press, London (1973) [7] Ionescu, A.D., Klainerman, S.: On the uniqueness of smooth, stationary black holes in vacuum. Invent. Math. 175, 35–102 (2009)
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[8] Ionescu, A.D., Klainerman, S.: Uniqueness results for ill-posed characteristic problems in curved space-times. Commun. Math. Phys. 285, 873–900 (2009) [9] Israel, W.: Event horizons in static electrovac space-times. Commun. Math. Phys. 8, 245–260 (1968) [10] Rendall, A.: Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations. Proc. R. Soc. Lond. A 427, 221–239 (1990) [11] Robinson, D.C.: Uniqueness of the Kerr black hole. Phys. Rev. Lett. 34, 905–906 (1975) Pin Yu Department of Mathematics Princeton University Princeton, NJ 08544, USA e-mail:
[email protected] Communicated by Piotr T. Chrusciel. Received: September 28, 2009. Accepted: February 19, 2010.
Ann. Henri Poincar´e 11 (2010), 23–67 c 2010 Springer Basel AG 1424-0637/10/010023-45 published online May 12, 2010 DOI 10.1007/s00023-010-0029-4
Annales Henri Poincar´ e
A Born–Oppenheimer Expansion in a Neighborhood of a Renner–Teller Intersection Mark S. Herman Abstract. We perform a rigorous mathematical analysis of the bending modes of a linear triatomic molecule that exhibits the Renner–Teller effect. Assuming the potentials are smooth, we prove that the wave functions and energy levels have asymptotic expansions in powers of , where 4 is the ratio of an electron mass to the mass of a nucleus. To prove the validity of the expansion, we must prove various properties of the leading order equations and their solutions. The leading order eigenvalue problem is analyzed in terms of a parameter ˜b, which is equivalent to the parameter originally used by Renner. For 0 < ˜b < 1, we prove self-adjointness of the leading order Hamiltonian, that it has purely discrete spectrum, and that its eigenfunctions and their derivatives decay exponentially. Perturbation theory and finite difference calculations suggest that the ground bending vibrational state is involved in a level crossing near ˜b = 0.925. We also discuss the degeneracy of the eigenvalues. Because of the crossing, the ground state is degenerate for 0 < ˜b < 0.925 and non-degenerate for 0.925 < ˜b < 1.
1. Introduction and Background In their original paper [1], Born and Oppenheimer let 4 be the ratio of the electron mass to the nuclear mass and expanded the wave functions and eigenvalues of the time independent Schr¨ odinger equation in powers of . We shall refer to such an expansion as a Born–Oppenheimer expansion. Since is small, the first few orders of the expansions are thought to provide reasonably accurate results for the bound states of the molecular system. Often only the lowest (or leading) order terms of the expansions are even considered. The focus of this paper is the Renner–Teller effect (also called the Renner effect), which is later described in more detail. In short, a symmetry induced degeneracy exists in the electron states at a particular nuclear configuration,
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but when the nuclei move away from this configuration the degeneracy splits. As a result one must use more than one electronic state when attempting to solve for the total wave function and energy using the Born–Oppenheimer approximation. This effect was first predicted in 1933 by Herzberg and Teller [14] and was analyzed one year later by Renner [27] in a simplified model.We consider the current paper as an extension of the mathematically rigorous works related to the Born–Oppenheimer approximation, such as [3,4,7,8,10, 12,19], to the model originally considered by Renner [27]. The main results are contained in Theorem 2.1. We show rigorously that a Born–Oppenheimer expansion exists to all orders of , with minimal mathematical assumptions. We prove that under our hypotheses, the molecular energy and wave function can be approximated by an asymptotic series in that is truncated at arbitrary order. The leading order equations we obtain are unitarily equivalent to those found by Renner in [27]. This is the first rigorous derivation of the leading order equations of which we are aware. We feel it is especially important to make contact with a rigorous Born–Oppenheimer expansion here, since the Renner–Teller effect is not a straightforward application of the Born– Oppenheimer approximation. In their extensive review of the subject [23], Peri´c and Peyerimhoff give several interpretations of the origin of the Renner–Teller effect, and in particular they state “from the quantum chemical standpoint, the R–T effect is a consequence of violation of validity of the Born–Oppenheimer approximation.” We will see that in the Renner–Teller case there is a valid Born–Oppenheimer expansion, but it differs significantly from the usual Born–Oppenheimer approximation since the degeneracy cannot be ignored. It must be analyzed in terms of degenerate perturbation theory. In recent years there have been several mathematically rigorous results justifying the validity of Born–Oppenheimer expansions under various hypotheses. The first rigorous proof related to the Born–Oppenheimer approximation in a physically realistic model was given by Combes et al. [3,4]. They proved the validity of the fourth order approximation for the eigenvalue and the leading order approximation for the eigenfunction. A few years later, Hagedorn proved [7] the existence of a Born–Oppenheimer expansion to all orders using the method of multiple scales, assuming that the potentials are smooth functions. In particular, he proved that for arbitrary K, there exist quasiK mode energies of the form EK () = k=0 k E (k) and quasimodes of the form K k (k) ΨK () = k=0 Ψ , that asymptotically approximate an exact eigenvalue and eigenfunction below the essential spectrum of a Hamiltonian H(), in the sense that ||H()ΨK () − EK ()ΨK ()|| ≤ CK K+1 ||ΨK ()|| .
(1.1)
The first five orders of E() were determined explicitly, and it is discussed how one could proceed to any arbitrary order K. These results were then extended to the case of Coulomb potentials for diatomic molecules in [8] and to general polyatomic molecules by Klein et al. [19]. Here, we will assume that the potentials are smooth, but we believe our results can be extended
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in a similar manner to the case of Coulomb potentials. There have been various mathematical works focusing on the time-dependent Born–Oppenheimer approximation, we only mention a few recent ones [9,21,24]. See [11] for a review of the mathematical aspects of the Born–Oppenheimer approximation.
2. Description of the Model and Statement of the Main Theorem Consider a triatomic molecule and fix the reference frame so that when the molecule is in the linear configuration, the middle nucleus is at the origin and the z-axis passes through all three nuclei. Let (0, 0, R1 ) and (0, 0, R2 ) be the coordinates of the upper and lower nuclei (so R1 > 0 and R2 < 0). We consider the bending modes by clamping the upper and lower nuclei to their fixed positions on the z-axis and allowing the middle nucleus to move in the perpendicular plane. Let (x, y, 0) be the cartesian coordinates of this middle nucleus, and let (˜ ρ, φ) be the usual polar coordinates associated with (x, y) (see Fig. 1). If (x1 , x2 , . . . , xN ) are the N three-dimensional electron coordinates, the electronic Hamiltonian is N 1 h(x, y) = − Δxj + V (x, y; x1 , x2 , . . . , xN ), 2 j=1 where we have taken the electron mass to be 1, and the potential V includes the repulsion forces between the nuclei, the attraction forces between the nuclei and electrons, and the repulsion forces between the electrons. We think of h(x, y) as having parametric dependence on (x, y) (i.e. it is a mapping from R2 to the linear operators on the electronic Hilbert space), and we assume it is a real symmetric operator. We assume that V is a smooth function in all variables. This assumption allows us to construct electronic basis vectors that are smooth in the nuclear coordinates. These are necessary to derive the
Figure 1. The reference frame for the middle nucleus
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M. S. Herman
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matrix elements of the electronic Hamiltonian in Sect. 3.2 and to construct the quasimodes of Theorem 2.1. Perhaps one way to account for Coulomb singularities is using distortion techniques as in [16]. If one makes a clever change of coordinates so the singularity locations are independent of the new nuclear coordinates, the electronic Hamiltonian will be analytic in terms of the nuclear coordinates. We believe many of our results extend to the case of Coulomb potentials (for example the form of the matrix potential derived in Sect. 3.2), but we choose not to focus on this matter here. Let 4 be the ratio of the mass of an electron to the mass of the middle nucleus. Then, the full hamiltonian of this model is given by 4 Δx,y + h(x, y). (2.2) 2 Let Hnuc = L2 (R2 , dx dy) and Hel = L2 (R3N ), so that H() acts on the Hilbert space Hnuc ⊗ Hel . We denote the inner product and norm on Hel by ·, ·el and ||·||el and similarly on Hnuc by ·, ·nuc and ||·||nuc . ∂ nuc = −i ∂φ be the operators associated with the projecLet Lel z and Lz tions of the electronic and nuclear angular momenta on the z-axis, respectively. The operator of total angular momentum about the z-axis is denoted nuc ⊗ I). We note that H() commutes with LTz OT . by LTz OT = (I ⊗ Lel z ) + (Lz We consider the electronic states when (x, y) = (0, 0). In this case the electronic hamiltonian h(0, 0) commutes with Lel z since the nuclei are in a linear arrangement. So, for |lzel | = 0 there are two-fold degenerate electronic vecH() = −
|lel |
|lel |
tors ψ1 , ψ2 ∈ Hel satisfying h(0, 0) ψ1 = E0 z ψ1 and h(0, 0) ψ2 = E0 z ψ2 , el el el where Lel z ψ1 = lz ψ1 and Lz ψ2 = −lz ψ2 . Then, if the molecule is bent so that (x, y) = (0, 0), this degeneracy splits since the nuclei are no longer in a linear arrangement, and h(x, y) no longer commutes with Lel z (see [17] for a discussion directly relating the breaking of symmetry with the breaking of the degeneracy). This is the Renner–Teller effect. As previously mentioned, the application of the Born–Oppenheimer approximation is not straightforward in this case. There have been numerous papers related to the Renner–Teller effect, few of which are relevant to our analysis here. We highlight one such paper by Brown and Jørgensen [2] for its completeness, and because it does discuss effects beyond the leading order. We encourage the reader interested to learn the historical development and recent findings of the theory to consult the review by Peri´c and Peyerimhoff [23]. Note that since changes in φ correspond to an overall molecular rotation, the eigenvalues of h(x, y) are independent of φ. Corresponding to the situation above where the electronic states at ρ˜ = 0 are linear combinations of el el eigenstates of Lel z with eigenvalues lz , −lz = 0, consider a pair of electronic ρ) and E2 (˜ ρ) of h(x, y) that are degenerate at ρ˜ = 0, but the eigenvalues E1 (˜ degeneracy breaks when ρ˜ = 0. We refer to two such electronic states as an R–T pair with value |lzel |. The eigenvalues of h(x, y) provide the usual potential energy surfaces for the nuclei, and there are several qualitatively different possibilities where the Renner Teller effect is important. See Fig. 2. We refer to [17,20] for further examples and discussion of Renner–Teller surfaces. We focus
Vol. 11 (2010)
(a)
Born–Oppenheimer Expansion Near a Renner–Teller
(b)
27
(c)
Figure 2. Potential energy surfaces of three qualitatively different cases corresponding to an R–T pair of electronic states strictly on an R–T pair of states corresponding to sketch (a) of Fig. 2, where both surfaces have local minima at ρ˜ = 0. In this case the optimal nuclear configuration, corresponding to both electronic states of the R–T pair, is linear. This was the situation considered by Renner [27] in 1934. The first experimental observation of the Renner–Teller effect came in the 1950s in a study of NH2 [5,6] and corresponded to the case in sketch (c). Throughout this paper, we assume the following hypotheses: There is an R–T pair of states, with eigenvalues E1 and E2 both having minima at ρ˜ = 0. This corresponds to case (a) in Fig. 2. This type of intersection is known to occur for example in the A3 Πu state of NCN [23]. We assume that for some neighborhood of ρ˜ = 0 no other crossings are present, so E1 and E2 are isolated from the rest of the spectrum of h(x, y). From our smoothness assumption on ρ) and h(x, y), it follows that E1 and E2 are C ∞ in (x, y). This implies that E1 (˜ ρ) have asymptotic expansions in powers of ρ˜2 . We assume that the splitE2 (˜ b 2 ting occurs at 2nd order, that is, that E1 and E2 are asymptotic to a + ˜ 2 ρ a−b 2 and 2 ρ˜ for small ρ˜, respectively, for some 0 < b < a (we have taken E1 (0) = E2 (0) = 0 for convenience). Renner [27] argued that an R–T pair with value |lzel | = 1 (also called Π states) will exhibit splitting at 2nd order, an R–T pair with value |lzel | = 2 (called Δ states) will exhibit splitting at 4th order, and in general an R–T pair with value |lzel | = n will exhibit splitting at order 2n. We instead assume 2nd order splitting occurs and later prove that the R–T pair has value |lzel | = 1, agreeing with Renner’s argument. We are now ready to state our main theorem. Theorem 2.1. Assume the hypotheses described above, in particular that the potentials are smooth and there is an R–T pair E1 (˜ ρ), E2 (˜ ρ) that are asympa−b 2 2 totic to a+b ρ ˜ and ρ ˜ respectively, where 0 < b < a. Then for arbitrary K, 2 2 there exist quasimode energies E,K =
K
k E (k)
k=0
and quasimodes Φ,K =
K k=0
k Φ(k)
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that satisfy || ( H() − E,K ) Φ,K ||Hnuc ⊗Hel ≤ CK K+1 || Φ,K ||Hnuc ⊗Hel . The quasimodes are associated with the local wells of E1 and E2 in a neighborhood of ρ˜ = 0. Remarks. 1. Quasimode estimates correspond to discrete eigenvalues of H() when E,K lies below the essential spectrum as characterized by the HVZ theorem [26]. 2. Since [LTz OT , H()] = 0, quasimodes can be constructed to be eigenfunctions of LTz OT , with eigenvalues lzT OT ∈ Z. The eigenstates of H() corresponding to lzT OT = 0 are non-degenerate, while the eigenstates corresponding to |lzT OT | = 0 are two-fold degenerate. In particular, there is an lzT OT state and a −lzT OT state, with eigenfunctions that are complex conjugates of one another, together forming a degenerate pair of states associated with H(). 3. The first two orders E (0) and E (1) are zero and the second order E (2) is determined by the leading order eigenvalue equation H2 Ψ = E (2) Ψ, on the Hilbert space L2 (R2 , dX dY ; C2 ), where 1 a−b 2 2 bX Y − 2 ΔX,Y + a+b 2 X + 2 Y H2 = a+b 2 2 bX Y − 12 ΔX,Y + a−b 2 X + 2 Y The higher order E (k) are determined through the perturbation formulas presented in Sect. 3. All odd order E (k) are zero (see Appendix). ρ) and E2 (˜ ρ) gives rise to twice the 4. The presence of the two levels E1 (˜ number of vibrational levels as usual in the following sense: If b = 0, the upper and lower component equations of the leading order equation which determines E (2) , are both two-dimensional harmonic oscillator equations. So, there will be two eigenfunctions, one associated with the each of the upper and lower components, for each of the usual eigenstates of the usual harmonic oscillator. Then for small b, this will give rise to two vibrational states via a perturbative approach, for each of the usual harmonic oscillator states. This is shown in detail in Sect. 5. 5. For small b and , the ground state of H() (meaning the lowest vibrational level corresponding to the R–T pair we are considering) is degenerate, corresponding to a pair of states which are eigenfunctions of LTz OT with eigenvalues lzT OT = ±1. In Sect. 5, we give plots which suggest that for approximately 0.925a < b < a the ground state is non-degenerate, corresponding to a state with lzT OT = 0. The paper is organized as follows: In Sect. 3, we derive perturbation formulas to construct the quasimodes that will enter in our main theorem. In Sect. 4, we prove various properties of the leading order Hamiltonian that are needed to prove the main theorem. In Sect. 5, we analyze the leading order eigenvalue problem. Only some of the eigenvalues and eigenfunctions of the
Vol. 11 (2010)
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leading order equation are solved for exactly. In Sect. 6, the degeneracy structure of the full Hamiltonian H() is discussed. In Sect. 7, we use the results of the previous sections to prove the main theorem.
3. The Construction of the Quasimodes Before we begin the formal expansion, we first look at some properties of the electronic eigenvectors and eigenvalues, construct electronic basis vectors that are smooth in terms of the nuclear coordinates, and derive the leading orders of the matrix elements of the electronic Hamiltonian in this basis. 3.1. The Two-Dimensional Electronic Basis Vectors For the N electrons, as well as the nuclei, we use the same fixed reference frame previously described. Let (rj , θj , zj ) be the cylindrical coordinates of the jth electron in this frame. Suppose that for ρ˜ > 0, ψ(x, y; θ1 , θ2 , . . . , θN ) : R2 → Hel is an electronic eigenvector of h(x, y). We have suppressed the dependence on rj and zj because it is irrelevant to the discussion here. The electronic eigenfunctions are invariant with respect to a rotation of the entire molecule. So, the eigenfunctions have the property ρ, 0; θ1 − φ, θ2 − φ, . . . , θN − φ) ψ(x, y; θ1 , θ2 , . . . , θN ) = ψ(˜ for ρ˜ > 0. It follows that if ψ(x, y) is continuous at ρ˜ = 0, then ψ(0, 0) has no θj dependence. Since an eigenvector corresponding to an R–T pair with positive |lzel | value must have θj dependence at ρ˜ = 0, we do not have well-defined continuous electronic eigenfunctions of h(x, y) in a neighborhood of ρ˜ = 0, that correspond to an R–T pair with value |lzel | > 0. We need basis vectors for the ρ) and E2 (˜ ρ) that are smooth in x and y. two-dimensional eigenspace of E1 (˜ The matrix elements of h(x, y) in our electronic basis determine the form of the leading order equations to follow. We note that in deriving these matrix elements, we do not use the matrix elements of Lel z . Only second order splitting in E1 and E2 is needed, as well as the fact that our smooth basis vectors are not eigenvectors of h(x, y). This gives rise to off-diagonal terms in the basis representation of h(x, y). In this sense, the unusual form of the leading order equations can be thought of as a result of the discontinuity of the electronic eigenvectors in the nuclear coordinates, i.e. there is no smooth electronic basis that diagonalizes the electronic hamiltonian. We note that matrix elements we derive here, are related by an (x, y)-independent unitary transformation to those given by Yarkony [29]. See also Worth and Cederbaum [28] for a general discussion of the topology and classification of different types of intersections of potential surfaces. We now describe our approach. Choose any two normalized orthogonal electronic vectors ψ1 and ψ2 that span the eigenvalue 0 eigenspace of h(0, 0). Let P (x, y) denote the two dimensional projection onto the electronic eigenspace associated to the two eigenvalues of h(x, y). For small x and y, define 1 P (x, y) ψ1 . Ψ1 (x, y) = (3.1) ψ1 , P (x, y) ψ1
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Let P1 (x, y) denote the orthogonal projection onto this vector, i.e., P1 (x, y) = |Ψ1 (x, y) Ψ1 (x, y)| . Next, define χ(x, y) = (1 − P1 (x, y))P (x, y) ψ2 , and Ψ2 (x, y) =
1 χ(x, y), χ(x, y)
χ(x, y).
(3.2)
Then {Ψ1 (x, y), Ψ2 (x, y)} is an orthonormal basis for the range of P (x, y). From the formula [26] 1 (λ − h(x, y))−1 dλ, P (x, y) = 2πi C
ρ) and E2 (˜ ρ) but where C is a closed path in the complex plane encircling E1 (˜ no other spectrum of h(x, y), we see that these vectors are smooth in x and y, since we have assumed that the potentials are smooth and hence the resolvent of h(x, y) is as well (recall we are only working in a neighborhood of the origin (x, y) = (0, 0)). Note that we can arrange for these vectors to be real, which we assume has been done. 3.2. The Matrix Elements of the Electronic Hamiltonian The span of {Ψ1 (x, y), Ψ2 (x, y)} is an invariant subspace for h(x, y). Using coordinates in this basis, the restriction of h(x, y) to this subspace is unitarily equivalent to the real symmetric matrix h11 (x, y) h12 (x, y) , h21 (x, y) h22 (x, y) where hjk (x, y) = Ψj (x, y), h(x, y)Ψk (x, y). Again, since we have smooth potentials, hij (x, y) can be expanded in powers of x and y. Since we assume the degeneracy splits at second order, the 2 2 ρ4 ) and E2 (x, y) = eigenvalues of this matrix are E1 (x, y) = a+b 2 (x + y ) + O(˜ a−b 2 2 4 ρ ). Using these expressions for the eigenvalues we show that 2 (x + y ) + O(˜ up to an (x, y)-independent unitary transformation, this matrix is a+b 2 a−b 2 ±bxy 2 x + 2 y (3.3) a−b 2 a+b 2 ±bxy 2 x + 2 y To show this, we consider a traceless, real symmetric matrix ˜ 11 (x, y) ˜ 12 (x, y) h h , ˜ 11 (x, y) ˜ 21 (x, y) −h h
(3.4)
˜± (x, y) = ± ρ˜2 + O(˜ ρ4 ). The form in (3.3) will follow from with eigenvalues E the analysis below.
Vol. 11 (2010)
Born–Oppenheimer Expansion Near a Renner–Teller
Using (3.4), we have the characteristic equation 2 ˜2 + h ˜2 . ˜± E + O(˜ ρ6 ) = h 11 12
31
(3.5)
By expanding in powers of x and y and equating orders in the above equation, ˜ 12 must ˜ 11 and h it can be easily shown that the constant and linear terms of h vanish. We then write, ˜ 12 (x, y) ˜ 11 (x, y) h h ρ3 ), (3.6) = A x2 + B y 2 + C xy + O(˜ ˜ ˜ h21 (x, y) −h11 (x, y) where A, B, and C are traceless 2 by 2 matrices with constant entries. We can apply a constant unitary transformation to (3.6) that diagonalizes A, which we assume has been done. An obvious consequence of (3.5) and ˜± (x, y) = ± ρ˜2 + O(˜ E ρ4 ) is that if A is diagonal, it must be 1 0 A= . 0 −1 We let B=
b11 b12
b12 , −b11
and C =
c12 , −c11
c11 c12
˜± (x, y) = ± ρ˜2 + O(˜ and use (3.5) with E ρ4 ) to solve for bij and cij by equating the powers of x and y. This gives us four equations, the first equation comes from the y 4 coefficients, the second comes from the x2 y 2 coefficients, etc. y 4 : 1 = b211 + b212
2 2
x y : 2 = 2 b11 + 3
(3.7)
c211
+
c212
(3.8)
xy : 0 = 2(b11 c11 + b12 c12 )
(3.9)
3
x y : 0 = 2c11
(3.10)
These equations have three solutions. Two of the solutions are (b11 , b12 , c11 , c12 ) = (−1, 0, 0, ±2), which give
˜ 11 (x, y) h ˜ 21 (x, y) h
˜ 12 (x, y) h ˜ 11 (x, y) −h
=
x2 − y 2 ± 2xy
± 2xy . −(x2 − y 2 )
These solutions give rise to (3.3). The only other solution of Eqs. (3.7)–(3.10) is (b11 , b12 , c11 , c12 ) = (1, 0, 0, 0), which gives rise to h11 (x, y) h21 (x, y)
h12 (x, y) h22 (x, y)
a+b =
2
0
ρ˜2
0 a−b 2
ρ˜2
.
We do not consider this case. Aside from being uninteresting, it implies that the basis vectors are the eigenfunctions of h(x, y) (at least to leading order). We assume that the off diagonal terms in (3.3) are bxy, since the −bxy case is related by the trivial change of coordinates y → −y.
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3.3. The Formal Expansion To construct the quasimodes in Theorem 2.1, we introduce the scaled variables (X, Y ) = (x/, y/). The intuition of the Born–Oppenheimer approximation suggests that the adiabatic effects will occur on the (x, y) = (X, Y ) scale, whereas the semi-classical motion of the nuclei is determined on the (X, Y ) scale. In terms of the (X, Y ) variables, the Hamiltonian in (2.2) is H() = −
2 ΔX,Y + h( X, Y ). 2
We define H to be the Hilbert space L2 (R2 , dX dY ; C2 ) and we denote the inner product on this space by ·, · H . We seek solutions to H() Ψ(, X, Y ) = E() Ψ(, X, Y ). The wave function Ψ(, X, Y ) can be written in terms of the orthonormal basis functions { Ψ1 (x, y), Ψ2 (x, y) } from (3.1) and (3.2) as Ψ(, X, Y ) = f (, X, Y ) Ψ1 ( X, Y ) +g(, X, Y )Ψ2 ( X, Y ) + ψ⊥ (, X, Y ),
(3.11)
where ψ⊥ , Ψi el = 0. Substituting (3.11) in H() Ψ(, X, Y ) = E() Ψ(, X, Y ) gives three equations; one along Ψ1 , one along Ψ2 , and one in span{Ψ1 , Ψ2 }⊥ . We denote the projection on span{Ψ1 , Ψ2 }⊥ by P⊥ . Along Ψ1 : −
2 2 4 ΔX,Y f + h11 f + h12 g − Ψ1 , ΔX,Y ψ⊥ el − f Ψ1 , Δx,y Ψ1 el 2 2 2 ∂g 4 ∂Ψ ∂g ∂Ψ2 2 3 Ψ1 , el + Ψ1 , el − gΨ1 , Δx,y Ψ2 el − 2 ∂X ∂x ∂Y ∂y = E()f. (3.12)
i Above we have used that Ψi , ∂Ψ ∂x el = 0, which we know from normalization and the fact that the electronic basis vectors were chosen real. Along Ψ2 we get a similar equation with f ↔ g, Ψ1 ↔ Ψ2 , h11 ↔ h22 , h12 ↔ h21 . In span{Ψ1 , Ψ2 }⊥ :
−
2 4 4 P⊥ [ΔX,Y ψ⊥ ] + (h P⊥ ) ψ⊥ − f P⊥ [Δx,y Ψ1 ] − g P⊥ [Δx,y Ψ2 ] 2 2 2
∂f ∂Ψ1 ∂Ψ1 ∂f 3 P⊥ P⊥ − + ∂X ∂x ∂Y ∂y
∂Ψ2 ∂Ψ2 ∂g ∂g P⊥ P⊥ + + ∂X ∂x ∂Y ∂y (3.13) = E() ψ⊥ .
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We adopt the following notation for simplicity: Tij (x, y) = Ψi , Δx,y Ψj el ,
∂Ψj , Aij (x, y) = Ψi , ∂x el
∂Ψj Bij (x, y) = Ψi , . ∂y el We have identities involving these quantities since {Ψ1 , Ψ2 } are orthonormal and real valued. For instance we know the diagonal elements of A and B are zero and A12 = −A21 , B12 = −B21 . Now we expand all ∞functions and operators with dependence. For example, f (, X, Y ) = k=0 k f (k) (X, Y ). For functions and operators with exclusively (x, y) dependence, we know the form of the expansions. For exam∞ k (k) (k) ple, Ψ1 (x, y) = Ψ1 ( X, Y ) = k=0 Ψ1 (X, Y ), where Ψ1 (X, Y ) = k k ∂ Ψ1 1 k k−j . Equations (3.12) and (3.13) become: j=0 j!(k−j)! ∂xj ∂y k−j (0, 0)X Y ∞
k
−
k=2
+
∞
k
∞
∞
ΔX, Y f (k−2) +
k
k
k=4
+
−
j=2
k=2
+
1 2
k
k=3
1 2
∞
k
k
(j−2)
Ψ1
(j)
j=0
k=0
(j)
h11 f (k−j) + h12 g (k−j)
(k−j)
, ΔX, Y ψ⊥
el
k 1 (j−4) (k−j) (j−4) T11 f + T12 g (k−j) − 2 j=4
∞ k
k (j−3) ∂ (j−3) ∂ − B12 k E (j) f (k−j) −A12 g (k−j) = ∂X ∂Y j=3 j=0 k=0
(3.14) and ∞ k=2
∞ k k 1 (j−2) (k−j) (j) (k−j) ΔX,Y ψ⊥ + k (hP⊥ ) ψ⊥ − P⊥ 2 j=2 j=0 k
+
∞
k
k=4
j k
−
j=4 l=4
k=0
(j−l)
P⊥
(l−4)
(Δx,y Ψ1 )
(l−4) (Δx,y Ψ2 ) g (k−j) (l−3) j ∞ k ∂Ψ ∂ 1 (j−l) + −P⊥ k ∂x ∂X j=3 l=3 k=3 (l−3) ∂Ψ1 ∂ (j−l) f (k−j) −P⊥ ∂y ∂Y (j−l) +P⊥
1 2
f (k−j)
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M. S. Herman
+
∞
k
(j−l) −P⊥ ∞ k=0
k
j=3 l=3
k=3
=
j k
k
∂Ψ2 ∂y
(j−l) −P⊥
(l−3)
(k−j)
E (j) ψ⊥
∂Ψ2 ∂x
∂ ∂Y
Ann. Henri Poincar´e
(l−3)
∂ ∂X
g (k−j)
.
(3.15)
j=0
We now collect terms at each order of . Recall there is an equation along Ψ2 analogous to (3.14). At each order, we will combine these two similar equations into one matrix equation. Order 0 The 0 terms require (0) (0) (0) (0) h11 f h12 (0) f = E , (0) (0) g (0) g (0) h21 h22 ( h P⊥ )
(0)
(0)
(3.16)
(0)
ψ⊥ = E (0) ψ⊥ .
(3.17)
The hij (x, y) vanish until second order, so this forces E (0) = 0 in (3.16), and (0) consequently ψ⊥ = 0 after applying the reduced resolvent of (hP⊥ )(0) in (3.17). Order 1 As above, the 1 terms reduce to (0) (1) f E = 0, g (0) (0)
(h P⊥ )
(1)
ψ⊥ = 0.
(1)
So we get E (1) = 0 and ψ⊥ = 0. Order 2 Using the known second order terms for the hij (x, y), the 2 terms require (0) (0) f (2) f H2 =E , g (0) g (0) (2)
(hP⊥ )(0) ψ⊥ = 0, where H2 =
− 12 ΔX,Y +
a+b 2 2 X
bX Y
+
a−b 2 2 Y
− 12 ΔX, Y
bX Y 2 + a−b 2 X +
a+b 2
Y2
Recall we have assumed the +bxy case for the off diagonal entries. By (2) again applying the reduced resolvent in the last equation we have ψ⊥ = 0. In Sect. 4 we show that H2 is self-adjoint (on the correct domain) and has purely discrete spectrum with infinitely many eigenvalues for a > b > 0. We are only able to solve for some of them exactly. In Sect. 5 we show that there is at most a two-fold degeneracy in the eigenstates of H2 , but that no splitting occurs in
Vol. 11 (2010)
Born–Oppenheimer Expansion Near a Renner–Teller
35
the quasimode eigenvalues, i.e., the degeneracy remains to all orders of . We can therefore proceed as if the eigenstates of H2 were non-degenerate, since we can take any linear combination of degenerate states for f (0) and g (0) , and we know it will lead to a valid quasimode and energy E(). Fix E (2) , f (0) and g (0) corresponding to one of the states of H2 . Order 3 The 3 terms require (0) (0) (1) f f (2) (3) f H3 + H2 − E =E , g (0) g (1) g (0)
(0)
(h P⊥ )
(3) ψ⊥
(3.18)
(0) (0) ∂Ψ1 ∂Ψ1 ∂ ∂ (0) + P⊥ = f (0) ∂x ∂X ∂y ∂Y (0) (0) ∂Ψ2 ∂Ψ2 ∂ ∂ (0) (0) + P⊥ g (0) , + P⊥ ∂x ∂X ∂y ∂Y (0) P⊥
(3.19) where
H3 =
(3)
h11 (3) h21
(3)
h12 (3) h22
+
0 (0) ∂ (0) ∂ −A21 ∂X − B21 ∂Y
(0) ∂ ∂X
−A12
(0)
− B12 0
∂ ∂Y
Since H2 is self-adjoint, we can take inner products of both sides in (3.18) with f (0) to obtain g (0) (0) (0) f f . E (3) = , H 3 g (0) g (0) H In the appendix we argue that all of the odd order E (k) are zero. Let Q⊥ be the projection in H onto the subspace perpendicular to the eigenspace of the eigenvalue E (2) of H2 . Adopting “intermediate normalization” we may choose the non-zero order wave functions perpendicular to the eigenspace of E (2) (note that this will produce a non-normalized quasimode), so that (k) (k) f f = Q⊥ , g (k) g (k) for k ≥ 1. Then from (3.18) we get (1) (0) −1 f f (2) Q⊥ H3 = − H2 − E . g (1) g (0) r From (3.19) we have (3) ψ⊥
(3.20)
(0) ∂Ψ1 ∂ ∂ (0) + P⊥ f (0) = (h P⊥ ) ∂X ∂y ∂Y r (0) (0) ∂Ψ2 ∂Ψ2 ∂ ∂ (0) (0) (0) + P⊥ g + P⊥ . (3.21) ∂x ∂X ∂y ∂Y
(0)
−1
(0) P⊥
∂Ψ1 ∂x
(0)
36
M. S. Herman
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Order 4 The 4 terms require (2) (1) (0) f f f (3) (4) H2 −E (2) −E − E + H + H = 0, 3 4 g (2) g (1) g (0) (0)
(4)
(1)
(3.22)
(3)
(hP⊥ ) ψ⊥ = − (hP⊥ ) ψ⊥ 1 (0) (0) (0) (0) f (0) +P⊥ (Δx,y Ψ2 ) g (0) P⊥ (Δx,y Ψ1 ) + 2 (l−3) (l−3) j 4 ∂Ψ1 ∂Ψ1 ∂ ∂ (j−l) (j−l) + P⊥ f (4−j) P⊥ + ∂x ∂X ∂y ∂Y j=3 l=3 (l−3) (l−3) j 4 ∂Ψ2 ∂Ψ2 ∂ ∂ (j−l) (j−l) +P⊥ g (4−j) , + P⊥ ∂x ∂X ∂y ∂Y j=3 l=3
(3.23) where H4 =
1 − 2
⎛ (0) T ⎝ 11 (0) T21
(0)
T12
(0)
⎞ ⎠+
T22
0 + (1) ∂ (1) ∂ −A21 ∂X − B21 ∂Y
(4) h11 (4) h21
(4)
(1) ∂ ∂X
−A12
h12 (4) h22 (1) ∂ ∂Y
− B12 0
.
Using what we know through order 3, we can solve (3.22) and (3.23). From (3.22) we obtain: (0) (0) (0) (1) f f f f (3) − E + E (4) = , H , H 3 4 (0) g (0) g (1) g g (0) H H and
(0) −1 f (1) f Q⊥ H3 − E (3) + H . f (2) g (2) = − H2 − E (2) 4 g (1) g (0) r
From (3.23) we get (4) ψ⊥
= (h P⊥ )
(0)
−1 r
− (h P⊥ )
(1)
(3)
ψ⊥
1 (0) (0) (0) (0) f (0) + P⊥ (Δx, y Ψ2 ) g (0) P⊥ (Δx,y Ψ1 ) 2 (l−3) (l−3) j 4 ∂Ψ1 ∂Ψ1 ∂ ∂ (j−l) (j−l) + P⊥ f (4−j) P⊥ + ∂x ∂X ∂y ∂Y j=3 l=3 ⎤ (l−3) (l−3) j 4 ∂Ψ2 ∂Ψ2 ∂ ∂ (j−l) (j−l) +P⊥ g (4−j) ⎦. + P⊥ ∂x ∂X ∂y ∂Y j=3
+
l=3
Vol. 11 (2010)
Born–Oppenheimer Expansion Near a Renner–Teller
37
Order k ≥ 5 We now show that we can proceed in this manner to any order of desired. In Sect. 4 we will show that all of the quantities involved exist in the relevant Hilbert space. If k ≥ 5, the k terms require
(k−3) f (k−2) f (3) H2 − E + H3 − E g (k−2) g (k−3) k−1 (0) f (k−j) f (j) (k) Hj − E + + Hk − E g (k−j) g (0) j=4 k−3 1 Ψ(j−2) , ΔX,Y ψ (k−j) el 1 ⊥ + = 0, − (j−2) (k−j) 2 Ψ2 , ΔX,Y ψ⊥ el j=2 (2)
(hP⊥ )
(0)
(k)
ψ⊥ =
k−3 j=2
(3.24)
k−3 1 (j−2) (k−j) (j) (k−j) ΔX,Y ψ⊥ − P⊥ (hP⊥ ) ψ⊥ 2 j=1
1 (j−l) (j−l) (l−4) (l−4) (k−j) (Δx,y Ψ1 ) f (k−j) +P⊥ (Δx,y Ψ2 ) g P⊥ + 2 j=4 l=4 (l−3) (l−3) j k ∂Ψ1 ∂Ψ1 ∂ ∂ (j−l) (j−l) +P⊥ f (k−j) + P⊥ ∂x ∂X ∂y ∂Y j=3 l=3 (l−3) (l−3) j k ∂Ψ2 ∂Ψ2 ∂ ∂ (j−l) (j−l) + P⊥ g (k−j) + P⊥ ∂x ∂X ∂y ∂Y j=3 j k
l=3
+
k−3
(k−j)
E (j) ψ⊥
,
(3.25)
j=2
where Hj =
1 − 2 ⎛
+⎝
(j−4) T11 (j−4) T21
(j−4)
T12 (j−4) T22
⎛
+⎝
−
(j)
h12
(j) h21
(j) h22
⎞ ⎠
(j−3) ∂ ∂X
−A12
0 (j−3) ∂ −A21 ∂X
(j)
h11
(j−3) ∂ B21 ∂Y
(j−3) ∂ ∂Y
− B12
⎞ ⎠,
0
for j ≥ 4. Following what we have seen through order 4, assume from previous orders that
f (j) g (j)
for j = 0, 1, . . . , k − 3,
E (j)
and
(j)
ψ⊥ for j = 0, 1, . . . , k − 1,
38
M. S. Herman
Ann. Henri Poincar´e
are already determined. Then, we can solve (3.24) and (3.25) for f (k−2) , g (k−2) , (k) ψ⊥ , and E (k) . From (3.24) we obtain: (0) (0) k−1 (k−j) f (0) f f f (j) − E + E (k) = , H , H j k (0) g (0) g (k−j) g g (0) H H j=3 k−3 (j−2) (k−j) 1 f (0) , ΔX,Y ψ⊥ el Ψ1 − (3.26) (0) (j−2) (k−j) g 2 j=2 Ψ2 , ΔX,Y ψ⊥ el H
and
(k−2)
f g (k−2)
= − H2 − E (2)
−1
⎡
k−1
Q⊥ ⎣
Hj − E (j)
f (k−j)
g (k−j) ⎤ (0) k−3 (j−2) (k−j) 1 Ψ1 , ΔX,Y ψ⊥ el ⎦ f + Hk . − (k−j) g (0) 2 j=2 Ψ(j−2) , Δ ψ el X,Y ⊥ 2 r
j=3
(3.27)
From (3.25) we get (k) ψ⊥
−1 k−3 k−3 1 (j−2) (k−j) (k−j) (0) ΔX,Y ψ⊥ − = (hP⊥ ) (h P⊥ )(j) ψ⊥ P⊥ 2 r j=2 j=1 j k 1 (j−l) (j−l) P⊥ (Δx,y Ψ1 )(l−4) f (k−j) +P⊥ (Δx,y Ψ2 )(l−4) g (k−j) 2 j=4 l=4 (l−3) (l−3) j k ∂Ψ1 ∂Ψ1 ∂ ∂ (j−l) (j−l) + P⊥ + P⊥ f (k−j) ∂x ∂X ∂y ∂Y j=3 l=3 (l−3) (l−3) j k ∂Ψ2 ∂Ψ2 ∂ ∂ (j−l) (j−l) + P⊥ + P⊥ g (k−j) ∂x ∂X ∂y ∂Y j=3 l=3 k−3 (j) (k−j) + E ψ⊥ . (3.28)
+
j=2
So we can proceed in this manner to obtain Ψ() and E() up to any order in .
4. Properties of the Leading Order Hamiltonian We adopt the following notation throughout: 1 0 . If A is an operator on the Hilbert space 1. We let I2 = 0 1 A 0 2 2 L (R , dX dY ), then A ⊗ I2 , is the operator on H given by . 0 A 2. If D(A) is the domain of the operator A on the Hilbert space L2 (R2 , dX dY ), then D(A ⊗ I2 ) = D(A) ⊕ D(A) ⊂ H.
Vol. 11 (2010)
Born–Oppenheimer Expansion Near a Renner–Teller
39
In what follows, we prove various needed properties for the expansion to all orders. Let a+b 2 a−b 2 1 bX Y 2 X + 2 Y H2 = − ΔX,Y ⊗ I2 + a−b a+b 2 2 . bX Y 2 2 X + 2 Y ˜ Y˜ ) = (a1/4 X, a1/4 Y ) and ˜b = b , then Note that if we let (X, a ⎡ √ 1 1 ˜2 ˜2 ( X H2 = a ⎣ − ΔX, + + Y ) ⊗ I2 ˜ Y˜ 2 2 ⎛ ⎞⎤ 1 ˜ 2 − Y˜ 2 ˜ Y˜ X X 2 ⎠⎦ . + ˜b ⎝ ˜ Y˜ ˜ 2 − Y˜ 2 X − 12 X
(4.1)
We now use the Kato–Rellich Theorem [25] to prove self-adjointness of H2 . Theorem 4.1. If a > b > 0, then H2 is self-adjoint on DHO ⊕ DHO , where DHO is the usual Harmonic oscillator domain in L2 (R2 , dXdY ), and essen˜ HO ⊕ D ˜ HO , where D ˜ HO is any core for the usual Hartially self-adjoint on D monic oscillator. Proof. Define
HHO =
and
1 1 − ΔX,Y + (X 2 + Y 2 ) ⊗ I2 2 2
1 V (˜b) = ˜b
2
X2 − Y 2 XY
XY
. − 12 X 2 − Y 2
We prove that for 0 < ˜b < 1, V (˜b) is relatively bounded with respect to HH0 , with relative bound ˜b. The conclusion then follows from the Kato–Rellich theorem [25] and (4.1). ˜ For each fixed X and Y , the eigenvalues of V (b) are ± 2b (X 2 + Y 2 ). It follows that 1 V (˜b) v (X 2 + Y 2 ) ⊗ I2 v , ≤ ˜b 2 e e where v ∈ C2 is any two component vector, and we use the usual Euclidean norm. This inequality implies the L2 (R2 , dX dY ; C2 ) = H norm estimate 1 2 2 ˜ ˜ V (b)ψ (X + Y ) ⊗ I2 ψ , ≤b 2 H H where ψ(X, Y ) ∈ H is a two-component vector-valued function. We now show that 1 2 (X + Y 2 ) ⊗ I2 ψ ≤ ˜bHHO ψH + ˜bψH . (4.2) V (˜b)ψH ≤ ˜b 2 H
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M. S. Herman
Ann. Henri Poincar´e
for all ψ ∈ DHO ⊕ DHO . We have already shown the first inequality. The hard part is the second estimate, which follows from 2 −ΔX,Y + X 2 + Y 2 ⊗ I2 ψ + 2ψ. X + Y 2 ⊗ I2 ψ ≤ This easily follows from 2 X + Y 2 ⊗ I2 ψ
2
≤
−ΔX,Y + X 2 + Y 2 ⊗ I2 ψ
2
+ 4ψ2 . (4.3)
Rather than proving this directly, let us first prove a simpler relative bound estimate for the operators on L2 (R, dx). We show that for 2 ∂ 2 , φ ∈ D − ∂x 2 + x 2 ∂2 2 − 2 + x2 φ + 2φ2 . x2 φ ≤ (4.4) ∂x ∂ To prove this, let p = −i ∂x , and calculate the commutators
[x, p ] = i
and
[x, p2 ] = 2 i p.
We have x2 φ
2
= φ, x4 φ = φ, (p2 + x2 )2 − x2 p2 − p2 x2 − p4 φ 2 ≤ p2 + x2 φ − φ, x2 p2 + p2 x2 φ.
(4.5)
In this last expression, we use the commutators above to write φ, x2 p2 + p2 x2 φ = φ, xp2 x + x[x, p2 ] + xp2 x + [p2 , x]x φ = 2φ, xp2 xφ + 2iφ, (xp − px)φ = 2φ, xp2 xφ − 2φ, φ. In this last expression, the first inner product is the expectation of a positive operator (since xp2 x has the form A∗ A with A = px). Using this and (4.5), we see that 2 2 x2 φ ≤ p2 + x2 φ + 2φ2 , and (4.4) is proved. Now we simply mimic the proof of (4.4) to prove (4.3). We write ! (X 2 + Y 2 )φ2 = φ, ((− ΔX,Y + X 2 + Y 2 )2 − Δ2X,Y " + ΔX,Y (X 2 + Y 2 ) + (X 2 + Y 2 )ΔX,Y )φ The operator Δ2X,Y is positive. The operator −ΔX Y 2 = −Y 2 ΔX is also positive since it equals A∗ A with A = pX Y . Similarly, −ΔY X 2 = −X 2 ΔY is positive. By the commutator tricks we used above, −ΔX X 2 − X 2 ΔX and −ΔY Y 2 − Y 2 ΔY each are positive operators minus twice the identity. Thus for all φ ∈ DHO , ! " φ, ((− ΔX,Y + X 2 + Y 2 )2 − Δ2X,Y + ΔX,Y (X 2 + Y 2 ) + (X 2 + Y 2 )ΔX,Y )φ " ! ≤ φ, (−ΔX,Y + X 2 + Y 2 )2 φ + 4 φ, φ
Vol. 11 (2010)
and hence,
Born–Oppenheimer Expansion Near a Renner–Teller
X2 + Y 2 φ
2
≤
−ΔX,Y + X 2 + Y 2 φ
2
41
+ 4φ2 .
It follows that (4.3) holds for all ψ ∈ DHO ⊕ DHO . This proves (4.2) and the theorem follows. Unless otherwise stated, it is assumed that by H2 we are referring to this operator with domain D(H2 ) = DH0 ⊕ DHO . We now show that H2 has purely discrete spectrum. Theorem 4.2. If a > b > 0, H2 has purely discrete spectrum, with countably many eigenvalues {μj (H2 )}∞ j=1 satisfying √ N a − b ≤ μN (N −1)+1 (H2 ) ≤ μN (N −1)+2 (H2 ) √ ≤ · · · ≤ μN (N +1) (H2 ) ≤ N a + b, for N = 1, 2, 3, . . . Proof. Let (ρ, φ) be the usual polar coordinates associated with (X, Y ) and let Δρ,φ denote the Laplacian in these coordinates. Define the unitary operators U, W : H → H by (defined as multiplication operators on H): iφ cos(φ) − sin(φ) 1 e−iφ e U= and W = √ . iφ −ie−iφ 2 ie sin(φ) cos(φ) Define
H0 = U
−1
H2 U =
and
H± 0 =
− 12 Δρ,φ + 2ρ12 + ∂ − ρ12 ∂φ
− 21 Δρ,φ + 2ρ12 + ∂ − ρ12 ∂φ
a±b 2 2 ρ
a+b 2 2 ρ
1 ∂ ρ2 ∂φ 1 − 2 Δρ,φ + 2ρ12
1 ∂ ρ2 ∂φ 1 − 2 Δρ,φ + 2ρ12
+ and note that H− 0 ≤ H0 ≤ H0 . Now we define 1 2 − 2 Δρ,φ + a±b −1 ± 2 ρ H± = W H W = 1 0 0
+
a−b 2 2 ρ
,
+
a±b 2 2 ρ
0 − 12 Δρ,φ +
,
a±b 2 2 ρ
.
In the context of the min/max principle [26], for all n ∈ N, − + + μn (H− 1 ) = μn (H0 ) ≤ μn (H0 ) = μn (H2 ) = μn (H0 ) ≤ μn (H0 ) = μn (H1 ). ± The operators H√ 1 have purely discrete spectrum, with 2N -fold degenerate eigenvalues of N a ± b for N = 1, 2, . . . So, H2 must have purely discrete spectrum with eigenvalues μ1 (H2 ) ≤ μ2 (H2 ) ≤ · · · satisfying the required bound.
To prove the quasimode can be expanded to any order in , we must show the terms arising at arbitrary order in the equations of Sect. 3 are in H. This follows from the propositions and lemmas we now prove. A similar analysis was needed in [12] and the proofs presented here are analogous to those found
42
M. S. Herman
Ann. Henri Poincar´e
in [12]. For our purposes it must be shown that the details can be extended to this situation on H, which is not obvious, and a careful analysis is necessary. Before we prove Proposition 4.3, we consider a different decomposition of H2 . We define H0 and V to be a+b 2 a−b 2 1 X + 2 Y bXY 2 H0 = − ΔX,Y ⊗ I2 and V = a−b a+b 2 2 , bXY 2 2 X + 2 Y so that H2 = H0 + V . Note that for any X, Y , the eigenvalues of V are a+b a−b 2 2 2 2 2 2 2 (X + Y ) and 2 (X + Y ). So for f, g ∈ L (R ), a−b f f (X 2 + Y 2 ) |f |2 + |g|2 dX dY, ,V ≥ g g 2 and V is a positive operator. Note: Using the notation introduced in the proposition below, the rest of the results of this section are only needed for some γ > 0, while we prove the results for any γ > 0. f Proposition 4.3. Let Ψ = ∈ H be a solution of H2 Ψ = EΨ, with E > 0. g Then, f, g ∈ C ∞ (R2 ), ∇f, ∇g ∈ L2 (R2 ), and for any γ > 0, f, g ∈ D(eγx ), ∇f, ∇g ∈ D(eγx ), √ where x = 1 + X 2 + Y 2 .
Δf, Δg ∈ D(eγx ),
a−b 2 2 Proof. Let V11 = a+b 2 X + 2 Y , V12 = V21 = bXY, and V22 = a+b 2 2 Y . Then, f, g satisfy the following pair of equations:
a−b 2
X2 +
(− Δ + V11 ) f + V12 g = Ef
(4.6)
(− Δ + V22 ) g + V21 f = Eg
(4.7)
To show that f, g ∈ C ∞ (R2 ), we follow the proof of Theorem IX.26 of [25]. Let Ω be a bounded open set in R2 . Since f, g ∈ L2 (R2 ) = W0 and the Vij ∈ C ∞ , we have V11 f, V21 f, V12 g, V22 g ∈ W0 (Ω). It follows from (4.6) and (4.7) that Δf, Δg ∈ W0 (Ω). Then by the Lemma on pg. 52 of [25], f, g ∈ W2 (Ω). Repeating the argument we get f, g ∈ Wm (Ω) ∀ m ∈ Z. It follows from Sobolev’s Lemma that f, g ∈ C ∞ on Ω. Since Ω was arbitrary f, g ∈ C ∞ (R2 ). We now show ∇f, ∇g ∈ L2 . We know Ψ ∈ D(H2 ). Let D(−Δ) and Q(−Δ) be the domain of self-adjointness and quadratic form domain of −Δ respectively. Then D(H2 ) ⊂ D(−Δ) ⊕ D(−Δ) ⊂ Q(−Δ) ⊕ Q(−Δ) # $ f = Ψ= : ∇f, ∇g ∈ L2 (R2 ) . g We now use the Combes–Thomas argument (see theorem XIII.39 of [26]) to prove that f, g ∈ D(eγ|X| ). The argument can be repeated for D(eγ|Y | ), and since
Vol. 11 (2010)
Born–Oppenheimer Expansion Near a Renner–Teller
eγx ≤ eγ eγ(|X|+|Y |) ≤ eγ e2γ max{|X|, |Y |} ≤ eγ
43
e2γ|X| + e2γ|Y | ,
we then have f, g ∈ D(eγx ). For α ∈ R, consider the unitary group W (α) = eiαX ⊗ I2 and the operator H2 (α) = W (α)H2 W (α)−1 . We have H2 (α) = H2 +
α2 ∂ ⊗ I2 + i α ⊗ I2 . 2 ∂X
∂ The operator i ∂X is form bounded with respect to −Δ with relative bound ∂ zero. Since V is positive, it follows that i ∂X ⊗ I2 is form bounded with respect to H2 with relative bound zero. So, H2 (α) is an entire analytic family of type (B), with domain D(H2 ). Furthermore, since H2 (α) is unitarily equivalent to H2 for α ∈ R, we know that H2 (α) is self-adjoint and σ(H2 ) = σ(H2 (α)) for α ∈ R. Since H2 (0) = H2 has purely discrete spectrum accumulating at infinity, it has compact resolvent by Theorem XIII.64 of [26]. Then by Theorem 4.3 of section VII of [18], we know H2 (α) has compact resolvent for all α ∈ C. Thus it has purely discrete spectrum for all α ∈ C. Since H2 (α) is an entire analytic family in the sense of Kato, the eigenvalues are analytic on C except possibly at isolated crossings [26]. W (α) unitary implies that the eigenvalues are constant in a neighborhood of the real axis and thus crossings will not be an issue. Therefore, the eigenvalues are entire functions and constant in α. Let P (α) be the projection onto the eigenspace corresponding to the eigenvalue E of H2 (α). Then P (α) is entire in α and has the form
P (α) =
−1 2πi
−1
(H2 (α) − λ)
dλ.
|λ−E|=
If α, α0 ∈ R, P (α + α0 ) =
−1 2πi
−1 = 2πi
−1
(H2 (α + α0 ) − λ)
dλ
|λ−E|=
−1
W (α0 ) (H2 (α) − λ)
W (α0 )−1 dλ
|λ−E|=
= W (α0 )P (α)W (α0 )−1 . For α0 ∈ R, the operator valued function f (α) = W (α0 )P (α)W (α0 )−1 − P (α + α0 ) is entire in α. Since it vanishes ∀ α ∈ R, it is zero ∀ α ∈ C. So P (α + α0 ) = W (α0 )P (α)W (α0 )−1 , for α0 ∈ R and α ∈ C. The hypotheses f of O’Connors lemma are satisfied [26]. So, for the eigenvector Ψ = , we g know Ψ(α) = W (α)Ψ has an analytic continuation to all of C. Therefore f, g ∈ D(eγ|X| ) for any γ > 0.
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M. S. Herman
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From this it now follows that Δf, Δg ∈ D(eγx ), for any γ > 0. To see this, consider %% γx %% γx %%2 %%2 %% e %% e Δ f %%%% [(V11 − E) f + V12 g] %%%% % %% % %% eγx Δg %% = 4 %% eγx [V21 f + (V22 − E) g] %% % % %% 2 %% %% = 4 %%eγx (V11 − E) f + eγx V12 g %% 2
%% %%2 %% %% + %% eγx V21 f + eγx (V22 − E) g %% 2
Let β > 0. Then, %% %% %% 2 −2βx %% e2γx |V21 f |2 dX dY ≤ %%V21 e %%
∞
%% %% %% 2x(γ+β) %%2 f %% < ∞ %%e 2
So, eγx V21 f ∈ L2 (R2 ) and by similar arguments eγx (V11 − E)f, eγx V12 g, eγx (V22 − E)g ∈ L2 (R2 ). Hence, %% γx %%2 %% e Δ f %%%% %% %% eγx Δ g %% < ∞ and Δf, Δg ∈ D(eγx ). For ∇f, ∇g ∈ D(eγx%), we apply Lemma 3.4 of [12]: Let p ∈ C 1 (RN ) and % & % ∇p(x) % suppose for some C < ∞, % p(x) % ≤ 2C ∀ x ∈ RN . If RN (|f |2 + |Δf |2 )pdx < ∞, then ⎛ ⎝
⎞1/2
|∇f |2 pdx⎠
⎛ ≤C⎝
RN
⎞1/2 |f |2 p dx⎠
RN
⎡⎛ ⎤1/2 ⎞1/2 ⎛ ⎞1/2 ⎢ ⎥ + ⎣⎝ |f |2 p dx⎠ ⎝ |Δf |2 p dx⎠ + C 2 |f |2 p dx⎦ . RN
RN
RN
% % % )% 2 We let p(X, Y ) = e2γx . Then % ∇p(X,Y p(X,Y ) % ≤ 2γ ∀(X, Y ) ∈ R . We have already shown that for f and g, the right hand side in the lemma is finite for any γ > 0. So, ∇f, ∇g ∈ D(eγx ) for any γ > 0. Corollary 4.4. Let R(λ) = (H2 − λ)−1 for λ ∈ ρ(H2 ). Let PE be the projection onto the eigenspace associated with E and define r(E) = [(H2 − E)|Ran(I−PE ) ]−1 , the reduced resolvent at E. Then, (eγx ⊗ I2 ) R(λ) (e−γx ⊗ I2 ) and (eγx ⊗ I2 ) r(E) (e−γx ⊗ I2 ) are bounded on H for any γ > 0. In particular, if Ψ ∈ D(eγx ⊗ I2 ), then R(λ) Ψ, r(E) Ψ ∈ D(eγx ⊗ I2 ). Note: See [13] for a proof.
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We need the following lemma for proposition 4.6: n ˜ > 0 and S(t) > 0, such that Lemma 4.5. For fixed exist K > K n t ∈ 2R , there n 2 if p ∈ R satisfies j=1 pj ≥ S(t) , then % % % % n n n % % ˜ K p2j ≤ %% (pj + itj )2 %% ≤ K p2j . % j=1 % j=1 j=1 Furthermore, S(t) is uniformly bounded for t in compact subsets of Rn . 1/2 1/2 n n 2 2 Proof. Let ||t|| = t , ||p|| = p . Taking S(t) = 1+4 ||t||, j=1 j j=1 j one can show 17 2 2 2 ||p|| , (4.8) ||p|| + ||t|| ≤ 16 and then take K = 17/16. Also using this S(t), one can show the above bounds ˜ = 7/16. hold with K f Proposition 4.6. Let Ψ = ∈ H be a solution of H2 Ψ = EΨ, with E > g 0. Then, for any γ > 0, and any α ∈ N2 , Dα f, Dα g ∈ D(eγx ), where α1 α2 ∂Y . D α = ∂X Proof. We use a Paley–Wiener Theorem, Theorem IX.13 of [25]: Let φ ∈ L2 (Rn ). Then eγ|x| φ ∈ L2 (Rn ) for all γ < γ if and only if φˆ has an analytic continuation to the set {p : |Im p| < γ } with the propn 2 n ˆ erty that for each %% t ∈ R %%with |t| < γ , φ(· + it) ∈ L (R ), and for any % % % % ˆ + it) %% < ∞. γ < γ , sup|t|≤γ %%φ(· 2 If a function φˆ satisfies the conditions in this theorem we will say that φˆ is “P-W”. Let pj = −i ∂xj . We present the proof for general n. In our case we have n = 2 with x1 = X and x2 = Y . Proposition 4 shows that fˆ and gˆ are P-W for any γ > 0. In particular we know that fˆ, gˆ are analytic everywhere. So the analyticity condition ) , ∇g, ) , Δg ) Δf ) are also will be a non-issue in the course of the proof. ∇f n P-W for any γ > 0. So p → pj fˆ(p), p → pj gˆ(p), p → j=1 pj2 fˆ(p), and n p → j=1 pj2 gˆ(p) are P-W for all γ > 0. Let S(t) = 1 + 4 ||t|| and BS be a ball of radius S centered at the origin. n ˜ as in the Since j=1 pj2 fˆ(p) is P-W, with Lemma 4.5 (choosing K and K proof above) we have 4 ||p|| |fˆ(p + it)|2 dp Rn \BS(t)
≤
16 7
<∞
2
Rn \BS(t)
% %2 % % % n % 2% % (pj + itj ) % |fˆ(p + it)|2 dp % % j=1 % (4.9)
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uniformly for t in compact subsets of Rn . We only show results involving f . The same results hold with f replaced by %g. % %% %% %% Note that since S(t) and %%fˆ(· + it)%% are uniformly bounded for t in 2
compact subsets of Rn , we only need to prove estimates for ||p|| ≥ S(t). All of the integral estimates that follow hold uniformly for t in compact subsets of Rn . From (4.8) and (4.9) we have
|pj + itj |2 |pk + itk |2 |fˆ(p + it)|2 dp
Rn \BS(t)
≤ Rn \BS(t)
≤
17 16
2
2
2
||p|| + ||t||
2
|fˆ(p + it)|2 dp
4
||p|| |fˆ(p + it)|2 dp
Rn \BS(t)
< ∞. It follows that ∂xj ∂xk f ∈ D(eγx ) for any γ > 0. Again the same will hold for g. We now start an induction on the length |α| in Dα f and Dα g. Assume that Dβ f, Dβ g ∈ D(eγx ) for any γ > 0 and any |β| ≤ m − 1. It suffices to prove that Dα f ∈ D(eγx ) for any γ > 0 and any |α| = m. Following the notation in the proof of Proposition 4, the eigenvalue equation gives us Δ f = V12 g + (V11 − E) f, Δ g = V21 f + (V22 − E) g. where V11 , V12 = V21 , and V22 are polynomials in xj . Let |α | = m − 2. Since the Vij are polynomial, our induction hypothesis gives us Dα Δ f ∈ D(eγx ) for any γ > 0. It follows that for jk ∈ {1, 2, . . . , n}
|pj1 + itj1 |2 |pj2 + itj2 |2 · · · |pjm−2 + itjm−2 |2
Rn \BS(t)
% %2 % n % % % 2% % ×% (pj + itj ) % |fˆ(p + it)|2 dp < ∞, % j=1 %
and from Lemma 4.5 we have 4 |pj1 +itj1 |2 |pj2 +itj2 |2 · · · |pjm−2 +itjm−2 | 2 ||p || |fˆ(p + it)|2 dp < ∞. Rn \BS(t)
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Since the jk are arbitrary, we have n 4 ∞> |pj1 + itj1 |2 |pj2 + itj2 |2 · · · |pjm−2 + itjm−2 |2 ||p|| j1 ,j2 ,...,jm−2 =1
=
Rn \BS(t)
×|fˆ(p + it)|2 dp n
4
j1 ,j2 ,...,jm−2 =1 n R \BS(t)
(p2j1 + t2j1 ) (pj22 + tj22 ) · · · (p2jm−2 + t2jm−2 ) ||p||
×|fˆ(p + it)|2 dp 2 2 4 = (||p|| + ||t|| )m−2 ||p|| |fˆ(p + it)|2 dp Rn \BS(t)
2(m−2)
≥
||p||
4
||p|| |fˆ(p + it)|2 dp
Rn \BS(t)
2m
||p||
=
|fˆ(p + it)|2 dp
Rn \BS(t)
Then using (4.8), we have for any jk ∈ {1, 2, . . . , n} |pj1 + itj1 |2 |pj2 + itj2 |2 · · · |pjm + itjm |2 |fˆ(p + it)|2 dp Rn \BS(t)
2
Rn \BS(t)
≤
17 16
2
(||p|| + ||t|| )m |fˆ(p + it)|2 dp
≤
m
2m
||p||
|fˆ(p + it)|2 dp
Rn \BS(t)
< ∞. So, for arbitrary jk ∈ {1, 2, . . . , n}, p → pj1 pj2 · · · pjm fˆ(p) is P-W and it follows that Dα f ∈ D(eγx ) for any γ > 0 and any |α| = m. The same argument will work with f replaced by g and the proposition is proved. f Lemma 4.7. Let Ψ = , R(λ) = (H2 − λ)−1 for λ ∈ ρ(H2 ), and g be the reduced resolvent at E. If f, g ∈ C ∞ and r(E) = (H2 − E)−1 r α γx (D ⊗ I2 )Ψ ∈ D(e ⊗ I2 ), for all α ∈ N2 and any γ > 0, then α α (D ⊗ I2 )R(λ)Ψ, (D ⊗ I2 )r(E)Ψ ∈ D(eγx ⊗ I2 ), for all α ∈ N2 and any γ > 0. Proof. First note that for any γ1 > γ2 > 0 and j, k = 0, 1, 2, . . ., there exists M > 0 such that %% %% %% %% %% %% γ2 x j k %% %% X Y φ%% ≤ M ||φ|| + %%eγ1 x φ%% . %%e
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This relative bound implies that if φ ∈ D(eγx ) for all γ > 0, then X j Y k φ ∈ D(eγx ) for all γ > 0, and arbitrary j, k = 0, 1, 2, . . .. By an argument similar to the one by which we obtained f, g ∈ C ∞ (R2 ) in the proof of Proposition 4.3, R(λ) and r(E) map functions from C ∞ (R2 ) ⊕ C ∞ (R2 ) to C ∞ (R2 ) ⊕ C ∞ (R2 ). The following identity holds as long as the terms on the right hand side are in L2 (R2 ) ⊕ L2 (R2 ): (∂X ⊗ I2 )R(λ)Φ = R(λ)(∂X ⊗ I2 )Φ − R(λ)[(∂X ⊗ I2 )(V )]R(λ)Φ,
(4.10)
(a + b) X bY . To see this, let R(λ) Φ = where [(∂X ⊗ I2 )(V )] = bY (a − b) X ψ1 and we compute [∂X ⊗ I2 , R(λ)]: ψ2 {(∂X ⊗ I2 )R(λ) − R(λ)(∂X ⊗ I2 )} Φ ψ1 ψ1 = (∂X ⊗ I2 ) − R(λ)(∂X ⊗ I2 )(H2 − λ) ψ2 ψ2 1 (− 2 Δ − λ)ψ1 V11 ψ1 + V12 ψ2 ∂X 0 ∂X ψ 1 = + − R(λ) ∂X ψ 2 0 ∂X V21 ψ1 + V22 ψ2 (− 1 Δ − λ)ψ2 =
∂X ψ 1
2
− R(λ)
1 (− 2 Δ − λ) ∂X ψ1
(− 12 Δ − λ)∂X ψ2 V11 ∂X (ψ1 ) + V12 ∂X (ψ2 ) + V21 ∂X (ψ1 ) + V22 ∂X (ψ2 ) ∂X ψ 2
+
∂X (V11 ) ψ1 + ∂X (V12 ) ψ2 ∂X (V21 ) ψ1 + ∂X (V22 ) ψ2
∂X ψ 1 ∂ ψ ψ1 − R(λ) (H2 − λ) X 1 + [(∂X ⊗ I2 )(V )] ∂X ψ 2 ∂X ψ 2 ψ2 ψ1 = −R(λ)[(∂X ⊗ I2 )(V )] ψ2 =
= −R(λ)[(∂X ⊗ I2 )(V )]R(λ)Φ. Clearly (4.10) holds with X replaced by Y . From the hypotheses on Ψ and Corollary 4.4, we know that for all γ > 0, R(λ)(∂X ⊗ I2 )Ψ ∈ D(eγx ⊗ I2 ) ⊂ L2 (R2 ) ⊕ L2 (R2 ). From Corollary 4.4 and the note above, we know that R(λ) [(∂X ⊗ I2 )(V )] R(λ)Ψ ∈ D(eγx ⊗ I2 ) ⊂ L2 (R2 ) ⊕ L2 (R2 ) for all γ > 0. From this we see that (4.10) holds when applied to Ψ and therefore (∂X ⊗ I2 )R(λ)Φ ∈ D(eγx ⊗ I2 ) for all γ > 0. Similarly, (∂Y ⊗ I2 )R(λ)Φ ∈ D(eγx ⊗ I2 ) for all γ > 0. By applying (4.10) repeatedly, we see that (Dα ⊗ I2 )R(λ)Ψ is a linear combination of terms of the form
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R(λ)[(Dα1 ⊗ I2 )(V )]R(λ)[(Dα2 ⊗ I2 )(V )] · · · R(λ)[(Dαm−1 ⊗ I2 )(V )]R(λ)(Dαm ⊗ I2 )Ψ,
m where j=1 |αj | = |α|. Since the [(Dαj ⊗ I2 )(V )] are matrices with polynomial entries, we use Corollary 4.4 and the note above to obtain (Dα ⊗ I2 ) R(λ)Ψ ∈ D(eγx ⊗ I2 ) for all α ∈ N2 and γ > 0. The conclusion involving (Dα ⊗ I2 )r(E)Ψ follows by writing the reduced resolvent in the form (see Theorem XII.5 in [26]) 1 1 dλ. R(λ) r(E) = 2πi λ−E |λ−E|=Γ>0
Theorem 4.8. For k ≥ 2, let Ψ(k−2) =
f (k−2) (k) , and ψ⊥ be determined by g (k−2) (k)
the perturbation of Sect. 3. Then, f (k−2) , %g% (k−2)%%, ψ⊥ ∈ C ∞ (R2 ), %% %% formulas %% (k) %% (k−2) (k−2) %% (k) %% f ,g , %%ψ⊥ %% ∈ L2 (R2 ) and f (k−2) , g (k−2) , %%ψ⊥ %% ∈ D(eγx ), for el %% el %% %% (k) %% any γ > 0. In addition, Dα f (k−2) , Dα g (k−2) , %%Dα ψ⊥ %% ∈ D(eγx ) for all α ∈ N2 and any γ > 0.
el
Proof. We refer to a function in D(eγx ) (or D(eγx ⊗ I2 )) for any γ > 0, as exponentially decaying with arbitrary γ. We first note that from the proof of Lemma 4.7, multiplication by polynomials in X and Y preserves exponential decay with arbitrary γ. f (0) (0) is determined at second order in as an eigenfuncSince Ψ = g (0) tion of H2 , we already know from Propositions 4.3 and 4.6 that Ψ(0) satisfies the conclusion. The f (1) and g (1) given by Eq. (3.20) are determined by H3 followed by a projection Q⊥ , and reduced resolvent H2 − E (2) , acting on Ψ(0) . By Corollary 4.4 we know that the reduced resolvent preserves exponential decay with arbitrary γ. The projection Q⊥ was the projection in H onto the subspace perpendicular to the eigenspace of the eigenvalue E (2) of H2 . From Proposition 4.3, we know that the eigenvectors of H2 have exponential decay with arbitrary γ, and so it follows that Q⊥ will preserve exponential decay with arbitrary γ. Since the matrix entries of H3 only contain polynomials and derivatives in X and Y , we know from Lemma 4.7 that (H3 Ψ(0) ) will have exponential decay with arbitrary γ. It follows that Ψ(1) will have exponential decay with arbitrary γ. By a similar argument, Ψ(1) ∈ C ∞ ⊕ C ∞ . From the definition of Q⊥ along with Proposition 4.6, we see that all of the derivatives of Q⊥ H3 Ψ(0) are exponentially decaying with arbitrary γ. It then follows from Lemma 4.7 that all of the derivatives of Ψ(1) are exponentially decaying with arbitrary γ.
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(1)
Ann. Henri Poincar´e
(2)
Recall that ψ⊥ = ψ⊥ = ψ⊥ = 0. From Eq. (3.21) we know that %% (0) %%%% % (0) % %% %% %% −1 ∂Ψ1 %% %% ∂f %% %% (3) %% %% (0) (0) P⊥ %% % %%ψ⊥ %% ≤ %% (h P⊥ ) %% %% ∂x ∂X % r el el % %% % (0) %% % (0) % % % −1 ∂Ψ1 %% %% %% ∂f %% (0) (0) + %% (h P⊥ ) P⊥ %% % %% %% ∂y ∂Y % r el % %% % (0) %% % (0) % % % −1 ∂Ψ2 %% %% %% ∂f %% (0) (0) + %% (h P⊥ ) P⊥ %% % %% %% ∂x ∂X % r el % %% % (0) %% % (0) % % % −1 ∂Ψ2 %% %% %% ∂f %% (0) (0) + %% (h P⊥ ) . P⊥ %% % %% %% ∂y ∂Y % r el
(0)
1 (0) By assumption, ( ∂Ψ ∈ Hel , and [(h P⊥ )(0) ]−1 r and P⊥ are bounded oper∂x ) ators on Hel . So we have % (0) % % (0) % %% %% % % % ∂f % %% (3) %% % + B % ∂f % , %%ψ⊥ %% ≤ A %% % % ∂X ∂Y % el %% %% %% (3) %% for some positive real numbers A and B and %%ψ⊥ %% is exponentially decay-
(3)
el
ing for arbitrary γ by Proposition 4.6. Also, ψ⊥ ∈ C ∞ (R2 ) since its (X, Y ) dependence comes%%strictly from derivatives of f (0) and g (0) . By a similar argu%% %% α (3) %% ment, we see that %% D ψ⊥ %% is exponentially decaying with arbitrary γ, from el Proposition 4.6. One can now use induction on k to show the conclusion. For the induc(k−1) tion hypothesis, assume that Ψ(k−3) and ψ⊥ given by the perturbation formulas of Sect. 3 satisfy the conclusions. Using Eqs. (3.27) and (3.28) to (k) determine Ψ(k−2) and ψ⊥ , the conclusion follows from the propositions and lemmas previously proved.
5. The Eigenstates of the Leading Order Hamiltonian We adopt the following notation throughout: 1. The operator of nuclear angular momentum about the z-axis is denoted by ∂ Lnuc = −i ∂φ . The operator of total electronic angular momentum about z the z-axis is denoted by Lel z . The operator of total angular momentum ⊗ I) + (I ⊗ Lel about the z-axis is denoted by LTz OT = (Lnuc z z ). k 2. We let Ln (x) be the associated Laguerre polynomials, as defined in [22]. f
(0)
The first non-vanishing terms in our perturbation expansion are E (2) , (X, Y ), g (0) (X, Y ) arising from the eigenvalue equation (0) (0) f (2) f H2 = E , g (0) g (0)
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where 1 H2 = − ΔX,Y ⊗ I2 + 2
a+b 2
2 X 2 + a−b 2 Y bX Y
a−b 2
bX Y 2 . X 2 + a+b 2 Y
Let (ρ, φ) be the usual polar coordinates associated with (X, Y ). Define the unitary operators U, Z : H → H by: U=
cos(φ) sin(φ)
− sin(φ) cos(φ)
and
1 Z=√ 2
1 i
1 . −i
Let r = a1/4 ρ, ˜b = ab , and 1 HU = √ U −1 H2 U a 2 1 ∂2 1 2 (Lnuc 1 ∂ z ) +1 = − + r − + ⊗ I 2 2 ∂r2 2r ∂r 2 2r2 ˜ i b 2 nuc 2 r r 2 Lz + ˜ i b 2 nuc − r2 Lz −2r 1 HU Z = √ (U Z)−1 H2 (U Z) a 1 ∂2 1 2 1 ∂ = − + r ⊗ I2 − 2 ∂r2 2r ∂r 2 ⎞ ⎛ nuc 2 ˜ (Lz −1) b 2 r 2 2 ⎟ ⎜ 2r +⎝ ⎠. nuc 2 ˜ (Lz +1) b 2 2r 2r 2 Both HU and HU Z commute with Lnuc ⊗ I2 . So, we search for eigenfunctions z ±i|l|φ e ψ1 (r) , |l| = 0, 1, 2, . . .. We warn the of these operators of the form e±i|l|φ ψ2 (r) reader that although l arises here as an eigenvalue of Lnuc z , at this point we should not associate any physical meaning to l. Here we are dealing with the operators HU and HU Z , which are related to H2 by the operations of U and Z. The physical meaning of l will become apparent in Theorem 6.1. We note that (U −1 H2 U )Ψ = EΨ was the leading order equation obtained by Renner [27], which is unitarily equivalent to our leading order equation H2 Ψ = EΨ. Renner showed that some of the eigenvalues can be solved for exactly, and used regular perturbation theory up to second order to approximate the other eigenvalues. These equations have been studied by several other authors, for instance [2,15]. We repeat some of Renner’s results here, but we calculate the perturbation series to much higher orders, demonstrating that many of the series are diverging inside the region of interest. We also illustrate that there is likely a crossing involving the ground state eigenvalue of H2 near
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b ≈ 0.925a. The ground state appears to be degenerate for 0 < b < 0.925a and non-degenerate for 0.925a < b < a. 5.1. The Exactly Solvable l = 0 States The l = 0 states (no angular dependence) are exactly solvable. In this case HU reduces to [l=0] HU =
∂2 1 ∂ 1+˜ b 2 1 − 12 ∂r 2 − 2r ∂r + 2 r + 2r 2 0
0 ∂2 1 ∂ 1−˜ b 2 1 − − 12 ∂r 2 2r ∂r + 2 r + 2r 2
.
We recognize that the component equations are of the same form as the radial equation for angular momentum 1 states of the two dimensional Isotropic Harmonic Oscillator. From the first component equation, the eigenvalues and eigenfunctions (non-normalized) are , EN+ = (2N+ + 2) 1 + ˜b,
r˜+ L1N+ (r˜+ 2 ) e−r˜+ 0
2
/2
,
N+ = 0, 1, 2, . . .
where r˜+ = (1 + ˜b)1/4 r. From the second component equation, the eigenvalues and eigenfunctions (non-normalized) are EN−
, = (2N− + 2) 1 − ˜b,
0 2 , r˜− L1N− (r˜− 2 ) e−r˜− /2
N− = 0, 1, 2, . . .
where r˜− = (1 − ˜b)1/4 r. √ Since H2 is unitarily equivalent to a HU , we see these states give rise to eigenvalues and eigenfunctions of H2 given by √ EN− = (2N− + 2) a − b 2 −r˜− sin(φ) L1N− (r˜− 2 ) e−r˜− /2 [l=0] , ΨN− (ρ, φ) = 2 r˜− cos(φ) L1N− (r˜− 2 ) e−r˜− /2
N− = 0, 1, 2, . . . (5.1)
where r˜− = (a − b)1/4 ρ, and √ EN+ = (2N+ + 2) a + b 2 r˜+ cos(φ) L1N+ (r˜+ 2 ) e−r˜+ /2 [l=0] , ΨN+ (ρ, φ) = 2 r˜+ sin(φ) L1N+ (r˜+ 2 ) e−r˜+ /2
N+ = 0, 1, 2, . . . (5.2)
where r˜+ = (a + b)1/4 ρ.
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5.2. The Perturbation Calculation for the l = 0 States In this case, HU Z reduces to [±|l|] HU Z
+
⎛ 2 −1 ∂ 2 − ⎝ 2 ∂r
˜b 0 r2 1 2
1 ∂ 2r ∂r
1 . 0
+ 12 r2 +
⎞
(|l|∓1)2 2r 2
0 2
∂ − 12 ∂r 2 −
0
1 ∂ 2r ∂r
+ 12 r2 +
(|l|±1)2 2r 2
⎠
[±|l|] f (r) [±|l|] . It is clear that if Denote the eigenfunctions of HU Z by g [±|l|] (r) [|l|] [−|l|] f (r) f (r) [|l|] is an eigenfunction of HU Z with eigenvalue E, then = g [|l|] (r) g [−|l|] (r) [|l|] g (r) [−|l|] is an eigenfunction of HU Z with eigenvalue E. So we only need f [|l|] (r) [|l|]
to find the eigenfunctions and eigenvalues of the HU Z . We have not been able to solve for the eigenvalues and eigenfunctions in this case exactly. We use regular perturbation theory with perturbation [|l|] [|l|] parameter ˜b, letting HU Z = H0 + ˜bV˜ , where (|l|−1)2 ∂2 1 ∂ 1 2 −12 ∂r 0 2 − 2r ∂r + 2 r + 2 2r (|l|+1)2 ∂2 1 ∂ 1 2 0 − 12 ∂r 2 − 2r ∂r + 2 r + 2r 2 1 0 1 V˜ = r2 . 1 0 2
[|l|] H0 =
One can show using the relative bound found in equation (4.2), that V˜ is rel[|l|] atively bounded with respect to H0 on H. So, we know that in terms of ˜b, [|l|] HU Z is an analytic family of type A for small ˜b [26]. Therefore, the eigenvalues and eigenfunctions will be analytic functions of ˜b in a neighborhood of ˜b = 0. [|l|] We expand the eigenvalues and eigenfunctions of HU Z in a series in ˜b: E
N,|l|
(˜b) =
∞
N,|l| Ek ˜bk ,
N,|l|
Ψ
(˜b) =
k=0
∞
N,|l| ˜k
Ψk
b
(5.3)
k=0 N,|l|
N,|l|
and solve for the coefficients Ek , Ψk recursively. Here N indexes the [|l|] energy levels of HU Z for fixed |l|. Again from the two-dimensional isotropic [|l|] oscillator, the eigenfunctions of H0 are known exactly. The lowest state is non-degenerate, with eigenvalue and eigenfunction given by 0,|l|
E0
= |l|,
0,|l|
Ψ0
=
r|l|−1 e−r 0
2
/2
.
(5.4)
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The rest of the states are two-fold degenerate, with eigenvalues and eigenfunctions given by 2 |l|−1 N,|l| N,|l| r|l|−1 LN (r2 )e−r /2 Ψ0, up = , E0 = 2N + |l|, 0 0 N,|l| Ψ0, dwn = , N = 1, 2, . . . (5.5) 2 |l|+1 r|l|+1 LN −1 (r2 )e−r /2 |l|
The functions {ei l φ r|l| LK (r2 ) e−r
2
/2
}
l∈Z K=0,1,2,...
form a basis for L2 (R2 ) by |l|
theorem XIII.64 of [26]. Then for fixed l ∈ Z, the functions {r|l| LK (r2 ) 2 e−r /2 }K=0,1,2,... form a basis for the projection of L2 (R2 ) onto r-dependent multiples of ei l φ . We can then use the following orthonormal basis for the perturbation expansion: - 2 |l|−1 BN,|l|−1 r|l|−1 LN (r2 ) e−r /2 , 0 . 0 |l|+1
BN,|l|+1 r|l|+1 LN
(r2 ) e−r
2
/2
N =0,1,2,...
where the BN,|l| are constants of normalization. The matrix elements of the perturbation V˜ in this basis can be obtained explicitly [13]. 5.2.1. The Non-Degenerate Perturbation Calculation. Recall from (5.4), for [|l|] 0,|l| fixed |l| = 0, the lowest lying eigenvalue of H0 is E0 = |l| (non-degenerate). |l| Since HU Z is an analytic family, we use non-degenerate, regular perturbation theory. Using the M athematica software package, we easily computed the exact perturbation coefficients up to 28th order for the non-degenerate, lowest lying eigenvalue E 0,|l| , for several values of |l| (see Fig. 3). Recall that we are concerned with the case where 0 < b < a, so that 0 < ˜b = ab < 1. The functions E 0,|l| (˜b) likely do not exist as eigenvalues of [|l|] HU Z if ˜b ≥ 1. Seemingly the radii of convergence of the series are smaller as |l| increases. It appears that for |l| = 1, 2, and 3 the radius of convergence is likely close to 1 (if not larger). For |l| > 4, the series are behaving erratically for values of ˜b < 1. The |l| = 4 case appears to be borderline, with radius of convergence possibly only slightly smaller than 1. This divergent behavior was seen even from the low order coefficients for the larger values of |l|. The singularities are likely caused by avoided crossings between two states with the same value of |l|, as suggested in [2,15,27]. We highlight the crossing between the |l| = 1 state and the lowest lying [|l|] l = 0 state near ˜b = 0.925. Recall that for l = 0, the eigenvalue E J,|l| of HU Z is [−|l|] also an eigenvalue of HU Z . Together these states correspond to a degenerate eigenvalue of the original operator H2 . The l = 0 states are all non-degenerate for b > 0. So, this crossing implies that the ground state of H2 is degenerate for approximately 0 < b < 0.925a and non-degenerate for 0.925a < b < 1.
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8
6
4
2
0.2
0.4
0.6
0.8
1.0
Figure 3. A plot of the perturbation series versus ˜b, of the [|l|] non-degenerate, lowest lying eigenvalue of HU Z up to order 28, for |l| = 1, 2, . . . , 8. The dashed curves are (2N +2) 1 − ˜b, for N = 0, 1, 2, 3, which are the l = 0 states that we have solved for exactly 5.2.2. The Degenerate Perturbation Calculation. Recall from (5.4), that only [|l|] the ground state of H0 is non-degenerate if |l| = 0. In the perturbation calculation described in Sect. 5.2.1, we used regular non-degenerate perturbation theory to obtain the perturbation coefficients for these eigenvalues. From (5.5), [|l|] we have that for fixed |l| = 0, H0 also has two-fold degenerate eigenvalues N,|l| = 2N + |l| for N = 1, 2, . . . So we must use degenerate perturbation of E0 theory to calculate the perturbation coefficients of these eigenvalues. Recall N,|l| are from (5.5), the degenerate pair of eigenfunctions corresponding to E0 N,|l| N,|l| given by Ψ0,up and Ψ0,dwn . Employing degenerate perturbation theory in the usual manner, we find there is splitting that occurs at first order (we omit the details). Armed with the proper linear combinations we can then proceed as in the non-degenerate case. Using the M athematica software package, we easily computed the exact perturbation coefficients up to 12th order for the first few eigenvalues E N,|l| that are degenerate at zeroth order, for several values of |l| (see Figs. 4, 5). While the splitting is nicely illustrated, we see that all of the series likely have radii of convergence well below 1. The radius of convergence appears to decrease as |l| or N increase. The divergent behavior was seen even at low orders of the perturbation coefficients. We also used an elementary finite difference scheme to approximate the eigenvalues at several values of ˜b, for 0 < ˜b < 1. The results are given in Fig. 6. The plot was generated by approximating the lowest lying 17 eigenvalues for a fixed ˜b value, then the value of ˜b was changed and the lowest 17 eigenvalues were calculated again. This was repeated at steps of Δ˜b = 0.01
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10 8 6 4 2
0.2
0.4
0.6
0.8
1.0
b
Figure 4. A plot of the perturbation series versus ˜b up to [|l|] order 12, of the first eight eigenvalues of HU Z that are degenerate at zeroth order, for |l| = 1. The dashed curves are (2N + ˜ 2) 1 − b and (2N + 2) 1 + ˜b, for N = 0, 1, 2, 3, 4, which are the l = 0 states that we have solved for exactly 12 10 8 6 4 2
0.2
0.4
0.6
0.8
1.0
b
Figure 5. A plot of the perturbation series versus ˜b up to [|l|] order 12, of the first eight eigenvalues of HU Z that are degenerate at zeroth order, for |l| = 2. The dashed curves are ˜ (2N + 2) 1 − b and (2N + 2) 1 + ˜b, for N = 0, 1, 2, 3, 4, which are the l = 0 states that we have solved for exactly
from 0 ≤ ˜b < 0.99. Recall that the l = 0 states were exactly solvable. For comparison, the exact values of the lowest lying l = 0 states were plotted as dotted curves. We see that the finite difference scheme approximates these eigenvalues
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4 3.5 3 2.5 2 1.5 1 0.5 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 6. A plot of lowest 17 eigenvalues of H2 as a function of ˜b, on 0 < ˜b < 1, as approximated by a finite difference ˜ scheme. The dotted curves are 2 1 − b and 2 1 + ˜b, which are the lowest of the l = 0 states that we have solved for exactly
so well that the dotted curve are hardly distinguishable from the finite difference approximation of these eigenvalues. Near ˜b = 0, the 17 eigenvalues that are being approximated can be identified by their values at ˜b = 0: 1. The curve that has value 1 at ˜b = 0 is actually two overlapping eigenvalues of H2 corresponding to the degenerate pair of lowest lying |l| = 1 states, one for l = 1 and l = −1. 2. There are three curves that have value 2 at ˜b = 0. Two of the curves are the non-degenerate l = 0 states (one increases with ˜b and one decreases with ˜b). The other curve is two overlapping eigenvalues corresponding to the degenerate pair of lowest lying |l| = 2 states, one for l = 2 and l = −2. These curves together account for four eigenvalues of H2 . 3. There are three curves that have value 3 at ˜b = 0. Two of the curves are overlapping degenerate |l| = 1 states, (one degenerate pair increases with ˜b and one degenerate pair decreases with ˜b). The other curve an overlapping degenerate pair of lowest lying |l| = 2 states. These curves together account for six eigenvalues of H2 . 4. There are three curves that have value 4 at ˜b = 0. One of the curves is a non-degenerate l = 0 state, one is a degenerate pair of |l| = 2 states, and one is a degenerate pair of |l| = 4 states. Together these curves account for five eigenvalues of H2 .
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This plot supports the claim that a crossing occurs involving the ground state near ˜b = 0.925. While the finite difference scheme is crude, we are inclined to trust the qualitative features of the results considering the lowest of the exactly solvable l = 0 eigenvalues were so well approximated, even near ˜b = 1 as seen in the figure. We note that as ˜b increases from zero, avoided crossings involving states with the same value of |l| occur, as well as crossings involving states with different values of |l|. When the uppermost curve is involved with such a phenomenon it will appear to change behavior suddenly without reason, but this is only because we can only see the lowest 17 eigenvalues at each ˜b.
6. Degeneracy of the Quasimode Energies The eigenfunctions of H2 provide the zeroth order states for the quasimode f (0) expansion. Recall that if is an eigenfunction of H2 , we have derived g (0) perturbation formulas in section 3 that determine the functions f (k) (X, Y ), (k) g (k) (X, Y ), and ψ⊥ (X, Y ) that enter in Eq. (3.11) as the asymptotic series Φ = Ψ1 ( X, Y ) +
∞
∞
f
(k)
k
(X, Y ) + Ψ2 ( X, Y )
k=0
∞
g (k) (X, Y )k
k=0
(k)
ψ⊥ (X, Y )k
k=0
=
∞ k=0
k
k j=0
(j) +Ψ2 (X, Y
=:
∞
(j)
Ψ1 (X, Y )f (k−j) (X, Y )
(k) )g (k−j) (X, Y ) + ψ⊥ (X, Y ) ,
k Φk ,
k=0
where {Ψ1 ( X, Y ), Ψ2 ( X, Y )} is the electronic eigenfunction basis. The f (k) and g (k) have no electronic dependence (they are scalar functions) and (k) ψ⊥ has both electronic and nuclear [|l|]dependence. f (r) [|l|] is an eigenfunction of HU Z with Recall that for |l| = 0, if g [|l|] (r) [|l|] [−|l|] g (r) f (r) [−|l|] = is an eigenfunction of HU Z eigenvalue E, then g [−|l|] (r) f [|l|] (r) with eigenvalue E. So if |l| = 0, we have two-fold degenerate eigenfunctions of H2 of the form (recall r = a1/4 ρ) i|l|φ [|l|] (0) f (r) e F =: U Z i|l|φ [|l|] (0) G g (r) e i(|l|−1)φ [|l|] i(|l|+1)φ [|l|] 1 e i(|l|−1)φf [|l|](r) + e i(|l|+1)φg [|l|](r) = √ (6.1) f (r) − e g (r) 2 i e
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e−i|l|φ g [|l|] (r) UZ e−i|l|φ f [|l|] (r) 1 e−i(|l|−1)φ f [|l|] (r) + e−i(|l|+1)φ g [|l|] (r) =√ −i(|l|−1)φ [|l|] f (r) − e−i(|l|+1)φ g [|l|] (r) 2 −i e F (0) = G(0)
59
(6.2)
By taking appropriate linear combinations, these degenerate zeroth order functions lead to two orthogonal quasimodes using the perturbation formulas of Sect. 3, possibly degenerate (no splitting) or non-degenerate (splitting). We adopt the following nomenclature: We refer to the eigenfunctions of [|l|] H2 that arise from the eigenfunctions of HU Z , where |l| = 0, as +|l| states. We [−|l|] refer to the eigenfunctions of H2 that arise from the eigenfunctions of HU Z , where |l| = 0, as −|l| states. We refer to the eigenfunctions of H2 that arise [|l|=0] from the eigenfunctions of HU as |l| = 0 states. Theorem 6.1. Let LTz OT be the operator of total angular momentum around the z-axis and 0 < ˜b < 1. Then: 1. For l = 0, each +|l| state generates a quasimode ΦA of H() that satisA fies LTz OT ΦA = |l|Φ . The corresponding degenerate −|l| state generates A B T OT B A Φ = −|l|ΦB a quasimode ΦB that satisfies Φ = Φ and Lz . The Φ B and Φ quasimodes are orthogonal, and asymptotic to two-fold degenerate eigenfunctions of H(). We see that linear combinations of the these two-fold degenerate ±|l| states also generate valid quasimodes. 2. Each |l| = 0 state generates a quasimode that is asymptotic to a non-degenerate eigenfunction of H(). In either case, the zeroth order of the electronic eigenfunction basis vectors Ψ1 (0, 0) and Ψ2 (0, 0) are linear combinations of eigenfunctions of Lel z with eigenvalues ±1. Remark. The physical meaning of l is now apparent. It corresponds to the total angular momentum about the z-axis of the wave function being approximated. From the proof to follow, it will be clear that the zeroth order Φ0 of a quasimode, can be constructed to satisfy LTz OT Φ0 = lzT OT Φ0 . In this case it is a linear combination of two states of the form ˜ + (rel ) and ΞlT OT +1 (rnuc ) Ψ ˜ − (rel ), ΞlT OT −1 (rnuc ) Ψ z
z
where ˜ ˜ Lel z Ψ+ = Ψ+ ,
˜ ˜ Lel z Ψ− = − Ψ− ,
ΞlzT OT −1 = (lzT OT − 1) ΞlzT OT −1 , Lnuc z
Lnuc ΞlzT OT +1 = (lzT OT + 1) ΞlzT OT +1 . z
Proof. Since [H(), LTz OT ] = 0, we know that the true eigenfunctions Ψ() of H() can be constructed to satisfy LTz OT Ψ() = lzT OT Ψ(), at each in a neighborhood of 0, for some lzT OT ∈ Z. This implies that LTz OT Ψ() = − lzT OT Ψ(),
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since LTz OT Ψ() =− LTz OT Ψ(). We can therefore arrange so that the asymp∞ totic series Φ = k=0 k Φk satisfies LTz OT Φ = lzT OT Φ at each order of . We then know that each order Φk of the quasimode, and its complex conjugate, are eigenfunctions of LTz OT with eigenvalues lzT OT and −lzT OT respectively. We now separate into two cases: Case 1: |l| = 0 In this case, we have degenerate zeroth order states of the form in (6.1) and (6.2). Regardless of whether splitting occurs, assume depart (0) that we (0) f F from zeroth order with a correct linear combination = α + g (0) G(0)
F (0) , so that this leads to a valid quasimode, which satisfies LTz OT Φ = G(0) lzT OT Φ . Then the zeroth order Φ0 must also be an eigenfunction of LTz OT with eigenvalue lzT OT . The Φ0 function is given by β
Φ0 = Ψ1 (0, 0)f (0) + Ψ2 (0, 0)g (0) = Ψ1 (0, 0) α F (0) + βF (0) + Ψ2 (0, 0) α G(0) + βG(0) = α ei(|l|−1)φ f [|l|] (Ψ1 (0, 0) + i Ψ2 (0, 0)) +ei(|l|+1)φ g [|l|] (Ψ1 (0, 0) − i Ψ2 (0, 0))
+β e−i(|l|−1)φ f [|l|] (Ψ1 (0, 0) − i Ψ2 (0, 0)) +e−i(|l|+1)φ g [|l|] (Ψ1 (0, 0) + i Ψ2 (0, 0)) . We now plug this into the equation LTz OT Φ0 − lzT OT Φ0 = 0, and for |l| ≥ 2, we project along ei(|l|−1)φ f [|l|] , ei(|l|+1)φ g [|l|] , e−i(|l|−1)φ f [|l|] , and e−i(|l|+1)φ g [|l|] , and obtain the following four equations: T OT − |l| + 1 (Ψ1 (0, 0) + i Ψ2 (0, 0)) (6.3) Lel z (Ψ1 (0, 0) + i Ψ2 (0, 0)) = lz T OT Lel − |l| − 1 (Ψ1 (0, 0) − i Ψ2 (0, 0)) (6.4) z (Ψ1 (0, 0) − i Ψ2 (0, 0)) = lz T OT Lel + |l| − 1 (Ψ1 (0, 0) − i Ψ2 (0, 0)) (6.5) z (Ψ1 (0, 0) − i Ψ2 (0, 0)) = lz T OT Lel + |l| + 1 (Ψ1 (0, 0) + i Ψ2 (0, 0)) (6.6) z (Ψ1 (0, 0) + i Ψ2 (0, 0)) = lz Equations (6.3) and (6.4) hold as long as α = 0 and Eqs. (6.5) and (6.6) hold as long as β = 0. By combining (6.3) and (6.6) we obtain lzT OT − |l| + 1 = lzT OT + |l| + 1 which contradicts our assumption that |l| = 0. So, either α = 0 or β = 0. Assume that β = 0 and take α = 1, so that Eqs. (6.3) and (6.4) still hold. Since LTz OT Φ0 = lzT OT Φ0 , we know that LTz OT Φ0 = −lzT OT Φ0 . Using this equation and projecting along e−i(|l|−1)φ f [|l|] and e−i(|l|+1)φ g [|l|] , we obtain
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equations similar to (6.5) and (6.6), but with lzT OT replaced by −lzT OT : T OT Lel + |l| − 1 (Ψ1 (0, 0) − i Ψ2 (0, 0)) (6.7) z (Ψ1 (0, 0) − i Ψ2 (0, 0)) = −lz T OT + |l| + 1 (Ψ1 (0, 0) + i Ψ2 (0, 0)) . (6.8) Lel z (Ψ1 (0, 0) + i Ψ2 (0, 0)) = −lz By combining (6.3) and (6.8) we obtain lzT OT = |l| and these equations now reduce to Lel z (Ψ1 (0, 0) − i Ψ2 (0, 0)) = − (Ψ1 (0, 0) − i Ψ2 (0, 0)) Lel z
(Ψ1 (0, 0) + i Ψ2 (0, 0)) = Ψ1 (0, 0) + i Ψ2 (0, 0).
(6.9) (6.10)
By repeating the argument with α = 0, β = 1, we would instead find lzT OT = −|l|. From this analysis we see that α = 1, β = 0 and α = 0, β = 1 are correct linear combinations that will generate two orthogonal quasimodes B T OT A Φ = |l|ΦA ΦA and Φ respectively. These quasimodes satisfy Lz and B = −|l| Φ and are asymptotic to eigenfunctions of H(). We note LTz OT ΦB A B that Φ0 = Φ0 . Since H() commutes with complex conjugation, we have A that Φ is also asymptotic to an eigenfunction with the same eigenvalue as ΦA . Since quasimodes are determined by their zeroth order eigenfunctions A through the perturbation formulas of Sect. 3, this implies that ΦB = Φ since A A B ΦB 0 = Φ0 . So, the Φ and Φ correspond to a degenerate pair and we see that no splitting occurs in the perturbation expansion. As a result, any linear A B combination would be a correct one. The ΦA and Φ = Φ generated by the combinations α = 1, β = 0 and α = 0, β = 1 respectively, are the quasimodes A T OT B Φ = −|l| ΦB that satisfy LTz OT ΦA = |l| Φ and Lz . T OT T OT If |l| = 1, we take projections of Lz Φ0 − lz Φ0 = 0 along f [1] , 2iφ [1] −2iφ [|1|] e g , and e g , and obtain three equations. By proceeding in a similar manner to the analysis in the |l| ≥ 2 case above, we would obtain lzT OT = 1 if α = 1, β = 0 and lzT OT = −1 if α = 0, β = 1. In either case, we would obtain (6.9) and (6.10) and the desired results follow as in the |l| ≥ 2 case above. Case 2: |l| = 0 If |l| = 0, we have non-degenerate eigenfunctions of H2 of the form in Eqs. (5.1) or (5.2). In any case, we see that Φ0 is real. Then from LTz OT Φ0 = lzT OT Φ0 and LTz OT Φ0 = −lzT OT Φ0 , it is clear that lzT OT = 0 in this case. By plugging into LTz OT Φ0 = 0 and taking projections along eiφ and e−iφ in a manner similar to the |l| = 0 case, we obtain the same relations for the electronic basis vectors at zeroth order given by Eqs. (6.9) and (6.10). Corollary 6.2. For all ˜b in some interval (0, δ), if is sufficiently small, the ground state of H() (corresponding to the R–T pair of states we are considering) is degenerate. Proof. From our perturbation analysis, we have that for all ˜b in some interval (0, δ), the ground state of H2 is degenerate, arising from the l = ±1 states of HU Z . The previous theorem tells us that these generate a degenerate
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B T OT A B pair of quasimodes ΦA Φ = ΦA and Φ that satisfy Lz , Φ = Φ and T OT B B Φ = − Φ . If the quasimode energy lies below the essential spectrum, Lz then this will correspond to the lowest lying eigenvalue of H() corresponding to the R–T pair.
Our perturbation calculations suggest that there is a crossing involving this eigenvalue with the lowest lying l = 0 eigenvalue, somewhere near ˜b = 0.925. The ground state seemingly corresponds to these l = ±1 states for 0 < ˜b < 0.925 and corresponds to the non-degenerate, lowest lying l = 0 state for 0.925 < ˜b < 1. We now prove that the ground state of H2 cannot arise from any other |l| states. Proposition 6.3. Let 0 < ˜b < 1. Then the ground state of HU Z is either an |l| = 1 state or an l = 0 state. Proof. previously mentioned, since [HU Z , Lnuc = 0 we assume z ] As ±i|l|φ e ψ (r) and then HU Z can be written in the form Ψ = ±i|l|φ 1 ψ2 (r) e 2 1 |l| ∓ 2|l| 0 [±|l|] [0] HU Z = H U Z + 2 . (6.11) 0 |l|2 ± 2|l| 2r [0]
have We now show that HU Z must aneigenvalue below the eigenvalues of [|l|] [−|l|] [|l|] if |l| ≥ 2. Recall that σ HU Z = σ HU Z , so we only consider HU Z . [|l|] [0] Since 2r12 |l|2 ± 2|l| > 0 for |l| > 2, we know from (6.11) that HU Z > HU Z [0] for all |l| > 2. It easily follows that the lowest eigenvalue of HU Z must lie [±|l|] below the eigenvalues of HU Z for all |l| > 2. [|l|] The presence of the off-diagonal terms in HU Z when ˜b = 0, implies that both components of the eigenvectors must be non-vanishing. Let Ψ be the [2] eigenvector corresponding to the lowest lying eigenvalue of HU Z . Then, / 0 / 0 1 0 0 [2] [0] Ψ, HU Z Ψ = Ψ, HU Z Ψ + Ψ, 2 Ψ 2r 0 8 / 0 [0] [0] > Ψ, HU Z Ψ ≥ inf σ HU Z . [±|l|] HU Z
[0]
[2]
We see that HU Z has at least one eigenvalue below the eigenvalues of HU Z . [0] So, the ground state of HU Z must correspond to the ground state of HU Z or [±1] HU Z .
7. Proof of the Main Theorem Here we use the quasimode expansion constructed in Sect. 3 to sketch the proof of Theorem 2.1. Our candidates for the approximate wave function and
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energy in the theorem are Φ,K = F (ρ) Ψ,K , where ⎛ K−2 Ψ,K = ⎝ j Ψ1 (X, Y )f (j) (X, Y ) + Ψ2 (X, Y )g (j) (X, Y ) j=0
+
K
⎞ (j)
j ψ⊥ (X, Y )⎠
j=0
K (j) and E,K = j=0 j E (j) , where the f (j) , g (j) , ψ⊥ , and E (j) are determined by the perturbation formulas in Sect. 3. The cut-off function F (ρ) is needed to restrict the analysis to a neighborhood of the local minimum of the electronic ρ) and E2 (˜ ρ) at ρ˜ = 0, where E1 and E2 are isolated from the eigenvalues E1 (˜ rest of the spectrum of h(X, Y ) and also where the functions and operators that we have expanded into powers of (such as Ψ1 (X, Y )) have asymptotic expansions (recall that (x, y) = (X, Y ) and ρ˜ = ρ). We require that the cut-off function F (˜ ρ) : R2 → [0, 1] be smooth in both variables x and y. It has support in some neighborhood where ρ˜ < S. Also, F (˜ ρ) = 1 for ρ˜ ≤ R, where 0 < R < S. So, the derivatives of F (˜ ρ) with respect to x and y vanish outside the region R ≤ ρ˜ ≤ S. We have (H() − E,K ) Φ,K = F (ρ) (H() − E,K ) Ψ,K −
2 [ΔX,Y , F (ρ) ] Ψ,K . 2 (7.1)
To prove the theorem, one can first show the norm of Φ,K is asymptotic to . It then suffices to prove that both terms on the right hand side of (7.1) are finite linear combinations of the form J G, where ||G||Hnuc ⊗Hel < ∞ and J ≥ K +2. ⊥ Recall from Sect. 3 that P⊥ was the projection in Hel onto {Ψ1 , Ψ2 } . We can write (H() − E,K )Ψ,K = χ1 (, X, Y )Ψ1 (X, Y ) + χ2 (, X, Y ) Ψ2 (X, Y ) + χ⊥ (, X, Y ), where χ⊥ = P⊥ [(H() − E,K )Ψ,K ] (so χ1 and χ2 have no electronic dependence, but χ⊥ does have electronic dependence). The analysis regarding χ2 and χ⊥ is similar to that of χ1 and will be omitted. Using (3.12) with our definition of Ψ,K , we have χ1 = − −
K−2 K−2 K−2 2 ΔX,Y j f (j) + h11 j f (j) + h12 j g (j) 2 j=0 j=0 j=0 K−2 2 4 j (j) Ψ1 , ΔX,Y ψ⊥ el − f Ψ1 , Δx, y Ψ1 el 2 2 j=0
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K−2 4 j (j) g Ψ1 , Δx, y Ψ2 el 2 j=0 ⎛ ⎞ K−2 K−2 (j) (j) ∂g ∂g ∂Ψ2 ∂Ψ2 ⎠ Ψ1 , el + Ψ1 , el −3 ⎝ j j ∂X ∂x ∂Y ∂y j=0 j=0
−
−
K
j E (j)
j=0
K−2
l f (l) .
l=0
To show ||F (ρ) χ1 Ψ1 ||Hnuc ⊗Hel ≤ C K+2 , we can consider the terms in the above equation separately and use the triangle inequality. Analogous to Eqs. (3.14) and (3.15), we expand all functions with (X, Y ) dependence into powers of , however, we truncate the series here and add an error term. For K (j) example, we can write h11 (X, Y ) = j=0 j h11 + K+1 herr 11 (X, Y ), where ∞ we know herr 11 (X, Y ) is in C (X, Y ) and is bounded by a polynomial in X and Y of order K + 1 on supp(F (ρ)). If we do this, we know all terms of order j , for j ≤ K, will cancel in the above equations, since the terms of f , g, E, and ψ⊥ were chosen using the perturbation formulas. We show how to deal with the h11 term arising in F (ρ) χ1 Ψ1 only, the rest of the terms are handled similarly. Considering only expressions of order K+1 or higher, this term can be written h11 (X, Y )
K−2
j f (j) =
(l)
l+j h11 f (j) +
0≤j≤K−2 0≤l≤K j+l≥K+1
j=0
K−2
(j) K+j+1 herr . 11 (X, Y )f
j=0
Then using the results of Sect. 4, in particular that f (j) ∈ D(eγx ), one can show that %% %% %% %% K−2 %% %% j (j) %% F (ρ) h11 (X, Y ) f Ψ1 (X, Y ) %%%% %% %% %% j=0 Hnuc ⊗Hel
≤
0≤j≤K−2 0≤l≤K j+l≥K+1
l+j+1 Cl,j +
K−2
K+j+2 Dl,j ,
j=0
where Cm , Dm , Cl,j , Dl,j < ∞. We see that this term is indeed of order greater or equal to O(K+2 ). All of the terms of χ1 and χ⊥ can be handled in a similar fashion using the results of Theorem 4.8. The term involving the derivatives of F in Eq. (7.1) are handled using Theorem 4.8 as well. The derivatives of F are supported away from the origin and the terms of Ψ,K are exponentially decaying. We consider the terms 2 involving derivatives with respect to X. Let M be larger than supR2 | ∂∂xF2 | and
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supR2 | ∂F ∂x |. Then, using Theorem 4.8, one can easily show that %% 2 %% %% ∂ %% %% %% , F (ρ) Ψ ,K %% %% ∂X 2 Hnuc ⊗Hel √ −γ 1+R2 /2 ≤Me %% %% %% %% % % % % ∂Ψ % % % % ,K %% × 2 %%eγx Ψ,K %% + 2 %%%%eγx ∂X %%Hnuc ⊗Hel Hnuc ⊗Hel ≤ O(∞ ). The conclusion of the theorem follows.
Acknowledgements It is a pleasure to thank Professor George A. Hagedorn for his advise and many useful comments. This research was supported in part by National Science Foundation Grant DMS–0600944 while at Virginia Polytechnic Institute and State University, and also by the Institute for Mathematics and its Applications at the University of Minnesota, with funds provided by the National Science Foundation.
Appendix We now argue that the odd terms in the E() series must be zero. See [13] for a detailed proof which utilizes the perturbation formulas derived in Sect. 3. The Hamiltonian of interest in terms of the scaled nuclear coordinates (X, Y ) = (x/, y/) is given by H() =
−2 ΔX,Y + h(X, Y ), 2
where h(x, y) is the electronic hamiltonian that also contains the nuclear repulsion terms. If E() is an eigenvalue of H(), then E(−) is an eigenvalue of −2 ΔX,Y + h(−X, −Y ). 2 ˜ = −X, Y˜ = −Y , we see that H(−) becomes Under the unitary change X H(−) =
H(−) =
−2 ˜ ˜ ΔX, ˜ Y˜ + h(X, Y ). 2
It is clear that H() and H(−) share the same eigenvalues. This does not immediately imply E() = E(−), since there could be a pair of eigenvalues (2) (2) related by EA () = EB (−). However, this would imply that EA = EB , and then Theorem 6.1 implies EA () = EB (). Therefore, E() = E(−), and as a result the odd terms in the expansion must vanish.
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References [1] Born, M., Oppenheimer, R.: Zur Quantentheorie der Molekeln. Ann. Phys. (Leipzig) 84, 457–484 (1927) [2] Brown, J. M., Jørgensen, F.: In: Prigogine, I., Rice, S.A. (eds.) Advances in Chemical Physics, vol. 52, p. 117. Wiley, New York (1983) [3] Combes, J.-M., Seiler, R.: In: Wooley, R.G. (ed.) Quantum Dynamics of Molecules: The New Experimental Challenge to Theorists. NATO Advanced Study Institutes Series, Series B, Physics, vol. 57, pp. 435–482. Plenum Press, New York (1980) [4] Combes, J.-M., Duclos, P., Seiler, R.: In: Velo, G., Wightman, A. (eds.) Rigorous Atomic and Molecular Physics, pp. 185–212. Plenum Press, New York (1981) [5] Dressler, K., Ramsay, D.A.: Renner Effect in Polyatomic Molecules. J. Chem. Phys. 27, 971 (1957) [6] Dressler, K., Ramsay, D.A.: The Electronic Absorption of NH2 and ND2 . Phil. Trans. R. Soc. Ser. A 251, 553 (1958) [7] Hagedorn, G.A.: High Order Corrections to the Time-Independent BornOppenheimer Approximation I: Smooth Potentials. Ann. Inst. H. Poincar´e Sect. A. 47, 1–16 (1987) [8] Hagedorn, G.A.: High Order Corrections to the Time-Independent BornOppenheimer Approximation II: Diatomic Coulomb Systems. Commun. Math. Phys. 116, 23–44 (1988) [9] Hagedorn, G.A., Joye, A.: A Time-Dependent Born-Oppenheimer Approximation with Exponentially Small Error Estimates. Commun. Math. Phys. 223, 583– 626 (2001) [10] Hagedorn, G.A., Toloza, J.H.: Exponentially Accurate Quasimodes for the TimeIndependent Born-Oppenheimer Approximation on a One-Dimensional Molecular System. Int. J. Quantum Chem. 105, 463–477 (2005) [11] Hagedorn, G.A., Joye, A.: In: Gesztesy, F., Deift, P., Galvez, C., Perry, P., Schlag, W. (eds.) Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday. Part I: Quantum Field Theory, Statistical Mechanics, and Nonrelativistic Quantum Systems. AMS Proc. of Symposia in Pure Math., vol. 76, part 1, pp. 203–226 (2007) [12] Hagedorn, G.A., Joye, A.: A Mathematical Theory for Vibrational Levels Associated with Hydrogen Bonds I: The Symmetric Case. Commun. Math. Phys. 274, 691–715 (2007) [13] Herman, M.S.: Born–Oppenheimer Corrections Near a Renner–Teller Crossing. Ph.D. Thesis, Virginia Polytechnic Institute and State University [14] Herzberg, G., Teller, E.: Schwingungsstruktur der Elektronen¨ ubergange bei mehratomigen Molek¨ ulen. Z. Phys. Chem. B 21, 410 (1933) [15] Herzberg, G.: Electronic Spectra of Polyatomic Molecules. D. Van Nostrand Company, Inc., Princeton (1966) [16] Hunziker, W.: Distortion analyticity and molecular resonance curves. Ann. Inst. H. Poincar´e Sect. A. 45, 339–358 (1986) [17] Jensen, P., Osmann, G., Bunker, P.R.: In: Jensen, P., Bunker, P.R. (eds.) Computational Molecular Spectroscopy, p. 485. Wiley, New York (2000)
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[18] Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Classics in Mathematics. Springer, Berlin (1980) [19] Klein, M., Martinez, A., Seiler, R., Wang, X.: On the Born-Oppenheimer expansion for polyatomic molecules. Commun. Math. Phys. 143, 607–639 (1992) [20] Lee, T.J., Fox, D.J., Schaefer, H.F. III., Pitzer, R.M.: Analytic second derivatives for Renner-Teller potential energy surfaces. Examples of the five distinct cases. J. Chem. Phys. 81, 356–361 (1984) [21] Martinez, A., Sordoni, V.: Math. Phys. Preprint Archive mp arc 08-171 (2008, to appear) Memoirs American Mathematical Society [22] Messiah, A.: Quantum Mechanics. Wiley, New York (1958) [23] Peri´c, M., Peyerimhoff, S.D.: In: Baer, M., Billing, G.D. (eds.) The Role of Degenerate States in Chemistry. In: Prigogine, I., Rice, S.A. (eds.) A Special Volume of Advances in Chemical Physics, vol. 124, pp. 583–658. Wiley, New York (2002) [24] Panati, G., Spohn, H., Teufel, S.: The Time-Dependent Born-Oppenheimer Approximation. ESAIM: Math. Model. Numer. Anal. 41(2), 297–314 (2007) [25] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness. Academic Press, New York (1975) [26] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978) [27] Renner, R.: Zur Theorie der Wechselwirkung zwischen Elektronen- und Kernbewegung bei dreiatomigen, stabf¨ ormigen Molek¨ ulen. Z. Phys. 92, 172 (1934) [28] Worth, G., Cederbaum, L.: Beyond Born-Oppenheimer: Molecular Dynamics Through a Conical Intersection. Annu. Rev. Phys. Chem. 55, 127–158 (2004) [29] Yarkony, D.: Diabolical Conical Intersections. Rev. Mod. Phys. 68, 985– 1013 (1996) Mark S. Herman Department of Mathematics University of Rochester Rochester, NY 14627, USA e-mail:
[email protected] Communicated by Claude Alain Pillet. Received: March 12, 2009. Accepted: January 11, 2010.
Ann. Henri Poincar´e 11 (2010), 69–99 c 2010 Springer Basel AG 1424-0637/10/010069-31 published online May 8, 2010 DOI 10.1007/s00023-010-0034-7
Annales Henri Poincar´ e
Tiling Groupoids and Bratteli Diagrams Jean Bellissard, Antoine Julien and Jean Savinien Abstract. Let T be an aperiodic and repetitive tiling of Rd with finite local complexity. Let Ω be its tiling space with canonical transversal Ξ. The tiling equivalence relation RΞ is the set of pairs of tilings in Ξ which are translates of each others, with a certain (´etale) topology. In this paper RΞ is reconstructed as a generalized “tail equivalence” on a Bratteli diagram, with its standard AF -relation as a subequivalence relation. Using a generalization of the Anderson–Putnam complex (Bellissard et al. in Commun. Math. Phys. 261:1–41, 2006) Ω is identified with the inverse limit of a sequence of finite CW -complexes. A Bratteli diagram B is built from this sequence, and its set of infinite paths ∂B is homeomorphic to Ξ. The diagram B is endowed with a horizontal structure: additional edges that encode the adjacencies of patches in T . This allows to define an ´etale equivalence relation RB on ∂B which is homeomorphic to RΞ , and contains the AF -relation of “tail equivalence”.
1. Introduction This article describes a combinatorial way to reconstruct a tiling space and its groupoid, using Bratteli diagrams. In a forthcomming paper [23], this description will be adapted and used for tilings with a substitution. 1.1. Results Given a repetitive, aperiodic tiling T with finite local complexity (FLC) in Rd , its tiling space Ω, or hull, is a compact space obtained by taking a suitable closure of the family of tilings obtained by translating T . By construction, the translation group Rd acts on Ω by homeomorphisms making the pair (Ω, Rd ) a topological dynamical system. It is well known that repetitivity (or uniform repetitivity, as it is called in the symbolic one-dimensional case) is equivalent to the minimality of this action, see [34,35]. Equivalently, this dynamical system can be described through a groupoid [10,11,36], denoted by Ω Rd , Work supported by the NSF grants no. DMS-0300398 and no. DMS-0600956.
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called the crossed product of the tiling space by the action. If the tiles of T are punctured, the subset of Ω made of tilings with one puncture at the origin of Rd is a compact subset Ξ called the canonical transversal [3]. For quasi-crystals this identifies with the so-called atomic surface [5,22]. It has been shown that FLC implies that the transversal is completely disconnected [26], while aperiodicity and minimality eliminate isolated points, making it a Cantor set. Similarly, there is an ´etale groupoid ΓΞ associated with it [10,36], which plays a role similar to the Poincar´e first return map in usual dynamical systems. This groupoid is defined by the equivalence relation RΞ identifying two tilings of the transversal differing by a space translation. While the existence of the hull does not require much information, it can be quite involved to have an effective description which allows computations. The first step in this direction came from the work of Anderson and Putnam for substitution tilings [2]. A substitution is a rule describing how each tile, after suitable rescaling, is covered by other tiles touching along their faces. The most publicized example is the Penrose tiling in its various versions, like kites and darts [32]. Anderson and Putnam built a CW -complex X of dimension d, with d-cells given by suitably decorated prototiles and showed that the substitution induces a canonical map φ : X → X. Then the hull can be recovered as the inverse limit Ω = lim(X, φ). This ←− construction has several generalizations for repetitive, aperiodic, and FLC tilings without a substitution rule, see [7,14,37]. It is proved that there is a sequence of finite CW -complexes (Xn )n∈N of dimension d and maps φn : Xn+1 → Xn so that the hull is given by the inverse limit Ω = lim(Xn , φn ). ←− It is even possible to choose these CW -complexes to be smooth branched manifolds [7]. The first goal of the present paper is to describe the construction of a Bratteli diagram from these data in Sect. 3.1. A Bratteli diagram is a graph with a marked vertex ◦ called the root. The set of vertices V is graded by a natural integer, called the generation, so that V = n∈N Vn with V0 = {◦} and Vn ∩ Vm = ∅ if n = m. Then edges exist only between Vn and Vn+1 . In the present construction, each vertex in Vn is given by a d-cell of Xn . Then there is an edge between a vertex v ∈ Vn and w ∈ Vn+1 if and only if the d-cell v can be found inside w. Each edge will be labeled by the translation vector between the puncture of w and the one of v inside w. Namely at each generation n, the Bratteli diagram encodes how a d-cell of Xn+1 is filled by d-cells of Xn . More precisely it encodes not only which cells in Xn occur but also their position relative to the d-cell of Xn+1 . As in [7], since the family (Xn )n∈N is not unique, there are several choices to build a Bratteli diagram associated with a tiling space. Conversely, starting from such a Bratteli diagram, a tiling can be described as an infinite path starting from the root. Such a path describes how a given patch at generation n, is embedded in a larger patch at generation n + 1. By induction on the generations, an entire tiling is reconstructed. In addition, the origin of the tiling is defined uniquely at each step, thanks to the label of the edges involved. If each patch involved in the construction is
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decorated by its collar, namely provided that the tiling forces its border1 in the sense of [7], the tiling obtained eventually covers Rd , even if the origin is at a fixed distance from the boundary of the patch, because the collar, beyond this boundary, increases in size as well. As a result the transversal Ξ is recovered, as a compact topological space, from the space of infinite rooted paths in the diagram (Theorem 3.6). In a Bratteli diagram, the tail equivalence identifies two infinite paths differing only on a finite number of edges (Definition 3.8). It gives rise to a groupoid called the AF-groupoid of the diagram. In the language of tiling spaces, this groupoid describes the translation structure inside the tiling, up to an important obstacle. Namely, a tiling built from an infinite path with the origin at a fixed distance from the boundary of the corresponding patches cannot be identified, modulo the tail equivalence, with a tiling built from a path with origin at a fixed distance from the same boundary but located on the other side of it. As a result, a tiling obtained in this way will be subdivided into regions, that will be called AF-regions here (Definition 3.12), separated by boundaries. Note that Matui has found similar features for a class of 2-dimensional substitution tilings in [31]. By contrast, the groupoid of the transversal ΓΞ (Definition 2.10) allows to identify two such regions through translation. So that recovering ΓΞ from the Bratteli diagram requires to change the definition of the tail equivalence. The present paper offers a solution to this problem by adding a horizontal structure to the diagram making it a collared Bratteli diagram (Definition 3.7). Its aim is to describe in a combinatorial way, from local data, how to locate a tile in a patch, relative to its boundary. Practically it consists in adding edges between two vertices of the same generation, describing pairs of tiles, in a pair of patches, each in the collar of one another (see Fig. 3). These edges are labeled by the translation between these tiles. In other words, the horizontal edges describe how to glue together two patches in a tiling across an AF-boundary. Then it becomes possible to extend the tail equivalence into an ´etale equivalence relation on the collared Bratteli diagram (Definition 3.18 and Theorem 2.9) in order to recover the groupoid of the transversal (Theorem 3.22 and Corollary 3.23), with the AF-groupoid as a sub-groupoid (Remark 3.19). In the particular case of 1-dimensional tilings, this larger equivalence relation is generated by the AF-relation and finitely many pairs of minimal and maximal paths in the diagram, which are derived from the collared diagram in a natural and explicit way (Proposition 4.1). The examples of the Fibonacci and Thue-Morse tilings are illustrated in detail in Sect. 4. This study of 1-dimensional tilings also allows to view the paths whose associated tilings have their punctures at a minimum distance to an AF-boundary (Corollary 3.11) as generalizations of extremal paths for 1-dimensional systems.
1
A notion initially introduced by Kellendonk for substitution tilings [26].
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1.2. Background This work gives one more way of describing tilings or their sets of punctures, and their groupoids, liable to help describing various properties of tiling spaces, such as their topology or their geometry. It benefited from almost 30 years of works with original motivation to describe more precisely the properties of aperiodic solids. In particular the notion of hull, or tiling space, was described very early as a fundamental concept [3] encoding their macroscopic translation invariance, called homogeneity. See for instance [4,6] for reviews and updates. During the eighties, the discovery of quasicrystals [40] was a landmark in this area and stimulated a lot of mathematical works to describe this new class of materials [22,39]. In particular, it was very convenient to represent the structure of these materials by various examples of tilings, such as the Penrose tiling or its 3-dimensional analogs [22,27,32]. It was shown subsequently [24] that such structures are liable to describe precisely the icosahedral phase of quasicrystalline alloys such as AlCuFe. In the end of the eighties, Connes attracted the attention of mathematicians to the subject by showing that the Penrose tiling was a typical example of a noncommutative space [11]. It is remarkable that already then, Connes used a Bratteli diagram to encode the combinatorics of patches between generations. Its is important to realize, though, that the construction given by Connes was based on the substitution proposed originally by Penrose [32] but ignored entirely the additional structure given by space translations. In the context of the present paper, including the translations is the key reason leading to the extension of the AF -relation. This leads to a non-AF C ∗ -algebra instead, a difficulty at the source of so many works during the last 20 years. Bratteli created his diagrams in [8] to classify AF -algebras, namely C ∗ -algebras obtained as the unions of finite dimensional C ∗ -algebras. It took two decades before it was realized that, through the notion of Vershik map [42] such diagrams could encode any minimal homeomorphism of the Cantor set [21,41]. The corresponding crossed product C ∗ -algebra was shown to characterize the homeomorphism up to orbit equivalence. This classification was a natural extension of a similar problem for ergodic actions of Z, a problem solved by Krieger and Connes within the framework of von Neumann algebras (see [9] for instance). This program was continued until recently and lead to the proof of a similar result for minimal actions of Zd on the Cantor set [12,13,15–18,33]. In particular, as a consequence of this construction, Giordano–Matui–Putnam–Skau proved that any minimal Zd -action on the Cantor set is orbit equivalent to a Z-action [19]. Moroever, the groupoid of the transversal of every aperiodic repetitive FLC tiling space is orbit equivalent to a minimal Z-action on the Cantor set [1]. As it turns out the formalism described in the present work is similar to and, to a certain extend inspired by, the construction of refined tessellations made by Giordano et al. in [19]. This paper is organized as follows: Sect. 2 contains the basic definitions about tilings, groupoids, and the constructions existing in the literature required in the present work. Section 3 is dedicated to the definition and properties of the Bratteli diagram associated with a tiling. The main reconstruction
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theorems are reproduced there. Section 4 illustrates the construction for 1dimensional substitution tilings, and treats the two examples of the Fibonacci and Thue-Morse tilings in detail.
2. Tilings and Tiling Groupoids This section is a reminder. The basic notions about tilings and their groupoids are defined and described. The finite volume approximations of tiling spaces by branched manifolds are also summarized [7]. 2.1. Tilings and Tiling Spaces All tilings in this work will be subsets of the d-dimensional Euclidean space Rd . Let B(x, r) denote the open ball of radius r centered at x. Definition 2.1. (i) A tile is a compact subset of Rd which is the closure of its interior. (ii) A punctured tile tx is an ordered pair consisting of a tile t and a point x ∈ t. (iii) A partial tilingis a collection {ti }i∈I of tiles with pairwise disjoint interiors. The set i∈I ti is called its support. (iv) A patch is a finite partial tiling. A patch is punctured by the puncture of one of the tiles that it contains. (v) A tiling is a countable partial tiling with support Rd . A tiling is said to be punctured if its tiles are punctured. (vi) The inner radius of a tile or a patch is the radius of the largest ball (centered at its puncture) that is contained in its support. The outer radius is the radius of the smallest ball (centered at its puncture) that contains its support. Hypothesis 2.2. From now on all tiles and tilings are punctured. In addition, each tile is assumed to have a finite CW -complex structure. The CW -complex structures of the tiles in a tiling are compatible: the intersection of two tiles is a subcomplex of both. In particular, the support of a tiling gives a CW -complex decomposition of Rd . Remark 2.3. (i) Definition 2.1 allows tiles and patches to be disconnected. (ii) Tiles or patch are considered as subsets of Rd . If t is a tile and p a patch, the notations t ∈ T and p ⊂ T mean t is a tile and p is a patch of the tiling T , at the positions they have as subsets of Rd . The results in this paper are valid for the class of tilings that are aperiodic, repetitive, and have finite local complexity. Definition 2.4. Let T be a tiling of Rd . (i) T has Finite Local Complexity (FLC) if for any ρ > 0 there are up to translation only finitely many patches of outer radius less than ρ. (ii) If T has FLC then it is repetitive if given any patch p, there exists ρp > 0 such that for every x ∈ Rd , there exists some u ∈ Rd such that p + u ∈ T ∩ B(x, ρp ).
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For a ∈ Rd let T + a = {t + a : t ∈ T } denote the translate of T by a. Then T is aperiodic if T + a = T , for all vectors a = 0.
From now on, we will only consider tilings which satisfy the conditions above. Note that FLC implies that there are only finitely many tiles up to translation. One of the authors defined in [5] a topology that applies to a large class of tilings (even without FLC). In the present setting, this topology can be adapted as follows. Let F be a family of tilings. Given an open set O in Rd with compact closure and an > 0, a neighborhood of a tiling T in F is given by UO, (T ) = {T ∈ F : ∃x, y ∈ B(0, ), (T ∩ O) + x = (T ∩ O) + y}, where T ∩ O is the notation for the set of all cells of T which intersect O. Definition 2.5. (i) The hull or tiling space of T , denoted Ω, is the closure of T + Rd for the topology defined above. (ii) The canonical transversal, denoted Ξ, is the subset of Ω consisting of tilings having one tile with puncture at the origin 0Rd . By FLC condition, the infimum of the inner radii of the tiles of T is r > 0. So the canonical transversal is actually an abstract transversal for the Rd action in the sense that it intersects every orbit, and (Ξ + u) ∩ Ξ = ∅ for all u ∈ Rd small enough (|u| < 2r). The hull of a tiling is a dynamical system (Ω, Rd ) which, for the class of tilings considered here, has the following well-known properties (see for example [7, section 2.3]). Theorem 2.6. Let T be a tiling of Rd . (i) Ω is compact. (ii) T is repetitive if and only if the dynamical system (Ω, Rd ) is minimal. (iii) If T has FLC, then its canonical transversal Ξ is totally disconnected [5,26]. (iv) If T is repetitive and aperiodic, then Ω is strongly aperiodic, i.e. contains no periodic points. (v) If T is aperiodic, repetitive, and has FLC, then Ξ is a Cantor set. By the minimality property, if Ω is the hull of a tiling T , any patch p of any tiling T ∈ Ω appears in T in some position. In case (iv), the sets Ξ(p) = {T ∈ Ξ : T contains (a translate of) p at the origin},
(1)
for p ⊂ T a patch of T , form a base of clopen sets for the topology of Ξ. Remark 2.7. A metric topology for tiling spaces has been used in the literature. Let T be a repetitive tiling or Rd with FLC. The orbit space of T under translation by vectors of Rd , T + Rd , is endowed with a metric as follows (see [7, Section 2.3]). For T1 and T2 in T + Rd , let A denote the set of ε in (0, 1) such that there exists a1 , a2 ∈ B(0, ε) for which T1 + a1 and T2 + a2 agree on B(0, 1/ε), i.e. their tiles whose punctures lie in the ball are matching, then δ(T1 , T2 ) = min (inf A, 1).
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Hence the diameter of T + Rd is bounded by 1. With this distance, the action of Rd is continuous. For the class of repetitive tilings with FLC, the topology of the hull given in Definition 2.5 is equivalent to this δ-metric topology [7]. 2.2. Tiling Equivalence Relations and Groupoids Let Ω be the tiling space of an aperiodic, repetitive, and FLC tiling of Rd , and let Ξ be its canonical transversal. Definition 2.8. The equivalence relation RΩ of the tiling space is the set (2) RΩ = (T, T ) ∈ Ω × Ω : ∃a ∈ Rd , T = T + a with the following topology: a sequence (Tn , Tn = Tn + an ) converges to (T, T = T + a) if Tn → T in Ω and an → a in Rd . The equivalence relation of the transversal is the restriction of RΩ to Ξ: (3) RΞ = (T, T ) ∈ Ξ × Ξ : ∃a ∈ Rd , T = T + a Note that the equivalence relations are not endowed with the relative topology of RΩ ⊂ Ω × Ω and RΞ ⊂ Ξ × Ξ. For example, by repetitivity, for a large, T and T + a might be close to each other in Ω, so that (T, T + a) is close to (T, T ) for the relative topology, but not for that from Ω × Rd . The map (T, a) → (T, T + a) from Ω × Rd to Ω × Ω has a dense image, coinciding with RΩ and is one-to-one because Ω is strongly aperiodic (contains no periodic points). The topology of RΩ is the topology induced by this map. Definition 2.9. An equivalence relation R on a compact metrizable space X is called ´etale when the following holds. (i)
(ii) (iii)
The set R2 = {((x, y), (y, z)) ∈ R × R} is closed in R × R and the maps sending ((x, y), (y, z)) in R × R to (x, y) in R, (y, z) in R, and (x, z) in R are continuous. The diagonal Δ(R) = {(x, x) : x ∈ X} is open in R. The range and source maps r, s : R → X given by r(x, y) = x, s(x, y) = y, are open and are local homeomorphisms.
A set O ⊂ R is called an R-set if O is open in R and r|O and s|O are homeomorphisms. It is proved in [25] that RΞ is an ´etale equivalence relation. A groupoid [36] is a small category (the collections of objects and morphisms are sets) with invertible morphisms. A topological groupoid, is a groupoid G whose sets of objects G0 and morphisms G are topological spaces, and such that the composition of morphisms G × G → G, the inverse of morphisms G → G, and the source and range maps G → G0 are all continuous maps. Given an equivalence relation R on a topological space X, there is a natural topological groupoid G associated with R, with objects G0 = X, and morphisms G = {(x, x ) : x ∼R x }. The topology of G is then inherited from that of R.
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Definition 2.10. The groupoid of the tiling space is the groupoid of RΞ , with set of objects Γ0Ξ = Ξ and morphisms (4) ΓΞ = (T, a) ∈ Ξ × Rd : T + a ∈ Ξ . There is also a notion of ´etale groupoids [36]. Essentially, this means that the range and source maps are local homeomorphisms. It can be shown that ΓΞ is an ´etale groupoid [25]. 2.3. Approximation of Tiling Spaces Let T be an aperiodic, repetitive and FLC (punctured) tiling of Rd . Since T has finitely many tiles up to translation, there exists R > r > 0, such that the minimum of the inner radii of its tiles equals r, and the maximum of the outer radii equals R. The set T punc of punctures of tiles of T is a Delone set (or (r, R)-Delone set). That is, T punc is uniformly discrete (or r-uniformly discrete): for any x ∈ Rd , B(x, r) ∩ T punc contains at most a point, and relatively dense (or R-relatively dense): for any x ∈ Rd , B(x, R) ∩ T punc contains at least a point. Remark 2.11. Properties of aperiodicity, repetitivity, and FLC can be defined for Delone sets as well. See [5,28–30] for instance. The study of Delone sets is equivalent to that of tilings with the same properties. Any tiling T defines the Delone set T punc . Conversely, a Delone set L defines its Voronoi tiling as follows. Let vx = y ∈ Rd : |y − x| ≤ |z − x| , ∀z ∈ L be the Voronoi tile at x ∈ L (it is a closed and convex polytope). The Voronoi tiling V (L) is the tiling with tiles vx , x ∈ L. Definition 2.12. Let T be a tiling such that T punc is r-uniformly discrete for some r > 0. (i) The collar of a tile t ∈ T is the patch Col(t) = {t ∈ T : dist(t, t ) ≤ r}. (ii) A prototile is an equivalence class of tiles under translation. A collared prototile is the subclass of a prototile whose representatives have the same collar up to translation. Then T(T ) and Tc (T ) will denote the set of prototiles and collared prototiles of T respectively. (iii) The support supp[t] of a prototile [t] is the set t − a with t a representative in the class of [t], and a ∈ Rd is the vector joining the origin to the puncture of t . As can be checked easily, the set t − a does not depend upon which tile t is chosen in [t]. Hence supp[t] is a set obtained from any tile in [t] by a translation with puncture at the origin. The support of a collared prototile is the union of the supports of the tiles it contains. A collared prototile is a prototile where a local configuration of its representatives has been specified: each representative has the same neighboring tiles up to translation. In general, Col(t) may contain more tiles than just the ones intersecting t, as illustrated in Fig. 1.
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Figure 1. Col(t) versus patch of tiles intersecting t Since T is FLC, it follows that it has only a finite number of collared prototiles. Moreover, any tiling in its hull has the same set of collared prototiles. The previous description allows to represent a tiling T in a more combinatorial way. Given a prototile tˆ (collared or not), let T punc (tˆ) be the subset of T punc made of punctures corresponding to tiles in tˆ (it should be noted that it is also an aperiodic repetitive FLC Delone set). Definition 2.13. The set I(T ) of pairs i = (a, tˆ), where tˆ ∈ T(T ) is a prototile and a ∈ T punc (tˆ), will be called the combinatorial representation of T . Similarly, the set Ic (T ) of pairs i = (a, tˆ), where tˆ ∈ Tc (T ) is a collared prototile and a ∈ T punc (tˆ), will be called the combinatorial collared representation of T . There is a one-to-one correspondence between the family of tiles of T and I(T ), namely, with each tile t ∈ T is associated the pair i(t) = (a(t), [t]) where a(t) is the puncture of t. By construction this map is one-to-one. Conversely, ti will denote the tile corresponding to i = (a, tˆ). Thanks to the Hypothesis 2.2, the tiles of T are finite and compatible CW -complexes. The next definition is a reformulation of the Anderson– Putnam space [2] that some of the authors gave in [38]. Definition 2.14. Let tˆi , i = 1, . . . p, be the prototiles of T . Let ti be the support of tˆi . The prototile space of T is the quotient CW -complex n K(T ) = ti / ∼, i=1
where two k-cells c ∈ tki and c ∈ tkj are identified if there exists u, u ∈ Rd for which ti + u, tj + u ∈ T , with c + u = c + u . The collared prototile space K c (T ) is built similarly out of the collared prototiles of T . We remind the following from [38, Proposition 1]. Proposition 2.15. There is a continuous map κ(c) : Ω → K (c) (T ). Proof. Let λ : tj → K (c) (T ) be the quotient map. And let ρ : Ω × Rd → tj be defined as follows. If a belongs to the intersection of k tiles tα1 , . . . tαk , in a tiling T ∈ Ω, with tαl = tjl + uαl (T ), l = 1, . . . k, then the point
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a − uαl (T ) belongs to tjl . Moreover, all these points are identified after taking the quotient, namely λ(a − uαl (T )) = λ(a − uαm (T )) if 1 ≤ l, m ≤ k. Therefore ρ(T , a) = λ(a − uαl (T )) is well defined. This allows to set κ(c) (T ) = ρ(T , 0Rd ). This map sends the origin of Rd , that lies in some tile of T , to the corresponding tile tj ’s at the corresponding position. In K (c) (T ), points on the boundaries of two tiles ti and tj are identified if there are neighboring copies of the tiles ti , tj somewhere in T such that the two associated points match. This ensures that the map κ(c) is well defined, for if in Rd tiled by T , the origin belongs to the boundaries of some tiles, then the corresponding points in tj given by ρ(T, 0Rd ) are identified by λ. Let a be a point in K (c) (T ), and Ua an open neighborhood of a. Say a belongs to the intersection of some tiles tj1 , . . . tjk . Let T be a preimage of a: κ(c) (T ) = a. The preimage of Ua is the set of tilings for which the origin lies in some neighborhood of tiles that are translates of tj1 , . . . tjk , and this defines a neighborhood of T in the Ω. Therefore κ(c) is continuous. A nested sequence of tilings (Tn )n , is a countable infinite sequence of tilings such that T1 = T and for all n ≥ 2 the tiles of Tn are (supports of) patches of Tn−1 . Without loss of generality, it will be assumed that T ∈ Ξ, i.e. has a puncture at the origin. To built such a sequence, the following procedure will be followed: assume that the tiling Tn−1 has been constructed and is aperiodic, repetitive and FLC with one puncture at the origin; then (i)
(ii) (iii)
let pn ⊂ Tn−1 be a finite patch with a puncture at the origin, and let Ln = {u ∈ Rd : pn + u ⊂ Tn−1 } be the Delone set of the punctures of the translated copies of pn within Tn−1 ; let V (Ln ) be the Voronoi tiling of Ln (Remark 2.11); to each tile of Tn−1 a tile v ∈ V (Ln ) will be assigned (see the precise definition below); (n) for each tile v ∈ V (Ln ), let tv be the union of the tiles of Tn−1 that (n) have been assigned to it; then Tn is defined as the tiling {tv }v∈V (Ln ) .
It is worth remarking that Ln is an aperiodic, repetitive, FLC, Delone set as well. Thus the Voronoi tiling inherits these properties. Note however (n) that in point (iii), two tiles tv may have the same shape. However, if they correspond to different patches of Tn−1 , they should be labeled as different. The second step of the construction above needs clarification since the tiles of V (Ln ) are not patches of Tn−1 , but convex polytopes built out of Ln (see Remark 2.11). First, the set of patches of T is countable, and second, for any patch p ⊂ T the Voronoi tiling of Lp = {u ∈ Rd : p + u ⊂ Tn−1 } has FLC, thus has finitely many prototiles. Hence there is a vector u ∈ Rd which is not in the span of any of the subspaces generated by the faces of the tiles Lp for all p. A point x ∈ Rd is called u-interior to a closed set X ⊂ Rd , and write u x ∈ X, if ∃δ > 0, ∀ε ∈ (0, δ), x + εu ∈ X. Given a Voronoi tile v ∈ V (Ln ) the patch associated with it is defined by
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u
tv(n) = ∪{t ∈ Tn−1 ; a(t) ∈ v}, where a(t) denotes the puncture of t. This gives an unambiguous assignment of tiles of Tn−1 to tiles of Tn . There is therefore a natural subdivision of each tile of Tn into tiles of Tn−1 . Let In = I(Tn ) be the combinatorial representation of Tn (see Definition 2.13). There is a map ln : In−1 → In , describing how to assign a tile in Tn−1 to the tile of Tn , namely to a patch of Tn−1 it belongs to. The “inverse map” defines a substitution, denoted by σn , namely a map from the set of tiles of Tn to the set of patches of Tn−1 defined by (n)
(n−1)
σn (ti ) = {tj
; ln (j) = i}
Such a substitution defines a map, also denoted by σn , from the CW -complex (n) Tn onto the CW -complex Tn−1 , if the tiles ti are given the CW -complex (n−1) structure inherited by the ones of the unions of the tj for ln (j) = i. In much the same way, this gives a map on the prototile space, and thus on the Anderson–Putnam complex as well. A similar construction holds if tiles and prototiles are replaced by collared tiles and collared prototiles. It leads to a canonical map σn : K c (Tn ) → K c (Tn−1 ), also denoted by σn . Let us denote by rn > 0 be the minimum of the inner radii of the tiles of Tn , and Rn > 0 the maximum of the outer radii. Definition 2.16. A nested sequence of tilings (Tn )n∈N with substitution maps σn : Tn → Tn−1 is called a proper nested sequence, if for all n the tiling Tn is aperiodic, repetitive, and has FLC, and the following holds for all n ≥ 2: (i) for each tile tn ∈ Tn , σn (tn ) is a patch of Tn−1 , (ii) for each tile tn ∈ Tn , σn (tn ) contains a tile of Tn−1 in the interior of its support, (iii) there exists ρ > 0 (independent of n), such that rn > rn−1 + ρ and Rn > Rn−1 + ρ. The nested sequence constructed above can be made proper by choosing, for each n, the patch pn to be large enough. These conditions are sufficient to ensure that K c (Tn ) is zoomed out of c K (Tn−1 ) for all n ≥ 2, in the terminology of [7, Definition 2.41]. The map σn : K c (Tn ) → K c (Tn−1 ) also forces the border (see [26], Definition 15) because it is defined on the collared prototile spaces. This suffices to recover the tiling space from the sequence (K c (Tn ))n∈N . Theorem 2.17. Let (Tn )n be a proper nested sequence of tilings. Then the tiling space Ω of T1 is homeomorphic to the inverse limit of the complexes K c (Tn ): (K c (Tn ), σn ) . Ω∼ = lim ←− n∈N
This Theorem was first proved in [7] and it was also proved that the Rd action could also be recovered from this construction (see also [38, Theorem 5]). The reader can also easily adapt the proof of Theorem 3.6 given in Sect. 3.1 to deduce Theorem 2.17.
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For each n, we can see the tiling Tn as a “subtiling” of T1 . The map κ(c) : Ω → K (c) (T1 ) of Proposition 2.15 can be immediately adapted to a map onto K (c) (Tn ) for each n. (c)
Proposition 2.18. There is a continuous map κn : Ω → K (c) (Tn ). Proof. Let T ∈ Ω. The origin of Rd lies in some patch pni ⊂ T which is a (n) (c) translate of σ1 ◦ σ2 ◦ · · · ◦ σn (ti ) for some i ∈ In . We set κn (T ) to be the (n) corresponding point in image of ti in K (c) (Tn ).
3. Tiling Groupoids and Bratteli Diagrams 3.1. Bratteli Diagrams Associated with a Tiling Space Bratteli diagrams were introduced in the seventies for the classification of AF algebras [8]. They were then adapted, in the topological setting, to encode Zactions on the Cantor set [13]. A specific case, close to our present concern, is the action of Z by the shift on some closed, stable, and minimal subset of {0, 1}Z (this is the one-dimensional symbolic analog of a tiling). Then, Bratteli diagrams were used to represent Z2 [17], and recently Zd [19] Cantor dynamical systems, or to represent the transversals of substitution tiling spaces [12]. Definition 3.1. A Bratteli diagram is an infinite directed graph B = (V, E, r, s) with sets of vertices V and edges E given by V= Vn , E = En , n∈N∪{0}
n∈N
where the Vn and En are finite sets. The set V0 consists of a single vertex, called the root of the diagram, and denoted ◦. The integer n ∈ N is called the generation index. And there are maps s : En → Vn−1 ,
r : En → Vn ,
called the source and range maps respectively. We assume that for all n ∈ N and v ∈ Vn one has s−1 (v) = ∅ and r−1 (v) = ∅ (regularity). We call B stationary, if for all n ∈ N, the sets Vn are pairwise isomorphic and the sets En are pairwise isomorphic. Regularity means that there are no “sinks” in the diagram, and no “sources” apart from the root. A Bratteli diagram can be endowed with a label, that is a map l : E → S to some set S. Definition 3.2. Let B be a Bratteli diagram. (i) A path in B is a sequence of composable edges (en )1≤n≤m , m ∈ N∪{+∞}, with en ∈ En and r(en ) = s(en+1 ). (ii) If m is finite, γ = (en )n≤m , is called a finite path, and m the length of γ. We denote by Πm the set of paths of length m. (iii) If m is infinite, x = (en )n∈N , is called an infinite path. We denote by ∂B the set of infinite paths.
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We extend the range map to finite paths: if γ = (en )n≤m we set s(γ) := s(em ).
In addition to Definition 3.1, we ask that a Bratteli diagram satisfies the following condition. Hypothesis 3.3. For all v ∈ V, there are at least two distinct infinite paths through v. The set ∂B is called the boundary +∞ of B. It has a natural topology inherited from the product topology on i=0 Ei , which makes it a compact and totally disconnected set. A base of neighborhoods is given by the following sets: [γ] = {x ∈ ∂B ; γ is a prefix of x}. Hypothesis 3.3 is the required condition to make sure that there are no isolated points. This implies the following. Proposition 3.4. With this topology, ∂B is a Cantor set. We now build a Bratteli diagram associated with a proper nested sequence of tilings. (n) Definition 3.5. Let (Tn )n∈N be a proper nested sequence of tilings. Let tˆi , i = (n) (n) 1, . . . , pm , be the collared prototiles of Tn , and ti the representative of tˆi that has its puncture at the origin. The Bratteli diagram B = (V, E, r, s, u) associated with (Tn )n∈N is given by the following: (n)
(i) (ii) (iii)
V0 = {◦}, and Vn = {ti , i = 1, . . . , pn }, n ∈ N, (1) E1 ∼ = V1 : e ∈ E1 if and only if s(e) = ◦ and r(e) = ti , (n−1) (n) e ∈ En with s(e) = ti and r(e) = tj , if and only if there exists
(iv)
a ∈ Rd such that ti + a is a tile of the patch σn (tj ), d a label u : E → R , with u(e) = −a for e ∈ En≥2 (and u = 0 on E1 ).
(n−1)
(n)
Figure 2 illustrates condition (iii). We extend the label as a map on finite paths u : Π → Rd : for γ = (e1 , . . . , en ) ∈ Πn we set u(γ) =
n
u(ei ).
i=1
We can associate to each finite path γ = (e1 , . . . , en ) in B, with s(γ) = t, the patches of T1 pγ = σ1 ◦ σ2 ◦ · · · σn (t) + u(γ) ,
and pcγ = σ1 ◦ σ2 ◦ · · · σn (Col(t)) + u(γ), (5)
where pcγ is a “collared patch”, in the sense that it is the set of tiles of T1 which make up the collar of a tile of Tn .
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Figure 2. Illustration of an edge e ∈ E, with s(e) = t and r(e) = t Theorem 3.6. Let Ξ be the transversal of the tiling space of T1 . There is a canonical homeomorphism ϕ : ∂B → Ξ. Proof. Let x ∈ ∂B, and set γn = x|Πn . Define Ξ(pcγn ), ϕ(x) = n∈N
Ξ(pcγn )
is the clopen set of tilings that have the patch pcγn at the origin where (see Eqs. (1) and (5)). Since Ξ is compact, and Ξ(pcγn ) is closed and contains Ξ(pcγn+1 ) for all n, by the finite intersection property ϕ(x) is a closed and non-empty subset of Ξ. Let us show it consists of a single tiling. Let then T, T ∈ ϕ(x). For all n, pcγn ⊂ T, T . And since pcγn is a collared patch, it contains a ball of radius rn (see Definition 2.16) centered at its puncture. Using the metric of Remark 2.7 this implies that δ(T, T ) ≤ 1/rn for all n. By condition (iii) in Definition 2.16, rn → ∞ as n → ∞, therefore δ(T, T ) = 0 and T = T . If x = x then γn = γn for some n, thus Ξ(pcγn ) ∩ Ξ(pcγn ) = ∅ and thus ϕ(x) = ϕ(x ). This proves that ϕ is injective. To prove that it is onto we exhibit an inverse. Let T ∈ Ξ. For each n, (n) κcn (T ) (Proposition 2.18) lies in the interior of some tile ti and therefore we can associate with T a sequence of edges through those vertices. This defines an inverse for ϕ. To prove that ϕ and ϕ−1 are continuous it suffices to show that the preimages of base open sets are open. Clearly ϕ([γ]) = Ξ(pcγ ), hence ϕ−1 is continuous. Conversely, since Ξ is compact and ∂B is Hausdorff, the continuity of ϕ is automatic. We now endow B with a horizontal structure to take into account the adjacency of prototiles in the tilings Tn , n ∈ N. Definition 3.7. A collared Bratteli diagram associated with a proper nested sequence (Tn )n∈N , is a graph B c = (B, H), with B = (V, E, r, s, u) the Bratteli diagram associated with (Tn )n∈N as in Definition 3.5, and where H is the set of horizontal edges:
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Figure 3. A horizontal edge h with s(h) = t and r(h) = t H=
Hn ,
with r, s : Hn → Vn ,
n∈N
given by h ∈ Hn with s(h) = t, r(h) = t , if and only if there exists a, a ∈ Rd , such that t + a, t + a ∈ Tn , with t + a ∈ Col(t + a ) ,
and t + a ∈ Col(t + a),
and we extend the label u to H, and set u(h) = a − a. There is a horizontal arrow in Hn between two tiles t, t ∈ Vn , if one can find “neighbor copies” in Tn where each copy belongs to the collar of the other. In other words there exists a patch p(t, t ) (i.e. its tiles have pairwise disjoint interiors) with p(t, t ) + a ⊂ Tn such that Col(t) ∪ (Col(t ) − u(h)) ⊂ p(t, t ), see Fig. 3 for an illustration. For h ∈ H, we define its opposite edge hop by s(hop ) = r(h), r(hop ) = s(h) , and u(hop ) = −u(h). Clearly, for all h in H, hop also belongs to H, and (hop )op = h. Also, the definition allows trivial edges, that is edges h for which s(h) = r(h) and u(h) = 0. 3.2. Equivalence Relations Definition 3.8. Let B be a Bratteli diagram and let Rn = {(x, γ) ∈ ∂B × Πn : r(x|Πn ) = r(γ)} . with the product topology (discrete topology on Πn ). The AF -equivalence relation is the direct limit of the Rn given by RAF = lim Rn = {((en )n∈N , (en )n∈N ) ∈ ∂B × ∂B : ∃n0 ∀n ≥ n0 en = en }, −→ n AF
with the direct limit topology. For (x, y) ∈ RAF we write x ∼ y and say that the paths are tail equivalent.
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It is well known that RAF is an AF -equivalence relation, as the direct limit of the compact ´etale relations Rn , see [33]. Assume now that B is a Bratteli diagram associated with a proper sequence of nested tilings (Tn )n∈N . We show now what this AF -equivalence relation represents for T1 and its transversal Ξ. For a finite path γ ∈ Πn , let us write tγ = r(γ) + u(γ),
(6)
which is the support of the patch pγ as defined in Eq. (5): σ1 ◦ σ2 ◦ · · · σn (tγ ) = pγ . For x ∈ ∂B, we can see t1 = tx|Π1 as a subset of tn = tx|Πn for all n ≥ 2. We characterize the subset of ∂B for which t1 stays close to the boundary of tn for all n, and its complement. Recall that a Gδ is a countable intersection of open sets, and an Fσ a countable union of closed sets. Lemma 3.9. For any n ∈ N, there exists a k > 0 such that for any v ∈ Vn and any v ∈ Vn+k , there is a path in ∂B from v to v . In particular, for any x ∈ ∂B, the AF -orbit of x is dense. Proof. The definition of the AF topology and the repetitivity of the underlying tilings are the two key elements of this proof. First, we prove that for any v ∈ Vn , there exists k ∈ N such that for any v ∈ Vn+k , there is a path from v to v . This is repetitivity: a vertex v corresponds to a tile in Tn . By repetitivity, any tile of Tn appears within a prescribed range, say R. Now, pick k such that the inner radius of the tiles of Tn+k is greater than R. It means that any tile of Tn appears in (the substitute) of any tile of Tn+k . This is exactly equivalent to the existence of a path from any v ∈ Vn to any v ∈ Vn+k . Now, consider x, y ∈ B. Let us show that y can be approximated by elements xn in B which are all AF -equivalent to x. Let xn be defined as follows: (xn )|Πn = y|Πn . We just proved that there is a k such that there is a path from r(y|Πn ) to r(x|Πn+k ). Continue xn with this path, and define its tail to be the tail of x. Then the sequence (xn )n∈N is the approximation we were looking for. Proposition 3.10. The subset G = x ∈ ∂B :
lim dist(t1 , ∂tn ) = +∞
n→+∞
is a dense Gδ in ∂B. Proof. For m ∈ N, let Gm = {x ∈ ∂B : ∃n0 ∈ N, ∀n ≥ n0 , dist(t1 , ∂tn ) > m}. Then G = m∈N Gm . Show that every Gm is a dense open set in ∂B. Remark that if for some n0 , dist(t1 , ∂tn0 ) > m, then this property holds for all n > n0 . Let us first prove that Gm is dense. Let n ∈ N. Then there is a k such that there is a path from any v ∈ Vn to any v ∈ Vn+k , by Lemma 3.9. Let l be such that Rn+k+l − Rn+k > m, where Rn is the outer radius of the tiles of Tn . Then, let γ be a path from Vn+k to Vn+k+l corresponding to the inclusion of a tile of Tn+k in the middle of a patch of Tn+k+l . Now, for any path η of length n, it is possible to join η to γ. Extend then this
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path containing η and γ arbitrarily to an infinite path x in ∂B. Then x satisfies dist(t1 , tn+k+l ) > dist(tn+k , tn+k+l ) > Rn+k+l − Rn+k > m, so x ∈ Gm . It proves that Gm is non-empty. Since we could do this construction for all n ∈ N and all η of length n, it proves that Gm is dense. Finally, Gm is open because if x ∈ Gm and satisfies dist(t1 , tn0 ) > m, then the tail of x after generation n can be changed without changing this property. It proves that Gm contains a neighborhood around all of its points, and so it is open. It proves that G is a Gδ as an intersection of dense open sets. Since ∂B is compact, it satisfies the Baire property and so G is dense. Corollary 3.11. The subset F = x ∈ ∂B :
lim dist(t1 , ∂tn ) < +∞
n→+∞
is a dense Fσ in ∂B. Proof. With the notation of Proposition 3.10 consider the closed set Fm = Gcm . We have Fm ⊂ Fm+1 and F = ∪m∈N Fm , thus F is an Fσ . The proof of the density of F in ∂B is similar to that for G in Proposition 3.10. Let x ∈ ∂B. Fix l ∈ N and consider the patch pl = px|Πl as in equation (5). By repetitivity of T1 there exists Rl such that T1 has a copy of pl in each ball of radius Rl . Hence there exists nl ∈ N such that for all n ≥ nl the tiles of Tn (viewed as patches of T1 under the map σ1 ◦ σ2 ◦ · · · σn ) contain a copy of pl that lies within a distance Rl to their boundaries : dist(tl , ∂tn ) ≤ dist(t1 , ∂tn ) < Rl . We can thus extend the finite path x|Πnl to x ∈ ∂B such that dist(t1 , ∂tn ) < Rl for all n ≥ nl . Set xl = x . We clearly have xl ∈ F . The sequence (xl )l∈N built in this way converges to x in ∂B. This proves that F is dense in ∂B. Notation. We will use now the following notation: Tx := ϕ(x) , for x ∈ ∂B ,
and xT := ϕ−1 (T ) , for T ∈ Ξ ,
(7)
where ϕ is the homeomorphism of Theorem 3.6. AF
For each T ∈ Ξ the equivalence relation ∼ induces an equivalence relation : on T punc
AF
a∼ ˙ b in T punc ⇐⇒ xT −a ∼ xT −b . Definition 3.12. An AF -region in a tiling T ∈ Ξ is the union of tiles whose punctures are ∼-equivalent ˙ in T punc . Proposition 3.13. A tiling T ∈ Ξ has a single AF -region if and only if xT ∈ G. Proof. Assume T ∈ Ξ has a single AF -region. Fix ρ > 0. Pick a ∈ T punc with AF |a| > ρ. Since xT ∼ xT −a , there exists n0 such that κcn (T ) and κcn (T − a) belong to the same tile in K c (Tn ) for all n ≥ n0 (see Proposition 2.18). Therefore dist(t1 , ∂tn ) > a > ρ for all n ≥ n0 . Since ρ was arbitrary this proves that limn→+∞ ρn = +∞, i.e. that xT ∈ G.
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Assume that x ∈ G. Choose a ∈ Txpunc . Since limn→+∞ dist(t1 , ∂tn ) = +∞, there exists n0 such that for all n ≥ n0 the patch pn = px|Πn ⊂ Tx contains a ball of radius 2aR/r around the origin (where r, R, are the parameters of the Delone set Txpunc ). Therefore Tx − a agree with pn − a on a ball of radius aR/r. Hence κcn (Tx ) and κcn (Tx − a) belong to the same tile in K c (Tn ) for all AF n ≥ n0 . Hence r(x|Πn ) = r(xTx −a |Πn ) for all n ≥ n0 . So we have x ∼ xTx −a , i.e. a ∼ ˙ 0. Since a was arbitrary, this shows that ξ has a single AF -region. = ϕ∗ (RAF ) be the equivalence relation on Ξ that is Remark 3.14. Let RAF the image of the equivalence relation RAF induced by the homeomorphism of -orbit of a tiling T ∈ Ξ is Theorem 3.6. Proposition 3.13 shows that the RAF the set of all translates of T by vectors linking to punctures that are in the AF -region of the origin:
˙ 0}, [T ]AF = {T − a : a ∈ T punc , a ∼ -orbit is So if T has more than one AF -region, i.e. if xT ∈ F , then its RAF only a proper subset of its RΞ -orbit (Definition 3). So we have:
[T ]AF = [T ]RΞ ⇐⇒ xT ∈ G. And for all T in the dense subset ϕ(F ) ⊂ Ξ we have [T ]AF [T ]RΞ . 3.3. Reconstruction of Tiling Groupoids As noted in Remark 3.14, the images in Ξ of the RAF -orbits do not always match those of RΞ . In this section, we build a new equivalence relation on ∂B that “enlarges” RAF , and from which we recover the full equivalence relation RΞ on Ξ. We consider a collared Bratteli diagram B c = (B, H) associated with a nested sequence (Tn )n∈N (Definitions 3.5 and 3.7), and we denote by Ξ the canonical transversal of the tiling space of T1 . Definition 3.15. A commutative diagram in B c is a closed subgraph h
/ e
e
h
/
where e, e ∈ En , h ∈ Hn−1 , and h ∈ Hn , for some n, and such that ⎧ s(h) = s(e) , ⎪ ⎪ ⎨ r(h) = s(e ) , and u(e) + u(h ) = u(h) + u(e ). r(e) = s(h ) , ⎪ ⎪ ⎩ r(e ) = r(h ) , Figure 4 illustrates geometrically the conditions of adjacency required for tiles to fit into a commutative diagram. With the notion of commutative diagram we can now define an equivalence relation on ∂B that contains RAF .
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Figure 4. Illustration of the commutative diagram in Definition 3.15 Definition 3.16. We say that two infinite paths x = (en )n∈N and y = (en )n∈N in ∂B are equivalent, and write x ∼ y, if there exists n0 ∈ N, and hn ∈ Hn for all n ≥ n0 , such that for each n > n0 the subgraph hn−1
/ en
en
hn
/
is a commutative diagram. It is not immediate that ∼ defines an equivalence relation. The following lemma proves that it is transitive. Lemma 3.17. x ∼ y if and only if Tx = Ty + a(x, y) for some a(x, y) ∈ Rd Proof. Assume x ∼ y in ∂B. Let n0 ∈ N be as in Definition 3.16, and for n > n0 set an = u(x|Πn ) − u(y|Πn ) + u(hn ). For all n > n0 + 1 we have an = an−1 − u(hn−1 ) + u(en ) − u(en ) + u(hn ) = an−1 , where the last equality occurs by commutativity of the diagram between generations n − 1 and n. Hence we have an = an0 for all n > n0 . Now for all n > n0 , the patches px|Πn and py|Πn + an0 belong to Tx (and similarly py|Πn , px|Πn − an0 ⊂ Ty ). Hence Tx = Ty + an0 , and set a(x, y) = −an0 . ˙ y) in Txpunc then Assume Tx = Ty +a(x, y) for some a(x, y) ∈ Rd . If 0∼a(x, x and y are AF -equivalent and thus x ∼ y (see Remark 3.19). If 0 and a(x, y) belong to different AF -regions, consider the nested sequence of tilings (Tn )n∈N corresponding to Tx = T1 . For all n, the two patches of Tn which respectively contain the points 0 and a(x, y) are within a distance less than |a(x, y)|. Hence, by hypothesis (iii) in Definition 2.16, their collars must intersect for n large enough. It follows that x ∼ y, and this completes the proof. Definition 3.18. We define the equivalence relation on ∂B RB = {(x, y) ∈ ∂B : x ∼ y},
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with the following topology: (xn , yn )n∈N converges to (x, y) in RB , if (xn )n∈N converges to x in ∂B, and a(xn , yn ) → a(x, y) in Rd . The FLC and repetitivity properties of T1 imply that for all T ∈ Ξ, the set of vectors linking its punctures, T punc −T punc , equals T1punc −T1punc , and is discrete and closed (see [5,28–30] for instance). The convergence (xn , yn ) → (x, y) in RB implies then that there exists n0 ∈ N such that a(xn , yn ) = a(x, y) for n ≥ n0 . Remark 3.19. If x ∼ y in RB are such that the horizontal edges hn ∈ Hn of the commutative diagrams are all trivial, then for all n ≥ n0 we have r(x|Πn ) = AF s(hn ) = r(hn ) = r(y|Πn ), i.e. x and y are tail equivalent in B : x ∼ y. Thus we have the inclusion RAF RB . In view of Remark 3.14, the two equivalence relation coincide on G, but differ on F (see Proposition 3.10 and Corollary 3.11) hence the inclusion is not an equality. We now give a technical lemma to exhibit a convenient base for the topology of RB . Given two paths γ, γ ∈ Πn , such that there exists h ∈ Hn with s(h) = r(γ) and r(h) = r(h) we define aγγ = u(γ) − u(γ ) + u(h) ,
and
[γγ ] = ϕ−1 (ϕ([γ]) ∩ (ϕ([γ ]) − aγγ )) , (8)
where ϕ is the homeomorphism of Theorem 3.6. So [γγ ] is the clopen set of tilings in Ξ which have the patch pγ at the origin and a copy of the patch pγ at position aγγ . Recall from Definition 2.9 that an RB -set is an open set in RB on which the source and range maps are homeomorphisms. Lemma 3.20. For γ, γ ∈ Πn , n ∈ N, the sets Oγγ = {(x, y) ∈ RB : x ∈ [γγ ] , a(x, y) = aγγ }, form a base of RB -sets for the topology of RB . Proof. We first prove that the Oγγ form a base for the topology of RB . A base open set in RB reads OU V = {(x, y) ∈ RB : x ∈ U, a(x, y) ∈ V } for a clopen U ⊂ Ξ and an open set V ⊂ Rd . As noted after Definition 3.18, the set of vectors a(x, y) is a subset of the countable, discrete, and closed set T1punc − T1punc . Hence we can write OU V as a countable union of open sets of the form OU,a = {(x, y) ∈ RB : x ∈ U, a(x, y) = a} for some a ∈ T1punc − T1punc . Since the sets [γ], γ ∈ Π, form a base for the topology of Ξ we can write OU,a as a (finite) union of open sets of the form Oγ,a = {(x, y) ∈ RB : x ∈ [γ], a(x, y) = a} for some γ ∈ Π. And we can choose those γ ∈ Πn for n large enough such that a belongs to a puncture of a tile in pγ , that is a = a(γ, γ ) for some γ ∈ Πn , and thus Oγ,a = Oγγ . Note that this proves that the sets [γγ ] also form a
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base for the topology of ∂B. Hence any open set in RB is a union of Oγγ , and therefore the sets Oγγ form a base for the topology of RB . We now prove that Oγγ is an RB -set. By definition Oγγ is open, so it suffices to show that the maps s|Oγγ and r|Oγγ are homeomorphisms. First note that s(Oγγ ) = [γγ ],
and
r(Oγγ ) = [γ γ].
(9)
Given (y, z) ∈ Oγγ , by Lemma 3.17, we have Tz = Ty + aγγ . Hence given y ∈ [γγ ], there is a unique z ∈ [γ γ] such that (y, z) ∈ Oγγ . And similarly, given z ∈ [γ γ], there is a unique y ∈ [γγ ] such that (y, z) ∈ Oγγ . Therefore the maps s|Oγγ and r|Oγγ are one-to-one. But Eq. (9) shows that they map base open sets in RB to base open set in ∂B. Hence those maps are homeomorphisms. We now state the main theorems, which characterize the equivalence relation RB , and compare it with RΞ . Theorem 3.21. The equivalence relation RB is ´etale. Proof. We check conditions (i), (ii), and (iii) of Definition 2.9. (iii) Let us show first that the maps r and s are continuous. It suffices to show that s−1 [γ] = {(x, y) : x ∈ [γ], x ∼ y} and r−1 ([γ]) = {(x, y) : y ∈ [γ], x ∼ y} are open in RB . Pick (x, y) ∈ s−1 [γ] (respectively (x, y) ∈ r−1 [γ]). Since x ∼ y, there exists γ such that x ∈ [γγ ] ⊂ [γ] (respectively y ∈ [γ γ] ⊂ [γ]). Since a(x, y) = a(γ, γ ) we have (x, y) ∈ Oγγ ⊂ s−1 ([γ]) (respectively (x, y) ∈ Oγγ ⊂ r−1 ([γ])). Thus s−1 ([γ]) (respectively r−1 ([γ])) is open. We have showed in Lemma 3.20 that sets Oγγ are base RB -sets. Hence the maps r and s are local homeomorphisms. From equation (9), we see that they are also open. 2 . Let (i) Pick w = ((x1 , x2 ), (x3 , x4 )), with x2 = x3 , in RB × RB \RB γ2 , γ3 ∈ Πn be such that for all (x, y) ∈ [γ2 ] × [γ3 ] we have x = y. Choose γ1 , γ2 ∈ Πm , m ≥ n, with γ2 |Πn = γ2 , such that (x1 , x2 ) ∈ Oγ1 γ2 . And choose similarly γ3 , γ4 ∈ Πl , l ≥ n, with γ3 |Πn = γ3 , such that (x3 , x4 ) ∈ Oγ3 γ4 . The 2 2 and contains w. Hence RB × RB \RB set Oγ1 γ2 × Oγ3 γ4 is open in RB × RB \RB 2 is open, and therefore RB is closed in RB × RB . Call p1 the map that sends ((x, y), (y, z)) to (y, z), p2 that which sends it to (x, z), and p3 that which sends it to (x, y). We have ⎧ −1 ⎨ p1 (Oγγ ) = s−1 ([γ γ]) × Oγγ p−1 (O ) = r−1 ([γγ ]) × s−1 ([γ γ]) , ⎩ 2−1 γγ p3 (Oγγ ) = Oγγ × r−1 ([γ γ]) 2 and the sets on the right hand sides are all open sets in RB . Hence the maps p1 , p2 , and p3 are continuous. (ii) Let (x, y) ∈ RB \Δ(RB ), so we have x = y. For each n pick xn ∈ [x|Πn ], and define yn ∈ ∂B to coincide with y on Πn , and with xn on its tail. Since x = y we have xn = yn , hence (xn , yn ) ∈ RB \Δ(RB ), for all n ≥ n0 for some n0 . Since (x, y) ∈ RB , by Lemma 3.17, there exists n1 such that for n > n1 we have a(x|Πn , y|Πn ) = a(x, y). Set n2 = max(n0 , n1 ) + 1. We have proved
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that the sequence (xn , yn )n≥n2 has all its elements in RB \Δ(RB ), and is such that: xn → x in ∂B, and a(xn , yn ) = a(x, y). Therefore it converges to (x, y) in RB \Δ(RB ). This proves that RB \Δ(RB ) is closed in RB , hence that Δ(RB ) is open in RB . Theorem 3.22. The two equivalence relations RB on ∂B, and RΞ on Ξ, are homeomorphic: RB ∼ = RΞ . The homeomorphism is induced by ϕ : ∂B → Ξ from Theorem 3.6. Proof. Consider the map ϕ∗ : RB → RΞ , given by ϕ∗ (x, y) = (Tx , Ty = Tx + a(x, y)). Since ϕ is a homeomorphism, ϕ∗ is injective. To prove that is surjective, consider (T, T = T + a) ∈ RΞ and let us show that (xT , xT ) belongs to RB , i.e. that xT ∼ xT . For each n ∈ N, call tn , tn , the tiles in Vn such that κcn (T ) ∈ tn and κcn (T ) ∈ tn (see Proposition 2.18). The nested sequence of tilings (Tn )n∈N induces a nested sequence (T˜n )n∈N with T˜1 = T . In T˜n the origin lies in a translate t˜n of tn , and the point a in a translate t˜n of tn . Since rn → ∞ (condition (iii) in Definition 2.16), there exists n0 such that for all n > n0 those two tiles t˜n and t˜n are within a distance rn to one another: dist(t˜n , t˜n ) ≤ rn . Therefore we have t˜n ∈ Col(t˜n ) and t˜n ∈ Col(t˜n ) (Definition 2.12). This means that for all n > n0 there exists a horizontal edge hn ∈ Hn with source tn and range tn (Definition 3.7). As xT and xT are the infinite paths in ∂B through the vertices tn and tn respectively, we have xT ∼ xT . This proves that ϕ∗ is a bijection. Now a sequence (xn , yn )n∈N converges to (x, y) in RB if and only if xn → x in ∂B, and a(xn , yn ) → a(x, y) in Rd . This is the case if and only if Txn → Tx = Ty + a(x, y) in Ξ and a(xn , yn ) → a(x, y) in Rd , since for all n one has Tyn = Txn + a(xn , yn ) by Lemma 3.17. Hence the map ϕ∗ and its inverse are continuous. Corollary 3.23. The groupoid of the equivalence relation RB is homeomorphic to ΓΞ .
4. Examples: One-Dimensional Tilings We illustrate here our construction for dimension 1 substitutions. In this case, if one chooses the proper sequence of tilings according to the substitution, then one recovers the usual formalism of Bratteli diagrams associated with the (Abelianization matrix of the) substitution. We show here that the horizontal structure introduced in Definition 3.7 allows to recover natural minimal and maximal infinite paths, and that the equivalence relation RB is then exactly / generated by RAF and the set of minimal and maximal paths (xmin , xmax ) ∈ RAF such that V (xmax ) = xmin , where V is the Vershik map on B (corresponding to an associated ordering of the edges). This will be shown carefully in examples, but this fact is general: the following result holds.
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Proposition 4.1. Let B be a collared Bratteli diagram associated with a 1-dimensional tiling, with labelled edges. Then there is a partial order on edges, which induces a partial order on infinite paths. Furthermore, there is a one-to-one map ψ from the set of maximal paths to the set of minimal paths, such that: RB = RAF ∧
(x, ψ(x)).
x maximal path
The fact that our labels on vertical edges give a partial ordering on edges is immediate: given v ∈ V, the set r−1 (v) is a set of edges encoding the inclusions of tiles in the substitution of tv . In dimension 1, it makes sense to define the edge of r−1 (v) which corresponds to the leftmost tile included in the substitution of tv (it is the edge with the maximal label in R). A minimal (respectively maximal) path is then a path made uniquely of minimal (respectively maximal) edges. It is then an exercise to show that given two paths in RB \RAF then they are tail-equivalent to a maximal path, respectively a minimal path. This gives a pairing of minimal with maximal paths, and thanks to the tiles decorations, this pairing is one-to-one. The fact that the groupoid relation can be recovered from the AF relation and a finite number of pairs is already known, and our formalism recovers this here. Furthermore, the pairing ψ corresponds actually to the translation of the associated tilings (more precisely to the action of the fist return map on Ξ). This map, the Vershik map, can be read from the Bratteli diagram (from the partial ordering of vertices), see [12] for example for the definition of this map. We treat in detail the cases of the Fibonacci and the Thue-Morse tilings. Those tilings have been extensively studied, and we refer the reader to [2] for a short presentation, and to [20] for further material. Both tiling spaces are strongly aperiodic, repetitive, and FLC [2]. This implies that the substitution induces a homeomorphism on the tiling space. Let (Ω, σ) denote either the Fibonacci or Thue-Morse tiling space, Ξ its canonical transversal, and λ the inflation constant of the substitution. Fix T ∈ Ξ. We can build a nested sequence of tilings associated with T . For k large enough, the nested sequence (λkn σ −kn (T ))n∈N is proper, because the k-th substitute of any tile contains one in its interior. One can easily see however that condition (ii) in Definition 2.16 can be replaced by the weaker assumption: For all n large enough and each tile tn ∈ Tn there exists m < n such that σnm (tn ) contains a tile of Tm in its interior. For example, it is straightforward to see that the proof of Theorem 3.6 goes through with this weaker condition. As noted earlier, we can take here m = n − k independently of the tile, for some fixed k. We will thus consider the sequence (Tn )n∈N , with Tn = λn σ −n (T ). The Bratteli diagram built in Definition 3.5, without the label u, is therefore exactly the usual Bratteli diagram associated with a substitution (its Abelianization matrix). For example, the encoding of words by paths, corresponds also exactly to the encoding of patches given in Eq. (5).
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4.1. The Fibonacci Tiling Let Ω be the Fibonacci tiling space, and Ξ its canonical transversal. Each tiling in Ω has two types of tiles up to translation, denoted 0 and 1. The prototile 0 is identified with the closed interval [−1/2, 1/2] with puncture at the origin, and the prototile 1 is identified with [−1/(2φ), 1/(2φ)] with puncture at the √ origin, where φ = (1 + 5)/2 is the golden mean. The substitution is given by 0 → 01, 1 → 0, and its inflation constant is φ. We write a, b, c, d, for the collared prototiles, where ˙ , Col(a) = 001
˙ , Col(b) = 100
˙ , Col(c) = 101
˙ Col(d) = 010,
and where the dot indicates the tile that holds the puncture. So a, b, and c correspond to the tile 0 but with different labels, while d corresponds to 1. The substitution on collared tiles reads then σ(a) = cd ,
σ(b) = ad ,
σ(c) = ad ,
σ(d) = b.
Let T ∈ Ξ, and consider the sequence (Tn )n∈N , with Tn = φn σ −n (T ). The Bratteli diagram B associated with this proper sequence has then the following form between two generations (excluding the root): an−1 Q cn−1 d bn−1 Q DD QQ QQQ m hhhm n−1 DD QQQQ QQQmmmmm hhmhmhmmmzzz h h h DD QQQ h mQ m DD mmm QQQ hhhhh mmm zzz QQQ QQmQmmmm hhhhhhQhQQQmQmmmm DD zz DD mmmm QQhQhhhh mmm QQQQQ zzz Q m h D m Q m h Q h m QQQ mmm zz QQQQ mm DD hhh QQQ 1 mhmhmhhhhhDDD 2 1 mmmmmQQQQQ 2 1 zzz m 2 mm h QQ Q m mhhh Q z m m m h 1 cn an bn dn
(10)
where tn is the support of Col(σ n (t)) for t = a, b, c, d. We write an arrow as entt ∈ En with s(entt ) = tn−1 and r(entt ) = tn . We have for n ≥ 2: 1 n−2 φ , 2φ 1 u(enda ) = u(endb ) = u(endc ) = − φn−2 , 2 u(enbd ) = 0. u(enab ) = u(enac ) = u(enca ) =
Note that given a path γ ∈ Π, the patch pγ of Eq. (5) corresponds exactly to the word associated with that path in the usual formalism of the Bratteli diagram of a substitution. For example, for the path (e1a , e2ac , e3ca , e4ab ), we ˙ → σ(cdb) ˙ → adbad. ˙ associate the patch or word σ 3 (b) → σ 2 (ad) The horizontal graph Hn has the following form. db ba cd
an
cn
bn
dn dc
ad
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Here, the indices show the patches corresponding to the edges (with left-right orientation). We have not shown trivial edges, and have identified an edge and its opposite in the drawing. If tt is one of the patches (with this orientation) in the above graph, we write an horizontal edge hntt ∈ Hn with s(hntt ) = tn−1 and r(hntt ) = tn . And we will simply use (hntt )op to avoid confusions (for example between (hncd )op and hndc ). For n ≥ 1 we have u(hnab ) = −φn−1 ,
φ u(hnad ) = u(hncd ) = u(hndc ) = u(hndb ) = − φn−1 . 2
There are only two commutative diagrams that one can write between two generations, namely bn−1 o
hn−1 ba
en bd
dn o
an−1 en ac
hn cd
cn
dn−1 o
hn−1 dc
en db
bn o
cn−1 (11) en ca
hn ab
an
The translation do match in those diagrams, we have: u(enbd ) + u(hn−1 ba ) = n n n−2 ) = u(e )+u(h ) = −φ (φ+ u(enac )+u(hncd ) = −φn−2 , and u(endb )+u(hn−1 ca ab dc 1)/2. Let us write Dn1 and Dn2 for the left and right above diagrams respec1 matches the bottom horizontal edge of tively. The top horizontal edge of Dn+1 2 2 matches the bottom one of Dn1 . We Dn , and the top horizontal edge of Dn+1 can thus “compose” those diagrams, and consider the two infinite sequences 1 2 2 1 , D2n+1 , . . .) and (D22 , D31 , D42 , . . . , D2n , D2n+1 , . . .). Each (D21 , D32 , D41 , . . . , D2n of those sequences contains exactly two infinite paths, namely: 2n+1 x1min = e1a , e2ac , e3ca , . . . , e2n ,... , ac , eca 2n+1 ,... , x1max = e1b , e2bd , e3db , . . . , e2n bd , edb 2n+1 x2min = e1c , e2ca , e3ac , . . . , e2n ,... , ca , eac 2n+1 ,... x2max = e1d , e2db , e3bd , . . . , e2n db , ebd (where we have added an edge to the root). If we order the edges in B as shown in Eq. (10), those are the two minimal and maximal infinite paths. And if we let V denote the Vershik map on ∂B, we have V (ximax ) = ximin , for i = 1, 2. All those paths are pairwise non-equivalent in RAF , but by definition they are equivalent in RB . Now given two infinite paths x, y ∈ ∂B such that AF AF x ∼ ximin and y ∼ ximax , for i = 1 or 2, we have x ∼ y in RB . We have thus shown that RB is generated by RAF and the two pairs (x1min , x1max ) and (x2min , x2max ): RB = RAF ∧ (x1min , x1max ), (x2min , x2max ) .
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4.2. The Thue-Morse Tiling Let Ω be the Thue-Morse tiling space, and Ξ its canonical transversal. Each tiling has two types of tiles up to translation, denoted 0 and 1. Each prototile is identified with the closed interval [−1/2, 1/2] with puncture at the origin. The substitution is given by 0 → 01, 1 → 10, and its inflation constant is 2. We write a, b, c, d, e, f, for the collared prototiles, where ˙ ˙ ˙ Col(a) = 001 Col(b) = 100 Col(c) = 011 ˙ ˙ ˙ Col(d) = 110 Col(e) = 101 Col(f ) = 010 and where the dot indicates the tile that holds the puncture. So a, b, and e correspond to the tile 0 but with different labels, while c, d, and f correspond to 1. The substitution on collared tiles reads then σ(a) = bf,
σ(b) = ec,
σ(c) = de,
σ(d) = f a,
σ(e) = bc,
σ(f ) = da.
Let T ∈ Ξ, and consider the nested sequence (Tn )n∈N , with Tn = 2n σ −n (T ). The Bratteli diagram B associated with that sequence has then the following form between two generations (excluding the root): cn−1 an−1 VV en−1 dn−1 h fn−1 PPPVVVVbn−1 PPP II II nnnuu hhhhuu h II PPPVVVVV PPP h I h n u h II uu PPPP VVVVVPPPP nnn uIuhIhhh PPP VVVVPVPVP InIhnIhnIhnhnuhuhuhuh IIII uuuu PPP IuIu hVnPhVnPhVnPhV uIuI PhPhPhPhPnhnhnhn PuVPuVPuVPVPVIIVIV uuuu IIII n h V I u h PPP IuVIuVVVV II hhhh nnn PPP uu hh2hhnnnnn 1 uuuPuPPP2P 1uuuPuPPIPIPI2 VVVVVVVIII1 h 1 h h h VV n 1 en cn 2 an hh 2 2 fn bn 1 dn
(12)
where tn is the support of Col(σ n (t)) for t = a, b, c, d, e, f . We write an arrow as entt ∈ En with s(entt ) = tn−1 and r(entt ) = tn . We have for n ≥ 2 1 n−2 2 , 2 1 u(enad ) = u(enaf ) = u(encb ) = u(ence ) = u(enec ) = u(enfa ) = − 2n−2 . 2 Note that given a path γ ∈ Π, the patch pγ of Eq. (5) corresponds exactly to the word associated with that path in the usual formalism of the Bratteli diagram of a substitution. For example, for the path (e1a , e2ad , e3dc , e4cb ), we ˙ → ecdef abc. associate the patch or word σ 3 (b) → σ 2 (ec) ˙ → σ(bcde) ˙ The horizontal graph Hn has the form u(enba ) = u(enbe ) = u(endc ) = u(endf ) = u(eneb ) = u(enfd ) =
bc cn an OO ab bn OOO ~ ~ ~ OOO ~~ ~~ O~ ~~ ~~OOOOOda ~ ~ ~ ~ OOO fa cd OOO~~~ec ~~ bf ~ ~ O ~ ~ O ~~ ~~ OOOOO ~~ f e O ~~ en fn dn ef
de
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where the indices show the patches corresponding to the edges (with left-right orientation). We have not shown trivial edges, and have identified an edge and its opposite in the drawing. If tt is one of the patches (with this orientation) in the above graph, we write an horizontal edge hntt ∈ Hn with s(hntt ) = tn−1 and r(hntt ) = tn . And we will simply use (hntt )op to avoid confusions (for example between (hnef )op and hnfe ). For n ≥ 1, we have u(hntt ) = −2n−1 . There are only four commutative diagrams that one can write between two generations, namely an−1 o
hn−1 ab
en af
fn o
bn−1 en be
hn fe
fn−1 o
hn−1 fe
en fa
en
an o
dn−1
en−1 o
en−1 (13) en eb
hn ab
bn
and cn−1 o
hn−1 cd
en df
en ce
en o
hn ef
fn
hn−1 ef
en fd
en ec
cn o
fn−1 (14)
hn cd
dn
The translation do match in those diagrams, for example we have: n n n−2 u(enaf ) + u(hn−1 , and u(ence ) + u(hn−1 ab ) = u(ebe ) + u(hf e ) = −(3/2)2 cd ) = u(endf ) + u(hnef ) = −(3/2)2n−2 . Let us write Dn1 and Dn2 for the left and right diagrams in Eq. (13) respectively, and Dn3 and Dn4 for those on the left and right of Eq. (14) respectively. The diagrams Dn1 and Dn2 , and Dn3 and Dn4 are “composable”: the top horizontal edge of one at generation n + 1 matches the bottom horizontal edge of the other at generation n. We consider the four infinite sequences: 1 2 1 2 1 2 1 2 2 1 D2 , D3 , D4 , . . . D2n , D2n+1 ,... , D2 , D3 , D4 , . . . D2n , D2n+1 ,... , 3 4 3 4 3 4 3 4 4 3 D2 , D3 , D4 , . . . D2n , D2n+1 ,... , D2 , D3 , D4 , . . . D2n , D2n+1 ,... . Each of those sequences contains exactly two infinite paths, namely: 2n+1 ,... , x1min = e1b , e2be , e3eb , . . . e2n be , eeb 2n+1 x1max = e1a , e2af , e3f a , . . . e2n ,... , af , ef a 2n+1 x2min = e1e , e2eb , e3be , . . . e2n ,... , eb , ebe
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2n+1 x2max = e1f , e2f a , e3af , . . . e2n , e , . . . , f a af 2n+1 x3min = e1d , e2df , e3f d , . . . e2n , e , . . . , df fd 2n+1 ,... , x3max = e1c , e2ce , e3ec , . . . e2n ce , eec 2n+1 x4min = e1f , e2f d , e3df , . . . e2n , e , . . . , f d df 2n+1 ,... x4max = e1e , e2ec , e3ce , . . . e2n ec , ece (where we have added edges to the root). If we order the edges in B as shown in Eq. (12), those are the four minimal and four maximal infinite paths. And if we let V denote the Vershik map on ∂B, we have V (ximax ) = ximin , for i = 1, 2, 3, 4. All those paths are pairwise non-equivalent in RAF , but by definition they are equivalent in RB . Now given two infinite paths x, y ∈ ∂B such that AF AF x ∼ ximin and y ∼ ximax , for i = 1, 2, 3 or 4, we have x ∼ y in RB . We have thus shown that RB is generated by RAF and the two pairs (ximin , ximax ) for i = 1, 2, 3, 4: RB = RAF ∧ (x1min , x1max ), (x2min , x2max ), (x3min , x3max ), (x4min , x4max ) .
Acknowledgements The three authors are indebted to Johannes Kellendonk for his invaluable input while starting this work. J.S. would like to thank John Hunton for discussions and for communicating information about his research using the notion of Bratteli diagrams. This work was supported by the NSF grants no. DMS0300398 and no. DMS-0600956, by the School of Mathematics at Georgia Tech in the Spring 2009 and by the SFB 701 (Universit¨ at Bielefeld, Germany) in the group led by Michael Baake.
References ´ Affability of Euclid[1] Alcade Cuesta, F., Gonz´ alez Sequiros, P., Lozano Rojo, A.: ean tilings. C. R. Acad. Sci. Paris, Ser. I 347, 947–952 (2009) [2] Anderson, J.E., Putnam, I.F.: Topological invariants for substitution tilings and their associated C ∗ -algebra. Ergod. Theory Dyn. Syst. 18, 509–537 (1998) [3] Bellissard, J.: K-Theory of C ∗ -algebras in solid state physics. In: Dorlas, T.C., Hugenholtz, M.N., Winnink, M. (eds.) Statistical Mechanics and Field Theory, Mathematical Aspects. Lecture Notes in Physics, vol. 257, pp. 99–156 (1986) [4] Bellissard, J.: Gap Labelling Theorems for Schr¨ odinger’s Operators, In: Luck, J.M., Moussa, P., Waldschmidt, M. (eds.) From Number Theory to Physics, pp. 538–630, Les Houches March 89. Springer, Berlin (1993)
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[5] Bellissard, J., Herrmann, D., Zarrouati, M.: Hull of Aperiodic Solids and Gap labeling Theorems, In: Baake, M.B., Moody, R.V. (eds.) Directions in Mathematical Quasicrystals, CRM Monograph Series, vol. 13, pp. 207–259. American Mathematical Society, Providence (2000) [6] Bellissard, J.: Noncommutative geometry of aperiodic solids. In: Geometric and Topological Methods for Quantum Field Theory, (Villa de Leyva, 2001), pp. 86–156, World Sci. Publishing, River Edge (2003) [7] Bellissard, J., Benedetti, R., Gambaudo, J.-M.: Spaces of tilings, finite telescopic approximations and gap-labelling. Commun. Math. Phys. 261, 1–41 (2006) [8] Bratteli, O.: Inductive limits of finite dimensional C ∗ -algebras. Trans. Am. Math. Soc. 171, 195–234 (1972) [9] Connes, A., Krieger, W.: Measure space automorphisms, the normalizers of their full groups, and approximate finiteness. J. Funct. Anal. 24, 336–352 (1977) [10] Connes, A.: Sur la th´eorie non commutative de l’int´egration, in Alg`ebres d’Op´erateurs. Lecture Notes in Mathematics, vol. 725, pp. 19–143. Springer, Berlin (1979) [11] Connes, A.: Noncommutative Geometry. Academic Press, San Diego (1994) [12] Durand, F., Host, B., Skau, C.: Substitutional dynamical systems, Bratteli diagrams and dimension groups. Ergod. Theory Dynam. Syst. 19(4), 953–993 (1999) [13] Forrest, A.H.: K-groups associated with substitution minimal systems. Israel J. Math. 98, 101–139 (1997) [14] G¨ ahler, F.: Unpublished work [15] Giordano, T., Putnam, I.F., Skau, C.F.: Affable equivalence relations and orbit structure of Cantor dynamical systems. Ergodic Theory Dyn. Syst. 24, 441– 475 (2004) [16] Giordano, T., Putnam, I.F., Skau, C.F.: The orbit structure of Cantor minimal Z2 -systems, Operator Algebras: The Abel Symposium 2004, pp. 145–160, Abel Symp. 1. Springer, Berlin (2006) [17] Giordano, T., Matui, H., Putnam, I.F., Skau, C.F.: Orbit equivalence for Cantor minimal Z2 -systems. J. Am. Math. Soc. 21(3), 863–892 (2008) [18] Giordano, T., Matui, H., Putnam, I.F., Skau, C.F.: The absorption theorem for affable equivalence relations. Ergodic Theory Dyn. Syst. 28, 1509–1531 (2008) [19] Giordano, T., Matui, H., Putnam, I.F., Skau, C.F.: Orbit equivalence for Cantor minimal Zd -systems. arXiv:0810.3957 [math.DS] [20] Gr¨ unbaum, B., Shephard, G.C.: Tilings and Patterns, 1st edn. W.H. Freemand and Co, New York (1987) [21] Herman, R.H., Putnam, I.F., Skau, C.F.: Ordered Bratteli diagrams, dimension groups and topological dynamics. Int. J. Math. 3, 827–864 (1992) [22] Hippert, F., Gratias, D. (eds): Lectures on Quasicrystals. Editions de Physique, Les Ulis (1994) [23] Julien, A., Savinien, J.: Tiling groupoids and Bratteli diagrams II: substitution tilings (in preparation) [24] Katz, A., Gratias, D.: In: Janot, C., Mosseri, R. (eds.) Proceeding of the 5th International Conference on Quasicrystals, pp. 164–167. World Scientific, Singapore (1995)
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[25] Kellendonk, J.: Noncommutative geometry of tilings and gap labelling. Rev. Math. Phys. 7, 1133–1180 (1995) [26] Kellendonk, J.: Local structure of tilings and their integer group of coinvariants. Commun. Math. Phys. 187, 115–157 (1997) [27] Kramer, P.: Nonperiodic central space filling with icosahedral symmetry using copies of seven elementary cells. Acta Cryst. Sect. A 38, 257–264 (1982) [28] Lagarias, J.C.: Geometric models for quasicrystals. I. Delone sets of finite type. Discrete Comput. Geom. 21, 161–191 (1999) [29] Lagarias, J.C.: Geometric models for quasicrystals. II. Local rules under isometries. Discrete Comput. Geom. 21, 345–372 (1999) [30] Lagarias, J.C., Pleasants, P.A.B.: Repetitive Delone sets and quasicrystals. Ergod. Theory Dyn. Syst. 23, 831–867 (2003) [31] Matui, H.: Affability of equivalence relations arising from two-dimensional substitution tilings. Ergod. Theory Dyn. Syst. 26, 467–480 (2006) [32] Penrose, R.: The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 10, 55–65 (1974) [33] Phillips, N.C.: Crossed products of the Cantor set by free minimal actions of Zd . Commun. Math. Phys. 256(1), 1–42 (2005) [34] Queff´elec, M.: Substitution dynamical systems—spectral analysis. Lecture Notes in Math., vol. 1294. Springer, Berlin (1987) [35] Radin, C., Wolff, M.: Space tilings and local isomorphism. Geom. Dedicata 42, 355–360 (1992) [36] Renault, J.: A groupoid approach to C ∗ -algebras, Lecture Notes in Math., vol. 793. Springer, Berlin (1980) [37] Sadun, L.: Tiling spaces are inverse limits. J. Math. Phys. 44(11), 5410–5414 (2003) [38] Savinien, J., Bellissard, J.: A Spectral Sequence for the K-theory of Tiling Spaces. Ergod. Thpry Dyn. Syst. 29, 997–1031 (2009) [39] Senechal, M.: Quasicrystals and Geometry. Cambridge University Press, Cambridge (1995) [40] Shechtman, D., Blech, I., Gratias, D., Cahn, J.W.: Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 51, 1951–1953 (1984) [41] Skau, C.: Minimal dynamical systems, ordered Bratteli diagrams and associated C ∗ -crossed products. Current topics in operator algebras (Nara, 1990), pp. 264– 280. World Sci. Publishing, River Edge (1991) [42] Vershik, A.M.: A Theorem on Periodical Markov Approximation in Ergodic Theory. Ergodic Theory and Related Topics (Vitte, 1981), pp. 195–206. Math. Res., vol. 12. Akademie-Verlag, Berlin (1981)
Jean Bellissard School of Mathematics Georgia Institute of Technology Atlanta, GA, USA e-mail:
[email protected]
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Antoine Julien, Jean Savinien Universit´e de Lyon Universit´e Lyon 1, Institut Camille Jordan UMR 5208 du CNRS, 42, boulevard du 11 novembre 1918 69622 Villeurbanne Cedex, France e-mail:
[email protected];
[email protected] Communicated by Jens Marklof. Received: November 16, 2009. Accepted: March 18, 2010.
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A Bijection Between Paths for the M(p, 2p + 1) Minimal Model Virasoro Characters Olivier Blondeau-Fournier, Pierre Mathieu and Trevor A. Welsh Abstract. The states in the irreducible modules of the minimal models can be represented by infinite lattice paths arising from consideration of the corresponding RSOS statistical models. For the M(p, 2p + 1) models, a completely different path representation has been found recently, this one on a half-integer lattice; it has no known underlying statistical-model interpretation. The correctness of this alternative representation has not yet been demonstrated, even at the level of the generating functions, since the resulting fermionic characters differ from the known ones. This gap is filled here, with the presentation of two versions of a bijection between the two path representations of the M(p, 2p + 1) states. In addition, a half-lattice path representation for the M(p + 1, 2p + 1) models is stated, and other generalisations suggested.
1. Introduction The corner-transfer matrix is a powerful tool for expressing the local state probabilities of the order variable in terms of weighted sums of one-dimensional configurations [2]. For the restricted-solid-on-solid (RSOS) models [1,11] in regime III (in the infinite length limit), each weighted configuration sum turns out to be the character of an irreducible module of the corresponding minimal model [5,19]. Each configuration is the specification of the order variable’s value at each position, with fixed extremities, and the differences between neighbouring values constrained. When dressed with edges linking adjacent points, it becomes a lattice path. Every state in the corresponding conformal theory is thus represented by a particular lattice path [7,21,22]. This path description of the states has proved to be a royal road for the construction of the fermionic characters [17], either using direct methods [16,21,22], or recursive ones [7,10,24]. For the special class of minimal models M(p, 2p+1), an alternative path representation was found in [15], and this led
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to novel fermionic expressions. These paths, however, were not deduced from a statistical model whose scaling limit was known to be the M(p, 2p+1) minimal model. Moreover, their generating functions have not, hitherto, been proved to be equivalent to either the bosonic expressions [20] or the usual fermionic expressions [8,10,16,17,24] for these characters (although the equivalence has been checked to high order, and was supported by an asymptotic analysis). Albeit formally ad hoc, the discovery of this new path representation relied on heuristic considerations which we briefly recall. Like their scaling limit, the RSOS models are parameterised by two relatively prime positive integers p, p with p > p ≥ 2.1 The corresponding paths are sequences of NE and SE edges restricted to the interior of an infinite strip of vertical width p − 2 (this definition is made more precise in Sect. 2.2). The weight of each path is the sum of the weights of its vertices. The weight of a vertex depends not only upon its shape, but also upon the two parameters p and p [11]. In the unitary cases (those cases where p = p + 1), a drastic simplification occurs in that the peaks and the valleys of a path do not contribute to its weight, while every other vertex contributes x/2, where x is its horizontal position. On the other hand, weighting the path in the dual way (assigning weight x/2 to the valleys and peaks and 0 to all other vertices) leads to the character of the Zp−1 parafermionic models [3,5,9,21,22]. These cases were then contrasted with the paths describing the states of the graded versions of these parafermionic models—equivalent u(1) [4,12]. These latter paths reside on a half-inteto the cosets o sp(1, 2)p−1 /ˆ ger lattice with only peaks and valleys contributing to the weight, each to the amount x/2, and with the special constraint that peaks are forced to occur at integer heights.2 Seeking their dual versions led to the new path description of the M(p, 2p + 1) models [15]. The demonstration of the correctness of this new path description of the M(p, 2p+1) states is the main subject of the present work. This is achieved by exhibiting a bijection between these paths and the RSOS paths. In fact, two equivalent but quite different-looking versions of this bijection are presented. After defining the two types of path in Sect. 2, the first version of the bijection is presented in Sect. 3. It relies on techniques developed in [7,24]. It essentially amounts to the deconstruction of an RSOS path, followed by a corresponding construction of a half-lattice path. The second version of the 1
For the RSOS models/paths, we follow the notation used in [7, 10, 11, 24]. Consequently, the corresponding notation for the minimal models differs from that of [6, 16] by the interchange of p and p . This interchange is of course irrelevant, the physical characteristics, the conformal dimensions hr,s =
[r max(p, p ) − s min(p, p )]2 − (p − p )2 4pp
of the highest-weight states in modules labelled by (r, s) for 1 ≤ r < min(p, p ) and 1 ≤ s < max(p, p ), remain unchanged. 2 The usual parafermionic states have two path representations, bijectively related in [13] (see also [18] and references therein). This is also true for the graded ones [15, Sect. 5]. We are referring here to the RSOS-type path representation (with no horizontal edges).
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bijection is presented in Sect. 4. It relies on an encoding of the vertex words for RSOS paths introduced in [23], and an analogous construction for the halflattice paths that is a modification of that given in [14]. In this approach, each path, with given extremity conditions, is constructed from a sequence of operators acting on a suitable ground-state path. The bijection is expressed as a rule for transforming between sequences of operators that describe the two paths. Although the two versions of the bijection were originally obtained independently, we show in Sect. 4.3 that the second is, in fact, a transformation of the first, thereby immediately obtaining its proof.
2. Defining the Two Types of Paths 2.1. α-Lattice Paths An infinite length α-lattice path h is a sequence (h0 , hα , h2α , . . .) for which |hx+α − hx | = α for x ∈ αZ≥0 . The path h is said to be (f, g)-restricted if f ≤ hx ≤ g for all x ≥ 0 and b-tailed if there exists L ≥ 0 such that hx ∈ {b, b − α} for all x ≥ L. In this work, we consider two cases: α = 1 (integer lattice paths) and α = 1/2 (half-lattice paths). By linking the points (0, h0 ), (α, hα ), (2α, h2α ), . . . on the x–y plane we obtain a graph that conveniently depicts the path: we refer to this graph as the path picture. For each x > 0, the position (x, hx ) on the path picture of h is referred to as a vertex. The shape of the vertex is either a peak, valley, straight-up or straight-down depending on whether the edges on the two sides of the vertex are NE-SE, SE-NE, NE-NE or SE-SE, respectively. On occasion, it will be necessary to refer also to the position (0, h0 ) as a vertex, and to have its shape determined. When required, this is done by specifying the value of h−α to be one of the two values h0 ± α. In effect, this defines the direction of a path pre-segment. 2.2. RSOS Paths and the M(p, p ) Models
p,p The set Pa,b of RSOS paths is defined to be the set of infinite3 length integer lattice paths h that are (1, p − 1)-restricted, b-tailed, with h0 = a. Each RSOS p,p is assigned a weight wt(h) which we now define.4 path h ∈ Pa,b For 1 ≤ k ≤ p − 1, we refer to the region of the x–y plane between y = k and y = k + 1 as the kth band. Thus, in the path picture, the paths from p,p lie in the p − 2 bands between y = 1 and y = p − 1. For 1 ≤ r < p, Pa,b 3 These paths were originally defined for finite length in [11] and studied extensively in [7, 10, 24]. The collection of finite paths (with specific boundary conditions) provides a finitized version of the Virasoro characters [19]. The finitization parameter enables the derivation of useful recurrence relations for the path generating functions. In particular, one can define the notion of dual finitized characters, which are obtained via the transformation q → 1/q [3, 7, 15, 21, 22]. 4 The definition given here is that of [7]. It differs considerably from that originally given
p,p in [11]. In fact, although this is not obvious, the weighting of the paths Pa,b defined in [11] differs from that of [7] by an overall constant depending upon a and b.
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4,9 Figure 1. A path h ∈ P1,3 . The unfilled circles indicate upscoring vertices, which have weight ux , and the filled circles indicate down-scoring vertices, which have weight vx , both values being defined in (1)
we shade the rp /pth band and refer to it as the rth dark band. Bands that are not dark are referred to as light bands.5 The band structure for the case 4,9 , is shown in Fig. 1. This (p, p ) = (4, 9), along with a typical path from P1,3 band pattern is typical of all models with p = 2p + 1: there are p − 1 dark bands separated from each other by a single light one. Those vertices which either are straight with the right edge in a dark band, or are not straight with the right edge in a light band, are referred to as scoring vertices. All other vertices are referred to as non-scoring vertices. Each scoring vertex is said to be up-scoring or down-scoring depending on whether the left edge is up or down respectively. In the path picture, we (often) highlight the up-scoring vertices with an unfilled circle, and the down-scoring vertices with a filled circle. This is done for the path of Fig. 1. After setting6 1 1 ux = (x − hx + a) vx = (x + hx − a), (1) 2 2 the weight wt(h) of a path h is defined to be: wt(h) = wx , (2) x>0
where we define
⎧ ⎪ ⎨ux wx = vx ⎪ ⎩ 0
if (x, hx ) is up-scoring; if (x, hx ) is down-scoring; if (x, hx ) is non-scoring.
(3)
For the path h depicted in Fig. 1, we obtain wt(h) = 0 + 0 + 3 + 1 + 1 + 2 + 9 + 9 + 6 + 11 + 11 + 9 + 12 = 74.
(4)
The tail condition of an RSOS path h indicates that after a certain position, the path forever oscillates within a single band. The definition (2) then 5
In [7, 10, 24], dark and light bands were referred to as odd and even bands, respectively. The point (x, hx ) has coordinates (ux , vx ) in the system where the axes are inclined at 45◦ , and the origin is at the path startpoint.
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implies that wt(h) is finite or infinite depending on whether this band is dark or light, respectively. Below, we restrict to those cases where that band is dark. p,p The paths in the set Pa,b provide a combinatorial description of the states in the irreducible module of the minimal model M(p, p ) that is labelled by (r, s), where the extremity points a and b are related to r and s by a = s and b = rp /p + 1.
(5)
(Thereupon, the heights b and b − 1 straddle the rth dark band of the path p,p of paths is then the Virasoro picture). The generating function of the set Pa,b p,p character χr,s [7]: q wt(h) = χp,p (6) r,s (q).
p,p h∈Pa,b
2.3. M(p, 2p + 1) Characters and Half-Lattice Paths The states in the irreducible modules of the M(p, 2p + 1) models turn out to have a path representation alternative to that specified above. This description was first given in [15]. ˆ For p ≥ 2 and a ˆ, ˆb ∈ Z, define Hapˆ,ˆb to be the set of all half-lattice paths h ˆ0 = a ˆ, and the additional restricthat are (0, p − 1)-restricted, ˆb-tailed, with h ˆ x+1 ∈ Z, then h ˆ x+1/2 = h ˆ x − 1/2. The additional restriction ˆx = h tion that if h here implies that peaks can only occur at integer heights. ˆ ∈ Hp , we specify the shape of the vertex at its startFor each path h a ˆ,ˆ b ˆ −1/2 = a ˆ − 1/2 (we do this even point (0, a ˆ) by adopting the convention that h if a ˆ = 0). ˆ of a path h ˆ ∈ Hp is defined to be half The unnormalised weight w ˆ ◦ (h) a ˆ,ˆ b ˆ x ) is a straight-vertex: the sum of those x ∈ 1 Z≥0 for which (x, h 2
ˆ =1 w ˆ ◦ (h) 2
x.
(7)
x∈ 12 Z≥0 ˆ x+1/2 ˆ hx−1/2 =h
ˆ gs ∈ Hp to be that path which has minimal Define the ground-state path h a ˆ,ˆ b
weight amongst all the elements of Hapˆ,ˆb . It is easily seen that this path has the ˆ gs ˆ gs = ˆb and h = ˆb − 1/2 for following shape: an oscillating part having h x
x+1/2
ˆ gs = (ˆ a, a ˆ ± 1/2, a ˆ± x ∈ Z≥|ˆa−ˆb| , preceded by an initial straight line having h x 1, . . . , ˆb ∓ 1/2) for x = (0, 1/2, 1, . . . , |ˆ a − ˆb| − 1/2), where the upper signs apply when ˆb ≥ a ˆ and the lower signs apply when ˆb < a ˆ. p ˆ ˆ The weight wt(h) of each h ∈ H is then defined by a ˆ,ˆ b
ˆ −w ˆ gs ). ˆ =w t(h) ˆ ◦ (h w ˆ ◦ (h)
(8)
ˆ ∈ H4 of Fig. 2, we have used dots to indiFor instance, in the path h 0,1 cate the vertices that contribute to (7). The corresponding ground-state path
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ˆ ∈ H5 (solid) and the corresponding Figure 3. A path h 3,1 ˆ gs (dotted) ground state h
ˆ gs = 1 for x ∈ Z>0 and h ˆ gs = 1/2 for x ∈ Z≥0 +1/2. Therefore, ˆ gs ∈ H4 has h h x x 0,1 from (8), we obtain: ˆ = 1 0 + 1 + 1 + 3 + 2 + 5 + 9 + 5 + 11 + 13 + 15 + 8 t(h) w 2 2 2 2 2 2 2 2
25 27 35 37 41 43 1 1 + + + + + + − 0+ = 74. (9) 2 2 2 2 2 2 2 2 ˆ t(h) In what follows, we make use of the following trick to calculate w p ˆ ∈ H . First note that the minimal weight directly from the path picture of h a ˆ,ˆ b p gs ˆ ∈ H extends between heights a ˆ and ˆb in its first 2e (half-integer) path h a ˆ,ˆ b
ˆ is to extend t(h), steps, where e = |ˆ a − ˆb|. An alternative to (8) for obtaining w ˆ ˆ ˆ h to the left by 2e steps, in such a way that h−e = b (overriding the above ˆ is then obtained by ˆ −1/2 ). The renormalised weight w t(h) convention for h summing the x-coordinates of all the straight vertices of this extended path, beginning with its first vertex at (−e, ˆb) whose nature is specified by setting ˆ −e−1/2 = ˆb − 1/2, and dividing by 2. h ˆ ∈ H5 represented by To illustrate this construction, consider the path h 3,1 ˆ gs ∈ H5 is shown dashed. the solid line in Fig. 3. The minimal weight path h 3,1
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Using (7) and (8), we obtain
ˆ = 1 2 + 5 + 9 + 5 + 11 + 6 + 13 + 7 + 15 + 17 = 55 , w ˆ ◦ (h) 2 2 2 2 2 2 2 2 (10)
3 7 55 7 1 1 ◦ ˆ gs ˆ w ˆ (h ) = +1+ +2 = =⇒ wt(h) = − = 24. 2 2 2 2 2 2 Alternatively, we may use the extended path shown in Fig. 4. From this, the ˆ is immediately obtained via t(h) renormalised weight w ˆ = 1 −2− 3 −1− 1 +2 + 5 + 9 t(h) w 2 2 2 2 2
13 15 17 11 + 6+ +7+ + +5+ = 24. (11) 2 2 2 2 In [15], the generating function for these paths was conjectured to be a Virasoro character: ˆ p,2p+1 q wt(h) = χr,s (q), (12) p ˆ h∈H a ˆ ,ˆ b
where the module labels r, s are given by s = 2ˆ a + 1 and r = ˆb.
(13)
2.4. Statement of the Results In the following two sections, we describe a weight-preserving bijection between the sets p,2p+1 ↔ Hp1 (a−1), 1 (b−1) , Pa,b 2
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(14)
for p ≥ 2, and a and b odd integers with 1 ≤ a < 2p and 3 ≤ b < 2p. In combination with the p = 2p + 1 case of (6), the establishment of this bijection proves (12). Note that for the two sets related by (14), the values of the module labels r and s, obtained for the two cases using (5) and (13), respectively, are in agreement. At first sight, the restriction on the parity of the values of a and b appears to constrain the applicability of our construction to those modules which are labelled by (r, s) with s odd. However, the equivalence of the modules labelled
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by (r, s) and (p − r, p − s) (a consequence of the identity hr,s = hp−r,p −s for conformal dimensions) implies that, in this p = 2p + 1 case, the construction applies to all inequivalent modules. In what follows, it is notationally convenient to make the identification p p,2p+1 Pa,b ≡ Pa,b .
(15)
3. The Combinatorial Bijection The description of the bijection given in this section is combinatorial because it is specified in terms of direct manipulations of the paths. This contrasts with the second description, presented in the next section, which is formulated in terms of (non-local) operators. Roughly, in the combinatorial scheme, a path p is transformed by first stripping off its (charge 1) particles, then from Pa,b reinterpreting the cut path as a Hapˆ,ˆb path through rescaling it by a factor of 1/2, and finally, reinserting the particles in a precise way. Throughout this section, we take a and b to be odd integers, with 1 ≤ a < 2p and 3 ≤ b < 2p, and set a ˆ=
1 (a − 1) 2
and
ˆb = 1 (b − 1). 2
(16)
3.1. Specifying the Bijection p Let h ∈ Pa,b . From h, repeatedly remove adjacent pairs of scoring vertices, in each case adjoining the loose ends (which will be at the same height), until no adjacent pair of scoring vertices remains. Let hcut denote the resulting path. 4 4 given in Fig. 1, the resulting hcut ∈ P1,3 is given in For the path h ∈ P1,3 Fig. 5, having removed n = 4 pairs of scoring vertices from the former. To describe the bijection, it is also required that, when carrying out the above removal process, we record the number of non-scoring vertices to the left of each pair of scoring vertices that is removed. This resulting list of n integers then encodes a partition, of at most n parts, which we denote λ(h). In the case of the path h of Fig. 1, the n = 4 removals result in the partition λ(h) = (9, 8, 5, 1).
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ˆ cut obtained from Fig. 5 Figure 6. h
Note that the removal process ensures that hcut has no peak at an even height, and no valley at an odd height. Therefore, if we shrink the whole path hcut by a factor of 2, and decrease the values of the heights by 1/2, we obtain a ˆ cut , whose peaks are all at integer heights and whose valleys half-lattice path, h ˆ cut does not extend above height are all at non-integer heights. In particular, h cut cut ˆ ˆ ˆ cut ∈ Hp . In the ˆ ˆ and h is b-tailed. Therefore, h p − 1. In addition, h0 = a a ˆ,ˆ b ˆ cut is given in Fig. 6. case of the hcut of Fig. 5, the resulting h ˆ cut occurs at a non-integer height. As already stressed, each valley of h cut ˆ ˆ by deepening n of these valleys to the next The path h is obtained from h integer height by inserting pairs of SE-NE edges. On labelling the valleys of ˆ cut from left to right by 1, 2, 3, . . . , the n valleys labelled μ1 , μ2 , . . . , μn , are h deepened, where we set μi = λi + n + 1 − i,
(17)
with λ = λ(h). Note that the partition μ = (μ1 , μ2 , . . . , μn ) has distinct parts. In our ongoing example, we have λ = (9, 8, 5, 1). From this (17) yields ˆ from h ˆ cut by deepening the μ = (13, 11, 7, 2), and therefore, we obtain h ˆ is given 13th, 11th, 7th and 2nd valleys of the latter. The resulting path h ˆ = 74 = wt(h) in this case. t(h) in Fig. 2. It may be checked that w ˆ cut , n, μ) → h ˆ is We claim that the combined map h → (hcut , n, λ) → (h p p a weight-preserving bijection between Pa,b and Haˆ,ˆb . That it is a bijection fol-
p lows because the inverse map from Hapˆ,ˆb to Pa,b , which is easily described, ˆ ∈ Hp arises from a is well-defined. This relies on the fact that each h a ˆ,ˆ b ˆ cut , whose characteristic property, we recall, is that its valunique path, h ˆ by making ˆ cut is thus recovered from h leys are all at non-integer heights; h shallow each integer valley. With n the number of such integer valleys, the partition μ = (μ1 , . . . , μn ) is determined by setting its parts to be the numberings of the integer valleys amongst all valleys, counted from the left. The parts of μ are necessarily distinct, and thus a genuine partition λ is recovˆ is weight-preserving is demonstrated ered via (17). That the map h → h below.
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3.2. Removing Basic Particles from the RSOS Paths7 Each pair of adjacent scoring vertices (which are necessarily of different types) p is identified with a particle:8 where there occur d ≥ 2 consecuin h ∈ Pa,b tive scoring vertices, we identify d/2 particles (when d is odd, the ambiguity over which pairs are the actual particles is immaterial). The excitation of each of these particles is defined to be the number of non-scoring vertices to its left. Thus, the partition λ(h), defined in Sect. 3.1, lists the excitations of the particles in h. A path which contains no particles is said to be particle-deficient. We p simply by removing all of the construct the particle-deficient path hcut ∈ Pa,b p particles from h ∈ Pa,b . To determine the weight of hcut , first consider the removal of one such particle from h. Let λi be the number of non-scoring vertices to its left, and let k be the total number of scoring vertices in h (k = 13 for the path of Fig. 1). The band structure for the p = 2p + 1 cases ensures that the first of the two scoring vertices that comprise the particle is necessarily a peak or valley. Let it be at position (x, hx ). If it is a peak, then the following vertex is at (x + 1, hx − 1), and together they contribute 1 ux + vx+1 = (x − hx + a + x + 1 + hx − 1 − a) = x (18) 2 to the weight. If it is a valley, then the following vertex is at (x + 1, hx + 1), and together they contribute 1 (19) vx + ux+1 = (x + hx − a + x + 1 − hx − 1 + a) = x 2 to the weight. There are x − 1 − λi scoring vertices to the left of the particle, and thus λi + k − x − 1 to its right. On removing the particle, the contribution of each of the latter to the weight decreases by one. Thus, the total weight reduction on removing the particle is λi + k − 1. Then, if h contains n particles, on removing all of them, noting that k decreases by two at each step, we obtain n cut wt(h ) = wt(h) − λi − (k − 1) − (k − 3) − (k − 5) − · · · − (k − 2n − 1) i=1
= wt(h) −
n
λi − n(k − n).
(20)
i=1
3.3. Mapping from RSOS Paths to Half-Lattice Paths p ˆ cut by , define a half-lattice path h Given a particle-deficient path hcut ∈ Pa,b shrinking hcut by a factor of 2, and decreasing all heights by 1/2. Note that ˆ cut is a element of Hp , whose valleys are at non-integer heights. In this h a ˆ,ˆ b ˆ cut ) = wt(hcut ). t(h section, we show that w 7
The process presented in this section is described, using similar terminology, in [23, Sect. 5] (cf. Eq. 15 therein), and previously in [7, Sect. 2] using the notions of B2 and B3 transforms (therein, h(0) is used to denote hcut ). 8 These are the particles of charge 1 in the terminology of [16].
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Figure 7. Example of segment matching for a path h First consider the case where a = b. (The path hcut then starts and ends at the same height—if truncated beyond the start of the final oscillation). In such a case, we may match (pair) each path segment with another at the same height, one NE and one SE (the order is immaterial). This matching process is illustrated in Fig. 7. Since hcut has no peak at an even height and no valley at an odd height, the scoring vertices of hcut occur at the right ends of all the segments in the light bands. Consider a specific matched pair of segments in a light band, with the left ends of the NE and SE segments at positions (x, y) and (x , y + 1) respectively. The two scoring vertices at (x + 1, y + 1) and (x + 1, y) together contribute 1 1 ux+1 + vx +1 = (x + 1 − (y + 1) + a + x + 1 + y − a) = (x + x + 1) (21) 2 2 to wt(hcut ). ˆ cut , the weight may be obtained For the corresponding half-lattice path h by considering the four straight vertices at each end of each of the segments of the matched pair (the vertex at (0, a ˆ) will be required here if it is straight). These four vertices contribute 1 1 1 1 1 1 x + (x + 1) + x + (x + 1) = (x + x + 1) (22) 2 2 2 2 2 2 ˆ cut ). Since this agrees with the contribution of the corresponding two to w ˆ ◦ (h ˆ cut ) = ˆ ◦ (h scoring vertices of hcut to the weight wt(hcut ), we conclude that w cut cut ◦ cut ˆ )=w ˆ ), thereby proving that w ˆ cut ) = t(h t(h ˆ (h wt(h ). But, for a = b, w cut wt(h ) in this a = b case. In the case that a = b, we make use of the trick described in Sect. 2.3 ˆ cut ) by extending the path h ˆ cut to the left. On the other hand, t(h to obtain w extending the path hcut to the left by 2e = | a − b | steps with hcut −2e = b, cut creates a path of unchanged weight wt(h ) because, via (1), the additional scoring vertices each contribute 0 to the weight. Then, upon applying the argument used above in the a = b case to these extended paths, we obtain ˆ cut ) = wt(hcut ) for a = b also. t(h w
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3.4. Deepening Valleys ˆ (0) ∈ Hp , having m straight vertices. This count Consider a half-lattice path h a ˆ,ˆ b includes consideration of the vertex at (0, a ˆ), which, through the convention ˆ (0) is stated in Sect. 2.3, is deemed straight if and only if the first segment of h (0) ˆ in the NE direction. The vertices of h that do not contribute to its weight are the peaks and valleys. We now determine the change in weight on deepening one of the valleys. Let the valley being deepened be the jth, counting from ˆ (0) the left, and let it be situated at (x, h x ). There are necessarily j peaks to the left of this valley (perhaps including one at (0, a ˆ)), and therefore 2x + 1 − 2j straight vertices. After this position, there are thus m − 2x + 2j − 1 straight vertices. The deepening moves each of these to the right by two (half-integer) ˆ (0) positions. It also introduces two straight vertices, at positions (x, h x ) and ˆ (0) ˆ (1) , (x + 1, h x ). Thus, if this resulting path is denoted h ˆ (1) ) = w ˆ (0) ) + 1 (x + (x + 1)) + 1 (m − 2x + 2j − 1)) t(h t(h w 2 2 1 (0) ˆ = wt(h ) + m + j. 2
(23)
Note that the deepening increases the value of m by 2. So if we obtain ˆ by performing a succession of deepenings to a path h ˆ cut at valleys the path h numbered μ1 , μ2 , . . . , μn , we have n ˆ =w ˆ cut ) + 1 (m + (m + 2) + · · · + (m + 2n − 2)) + t(h) t(h μi w 2 i=1 n ˆ cut ) + n (m + n − 1) + t(h =w μi . 2 i=1
(24)
3.5. Altogether Now ˆ cut , n, μ) → h ˆ defined in Consider the combined map h → (hcut , n, λ) → (h Sect. 3.1 above. Let k and k be the number of scoring vertices in h and hcut , respectively, and let m be the number of straight vertices (including considˆ cut . The pair removal process in Sect. 3.2 eration of the vertex at (0, a ˆ)) in h shows that k = k − 2n. The matching of edges described in Sect. 3.3 shows that m = 2k and thus m = 2k − 4n. Then, using Eqs. (20) and (24), and the ˆ cut ) = wt(hcut ), we obtain t(h fact that w ˆ − wt(h) = t(h) w
n
μi −
i=1
=
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= 0,
n
λi +
n (m + n − 1) − n(k − n) 2
λi −
n (n + 1) 2
i=1
μi −
n i=1
(25)
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where the final equality follows because, using (17), n
μi −
i=1
n i=1
λi =
n i=1
i=
n (n + 1). 2
(26)
Thus the weight-preserving nature of the bijection has been proved.
4. The Path-Operator Bijection In this section, we introduce natural descriptions of each of the two types of paths in terms of sequences of (local and non-local) operators. In the case of p , these sequences are encoding of the vertex words introthe RSOS paths Pa,b duced in [23]. For the Hapˆ,ˆb paths, the sequences are simplified versions of a
p modification of those already presented in [14]. The bijection between Pa,b p and Haˆ,ˆb , described in the previous section, is then formulated as a rule for transforming between such sequences.
4.1. P p Paths as Sequences of Operators
p,p , the vertex word w(h) is defined to be the sequence of For each path h ∈ Pa,b letters S and N which indicate the sequence of vertices, scoring or non-scoring, of h, read from the left [23, Sect. 5]. For example, for the path h given in Fig. 1,
w(h) = SN SSSN SN N N SSN SN N SSN SSSN N N N N N N N N N N · · · . (27) Each vertex word w(h) is of infinite length and, because the tail of h lies in a dark band, w(h) contains only a finite number of letters S. As indicated in [23], every infinite length word w in S and N having only finite number of entries S, is the vertex word w = w(h) of at most one path h. In the cases where p = 2p + 1, and a and b are odd integers with 1 ≤ a < 2p and 3 ≤ b < 2p, we encode w(h) as follows. After indexing the letters of the vertex word of w(h) by the x-coordinates of the corresponding vertices, drop all letters N . Then, reading from the left, replace each consecutive pair Sx+1 Sx+2 by dx . Then replace each remaining Sy+1 by cy or c∗y depending on whether y is even or odd. This yields a finite sequence of symbols dx , cy and c∗z which uniquely represents the original path h. We denote it π(h) and refer to it as the operator word of h. In the case of the path h of Fig. 1, from (27), we obtain π(h) = c0 d2 c4 c6 d10 c∗13 d16 d19 c∗21 . c∗z
(28)
Each symbol dx , cy or may be viewed as an operator which changes the nature of three or two vertices of an oscillating portion of a path near position x, y or z respectively. These actions are illustrated in Fig. 8. The path h then results from the action of the operator word π(h) on the purely oscillating vacvac(b) vac(b) p , defined by hx = b for x even and hx = b−1 uum path hvac(b) ∈ Pb,b for x odd.
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(a)
(b)
(c)
(d)
Figure 8. The action of the operators dx , cx and c∗x on a portion of a path that oscillates in a dark band. Note that dx either creates a valley (cf. a) or a peak (cf. b) in a light band according to the odd/even parity of x
Note that the operator dx acts locally to generate either a peak or a valley in a light band, changing neither the path’s startpoint nor tail. In contrast, the actions of the operators cx and c∗x are non-local in the sense that they change the portion of the path lying between 0 and x + 1. In particular, they change the startpoint of the path, decreasing and increasing it by 2, respectively.9 Therefore, because the operator word π(h) maps between elements of p p and Pa,b , it follows that the difference between the number of operators Pb,b cy and c∗z in the operator word π(h) is (b − a)/2. 9
The operators cx and c∗x map between paths which label states of different modules. Their actions could be defined either to maintain the tail and change the initial point p p (Pa,b → Pa∓2,b ), as described in the main text, or, alternatively, to change the tail and p p → Pa,b±2 ), as in [14] for unitary RSOS paths. Under the maintain the initial point (Pa,b (∗)
action of cx , the paths are then modified either in the interval [0, x + 1], as in the main text, or, alternatively, in the interval [x, ∞]. Either choice corresponds to a genuine non-local action. However, the first one is somewhat more localized. This is one motivation for the choice made here. But the ultimate reason lies in the expected greater simplicity of this choice in the treatment of other (than the p = 2p + 1) classes of models.
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Obtained as described above, the subscripts of neighbouring pairs of operators in π(h) naturally satisfy certain constraints. They are: (∗)
dx cx , cx(∗) dx , dx dx , cx cx , c∗x c∗x cx c∗x , c∗x cx
=⇒
x ≥ x + 2,
=⇒
x ≥ x + 3,
(29)
where c(∗) denotes either c or c∗ . A word π in the symbols dx , cy , c∗z is called standard if every neighbouring pair in π respects the constraints (29). In the operator word π(h), the particles described in Sect. 3.2 correspond to the symbols dx . The excitation of each such particle is given by the number of non-scoring vertices in h to its left. Therefore, for the ith operator dx in π(h), counted from the right, this excitation is given by λi = x − 2#{d · · · dx } − #{c(∗) · · · dx },
(30)
where #{A · · · B} denotes the number of pairs A and B of operators in π(h) with A to the left of B. For instance, consider the excitation λ1 of the particle corresponding to d19 in the word (28): there are three pairs of the first type: (d2 , d19 ), (d10 , d19 ), and (d16 , d19 ), and four pairs of the second type: (c0 , d19 ), (c4 , d19 ), (c6 , d19 ), and (c∗13 , d19 ). Thus, in this case, the excitation is λ1 = 19 − 6 − 4 = 9. For an arbitrary word π in the symbols dx , cy , c∗z , we use (30) to define λi for the ith operator dx , counted from the right. Then define the vector λ(π) = (λ1 , λ2 , . . . , λn ), where n is the number of operators dx in π. If λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0 (so that λ(π) is a partition), then we say that π is physical. Otherwise, we say that π is unphysical. The constraints (29) guarantee that every standard word π(h) is physical. We now consider the effect on π(h) of changing the excitations of the particles in h. First consider incrementing or decrementing the excitation of the particle corresponding to an operator dx . This is possible if and only if we obtain a physical word π on replacing the operator dx in π(h) by dx+1 or dx−1 , respectively (otherwise, the particle is ‘blocked’: two particles cannot occupy the same position). Even if physical, the word π might not be standard. However, if standard, the resulting path is readily obtained from π via the actions given in Fig. 8. If it is not standard, then necessarily we would have made one of the local changes: (∗)
(∗)
dx cx+2 → dx+1 cx+2 ,
(31a)
(∗) cx−2
(31b)
dx →
(∗) cx−2
dx−1 ,
depending on whether we incremented or decremented the excitation. These violations of (29) may be seen to arise because, within an odd length sequence of scoring vertices, there is ambiguity over which pairs correspond to the particles. For example, the sequence N SSSN might be interpreted with the first two Ss being the particle, thereby yielding the operators d1 c3 , or the latter two Ss being the particle, thereby yielding the operators c1 d2 . Thus, we should impose the equivalence (∗)
dx cx+2 ≡ cx(∗) dx+1 .
(32)
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In the case of the moves (31), this equivalence should be applied before increasing the excitation in the case (31a), and after decreasing the excitation in the case (31b). To excite the particle corresponding to a particular dx by more than 1, we can proceed as above, one step at a time. However, we easily see that the process can be streamlined by first adding the required excitement to the subscript of dx (this excitation being possible if and only if the resulting word (∗) is physical), and then for each pair dy cy with y < y + 2, imposing the equivalence (∗)
(∗)
dy cy ≡ cy −2 dy+1 .
(33)
For example, consider the operator word d0 d2 d4 d6 c8 c10 c12 c∗17 c∗21 .
(34)
Exciting the rightmost particle (represented by d6 ) by 9 yields first d0 d2 d4 d15 c8 c10 c12 c∗17 c∗21 .
(35)
Repeatedly applying (33) yields the following sequence of words: d0 d2 d4 (d15 c8 ) c10 c12 c∗17 c∗21 → d0 d2 d4 c6 (d16 c10 ) c12 c∗17 c∗21 → d0 d2 d4 c6 c8 (d17 c12 ) c∗17 c∗21 → d0 d2 d4 c6 c8 c10 (d18 c∗17 ) c∗21 → d0 d2 d4 c6 c8 c10 c∗15 d19 c∗21 ,
(36)
where, in each line, we have used parentheses to indicate the pair of operators affected. Conversely, we can reduce the excitation of a dx by decreasing its subscript by the required amount (again, this is possible if and only if the resulting word is physical), and then repeatedly using (33) in the cases y ≥ y + 2 to reexpress the right side as the left side. In either case, once the standard operator word has been obtained, the corresponding RSOS path can be readily constructed using Fig. 8. The construction may be applied to excite a number of particles simultaneously. For example, applying the excitations 1, 5, 8 and 9 to the particles of the operator word (34) yields the non-standard word d1 d7 d12 d15 c8 c10 c12 c∗17 c∗21 .
(37)
It may be checked that repeated use of (33) then yields the standard word given in (28), which corresponds to the path of Fig. 1.10 Finally, for this section, we note that the excitation of a particle corresponding to an operator dx in an operator word π is given by (30), even when the word is non-standard. This is so because the value of the expression (30) is unchanged for operator words obtained from one another through the equivalence (33). 10
Operator words of the form (34) which begin with the n operators d0 d2 d4 · · · d2n−2 are, in the language of [7], the result of a B2 (n)-transform on a particle-deficient path. The process of exciting these particles corresponds to the B3 (λ)-transform, with the excitation of the ith particle, counted from the right, specified by the part λi of the partition λ.
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4.2. Hp Paths as Sequences of Operators Here, we provide an encoding of the half-lattice paths of Hapˆ,ˆb that is analogous to that of the previous section. ˆ has no peak at a non-integer height, ˆ ∈ Hp . Because h For a ˆ, ˆb ∈ Z, let h a ˆ,ˆ b it follows that if there is a straight-up vertex at position x ∈ Z, there must also be one at position x + 1/2, and if there is a straight-down vertex at position x ∈ Z + 12 , there must either be a straight-down vertex at position x + 1/2 or a straight-up vertex at position x + 1. Similarly, if there is a straight-down vertex at position x ∈ Z, there must also be one at position x − 1/2, and if there is a straight-up vertex at position x ∈ Z + 12 , there must either be a straight-up vertex at position x − 1/2 or a straight-down vertex at position ˆ we may consecutively pair the x − 1. Consequently, working from the left of h, straight vertices such that each pair occurs at neighbouring positions x and x + 1/2, or next neighbouring positions x and x + 1 (through the convention stated in Sect. 2.3, the vertex at (0, a ˆ) is deemed straight if and only if the ˆ is in the NE direction). We encode each of those pairs that first segment of h occur at positions x and x + 1/2 using cˆx or cˆ∗x depending on whether x is integer or non-integer, respectively. In the other case, where the pair occurs at positions x and x+1, we encode the pair using dˆx . In this latter case, note that x ∈ Z + 12 and that there is a valley at the intermediate position x + 1/2. Let ˆ denote the word in the symbols dˆx , cˆy , cˆ∗ obtained in this way, ordered π ˆ (h) z ˆ Of course, with increasing subscripts. We refer to it as the operator word of h. 11 ˆ ˆ the half-lattice path h can be immediately recovered from π ˆ (h). 11
As mentioned in the introduction of the section, this operator construction of the Hp paths is a modified version of the one already presented in [14]. To substantiate this statement, let us recall briefly the operator construction of Hp paths in [14]. The paths are constructed from the appropriate ground state by the action of a sequence of non-local operators bx , b∗x : bx transforms a peak at x into a straight-up segment and b∗x transforms a valley at x into a straight-down segment (both segments linking the points x and x + 1/2). These actions modify the path for all x ≥ x + 1/2 (in contradistinction with the actions of the operators defined in the main text, which are limited to the initial portion of the path, namely x ≤ x). The constraint on the integrality of the peak positions shows that the operators b and b∗ always occur in successive pairs that are either of the types (b b), (b∗ b∗ ) or (b∗ b), with subindices differing by a half-integer in the first two cases and by an integer in the third one. In terms of these operators, the operators c˜, c˜∗ and dˆ are expressed , dˆx = b∗ bx +1 . c˜x = bx bx+1/2 , c˜∗ = b∗ b∗ x
x
x +1/2
x
dˆ is the same as the operator defined in the main text, while c˜ and c˜∗ are roughly the operators cˆ and cˆ∗ but acting instead to maintain the path’s initial portion, while changing its tail. Summing up, our present operator construction is different from that introduced in [14] in two ways: it uses a reduced number of operators and the operator action modifies the initial rather than the final portion of the path. Let us now turn to the interpretation where particles are viewed as the path’s basic constituents [14, 16, 21, 22]. Given that every path is described by a sequence composed of an equal number of operators b and b∗ , it can be seen, from the decomposition of a path into charged peaks [15], that there are actually 2p − 3 combinations of the b and b∗ that are allowed. These are the l-blocks—the particles whose numbers are the summation variables in the fermionic character—defined by [14]: bl−1 b∗l b
for l odd
and
bl b∗l
for l even,
where
1 ≤ l ≤ 2p − 3.
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(a)
(b)
(c)
Figure 9. The action of the operators dˆx , cˆx and cˆ∗x on a portion of a path that oscillates between heights l and l − 1/2, where l ∈ Z. The black dots are the vertices that contribute to the weight ˆ of Fig. 2. Here, the To illustrate this construction, consider the path h ˆ is found to be operator word π ˆ (h) ˆ = cˆ0 cˆ1 cˆ2 cˆ∗9 dˆ11 cˆ∗15 dˆ25 dˆ35 dˆ41 . π ˆ (h) 2 2 2 2 2
2
(38)
Each symbol dˆx , cˆy or cˆ∗z may be viewed as an operator which changes the nature of two vertices of an oscillating portion of a path near position x, y or ˆ then results z, respectively. These actions are illustrated in Fig. 9. The path h ˆ from the action of the operator word π ˆ (h) on the purely oscillating vacuum ˆ b) ˆ vac( ˆb and h ˆ vac(ˆb) = ˆb − 1/2 for x ∈ Z≥0 . ˆ vac(ˆb) ∈ Hp , defined by h = path h x ˆ b,ˆ b
x+1/2
Note that the actions of the operators cˆy and cˆ∗z are non-local in the sense that they change the starting height of the path, decreasing and increasing it by 1, respectively. In contrast, dˆx does not affect the path’s starting point, acting locally to change an integer peak into an integer valley.
Footnote 11 continued In terms of the new operators, the path building blocks are seen to be simply the p − 1 combinations: dˆ and c˜ c˜∗ , or equivalently cˆ cˆ∗ , for 1 ≤ ≤ p − 2. (In the terminology of [16], these particles are interpreted as a breather and pairs of kinks-antikinks of topological charge respectively.) This particle interpretation is the starting point for a direct derivation of the characters (in the line of [21, 22]) that matches the usual expressions given in [3, 7, 8, 16, 17, 24].
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ˆ naturally satisfy certain The subscripts of the symbols in the word π ˆ (h) constraints. They are: cˆx cˆx , cˆ∗x cˆ∗x , cˆ∗x dx =⇒ x ≥ x + 1, 3 cˆ∗x cˆx , cˆx cˆ∗x , dˆx cˆx , cˆx dˆx =⇒ x ≥ x + , 2 dˆx dˆx , dˆx cˆ∗x =⇒ x ≥ x + 2.
(39)
Later, we consider arbitrary words in the operators dˆx , cˆy and cˆ∗z , with each y ∈ Z and each x, z ∈ Z + 12 . Such a word π ˆ is called standard if every neighbouring pair in π ˆ respects the constraints (39). ˆ corresponds to an integer valley of h ˆ at position Each operator dˆx in π ˆ (h) x + 1/2 ∈ Z. We define the excitation of this integer valley to be the number of non-integer valleys to its left. To express this excitation in terms of the ˆ let #{A · · · B} denote the number of pairs A and B of operator word π ˆ (h), ˆ with A to the left of B. The number of straight vertices operators in π ˆ (h) c(∗) · · · dˆx } + 1, strictly to the left of position x + 1 is then 2#{dˆ· · · dˆx } + 2#{ˆ (∗) ∗ where cˆ denotes either cˆ or cˆ (the +1 accounting for the straight-down vertex of dˆx at x). Thus, the number of valleys and peaks strictly to the left of c(∗) · · · dˆx }. Exactly half of these are valleys. x+1 is 2x+1−2#{dˆ· · · dˆx }−2#{ˆ Of those, #{dˆ· · · dˆx } + 1 are integer valleys. Therefore, the excitation of the ˆ counted from the ˆ (h), integer valley corresponding to the ith operator dˆx in π right, is given by ˆ i = x − 1 − 2#{dˆ· · · dˆx } − #{ˆ λ c(∗) · · · dˆx }. 2
(40)
Proceeding as in Sect. 4.1, for an arbitrary word π ˆ in the symbols dˆx , cˆy , ˆ ˆ we use (40) to define λi for the ith operator dx , counted from the right. ˆ2, . . . , λ ˆ n ), where n is the number of operˆ π ) = (λ ˆ1, λ Then define the vector λ(ˆ ˆ ˆ ˆ ˆ ˆ . If λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0 then we say that π ˆ is physical, and ators dx in π unphysical otherwise. Again, the constraints (39) ensure that every standard ˆ is physical. word π ˆ (h) ˆ of changing the excitations of the We now consider the effect on π ˆ (h) ˆ particles in h, mirroring the analysis of the previous section with some adjustments. First consider incrementing the excitation of the integer valley corresponding to an operator dˆx . This is possible if and only if we obtain a physical ˆ by dˆx+1 (otherwise, the integer ˆ (h) word π ˆ on replacing the operator dˆx in π valley is ‘blocked’). Here again, even if physical, the word π ˆ might not be standard. ˆ of incrementing the excitation is to make shallow the The effect on h ˆ ˆ be valley of h at position x + 1/2, and to deepen the subsequent valley. Let h ˆ the resulting path. We claim that if π ˆ is standard then π ˆ (h ) = π ˆ . This is so ˆ the operator dˆx is immediately followed ˆ (h), because, if π ˆ is standard then, in π (∗) by either an operator cˆx or an operator dˆx (or nothing) with x > x + 2. cˆ∗z ,
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Thus, the subsequent valley to be deepened is at position x + 2. We then see ˆ ) = π ˆ , as claimed. that π ˆ (h ˆ is non-standard. In this On the other hand, if x ≤ x + 2, the word π ˆ from the sequence case, we ascertain the position of the subsequent valley in h ˆ ˆ ˆ ˆ (h). In this word, dx and subsequent operators of operators that follow dx in π necessarily form the standard subword dˆx cˆx+3/2 cˆx+5/2 · · · cˆx+t+1/2 cˆ∗x+t+2 cˆ∗x+t+3 · · · cˆ∗x+t+t∗ +1 ,
(41)
∗
where t ≥ 0, t ≥ 0, with at least one of these non-zero, and any subsequent operators have subscripts at least x+t+t∗ +5/2 (there cannot be a subsequent operator dˆx+t+t∗ +2 because, as is readily checked using (40), with such an operator, the word π ˆ would be unphysical). The subsequent valley is then at ˆ ) is obtained from π ˆ by ˆ (h ˆ (h) position x + t + t + 2. Therefore, the word π replacing the subword (41) by cˆx+1/2 cˆx+3/2 · · · cˆx+t−1/2 cˆ∗x+t+1 cˆ∗x+t+2 · · · cˆ∗x+t+t∗ dˆx+t+t∗ +1 .
(42)
Alternatively, we may proceed in a similar way to that in Sect. 4.1, using non-standard words. Then, to effect the excitement, first increment the subˆ regardless of subsequent operators. The excitement is ˆ (h), script of dˆx in π possible if the resulting word π ˆ is physical. Then, if the word is standard, it is ˆ ˆ ). Otherwise, a subword of the form (41) must have been present in π ˆ (h). π ˆ (h We then proceed by imposing the equalities dˆy cˆy+1/2 ≡ cˆy−1/2 dˆy+1 , dˆy cˆ∗y+1 ≡ cˆ∗y dˆy+1 ,
(43a) (43b)
ˆ ) because, after replacuntil a standard word results. This standard word is π ˆ (h ing dˆx by dˆx+1 in (41), this procedure produces (42). As in Sect. 4.1, we can streamline the process of exciting a particular dˆx , by adding the required excitation to the subscript (this excitation being possible if and only if the resulting word is physical) and then repeatedly imposing (∗) (∗) dˆy cˆy ≡ cˆy −1 dˆy+1 ,
(44)
until the word is standard. For example, consider the standard word cˆ1 cˆ2 dˆ72 cˆ∗11 cˆ8 cˆ∗19 cˆ∗21 cˆ14 . 2
2
(45)
2
Increasing the excitation of the dˆ72 by 5, and standardising the resulting nonstandard word using (44), results in the sequence: cˆ∗11 ) cˆ8 cˆ∗19 cˆ∗21 cˆ14 → cˆ1 cˆ2 cˆ∗9 (dˆ19 cˆ8 ) cˆ∗19 cˆ∗21 cˆ14 cˆ1 cˆ2 (dˆ17 2 2 2
2
2
2
2
2
∗
→ cˆ1 cˆ2 cˆ 9 cˆ7 (dˆ21 cˆ∗19 ) cˆ∗21 cˆ14 2 2
2
2
→ cˆ1 cˆ2 cˆ∗9 cˆ7 cˆ∗17 (dˆ23 cˆ∗21 ) cˆ14 2 2
2
2
→ cˆ1 cˆ2 cˆ∗9 cˆ7 cˆ∗17 cˆ∗19 dˆ25 cˆ14 . 2 2
2
2
(46)
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where in each line, we have used parentheses to indicate the pair of symbols affected. To excite a number of integer valleys simultaneously, we simply add the ˆ and standardise the required excitement to the subscript of each dˆx in π ˆ (h), resulting word by repeatedly using (44). We note that, as in the last section, the excitation of an integer valley ˆ is given by (40), even corresponding to an operator dˆx in an operator word π when the word is non-standard. This is so because the value of the expression (40) is unchanged for operator words obtained from one another through the equivalence (44). 4.3. Bijection Relating the P p and Hp Paths The bijection between these two path representations of the M(p, 2p+1) models can now be accomplished by mapping from the standard operator word corresponding to one, to an operator word (which may be non-standard) for the other. The appropriate equivalences, either (33) or (44), are then used to obtain the standard operator word, from which the path of the bijective image is readily read. p and For odd integers a and b, with 1 ≤ a < 2p and 3 ≤ b < 2p, let h ∈ Pa,b p 1 ˆ ˆ = 2 (a − 1) let h ∈ Haˆ,ˆb be its image under the bijection of Sect. 3.1, where a 1 ˆ ˆ is an and b = 2 (b − 1). We claim that if π is an operator word for h, then π ˆ where we obtain π operator word for h, ˆ from π by transforming each operator within according to cx → cˆx/2 ,
c∗y → cˆ∗y/2 ,
dx → dˆx+1/2 .
(47)
p ˆ ∈ Hp , that To verify this claim, we show that the paths h ∈ Pa,b and h a ˆ,ˆ b correspond to the operator words π and π ˆ respectively, are related through the ˆ defined in Sect. 3.1. map h → h Let the triple (hcut , n, λ) be that corresponding to h, as defined in Sect. 3.1. Then h contains n particles and, consequently, π contains n operators dx . Whether π is standard or non-standard, the excitation λi of the particle corresponding to the ith operator dx , counting from the right, is given by (30). In the case of a standard word π, the standard word π = π(hcut ) of hcut is obtained from π (as may be seen from Fig. 8) simply by reducing the subscript (∗) (∗) of each term cz by 2#{d · · · cz } (i.e., by twice the number of particles to its left), and dropping all operators dx . This also holds for non-standard words π, because π , obtained in this way, is unchanged under the equivalence (33). We now proceed analogously for the operator word π ˆ , determining μ and ˆ cut . Since π ˆ has n integer valleys. h ˆ has n operators dx , the half-lattice path h ˆ i of the ith of these integer valleys, counting from the right, is The excitation λ given by (40), whether π ˆ is standard or non-standard. Since the transformaˆ i = λi . Thereupon, the value tion (47) specifies that x = x + 1/2, we have λ ˆ amongst all its valleys, is of μi , the numbering of the ith integer valley of h
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ˆ i + #{dˆ· · · dˆx } + 1 = λi + n − i + 1. Thus, λ and μ are related given by μi = λ by (17), as required. ˆ cut ) of h ˆ cut ˆ (h In the case of a standard word π ˆ , the standard word π ˆ = π is obtained from π ˆ (as may be seen from Fig. 9) simply by reducing the sub(∗) (∗) script of each term cˆz by #{dˆ· · · cˆz } (i.e., by the number of integer valleys to its left), and dropping all operators dˆx . This also holds for non-standard words π ˆ , because π ˆ , obtained in this way, is unchanged under the equivalence (44). Now, in view of the transformation (47), we see that the subscripts of the ˆ cut ˆ . Thus, the particle-deficient path h terms in π are precisely twice those in π cut is obtained from h by shrinking by a factor of 2. Therefore, the combined ˆ cut , n, μ) → h ˆ is precisely as specified in Sect. 3.1. map h → (hcut , n, λ) → (h This completes the verification of this operator description of the bijection. 4 of Fig. 1. Its To illustrate the description, consider the path h ∈ P1,3 standard operator word π(h) is given in (28). Using the transformation (47) and the equivalences (44), we obtain: c0 d2 c4 c6 d10 c∗13 d16 d19 c∗21 → cˆ0 dˆ52 cˆ2 cˆ3 dˆ21 cˆ∗13 dˆ33 (dˆ39 cˆ∗21 ) 2 2 2 2
2
→ cˆ0 dˆ52 cˆ2 cˆ3 dˆ21 cˆ∗13 (dˆ33 cˆ∗19 ) dˆ41 2 2 2 2
2
→ cˆ0 dˆ52 cˆ2 cˆ3 (dˆ21 cˆ∗13 ) cˆ∗17 dˆ35 dˆ41 2 2 2 2
2
→ cˆ0 dˆ52 cˆ2 cˆ3 cˆ∗11 (dˆ23 cˆ∗17 ) dˆ35 dˆ41 2 2 2 2
2
→ cˆ0 (dˆ52 cˆ2 ) cˆ3 cˆ∗11 cˆ∗15 dˆ25 dˆ35 dˆ41 2 2 2 2
2
→ cˆ0 cˆ1 (dˆ72 cˆ3 ) cˆ∗11 cˆ∗15 dˆ25 dˆ35 dˆ41 2 2 2 2
2
→ cˆ0 cˆ1 cˆ2 (dˆ92 cˆ∗11 ) cˆ∗15 dˆ25 dˆ35 dˆ41 2 2 2 2
2
∗
→ cˆ0 cˆ1 cˆ2 cˆ 9 dˆ11 cˆ∗15 dˆ25 . dˆ35 dˆ41 2 2 2 2 2
2
Here, we have used parentheses to indicate the operators reordered by means of (44). We recognize the final word here to be (38), the standard operator word corresponding to the path of Fig. 2. To illustrate the inverse mapping, begin with the standard operator word (38). Applying the reverse of the transformations (47), and standardising the result using (33) produces cˆ∗15 dˆ25 → c0 (c2 c4 c∗9 d5 ) c∗15 d12 d17 d20 dˆ35 dˆ41 cˆ0 cˆ1 cˆ2 cˆ∗9 dˆ11 2 2 2 2 2
2
→ c0 d2 c4 c6 (c∗11 c∗15 d12 ) d17 d20 → c0 d2 c4 c6 d10 c∗13 (c∗17 d17 ) d20 → c0 d2 c4 c6 d10 c∗13 d16 (c∗19 d20 ) → c0 d2 d4 c6 d10 c∗13 d16 d19 c∗21 ,
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thus recovering the operator sequence (28) of our original path, Fig. 1. Here, in each line, we have used parentheses to indicate a number of applications of (33) which are applied successively within the included subword to shift the operator d completely to the left.
5. Outlook There is a well-known duality between the characters of the M(p, p ) and M(p − p, p ) models [3]. In terms of RSOS paths, this duality involves interchanging the role of the light and dark bands [7]. This hints at a description of the M(p + 1, 2p + 1) states in terms of some sort of dual half-lattice paths. It turns out that such a description does exist. Roughly, these are half-lattice paths with half-integer extremity conditions (instead of integer ones) and with peaks at half-integer heights. ˜ p , where To be more precise, let us denote the set of new paths by H a ˜,˜ b 1 ˜ ˆ a ˜, b ∈ Z + 2 . This set is defined to contain all half-lattice paths h that are ˆ0 = a (0, p−1/2)-restricted, ˜b-tailed, with h ˜, and the additional restriction that 1 ˆ x+1 ∈ Z + , then h ˆ x+1/2 = h ˆ x −1/2. This additional restriction forces ˆx = h if h 2 the peaks to occur at non-integer heights (however, the half-integer initial point ensures that all peaks occur at integer x-positions). These paths are weighted similarly to their dual versions, using (7) and (8). As announced, these new paths describe the states in the (r, s) irreducible module of M(p + 1, 2p + 1) where r=a ˜+
1 2
and
s = 2˜b.
(48)
Note that, in comparison with the Hp paths (see (13)), the roles of a ˜ and ˜b ˜ have interchanged in that, here, a ˜ and b are related to r and s respectively. ˜ p than for those in Note also that the vertical range is larger for paths in H a ˜,˜ b p Haˆ,ˆb , being p − 1/2 and p − 1 respectively. However, as previously mentioned, in the latter case, the maximal height could be augmented to p − 1/2 without affecting the set of paths since the constraints on the integrality of the peaks’ heights prevents the extra portion from being reached. Therefore, both the Hp ˜ p paths can be defined in the same strip, enhancing their duality and the H relationship. Following the analysis of [15], this path representation leads to new expressions for the M(p + 1, 2p + 1) fermionic characters and with a clear particle content. This will be detailed elsewhere. Finally, aspects of the present bijection can be turned into a well-controlled exploratory tool for an alternative path description of the M(p, f p + 1) models on a (1/f )-lattice, with special restrictions on the positions of the peaks and valleys. As already stressed, this is interesting in that it could lead to novel fermionic forms. Such a study is left to a future work.
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Acknowledgements OBF acknowledges a NSERC student fellowship and thanks the Institut Henri ´ Poincar´e and the Centre Emile Borel for its hospitality in the course of the thematic semester Statistical physics, combinatorics and probability: from discrete to continuous models, during which part of this work was done. This work was supported by NSERC.
References [1] Andrews, G.E., Baxter, R.J., Forrester, P.J.: Eight-vertex SOS model and generalized Rogers–Ramanujan-type identities. J. Stat. Phys. 35, 193–266 (1984) [2] Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London (1982) [2007 (Dover, New York)] [3] Berkovich, A., McCoy, B.M.: Continued fractions and fermionic representations for characters of M(p, p ) minimal models. Lett. Math. Phys. 37, 49–66 (1996) [4] Camino, J.M., Ramallo, A.V., Sanchez de Santos, J.M.: Graded parafermions. Nucl. Phys. B 530, 715–741 (1998) [5] Date, E., Jimbo, M., Kuniba, A., Miwa, T., Okado, M.: Exactly solvable SOS models: local height probabilities and theta function identities. Nucl. Phys. B 290, 231–273 (1987) [6] Di Francesco, P., Mathieu, P., S´en´echal, D.: Conformal Field Theory. Springer, New York (1997) [7] Foda, O., Lee, K.S.M., Pugai, Y., Welsh, T.A.: Path generating transforms. Contemp. Math. 254, 157–186 (2000) [8] Foda, O., Quano, Y.-H.: Virasoro character identities from the Andrews–Bailey construction. Int. J. Mod. Phys. A 12, 1651–1675 (1997) [9] Foda, O., Welsh, T.A.: Melzer’s identities revisited. Contemp. Math. 248, 207– 234 (1999) [10] Foda, O., Welsh, T.A.: On the combinatorics of Forrester–Baxter models. In: Kashiwara, M., Miwa, T. (eds.) Proceedings of “Physical Combinatorics”, Kyoto 1999. Progress in Mathematics, vol. 191, pp. 49–103. Birkh¨ auser, Boston (2000) [11] Forrester, P.J., Baxter, R.J.: Further exact solutions of the eight-vertex SOS model and generalizations of the Rogers-Ramanujan identities. J. Stat. Phys. 38, 435–472 (1985) [12] Jacob, P., Mathieu, P.: Graded parafermions: standard and quasi-particle bases. Nucl. Phys. B 630, 433–452 (2002) [13] Jacob, P., Mathieu, P.: Paths for Zk parafermionic models. Lett. Math. Phys. 81, 211–226 (2007) [14] Jacob, P., Mathieu, P.: Nonlocal operator basis from the path representation of the M(k + 1, k + 2) and the M(k + 1, 2k + 3) minimal models. J. Phys. A 41, 385201–385221 (2008) [15] Jacob, P., Mathieu, P.: A new path description for the M(k + 1, 2k + 3) models and the dual Zk graded parafermions. J. Stat. Mech. P11005, (43 pp.) (2007) [16] Jacob, P., Mathieu, P.: Particles in RSOS paths. J. Phys. A 42, 122001–122016 (2009)
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[17] Kedem, R., Klassen, T.R., McCoy, B.M., Melzer, E.: Fermionic sum representations for conformal field theory characters. Phys. Lett. B 307, 68–76 (1993) [18] Mathieu, P.: Paths and partitions: combinatorial descriptions of the parafermionic states. J. Math. Phys. 50, 095210 (43 pp.) (2009) [19] Melzer, E.: Fermionic character sums and the corner transfer matrix. Int. J. Mod. Phys. A 9, 1115–1136 (1994) [20] Rocha-Caridi, A.: Vacuum vector representations of the Virasoro algebra. In: Lepowsky, J., Mandelstam, S., Singer, I.M. (eds.) Proceedings of “Vertex Operators in Mathematics and Physics”, pp. 451–473. Springer, New York (1985) [21] Warnaar, S.O.: Fermionic solution of the Andrews–Baxter–Forrester model. I. Unification of CTM and TBA methods. J. Stat. Phys. 82 (1996), 657–685 [22] Warnaar, S.O.: Fermionic solution of the Andrews-Baxter-Forrester model. II. Proof of Melzer’s polynomial identities. J. Stat. Phys. 84, 49–83 (1996) [23] Welsh, T.A.: Paths, Virasoro characters and fermionic expressions. Proceedings of “Symmetry and Structural Properties of Condensed Matter”, Myczkowce, Poland, September 2005. J. Phys. Conf. Ser. 30, 119–132 (2006) [24] Welsh, T.A.: Fermionic expressions for minimal model Virasoro characters. Mem. Am. Math. Soc. 175(827) (2005) Olivier Blondeau-Fournier and Pierre Mathieu D´epartement de physique, de g´enie physique et d’optique Universit´e Laval Qu´ebec, QC G1K 7P4, Canada e-mail:
[email protected];
[email protected] Trevor A. Welsh Department of Physics University of Toronto Toronto, ON M5S 1A7, Canada e-mail:
[email protected] Communicated by Bernard Nienhuis. Received: December 4, 2009. Accepted: February 16, 2010.
Ann. Henri Poincar´e 11 (2010), 127–149 c 2010 Springer Basel AG 1424-0637/10/010127-23 published online May 11, 2010 DOI 10.1007/s00023-010-0030-y
Annales Henri Poincar´ e
Localization for the Random Displacement Model at Weak Disorder Fatma Ghribi and Fr´ed´eric Klopp Abstract. This paper is devoted to the study of the random displacement model on Rd . We prove that, in the weak displacement regime, Anderson and dynamical localization hold near the bottom of the spectrum under a generic assumption on the single-site potential and a fairly general assumption on the support of the possible displacements. This result follows from the proof of the existence of Lifshitz tails and of a Wegner estimate for the model under scrutiny. R´esum´e Cet article est consacr´e ` a l’´etude d’un mod`ele de petits d´eplacements al´eatoires. Sous une hypoth`ese g´en´erique sur le potentiel de simple site et des hypoth`eses assez g´en´erales sur les d´eplacements autoris´es, on d´emontre que le bas du spectre est exponentiellement et dynamiquement localis´e dans la limite des petits d´eplacements. La preuve repose sur la preuve d’une estim´ee de Lifshitz et d’une estim´ee de Wegner pour le mod`ele ´etudi´e.
0. Introduction We consider the following random displacement model: q(x − γ − λωγ ), Hλ,ω = −Δ + p + qλ,ω where qλ,ω (x) =
(0.1)
γ∈Zd
acting on L2 (Rd ). We assume the following: (H.0.0). The potential p is a real valued, Zd -periodic function. (H.0.1). The single-site potential q is a twice continuously differentiable, compactly supported real-valued function. The work of F. Klopp was supported by the grants DGRS-CNRS 08-R 15-01 and ANR08-BLAN-0261-01. The work of F. Ghribi was supported by the grant DGRS-CNRS 08-R 15-01.
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(H.0.2). ω := (ωγ )γ∈Zd is a collection of non trivial, independent, identically distributed, bounded random variables; let K ⊂ Rd be the support of their common distribution. (H.0.3). λ is a small positive coupling constant. Under these assumptions, Hλ,ω is ergodic and, for all ω, Hλ,ω is self-adjoint on the standard Sobolev space H2 (Rd ). The theory of ergodic operators teaches us that the spectrum of Hλ,ω is ω-almost surely independent of ω (see e.g. [12,21]); we denote it by Σλ . Our assumptions on qλ,ω imply that Σλ is bounded below. Define Eλ := inf Σλ . The goal of the present paper is to study the nature of the spectrum of Hλ,ω near Eλ . A result typical of the class of results that we will prove is Theorem 0.1. Assume p is not constant and that the random variables (ωγ )γ∈Zd are uniformly distributed in the unit ball in Rd . Then, there exists ε0 > 0 and λ0 > 0 such that, for a generic single site potential q such that q∞ ≤ ε0 , for λ ∈ (0, λ0 ], Anderson and strong dynamical localization hold near the bottom of the spectrum Eλ . Namely, there exist Eλ,1 > Eλ such that Hλ,ω has dense pure point spectrum on [Eλ , Eλ,1 ] almost surely, and each eigenfunction associated with an energy in this interval decays exponentially as |x| → ∞, and strong dynamical localization holds in the same region. For details on strong dynamical localization, we refer to [7]. When studying its spectral properties, an important feature of Hλ,ω is that it depends non monotonically (see e.g. [19]) on the random variables (ωγ )γ∈Zd , even if q is assumed to be sign-definite. As each of the random variables (ωγ )γ∈Zd is multidimensional, there cannot be a real monotonicity. Nevertheless, we exhibit a set of assumptions on the single-site potential q and on the random variables (ωγ )γ∈Zd that guarantee that, for sufficiently small disorder λ, • •
there exists a neighborhood of Eλ where Hλ,ω admits a Wegner estimate, Hλ,ω exhibits a Lifshitz tails at Eλ .
It is well known that such results then entail Anderson and dynamical localization near Eλ (see e.g. [7]). Our assumptions are presumably not optimal; we show that they hold for a small generic q. We need to assume some regularity for the distribution of the random variables. As they are multi-dimensional, absolute continuity with respect to the d-dimensional Lebesgue measure is not necessary; actually, they can be concentrated on subsets of dimension one (see Sect. 1.2.3). As for the support of the single site random variable, they can have a wide variety of shapes but need to satisfy a type of strict convexity condition at certain points; we refer to Sect. 1.3 for more details. Due to the non monotonicity of Hλ,ω , few rigorous results are known for the random displacement model in dimension larger than 1.
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For the one-dimensional displacement model, localization at all energies was proven in [2] and with different methods and under more general assumptions in [6]. These proofs establish the Wegner estimate using two-parameter spectral averaging and use lower bounds on the Lyapunov exponent to replace the Lifshitz tails behavior. For the multi-dimensional random displacement model, to the best of our knowledge, the only available result on localization prior to the present paper was [16] establishing the existence of a localized region for the semi-classical operator −h2 Δ + p + qλ,ω when h is sufficiently small. The Wegner estimate was established through a careful analysis of quantum tunneling. The Lifshitz tails behavior was neither proved nor used in the energy region under consideration as, thanks to the semi-classical regime, the model is in a large disorder regime. It has been discovered recently that, for random displacement models, Lifshitz tails need not hold (see [4,18]). Related to the study of the occurrence of the Lifshitz tails, an important point is the study of the infimum of the almost sure spectrum, and in particular of the finite volume configurations of the random parameter, if any, that give rise to the same ground state energy. Such a study for non monotonous models has been undertaken recently in [3,19]. In the present paper, we give an analysis of those configuration in the small displacement case. Another model of random operators related to the displacement model is the Poisson model for which localization has been proved recently [8]. It is to be noted that as far as Lifshitz tails are concerned this model behaves like a monotonous operator in the sense that the configurations of potential giving rise to the bottom of the almost sure spectrum is unique.
1. The Main Results For n ≥ 0, let Λn = [−n − 1/2, n + 1/2]d . For (ωγ )γ∈Z , define the differential expression q(x − β − γ − λωγ ). (1.1) Hλ,ω,n = −Δ + p + β∈(2n+1)Zd
γ∈Zd /(2n+1)Zd
P Let Hλ,ω,n be restriction of Hλ,ω,n to the cube Λn with periodic boundary P has only discrete spectrum and is bounded from below. For conditions. Hλ,ω,n E ∈ R, the integrated density of states is, as usual, defined by 1 P Nλ (E) = lim #{eigenvalues of Hλ,ω,n in (−∞, E]}. n→+∞ (2n + 1)d
We refer to [12,21] for details on this function and the proofs of various standard results. 1.1. The Assumptions We now state our assumptions on the random potential. Therefore, we introduce the periodic operator obtained by shifting all the single-site potentials by
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exactly the same amount, i.e., for ζ ∈ K (see assumption (H.0.2)), let Hζ = Hλ,ζ = −Δ + p + q(x − γ − λζ).
(1.2)
γ∈Zd
Here and in the sequel, ζ denotes the constant vector with entries all equal to ζ i.e. ζ = (ζ)γ∈Zd . The spectrum of the Zd -periodic operator Hζ is purely absolutely continuous; it is a union of intervals (see e.g. [22]). Let E(λ, ζ) be the infimum of this spectrum. As E(λ, ζ) is the bottom of the spectrum of the periodic operator Hζ , we know that it is a simple Floquet eigenvalue associated with the Floquet quasi-momentum θ = 0 (see Sect. 2.1 for more details); hence, it is a twice continuously differentiable function of ζ. We assume that (H.1.1). there exits λ0 > 0 such that, for λ ∈ (0, λ0 ), there exists a unique point ζ(λ) ∈ K so that E(λ, ζ(λ)) = min E(λ, ζ); ζ∈K
(H.1.2). there exists α0 > 0 such that, for λ ∈ (0, λ0 ) and ζ ∈ K, one has ∇ζ E(λ, ζ(λ)) · (ζ − ζ(λ)) ≥ α0 λ |ζ − ζ(λ)|2 .
(1.3)
Remark 1.1. Note that ζ → E(λ, ζ) is λ−1 Zd -periodic. Under assumption (H.1.3), we see that ζ(λ) cannot be a critical point of this function as, by (1.3), ∇ζ E(λ, ζ(λ)) = 0. In Sect. 1.3, we discuss concrete conditions on p, q and K that ensure that assumption (H.1) is valid. We now turn to our main results. 1.2. The Results We start with a description of the realizations of the random potential where the infimum of the almost sure spectrum is attained. Then, we state our results on Lifshitz tails, a Wegner estimate, and the result on localization. 1.2.1. The Infimum of the Almost Sure Spectrum. Of course, as Σλ is the almost sure spectrum, almost all realizations have their infimum as the infimum of the spectrum. The realizations we are interested in are those that attain this infimum when restricted to a finite volume. In the present paper, we construct these restrictions using periodic boundary conditions, actually considering periodic realizations of the random potential. In [3,4,18,19], the restrictions were performed using Neumann boundary conditions. We define periodic configurations of the random potential. Fix n ≥ 0 and, for (ωγ )γ∈Zd /(2n+1)Zd , consider the differential operator Hλ,ω,n defined by (1.1) with domain H2 (Rd ). It is (2n + 1)Zd -periodic; let E0n (λω) be its ground state energy, i.e., the infimum of its spectrum.
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One has Theorem 1.1. Under assumptions (H.0) and (H.1), there exists λ0 > 0 d such that, for any n ≥ 0, for λ ∈ (0, λ0 ], on K (2n+1) , the function ω → E0n (λω) reaches its infimum E(λ, ζ(λ)) at a single point, the point ω = (ζ(λ))γ∈Zd /(2n+1)Zd . So, when it comes to finding the “ground state” of our random system, for small λ, the Hamiltonian behaves as if it were monotonous in the random variables (ωγ )γ in the sense that it reaches its global minimum at a single point. By the standard characterization of the almost sure spectrum in terms of the spectra of the periodic approximations (see e.g. [21]), for λ sufficiently small, one has that Eλ = inf Σλ = E(λ, ζ(λ)). 1.2.2. The Lifshitz Tails. As a consequence of the determination of the minimum, we obtain Theorem 1.2. Under assumptions (H.0) and (H.1), there exists λ0 > 0 such that for all λ ∈ (0, λ0 ], lim
E→Eλ
d log | log(Nλ (E) − Nλ (Eλ )| ≤− . log(E − Eλ ) 2
Moreover, if the common distribution of the random variables (ωγ )γ is such that, for all λ, ε and δ positive sufficiently small, one has −δ
P({|ω0 − ζ(λ)| ≤ ε}) ≥ e−ε , then lim
E→Eλ
d log | log(Nλ (E) − Nλ (Eλ )| =− . log(E − Eλ ) 2
The Lifshitz tail behavior is well known for monotonous alloy type models [13]. It has also been discovered recently that, for general displacement or non monotonous alloy type models, this behavior need not hold (see [4,18,19]). 1.2.3. The Wegner Estimate. A Wegner estimate is an estimate on the probability that a restriction of the random Hamiltonian to a cube admits an eigenvalue in a fixed energy interval. Clearly, the estimate should grow with the size of the cube and decrease with the length of the interval in which one looks for eigenvalues. The restrictions we choose are the periodic ones, i.e., those defined at the beginning of Sect. 1.2. We assume that (H.2). There exists C > 0 such that, for λ sufficiently small, one has Eλ ≤ E0 − λ/C. Clearly, Theorem 1.1 shows that this assumption is a consequence of assumptions (H.0) and (H.1). For the alloy type models, it is well known that a Wegner estimate will hold only under a regularity assumption. We now turn to the corresponding assumption for our displacement model. We keep the notations of
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Sect. 1.2.1. Consider the polar decomposition of the random variable ω0 , say ω0 = r(ω0 )σ(ω0 ). For σ ∈ Sd−1 , define rσ (ω0 ), the random variable r(ω0 ) conditioned on σ(ω0 ) = σ. We assume that (H.3). for almost all σ ∈ Sd−1 , the distribution of rσ (ω0 ) admits a density with respect to the Lebesgue measure, say, hσ that itself is absolutely continuous with respect to the Lebesgue measure; moreover, one has ess-supσ∈Sd−1 hσ ∞ < +∞.
(1.4)
Remark 1.2. Assumption (H.3) will hold, for example, if • •
the random variable admit a density that is continuously differentiable on its support; the random variable is supported on a smooth submanifold of dimension 1 ≤ d ≤ d, and on this submanifold, it admits continuously differentiable density. We prove
Theorem 1.3. Under assumptions (H.0), (H.2) and (H.3), for any ν ∈ (0, 1), there exists λ0 > 0 such that, for λ ∈ (0, λ0 ], there exists Cλ > 0 such that, for all E ∈ [Eλ , Eλ + λ/C] and ε > 0 such that P P(dist(σ(Hλ,ω,n ), E) ≤ ε) ≤ Cλ εν nd .
(1.5)
The result is essentially a quite simple consequence of Theorem 6.1 of [10]; the modifications are indicated in Sect. 2.5. In the case of monotonous random operators, under our smoothness assumptions for the distribution of the random variables, the estimate (1.5) can be improved in the sense that the power ν can be taken equal to 1 (see [5]). It seems reasonable to think that the same holds true for most non-monotonous models; to our knowledge, no proof of this fact exists. A Wegner estimate of the type (1.5) implies a minimal regularity for Nλ , the integrated density of states of Hλ,ω in the low-energy region. Indeed, one proves Corollary 1.1. Under the assumptions of Theorem 1.3, for any ν ∈ (0, 1), the older continuous is the region [Eλ , Eλ + integrated density of states Nλ is ν-H¨ λ/C] defined in Theorem 1.3. 1.2.4. Localization. Once Theorems 1.2 and 1.3 are proved, localization follows by the now standard multiscale argument (see e.g. [7]) Theorem 1.4. Under assumptions (H.0), (H.1) and (H.3), there exists λ0 > 0 such that, for λ ∈ (0, λ0 ], Anderson and strong dynamical localization hold near the bottom of the spectrum. Namely, there exist Eλ,1 > Eλ such that Hλ,ω has dense pure point spectrum on [Eλ , Eλ,1 ] almost surely, and each eigenfunction associated with an energy in this interval decays exponentially as |x| → ∞, and strong dynamical localization holds in the same region.
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We omit the details of the proofs of this result. We only note that the Combes–Thomas estimate and the decomposition of resolvents in the multiscale argument work for the random displacement model in the same way as for alloy type models. 1.3. The Validity of Assumption (H.1) Let us now describe some concrete conditions on q and K that ensure that assumption (H.1) does hold. Let H0 = Hλ,0 be defined by (1.2) for ζ = 0. The spectrum of this operator is purely absolutely continuous; it is a union of intervals (see e.g. [22]). Let E0 be the infimum of this spectrum and ϕ0 be the solution to the following spectral problem: H0 ϕ0 = E0 ϕ0 , (1.6) ∀γ ∈ Zd , ϕ0 (x + γ) = ϕ0 (x). This solution is unique up to a constant; it can be chosen positive and normalized (see [14,23]). We call it the ground state for H0 . Recall that K is the essential support of the random variables (ωγ )γ ; thus K ⊂ Rd . We prove Proposition 1.1. Assume that K is • either a convex set with C 2 -boundary such that all its principal curvatures are positive at all points, • or the boundary of such a convex set, and that (1.7) v(q) := − ∇q(x)|ϕ0 (x)|2 dx = 0, Rd
Then, assumption (H.1) holds. For a fixed periodic potential p that is not constant, by perturbation theory, it is not difficult to see that condition (1.7) is satisfied for a generic small q. Indeed, if ψ0 is the ground state for −Δ + p (in the sense defined above), as ψ0 is positive, its modulus is constant if and only if it is constant. In which case, the eigenvalue equation (1.6) tells us that p is constant, identically equal to E0 . So we may assume that ψ0 is not constant, one can then find q smooth and compactly supported such that (1.7) holds. Indeed, by integration by parts, w(q) := ∂i q(x)ψ02 (x)dx = 2 q(x)ψ0 (x)∂i ψ0 (x)dx Rd
Rd
which vanishes for all smooth compactly supported functions if and only if ∂i ψ0 vanishes identically. Hence, w(q) vanishes for all q small, smooth, and compactly supported functions if and only if ψ0 is a constant (as q → w(q) is linear). As ϕ0 is the ground state for the operator −Δ + p + γ q(· − γ) and this ground state is a real analytic function of the potential q, the difference
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ψ0 − ϕ0 is small for q small. So, if we pick q0 such that w(q0 ) = 0, for ε small and q = εq0 , we know that v(q) does not vanish, i.e., (1.7) is satisfied. By Proposition 1.1 and Remark 1.2, it is clear now that Theorem 0.1 is a consequence of Theorem 1.4. Let us now give another assumption on K under which (H.1) holds. We prove Proposition 1.2. Assume that (1.7) is satisfied and that the set K satisfies that there exists ε > 0 and ζ0 ∈ K, such that, for all ζ ∈ K and |v − v(q)| < ε, one has v · (ζ − ζ0 ) ≥ 0. Then, assumption (H.1) holds. Moreover, for λ small, the minimum ζ(λ) satisfies ζ(λ) = ζ0 . Before we proceed to the proofs of Propositions 1.1 and 1.2, let us compare our setting with the one studied in [3,4,18]. In those studies, assumption (1.7) but also assumption (H.1) are not fulfilled. Indeed, there, p and q are assumed to be reflection symmetric with respect to the coordinate planes, i.e., for any σ = (σ1 , . . . , σd ) ∈ {0, 1}d and any x = (x1 , . . . , xd ) ∈ Rd , q(x1 , . . . , xd ) = q((−1)σ1 x1 , . . . , (−1)σd xd ). Hence, the potential p(·) + γ q(· − γ) and the ground state ϕ0 satisfy the same reflection symmetry. This implies that ∇q(x)|ϕ0 (x)|2 dx = − ∇q(x)|ϕ0 (x)|2 dx = 0. Rd
Rd
The fact that, in the setting of [3,4,18], assumption (H.1.1) is not satisfied is seen directly from those papers as the ground state of the periodic operator Hζ reaches its minimum at 2d values as soon as K is reflection symmetric. 1.4. The Proofs of Propositions 1.1 and 1.2 Consider the mapping ζ → F (λ, ζ) = λ−1 E(λ, ζ) on some ball B containing K. As E(λ, ζ) is a simple Floquet eigenvalue associated with the normalized Floquet eigenvector ϕ0 (λ, ζ, 0) (see Sect. 2.1), we can compute the gradient of F in the ζ-variable using the Feynman–Hellmann formula to obtain ∇ζ F (λ, ζ) = − ∇q(x − λζ)|ϕ0 (λ, ζ, 0; x)|2 dx. Rd
Hence, sup |∇ζ F (λ, ζ) − v(q)| → 0
ζ∈B
λ→0
(1.8)
Proof of Proposition 1.1. Assume first that K is a convex set satisfying the assumptions of Proposition 1.1. Using the rectification theorem (see e.g. [1]), assumption (1.7) and equation (1.8) guarantee that, for λ small, one can find
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a C 2 -diffeomorphism, say Ψλ , from B to Ψλ (B) such that |Ψλ − Id|C 2 → 0 when λ → 0 and ∇ζ (F (λ, Ψλ (ζ))) = v(q). Now, assume that K is a convex set with a C 2 -boundary having all its principal curvatures positive at all points. Then, for λ small, the set Kλ = Ψ−1 λ (K) also is convex with a C 2 -boundary having all its principal curvatures positive at all points; moreover, for λ small, the curvatures are bounded away from 0 independently of λ. On the convex set Kλ , the affine function G(ζ) := F (λ, Ψλ (ζ)) = v(q) · ˜ = Ψ−1 ζ + Cλ reaches its infimum at a single point, say ζ(λ) λ (ζ(λ)), ζ(λ) ∈ ∂K. Hence, we have that, for ζ ∈ K\{ζ(λ)}, F (λ, ζ) > F (λ, ζ(λ)). The convexity of K ensures that, for ζ ∈ K\{ζ(λ)}, one has ∇ζ F (λ, ζ(λ)) · (ζ − ζ(λ)) ≥ 0.
(1.9)
Indeed, as K is convex, for ν ∈ (0, 1) and ζ ∈ K\{ζ(λ)}, one has ζν = νζ + (1 − ν)ζ(λ) ∈ K\{ζ(λ)}; thus F (λ, ζν ) > F (λ, ζ(λ)). Taking the righthand side derivative of ν → F (λ, ζν ) at ν = 0 yields (1.9). The strict convexity of K, guaranteed by the positivity of the principal curvatures of ∂K, ensures that, for ζ ∈ K\{ζ(λ)}, one has ∇ζ F (λ, ζ(λ)) · (ζ − ζ(λ)) > 0.
(1.10)
Indeed, assume that for some ζ0 ∈ K\{ζ(λ)}, (1.10) is not satisfied i.e ∇ζ F (λ, ζ(λ)) · (ζ0 − ζ(λ)) = 0. As K is strictly convex, K contains a cone of the form {ζ(λ) + r(ζ0 − ζ(λ)) + rw; |w| ≤ 1, r ∈ [0, r0 ]} for some small r0 > 0. Picking w such that ∇ζ F (λ, ζ(λ)) · w < 0, one constructs ζ ∈ K such that ∇ζ F (λ, ζ(λ)) · (ζ − ζ(λ)) < 0 which contradicts (1.9). To show (1.3), it suffices to show that, for ζ ∈ K, ∇ζ F (λ, ζ(λ)) · (ζ − ζ(λ)) ≥
1 |ζ − ζ(λ)|2 . C0
(1.11)
Let Hλ be the hyperplane orthogonal to ∇ζ F (λ, ζ(λ)) at ζ(λ). It intersects K at ζ(λ) and K is contained in one of the half-spaces defined by this hyperplane. Thus, the hyperplane is tangent to K at ζ(λ) (see e.g. [11]). Hence, there exists α0 > 0 such that, for ζ ∈ K, one has ∇ζ F (λ, ζ(λ)) · (ζ − ζ(λ)) ≥ α0 d(ζ, Hλ )2
(1.12)
where d(ζ, Hλ ) denotes the distance from ζ to Hλ . The constant α0 can be chosen independent of λ for λ small as the principal curvatures of ∂K are uniformly positive. Now, if u = ∇ζ F (λ, ζ(λ))−1 ∇ζ F (λ, ζ(λ)), for ζ ∈ K as K is compact, one has ∇ζ F (λ, ζ(λ)) · (ζ − ζ(λ)) = ∇ζ F (λ, ζ(λ)) [u · (ζ − ζ(λ))] ≥ α0 [u · (ζ − ζ(λ))]2 .
(1.13)
As |ζ − ζ(λ)|2 = d(ζ, Hλ )2 + [u · (ζ − ζ(λ))]2 , the lower bounds (1.12) and (1.13) imply (1.11).
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To deal with the case when K is the boundary of a convex set, we only need to do the analysis done above for the convex hull of K and notice that the minimum is attained on K the boundary of this convex hull. This completes the proof of Proposition 1.1. Proof of Proposition 1.2. By assumption, for ζ ∈ K and |v − v(q)| < ε, one has v · (ζ − ζ0 ) ≥ 0. Hence, as K is compact, there exists c > 0 such that, for all ζ ∈ K and |v − v(q)| < ε/2, one has v · (ζ − ζ0 ) ≥ c|ζ − ζ0 |.
(1.14)
Let B be a closed ball centered in ζ0 such that K ⊂ B. By (1.8), (1.14) implies that, for λ sufficiently small, for all ζ˜ ∈ B and ζ ∈ K, one has ˜ · (ζ − ζ0 ) ≥ c|ζ − ζ0 |. ∇ζ F (λ, ζ) Hence, 1 ∇ζ F (λ, ζ0 + t(ζ − ζ0 )) · (ζ − ζ0 )dt
F (λ, ζ) − F (λ, ζ0 ) = 0
≥ c|ζ − ζ0 |. So ζ0 is the unique minimum of ζ → F (λ, ζ) in K, i.e., for λ sufficiently small, ζ(λ) = ζ0 . Using again the boundedness of K, we get the estimate (1.3) of assumption (H.1). This completes the proof of Proposition 1.2.
2. The Reduction to a Discrete Model In this section, we prove the results announced in Sect. 1.2.1. Therefore, we will use the Floquet decomposition for periodic operators to reduce our operator to some discrete model in the way it was done in [9,17]. 2.1. Floquet Theory Pick ζ ∈ K and let Hζ be the Zd -periodic operator defined by (1.2). For θ ∈ T∗ := Rd /(2πZd ) and u ∈ S(Rd ), the Schwartz space of rapidly decaying functions, following [22], we define (U u)(θ, x) = eiγ·θ u(x − γ) γ∈Zd
which can be extended as a unitary isometry from L2 (Rd ) to H := L2 (K0 ×T∗ ) where K0 = (−1/2, 1/2]d is the fundamental cell of Zd . The inverse of U is given by 1 ∗ v(θ, x)dθ. for v ∈ H, (U v)(x) = Vol(T∗ ) T∗
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As Hλ,ζ is Zd -periodic, Hλ,ζ admits the Floquet decomposition ∗
⊕
U Hλ,ζ U =
Hλ,ζ (θ)dθ T∗
where Hλ,ζ (θ) is the differential operator Hλ,ζ acting on Hθ with domain Hθ2 where • for v ∈ Rd , τv : L2 (Rd ) → L2 (Rd ) denotes the “translation by v” operator, i.e., for ϕ ∈ L2 (Rd ) and x ∈ Rd , (τv ϕ)(x) = ϕ(x − v); • Dθ is the space θ-quasi-periodic distribution in Rd , i.e., the space of distributions u ∈ D (Rd ) such that, for any γ ∈ Zd , we have τγ u = e−iγ·θ u. Here θ ∈ T∗ ; k (Rd ) is the space of distributions that locally belong to Hk (Rd ) and • Hloc k we define Hθk = Hloc (Rd ) ∩ Dθ ; • for k = 0, we define Hθ = Hθ0 and identify it with L2 (K0 ); equipped with the L2 -norm over K0 , it is a Hilbert space; the scalar product will be denoted by ·, ·θ . We know that Hλ,ζ (θ) is self-adjoint and has a compact resolvent; hence, its spectrum is discrete. Its eigenvalues repeated according to multiplicity, called Floquet eigenvalues of Hλ,ζ , are denoted by E0 (λ, ζ, θ) ≤ E1 (λ, ζ, θ) ≤ · · · ≤ En (λ, ζ, θ) → +∞. The functions ((λ, ζ, θ) → En (λ, ζ, θ))n∈N are Lipschitz-continuous in the variable θ; they are even analytic in (λ, ζ, θ) when they are simple eigenvalues. Define ϕn (λ, ζ, θ) to be a normalized eigenvector associated to the eigenvalue En (λ, ζ, θ). The family (ϕn (λ, ζ, θ))n≥0 is chosen so as to be a Hilbert basis of Hθ . If En (λ0 , ζ0 , θ0 ) is a simple eigenvalue, the function (λ, ζ, θ) → ϕn (λ, ζ, θ) is analytic near (λ0 , ζ0 , θ0 ). It is well known (see, e.g. [14]) that, for given λ and ζ, the eigenvalue E0 (λ, ζ, θ) reaches its minimum at θ = 0, and that it is simple for θ small. 2.2. The Reduction Procedure Recall that the (ϕn (λ, ζ, θ))n≥0 are the Floquet eigenvectors of Hλ,ζ . Let Πλ,ζ,0 (θ) and Πλ,ζ,+ (θ), respectively, denote the orthogonal projections in Hθ on the vector spaces, respectively, spanned by ϕ0 (λ, ζ, θ) and (ϕn (λ, ζ, θ))n≥1 . Obviously, these projectors are mutually orthogonal and their sum is the identity for any θ ∈ T∗ . Define Πλ,ζ,α = U ∗ Πλ,ζ,α (θ)U where α ∈ {0, +}. Πλ,ζ,α is an orthogonal projector on L2 (Rd ) and, for γ ∈ Zd , we have τγ∗ Πλ,ζ,α τγ = Πλ,ζ,α . It is clear that Πλ,ζ,0 + Πλ,ζ,+ = IdL2 (Rd ) and Πλ,ζ,0 , and Πλ,ζ,+ are mutually orthogonal. For α ∈ {0, +}, we set Eλ,ζ,α = Πλ,ζ,α (L2 (Rd )). These spaces are invariant under translations by vectors in Zd and Eλ,ζ,0 is of finite energy (see [17]). For u ∈ L2 (T∗ ), we define Pλ,ζ (u) = U ∗ (u(θ)ϕ0 (λ, ζ, θ)).
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The mapping Pλ,ζ : L2 (T∗ ) → Eλ,ζ,0 defines a unitary equivalence (see [17]); its inverse is given by ∗ (v) = (U v)(θ), ϕ0 (λ, ζ, θ), Pλ,ζ
v ∈ Eλ,0 .
∗ ∗ One checks that Pλ,ζ Pλ,ζ = Πλ,ζ,0 and Pλ,ζ Pλ,ζ = IdL2 (T∗ ) . The main result of this section is
Theorem 2.1. Under assumptions (H.0) and (H.1), there exists C0 > 0 such that, for any α > 0, there exists λ0 > 0 such that, for λ ∈ (0, λ0 ), for any d ζ ∈ K and any ω = (ωγ )γ∈Zd ∈ K Z , one has 1 ∗ Pλ,ζ h− λ,ω,ζ Pλ,ζ + Πλ,ζ,+ C0 ≤ Hλ,ω − E(λ, ζ) ∗ ˜ (2.1) P + H ≤ C0 Pλ,ζ h+ λ,ζ,+ λ,ω,ζ λ,ζ where ˜ • H •
λ,ζ,+ = (Hλ,ζ − E(λ, ζ))Πλ,ζ,+ . 2 ∗ h± λ,ω,ζ is the random operator acting on L (T ) defined by h+ v(λ, ζ) · (ωγ − ζ) + C0 α |ωγ − ζ|2 Πγ , λ,ω,ζ = C0 (·) + λ γ∈Zd
h− λ,ω,ζ =
1 v(λ, ζ) · (ωγ − ζ) − C0 α |ωγ − ζ|2 Πγ . (·) + λ C0 d γ∈Z
•
(·) is the multiplication operator by the function (θ) =
d
(1 − cos(θj )),
(2.2)
j=1
• •
Πγ is the orthogonal projector on eiγθ , the vector v(λ, ζ) is given by 1 v(λ, ζ) = − ∇q(x − λζ)|ϕ0 (λ, ζ, 0; x)|2 dx = ∇ζ E(λ, ζ). λ
(2.3)
Rd
The proof of Theorem 2.1 is the content of Sect. 3. We now use this result to derive Theorem 1.1 and 1.2. 2.3. The Characterization of the Infimum of the Almost Sure Spectrum We now prove Theorem 1.1. Using Theorem 2.1 for ζ = ζ(λ), we see that, for d λ sufficiently small, for ω ∈ K (2n+1) , one has 1 ∗ Hλ,ω,n − Eλ ≥ Pλ,ζ(λ) h− (2.4) Pλ,ζ(λ) + Πλ,ζ(λ),+ . λ,ω,ζ(λ),α,n C0
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Using (1.3) and (2.3), taking C0 α ≤ α0 /2 (where α0 is defined in (1.3)), we get C0 λ α0 C0 h− |ωγ − ζ(λ)|2 Πγ+β . λ,ω,ζ(λ),n ≥ (·) + 2 d d d β∈(2n+1)Z
γ∈Z /(2n+1)Z
h− λ,ω,ζ(λ),n
As the spectrum of is non negative, the operator in the left-hand ∗ side of (2.4) is clearly non negative; recall that Pλ,ζ Pλ,ζ +Πλ,ζ,+ = IdL2 , Πλ,ζ,+ ∗ is an orthogonal projector and Pλ,ζ is a partial unitary equivalence. To prove Theorem 1.1, we will show that, if ω = (ζ(λ))γ∈Zd /(2n+1)Zd , then, there exists c(ω) > 0 such that h− λ,ωζ(λ),n ≥ c(ω). Therefore, recall that − hλ,ω,ζ(λ),n is a periodic operator so we can do its Floquet decomposition in the same way as in Sect. 2.1. In the present case, as we deal with a discrete model, the fiber operators will be finite dimensional matrices (see, e.g. [15]); they can also be represented as the operator h− λ,ω,ζ(λ),n acting on the finite dimensional space of linear combinations of the Dirac masses (δ2πk/(2n+1)+θ )k∈Zd /(2n+1)Zd ; the Floquet parameter θ belongs to (2n + 1)−1 T∗ . As ≥ 0 and h− λ,ω,ζ(λ),n − ≥ 0, the energy 0 is in the spectrum of − hλ,ω,ζ(λ),n if and only if there exists θ ∈ (2n + 1)−1 T∗ and v, a linear combination of the Dirac masses (δ2πk/(2n+1)+θ )k∈Zd /(2n+1)Zd (seen as distributions on T∗ ) such that · v = 0 and h− λ,ω,ζ(λ),n v = 0. Now, · v = 0 implies that − θ = 0 and v = cδ0 . Hence, hλ,ω,ζ(λ),n v = 0 implies that |ωγ − ζ(λ)|2 = 0 γ∈Zd /(2n+1)Zd
i.e., ω = (ζ(λ))γ∈Zd /(2n+1)Zd . So we see that the function ω → E0n (λω) reaches its infimum only at the point ω = (ζ(λ))γ∈Zd /(2n+1)Zd . This completes the proof of Theorem 1.1. 2.4. The Lifshitz Tails We now prove Theorem 1.2. Therefore, we again use the reduction given by Theorem 2.1. Fix ζ = ζ(λ). First, the operators h± λ,ω,ζ(λ) are both standard discrete Anderson models, and as such, admit each integrated density of states that we denote by Nr± . As we have seen in the previous section, their spectra are contained in R+ . The inequality (2.1) implies that, for λ sufficiently small and E ∈ [0, 1/C02 ] where C0 is the constant given in Theorem 2.1, one has Nr+ (E/C0 ) ≤ Nλ (Eλ + E) ≤ Nr− (C0 E). Now, Theorem 1.2 immediately follows from the existence of Lifshitz tail for the Anderson models h± λ,ω,ζ(λ) (see e.g. [21]) which, in turn, follows from the facts that, for λ sufficiently small, under our assumptions, if C0 α ≤ α0 /2, by (1.3), the random variables ωγ± = [v(λ, ζ(λ)) · (ωγ − ζ(λ))] ± C0 α|ωγ − ζ(λ)|2
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are i.i.d, non negative, non trivial, and 0 belongs to their support (see, e.g. [21, 24]). Now if the common distribution of the random variables (ωγ )γ is such that, for all λ, ε and δ positive sufficiently small, one has −δ
P({|ω0 − ζ(λ)| ≤ ε}) ≥ e−ε , then, by virtue of (1.3), for all λ, ε and δ positive sufficiently small, one has −δ
P({ω0± ≤ ε}) ≥ e−ε . It is well known that, under this assumption, the Lifshitz exponent for the density of states of the discrete Anderson model is equal to d/2 (see, e.g. [21]). This completes the proof of Theorem 1.2. 2.5. The Wegner Estimate P We now prove Theorem 1.3 using the results of [10]. Let Hλ,r,σ,n be the operP ator Hλ,ω,n where the random variables (ωγ )γ∈Zd are written in polar coordinates i.e. (ωγ )γ∈Zd = (rγ (ω) σγ (ω))γγ ∈Zd where r = (rγ (ω))γ∈Zd has only
non-negative components and σ = (σγ (ω))γ∈Zd ∈ [Sd−1 ]Z . Then, the basic observation is that
P P ), E) ≤ ε) = Eσ Pr (dist(σ(Hλ,r,σ,n ), E) ≤ ε| σ) (2.5) P(dist(σ(Hλ,ω,n d
where Pr (·| σ) denotes the probability in the r-variable conditioned on σ, and Eσ , the expectation in the σ-variable. d Now, fix σ ∈ [Sd−1 ]Z . Using the notations of Sect. 1.2.3, we write Hλ,ω = H0 + λ rσ (ωγ )vσγ (· − γ) + λ2 V2,ω,λ (2.6) γ∈Zd
where • •
vσγ = −σγ · ∇q, V2,ω,λ is a potential bounded uniformly in λ and ω.
As q is C 2 with compact support, for any σ0 ∈ Sd−1 , vσ0 is C 1 with compact support and does not vanish identically. Assumptions (H.0.2) and (H.3) guarantee that the random variables (rγ (ω))γ∈Zd are independent and nicely distributed. Hence, the model (2.6) satisfies the assumptions considered in section 6 of [10] except for the fact that, in the present case, V2,ω,λ depends on λ. This does not matter as it is bounded uniformly in λ. In particular, Theorem 6.1 of [10] asserts that there exists λ0 > 0 such that, for λ ∈ (0, λ0 ], there exists Cλ > 0 such that, for all E ∈ [Eλ , Eλ + λ/C] and ε > 0 such that
P P(dist(σ(Hλ,ω,σ,n ), E) ≤ ε| σ) ≤ Cλ sup hσγ ∞ εν nd .
(2.7)
γ∈Zd
The explicit form of the constant appearing in the right side of formula (2.7) is obtained by following the proof of Theorem 6.1 in [10].
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The bound (1.4) then guarantees that the supremum taken in the righthand side of (2.7) is essentially bounded as a function of σ. We complete the proof of Theorem 1.3 by integrating (2.7) with respect to σ and using (2.5).
3. Proof of Theorem 2.1 We now turn to the proof of Theorem 2.1. The proof follows the spirit of [9,17]. For γ ∈ Zd , define ω ˜ = (˜ ωγ )γ∈Zd = (ωγ − ζ)γ∈Zd . Write Vλ,ω = Vλ,ζ + λ δVλ,˜ω = Vλ,ζ + λV1,λ,˜ω + λ2 V2,λ,˜ω where V1,λ,˜ω = −
∇q(x − γ − λζ) · ω ˜γ .
(3.1)
(3.2)
γ∈Zd
We note that V1,λ,˜ω and V2,λ,˜ω are bounded uniformly in λ and ω for |λ| ≤ 1 and ω in the support of the random variables. We decompose our random Hamiltonian Hλ,˜ω := Hλ,ω on the translation-invariant subspaces Eλ,ζ,0 and Eλ,ζ,+ defined in the Sect. 2.2. Thus, we obtain the random operators Hλ,˜ω,0 = Πλ,ζ,0 Hλ,˜ω Πλ,ζ,0
and
Hλ,˜ω,+ = Πλ,ζ,+ Hλ,˜ω Πλ,ζ,+ . ⊥
In the orthogonal decomposition of L2 (Rd ) = Eλ,ζ,0 ⊕ Eλ,ζ,+ , Hλ,˜ω is represented by the matrix Hλ,˜ω,0 λ Πλ,ζ,0 δVλ,˜ω Πλ,ζ,+ (3.3) λΠλ,ζ,+ δVλ,˜ω Πλ,ζ,0 Hλ,˜ω,+ . In Sect. 3.1, we give lower and upper bounds on Hλ,˜ω,0 which we prove in Sect. 3.4. Theorem 2.1 then follows from the fact that the off-diagonal terms in (3.3) are controlled by the diagonal ones; this is explained in Sect. 3.2. 3.1. The Operator Hλ,ω,0 ˜ In this section, using the non-degeneracy for the density of states of Hλ,ζ at E(λ, ζ), we give lower and upper bounds on Hλ,˜ω,0 . As seen in Sect. 2.2, the operator Hλ,˜ω,0 is unitarily equivalent to the operator hλ,˜ω acting on L2 (T∗ ) and defined by hλ,˜ω = hλ + λv1,λ,˜ω + λ2 v2,λ,˜ω , where • •
hλ is the multiplication by E0 (λ, ζ, θ), the operator v1,λ,˜ω has the kernel v1,λ,˜ω (θ, θ ) = V1,λ,˜ω ϕ0 (λ, ζ, θ, ·), ϕ0 (λ, ζ, θ , ·)L2 (K0 ) ,
•
the operator v2,λ,˜ω has the kernel v2,λ,˜ω (θ, θ ) = V2,λ,˜ω ϕ0 (λ, ζ, θ, ·), ϕ0 (λ, ζ, θ , ·)L2 (K0 ) .
(3.4)
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The potential V1,λ,˜ω and V2,λ,˜ω are defined in (3.1) and (3.2). They are bounded uniformly in all parameters. This will be used freely without special mention. We now recall a number of facts and definitions taken from [17]. Let t ∈ L2 (T∗ , Hθ ). We define the operator Pt : L2 (T∗ ) → L2 (Rd ) by ∀u ∈ L2 (T∗ ), [Pt (u)](x) = t(θ, x)u(θ)dθ. T∗
It satisfies Pt L2 (T∗ )→L2 (Rd ) ≤ tL2 (T∗ ,Hθ ) .
(3.5)
As the Floquet eigenvalue E0 (λ, ζ, θ) is simple in a neighborhood of 0, the Floquet eigenvector ϕ0 (λ, ζ, θ, ·) is analytic in this neighborhood. Recall that is defined in (2.2). We define the functions ϕ0,λ,ζ , ϕ˜0,λ,ζ and δϕ0,λ,ζ in L2 (T∗ , Hθ ) by ϕ0,λ,ζ (θ, x) = ϕ0 (λ, ζ, θ; x), ϕ˜0,λ,ζ (θ, x) = ϕ0,λ,ζ (0, x)eiθ·x 1 (ϕ0 (λ, .ζ, θ; x) − ϕ˜0,λ,ζ (θ, x)). δϕ0,λ,ζ (θ, x) = (θ) Furthermore, these functions are bounded in L2 (T∗ , Hθ ) uniformly in ζ and λ small. Finally, we note that, for u ∈ L2 (T∗ ), √ Pϕ0,λ,ζ (u) = Pϕ˜0,λ,ζ (u) + Pδϕ0,λ,ζ ( u). (3.6) Remark 3.1. It is proved in [17] that there exits C > 1 such that, as operators on L2 (T), one has 1 ≤ hλ − E(λ, ζ) ≤ C . C 3.1.1. Lower and Upper Bounds on v1,λ,ω˜ and v2,λ,ω˜ . Proposition 3.1. Recall that v(λ, ζ) is defined in (2.3). There exists C > 0 such that, for u ∈ L2 (T∗ ) and α > 0, we have 2 v1,λ,˜ω u, u − [v(λ, ζ) · ω ˜ ] · |ˆ u (γ)| γ γ∈Zd ⎛ ⎞ 1 ≤ C ⎝α ˜ ωγ 2 · |ˆ u(γ)|2 + 1 + u, u⎠ (3.7) α d γ∈Z
and |v2,λ,˜ω u, u| + V1,λ,˜ω Pϕ0,λ,ζ (u)2 + V2,λ,˜ω Pϕ0,λ,ζ (u)2 ⎛ ⎞ ≤C⎝ ˜ ωγ 2 · |ˆ u(γ)|2 + u, u⎠ .
(3.8)
γ∈Zd
Proposition 3.1 is proved in Sect. 3.4. We now use these results to give lower and upper bounds on Hλ,˜ω,0 .
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3.1.2. Lower and Upper Bounds for Hλ,ω,0 − E(λ, ζ). We prove ˜ Proposition 3.2. Under assumptions (H.0) and (H.1), there exists C0 > 0 such that, for α > 0, there exists λα > 0 and Cα > 0 such that, for all λ ∈ [0, λ0 ], on Eλ,ζ,0 , one has 1 ∗ ˜ ω,0 := Hλ,˜ω,0 − E(λ, ζ) ≤ C0 Pλ,ζ h+ P ∗ , Pλ,ζ h− λ,˜ ω ,ζ Pλ,ζ ≤ Hλ,˜ λ,˜ ω ,ζ λ,ζ C0 where h± λ,˜ ω ,ζ are the random operators defined in Theorem 2.1. Proof of Proposition 3.2. For λ small, Proposition 3.1 and Remark 3.1 imply that, there exists C0 > 0 such that, for α > 0, there exists λα > 0 such that, for all λ ∈ [0, λα ], one has 1 +λ [v(λ, ζ) · ω ˜ γ − C0 α˜ ωγ 2 ] · Πγ ≤ hλ,˜ω − E(λ, ζ) C0 d γ∈Z
and hλ,˜ω − E(λ, ζ) ≤ C0 + λ
[v(λ, ζ) · ω ˜ γ + C0 α˜ ωγ 2 ] · Πγ .
γ∈Zd
As Hλ,˜ω,0 and hλ,˜ω are unitarily equivalent, this completes the proof of Proposition 3.2. 3.2. The Operator Hλ,ω,+ ˜ By the definition of Πλ,ζ,+ , there exists η > 0 such that, for λ sufficiently small, (E(λ, ζ) + η)Πλ,ζ,+ ≤ Πλ,ζ,+ Hλ,ζ Πλ,ζ,+ . ˜ ω,+ = Hλ,˜ω,+ − E(λ, ζ) (see (3.3)). As ˜ Let H λ,ζ,+ = Hλ,ζ,+ − E(λ, ζ) and Hλ,˜ |V1,λ,˜ω | and |V1,λ,˜ω | are bounded, for λ sufficiently small, one has 1˜ η ˜ λ,˜ω,+ ≤ 2H ˜ Πλ,ζ,+ ≤ H ≤H λ,ζ,+ . 2 2 λ,ζ,+
(3.9)
3.3. The Proof of Theorem 2.1 ⊥
For ϕ = ϕ0 + ϕ+ ∈ Eλ,ζ,0 ⊕ Eλ,ζ,+ , by (3.3), one has ˜ ˜ λ,˜ω,0 ϕ0 , ϕ0 − H ˜ λ,˜ω,+ ϕ+ , ϕ+ Hλ,˜ω ϕ, ϕ − H ≤ 2λ|V1,λ,˜ω ϕ+ , ϕ0 | + 2λ2 |V2,λ,˜ω ϕ+ , ϕ0 |. Using the Cauchy–Schwarz inequality, we get |V1,λ,˜ω ϕ+ , ϕ0 | + |V2,λ,˜ω ϕ+ , ϕ0 | ≤ 2V1,λ,˜ω ϕ0 2 + 2V2,λ,˜ω ϕ0 2 + 4ϕ+ 2 . Then, the decomposition (3.3) and (3.9) give ˜ λ,˜ω,0 − CλK0 1 H 0 ˜ ˜ λ,˜ω,+ − CλΠλ,ζ,+ ≤ Hλ,˜ω H 0 C ˜ 0 ˜ λ,˜ω ≤ C Hλ,˜ω,0 + CλK0 =H ˜ λ,˜ω,+ + CλΠλ,ζ,+ 0 H
(3.10)
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where 2 2 K0 = Πλ,ζ,0 (V1,λ,˜ ω + V2,λ,˜ ω )Πλ,ζ,0 .
The estimate (3.8) of Proposition 3.1 implies that ⎛ ⎞ ∗ K0 ≤ CPλ,ζ ⎝ + ˜ ωγ 2 Πγ ⎠ Pλ,ζ . γ∈Zd
On the other hand, (3.9) implies that, for λ sufficiently small, 1˜ ˜ λ,˜ω,+ − CλΠλ,ζ,+ ≤ H ˜ λ,˜ω,+ + CλΠλ,ζ,+ ≤ 2H ˜ λ,˜ω,+ . Hλ,˜ω,+ ≤ H 2 Combining these two estimates with (3.10) and Proposition 3.2, we complete the proof of Theorem 2.1. 3.4. The Proof of Propositions 3.1 We first prove Lemma 3.1. There exists a constant C > 0 such that, for all u ∈ L2 (Td ) and α > 0, one has 2 V1,λ,˜ω Pϕ˜ (u), P (u) − [v(λ, ζ) · ω ˜ ] |ˆ u (γ)| ϕ ˜0,λ,ζ γ 0,λ,ζ γ∈Zd ⎛ ⎞ 1 ≤ C ⎝α ˜ ωγ 2 · |ˆ u(γ)|2 + 1 + u, u⎠ , (3.11) α d γ∈Z
V1,λ,˜ω Pϕ˜0,λ,ζ (u)2 + V2,λ,˜ω Pϕ˜0,λ,ζ (u)2 ⎛ ⎞ 1 ≤ C ⎝α ˜ ωγ 2 · |ˆ u(γ)|2 + 1 + u, u⎠ , α d
(3.12)
γ∈Z
Proof of Lemma 3.1. We compute V1,λ,˜ω Pϕ˜0,λ,ζ (u), Pϕ˜0,λ,ζ (u) =− ω ˜ γ · ∇q(x − λζ)|ϕ0 (λ, ζ, 0; x)|2 · |φγ (u)(x)|2 dx, γ∈Zd
Rd
· eiθ·x u(θ)dθ. where φγ (u)(x) = T e Recall that 0 is the unique zero of on T∗ and it is non-degenerate. Thus, the function g(θ, x) = (θ)−1/2 (eiθ·x − 1) is defined on T × Rd and iθ·γ
sup
(1 + |x|)−1 |g(θ, x)| < +∞.
(θ,x)∈T∗ ×R
For γ ∈ Zd , u ∈ L2 (T∗ ) and x ∈ Rd , one has ˆ(γ) = g(θ, x)eiγ·θ (θ)u(θ)dθ. ψγ (u)(x) = φγ (u)(x) − u T∗
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Note that
|ψγ (u)(x)|2 =
γ∈Zd
|g(θ, x)|2 | (θ)u(θ)|2 dθ
T∗
≤ C(1 + |x|)2 u, u.
(3.13)
Recall that v(λ, ζ) is defined by (2.3). We define v1,λ,˜ ω [u]
=−
ω ˜γ ·
γ∈Zd v1,λ,˜ ω [u]
∇q(x − λζ)|ϕ0 (λ, ζ, 0; x)|2 |ψγ (u)(x)|2 dx,
Rd
= V1,λ,˜ω Pϕ˜0,λ,ζ (u), Pϕ˜0,λ,ζ (u) −
v(λ, ζ) · ω ˜ γ |ˆ u(γ)|2 − v1,λ,˜ ω [u].
γ∈Zd
As the random variables (˜ ωγ )γ∈Zd are bounded, by (3.13), we compute
|v1,λ,˜ ω [u]| ≤ C
∇q(x − λζ)|ϕ0 (λ, ζ, 0; x)|2
Rd
≤ C u, u
|ψγ (u)(x)|2 dx
γ∈Zd
∇q(x − λζ)|ϕ0 (λ, ζ, 0; x)|2 (1 + |x|)2 dx
Rd
≤ C u, u. By the Cauchy–Schwarz inequality, one has |v1,λ,˜ ω [u]| ⎛ ⎞ 2 ⎝ ⎠ u ˆ(γ)˜ ωγ · ∇q(x − λζ)|ϕ0 (λ, ζ, 0; x)| ψγ (u)(x)dx = 2 Re γ∈Zd Rd ⎡ ⎤ ≤ α⎣ ∇q(x − λζ) |ϕ0 (λ, ζ, 0; x)|2 dx⎦ ˜ ωγ 2 |ˆ u(γ)|2 γ∈Zd
Rd
4 + ∇q(x − λζ) |ϕ0 (λ, ζ, 0; x)|2 |ψγ (u)(x)|2 dx. α d γ∈Z Rd
Using (3.13), we obtain that ⎛ ⎝α |v1,λ,˜ ω [u]| ≤ C
⎞ 1 ˜ ωγ 2 · |ˆ u(γ)|2 + u, u⎠ . α d
γ∈Z
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Finally, adding this to the estimate for |v1,λ,˜ ω [u]|, we get 2 V1,λ,˜ω Pϕ˜ (u), P (u) − v(λ, ζ) · ω ˜ |ˆ u (γ)| ϕ ˜ γ 0,λ,ζ 0,λ,ζ γ∈Zd ⎛ ⎞ 1 ≤ C ⎝α ˜ ωγ 2 · |ˆ u(γ)|2 + 1 + u, u⎠ . α d γ∈Z
This completes the proof of (3.11). The two terms in the left-hand side of (3.12) are dealt with in the same way; so, we only give the details for V1,λ,˜ω Pϕ˜0,λ,ζ (u)2 . We compute 2 2 V1,λ,˜ω Pϕ˜0,λ,ζ (u) = |V1,λ,˜ω (x) ϕ0 (λ, ζ, 0; x) φ(u)(x)| dx Rd
2 ∇q(x − λζ − γ) · ω ˜ γ |ϕ0 (λ, ζ, 0; x) φ(u)(x)|2 dx = d Rd γ∈Z ≤C ˜ ωγ 2 ∇q(x − λζ)2 |ϕ0 (λ, ζ, 0; x) φγ (u)(x)|2 dx γ∈Zd
Rd
where, in the last step, as q is compactly supported, the number of non-vanishing terms of the sum inside the integral is bounded uniformly. Now, by the definition of φγ and ψγ , we have |∇q(x − λζ)|2 |ϕ0 (λ, ζ, 0; x)|2 |φγ (u)(x)|2 dx Rd
≤2
u(γ)|2 + |ψγ (u)(x)|2 dx |∇q(x − λζ)|2 |ϕ0 (λ, ζ, 0; x)|2 |ˆ
Rd 2
≤ C|ˆ u(γ)| + C
|∇q(x − λζ)|2 |ϕ0 (λ, ζ, 0; x)|2 |ψγ (u)(x)|2 dx.
Rd
We plug this into the estimate for V1,λ,˜ω Pϕ˜0,λ,ζ (u)2 and (3.13) yields V1,λ,˜ω Pϕ˜0,λ,ζ (u)2 ≤ C |∇q(x − λζ)|2 |ϕ0 (λ, ζ, 0; x)|2 · |φγ (u)(x)|2 dx Rd
+C
γ∈Zd
˜ ωγ 2 |ˆ u(γ)|2
γ∈Zd
≤ C˜ ωγ 2 |ˆ u(γ)|2 + Cu, u. The computation for V2,λ,˜ω Pϕ˜0,λ,ζ (u)2 is the same as V2,λ,˜ω = q(x − γ, ζ, ω ˜γ ) γ∈Zd
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where, denoting the Hessian of q at x by Q, we have 1 Q(x − λ(ζ + t˜ ωγ ))˜ ωγ , ω ˜ γ (1 − t)dt.
q(x, ζ, ω ˜γ ) = 0
So (3.12) is proved and the proof of Lemma 3.1 is complete.
Proof of Proposition 3.1. Using (3.4) and (3.6), we write √ √ v1,λ,˜ω,ζ u, u = V1,λ,˜ω Pδϕ0,λ,ζ ( u), Pδϕ0,λ,ζ ( u) + V1,λ,˜ω Pϕ˜0,λ,ζ (u), Pϕ˜0,λ,ζ (u)
√ + 2Re(V1,λ,˜ω Pϕ˜0,λ,ζ (u), Pδϕ0,λ,ζ ( u)). Hence, as V1,λ,˜ω is bounded, 2 v1,λ,˜ω u, u − v(λ, ζ) · ω ˜ |ˆ u (γ)| γ γ∈Zd ≤ V1,λ,˜ω Pϕ˜0,λ,ζ (u), Pϕ˜0,λ,ζ (u) − v(λ, ζ) · ω ˜ γ |ˆ u(γ)|2 γ∈Zd 2 √ √ +2 V1,λ,˜ω Pϕ˜0,λ,ζ (u), Pδϕ0,λ,ζ ( u) + Pδϕ0,λ,ζ ( u) .
(3.14)
The Cauchy–Schwarz inequality then yields √ |V1,λ,˜ω Pϕ˜0,λ,ζ (u), Pδϕ0,λ,ζ ( u)| √ 4 ≤ αV1,λ,˜ω Pϕ˜0,λ,ζ (u)2 + Pδϕ0,λ,ζ ( u)2 . α Combining this with (3.14), (3.12) and (3.11), we obtain 2 v1,λ,˜ω u, u − [v(λ, ζ) · ω ˜ ] · |ˆ u (γ)| γ γ∈Zd ⎛ ⎞ 1 ≤ C ⎝α ˜ ωγ 2 · |ˆ u(γ)|2 + 1 + u, u⎠ . α d γ∈Z
This completes the proof of (3.7). Using (3.6) and the expansion done above for v1,λ,˜ω u, u , we compute √ |v2,λ,˜ω u, u| ≤ 2V2,λ,˜ω Pϕ˜0,λ,ζ (u)2 + 2Pδϕ0,λ,ζ ( u)2 . Combining this with (3.12) for α = 1, we get ⎛ ⎞ ˜ ωγ 2 · |u(γ)|2 + u, u⎠ . |v2,λ,˜ω u, u| ≤ C ⎝ γ∈Zd
The estimates for V1,λ,˜ω Pϕ0,λ,ζ (u)2 and V2,λ,˜ω Pϕ0,λ,ζ (u)2 are obtained in the same way. This completes the proof of (3.8)and hence, of Proposition 3.1.
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References [1] Arnold, V.I.: Ordinary Differential Equations. Universitext. Springer-Verlag, Berlin, 2006. Translated from the Russian by Roger Cooke, Second printing of the 1992 edition [2] Buschmann, D., Stolz, G.: Two-parameter spectral averaging and localization for non-monotonic random Schr¨ odinger operators. Trans. Am. Math. Soc. 353, 635–653 (2001) [3] Baker, J., Loss, M., Stolz, G.: Minimizing the ground state energy of an electron in a randomly deformed lattice. Comm. Math. Phys. 283(2), 397–415 (2008) [4] Baker, J., Loss, M., Stolz, G.: Low energy properties of the random displacement model. J. Funct. Anal. 256(8), 2725–2740 (2009) [5] Combes, J., Hislop, P., Klopp, F.: An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schr¨ odinger operators. Duke Math. J. 140(3), 469–498 (2007) [6] Damanik, D., Sims, R., Stolz, G.: Localization for one-dimensional, continuum, Bernoulli-Anderson models. Duke Math. J. 114, 59–100 (2002) [7] Germinet, F., Klein, A.: Bootstrap multiscale analysis and localization in random media. Comm. Math. Phys. 222(2), 415–448 (2001) [8] Germinet, F., Hislop, P., Klein, A.: Localisation for Schr¨ odinger operators with Poisson random potential. J. Eur. Math. Soc. 9, 577–607 (2007) [9] Ghribi, F.: Internal Lifshits tails for random magnetic Schr¨ odinger operators. J. Funct. Anal. 248, 387–427 (2007) [10] Hislop, P., Klopp, F.: The integrated density of states for some random operators with nonsign definite potentials. J. Funct. Anal. 195(1), 12–47 (2002) [11] H¨ ormander, L.: Notions of Convexity. Modern Birkh¨ auser Classics. Birkh¨ auser Boston Inc., Boston, 2007. Reprint of the 1994 edition [12] Kirsch, W.: Random Schr¨ odinger oerators: a course in Schr¨ odinger Operators (Sonderborg, 1989). In: Jensen, A., Holden, H. (eds.) Lecture Notes in Phys., vol. 345. Springer-Verlag, Berlin (1989) [13] Kirsch, W., Metzger, B.: The integrated density of states for random Schr¨ odinger operators. In: Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, vol. 76 of Proc. Sympos. Pure Math., pp. 649– 696. Amer. Math. Soc., Providence (2007) [14] Kirsch, W., Simon, B.: Comparison theorems for the gap of Schr¨ odinger operators. J. Funct. Anal. 75(2), 396–410 (1987) [15] Klopp, F.: Weak disorder localization and Lifshitz tails. Comm. Math. Phys. 232(1), 125–155 (2002) [16] Klopp, F.: Localization for semiclassical continuous random Schr¨ odinger operators. Part II: the random displacement model. Helv. Phys. Acta 66, 810– 841 (1993) [17] Klopp, F.: Internal Lifshits tails for random perturbations of periodic Schr¨ odinger operators. Duke Math. J. 98, 335–396 (1999) [18] Klopp, F., Nakamura, S.: Lifshitz tails for generalized alloy type random Schr¨ odinger operators. http://arxiv.org/abs/0903.2105 [19] Klopp, F., Nakamura, S.: Spectral extrema and Lifshitz tails for non monotonous alloy type models. Comm. Math. Phys. 287(1), 1133–1143 (2009)
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[20] Lott, J., Stolz, G.: The spectral minimum for random displacement models. J. Comput. Appl. Math. 148, 133–146 (2002) [21] Pastur L., Figotin, A.: Spectra of random and almost-periodic operators. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 297. Springer-Verlag, Berlin (1992) [22] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Analysis of Operators, vol. IV. Academic Press, New York (1978) [23] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. III. Scattering Theory. Academic Press, New York (1979) [24] Stollmann, P.: Caught by Disorder, Bound States in Random Media. Birkh¨ auser, Boston (2001) Fatma Ghribi D´epartement de Math´ematiques Facult´e des Sciences de Monastir Avenue de l’Environnement 5019 Monastir, Tunisia e-mail:
[email protected] Fr´ed´eric Klopp LAGA Institut Galil´ee Universit´e de Paris-Nord U.R.A 7539 C.N.R.S Avenue J.-B. Cl´ement 93430 Villetaneuse, France and Institut Universitaire de France Paris, France e-mail:
[email protected]
Communicated by Bernard Nienhuis. Received: September 9, 2009. Accepted: January 15, 2010.
Ann. Henri Poincar´e 11 (2010), 151–350 c 2010 Springer Basel AG 1424-0637/10/010151-200 published online May 12, 2010 DOI 10.1007/s00023-010-0028-5
Annales Henri Poincar´ e
The Temporal Ultraviolet Limit for Complex Bosonic Many-body Models Tadeusz Balaban, Joel Feldman, Horst Kn¨orrer and Eugene Trubowitz Abstract. We consider the partition function for a many-body model consisting of a weakly coupled gas of bosons at any temperature T > 0 and any chemical potential μ, but with both infrared and ultraviolet cutoffs imposed in both temporal and spatial directions. We take the limit as the ultraviolet cutoff in the temporal direction is removed and develop a representation for the limit that, hopefully, provides a suitable starting point for controlling the limit as the infrared cutoffs are removed.
1. Introduction We are developing a set of tools and techniques for analyzing the large distance/infrared behaviour of a system of identical bosons, as the temperature tends to zero. In this paper we retain an infrared cutoff. That is, we consider bosons moving in the discrete torus X = ZD /LZD , endowed with the standard Euclidean metric d(x, y). Our long term goal is to rigorously treat the infrared limit L → ∞. See the introduction of [1]. The total energy of our many boson systems has two sources. First, each particle in the system has a kinetic energy. We shall denote the corresponding 1 Δ, but, in this quantum mechanical observable by h. The most common is − 2m paper, more general operators are allowed. We assume that h = ∇∗ H∇ where H : L2 (X ∗ ) → L2 (X ∗ ) is a translation invariant, real, strictly positive, operator, X ∗ is the set of all bonds (ordered pairs of nearest neighbour points of X, but with the pair (y, x) viewed as −(x, y)) and the gradient (∇f )((x, y)) = f (y) − f (x). Second, the particles interact with each other through a two– body potential, 2v(x, y), which is assumed to be real, symmetric, translation invariant and exponentially decaying. For stability, v is also required to be repulsive, in the sense that, viewed as the kernel of a convolution operator, it is strictly positive.
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We assume that the system is in thermodynamic equilibrium and that expectations of observables are given by the grand canonical ensemble at tem1 > 0 and chemical potential μ. We concentrate on the partition perature T = kβ function Tr e− kT (H−μN ) . Here, H is the Hamiltonian and N is the number operator. The techniques developed here can also be applied to correlations functions. See (1.8). In [2], we developed a functional integral representation for the partition function. See (1.2) below. The integration variable of this functional integral is a complex field ατ (x) depending on position x ∈ X and time/temperature 1 1 . ]. Here, we have periodic boundary conditions, that is α0 = α kT τ ∈ (0, kT The representation of [2] can be viewed as a rigorous version of the formal representation dα∗ (x) dατ (x) ∗ 1 τ Tr e− kT (H−μN ) = · · · eAX (α ,α) 2πi 1
x∈X 1 0≤τ ≤ kT
where AX (α∗ , α) 1
=
kT dτ 0
dx
ατ (x)∗
∂ ατ (x) − ατ (x)∗ (hατ )(x) + ματ (x)∗ ατ (x) ∂τ
X 1 kT
−
dτ 0
dx dy ατ (x)∗ ατ (y)∗ v(x, y) ατ (x)ατ (y)
(1.1)
X2
This formal representations is frequently used in the physics literature. See [11, (2.66)]. The ultraviolet problem is to integrate out, in this representation, all variables ατ (x) except for those having τ in a lattice with spacing of order one. In this paper, we treat the ultraviolet problem for our rigorous version of the formal representation above. As pointed out in the introduction to [1], it is not possible to give rigorous mathematical meaning to the functional integral above in a straightforward way. For this reason, we derived (in Theorem 2.2 of [2]) the representation ) Tr e− kT (H−μN = lim 1
ε→0
dμR(ε) (ατ∗ , ατ )ζε (ατ −ε , ατ )
1 τ ∈εZ∩(0, kT ]
∗ ∗ ∗ ×eατ −ε ,j(ε)ατ −εατ −ε ατ v ατ −ε ατ
Here, for any r > 0, dμr (α∗ , α) =
dα∗ (x) ∧ dα(x) ∗ e−α (x)α(x) χ (|α(x)| < r) 2πı
x∈X
(1.2)
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denotes the unnormalized Gaussian measure, cut off at radius r, and ζε (α, β) is the characteristic function of
α, β : CX → C α − β ∞ < p0 (ε) The cutoffs R(ε) > 0 and p0 (ε) ≥ ln 1ε are decreasing functions of ε defined for all 0 < ε ≤ 1 that obey √ 1 R(ε) ≥ √ p0 (ε) and lim ε R(ε) = 0 4 ε→0 ε −t(h−μ) . We write the (R– Furthermore, for any t > 0, the operator j(t) = e style) scalar product, f, g = x∈X f (x)g(x) for any two fields f, g : X → C.1 In this paper, we treat the ultraviolet problem in the representation (1.2). The final result is to write the partition function as a functional integral which involves ατ for only finitely many values of τ , independent of ε. To achieve this, we have to integrate out all but a fixed number of fields ατ in the representation (1.2). For n ≥ 1 and ε > 0, set
In (ε; α∗ , β) =
dμR(ε) (ατ∗ , ατ )
τ ∈εZ∩(0,2n ε)
×
∗ ∗ ∗ ζε (ατ −ε , ατ ) eατ −ε ,j(ε)ατ −ε ατ −ε ατ v ατ −ε ατ
τ ∈εZ∩(0,2n ε]
(1.3) with α0 = α and α2n ε = β. If ε = , for m, p ∈ N, then dμR(ε) (ατ∗ , ατ ) ζε (ατ −ε , ατ ) 1 1 2m pkT
1 τ ∈εZ∩(0, kT ]
∗ ∗ ∗ ×eατ −ε ,j(ε)ατ −ε ατ −ε ατ v ατ −ε ατ p
dμR(ε) (φ∗n , φn ) Im (ε; φ∗n−1 , φn ) =
(1.4)
n=1
with the convention φ0 = φp . Combining (1.2) and (1.4) we get p 1 1 − kT (H−μN ) ∗ ∗ ;φ Tr e = lim , φn dμR( 2m1pkT ) (φn , φn ) Im m→∞ 2m pkT n−1 n=1 (1.5) For a treatment of the ultraviolet problem in some related models, see [6,7,9,10]. In this paper we show that, that for all sufficiently small2 θ > 0, Iθ (α∗ , β) = lim Im (2−m θ; α∗ , β) m→∞
1 2
Thus the usual scalar product over C|X| is f ∗ , g. The smallness condition on θ does not depend on the interaction v.
(1.6)
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exists and we also exhibit properties of Iθ that we deem useful for a potential infrared analysis. If θ was chosen sufficiently small, we will write Iθ as the sum of a dominant part (which is shown to have a logarithm) and terms indexed by proper subsets of X and which are exponentially small in the size of the subsets. The partition function can be written as p dφn (x)∗ φn (x) 1 − kT (H−μN ) −φn (x)∗ φn (x) e = Iθ (φ∗n−1 , φn ) Tr e 2πı n=1 x∈X
More generally, one also gets a representation for the Green’s functions3 1 Tr e− kT (H−μN ) T ψ (†) (βj , xj ) j=1
(1.7) 1 Tr e− kT (H−μN ) 1 . To this end, choose a sufficiently with x1 , . . . , x ∈ X and 0 ≤ β1 , . . . , β ≤ kT 1 fine partition, 0 = τ0 < τ1 < · · · < τp = kT that contains β1 , . . . , β . It follows from [2, Theorem 3.7] that the numerator of (1.7) is equal to p dφτ (x)∗ φτ (x) n n −φτn (x)∗ φτn (x) ∗ e Iτn −τn−1 (φτn−1 , φτn ) 2πı n=1 x∈X
×
φβj (xj )(∗)
(1.8)
j=1 1 . with φ0 = φ kT The functions In (ε; α∗ , β) can also be defined recursively by ∗ ∗ I1 (ε; α∗ , β) = dμR(ε) (φ∗ , φ) ζε (α∗ , φ) eα , j(ε)φ +φ , j(ε)β ∗
×e−ε(α and ∗
In+1 (ε; α , β) =
φ, v α∗ φ +φ∗ β, v φ∗ β )
ζε (φ∗ , β)
dμR(ε) (φ∗ , φ) In (ε; α∗ , φ)In (ε; φ∗ , β)
(1.9)
(1.10)
This recursive definition is called decimation, because we successively integrate out every second field. The description of and estimates on In will be obtained inductively. The integrals (1.9) and (1.10) are oscillatory. Their dominant contributions are extracted by stationary phase. In [5], we describe the construction and estimates that one obtains if, in (1.9) and (1.10), one always ignores contributions far away from the critical point of the (“free part”) of the exponent. We call this the “stationary phase approximation”. To be somewhat more precise, fix a suitable (see [5, Hypothesis I.1]) non negative decreasing function r(t), and assume that one keeps in (1.9) and (1.10) only the integral over fields that 3
Here ψ(βj , xj ) and ψ † (βj , xj ) are annihilation and creation operators, conjugated by e , and T denotes time ordering. βj (H−μN )
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are within distance r(2n ε) from the critical point of the exponent, evaluated at v ≡ 0. Then the dominant contribution to In (ε; α∗ , β) is ∗
In(SP) (ε; α∗ , β) = Z2n ε (ε)|X| eα
, j(2n ε)β +V2n ε (ε; α∗ ,β)+E2n ε (ε; α∗ ,β)
where, for every δ that is an integer multiple of ε, Vδ (ε; α∗ , β) = −ε [j(τ )α∗ ] [j(δ − τ − ε)β] , v [j(τ )α∗ ] [j(δ − τ − ε)β] τ ∈εZ∩[0,δ)
and the functions Eδ are recursively defined by Eε (ε; α∗ , β) = 0 E2δ (ε; α∗ , β) = Eδ (ε; α∗ , j(δ)β) + Eδ (ε; j(δ)α∗ , β) ∗ ∗ dμr(δ) (z ∗ , z) e∂Aδ (ε; α ,β;z ,z) + log dμr(δ) (z ∗ , z)
(1.11)
with ∂Aδ (ε; α∗ , β; z∗ , z) = [Vδ (ε; α∗ , j(δ)β + z) − Vδ (ε; α∗ , j(δ)β)] + [Vδ (ε; j(δ)α∗ + z∗ , β) − Vδ (ε; j(δ)α∗ , β)] + [Eδ (ε; α∗ , j(δ)β + z) − Eδ (ε; α∗ , j(δ)β)] + [Eδ (ε; j(δ)α∗ + z∗ , β) − Eδ (ε; j(δ)α∗ , β)] The normalization constant Zδ (ε), which is extremely close to one, is chosen so that Eδ (ε; 0, 0) = 0. See (1.7), (1.8) and (1.9) in [5]. The motivation for this recursion relation comes from a stationary phase construction and is given below and also in the section 2 of [5]. Theorem I.4 of [5] shows that, under suitable assumptions on the function r(t) and the number θ, the logarithm in (1.11) always exists as an analytic function of the fields, and that one can get a good estimate on Eθ (α∗ , β) = lim Eθ 2−m θ; α∗ , β m→∞
To motivate the recursive definition (1.11) of Eδ (ε; α∗ , β) we replace In by ∗
In(SP) (ε; α∗ , β) = Zεn (ε)|X| eα
, j(εn )β +Vεn (ε; α∗ ,β)+Eεn (ε; α∗ ,β)
in the recursion relation (1.10). Here, εn = 2n ε. The resulting integral dμR(ε) (φ∗ , φ) In(SP) (ε; α∗ , φ) In(SP) (ε; φ∗ , β) ∗ ∗ ∗ ∗ = Zεn (ε)2|X| dμR(ε)(φ∗, φ) eα , j(εn )φ +φ , j(εn )β eVεn (ε; α ,φ)+Vεn (ε; φ ,β) ∗
∗
× eEεn (ε; α ,φ)+Eεn (ε; φ ,β) ∗ ∗ dφ∗ (x)dφ(x) 2|X| χ (|φ(x)| < R(ε)) eA(α ,β ; φ ,φ) = Zεn (ε) 2πı x∈X
(1.12)
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with A(α∗ , β ; φ∗ , φ) = −φ∗ , φ + α∗ , j(εn )φ + φ∗ , j(εn )β +Vεn (ε; α∗ , φ)+Vεn (ε; φ∗ , β)+Eεn (ε; α∗ , φ)+Eεn (ε; φ∗ , β) Here we have written A as a function of four independent complex fields α∗ , β, φ∗ and φ. The activity in (1.12) is obtained by evaluating A(α∗ , β; φ∗ , φ) with φ∗ = φ∗ , the complex conjugate of φ. The reason for introducing independent complex fields φ∗ and φ lies in the fact that the critical point (with respect to the variables φ∗ , φ) of the quadratic part −φ∗ , φ + j(εn )α∗ , φ + φ∗ , j(εn )β = −φ∗ − j(εn )α∗ , φ − j(εn )β + α∗ , j(εn+1 )β of A is “not real”. Precisely, the critical point is φcrit = j(εn ) α∗ ∗ ∗ (φcrit ∗ )
φcrit = j(εn ) β
(1.13)
and in general = φ . It is reasonable to expect that the dominant contribution to the integral in (1.12) comes from the fields φ(x) in a neighbourhood of the critical point. We now sketch, approximately, the strategy that we use to verify that this is indeed the case. We decompose, for each x ∈ X, the domain of integration {|φ(x)| < R(ε)} into the “small field region”, where φ∗ (x) is close to φcrit ∗ (x) and φ(x) is close to φcrit (x), and the “large field region” where this is not the case. Precisely, write crit
χ (|φ(x)| < R(ε)) = χx, small (φ(x), φ∗ (x)) + χx, large (φ(x)) where
⎧ ∗ ⎪ ⎨1 if φ∗ = φ , |φ| < R(ε) crit χx, small (φ, φ∗ ) = (x) < r(εn ) and φ∗ − φcrit ∗ (x) < r(εn ), φ − φ ⎪ ⎩ 0 otherwise
and χx, large (φ) = χ (|φ| < R(ε)) (1 − χx, small (φ, φ∗ )) We multiply out the products of sums of characteristic functions and get that (1.12) is equal to dφ∗ (x) ∧ dφ(x) 2|X| χx, small (φ(x), φ∗ (x)) Zεn (ε) 2πı Λ⊂X x∈Λ ⎡ ⎤ dφ∗ (x) ∧ dφ(x) ∗ χx, large (φ(x))⎦ eA(α ,β ; φ∗ ,φ) ×⎣ φ∗ (x)=φ∗ (x) 2πı x∈X\Λ
for x∈X\Λ
(1.14) Select a term of (1.14), that is, a subset Λ of X. For points x ∈ Λ, we introduce the “fluctuation variables” z∗ (x), z(x) by the change of variables φ∗ (x) = φcrit ∗ (x) + z∗ (x),
φ(x) = φcrit (x) + z(x)
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For points outside Λ, we do not perform any change of variables. So the fluctuation fields z∗ , z are supported on Λ, and the change of variables is φ∗ = Λφcrit + z∗ + Λc φ∗ = Λj(εn )α∗ + z∗ + Λc φ∗ ∗ φ = Λφcrit + z + Λc φ = Λj(εn )β + z + Λc φ
(1.15)
Here, we also denote by Λ the operator “multiplication by the characteristic function of the set Λ”. Under this change of variables the domain of integration
(φ(x), φ∗ (x)) χx, small (φ(x), φ∗ (x)) = 1 is transformed into " ∗ = φcrit (x) + z(x), D(x) = (z∗ (x), z(x)) φcrit ∗ (x) + z∗ (x) # crit φ (x) + z(x) < R(ε) and |z∗ (x)| ≤ r(εn ), |z(x)| ≤ r(εn ) (1.16) ∗
Observe that for (z∗ (x), z(x)) ∈ D(x), in general z∗ (x) = z (x). The quadratic part of the effective action + z∗ + Λc φ∗ , Λφcrit + z + Λc φ A α∗ , β; Λφcrit ∗ in the new variables is − Λj(εn )α∗ + z∗ + Λc φ∗ , Λj(εn )β + z + Λc φ + α∗ , j(εn ) (Λj(εn )β + z + Λc φ) + j(εn ) (Λj(εn )α∗ + z∗ + Λc φ∗ ) , β = −z∗ , z − Λc φ∗ , Λc φ + QΛ (α∗ , β; Λc φ∗ , Λc φ) where QΛ (α∗ , β; Λc φ∗ , Λc φ) = Λj(εn )α∗ , Λj(εn )β + α∗ , j(εn )Λc φ + j(εn )Λc φ, β
(1.17)
Observe that the terms linear in z∗ , z cancelled, because we centered the change of variables at the critical point. Inserting this change of variables in (1.14), we see that (1.12) is equal to ⎡ ⎤ ⎢ dz∗ (x) ∧ dz(x) −z∗ (x)z(x) ⎥ e Zεn (ε)2|X| ⎣ ⎦ 2πı Λ⊂X
×
x∈Λ D(x)
dφ∗ (x) dφ(x) ∗ ˜ −φ∗ (x)φ(x) e χx, large (φ(x)) eAΛ (α ,β; φ,z∗ ,z) 2πı c
x∈Λ
(1.18) where A˜Λ (α∗ , β; φ, z∗ , z) = QΛ (α∗ , β; Λc φ∗ , Λc φ) +Vεn (ε; α∗ , Λφcrit + z + Λc φ) +Vεn (ε; Λφcrit + z∗ + Λc φ∗ , β) ∗
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+Eεn (ε; α∗ , Λφcrit + z + Λc φ) +Eεn (ε; Λφcrit + z∗ + Λc φ∗ , β) ∗
(1.19)
If we apply Stokes’ Theorem (Lemma A.1 of [5]) with X replaced by Λ, r = ∗ crit to (1.18) we see that (1.12) is equal r(εn ), σ = σ∗ = 0 and ρ = (φcrit ∗ ) −φ to ⎡ ⎤ ⎢ dz ∗ (x)dz(x) −|z(x)|2 ⎥ e Zεn (ε)2|X| ⎣ ⎦ 2πı Ω⊂Λ⊂X
⎡ ⎢ ×⎣
x∈Λ\ΩC(x)
x∈Ω |z(x)|≤r(εn )
⎤
dz∗ (x) ∧ dz(x) −z∗ (x)z(x) ⎥ e ⎦ 2πı
dφ∗ (x)dφ(x) 2 ∗ ˜ e−|φ(x)| χx,large (φ(x)) eAΛ (α ,β;φ,z∗ ,z) z∗ (x)=z∗ (x) × 2πı c for x∈Ω
x∈Λ
(1.20) 2 where, for each x ∈ X, C(x) is a two real dimensional submanifold of C whose 2 ∗ boundary is the union of “circles” ∂D(x) and {(z∗ (x), z(x)) ∈ C z∗ (x) = z(x), |z(x)| = r(εn )}. For points x ∈ Ω, the domain of integration has been moved “back to the reals”. Whenever χx, large (φ(x)) = 1 or (z∗ (x), z(x)) ∈ C(x) for some x ∈ X, then A˜Λ (α∗ , β; φ, z∗ , z) or −z∗ (x)z(x) has extremely large negative real part and the contribution to the integral is very small. (See the discussion following Theorem 3.35.) For this reason we kept only the term of (1.20) with Ω = Λ = X for the “stationary phase approximation” in [5]. In this case, QX (α∗ , β; Λc φ∗ , Λc φ) = α∗ , j(εn+1 )β . Thus, the stationary phase approximation to dμR(ε) (φ∗ , φ) In(SP) (ε; α∗ , φ) In(SP) (ε; φ∗ , β)
is ∗
Zεn (ε)2|X| eα ⎡ ⎢ ×⎣
,j(εn+1 )β
x∈X |z(x)|
= Zn2|X| eα
,j(εn+1 )β
⎤ dz ∗ (x) ∧ dz(x) −|z(x)|2 ⎥ A(α ˜ ∗ ,β;z ∗ ,z) e ⎦ e 2πı
˜
∗
dμr(εn ) (z ∗ , z) eAX (α
,β;z ∗ ,z)
where A˜X (α∗ , β; z∗ , z) = Vεn (ε; α∗ , φcrit + z) + Vεn (ε; φcrit + z∗ , β) ∗
+Eεn (ε; α∗ , φcrit + z) + Eεn (ε; φcrit + z∗ , β) ∗
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By construction Vεn (ε; α∗ , φcrit ) + Vεn (ε; φcrit ∗ , β) ∗ = Vεn (ε; α , j(εn )β) + Vεn (ε; j(εn )α∗ , β) = Vεn+1 (ε; α∗ , β)
so that + z ∗ , β) Vεn (ε; α∗ , φcrit + z) + Vεn (ε; φcrit ∗
= Vεn+1 (ε; α∗ , β) + [Vεn (ε; α∗ , j(εn )β + z) − Vεn (ε; α∗ , j(εn )β)] + [Vεn (ε; j(εn )α∗ + z ∗ , β) − Vεn (ε; j(εn )α∗ , β)]
Consequently, the stationary phase approximation to dμR(ε) (φ∗ , φ) In(SP) (ε; α∗ , φ) In(SP) (ε; φ∗ , β) can also be written as ∗
Zεn (ε)2|X| eα
∗
× eEεn (ε; α
,j(εn+1 )β +Vεn+1 (ε; α∗ ,β)
,j(εn )β) +Eεn (ε; j(εn )α∗ ,β)
This is compatible with (1.11) if we take 2 Zεn+1 (ε) = Zεn (ε) |z|
∗
dμr(εn ) (z ∗ , z) e∂Aεn (ε; α dz ∗ ∧ dz −|z|2 e 2πi
,β;z ∗ ,z)
(1.21)
It turns out that, for each fixed Ω ⊂ X, the sum over all sets Λ with Ω ⊂ Λ ⊂ X in (1.20) can be written in the form ∗
Zεn+1 (ε)|X| eα
,j(εn+1 )β Ω VΩ;εn+1 (ε; α∗ ,β)+EΩ;εn+1 (ε; α∗ ,β)
e
ϕΩ;εn+1 (ε; α∗ , β) (1.22)
where VΩ;εn+1 and EΩ;εn+1 depend only on α∗ (x) and β(x) with x ∈ Ω, and are given by the same formulae as Vεn+1 and Eεn+1 but with the total space X replaced by Ω everywhere, and where ϕΩ;εn+1 is a very small function that encapsulates the sum over Λ and various integral operators. The sets Λ, resp. Ω, introduced above are called “small field sets” of the first, resp. second, kind. The discussion of the previous paragraph shows that (1.20) is equal to dμR(ε) (φ∗ , φ) In(SP) (ε; α∗ , φ) In(SP) (ε; φ∗ , β) and it indicates that In+1 (ε; α∗ , φ) is its dominant contribution. The next decimation step would be to consider dμR(ε) (φ∗1 , φ1 ) In(SP) (ε; α∗ , φ1 ) In(SP) (ε; φ∗1 , φ) dμR(ε) (φ∗ , φ) × dμR(ε) (φ∗1 , φ2 ) In(SP) (ε; φ∗ , φ2 ) In(SP) (ε; φ∗2 , β) (SP)
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Inserting (1.20) and (1.22) for the two integrals in brackets gives a normalization constant times a sum over small field sets Ω1 and Ω2 of integrals dφ∗ (x) dφ(x) χ (|φ(x)) < R(ε)) 2πı x∈X
∗
×eA (α
,β;φ∗ ,φ)
ϕΩ1 ;εn+1 (ε; α∗ , φ) ϕΩ2 ;εn+1 (ε; φ∗ , β)
(1.23)
with A (α∗ , β; φ∗ , φ) = − φ∗ , φ + α∗ , j(εn+1 )φΩ1 + φ∗ , j(εn+1 )βΩ2 +VΩ1 ;εn+1 (ε; α∗ , φ) + VΩ2 ;εn+1 (ε; φ∗ , β) +EΩ1 ;εn+1 (ε; α∗ , φ) + EΩ2 ;εn+1 (ε; φ∗ , β)
This oscillatory integral is similar to (1.12). The small factors for points x ∈ (X\Ω1 ) ∪ (X\Ω2 ) mentioned above are so strong that we only have to perform a stationary phase argument for points inside Ω1 ∩ Ω2 . That is, we would, as in (1.18), write (1.23) as ⎡ ⎤ ⎢ dz∗ (x) ∧ dz(x) ⎥ e−z∗ (x)z(x) ⎦ ⎣ 2πı
Λ ⊂Ω1 ∩Ω2
⎡
x∈Λ
D (x)
⎤ dφ∗ (x)dφ(x) 2 ˜ ∗ e−|φ(x)| χ x,large (φ(x))⎦eA (α ,β;z∗ ,z) f (α∗ , β; z∗ , z) ×⎣ 2πı x∈X\Λ
where D (x) and χ x, large are defined as were D(x) and χx, large , but with εn replaced by εn+1 , and where A (α∗ , β; φ, z∗ , z) and f (α∗ , β; φ, z∗ , z) are obtained using the change of variables around the critical point of the quadratic form. The next step would be to again apply Stokes’ Theorem for the variables z∗ (x), z(x) with x ∈ Λ. Here, a small technical difficulty arises. Namely, the factor f (α∗ , β; z∗ , z) in the integrand need not be analytic in z∗ , z and the version of Stokes’ Theorem presented in Appendix A of [5] cannot be applied directly. To circumvent this difficulty, we introduce a constant c > 0 and define the cut-off propagator 1 if d(x, y) ≤ c (1.24) jc (τ )(x, y) = j(τ )(x, y) · 0 if d(x, y) > c where d(x, y) is the distance on the torus X, replace j(t) by jc (t) in the formulae above and control the error terms. We apply Stokes’ theorem only for points x ∈ Λ that have distance at least c from (X\Ω1 ) ∪ (X\Ω2 ). That is, we apply Stokes’ theorem only in
Λ = 1x ∈ Λ d(x, y) ≥ c for all y ∈ (X\Ω1 ) ∪ (X\Ω2 ) With this construction, the analogue of f (α∗ , β; z∗ , z) will not depend on the variables z∗ (x), z(x) with x ∈ Λ. The integrals for points in the “corridor”
x ∈ Λ d(x, y) ≤ c for some y ∈ (X\Ω1 ) ∪ (X\Ω2 )
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can again be controlled by the small factors from the points y ∈ X\Λ . This modification leads to the somewhat more involved formulae described in Sect. 2 below. Obviously we want to iterate the procedure described above, starting with n = 1. In this way one creates a “hierarchy” of small field sets of the first and second kind. In Definition 2.4, this is made more precise, and enriched with more sets that are used to describe the various “large field conditions”. The main results of this paper are estimates on all of the functions appearing in the functional integral representation of the partition function. For these estimates we use the norms developed in [3,4]. One of the simplest versions of such a norm is defined as follows. Let κ, m > 0. We define the norm of the power series f (α∗ , β) = k,≥0 x1 ,...,xk ∈X y1 ,...,y ∈X
× a(x1 , . . . , xk ; y1 , . . . , y ) α(x1 )∗ · · · α(xk )∗ β(y1 ) · · · β(y ) (with the coefficients a(x1 , . . . , xk ; y1 , . . . , y ) invariant under permutations of x1 , . . . , xk and of y1 , . . . , y ) to be f (α∗ , β) κ,m = max max κk+ emτ ( x, y) ) |a(x ; y)| k,≥0
x∈X 1≤i≤k+
( x, y)∈X k ×X ( x, y)i =x
(1.25) where τ (x, y) is the minimal length of a tree which contains vertices at the points of the set {x1 , . . . , xk , y1 , . . . , y }. Our main results, the description of and bounds on Iθ , are stated in Sect. 2.6 (Theorems 2.16 and 2.18). The subsections of Sect. 2 introduce the notation used in these Theorems. Section 3 gives an outline of the proof and contains discussions which might illuminate the concepts introduced in Sect. 2. The proof of Theorem 2.16 is split over Sects. 3–5. The proof of Theorem 2.18 is split over Sects. 3 and 6.
2. Formulation of the Main Theorem Our main result is a representation of the “effective density” Iθ of (1.6) as a sum over subsets Ω of X. For each Ω ⊂ X the corresponding summand is the product of • •
the exponential of a function that is analytic in the fields and that depends only on Ω. A function that involves all possible “hierarchies” (collections of large and small field sets—see Definition 2.4) that lead to Ω, which is not necessarily analytic, but can be proven to be very small (unless Ω = X). Indeed, if Ωc = ∅, this function is O(|||v|||n ) (this norm will be defined in (2.5)) for all n ∈ N and also decreases exponentially quickly with |Ωc |.
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The first factor will be called the “small field part”. It is described in Sect. 2.1 below. 2.1. The Small Field Parts Let Ω be a subset of X. For any kernel w(x, y) on X denote by w(x, y) if x, y ∈ Ω wΩ (x, y) = 0 otherwise its truncation to Ω. For any t > 0, set ∞ 1 j(Ω) (t) = 1Ω + − t(h − μ)Ω ) ! =1 −t(h−μ)Ω
=e and
− 1X\Ω
j(Ω) (t)(x, y) j(Ω),c (t)(x, y) = 0
if d(x, y) ≤ c if d(x, y) > c
1 For each 0 < t ≤ 2kT and each analytic function V (α∗ , β), that depends only on the variables α∗ (x), β(x) with x ∈ Ω, consider the formal renormalization group operator RΩ;t (V ; ·, · ) at scale t with “principal interaction V ” that is defined as the following generalization of (1.11). It associates to any two analytic functions4 f1 (α∗ , β), f2 (α∗ , β) that depend only on the variables α∗ (x), β(x) with x ∈ Ω the function
RΩ;t (V ; f1 , f2 )(α∗ , β)
= f1 α∗ , j(Ω),c (t)β + f2 j(Ω),c (t)α∗ , β + log & ' − [j(Ω) (t) − j(Ω),c (t)]α∗ , [j(Ω) (t) − j(Ω),c (t)]β +V α∗ , j(Ω),c (t)β − V α∗ , j(Ω) (t)β +V j(Ω),c (t)α∗ , β − V j(Ω) (t)α∗ , β
dμΩ,r(t) (z ∗ , z) eA(α∗ ,β;z dμΩ,r(t) (z ∗ , z)
∗
,z)
where A(α∗ , β; z∗ , z) is f1 α∗ , z + j(Ω),c (t)β − f1 α∗ , j(Ω),c (t)β + f2 z∗ + j(Ω),c (t)α∗ , β − f2 j(Ω),c (t)α∗ , β & ' & ' + [j(Ω) (t) − j(Ω),c (t)]α∗ , z + z∗ , [j(Ω) (t) − j(Ω),c (t)]β + V α∗ , z + j(Ω),c (t)β − V α∗ , j(Ω),c (t)β + V z∗ + j(Ω),c (t)α∗ , β − V j(Ω),c (t)α∗ , β and dμΩ,r (z ∗ , z) =
dz(x)∗ ∧ dz(x) ∗ e−z(x) z(x) χ (|z(x)| < r) 2πı
x∈Ω 4
We introduce the complex field α∗ in order to clarify the analyticity properties of the functions f1 , f2 . We shall usually evaluate α∗ at α∗ .
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As “principal interaction” V in the renormalization group map, we use the dominant part of the interaction at the corresponding scale. Precisely, for every δ that is an integer multiple of ε, define VΩ,δ (ε; α∗ , β) &
' j(Ω) (τ )α∗ j(Ω) (δ−τ −ε)β , vΩ j(Ω) (τ )α∗ j(Ω) (δ−τ −ε)β = −ε τ ∈εZ∩[0,δ)
(2.1) Define recursively DΩ;0 (ε; α∗ , β) = 0 DΩ;n+1 (ε; α∗ , β) = RΩ;2n ε (VΩ;2n ε (ε; ·, · ); DΩ;n (ε; ·, · ), DΩ;n (ε; ·, · )) If θ is small enough, this recursion defines analytic functions DΩ;n (ε; α∗ , β), when 2n ε ≤ θ, and furthermore DΩ;θ (α∗ , β) = lim DΩ;m (2−m θ; α∗ , β) m→∞
(2.2)
exists and fulfills the estimates of Proposition 2.1 and Theorem 2.16 below. Our representation of the effective density will be of the form Iθ (α∗ , β) |Ω| ∗ ∗ ∗ Zθ eα , j(Ω) (θ)β +VΩ;θ (α ,β)+DΩ;θ (α ,β) χθ (Ω; α, β) ϕΩ;θ (α∗ , β) = Ω⊂X
(2.3) where the normalization constant Zθ , which will be defined in Lemma 2.7, is very close to one, VΩ;θ (α∗ , β) = lim VΩ;θ (2−m θ; α∗ , β) m→∞
θ =−
dt
&
' j(Ω) (t)α∗ j(Ω) (θ − t)β , vΩ j(Ω) (t)α∗ j(Ω) (θ − t)β
0
(2.4) and χθ (Ω; α, β) implements the small field conditions for the set Ω (see Theorem 2.16). For our construction to work, we need exponential decay of the interaction v. Precisely, we assume that there is a “mass” m > 0 such that e5m d(x,y) |v(x, y)| (2.5) |||v||| = sup x∈X
y∈X
is finite. Proposition 2.1. There are constants const , κ > 0 such that for all sufficiently small θ and interactions v, and all Ω ⊂ X, the function DΩ;θ (α∗ , β) of (2.2)
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is well defined,5 and obeys DΩ;θ κ,m ≤ const |||v||| Here, we use the norm (1.25). The proof of this Proposition is much the same as the proof of [5, Theorem I.4]. It is also a consequence of our complete analysis of Iθ in Theorem 2.16. There, we shall also state more properties of DΩ;θ . 2.2. Decomposition of Space into Large and Small Field Subsets As indicated in the introduction, the functions ϕΩ of (2.3) are expressed in terms of “hierarchies” of large and small field subsets of X. Recall that we have chosen a function r(t) that measures the size of the neighbourhood of a critical point at scale t in which Stokes’ argument is applied to move the domain of integration “back to the reals”. We fix, in addition to the functions R(t) and r(t) of Sect. 1, another decreasing positive function R (t). If we are at scale t and, for some point x, one of the fields α∗ (x) or β(x) is larger than R(t) we shall get a controllable small factor. This leads us to introduce a further decomposition of X where the corresponding large field sets will be denoted by Pα and Pβ , respectively. Also large values of the spatial gradients of the fields give rise to small factors. Spatial gradients are controlled in terms of the function R (t). The corresponding large field sets will be sets Pα and Pβ of bonds in the lattice X. Similarly, large time derivatives cause small factors. The corresponding large field sets will be denoted by the letter Q. The sets where, in the application of Stokes’ theorem, the “side” C(x) of the cylinder is chosen as the domain of integration, will be labelled by R. All the sets in our construction should be separated by corridors. The width of these corridors is scale dependent and will be measured by a function c(t). Later, in (2.18), we will make specific choices of all these functions. Their properties will be proven in Appendix F. Here, we concentrate on the purely set theoretic picture. As indicated above, we will have to deal with bonds of the lattice X. We denote by X ∗ the set of all bonds (i.e. pairs of neighbouring points). Furthermore, for a subset Y of X, we set
Y = x ∈ X x is connected by a bond to a point of Y
Y ∗ = b ∈ X ∗ b has at least one end point in Y For P ⊂ X ∗ , we denote by supp P the set of all end points of all bonds in P . The large and small field sets will be conveniently indexed by intervals whose length is related to the scale at which they were created. Notation 2.2. (i) A decimation point for the interval [0, δ] is a point τ = 2pk δ with integers k ≥ 0 and 0 ≤ p ≤ 2k . δ] (ii) A decimation interval in [0, δ] is an interval of the form J = [ 2pk δ, p+1 2k k with k ≥ 0 and 0 ≤ p ≤ 2 − 1. 5
That is, it is possible to take the logarithms involved and the limit.
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(iii) For a decimation point τ = 0, δ, there is a unique k ≥ 1 such that δ Z. This number k is called the decimation index d(τ ) of τ τ ∈ 2δk Z\ 2k−1 in [0, δ]. We also set d(0) = d(δ) = 0. We call s = 2−d(τ ) δ the scale of τ . The unique decimation interval that has τ as its midpoint is Jτ = [τ − s, τ + s] Its left and right halves Jτ− = [τ − s, τ ],
Jτ+ = [τ, τ + s]
are also decimation intervals. 3 δ, then τ ∈ 2δ4 Z\ 2δ3 Z so that d(τ ) = 4, Example 2.3. For example, if τ = 16 δ 2 4 2 3 3 4 − s = 16 , Jτ = [ 16 δ, 16 δ], Jτ = [ 16 δ, 16 δ] and Jτ+ = [ 16 δ, 16 δ].
If τ is a decimation point, then the field ατ appears as an integration variable in the construction of Im 2−m δ; α∗ , β) for all m ≥ d(τ ). When this variable is integrated, large and small field sets are introduced. We choose to label them by Jτ , because Jτ carries the information about τ , and through its length, also about the scale 2−d(τ ) δ. 1 , of large and Definition 2.4 (Hierarchy). A hierarchy, S, for scale 0 < δ ≤ kT small field sets is a collection ◦ Pα (J ), Pβ (J ), Q(J ) of subsets of X, called large field sets of the first kind ◦ Pα (J ), Pβ (J ) of subsets of X ∗ , also called large field sets of the first kind ◦ R(J ) of subsets of X, called large field sets of the second kind ◦ Λ(J ), Ω(J ) of subsets of X, called the small field sets of the first and second kind respectively. These sets ◦ are indexed by all decimation intervals J in [0, δ], ◦ and obey the following “large/small field set” compatibility conditions. Let J be a decimation interval in [0, δ], and J − and J + be its left and right halves. Let t = length(J + ) = length(J − ) = 12 length(J ). Then Pα (J ), Pβ (J ) ⊂ Ω(J − ) ∩ Ω(J + ) and Pα (J ), Pβ (J ) ⊂ (Ω(J − ) ∩ − + Ω(J + ))∗ and Q(J ) ⊂ (Ω(J ) ∩ Ω(J )) . Λ(J ) = x ∈ X d (x, Pα (J ) ∪ Pβ (J ) ∪ Q(J )) > c(t), ( ) d x, supp Pα (J ) ∪ supp Pβ (J ) > c(t),
d (x, Ω(J − )c ∪ Ω(J + )c ) > c(t) R(J ) ⊂ Λ(J ) Ω(J ) = {x ∈ Λ(J ) d (x, R(J )) > c(t)}
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◦ There is a non-negative integer k0 such that Λ(J ) = Ω(J ) = X, and consequently Pα (J ) = Pβ (J ) = Pα (J ) = Pβ (J ) = Q(J ) = R(J ) = ∅, for all decimation intervals of lengths 2−k δ with k ≥ k0 . The smallest such k0 is called depth(S). The following figure schematically illustrates the set relations amongst the various large and small field sets at a single scale, but is not metrically accurate.
Notation 2.5. Let S be a hierarchy of scale δ. (i) We also denote, for example, Λ(J ) by ΛS (J ), when we wish to emphasize its dependence on the hierarchy S. (ii) The “summits” ΩS ([0, δ]) and ΛS ([0, δ]) are also denoted ΩS and ΛS , respectively. (iii) The decimation points, resp. intervals, in [0, δ] are called the decimation points, resp. intervals, for the hierarchy S. (iv) For a decimation point τ , we set Λ(Jτ ) if τ = 0, δ Λτ = ∅ if τ = 0, δ Here, Jτ is the unique decimation interval centered on τ . See Notation 2.2. Observe that Λτ = X
if d(τ ) > depth(S)
Remark 2.6. It follows from the definition that, for decimation intervals J J ◦ Λ(J ) ⊃ Ω(J ) ⊃ Λ(J ) ⊃ Ω(J ) ◦ Ω(J )c ∪ Pα (J ) ∪ Pβ (J ) ∪ supp Pα (J ) ∪ supp Pβ (J ) ∪ Q(J ) ⊂ Λ(J )c and Λ(J )c ∪ R(J ) ⊂ Ω(J )c and indeed if, ∅ = Ω(J ) = X, then d(Ω(J )c , Ω(J )) ≥ d(Ω(J )c , Λ(J )) > c(|J |). Since X is a finite set and c(t) ≥ 1 for all t, it then follows that there is a natural number kX with the property that ◦ if J is a decimation interval with Ω(J ) = X, then Ω(J ) = Λ(J ) = Pα (J ) = Pβ (J ) = Pα (J ) = Pβ (J ) = Q(J ) = R(J ) = ∅ for all decimation intervals J J with
|J | |J |
≥ 2kX .
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In particular, for each δ > 0, there are only finitely many hierarchies S for scale δ that have ΩS ([0, δ]) = ∅. Each such hierarchy has depth at most kX . 2.3. The Large Field Integral Operator The functions ϕΩ;θ (α∗ , β) of (2.3) will be written as a sum ϕΩ;θ = ϕS;θ
(2.6)
S hierarchy for scale θ ΩS ([0,θ])=Ω
Each of the functions ϕS;θ (α∗ , β) will be an integral over variables in the large field regions determined by the hierarchy S. As we saw in the discussion of the stationary phase approximation in Sect. 1, we shall need normalization constants in the representation (2.3) of the effective densities. They should obey the recursion relation (1.21). We make a particular choice of normalization constants Zδ , by prescribing their asymptotic behaviour as δ → 0. This choice is made to simplify the proof that limm→∞ Im (2−m θ; · · · ) exists. Lemma 2.7. There is a unique function δ ∈ (0, 1) → Zδ ∈ (0, 1) that obeys dz ∗ ∧ dz −|z|2 1 2 e Z2δ = Zδ lim log Zε = 0 ε→0+ ε 2πi |z|≤r(δ)
Furthermore, |ln Zδ | ≤ e−r(δ)
2
This lemma is proven in Appendix C. The large field integral operator arises from the “left over” fields in the decimation procedure outlined in and after (1.12). The decimation steps are indexed by decimation points τ ∈ (0, δ). When the field φ is being integrated out in such a step, one gets, as in (1.20), a sum over pairs of small field sets Ω(Jτ ) ⊂ Λ(Jτ ) and (i) the fluctuation integral with variables |z(x)| ≤ r 12 |Jτ | for x ∈ Ω(Jτ ) (ii) the integral over the Stokes’ cylinders C(x), x ∈ Λ(Jτ )\Ω(Jτ ), with the variables z∗ (x), z(x) (iii) the “large field integral of the first kind” for points x ∈ Λ(Jτ )c , with variables φ(x) which violated at least one of the small field conditions of the first kind. In the decimation step, the fluctuation integral (i) is performed, while the integrals (ii) and (iii) are are not performed explicitly and form part of the large field integral operator. For labelling purposes, the integration variables of (ii) are renamed z∗τ (x) and zτ (x), and the integration variables of (iii) are renamed ατ (x). We call the fields ατ (x), x ∈ Λ(Jτ )c , and z∗τ (x), zτ (x), x ∈ Λ(Jτ )\Ω(Jτ ), the residual fields. The integral operator associated to a full hierarchy, S, is the concatenation of all integral operators associated to all decimation points τ for the hierarchy with decimation index d(τ ) < depth(S). The definition of the integral operators involves the constant c and the cut-off propagator jc (τ ) of (1.24).
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Definition 2.8. (Large field integral operator). Let S be a hierarchy for scale δ > 0. (i) Let τ be a decimation point for S with d(τ ) ≤ depth(S). The scale of τ is s = 2−d(τ ) δ, and its corresponding decimation interval is J = Jτ = [τ , τr ] with τ = τ − s and τr = τ + s. The integral operator associated to the decimation point τ is I(J ; α∗ ,β) = I(J ,S ; α∗ ,β) ⎛ ⎜ =⎝
x∈Λ(J )\(R(J )∪Ω(J ))|z (x)|≤r(s) τ
⎛
⎜ ×⎝ ⎛ ×⎝
x∈R(J )C (x;α∗ ,β) s
x∈X\Λ(J )
×Zs|Ω(J
−
)\Ω(J )|
⎞ ∗
dzτ (x) ∧ dzτ (x) −zτ (x)∗ zτ (x) ⎟ e ⎠ 2πi ⎞
dz∗τ (x) ∧ dzτ (x) −z∗τ (x)zτ (x) ⎟ e ⎠ 2πi
⎞ dατ (x)∗ ∧ dατ (x) ⎠ χJ (α, ατ , β) 2πi
Zs|Ω(J
+
)\Ω(J )|
Here, for each x ∈ R(J ), Cs (x; α∗ , β) is a two real dimensional surface in
(z∗ , z) ∈ C2 |z∗ |, |z| < R(s) # " whose boundary is the union of the circle (z∗ , z) ∈ C2 z∗∗ = z, |z| = r(s) and the curve bounding6 " (z∗ , z) ∈ C2 |z∗ −([1−jc (s)]α∗ ) (x)| ≤ r(s),
# |z−([1−jc (s)]β) (x)| ≤ r(s), z∗∗ − z = (jc (s)[β − α]) (x)
(2.7)
Analyticity and Stokes’ theorem ensures the action of the integral operator is independent of the choice of the surfaces Cs (x; α∗ , β). See [5, §II and Lemma A.1]. We choose Cs (x; α∗ , β) to depend only on the values of the fields α and β at points y ∈ X with d(x, y) ≤ c. This is possible because the boundary curves have the same property, since jc (s) has range c. The characteristic function χJ (α, ατ , β) implementing the large and small field conditions of the first kind is given in Appendix A. If Ω(J ) = X, we set I(J ; α∗ ,β) = 1. (ii) The integral operator associated to the hierarchy S is I(S;α∗ ,β) = I(J ; α∗τ ,ατr ) n=0,··· ,depth(S) decimation intervals J =[τl ,τr ]⊂[0,δ] of length 2−n δ 6
The set (2.7) is a technically precise variant of (1.16).
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Observe that the arguments ατ∗l , ατr in each I(J ; α∗τ ,ατr ) are the integra tion variables for an integral appearing to its left. This is the reason for 0 ordering the product n=0,...,depth(S) with larger values of n to the right. (iii) We will bound, in Theorem 2.18, the “absolute value” |I(S;α∗ ,β) | = I(J ; α∗τ ,ατr )
n=0,...,depth(S) time intervals J =[τl ,τr ]⊂[0,δ] of length 2−n δ
of the integral operator. Here I(J ;α∗ ,β) is constructed by replacing dz∗τ (x) ∧ dzτ (x) −z∗τ (x)zτ (x) e 2πi Cs (x;α∗ ,β)
by
Cs (x;α∗ ,β)
dz∗τ (x) ∧ dzτ (x) −Re z (x)z (x) ∗τ τ e 2πi
in the formula for I(J ;α∗ ,β) of part (i). The integral operator IS integrates over the fields ατ , z∗τ , zτ with τ ∈ εZ ∩ (0, δ), where ε = 2−depth(S) δ. We introduce the shorthand notation (2.8) α = ατ )τ ∈εZ∩(0,δ) , z = zτ )τ ∈εZ∩(0,δ) , z∗ = z∗τ )τ ∈εZ∩(0,δ) for these “residual” fields. In Theorem 2.18, we give an estimate on the integral operators IS . 2.4. The Background Field In (1.17), we described the change of the quadratic part of the effective interaction after one decimation step. We iterate this procedure and are led to explicit, but relatively complicated expressions for the quadratic part of the effective action at a given scale. To organize the description of the quadratic part and also of the dominant quartic part, we introduce “background fields”. The effective action depends on the fields ατ both directly and through their complex conjugates, but is an analytic function if we treat the complex conjugates as independent variables. Consequently we introduce new complex fields α∗τ that will often be evaluated at ατ∗ . Definition 2.9 (The background field). Let S be a hierarchy for scale δ. Set ε = ∗ = (α∗τ (x))τ ∈εZ∩(0,δ) 2−n δ with the integer n ≥ depth(S). Given fields α∗ , β, α and α = (ατ (x))τ ∈εZ∩(0,δ) , we define the background fields7 for S by x∈X
Γ∗S (τ ; α∗ , α ∗ ) = Γ0∗τ (S) α∗ +
x∈X
Γτ∗τ (S) α∗τ
τ ∈εZ∩(0,δ) 7
For each fixed τ ∈ (0, δ), α∗ and α ∗ , the background field Γ∗S (τ ; α∗ , α ∗ ) is a function of x ∈ X.
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ΓS (τ ; α , β) =
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Γττ (S) ατ + Γδτ (S) β
τ ∈εZ∩(0,δ)
For τ ∈ (0, δ) and decimation points τ ∈ [0, δ], the coefficients Γτ∗τ (S) = Γτ∗τ , with τ = δ, and Γττ (S) = Γττ , with τ = 0, are defined as follows: ◦ For τ = τ ∈ (0, δ), Γτ∗τ = Γττ = Λcτ
◦
◦
Here, we use the following notation. If Y is a subset of X, the operator “multiplication by the characteristic function of Y ” is also denoted by Y . For τ = τ , Γτ∗τ = 0 unless τ > τ and [τ , τ ] is strictly contained in a decimation interval with τ as its left endpoint.8 If J is the smallest such decimation interval and δ its length, then δ δ τ Γ∗τ = j τ − τ − Λ(J ) j Λcτ 2 2
Similarly for τ = τ , Γττ = 0 unless τ < τ and [τ, τ ] is strictly contained in a decimation interval with τ as its right endpoint. If J is the smallest such interval and δ its length, then δ δ Γττ = j τ − τ − Λ(J ) j Λcτ 2 2
Remark 2.10. The Definition 2.9, of the background field, is independent of the choice of integer n ≥ depth(S). To see this, let εS = 2−depth(S) δ. The only place in the definition where ε appears is in the range of summation c τ ∈εZ∩(0,δ) . If τ ∈ (εZ\εS Z) ∩ (0, δ), then d(τ ) > depth(S) so that Λτ = ∅
and Γττ (S) = Γτ∗τ (S) = 0. The dominant contributions to the quadratic part of the effective action associated to the hierarchy S for scale δ will be QS (α∗ , β; α ∗, α ) = Qε,δ (α∗ , β; Γ∗S ( · ; α∗ , α ∗ ) , ΓS ( · ; α , β) ) −depth(S)
with ε = 2
(2.9)
δ, where
Qε,δ (α∗ , β; γ∗ , γ ) γ∗τ , γτ − α∗ , j(ε)γε − = τ ∈εZ∩(0,δ)
γ∗τ , j(ε) γτ +ε
τ ∈εZ∩(0,δ−ε)
− γ∗ δ−ε , j(ε) β 8
This implies that d(τ ) > d(τ ) whenever τ is a decimation point. Observe that there is a maximal decimation interval with τ as its left endpoint. If τ = 0, it is [τ , τ + 2−d(τ ) δ]. If τ = 0 it is [0, δ].
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γ∗τ , γτ − j(ε) γτ +ε
τ ∈εZ∩(0,δ−ε)
+ γ∗ δ−ε , γδ−ε − j(ε) β = γ∗ε − j(ε) α∗ , γε +
γ∗τ − j(ε) γ∗τ −ε , γτ
τ ∈εZ∩(ε,δ)
− γ∗ δ−ε , j(ε) β
(2.10)
The dominant part to the quartic part of the effective action will be VS (α∗ , β; α ∗, α ) δ ∗ )ΓS (τ ; α , β), v Γ∗S (τ ; α∗ , α ∗ )ΓS (τ ; α , β) = − dτ Γ∗S (τ ; α∗ , α 0
(2.11) The contributions characteristic of the small field set ΩS ([0, δ]) are not being integrated over. Therefore we set Qres ∗, α ) = QS (α∗ , β; α ∗, α ) − α∗ , j(Ω) (δ)β S (α∗ , β; α res VS (α∗ , β; α ∗, α ) = VS (α∗ , β; α ∗, α ) − VΩ;δ (α∗ , β)
(2.12)
2.5. Norms 1 Our main result will be, that for sufficiently small 0 < θ ≤ kT , the effective density can be represented in the form |Ω| ∗ ∗ ∗ Zθ eα , j(Ω) (θ)β +VΩ;θ (α ,β)+DΩ;θ (α ,β) χθ (Ω; α, β) Iθ (α∗ , β) = Ω⊂X
×
I(S;α∗ ,β)
S hierarchy for scale θ ΩS =Ω
) ( res ∗ ∗ res ∗ ∗ ∗ ∗ × e−QS (α ,β; α , α)+VS (α ,β; α , α) eBS (α ,β; ρ )+LS (α ,β; ρ ) (2.13) Qres S
res VS
In this formula, VΩ;θ , and are explicit functions; their definitions have been given in (2.4), (2.10), (2.11) and (2.12). Observe that they are evaluated with α∗ = α∗ . The pure small field part DΩ;θ has been constructed in , z∗ , z that (2.2). The functions LS and BS depend on the “residual fields” α are the integration variables of IS . Again we choose to write them as analytic functions of ρ = ( α∗ , α , z∗ , z) ∗
as well as α and β. When they appear inside the integral operator we evaluate them at α∗ , α , z∗ , z) α ∗ = α∗ ρ = ( z∗τ (x)=zτ (x)∗ for x∈Λ(Jτ )\(R(Jτ )∩Ω(Jτ ))
The function LS (α∗ , β; ρ ) will be analytic in the fields and depends only on the values of the fields α∗τ (x), ατ (x) for points x ∈ X\Ω. It is called the “pure
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large field contribution”. The function BS (α∗ , β; ρ ) depends on the fields at points x both inside and outside Ω and is called the “boundary contribution”. In Proposition 2.1, we gave estimates on DΩ;θ , expressed in terms of the norms (1.25). The norms that we use to measure LS and BS are similar to the ones introduced in (1.25), but are more sophisticated. They weight the variables α∗τ (x), ατ (x) so as to take into account their maximum possible magnitudes on IS ’s domain of integration. The abstract framework for these norms was developed in [4, §II]. For the convenience of the reader, we review it. In Definition 2.13, we introduce the concrete weight factors used in this paper. Definition 2.11. (i) A weight factor on X is a function κ : X → (0, ∞]. (ii) Let n1 , . . . , ns be nonnegative integers and x1 ∈ X n1 , . . . , xs ∈ X ns . If δ is any metric on X, we define the tree size τδ (x1 , . . . , xs ) as the length (with respect to the metric δ) of the shortest tree in X whose set of vertices contains x1,1 , x1,2 , . . . , x1,n1 , . . . , xs,ns . (iii) For any subset Ω of X we construct a metric dΩ on X as follows: Denote ¯ the union of closed unit cubes centered at the points of Ω. For a by Ω curve γ in Rn we set ¯ + length γ ∩ (Rn \Ω) ¯ lengthΩ (γ) = 2 · length(γ ∩ Ω) where length is the ordinary length in X. For any two points x, y ∈ X define
dΩ (x, y) = inf m lengthΩ (γ) γ a curve joining x to y where m is the “mass” introduced just before (1.25). Clearly, m d ≤ dΩ ≤ 2m d
(2.14)
Recall that d is the standard metric on X. If Ω ⊂ Ω ⊂ X and the set S = {x1,1 , x1,2 , . . . , x1,n1 , . . . , xs,ns } contains both a point of Ω and of X\Ω then ¯ Rn \Ω ¯ ) (2.15) τdΩ (x1 , . . . , xs ) ≤ τdΩ (x1 , . . . , xs ) − m dist(Ω, where, for subsets U, V of Rn
dist(U, V ) = inf length(γ) γ a curve joining a point of U to a point of V Definition 2.12. Let φ1 , . . . , φs be a collection of fields on X. (i) Let f (φ1 , . . . , φs ) be a function which is defined and analytic on a neighbourhood of the origin in Cs|X| . Then f has a unique expansion of the form a(x1 , . . . , xs ) φ1 (x1 ) · · · φs (xs ) f (φ1 , . . . , φs ) = n1 ,...,ns ≥0 ( x1 ,..., xs )∈X n1 ×···×X ns
with the coefficients a(x1 , . . . , xs ) invariant under permutations of the components of each vector xj . The functions a(x1 , . . . , xs ) are called the (symmetric) coefficient system for f . (ii) For any n1 , . . . , ns ≥ 0 and any function b(x1 , . . . , xs ) on X n1 ×· · ·×X ns , we define the norm b n1 ,...,ns as follows:
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s If there is at least one field, that is if j=1 nj = 0, then |b(x1 , . . . , xs )| b n1 ,...,ns = max max max x∈X 1≤j≤s 1≤i≤nj x ∈X n nj =0 1≤≤s ( xj )i =x
◦
For the constant term, that is if
s
j=1
nj = 0,
b n1 ,...,ns = |b(−, · · · , −)| (iii) Given weight factors κ1 , . . . , κs , and a metric δ on X, the weight system with metric δ that associates the weight factor κj to the field φj is defined by wδ (x1 , . . . , xs ) = eτδ ( x1 ,..., xs )
nj s
κj (xj, )
j=1 =1
for all (x1 , . . . , xs ) ∈ X n1 × · · · × X ns and all nonnegative integers n1 , . . . , ns . If Ω is a subset of X, the weight system with core Ω that associates the weight factor κj to the field φj (and the weight factor one to the history field) is wdΩ . (iv) Let f (φ1 , . . . , φs ) be a function which is defined and analytic on a neighbourhood of the origin in Cs|X| and a the symmetric coefficient system of f . We define the norm, with weight w, of f to be w(x1 , . . . , xs ) a(x1 , . . . , xs ) n1 ,...,ns f w = n1 ,...,ns ≥0
The functions BS (α∗ , β; ρ ) and LS (α∗ , β; ρ ) in (2.13) depend on the = ( α∗ , α , z∗ , z) that fields α∗ , β and, in addition, on the residual fields ρ are integrated over in the large field integral operator IS . The weight factors that we associate to these variables depend on the functions r(t) and R(t) introduced before. Recall that r(t) measures the size of the region close to a critical point where the stationary phase construction at scale t is performed (see the Introduction just after (1.10)). R(t) is the threshold between “large” and “small” fields for scale t, see the beginning of Sect. 2.2. Definition 2.13 (Weight factors). Let S be a hierarchy for scale δ. (i) We define the weight factor κ∗S,0 for the field α∗ by κ∗S,0 (x)
# " = min 2 R(t) x ∈ Λ ([0, t]) such that [0, t] is a decimation interval
and, for τ a decimation point in (0, δ), the weight factor κ∗S,τ for the field α∗τ by κ∗S,τ (x) ∞ " # = min R(t) x ∈ Λ([τ, τ +t]), [τ, τ +t] a decimation interval
if x ∈ Λτ otherwise
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Similarly we define the weight factor κS,δ for the field β by # " κS,δ (x) = min 2 R(t) x ∈ Λ ([δ − t, δ]) , [δ−t, δ]a decimation interval and, for τ a decimation point in (0, δ), the weight factor κS,τ for the field ατ by κS,τ (x) ∞ " # = min R(t) x ∈ Λ([τ −t, τ ]), [τ −t, τ ]a decimation interval
if x ∈ Λτ otherwise
The significance of the “∞” lines is the following. If d(τ ) ≤ depth(S) and x ∈ Λτ , then the integration variables α∗τ (x), ατ (x) have been replaced by the integration variable z∗τ (x), zτ (x) during the decimation step for τ . Thus α∗τ (x), ατ (x) no longer appear as arguments. If d(τ ) > depth(S), that is if α∗τ and ατ do not appear as integration variables in IS at all, then Λτ = X and κ∗,τ (x) = κτ (x) = ∞ for all x. The spatial decay of these weight factors is discussed in Appendix B. (ii) We define weight factors λτ for the “residual” fields z∗τ , zτ , τ a decimation point in (0, δ) by 32 r(2−d(τ ) δ) if x ∈ Λτ \Ω (Jτ ) λS,τ (x) = ∞ otherwise (iii) We denote by wS the weight system with core ΩS that associates the weight factor κ∗S,0 to the field α∗ , the weight factor κS,δ to the field β, and, for τ ∈ (0, δ), the weight factors κ∗S,τ , κS,τ , λS,τ and λS,τ to the fields α∗τ , ατ , z∗τ and zτ , respectively. We will generally write · S in place of · wS . Our main result (Theorem 2.16 below) states, that, under suitable assumptions on the functions R(t) and r(t) the decomposition (2.13) of Iθ exists, and gives bounds on BS S and LS S . 2.6. Summary and Statement of the Main Theorems We are studying many particle systems of Bosons on the finite lattice X whose single particle Hamiltonianh is of the form h = ∇∗ H∇ with a translation invariant, strictly positive operator H : L2 (X ∗ ) → L2 (X ∗ ). For our construction, we assume that there are constants 0 < cH < CH such that all of its eigenvalues lie between cH and CH . Also we assume that e6 md(x,0) |H (bi (0), bj (x))| < ∞ (2.16) DH = x∈X 1≤i,j≤d
where m is the mass used in (2.5). Here, for each 1 ≤ i ≤ D and x ∈ X, bi (x) = x, x + ei ) denotes the bond with base point x and direction ei . The interactions v(x, y) we allow are assumed to be translation invariant, repulsive
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and exponentially decaying in that the norm e5m d(x,y) |v(x, y)| |||v||| = sup x∈X
y∈X
introduced in (2.5), is sufficiently small. We discuss the system at temperature 1 > 0 and chemical potential μ. T = kβ The representation (2.13) of the effective density that we want to achieve depends on • the functions R(t) and R (t) that implement the large field conditions at scale t. • the function r(t) that gives the size of the region near the critical point at scale t where stationary phase is applied. • the function c(t) that measures the size of the “corridors” in the hierarchies. • the constant c that measures the size of the cut off of the single particle operator that is needed for the analyticity in Stokes’ argument (see (1.24)). • a constant v > 0 that measures, roughly speaking, the size of the interaction v. The precise conditions relating v and v are given in Hypothesis 2.14, below. • a constant cv > 0 that, roughly speaking, imposes a lower bound on the smallest eigenvalue, v1 , of v, viewed as the kernel of a convolution operator acting on L2 (X). Again, see Hypothesis 2.14, below. Clearly v1 ≤ v ≤ |||v|||. • the chemical potential μ Hypothesis 2.14. The two-body potential v(x, y) is a real, symmetric, translation invariant function on X × X that obeys 1 1 v ≤ |||v||| ≤ v 4 2
and
v1 ≥ cv |||v|||
For our construction, we fix strictly positive exponents er , eR , eR and eμ that obey 1 ≤ 4eR + 2er 2(eR + er ) < eμ ≤ 1 1 ≤ eR eR + er < 1 (2.17) 2 and a constant Kμ > 0. We make the particular, v–dependent, choices er eR eR 1 1 1 r(t) = R(t) = r(t) R (t) = r(t) tv tv t 1 1 c(t) = log2 (2.18) c = log2 v tv and assume that 3eR + 4er < 1
|μ| ≤ Kμ veμ
(2.19)
Example 2.15. Natural choices for eR and eR are eR = 14 , eR = 12 . It is also natural to choose eμ bigger than, but close to 12 .
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We are working with a Riemann sum approximation to the quartic
1term of in (1.1) which is, roughly speaking, (−2 times) a sum over τ ∈ εZ ∩ 0, kT
ε ατ∗ ατ , v ατ∗ ατ ≥ εv1 |ατ (x)|4 ≥ εv1 R(ε)4 x ∈ X |ατ (x)| > R(ε) x∈X
The coefficient εv1 R(ε)4 = vv1 εvR(ε)4 would be exactly vv1 , which is of order one, for the choice eR = 14 , if er were zero. We think of er as being small. Similarly, the h term in (1.1) is, roughly speaking, a sum over τ ∈ εZ ∩ 1 [0, kT ] of (minus)
ε ∇ατ L2 (X ∗ ) ≥ εR (ε)2 b ∈ X ∗ |∇ατ (b)| > R (ε) The coefficient εR (ε)2 would be exactly one, for the choice eR = 12 , if er were zero. The combined v and μ terms in (1.1) are, roughly speaking, a sum over
1 of (−ε times) τ ∈ εZ ∩ 0, kT 1 1 ∗ ατ ατ , v ατ∗ ατ − μ ατ 2 ≥ v1 |ατ (x)|4 − μ|ατ (x)|2 2 2 x∈X 2 1 μ μ2 2 = v1 |ατ (x)| − − 2 v1 2v1 x∈X
≥−
μ2 2v1
x∈X
With this choice of eμ , the right hand side is small. Making eμ larger would make the right hand side smaller, but would also make the critical value, vμ1 , of |ατ (x)| smaller too. This is not desirable for generating symmetry breaking in the infrared regime. With these choices, (2.17) is satisfied if er > 0 is small enough. Our main theorems are Theorems 2.16 and 2.18, below. They are proven in Sect. 3.8. Theorem 2.16. There is a constant K, such that for all sufficiently9 small v, θ > 0, the limit Iθ (α∗ , β) = limm→∞ Im (2−m θ; α∗ , β) exists and has the representation |Ω| ∗ ∗ ∗ Zθ eα , j(Ω) (θ)β +VΩ;θ (α ,β)+DΩ;θ (α ,β) χθ (Ω; α, β) Iθ (α∗ , β) = Ω⊂X
×
S hierarchy for scale θ ΩS =Ω
I(S;α∗ ,β)
) ( res ∗ ∗ res ∗ ∗ ∗ ∗ × e−QS (α ,β; α , α)+VS (α ,β; α , α) eBS (α ,β; ρ )+LS (α ,β; ρ ) with the following properties. 9
See Hypothesis F.7.
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VΩ;θ and DΩ;θ are the functions defined in (2.4) and (2.2). Furthermore, DΩ;θ can be decomposed in the form DΩ;θ (α∗ , β) = RΩ;θ (α∗ , β) + EΩ;θ (α∗ , β) with a function RΩ;θ (α∗ , β) that is bilinear in α∗ and β, is independent10 of the interaction v, and fulfills the estimate RΩ;θ 2R(θ), 2m ≤ K e−2mc θ2 r(θ)2 R(θ)2 ≤ K θ vm log v 1
and a function EΩ;θ (α∗ , β) that has degree at least two11 both in α∗ and in β and fulfills the estimate 2 |||v||| EΩ;θ 2R(θ), 2m ≤ K θ2 |||v|||2 r(θ)2 R(θ)6 ≤ K v ◦
χθ (Ω; α, β) is the characteristic function imposing the small field conditions. It is one if |α(x)|, |β(x)| ≤ R(θ) for all x ∈ Ω and |∇α(b)|, |∇β(b)| ≤ R (θ) for all bonds b on X that have at least one end in Ω and |α(x) − β(x)| ≤ r(θ) for all x within a distance one from Ω and it is zero otherwise ) is an analytic function of ◦ For each hierarchy S for scale θ, BS (α∗ , β; ρ its arguments and fulfills the estimate |||v||| BS S ≤ K θ |||v||| r(θ) R(θ)3 ≤ K v ◦ For each hierarchy S for scale θ, LS has the decomposition LS (J ; α∗ , β; ρ ) LS (α∗ , β; ρ ) = decimation intervals J ⊂[0,θ]
where, for each decimation interval J ⊂ [0, θ], the function LS (J ; α∗ , β; ρ ) is an analytic function of its arguments that is “large field with respect to the interval J ” (that is, it depends only on values of the fields at points x ∈ X\ΩS (J ) and depends only on the variables α∗τ , ατ , z∗τ , zτ with τ ∈ J . (If τ = 0 ∈ J , then replace α∗0 (x) by α∗ (x). If τ = θ ∈ J , then replace αθ (x) by β(x) ) and fulfills the estimate |||v||| (vδ)1−3eR −4er LS (J ; · ) S ≤ K δ |||v||| r(δ) R(δ)3 = K v where δ is the length of the time interval J . ◦ The functions RΩ;θ , EΩ;θ , BS and LS (J ) are all invariant under α∗ → e−iθ α∗ , β → eiθ β, ρ = ( α∗ , α , z∗ , z) → (e−iθ α ∗ , eiθ α , e−iθ z∗ , eiθ z). ◦ For each hierarchy S for scale θ, IS is the integral operator of Definition 2.8. Its properties are described in Theorem 2.18, below. 10
RΩ;θ is constructed like DΩ;θ , but with v = 0. By this we mean that every monomial appearing in the power series expansion of these functions contains a factor of the form α∗ (x1 ) α∗ (x2 ) β(x3 ) β(x4 ). 11
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Definition 2.17. Let Ω ⊂ X and θ, v > 0. We define the technical small field regulator (2)
(4)
RegSF (Ω; α, β) = RegSF (Ω; α, β) + RegSF (Ω; α, β) with
(2) RegSF (Ω; α, β) = Kreg θ|μ| α 2Ω + β 2Ω # " " (4) α − β L4 (Ω) RegSF (Ω; α, β) = Kreg θv α 3L4 (Ω) + β 3L4 (Ω) ˜ + θ μ+e−5mc(θ) α L4 (Ω) ˜ # 1 1 + θ ∇α1L4 (Ω˜ ∗ ) + ∇β 1L4 (Ω˜ ∗ ) + β L4 (Ω) ˜
˜ is the set of points of X that are within a distance c(θ) of Ω, and Here Ω
Kreg = 29 exp 20e12m DDH . In addition to the constants of (2.17), we choose 0 < e < 2er and set e 1 (t) = (2.20) tv Theorem 2.18. Let Ω ⊂ X and assume that α and β obey the small field conditions χθ (Ω; α, β) = 1. Then, e− 2 α 1
×
2
− 12 β2 Re (α∗ , j(Ω) (θ)β +VΩ;θ (α∗ ,β)) −RegSF (Ω;α,β)
e
e ) ( res I(S;α∗ ,β) eRe (−Qres S +VS )+|BS |+|LS |
hierarchies S for scale θ with ΩS =Ω
− 14 (θ)|Ωc |
≤e
x∈Ωc
1 1 1 + |α(x)|3 1 + |β(x)|3
In the above theorem, 0 1 1 ◦ the factor x∈Ωc 1+|α(x)| 3 1+|β(x)|3 on the right hand side ensures the left hand side, which is a function of {α(x), β(x)}x∈X is integrable, c ◦ the factor e−(θ)|Ω | on the right hand side tells us that when the large field region Ωc is large, then the left hand side is small and A stronger bound than that of Theorems 2.18 is given in (3.22).
3. Strategy of the Proof Our proof of Theorem 2.16 goes roughly as follows. In a first step, we fix ε = 2−k θ for some k ∈ N and show that there is a representation for Ik (ε; α∗ , β) similar to (2.13), but with a sum over hierarchies for scale θ which have depth at most k. In a second step we compare the resulting representations for Ik (2−k θ; α∗ , β) and Ik+1 (2−(k+1) θ; α∗ , β) in order to take the limit k → ∞. For the first step, we use the decimation strategy sketched in the introduction. We construct, for n = 1, 2, . . . , k a representation of In (ε; α∗ , β) similar
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to (2.13), but with a sum over hierarchies for scale 2n ε = 2n−k θ. In the step (1.10) from n to n + 1, In+1 (ε; α∗ , β) = dμR(ε) (φ∗ , φ) In (ε; α∗ , φ)In (ε; φ∗ , β) write the representation of In (ε; α∗ , φ) resp. In (ε; φ∗ , β) as a sum over such hierarchies S1 resp. S2 and write the integral as a sum of integrals, indexed by (S1 , S2 ). One such integral leads to the sum of terms in the representation of In+1 that are associated to the hierarchies for scale 2n+1 ε = 2n−k+1 θ that are preceded by (S1 , S2 ) in the following sense. Definition 3.1. A pair (S1 , S2 ) of hierarchies for scale δ is said to precede the hierarchy S for scale 2δ if SS (J ) = SS1 (J ) and SS (δ + J ) = SS2 (J ) for all S = Λ, Ω, Pα , Pβ , Pα , Pβ , Q, R and all decimation intervals J in [0, δ]. In this case, we write (S1 , S2 ) ≺ S We also denote, for any field α = (ατ (x))τ ∈εZ∩(0,2δ) , the left and right half x∈X
fields to be α l = (ατ (x))τ ∈εZ∩(0,δ) x∈X
α r = (ατ +δ (x))τ ∈εZ∩(0,δ) x∈X
Given hierarchies S1 , S2 for scale δ, the choice of a hierarchy S with (S1 , S2 ) ≺ S amounts to the choice of (i) the small/large field sets of the first kind Pα ([0, 2δ]) ,
Pβ ([0, 2δ]) ⊂ ΩS1 ∩ ΩS2
Pα
Pβ ([0, 2δ]) ⊂ (ΩS1 ∩ ΩS2 )
([0, 2δ]) ,
∗
Q ([0, 2δ]) ⊂ (ΩS1 ∩ ΩS2 ) (ii)
By Definition 2.4, these sets determine Λ ([0, 2δ]). the large field set of the second kind R([0, 2δ]) ⊂ Λ([0, 2δ]). Again, by Definition 2.4, this set determines the new small field set of the second kind Ω([0, 2δ]). In a decimation step as outlined above, we start with two hierarchies S1 , S2 for scale δ. Then we first decompose ΩS1 ∩ ΩS2 into large/small field sets of the first kind, and afterwards decompose the resulting small field sets Λ according to the choice of the regions where “stationary phase” is applied. To formalize the first step, we use
Definition 3.2. Let Ω0 ⊂ X and δ > 0. We denote by Fδ (Ω0 ) the set of all choices of “small/large field sets of the first kind” A = (Λ, Pα , Pβ , Pα , Pβ , Q) with ◦ Λ, Pα , Pβ ⊂ Ω0 , Pα , Pβ ⊂ Ω∗0 and Q ⊂ Ω 0
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( ) # x ∈ X d x, Pα ∪ Pβ ∪ Q ∪ supp Pα ∪ supp Pβ ∪ Ωc0 > c(δ)
3.1. History Fields As described in the previous section, the decimation step from scale δ to scale 2δ involves integrals of products of pairs of terms like in (2.13), indexed by two hierarchies of S1 , S2 for scale δ. The result of this integral will be represented as a sum over all hierarchies S of scale 2δ that are preceded by (S1 , S2 ). Each such term should contain a factor ∗
eα
, j(ΩS ) (2δ)β +VΩS ;2δ (ε; α∗ ,β)+DΩS ;2δ (ε; α∗ ,β)
For the construction of DΩS ;2δ (ε; ·, · ) out of DΩS1 ;δ (ε; ·, · ) and DΩS2 ;δ (ε; ·, · ) we need to know which contributions to DΩSi ;δ (ε; ·, · ) involved points outside the new small field region ΩS . To keep track of precisely which points were involved in the construction of each contribution, we introduce the concept of a history field. This is a field on X that takes only the values 0 and 1. In particular h2 = h The history field is never integrated over. We put in a history field at each point where some construction is performed. That is, we shall always work with “history complete” functions in the following sense. Definition 3.3. (i) A function f (φ1 , . . . , φs ; h) in the fields φ1 , . . . , φs , h is called history complete if it is in fact a function of φ1 h, · · · , φs h, h. If f is a history complete analytic function, any non trivial monomial in its power series expansion that contains a factor φi (x) automatically also contains a factor h(x). (ii) Given a power series f (φ1 , . . . , φs ; h) and a subset Ω of X we set f Ω = f (φ1 , . . . , φs ; h) h(x)=0 for x∈X\Ω
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If f is history complete, f Ω depends only on the fields φ1 (x), . . . , φs (x), h(x) with x ∈ Ω. The starting points of our construction are the single particle Hamiltonian h = ∇∗ H∇ and the interaction v. Already at this point, we have to monitor the points in space involved in the construction, for example in the exponential j(t) = et(h−μ) . This is the motivation for the notion of h-operator introduced in [4, Definition IV.1]. For the convenience of the reader, we repeat some of the concepts introduced in [4]. Definition 3.4. (i) An h-operator or h-linear map A on CX is a linear operator on CX whose kernel is of the form ∞ A(x; x1 , . . . , x ; y) h(x) h(x1 ) · · · h(x ) h(y) A(x, y) = =0 (x1 ,...,x )∈X
(ii) The composition A ◦ B of two h–operators A, B on CX is by definition the h–operator with kernel (A ◦ B)(x, y) = A(x, z) B(z, y) z∈X
=
A(x; x1 , . . . , x ; z) B(z; y1 , . . . , y ; y)
z∈X , ≥0 x1 ,...,x y1 ,...y
×h(x) h(x1 ) · · · h(x ) h(z) h(y1 ) · · · h(y ) h(y) Here we used that h2 = h. (iii) For an “ordinary” linear operator J on CX with kernel J(x, y), we define the associated h–operator by ¯ y) = h(x) J(x, y) h(y) J(x, and the associated h-exponential as ∞ 1 ¯ J = hehJh = ehJh h exph (J) = h + ! =1
The h-exponential obeys the product rule exph (J1 ) exph (J2 ) = exph (J1 + J2 ), provided the operators J¯1 and J¯2 commute. (iv) If φ is any field on X and A an h-operator, we set A(x, y) φ(y) (Aφ)(x) = =
y∈X ∞
A(x; x1 , . . . , x ; y) h(x) h(x1 ) · · · h(x ) h(y) φ(y)
=0 x1 ,...,x ,y
To keep the notation simple, we shall frequently use the same symbol for a history complete function or h-operator as we used in Sect. 2 for the function or operator one gets if one sets h(x) = 1 for all x ∈ X. So we shall again write v for the operator v¯, and set j(t) = exph (−t(h − μ))
(3.1)
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j(t) h(x)=0 for
x∈X\Ω h(x)=1 for x∈Ω
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= j(Ω) (t)
with the operator j(Ω) (t) introduced at the beginning of Sect. 2.1. Again we define 1 if d(x, y) ≤ c (3.2) jc (τ )(x, y) = j(τ )(x, y) · 0 if d(x, y) > c 3.2. Properties of the Background Fields Here, and through the rest of the paper, use the same definition as before for background fields (that is Definition 2.9, just with j(t) being interpreted as the h–operator of (3.1). Here, we want to study some of their properties. In particular, we develop a recursion relation (Proposition 3.6). Remark 3.5. (The structure of the background fields) Let S be a hierarchy for scale δ. (i) When the history field is identically zero, the background field becomes ∗ ) h=0 = Λcτ α∗τ ΓS (τ ; α , β) h=0 = Λcτ ατ Γ∗S (τ ; α∗ , α The differences Γ∗S (τ ; α∗ , α ∗ ) − Λcτ α∗τ and ΓS (τ ; α , β) − Λcτ ατ are history complete. Their restrictions to the small field region ΩS are ∗ ) − Λcτ α∗τ = j(τ )α∗ , Γ∗S (τ ; α∗ , α ΩS ΩS c ΓS (τ ; α , β) − Λτ ατ = j(δ − τ )β ΩS
ΩS
d(τ )
(ii) Let τ = δ k=1 a2kk , ak ∈ {0, 1}, be the “binary expansion” of the decimation point τ ∈ (0, δ), and let τ ∈ (0, δ) be another decimation point different from τ . Then Γτ∗τ = 0 unless τ is one of the “binary approx d ak imations” δ k=1 2k (1 ≤ d < d(τ ), ad = 1) of τ . In this case, let d = min k > d ak = 1 . Then δ δ δ Γτ∗τ = j τ − τ − d Λ [τ , τ + d −1 ] j Λcτ 2 2 2d
In particular, Γτ∗τ = 0 whenever τ > τ or d(τ ) > d(τ ). Analogous statements hold for Γττ , in terms of the binary expansions of τ δ − τ and δ − τ . In particular, Γτ = 0 whenever τ < τ or d(τ ) > d(τ ). (iii) If τ = τ , then Γτ∗τ and Γττ are always of the form j(τ1 )Λ1 j(τ2 )Λc2 with τ1 , τ2 ≥ 0, τ1 + τ2 = |τ − τ | and Λ1 and Λc2 being (possibly trivial) characteristic functions. (iv) Let τ ∈ (0, δ) and τ ∈ [0, δ] be decimation points with d(τ ) > d(τ ). (For δ τ = 0, δ, set d(τ ) = 0.) Furthermore let 0 < t ≤ 2d(τ ) . If τ is not of
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the form τ + 2δd for any d(τ ) ≤ d ≤ depth(S), then Γτ∗τ = j(t) Γτ∗τ −t . If τ is not of the form τ − 2δd for any d(τ ) ≤ d ≤ depth(S), then Γττ = j(t) Γττ +t . For more information, see Lemma E.16. (v) If d(τ ) > depth(S) then Γτ∗τ = Γττ = 0 for all τ ∈ (0, δ).
Proof. (i) Since j(t) vanishes when h is identically zero, Γτ∗τ |h=0 = Γττ |h=0 = 0 for all τ = τ . For all τ ∈ (0, δ), ΩS ∩ Λcτ = ∅ and hence Γτ∗τ |ΩS = Γττ |ΩS = 0 whenever τ = τ . Finally, δ δ δ δ Γ0∗τ Ω = j τ − = j τ − = j(τ ) Ω ΩS j j Ω Ω S S S S 2 2 2 2 and Γδτ = j(δ − τ ) . ΩS
ΩS
(ii) Assume that Γτ∗τ = 0. Denote by J the smallest decimation interval with τ as its left endpoint that strictly contains [τ , τ ]. As J is a decimation interval, there is d ≥ d(τ ) + 1 such that J = [τ , τ + 2dδ−1 ]. Since [τ , τ + 2δd ] is also a decimation interval, but does not contain τ in its interior, we have τ + 2δd ≤ τ < τ + 2dδ−1 . d Set d = d(τ ) < d(τ ) and let τ = δ k=1 2bkk , bk ∈ {0, 1} be the “binary expansion” of τ . As 2δd ≤ τ − τ < 2dδ−1 and d > d we have ak = bk for k = 1, . . . , d,
ad = 1,
ad+1 = · · · = ad −1 = 0,
ad = 1
So τ is a binary approximation of τ . The remaining claims follow directly from the Definition 2.9 of the background fields. (iii) follows by inspection. Recall that j(0) = h. (iv) We consider Γτ∗τ . By part (ii) we may assume that τ > τ . Observe that δ [τ , τ + 2d(τ ) ] is the maximal decimation interval with τ as left endpoint. δ δ If τ does not lie in this interval, then τ − t ≥ τ − 2d(τ ) ≥ τ + d(τ ) , and 2 δ consequently Γτ∗τ = Γτ∗τ −t = 0. So we may assume that τ ∈ (τ , τ + 2d(τ ) ). δ Let J = [τ , τ + 2d−1 ], d > d(τ ) be the smallest decimation interval with δ τ as left endpoint such that [τ , τ ] J . Then τ + 2δd ≤ τ < τ + 2d−1 . δ δ δ If τ = τ + 2d then, again, τ − t ≥ τ − 2d(τ ) ≥ τ + 2d and consequently δ δ δ Γτ∗τ −j(t) Γτ∗τ −t = j τ −τ − d −j(t)j τ −t−τ − d Λ(J ) j Λcτ 2 2 2d =0
If τ = τ +
Γτ∗τ
δ , 2d
but d > depth(S), then Λ(J ) = X and δ = Λ(J ) j Λcτ = j(τ − τ ) Λcτ , Γτ∗τ −t = j(τ − t − τ ) Λcτ d 2
(v) In this case Λcτ = ∅. In the decimation step from scale δ to scale 2δ, we are passing from the product of the terms indexed by hierarchies S1 , S2 for scale δ to a sum of terms indexed by hierarchies S that are preceded by (S1 , S2 ) in the sense of Definition 3.1. For this reason, we need
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Proposition 3.6. (Recursion relation for the background fields) Let (S1 , S2 ) be a pair of hierarchies for scale δ that precede the hierarchy S for scale 2δ. Let τ ∈ (0, 2δ) be a decimation point. Then Γ∗S (τ ; α∗ , α ∗) ⎧ ⎪ ∗l ) ⎨Γ∗S1 (τ ; α∗ , α = ΛS j(δ)α∗ + ΛcS α∗δ ⎪ ⎩ ∗r ) + ∂Γ∗τ α∗ Γ∗S2 (τ −δ; ΛS j(δ)α∗ +ΛcS α∗δ , α
if τ ∈ (0, δ) if τ = δ if τ ∈ (δ, 2δ)
Here, for τ ∈ (δ, 2δ), the difference operator ∂Γ∗τ is defined as follows. Let J be the smallest decimation interval for S with left endpoint δ such that [δ, τ ] is strictly contained in J , and let δ be the length of J . Then δ δ ∂Γ∗τ = j τ − δ − ΛS (J )c j ΛS j(δ) 2 2 Similarly,
⎧ ⎪ l , ΛS j(δ)β + ΛcS αδ ) + ∂Γτ β ⎨ΓS1 (τ ; , α ΓS (τ ; α , β) = ΛS j(δ)β + ΛcS αδ ⎪ ⎩ r , β) ΓS2 (τ − δ; α
if τ ∈ (0, δ) if τ = δ if τ ∈ (δ, 2δ)
Here, for τ ∈ (0, δ), the difference operator ∂Γτ is defined as follows. Let J be the smallest decimation interval for S with right endpoint δ such that [τ, δ] is strictly contained in J , and let δ be the length of J . Then δ δ ∂Γτ = j δ − τ − ΛS (J )c j ΛS j(δ) 2 2
Proof. If τ ∈ (0, δ) and 0 ≤ τ ≤ τ is a decimation point, then Γτ∗τ (S) = Γτ∗τ (S1 ) by Definition 2.9. By Remark 3.5(ii), Γτ∗τ (S) = 0 and Γτ∗τ (S1 ) = 0 whenever τ > τ . If τ = δ then Γ∗S (τ ; α∗ , α ∗ ) = ΛS j(δ)α∗ + ΛcS α∗δ directly by Definition 2.9. Now let τ ∈ (δ, 2δ). Directly from Definition 2.9, ⎧ 0 c ⎪ ⎨Γ∗τ −δ (S2 ) ΛS if τ = δ τ τ −δ Γ∗τ (S) = Γ∗τ −δ (S2 ) if δ < τ < 2δ ⎪ ⎩ 0 if 0 < τ < δ so that Γ∗S (τ ; α∗ , α ∗ ) − Γ∗S2 (τ −δ; ΛS j(δ)α∗ + ΛcS α∗δ , α ∗r ) = Γ0∗τ (S)α∗ − Γ0∗τ −δ (S2 )ΛS j(δ)α∗
Let J be the interval defined in the Proposition, and set J = t − δ t ∈ J . Then ΛS (J ) = ΛS2 (J ) and Γ0∗τ (S) − Γ0∗τ −δ (S2 ) ΛS j(δ) δ δ = j(τ − δ) ΛS j(δ) − j τ − δ − ΛS2 (J j ΛS j(δ) 2 2
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ΛS2 (J ) j
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ΛS j(δ) = ∂Γ∗τ
Here, we used j(τ − δ) = j(τ − δ − δ2 ) (ΛS2 (J )c + ΛS2 (J )) j( δ2 ). The recursion representation for ΓS is proven similarly.
Further properties and alternative definitions of the background fields are given in Appendix E. 3.3. The Explicit Quadratic and Quartic Terms in the Effective Action In (2.9), we defined the prospective dominant contribution to the quadratic part of the effective action associated to a hierarchy S for scale δ to be QS (α∗ , β; α ∗, α ) = Qε,δ (α∗ , β; Γ∗S ( · ; α∗ , α ∗ ) , ΓS ( · ; α , β) )
(3.3)
where Qε,δ was defined in (2.10) and ε was chosen to be 2−depth(S) δ. Here, and through the rest of the paper, we use the same definition but with j(t) being interpreted as the h-operator of (3.1) and with the background fields Γ∗S , ΓS of Sect. 3.2. Part (i) of the following Lemma shows that, in (3.3), we could have chosen ε = 2−n δ for any n ≥ depth(S). Part (ii) isolates the pure small field part and the history–independent part. Part (iii) gives a recursion relation. Lemma 3.7.
(i) Let S be a hierarchy at scale δ. For all k, n ≥ depth(S) Q
δ 2k
,δ
(α∗ , β; Γ∗S ( · ; α∗ , α ∗ ) , ΓS ( · ; α , β) )
=Q
δ 2n
,δ
(α∗ , β; Γ∗S ( · ; α∗ , α ∗ ) , ΓS ( · ; α , β) )
(ii) Let S be a hierarchy at scale δ. When the history field is identically zero, QS (α∗ , β; α ∗, α ) h=0 = Λcτ α∗τ , Λcτ ατ τ ∈εZ∩(0,δ)
where ε = 2 δ. The difference QS (α∗ , β; α ∗, α )− τ Λcτ α∗τ , Λcτ ατ is history complete. Its restriction to the small field region ΩS is ∗, α ) Ω − Λcτ α∗τ , Λcτ ατ = −α∗ , j(δ)β Ω QS (α∗ , β; α −depth(S)
S
τ ∈εZ∩(0,δ)
S
(iii) Let S1 , S2 be hierarchies of scale δ such that (S1 , S2 ) ≺ S, where S is a hierarchy of scale 2δ. Then QS (α∗ , β; α ∗, α ) = QS1 (α∗ , Λj(δ)β + Λc αδ ; α ∗l , α l) c + QS2 (Λj(δ)α∗ + Λ α∗δ , β; α ∗r , α r) + Λj(δ)α∗ , Λj(δ)β + Λc α∗δ , Λc αδ Γ∗S1 (τ ; α∗ , α ∗l ), (∂Γτ − j(ε)∂Γτ +ε ) β + τ ∈εZ∩[0,δ)
+
τ ∈εZ∩(0,δ]
(∂Γ∗δ+τ − j(ε)∂Γ∗δ+τ −ε ) α∗ , ΓS2 (τ ; α r , β)
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where Λ = ΛS , ε = 2−depth(S) and the differences ∂Γ∗τ , ∂Γτ were defined in Proposition 3.6 for τ ∈ (δ, 2δ) and τ ∈ (0, δ), respectively. Here, we have also set ∂Γ∗δ = ∂Γ∗2δ = ∂Γ0 = ∂Γδ = 0 and Γ∗S1 (0; ·) = α∗ , ΓS2 (δ; ·) = β. (iv) Let S be a hierarchy of scale δ. Then for any fields φ∗ , φ QS (α∗ , Λj(δ)β + Λc φ ; α ∗, α ) − QS (α∗ , Λjc (δ)β + Λc φ; α ∗, α ) & δ ' δ Γ∗S (τ ; α∗ , α ∗ ), Γτ − j(ε)Γτ +ε Λ (j(δ) − jc (δ)) β = τ ∈εZ∩[0,δ)
∗, α ) − QS (Λjc (δ)α∗ + Λc φ∗ , β; α ∗, α ) QS (Λj(δ)α∗ + Λc φ∗ , β; α & ' 0 0 Γ∗τ − j(ε)Γ∗τ −ε Λ (j(δ) − jc (δ)) α∗ , ΓS (τ ; α , β) = τ ∈εZ∩(0,δ]
where Λ = ΛS and, again, ε = 2−depth(S) . Here, we have also set Γ0∗δ = Γδ0 = 0 and Γ∗S (0; · ) = α∗ , ΓS (δ; · ) = β and, correspondingly, Γ0∗0 = Γδδ = 1. Proof. (i) It suffices to prove this in the case that k > depth(S) and n = k − 1. To simplify notation, set ε = 2−k δ, Γ∗S (τ ; α∗ , α ΓS (τ ; α ∗ ) if τ = 0 , β) if τ = δ γ∗τ = γτ = if τ = 0 α∗ β if τ = δ If τ, τ are decimation points with d(τ ) = k and d(τ ) ≤ depth(S), then by Remark 3.5(iv), Γτ∗τ = j(ε) Γτ∗τ −ε . Combining this with Remark 3.5(v), we see that γ∗τ = j(ε) γ∗τ −ε
if d(τ ) = k
So by (2.10) Qε,δ (α∗ , β; γ∗ , γ ) γ∗τ − j(ε) γ∗τ −ε , γτ − γ∗ δ−ε , j(ε) β = τ ∈εZ∩(0,δ)
=
γ∗τ − j(ε) γ∗τ −ε , γτ − j(ε) γ∗ δ−2ε , j(ε) β
τ ∈2εZ∩(0,δ)
=
γ∗τ − j(ε)j(ε) γ∗τ −2ε , γτ − γ∗ δ−2ε , j(ε)j(ε) β
τ ∈2εZ∩(0,δ)
= Q2ε,δ (α∗ , β; γ∗ , γ ) (ii) Define γ∗τ and γτ as above. By Remark 3.5(i), j(ε)γ∗τ −ε h=0 = 0 j(ε)γ∗τ −ε Ω = j(τ )α∗ Ω S
S
Both equations follow from the definition (2.10) and Remark 3.5(i).
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(iii) Let ε = 2−depth(S) (2δ) = 2−(depth(S)−1) δ. Define γ∗τ and γτ as above, but with δ replaced by 2δ. By Proposition 3.6 ⎧ (1) ⎪ if τ ∈ (0, δ) ⎨γ∗τ c γ∗τ = Λj(δ)α∗ + Λ α∗δ if τ = δ ⎪ ⎩ (2) γ∗τ −δ + ∂Γ∗τ α∗ if τ ∈ (δ, 2δ) ⎧ (1) ⎪ if τ ∈ (0, δ) ⎨γτ + ∂Γτ β c γτ = Λj(δ)β + Λ αδ if τ = δ ⎪ ⎩ (2) γτ −δ if τ ∈ (δ, 2δ) where Λ = ΛS and (1)
γτ(1) = ΓS1 (τ ; α l , Λj(δ)β + Λc αδ )
(2)
γτ(2) = ΓS2 (τ ; α r , β)
γ∗τ = Γ∗S1 (τ ; α∗ , α ∗l )
γ∗τ = Γ∗S2 (τ ; Λj(δ)α∗ +Λc α∗δ , α ∗r ) Then
QS =
(1)
(1)
γ∗τ − j(ε) γ∗ τ −ε , γτ(1) + ∂Γτ β
τ ∈εZ∩(0,δ) (1)
+ Λj(δ)α∗ + Λc α∗δ − j(ε) γ∗ δ−ε , Λj(δ)β + Λc αδ 2 3 (2) + γ∗ε + ∂Γ∗δ+ε α∗ − j(ε)(Λj(δ)α∗ + Λc α∗δ ), γε(2) +
2 3 (2) (2) γ∗τ + ∂Γ∗δ+τ α∗ − j(ε)(γ∗τ −ε + ∂Γ∗δ+τ −ε α∗ ), γτ(2)
τ ∈εZ∩(ε,δ)
2 3 (2) − γ∗ δ−ε + ∂Γ∗ 2δ−ε α∗ , j(ε)β
=
(1)
(1)
(1)
γ∗τ −j(ε) γ∗ τ −ε , γτ(1) − j(ε) γ∗ δ−ε , Λj(δ)β + Λc αδ
τ ∈εZ∩(0,δ)
2 3 (2) + γ∗ε − j(ε)(Λj(δ)α∗ + Λc α∗δ ), γε(2)
+
3 2 3 2 (2) (2) (2) γ∗τ −j(ε)γ∗τ −ε , γτ(2) − γ∗ δ−ε , j(ε)β
τ ∈εZ∩(ε,δ)
+Λj(δ)α∗ , Λj(δ)β + Λc α∗δ , Λc αδ (1) (1) γ∗τ −j(ε) γ∗ τ −ε , ∂Γτ β + τ ∈εZ∩(0,δ)
2 3 + ∂Γ∗δ+ε α∗ , γε(2) +
∂Γ∗δ+τ α∗ −j(ε)∂Γ∗δ+τ −ε α∗ , γτ(2)
'
τ ∈εZ∩(ε,δ)
− ∂Γ∗ 2δ−ε α∗ , j(ε)β
∗l , α
l )+QS2 (Λj(δ)α∗ +Λc α∗δ , β; α
∗r , α
r) = QS1 (α∗ , Λj(δ)β +Λc αδ ; α +Λj(δ)α∗ , Λj(δ)β + Λc α∗δ , Λc αδ
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3 2 (1) γ∗τ , (∂Γτ − j(ε)∂Γτ +ε ) β
τ ∈εZ∩[0,δ)
+
2
(∂Γ∗δ+τ − j(ε)∂Γ∗δ+τ −ε ) α∗ , γτ(2)
3
τ ∈εZ∩(0,δ]
with ∂Γ0 = ∂Γδ = ∂Γ∗δ = ∂Γ∗2δ = 0. (iv) The first equality follows from (2.10) and the fact that , Λj(δ)β + Λc φ ; α ) − ΓS (τ ; α , Λjc (δ)β + Λc φ ; α ) ΓS (τ ; α = Γδτ Λ (j(δ) − jc (δ)) β The second equality follows from (2.10) and the fact that ∗ ) − Γ∗S (τ ; Λjc (δ)α∗ + Λc φ∗ ; α ∗) Γ∗S (τ ; Λj(δ)α∗ + Λc φ∗ ; α = Γ0∗τ Λ (j(δ) − jc (δ)) α∗ The quartic part of the interaction for a given decimation step depends on the scale at which the process has been started. To keep track of this we need the Riemann sums approximating (2.11). Definition 3.8. (i) Let S be a hierarchy for scale δ, and let ε = 2−n δ with n ≥ depth(S). We define ∗, α ) = ε Vn (α∗ , β; Γ∗S ( · ; α∗ , α ∗ ) , ΓS ( · ; α , β) ) VS (ε; α∗ , β; α where
⎡
Vn (α∗ , β; γ∗ , γ ) = − ⎣α∗ γε , v α∗ γε +
γ∗τ γτ +ε , v γ∗τ γτ +ε
τ ∈εZ∩(0,δ−ε)
⎤
+ γ∗ δ−ε β, v γ∗ δ−ε β ⎦ and Γ∗S (τ ; α∗ , α ∗ ), ΓS (τ ; α , β) are the background fields. (ii) In the remark below we shall identify the small field part of VS . For that purpose, we define for any subset Ω of X and every δ that is an integer multiple of ε VΩ,δ (ε; α∗ , β) [j(τ )α∗ ] [j(δ − τ − ε)β] , v [j(τ )α∗ ] [j(δ − τ − ε)β] = −ε τ ∈εZ∩[0,δ)
Ω
(Evaluating this at h = 1 gives the VΩ,δ (ε; α∗ , β) of (2.1).) Clearly, VS is history complete (in particular VS h=0 = 0). It follows from Remark 3.5(i) that Remark 3.9. (i) If ε = 2−n δ with n ≥ depth(S) then VS (ε; α∗ , β; α ∗, α ) Ω = VΩ,δ (ε; α∗ , β) for all Ω ⊂ ΩS . (ii) VΩ,δ (ε; α∗ , j(δ)β) + VΩ,δ (ε; j(δ)α∗ , β) = VΩ,2δ (ε; α∗ , β)
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3.4. Properties of the Large Field Integral Operator In our representation of In (ε; α∗ , β), we persist in using the integral operator I(S;α∗ ,β) of Definition 2.8, except that the surface Cs (x; α∗ , β) is now h–dependent through the operators jc (s) of (2.7). Note that the characteristic functions χJ (α, ατ , β) and the cutoff Gaussian measures do not depend on h. Lemma 3.10. (i) Let (S1 , S2 ) be a pair of hierarchies for scale δ that precede the hierarchy S for scale 2δ. We have the recursion relation I(S;α∗ ,β) [dμ( α, z∗ , z)] = I([0,2δ],S ; α∗ ,β) I(S1 ;α∗ ,αδ ) [dμ( αl , z∗l , zl )] I(S2 ;α∗δ ,β) [dμ( αr , z∗r , zr )] (ii) Let S be a hierarchy for scale δ and set ε = 2−depth(S) δ. The large α, z∗ , z)] depends on α(y) and β(y) only for field operator I(S;α∗ ,β) [dμ( y ∈ ΩcS . Proof. Part (i) is trivial. To prove part (ii) by induction, it suffices to show that I([0,2δ],S;α∗ ,β) depends on α(y) and β(y) only for y ∈ Ω([0, 2δ])c . The dependence of I([0,2δ],S;α∗ ,β) on α(y) and β(y) arises only through two mechanisms. First, through the integration domain Cδ (x; α∗ , β) with x ∈ R([0, 2δ]). By construction, this integration domain depends only on α(y) and β(y) for y within a distance c of R([0, 2δ]). Since Ω([0, 2δ]) is the set of all points of Λ([0, 2δ]) whose distance from R([0, 2δ]) is at least c(δ), any such y is in Ω([0, 2δ])c . The second dependence is through the characteristic function χ[0,2δ] (α, αδ , β). One sees by direct inspection of its definition in Appendix A that this characteristic function is independent of α(y) and β(y) for all y ∈ Ω([0, 2δ]). Remark 3.11. Let S be a hierarchy for scale δ and set ε = 2−depth(S) δ. The α, z∗ , z)] acts on the space of functions that are integral operator I(S;α∗ ,β) [dμ( defined and measurable in the variables zτ (x) z∗τ (x), zτ (x) ατ (x)
x ∈ Λ(J )\ (R(J ) ∪ Ω(J )) x ∈ R(J ) x ∈ Λ(J )c
with|zτ (x)| ≤ r(s) with|z∗τ (x)|, |zτ (x)| ≤ R(s) with|ατ (x)| ≤ R(ε)
and analytic in the variables z∗τ (x), zτ (x), x ∈ R(Jτ ). Here J = Jτ runs over all decimation intervals in [0, δ] of length 2ε ≤ 2s ≤ δ. Proof. The first two restrictions appear explicitly in the limits of integration in Definition 2.8. The restriction |ατ (x)| ≤ R(ε) is an immediate consequence of Lemma A.4(a). We shall prove a bound on these large field integral operators, with h ≡ 1, in Theorem 2.18. To obtain a good bound, we make a specific choice of Cs (x; α∗ , β). Roughly speaking, each point x ∈ R(J ) will provide a very small factor for the size of the integral over Cs (x; α∗ , β) which arises from the factor e−z∗τ (x)zτ (x) . This will be established in Proposition 3.38, below.
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3.5. Norms and the Renormalization Group Map In our description of the effective density in Theorem 2.16, the non-explicit quantities DΩS , BS and LS are estimated with the help of the norms · S . These norms were defined abstractly in Definition 2.12 and then made concrete by the choice of the weight factors in Definition 2.13. In the construction itself, all functions involve history fields. For this reason, we extend, in Definition 3.12 below, the abstract definition of norms to the case of functions that depend on a history field too. In this abstract setting, we recall the main theorem from [4, §II] that will allow us to control the fluctuation integrals (Theorem 3.14. A fluctuation integral is performed for every triple S1 , S2 , S of hierarchies with (S1 , S2 ) ≺ S. (See the discussion before Definition 3.1.) Therefore, we discuss, in Remark 3.17, how the weight factors for S1 , S2 and S are related. In each decimation step, the fluctuation integral is introduced through coordinates that are centred on the critical point of the quadratic part of the effective interaction. This change of variables affects our norms and is controlled using the operator norm of Definition 3.18, below. Definition 3.12. Let φ1 , . . . , φs be a collection of fields on X and let h be a history field on X. (i) Let f (φ1 , . . . , φs ; h) be a function which is defined and analytic on a neighbourhood of the origin in C(s+1)|X| . Then f has a unique expansion of the form f (φ1 , . . . , φs ; h) = n1 ,...,ns+1 ≥0 ( x1 ,..., xs+1 )∈X n1 ×···×X ns+1
×a(x1 , . . . , xs ; xs+1 )φ1 (x1 ) · · · φs (xs )h(xs+1 ) with the coefficients a(x1 , . . . , xs ; xs+1 ) invariant under permutations of the components of each vector xj . The functions a(x1 , . . . , xs ; xs+1 ) are called the (symmetric) coefficient system for f . (ii) For any n1 , . . . , ns+1 ≥ 0 and any function b(x1 , . . . , xs ; xs+1 ) on X n1 × · · · × X ns+1 , we define the norm b n1 ,...,ns+1 as follows: s ◦ If there is at least one nonhistory field, that is if j=1 nj = 0, then |b(x1 , . . . , xs ; xs+1 )| b n1 ,...,ns+1 = max max max x∈X 1≤j≤s 1≤i≤nj x ∈X n nj =0 1≤≤s+1 ( xj )i =x
◦
Here (xj )i is the ith component of the nj -tuple xj . s If there are only history fields, that is if j=1 nj = 0, but ns+1 = 0, then we take the pure L1 norm |b(−, · · · , −; xs+1 )| b n1 ,...,ns+1 = xs+1 ∈X ns+1
◦
Finally, for the constant term, that is if
s+1 j=1
b n1 ,...,ns+1 = |b(−, · · · , −)|
nj = 0,
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(iii) Given weight factors κ1 , . . . , κs , and a metric d on X, the weight system with metric d that associates the weight factor κj to the field φj (and the weight factor one to the history field) is defined by τd ( x1 ,..., xs+1 )
wd (x1 , . . . , xs ; xs+1 ) = e
nj s
κj (xj, )
j=1 =1
for all (x1 , . . . , xs ; xs+1 ) ∈ X n1 × · · · × X ns × X ns+1 and all nonnegative integers n1 , . . . , ns+1 . If Ω is a subset of X, the weight system with core Ω that associates the weight factor κj to the field φj (and the weight factor one to the history field) is wdΩ . The metric dΩ was specified in Definition 2.11(iii). (iv) Let f (φ1 , . . . , φs ; h) be a function which is defined and analytic on a neighbourhood of the origin in C(s+1)|X| and a the symmetric coefficient system of f . We define the norm, with weight w, of f to be f w = w(x1 , . . . , xs , xs+1 ) a(x1 , . . . , xs , xs+1 ) n1 ,...,ns+1 n1 ,...,ns+1 ≥0
Remark 3.13. Let f (φ1 , . . . , φs ; h) be a power series and Ω a subset of X. (i) For any weight system w 1 1 1f (φ1 , . . . , φs ; h) h(x)=1
1 1
for x∈Ω 1 w h(x)=0 for x∈X\Ω
1 1 ≤ 1f Ω 1w ≤ f w
(ii) For any measure dμ(φs ) that is independent of h, dμ(φs ) f (φ1 , . . . , φs ; h) dμ(φs ) f (φ1 , . . . , φs ; h) Ω =
Ω
Theorem III.4 of [4, §II] uses norms as in Definitions 2.12 and 3.12 to control a renormalization group step. For the reader’s convenience, we repeat its statement as well as the main result of [4, Corollary III.5] here. Theorem 3.14. Let f (φ1 , . . . , φs ; z∗ , z; h) be a function which is defined and analytic on a neighbourhood of the origin in C(s+3)|X| . Let r ≥ 0 and denote by dμ(z ∗ , z) the measure dμ(z ∗ , z) =
x∈X
χ (|z(x)|) ≤ r) e−z(x)
∗
z(x)
dz ∗ (x) ∧ dz(x) 2πı
Furthermore let κ1 , . . . , κs+2 be weight factors such that κs+1 (x), κs+2 (x) ≥ 4r for all x ∈ X, and let m ≥ 0 and Ω ⊂ X. Denote by w the weight system with core Ω that associates the weight factor κj to the field φj , and the weight factors κs+1 , κs+2 to the fields z ∗ and z respectively. 1 If f w < 16 , then there is an analytic function g(φ1 , . . . , φs ) such that f (φ ,...,φ ;z∗ ,z) s e 1 dμ(z ∗ , z) (3.4) = eg(φ1 ,...,φs ) ef (0,...,0;z∗ ,z) dμ(z ∗ , z)
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and g w ≤
f w 1 − 16 f w
1 If, in addition, f w < 32 and the symmetric coefficient system a(x1 , . . . , xs ; y∗ , y; x) of f obeys a(x1 , . . . , xs ; y∗ , y; x) = 0 whenever y = y∗ , then
g w ≤
f w 1 20 − f w
2 (3.5)
Definition 3.15. (i) We also use the weight system wS of Definition 2.13(iii) for functions that depend on the history field, giving weight one to the history field as in Definition 3.12(iii). We again write · S for · wS . (ii) For any κ, m > 0, we use wκ,m to denote the weight system with metric m d the associates the constant weight factor κ to the fields α∗ and β and the weight factor 1 to the history field. The norm f (α∗ , β ; h) κ,m = f wκ,m . Remark 3.16. For a function f (α∗ , β) that depends only on α∗ (x) and β(x) with x ∈ ΩS , f 2R(δ), m ≤ f S ≤ f 2R(δ), 2m For the decimation step, we need Remark 3.17 (Weight factor recursion relation). Let (S1 , S2 ) be a pair of hierarchies for scale δ that precede the hierarchy S for scale 2δ. Then weight factors of Definition 2.13 obey ⎧ 2 R(2δ) = R(2δ) ⎪ R(δ) κ∗S1 ,0 (x) ⎪ ⎪ ⎪ ⎪ κ∗S1 ,0 (x) ⎪ ⎪ ⎪ ⎨ κ∗S1 ,τ (x) κ∗S,τ (x) = ⎪ ∞ ⎪ ⎪ ⎪ ⎪ ⎪ 21 κ∗S2 ,0 (x) ⎪ ⎪ ⎩ κ∗S2 ,τ −δ (x)
if τ = 0, x ∈ ΛS if if if if if
τ = 0, x ∈ / ΛS 0<τ <δ τ = δ, x ∈ ΛS τ = δ, x ∈ / ΛS δ < τ < 2δ
and ⎧ 2 R(2δ) = R(2δ) ⎪ R(δ) κS2 ,δ (x) ⎪ ⎪ ⎪ ⎪ κS2 ,δ (x) ⎪ ⎪ ⎪ ⎨ κS2 ,τ −δ (x) κS,τ (x) = ⎪ ∞ ⎪ ⎪ ⎪ ⎪ ⎪ 21 κS1 ,δ (x) ⎪ ⎪ ⎩ κS1 ,τ (x)
if τ = 2δ, x ∈ ΛS if if if if if
τ = 2δ, x ∈ / ΛS δ < τ < 2δ τ = δ, x ∈ ΛS τ = δ, x ∈ / ΛS 0<τ <δ
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⎧ λS1 ,τ (x) ⎪ ⎪ ⎪ ⎨32 r(δ) λS,τ (x) = ⎪ ∞ ⎪ ⎪ ⎩ λS2 ,τ −δ (x)
if if if if
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0<τ <δ τ = δ, x ∈ ΛS \ΩS τ = δ, x ∈ / ΛS \ΩS δ < τ < 2δ
We will deal with linear changes in the φ-fields which may be compositions of several such changes of variables. For the convenience of the reader we repeat the [4, Definition IV.2] of the operator norms we use. Definition 3.18 (Weighted L1 –L∞ operator norms). Let κ, κ : X → (0, ∞] be weight factors and δ an arbitrary metric and let A be an h-linear map on CX . We define the operator norm 1 1 1 1 Nδ (A; κ, κ ) = 1 A(x; x; y) h(x)βl (x) h(x) h(y)βr (y)1 ω
x,y∈X x∈X (1)
where ω is the weight system with metric δ that associates the weight κ1 to βl , the weight κ to βr and the weight 1 to h. ¯ κ, κ ). For an ordinary operator J on CX , we set Nδ (J; κ, κ ) = Nδ (J; Observe that if J is multiplication by the characteristic function of a set Y , then κ (x) (3.6) Nδ (J; κ, κ ) = sup x∈Y κ(x) In [4, Remark IV.3], we gave a more explicit reformulation of the definition of Nδ (A; κ, κ ). The main change of variables formula [4, Proposition IV.4] is Proposition 3.19. Let Aj , 1 ≤ j ≤ s, be h-operators on CX , and let f (φ1 , . . . , φs ; h) be an analytic function on a neighbourhood of the origin in C(s+1)|X| . Define f˜ by f˜(φ1 , . . . , φs ; h) = f (A1 φ1 , . . . , As φs ; h) Let κ1 , . . . κs , κ ˜1, . . . κ ˜ s be weight factors. Denote by w and w ˜ the weight sys˜j tems with metric δ that associate to the field φj the weight factor κj and κ respectively. ˜ j ) ≤ 1 for 1 ≤ j ≤ s, then If Nδ (Aj ; κj , κ f˜ w˜ ≤ f w In [4] and Appendix G, we also state variants and corollaries of this Proposition. Most of the times, the metric δ used in Definition 3.18 will be a multiple of the standard metric d. Therefore we introduce the notation Nμ (A; κ, κ ) = Nμd (A; κ, κ )
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for any μ ≥ 0, any h–operator A and any weight factors κ, κ . By (2.14), for any Ω ⊂ X NdΩ (A; κ, κ ) ≤ N2m (A; κ, κ )
(3.7)
Specifically, we shall use Definition 3.20. |||A||| = N5m (A; 1, 1) This is consistent with the definition of |||v||| of (2.5). The most important operators for our considerations are the propagator j(t) = exph (−t(h − μ)) introduced in (3.1), and the cutoff propagator 1 if d(x, y) ≤ c jc (t)(x, y) = j(t)(x, y) · 0 if d(x, y) > c introduced in (3.2). They fulfill the estimates Lemma 3.21. Set Kj = N6m (h − μ; 1, 1) For all t ≥ 0, (i) |||j(t)||| ≤ N6m (j(t); 1, 1) ≤ eKj t (ii) |||j(t) − h||| ≤ N6m (j(t) − h; 1, 1) ≤ tKj eKj t (iii) |||jc (t) − j(t)||| ≤ tKj eKj t e−m c . Proof. These estimates follow directly from parts (ii), with A = j(t) − h, and (iv) of Remark G.4. Further estimates on j(t) are given in Appendix D. 3.6. Decimation The first step in our proof of Theorem 2.16 is the construction of the “large field/ small field” decomposition of In (ε; α∗ , β)(n = 0, 1, . . . , [log2 θ/ε]) for fixed ε > 0. The recursion step for this construction is Theorem 3.26, below. Recall that we are assuming that the two–body potential v satisfies Hypothesis 2.14, and in particular that |||v||| ≤ 12 v. As in Sect. 2.1, the small field parts of the term associated to a hierarchy will depend only on the small field set, not on the hierarchy. Its description is similar to that in Sect. 2.1. First, we introduce the analogue of the renormalization group map RΩ,δ for history complete functions: Definition 3.22. Let Ω ⊂ X. If f1 (α∗ , β; h), f2 (α∗ , β; h), V (α∗ , β; h) are history complete functions that are supported in Ω (i.e.fi Ω = fi ) we define
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¯ Ω;t (V ; f1 , f2 )(α∗ , β; h) R =
f1 (α∗ , jc (t)β) + f2 (jc (t)α∗ , β) + log
195
dμΩ,r(t) (z ∗ , z) eA(α∗ ,β;z dμΩ,r(t) (z ∗ , z)
∗
,z)
− [j(t) − jc (t)]α∗ , [j(t) − jc (t)]β
+ V (α∗ , jc (t)β) − V (α∗ , j(t)β) + V (jc (t)α∗ , β) − V (j(t)α∗ , β)
Ω
where A(α∗ , β; z∗ , z) is [f1 (α∗ , z + jc (t)β) − f1 (α∗ , jc (t)β)] + [f2 (z∗ + jc (t)α∗ , β) − f2 (jc (t)α∗ , β)] + [j(t) − jc (t)]α∗ , z + z∗ , [j(t) − jc (t)]β + [V (α∗ , z + jc (t)β) − V (α∗ , jc (t)β)] + [V (z∗ + jc (t)α∗ , β) − V (jc (t)α∗ , β)] Since j(t) Ω h=1 = j(Ω) (t) and jc (t) Ω h=1 = j(Ω),c (t), ( ) ¯ Ω;t (V ; f1 , f2 ) = RΩ;t V h=1 ; f1 h=1 , f2 h=1 R h=1
Remark 3.23. If Ω ⊂ Ω ¯ Ω;t (V ; f1 , f2 ) ¯ Ω ;t V ; f1 , f2 = R R Ω Ω Ω
Ω
We control the small field part in one decimation step by the following Theorem 3.24. Set KR = 212 Kj2 and KE = 223 . There are constants Θ, v0 > 0 such that, for all δ ≤ 12 Θ and v ≤ v0 , the following holds for all Ω ⊂ X: Let R1 (α∗ , β; h ), R2 (α∗ , β; h ) and E1 (α∗ , β; h ), E 2 (α∗ , β; h ) be history complete functions that are supported on Ω (that is Ri Ω = Ri , Ei Ω = Ei for i = 1, 2) with the following properties ◦ R1 and R2 are both bilinear in α∗ and β and fulfill the estimates Ri 2R(δ),2m ≤ KR δ 2 r(δ)2 R(δ)2 e−2m c ◦
E1 and E2 both have degree at least two both in α∗ and in β and fulfill the estimates Ei 2R(δ),2m ≤ KE (δv)2 r(δ)2 R(δ)6
◦
◦
All four of these functions are invariant under α∗ → e−iθ α∗ , β → eiθ β. Let ε be a divisor of δ. Then there are history complete functions R(α∗ , β; h) and E(α∗ , β; h) that are supported on Ω such that ¯ Ω;δ (VΩ,δ (ε; · ); R1 + E1 , R2 + E2 ) = R + E R and which have the following properties: R is bilinear in α∗ and β and fulfills the estimates R 2R(2δ),2m ≤ KR (2δ)2 r(2δ)2 R(2δ)2 e−2m c ¯ Ω;δ (0; R1 , R2 ) and, in particular, Furthermore R is the quadratic part of R is independent of VΩ,δ (ε; · ), E1 and E2 .
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E has degree at least two both in α∗ and in β and fulfills the estimates E 2R(2δ),2m ≤ KE (2δv)2 r(2δ)2 R(2δ)6
◦
Both functions are invariant under α∗ → e−iθ α∗ , β → eiθ β.
The proof of Theorem 3.24 is similar to that of [5, Proposition III.3] and is given in Sect. 4. Remark 3.25. By the definitions (2.18) δ 2 r(δ)2 R(δ)2 e−2m c = (δv)2(1−eR −2er )
e−2m c v2
(δv)2 r(δ)2 R(δ)6 = (δv)2(1−3eR −4er ) Both quantities go to zero with δ by (2.17).
The last Theorem allows us to recursively control the “small field parts”. We now consider the full model. Theorem 3.26. Set KD = 235 e6Kj and KL = 248 e6Kj . There are constants Θ, v0 > 0 such that, for all δ ≤ 12 Θ and v ≤ v0 , the following holds: Let S1 and S2 be hierarchies for scale δ with summits Ω1 = ΩS1 and Ω2 = ΩS2 . Let ε = 2−n δ with n ≥ max{depth(S1 ), depth(S2 )}. Furthermore let D1 (α∗ , β; ρ; h), D2 (α∗ , β; ρ; h), b1 (α∗ , β; ρ; h) and b2 (α∗ , β; ρ ; h) be history complete functions with the following properties ◦ For i = 1, 2, Di (0, 0; 0; 0) = 0 and 1 1 1Di 1 ≤ 2−20 Di Si ≤ 1 Ω i Si ◦ The “pure large field parts” D1 Ωc and D2 Ωc vanish. On the other hand, 1 2 b1 and b2 are purely large field. That is, b1 = b1 c and b2 = b2 c . ◦
Ω1
Ω2
= ( α∗ , α , z∗ , z) → D1 , D2 are invariant under α∗ → e−iθ α∗ , β → eiθ β, ρ ∗ , eiθ α , e−iθ z∗ , eiθ z) (e−iθ α
Set, for i = 1, 2, |Ω |
Ii (α∗ , β) = Zδ i χδ (Ωi ; α, β)I(Si ;α∗ ,β) ( ) ∗ ∗ ∗ ∗ ∗ × e−QSi (α ,β; α , α)+VSi (ε;α ,β; α , α)+Di (α ,β;ρ ) bi (α∗ , β; ρ ) where χδ (Ωi ; α, β) is the characteristic function introduced in Theorem 2.16. Also set I(α∗ , β) = dμR(ε) (φ∗ , φ) I1 (α∗ , φ) I2 (φ∗ , β) Then I(α∗ , β) =
hierarchies S for scale 2δ (S1 ,S2 )≺S
|Ω |
Z2δ S χ2δ (ΩS ; α, β) I(S;α∗ ,β)
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( ∗ ∗ ∗ ∗ ∗ × e−QS (α ,β; α , α)+VS (ε; α ,β; α , α) eDS (α ,β; ρ ) ) ∗ b1 (α∗ , αδ ; ρl ) b2 (αδ∗ , β; ρr ) eLS (α ,β;ρ ) where, for each hierarchy S that is preceded by (S1 , S2 ) as in Definition 3.1, DS and L S are analytic history complete functions that fulfill ◦ The “pure large field part” DS c of DS vanishes. Also, ΩS
1 KD (2δv) r(2δ) R(2δ)3 + 214 e−mc(δ) ( D1 S1 + D2 S2 ) 2 +214 D1 |Ω1 S1 + D2 |Ω2 S2 The “pure small field part” DS Ω is determined by D1 Ω and D2 Ω S S S through ( ) ¯ Ω ;δ VΩ ;δ (ε; · ); D1 , D2 D S Ω = R S S ΩS ΩS S LS is “pure large field” (that is LS = LS Ωc ) and fulfills the estimate DS S ≤
◦
S
1 L S S ≤ KL (2δv) r(2δ) R(2δ)3 + 28 ( D1 S1 + D2 S2 ) 2 = ( α∗ , α , ◦ DS and L S are invariant under α∗ → e−iθ α∗ , β → eiθ β, ρ z∗ , z) → (e−iθ α ∗ , eiθ α , e−iθ z∗ , eiθ z) This Theorem allows the construction of Im (2−m θ; ·, · ) for each fixed m, and Theorem 3.24 provides control of its small field parts. See Propositions 3.32 and 3.29, with n = m, below. The second step in the proof of Theorem 2.16 is the comparison of Im (2−m θ; ·, ·) and Im+1 (2−(m+1) θ; ·, ·), leading to the limit m → ∞. To do this, we compare In (2−m θ; ·, ·) and In+1 (2−(m+1) θ; ·, ·) for 1 ≤ n ≤ m. For the pure small field part the essential comparison step is Theorem 3.27. Under the hypotheses of Theorem 3.24, assume that there is a ˜ 2 (α∗ , β; h ) and E˜1 (α∗ , β; h ), E˜2 (α∗ , β; h ) of history ˜ 1 (α∗ , β; h ), R second set R complete functions that have similar properties to R1 , R2 , E1 , E2 and are close to these functions. Precisely, we assume that ˜ 1 and R ˜ 2 are both bilinear in α∗ and β and fulfill the estimates ◦ R ˜ i 2R(δ),2m ≤ KR δ 2 r(δ)2 R(δ)2 e−2m c R ◦
E1 and E2 both have degree at least two both in α∗ and in β and fulfill the estimates E˜i 2R(δ),2m ≤ KE (δv)2 r(δ)2 R(δ)6 Let
) ( ( ) ¯ Ω;δ VΩ,δ ε ; · ; R ˜ 1 + E˜1 , R ˜ 2 + E˜2 = R ˜ + E˜ R 2 be the decomposition of Theorem 3.24. Then ˜ − R 2R(2δ),2m ≤ 2ΔR R
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E˜ − E 2R(2δ),2m ≤ ΔE + KΔ (2δv) r(2δ)2 R(2δ)2 ΔR + ε(2δv) R(2δ)4 where KΔ = 240 e10Kj and ) 1( ˜ ˜ 2 − R2 2R(δ),2m R1 − R1 2R(δ),2m + R ΔR = 2 ) 1( ˜ ΔE = E1 − E1 2R(δ),2m + E˜2 − E2 2R(δ),2m 2 For the full model the essential comparison step is Theorem 3.28. Under the hypotheses of Theorem 3.26, assume that there is a ˜b2 (α∗ , β; ρ; h ) of ˜ 1 (α∗ , β; ρ ˜ 2 (α∗ , β; ρ second set D ; h ), D ; h ), ˜b1 (α∗ , β; ρ; h ) and ˜ ˜ history complete functions such that Di Si ≤ 1 and Di Ω Si ≤ 2−20 . Set i ( ) ˜i ˜ |Ω | i ∗ −Q +V (ε/2; · )+D I˜i (α , β) = Zδ χδ (Ωi ; α, β) I(Si ;α∗ ,β) e Si Si bi ∗ ˜ I(α , β) = dμR( 2ε ) (φ∗ , φ) I˜1 (α∗ , φ) I˜2 (φ∗ , β) and let ˜ ∗ , β) I(α =
( ) ˜ ˜ |Ω | Z2δ S χ2δ (ΩS ; α, β) I(S;α∗ ,β) e−QS +VS (ε/2; · ) eDS ˜b1 ˜b2 eLS
hierarchies S for scale 2δ (S1 ,S2 )≺S
be its representation as in Theorem 3.26. Then 1 1 ( 1 1 ˜S1 ˜ 1 S 1DS − D 1 ≤ KD ε(2δv)r(2δ)R(2δ)3 + 216 e−mc(δ) D1 − D 1 2 S ) ˜ 2 S + D2 − D 2 1 1 1 1 1 1 ˜ 1 ) 1 ˜ 2 ) 1 (D +216 1(D1 − D + − D 1 1 1 2 Ω1 S Ω2 S 1 2 1 1 1 1 1 1LS − L˜ S 1 ≤ KL ε(2δv) r(2δ) R(2δ)3 2 ( S ) ˜ 1 S + D2 − D ˜ 2 S + 28 D1 − D 1 2 Theorems 3.24 and 3.27 will be proven in Sect. 4. Theorems 3.26 and 3.28 will be proven in Sect. 5. In the proof of Theorem 2.16, the analysis of the “pure small field part” can be treated almost independently from the rest. To do this, we define, as in Sect. 2.1 (but now with history complete functions) DΩ;0 (ε; α∗ , β) = 0 ¯ Ω;2n ε (VΩ;2n ε (ε; ·, · ); DΩ;n (ε; ·, · ), DΩ;n (ε; ·, · )) DΩ;n+1 (ε; α∗ , β) = R (3.8) By Remark 3.23
DΩ ;n (ε; · ) = DΩ;n (ε; · )
Ω
when Ω ⊂ Ω
(3.9)
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Proposition 3.29. There are constants Θ, v0 > 0 such that, for all θ ≤ Θ and v ≤ v0 , the following holds for all Ω ⊂ X: For each ε > 0 and each integer 1 ≤ n ≤ log2 θε the function DΩ;n (ε; · ) is the sum of a function RΩ;n (ε; · ) that is bilinear in α∗ and β and an analytic function EΩ;n (ε; · ) that has degree at least two both in α∗ and in β. They fulfill the estimates RΩ;n (ε; · ) 2R(2n ε),2m ≤ KR (2n ε)2 r(2n ε)2 R(2n ε)2 e−2m c EΩ;n (ε; · ) 2R(2n ε),2m ≤ KE (2n ε v)2 r(2n ε)2 R(2n ε)6 Furthermore 1 (ε 1 ; 1RΩ;n (ε; · ) − RΩ;n+1 2 1 (ε 1 ; 1EΩ;n (ε; · ) − EΩ;n+1 2
)1 1 · 1 ≤ KR 2n ε2 r(ε)2 R(ε)2 e−2m c 2R(2n ε),2m )1 (3.10) 1 · 1 ≤ 2KE (εv)2 r(ε)2 R(ε)6 n 2R(2 ε),2m
Proof. We first prove, by induction on n, the statement of the Proposition but with the second line of (3.10) replaced by 1 ( ε )1 1 1 ;· 1 1EΩ;n (ε; ·) − EΩ;n+1 2 2R(2n ε),2m 2 2 6 ≤ KE (εv) r(ε) R(ε) 4 n 5 1−2eR −4er k 2 k 2 k 6 +KΔ ε (2 εv) r(2 ε) R(2 ε) (3.11) k=1
This induction argument is similar to that of Theorems I.3 and I.4 in [5]. The induction starts with n = 0. Observe that RΩ;0 (ε; · ) = EΩ;0 (ε; · ) = 0 while
(ε ) ( ( ) ) ) ¯ Ω; ε VΩ, ε ε ; · ; 0, 0 ; · + EΩ;1 ;· =R 2 2 2 2 2 ε By Theorem 3.24, with δ = 2 , 1 ( ε )1 1 1 ;· 1 ≤ KR ε2 r(ε)2 R(ε)2 e−2m c 1RΩ;1 2 2R(ε),2m 1 ( ε )1 1 1 ;· 1 ≤ KE (εv)2 r(ε)2 R(ε)6 1EΩ;1 2 2R(ε),2m RΩ;1
(ε
For the induction step from n to n + 1 we apply Theorems 3.24 and 3.27 with δ = 2n ε ( ) ˜1 = R ˜ 2 = RΩ;n+1 ε ; ·, · R1 = R2 = RΩ;n (ε; ·, · ) R (ε 2 ) ˜ ˜ ; ·, · E1 = E2 = EΩ;n (ε; ·, · ) E1 = E2 = EΩ;n+1 2 By the inductive hypothesis and Theorem 3.24, ¯ Ω;2n ε (VΩ;2n ε (ε; ·, · ); RΩ;n (ε; ·, · ) + EΩ;n (ε; ·, · ), DΩ;n+1 (ε; α∗ , β) = R RΩ;n (ε; ·, · ) + EΩ;n (ε; ·, · ))
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has the decomposition DΩ;n+1 (ε; α∗ , β) = RΩ;n+1 (ε; α∗ , β) + EΩ;n+1 (ε; α∗ , β) where RΩ;n+1 (ε; · ) and EΩ;n+1 (ε; · ) have all of the required properties. Furthermore, by Theorem 3.27, 1 ( ε )1 1 1 ;· 1 1RΩ;n+1 (ε; · ) − RΩ;n+2 2 2R(2n+1 ε),2m 1 ( ε )1 1 1 ;· 1 ≤ 2 1RΩ;n (ε; · ) − RΩ;n+1 2 2R(2n ε),2m n+1 2 2 2 −2m c ≤ KR 2 ε r(ε) R(ε) e and 1 ( ε )1 1 1 ;· 1 1EΩ;n+1 (ε; · ) − EΩ;n+2 2 2R(2n+1 ε),2m 1 ( ε )1 1 1 ;· 1 ≤ 1EΩ;n (ε; · ) − EΩ;n+1 2 2R(2n ε),2m 1 ( ε )1 1 1 ;· 1 +KΔ (2δv) r(2δ)2 R(2δ)2 1RΩ;n (ε; · ) − RΩ;n+1 2 2R(2n ε),2m + ε(2δv) R(2δ)4 1 ( ε )1 1 1 ≤ 1EΩ;n (ε; · ) − EΩ;n+1 ;· 1 2 2R(2n ε),2m +KΔ (2δv) r(2δ)2 R(2δ)2 KR δε r(ε)2 R(ε)2 e−2m c + ε(2δv) R(2δ)4 1 ( ε )1 1 1 ;· 1 ≤ 1EΩ;n (ε; · ) − EΩ;n+1 2 2R(2n ε),2m −2m c 1 2 2 2 2 2 e 4 KR r(ε) R(ε) + R(2δ) +KΔ ε (2δv) r(2δ) R(2δ) 2 v 1 ( ε )1 1 1 ≤ 1EΩ;n (ε; · ) − EΩ;n+1 ;· 1 2 2R(2n ε),2m 1 2 2 2 4 +KΔ ε (2δv) r(2δ) R(2δ) + R(2δ) ε2eR +4er 4 n 5 2 2 6 1−2eR −4er k 2 k 2 k 6 (2 εv) r(2 ε) R(2 ε) ≤ KE (εv) r(ε) R(ε) + KΔ ε k=1
+KΔ ε1−2eR −4er (2δv)2 r(2δ)2 R(2δ)6 2
2
6
= KE (εv) r(ε) R(ε) + KΔ ε
1−2eR −4er
4n+1 k=1
5 k
2
k
2
k
6
(2 εv) r(2 ε) R(2 ε)
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For the fourth inequality, we used that, by (F.4b) and Hypothesis F.7(ii), 1 1 e−2m c 1 e−2m c 1 KR r(ε)2 R(ε)2 = 2eR +4er KR v1−2eR −4er ≤ 2eR +4er 2 v ε 2 v2 ε This completes the induction argument. It remains only to prove that (3.11) implies the second line of (3.10). Summing the geometric series in (3.11) gives 1 ( ε )1 1 1 ;· 1 1EΩ;n (ε; · ) − EΩ;n+1 2 2R(2n ε),2m KΔ ≤ KE (εv)2−6eR −8er + ε1−2eR −4er (2n εv)2−6eR −8er 1 − 2−(2−6eR −8er ) 217 e10Kj KE 1−2eR −4er n 2−6eR −8er = (εv)2−6eR −8er KE + ε (2 ) 1 − 2−(2−6eR −8er ) 217 e10Kj n 1−2eR −4er ≤ (εv)2−6eR −8er KE 1 + (2 ε) 1 − 2−(2−6eR −8er ) ≤ 2KE (εv)2−6eR −8er
by (2.17) and Hypothesis F.7(i). Corollary 3.30. For each subset Ω of X DΩ,θ (α∗ , β) = lim DΩ,m (2−m θ; α∗ , β) m→∞
12
exists.
It has a decomposition DΩ,θ = RΩ,θ + EΩ,θ
where ◦
RΩ;θ (α∗ , β) is bilinear in α∗ and β and fulfills the estimate 1 RΩ;θ 12R(θ),2m ≤ KR θ2 r(θ)2 R(θ)2 e−2m c
◦
EΩ;θ (α∗ , β) has degree at least two both in α∗ and in β and fulfills the estimate EΩ;θ 2R(θ),2m ≤ KE (θv)2 r(θ)2 R(θ)6
Proof. As 2n ε2 r(ε)2 R(ε)2 e−2m c = 2n (εv)2−2eR −4er (εv)2 r(ε)2 R(ε)6 = (εv)2−6eR −8er
e−2m c v2
and the exponents 2 − 2eR − 4er and 2 − 6eR − 8er are strictly positive, Proposition 3.29, with n = m, implies that the limits lim RΩ;m (2−m θ; · )
m→∞
lim EΩ;m (2−m θ; · )
m→∞
exist. 12
The convergence is with respect to the norm · 2R(θ),2m .
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We now describe the construction of the functions DΩ;θ , BS and LS of Theorem 2.16 from Theorems 3.24, 3.26, 3.27 and 3.28. In each of the four theorems mentioned above we assert the existence of constants v0 and Θ such that for all interactions and “times” bounded by v0 resp. Θ, the conclusions are true. Now choose v0 and Θ as the smallest of the constants from the three theorems. Fix an interaction v obeying Hypothesis 2.14. For each 0 < ε < Θ/2 and each natural number 1 ≤ n ≤ log2 Θε define, for each α and β obeying |α(x)|, |β(x)| ≤ R(ε) for all x ∈ X, the effective density In• (ε; α∗ , β) recursively by ∗ ∗ • ∗ 2|X| dμR(ε) (φ∗ , φ) ζε (α∗ , φ) eα , j(ε)φ +φ , j(ε)β I1 (ε; α , β) = Zε ∗
∗
∗
∗
×e−ε(α φ, v α φ +φ β, v φ β ) ζε (φ∗ , β) • • • In+1 (ε; α∗ , β) = dμR(ε) (φ∗ , φ) In (ε; α∗ , φ) In (ε; φ∗ , β)
(3.12)
with the Zε of Lemma 2.7. Remark 3.31. For each n ≥ 1 we have n • In (ε; α∗ , β) = Zε2 |X|
τ ∈εZ∩(0,2n ε)
× ζε (ατ −ε , ατ ) e
dμR(ε) (ατ∗ , ατ ) α∗ τ −ε ,j(ε)ατ
τ ∈εZ∩(0,2n ε]
−ε
α∗ τ −ε ατ
v α∗ τ −ε ατ
with α0 = α and α2n ε = β. Comparing this with (1.3), we see that n • = Zε2 |X| In (ε; α∗ , β) In (ε; α∗ , β) h=1
In Sect. 3.7, we shall prove that
• • Iθ (α∗ , β) = lim Im (2−m θ; α∗ , β) h=1 m→∞
exists. Using the initial condition in Lemma 2.7, it will then follow that Iθ (α∗ , β) = limm→∞ Im (2−m θ; α∗ , β) also exists and Iθ (α∗ , β) = Iθ (α∗ , β) •
Theorem 3.26 allows us to recursively construct a representation of In• (ε; · ) similar to that of Theorem 2.16. To do so, fix v ≤ v0 and an interaction v obeying Hypothesis 2.14. In the following, we use the constants KD and KL of Theorem 3.26. Proposition 3.32. For each ε, sufficiently small, and integer n ≥ 0 with 2n ε ≤ Θ, the effective density In• (ε; α∗ , β) has, for all α, β obeying supx∈X |α(x)|, supx∈X |β(x)| ≤ R(ε), a representation |Ω | • Z2n S In (ε; α∗ , β) = ε χ2n ε (ΩS ; α, β) S hierarchy for scale 2n ε of depth at most n
( ) ∗ ∗ ∗ ∗ ∗ ∗ I(S;α∗ ,β) e−QS (α ,β; α , α)+VS (ε; α ,β; α , α) eDS (ε; α ,β; ρ )+LS (ε; α ,β; ρ )
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Here, for each ε > 0, sufficiently small, and hierarchy S for scale 2n ε ≤ Θ, ) and LS (ε; α∗ , β; ρ ) are analytic funcwith depth at most n, DS (ε; α∗ , β; ρ tions that have the following properties ◦ The “pure large field part of DS (ε; · ) vanishes, that is DS c = 0. The ΩS
“pure small field part”
DS (ε; · )
ΩS
= DΩS ,n (ε; · )
as in (3.8) and Proposition 3.29. Also DS (ε; · ) S ≤ KD (2n ε v) r(2n ε) R(2n ε)3 ◦
The “pure large field part” LS has the decomposition LS (J , ε; α∗ , β; ρ ) LS (ε; α∗ , β; ρ ) = decimation intervals J for [0,2n ε] of length at least 2n−depth(S)+1 ε
where, for each decimation interval J in the sum, the function LS (J , ε; α∗ , β; ρ ) is an analytic function of its arguments that is “large field with respect to the interval J ” (that is, it depends only on values of the fields at points x ∈ X\ΩS (J ) and depends only on the variables α∗τ , ατ , z∗τ , zτ with τ ∈ J ∩ (2−depth(S) θ) Z. and fulfills the estimate LS (J , ε; · ) S ≤ KL δ v r(δ) R(δ)3 where δ is the length of the time interval J . The functions DS and LS (J ) are all history complete and invariant under = ( α∗ , α , z∗ , z) → (e−iθ α ∗ , eiθ α , e−iθ z∗ , α∗ → e−iθ α∗ , β → eiθ β, ρ iθ e z) Furthermore, for each hierarchy S for scale 2n ε, of depth at most n, and each decimation interval J ⊂ [0, 2n ε], of length at least 2n−depth(S)+1 ε, 1 ( ε )1 1−3eR −4er 1 1 ; · 1 ≤ ε2 v 1DS (ε; · ) − DS 2 1S 1 1−3eR −4er ε 1 1 1LS (J , ε; · ) − LS (J , ; · )1 ≤ 210 ε2 v 2 S
◦
Proof of Proposition 3.32 from Theorems 3.24, 3.26, 3.27 and 3.28. We introduce, in addition to the effective densities I1 , I2 , · · · of (3.12), the initial effective density ∗
I0 (ε; α∗ , β) = Zε|X| χR(ε) (X, α)χR(ε) (X, β)ζε (α∗ , β) eα •
, j(ε)β −εα∗ β, v α∗ β
e
where χR (X, α) is the characteristic function which restricts |α(x)| ≤ R for each x ∈ X. Then the recursion relation of (3.12) still holds for the step from n = 0 to n = 1. We prove the Proposition by induction on n, starting with n = 0. There is only a single hierarchy S of scale ε and depth 0, namely that with ΩS = X.
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For ε sufficiently small, the initial effective density I0 may also be written ∗
I0 (ε; α∗ , β) = Zε|X| χε (X; α, β) eα •
, j(ε)β −εα∗ β, v α∗ β
e
because |∇α(b)|, |∇β(b)| ≤ 2R(ε) ≤ R (ε) for all b ∈ X ∗ . It satisfies the conclusions of the Proposition, with DS (ε) = LS (ε) = 0 and I(S;α∗ ,β) the identity operator. Since ΩS = X, we have LS ( 2ε ) = 0 and, in the notation of Proposition 3.29, (ε) (ε) (ε) (ε) = DX,1 = RX,1 + EX,1 DS 2 2 2 2 so that (ε) S DS (ε) − DS 1 ( ε )1 2 1 1 = 1DS ≤ KR ε2 r(ε)2 R(ε)2 e−2mc + KE (εv)2 r(ε)2 R(ε)6 1 2 2R(ε),2m e−2mc = KR (εv)2−2eR −4er + KE (εv)2−6eR −8er v2 1−3eR −4er ≤ ε2 v by (F.6a) and (F.4b), if ε is small enough. For the induction step from n to n + 1, set δ = 2n ε. By definition and the induction hypothesis • In+1 (ε; α∗ , β) = dμR(ε) (φ∗ , φ) IS1 (ε; α∗ , φ) IS2 (ε; φ∗ , β) S1 ,S2 hierarchies for scale δ and depth at most n
where |ΩS |
ISi (ε; α∗ , β) = Zδ i χδ (ΩSi ; α, β) ( ) ∗ ∗ ∗ ∗ ∗ ∗ I(Si ; α∗ ,β) e−QSi (α ,β; α , α)+VSi (ε; α ,β; α , α)+DSi (ε; α ,β; ρ ) eLSi (ε; α ,β; ρ ) By the induction hypothesis, Remark 3.16, Proposition 3.29, (F.4b) and (F.6a), DSi (ε; · ) Si ≤ KD (δ v) r(δ) R(δ)3 ≤ 1 1 1 1 1 1 1 1 1 1DSi (ε; · ) ΩS 1 ≤ 1DΩSi ,n (ε; · )1 i Si 2R(δ),2m 1 1 1 1 1 1 1 1 ≤ 1RΩSi ,n (ε; · )1 + 1EΩSi ,n (ε; · )1 2R(δ),2m 2 −2m c
≤ KR δ r(δ) R(δ) e 2
−17
≤2
2
2
2R(δ),2m 2 6
+ KE (δ v) r(δ) R(δ)
KD (2δv) r(2δ) R(2δ)3 ≤ 2−20
Hence, for each pair of hierarchies S1 and S2 , we may apply Theorem 3.26 to the integral dμR(ε) (φ∗ , φ) IS1 (ε; α∗ , φ) IS2 (ε; φ∗ , β). We apply it with Di = DSi and bi = eLi . It gives the representation |Ω | • Z2δ S χ2δ (ΩSi ; α, β) In+1 (ε; α∗ , β) = S1 ,S2 hierarchies hierarchies S for scale 2δ for scale δ and depth at most n with (S1 ,S2 )≺S
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( ∗ ∗ ∗ ∗ ∗ I(S; α∗ ,β) e−QS (α ,β; α , α)+VS (ε; α ,β; α , α)+DS (ε; α ,β; ρ ) ) ∗ ∗ ∗ eLS (ε; α ,β; ρ )+LS1 (ε; α ,αδ ; ρ l )+LS2 (ε; αδ ,β; ρ r ) The resulting functions DS fulfill DS (ε; · ) Ωc = 0, S ( ) ¯ Ω ;δ VΩ ;δ (ε; · ); DΩ ,n (ε; · ) , DΩ ,n (ε; · ) DS (ε; · ) Ω = R S S S1 S Ω Ω 2 S
S
S
¯ Ω ;δ (VΩ ;δ (ε; · ); DΩ ,n (ε; · ), DΩ ,n (ε; · )) =R S S S S = DΩS ,n+1 (ε; · ) by the inductive hypothesis, (3.9) and (3.8), and DS (ε; ·) S 1 ≤ KD (2δv)r(2δ)R(2δ)3 + 214 e−mc(δ) ( DS1 (ε; ·) S1 + DS2 (ε; ·) S2 ) 2 (1 1 1 1 ) + 214 1DS (ε; · )|Ω 1 + 1DS (ε; · )|Ω 1 1
S1
S1
2
S2
S2
1 ≤ KD (2δv) r(2δ) R(2δ)3 + 215 e−mc(δ) KD (δ v) r(δ) R(δ)3 2 1 + KD (2δv) r(2δ) R(2δ)3 4 ≤ KD (2δv) r(2δ) R(2δ)3 by (F.4c). The resulting functions L S fulfill L S = L S Ωc and S
L S S
1 ≤ KL (2δv) r(2δ) R(2δ)3 + 28 ( D1 S1 + D2 S2 ) 2 1 ≤ KL (2δv) r(2δ) R(2δ)3 + 29 KD (δ v) r(δ) R(δ)3 2 ≤ KL (2δv) r(2δ) R(2δ)3
since KL ≥ 213 KD . If J is a decimation interval for S, ⎧ ⎪ l ) ⎨LS1 (J , ε; α∗ , αδ ; ρ LS (J , ε; α∗ , β; ρ ) = LS2 (J − δ, ε; α∗δ , β; ρr ) ⎪ ⎩ LS (ε; α, β; ρ )
we set if J ⊂ [0, δ] if J ⊂ [δ, 2δ] if J = [0, 2δ]
Now let S, S1 and S2 be hierarchies with (S1 , S2 ) ≺ S. By the induction hypothesis and Proposition 3.29, 1 ( ε )1 1 1 ;· 1 1DSi (ε; · ) − DSi 2 Si 1 1 ( ε ) 1 2 1−3eR −4er 1 1 1 ≤ ε v 1DSi (ε; · ) ΩSi − DSi 2 ; · ΩS 1 i Si 1 ( ε )1 1 1 ;· 1 ≤ 1RΩSi ,n (ε; · ) − RΩSi ,n 2 2R(δ),2m 1 ( ε )1 1 1 ;· 1 + 1EΩSi ,n (ε; · ) − EΩSi ,n 2 2R(δ),2m
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≤ KR εδ r(ε)2 R(ε)2 e−2m c + 2KE (ε v)2 r(ε)2 R(ε)6 e−2m c + 2KE (ε v)2−6eR −8er = KR δ (εv)1−2eR −4er v ≤ 3KE (ε v)2−6eR −8er ˜i = if ε is small 3.28 with Di = DSi (ε; · ), D enough. We apply Theorem ε DSi 2ε ; · , bi = eLSi (ε; · ) and ˜bi = eLSi ( 2 ; · ) . It gives 1 ( ε )1 1 1 ;· 1 1DS (ε; · ) − DS 2 S 1−3eR −4er 1 ≤ KD ε(2δv)r(2δ)R(2δ)3 + 217 e−mc(δ) ε2 v 2 +217 3KE (ε v)2−6eR −8er 1−3eR −4er 1 = KD (2δ)1−3eR −4er ε v1−3eR −4er + 217 e−mc(δ) ε2 v 2 +3 · 217 KE v1−3eR −4er (ε2 v)1−3eR −4er 1−3eR −4er ≤ ε2 v by (F.4c) and (F.6a), if ε is small enough, and 1 1 ε 1 1 1LS ([0, 2δ], ε; · ) − LS ([0, 2δ], ; · )1 2 S 2 1−3eR −4er 1 3 9 ≤ KL ε(2δv) r(2δ) R(2δ) + 2 ε v 2 1−3eR −4er ≤ 210 ε2 v
if ε is small enough. Corollary 3.33. Let S be a hierarchy of scale θ. Then the limit
DS ( α∗ , β; ρ ) = lim DS;m (2−m θ; α∗ , β; ρ ) m→∞ exists.13 Its “pure large field part” vanishes, that is DS Ωc = 0. The “pure S small field part” DS = DΩ ,θ ΩS
S
as in Corollary 3.30. Also DS S ≤ KD (θ v) r(θ) R(θ)3 For each decimation interval J in [0, θ], the limit LS ( J ; α∗ , β; ρ ) = lim LS;m (J , 2−m θ; α∗ , β; ρ ) m→∞
exists(2) . The function LS (J ; α∗ , β; ρ ) is an analytic function of its arguments that is “large field with respect to the interval J ” (that is, it depends only on values of the fields at points x ∈ X\ΩS (J ) and depends only on the variables α∗τ , ατ , z∗τ , zτ with τ ∈ J ∩ (2−depth(S) θ) Z. 13
The convergence is with respect to the norm · S .
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and fulfills the estimate LS (J ; · ) S ≤ KL δ |||v||| r(δ) R(δ)3
where δ is the length of the time interval J . The functions DS and LS (J ) are all history complete and invariant under = ( α∗ , α , z∗ , z) → (e−iθ α ∗ , eiθ α , e−iθ z∗ , eiθ z). α∗ → e−iθ α∗ , β → eiθ β, ρ The remaining ε dependent term in the representation of Proposition 3.32 is VS (ε; α∗ , β; α ∗, α ). We now show that its limit as ε tends to zero is the interaction ∗, α ) VS (α∗ , β; α δ ∗ )ΓS (τ ; α , β), v Γ∗S (τ ; α∗ , α ∗ )ΓS (τ ; α , β) = − dτ Γ∗S (τ ; α∗ , α 0
This agrees with (2.11), though of course v now implicitly depends on h. Lemma 3.34. Let S be a hierarchy for scale θ. Then, for each α∗ , β, α ∗, α , lim VS (2−m θ; α∗ , β; α ∗, α ) = VS ( α∗ , β; α ∗, α )
m→∞
The convergence is uniform on compact sets. Furthermore VS ( α∗ , β; α ∗, α ) Ω = VΩS ;θ (α∗ , β) S
where VΩ;θ (α∗ , β) is defined as in (2.4), but with history fields included. Proof. We use the shorthand notations Γ∗S (τ ; α∗ , α ∗ ) if τ ∈ (0, θ) γ∗τ = if τ = 0 α∗0 = α∗
ΓS (τ ; α , β) γτ = αθ = β
if τ ∈ (0, θ) if τ = θ
Set ε = 2−m θ and write VS ( α∗ , β; α ∗, α ) − VS (ε; α∗ , β; α ∗, α ) θ = − dτ γ∗τ γτ , v γ∗τ γτ + ε
=
γ∗τ γτ +ε , v γ∗τ γτ +ε
τ ∈εZ∩[0,θ)
0
ε dt [γ∗τ γτ +ε , v γ∗τ γτ +ε − γ∗τ +t γτ +t , v γ∗τ +t γτ +t ]
τ ∈εZ∩[0,θ) 0
Consequently ∗, α ) − VS (ε; α∗ , β; α ∗, α )| |VS ( α∗ , β; α ≤θ ≤θ
sup τ ∈εZ∩[0,θ) 0≤t≤ε
sup τ ∈εZ∩[0,θ) 0≤t≤ε
|γ∗τ γτ +ε , vγ∗τ γτ +ε − γ∗τ +t γτ +t , vγ∗τ +t γτ +t | "
|(γ∗τ − γ∗τ +t )γτ +ε , vγ∗τ γτ +ε |
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& ' + γ∗τ +t (γτ +ε − γτ +t , vγ∗τ γτ +ε + |γ∗τ +t γτ +t , v(γ∗τ − γ∗τ +t )γτ +ε | & ' # + γ∗τ +t γτ +t , vγ∗τ +t (γτ +ε − γτ +t We bound the first term. The bounds on the remaining three terms are virtually identical. For all τ ∈ ε ∩ [0, θ), |(γ∗τ − γ∗τ +t )γτ +ε , v γ∗τ γτ +ε | = |([j(t) − h]γ∗τ ) γτ +ε , v γ∗τ γτ +ε | 2
≤ Kj teKj t |||v||| |X| max |γ∗τ (x)| max |γτ +ε (x)| x∈X
2
x∈X
by Lemmas E.14 and 3.21(ii), and max |γτ +ε (x)| ≤ Kα ≡ 2 + 5eKj x∈X
max |γ∗τ (x)| ≤ Kα∗ ≡ 2 + 5eKj x∈X
max
max |ατ (x)|
τ ∈∩(0,θ] x∈Λcτ
max
max |α∗τ (x)|
τ ∈∩[0,θ) x∈Λcτ
by Lemma E.8. Hence
2 ε |VS ( α∗ , β; α ∗, α ) − VS (ε; α∗ , β; α ∗, α )| ≤ 4θe2Kj |||v||| |X| Kα2 Kα∗
clearly converges to zero, uniformly on compacta, as ε → 0.
3.7. Bounds on the Large Field Integral Operator In this section, we bound the large field integral operators I(S;α∗ ,β) when the history field is identically one. So we set h ≡ 1 throughout this section. Recall that I(S;α∗ ,β) and its absolute value |I(S;α∗ ,β) | were defined in Definition 2.8. • Proposition 3.32 gives a representation for Im (2−m θ; α∗ , β) that is analo• ∗ (2−m θ; α∗ , β) h=1 given gous to the representation for Iθ (α , β) = limm→∞ Im in our main Theorem 2.16. To take the limit and prove Theorem 2.16, we shall apply the dominated convergence theorem to the sum over hierarchies in Proposition 3.32. To do so, we apply a result, Theorem 3.35, below, that is slightly more general than Theorem 2.18. Fix 0 < θ ≤ Θ. Let m ∈ N and set ε = 2−m θ. Assume that α and β obey the small field conditions χθ (Ω; α, β) = 1. Theorem 3.35. For any bounded measurable function f (α, β; ρ) and subset Ω ⊂ X, e− 2 α 1
×
2
− 12 β2 −RegSF (Ω;α,β)
e
−k+1
θ)
e
I(S;α∗ ,β) eRe (−QS +VS (ε)) |f |
1 1 sup |f | 1 + |α(x)|3 1 + |β(x)|3
ωS
hierarchies S for scale θ of depth k with ΩS =Ω
≤e
e(2
k=1
− 14 (θ)|Ωc |
m
x∈Ωc
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with the RegSF (Ω; α, β) of Definition 2.17 and 1 |Ω(J )c | ωS = 2 decimation intervals J ⊂[0,θ]
The small factors that lead to Theorem 3.35 arise from three sources. ◦ For each decimation interval J , there is a large positive contribution to the main quadratic part of the action, QS (α∗ , β; α ∗, α ), for each point of the “large field sets of the first kind” Pα (J ), Pβ (J ) and Q(J ) that were introduced in the Definition 2.4 of a hierarchy. This is made precise in Proposition 3.36, below. ◦ For each decimation interval J , there is a large negative contribution to the main quartic part of the action, VS (ε; α∗ , β; α ∗, α ), for each point of the “large field sets of the first kind” Pα (J ) and Pβ (J ), that were introduced in the Definition 2.4 of a hierarchy. This is made precise in Proposition 3.37, below. ◦ For each decimation interval J of length 2s, the integral operator I(J ,S ; α∗ ,β) , of Definition 2.8, includes an integral dz∗τ (x) ∧ dzτ (x) −z∗τ (x)zτ (x) e 2πi Cs (x;α∗ ,β)
for each point x in the “large field set of the second kind” R(J ). Proposition 3.38, below, shows that we may choose the domain of integration Cs (x; α∗ , β) in such a way that Re z∗τ (x)zτ (x) is large on the entire surface Cs (x; α∗ , β). At the end of this section, we show how Theorem 3.35 is deduced from these three propositions, which, in turn, will be proven in Sect. 6. The three propositions involve a constant CL > 0, that is defined in Lemma F.5 and depends only on h and the constants of Hypothesis 2.14, (2.18), (2.19) and (2.20). For these three propositions, we fix an integer 0 ≤ n ≤ m and a hierarchy S for scale δ = 2n ε of depth at most n and write ⎧ ⎧ ⎫ ⎫ ∗ ⎪ ⎪ ⎪ ⎪ if τ = 0 if τ = 0 ⎨α ⎨α ⎬ ⎬ ∗ ∗ γτ = ΓS (τ ; α γ∗τ = Γ∗S (τ ; α , α ) if τ ∈ (0, δ) , β) if τ ∈ (0, δ) ⎪ ⎪ ⎪ ⎪ ⎩ ∗ ⎩ ⎭ ⎭ if τ = δ β β if τ = δ Proposition 3.36. 1 1 2 ∗ ∗ 2 α + QS (α , β; α ,α ) + β −Re 2 2 " # ˜ )| + |P˜α (J )| + |P˜β (J )| ≤ −CL r(|J |)2 |Q(J decimation intervals J ⊂[0,θ]
−
1 ∗ γ∗τ − γτ +ε 2Λcτ ∪Λcτ +ε 16 τ ∈[0,δ)
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+
decimation intervals J ⊂[0,δ]
Ann. Henri Poincar´e
1 |Ω(J )c | + |eεμ − 1| γτ 2Ωc + γ∗τ 2Ωc 8 τ ∈[0,δ]
(2)
+ RegSF (Ω; α, β) (2)
where RegSF (Ω; α, β) is given in Definition 2.17 and, for each decimation interval J ,
P˜α (J ) = b ∈ Pα (J ) d b, Λ(J − )c ∪ Λ(J + )c > 2c |J ± |
P˜β (J ) = b ∈ Pβ (J ) d b, Λ(J − )c ∪ Λ(J + )c > 2c |J ± |
˜ (J ) = x ∈ Q (J ) d x, Λ(J − )c ∪ Λ(J + )c > 2c |J ± | Q Proposition 3.37. (4) ∗, α ) − RegSF (α, β) Re VS (ε; α∗ , β; α " # r(|J |)2 |P˜α (J )| + |P˜β (J )| ≤ − CL decimation intervals J ⊂[0,δ]
−
1 1 ∗ ∗ ε γ∗τ γ∗τ , v γ∗τ γ∗τ − ε γτ∗ γτ , v γτ∗ γτ 4 4 τ ∈[0,δ)
+
decimation intervals J ⊂[0,δ]
τ ∈(0,δ]
1 1 2 |Ω(J )c | + γ ∗ − γτ +ε Λcτ ∪Λc τ +ε 8 16 ∗τ τ ∈[0,δ)
(4)
where RegSF (α, β) is given in Definition 2.17 and, for each decimation interval J, 1 P˜α (J ) = x ∈ Pα (J ) d x , Λ(J − )c ∪ Λ(J + )c > 2 c |J | 2 1 P˜β (J ) = x ∈ Pβ (J ) d x , Λ(J − )c ∪ Λ(J + )c > 2 c |J | 2 Proposition 3.38. Let J be a decimation interval for S with length 2s and centre τ , and x ∈ Λ(J ). We may choose the surface Cs (x; α∗ , β) of Definition 2.8 so that the following holds. Assume that α and β are such that the characteristic functions χ2s (Λ(J ); α, β) and χJ (α, ατ , β) are nonzero. Then
(i) Cs (x; α∗ , β) ⊂ (z∗ , z) ∈ C2 |z∗ |, |z| ≤ 2r(s) . (ii) For all (z∗ , z) ∈ Cs (x; α∗ , β), Re (z∗ z) ≥ CL r(s)2 (iii) The area of Cs (x; α∗ , β) is bounded by 40π r(s)2 . Proof of Theorem 3.35. We still fix a hierarchy S for scale δ = 2n ε. For bounding the “absolute value” |IS | = |I(S;α∗ ,β) |, we introduce the auxiliary integral
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operator
I¯S =
I¯(J ; α∗τ
k=0,...,depth(S) decimation intervals J =[τl ,τr ]⊂[0,δ] of length 2−k δ
where I¯(J ;α∗ ,β) is constructed by replacing dz∗τ (x) ∧ dzτ (x) −z∗τ (x)zτ (x) e by 2πi Cs (x;α∗ ,β)
211
Cs (x;α∗ ,β)
,ατr )
dz∗τ (x) ∧ dzτ (x) 2πi
in the formula for I(J ;α∗ ,β) in Definition 2.8. Then, |IS | = I¯S e−Re z∗τ (x)zτ (x)
(3.13a)
decimation intervals x∈R(J ) J ⊂[0,δ] with centre τ
By Proposition 3.38,
e−Re z∗τ (x)zτ (x) ≤
decimation intervals x∈R(J ) J ⊂[0,δ] with centre τ
e−CL |R(J )| r( 2 |J |) 1
2
decimation intervals J ⊂[0,δ]
(3.13b) on the domain of integration.
Lemma 3.39. 1 1 ∗, α ) − β 2 + VS (ε; α∗ , β; α ∗, α ) Re − α 2 − QS (α∗ , β; α 2 2 −RegSF (Ω; α, β) # " ˜ )|+|P˜α (J )|+|P˜β (J )|+|P˜α (J )|+|P˜β (J )| ≤ −CL r(|J |)2 |Q(J decimation intervals J ⊂[0,δ]
+
decimation intervals J ⊂[0,δ]
1 |Ω(J )c | 2
Proof. We assume that ε is sufficiently small that |eεμ − 1| ≤ 18 , which implies that |eεμ − 1| ≤ 2ε|μ|. It suffices to apply Propositions 3.36 and 3.37 together with 1 γτ∗ γτ 2Ωc + 2|eεμ − 1| γτ 2Ωc − εv1 4 τ ∈(0,δ] τ ∈(0,δ] 1 |eεμ − 1| 4 2 = − εv1 |γτ (x)| |γτ (x)| − 8 4 εv1 τ ∈(0,δ] x∈Ωc 2 |eεμ − 1| |eεμ − 1|2 1 2 − 16 |γτ (x)| − 4 = − εv1 4 εv1 ε2 v12 c τ ∈(0,δ] x∈Ω
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≤ 4εv1
Ann. Henri Poincar´e
|eεμ − 1|2 |eεμ − 1|2 c μ2 c = 4δ |Ω | ≤ 16δ |Ω | ε2 v12 ε2 v1 v1 c
τ ∈(0,δ] x∈Ω
≤ 64 ≤
Kμ2 cv
δv2eμ −1 |Ωc |
1 c |Ω | 8
and
by Hypothesis II(14)
by Hypothesis F.7(i) and the fact that eμ >
1 2
1 ∗ − εv1 γ∗τ γ∗τ 2Ωc + 2|eεμ − 1| γ∗τ 2Ωc 4 τ ∈[0,δ)
τ ∈[0,δ)
|eεμ − 1|2 c 1 ≤ 4δ |Ω | ≤ |Ωc | 2 ε v1 8 Consequently, by (3.13) and Lemma 3.39, 2 2 1 1 e− 2 α − 2 β e−RegSF (Ω;α,β) eωS I(S;α∗ ,β) eRe (−QS +VS (ε)) |f | ≤ e−L(S) I¯S |f | (3.14) where L(S) = CL
r(|J |)2
decimation intervals J ⊂[0,δ]
" # ˜ )| + |P˜α (J )| + |P˜β (J )| + |P˜α (J )| + |P˜β (J )| + |R(J )| × |Q(J |Ω(J )c | − decimation intervals J ⊂[0,δ]
The quantity L(S) is defined in terms of the number of points at which there are violations of the various small field conditions (the sets Q(J ), etc.). Lemma 3.40, which is proven in Sect. 6, provides a lower bound for L(S) purely in terms of the large field sets Ω(J )c . Lemma 3.40. We have " # ˜ )|+|P˜ (J )|+|P˜ (J )|+|P˜α (J )|+|P˜β (J )|+|R(J )| CL r(|J |)2 |Q(J α β decimation intervals J ⊂[0,δ]
≥ (S) +
|Ω(J )c |
decimation intervals J ⊂[0,δ]
where (S) =
decimation intervals J ⊂[0,δ]
(|J |) |Ω(J )c |
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Applying (3.14) and Lemma 3.40, we have 2 2 1 1 e− 2 α − 2 β e−RegSF (Ω;α,β) eωS I(S;α∗ ,β) eRe (−QS +VS (ε)) |f | ≤ e−(S) I¯S |f | (3.15) Theorem 3.35 is an immediate consequence of (3.15) and Proposition 3.41. For any (small field) subset Ω ⊂ X, any 0 ≤ n ≤ m, and any bounded function f (α, β; ρ), eλS e−(S) I¯S |f | hierarchies nS for scale 2 ε with ΩS =Ω and 0<depth(S)≤n
− 14 (2n ε)|Ωc |
≤e
x∈Ωc
where
1 1 sup |f | 1 + |α(x)|3 1 + |β(x)|3
(2n−depth(S)+1 ε) λS = 0
if depth (S) > 0 if depth (S) = 0
Here 2n−depth(S)+1 ε is the length of the shortest decimation interval J with ΩS (J ) = X. Proof. The proof is by induction on n. The case n = 0 is trivial as is the case that Ωc = ∅. Assume that the statement holds for some 0 ≤ n < m and that Ωc = ∅. Set δ = 2n ε. Given subsets Ω1 , Ω2 , Λ of X containing Ω and hierarchies S1 , S2 for scale δ with ΩS1 = Ω1 and ΩS2 = Ω2 , a hierarchy S with ΛS = Λ, ΩS = Ω and (S1 , S2 ) ≺ S is specified by the sets Pα , Pβ ⊂ Ω1 ∩ Ω2 ∩ Λc and
Pα , Pβ ⊂ (Ω1 ∩ Ω2 ∩ Λc )∗
Q ⊂ (Ω1 ∩ Ω2 ) ∩ Λc
R ⊂ Λ ∩ Ωc
See Definition 2.4. There are at most 22|Ω1 ∩Ω2 ∩Λ | 24D|Ω1 ∩Ω2 ∩Λ | 22|Ω1 ∩Ω2 ∩Λ | 2|Λ∩Ω | ≤ e(5+4D) |Ω c
c
c
c
c
|
(3.16)
(where D is the dimension of space) such choices. For each hierarchy S as above (see Definition 2.8 and Appendix A) ⎞ ⎛ ∗ dzδ (x) ∧ dzδ (x) −zδ (x)∗ zδ (x) ⎟ ⎜ I¯S |f | = ⎝ e ⎠ 2πi x∈Λ\(R∪Ω)|z (x)|≤r(δ) δ
⎛
⎜ ×⎝
x∈R Cδ (x;α∗ ,β)
⎞ dz∗δ (x) ∧ dzδ (x) ⎟ ⎠ 2πi
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⎛
⎞
⎜ ×⎝
∗
x∈X\Λ|α (x)|≤R(ε) δ |Ω1 \Ω|
×Zδ
|Ω2 \Ω|
Zδ
2 |R|
≤ (40πr(δ) )
Ann. Henri Poincar´e
dαδ (x) ∧ dαδ (x) ⎟ ⎠ χ[0,2δ] (α, αδ , β) 2πi
I¯S1 I¯S2 sup |f |
χR(δ) (Ω1 ∩ Λc , α)χR(2δ) (Λ ∩ Ωc , α)
×χR(δ) (Ω2 ∩ Λc , β)χR(2δ) (Λ ∩ Ωc , β) ⎞ ⎛ ∗ dαδ (x) ∧ dαδ (x) ⎟ ⎜ ×⎝ ⎠ 2πi x∈X\Λ|α (x)|≤R(ε) δ
×χR(δ) ((Ω1 ∪ Ω2 ) ∩ Λc , αδ ) I¯S1 I¯S2 sup |f |
(3.17)
Here we used Lemma A.4.a, which ensures that the integral over αδ (x) is ∗ ∧dzδ (x) restricted to |αδ (x)| ≤ R(ε). For the inequality, we used that C dzδ (x)2πi ∗ e−zδ (x) zδ (x) = 1, that the area of Cδ (x; α∗ , β) is at most 40πr(δ)2 (by Lemma 3.38(iii)), that Zδ ≤ 1, and the definition of χ[0,2δ] (α, αδ , β) (given in Appendix A). Since λS − (S) = −(2δ)|Ωc |−(S1 )−(S2 ) max {λS1 , λS2 } if depth (S) > 1 + (2δ) if depth (S) = 1 ≤ −(2δ) [|Ωc | − 1] + λS1 − (S1 ) + λS2 − (S2 ) (3.16), (3.17) and the induction hypothesis give that
eλS −(S) I¯S |f |
S1 ,S2 S (S1 ,S2 )≺S ΩS1 =Ω1 =Ω, ΛS =Λ Ω S ΩS2 =Ω2 |R|
≤ e−(2δ)[|Ω |−1] e(5+4D) |Ω | (40πr(δ)2 ) ×χR(δ) (Ω1 ∩ Λc , α)χR(2δ) (Λ ∩ Ωc , α) χR(δ) (Ω2 ∩ Λc , β)χR(2δ) (Λ ∩ Ωc , β) ⎞ ⎛ ∗ dαδ (x) ∧ dαδ (x) ⎟ ⎜ c ×⎝ ⎠ χR(δ) ((Ω1 ∪ Ω2 ) ∩ Λ , αδ ) 2πi c
c
x∈X\Λ|α (x)|≤R(ε) δ
×
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
eλS1 −(S1 ) I¯S1
S1 ΩS1 =Ω1
≤ e−(2δ)[|Ω
c
⎫⎧ ⎪ ⎪ ⎪ ⎨ ⎬⎪ ⎪ ⎪ ⎪ ⎩ ⎭⎪
S2 ΩS2 =Ω2
eλS2 −(S2 ) I¯S2
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
sup |f |
|−1] (5+4D) |Ωc | [5+2 ln r(δ)]|Ωc |
e
e
χR(δ) (Ω1 ∩ Λ , α)χR(2δ) (Λ ∩ Ωc , α) χR(δ) (Ω2 ∩ Λc , β)χR(2δ) (Λ ∩ Ωc , β) c
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⎛
⎞
⎜ ×⎝
∗
x∈X\Λ|α (x)|≤R(ε) δ
⎡
215
dαδ (x) ∧ dαδ (x) ⎟ c ⎠ χR(δ) ((Ω1 ∪ Ω2 ) ∩ Λ , αδ ) 2πi
⎤ c 1 1 ⎣ ⎦ e− 14 (δ)|Ω2 | ×e 3 1 + |α (x)|3 1 + |α(x)| δ x∈Ωc1 ⎤ ⎡ 1 1 ⎦ sup |f | ×⎣ 3 1 + |β(x)|3 1 + |α (x)| δ c − 14 (δ)|Ωc1 |
(3.18)
x∈Ω2
For any t > 1, χt (α) ≤
1 (1 + t)3 8t3 ≤ ≤ eln 8+3 ln t 1 + |α|3 1 + |α|3 1 + |α|3
which yields the bound χR(δ) (Ω1 ∩ Λc , α)χR(2δ) (Λ ∩ Ωc , α) ≤ χR(δ) (Ω1 ∩ Ωc , α)
x∈Ωc1
x∈Ωc1
≤ e[ln 8+3 ln R(δ)]
|Ωc |
x∈Ωc
1 1 + |α(x)|3
1 1 + |α(x)|3 1 1 + |α(x)|3
Similarly χR(δ) (Ω2 ∩ Λc , β)χR(2δ) (Λ ∩ Ωc , β) ≤ e[ln 8+3 ln R(δ)]
|Ω | c
x∈Ωc
x∈Ωc2
1 1 + |β(x)|3
1 1 + |β(x)|3
To bound the αδ integrals, we use ⎞ ⎛ ∗ dαδ (x) ∧ dαδ (x) ⎟ ⎜ c ⎠ χR(δ) ((Ω1 ∪ Ω2 ) ∩ Λ , αδ ) ⎝ 2πi c x∈Λ
|αδ (x)|≤R(ε)
× ≤
1 1 3 3 1 + |α (x)| 1 + |α δ δ (x)| x∈Ωc1 x∈Ωc2 1 d2 αδ (x) π 1 + |αδ (x)|3 c
x∈Ω1 ∩Ω2
x∈(Ω1 ∩Ω2 )
|(Ω1 ∩Ω2 ) |
≤3
c
R(δ)
[ln 3+2 ln R(δ)] |Λc |
≤e
2|Ω1 ∩Ω2 ∩Λ | c
∩Λc
1 π
d2 αδ (x) |αδ (x)|≤R(δ)
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Inserting the last estimates into (3.18) gives eλS −(S) I¯S |f | S ΩS =Ω, ΛS =Λ ΩS ([0,δ])=Ω1 ΩS ([δ,2δ])=Ω2
≤ e−(2δ)[|Ω |−1] e(5+4D) |Ω | e[5+2 ln r(δ)]|Ω | e2[ln 8+3 ln R(δ)] |Ω | c 1 1 × e[ln 3+2 ln R(δ)] |Λ | sup |f | 3 3 1 + |α(x)| 1 + |β(x)| x∈Ωc c c 1 1 1 sup |f | ≤ e− 2 (2δ)|Ω | e[16+4D+10 ln R(δ)] |Ω | 3 1 + |α(x)| 1 + |β(x)|3 c c
c
c
c
x∈Ω
Summing over the subsets Ωc1 , Ωc2 , Λc ⊂ Ωc , c c 1 eλS −(S) I¯S |f | ≤ e− 2 (2δ)|Ω | e[19+4D+10 ln R(δ)] |Ω | S ΩS =Ω
1 1 sup |f | 3 1 + |α(x)| 1 + |β(x)|3 x∈Ωc c 1 1 1 sup |f | ≤ e− 4 (2δ)|Ω | 3 1 + |β(x)|3 1 + |α(x)| c ×
x∈Ω
by (F.6d). 3.8. Proof of Theorems 2.16 and 2.18
As we saw in Remark 3.31, it suffices, for the proof of Theorems 2.16, to prove • the convergence of Im (2−m θ; · ) h=1 to the desired limit. By Proposition 3.32, with ε = 2−m θ and n = m, |Ω| ∗ ∗ • Im (2−m θ; α∗ , β) = Zθ eα∗ , j(θ)β |Ω +VΩ,θ (ε;α ,β)+DΩ,m (ε;α ,β) χθ (Ω; α, β) Ω⊂X
×
( res ∗ ∗ res ∗ ∗ I(S;α∗ ,β) e−QS (α ,β; α , α)+VS (ε; α ,β; α , α)
S hierarchy for scale θ depth(S)≤m with ΩS =Ω ∗
× eBS (ε; α where
,β; ρ )+LS (ε; α∗ ,β; ρ )
)
(3.19)
ε=2−m θ
∗ ∗, α ) = QS (α∗ , β; α ∗, α ) + α∗ , j(θ)β Ω Qres S (α , β; α
S
res VS (ε; α∗ , β; α ∗, α ) = VS (ε; α∗ , β; α ∗, α ) − VΩS ,θ (ε; α∗ , β)
BS (ε; α∗ , β; ρ ) = DS (ε; α∗ , β; ρ ) − DΩS ,m (ε; α∗ , β)
with the VΩ,θ (ε; · ) of Definition 3.8.ii and the DΩ,m (ε; · ) of (3.8). By (2.4) and Corollary 3.30, for any fixed Ω ⊂ X, lim χθ (Ω; α, β)eVΩ,θ (2
−m
θ;α∗ ,β)+DΩ,m (2−m θ;α∗ ,β)
m→∞
∗
= χθ (Ω; α, β) eVΩ;θ (α
,β)+DΩ,θ (α∗ ,β)
(3.20)
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uniformly in α and β. Fix any hierarchy S for scale θ with ΩS = Ω and set res (α∗ , β; α ∗, α ) = VS (α∗ , β; α ∗, α ) − VΩS ,θ (α∗ , β) VS ∗ ∗ BS (α , β; ρ ) = DS (α , β; ρ ) − DΩS ,θ (α∗ , β)
By Lemma 3.34 and Corollary 3.33, χθ (Ω; α, β) eVS
res
(2−m θ; α∗ ,β; α ∗ , α) BS (2−m θ; α∗ ,β; ρ )+LS (2−m θ; α∗ ,β; ρ )
e
converges as m → ∞ to χθ (Ω; α, β) eVS
res
(α∗ ,β; α ∗ , α) BS (α∗ ,β; ρ )+LS (α∗ ,β; ρ )
e
uniformly for ρ in the domain of integration of I(S;α∗ ,β) and (α, β) in the support of χθ (Ω; α, β). By Remark 3.11, the domain of integration for I(S;α∗ ,β) is compact. Consequently, as Qres S is a polynomial, ( res ∗ ∗ res −m ∗ ∗ lim χθ (Ω; α, β) I(S;α∗ ,β) e−QS (α ,β; α , α)+VS (2 θ; α ,β; α , α) m→∞ ) −m ∗ −m ∗ × eBS (2 θ; α ,β; ρ )+LS (2 θ; α ,β; ρ ) ( res ∗ ∗ res ∗ ∗ = χθ (Ω; α, β) I(S;α∗ ,β) e−QS (α ,β; α , α)+VS (α ,β; α , α) ) ∗ ∗ (3.21) ×eBS (α ,β; ρ )+LS (α ,β; ρ ) uniformly for (α, β) in the support of χθ (Ω; α, β). By (3.20) and (3.21), the (Ω, S) term on the right hand side of (3.19) converges to ∗
|Ω|
∗
Zθ eα∗ , j(θ)β |Ω +VΩ,θ (α ,β)+DΩ,θ (α ,β) χθ (Ω; α, β) ( ) res ∗ ∗ res ∗ ∗ ∗ ∗ × I(S;α∗ ,β) e−QS (α ,β; α , α)+VS (α ,β; α , α) eBS (α ,β; ρ )+LS (α ,β; ρ ) as m → ∞. Setting h = 1 gives the (Ω, S) term on the right hand side of the representation of Iθ (α∗ , β) in Theorem 2.16. The properties and estimates of the various functions in Theorem 2.16 follow from Corollaries 3.30 and 3.33. It remains to prove the convergence of the sum over Ω and S. Recall that X is a finite set. We saw in Remark 2.6 that, for any ∅ = Ω ⊂ X, the corridor condition in the definition of a hierarchy ensures that there are only finitely many hierarchies S with ΩS = Ω. If Ω = ∅, then Qres S = QS , res = VS and BS = DS = 0. Therefore it suffices to prove the convergence, VS as m → ∞, of m Fk,m (α∗ , β) k=1
where Fk,m (α∗ , β) =
S hierarchy for scale θ depth(S)=k with ΩS =∅
( ) ∗ ∗ −m ∗ ∗ −m ∗ ×I(S;α∗ ,β) e−QS (α ,β; α , α)+VS (2 θ;α ,β; α , α) eLS (2 θ;α ,β;ρ )
h=1
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We already know the convergence, for each fixed k, of Fk,m (α∗ , β) as m → ∞. By Theorem 3.35 and Lemma 3.42, below, |Fk,m (α∗ , β)| ≤ e 2 α 1
2
+ 12 β2 −(2−k+1 θ)
e
As this bound is summable in k, the dominated convergence theorem gives the desired limit. This completes the proof of Theorems 2.16. Lemma 3.42. Let S be a hierarchy for scale θ. Then, on the domain of integration for IS , |BS | + |LS | ≤ ωS where ωS was defined in Theorem 3.35. Proof. Let ε = 2−depth(S) θ. On the domain of integration, |ατ (x)| ≤ min{κS,τ (x), κ∗S,τ (x)}
|α(x)| ≤ κ∗S,0 (x)
|β(x)| ≤ κS,θ (x)
for all τ ∈ εZ ∩ (0, θ) and x ∈ X. By Proposition 3.32 and [4, Lemma B.1], |BS | ≤ Kd KD (θv) r(θ)R(θ)3 |ΩcS | |LS | ≤ Kd KL (|J |v) r(|J |)R(|J |)3 |ΩS (J )c | decimation intervals J ⊂[0,θ]
where Kd = supy∈X
x∈X
e−d(x,y) . The result now follows by (F.6a).
Proof of Theorem 2.18. Set h = 1 and let f be any bounded measurable function. First consider any fixed hierarchy S for scale θ with ΩS = Ω. By Remark 3.11, the domain of integration for the large field integral operator I(S;α∗ ,β) is compact. By Lemma 3.34, VS (2−m θ; α∗ , β; α ∗, α ) converges uniformly on this domain to res VS ( α∗ , β; α ∗, α ) = VΩ;θ (α∗ , β) + VS ( α∗ , β; α ∗, α )
as m → ∞. Consequently, −m lim I(S;α∗ ,β) eRe (−QS +VS (2 θ)) |f | m→∞ ∗ res res = eRe (α , j(Ω) (θ)β +VΩ;θ (α∗ ,β)) I(S;α∗ ,β) eRe (−QS +VS ) |f | For any natural number k, there are only finitely many hierarchies for scale θ of depth k. So, by Theorem 3.35, e− 2 α 1
2
− 12 β2 −RegSF (Ω;α,β)
e
×
hierarchies S for scale θ of depth k with ΩS =Ω − 14 (θ)|Ωc |
≤e
k0
e(2
−k+1
θ)
k=1 ωS
e
Re (α∗ , j(Ω) (θ)β +VΩ;θ (α∗ ,β))
e
x∈Ωc
res I(S;α∗ ,β) eRe (−Qres S +VS ) |f |
1 1 sup |f | 1 + |α(x)|3 1 + |β(x)|3
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for all k0 ∈ N. As all terms on the left hand side are nonnegative, ∗
e− 2 α − 2 β eRe (α ∞ −k+1 θ) × e(2 2
1
1
2
k=1
− 14 (θ)|Ωc |
≤e
, j(Ω) (θ)β +VΩ;θ (α∗ ,β)) −RegSF (Ω;α,β)
e res I(S;α∗ ,β) eRe (−Qres S +VS )+ωS |f |
hierarchies S for scale θ of depth k with ΩS =Ω
x∈Ωc
1 1 sup |f | 1 + |α(x)|3 1 + |β(x)|3
(3.22)
The theorem now follows by setting f ≡ 1 and applying Lemma 3.42.
4. The “Small Field” Part of the Decimation Step The “small field decimation step” is formulated in Theorems 3.24 and 3.27, which we are going to prove in this section. It deals with the “stationary phase approximation” to the construction on a fixed subset Ω of X. As the estimates do not depend on Ω, we may, for simplicity of notation, assume that Ω = X. ¯ Ω;δ and Vδ in place of VΩ,δ . Also write ¯ δ in place of R We shall write R r = r(δ) r+ = r(2δ)
R = R(δ) R+ = R(2δ)
Observe that, by (2.18) 1 r+ = er r 2
1 R+ = eR +er R 2
(4.1)
We also write · δ for · 2R(δ),2m . The proofs of Theorem 3.24 and 3.27 are similar to [5]. The main technical differences are the presence of the history field and the additional terms created by the fact that the cut off propagator jc (t) of (3.2) is not a semigroup—in contrast to the original propagator j(t) = exph(−t(h − μ)). ¯ δ (Vδ (ε; · ); f1 , f2 ) in It turns out that the dominant contribution to R Definition 3.22 is f1 (α∗ , jc (t)β) + f2 (jc (t)α∗ , β) To estimate it, we use Lemma 4.1. Let f (α∗ , β; h) be an analytic function. Let δ ≤ Θ. (i) If f is bilinear in α∗ , β then f (α∗ , jc (δ)β) 2δ , f (jc (δ)α∗ , β) 2δ ≤ eδKj
R+ R
2 f δ
(ii) If f has degree at least two both in α∗ and β f (α∗ , jc (δ)β) 2δ , f (jc (δ)α∗ , β) 2δ ≤ e2δKj
R+ R
4 f δ
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Proof. Both in cases (i) and (ii), we prove the first inequality. To do so, we introduce the auxiliary weight system waux with metric 2md that associates the constant weight factor 2R to the field α∗ and the constant weight factor 2e−δKj R to the field β. To control the change of variables from f (α∗ , β) to f (α∗ , jc (δ)β), we use the weighted L1 –L∞ operator norm N2md (jc (δ); 2R, 2e−δKj R) of [4, Definition IV.2]. In the notation of this paper, N2md j(δ); 2R, 2e−δKj R ≤ e−δKj |||j(δ)||| ≤ 1 by Lemma 3.21.i. Hence, by [4, Proposition IV.4] f (α∗ , jc (δ)β) waux ≤ f δ If f is bilinear in α∗ , β then f (α∗ , jc (δ)β) 2δ 2 2R+ R+ 2R+ δKj = f δ f (α∗ , jc (δ)β) waux ≤ e 2R 2e−δKj R R If f has degree at least two both in α∗ and β 2 2 2R+ 2R+ f (α∗ , jc (δ)β) 2δ ≤ f (α∗ , jc (δ)β) waux 2R 2e−δKj R 4 R+ 2δKj ≤e f δ R since e−δKj ≥
R+ R
=
1 2eR +er
by Hypothesis F.7(i).
To treat the fluctuation integral in Definition 3.22, we introduce a second auxiliary weight system wfluct with metric 2md that gives weight 2R+ both to α∗ and β, and weight 32r to the fields z∗ and z. We write · fluct for · wfluct . If f (α∗ , β, z∗ , z) happens to be independent of z∗ and z, then f fluct = f 2δ . For the first two terms in the effective action A of Definition 3.22, we have Lemma 4.2. For any history complete analytic function f (α∗ , β) f (α∗ , z + jc (δ)β) − f (α∗ , jc (δ)β) fluct ≤ f δ f (z∗ + jc (δ)α∗ , β) − f (jc (δ)α∗ , β)) fluct ≤ f δ for all δ ≤ Θ. Proof. We prove the first inequality. By [4, Corollary IV.6], f (α∗ , z + jc (δ)β) − f (α∗ , jc (δ)β) fluct ≤ f (α∗ , z + jc (δ)β) fluct ≤ f (α∗ , β) δ since, by Lemma 3.21(i) N2md (1; 2R, 32 r) + N2md (jc (δ); 2R, 2R+ ) 32r R+ + |||jc (δ)||| ≤ |||1||| 2R R R 16r eδKj + + eδKj = 16(δv)eR + eR +er ≤ 1 ≤ R R 2 by Hypothesis F.7(i).
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The remaining summands in Definition 3.22 are explicit quadratic and quartic terms. The quadratic terms all involve the difference jc (δ) − j(δ) and are consequently exponentially small with c. Lemma 4.3. [j(δ) − jc (δ)]α∗ , [j(δ) − jc (δ)]β 2δ ≤ 4δ 2 Kj2 e2Kj δ R2+ e−2m c [j(δ) − jc (δ)]α∗ , z fluct , z∗ , [j(δ) − jc (δ)]β fluct ≤ 64 δ Kj eKj δ r R+ e−m c Proof. By definition α∗ , β 2δ = 4R2+ Therefore, by Lemma G.2(a), [j(δ) − jc (δ)]α∗ , [j(δ) − jc (δ)]β 2δ ≤ 4R2+ N2md (j(δ) − jc (δ); 2R+ , 2R+ )2 = 4R2+ N2md (j(δ) − jc (δ); 1, 1)2 ≤ 4R2+ |||j(δ) − jc (δ)|||2 ≤ 4δ 2 Kj2 e2Kj δ R2+ e−2m c In the last line, we used Lemma 3.21(iii). Similarly [j(δ) − jc (δ)]α∗ , z fluct ≤ 64 R+ r N2md (j(δ) − jc (δ); 2R+ , 2R+ ) ≤ 64 δ Kj eKj δ r R+ e−m c ¯ δ (Vδ (ε; · ); R1 + E1 , R2 + E2 ) are The explicit quartic terms in R [Vδ (ε; α∗ , jc (δ)β) − Vδ (ε; α∗ , j(δ)β)] + [Vδ (ε; jc (δ)α∗ , β) − Vδ (ε; j(δ)α∗ , β)] “downstairs”, and [Vδ (ε; α∗ , z + jc (δ)β) − Vδ (ε; α∗ , jc (δ)β)] + [Vδ (ε; z∗ + jc (δ)α∗ , β) − Vδ (ε; jc (δ)α∗ , β)] as a contribution to the effective action. The term “downstairs” again involves the difference jc (δ) − j(δ) and is exponentially small with c. Lemma 4.4. If δ ≤ Θ, then Vδ (ε; α∗ , jc (δ)β) − Vδ (ε; α∗ , j(δ)β) 2δ ≤ 64Kj e8Kj δ δ 2 |||v||| R4+ e−m c 1 1Vδ (ε; jc (δ)α∗ , β) − Vδ (ε; j(δ)α∗ , β) 2δ ≤ 64Kj e8Kj δ δ 2 |||v||| R4+ e−m c and Vδ (ε; α∗ , z + jc (δ)β) − Vδ (ε; α∗ , jc (δ)β) fluct ≤ 211 δ|||v||| r R3+ Vδ (ε; z∗ + jc (δ)α∗ , β) − Vδ (ε; jc (δ)α∗ , β) fluct ≤ 211 δ|||v||| r R3+
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Proof. We prove the first in each pair of inequalities. By Definition 3.8 Vδ (ε; α∗ , β) = −ε γ∗τ γτ +ε , v γ∗τ γτ +ε τ ∈εZ∩[0,δ)
with γ∗τ = j(τ )α∗
γτ = j(δ − τ )β
For the first inequality, we write Vδ (ε; α∗ , jc (δ)β) − Vδ (ε; α∗ , j(δ)β) =ε [ γ∗τ gτ +ε , v γ∗τ gτ +ε − γ∗τ g˜τ +ε , v γ∗τ g˜τ +ε ] τ ∈εZ∩[0,δ)
with gτ = j(δ − τ )j(δ)β
g˜τ = j(δ − τ )jc (δ)β
By definition α∗ β, v α∗ β 2δ ≤ 16|||v|||R4+ We apply Corollary G.3.ii, with d and d both replaced by 2md, δ = 0, r = 4, s = 2, h(γ1 , . . . , γ4 ) = γ1 γ2 , v γ3 γ4 , α1 = α∗ , α2 = β and Γ11 = Γ13 = j(τ ) ˜ 11 = Γ ˜ 13 = j(τ ) Γ
Γ21 = Γ23 = 0 Γ12 = Γ14 = 0 ˜ 21 = Γ ˜ 23 = 0 Γ ˜ 12 = Γ ˜ 14 = 0 Γ
Γ22 = Γ24 = j(δ − τ − ε)j(δ) ˜ 22 = Γ ˜ 24 = j(δ − τ − ε)jc (δ) Γ
As σ = max {N2md (j(τ ); 2R+ , 2R+ ), N2md (j(δ − τ − ε)j(δ); 2R+ , 2R+ ), N2md (j(δ − τ − ε)jc (δ); 2R+ , 2R+ )} ≤ max {|||j(τ )|||, |||j(δ − τ − ε)j(δ)|||, |||j(δ − τ − ε)jc (δ)|||} ≤ e2Kj δ σδ = N2md (j(δ − τ − ε)[j(δ) − jc (δ)]; 2R+ , 2R+ ) ≤ |||j(δ − τ − ε)||| |||j(δ) − jc (δ)||| ≤ δKj e2Kj δ e−m c by Lemma 3.21, it gives, for each τ ∈ εZ ∩ [0, δ) γ∗τ gτ +ε , v γ∗τ gτ +ε − γ∗τ g˜τ +ε , v γ∗τ g˜τ +ε 2δ ≤ 64|||v|||R4+ σδ σ 3 ≤ 64|||v|||R4+ δKj e8Kj δ e−m c Summing over τ and multiplying with ε gives the desired estimate. For the third inequality, we write Vδ (ε; α∗ , z + jc (δ)β) − Vδ (ε; α∗ , jc (δ)β) [ γ∗τ gˆτ +ε , v γ∗τ gˆτ +ε − γ∗τ g˜τ +ε , v γ∗τ g˜τ +ε ] = −ε τ ∈εZ∩[0,δ)
with gˆτ = j(δ − τ ) (z + jc (δ)β) = j(δ − τ )z + j(δ − τ )jc (δ)β
(4.2)
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This time, we apply Corollary G.3(ii), with d and d both replaced by 2md, δ = 0, r = 4, s = 3, h(γ1 , . . . , γ4 ) = γ1 γ2 , v γ3 γ4 , α1 = α∗ , α2 = β, α3 = z and ˆ 11 = Γ ˆ 13 = j(τ ) Γ ˆ ˆ ˜1 = Γ ˜ 1 = j(τ ) Γ 1 3
ˆ 22 = Γ ˆ 24 = j(δ − τ − ε)jc (δ) Γ ˆ ˆ ˜2 = Γ ˜ 2 = j(δ − τ − ε)jc (δ) Γ 2 4
ˆ 32 = Γ ˆ 34 = j(δ − τ − ε) Γ ˆ˜ 3 = Γ ˆ˜ 3 = 0 Γ (4.3) 2 4
ˆ j ’s being zero. Then ˜ ˆ j ’s and Γ with all other Γ i i σ ˆ = max {N2md (j(τ ); 2R+ , 2R+ ), N2md (j(δ − τ − ε)jc (δ); 2R+ , 2R+ ), N2md (j(δ − τ − ε); 2R+ , 32r) + N2md (j(δ − τ − ε)jc (δ); 2R+ , 2R+ )} ≤ max |||j(τ )|||, |||j(δ − τ − ε)jc (δ)|||, r |||j(δ − τ − ε)||| + |||j(δ − τ − ε)jc (δ)||| 16 R+ ≤ (1 + 16(δv)eR ) e2Kj δ σ ˆδ = N2md (j(δ − τ − ε); 2R+ , 32r) r |||j(δ − τ − ε)||| ≤ 16 R+ r Kj δ e ≤ 16 R+
(4.4)
So, as before, for each τ ∈ εZ ∩ [0, δ) γ∗τ gˆτ +ε , v γ∗τ gˆτ +ε − γ∗τ g˜τ +ε , v γ∗τ g˜τ +ε fluct ≤ 64|||v|||R4+ σ ˆδ σ ˆ3 ≤ 211 |||v||| r R3+ by Hypothesis F.7(i). Summing over τ gives the desired result.
Proof of Theorem 3.24. Set R(1) (α∗ , β) = R1 (α∗ , jc (δ)β) + R2 (jc (δ)α∗ , β) − [j(δ) − jc (δ)]α∗ , [j(δ) − jc (δ)]β E
(1)
(α∗ , β) = E1 (α∗ , jc (δ)β) + E2 (jc (δ)α∗ , β) + Vδ (ε; α∗ , jc (δ)β) −Vδ (ε; α∗ , j(δ)β) + Vδ (ε; jc (δ)α∗ , β) − Vδ (ε; j(δ)α∗ , β)
and Rl (α∗ , z) = R1 (α∗ , z) + [j(δ) − jc (δ)]α∗ , z Rr (α∗ , z) = R2 (z∗ , β) + z∗ , [j(δ) − jc (δ)]β El (α∗ , β, z) = [E1 (α∗ , z + jc (δ)β) − E1 (α∗ , jc (δ)β)] + [Vδ (ε; α∗ , z + jc (δ)β) − Vδ (ε; α∗ , jc (δ)β)] Er (z∗ , α∗ , β) = [E2 (z∗ + jc (δ)α∗ , β) − E2 (z∗ + jc (δ)α∗ , β)]
+ Vδ (ε; z∗ + jc (δ)α∗ , β) − Vδ ε; jc (δ)α∗ , β Clearly, R(1) (α∗ , β) is bilinear in α∗ , β and E (1) (α∗ , β) has degree at least two both in α∗ and β. Furthermore, by Lemmas 4.1, 4.3 and 4.4
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2 R+ ( R1 δ + R2 δ ) + 4δ 2 Kj2 e2Kj δ R2+ e−2m c R (4.5) 4 R+ 2δKj 8Kj δ 2 4 −m c ≤e ( E1 δ + E2 δ ) + 128 Kj e δ |||v||| R+ e R
R(1) 2δ ≤ eδKj E (1) 2δ
Also, Rl is bilinear in α∗ , z, and Rr is bilinear in z∗ , β. By Lemma 4.2 and Lemma 4.3 Rl fluct ≤ R1 δ + 64 δ Kj eKj δ r R+ e−m c Rr fluct ≤ R2 δ + 64 δ Kj eKj δ r R+ e−m c
(4.6)
Furthermore, El (α∗ , β, z) has degree at least two in the variable α∗ ; and the sums of the degrees of the variables β and z in each monomial of its power series expansion is also at least two. Also, by Lemmas 4.2 and 4.4 El fluct ≤ E1 δ + 211 δ|||v||| r R3+
(4.7)
There is the analogous statement for Er . By construction, the effective density in Definition 3.22 in the situation of Theorem 3.24 is A(α∗ , β; z∗ , z) = Rl (α∗ , z) + El (α∗ , β, z) + Rr (z∗ , β) + Er (z∗ , α∗ , β) so that ¯ δ (Vδ (ε; · ); R1 + E1 , R2 + E2 ) R ∗ ∗ 1 = R(1) + E (1) + log dμr (z ∗ , z) eRl (α∗ ,β,z)+El (α∗ ,β,z)+Rr (z ,β)+Er (z ,α∗ ,β) Z where Z = dμr (z ∗ , z). Set 1 (2) R (α∗ , β) = dμr (z ∗ , z) Rl (α∗ , z) Rr (z ∗ , β) Z By [4, Remark III.3.ii] 1 1 1 (2) 1 1R 1
2δ
1 1 1 1 = 1R(2) 1
fluct
≤ Rl fluct Rr fluct
Expanding the exponential in Definition 3.22 and using the fact that the dμr (z ∗ , z) integral is zero unless there are the same number of z’s and z ∗ ’s, one sees that ∗ 1 E (2) (α∗ , β) = log dμr (z ∗ , z) eA(α∗ ,β;z ,z) − R(2) (α∗ , β) Z has degree at least two both in α∗ and in β. By [4, Corollary III.5], with n = 2, there is a function E (α∗ , β) such that ∗ 1 log dμr (z ∗ , z) eA(α∗ ,β;z ,z) Z 1 2 = E (α∗ , β) + dμr (z ∗ , z) [Rl + El + Rr + Er ] 2Z 1 dμr (z ∗ , z) [Rl Er + El Rr + El Er ] = E (α∗ , β) + R(2) (α∗ , β) + Z
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225
3 (4.8)
1 is verified below.) Clearly (The hypothesis that A fluct < 32 1 E (2) = E + dμr (z ∗ , z) [Rl Er + El Rr + El Er ] Z
and, by [4, Remark III.3.ii] E (2) 2δ ≤ E 2δ + Rl fluct Er fluct + El fluct Rr fluct + El fluct Er fluct We set R = R(1) + R(2) ,
E = E (1) + E (2)
By construction ¯ δ (Vδ (ε; · ); R1 + E1 , R2 + E2 ) = R + E R Furthermore, R(α∗ , β) is bilinear in α∗ , β, and E(α∗ , β) has degree at least two ¯ δ (Vδ (ε; · ); R1 + E1 , R2 + both in α∗ and β. Hence R is the quadratic part of R ¯ E2 ), which coincides with the quadratic part of Rδ (0; R1 , R2 ). Also R 2δ ≤ R(1) 2δ + R(2) 2δ ≤ R(1) 2δ + Rl fluct Rr fluct 2 R+ ≤ eδKj ( R1 δ + R2 δ ) + (2δ)2 Kj2 e2Kj δ R2+ e−2m c R + R1 δ + 64 δ Kj eKj δ r R+ e−m c R2 δ + 64 δ Kj eKj δ r R+ e−m c The hypotheses on R1 δ and R2 δ imply that 2 R+ R 2δ ≤ 2 eδKj KR δ 2 r2 R2 e−2m c + (2δ)2 Kj2 e2Kj δ R2+ e−2m c R 2 + KR δ 2 r2 R2 e−2m c + 64 δ Kj eKj δ r R+ e−m c Kj2 e2Kj δ eδKj ≤ KR (2δ)2 r2+ R2+ e−2m c 1−2er + (2δv)2er 2 KR √ 2 KR δ e−mc 25+er Kj eKj δ √ + + 21−eR −2er (δv)eR +2er KR ≤ KR (2δ)2 r2+ R2+ e−2m c by (F.8a). Similarly E 2δ ≤ E (1) 2δ + E (2) 2δ ≤ E (1) 2δ + E 2δ + Rl fluct Er fluct + El fluct Rr fluct + El fluct Er fluct
(4.9)
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By (4.5) and the hypotheses in Theorem 3.24 4 R+ E (1) 2δ ≤ 2 e2δKj KE (δv)2 r2 R6 + 64 Kj e8Kj δ δ 2 v R4+ e−m c R e2δKj 16Kj e8Kj δ e−mc ≤ (2δv)2 r2+ R6+ KE 1−(2e +4e ) + R r r2+ R2+ v 2 By (4.6), (4.7) and the hypotheses of the Theorem Rl fluct , Rr fluct ≤ KR δ 2 r2 R2 e−2m c + 64 δ Kj eKj δ r R+ e−m c δ e−mc er δKj ≤ δ r+ R+ e−m c 2eR +2er KR + 64K 2 e j (δv)eR +2er ≤ δ r+ R+ e− 2 m c ≤ KE (δv)2 r2 R6 + 211 δ|||v||| r R3+ ≤ (2δv) r+ R3+ KE 23eR +4er −1 (δv) r R3 + 29 eer 1
El fluct , Er fluct
≤ 210 (2δv) r+ R3+ by (F.4b), Hypothesis F.7(i,ii), (F.6a) and the fact that er ≤ 0.1. Consequently, by (F.4b), 4 1 5 e− 2 m c 3 11 +2 A fluct = Rl + Rr + El + Er fluct ≤ (2δv) r+ R+ vR2+ ≤ 212 (2δv) r+ R3+ By (F.6a), this number is smaller than
(4.10) 1 60 .
Therefore, by (4.8),
E 2δ ≤ (30)3 A 3fluct ≤ 251 (2δv)3 r3+ R9+ Inserting all these estimates into (4.9) gives e2δKj 16Kj e8Kj δ e−mc 2 2 6 E 2δ ≤ (2δv) r+ R+ KE 1−(2e +4e ) + R r r2+ R2+ v 2 1 210 e− 2 mc + 220 + 251 (2δv)r+ R3+ + vR2+ ≤ KE (2δv)2 r2+ R6+
by (F.8b). The additional ingredient that we need for the proof of Theorem 3.27 is Lemma 4.5. Set W (α∗ , β) = Vδ (ε; α∗ , β) − Vδ
(ε 2
) ; α∗ , β
Then W (α∗ , j(δ)β) − W (α∗ , jc (δ)β) 2δ ≤ 28 e10Kj εδ 2 |||v|||R4+ e−m c W ( j(δ)α∗ , β) − W ( jc (δ)α∗ , β) 2δ ≤ 28 e10Kj εδ 2 |||v|||R4+ e−m c
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and W (α∗ , z + jc (δ)β) − W (α∗ , jc (δ)β) fluct ≤ 212 e9Kj εδ|||v|||rR3+ W (z∗ + jc (δ)α∗ , β) − W (jc (δ)α∗ , β) fluct ≤ 212 e9Kj εδ|||v|||rR3+ Proof. We prove the first of each of the two pairs of inequalities and use the same notation as in Lemma 4.4. By definition W (α∗ , β) = W1 (α∗ , β) + W2 (α∗ , β) with W1 (α∗ , β) =
ε 2
ε W2 (α∗ , β) = 2
τ ∈εZ∩[0,δ)
&
'
γ∗τ γτ + 2ε , vγ∗τ γτ + 2ε − γ∗τ γτ +ε , vγ∗τ γτ +ε
&
'
γ∗τ + 2ε γτ +ε , vγ∗τ + 2ε γτ +ε − γ∗τ γτ +ε , vγ∗τ γτ +ε
τ ∈εZ∩[0,δ)
Now, using the gτ and g˜τ of Lemma 4.4, W1 (α∗ , j(δ)β) − W1 (α∗ , jc (δ)β) ' ε & γ∗τ gτ + 2ε , v γ∗τ gτ + 2ε − γ∗τ gτ +ε , v γ∗τ gτ +ε = 2 τ ∈εZ∩[0,δ) & '
− γ∗τ g˜τ + 2ε , v γ∗τ g˜τ + 2ε + γ∗τ g˜τ +ε , v γ∗τ g˜τ +ε 3 (ε) (ε) ε 2 γ∗τ j gτ +ε , v γ∗τ j gτ +ε − γ∗τ gτ +ε , v γ∗τ gτ +ε = 2 2 2 τ ∈εZ∩[0,δ) 3 (ε) (ε) 2 g˜τ +ε , v γ∗τ j g˜τ +ε + γ∗τ g˜τ +ε , v γ∗τ g˜τ +ε − γ∗τ j 2 2 We apply Corollary G.3(iii), with the metrics d and d both replaced by 2md, the metric δ = 0, and with the same substitutions as in the first bound on Lemma 4.4, namely, Γ11 = Γ13 = j(τ ) ˜1 = Γ ˜ 1 = j(τ ) Γ 1 3
Γ21 = Γ23 = 0 Γ12 = Γ14 = 0 ˜2 = Γ ˜2 = 0 Γ ˜1 = Γ ˜1 = 0 Γ 1 3 2 4
Γ22 = Γ24 = j(δ − τ − ε)j(δ) ˜2 = Γ ˜ 2 = j(δ − τ − ε)jc (δ) Γ 2 4
and, in addition,
(ε) A˜2 = A˜4 = h (4.11) A1 = A˜1 = A3 = A˜3 = 1 A2 = A4 = j 2 The Corollary bounds the · 2δ norm of the τ term in terms of the σ and σδ of (4.2) and " (ε) # ε ||| ≤ eKj 2 a ≤ max |||1|||, |||j 2 ( (ε) ) (ε) (4.12) ε ε − h; 2R+ , 2R+ ≤ |||j − h||| ≤ Kj eKj 2 aδ = N2md j 2 2 2 by Lemma 3.21. Summing over τ and multiplying by 2ε , we get ε δ 2 4 16|||v|||R4+ σδ aδ (σa)3 2 ε ≤ 27 e10Kj εδ 2 |||v|||R4+ e−m c
W1 (α∗ , jc (δ)β) − W1 (α∗ , j(δ)β) 2δ ≤
The same estimate holds for W2 (α∗ , jc (δ)β) − W2 (α∗ , j(δ)β) 2δ .
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For the first of the second pair of inequalities, write, using the gˆτ and g˜τ of Lemma 4.4, W1 (α∗ , z + jc (δ)β) − W1 (α∗ , jc (δ)β) ' ε & γ∗τ gˆτ + 2ε , v γ∗τ gˆτ + 2ε − γ∗τ gˆτ +ε , v γ∗τ gˆτ +ε = 2 τ ∈εZ∩[0,δ) & '
− γ∗τ g˜τ + 2ε , v γ∗τ g˜τ + 2ε + γ∗τ g˜τ +ε , v γ∗τ g˜τ +ε 3 (ε) (ε) ε 2 γ∗τ j gˆτ +ε , v γ∗τ j gˆτ +ε − γ∗τ gˆτ +ε , v γ∗τ gˆτ +ε = 2 2 2 τ ∈εZ∩[0,δ) 3 (ε) (ε) 2 g˜τ +ε , v γ∗τ j g˜τ +ε + γ∗τ g˜τ +ε , v γ∗τ g˜τ +ε − γ∗τ j 2 2 ˆ˜ j ’s of (4.3) together ˆ j ’s and Γ This time, we apply Corollary G.3(iii) using the Γ i
with the Ai ’s and A˜i ’s of (4.11). By (4.4) and (4.12)
i
ε δ 2 4 16|||v|||R4+ σ ˆδ aδ (ˆ σ a)3 2 ε ≤ 211 e9Kj εδ|||v|||rR3+
W1 (α∗ , z + jc (δ)β) − W1 (α∗ , jc (δ)β) fluct ≤
The same estimate holds for W2 (α∗ , z + jc (δ)β) − W2 (α∗ , jc (δ)β) fluct .
Proof of Theorem 3.27. We use the same notation as in the proof of Theo˜ (1) , E˜(1) , R ˜ l, R ˜ r , E˜l , E˜r and A˜ in the same way as the rem 3.24. Define R corresponding “untitled” quantities were defined at the beginning of the proof of Theorem 3.24. They fulfill the same bounds as their untitled siblings. Furthermore 1 1 1 1 ˜ (1) 1R − R(1) 1 2δ 1 1 1 1 1 ˜ 1 1 ˜ 1 ≤ 1(R1 − R1 ) (α∗ , jc (δ)β)1 + 1(R 2 − R2 ) (jc (δ)α∗ , β)1 2δ 2δ 2 R+ ≤ 2eδKj ΔR R 1 1 1 ˜(1) 1 1E − E (1) 1 2δ 1 1 1 1 1 1 1 1 ≤ 1(E˜1 − E1 ) (α∗ , jc (δ)β)1 + 1(E˜2 − E2 ) (jc (δ)α∗ , β)1 2δ
+ W (α∗ , jc (δ)β) − W (α∗ , j(δ)β) 2δ + W (jc (δ)α∗ , β) − W (j(δ)α∗ , β) 2δ 4 R+ 2δKj ≤ 2e ΔE + 29 e10Kj εδ 2 |||v|||R4+ e−m c R by Lemmas 4.1 and 4.5. Also 1 1 1 1 1 1˜ 1 1˜ = 1R 1Rl − Rl 1 1 (α∗ , z) − R1 (α∗ , z)1 fluct fluct 1 R+ 16r 1 1˜ 1 = 1R1 (α∗ , β) − R1 (α∗ , β)1 R R δ
2δ
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1 1 1 1 ≤ 1(E˜1 − E1 ) (α∗ , z + jc (δ)β) − (E˜1 − E1 ) (α∗ , jc (δ)β)1
fluct
+ W (α∗ , z + jc (δ)β) − W (α∗ , j(δ)β) fluct 1 1 1 1˜ ≤ 1E1 − E1 1 + 212 e9Kj εδ|||v|||rR3+ δ
by Lemmas 4.2 and 4.5. Similarly 1 1 1 R+ 16r 1 1˜ 1 1 1˜ = 1R2 − R2 1 1Rr − Rr 1 R R fluct δ 1 1 1 1 1˜ 1 1 1˜ 12 9Kj ≤ 1E2 − E2 1 + 2 e εδ|||v|||rR3+ 1Er − Er 1 fluct
Consequently 1 1 1 1˜ 1A − A1
δ
32r ΔR + 2ΔE + 213 e9Kj εδ|||v|||rR3+ R For the fluctuation integral, we have fluct
≤
˜ (2) − R(2) R 1 ˜ l −Rl )(α∗ , z) R ˜ r (z ∗ , β)+Rl (α∗ , z) (R ˜ r −Rr )(z ∗ , β) = dμr (z ∗ , z) (R Z By [4, Remark III.3.ii] and (4.6), 1 1 1 1 1 ˜ (2) 1 1 1 ˜ (2) − R(2) 1 1R − R(2) 1 = 1R 2δ
fluct
˜ l − Rl fluct R ˜ r fluct + Rl fluct R ˜ r − Rr fluct ≤ R 32r ΔR ≤ KR δ 2 r2 R2 e−2mc + 64 δ Kj eKj δ r R+ e−m c R 12 2Kj 2 −m c ≤2 e δ r+ e ΔR by (F.4b) and (F.6a). Furthermore ) ( E˜(2) − E (2) (α∗ , β) ∗ ∗ 1 1 ˜ dμr (z ∗ , z) eA(α∗ ,β;z ,z) − log dμr (z ∗ , z) eA(α∗ ,β;z ,z) = log Z Z (2) (2) ˜ −(R − R )(α∗ , β) We apply [4, Corollary III.6] to the difference of the two logarithms of integrals. Since each monomial of A and A˜ contains either only z’s or only z ∗ ’s, and hence integrates to zero, the first hypothesis of this corollary is satisfied. For the second hypothesis, we have, by (4.10), A fluct + A˜ − A fluct ) 16r ( ˜ ˜ 2 − R2 δ R1 − R1 δ + R ≤ 212 (2δv)r+ R3+ + R ( ) + E˜1 − E1 δ + E˜2 − E2 δ + 213 e9Kj εδ|||v|||rR3+ ≤ 213 (2δv)r+ R3+ +
) 16r ( ˜ ˜ 2 δ + R2 δ R1 δ + R1 δ + R R
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( ) + E˜1 δ + E1 δ + E˜2 δ + E2 δ ≤ 214 (2δv)r+ R3+ ≤
1 34
by Hypothesis F.7(i) and (F.6a). The corollary gives 1 1 1 1 ˜(2) 1E − E (2) 1 2δ 32r 2 14 3 13 9Kj 3 ΔR + 2ΔE + 2 e ≤ 4(34) 2 (2δv)r+ R+ εδ|||v|||rR+ R 1 1 1 ˜ (2) 1 + 1R − R(2) 1 2δ
≤ 233 e2Kj (2δv)r2+ R2+ ΔR + 228 (2δv)r+ R3+ ΔE + 239 e9Kj ε(2δv)2 r2+ R6+ Combining the above bounds, we have 1 1 1 1 1 1 1 1 1 1 ˜ (1) 1 ˜ (2) 1˜ − R(1) 1 + 1R − R(2) 1 1R − R1 ≤ 1R 2δ 2δ 2δ 4 5 2 R+ δKj 12 2Kj 2 −m c ≤ 2e +2 e δ r+ e ΔR R 1 1 1 1˜ 1E − E 1
2δ
≤ 2ΔR 1 1 1 1 1 1 1 1 ≤ 1E˜(1) − E (1) 1 + 1E˜(2) − E (2) 1 2δ 2δ 4 5 4 R + ≤ 2e2δKj +228 (2δv)r+ R3+ ΔE +233 e2Kj (2δv)r2+ R2+ ΔR R +29 e10Kj εδ 2 |||v|||R4+ e−m c + 239 e9Kj ε(2δv)2 r2+ R6+ ≤ ΔE + 233 e2Kj (2δv)r2+ R2+ ΔR + 240 e10Kj ε(2δv)2 r2+ R6+
by (F.4b), (F.6a), Hypothesis F.7(i) and (F.8c).
5. The Decimation Step in all of Space The “decimation step” for all large and small field regions is formulated in Theorems 3.26 and 3.28. We shall prove them in this section. As in Sect. 4, write r = r(δ)
R = R(δ)
R = R (δ)
r+ = r(2δ)
R+ = R(2δ)
R + = R (2δ)
Recall that, by (2.18), 1 R+ 1 R + 1 r+ = er = eR +er = e +er R r 2 R 2 R 2
(5.1)
The hierarchies S1 and S1 of Theorem 3.26 each specify the large and small field sets for decimation intervals of length at most δ. For a decimation interval
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J of length at most δ contained in [0, 2δ], we set, for X = Λ, Ω, Pα , Pβ , Pα , Pβ , Q, R XS1 (J ) if J ⊂ [0, δ] X (J ) = XS2 (J − δ) if J ⊂ [δ, 2δ] Then X (J ) = XS (J ) for all hierarchies S of scale 2δ with (S1 , S2 ) ≺ S and all J contained in either [0, δ] or [δ, 2δ]. We also set Ω0 = Ω1 ∩ Ω2 . 5.1. The New Small Field/Large Field Decomposition of the First Kind By hypothesis, the integrand of the integral defining I(α∗ , β) in Theorem 3.26 contains the product χδ (Ω1 ; α∗ , φ) χδ (Ω2 ; φ∗ , β) of characteristic functions. These characteristic functions impose small field conditions for the decimation intervals [0, δ] and [δ, 2δ]. For example, the first characteristic function contains a factor which vanishes unless |α(x)| ≤ R(δ) for each x ∈ Ω1 . The representation for I(α∗ , β) in the conclusion of Theorem 3.26 contains the characteristic function χ2δ (ΩS ; α, β) which imposes small field conditions for the decimation interval [0, 2δ]. For example, it vanishes unless |α(x)| ≤ R(2δ) for each x ∈ ΩS . The first step in the proof of Theorem 3.26 builds χ2δ (ΩS ; α, β) from the product χδ (Ω1 ; α∗ , φ) χδ (Ω2 ; φ∗ , β). To illustrate the construction procedure, we consider the conditions on |α(x)|. We expand the existing conditions on |α(x)|, for x ∈ Ω0 = Ω1 ∩ Ω2 , to χR(δ) (Ω0 , α) = χR(δ) (α(x)) x∈Ω0
=
χR(2δ) (α(x)) + χR(2δ),R(δ) (α(x))
x∈Ω0
=
χR(2δ) (Ω0 \Pα , α) χR(2δ),R(δ) (Pα , α)
Pα ⊂Ω0
where we are using Notation 5.1. Set, for 0 < r < R, t ∈ C, any set Y and any complex valued function f on Y 1 if |t| ≤ r 1 if r < |t| ≤ R χr (t) = χr,R (t) = 0 otherwise 0 otherwise and χr (Y, f ) =
x∈Y
χr (f (x))
χr,R (Y, f ) =
χr,R (f (x))
x∈Y
The characteristic function χR(2δ) (Ω0 \Pα , α) successfully imposes the new small field condition of scale 2δ on |α(x)| at each point of Ω0 \Pα . The characteristic function χR(2δ),R(δ) (Pα , α) says that |α(x)| violates the new small
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field condition at each x ∈ Pα . In Lemma A.3, we perform a similar expansion for the other small field conditions as well. The conclusion of that Lemma is χδ (Ω1 ; α, φ) χδ (Ω2 ; φ, β) χ2δ (ΛA ; α, β) χr (ΛA , α − φ) χr (ΛA , φ − β) = A∈Fδ (Ω0 )
×χA,δ (Ω1 , Ω2 ; α, φ, β)
(5.2)
where Fδ (Ω0 ) is the set of possible configurations of large field/small field sets for the decimation interval [0, 2δ] that are compatible with (S1 , S2 ) (see Definition 3.2) and the associated characteristic function χA,δ (Ω1 , Ω2 ; α, φ, β) is given in Definition A.1. Note, for future reference, that the factor χA,δ (Ω1 , Ω2 ; α, φ, β) does not depend on the values of the fields α, φ and β at points of Λ. Recall, from Definition 3.1, the notation α l = (ατ (x))τ ∈εZ∩(0,δ) x∈X
α r = (ατ +δ (x))τ ∈εZ∩(0,δ) x∈X
for a system α = (ατ (x))τ ∈εZ∩(0,2δ) of fields. x∈X
As an immediate consequence of (5.2) we have that the I(α∗ , β) of Theorem 3.26 is I(α∗ , β) =
2|ΛA |
Zδ
A∈Fδ (Ω0 )
∗ dφ(x)∗ ∧ dφ(x) Int(A;α∗ ,β) e−φ ,φ 2πi x∈Λ A
×χ2δ (ΛA ; α, β)χr (ΛA , α − φ)χr (ΛA , φ − β) ∗
×e−QS1 (α ∗
×eD1 (α
,φ; α ∗ αl )−QS2 (φ∗ ,β; α ∗ αr ) l , r ,
∗ ,φ; ρ r ) l )+D2 (φ ,β; ρ
∗
eVS1 (ε; α
,φ; α ∗ αl )+VS2 (ε; φ∗ ,β; α ∗ αr ) l , r ,
b1 (α∗ , φ ; ρ l )b2 (φ∗ , β; ρ r )
where Int(A;α∗ ,β) is the integral operator Int(A;α∗ ,β) [dμ(φ∗ , φ; α , z∗ , z)] dφ(x)∗ ∧ dφ(x) |Ω1\ΛA | |Ω2\ΛA | χR(ε) (φ(x)) χA,δ (Ω1 , Ω2 ; α, φ, β) Zδ = Zδ 2πi x∈X\ΛA
αl , z∗l , zl )] I(S2 ;φ∗ ,β) [dμ( αr , z∗r , zr )] × I(S1 ;α∗ ,φ) [dμ( ∗ dφ(x) ∧ dφ(x) |Ω \Λ | |Ω \Λ | χA,δ (Ω1 , Ω2 ; α, φ, β) = Zδ 1 A Zδ 2 A 2πi x∈X\ΛA
αl , z∗l , zl )] I(S2 ;φ∗ ,β) [dμ( αr , z∗r , zr )] × I(S1 ;α∗ ,φ) [dμ( by Lemma A.4(b) By Lemma 3.10(ii), I(S1 ;α∗ ,φ) and I(S2 ;φ∗ ,β) are independent of φ(x) for all x ∈ ΛA . Inspection of the definition of χA,δ (Ω1 , Ω2 ; α, φ, β) in Definition A.1 shows that it is also independent of φ(x) for all x ∈ ΛA . By l ) and b2 (φ∗ , β; ρr ) are pure large field and thus indehypothesis, b1 (α∗ , φ; ρ pendent of φ∗ (x), φ(x) for all x ∈ ΛA . Therefore we may move the integral 0 dφ(x)∗ ∧dφ(x) to the right to give x∈ΛA 2πi
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Corollary 5.2. I(α∗ , β) =
Int(A;α∗ ,β) b1 (α∗ , φ ; ρ l )b2 (φ∗ , β; ρ r ) χ2δ (ΛA ; α, β)
A∈Fδ (Ω0 ) 2|ΛA |
×Zδ
dφ(x)∗ ∧ dφ(x) −φ∗ ,φ χr (ΛA , α − φ)χr (ΛA , φ − β) e 2πi
x∈ΛA
−QS1 (α∗ ,φ; α ∗ αl )−QS2 (φ∗ ,β; α ∗ αr ) l , r ,
×e
∗
×eD1 (α
∗
∗
eVS1 (ε; α
,φ; α ∗ αl )+VS2 (ε; φ∗ ,β; α ∗ αr ) l , r ,
,φ; ρ l )+D2 (φ ,β; ρ r )
Observe that the integrand above is independent of the variables z∗δ , zδ and α∗δ , αδ . We will shortly introduce z∗ = z∗δ and z = zδ as the fluctuation fields associated to φ∗ , φ, and α∗δ , αδ as the values of φ∗ , φ in the new large field region of the second kind where no fluctuation fields are introduced. 5.2. Approximate Diagonalization of the Quadratic Form In this subsection, we fix A ∈ Fδ (Ω0 ) and study the small field integral, that is the integral over the variables φ(x), x ∈ ΛA , in the conclusion of Corollary 5.2. To simplify notation, write Λ = ΛA . So, we study the integral JA (α∗ , β; ρ l , ρ r ; Λ c φ∗ , Λ c φ ) dφ(x)∗ ∧ dφ(x) χr (Λ, α − φ)χr (Λ, φ − β) = 2πi x∈Λ
∗
× e−QS1 (α∗ ,φ; α ∗l , αl )−φ
,φ −QS2 (φ∗ ,β; α ∗r , αr ) ∗
× eVS1 (ε; α∗ ,φ; α ∗l , αl )+VS2 (ε; φ
,β; α ∗r , αr )
∗
eD1 (α∗ ,φ; ρ l )+D2 (φ
,β; ρ r )
(5.3)
To “compute” this integral, we use “stationary phase” in many complex variables as discussed in Sect. 1. As we saw there, if we treat φ∗ and φ as independent variables, the critical value of φ∗ for the quadratic part of the effective action is not the complex conjugate of the critical value of φ. To produce a mathematically rigorous argument we introduce independent complex variables φ∗ (x), φ(x) in CΛ and write (5.3) as dφ∗ (x) ∧ dφ(x) e−QS1 (α∗ ,φ; α∗l , αl )−φ∗ ,φ −QS2 (φ∗ ,β; α∗r , αr ) JA = 2πi D(α∗ ,β)
x∈Λ
×eVS1 (ε;α∗ ,φ; α∗l , αl )+VS2 (ε;φ∗ ,β; α∗r , αr ) eD1 (α∗ ,φ; ρl )+D2 (φ∗ ,β; ρr )
(5.4)
where the domain of integration is D(α∗ , β) = Xx∈Λ D(x; α∗ , β) with 2 ∗ D(x; α∗ , β) = (φ∗ (x), φ(x)) ∈ C φ∗ (x) = φ(x) (5.5) |φ∗ (x) − α∗ (x)| ≤ r, |φ(x) − β(x)| ≤ r A good approximation to the critical point of the bilinear form QS1 (α∗ , φ; α ∗l , α l ) + φ∗ , φ + QS2 (φ∗ , β; α ∗r , α r)
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(in the variables φ∗ , φ, with α∗ , α ∗, α , β considered as parameters) is14 φcr ∗ = Λjc (δ) α∗
φcr = Λjc (δ) β
where the cut off propagator jc was defined in (1.24). We shall expand around the approximate critical point. Therefore we make a change of variables from φ∗ (x), φ(x), x ∈ Λ to variables z∗ (x), z(x) with x ∈ Λ by φ∗ (x) = z∗ (x) + [jc (δ) α∗ ](x),
φ(x) = z(x) + [jc (δ) β](x)
when x ∈ Λ
Under this change of variables, we have, on Λ, φ∗∗ − φ = z∗∗ − z + jc (δ) [α∗∗ − β] φ∗ − α∗ = z∗ − [1 − jc (δ)] α∗ φ − β = z − [1 − jc (δ)] β Thus the change of variables transforms the domain D(x; α∗ , β) of (5.5) into " D (x; α∗ , β) = (z∗ (x), z(x)) ∈ C2 |z∗ (x) − ([1 − jc (δ)]α∗ ) (x)| ≤ r, |z(x) − ([1 − jc (δ)]β) (x)| ≤ r,
# z(x) − z∗ (x)∗ = (jc (δ)[α∗∗ − β]) (x)
(5.6)
Observe that D (x; α∗ , β) depends only on the values of the fields α∗ and β at points y ∈ X with d(x, y) ≤ c. Also D (x; α∗ , β) depends on h through jc (δ). For convenience, we rename φ(x) = αδ (x) and φ∗ (x) = α∗δ (x) when x ∈ Λc , and also define z(x) = z∗ (x) = 0 for all x ∈ Λc and α∗δ (x) = αδ (x) = 0 for x ∈ Λ. To this point, we have obtained the following expression for the function JA of (5.3). Lemma 5.3. Set D (α∗ , β) = Xx∈Λ D (x; α∗ , β). Then l , ρ r , α∗δ , αδ ) JA (α∗ , β; ρ dz∗ (x) ∧ dz(x) = e−z∗ (x)z(x) efA (α∗ ,β; ρ ; z∗ ,z) 2πi D (α∗ ,β)
x∈Λ
where ; z∗ , z) = fA (α∗ , β; α ∗, α , z∗ , z ; z∗ , z) fA (α∗ , β; ρ cr = − z∗ , Λjc (δ)β − Λjc (δ)α∗ , z − α∗δ , αδcr cr ∗ cr −QS1 (α∗ , z + αδ ; α l , α l ) − QS2 (z∗ + α∗δ , β; α ∗r , α r )
cr +VS1 (ε; α∗ , z+αδcr ; α l∗ , α l ) + VS2 (ε; z∗ +α∗δ , β; α ∗r , α r ) cr cr +D1 (α∗ , z + αδ ; ρ l ) + D2 (z∗ + α∗δ , β; ρ r )
and cr cr = α∗δ (α∗ , α∗δ ) = Λjc (δ)α∗ + Λc α∗δ α∗δ 14
αδcr = αδcr (αδ , β) = Λjc (δ)β + Λc αδ
If one approximated QS1 (α∗ , φ; · ) and QS2 (φ∗ , β; · ) by their pure small field parts − α∗ , j(δ)φ and − φ∗ , j(δ)β, respectively (see Lemma 3.7(ii)), the critical point would be exactly φcrit = j(δ) α∗ , φcrit = j(δ) β as in (1.13). The cutoff jc (δ) was motivated near ∗ (1.24). See also Lemma 3.10(ii), which was used to give Corollary 5.2.
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By Lemma 3.7(ii), fA h=0 = − Furthermore, fA +
τ
235
Λcτ α∗τ , Λcτ ατ
τ ∈εZ∩(0,2δ)
Λcτ α∗τ , Λcτ ατ is history complete.
5.3. Stokes’ Theorem and the Small Field/Large Field Decomposition of the Second Kind We apply [5, Lemma A.1] to the integral in Lemma 5.3 with X replaced by Λ, with ρ(x) = (jc (δ)[α∗∗ − β])(x), σ∗ (x) = ([1 − jc (δ)]α∗ )(x), σ(x) = ([1 − jc (δ)]β)(x) and with r = r. We block the resulting sum over R into = R⊂Λ
Ω⊂Λ
R⊂Λ Ω=Λ\L(c(δ),R)
where L(c(δ), R) is the set of points of X that are within a distance c(δ) of R. This gives ⎛ ⎞ ⎜ dz∗ (x) ∧ dz(x) −z∗ (x)z(x) ⎟ JA = e ⎝ ⎠ 2πi Ω⊂Λ
R⊂Λ x∈R Ω=Λ\L(c(δ),R)
⎛
×
⎜ ⎝ x∈Λ\R
|z(x)|≤r
C(x;α∗ ,β)
⎞
dz(x) ∧ dz(x) −z(x)∗ z(x) ⎟ fA (α∗ ,β; ρ ; z∗ ,z) e z∗ (x)=z(x)∗ ⎠e 2πi for x∈Λ\R ∗
(5.7) with, for each x ∈ Λ, C(x; α∗ , β) a two real dimensional surface in
P(x) = (z∗ (x), z(x)) ∈ C2 |z∗ (x)|, |z(x)| < R whose boundary is the union of the circle {(z∗ (x), z(x)) ∈ C2 | z∗∗ (x) = z(x), |z(x)| = r} and the curve bounding D (x; α∗ , β). We now verify the hypotheses of [5, Lemma A.1], for arguments that appear in the integral operator. For this, we must show that the function fA (α∗ , β; ρ ; z∗ , z) is analytic in Xx∈Λ P(x), for all allowed (α∗ , β; ρ ), and that, for each x ∈ Λ, the two boundary curves are contained in P(x). ◦ As Di Si , i = 1, 2 are finite, Xx∈Λ P(x) will be contained in the domain ; z∗ , z) provided of analyticity of fA (α∗ , β; ρ cr |z∗ (x) + α∗δ (x)| < κ∗S2 ,0 (x) = 2R
|z(x) + αδcr (x)| < κS1 ,δ (x) = 2R
for all x ∈ Λ and (z∗ (x), z(x)) ∈ P(x). This is the case because cr |α∗δ (x)| = |(jc (δ)α∗ ) (x)|
≤ |||jc (δ)|||
sup
y∈X d(x,y)≤c
|α(y)| ≤ |||jc (δ)|||R+ ≤ eKj δ R+ < R
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The second inequality follows from the observation that |α(y)| ≤ R+ for all y within a distance c of Λ. When y ∈ Λ, this is enforced by the characteristic function χ2δ (ΛA ; α, β) in Corollary 5.2. When y is not in Λ, but is within a distance c of Λ, this is enforced by the factor of χR2δ (Ω0 \(Pα ∪ Λ), α) in the characteristic function15 χA,δ (Ω1 , Ω2 ; α, αδ , β) appearing in the definition of IntA,ε in Corollary 5.2. (Recall that d(Λ, Pα ∪ Ωc0 ) > c(δ) ≥ c. The third inequality follows from Lemma 3.21(i). The fourth is implied by Hypothesis F.7(i). ◦ Since r < R, the boundary circle {z∗∗ (x) = z(x), |z(x)| = r} ⊂ P(x). ◦ The boundary of D (x; α∗ , β) is contained in P(x) provided |([1 − jc (δ)]α∗ ) (x)| + r , |([1 − jc (δ)]β) (x)| + r < R If h(x) = 0, this condition reduces to |α(x)| + r < R and |β(x)| + r < R. Since x ∈ Λ, both follow from R+ + r < R. See (F.3d) in Appendix F. If h(x) = 1, |([1−jc (δ)]α∗ ) (x)| ≤ |||h − jc (δ)|||
sup
y∈X d(x,y)≤c
|α∗ (y)| ≤ δKj eKj δ R+ < R − r
by Lemma 3.21(ii), (F.3d) and Hypothesis F.7(i). Under Stokes’ theorem we may choose any surface C(x; α∗ , β) that has the specified boundary and lies in the domain of analyticity of the integrand. We choose C(x; α∗ , β) to depend only on the values of the fields α and β at points y ∈ X with d(x, y) ≤ c. This is possible because D (x; α∗ , β) has the same property. Combining Corollary 5.2, (5.3), and (5.7) gives I(α∗ , β) = Int(A;α∗ ,β) b1 (α∗ , αδ , ρl ) b2 (αδ∗ , β, ρr ) χ2δ (ΛA ; α, β) A∈Fδ (Ω0 )
⎛
2|ΛA |
× Zδ
Ω⊂ΛA
×
⎜ ⎝
⎞ dz∗ (x)∧dz(x) −z∗ (x)z(x)⎟ e ⎠ 2πi
R⊂ΛA x∈R C(x;α∗ ,β) Ω=ΛA \L(c(δ),R) ∗
dμ(ΛA \R),r (z ∗ , z) efA (α
z∗ (x)=z(x)∗
,β; ρ ;z∗ ,z)
for x∈ΛA \R
with, for each x ∈ X\ΛA , the integration variable φ(x) of Int(A;α∗ ,β) renamed to αδ (x). Renaming the fields z∗ (x), z(x) with x ∈ ΛA \Ω to z∗δ (x), zδ (x), I= Int(Ω,R,A;α∗ ,β) A∈Fδ (Ω0 ) Ω⊂ΛA
R⊂ΛA Ω=ΛA \L(c(δ),R) ×b1 (α∗ , αδ , ρl )b2 (αδ∗ , β, ρr )χ2δ (Ω; α, β) 2|Ω|
×Zδ 15
∗
dμΩ,r (z ∗ , z)ef(Ω,R,A) (α
z∗δ (x)=zδ (x)∗
,β; ρ ;z∗ ,z)
for x∈ΛA \R
The characteristic function χA,δ (Ω1 , Ω2 ; α, αδ , β) is defined in Definition A.1.
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where 2|ΛA\Ω|
Int(Ω,R,A;α∗ ,β) = Int(A;α∗ ,β) Zδ ⎛ ⎜ ⎜ × ⎝ x∈R
χ2δ (ΛA \Ω; α, β)
⎞ dz∗δ (x) ∧ dzδ (x) −z∗δ (x)zδ (x)⎟ ⎟ dμΛ \(R∪Ω), r (z ∗ , zδ ) e δ ⎠ A 2πi
C(x;α∗ ,β)
and f(Ω,R,A) is obtained from the function fA of Lemma 5.3 by f(Ω,R,A) (α∗ , β; ρ ; z∗ , z) = fA (α∗ , β; ρ ; (Λ\Ω)z∗δ + Ωz∗ , (Λ\Ω)zδ + Ωz) (5.8) z∗δ (x), zδ (x) with x ∈ Λ\Ω are also “residual variables” and subsumed in ρ . The fields z∗ (x), z(x) with x ∈ Ω are the “fluctuation fields” to be integrated out. For each Ω, R, A = (Λ, Pα , · · · , Q) in the above sum, we define the hierarchy S with (S1 , S2 ) < S by setting X ([0, 2δ]) = X for X = Ω, R, Λ, . . ., Q. Using the recursion relation of Lemma 3.10(i), we have Int(Ω,R,A;α∗ ,β) = I(S; α∗ ,β) . We set fS = f(Ω,R,A) . Then Lemma 5.4. I(α∗ , β) =
I(S;α∗ ,β) b1 (α∗ , αδ , ρl ) b2 (αδ∗ , β, ρr ) χ2δ (ΩS ; α, β)
hierarchies S for scale 2δ (S1 ,S2 )≺S 2|ΩS |
×Zδ
∗
dμΩS ,r (z ∗ , z) efS (α
z∗δ (x)=zδ (x)∗
,β; ρ ;z ∗ ,z)
for x∈ΛS \R
where fS is given by (5.8). Under the hypotheses of Theorem 3.28, we have the analogous representation ˜ ∗ , β) = I(α I(S;α∗ ,β) ˜b1 (α∗ , αδ , ρl ) ˜b2 (αδ∗ , β, ρr ) χ2δ (ΩS ; α, β) hierarchies S for scale 2δ (S1 ,S2 )≺S 2|ΩS |
×Zδ
˜
∗
dμΩS ,r (z ∗ , z) efS (α
z∗δ (x)=zδ (x)∗
,β; ρ ;z ∗ ,z)
(5.9)
for x∈ΛS \R
In Remark 5.5, below, we give explicit descriptions of fS and f˜S . 5.4. Preparing for the Analysis of the Fluctuation Integral Fix a hierarchy S of scale 2δ. If J is a decimation interval for S, write Ω(J ) = ΩS (J ) and Λ(J ) = ΛS (J ). Again, we use the notation Ω = ΩS ([0, 2δ]), Λ = ΛS ([0, 2δ]). We will sometimes also shorten Ωz∗ + (Λ\Ω)z∗δ to z∗ + z∗δ and Ωz + (Λ\Ω)zδ to z + zδ . Remark 5.5. The function fS (α∗ , β; ρ ; z∗ , z) of (5.8) that appears in Lemma 5.4 is the sum of
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(i) the quadratic part cr , αδcr − z∗ + z∗δ , Λjc (δ)β − Λjc (δ)α∗ , z + zδ − α∗δ cr cr −QS1 (α∗ , z + zδ + αδ ; α ∗l , α l ) − QS2 (z∗ + z∗δ + α∗δ , β; α ∗r , α r )
which, by the bilinearity of Q and the observation that cr cr ,α ∗r ) = Γ∗S2 ( · ; z∗ , 0) + Γ∗S2 ( · ; z∗δ + α∗δ ,α ∗r ) Γ∗S2 ( · ; z∗ + z∗δ + α∗δ cr cr Γ S1 ( · ; α l , z + zδ + αδ ) = ΓS1 ( · ; 0, z) + ΓS1 ( · ; α l , zδ + αδ )
can be written in the form −QS (α∗ , β; α
∗, α
)+δQ(α∗ , β; ρ
) − [z∗ , Ωjc (δ)β + QS2 (z∗ , β; 0, α
r )] − [Ωjc (δ)α∗ , z + QS1 (α∗ , z; α
∗l , 0)]
where δQ(α∗ , β; ρ
) cr = QS (α∗ , β; α
∗, α
)−QS1 (α∗ , zδ + αδcr ; α
∗l , α
l )−QS2 (z∗δ + α∗δ , β; α
∗r , α
r) cr − α∗δ , αδcr − z∗δ , (Λ\Ω)jc (δ)β − (Λ\Ω)jc (δ)α∗ , zδ
= [QS (α∗ , β; α
∗, α
) − QS1 (α∗ , αδcr ; α
∗l , α
l) cr cr − QS2 (α∗δ , β; α
∗r , α
r ) − α∗δ , αδcr ]
− [z∗δ , (Λ\Ω)jc (δ)β + QS2 (z∗δ , β; 0, α
r )] − [(Λ\Ω)jc (δ)α∗ , zδ + QS1 (α∗ , zδ ; α
∗l , 0)]
(5.10)
(ii) the quartic part VS1 (ε; α∗ , z + zδ + α ∗l , α l ) + VS2 (ε; z∗ + z∗δ + cr , β; α ∗r , α r ) and α∗δ cr l ) + D2 (z∗ + z∗δ + α∗δ , β; ρ r ) from the non large (iii) D1 (α∗ , z + zδ + αδcr ; ρ field terms. αδcr ;
Remark 5.6. fS Ω = α∗ , j(2δ)β Ω − [j(δ) − jc (δ)]α∗ , [j(δ) − jc (δ)]β Ω + z∗ , [j(δ) − jc (δ)]β Ω + [j(δ) − jc (δ)]α∗ , z Ω − Λcτ α∗τ , Λcτ ατ τ ∈εZ∩(0,2δ)
+ [VΩ,δ (ε; α∗ , z + jc (δ)β) + VΩ,δ (ε; z∗ + jc (δ)α∗ , β)] Ω + [D1 (α∗ , z + jc (δ)β; 0 ) + D2 (z∗ + jc (δ)α∗ , β; 0 )] Ω
Proof. By its definition in Lemma 5.3 cr α∗δ = jc (δ)α∗ |Ω + Λc α∗δ
αδcr Ω = jc (δ)β|Ω + Λc αδ (5.11) By Lemma 3.7(ii), the quadratic part of fS Ω is equal to − z∗ , jc (δ)β Ω − jc (δ)α∗ , z |Ω − jc (δ)α∗ , jc (δ)β|Ω − Λc α∗δ , Λc αδ + α∗ , j(δ) (z + jc (δ)β) Ω − Λcτ α∗τ , Λcτ ατ Ω
τ ∈(0,δ)
' + z∗ + jc (δ)α∗ , j(δ)β Ω − Λcτ α∗τ , Λcτ ατ &
τ ∈(δ,2δ)
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= α∗ , j(2δ)β Ω − [j(δ) − jc (δ)]α∗ , [j(δ) − jc (δ)]β Ω +z∗ , [j(δ) − jc (δ)]β Ω +[j(δ) − jc (δ)]α∗ , z Ω − Λcτ α∗τ , Λcτ ατ τ ∈(0,2δ)
By Remark 3.9.i and (5.11), the restriction of the explicit quartic part in Remark 5.5(ii) is [VΩ,δ (ε; α∗ , z + jc (δ)β) + VΩ,δ (ε; z∗ + jc (δ)α∗ , β)] Ω , and the restriction of the term in Remark 5.5.iii is obvious. This enables us to control the small field part in Lemma 5.4. Corollary 5.7. ∗ dμΩ,r (z ∗ , z) efS (α∗ ,β; ρ ;z ,z) α∗ , j(2δ)β |Ω +VΩ,2δ (ε; α∗ ,β)
=e
Ω
eD(α∗ ,β)
e−Λτ α∗τ , Λτ ατ c
c
τ ∈(0,2δ)
where ¯ Ω;δ (VΩ,δ (ε; · ); D1 ( · ; 0) |Ω , D2 ( · ; 0)| ) D=R Ω Proof. By Remark 5.6 and Definition 3.22, ∗ dμΩ,r (z ∗ , z) efS (α∗ ,β; ρ;z ,z) Ω
α∗ ,j(2δ)β |Ω +VΩ,δ (ε;α∗ ,j(δ)β)+VΩ,δ (ε;j(δ)α∗ ,β)
=e
eD(α∗ ,β)
e−Λτ α∗τ , Λτ ατ c
c
τ ∈(0,2δ)
Now apply Remark 3.9(ii).
From the contributions in Remark 5.5, we shall split off the part that is independent of the fluctuation fields z∗ , z. The remaining parts, that truly contain fluctuation fields, will be integrated out. Proposition 5.14 below gives the decomposition just mentioned and thus prepares for the fluctuation integral. Lemmas 5.10 and 5.11 and Propositions 5.12 and 5.13 below are used in the proof of Proposition 5.14. To estimate the fluctuation integral we shall apply Theorem 3.14 with the weight system of the following definition. Definition 5.8 (Fluctuation integral weight systems). Let wfluct be the weight system with core Ω that associates ∗ , the weight factors ◦ the weight factors (κ∗τ )τ ∈[0,2δ) to the fields α∗ , α (κτ )τ ∈(0,2δ] to the fields α , β, ◦ the weight factors λτ to the fields z∗τ and zτ with τ ∈ (0, 2δ), and ◦ to the fluctuation fields z∗ , z the weight factor 32 r if x ∈ Ω ˜ λ(x) = (5.12) ∞ otherwise ◦ and the constant weight factor 1 to the history field h. We write f fluct instead of f wfluct .
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Remark 5.9. wfluct extends the weight system wS of Definition 3.15.i by weight factors for the fluctuation fields. Thus, for a function h that is independent of the fluctuation fields, h fluct = h S . Also, it is an extension of the weight system wfluct introduced just before Lemma 4.2. ˜i. The following Lemma shall be applied with gi = Di or gi = Di − D Lemma 5.10. Let g1 (α∗ , β; ρl ) and g2 (α∗ , β; ρr ) be history complete analytic functions. Then 1 1 1g1 (α∗ , zδ + αδcr ; ρ l ) Ωc 1S ≤ 28 g1 S1 1 1 cr 1g2 (z∗δ + α∗δ , β; ρ r ) Ωc 1S ≤ 28 g2 S2 and
1 1 1g1 (α∗ , zδ + αδcr ; ρ l ) − g1 (α∗ , zδ + αδcr ; ρ l ) Ωc 1S 1 1 1 1 8 −mc(δ) ≤2 e g1 S1 + 1g1 Ω1 1 S1 1 1 1 1 cr cr r ) − g2 (z∗δ + α∗δ , β; ρ r ) Ωc 1 1g2 (z∗δ + α∗δ , β; ρ S 1 1 1 1 8 −mc(δ) ≤2 e g2 S2 + 1g2 Ω 1 2 S2
Also g1 (α∗ , z + zδ + αδcr ; ρ l ) − g1 (α∗ , zδ + αδcr ; ρ l ) fluct 1 1 1 1 ≤ 28 e−mc(δ) g1 S1 + 1g1 Ω 1 1
S1
cr g2 (z∗ + z∗δ + ρ r ) − g2 (z∗δ + α∗δ , β; ρ r ) fluct 1 1 1 1 ≤ 28 e−mc(δ) g2 S2 + 1g2 Ω 1 cr α∗δ , β;
2
S2
cr Proof. We estimate the g2 -terms. Observe that g2 (z∗ + z∗δ + α∗δ , β; ρ r ) is r ) by substituting obtained from g2 (α∗ , β; ρ
h [Ωz∗ + (Λ\Ω)z∗δ + Λ jc (δ) α∗ + Λc α∗δ ] for α∗ . Introduce the auxiliary weight system w ˜aux that has the same weight factors as wfluct , but core Ω2 instead of Ω = ΩS . We shall apply Proposi4ν 8 tion G.1, with ν = 20 19 and Cν = (e ln ν)2 < 2 , to prove r ) w˜aux ≤ 28 g2 S2 g2 (z∗ + z∗δ + α∗cr , β; ρ
(5.13)
˜aux (namely κS,2δ and κ∗S,τ , κS,τ , The weight factors for the fields β, ρ r in w λS,τ with δ < τ < 2δ) are smaller than the weight factors for the corresponding fields in wS2 (namely κS2 ,δ and κ∗S2 ,τ , κS2 ,τ , λS2 ,τ with 0 < τ < δ). Consequently, the hypothesis of Proposition G.1 is satisfied if ˜ + Nd (Λ\Ω ; κ∗S ,0 , λS,δ ) Nd (Ω ; κ∗S ,0 , λ) Ω2
+NdΩ2
2
Ω2
2
19 (Λ jc (δ) ; κ∗S2 ,0 , κ∗S,0 ) ≤ 20
(5.14)
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NdΩ2 (Λc ; κ∗S2 ,0 , κ∗S,δ ) ≤
241
19 20
˜ Since λ(x) = 32r for all x ∈ Ω, κ∗S2 ,0 (x) = 2R for all x ∈ ΩS2 , λS,δ = 32r for all x ∈ Λ\Ω, κ∗S,0 (x) = 2R+ for all x ∈ Λ and, by Remark 3.17, κ∗S,δ (x) = 12 κ∗S2 ,0 (x) for all x ∈ Λc , (3.6) yields ˜ ≤ 32r N2m (Λ\Ω ; κ∗S ,0 , λS,δ ) ≤ 32r N2m (Ω ; κ∗S2 ,0 , λ) 2 2R 2R R+ 1 c N2m (Λ ; κ∗S2 ,0 , κ∗S,δ ) ≤ N2m (Λh; κ∗S2 ,0 , κ∗S,0 ) ≤ R 2 By Remark G.4(iii), Lemma B.1(i) and Lemma 3.21(ii), R+ N2m (Λ (jc (δ) − h) ; κ∗S,0 , κ∗S,0 ) R R+ ≤2 N 7 (Λ (jc (δ) − h) ; 1, 1) R 3m R+ ≤2 Kj δ eKj δ R ≤ 2md, the left hand side of the top inequality in (5.14) is bounded
N2m (Λ (jc (δ) − h) ; κ∗S2 ,0 , κ∗S,0 ) =
Since dΩ2 by 16r 16r + + N2m (Λh; κ∗S2 ,0 , κ∗S,0 ) + N2m (Λ (jc (δ) − h) ; κ∗S2 ,0 , κ∗S,0 ) R R 32r R+ R+ ≤ + +2 Kj δ eKj δ R R R 19 ≤ 20
1 1 < 19 by (F.3f) and Hypothesis F.7(i), since RR+ = 2eR1+er ≤ √ 4 20 − 16 , by (2.17). 2 The left hand side of the bottom inequality in (5.14) is bounded by 12 < 19 20 . This proves (5.13). Since dΩ ≤ dΩ2 , (5.13) gives that 1 1 1 1 1 1 1 1 cr cr , β; ρ r ) c 1 = 1g2 (z∗ + z∗δ + α∗δ , β; ρ r ) c 1 1g2 (z∗δ + α∗δ Ω
Ω
S
fluct
cr ≤ g2 (z∗ + z∗δ + α∗δ , β; ρ r ) w˜aux ≤ 28 g2 S2
As cr cr g2 (z∗ + z∗δ + α∗δ , β; ρ r ) − g2 (z∗δ + α∗δ , β; ρ r ) fluct 1 1 1 1 cr cr + 1g2 (z∗δ + α∗δ , β; ρ r ) − g2 (z∗δ + α∗δ , β; ρ r ) c 1 Ω fluct 1 1 1 1 cr cr = 1g2 (z + z∗δ + α∗δ , β; ρ r ) − g2 (z∗δ + α∗δ , β; ρ r ) 1
Ωc fluct
it now suffices to prove that 1 1 1 1 cr cr , β; ρ r ) − g2 (z∗δ + α∗δ , β; ρ r ) c 1 1g2 (z∗ + z∗δ + α∗δ Ω fluct 1 1 1 1 8 −mc(δ) ≤2 e g2 S2 + 1g2 Ω 1 2
S2
(5.15)
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Write g2 = B + S with B = g2 − g2 Ω2 its “S2 –boundary part” and S = g2 Ω2 “its “S2 –small field part”. Each nonzero monomial in the power series expansion of B contains at least one factor h(x) with x ∈ Ωc2 . Also, each nonzero monomial in cr cr the power series expansion of B(z∗ + z∗δ + α∗δ , β; ρ r ) − B(z∗δ + α∗δ , β; ρ r ) Ωc
contains at least one factor h(y) with y ∈ Ω. Therefore, by (2.15) and (5.13) 1 1 1 1 cr cr , β; ρ r ) − B(z∗δ + α∗δ , β; ρ r ) 1 1B(z∗ + z∗δ + α∗δ Ωc fluct 1 1 c 1 ¯ ¯ 1 cr cr ≤ e−m d(Ω,Ω2 ) 1B(z∗ + z∗δ + α∗δ , β; ρ r ) − B(z∗δ + α∗δ , β; ρ r ) 1 Ωc w ˜aux
¯ Ω ¯ c) −m d(Ω, 2
B(z∗ + z∗δ +
¯ Ω ¯ c) −m d(Ω, 2
cr g2 (z∗ + z∗δ + α∗δ , β; ρ r ) w˜aux
≤e ≤e
cr α∗δ , β;
ρ r ) w˜aux
≤ 28 e−mc(δ) g2 S2 Finally
1 1 1 1 cr cr , β; ρ r ) − S(z∗δ + α∗δ , β; ρ r ) c 1 1S(z∗ + z∗δ + α∗δ Ω
fluct
cr ≤ S(z∗ + z∗δ + α∗δ , β; ρ r ) waux
≤ 28 S S2 by (5.13), with g2 replaced by S.
Parts (i) and (ii) of Remark 5.5 (quadratic and quartic terms) implicitly cr (defined in Lemma 5.3) involve the substitution of the critical fields αδcr , α∗δ in the concrete background fields Γ∗S1 , ΓS1 and Γ∗S2 , ΓS2 . To control it, set ⎧ ⎪ ∗l ) if τ ∈ εZ ∩ (0, δ) ⎨Γ∗S1 (τ ; α∗ , α cr ˜ Γ∗ (τ ; α∗ , α ∗ ) = α∗δ if τ = δ ⎪ ⎩ cr ,α ∗r ) if τ ∈ εZ ∩ (δ, 2δ) Γ∗S2 (τ − δ; α∗δ ⎧ ⎪ l , αδcr ) ⎨ΓS1 (τ ; , α ˜ ;α Γ(τ , β) = αδcr ⎪ ⎩ r , β) ΓS2 (τ − δ; α
(5.16) if τ ∈ εZ ∩ (0, δ) if τ = δ if τ ∈ εZ ∩ (δ, 2δ)
Observe that (5.16) is very similar to the recursion relation of Proposition 3.6. cr Indeed, replacing α∗δ and αδcr by cr Λj(δ)α∗ + Λc α∗δ = α∗δ + Λ[j(δ) − jc (δ)]α∗ c and Λj(δ)β + Λ αδ = αδcr + Λ[j(δ) − jc (δ)]β
respectively, in (5.16) gives the recursion relation of Proposition 3.6, except for the ∂Γ∗τ α∗ and ∂Γτ β terms. For convenience, we set jδ = j(δ) − jc (δ)
(5.17)
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Lemma 3.21(iii) gives the estimate |||jδ ||| ≤ δKj eKj δ e−mc . This discussion shows that ˜ ∗ (τ ; α∗ , α ∗) − Γ ∗ ) = ∂c Γ∗τ α∗ Γ∗S (τ ; α∗ , α (5.18) ˜ ΓS (τ ; α , β) − Γ(τ ; α , β) = ∂c Γτ β where we set ∂c Γ∗τ
⎧ ⎪ ⎨0 = Λ jδ ⎪ ⎩ ∂Γ∗τ + Γ0∗τ −δ (S2 )Λ jδ
⎧ δ ⎪ ⎨∂Γτ + Γτ (S1 )Λ jδ ∂c Γτ = Λ jδ ⎪ ⎩ 0
if τ ∈ [0, δ) or τ = 2δ if τ = δ if τ ∈ (δ, 2δ) (5.19)
if τ ∈ (0, δ) if τ = δ if τ ∈ (δ, 2δ] or τ = 0
Under the conventions of parts (iii) and (iv) of Lemma 3.7, we may also write ∂c Γ∗τ = ∂Γ∗τ + Γ0∗τ −δ (S2 )Λ jδ for τ = δ, 2δ and ∂c Γτ = ∂Γτ + Γδτ (S1 )Λ jδ for τ = 0, δ. The operators ∂c Γ∗τ , ∂c Γτ are estimated in Lemma E.18. We use the shorthand notations Γ∗S (τ ; α∗ , α ΓS (τ ; α ∗ ) if τ ∈ (0, 2δ) , β) if τ ∈ (0, 2δ) γ∗τ = γτ = if τ = 0 α∗ β if τ = 2δ (5.20) By the recursion relation Proposition 3.6, ∗ ) = γ∗τ Γ∗S1 (τ ; α∗ , α
ΓS2 (τ ; α , β) = γτ +δ
for all 0 < τ < δ
(5.21)
Bound on the Quadratic Part of the Fluctuation Action In part (i) of Remark 5.5 we wrote the “quadratic part” as the sum of −QS , δQ, r) − z∗ , Ωjc (δ)β − QS2 (z∗ , β; 0, α = z∗ , Ω jδ β − z∗ , Ωj(δ)β − QS2 (z∗ , β; 0, α r) and ∗l , 0) − Ωjc (δ)α∗ , z − QS1 (α∗ , z; α = Ω jδ α∗ , z − Ωj(δ)α∗ , z − QS1 (α∗ , z; α ∗l , 0) We bound these quantities in Proposition 5.12, below. By way of preparation for the bound on δQ, recall from (5.10) that δQ cr cr ∗, α )−QS1 (α∗ , αδcr ; α ∗l , α l )−QS2 (α∗δ , β; α ∗r , α r )−α∗δ , αδcr = QS (α∗ , β; α − [z∗δ , (Λ\Ω)j(δ)β + QS2 (z∗δ , β; 0, α r )] + z∗δ , (Λ\Ω)jδ (δ)β − [(Λ\Ω)j(δ)α∗ , zδ + QS1 (α∗ , zδ ; α ∗l , 0)] + (Λ\Ω)jδ (δ)α∗ , zδ We start by rewriting the first line of the right hand side.
(5.22)
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Lemma 5.11. cr cr QS (α∗ , β; α ∗, α )−QS1 (α∗ , αδcr ; α ∗l , α l )−QS2 (α∗δ , β; α ∗r , α r ) − α∗δ , αδcr ∂j Γ∗τ α∗ , γτ + γ∗τ , ∂j Γτ β − Λjδ α∗ , Λjδ β × τ ∈(0,2δ]
τ ∈[0,2δ)
where ∂j Γ∗τ = ∂c Γ∗τ − j(ε)∂c Γ∗τ −ε
∂j Γτ = ∂c Γτ − j(ε) ∂c Γτ +ε
Proof. By parts (iii) and (iv) of Lemma 3.7 and (5.21), cr cr QS (α∗ , β; α ∗, α ) − QS1 (α∗ , αδcr ; α ∗l , α l ) − QS2 (α∗δ , β; α ∗r , α r ) − α∗δ , αδcr = QS1 (α∗ , Λj(δ)β + Λc αδ ; α ∗l , α l ) − QS1 (α∗ , Λjc (δ)β + Λc αδ ; α ∗l , α l)
+ QS2 (Λj(δ)α∗ + Λc α∗δ , β; α ∗r , α r )−QS2 (Λjc (δ)α∗ + Λc α∗δ , β; α ∗r , α r) c c + Λj(δ)α∗ , Λj(δ)β + Λ α∗δ , Λ αδ − Λjc (δ)α∗ + Λc α∗δ , Λjc (δ)β + Λc αδ + γ∗τ , (∂Γτ − j(ε)∂Γτ +ε ) β τ ∈εZ∩[0,δ)
+
(∂Γ∗δ+τ − j(ε)∂Γ∗δ+τ −ε ) α∗ , γδ+τ
τ ∈εZ∩(0,δ]
=
&
' γ∗τ , Γδτ (S1 ) − j(ε)Γδτ +ε (S1 ) Λjδ β
τ ∈εZ∩[0,δ)
+
&
' Γ0∗τ (S2 ) − j(ε)Γ0∗τ −ε (S2 ) Λjδ α∗ , γδ+τ
τ ∈εZ∩(0,δ]
+ Λj(δ)α∗ , Λj(δ)β − Λjc (δ)α∗ , Λjc (δ)β γ∗τ , (∂Γτ − j(ε)∂Γτ +ε ) β + τ ∈εZ∩[0,δ)
+
(∂Γ∗δ+τ − j(ε)∂Γ∗δ+τ −ε ) α∗ , γδ+τ
τ ∈εZ∩(0,δ]
=
γ∗τ , (∂c Γτ − j(ε)∂c Γτ +ε ) β
τ ∈εZ∩[0,δ)
+
(∂c Γ∗τ − j(ε)∂c Γ∗τ −ε ) α∗ , γτ
τ ∈εZ∩(δ,2δ]
+ Λj(δ)α∗ , Λj(δ)β − Λjc (δ)α∗ , Λjc (δ)β This is the desired equation, since 0 if 0 < τ < δ ∂j Γ∗τ = Λjδ if τ = δ γδ = Λc αδ + Λj(δ)β
0 ∂j Γτ = Λjδ
if δ < τ < 2δ if τ = δ
γ∗δ = Λc α∗δ + Λj(δ)α∗
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so that
The Temporal Ultraviolet Limit
0 ∂j Γ∗τ α∗ , γτ = Λjδ α∗ , Λj(δ)β 0 γ∗τ , ∂j Γτ β = Λj(δ)α∗ , Λjδ β
if 0 < τ < δ if τ = δ
245
if δ < τ < 2δ if τ = δ
(5.23)
Proposition 5.12. Set KQ = 29 e2Kj . (i)
z∗ , Ωjc (δ)β + QS2 (z∗ , β; 0, α r ) fluct ≤ KQ e−mc(δ) + δ rR+ e−m c ∗l , 0) fluct ≤ KQ e−mc(δ) + δ rR+ e−m c Ωjc (δ)α∗ , z + QS1 (α∗ , z; α
(ii)
1 δQ(α∗ , β; ρ ) S ≤ KQ e− 4 mc(δ) + δ rR+ e−m c
Proof. (i) We prove the second inequality. The proof of the first is analogous. Observe that, by the definition (3.3), QS1 (α∗ , z; α ∗l , 0) = Qε,δ (α∗ , z; Γ∗S1 ( · ; α∗ , α ∗l ), ΓS1 ( · ; 0, z)) with the Qε,δ of (2.10). Since z is supported in Ω, jδ α∗ , z + [Ωjc (δ)α∗ , z + Qε,δ (α∗ , z; Γ∗S1 ( · ; α∗ , α ∗l ), ΓS1 ( · ; 0, z))] = Ωj(δ)α∗ , z + Qε,δ (α∗ , z; Γ∗S1 ( · ; α∗ , α ∗l ), ΓS1 ( · ; 0, z)) & δ ' δ γ∗τ , Γτ (S1 ) − j(ε)Γτ +ε (S1 ) Ωz = τ ∈[0,δ)
evaluated at γ∗τ
α∗ = ∗l ) = Γ0∗τ (S1 )α∗ + τ ∈(0,τ ]∩εZ Γτ∗τ (S1 )α∗τ Γ∗S1 ( τ ; α∗ , α
if τ = 0 if τ ∈ (0, δ)
and with the conventions that Γδ0 (S1 ) = j(δ) and Γδδ (S1 ) = h. To bound this, we first claim that for any two h–operators A and A ( ) ( ) ˜ A α∗τ , A z fluct ≤ NdΩ A ; em d(x,Ω) , κ∗τ NdΩ A ; e−m d(x,Ω) , λ (5.24) As A α∗τ , A z is obtained from ϕ, ψ by the substitution ϕ = A α∗τ , ψ = A z, this inequality follows Lemma G.2.a with s = 2, d = dΩ , κ 1 = κ∗τ , ˜ κ1 (x) = emd(x,Ω) and κ2 (x) = e−md(x,Ω) . κ 2 = λ,
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Applying the estimates (5.24) and (3.7) we get ∗l ), ΓS1 ( · ; 0, z)) fluct Ωj(δ)α∗ , z + Qε,δ (α∗ , z; Γ∗S1 ( · ; α∗ , α ( ) ˜ ≤ aτ N2m Γδτ (S1 ) − j(ε)Γδτ +ε (S1 ) Ω ; e−m d(x,Ω) , λ τ ∈[0,δ)
where
, κ∗0 N2m 1 ; em d(x,Ω) ( ) aτ = τ m d(x,Ω) , κ∗τ τ ∈[0,δ) N2m Γ∗τ (S1 ) ; e
if τ = 0 if τ ∈ (0, δ)
By Lemma B.1(ii), Lemma E.13 and the observation that e−md(x,Ω) ≤ m m e− 2 d(x,Ω) ≤ e− 2 d(x,ΛS1 ) , we have aτ ≤ 16eKj R, for all τ ∈ [0, δ). There˜ O = Ω and ΩS replaced by Ω1 fore, by Lemma E.15(ii) with λ = λ, jδ α∗ , z + [Ωjc (δ)α∗ , z + Qε,δ (α∗ , z; Γ∗S1 (·; α∗ , α ∗l ), ΓS1 (·; 0, z))] fluct ( ) ˜ ≤ 16eKj R N2m Γδτ (S1 ) − j(ε)Γδτ +ε (S1 ) Ω ; e−m d(x,Ω) , λ τ ∈[0,δ)
≤2 e
11 2Kj
rRe−2md(Ω,Ω1 ) c
≤ e−mc(δ) 1
since 8 ≤ 16eKj r ≤ 16eKj R ≤ e 4 mc(δ) , by (F.6b), and d(Ω, Ωc1 ) ≥ c(δ). The function jδ α∗ , z is obtained from ϕ, z by the linear substitution ϕ = Ω jδ α∗ . Let ω be the weight system with core Ω that associates the constant weight factor 1 to the fields ϕ and z. Clearly, ϕ, z ω = 1 . It follows from Lemma G.2.a (with s = 2, w ˜ = wfluct , w = ω) that ( ) ˜ jδ α∗ , z fluct = Λ jδ α∗ , Ω z fluct ≤ NdΩ (Λ jδ ; 1, κ∗S,0 ) NdΩ Ω ; 1, λ ≤ 4Kj eKj δ δ R+ e−m c 32r since, by (3.7), Remark G.4(i), Lemma B.1(i) and Lemma 3.21(iii), NdΩ (Λ jδ ; 1, κ∗S,0 ) ≤ N2m (Λ jδ ; 1, κ∗S,0 ) ≤ 4||| jδ ||| R+ ≤ 4Kj eKj δ δ R+ e−m c This completes the proof of the second inequality. As in part (i) one shows that z∗δ , (Λ\Ω)jδ β S ≤ 27 Kj eKj δ rR+ e−m c z∗δ , (Λ\Ω)j(δ)β + QS2 (z∗δ , β; 0, α r ) S ≤ e−m c(δ) (Λ\Ω)jδ α∗ , zδ S ≤ 2 Kj e 7
Kj
(5.25) −m c
δ rR+ e
−m c(δ)
(Λ\Ω)j(δ)α∗ , zδ + QS1 (α∗ , zδ ; α ∗l , 0) S ≤ e
This bounds the last four terms of (5.22). It remains to bound the first four terms of (5.22), which form the left hand side of Lemma 5.11. By (5.23), the first sum on the right hand side of Lemma 5.11 is
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∂j Γ∗τ α∗ , γτ = j(δ)Λjδ α∗ , β τ ∈(0,2δ]
+
∂j Γ∗τ α∗ , γτ + ∂j Γ∗2δ α∗ , γ2δ
τ ∈(δ,2δ)
=
∂j Γ∗τ α∗ , γτ
τ ∈(δ,2δ)
+ (∂j Γ∗2δ + j(δ)Λjδ ) α∗ , γ2δ
(5.26)
As in (5.24) one sees for all h–operators A and B ( ) ( ) m m Aα∗ , B ατ S ≤ NdΩ A ; e− 2 d(x,Λ) , κ∗0 NdΩ B ; e 2 d(x,Λ) , κτ Hence,
1 1 1 1 1 1 1 1 ∂ Γ α , γ + (∂ Γ + j(δ)Λj ) α , γ j ∗τ ∗ τ j ∗2δ δ ∗ 2δ 1 1 1 1τ ∈(δ,2δ) S ( ) m − 2 d(x,Λ) ≤ aτ N2m ∂j Γ∗τ ; e , κ∗0 τ ∈(δ,2δ)
( ) m + a2δ N2m ∂j Γ∗2δ + j(δ)Λjδ ; e− 2 d(x,Λ) , κ∗0 where
m , κ2δ N2m 1 ; e 2 d(x,Λ) ( ) aτ = m τ 2 d(x,Λ) , κ N ; e Γ 2m τ τ τ ∈(0,2δ]
if τ = 2δ if τ ∈ (δ, 2δ)
By Lemma B.1(ii) and Lemma E.13, we have aτ ≤ 16eKj R+ for all τ ∈ (0, 2δ]. Therefore, by (5.26), and Lemma E.20(ii), the first sum on the right hand side of Lemma 5.11 is bounded by 1 1 1 1 1 1 1 Kj − 12 mc(δ) 1 ∂j Γ∗τ α∗ , γτ 1 ≤ e− 4 mc(δ) 1 ≤ 16e R+ e 1 1 1τ ∈(0,2δ] S
by (F.6b). The second sum is bounded in the same way. Combining this with (5.22) and (5.25) gives δQ(α∗ , β; ρ) + Λjδ α∗ , Λjδ β S # " 1 ≤ 2 e− 4 mc(δ) + e−mc(δ) + 27 Kj eKj δ rR+ e−mc
(5.27)
As Λjδ α∗ , Λjδ β is obtained from ϕ, ψ by the substitution ϕ = Λjδ α∗ , ψ = Λjδ β, an application of Lemma G.2(a) with s = 2, d = dΩ , κ 1 = κ∗0 , κ 2 = κ2δ , and κ1 (x) = κ2 (x) = 1 yields Λjδ α∗ , Λjδ β S ≤ NdΩ (Λjδ ; 1, κ∗0 ) NdΩ (Λjδ ; 1, κ2δ ) ≤ (4R+ |||jδ |||) ≤ 16Kj2 e2Kj δ 2 e−2mc R2+ 2
by (3.7), Remark G.4(i), Lemma B.1(i) and Lemma 3.21(iii).
(5.28)
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Combining (5.27) and (5.28), we have
1 δQ(α∗ , β; ρ) S ≤ 4 e− 4 mc(δ) + 28 Kj eKj + 16Kj2 e2Kj δ R+ e−mc δ rR+ e−mc 1 1 ≤ 4 e− 4 mc(δ) + 28 Kj eKj + δ rR+ e−mc 40
by (F.4b) and (F.6a).
Bound on the Quartic Part of the Fluctuation Action. Finally, we treat the quartic contribution to fS (α∗ , β; ρ ; z∗ , z) identified in Remark 5.5(ii). We write it as cr VS1 (ε; α∗ , z + zδ + αδcr ; α ∗l , α l ) + VS2 (ε; z∗ + z∗δ + α∗δ , β; α ∗r , α r)
= VS (ε; α∗ , β; α ∗, α ) + δV(ε; α∗ , β; ρ) + Vl (ε; α∗ , β; ρ; z) +Vr (ε; α∗ , β; ρ; z∗ )
(5.29)
where δV = −VS (ε; α∗ , β; α ∗, α ) + VS1 (ε; α∗ , zδ + αδcr ; α ∗l , α l ) cr +VS2 (ε; z∗δ + α∗δ , β; α ∗r , α r ) Vl = VS1 (ε; α∗ , z + zδ + αδcr ; α ∗l , α l ) − VS1 (ε; α∗ , zδ + αδcr ; α ∗l , α l ) cr cr Vr = VS2 (ε; z∗ + z∗δ + α∗δ , β; α ∗r , α r ) − VS2 (ε; z∗δ + α∗δ , β; α ∗r , α r ) Similarly,
(ε ) ) cr ; α∗ , z + zδ + αδcr ; · · · + VS2 ; z∗ + z∗δ + α∗δ , β; · · · 2 ( ) 2 ε 9 + δV + V˜l + V˜r = VS 2
VS1
(ε
where
( (ε ) ) 9 = −VS ε ; α∗ , β; α ; α∗ , zδ + αδcr ; α δV ∗, α + VS1 ∗l , α l (2ε )2 cr ; z∗δ + α∗δ , β; α +VS2 ∗r , α r (ε 2 ) (ε ) ; α∗ , z + zδ + αδcr ; α ; α∗ , zδ + αδcr ; α V˜l = VS1 ∗l , α l − VS1 ∗l , α l 2( ( 2ε ) ) ε cr cr ˜ ; z∗ + z∗δ + α∗δ , β; α ; z∗δ + α∗δ Vr = VS2 ∗r , α r − VS2 , β; α ∗r , α r 2 2
Proposition 5.13. Set KV = 225 e6Kj . (i) Vl fluct , Vl − V˜l fluct ,
Vr fluct ≤ KV δ|||v||| r R3+ Vr − V˜r fluct ≤ KV εδ|||v||| r R3+
(ii) δV S ≤ KV δ|||v||| r R3+ 9 S ≤ KV εδ|||v||| rR3 δV − δV +
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Proof. (i) We treat Vl . By (5.16), (5.18) and (5.20) ˜ ∗ (τ ; α∗ , α ∗l ) = Γ ∗ ) = Γ∗S (τ ; α∗ , α ∗ ) = γ∗τ Γ∗S (τ ; α∗ , α 1
ΓS1 (τ ; α l , z + zδ + αδcr ) = Γδτ (S1 )z + Γδτ (S1 )zδ + ΓS1 (τ ; α l , αδcr ) ˜ ;α , β) = Γδτ (S1 )z + Γδτ (S1 )zδ + Γ(τ , β) − ∂c Γτ β = Γδτ (S1 )z + Γδτ (S1 )zδ + ΓS (τ ; α = Γδτ (S1 )z + Γδτ (S1 )zδ + γτ − ∂c Γτ β for all τ ∈ (0, δ). When τ = 0, the right hand side γ∗τ = α∗ and, with the convention Γδδ (S1 ) = 1, when τ = δ the right hand side Γδτ (S1 )z + Γδτ (S1 )zδ + ˜ α Γ(δ; , β) = z + zδ + αδcr . Similarly, cr ,α ∗r ) = Γ0∗τ −δ (S2 )z∗ +Γ0∗τ −δ (S2 )z∗δ +γ∗τ −∂c Γ∗τ α∗ Γ∗S2 (τ − δ; z∗ +z∗δ +α∗δ ΓS2 (τ − δ; α r , β) = γτ
for all τ ∈ (δ, 2δ). So, by the Definition 3.8.i of VS1 , ∗l , α l ) VS1 (ε; α∗ , z + zδ + αδcr ; α & δ ' γ∗τ Γτ +ε (S1 )z + gτ +ε , v γ∗τ Γδτ +ε (S1 )z + gτ +ε = −ε τ ∈εZ∩[0,δ)
where gτ = Γδτ (S1 )zδ + γτ − ∂c Γτ β g∗τ = Γ0∗τ −δ (S2 )z∗δ + γ∗τ − ∂c Γ∗τ α∗ Consequently, Vl = −ε
&
(5.30)
' γ∗τ Γδτ +ε (S1 )z + gτ +ε , v γ∗τ Γδτ +ε (S1 )z + gτ +ε
τ ∈εZ∩[0,δ)
− γ∗τ gτ +ε , v γ∗τ gτ +ε ]
(5.31)
To each term, we apply Corollary G.3(ii), r = 4, h(γ1 , . . . , γ4 ) = γ1 γ2 , v γ3 γ4
λ1 = λ2 = λ3 = λ4 = 1
and the s fields α1 , . . . , αs being z, zδ , β, α∗ , α∗ε , . . . , α∗δ−ε , αε , . . . , α2δ−ε . Recalling that γ∗τ = Γ0∗τ (S)α∗ + Γτ∗τ (S)α∗τ τ ∈εZ∩(0,δ)
gτ +ε =
Γδτ +ε (S1 ) zδ
+
Γττ +ε (S)ατ + Γ2δ τ +ε (S)β − ∂c Γτ +ε β
τ ∈εZ∩(0,2δ)
we have, as coefficients for the substitution, ∗ Γα 1 α∗τ Γ1 Γz2 Γz2δ Γβ2 ατ Γ2
= = = = = =
0 ∗ Γα 3 = Γ∗τ (S) α∗τ Γ3 = Γτ∗τ (S) Γz4 = Γδτ +ε (S1 ) Ω Γz4δ = Γδτ +ε (S1 ) (Λ\Ω) Γβ4 = Γ2δ τ +ε (S) − ∂c Γτ +ε ατ Γ4 = Γττ +ε (S)
˜ α∗ = Γ ˜ α∗ Γ 1 3 α ˜ ∗τ = Γ ˜ α∗τ Γ 1 3 ˜z = Γ ˜z = 0 Γ 2 4 ˜ zδ = Γ ˜ zδ = Γ 2 4 ˜β = Γ ˜β = Γ 2 4 ˜ ατ = Γ ˜ ατ = Γ 2 4 = =
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˜ j ’s being with τ ∈ εZ ∩ (0, δ), τ ∈ εZ ∩ (0, 2δ) and with all other Γji ’s and Γ i zero. We apply Corollary G.3(ii) with L = Λ, d replaced by 4md, d = dΩ and δ=m 2 d. The Corollary gives 1& ' 1 1 γ∗τ Γδτ +ε (S1 )z + gτ +ε , v γ∗τ Γδτ +ε (S1 )z + gτ +ε 1 1 − γ∗τ gτ +ε , v γ∗τ gτ +ε 1 ≤ 4 |||v||| σδ σ 3 fluct
where σ ≤ max {σ∗τ , στ +ε } ( ) 3 ˜ σδ ≤ NdΩ Γδτ +ε (S1 ) Ω ; e− 2 m d(x,Λ) , λ with
σ∗τ =
( ) m NdΩ Γτ∗τ (S); e 2 d(x,Λ) , κ∗τ
τ ∈[0,δ)
( ( ) ) m ˜ + Nd Γδ (S1 ) (Λ\Ω) ; e m2 d(x,Λ) , λδ στ = NdΩ Γδτ (S1 ) Ω ; e 2 d(x,Λ) , λ τ Ω ( ) ( ) m m NdΩ Γττ (S); e 2 d(x,Λ) , κτ +NdΩ ∂c Γτ ; e 2 d(x,Λ) , κ2δ + τ ∈(0,2δ]
Below we prove that σδ ≤ 32eKj r σ∗τ ≤ 16eKj τ R+ στ ≤ 32e
Kj
(5.32)
R+
Consequently, the fluctuation norm of each term in (5.31) is bounded by 3 4 |||v||| 32eKj r 32eKj R+ ≤ 222 e4Kj |||v||| r R3+ Summing over τ and multiplying by ε gives the desired bound on Vl . By (5.31), Vl − V˜l = W1 + W2 where
& ' ε γ∗τ Γδτ +ε (S1 )z + gτ +ε , v γ∗τ Γδτ +ε (S1 )z + gτ +ε 2 τ ∈εZ∩[0,δ) ( ) ( )3 2 − γ∗τ Γδτ + 2ε (S1 )z + gτ + 2ε , v γ∗τ Γδτ + 2ε (S1 )z + gτ + 2ε ' & − γ∗τ gτ +ε , v γ∗τ gτ +ε + γ∗τ gτ + 2ε , v γ∗τ gτ + 2ε
W1 = −
and
& ' ε γ∗τ Γδτ +ε (S1 )z + gτ +ε , v γ∗τ Γδτ +ε (S1 )z + gτ +ε 2 τ ∈εZ∩[0,δ) & ' − γ∗τ + 2ε Γδτ +ε (S1 )z + gτ +ε , v γ∗τ + 2ε Γδτ +ε (S1 )z + gτ +ε ' & − γ∗τ gτ +ε , v γ∗τ gτ +ε + γ∗τ + 2ε gτ +ε , v γ∗τ + 2ε gτ +ε
W2 = −
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By Lemma E.14, (5.30), (5.19) and Remark E.19, (ε) (ε) (ε) Γδτ +ε (S1 ) gτ +ε γ∗τ Γδτ + 2ε (S1 ) = j gτ + 2ε = j γ∗τ + 2ε = j 2 2 2 for all τ ∈ εZ ∩ [0, δ). Therefore the τ term of W1 is & ' γ∗τ Γδτ +ε (S1 )z + gτ +ε , v γ∗τ Γδτ +ε (S1 )z + gτ +ε 2 (ε) ( (ε) ) − γ∗τ j Γδτ +ε (S1 )z + j gτ +ε , ( ( ε )2 )3 ( ε )2 δ Γτ +ε (S1 )z + j gτ +ε v γ∗τ j 2 3 2 2 (ε) (ε) gτ +ε , v γ∗τ j gτ +ε − γ∗τ gτ +ε , v γ∗τ gτ +ε + γ∗τ j 2 2 We again apply Corollary G.3, this time part (iii), but with the same metrics, ˜ j ’s as before and with, in addition, h, Γji ’s and Γ i (ε) A1 = A˜1 = A3 = A˜3 = 1 A2 = A4 = h A˜2 = A˜4 = j 2 The corollary bounds the fluctuation norm of the τ term by 16 |||v||| σδ aδ (σa)3 where " (ε) # ε ||| ≤ eKj 2 a ≤ max |||1|||, |||j 2 (ε) ε ε − h||| = Kj eKj 2 aδ ≤ |||j (5.33) 2 2 by Lemma 3.21. By this and (5.32) ) ( ε ε ε 3 16 |||v||| σδ aδ (σa)3 ≤ 16 |||v||| 32eKj r Kj eKj 2 32eKj R+ eKj 2 2 ≤ 223 e6Kj ε|||v||| r R3+ ε 2 shows 22 6Kj
Summing over τ and multiplying by
W1 fluct ≤ 2 e
that
εδ|||v||| r R3+
The same bound applies to W2 fluct . This gives the desired bound on Vl − V˜l . Finally, we prove (5.32). By (3.7) and Lemma E.13, ( ) m NdΩ Γτ∗τ (S); e 2 d(x,Λ) , κ∗τ ≤ 16eKj τ R+ τ ∈[0,2δ)
( ) m NdΩ Γττ (S); e 2 d(x,Λ) , κτ ≤ 16eKj τ R+
(5.34)
τ ∈(0,2δ]
This gives the second line of (5.32). If J is the smallest decimation interval that strictly contains [τ, δ] and has δ as its right endpoint and if τJ denotes the midpoint of J , then ( ) 3 ˜ NdΩ Γδτ (S1 )Ω ; e− 2 m d(x,Λ) , λ ) ( 3 ˜ = NdΩ j(τJ − τ ) Λ(J ) j(δ − τJ )Ω ; e− 2 m d(x,Λ) , λ ≤ 32r |||j(τJ − τ )||| |||j(δ − τJ )||| ≤ 32 eKj r
(5.35)
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by (3.7) and Lemma G.5(ii) with d replaced by dΩ , R replaced by 32r and L1 = X, L2 = Λ(J ), L3 = Ω, O1 = Λ, O2 = X 3 ˜ δ1 = md, δ2 = 0, δ = 0, d˜ = 4md, κ = λ 2 ˜ (The hypothesis that κ(x) = λ(x) ≤ R = 32r for all x ∈ L3 = Ω is fulfilled by ˜ the definition (5.12) of λ). This gives the first line of (5.32). Similarly ( ) m NdΩ Γδτ (S1 ) (Λ\Ω) ; e 2 d(x,Λ) , λδ ( ) 3 (5.36) ≤ NdΩ Γδτ (S1 ) (Λ\Ω) ; e− 2 m d(x,Λ) , λδ ≤ 32 eKj r By Lemma E.18 ( ) ) m 3 NdΩ ∂c Γτ ; e 2 d(x,Λ) , κ2δ ≤ NdΩ (∂c Γτ ; e− 2 m d(x,Λ) , κ2δ ( ) ≤ 4 e2Kj R+ δ e−m c + e−mc(δ)
(5.37)
Combining these two bounds with (5.34) and (5.35) gives ( ) στ ≤ 32 eKj r + 32 eKj r + 4 e2Kj R+ δ e−m c + e−mc(δ) + 16eKj τ R+ ≤ 32eKj R+ This completes the proof of part (i) of the proposition. (ii) With the notation (5.30) we have δV = δV1 (ε) + δV2 (ε) where δV1 (ε) = ε
{γ∗τ γτ +ε , v γ∗τ γτ +ε − γ∗τ gτ +ε , v γ∗τ gτ +ε }
τ ∈[0,δ)
δV2 (ε) = ε
{γ∗τ γτ +ε , v γ∗τ γτ +ε − g∗τ γτ +ε , v g∗τ γτ +ε }
τ ∈[δ,2δ)
We bound δV1 . To treat the τ th term, we again apply Corollary G.3.ii with r = 4, h(γ1 , . . . , γ4 ) = γ1 γ2 , v γ3 γ4
λ1 = λ2 = λ3 = λ4 = 1
This time the s fields α1 , . . . , αs are zδ , β, α∗ , α∗ε , . . . , α∗δ−ε , αε , . . ., α2δ−ε . Recalling that γ∗τ = Γ0∗τ (S)α∗ + Γτ∗τ (S)α∗τ γτ +ε =
τ ∈εZ∩(0,δ)
Γττ +ε (S)ατ + Γ2δ τ +ε (S)β
τ ∈εZ∩(0,2δ)
gτ +ε = Γδτ +ε (S1 ) zδ +
τ ∈εZ∩(0,2δ)
Γττ +ε (S)ατ + Γ2δ τ +ε (S)β − ∂c Γτ +ε β
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we have, as coefficients for the substitution, ∗ Γα 1 α∗τ Γ1 Γz2δ Γβ2 ατ Γ2
0 ∗ ˜ α∗ = Γ ˜ α∗ = Γα =Γ 3 = Γ∗τ (S) 1 3 α∗τ α ˜ ∗τ = Γ ˜ α∗τ = Γ3 = Γτ∗τ (S) = Γ 1 3 ˜ zδ = Γ ˜ zδ = Γδτ +ε (S1 ) (Λ\Ω) = Γz4δ = 0 Γ 2 4 ˜β = Γ ˜ β = Γ2δ (S) − ∂c Γτ +ε = Γβ4 = Γ2δ Γ τ +ε (S) τ +ε 2 4 ατ α τ τ ˜ ˜ ατ = Γ4 = Γτ +ε (S) = Γ =Γ 2 4
(5.38)
˜ j ’s being with τ ∈ εZ ∩ (0, δ), τ ∈ εZ ∩ (0, 2δ) and with all other Γji ’s and Γ i zero. We apply Corollary G.3(ii) with L = Λ, d replaced by 4md, d = dΩ and δ=m 2 d. The Corollary gives γ∗τ γτ +ε , v γ∗τ γτ +ε − γ∗τ gτ +ε , v γ∗τ gτ +ε fluct ≤ 4 |||v||| σδ σ 3 where
, στ +ε σ ≤ max σ∗τ ( ) ( ) 3 3 σδ = NdΩ Γδτ +ε (S1 ) (Λ\Ω) ; e− 2 m d(x,Λ) , λδ + NdΩ ∂c Γτ ; e− 2 m d(x,Λ) , κ2δ ( ) by (5.36) and (5.37) ≤ 32eKj r + 4e2Kj R+ δe−mc + e−mc(δ)
with σ∗τ =
( ) m NdΩ Γτ∗τ (S); e 2 d(x,Λ) , κ∗τ = σ∗τ ≤ 16eKj τ R+
τ ∈[0,δ)
( ) ( ) m m στ = NdΩ Γδτ (S1 ) (Λ\Ω) ; e 2 d(x,Λ) , λδ + NdΩ ∂c Γτ ; e 2 d(x,Λ) , κ2δ ( ) m NdΩ Γττ (S); e 2 d(x,Λ) , κτ + τ ∈(0,2δ]
≤ στ ≤ 32eKj R+ by (5.32), twice. Hence γ∗τ γτ +ε , v γ∗τ γτ +ε − γ∗τ gτ +ε , v γ∗τ gτ +ε fluct ( )
3 32eKj R+ ≤ 4 |||v||| 32eKj r + 4e2Kj R+ δe−mc + e−mc(δ) ( ) R3+ ≤ 222 e4Kj |||v||| r + eKj R+ δe−mc + e−mc(δ) ≤ 223 e4Kj |||v||| rR3+
(5.39)
by (F.4b), (F.6a) and (F.6b). Summing over τ and multiplying by ε bounds δV1 fluct by 223 e4Kj δ|||v||| rR3+ . The δV2 contribution obeys the same bound. 9 S . Then We now move onto the bound of δV − δV (ε) (ε) 9 = δV1 (ε) − δV1 + δV2 (ε) − δV2 δV − δV 2 2 Write (ε) δV1 (ε) − δV1 = δW1 + δW2 2
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where ∗, α ) δW1 (α∗ , β; α ε [γ∗τ γτ +ε , v γ∗τ γτ +ε = 2 τ ∈εZ∩[0,δ) & ' ' & − γ∗τ γτ + 2ε , v γ∗τ γτ + 2ε −γ∗τ gτ +ε , v γ∗τ gτ +ε + γ∗τ gτ + 2ε , v γ∗τ gτ + 2ε ∗, α ) δW2 (α∗ , β; α ' & ε γ∗τ γτ +ε , v γ∗τ γτ +ε − γ∗τ + 2ε γτ +ε , v γ∗τ + 2ε γτ +ε = 2 τ ∈εZ∩[0,δ) ' & − g∗τ γτ +ε , v g∗τ γτ +ε + g∗τ + 2ε γτ +ε , v g∗τ + 2ε γτ +ε We estimate δW1 . The term of δW1 with index τ is 3 2 (ε) (ε) γ∗τ γτ +ε , v γ∗τ γτ +ε − γ∗τ j γτ +ε , v γ∗τ j γτ +ε 2( ) 3 2 2 (ε) ε gτ +ε , v γ∗τ j gτ +ε − γ∗τ gτ +ε , v γ∗τ gτ +ε + γ∗τ j 2 2 since, by Lemma E.14, (5.30), (5.19) and Remark E.19, (ε) (ε) γτ +ε gτ +ε γτ + 2ε = j gτ + 2ε = j 2 2 ˜ j ’s of (5.38). This time we We again apply Corollary G.3 with the Γji ’s and Γ i use part (iii) with (ε) A1 = A˜1 = A3 = A˜3 = 1 A2 = A4 = h A˜2 = A˜4 = j 2 as coefficients for the substitution. The Corollary, with L = Λ, d replaced by 4md, d = dΩ and δ = m 2 d, gives 1 3 2 (ε) (ε) 1 γτ +ε , v γ∗τ j γτ +ε 1 γ∗τ γτ +ε , v γ∗τ γτ +ε − γ∗τ j 2 ( ) 31 2 2 (ε) ε 1 gτ +ε , v γ∗τ j gτ +ε 1 − γ∗τ gτ +ε , v γ∗τ gτ +ε + γ∗τ j 2 2 fluct ≤ 16 |||v||| σδ aδ (σ a)3 where, by Lemma 3.21,
" (ε) # ε ||| ≤ eKj 2 a ≤ max |||1|||, |||j 2 (ε) ε ε − h||| = Kj eKj 2 aδ ≤ |||j 2 2
Thus ε δ 16 |||v||| σδ aδ (σ a)3 2 ε ≤ εδKj e2Kj ε 4 |||v||| σδ σ 3 ≤ 223 e6Kj εδ|||v||| rR3+
δW1 fluct ≤
since the right hand side of (5.39) is a bound on 4 |||v|||σδ σ 3 . The bounds on δW1 and the two corresponding terms of δV2 (ε) − δV2 2ε are the same.
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The Structure of the Fluctuation Integrand Proposition 5.14. The function fS (α∗ , β; ρ ; z∗ , z) of (5.8) that appears in Lemma 5.4 and the function f˜S of (5.9) can be written in the form fS (α∗ , β; ρ ; z∗ , z) = −QS (α∗ , β; α ∗, α ) + VS (ε; α∗ , β; α ∗, α ) + Fl (α∗ , β; ρ ; z) + Fr (α∗ , β; ρ ; z∗ ) + D(1) (α∗ , β; ρ ) + L S (α∗ , β; ρ ) and
(ε ) ˜ (1) + L˜ S ; · · · + F˜l + F˜r + D f˜S = −QS + VS 2
respectively, with history complete analytic functions Fl , Fr , D(1) , L S , F˜l , F˜r , ˜ (1) , L˜ that have the following properties. D S ; 0) = Fr (α∗ , β; ρ ; 0) = 0 and (i) Fl (α∗ , β; ρ 1 1 1 1 Fl fluct ≤ KV δv rR3+ + 28 e−mc(δ) D1 S1 + 1D1 Ω1 1 S1 1 1 1 1 Fr fluct ≤ KV δv rR3+ + 28 e−mc(δ) D2 S2 + 1D2 Ω 1 2 S2
; 0) = F˜r (α∗ , β; ρ ; 0) = 0 and Similarly, F˜l (α∗ , β; ρ 1 1 1 1 1˜ 1 1˜ 1 + 1D F˜l fluct ≤ KV δv rR3+ + 28 e−mc(δ) 1D 11 1 Ω 1 1 S S1 1 1 1 1 1 1˜ 1 1˜ 1 F˜r fluct ≤ KV δv rR3+ + 28 e−mc(δ) 1D + 1D 21 2 Ω2 1 S2
Furthermore Fl − F˜l fluct
1 1 1 ˜1 1 ≤ KV εδv rR3+ + 28 e−mc(δ) 1D1 − D 1
S1
Fr − F˜r fluct ≤
KV εδv rR3+
˜ (1) (ii) D(1) Ωc = D
8
+2
Ωc
−mc(δ)
e
1 1 1 ˜2 1 1D2 − D 1
S2
S2
1 1 1 ˜ 1 ) 1 + 1(D1 − D 1 Ω1
1 1 1 ˜ 2 ) 1 + 1(D2 − D 1 Ω2
S1
S2
= 0 and
D(1) S ≤ KV (2δv) r+ R3+ + 28 e−mc(δ) ΣD + 28 ΣD
(SF )
˜ D + 28 Σ ˜ (SF ) ˜ (1) S ≤ KV (2δv) r+ R3 + 28 e−mc(δ) Σ D + D where (SF )
ΣD = D1 S1 + D2 S2
ΣD
˜ D = D ˜ 1 S + D ˜ 2 S Σ 1 2
˜ (SF ) Σ D
1 1 1 1 1 1 1 1 = 1D1 Ω 1 + 1D2 Ω 1 1 S 2 S 1 1 1 1 1 2 1˜ 1 1˜ 1 = 1D1 Ω 1 + 1D 2 Ω 1 1 2 S1
Furthermore ˜ (1) S ≤ KV εδv r R3+ + 28 e−mc(δ) ΔD + 28 Δ(SF ) D(1) − D D
S2
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where ˜ 1 S + D2 − D ˜ 2 S ΔD = D1 − D 1 2 1 1 1 1 1 1 1 (SF ) ˜ ˜ 2 ) 1 ΔD = 1(D1 − D1 ) Ω 1 + 1(D2 − D 1 Ω2 1 S1
(iii) L S Ωc = L S and L˜ S Ωc = L˜ S and
S2
L S S ≤ KV (2δv) r+ R3+ + 28 ΣD ˜D L˜ S S ≤ KV (2δv) r+ R3+ + 28 Σ and
1 1 1 1 1LS − L˜ S 1 ≤ KV εδv r R3+ + 28 ΔD S
Proof. We set ; z) = − [Ωjc (δ)α∗ , z + QS1 (α∗ , z; α ∗l , 0)] + Vl (ε; α∗ , β; ρ ; z) Fl (α∗ , β; ρ +D1 (α∗ , z + zδ + αδcr ; ρ l ) − D1 (α∗ , zδ + αδcr ; ρ l ) Fr (α∗ , β; ρ ; z∗ ) = − [z∗ , Ωjc (δ)β + QS2 (z∗ , β; 0, α r )] + Vr (ε; α∗ , β; ρ ; z∗ ) cr cr +D2 (z∗ + z∗δ + α∗δ , β; ρ r ) − D2 (z∗δ + α∗δ , β; ρ r ) D (α∗ , β; ρ ) = δQ(α∗ , β; ρ ) + δV(ε; α∗ , β; ρ ) cr +D1 (α∗ , zδ+ αδcr ; α ∗l , α l ) + D2 (z∗δ+ α∗δ , β; α ∗r , α r )
and
L S = D Ωc ,
D(1) = D − L S
The fact that fS = −QS + VS + Fl + Fr + D(1) + L S is immediate from Remark 5.5 and (5.29). Similarly, we set (ε ) F˜l (α∗ , β; ρ ; α∗ , β; ρ ; z) = − [Ωjc (δ)α∗ , z + QS1 (α∗ , z; α ∗l , 0)] + Vl ; z 2 ˜ 1 (α∗ , z + zδ + αδcr ; ρ ˜ 1 (α∗ , zδ + αδcr ; ρ +D l ) − D l ) (ε ) ; α∗ , β; ρ F˜r (α∗ , β; ρ ; z∗ ) = − [z∗ , Ωjc (δ)β + QS2 (z∗ , β; 0, α r )] + Vr ; z∗ 2 cr cr ˜ ˜ +D2 (z∗ + z∗δ + α∗δ , β; ρ r ) − D2 (z∗δ + α∗δ , β; ρ r ) (ε ) 9 ˜ ; α∗ , β; ρ D (α∗ , β; ρ ) = δQ(α∗ , β; ρ ) + δV 2 ˜ 1 (α∗ , zδ+ αcr ; α ˜ 2 (z∗δ+ αcr , β; α +D l ) + D ∗r , α r ) δ ∗l , α ∗δ ˜ c , D ˜ (1) = D ˜ − L˜ . and L˜ S = D S Ω (i) By Propositions 5.12(i), 5.13(i) and Lemma 5.10, with g1 = D1 , Fl fluct ≤ KQ e−mc(δ) + δrR+ e−m c + KV δ|||v|||rR3+ 1 1 1 1 +28 e−mc(δ) D1 S1 + 1D1 Ω 1 1 S 1 1 1 1 1 ≤ KV δrR3+ v + 28 e−mc(δ) D1 S1 + 1D1 Ω 1 1 S1
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by (F.4b), (F.4c) and the hypothesis that |||v||| ≤ v2 . Also, by construction, (ε ) ; α∗ , β; ρ ; z) − Vl ; z Fl − F˜l = Vl (ε; α∗ , β; ρ 2 ˜ ˜ 1 )(α∗ , zδ + αδcr ; ρ +(D1 − D1 )(α∗ , z + zδ + αδcr ; ρ l ) − (D1 − D l ) ˜1, By Proposition 5.13(i) and Lemma 5.10, with g1 = D1 − D 1 1 1 1 1 1 1 1 1 1 ˜1 1 ˜ 1 ) 1 ≤ KV εδv rR3+ +28 e−mc(δ) 1D1 − D 1 + 1(D1 − D 1 1Fl − F˜l 1 Ω1 fluct
D(1) D(1)
S1
S1
and L S are constructed so that D(1) Ωc = 0 and L S = L S Ωc . Also = δQ − δQ Ωc + δV − δV Ωc + D1 (α∗ , zδ + αδcr ; ·) − D1 (α∗ , zδ + αδcr ; ·) Ωc cr cr + D2 (z∗δ + α∗δ , β; ·) − D2 (z∗δ + α∗δ , β; ·) c Ω
Therefore, by Propositions 5.12(ii), 5.13(ii) and Lemma 5.10 1 D(1) S ≤ KQ e− 4 mc(δ) + δrR+ e−m c + KV δ|||v||| r R3+ 1 1 1 1 1 1 1 1 8 −mc(δ) +2 e ( D1 S1 + D2 S2 ) + 1D1 Ω 1 + 1D2 Ω 1 1 S 2 S 1 2 ( ) (SF ) 3 8 −mc(δ) ≤ KV 2δr+ R+ v + 2 e ΣD + ΣD by (F.3b), (F.4b),(F.4c). Also ( ) ( ) 9 − δV − δV 9 c ˜ (1) = δV − δV D(1) − D Ω
˜ 1 )(α∗ , zδ + αcr ; ·) − (D1 − D ˜ 1 )(α∗ , zδ + αcr ; ·) c +(D1 − D δ δ Ω cr cr ˜ 2 )(z∗δ + α∗δ ˜ 2 )(z∗δ + α∗δ +(D2 − D , β; ·) − (D2 − D , β; ·) c Ω
˜ (1) S follows from Proposition Therefore, the desired estimate on D(1) − D 5.13.ii and Lemma 5.10. By Propositions 5.12(ii), 5.13(ii) and Lemma 5.10 1 1 L S S = 1D Ωc 1S 1 ≤ KQ e− 4 mc(δ) + δrR+ e−mc + KV δ|||v|||rR3+ + 28 D1 S1 + 28 D2 S2 ≤ KV 2δr+ R3+ v + 28 ΣD by (F.3b), (F.4b), (F.4c). Furthermore, ˜ D − D
9 + (D1 − D ˜ 1 )(α∗ , zδ+ αcr ; α = δV − δV l ) δ ∗l , α cr ˜ ∗ , α r ) +(D2 − D2 )(z∗δ+ α∗δ , β; α r
The desired bound on L S − L˜ S S now follows from Proposition 5.13(ii) and Lemma 5.10.
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5.5. The Fluctuation Integral: Proofs of Theorems 3.26 and 3.28 Recall from Lemma 5.4 that I(S;α∗ ,β) b1 (α∗ , αδ , ρl ) b2 (αδ∗ , β, ρr ) χ2δ (ΩS ; α, β) I(α∗ , β) = hierarchies S for scale 2δ (S1 ,S2 )≺S 2|ΩS |
×Zδ
∗
dμΩS ,r (z ∗ , z) efS (α
z∗δ (x)=zδ (x)∗
,β; ρ ;z ∗ ,z)
for x∈ΛS \R
In this section, we perform the fluctuation integral to give the proofs of Theorems 3.26 and 3.28. For each hierarchy S preceded by S1 and S2 , by Proposition 5.14 and Lemma 2.7, ∗ ∗ 2|ΩS | Zδ dμΩS ,r (z ∗ , z) efS (α ,β; ρ ;z ,z) (1) ∗ 2|Ω | = e−QS +VS +D +LS Zδ S dμΩS ,r (z ∗ , z) eFl (α∗ ,β; ρ ;z)+Fr (α∗ ,β; ρ ;z ) ∗ (1) dμΩS ,r (z ∗ , z) eFl (α∗ ,β; ρ ;z)+Fr (α∗ ,β; ρ ;z ) |Ω | = e−QS +VS +D +LS Z2δ S dμΩS ,r (z ∗ , z) By Proposition 5.14.i and the hypotheses of Theorem 3.26, Fl + Fr fluct ≤ 2KV δv rR3+ + 28 e−mc(δ) ΣD + 28 ΣD 1 (5.40) ≤ 40 by (F.4c) and (F.6a). So we can apply Theorem (3.14) with w = wfluct and f = Fl + Fr . It gives the existence of a function D(2) (α∗ , β; ρ ) such that ∗ (2) dμΩS ,r (z ∗ , z) eFl (α∗ ,β; ρ ;z)+Fr (α∗ ,β; ρ ;z ) (5.41) = eD (α∗ ,β; ρ ) dμΩS ,r (z ∗ , z) (SF )
since Fl (0, 0; 0 ; z) = Fr (0, 0; 0 ; z ∗ ) = 0. The estimate (3.5) in Theorem 3.14 applies and 2
D(2) S = D(2) fluct ≤ 1600 ( Fl + Fr fluct ) ( ) (SF ) ≤ 40 2KV δv rR3+ + 28 e−mc(δ) ΣD + 28 ΣD by (5.40). As [Fl + Fr ] = 0, also D(2) Ωc = 0. We set
(5.42)
Ωc
DS = D(1) + D(2) By Proposition 5.14(ii) and (5.42) DS S ≤ 27 KV (2δ) r+ R3+ v + 214 e−mc(δ) ΣD + 214 ΣD
(SF )
By Corollary 5.7, ¯ Ω;δ VΩ,δ (ε; · ); D1 ( · ; 0) , D2 ( · ; 0) DS Ω = R Ω Ω The desired bound on L S was proven in Proposition 5.14. Since KD ≥ 28 KV and KL ≥ 2KV the proof of Theorem 3.26 is now complete.
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˜ (2) and We now move on to the proof of Theorem 3.28. The functions D ˜ DS are constructed as above and obey the corresponding estimates. By Proposition 5.14(i) and the hypotheses of Theorem 3.28, (SF ) (Fl + Fr ) − (F˜l + F˜r ) fluct ≤ 2KV εδrv R3+ + 28 e−mc(δ) ΔD + 28 ΔD 1 ≤ 8 (5.43) 2
by (F.4c) and (F.6a). We apply [4, Corollary III.6] with f = Fl + Fr and f = F˜r + F˜r . Since 1 1 1 + ≤ Fl + Fr fluct + (Fl + Fr ) − (F˜l + F˜r ) fluct ≤ 40 28 34 it shows that
) ( ˜ (2) S ≤ 4(34) 2KV εδv rR3+ + 28 e−mc(δ) ΔD + 28 Δ(SF ) D(2) − D D
We combine this with the estimate of Proposition 5.14(ii) to give 1 1 1 ˜S1 1 1DS − D S ( ) (SF ) ≤ [(4)(34)(2) + 1]KV εδv rR3+ + [(4)(34) + 1]28 e−mc(δ) ΔD + ΔD ≤ 29 KV ε(2δv) r+ R3+ + 216 e−mc(δ) ΔD + 216 ΔD
(SF )
The desired bound on L S − L˜ S S was proven in Proposition 5.14(iii). Since KD ≥ 210 KV and KL ≥ 2KV the proof of Theorem 3.28 is now complete too.
6. Large Field Bounds In this section, we prove the large field bounds stated in Propositions 3.36, 3.37 and 3.38 and Lemma 3.40. Fix a 0 < δ ≤ Θ and an integer m ≥ 0 and set ε = 2−m δ, as in Theorem 3.35. We shall assume that ε is small enough that |eεμ − 1| ≤ 18 . For notational compactness, we set Rn = R(2n ε)
R n = R (2n ε)
rn = r(2n ε)
cn = c(2n ε)
as well as χn (Ω; α, β) = χ2n ε (Ω; α, β) and Fn (Ω0 ) = F2n ε (Ω0 ). Fix a hierarchy S for scale δ = 2m ε with depth at most m. For a decimation interval J ⊂ [0, δ], write Ω(J ) = ΩS (J ) and Λ(J ) = ΛS (J ). Similarly, ∗ ) = Γ∗ (τ ; α∗ , α ∗ ), ΓS (τ ; α , β) = Γ(τ ; α , β) and Ω = ΩS ([0, δ]), Γ∗S (τ ; α∗ , α Λ = ΛS ([0, δ]). Recall that, for each decimation interval J = [t− , t+ ] ⊂ [0, δ], with midpoint t, J − = [t− , t] and J + = [t, t+ ]. Also recall, from Notation 2.5, that Λ0 = Λδ = ∅ and, for each τ ∈ εZ ∩ (0, δ), Λτ = Λ(Jτ ) = Λ [τ − 2−d(τ ) δ, τ + 2−d(τ ) δ] , where Jτ is the unique decimation interval centred on τ and the decimation index d(τ ) is the smallest integer k ≥ 0 such that τ ∈ 2δk Z.
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We shall consistently ⎧ ∗ ⎪ if ⎨α = Γ∗ (τ ; α∗ , α ∗ ) if ⎪ ⎩ ∗ if β
Ann. Henri Poincar´e
use the notation ⎧ ⎫ ⎪ ⎪ τ =0 ⎨α ⎬ γτ = Γ(τ ; α , β) τ ∈ (0, δ) ⎪ ⎪ ⎩ ⎭ β τ =δ
⎫ ⎪ if τ = 0 ⎬ if τ ∈ (0, δ) ⎪ ⎭ if τ = δ (6.1)
Using this notation 1 1 2 ∗ ∗ 2 − α + QS (α , β; α ,α ) + β 2 2 1 1 = − γ∗τ , γτ + γ∗τ , j(ε)γτ +ε − γ∗τ +ε , γτ +ε 2 2
(6.2)
τ ∈εZ∩[0,δ)
and VS (ε; α∗ , β; α ∗, α ) = −ε
γ∗τ γτ +ε , v γ∗τ γτ +ε
(6.3)
τ ∈εZ∩[0,δ)
Throughout this section, we assume that the field ατ is compatible with S in the sense of Definition E.1, as is the case in the domain of the integral operator I(S,ε;α∗ ,β) . In particular, |ατ (x)| ≤ min{κτ (x), κ∗τ (x)}
|α(x)| ≤ κ∗0 (x)
|β(x)| ≤ κδ (x) (6.4)
for all τ ∈ εZ ∩ (0, δ) and x ∈ X. We also assume that h ≡ 1, as we did in Sect. 3.7 and, in particular, in Theorem 3.35. In Propositions 3.36 and 3.37, we introduced restricted large fields regions ˜ ). The reason for introducing these smaller large field sets is P˜α (J ), . . . , Q(J the following. When we are decimating at time t, the centre of J , we need to extract a small factor for certain points in Λ(J )c that are not in Λ(J − )c ∪ Λ(J + )c . Small factors were already extracted from the latter regions in previous decimation steps. Each time we extract a small factor associated with a point x we will distribute it amongst all nearby points y. As a result, when we are decimating at time t, it is not necessary to extract small factors from points that are within a distance 2c(|J ± |) of Λ(J − )c ∪Λ(J + )c . So, for example, P˜α (J ) consists of those points of Pα (J ) whose distance from Λ(J − )c ∪Λ(J + )c is greater than 2c(|J ± |). Remark 6.1. For each decimation interval J = [t− , t+ ], " P˜α (J ) = x ∈ Ω0 (J ) αt− (x) > R(|J |) , # 1 d(x,Λ(J ))> c( 2 |J |) d(x , Λ(J − )c ∪Λ(J + )c )> 2 c( 12 |J |)
" P˜β (J ) = x ∈ Ω0 (J ) αt+ (x) > R(|J |), # 1 d(b,Λ(J ))>c( 2 |J |) d(b , Λ(J − )c ∪Λ(J + )c )> 2 c( 12 |J |)
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" P˜α (J ) = b ∈ Ω0 (J )∗ (∇αt− )(b) > R (|J |) , # 1 d(b,Λ(J ))>c( 2 |J |) d(b, Λ(J − )c ∪Λ(J + )c ) > 2 c( 12 |J |)
" P˜β (J ) = b ∈ Ω0 (J )∗ (∇αt+ )(b) > R (|J |) , # 1 d(b,Λ(J ))>c( 2 |J |) d(b, Λ(J − )c ∪Λ(J + )c ) > 2 c( 12 |J |)
" ˜ (J ) = x ∈ Ω0 (J ) αt (x) − αt (x) > r(|J |) , Q + − # 1 d(x,Λ(J ))>c( 2 |J |) d(x, Λ(J − )c ∪Λ(J + )c ) > 2 c( 12 |J |)
where Ω0 (J ) = Ω(J − ) ∩ Ω(J + ). 6.1. Extracting Small Factors from the Quadratic Form In this subsection, we prove Proposition 3.36. The main ingredient is Lemma 6.2. Set j(ε) = e−εμ j(ε) = e−εh and cj,ε = 1 − j(ε) . Assume that cj,ε ≤ 19 . Then 1 1 Re − γ∗τ , γτ + γ∗τ , j(ε)γτ +ε − γ∗τ +ε , γτ +ε 2 2 τ ∈εZ∩[0,δ) ⎧ 1⎨ 1 ∗ γ∗τ −γτ +ε 2Λcτ ∪Λcτ +ε + ≤− γτ∗ , [1−j(ε)]γτ 4⎩ 2 τ ∈[0,δ) τ ∈(0,δ] ⎫ ⎬ ∗ + γ∗τ , [1−j(ε)]γ∗τ ⎭ τ ∈[0,δ)
1 + |γ∗τ − γτ∗ , Λτ (γτ − j(ε)γτ +ε )| 2 τ ∈(0,δ)
+
1 ∗ |γτ − γ∗τ , Λτ (γ∗τ − j(ε)γ∗τ −ε )| 2 τ ∈(0,δ)
+
1 2 ∗ γτ 2 + γ∗τ 2 Λcτ (γ∗τ − γτ ) + |eεμ − 1| 2 τ ∈(0,δ)
τ ∈[0,δ]
where, for each subset S ⊂ X, u 2S =
x∈S
|u(x)|2 = Su∗ , Su.
Proof. Recall that α ∗ = α∗ , α and γ∗0 = γ0∗ , γ∗δ = γδ∗ so that γ∗0 , γ0 = ∗ 2 and γ∗δ , γδ = γδ 2 . The real part of γ0 2 = γ∗0 1 1 − γ∗τ , γτ + γ∗τ , j(ε)γτ +ε − γ∗τ +ε , γτ +ε 2 2 τ ∈[0,δ)
1 1 = − γ0 2 − γ∗τ , γτ − γδ 2 + γ∗τ , j(ε)γτ +ε 2 2 τ ∈(0,δ)
τ ∈[0,δ)
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1 1 = − γ0 2 − γ∗τ , γτ − γδ 2 + γ∗τ , γτ +ε 2 2 τ ∈(0,δ) τ ∈[0,δ) γ∗τ , [1 − j(ε)]γτ +ε − τ ∈[0,δ)
1 1 ∗ 2 γ∗τ − γ∗τ , γτ + γτ 2 = 2 2 τ ∈(0,δ) 1 1 ∗ 2 2 + − γ∗τ + γ∗τ , γτ +ε − γτ +ε 2 2 τ ∈[0,δ) γ∗τ , [1 − j(ε)]γτ +ε − τ ∈[0,δ)
is
1 1 Re − γ∗τ , γτ + γ∗τ , j(ε)γτ +ε − γ∗τ +ε , γτ +ε 2 2 τ ∈[0,δ)
=
1 1 ∗ ∗ γ∗τ − γτ 2 − γ∗τ − γτ +ε 2 2 2 τ ∈(0,δ) τ ∈[0,δ) − Re γ∗τ , [1 − j(ε)]γτ +ε τ ∈[0,δ)
1 1 ∗ ∗ 2 2 γ − γτ − γ∗τ − γτ +ε − Re γ∗τ , [1 − j(ε)]γτ +ε = 2 ∗τ 2 τ ∈[0,δ)
because α − β 2 = α 2 + β 2 − 2Re α∗ , β. Now ∗ ∗ γ∗τ − γτ 2Λτ − γ∗τ − γτ +ε 2Λτ
12 ∗ ∗ = Λτ γ∗τ − Λτ j(ε)γτ +ε − Λτ (γτ − j(ε)γτ +ε ) 1 − γ∗τ − γτ +ε 2Λτ ∗ ∗ = Λτ γ∗τ − Λτ j(ε)γτ +ε 2 − γ∗τ − γτ +ε 2Λτ 12 & ' + Λτ (γτ − j(ε)γτ +ε ) 1 − 2Re γ∗τ − j(ε)γτ∗+ε , Λτ (γτ − j(ε)γτ +ε ) ∗ ∗ − γτ +ε ) + Λτ [1 − j(ε)]γτ +ε 2 − γ∗τ − γτ +ε 2Λτ = Λτ (γ∗τ 12 & ' + Λτ (γτ − j(ε)γτ +ε ) 1 − 2Re γ∗τ − j(ε)γτ∗+ε , Λτ (γτ − j(ε)γτ +ε ) ' & 2 = 2Re Λτ γ∗τ − γτ∗+ε , [1 − j(ε)]γτ +ε + Λτ [1−j(ε)]γτ +ε 12 + Λτ (γτ −j(ε)γτ +ε ) 1 & ' −2Re γ∗τ −γτ∗ + γτ∗ −j(ε)γτ∗+ε , Λτ (γτ −j(ε)γτ +ε ) ' & 2 = 2Re Λτ γ∗τ − γτ∗+ε , [1 − j(ε)]γτ +ε + Λτ [1 − j(ε)]γτ +ε 12 − Λτ (γτ − j(ε)γτ +ε ) 1 − 2Re γ∗τ − γτ∗ , Λτ (γτ − j(ε)γτ +ε )
so that 1 ∗ 1 ∗ γ∗τ − γτ 2 − γ∗τ − γτ +ε 2 − Re γ∗τ , [1 − j(ε)]γτ +ε 2 2
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' & 1 ∗ = − γ∗τ − γτ +ε 2Λcτ − Re Λτ γτ∗+ε + Λcτ γ∗τ , [1 − j(ε)]γτ +ε 2 1 2 + Λτ [1 − j(ε)]γτ +ε 2 1 1 − Λτ (γτ − j(ε)γτ +ε ) 12 − Re γ∗τ − γτ∗ , Λτ (γτ − j(ε)γτ +ε ) 2 1 ∗ + γ∗τ − γτ 2Λcτ 2 ' 1 & 1 ∗ = − γ∗τ − γτ +ε 2Λcτ − γτ∗+ε , [1 − j(ε)]γτ +ε − γτ − j(ε)γτ +ε 2Λτ 2 2 ' 1 & c 2 ∗ −Re Λτ γ∗τ − γτ +ε , [1 − j(ε)]γτ +ε + Λτ [1 − j(ε)]γτ +ε 2 1 ∗ −Re γ∗τ − γτ∗ , Λτ (γτ − j(ε)γτ +ε ) + γ∗τ − γτ 2Λcτ 2 & ' 1 ∗ = − γ∗τ − γτ +ε 2Λcτ − eεμ γτ∗+ε , [1 − j(ε)]γτ +ε 2 ' 1 & 2 −eεμ Re Λcτ γ∗τ − γτ∗+ε , [1 − j(ε)]γτ +ε + Λτ [1 − j(ε)]γτ +ε 2 ' & +(eεμ − 1) γτ +ε 2 + (eεμ − 1)Re Λcτ γ∗τ − γτ∗+ε , γτ +ε 12 1 − γτ − j(ε)γτ +ε 1Λ − Re γ∗τ − γτ∗ , Λτ (γτ − j(ε)γτ +ε ) τ 2 1 ∗ 2 + γ∗τ − γτ Λcτ 2 Using Cauchy–Schwarz and |AB| ≤ 12 (A2 + B 2 ), & c ' Λτ γ∗τ − γτ∗+ε , [1 − j(ε)]γτ +ε 1 1 1 1 ∗ ≤ γ∗τ − γτ +ε Λcτ 1 − j(ε) 1/2 1[1 − j(ε)]1/2 γτ +ε 1 ') & 1( ∗ √ ≤ cj,ε γ∗τ − γτ +ε 2Λcτ + γτ∗+ε , [1 − j(ε)]γτ +ε 2& εμ ' (e − 1)Re Λcτ γ∗τ −γτ∗+ε , γτ +ε ) ( 1 ∗ − γτ +ε 2Λcτ + γτ +ε 2 ≤ |eεμ − 1| γ∗τ 2 1 2 Λτ [1 − j(ε)]γτ +ε 2 2 2 ≤ [1 − j(ε)]γτ +ε + [j(ε) − j(ε)]γτ +ε 1 12 1 1 1 2 ≤ 1−j(ε) 1[1−j(ε)] 2 γτ +ε 1 +[eεμ −1]2 j(ε)γτ +ε & ' 2 ≤ cj,ε γτ∗+ε , [1 − j(ε)]γτ +ε + [eεμ − 1]2 γτ +ε Thus we have 1 1 − γ∗τ , γτ + γ∗τ , j(ε)γτ +ε − γ∗τ +ε , γτ +ε Re 2 2 τ ∈[0,δ)
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1 ∗ √ 1 − eεμ cj,ε − |eεμ − 1| γ∗τ − γτ +ε 2Λcτ 2 τ ∈[0,δ) & ' 1 εμ √ εμ − cj,ε − cj,ε γτ∗+ε , [1 − j(ε)]γτ +ε e − e 2 τ ∈[0,δ) 3 εμ 2 |e − 1| + [eεμ − 1]2 γτ +ε + 2
≤−
τ ∈[0,δ)
−
1 12 γτ − j(ε)γτ +ε 1Λ − Re γ∗τ − γτ∗ , Λτ (γτ − j(ε)γτ +ε ) τ 2
τ ∈[0,δ)
τ ∈[0,δ)
1 ∗ + γ∗τ − γτ 2Λcτ 2 τ ∈[0,δ)
√ √ and hence, since 1 − eεμ cj,ε − |eεμ − 1| ≥ 12 and eεμ − 12 eεμ cj,ε − cj,ε ≥ 12 , 1 1 Re − γ∗τ , γτ + γ∗τ , j(ε)γτ +ε − γ∗τ +ε , γτ +ε 2 2 τ ∈[0,δ) 1 ' 1& ∗ ∗ ≤− γ∗τ γτ +ε , [1 − j(ε)]γτ +ε − γτ +ε 2Λcτ + 4 2 τ ∈[0,δ) Re γ∗τ − γτ∗ , Λτ (γτ − j(ε)γτ +ε ) + 2 |eεμ − 1| γτ 2 − τ ∈(0,δ)
τ ∈[0,δ]
1 ∗ + γ∗τ − γτ 2Λcτ 2 τ ∈(0,δ)
The bound
1 1 Re − γ∗τ , γτ + γ∗τ , j(ε)γτ +ε − γ∗τ +ε , γτ +ε 2 2 τ ∈[0,δ) 1 ' 1& ∗ ∗ 2 ≤− γ∗τ γ − γ + , [1 − j(ε)]γ c τ ∗τ −ε −ε Λτ 4 2 ∗τ −ε τ ∈(0,δ] ∗ Re Λτ (γ∗τ − j(ε)γ∗τ −ε ), γτ − γ∗τ − τ ∈(0,δ)
+ 2 |eεμ − 1|
γ∗τ 2
τ ∈[0,δ]
1 ∗ + γ∗τ − γτ 2Λcτ 2 τ ∈(0,δ)
is proven similarly. Taking the average of these two bounds and using 1 ∗ 1 ∗ 1 ∗ γ − γτ +ε 2Λcτ + γ∗τ − γτ +ε 2Λcτ +ε ≥ γ∗τ − γτ +ε 2Λcτ ∪Λcτ +ε 8 ∗τ 8 8 gives the bound of the Lemma.
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The first line of the right hand side of the conclusion of Lemma 6.2, consists of terms that are invariably negative. The first can be thought of as a time derivative term and the other two as space derivative terms. These three terms ˜ )| + |P˜ (J )| + |P˜ (J )|} to are responsible for the contributions r(|J |)2 {|Q(J α β Proposition 3.36. See (6.8) a,b,c. The terms on the other two lines are all positive. The terms on the second line will be controlled using the “smallness” of γτ − j(ε)γτ +ε and γ∗τ − j(ε)γ∗τ −ε . See Lemma E.17. The first term on the third line is controlled in ∗ − γτ . The small field part of Proposition E.11(ii), using the smallness of γ∗τ the second term in the third line is bounded in the following lemma and gives (2) the small field regulator RegSF (Ω; α, β) in Proposition 3.36. The large field part is left explicitly in Proposition 3.36. In the proof of Theorem 3.35, it is canceled by quartic contributions. Lemma 6.3. |eεμ − 1|
1 γτ 2Ω + γ∗τ 2Ω ≤ Kreg δ|μ| α 2Ω + β 2Ω + |Ωc | 16
τ ∈[0,δ]
with the Kreg of Definition 2.17. Proof. Write, using the notation of Definition E.3, γτ 2Ω = (Γττ1 ατ1 ) (x) (Γττ2 ατ2 ) (x) x∈Ω τ ∈(0,δ) τ1 ,τ2 ∈Tr (τ,δ)
τ ∈(0,δ)
We bound the terms with (τ1 , τ2 ) = (δ, δ) using τ1 τ2 (Γτ ατ1 ) (x) (Γτ ατ2 ) (x) x∈Ω τ ∈(0,δ) τ1 ,τ2 ∈Tr (τ,δ) (τ1 ,τ2 )=(δ,δ)
≤
|Γττ1 (x, y)| κτ1 (y) |Γττ2 (x, z)| κτ2 (z)
x∈Ω τ ∈(0,δ) c c τ1 ,τ2 ∈Tr (τ,δ) y∈Λτ1 ,z∈Λτ2 (τ1 ,τ2 )=(δ,δ)
≤
( ) ( ) m m N0 Γττ1 ; e 2 d(x,Λ) , κτ1 N0 Γττ2 ; e 2 d(x,Λ) , κτ2
τ ∈(0,δ) τ1 ,τ2 ∈Tr (τ,δ) (τ1 ,τ2 )=(δ,δ)
× min |Λcτ1 |, |Λcτ2 | ⎤2 ⎡ ( m ) ⎣ ≤ |Ωc | N0 Γττ ; e 2 d(x,Λ) , κτ ⎦ τ ∈(0,δ)
τ ∈(0,δ]
2 δ ≤ |Ωc | 16eKj R(δ) ε
by Lemma E.13
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and we bound the term with τ1 = τ2 = δ using (Γδτ αδ ) (x) Γδτ αδ (x) τ ∈(0,δ) x∈Ω
=
Γδτ (x, y)β(y) Γδτ (x, z)β(z)
τ ∈(0,δ) x∈Ω y,z∈X
≤
Γδτ (x, y)β(y) Γδτ (x, z)β(z) + 2|Ωc |
τ ∈(0,δ) x,y,z∈Ω
×
( )2 m N0 Γδτ ; e 2 d(x,Λ) , κδ
τ ∈(0,δ)
≤
( )2 2 m N0 Γδτ ; 1, 1 β 2Ω + 2|Ωc | N0 Γδτ ; e 2 d(x,Λ) , κδ
τ ∈(0,δ)
τ ∈(0,δ)
2 δ δ ≤ e2Kj β 2Ω + 2 |Ωc | 16eKj R(δ) ε ε 2 δ We used that the operator on L (Ω) with kernel Γτ (x, y) has norm at most δ N0 Γτ ; 1, 1 . We have also used that |eεμ − 1| ≤ 18 . Consequently |eεμ − 1| ≤ 2ε|μ| and, all together,
γτ 2Ω + γ∗τ 2Ω |eεμ − 1| τ ∈[0,δ]
δ δ ≤ |e − 1| 2 α 2Ω + 2 β 2Ω + e2Kj α 2Ω + e2Kj β 2Ω ε ε 2 δ + 6 |Ωc | 16eKj R(δ) ε
≤ (4 + 2e2Kj )δ|μ| α 2Ω + β 2Ω + 212 e2Kj δ|μ|R(δ)2 |Ωc | εμ
By Hypothesis F.7.i, (2.19) and (2.17), 212 e2Kj δ|μ|R(δ)2 = 212 e2Kj Kμ δ 1−2eR −2er veμ −2eR −2er ≤ The claim now follows from Remark D.3. Proof of Proposition 3.36. Define, for each t ∈ (0, δ) ∩ εZ, Bt = Λ(Jt )c ∩ Λ Jt− ∩ Λ Jt+
1 16
(6.5)
Observe that, for all t ∈ (0, δ) ∩ εZ, supp P˜α (Jt ) ⊂ Bt supp P˜β (Jt ) ⊂ Bt
(6.6)
˜ (Jt ) ⊂ Bt Q Bt ∩ Bt = ∅
for all t = t ∈ Jt◦
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where Jt◦ = Jt \{t ± 12 |Jt |} is the interior of the interval Jt . To see the last line, observe that if t is strictly between t and t ± 12 |Jt |, then Λ(Jt± ) ⊂ Λt so that Bt ⊂ Λ(Jt± ) cannot intersect Bt ⊂ Λct . Now fix any 0 ≤ p, p < m. Suppose that t, t ∈ εZ ∩ (0, δ] have d(t) = p, d(t ) = p and t = t and suppose that Bt ∩ Bt = ∅. We claim that
τ ∈ εZ τ ∈ Jt− , τ = t ∩ τ ∈ εZ τ ∈ Jt− =∅ , τ = t (6.7)
+ + τ ∈ εZ τ ∈ Jt , τ = t ∩ τ ∈ εZ τ ∈ J , τ = t = ∅ t
1 To see the upper claim, first consider p = p , so that |Jt± | = |Jt± | = 2p δ. Then either t = t or (6.7) is satisfied. So, without loss of generality, we may assume − that p < p so that |Jt− | < |Jt |. If the upper claim of (6.7) is to be violated, then it is necessary that t be in the interior of Jt− . But then (6.6) provides the contradiction that Bt ∩ Bt = ∅. By Lemma D.4, γτ∗ , [1 − j(ε)]γτ
τ ∈(0,δ]
=
2
γτ∗ , [1 − e−ε∇
τ ∈(0,δ] −4DCH
= cH e
∗
H∇
3 ]γτ
≥ cH e−4DCH
ε ∇γτ
2
τ ∈(0,δ]
ε |∇γτ (b)|
2
b∈X ∗ τ ∈(0,δ]
Applying (6.7),
γτ∗ , [1
−4DCH
− j(ε)]γτ ≥ cH e
b∈X ∗
τ ∈(0,δ]
= cH e
t+|Jt+ |
t∈(0,δ) τ =t+ε
≥ cH e−4DCH
ε |∇γτ (b)|
2
τ =t+ε t∈(0,δ) supp b⊂Bt
−4DCH
t+|Jt+ |
ε |∇γτ (b)|
2
b∈X ∗ supp b⊂Bt
1 |Jt | R (|Jt |)2 P˜β (Jt ) 8
t∈(0,δ)
(6.8a) by Lemma E.12(ii). Similarly ∗ γ∗τ , [1 − j(ε)]γ∗τ τ ∈[0,δ)
≥ cH e−4DCH
t−ε
t∈(0,δ) τ =t−|Jt− |
≥ cH e−4DCH
ε |∇γ∗τ (b)|
b∈X ∗ supp b⊂Bt
1 |Jt | R (|Jt |)2 P˜α (Jt ) 8
t∈(0,δ)
2
(6.8b)
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and
τ ∈[0,δ)
≥
Ann. Henri Poincar´e
∗ 2 γ∗τ −γτ +ε 2Λcτ ∪Λcτ +ε
# " ∗ ∗ 2 γ∗τ − γτ +ε 2Λcτ ∪Λcτ +ε + γ∗τ −ε − γτ Λcτ −ε ∪Λcτ
τ ∈(0,δ)
# " ∗ ∗ γ∗t − γt+ε 2Λct ∪Λct+ε + γ∗t−ε − γt 2Λct−ε ∪Λct
=
t∈(0,δ)
# " ∗ ∗ 2 γ∗t − γt+ε 2Q(J + γ − γ t ˜ t) ˜ t) ∗t−ε Q(J
≥
t∈(0,δ)
≥
t∈(0,δ)
1 ˜ r(|Jt |)2 Q(J ) t 32
(6.8c)
Hence, by Lemmas 6.2 and E.17 and Proposition E.11(ii), Re
1 1 − γ∗τ , γτ + γ∗τ , j(ε)γτ +ε − γ∗τ +ε , γτ +ε 2 2
τ ∈[0,δ)
⎧ ⎨ & ∗ ' 1 1 ∗ ≤− γτ , [1−j(ε)]γτ γ∗τ −γτ +ε 2Λcτ ∪Λcτ +ε + ⎩ 4 2 τ ∈[0,δ)
+
&
∗ , [1−j(ε)]γ∗τ γ∗τ
τ ∈[0,δ)
+
⎫ '⎬
τ ∈(0,δ]
⎭
e−2mc( 2 |Jτ |) |Λcτ | + |eεμ − 1| 1
τ ∈(0,δ)
γτ 2 + γ∗τ 2
τ ∈[0,δ]
1 ∗ ≤− γ∗τ − γτ +ε 2Λcτ ∪Λcτ +ε − KL2 16 τ ∈[0,δ)
# " ˜ 2 ˜ 2 ˜ × r(|Jt |)2 Q(J t ) + |Jt | R (Jt ) Pα (Jt ) + |Jt | R (Jt ) Pβ (Jt ) t∈(0,δ)
+
e−2mc( 2 |Jτ |) |Λcτ | + |eεμ − 1| 1
τ ∈(0,δ)
γτ 2 + γ∗τ 2
τ ∈[0,δ]
1 1 where KL2 = 32 min{cH e−4DCH , 32 } with cH and CH being the smallest and largest eigenvalues of H, respectively. The claim follows from (F.4c), (F.7a),(F.7b) and Lemma 6.3.
6.2. Extracting Small Factors from the Quartic Form In this section, we prove Proposition 3.37. Recall that VS = −ε γ∗τ γτ +ε , v γ∗τ γτ +ε τ ∈εZ∩[0,δ)
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If we could replace γ∗τ by γτ∗ and γτ +ε by γτ , we would have γτ∗ γτ , v γτ∗ γτ ≤ −ε v1 γτ 2L4 (X) −ε τ ∈εZ∩[0,δ)
τ ∈εZ∩[0,δ)
= −εv1
|γτ (x)|
4
τ ∈εZ∩[0,δ) x∈X
which is very negative when some x’s are in large field regions. The following lemma expresses the error introduced by such a replacement as a sum of two ;S . The first, E:4,S is a pure small field contribution, which terms, E:4,S and V (4) will be bounded by the “small field regulator” RegSF . See Lemma 6.7. The ;S , is a large field contribution and is bounded by the two terms in second, V the third line of the right hand side in Proposition 3.37. Lemma 6.4. VS (α∗ , β; α ∗, α ) = −
1 1 ∗ ∗ ε γ∗τ γ∗τ , v γ∗τ γ∗τ − ε γτ∗ γτ , v γτ∗ γτ 2 2 τ ∈[0,δ)
τ ∈(0,δ]
;S (α∗ , β; α + E:4,S (α, β) + V ∗, α ) ;S (α∗ , β; α where V ∗, α ) is defined in (6.14) and bounded in Lemma 6.6 and E:4,S (α, β) is defined in (6.12) and bounded in Lemma 6.7. The first two terms on the right hand side are bounded in Lemma 6.5. Proof. We start with the difference ⎡ ⎤ 1 1 ∗ ∗ VS − ⎣− ε γ∗τ γ∗τ , v γ∗τ γ∗τ − ε γτ∗ γτ , v γτ∗ γτ ⎦ 2 2 τ ∈[0,δ)
τ ∈(0,δ]
between VS and the expressions which are manifestly large and negative in the large field region. (See Lemma 6.5). From this, we successively pull off ;4,S , leaving the small field contribution ;1,S , . . . , V four controllable pieces, V : E4,S (α, β). The first step is VS (α∗ , β; α ∗, α ) +
1 1 ∗ ∗ ε γ∗τ γ∗τ , vγ∗τ γ∗τ + ε γτ∗ γτ , vγτ∗ γτ 2 2 τ ∈[0,δ)
τ ∈(0,δ]
ε ∗ ∗ ∗ =− {γ∗τ (γτ +ε − γ∗τ ), vγ∗τ γτ +ε + γ∗τ γ∗τ , vγ∗τ (γτ +ε − γ∗τ )} 2 τ ∈[0,δ) ' ε & (γ∗τ − γτ∗+ε )γτ +ε , vγ∗τ γτ +ε − 2 τ ∈[0,δ) '
& ∗ + γτ +ε γτ +ε , v(γ∗τ − γτ∗+ε )γτ +ε ;1,S (α∗ , β; α ∗, α ) + V ∗, α ) = V (α∗ , β; α S
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where VS (α∗ , β; α ∗, α ) ε ∗ {Λγ∗τ (γτ +ε − γ∗τ ), v Λγ∗τ γτ +ε =− 2 τ ∈[0,δ)
∗ ∗ , v Λγ∗τ (γτ +ε − γ∗τ )} + Λγ∗τ γ∗τ & ' ε ∗ Λ(γ∗τ −γτ +ε )γτ +ε , v Λγ∗τ γτ +ε − 2 τ ∈[0,δ) '
& + Λγτ∗+ε γτ +ε , v Λ(γ∗τ −γτ∗+ε )γτ +ε
and ;1,S (α∗ , β; α V ∗, α ) ε ∗ =− {Λc γ∗τ (γτ +ε − γ∗τ ), v γ∗τ γτ +ε 2 τ ∈[0,δ)
∗ ∗ , v Λc γ∗τ (γτ +ε − γ∗τ )} + γ∗τ γ∗τ & c ' ε Λ (γ∗τ − γτ∗+ε )γτ +ε , v γ∗τ γτ +ε − 2 τ ∈[0,δ) '
& ∗ + γτ +ε γτ +ε , v Λc (γ∗τ − γτ∗+ε )γτ +ε ε ∗ {Λγ∗τ (γτ +ε − γ∗τ ), v Λc γ∗τ γτ +ε − 2 τ ∈[0,δ) ∗ , Λc γ∗τ γ∗τ
∗ v Λγ∗τ (γτ +ε − γ∗τ )} ' ε & ∗ Λ(γ∗τ −γτ +ε )γτ +ε , v Λc γ∗τ γτ +ε − 2 τ ∈[0,δ) '
& c ∗ + Λ γτ +ε γτ +ε , v Λ(γ∗τ −γτ∗+ε )γτ +ε
+
(6.9)
Next write ;2,S (α∗ , β; α VS (α∗ , β; α ∗, α ) = VS (α∗ , β; α ∗, α ) + V ∗, α )
where, in the notation of Definition 2.9 and Lemma E.4(i), ' ε & 0 ∗ ∗ Λ Γ∗τ α (γτ +ε − γ∗τ (α∗ , β; α ∗, α ) = − ), v Λ Γ0∗τ α∗ Γδτ +ε β VS 2 τ ∈[0,δ)
−
' ε & 0 ∗ 0 ∗ Λ Γ∗τ α Γ∗τ α , v Λ Γ0∗τ α∗ (γτ +ε − γ∗τ ) 2 τ ∈[0,δ)
−
' ε & Λ(γ∗τ − γτ∗+ε ) Γδτ +ε β , v Λ Γ0∗τ α∗ Γδτ +ε β 2 τ ∈[0,δ)
' ε & δ Λ Γτ +ε β ∗ Γδτ +ε β , v Λ(γ∗τ −γτ∗+ε ) Γδτ +ε β − 2 τ ∈[0,δ)
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and, using the notation of Definition E.3, ;2,S (α∗ , β; α V ∗, α ) ε =− 2 τ ∈[0,δ)
& τ ∗ ' ∗ ), vΛ Γτ∗τ2 α∗τ2 Γττ3+ε ατ3 Λ Γ∗τ1 ατ1 (γτ +ε − γ∗τ
τ1 ,τ2 ∈Tl (τ,δ)
τ3 ∈Tr (τ +ε,δ) (τ1 ,τ2 ,τ3 ) =(0,0,δ)
ε − 2
τ ∈[0,δ) τ1 ,τ2 ,τ3 ∈Tl (τ,δ)
& τ ∗ τ ' ∗ Λ Γ∗τ1 ατ1 (Γ∗τ2 ατ2 ) , vΛ Γτ∗τ3 α∗τ3 (γτ +ε − γ∗τ )
(τ1 ,τ2 ,τ3 ) =(0,0,0)
ε − 2
&
τ ∈[0,δ) τ1 ,τ2 ∈Tr (τ +ε,δ)
' Λ(γ∗τ − γτ∗+ε ) Γττ1+ε ατ1 , vΛ Γτ∗τ3 α∗τ3 Γττ2+ε ατ2
τ3 ∈Tl (τ,δ) (τ1 ,τ2 ,τ3 ) =(δ,δ,0)
−
ε 2
τ ∈[0,δ) τ1 ,τ2 ,τ3 ∈Tr (τ +ε,δ)
& τ1 ' Λ Γτ +ε α∗τ1 Γττ2+ε ατ2 , vΛ(γ∗τ − γτ∗+ε ) Γττ3+ε ατ3
(τ1 ,τ2 ,τ3 ) =(δ,δ,δ)
(6.10)
Next write ;3,S (α∗ , β; α VS (α∗ , β; α ∗, α ) = VS (α∗ , β; α ∗, α ) + V ∗, α )
where VS (α∗ , β; α ∗, α ) ' ε & 0 ∗ Λ Γ∗τ Ωα∗ (γτ +ε − γ∗τ ), v Λ Γ0∗τ Ωα∗ Γδτ +ε Ωβ =− 2 τ ∈[0,δ) ' ε & 0 ∗ Λ Γ∗τ Ωα∗ Γ0∗τ Ωα , v Λ Γ0∗τ Ωα∗ (γτ +ε − γ∗τ − ) 2 τ ∈[0,δ) ' ε & Λ(γ∗τ − γτ∗+ε ) Γδτ +ε Ωβ , v Λ Γ0∗τ Ωα∗ Γδτ +ε Ωβ − 2 τ ∈[0,δ) ' ε & δ Λ Γτ +ε Ωβ ∗ Γδτ +ε Ωβ , v Λ(γ∗τ − γτ∗+ε ) Γδτ +ε Ωβ − 2 τ ∈[0,δ)
and, using Ω(0) = Ω and Ω(1) = Ωc , ;3,S (α∗ , β; α V ∗, α ) ε =− 2 τ ∈[0,δ)
i,j,k∈{0,1} (i,j,k)=(0,0,0)
2 ( ) ∗ Λ Γ0∗τ Ω(i) α∗ (γτ +ε − γ∗τ ),
)( ( )3 × v Λ Γ0∗τ Ω(j) α∗ Γδτ +ε Ω(k) β ) 2 ( )( ε Λ Γ0∗τ Ω(i) α∗ Γ0∗τ Ω(j) α , − 2 τ ∈[0,δ)
i,j,k∈{0,1} (i,j,k)=(0,0,0)
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( 3 ) ∗ × v Λ Γ0∗τ Ω(k) α∗ (γτ +ε − γ∗τ ) 2 ( ) ε Λ(γ∗τ − γτ∗+ε ) Γδτ +ε Ω(i) β , − 2 τ ∈[0,δ)
i,j,k∈{0,1} (i,j,k)=(0,0,0)
)( )3 × v Λ Γ0∗τ Ω(j) α∗ Γδτ +ε Ω(k) β 2 ( )( ) ε Λ Γδτ +ε Ω(i) β ∗ Γδτ +ε Ω(j) β , − 2 (
τ ∈[0,δ)
i,j,k∈{0,1} (i,j,k)=(0,0,0)
( )3 × v Λ(γ∗τ − γτ∗+ε ) Γδτ +ε Ω(k) β
(6.11)
(α∗ , β; α ∗, α ), Finally substitute, in VS ∗ (1) (2) γ∗τ − γτ +ε = γ˜ ˜τ,ε + γ˜τ,ε + γ˜τ,ε
where ˜ ˜ − τ − ε)Ωβ γ˜˜τ,ε = j(τ )Ωα−j(δ
(1) ∗ γ˜τ,ε = γ∗τ − γτ +ε − γ˜τ,ε
(2) γ˜τ,ε = γ˜τ,ε − γ˜˜τ,ε
˜ the set of all point in X that are within a distance c(δ) of Ω and γ˜τ,ε where Ω was defined in Corollary E.9, and write ;4,S (α∗ , β; α VS (α∗ , β; α ∗, α ) = E:4,S (α, β) + V ∗, α )
where E:4,S (α, β) =
' ε & 0 Λ Γ∗τ Ωα∗ γ˜ ˜τ,ε , v Λ Γ0∗τ Ωα∗ Γδτ +ε Ωβ 2 τ ∈[0,δ) ' ε & 0 Λ Γ∗τ Ωα∗ Γ0∗τ Ωα , v Λ Γ0∗τ Ωα∗ γ˜˜τ,ε + 2 τ ∈[0,δ) ' ε & ˜∗ δ − Λγ˜τ,ε Γτ +ε Ωβ , v Λ Γ0∗τ Ωα∗ Γδτ +ε Ωβ 2 τ ∈[0,δ) δ ' ε & δ ∗ Λ Γτ +ε Ωβ ∗ Γδτ +ε Ωβ , v Λγ˜˜τ,ε Γτ +ε Ωβ − (6.12) 2 τ ∈[0,δ)
and ;4,S (α∗ , β; α V ∗, α ) (1) 3 ε 2 0 (2) γτ,ε + γ˜τ,ε ), v Λ Γ0∗τ Ωα∗ Γδτ +ε Ωβ = Λ Γ∗τ Ωα∗ (˜ 2 τ ∈[0,δ) 3 (1) ε 2 0 (2) Λ Γ∗τ Ωα∗ Γ0∗τ Ωα , v Λ Γ0∗τ Ωα∗ (˜ γτ,ε + γ˜τ,ε ) + 2 τ ∈[0,δ) )∗ 3 ε 2 ( (1) (2) Λ γ˜τ,ε + γ˜τ,ε Γδτ +ε β , v Λ Γ0∗τ Ωα∗ Γδτ +ε Ωβ − 2 τ ∈[0,δ) ( )∗ 3 ε 2 δ (1) (2) Γδτ +ε Ωβ + γ˜τ,ε − Λ Γτ +ε Ωβ ∗ Γδτ +ε Ωβ , v Λ γ˜τ,ε (6.13) 2 τ ∈[0,δ)
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Of course ;S (α∗ , β; α V ∗, α ) =
4
;i,S (α∗ , β; α V ∗, α )
(6.14)
i=1
Lemma 6.5. We have ε γτ∗ γτ , v γτ∗ γτ ≥ 4CL
τ ∈(0,δ]
2
r (|J |) #P˜β (J )
decimation intervals J ⊂[0,δ]
∗ ∗ ε γ∗τ γ∗τ , v γ∗τ γ∗τ ≥ 4CL
τ ∈[0,δ)
2
r (|J |) #P˜α (J )
decimation intervals J ⊂[0,δ]
with the CL of Lemma F.5. Proof. We again use, for each t ∈ (0, δ) ∩ εZ, the notation Bt = Λ(Jt )c ∩ Λ Jt− ∩ Λ Jt+ of (6.5). Since v is repulsive, its smallest eigenvalue v1 > 0, so that ε γτ∗ γτ , v γτ∗ γτ τ ∈(0,δ]
1 4 1γτ∗ γτ 2 = εv1 |γτ (x)|
≥ εv1
x∈X τ ∈(0,δ]
τ ∈(0,δ]
≥
t+|Jt+ |
εv1 |γτ (x)|
4
by (6.7)
x∈X t∈(0,δ) τ =t+ε x∈Bt
=
t+|Jt+ |
εv1 |γτ (x)|
4
t∈(0,δ) τ =t+ε x∈Bt
≥
1 4 |Jt+ | v1 R (|Jt |) #P˜β (Jt ) 16
by Lemma E.12(v)
t∈(0,δ)
Similarly
∗ ∗ ε γ∗τ γ∗τ , v γ∗τ γ∗τ ≥
τ ∈[0,δ)
By Hypothesis 2.14, follows from (F.7c).
1 4 |Jt− | v1 R (|Jt |) #P˜α ([Jt ) 16 t∈(0,δ)
± 1 16 |Jt |v1
≥
± cv 16 |Jt ||||v|||
≥
± cv 64 |Jt |v,
and the claim now
;2,S , V ;3,S and V ;4,S were defined in (6.9), (6.10), ;1,S , V Lemma 6.6. Recall that V (6.11) and (6.13) respectively.
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; 2 ∗ ∗, α ) ≤ 219 e3Kj εvR(ε)3 γ∗τ − γτ +ε Λc ∪Λc V1,S (α∗ , β; α τ τ +ε τ ∈[0,δ)
+
221 e3Kj εvR(ε)3
τ ∈(0,δ)
+ 228 e4Kj |Jτ | v r (|Jτ |) R (|Jτ |)
3
)
; ∗, α ) ≤ 217 e4Kj δv r(δ)R(δ)3 |Λc | V2,S (α∗ , β; α ; ∗, α ) ≤ 217 e4Kj δv r(δ)R(δ)3 |Ωc | V3,S (α∗ , β; α ; ∗, α ) ≤ 216 e4Kj δv r(δ)R(δ)3 |Ωc | V4,S (α∗ , β; α
|Λcτ |
Proof. We prove that & ' Λc (Λcτ ∪ Λcτ +ε )(γ∗τ − γτ∗+ε )γτ +ε , v γ∗τ γτ +ε ε τ ∈[0,δ)
≤
∗ 218 e3Kj εvR(ε)3 γ∗τ − γτ +ε Λcτ ∪Λc 2
τ +ε
τ ∈[0,δ)
+
220 e3Kj εvR(ε)3 |Λcτ |
(6.15)
τ ∈(0,δ)
and
ε
& c ' Λ Λτ Λτ +ε (γ∗τ − γτ∗+ε )γτ +ε , v γ∗τ γτ +ε
τ ∈εZ∩[0,δ)
≤
3
226 e4Kj |Jτ |v r (|Jτ |) R (|Jτ |) |Λcτ |
(6.16)
τ ∈(0,δ)
and ε
& ' Λ(γ∗τ − γτ∗+ε )γτ +ε , v Λc γ∗τ γτ +ε ≤ 214 e4Kj δv r(δ)R(δ)3 |Λcδ | 2
τ ∈[0,δ)
(6.17) and ε
& ' Λ(γ∗τ − γτ∗+ε ) Γττ1+ε ατ , vΛ Γτ∗τ3 ατ∗ Γττ2+ε ατ 1 2 3
τ ∈[0,δ) τ1 ,τ2 ∈Tr (τ +ε,δ) τ3 ∈Tl (τ,δ) (τ1 ,τ2 ,τ3 )=(δ,δ,0)
≤ 216 e4Kj δv r(δ)R(δ)3 |Λc | and ε
(6.18)
& ' Λ|γ∗τ − γτ∗+ε | Γδτ +ε Ωc |β| , v Λ Γ0∗τ |α| Γδτ +ε |β| τ ∈[0,δ)
≤ 214 e4Kj δv r(δ)R(δ)3 |Ωc |
(6.19)
Vol. 11 (2010)
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275
2 (1) 3 , vΛ Γ0∗τ Ωα∗ Γδτ +ε Ωβ Λ Γ0∗τ Ωα∗ γ˜τ,ε τ ∈[0,δ)
≤ 214 e4Kj δvr(δ)R(δ)3 |Λc | and ε
(6.20)
2 (2) 3 1 ˜ c| , v Λ Γ0∗τ Ωα∗ Γδτ +ε Ωβ ≤ e−2mc(δ) δv|Ω Λ Γ0∗τ Ωα∗ γ˜τ,ε 8
τ ∈[0,δ)
(6.21) 1 2 [(6.15)+(6.16)]
bounds the first two lines of Four copies of (minor variants of) ; the definition of V1,S (α∗ , β; α ∗, α ) in (6.9). Four copies of (minor variants of) 1 ;1,S (α∗ , β; α (6.17) bounds the last two lines of the definition of V ∗, α ) in (6.9). 2 1 ; Four copies of (minor variants of) 2 (6.18) bounds V2,S (α∗ , β; α ∗, α ), which was defined in (6.10). Twelve copies of (minor variants of) 12 (6.19) bounds ;3,S (α∗ , β; α V ∗, α ), which was defined in (6.11). Four copies of (minor variants ;4,S (α∗ , β; α ∗, α ), which was defined in (6.13). of) 12 [(6.20) + (6.21)] bounds V Proof of (6.15). By Proposition E.11(i,ii) with J = Jτ , and (6.4), |γτ (y)| , |γ∗τ (y)| ≤ 26 eKj R(ε) for all y ∈ X for all τ ∈ εZ ∩ [0, δ]. Hence, since y∈X |v(x, y)| ≤ v, & ' Λc (Λcτ ∪ Λcτ +ε )(γ∗τ − γτ∗+ε )γτ +ε , v γ∗τ γτ +ε ε τ ∈[0,δ)
≤
∗ εv 218 e3Kj R(ε)3 |γ∗τ (x) − γτ +ε (x)|
τ ∈[0,δ) x∈(Λcτ ∪Λcτ +ε )∩Λc
≤
218 e3Kj εvR(ε)3
" # 2 ∗ |γ∗τ (x) − γτ +ε (x)| + 1
τ ∈[0,δ) x∈(Λcτ ∪Λcτ +ε )∩Λc
≤
" 2 ∗ 218 e3Kj εvR(ε)3 γ∗τ − γτ +ε Λc ∪Λc τ
τ ∈[0,δ)
≤
τ +ε
∗ 218 e3Kj εvR(ε)3 γ∗τ − γτ +ε Λc ∪Λc 2
τ
τ ∈[0,δ)
τ +ε
+
# + |Λcτ ∩ Λcδ |+|Λcτ +ε ∩ Λcδ | 2
2
220 e3Kj εvR(ε)3 |Λcτ |
τ ∈(0,δ)
Proof of (6.16). We actually prove that & ' Λc Λτ Λτ +ε (γ∗τ − γτ∗+ε )γτ +ε , v γ∗τ γτ +ε ε τ ∈[0,δ) τ ∈2εZ
≤
τ ∈(0,δ)
3
225 e4Kj |Jτ |v r (|Jτ |) R (|Jτ |) |Λcτ |
(6.22a)
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The proof that
ε
& c ' Λ Λτ Λτ +ε (γ∗τ − γτ∗+ε )γτ +ε , v γ∗τ γτ +ε
τ ∈[0,δ) τ ∈εZ\2εZ
=ε
|Λc Λτ Λτ −ε (γ∗τ −ε − γτ∗ )γτ , v γ∗τ −ε γτ |
τ ∈(0,δ] τ ∈2εZ
≤
3
225 e4Kj |Jτ |v r (|Jτ |) R (|Jτ |) |Λcτ |
(6.22b)
τ ∈(0,δ)
is similar. Now for (6.22a). For any τ ∈ 2εZ ∩ (0, δ), we necessarily have Jτ +ε = [τ, τ + 2ε] ⊂ Jτ , so that Λτ +ε ⊃ Λτ . Hence, by Lemma E.4(ii), and recalling that Λ0 = ∅, & c ' Λ Λτ Λτ +ε (γ∗τ − γτ∗+ε )γτ +ε , v γ∗τ γτ +ε ε τ ∈2εZ∩[0,δ)
=ε
& c ' Λ Λτ (γ∗τ − γτ∗+ε )γτ +ε , v γ∗τ γτ +ε
τ ∈2εZ∩[0,δ)
≤ε
τ ∈2εZ∩(0,δ) τr ∈Tr (τ,δ) τl ∈Tl (τ,δ)
& ' × Λc Λπ(τr ) Λcτr Λσ(τl ) Λcτl (γ∗τ − γτ∗+ε )γτ +ε , v γ∗τ γτ +ε ≤ε τ ∈2εZ∩(0,δ) [τr ,τl ]∈Tlr (τ,δ)
& ' × Λc Λ([τl , τr ])Λcτr Λcτl (γ∗τ − γτ∗+ε )γτ +ε , v γ∗τ γτ +ε where Tlr (τ, δ) is the set of all decimation intervals (in the sense of Notation 2.2 [τl , τr ] with τl ∈ Tl (τ, δ), τr ∈ Tr (τ, δ) and Λπ(τr ) ∩ Λcτr ∩ Λσ(τl ) ∩ Λcτl = ∅. If [τl , τr ] ∈ Tlr (τ, δ), then τ ∈ (τl , τr ) so, given any decimation interval [τl , τr ], −τl and the number of τ ∈ 2εZ ∩ (0, δ) with [τl , τr ] ∈ Tlr (τ ) is less than τr2ε & c ' Λ Λ ([τl , τr ]) Λcτ Λcτ (γ∗τ − γτ∗+ε )γτ +ε , vγ∗τ γτ +ε ε r l τ ∈2εZ∩(0,δ) [τl ,τr ]∈Tlr (τ,δ)
≤
τr − τ l 2
sup τ ∈2εZ∩(0,δ) [τl ,τr ]∈Tlr (τ,δ)
& c ' Λ Λ ([τl , τr ]) Λcτ Λcτ (γ∗τ − γτ∗+ε )γτ +ε , vγ∗τ γτ +ε r l
We now fix any decimation interval [τl , τr ] and any τ ∈ 2εZ ∩ (0, δ) for which [τl , τr ] ∈ Tlr (τ, δ) and bound ' τr − τl & c Λ Λ ([τl , τr ]) Λcτl Λcτr (γ∗τ − γτ∗+ε )γτ +ε , v γ∗τ γτ +ε 2 τr − τ l c |Λ ∩ Λcτl ∩ Λcτr | (4eKj + 3)r(τr − τl ) ≤ 2
Vol. 11 (2010)
The Temporal Ultraviolet Limit
× ≤
sup
x∈Λ([τl ,τr ]) y∈X
277
|γτ +ε (x)v(x, y)γ∗τ (y)γτ +ε (y)|
τr − τ l c |Λ ∩ Λcτl ∩ Λcτr |(4eKj + 3)r(τr − τl )|||v||| 2 ⎤2 ⎡ ( ) m ×⎣ N0 Γττ +ε (S); e 2 d(x,Λ([τl ,τr ])) , κτ ⎦ τ ∈(0,δ]
⎡ ×⎣
(
N0 Γτ∗τ (S); e
m 2
d(x,Λ([τl ,τr ]))
)
⎤
, κ∗τ ⎦
τ ∈[0,δ)
by Corollary E.9(b). Since τ is in the interior of [τl , τr ] and τ + ε is its neighbour and has d(τ + ε) = m > d(τr ), τ + ε is also in the interior of [τl , τr ]. Hence by Lemma E.13, ' τr − τl & c Λ Λ ([τl , τr ]) Λcτl Λcτr (γ∗τ − γτ∗+ε )γτ +ε , v γ∗τ γτ +ε 2 ≤
τr − τ l c |Λ ∩ Λcτl ∩ Λcτr | (4eKj + 3)r(τr −τl ) |||v||| 403 e3Kj R(τr −τl )3 2
≤ 220 e4Kj (τr −τl )v r(τr −τl )R(τr −τl )3 |Λc ∩ Λcτl ∩ Λcτr | and
ε
& c ' Λ Λτ Λτ +ε (γ∗τ − γτ∗+ε )γτ +ε , v γ∗τ γτ +ε
τ ∈ε2Z∩[0,δ)
≤
220 e4Kj (τr −τl )v r(τr −τl )R(τr −τl )3 |Λc ∩ Λcτl ∩ Λcτr |
decimation intervals τl ,τr
≤
3
225 e4Kj |Jτ |v r (|Jτ |) R (|Jτ |) |Λcτ |
τ ∈εZ∩(0,δ)
For the last inequality, each decimation interval [τl , τr ] was assigned to a τ ∈ εZ ∩ (0, δ) by ◦ if [τl , τr ] = [0, δ], the assigned τ is 2δ . ◦ if [τl , τr ] = [0, δ], then [τl , τr ] is an interval of length 21p δ for some scale 1 ≤ p < m. In this case τ is assigned to the unique end point, τl or τr , whose decimation index is p. At most 3 ≤ 22 intervals are assigned to each τ and, for each decimation interval, 3
(τr −τl )v r(τr −τl )R(τr −τl )3 |Λc ∩ Λcτl ∩ Λcτr | ≤ 23 |Jτ |v r (|Jτ |) R (|Jτ |) |Λcτ |
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Proof of (6.17). & ' Λ(γ∗τ − γτ∗+ε )γτ +ε , v Λc γ∗τ γτ +ε ε τ ∈[0,δ)
& ' ≤ max δ Λ(γ∗τ − γτ∗+ε )γτ +ε , v Λc γ∗τ γτ +ε τ ∈[0,δ) γ∗τ (x) − γτ∗+ε (x) γτ +ε (x)v(x, y)γ∗τ (y)γτ +ε (y) ≤ max δ|Λc | sup τ ∈[0,δ)
y∈Λc
x∈Λ
⎡
≤ max δ|Λc | (4eKj + 3)r(δ) |||v||| ⎣ τ ∈[0,δ)
⎡
N0 Γττ +ε (S); e
m 2
d(x,Λ)
)
⎤2
, κτ ⎦
τ ∈(0,δ]
(
×⎣
(
N0 Γτ∗τ (S); e
m 2
d(x,Λ)
)
⎤
, κ∗τ ⎦
τ ∈[0,δ)
3 v 4 Kj 2 e R(δ) |Λc | 2 δv r(δ)R(δ)3 |Λc |
≤ δ(4eKj + 3)r(δ) ≤ 214 e4Kj
by Corollary E.9(b) and Lemma E.13 (When τ + ε = δ, use Γδδ (S) = 1 and Γτδ (S) = 0 for τ ∈ (0, δ), and when τ = 0, use Γ0∗0 (S) = 1 and Γτ∗0 (S) = 0 for τ ∈ (0, δ). In both cases, apply Lemma B.1). Proof of (6.18). We are to sum over (τ1 , τ2 , τ3 ) excluding (δ, δ, 0). We treat the case τ3 = 0. The other cases are similar. & ' Λ(γ∗τ − γτ∗+ε ) Γττ1+ε ατ1 , vΛ Γτ∗τ3 ατ∗ Γττ2+ε ατ2 ε 3 τ ∈[0,δ) τ1 ,τ2 ∈Tr (τ +ε,δ) τ3 ∈Tl (τ,δ)\{0}
≤ max δ τ ∈[0,δ)
& ' Λ(γ∗τ −γτ∗+ε ) Γττ1+ε ατ , vΛ Γτ∗τ3 ατ∗ Γττ2+ε ατ 1 2 3
τ1 ,τ2 ∈Tr (τ +ε,δ) τ3 ∈Tl (τ,δ)\{0}
≤ max δ τ ∈[0,δ)
(4eKj + 3)r(δ) Γττ1+ε ατ1 (x) |v(x, y)
τ1 ,τ2 ∈Tr (τ +ε,δ) x,y∈Λ c τ3 ∈Tl (τ,δ)\{0} z∈Λτ3
× |Γτ∗τ3 (y, z)| κ∗τ3 (z) Γττ2+ε ατ2 (y) ⎤2 ⎡ ( ) m ≤ (4eKj + 3)δr(δ) max |||v||| ⎣ N0 Γττ +ε (S); e 2 d(x,Λ) , κτ ⎦ τ ∈[0,δ)
⎡ ×⎣
(
τ ∈(0,δ] m
|Λcτ3 |N0 Γτ∗τ3 (S); e 2
d(x,Λ))
)
⎤
, κ∗τ3 ⎦
τ3 ∈(0,δ)
3 v 4 Kj 2 e R(δ) |Λc | 2 δvr(δ)R(δ)3 |Λc |
≤ (4eKj + 3)δr(δ) ≤ 214 e4Kj
by Lemma E.13 and Corollary E.9(b).
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279
Proof of (6.19). By Corollary E.9(b) and Lemma E.13, & ' Λ|γ∗τ − γτ∗+ε | Γδτ +ε Ωc |β| , v Λ Γ0∗τ |α| Γδτ +ε |β| ε τ ∈[0,δ)
& ' ≤ max δ Λ|γ∗τ − γτ∗+ε | Γδτ +ε Ωc |β| , v Λ Γ0∗τ |α| Γδτ +ε |β| τ ∈[0,δ)
≤ max δ τ ∈[0,δ)
(4eKj + 3)r(δ) Γδτ +ε (x, z) κδ (z) |v(x, y)|
x,y∈Λ z∈Ωc
z ,z ∈X
× Γ0∗τ (y, z ) κ∗0 (z ) Γδτ +ε (y, z ) κδ (z ) ( ) m ≤ (4eKj + 3) δr(δ) max |||v||| |Ωc |N0 Γδτ +ε (S); e 2 d(x,Λ) , κδ
τ ∈[0,δ)
(
m
× N0 Γ0∗τ (S); e 2 ≤ (4eKj + 3)δr(δ)
d(x,Λ))
) , κ∗0
( ) m N0 Γδτ +ε (S); e 2 d(x,Λ)) , κδ
3 v 4 Kj 2 e R(δ) |Ωc | 2
≤ 214 e4Kj δvr(δ)R(δ)3 |Ωc | Proof of (6.20). By Corollary E.9.c 2 (1) 3 ε , v Λ Γ0∗τ Ωα∗ Γδτ +ε Ωβ Λ Γ0∗τ Ωα∗ γ˜τ,ε τ ∈[0,δ)
≤ max δ τ ∈[0,δ)
∗ |(γτ +ε − γ∗τ + γ˜τ,ε )(x)| Γ0∗τ Ωα∗ (x) |v(x, y)|
x,y∈Λ
× Γ0∗τ Ωα∗ (y) Γδτ +ε Ωβ (y) )2 ( m ≤ δ 3eKj r(δ) |Λc | max |||v||| N0 Γ0∗τ (S); e 2 d(x,Λ) , κ∗0 τ ∈[0,δ) δ m ×N0 Γτ +ε (S); e 2 d(x,Λ)) , κδ 3 ≤ 3eKj δr(δ)v 24 eKj R(δ) |Λc | ≤ 214 e4Kj δv r(δ)R(δ)3 |Λc |
by Lemma E.13. Proof of (6.21). By Corollary E.9(c) 2 (2) 3 ε , v Λ Γ0∗τ Ωα∗ Γδτ +ε Ωβ Λ Γ0∗τ Ωα∗ γ˜τ,ε τ ∈[0,δ)
≤ max δ τ ∈[0,δ)
×
γτ,ε − γ˜ e−4md(x,Ω) (˜ ˜τ,ε )(x) e4md(x,Ω)
x,y∈Λ
Γ0∗τ Ωα∗
(x) |v(x, y)| Γ0∗τ Ωα∗ (y) Γδτ +ε Ωβ (y)
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( ) 1 ˜ c | max |||v||| N4m Γ0∗τ (S); e m2 d(x,Λ) , κ∗0 ≤ δ e−3mc(δ) |Ω 8 τ ∈[0,δ) ( ) ( ) m m ×N0 Γ0∗τ (S); e 2 d(x,Λ) , κ∗0 N0 Γδτ +ε (S); e 2 d(x,Λ)) , κδ 3 1 −3mc(δ) ˜ c| e δv 24 eKj R(δ) |Ω 8 1 ˜ c| ≤ e−2mc(δ) δv|Ω 8 by Lemma E.13, followed by (F.6b). ≤
Lemma 6.7.
: (4) E4,S (α, β) ≤ RegSF (α, β)
(4) with the E:4,S (α, β) defined in (6.12) and the RegSF (Ω; α, β) and Kreg of Definition 2.17.
Proof. Applying that f, vg ≤ |||v||| f L2 (X) g L2 (X) and that
1 1 1 1 1 5m d(x,Ω) 1 1 1 f g L2 (X) ≤ 1e−5m d(x,Ω) f 1 4 g1 4 1e L (X) L (X) 1 1 1 1 1 5m d(x,Ω) 1 1 5m d(x,Ω) 1 ≤ 1e f1 g1 1e 4 4 L (X)
L (X)
we have 1 1 : 1 1 ˜τ,ε 1 4 E4,S (α, β) ≤ 2δ max |||v||| 1e−5m d(x,Ω) γ˜ τ ∈[0,δ) L (X) 1 13 1 13 1 1 1 1 × max 1e5m d(x,Ω) Γ0∗τ Ωα∗ 1 4 , 1e5m d(x,Ω) Γδτ +ε Ωβ 1 4 L (X)
L (X)
Next use the fact [8, Theorem 9.5.1] that, for any function A : X × X → C, the norm of the operator f ∈ L4 (X) → (Af )(x) = y∈X A(x, y)f (y) ∈ L4 (X) is bounded by ⎫ ⎧ ⎬ ⎨ |A(x, y)| , max |A(x, y)| max max y∈X ⎭ ⎩ x∈X y∈X
x∈X
and the trivial observation that, for y ∈ Ω, e5m d(x,Ω) A(x, y) ≤ e5m d(x,y) A(x, y)|. Consequently 1 1 1 5m d(x,Ω) 0 1 Γ∗τ Ωα∗ 1 4 ≤ |||Γ0∗τ ||| α L4 (Ω) ≤ eKj δ α L4 (Ω) 1e L (X) 1 1 1 5m d(x,Ω) δ 1 Γτ +ε Ωβ 1 ≤ |||Γδτ +ε ||| β L4 (Ω) ≤ eKj δ β L4 (Ω) 1e L4 (X)
To bound e−5m d(x,Ω) γ˜ ˜τ,ε L4 (X) , write ˜ − β) + (j(τ ) − 1) Ωα ˜ − (j(δ − τ − ε) − 1) Ωβ ˜ γ˜ ˜τ,ε = Ω(α
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" ˜∗ B (0) = b = (u, v) ∈ Ω " ˜∗ Bb1 = b = (u, v) ∈ Ω " ˜∗ Bb2 = b = (u, v) ∈ Ω
281
# ˜ u, v ∈ Ω # ˜ v∈Ω ˜c u ∈ Ω, # ˜ c, v ∈ Ω ˜ u∈Ω
and write τ μτ
e
dτ
(
e−τ
∇∗ H∇
) ˜ ∇∗ H∇ Ωα (x) = a1 (x) + a2 (x) + a3 (x)
0
with a1 (x) =
τ
μτ
e
z∈X b∈B(0)
a2 (x) =
∇∗ H∇
(x, z) (∇∗ H)(z, b) (∇α)(b)
0
τ
μτ
e
z∈X b=(u,v)∈Bb1
a3 (x) =
dτ e−τ
∇∗ H∇
(x, z) (∇∗ H)(z, b) {−α(u)}
dτ e−τ
∇∗ H∇
(x, z) (∇∗ H)(z, b) {α(v)}
0
τ
dτ e−τ
μτ
e
z∈X b=(u,v)∈Bb2
0
By (D.4) 1 1 1 ˜ 1 1 1(1 − eμτ )Ωα
L4 (X)
≤ |μ|δeKj α L4 (Ω) ˜
By (D.3), and the corresponding bounds with the sup over the right argument and the sum over the left argument,
a1 L4 (X) ≤ δKj eKj ∇α L4 (Ω˜ ∗ ) ˜c
˜c
e−5md(x,Ω) a2 L4 (X) ≤ e−5md(Ω,Ω ) δKj eKj α L4 (Ω) ˜ e−5md(x,Ω) a3 L4 (X) ≤ e−5md(Ω,Ω ) δKj eKj α L4 (Ω) ˜ All together, using (D.2), ˜ L4 (X) ≤ α − β L4 (X) e−5md(x,Ω) γ˜ ˜c +2Kj eKj δ μ+e−5md(Ω,Ω ) α L4 (Ω) ˜ + β L4 (Ω) ˜ 1 1 1 1 + δ ∇α 1L4 (Ω˜ ∗ ) + ∇β 1 4 ˜∗ L (Ω )
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and # " : E4,S (α, β) ≤ δv 2Kj e3Kj +Kj max α 3L4 (Ω) , β 3L4 (Ω) × α − β L4 (X) + δ μ+e−5mc(δ) α L4 (Ω) ˜ + β L4 (Ω) ˜ 1 1 1 1 + δ ∇α 1L4 (Ω˜ ∗ ) + ∇β 1 4 ∗ ˜ L (Ω )
The lemma now follows by Remark D.3.
Proof of Proposition 3.37. It suffices to apply Lemmas 6.4, 6.5, 6.6 and 6.7 and the observations, from (F.6a), that 1 219 e3Kj εvR(ε)3 ≤ 16 1 3 221 e3Kj εvR(ε)3 + 228 e4Kj + 219 e4Kj |J |v r (|J |) R (|J |) ≤ 8 Remark 6.8. Since Proposition 3.37 is an upper bound, rather than an equal(4) ity, the specific choice of RegSF (α, β) that we have made is far from the only possibility. We have chosen it to be relatively simple. 6.3. Extracting Small Factors from the Stokes’ Cylinder Proof of Proposition 3.38. In this section we prove the required bounds involving the Stokes’ cylinder Cs (x; α∗ , β) introduced in Definition 2.8. For simplicity of notation, write r = r(s) R = R(s) R = R (s)
r+ = r(2s)
R+ = R(2s)
R + = R (2s)
and Ω1 = Ω(J − )
Ω2 = Ω(J + ) Ω0 = Ω1 ∩ Ω2
Λ = Λ(J ) Pα = Pα (J )
···
As pointed out in Definition 2.8, we may choose for this cylinder any two real dimensional surface in
(z∗ , z) ∈ C2 |z∗ |, |z| < R
whose boundary is the union of the circle (z∗ , z) ∈ C2 z∗∗ = z, |z| = r and the curve bounding " # D = (z∗ , z) ∈ C2 |z∗ − σ∗ | ≤ r, |z − σ| ≤ r, z − z∗∗ = ρ where σ = ([1 − jc (s)]β) (x) σ∗ = ([1 − jc (s)]α∗ ) (x) ρ = (jc (s)[α − β]) (x) For the estimates, we make the special choice suggested by [5, Example A.2]. < We introduce the interpolating set B = 0≤t≤1 Dt where
Dt = (z∗ , z) ∈ C2 |z∗ − tσ∗ | ≤ r, |z − tσ| ≤ r, z − z∗∗ = tρ
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< ∗ Set Cs (x; α∗ , β) = 0
|σ∗ |, |σ| ≤ se2Kj s |μ| + e−5mc R+ + R + ≤ r (6.23) |ρ| ≤ r+ + |σ| + |σ∗ | For the first bound, by Corollary D.2, |σ∗ | = |([1 − jc (s)]α∗ ) (x)| ⎛ ⎞
≤ sKj eKj s ⎝ |μ| + e−5mc max |α(y)| + max∗ |∇α(b)|⎠ y∈X d(x,y)≤c
≤ se2Kj s
|μ| + e−5mc R+ + R + ≤ r
b∈X d(x,b)≤c
The second inequality follows from the observations that |α(y)| ≤ R+ for all y within a distance c of Λ and |∇α(b)| ≤ R + for all bonds b within a distance c of Λ. When y ∈ Λ, this is enforced by the characteristic function χ2s (Λ; α, β). When y is not in Λ, but is within a distance c of Λ, this is enforced by the factor of χR+ ( Ω0 \(Pα ∪ Λ), α) in the characteristic function χA,s (Ω1 , Ω2 ; α, φ, β). (Recall that d(Λ , Pα ∪Ωc0 ) > c(s) > c, by (F.4a).) When b ∈ Λ∗ , this is enforced by the characteristic function χ2s (Λ; α, β). When b is not in Λ∗ , but is within a distance c of Λ, this is enforced by the factor of χR + (Ω∗0 \(Pα ∪ Λ∗ ), ∇α) in the characteristic function χA,s (Ω1 , Ω2 ; α, φ, β). The third inequality follows immediately from (F.7d). For the bound on ρ observe |ρ| ≤ |β(x) − α(x)| + |([1 − jc (s)][β − α]) (x)| ≤ r+ + |σ∗ | + |σ| < Since Cs (x; α∗ , β) ⊂ 0≤t≤1 Dt , part (i) follows from (6.23) and (F.3f). By [5, Remark A.3], on Cs (x; α∗ , β), 1 2 r − |ρ|2 − r (|σ| + |σ∗ |) 2 |σ| + |σ∗ | 1 2 r − r2+ − r+ + + r (|σ| + |σ∗ |) ≥ 2 2 1 ≥ (r − r+ ) (r + r+ ) − 3r (|σ| + |σ∗ |) 2
3 ≥ r (r − r+ ) − 6rse2Kj |μ| + e−5mc R+ + R + 4 ≥ CL r2
Re (z∗ z) ≥
(6.24)
The second and third inequality follow from (6.23), the fourth inequality from (F.3b) and (6.23) and the last from (F.7d). By [5, Remark A.3], the area of Cs (x; α∗ , β) is bounded by 8πr [|σ| + |σ∗ | + |ρ|] ≤ 40πr2
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6.4. Relative Sizes of Large Field Sets In this subsection, we prove Lemma 3.40, which is used in the proof of Theorem 3.35. As in the part of the proof of Theorem 3.35 where Lemma 3.40 is used, we fix a hierarchy for scale δ = 2n ε. By way of preparation, we have Lemma 6.9. Let J be a decimation interval of length 2p ε, for some 0 ≤ p ≤ n, and x ∈ Ω(J )c . Then there is a decimation interval J ⊂ J , of length 2q ε, with 1 ≤ q ≤ p, such that ) ( ˜ ) ∪ supp P˜α (J ) ∪ supp P˜β (J ) ∪ P˜α (J ) ∪ P˜β (J ) ∪ R(J ) d x , Q(J ≤4
p−1
c(2k ε)
k=q−1
Proof. The proof is by induction on p. If p = 0, then Ω(J )c = ∅ and the statement is vacuous. Assume that the statement is satisfied for some p ≥ 0. Let J be a decimation interval of length 2p+1 ε and x ∈ Ω(J )c . By Definition 2.4, there is a point y1 ∈ Ω(J − )c ∪ Ω(J + )c ∪ Pα (J ) ∪ Pβ (J ) ∪ supp Pα (J ) ∪ supp Pβ (J ) ∪ Q(J ) ∪ R(J ) such that d(x, y1 ) ≤ c(2p ε). •
If y1 ∈ Ω(J − )c ∪ Ω(J + )c , then by the induction hypothesis, there is a decimation interval J of length 2q ε, with 1 ≤ q ≤ p, that is contained either in J − or J + , such that p−1 ) ˜ ˜ c(2k ε) d y1 , Q(J ) ∪ · · · ∪ Pβ (J ) ∪ R(J ) ≤ 4
(
k=q−1
• •
As d(x, y1 ) ≤ c(2p ε), we are finished. ˜ ) ∪ R(J ), we set If y1 ∈ P˜α (J ) ∪ P˜β (J ) ∪ supp P˜α (J ) ∪ supp P˜β (J ) ∪ Q(J J = J and are finished. If y1 ∈ X (J )\X˜ (J ), for some X ∈ {Pα , Pβ , supp Pα , supp Pβ , Q}, then there is a y2 ∈ Λ(J − )c ∪ Λ(J + )c ⊂ Ω(J − )c ∪ Ω(J + )c with d(y1 , y2 ) ≤ 2c(2p ε) + 1 ≤ 3c(2p ε). By the induction hypothesis, there is a decimation interval J of length 2q ε, with 1 ≤ q ≤ p, that is contained either in J − or J + , such that p−1 ) ˜ ˜ c(2k ε) d y2 , Q(J ) ∪ · · · ∪ Pβ (J ) ∪ R(J ) ≤ 4
(
k=q−1
As d(x, y2 ) ≤ 4c(2p ε), we are finished.
Proof of Lemma 3.40. By Lemma 6.9, for each decimation interval J of length 2p ε, with 1 ≤ p ≤ n, volume of ball of radius " p p−1 ˜ )| + |P˜ (J )| k |Q(J |Ω(J )c | ≤ 1+4
q=1
decimation intervals J ⊂J of length 2q ε
c(2 ε)
k=q−1
# + |P˜β (J )| + |P˜α (J )| + |P˜β (J )| + |R(J )|
α
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Since, for each q ≤ p, any decimation interval J of length 2q ε is contained in a unique decimation interval of length 2p ε, n ((2p ε) + 1) |Ω(J )c | p=1
≤
decimation intervals J of length 2p ε n q=1 decimation intervals J of length 2q ε n p
× ≤
" # ˜ )| + · · · + |R(J )| |Q(J
((2 ε) + 1)
p=q n
volume of ball p−1
of radius
1+4
k
c(2 ε) k=q−1
CL r (2q ε)
2
" # ˜ )| + · · · + |R(J )| |Q(J
q=1 decimation intervals J of length 2q ε
by (F.7e).
Appendix A. Large/Small Field Characteristic Functions The representation of the effective density given in Theorem 2.16 involves the “large field integral operator” introduced in Definition 2.8. In part (i) of this definition we associate to a hierarchy S and a decimation interval J an integral operator I(J ; α∗ ,β) . Its definition involves a characteristic function χJ (α, ατ , β) = χJ ,S (α, ατ , β) implementing large and small field conditions. Here, we are going to define this characteristic function. For this definition, we use the notation that, for 0 < r < R, z ∈ C, any set Y and any complex valued function f on Y 1 if |z| ≤ r 1 if r ≤ |z| ≤ R χr (z) = χr,R (t) = 0 otherwise 0 otherwise and χr (Y, f ) =
x∈Y
χr (f (x))
χr,R (Y, f ) =
χr,R (f (x))
x∈Y
As pointed out in the leadup to Definition 3.2, the data associated to an interval in a hierarchy naturally split into two parts, the “first kind” of natural large/small field conditions and the “second kind” associated to the stationary phase construction. The following definition collects the conditions of the “first kind” that arise in Lemma A.3, below. Definition A.1. Let Ω1 , Ω2 ⊂ X, δ > 0 and let A = (Λ, Pα , Pβ , Pα , Pβ , Q) ∈ Fδ (Ω1 ∩ Ω2 ) be a choice of “small/large field sets of the first kind” as in Definition 3.2. The characteristic function for the small/large field sets of the first kind is (2)
(3)
χA,δ (Ω1 , Ω2 ; α, φ, β) = χ(1) (Ω1 , Ω2 ; α, φ, β) χA (α, β) χA (α, φ, β)
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where χ(1) (Ω1 , Ω2 ; α, φ, β) = χR(δ) (Ω1 \Ω0 , α) χR(δ) (Ω2 \Ω0 , β) χR(δ) ((Ω1 ∪ Ω2 )\Ω0 , φ) χR (δ) (Ω∗1\Ω∗0 , ∇α) χR (δ) (Ω∗2\Ω∗0 , ∇β) χR (δ) ((Ω∗1 ∪ Ω∗2 )\Ω∗0 , ∇φ) χr(δ) (Ω 1 \Ω 0 , α − φ) χr(δ) (Ω 2 \Ω 0 , φ − β) and (2)
χA (α, β) = χR(2δ) (Ω0 \(Pα ∪ Λ), α)
χR(2δ),R(δ) (Pα , α)
χR(2δ) (Ω0 \(Pβ ∪ Λ), β)
χR(2δ),R(δ) (Pβ , β)
χR (2δ) (Ω∗0 \(Pα ∪ Λ∗ ), ∇α) χR (2δ) (Ω∗0 \(Pβ ∪ Λ∗ ), ∇β) χr(2δ) (Ω 0 \(Q ∪ Λ ), α − β
χR (2δ),R (δ) (Pα , ∇α) χR (2δ),R (δ) (Pβ , ∇β) χr(2δ),∞ (Q, α − β)
and χA (α, φ, β) = χR(δ) (Ω0 \Λ, φ) χR (δ) (Ω∗0 \Λ∗ , ∇φ) (3)
χr(δ) (Ω 0 \Λ, α − φ) χr(δ) (Ω 0 \Λ, φ − β) with Ω0 = Ω1 ∩ Ω2 . If A is part of the data associated to a decimation interval of length 2δ in a hierarchy S, and Ω1 = Ω(J − ), Ω1 = Ω(J + ), then ◦ χ(1) imposes the “old” small field conditions on α, β and φ in (Ω(J − ) ∪ Ω(J + )) \ (Ω(J − ) ∩ Ω(J + )) (2) ◦ χA imposes the “new” small and large field conditions on α and β in the region (Ω(J − ) ∩ Ω(J + )) \Λ(J ), the complement of the “small field region of the first kind” for J in the previous “small field regions” (3) ◦ χA imposes small field conditions on φ in (Ω(J − ) ∩ Ω(J + )) \Λ(J ) Observe that χA,δ does not depend on the values of the fields on the set Λ. For the small/large field conditions of the second kind, we use (as in Theorem 2.16 the characteristic function χδ (Y ; α, β) (defined for δ > 0, a subset Y ⊂ X and fields α, β) which takes the value one if · |α(x)|, |β(x)| ≤ R(δ) for all x ∈ Y and · |∇α(b)|, |∇β(b)| ≤ R (δ) for all bonds b on X that have at least one end in Y and · |α(x) − β(x)| ≤ r(δ) for all x within a distance one of Y and which takes the value zero otherwise. Definition A.2. Let S be a hierarchy in the sense of Definition 2.4, and let J be a decimation interval for S of scale 2t. Define χJ (α, φ, β) = χA,t Ω(J − ), Ω(J + ); α, φ, β χ2t (Λ(J )\Ω(J ) ; α, β) where
A = Λ(J ), Pα (J ), Pβ (J ), Pα (J ), Pβ (J ), Q(J ) ∈ Fδ Ω(J − ) ∩ Ω(J + )
The second factor imposes the small field conditions in the difference between the small field regions of “the first and the second” kind for J .
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Lemma A.3. Let Ω1 , Ω2 ⊂ X and δ > 0. Then χδ (Ω1 ; α, φ) χδ (Ω2 ; φ, β) = χ2δ (ΛA ; α, β) χr(δ) (ΛA , α − φ) A∈Fδ (Ω1 ∩Ω2 )
× χr(δ) (ΛA , φ − β) χA,δ (Ω1 , Ω2 ; α, φ, β) Proof. Again set Ω0 = Ω1 ∩ Ω2 . By definition, χδ (Ω1 ; α, φ) χδ (Ω2 ; φ, β) = χR(δ) (Ω1 , α) χR(δ) (Ω1 , φ)
=
χR (δ) (Ω∗1 , ∇α) χR (δ) (Ω∗1 , ∇φ) χr(δ) (Ω 1 , α − φ) χ(1) (Ω1 , Ω2 ; α, φ, β) · χR(δ) (Ω0 , α)
χR(δ) (Ω2 , φ) χR(δ) (Ω2 , β) χR (δ) (Ω∗2 , ∇φ) χR (δ) (Ω∗2 , ∇β) χr(δ) (Ω 2 , φ − β) χR(δ) (Ω0 , β) χR (δ) (Ω∗0 , ∇α) χR (δ) (Ω∗0 , ∇β)
· χR(δ) (Ω0 , φ) χR (δ) (Ω∗0 , ∇φ)
χr(δ) (Ω 0 , α−φ) χr(δ) (Ω 0 , φ−β) (A.1)
The first factor, χ(1) (Ω1 , Ω2 ; α, φ, β), in (A.1) was defined in Definition A.1, and involves only fields at points of (Ω 1 ∪Ω 2 )\Ω0 . The next four factors involve only the external fields α and β at points x ∈ Ω 0 . To introduce the more restrictive small field conditions of scale 2δ, we expand
χR(2δ) (α(x)) + χR(2δ),R(δ) (α(x)) χR(δ) (Ω0 , α) = x∈Ω0
=
χR(2δ) (Ω0 \Pα , α) χR(2δ),R(δ) (Pα , α)
Pα ⊂Ω0
χR(δ) (Ω0 , β) =
χR(2δ) (Ω0 \Pβ , β) χR(2δ),R(δ) (Pβ , β)
Pβ ⊂Ω0
χR (δ) (Ω∗0 , ∇α) =
χR (2δ) (Ω∗0 \Pα , ∇α) χR (2δ),R (δ) (Pα , ∇α)
Pα ⊂Ω∗ 0
χR (δ) (Ω∗0 , ∇β) =
χR (2δ) (Ω∗0 \Pβ , ∇β) χR (2δ),R (δ) (Pβ , ∇β)
Pβ ⊂Ω∗ 0
1=
χr(2δ) (Ω 0 \Q, α − β χr(2δ),∞ (Q, α − β)
Q⊂Ω 0
and get, for the product of the four factors in (A.1) that depend only on the external fields on Ω 0 , χR(δ) (Ω0 , α) χR(δ) (Ω0 , β) χR (δ) (Ω∗0 , ∇α) χR (δ) (Ω∗0 , ∇β) = χR(2δ) (Ω0 \Pα , α) χR(2δ),R(δ) (Pα , α) (Λ,Pα ,Pβ ,Pα ,Pβ ,Q)∈Fδ (Ω0 )
χR(2δ) (Ω0 \Pβ , β)
χR(2δ),R(δ) (Pβ , β)
χR (2δ) (Ω∗0 \Pα , ∇α)
χR (2δ),R (δ) (Pα , ∇α)
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Ann. Henri Poincar´e
χR (2δ) (Ω∗0 \Pβ , ∇β) χR (2δ),R (δ) (Pβ , ∇β) χr(2δ) (Ω 0 \Q, α − β χr(2δ),∞ (Q, α − β)
(2)
χA (α, β) χ2δ (ΛA ; α, β)
(A.2)
A∈Fδ (Ω0 )
For each A = (Λ, Pα , Pβ , Pα , Pβ , Q) ∈ Fδ (Ω0 ) we write the last four factors of (A.1) χR(δ) (Ω0 , φ) χR (δ) (Ω∗0 , ∇φ) χr(δ) (Ω 0 , α − φ) χr(δ) (Ω 0 , φ − β) = χA (α, φ, β) · χR(δ) (Λ, φ) χR (δ) (Λ∗ , ∇φ) χr(δ) (Λ , α − φ) (3)
×χr(δ) (Λ , φ − β)
(A.3)
Inserting (A.2) and (A.3) into (A.1) we get χ2δ (ΛA ; α, β) χr(δ) (Λ A , α − φ) χδ (Ω1 ; α, φ) χδ (Ω2 ; φ, β) = A∈A
χr(δ) (Λ A , φ−β) χR(δ) (ΛA , φ) χR (δ) (Λ∗A , ∇φ) χA,δ (Ω1 , Ω2 ; α, φ, β) (A.4) In Lemma A.5 below we show that χ2δ (Λ; α, β) χR(δ) (Λ, φ) χR (δ) (Λ∗ , ∇φ) χr(δ) (Λ , α − φ) χr(δ) (Λ , φ − β) = χ2δ (Λ; α, β) χr(δ) (Λ , α − φ) χr(δ) (Λ , φ − β) If we insert this into (A.4) we get χδ (Ω1 ; α, φ) χδ (Ω2 ; φ, β) χ2δ (ΛA ; α, β) χr(δ) (Λ A , α − φ) χr(δ) (Λ A , φ − β) = A∈A
×χA,δ (Ω1 , Ω2 ; α, φ, β) χ2δ (ΛA ; α, β) χr(δ) (ΛA , α − φ) χr(δ) (ΛA , φ − β) = A∈A
×χA,δ (Ω1 , Ω2 ; α, φ, β) We were free to drop the factors χr(δ) (Λ A \ΛA , α − φ)χr(δ) (Λ A \ΛA , φ − β) from the term χr(δ) (Λ A , α − φ)χr(δ) (Λ A , φ − β) because they also appear in χA,δ (Ω1 , Ω2 ; α, φ, β). Lemma A.4. Let S be a hierarchy for scale 2δ. Set ε = 2−depth(S) (2δ). (a) Let J be a decimation interval centred on τ . If ατ (x)| > R(ε), for some x ∈ X\Λ(J ), then χJ˜ (ατl , ατc , ατr ) = 0 decimation intervals J˜=[τl ,τr ]⊂J
where τc =
τl +τr 2 .
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(b) If αδ (x)| > R(ε), for some x ∈ X\ΛS , then χA,δ (Ω([0, δ]), Ω([δ, 2δ]); α, αδ , β)
289
χJ˜ (ατl , ατc , ατr ) = 0
decimation intervals J˜=[τl ,τr ][0,2δ]
where A = (ΛS , Pα ([0, 2δ]), . . . , Q([0, 2δ])), τc =
τl +τr 2 .
Proof. We prove part (a). In the event that x ∈ (Ω(J − ) ∪ Ω(J + ))\Λ(J ), the factor χJ (ατl , ατ , ατr ) vanishes ◦ when x ∈ (Ω(J − ) ∪ Ω(J + ))\(Ω(J − ) ∩ Ω(J + )) because of the χR(δ) (Ω1 ∪ Ω2 )\Ω0 , φ) in χ(1) (Ω1 , Ω2 ; α, φ, β), with Ω1 = Ω(J − ), Ω2 = Ω(J + ), Ω0 = Ω(J − ) ∩ Ω(J + ), δ = 12 |J |, φ = ατ ◦ when x ∈ (Ω(J − ) ∩ Ω(J + ))\Λ(J ) because of the factor χR(δ) (Ω0 \Λ, φ) in (3) χA (α, φ, β), with Ω0 = Ω(J − ) ∩ Ω(J + ), Λ = Λ(J ), δ = 12 |J |, φ = ατ Observe that, in these two cases Λ(J ) = X, so that we necessarily have δ ≥ ε. In the event that x ∈ X\(Ω(J − ) ∪ Ω(J + )), there is a unique decimation interval J˜ = [τl , τr ] ⊂ J having τr = τ with x ∈ Ω(J˜+ )\Ω(J˜). In this case, the factor χJ˜ (ατl , ατc , ατr ) vanishes ◦ when x ∈ Ω(J˜+ )\(Ω(J˜− ) ∩ Ω(J˜+ )) because of the factor χR(δ) (Ω2 \Ω0 , β) in χ(1) (Ω1 , Ω2 ; α, φ, β), with Ω2 = Ω(J˜+ ), Ω0 = Ω(J˜− )∩Ω(J˜+ ), δ = 12 |J˜|, β = ατ ◦ when x ∈ (Ω(J˜− ) ∩ Ω(J˜+ ))\Λ(J˜) because of the factors χR(2δ) (Ω0 \(Pβ ∪ (2) Λ), β) and χR(2δ),R(δ) (Pβ , β) in χA (α, β), with Ω0 = Ω(J˜− ) ∩ Ω(J˜+ ), Pβ = Pβ (J˜), Λ = Λ(J˜), δ = 12 |J˜|, β = ατ ◦ when x ∈ Λ(J˜)\Ω(J˜) because of the factor χ2t (Λ(J )\Ω(J ); α, β) in the χJ (α, φ, β) of Definition A.2, with J replaced by J˜, 2t = |J˜|, β = ατ . Observe that, in these three cases Ω(J˜) = X, so that we necessarily have t, δ ≥ ε. Lemma A.5. Let δ > 0 be sufficiently small and Λ ⊂ Ω0 . Assume that |α|Λ , |β|Λ ≤ R(2δ) |α − φ|Λ ≤ r(δ)
|∇α|Λ∗ , |∇β|Λ∗ ≤ R (2δ) |φ − β|Λ ≤ r(δ)
Then |φ|Λ ≤ R(δ),
|∇φ|Λ∗ ≤ R (δ)
Proof. It follows from our assumptions that |∇φ − ∇α|Λ∗ ≤ ∇ |φ − α|Λ ≤ ∇ r(δ) where ∇ is the operator norm of the gradient. Consequently, by (F.3d,e), |φ|Λ ≤ R(2δ) + r(δ) ≤ R(δ) and |∇φ|Λ∗ ≤ R (2δ) + ∇ r(δ) ≤ R (δ)
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Appendix B. Bounds on Weight Factors Lemma B.1. Let S be a hierarchy for scale δ and ε = 2−n δ with n ≥ depth(S). Let τ ∈ εZ ∩ [0, δ] and τ ∈ εZ ∩ (0, δ). (i) e− 3 d(x,y) m
sup
x,y∈X κ∗τ (y)<∞
κ∗τ (y) ≤2, κ∗τ (x)
sup
e− 3 d(x,y) m
x,y∈X κτ (y)<∞
(ii) Let 0 ≤ k ≤ n and set ⎧ −k ⎪ ⎨[τ , τ + 2 δ] J∗k = the unique decimation interval ⎪ ⎩ of length 2−k δ that contains τ ⎧ −k ⎪ ⎨[τ − 2 δ, τ ] Jk = the unique decimation interval ⎪ ⎩ of length 2−k δ that contains τ
κτ (y) ≤2 κτ (x)
if k ≥ d(τ ) if k < d(τ ) if k ≥ d(τ ) if k < d(τ )
Then κ∗τ (y) ≤ 4e 3 d(y,Λ(J∗k )) R(2−k δ) m
κτ (y) ≤ 4e
m 3 d(y,Λ(Jk ))
−k
R(2
δ)
and for all y ∈ Λcτ
The κ∗τ (y) bound also applies when τ = 0 if we take d(0) = 0 and Λ0 = ∅. The κτ (y) bound also applies when τ = δ if we take Λδ = ∅. Proof. (i) We prove the first inequality. Let x, y ∈ X. It suffices to consider the case that κ∗τ (x) and κ∗τ (y) are both finite. Let J = [τ, τ + 2k ε] and J = [τ, τ + 2 ε] be the maximal decimation intervals with τ as left endpoint such that x ∈ Λ(J ), y ∈ Λ(J ). Then κ∗τ (x) = R(2k ε) and κ∗τ (y) = R(2 ε) if τ = 0 and κ∗τ (x) = 2R(2k ε) and κ∗τ (y) = 2R(2 ε) if τ = 0. If k ≤ + 1 (y) R(2 ε) R(2 ε) then κκ∗τ = k ε) ≤ R(2+1 ε) ≤ 2 by (F.3a). If k > + 1, then x ∈ Λ(J ), (x) R(2 ∗τ while y ∈ Λ(J )c for J = [τ, τ + 2+1 ε] J . Hence, in this case, we have κ∗τ (y) κ∗τ (x)
m
R(2 ε) 3 d(x,y) by (F.5). = R(2 k ε) ≤ e (ii) Again, we prove the first inequality. Let J = [τ , τ + t] be the maximal decimation interval with τ as left hand end point such that y ∈ Λ(J ). δ We automatically have t ≥ ε, since Λ([τ , τ + ε]) = X, and t ≤ 2d(τ ) , since no δ decimation interval having τ as an endpoint has length longer than 2d(τ ) . By definition, κ∗τ (y) = R(t) if τ = 0 and κ∗τ (y) = 2R(t) if τ = 0. The desired bound is trivial if 2−k δ ≤ 2t, so assume that 2−k δ > 2t. Set δ [τ , τ + 2t] if t < 2d(τ ) J = δ [τ − t, τ + t] = Jτ if t = 2d(τ )
Then y ∈ Λ(J )c and J J∗k , so that κ∗τ (y) ≤ 2R(t) ≤ 2e 3 d(x,y) R(2−k δ) m
for all x ∈ Λ(J∗k ), by (F.5).
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Appendix C. Normalization Constants In this appendix, we prove Lemma C.1. There is a unique function δ ∈ (0, 1) → Zδ ∈ (0, 1) that obeys dz ∗ ∧ dz −|z|2 1 e Z2δ = Zδ2 lim log Zδ = 0 δ→0+ δ 2πi |z|≤r(δ)
Furthermore, |ln Zδ | ≤ e−r(δ) Proof. Define, for all 0 < δ < 1, 1 log i(δ) = 2δ
|z|≤r(δ)
2
dz ∗ ∧ dz −|z|2 e 2πi
and observe that i(δ) < 0. The condition relating Z2δ to Zδ is equivalent to 1 1 log Z2δ = log Zδ + i(δ) 2δ δ Iterating gives n 1 1 log Zδ = −n log Z2−n δ + i 2− δ δ 2 δ =1
Existence and uniqueness will follow from convergence of the series ∞ − i(2 δ), which we now prove. =1 Since dz ∗ ∧ dz −|z|2 dx dy −(x2 +y2 ) e e = 1− 2πi π |z|≤r(δ)
|(x,y)|≥r(δ)
1 = 1− π
∞
2π dr
and |ln(1 − x)| ≤ the series
|x| 1−|x|
δer(δ)
2
dθ re
∞ =1−
0
r(δ)
= 1 − e−r(δ)
−r 2
ds e−s
r(δ)2
2
≤ 2|x| for all |x| ≤ 12 , we have |i(δ)| ≤ 1δ e−r(δ) . Hence 2
∞ ∞ − − 2 2 i 2 δ ≤ 2 e−(r(2 δ) −r(δ) ) =1
=1
By (F.7d), r(2− δ)2 − r(δ)2 =
r(2−p δ)2 − r(2−p+1 δ)2
p=1
≥
p=1
r(2−p δ) r(2−p δ) − r(2−p+1 δ)
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≥
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3 L(2−p δ) ≥ 3
p=1
so the series δer(δ)
2
∞ ∞ − i 2 δ ≤ 2 e−3 = =1
=1
does indeed converge and ∞ ∞ log Zδ = δ i 2− δ = 2−1 log =1
=1
2/e3 ≤1 1 − 2/e3
dz ∗ ∧ dz −|z|2 e 2πi
|z|≤r(2− δ)
is bounded in absolute value by e−r(δ) . 2
Appendix D. Bounds on the Propagator Throughout this appendix we assume that h ≡ 1. Then Recall that X = ZD /LZD . In (2.16), we assumed that the one-particle operator h = ∇∗ H∇ where H : L2 (X ∗ ) → L2 (X ∗ ) is a translation invariant, self–adjoint operator all of whose eigenvalues lie between cH > 0 and CH > 0 and for which e6md(x,0) |H (bi (0), bj (x))| < ∞ DH = x∈X 1≤i,j≤D
Here, for each 1 ≤ i ≤ D and x ∈ X, bi (x) = x, x + ei ) denotes the bond with base point x and direction ei . Under this hypothesis, the kernel of h is [H (bi (x − ei ), bj (y − ej )) − H (bi (x − ei ), bj (y)) h(x, y) = 1≤i,j≤D
− H (bi (x), bj (y − ej )) + H (bi (x), bj (y))] The norm |||h||| ≤ N6m (h; 1, 1) ≤ 4e12m DH and the constant Kj = N6m (h − μ; 1, 1) of Lemma 3.21 obeys Kj ≤ 4e12m DH + |μ|
(D.1)
Furthermore, we have the following bounds.
Lemma D.1. Set Kj = max |μ| + 4DDH e10m , 1 . Let S ⊂ X and A : L2 (X) → L2 (X). For all τ ≥ 0 and x ∈ X, |A ([1 − j(τ )]Sα) (x)| c ≤ τ Kj eKj τ |||A||| |μ| + e−5md(x,S ) max |α(y)| + max∗ |∇α(b)| y∈S
b∈S
For all τ ≥ 0 and S ⊂ X, |A ([1 − j(τ )]Sα) (x)| x∈S
≤ τ Kj eKj τ |||A|||
c |μ| + e−5md(S ,S ) max |α(y)| + max∗ |∇α(b)| |S| y∈S
b∈S
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Proof. Write ([1 − j(τ )]Sα) (y)
) ( ∗ = (1 − eμτ ) Sα(y) + eμτ [1 − e−τ ∇ H∇ ]Sα (y) τ = (1 − e ) Sα(y) + e μτ
μτ
dτ
(
e−τ
∇∗ H∇
) ∇∗ H∇ Sα (y)
(D.2)
0
As above
( ) ∗ 10m e5md(x,z) Ae−τ ∇ H∇ (x, z) ≤ e4τ DH e |||A|||
z∈X
sup
z∈X
(D.3)
e5m[d(z,b)+1] |(∇∗ H)(z, b)| ≤ 2DH e10m
b∈X ∗
For any bond b = (y, y ) ∇(Sα)(b) = S(y )α(y ) − S(y)α(y) = [S(y ) − S(y)]α(y ) + S(y)[α(y ) − α(y)] = α(y )(∇S)(b) + S(y)(∇α)(b) The second term is nonzero only for y ∈ S and hence for b ∈ S ∗ . The first term is bounded in magnitude by |α(y )| and is nonzero only if b connects a point of S to a point of S c . In this case, possibly replacing b by −b, we can always arrange that y is in S. The part of (A[1 − j(τ )]Sα)(x) in which the last ∇ of (D.2), multiplied on the left by A, acts on α is bounded by eμτ τ |||A|||e4τ DH e
10m
2DH e10m max∗ |∇α(b)| ≤ τ Kj eKj τ |||A||| max∗ |∇α(b)| b∈S
b∈S
The part in which the last ∇ of (D.2) acts on the characteristic function is bounded by eμτ τ |||A|||e4τ DH e
10m
2DH e10m e−5md(x,S
c
)
max
y ∈S d(y ,S c )≤1
|α(y )|
≤ τ Kj eKj τ |||A|||e−5md(x,S ) max |α(y)| c
y∈S
The first bound now follows from
|1 − eμτ | ≤ |μ|τ e|μ|τ ≤ |μ|τ eKj τ The proof of the second bound is similar. Corollary D.2. Let c > 0 and recall, from (3.2), that j(τ )(x, y) if d(x, y) ≤ c jc (τ )(x, y) = 0 otherwise
(D.4)
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For all τ ≥ 0 and x ∈ X,
Ann. Henri Poincar´e
⎛ ⎞ ⎝ |μ|+e−5mc max |α(y)|+ max∗ |∇α(b)|⎠
Kj τ
|([1 − jc (τ )]α) (x)| ≤ τ Kj e
y∈X d(x,y)≤c
b∈X d(x,b)≤c
Proof. Just apply the previous lemma with A being the identity operator and S the set of points y ∈ X that are within a distance c of x. Remark D.3. Recall, from 2.17, that Kreg = 29 exp{20e12m DDH }. In Lemmas 6.3 and 6.7, we used
4 + 2e2Kj ≤ 4 + 2e2 exp 4e12m DH ≤ 25 exp 4e12m DH ≤ Kreg and
2Kj e3Kj +Kj ≤ 2e3Kj +2Kj ≤ 2e5 exp 20De12m DH ≤ Kreg
Here, we have used that |μ| ≤ 1. Lemma D.4. For all α ∈ L2 (X) and all 0 ≤ ε ≤ 1, 3 2 ∗ cH e−4DCH ε ∇α 2 ≤ α∗ , 1 − e−ε∇ H∇ α ≤ CH ε ∇α 2 Proof. We have −ε∇∗ H∇
1−e
1 1 ∗ d −εt∇∗ H∇ = − dt e = dt ε∇∗ H∇ e−εt∇ H∇ dt 0
−εt∇∗ H∇
0 ∗
Since e commutes with ε∇ H∇ and all of the eigenvalues of e−εt∇ −εtCH ∇2 lie between e ≥ e−4εtDCH and one
∗
H∇
3 1 2 ∗ −ε∇∗ H∇ α ≥ dt εe−4εtDCH α∗ , ∇∗ H∇α α , 1−e 0
≥ εe−4εDCH α, ∇∗ H∇α ≥ cH εe−4εDCH ∇α
2
Similarly, 2 3 1 2 ∗ −ε∇∗ H∇ α , 1−e α ≤ dt ε α∗ , ∇∗ H∇α ≤ CH ε ∇α 0
Appendix E. Bounds on the Background Field In this appendix we provide both pointwise and norm bounds on the background field as well as comparisons between the background field and the
Vol. 11 (2010)
The Temporal Ultraviolet Limit
original field ατ . Throughout chy S for scale δ and write ⎧ ∗ ⎪ if τ ⎨α γ∗τ = Γ∗S (τ ; α∗ , α ∗ ) if τ ⎪ ⎩ ∗ if τ β
295
Sects. E.1–E.4 of the appendix, we fix a hierar⎧ ⎫ ⎪ ⎪ =0 ⎨α ⎬ ∈ (0, δ) γτ = ΓS (τ ; α , β) ⎪ ⎪ ⎩ ⎭ =δ β
⎫ ⎪ if τ = 0 ⎬ if τ ∈ (0, δ) ⎪ ⎭ if τ = δ
We also fix an integer n ≥ depthS and set ε = 2−n δ. Recall, from Notation 2.2, that, for each decimation point τ ∈ ∩(0, δ), δ 2δ ◦ the decimation index, d(τ ), of τ is determined by τ ∈ 2d(τ ) Z\ 2d(τ ) Z. By convention, we also set d(0) = d(δ) = 0 and, if τ is not a decimation point, d(τ ) = ∞. δ δ δ δ − + ◦ Jτ = [τ − 2d(τ = [τ − 2d(τ = [τ, τ + 2d(τ ) , τ + 2d(τ ) ], Jτ ) , τ ] and Jτ )] δ δ ◦ Λτ = Λ([τ − 2d(τ ) , τ + 2d(τ ) ]). By convention, we also set Λ0 = Λδ = ∅, and, if τ is not a decimation point, Λcτ = ∅. δ 1 − + ◦ In this appendix, we use ετ = 2d(τ ) = 2 |Jτ | = |Jτ | = |Jτ | to denote the lattice spacing of the coarsest lattice 2δk Z that contains τ . By convention, we set ε0 = εδ = δ.
In Sect. E.5, we fix a hierarchy S for scale 2δ. We further assume, throughout this appendix, that the field ατ is compatible with S in the following sense. (In the integral operator IS , the field ατ is compatible with S.) Definition E.1. A field configuration ατ (x) is said to be compatible with the hierarchy S if, for each decimation interval [τ− , τ+ ], ατ− (x) , ατ+ (x) ≤ R(τ+ −τ− )for all x ∈ Λ([τ− , τ+ ]) ∇ατ− (b) , ∇ατ+ (b) ≤ R (τ+ −τ− )for all b ∈ Λ([τ− , τ+ ])∗ ατ (x) − ατ (x) ≤ r(τ+ −τ− )for all x ∈ Λ([τ− , τ+ ]) +
−
where we recall that, for each S ⊂ X, S ∗ is the set of bonds with at least one end in S and S is the set of points in X that are connected to some point of S by some bond. Remark E.2. Compatibility implies |ατ (x)| ≤ min{κτ (x), κ∗τ (x)}
|α(x)| ≤ κ∗0 (x)
|β(x)| ≤ κδ (x)
for all τ ∈ εZ ∩ (0, δ) and x ∈ X. E.1. Additional Descriptions We now rewrite the definition, Definition 2.9, of the background field (still with j(t) being interpreted as the h-operator of (3.1)) in a way that makes more clear which coefficients Γττ (S), Γττ (S) are nonzero. The coefficients Γτ∗τ (S) = Γτ∗τ and Γττ (S) = Γττ were defined as follows. For τ ∈ (0, δ), ◦ For τ = τ , Γτ∗τ = Γττ = Λcτ
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◦ For τ = τ , Γτ∗τ = 0 unless τ > τ and [τ , τ ] is strictly contained in a decimation interval with τ as its left endpoint. If J is the smallest such decimation interval and δ its length, then δ δ Γτ∗τ = j τ − τ − Λ(J ) j Λcτ 2 2
◦ Similarly for τ = τ , Γττ = 0 unless τ < τ and [τ, τ ] is strictly contained in a decimation interval with τ as its right endpoint. If J is the smallest such interval and δ its length, then δ δ Γττ = j τ − τ − Λ(J ) j Λcτ 2 2 Observe that if J is a decimation interval, with τ as right hand endpoint and which contains τ in its interior, then d(τ ) > d(τ ) and τ is the smallest element of ετ Z that is above τ . Also observe that if d(τ ) > n, then Λcτ = ∅ so that Γτ∗τ = Γττ = 0. Definition E.3. For each τ ∈ (0, δ), set "
# Tr (τ, δ) = τ ∈ (τ, δ] ∩ εZ d(τ ) < d(τ ), τ = min τ ∈ ετ Z τ > τ "
# Tl (τ, δ) = τ ∈ [0, τ ) ∩ εZ d(τ ) < d(τ ), τ = max τ ∈ ετ Z τ < τ The figure below provides an example. In it, ε = 1, n = 5 so that δ = 32ε, τ = 14ε so that d(τ ) = 4, ετ = 2ε, Tr (τ, δ) = {16ε, 32ε} and Tl (τ, δ) = {12ε, 8ε, 0}.
As δ ∈ Tr (τ, δ) and 0 ∈ Tl (τ, δ) both Tr (τ, δ) and Tl (τ, δ) are always nonempty. When τ and δ are clear from the context, we drop them from the notation. Also, for each τ ∈ Tr , let π(τ ) denote the predecessor element of τ in Tr , which is the largest element of Tr that is strictly smaller than τ . When τ is the smallest element of Tr , set π(τ ) = τ . Similarly, for each τ ∈ Tl , let σ(τ ) be the successor element of τ , which is the smallest element of Tl that is strictly larger than τ . When τ is the largest element of Tl , σ(τ ) = τ . Lemma E.4. Let τ ∈ εZ ∩ (0, δ). (i) γτ = Λcτ ατ + j (π(τ ) − τ ) Λπ(τ ) j (τ − π(τ )) Λcτ ατ τ ∈Tr (τ,δ)
γ∗τ = Λcτ ατ∗ +
j (τ − σ(τ )) Λσ(τ ) j (σ(τ ) − τ ) Λcτ ατ∗
τ ∈Tl (τ,δ)
(ii) Λτ =
= τ ∈Tr
Λπ(τ ) ∩ Λcτ
Λτ =
= τ ∈Tl
Λσ(τ ) ∩ Λcτ
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provides two partitions of Λτ into disjoint subsets. Let τr ∈ Tr and τl ∈ Tl . Then we have σ(τl ) − εσ(τl ) = τl and π(τr ) + επ(τr ) = τr . If Λπ(τr ) ∩ Λcτr ∩ Λσ(τl ) ∩ Λcτl = ∅ then [τl , τr ] is a decimation interval of length min {ετl , ετr } and Λπ(τr ) Λcτr Λσ(τl ) Λcτl = Λ ([τl , τr ]) Λcτl Λcτr as well. In the figures below, the partitions of Λτ , stated in Lemma E.4(ii), are illustrated for the example inside Definition E.3.
In the proof of Lemma E.4 we need Remark E.5. (i) If τ , τ˜ ∈ Tr (τ, δ) and τ < τ˜ then d(τ ) > d(˜ τ ). τ ). If τ , τ˜ ∈ Tl (τ, δ) and τ < τ˜ then d(τ ) < d(˜ For each 0 < d < d(τ ) there is exactly one τ ∈ Tl (τ, δ) ∪ Tr (τ, δ) with d(τ ) = d. τ ), then (ii) If τ , τ˜ ∈ Tl (τ, δ) ∪ Tr (τ, δ) and d(τ ) > d(˜ Λτ˜ ⊂ Λτ ⊂ Λτ Proof. (i) Let τ < τ˜ both be in Tr (τ, δ) and suppose that d(τ ) ≤ d(˜ τ ). Then both τ and τ˜ are in ετ˜ Z and τ˜ cannot be the element of ετ˜ Z closest to τ , which is a contradiction. Let τ ∈ Tl (τ, δ)\{0} and τ˜ ∈ Tr (τ, δ)\{δ} and suppose that d(τ ) = ). τ − ετ˜ , τ˜ + ετ˜ ] are both decimation interd(˜ τ Then [τ − ετ , τ + ετ ] and [˜ vals of length 2ετ and both contain τ in their interiors. So they must be identical. That’s impossible since τ < τ < τ˜ forces τ = τ˜ . Let 0 ≤ d < d(τ ). There is exactly one decimation interval of length 2δd that contains τ . Call it [τ , τ˜ ]. If d > 0, then exactly one of τ , τ˜ has decimation index d. If d = 0, then both of τ , τ˜ have decimation index d. If d(τ ) = d, τ ) = d, then τ˜ ∈ Tr . then τ ∈ Tl . If d(˜ (ii) All of [τ − ετ , τ + ετ ], [τ − ετ , τ + ετ ] and [˜ τ − ετ˜ , τ˜ + ετ˜ ] are decτ ), imation intervals that contain τ in their interiors. Since d(τ ) > d(τ ) > d(˜ the first is contained in the second, which, in turn is contained in the third. Consequently, τ − ετ˜ , τ˜ + ετ˜ ]) Λ ([˜ τ − ετ˜ , τ˜ + ετ˜ ]) ⊂ Λ ([τ − ετ , τ + ετ ]) ⊂ Λ ([˜ (If d(˜ τ ) = 0, drop consideration of [˜ τ − ετ˜ , τ˜ +ετ˜ ] and use that Λτ˜ = ∅.)
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Proof of Lemma E.4. (i) We give the proof for γτ . We have already observed that Γττ may be nonzero only for τ = τ or τ ∈ Tr . Now fix any τ ∈ Tr . Then [τ, τ ] is strictly contained in a decimation interval with τ as its right endpoint, namely [τ − ετ , τ ]. Denote by J the smallest such interval and by δ its length, so that δ δ τ Γτ = j τ − τ − Λ([τ − δ , τ ]) j Λcτ 2 2
If τ − δ2 = τ , then all elements of (τ, τ ) have decimation index strictly larger than d(τ ) and τ is the smallest element of Tr . If τ − δ2 > τ , then τ is contained in the interior of the decimation inter val [τ − δ , τ − δ2 ] so that τ − δ2 is the smallest element of δ2 Z above τ . As τ − δ2 has the same decimation index as δ2 , we have τ − δ2 ∈ Tr . All elements of (τ − δ2 , τ ) have decimation index strictly larger than that of δ2 , and so cannot be in Tr (because τ − δ2 has smaller decimation index and is closer to τ ), we have π(τ ) = τ − δ2 . In both cases, Λ([τ − δ , τ ]) = Λπ(τ ) so that
Γττ = j (π(τ ) − τ ) Λπ(τ ) j (τ − π(τ )) Λcτ as desired. If τ − δ2 < τ , then [τ − δ2 , τ ] is a decimation interval that contains τ in its interior, contradicting the assumption that [τ − δ , τ ] is the shortest such decimation interval. The proof for γ∗τ is similar. (ii) Observe that Λτ ⊂ Λπ(τ ) for all τ ∈ Tr and Λτ ⊂ Λσ(τ ) for all τ ∈ Tl and that τ = δ ∈ Tr has Λcδ = X and τ = 0 ∈ Tl has Λc0 = X. Consequently = = Λπ(τ ) ∩ Λcτ Λτ = Λσ(τ ) ∩ Λcτ Λτ = τ ∈Tr
τ ∈Tl
provides two partitions of Λτ into disjoint subsets. We have already shown, in part (i), that π(τr ) + τr ∈ Tr . Now fix any τr ∈ Tr and τl ∈ Tl and assume that
δ 2d(π(τr ))
x ∈ Λπ(τr ) ∩ Λcτr ∩ Λσ(τl ) ∩ Λcτl = ∅ Write, as in part (i), 1 π(τr ) = τr − δr 2 Both
1 σ(τl ) = τl + δl 2
1 1 π(τr ) − δr , π(τr ) + δr = [τr − δr , τr ] 2 2 1 1 σ(τl ) − δl , σ(τl ) + δl = [τl , τl + δl ] 2 2
= τr for all
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are decimation intervals that contain τ in their interiors. Hence one must be contained in the other. Say that δr ≤ δl so that the first is contained in the second and Λσ(τl ) ⊂ Λπ(τr ) . To prove the claim, it suffices to show that τl + δl = τr —i.e. that the right hand ends of the two intervals coincide—since then [τl , τr ] is a decimation interval and Λ([τl , τr ]) = Λσ(τl ) ⊂ Λπ(τr ) . Suppose that they do not coincide. That is, 1 1 1 1 σ(τl ) − δl ≤ π(τr ) − δr < π(τr ) + δr < σ(τl ) + δl 2 2 2 2 or equivalently, τl ≤ τr − δr < τ < τr < τl + δl Since τr ∈ (τl , τl + δl ), we have Jτr ⊂ [τl , τl + δl ] leading to the contradiction Λcτr ∩ Λσ(τl ) = Λ (Jτr ) ∩ Λ ([τl , τl + δl ]) = ∅ c
The argument that the left hand ends of the intervals coincide when δr ≥ δl is similar. E.2. Comparison to the Case “j(τ ) = h” Recall, from (2.16), that j(τ ) = e−τ (h−μ) with h = ∇∗ H∇ where H : L2 (X ∗ ) → L2 (X ∗ ) is a translation invariant, self–adjoint operator all of whose eigenvalues lie between cH > 0 and CH > 0 and for which DH = e6md(x,0) |H (bi (0), bj (x))| < ∞ x∈X 1≤i,j≤d
Here, for each 1 ≤ i ≤ d and x ∈ X, bi (x) = x, x + ei ) denotes the bond with base point x and direction ei . If j(τ ) were the j(0) = h, the background field would reduce to Definition E.6. Define, for each 0 < τ < δ, Λσ(τ ) Λcτ hατ∗ γˆ∗τ = Λcτ ατ∗ + τ ∈Tl (τ,δ)
γˆτ = Λcτ ατ +
Λπ(τ ) Λcτ hατ
τ ∈Tr (τ,δ)
We now prove some bounds on the difference between γ(∗)τ and γˆ(∗)τ and derive some consequences of the bounds. Proposition E.7. Assume that h ≡ 1. Let 0 < τ < δ. If x ∈ Λcτ , then 1 |γ∗τ (x) − γˆ∗τ (x)| , |γτ (x) − γˆτ (x)| ≤ e−mc(ετ ) 2 If J is a decimation interval that contains τ and x ∈ Λ(J ), then |γ∗τ (x) − γˆ∗τ (x)| , |γτ (x) − γˆτ (x)| ≤ 2eKj + 1 r(|J |) The proof of this proposition uses16 Lemma E.8. Let 0 < τ < δ and Tr = {τ1 < τ2 < · · · < τp = δ}. 16
In Lemma E.8, we do not assume that h is identically one.
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If x ∈ Λcτ , then γτ (x) = ατ (x) + Ea (τ, x) with |Ea (τ, x)| ≤
1 −mc(ετ ) e 2
If x ∈ Λτ−1 ∩ Λcτ , for some 1 ≤ ≤ p (with the convention that τ0 = τ ), then (b) when d(x, Λτ ) ≥ 12 c(ετ ) (automatic for = p) and d(x, Λcτ−1 ) ≥ 1 2 c(2ετ−1 ), γτ (x) = hατ (x) + [j (τ − τ ) − h] Λτ−1 Λcτ ατ (x) + Eb (τ, x) with |Eb (τ, x)| ≤ 2e−mc(ετ ) (c)
when d(x, Λcτ−1 ) ≤ 12 c(2ετ−1 ) γτ (x)
( ) = hατ (x) − j (τ−1 − τ ) Λcτ−1 Λ ([τ−1 , τ ]) [ατ − ατ−1 ] (x) (" # + [j (τ − τ ) − h] − j (τ−1 − τ ) Λcτ−1 [j (τ − τ−1 ) − h] ) × Λ ([τ−1 , τ ]) Λcτ ατ (x) +Ec (τ, x) with |Ec (τ, x)| ≤ 2e−mc(ετ )
(d) when < p and d(x, Λτ ) ≤ 12 c(ετ ) γτ (x)
= hατ (x) + [j (τ − τ ) − h]Λ ([τ , τ+1 ]) Λcτ [ατ − ατ+1 ] (x) ( + {[j (τ − τ ) − h] + j (τ − τ ) Λτ [j (τ+1 − τ ) − h]} Λ ([τ , τ+1 ]) ) × Λcτ+1 ατ+1 (x) + Ed (τ, x) with |Ed (τ, x)| ≤ 2e−mc(ετ )
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(e)
For x running over Λ = Λ([0, δ]), we also have that if 0 < τ < 2δ , then
(f)
γ∗τ (x) = (j(τ )α∗ ) (x) + E∗e (x) δ δ δ δ c γτ (x) = j(δ−τ )Λ , δ β −j −τ Λ [j , δ β (x) −h] Λ 2 2 2 2 δ δ − τ Λc Λ , δ [α δ − β] (x) + Ee (x) + j 2 2 2 with x∈Λ |E∗e (x)| ≤ e−mc(δ) |Λc | and x∈Λ |Ee (x)| ≤ e−mc(δ) |Λc | if τ = 2δ , then
(g)
γ∗τ (x) = (j(τ )α∗ ) (x) + E∗f (x) γτ (x) = (j(δ − τ )β) (x) + Ef (x) with x∈Λ |E∗f (x)| ≤ e−mc(δ) |Λc | and x∈Λ |Ef (x)| ≤ e−mc(δ) |Λc | if 2δ < τ < δ, then δ δ δ δ γ∗τ (x) = j(τ )Λ 0, α∗ − j τ − Λc [j −h] Λ 0, α∗ (x) 2 2 2 2 δ δ α∗δ − α∗ (x) + E∗g (x) + j τ− Λc Λ 0, 2 2 2 γτ (x) = (j(δ − τ )β) (x) + Eg (x)
with
x∈Λ
|E∗g (x)| ≤ e−mc(δ) |Λc | and
x∈Λ
|Eg (x)| ≤ e−mc(δ) |Λc |
Proof. Recall from Lemma E.4 that γτ = Λcτ ατ +
p
j (τk−1 − τ ) Λτk−1 j (τk − τk−1 ) Λcτk ατk
(E.1)
k=1
For each 1 ≤ k ≤ p, j (τk−1 − τ ) Λτ j (τk − τk−1 ) Λcτk ατk (x) k−1 ≤ N0 χ{x} j (τk−1 − τ ) Λτk−1 j (τk − τk−1 ) Λcτk ; 1, κτk ≤ 4R(2ετk ) e− 2 m max{d(x, Λτk−1 ) 9
≤ 4e
Kj
, d(x, Λcτ )}
− 92 m max{d(x, Λτk−1 )
R(ετk ) e
k
|||j (τk−1 − τ ) ||| |||j (τk −τk−1 ) |||
, d(x, Λcτ )} k
(E.2)
where χ{x} is the characteristic function of the set {x} and k is the maximum of k − 1 and the largest with x ∈ Λτ . If x ∈ Λcτ , then k = k − 1. For the second inequality we used Lemma G.5(ii) with ◦ L1 being the single point set {x}, L2 = Λτk−1 = Λ [τk − 2ετk−1 , τk ] , L3 = Λcτk , 9 ˜ ◦ with d replaced by 0, δ1 = 0, δ2 = m 3 d, δ = 2 md, d = 5md, ◦ O2 = Λτk , κ = κτk and R = 4R(2ετk ). This choice of R is justified by Lemma B.1.ii.
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(a) First consider the case that x ∈ Λcτ (which forces τ ∈ εZ). Then, the k = 1 term in (E.1) vanishes so that |γτ (x) − ατ (x)| p c j (τk−1 − τ ) Λτk−1 j (τk − τk−1 ) Λτk ατk (x) = k=2
≤
p
4eKj R(ετk−1 ) e− 2 md(x, Λτk−1 ) 9
k=2
≤
p
4eKj R(ετk−1 ) e− 2 mc(2ετ ) 9
(put = 1 in the figure below)
k=2
≤
p 1 k=2
4
e−4mc(2ετ )
by (F.6b)
1 −2mc(2ετ ) −2mc(2ετk−1 ) e e 4 p
≤
k=2
≤
1 −mc(ετ ) e 2
by (F.4a,d)
(b, c, d) Now we consider the case x ∈ Λτ−1 ∩Λcτ with 1 ≤ ≤ p. In particular, x ∈ Λτ . Set ⎧ {} in case (b) ⎪ ⎪ ⎨ K = {, − 1} in case (c) ⎪ ⎪ ⎩ {, + 1} in case (d) As in the last paragraph c j (τk−1 − τ ) Λτk−1 j (τk − τk−1 ) Λτk ατk ≤ e−mc(ετ ) γτ (x) − k∈K
In case (b), when d(x, Λcτ−1 ) ≥ write the k = term as
1 2 c(2ετ−1 )
and d(x, Λτ ) ≥
1 2 c(ετ ),
j (τ−1 − τ ) Λτ−1 j (τ − τ−1 ) Λcτ ατ = j (τ − τ ) Λcτ ατ − j (τ−1 − τ ) Λcτ−1 j (τ − τ−1 ) Λcτ ατ = Λτ−1 Λcτ hατ + [j (τ − τ ) − h] Λτ−1 Λcτ ατ +j (τ − τ ) Λcτ−1 ατ − j (τ−1 − τ ) Λcτ−1 j (τ − τ−1 ) Λcτ ατ
we
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and bound, as in (E.2), ( ) c j (τ − τ−1 ) Λcτ ατ (x) j (τ−1 − τ ) Λτ−1 ≤ 4eKj R(2ετ−1 ) e− 4 mc(2ετ−1 ) 1 ≤ e−mc(ετ−1 ) 4 1 ( ) −mc(ετ−1 ) j (τ − τ ) Λcτ−1 ατ (x) ≤ e 4 For case (c), when d(x, Λcτ−1 ) ≤ 12 c(2ετ−1 ) and hence d(x, Λτ ) ≥ 1 2 c(2ετ−1 ), we first observe that Λτ Λτ−1 = Λ [τ−1 − ετ−1 , τ ] Λ ([τ−1 , τ ]) 9
Here the symbol A B signifies that A is a proper subset of B unless B = X. So the k = term is j (τ−1 − τ ) Λτ−1 j (τ − τ−1 ) Λcτ ατ = j (τ − τ ) Λ ([τ−1 , τ ]) Λcτ ατ −j (τ−1 − τ ) Λcτ−1 j (τ − τ−1 ) Λ ([τ−1 , τ ]) c
×Λcτ ατ + j (τ−1 − τ ) Λτ−1 j (τ − τ−1 ) Λ ([τ−1 , τ ]) ατ = Λ ([τ−1 , τ ]) Λcτ hατ − j (τ−1 − τ ) Λcτ−1 Λ ([τ−1 , τ ]) Λcτ ατ " # + [j (τ −τ )−h]−j (τ−1 −τ ) Λcτ−1 [j (τ −τ−1 )−h] Λ ([τ−1 , τ ]) Λcτ ατ c
+j (τ−1 − τ ) Λτ−1 j (τ − τ−1 ) Λ ([τ−1 , τ ]) ατ and the k = − 1 term, which is only present when ≥ 2, is j (τ−2 − τ ) Λτ−2 j (τ−1 − τ−2 ) Λcτ−1 ατ−1 = j (τ−1 − τ ) Λcτ−1 Λ ([τ−1 , τ ]) ατ−1 −j (τ−2 − τ ) Λcτ−2 j (τ−1 − τ−2 ) Λcτ−1 Λ ([τ−1 , τ ]) ατ−1 c
+j (τ−2 − τ ) Λτ−2 j (τ−1 − τ−2 ) Λ ([τ−1 , τ ]) ατ−1
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Ann. Henri Poincar´e
As in (E.2), j (τ−1 − τ ) Λτ j (τ − τ−1 ) Λ ([τ−1 , τ ])c ατ (x) ≤ 1 e−4mc(ετ−1 ) −1 4 ( 1 ) −2mc(ετ−2 ) c c j (τ−2 −τ) Λτ−2 j (τ−1 −τ−2 ) Λτ−1 Λ ([τ−1 , τ ]) ατ−1 (x) ≤ e 4 1 j (τ−2 − τ ) Λτ j (τ−1 − τ−2 ) Λ ([τ−1 , τ ])c ατ (x) ≤ e−4mc(ετ−1 ) −2 −1 4 Adding the representations of the k = and k = − 1 terms and using the fact that Λcτ−1 Λcτ = Λcτ−1 , we see that—up to an error of at most 2e−mc(ετ ) ( ) γτ (x) = hατ (x) − j (τ−1 − τ ) Λcτ−1 Λ ([τ−1 , τ ]) [ατ − ατ−1 ] (x) (" # + [j (τ − τ )−h] − j (τ−1 − τ ) Λcτ−1 [j (τ − τ−1 )−h] ) ×Λ ([τ−1 , τ ]) Λcτ ατ (x) (When = 1, (j(τ−1 − τ ) Λcτ−1 Λ([τ−1 , τ ]))(x) = 0, since x ∈ Λτ .) For case (d), when < p and d(x, Λτ ) ≤ 12 c(ετ ) so that d(x, Λcτ−1 ) ≥ 1 2 c(2ετ−1 ), we first observe that x ∈ Λ([τ , τ+1 ]) and that Λτ+1 Λτ = Λ ([τ − ετ , τ+1 ]) Λ ([τ , τ+1 ]) So the k = term is j (τ−1 − τ ) Λτ−1 j (τ − τ−1 ) Λcτ ατ c
= j (τ − τ ) Λ ([τ , τ+1 ]) Λcτ ατ + j (τ − τ ) Λ ([τ , τ+1 ]) ατ −j (τ−1 − τ ) Λcτ−1 j (τ − τ−1 ) Λcτ ατ and the k = + 1 term is j (τ −τ ) Λτ j (τ+1 −τ ) Λcτ+1 ατ+1 = j (τ −τ ) Λτ Λ ([τ , τ+1 ]) Λcτ+1 ατ+1 +j (τ − τ ) Λτ (j (τ+1 − τ ) − h) Λ ([τ , τ+1 ]) Λcτ+1 ατ+1 c
+j (τ − τ ) Λτ j (τ+1 − τ ) Λ ([τ , τ+1 ]) ατ+1 As in (E.2) 1 c |(j (τ − τ ) Λ ([τ , τ+1 ]) ατ ) (x)| ≤ e−2mc(ετ ) 4 1 ( ) −mc(ετ−1 ) c c j (τ−1 − τ ) Λτ−1 j (τ − τ−1 ) Λτ ατ (x) ≤ e 4 1 j (τ − τ ) Λτ j (τ+1 − τ ) Λ ([τ , τ+1 ])c ατ (x) ≤ e−2mc(ετ ) +1 4 Adding the representations of the k = and k = + 1 terms and using the fact that Λcτ Λcτ+1 = Λcτ , we see that—up to an error at x of at most 2e−mc(ετ ) γτ = j (τ − τ ) Λ ([τ , τ+1 ]) Λcτ ατ + j (τ − τ ) Λτ Λ ([τ , τ+1 ]) Λcτ+1 ατ+1 +j (τ − τ ) Λτ (j (τ+1 − τ ) − h) Λ ([τ , τ+1 ]) Λcτ+1 ατ+1
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= j (τ − τ ) Λ ([τ , τ+1 ]) Λcτ+1 ατ+1 + Λ ([τ , τ+1 ]) Λcτ [ατ − ατ+1 ] +j (τ − τ ) Λτ (j (τ+1 − τ ) − h) Λ ([τ , τ+1 ]) Λcτ+1 ατ+1 Writing, in the first term, j(τ − τ ) = h + [j(τ − τ ) − h] and evaluating at x, we have ) ( γτ (x) = hατ (x) + [j (τ − τ ) − h]Λ ([τ , τ+1 ]) Λcτ+1 ατ+1 (x) + [j (τ − τ ) − h]Λ ([τ , τ+1 ]) Λcτ [ατ − ατ+1 ] (x) ) ( + j (τ − τ ) Λτ [j (τ+1 − τ ) − h] Λ ([τ , τ+1 ]) Λcτ+1 ατ+1 (x) up to an error of at most 2e−mc(ετ ) . (e, f, g) We give the proof for γ∗τ . Recall from Lemma E.4(i), that, on Λ, ∗ γ∗τ = j (τ − σ(τ )) Λσ(τ ) j (σ(τ ) − τ ) Λcτ ατ τ ∈Tl (τ,δ)
As in (E.2), if 0 < τ < δ (otherwise Tl (τ, δ) is empty), j (τ − σ(τ )) Λσ(τ ) j (σ(τ ) − τ ) Λcτ ατ (x) x∈Λ τ ∈Tl (τ,δ)\{0, δ2 }
≤
N0 Λj (τ − σ(τ )) Λσ(τ ) j (σ(τ ) − τ ) Λcτ ; 1, κ∗τ |Λcτ |
τ ∈Tl (τ,δ) τ =0, δ2
≤
4eKj R(δ) e− 2 md(Λ, Λτ ) |Λc | c
9
τ ∈Tl (τ,δ) τ =0, δ2
≤
4eKj R(δ) e− 2 mc(2ετ ) |Λc | 9
τ ∈Tl (τ,δ)\{0}
≤
τ ∈Tl (τ,δ)\{0}
≤
1 −4mc(2ετ ) c e |Λ | 4
1 −mc(δ) c e |Λ | 4
by (F.6b)
by (F.4d)
Case 0 < τ ≤ 2δ : In this case τ = ∗ is contribution to γ∗τ
δ 2
∈ / Tl (τ, δ) and the τ = 0 ∈ Tl (τ, δ)
j (τ − σ(0)) Λσ(0) j (σ(0)) α = j(τ )α − j (τ − σ(0)) Λcσ(0) j (σ(0)) α For τ = 2δ , σ(0) = τ and the second term vanishes on Λ. For 0 < τ < 2δ , by Lemma B.1(i), ) ( j (τ − σ(0)) Λcσ(0) j (σ(0)) α (x) x∈Λ
≤
|j (τ − σ(0)) (x, y)| |j (σ(0)) (y, z)| κ∗0 (z)
x∈Λ,z∈X y∈Λcσ(0)
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≤ 4R(δ)e− 2 md(Λ, Λσ(0) ) 9 1 × |j (τ − σ(0)) (x, y)| e 2 md(x, y) |j (σ(0)) (y, z)| e 3 md(x, z) 9
c
x∈Λ,z∈X y∈Λcσ(0)
c 9 ≤ 4R(δ)e− 2 md(Λ, Λσ(0) ) Λcσ(0) |||j (τ − σ(0)) ||| |||j (σ(0)) ||| 9 δ δ δ ≤ 4eKj R(δ)e− 2 mc(δ/2) Λcσ(0) since τ < , so that σ(0) < and εσ(0) ≤ 2 2 4 1 ≤ e−4mc(δ) |Λc | by (F.6b) (E.3) 4 Case
δ 2
< τ < δ: In this case τ =
∈ Tl (τ, δ). The τ = 0 term is δ δ j (τ − σ(0)) Λσ(0) j (σ(0)) α = j τ − Λj α 2 2 δ δ δ δ c α−j τ − Λ j Λ 0, α = j(τ ) Λ 0, 2 2 2 2 c δ δ δ α Λj Λ 0, +j τ − 2 2 2 δ δ δ δ c = j(τ )Λ 0, α−j τ − Λ [j − h] Λ 0, α 2 2 2 2 δ δ Λc Λ 0, α −j τ − 2 2 c δ δ δ α Λj Λ 0, +j τ − 2 2 2 δ 2
and the τ = 2δ term is δ δ δ j τ −σ Λσ ( δ ) j σ − Λcδ α δ 2 2 2 2 2 2 δ δ =j τ− Λc Λ 0, αδ 2 2 2 δ δ δ δ −j τ − σ Λcσ( δ ) j σ − Λc Λ 0, αδ 2 2 2 2 2 2 c δ δ δ δ +j τ − σ αδ Λσ ( δ ) j σ − Λ 0, 2 2 2 2 2 2 As in (E.3), c 1 −4mc(δ/2) c j τ − δ Λ j δ Λ 0, δ ≤ e α (x) |Λ | 4 2 2 2 x∈Λ δ δ δ c c j τ −σ δ j σ Λ 0, (x) Λ − Λ α δ δ σ (2) 2 2 2 2 2 x∈Λ
≤
1 −4mc(δ/2) c e |Λ | 8
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c δ δ δ j τ −σ δ α δ (x) Λσ ( δ ) j σ − Λ 0, 2 2 2 2 2 2
x∈Λ
≤
1 −4mc(δ/2) c e |Λ | 8
Proof of Proposition E.7. For x ∈ Λcτ , γˆτ (x) = ατ (x) and γˆ∗τ = ατ (x)∗ and the desired bound follows immediately from Lemma E.8 and its analog for γ∗τ . For the rest of the proof we restrict to x ∈ Λτ . We prove the bound on |γτ (x) − γˆτ (x)|. The bound on |γ∗τ (x) − γˆ∗τ (x)| is proven similarly. Write Tr = {τ1 < τ2 < · · · < τp = δ} and fix any 1 ≤ ≤ p, any decimation interval J containing τ and any x ∈ Λτ−1 ∩ Λcτ ∩ Λ(J ). Note that Λcτ ∩ Λ(J ) is empty unless |J | ≤ ετ and that 2e−mc(ετ ) ≤ 18 ≤ 18 r(|J |) by (F.4c). For case (b) of Lemma E.8, it suffices to bound [j (τ − τ ) − 1] Λτ Λcτ ατ (x) −1 5 4 −5md(x,Λcτ ∪Λτ ) Kj −1 |μ|+e ≤ (τ − τ )Kj e max |ατ (y)|+ max |∇ατ (b)| ∗ (
y∈Λτ−1
b∈Λτ
−1
) 5 ≤ 2ετ−1 Kj eKj |μ| + e− 2 mc(ετ ) R(2ετ−1 ) + R (2ετ−1 )
by (F.3a,e), (F.4a) ≤ ετ Kj eKj |μ| + e−mc R(ετ ) + R (ετ ) 1 1 r(ετ ) ≤ r(|J |) by (F.6c), (F.3b) ≤ 32 32
In the first inequality, we used Lemma D.1. In the second inequality we used that both τ and τ are in [τ−1 − ετ−1 , τ−1 + ετ−1 ] to bound τ − τ by 2ετ−1 . In the third inequality we repeatedly used tR( ) (t) ≤ 2tR( ) (2t), which gives tR( ) (t) ≤ (2 t) R( ) (2 t) for all ∈ N. For case (c), we use the bounds [j (τ − τ ) − 1] Λ ([τ−1 , τ ]) Λcτ ατ (x) ≤ 1 r(|J |) 16 ( ) 1 r(|J |) j (τ−1 − τ ) Λcτ−1 [j (τ − τ−1 ) − 1] Λ ([τ−1 , τ ]) Λcτ ατ (x) ≤ 32 which are proven as above, together with ( ) j (τ−1 − τ ) Λcτ−1 Λ ([τ−1 , τ ]) [ατ − ατ−1 ] (x) −md(Λ(J ),Λcτ
≤e
−1
)
|||j (τ−1 − τ ) |||
sup y∈Λcτ
−1
∩Λ([τ−1 , τ ])
ατ (y) − ατ (y) −1
−md(Λ(J ),Λcτ ) −1
≤e
eKj r(ετ−1 ) r(ετ−1 ) ≤ 2r(|J |) ≤ eKj e−mc(2ετ−1 ) r(ετ−1 ) ≤ 2ετ−1 v r(ετ−1 ) ≤ 1 ≤ r(|J |)
if |J | ≤ 2ετ−1 if |J | > 2ετ−1
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by (F.3b), (F.4c) and (F.6a). For case (d) of Lemma E.8, it suffices to bound [j (τ − τ ) − 1]Λ ([τ , τ+1 ]) Λcτ [ατ − ατ ] (x) ≤ eKj r(ετ ) +1 ( ) 1 r(ετ ) [j (τ − τ ) − 1]Λ ([τ , τ+1 ]) Λcτ+1 ατ+1 (x) ≤ 32 ( ) 1 r(ετ ) j (τ − τ ) Λτ [j (τ+1 − τ ) − 1]Λ ([τ , τ+1 ]) Λcτ+1 ατ+1 (x) ≤ 32 since, as we have observed, |J | ≤ ετ . These bounds are proven as above.
Corollary E.9. Assume that h ≡ 1. (a)
Let 0 < τ < δ. If x ∈ Λcτ , then |γ∗τ (x) − γτ (x)∗ | ≤ e−mc(ετ ) Let J be a decimation interval of that contains τ . If x ∈ Λ(J ), then |γ∗τ (x) − γτ (x)∗ | ≤ 4eKj + 3 r(|J |)
(b) Let 0 < τ < δ. If τ ∈ 2εZ and x ∈ Λ(J ), then |γ∗τ (x) − γτ ±ε (x)∗ | ≤ 4eKj + 3 r(|J |) |γτ (x) − γ∗τ ±ε (x)∗ | ≤ 4eKj + 3 r(|J |) If τ ± ε ∈ {0, δ} and x ∈ Λ(J ), then
|γ∗τ (x) − γτ ±ε (x)∗ | ≤ 2 eKj + 1 r(|J |) |γτ (x) − γ∗τ ±ε (x)∗ | ≤ 2 eKj + 1 r(|J |)
(c)
Let 0 ≤ τ < δ and set ⎧ δ ⎪ ⎨j(τ )α − j(δ − τ − ε)Λ 2 , δ β γ˜τ,ε = j(τ )α − j(δ − τ − ε)β ⎪
⎩ j(τ )Λ 0, 2δ α − j(δ − τ − ε)β
if 0 ≤ τ < 2δ − ε if τ = 2δ − ε, 2δ if 2δ < τ < δ
and ˜ − j(δ − τ − ε)Ωβ ˜ γ˜ ˜τ,ε = j(τ )Ωα ˜ the set of all point in X that are within a distance c(δ) of where Ω Ω ([0, δ]). Then |γ∗τ (x)∗ − γτ +ε (x) − γ˜τ,ε (x)| ≤ 3eKj r(δ) |Λc | x∈Λ
and
x∈Λ
1 ˜ c| e−4md(x,Ω) γ˜τ,ε (x) − γ˜ ˜τ,ε (x) ≤ e−3mc(δ) |Ω 8
Proof. (a) The result for x ∈ Λcτ follows immediately from Proposition E.7, so we restrict to x ∈ Λτ for the rest of the proof.
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To prove the bound on |γ∗τ (x)−γτ (x)∗ | it suffices to prove that |ˆ γ∗τ (x)− γˆτ (x)∗ | ≤ r(|J |), which we now proceed to do. By Definition E.6 and Lemma E.4.ii, for x ∈ Λ(J ) ⊂ Λτ , γˆ∗τ (x) − γˆτ (x)∗= Λπ(τr ) (x)Λcτr (x)Λσ(τl ) (x)Λcτl (x) [ατl (x)∗ − ατr (x)∗ ] τr ∈Tr (τ,δ) τl ∈Tl (τ,δ)
and furthermore, since
<
τr ∈Tr τl ∈Tl
Λπ(τr ) ∩ Λcτr ∩ Λσ(τl ) ∩ Λcτl is a partition of Λτ
into disjoint subsets, at most one term in this sum is nonzero and it is Λ ([τl , τr ]) (x)Λcτl (x)Λcτr (x) [ατl (x)∗ − ατr (x)∗ ] This term is bounded by r(τr − τl ). If |J | > τr − τl , then at least one of τl , τr must be in the interior of J . If, for example, τl is in the interior of J , then [τl − ετl , τl + ετl ] ⊂ J so that Λ(J ) ⊂ Λτl and our one potentially nonzero term is in fact zero for all x ∈ Λ(J ). Hence the one possibly nonzero term is bounded by r(|J |). (b) To prove the bound on |γ∗τ (x) − γτ ±ε (x)∗ | in the case that τ ∈ 2εZ, we prove that |ˆ γ∗τ (x) − γˆτ ±ε (x)∗ | ≤ r(|J |). Again, by Definition E.6 and Lemma E.4(ii), for x ∈ Λ (J ) ⊂ Λτ ⊂ Λτ ±ε , γˆ∗τ (x) − γˆτ ±ε (x)∗ = Λπ(τr ) (x)Λcτr (x)Λσ(τl ) (x)Λcτl (x) [ατl (x)∗ − ατr (x)∗ ] τr ∈Tr (τ ±ε,δ) τl ∈Tl (τ,δ)
◦ If τr ∈ Tr (τ ± ε, δ) happens to have ετr < ετ , then, Λτr has τ as an end point and is properly contained in [τ − ετ , τ + ετ ]. Hence Λτ ⊂ Λτr and Λcτr Λσ(τl ) = 0 for all τl ∈ Tl (τ, δ). ◦ No τr ∈ Tr (τ + ε, δ) can have ετr = ετ , because the first element of ετr Z = ετ Z above τ + ε is τ + ετ , which does not satisfy ετ +ετ = ετ . ◦ If τr ∈ Tr (τ − ε, δ) happens to have ετr = ετ , then τr = τ and Λcτr Λσ(τl ) = Λcτ Λσ(τl ) = 0 for all τl ∈ Tl (τ, δ). ◦ If τr ∈ Tr (τ ±ε, δ) happens to have ετr > ετ , then, because τ ±ε is a nearest neighbour of τ and is in εZ\2εZ, it is necessary that τr > τ and indeed τr is the smallest element of ετr Z above τ too so that τr ∈ Tr (τ, δ) too. Conversely, every τr ∈ Tr (τ, δ) is also in Tr (τ ± ε, δ). Denote by π (τr ) and π(τr ) the predecessor elements of τr in Tr (τ ±ε, δ) and Tr (τ, δ) respectively. If επ (τr ) ≥ επ(τr ) , then π (τr ) = π(τr ). If επ (τr ) < επ(τr ) , then π(τr ) = τ and Λπ (τr ) (x) = Λπ(τr ) (x) = 1. Hence, once again, γˆ∗τ (x) − γˆτ ±ε (x)∗ = Λπ(τr ) (x)Λcτr (x)Λσ(τl ) (x)Λcτl (x) [ατl (x)∗ − ατr (x)∗ ] τr ∈Tr (τ,δ) τl ∈Tl (τ,δ)
= γˆ∗τ (x) − γˆτ (x)∗ and the desired result follows from |ˆ γ∗τ (x) − γˆτ (x)∗ | ≤ r(|J |).
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Finally, we prove the bound on |γ∗τ (x) − γτ ±ε (x)∗ | in the case that τ ±ε ∈ {0, δ}. In this case, either τ = ε so that γτ ±ε (x)∗ = α0 (x)∗ ≡ α(x)∗ or τ = δ−ε γ∗τ (x) − ατ ±ε (x)∗ | ≤ so that γτ ±ε (x)∗ = αδ (x)∗ ≡ β(x)∗ . We prove that |ˆ r(|J |). Again, by Definition E.6 and Lemma E.4(ii), for x ∈ Λ(J ) ⊂ Λτ , Λσ(τl ) (x)Λcτl (x) [ατl (x)∗ − ατ ±ε (x)∗ ] γˆ∗τ (x) − γˆτ ±ε (x)∗ = τl ∈Tl (τ,δ)
In the case that τ = ε, Tl (τ, δ) contains exactly one element, namely τl = 0 = τ ± ε, and the right hand side is exactly zero. In the case that τ = δ − ε, Tl (τ, δ) = δ − 2j ε 1 ≤ j ≤ n so that γˆ∗τ (x) − γˆτ ±ε (x)∗ = =
n j=1 n
Λδ−2j−1 ε (x)Λcδ−2j ε (x) [αδ−2j ε (x)∗ − αδ (x)∗ ] Λ [δ − 2j ε, δ] (x) Λcδ−2j ε (x) [αδ−2j ε (x)∗ − αδ (x)∗ ]
j=1
c j=1 Λδ−2j−1 ε Λδ−2j ε
Since is a partition of Λδ−ε = Λτ into disjoint subsets, at most one term in this sum is nonzero. If this term is the jth, it is bounded by r(2j ε). If |J | > 2j ε, then Λ(J ) ⊂ Λδ−2j ε and our one potentially nonzero term is in fact zero for all x ∈ Λ(J ). Hence the one possibly nonzero term is bounded by r(|J |). (c, first bound) It suffices to combine parts (e), (f) and (g) of Lemma E.8 with the bounds δ δ c j −τ −ε Λ Λ , δ [α δ − β] (x) 2 2 2 x∈Λ
≤
x∈Λ
j δ − τ − ε (x, y) [α δ − β](y) 2 2
y∈Λ Λ([ δ2 ,δ ]) c
≤
x∈Λ y∈Λc Λ([ δ2 ,δ ])
≤ |||j
j δ − τ − ε (x, y) r δ 2 2
δ δ − τ − ε ||| |Λc | r ≤ 2eKj r(δ) |Λc | 2 2
and, using Lemma D.1, δ δ δ c j − τ − ε Λ [j , δ β (x) − 1] Λ 2 2 2 x∈Λ
≤
δ j − τ − ε (x, y) 2
x∈Λ y∈Λc
δ [j δ − 1] Λ , δ β (y) 2 2
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δ 5md(x,y) δ K δ −5md(Λ,y) K e j2e ≤ j 2 − τ − ε (x, y) e 2 j x∈Λ y∈Λc
4 ×
−5md(y,Λ([ δ2 ,δ])c )
|μ|+e
5
max
z∈Λ([ δ2 ,δ])
|β(z)|+
max
b∈Λ([ δ2 ,δ])∗
|∇β(b)|
δ δ c δ 2Kj −5md(Λ,Λ([ δ2 ,δ])c ) − τ − ε ||| |Λ | e ≤ |||j |μ| + e R 2 2 2 δ + R 2 δ δ δ Kj 2Kj −5mc(δ/2) |μ| + e R ≤ e e +R |Λc | 2 2 2 δ 1 1 r r(δ)|Λc | by (F.4a), (F.6c), (F.3b) ≤ |Λc | ≤ 32 2 16
− ε and the similar bounds j τ − δ Λc Λ 0, δ ≤ 2eKj r(δ) |Λc | (x) [α δ − α] 2 2 2 x∈Λ j τ − δ Λc j δ − 1 Λ 0, δ ≤ 1 r(δ) |Λc | α (x) 16 2 2 2
when 0 ≤ τ <
δ 2
x∈Λ
when 2δ < τ < δ. We have used the compatibility assumption of Definition E.1. 1 r(δ) |Λc | ≤ 3eKj r(δ) |Λc |, by (F.4c), the desired Since 2eKj + 2e−mc(δ) + 16 results follow. (c, second bound) We have ⎧
˜ c α − j(δ − τ − ε)Ω ˜ cΛ δ , δ β ⎪ j(τ )Ω if 0 ≤ τ < 2δ − ε ⎪ 2 ⎪ ⎨ ˜ c α − j(δ − τ − ε)Ω ˜ cβ γ˜τ,ε − γ˜ ˜τ,ε = j(τ )Ω if τ = 2δ − ε, 2δ ⎪ ⎪
⎪ ⎩ ˜ cβ ˜ c Λ 0, δ α − j(δ − τ − ε)Ω j(τ )Ω if 2δ < τ < δ 2 All terms are bounded in the same way. For example, by Lemma B.1(i) and Definition 2.13, ( ) ˜ c α (x) e−4md(x,Ω) j(τ )Ω x∈Λ
≤
e−4md(x,Ω) |j(τ )(x, y)| κ∗0 (y)
x∈Λ ˜c y∈Ω
≤ 4R(δ)
|j(τ )(x, y)| e 3 d(x,y) e4md(x,y) e−4md(y,Ω) m
x∈Λ ˜c y∈Ω ˜c
˜ c| ≤ 4|||j(τ )|||e−4md(Ω,Ω ) R(δ)|Ω
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˜ c| ≤ 4eKj e−4mc(δ) R(δ)|Ω ≤
1 −3mc(δ) ˜ c e |Ω | 16
by (F.6b)
Lemma E.10. Assume that h ≡ 1. (i) Let t ∈ εZ ∩ (0, δ) and x ∈ Λct with d (x, Λt ) ≥ c(εt )
c
d (x, Λ ([t − εt , t]) ) ≥ c(εt )
c
d (x, Λ ([t, t + εt ]) ) ≥ c(εt )
c
c
(If t ∈ / 2εZ, then Λ ([t − εt , t]) = Λ ([t − εt , t]) = ∅ and the conditions c c d (x, Λ ([t − εt , t]) ) ≥ c(εt ) and d (x, Λ ([t, t + εt ]) ) ≥ c(εt ) are vacuous.) Then γτ (x)− j(t + εt −τ )Bx,ε Λct+ε αt+ε (x) ≤ e−mc(εt ) if t < τ ≤ t + εt t
t
if t−εt < τ ≤ t
|γ∗τ (x)−(j(τ −t)Bx,εt Λct αt∗ ) (x)| ≤ e−mc(εt ) ∗ γ∗τ (x)− j(τ −t+εt )Bx,εt Λct−ε αt−ε (x) ≤ e−mc(εt ) t t
if t ≤ τ < t + εt
for any set
(ii)
t
|γτ (x)−(j(t−τ )Bx,εt Λct αt ) (x)| ≤ e−mc(εt )
if t−εt ≤ τ < t
Bx,εt ⊃ y ∈ X d(x, y) ≤ c(εt ) c
Let x ∈ X obey d (x, Λ ([0, δ]) ) ≥ c(δ). Then |γτ (x) − (j(δ − τ )Bx,δ β) (x)| ≤ e−mc(δ) ∗
−mc(δ)
|γ∗τ (x) − (j(τ )Bx,δ α ) (x)| ≤ e for any set
if 0 < τ < δ if 0 < τ < δ
Bx,δ ⊃ y ∈ T d(x, y) ≤ c(δ)
Proof. (i) We prove the first bound. The proofs of the other three bounds are very similar. Definition 2.9, γτ (x) = ( Fix ) any τ ∈ εZ obeying t < τ ≤ t + εt . By τ τ Γ (x). Fix any τ ≤ τ α ≤ δ with Γ = 0. By Lemma E.4.i, τ τ τ τ ≥τ
τ ∈ {τ } ∪ Tr (τ, δ). Set
c Lx = x ∈ X d (x, Λt ) ≥ c(εt ), d (x, Λ ([t − εt , t]) ) ≥ c(εt ), c
d (x, Λ ([t, t + εt ]) ) ≥ c(εt )
when t ∈ 2εZ and Lx = x ∈ X d(x, Λt ) ≥ c(ε) when t ∈ / 2εZ. Case 1: τ ≤ τ < t + εt . In this case ετ < εt , (as is true for all times in (t, t + εt )) so that any x ∈ Lx obeys x ∈ Λ([t, t + εt ]) ⊂ Λτ By our rules for constructing small field sets, either Λcτ = ∅, or Λ([t, t + εt ]) = Λτ (in which case ετ = 12 εt and the distance from x to Λcτ is at least c(εt ) = c(2ετ )) or d(Λ([t, t + εt ]), Λcτ ) ≥ c(2ετ ). So the distance from Lx to Λcτ is at least c(2ετ ), unless Λcτ ≡ 0. Therefore, for τ = τ , (Γττ ατ )(x) = 0 and for τ < τ ,
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( ) τ Γτ ατ (x) = j (π(τ ) − τ ) Λπ(τ ) j (τ − π(τ )) Λcτ ατ (x) ≤ N0 Lx j (π(τ ) − τ ) Λπ(τ ) j (τ − π(τ )) Λcτ ; 1, κτ ≤ 4R(εt ) e−4mc(2ετ ) eKj (τ
−τ )
by Lemma G.5(ii) with κ = κτ , R = 4R(εt ), L1 = Lx ⊂ Λ([t, t + εt ]) = O2 , L2 = Λπ(τ ) and L3 = Λcτ . For this and all other applications of the lemma in ˜ this proof, d is replaced by 0, δ1 = 0, δ2 = m 3 d, δ = 4md and d = 5md. Case 2: τ > t + εt . In this case 2εt < 2επ(τ ) ≤ ετ (since all times t ∈ (τ − 2επ(τ ) , τ ), including t + εt have εt < 2επ(τ ) ). Now t − εt is the element of 2εt Z below t + εt that is nearest t + εt . As τ − 2επ(τ ) is an element of 2εt Z that is below t + εt , we have τ − 2επ(τ ) ≤ t − εt < t + εt < τ . Hence Λπ(τ ) = Λ([τ − 2επ(τ ) , τ ]) ⊂ Λt . So the distance from Lx to Λπ(τ ) is at least c(εt ). Therefore, ( ) τ Γτ ατ (x) = j (π(τ ) − τ ) Λπ(τ ) j (τ − π(τ )) Λcτ ατ (x) ≤ N0 Lx j (π(τ ) − τ ) Λπ(τ ) j (τ − π(τ )) Λcτ ; 1, κτ ≤ 4R(2επ(τ ) )e−4mc(εt ) eKj (τ
−τ )
by Lemma G.5(ii) with R = 4R(2επ(τ ) ), L1 = Lx , O2 = L2 = Λπ(τ ) and L3 = Λcτ . Case 3: τ = t + εt . If τ = t + εt too, then ( ) Γττ ατ (x) = (Λcτ ατ ) (x) = j(t + εt − τ )Bx,εt Λct+εt αt+εt (x) In general ετ ≥ 2επ(τ ) and ετ ≥ 2εt . Furthermore, since t = τ − εt < τ < τ , we have 2επ(τ ) ≤ ετ . Hence Λ([t, t + εt ]) ⊂ Λ([τ − 2επ(τ ) , τ ]) = Λπ(τ ) . So the distance from x to Λcπ(τ ) is at least c(εt ). As well, by definic tion, the distance from x to Bx,ε is at least c(εt ). Therefore, setting L1 = t
c c ≥ c(εt ) , y ∈ X d(y, Λ([t, t + εt ]) ) ≥ c(εt ), d(y, Bx,ε t ( ) τ Γτ ατ (x) − j(t + εt − τ )Bx,εt Λct+εt αt+εt (x)
= j (π(τ ) − τ ) Λπ(τ ) j (τ − π(τ )) − j(τ − τ )Bx,εt Λcτ ατ (x) (" ) # c c c = j(τ − τ )Bx,ε (x) − j (π(τ ) − τ ) Λ j (τ − π(τ )) Λ α τ τ π(τ ) t c ≤ N0 L1 j(τ − τ )Bx,ε Λcτ ; 1, κτ t ( ) +N0 L1 j (π(τ ) − τ ) Λcπ(τ ) j (τ − π(τ )) Λcτ ; 1, κτ ≤ 8R(εt )e−4mc(εt ) eKj (τ
−τ )
by Lemma G.5(ii) with R = 4R(εt ), L3 = Λcτ , O2 = Λ ([τ − εt , τ ]) = Λ ([t, t + εt ]) ⊃ L1 c and L2 being Bx,ε for the first bound and Λcπ(τ ) for the second bound. t We have now shown that |γτ (x) − (j(t + εt − τ )Bx,εt Λct+εt αt+εt )(x)| is bounded by
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8R(εt )e−4mc(min{εt ,2ετ }) eKj
τ ∈{τ }∪Tr (τ,δ)
≤
τ ∈{τ }∪Tr (τ,δ)
≤ e−mc(εt )
1 − 15 mc(min{εt ,2ετ }) e 4 2
e− 8 mc(ετ ) ≤ e−mc(εt ) 11
n
e− 8 mc(2 11
d
ε)
d =0
τ ∈{τ }∪Tr (τ,δ)
≤ e−mc(εt )
(E.4)
by (F.6b) and (F.4a,d). (ii) We prove the first bound. The proof of the second bound is similar. Fix any τ ∈ εZ obeying 0 < τ < δ. By definition γτ (x) = τ ≥τ (Γττ ατ )(x). Fix any τ ≤ τ ≤ δ with Γττ = 0. By Lemma E.4(i), τ ∈ {τ } ∪ Tr (τ, δ). Set
c Lx = x ∈ X d (x, Λ ([0, δ]) ) ≥ c(δ) Case 1: τ ≤ τ < δ. By our rules for constructing small field sets, either Λcτ = ∅, or Λ([0, δ]) = Λτ (in which case ετ = 2δ ) or d(Λ([0, δ]), Λcτ ) ≥ c(2ετ ). Consequently, if Λcτ = ∅, the distance from Lx to Λcτ is at least c(2ετ ). Therefore, for τ = τ , (Γττ ατ )(x) = 0 and for τ < τ , ( ) τ Γτ ατ (x) = j (π(τ ) − τ ) Λπ(τ ) j (τ − π(τ )) Λcτ ατ (x) ≤ N0 Lx j (π(τ ) − τ ) Λπ(τ ) j (τ − π(τ )) Λcτ ; 1, κτ ≤ 4R(δ)e−4mc(2ετ ) eKj (τ
−τ )
by Lemma G.5(ii) with κ = κτ , R = 4R(δ), O2 = Λ([0, δ]) ⊃ Lx = L1 , L2 = Λπ(τ ) and L3 = Λcτ . Case 2: τ = δ. As Λ([0, δ]) ⊂ Λ([δ − 2επ(τ ) , δ]) = Λπ(τ ) the distance from Lx c to Λcπ(τ ) is at least c(δ). We are assuming that the distance from x to Bx,δ is " # c also at least c(δ) so that, setting L1 = y ∈ Lx d(y, Bx,δ ) ≥ c(δ) , we have δ Γτ β (x) − (j(δ − τ )Bx,δ β) (x)
= j(π(τ ) − τ )Λπ(τ ) j (τ − π(τ )) − j(δ − τ )Bx,δ β (x) (" # # c = j(δ − τ )Bx,δ − j(π(τ ) − τ )Λcπ(τ ) j (τ − π(τ )) β (x) c ≤ N0 L1 j(δ − τ )Bx,δ ; 1, κδ ( ) +N0 L1 j(π(τ ) − τ )Λcπ(τ ) j (τ − π(τ )) ; 1, κδ ≤ 8R(δ)e−4mc(δ) eKj (δ−τ ) by Lemma G.5(ii) with R = 4R(δ), L3 = X O2 = Λ ([0, δ]) ⊃ L1 c Bx,δ
and L2 being for the first bound and Λcπ(τ ) for the second bound. The desired bound now follows from (E.4) with εt = δ.
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E.3. The Size of the Background Field In this subsection, we show that the background fields γ∗τ (x), γτ (x) obey roughly the same bounds as the large and small field conditions impose on the integration variables ατ (x). Proposition E.11. Assume that h ≡ 1. (i) Let τ ∈ εZ ∩ (0, δ) and J be a decimation interval that contains τ in its interior. If x ∈ Λ(J ), then |γ∗τ (x)| , |γτ (x)| ≤ 40 eKj R(|J |) (ii)
For all τ ∈ εZ ∩ (0, δ) and x ∈ Λcτ |γ∗τ (x) − ατ (x)∗ | , |γτ (x) − ατ (x)| ≤
1 −m c(ετ ) e 2
Proof. (i) The proof follows Lemma E.13, below. (ii) follows immediately from Proposition E.7.
Lemma E.12. Assume that h ≡ 1. Let [t− , t+ ] be a decimation interval in [0, δ] ˜ ) were with centre t = 12 (t+ + t− ). Recall that the sets P˜α (J ), P˜β (J ), Q(J ˜ ˜ defined in Proposition 3.36 and that the sets Pα (J ), Pβ (J ) were defined in Proposition 3.37. (i) If b ∈ P˜α ([t− , t+ ]), then 1 for all t− ≤ τ < t |∇γ∗τ (b)| ≥ R (t+ − t− ) 2 (ii) If b ∈ P˜β ([t− , t+ ]), then 1 R (t+ − t− ) 2 ˜ − , t+ ]), then (iii) If x ∈ Q([t |∇γτ (b)| ≥
for all
t < τ ≤ t+
|γ∗t (x)∗ − γt+ε (x)| + |γ∗t−ε (x)∗ − γt (x)| ≥
1 r(t+ − t− ) 4
(iv) If x ∈ P˜α ([t− , t+ ]), then 1 for all t− ≤ τ < t |γ∗τ (x)| ≥ R(t+ − t− ) 2 (v) If x ∈ P˜β ([t− , t+ ]), then 1 |γτ (x)| ≥ R(t+ − t− ) for all t < τ ≤ t+ 2 Proof. Set c± = c t − t− ) = c(t+ − t) and j(τ )(x, y) if d(x, y) ≤ c± Jτ,± (x, y) = jc± (τ )(x, y) = 0 otherwise ∗
Recall that j(τ ) = eτ μ e−τ ∇ H∇ with H obeying (2.16). (i), (ii) We give the proof for part (ii). The proof of part (i) is similar. Let t < τ ≤ t+ . As b ∈ P˜β ([t− , t+ ]), both end points x ∈ b satisfy the hypotheses,
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d (x, Λ([t− , t+ ])) , d (x, Λ([t− , t])c ) , d (x, Λ([t, t+ ])c ) > c± of Lemma E.10(i). (Again, when t+ − t− ≤ 2ε, Λ([t− , t]) = Λ([t, t+ ] = X and the conditions d(x, Λ([t− , t])c ), d(x, Λ([t, t+ ])c ) > c± are vacuous.) Furthermore, all points y within a distance c± of either
in end point x ∈ b, are Λct ⊂ Λct+ . Hence, by Lemma E.10.i, with Bx,εt = y ∈ X d(x, y) ≤ c± , ∇ γτ − (Jt −τ,± αt ) (b) ≤ 2 max γτ (x) − (Jt −τ,± αt )(x) ≤ 2e−mc± + + + + x∈b Write b = x, x + ei ) = bi (x). As Jt+ −τ,± is translation invariant ∇ (Jt+ −τ,± αt+ ) (b) = (Jt+ −τ,± αt+ )(x + ei ) − (Jt+ −τ,± αt+ )(x) Jt+ −τ,± (x + ei , y) αt+ (y) − Jt+ −τ,± (x, y) αt+ (y) = y
=
y
Jt+ −τ,± (x, y − ei ) αt+ (y) −
y
=
Jt+ −τ,± (x, y) αt+ (y)
y
Jt+ −τ,± (x, y) αt+ (y + ei ) −
y
=
Jt+ −τ,± (x, y) αt+ (y)
y
Jt+ −τ,± (x, y) (∇αt+ )(bi (y))
(E.5)
y
Since b ∈ P˜β ([t− , t+ ]), we have |∇αt+ (b)| > R (t+ − t− ) and |∇γτ (b)| = Jt+ −τ,± ∇αt+ (b) + ∇ γτ − Jt+ −τ,± αt+ (b) ≥ Jt+ −τ,± ∇αt+ (b) − 2e−mc± ≥ ∇αt+ (b) − (1 − Jt+ −τ,± )∇αt+ (b) − 2e−mc± ≥ R (t+ − t− ) − (1 − Jt+ −τ,± )∇αt+ (b) − 2e−mc± where we are using (Jt+ −τ,± ∇αt+ )(b) to refer to the last line of (E.5). By Lemma 3.21(ii), (1 − Jt −τ,± )∇αt (b) + +
≤ |||1 − j(t+ − τ )||| sup ∇αt+ (bi (y)) d(x, y) ≤ c±
≤ (t+ − t)Kj eKj (t+ −t) sup ∇αt (bi (y)) d(x, y) ≤ c± +
For all y with d(x, y) ≤ c± , bi (y) is necessarily in Λ([t, t+ ])∗ . Hence, ∇αt (bi (y)) ≤ R (t+ − t) +
so that
(1 − Jt
+ −τ,±
)∇αt+ (b) ≤ Kj eKj (t+ −t) (t+ − t)R (t+ − t)
and |∇γτ (b)| ≥ R (t+ − t− ) − Kj eKj (t+ −t) (t+ − t)R (t+ − t) − 2e−mc± 1 ≥ R (t+ − t− ) 2 by (F.3e), (F.4c) and Hypothesis F.7(i).
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˜ − , t+ ]), we have |αt (x) − αt (x)| > r(t+ − t− ). Hence (iii) Since x ∈ Q([t + − at least one of |αt+ (x) − αt (x)|, |αt (x) − αt− (x)| must be at least 12 r(t+ − t− ). We prove that in the former case |γ∗t (x)∗ − γt+ε (x)| ≥ 14 r(t+ − t− ). The proof that in the latter case |γ∗t−ε (x)∗ − γt (x)| ≥ 14 r(t+ − t− ) is similar. So assume that |αt+ (x) − αt (x)| ≥ 12 r(t+ − t− ). As in the proof of part (ii), using the third and first bounds of Lemma E.10(i), |γ∗t (x)∗ − γt+ε (x)| ∗ = αt (x) − αt+ (x) + (γ∗t − αt ) (x) + 1 − Jt+ −t−ε,± αt+ (x) − γt+ε − Jt+ −t−ε,± αt+ (x) 1 ≥ r(t+ − t− ) − 1 − Jt+ −t−ε,± αt+ (x) − 2e−mc± 2 since Λt+ ⊂ Λt so that
that d(x, Λt+ ) > c± too. By d(x, Λ t ) > c± implies Lemma D.1 with S = y ∈ X d(x, y) ≤ c± 1 − Jt+ −t−ε,± αt+ (x)
Kj (t+ −t) −5mc± |μ| + e max |αt+ (y)| + max∗ |∇αt+ (b)| ≤ (t+ −t)Kj e y∈S
b∈S
˜ − , t+ ]) is necessarily in Λ([t, t+ ]). Again, any y within a distance c± of x ∈ Q([t Hence max∗ ∇αt+ (b) ≤ R (t+ − t) max αt+ (y) ≤ R(t+ − t) y∈S
b∈S
so that 1 − Jt −t−ε,p ατ (x) + + ≤ Kj eKj (t+−t) (t+ −t) |μ|R(t+ −t)+e−5mc± R(t+ −t)+R (t+ −t) 1 r(t+ −t) by (F.6c) and (F.4a) ≤ 32 and 1 1 |γ∗t (x)∗ − γt+ε (x)| ≥ r(t+ − t− ) − r(t+ − t) − 2e−mc± 2 32 1 ≥ r(t+ − t− ) by (F.3b) and (F.4c) 4 (iv, v) We give the proof for part (v). The proof of part (iv) is similar. Let t < τ ≤ t+ . As x ∈ P˜β ([t− , t+ ]), it satisfies the hypotheses d (x, Λ([t− , t+ ])) , d (x, Λ([t− , t])c ) , d (x, Λ([t, t+ ])c ) > c± of Lemma E.10(i) (again, when t+ − t− ≤ 2ε, Λ([t− , t]) = Λ([t, t+ ] = X and the conditions d (x, Λ([t− , t])c ) , d (x, Λ([t, t+ ])c ) > c± are vacuous). As well, all points y within a distance c± of x, are in Λct ⊂ Λct+ . Hence, by Lemma E.10.i,
with Bx,εt = y ∈ X d(x, y) ≤ c± , γτ (x) − (Jt −τ,± αt )(x) ≤ e−mc± + +
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Since x ∈ P˜β ([t− , t+ ]), we have |αt+ (x)| > R(t+ − t− ) and |γτ (x)| = Jt+ −τ,± αt+ (x) + γτ − Jt+ −τ,± αt+ (x) ≥ Jt+ −τ,± αt+ (x) − e−mc± ≥ αt+ (x) − (1 − Jt+ −τ,± )αt+ (x) − e−mc± ≥ R(t+ − t− ) − (1 − Jt −τ,± )αt (x) − e−mc± +
+
By Corollary D.2, (1 − Jt+ −τ,± )αt+ (x) ⎛ ⎞
≤ (t+ − t)Kj eKj (t+ −t) ⎝ |μ| + e−5mc± max |αt+ (y)|+ max∗ |∇αt+ (b)|⎠ y∈X d(x,y)≤c±
b∈X d(x,b)≤c±
All y ∈ X with d(x, y) ≤ c± , are necessarily in Λ([t, t+ ]) and all bonds b with d(x, b) ≤ c± , are necessarily in Λ([t, t+ ])∗ . Hence, αt (y) ≤ R(t+ − t) ∇αt b) ≤ R (t+ − t) and +
+
so that, by (F.4a), (F.6c) and (F.3b), (1 − Jt −τ,± )αt (x) + +
≤ Kj eKj (t+ −t) ε(t+ − t) |μ| + e−5mc± R(t+ − t) + R (t+ − t) 1 r(t+ − t− ) ≤ 16 and, by (F.3d) and (F.4c), |γτ (b)| ≥ R(t+ − t− ) −
1 1 r(t+ − t− ) − e−mc± ≥ R(t+ − t− ) 16 2
Lemma E.13. Let τ ∈ εZ ∩ (0, δ) and J be a decimation interval that contains τ in its interior. Then ( m ) 40eKj τ R(|J |) if J [0, δ] τ d(x,Λ(J )) 2 N4m Γ∗τ ; e , κ∗τ ≤ if J = [0, δ] 16eKj τ R(δ) τ ∈[0,δ) ( m ) 40eKj (δ−τ ) R(|J |) if J [0, δ] N4m Γττ ; e 2 d(x,Λ(J )) , κτ ≤ if J = [0, δ] 16eKj (δ−τ ) R(δ) τ ∈(0,δ] Proof. We prove the first bound. The proof of the second is virtually identical. Write J = [τ− , τ+ ] and let τ+ − τ− = 21p δ. We first consider the case that τ ∈ Tl (τ, δ) with d(σ(τ )) > p. We claim that J is exactly the decimation interval J∗p of Lemma B.1. ◦ If d(τ ) > p, then, since τ is the largest element of ετ Z below τ and τ− is an element of ετ Z below τ , we have τ− < τ < τ < τ+ so that J the unique decimation interval of length 21p δ that contains τ .
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◦ If d(τ ) ≤ p < d(σ(τ )), then τ− < σ(τ ), since σ(τ ) is the largest element δ δ of 2d(σ(τ )) Z below τ . And τ− ≥ τ , since τ− is the largest element of 2p Z below τ . And it is impossible to have τ < τ− < σ(τ ) since then τ− would be in Tl (τ, δ) and σ(τ ) would not be the successor of τ . So τ− = τ and J = [τ , τ + 2δp ]. Consequently, ( m ) N4m Γτ∗τ ; e 2 d(x,Λ(J )) , κ∗τ ( ) m = N4m j(τ −σ(τ )) Λσ(τ ) j(σ(τ )−τ )Λcτ ; e 2 d(x,Λ(J )) , κ∗τ ≤ 4R(|J |) e− 6 d(Λ(J ),Λτ ) N5m (j(τ − σ(τ )); 1, 1) N5m (j(σ(τ ) − τ ); 1, 1) m
c
≤ 4R(|J |) e− 6 d(Λ(J ),Λτ ) |||j(τ − σ(τ ))||| |||j(σ(τ ) − τ )||| m e− 6 c(|Jτ |) if |Jτ | < |J | Kj (τ −τ ) R(|J |) ≤ 4e 1 otherwise m
c
(E.6)
For the first inequality we used Lemma G.5((i)c) with d replaced by 4md, L1 = X, L2 = Λσ(τ ) , L3 = Λcτ , O1 = O2 = Λ(J ) m m m δ1 = d, δ2 = d, δ = d, d˜ = 5md, κ = κ∗τ , R = 4R(|J |) 2 3 6 and DO ≥ δ(L3 , O1 ) = 16 md(Λ(J ), Λcτ ). (The hypothesis that κ(x) = m κ∗τ (x) ≤ Reδ2 (x,O2 ) = 4R(|J |) e 3 d(x,Λ(J )) for all x ∈ X is fulfilled by Lemma B.1(ii), since, as we observed above, J is the decimation interval J∗p of that Lemma). The last inequality follows from Lemma 3.21(i). We next consider the case that τ ∈ Tl (τ, δ) with p ≥ d(σ(τ )). In this case Λσ(τ ) ⊂ Λ(J ) and ( m ) N4m Γτ∗τ ; e 2 d(x,Λ(J )) , κ∗τ ≤ N4m j(τ −σ(τ )) Λσ(τ ) j(σ(τ )−τ )Λcτ ; 1, κ∗τ ≤ 4R |Jσ(τ ) | N5m (j(τ − σ(τ )); 1, 1) N5m (j(σ(τ ) − τ ); 1, 1) ≤ 4R |Jσ(τ ) | |||j(τ − σ(τ ))||| |||j(σ(τ ) − τ )||| ≤ 4eKj (τ −τ ) R |Jσ(τ ) | (E.7) by Lemma G.5(ii) with d replaced by 4md, L1 = X, L2 = Λσ(τ ) , L3 = Λcτ , O2 = Λσ(τ ) m δ = δ1 = 0, δ2 = d, d˜ = 5md, κ = κ∗τ , R = 4R |Jσ(τ ) | 3 Finally, as Γτ∗τ = Λcτ , ( ) ( ) N4m Γτ∗τ ; em/2 d(x,Λ(J )) , κ∗τ = N4m Λcτ ; 1, κ∗τ e−m/2 d(x,Λ(J )) ≤ 4R (|J |) N4m (Λcτ ; 1, 1) = 4R (|J |) Again we used Lemma B.1(ii).
(E.8)
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Using (F.4d) to sum (E.6) over τ ∈ Tl (τ, δ) with d(τ ) > p + 1 (so that |Jτ | < |J |), using (E.6) up to twice more for the cases d(τ ) = p + 1 (so that |Jτ | = |J |), and d(τ ) ≤ p < d(σ(τ )), using (F.3c) to sum (E.7) over τ ∈ Tl (τ, δ) with d(σ(τ )) ≤ p and finally adding (E.8) gives the bound for J [0, δ]. When J = [0, δ], the case d(σ(τ )) ≤ p is absent. Proof of Proposition E.11(i). For all x ∈ Λ(J ) ) ( m ) ( τ |γ∗τ (x)| ≤ N0 Γτ∗τ ; e 2 d(x,Λ(J )) , κ∗τ Γ∗τ ατ (x)∗ ≤ τ ≤τ τ ≤τ by Remark E.2 and the assumption that d(x, Λ(J )) = 0. Now just apply Lemma E.13. The proof of the bound on γτ (x) is similar. E.4. Comparison of γτ +ε and j(ε)γτ Lemma E.14. (a) Let τ, τ ∈ (0, δ] and t > 0. If [τ − t, τ ) ∩ εZ = ∅, then
Γττ −t = j(t) Γττ
Γ(τ − t; α , β) = j(t) Γ(τ ; α , β)
(b) Let τ, τ ∈ [0, δ) and t > 0. If (τ, τ + t] ∩ εZ = ∅, then
Γτ∗τ +t = j(t) Γτ∗τ
Γ(τ + t; α∗ , α ∗ ) = j(t) Γ(τ ; α∗ , α )
Proof. We prove part (a). It suffices to prove the τ formula, since the other one follows from it. Since Λcτ = ∅ for all τ with d(τ ) > n, it suffices to con sider τ ∈ εZ ∩ (0, δ]. If τ > τ , then τ − t > τ too, so that Γττ −t = Γττ = 0. So it suffices to consider τ ≤ τ . Case τ − τ ≥ ε: Let [τ − 2δd , τ ] be the shortest decimation interval with τ as right hand end point that properly contains [τ, τ ]. (If no such decimation interval exists, then Γττ −t = Γττ = 0.) Since τ − τ ≥ ε, we have d ≤ n and τ − 2δd ∈ εZ. Hence [τ − 2δd , τ ] also properly contains [τ − t, τ ] and δ δ δ τ j Λcτ Γτ = j τ − τ − d+1 Λ τ − d , τ 2 2 2d+1 δ δ δ τ Γτ −t = j τ − τ + t − d+1 Λ τ − d , τ j Λcτ = j(t) Γττ 2 2 2d+1 Case 0 ≤ τ − τ < ε: In this case τ − τ + t < ε too, since otherwise τ − ε would be an element of εZ in [τ − t, τ ). Let Jτ and Jτ −t be the shortest decimation intervals with τ as right hand end point that properly contain [τ, τ ] and [τ − t, τ ], respectively. Both are contained in [τ − ε, τ ]. Hence Λ(Jτ ) = Λ(Jτ −t ) = X and 1 1 τ |Jτ | Λcτ = j(τ − τ ) Λcτ Γτ = j τ − τ − |Jτ | Λ(Jτ ) j 2 2 1 1 τ |Jτ −t | Λcτ = j(τ −τ + t)Λcτ Γτ −t = j τ −τ + t − |Jτ −t | Λ(Jτ −t ) j 2 2 = j(t) Γττ
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Lemma E.15. (i)
321
Let τ ∈ εZ ∩ (0, δ). If d(τ ) = n (that is, τ ∈ εZ\2εZ), then
Λτ [γ∗τ − j(ε)γ∗τ −ε ] = Λτ [γτ − j(ε)γτ +ε ] = 0 If 0 < d(τ ) < n (that is, τ ∈ 2εZ), then ( ) N3m Λτ [Γτ∗τ − j(ε)Γτ∗τ −ε ] ; 1, κ∗τ ≤ e−mc(ετ ) τ ∈[0,δ)
( ) N3m Λτ [Γττ − j(ε)Γττ +ε ] ; 1, κτ ≤ e−mc(ετ )
τ ∈(0,δ]
(ii)
Let O ⊂ ΩS = Ω ([0, δ]), r > 0 and define the weight factor r if x ∈ O λ(x) = ∞ if x ∈ /O Then ( )
c N2m Γ0∗τ − j(ε)Γ0∗τ −ε O ; e−m d(x,O) , λ ≤ 4eKj r e−2m d(O,ΩS ) τ ∈(0,δ]
N2m
( )
c Γδτ − j(ε)Γδτ +ε O ; e−m d(x,O) , λ ≤ 4eKj r e−2m d(O,ΩS )
τ ∈[0,δ)
Here we set Γ0∗δ = Γδ0 = j(δ) and Γ0∗0 = Γδδ = 1. Proof. (i) The vanishing when d(τ ) = n is proven in Lemma E.16, below. Now assume that 0 < d(τ ) < n and write ετ = 2d ε. (That is, d = n − d(τ ).) The same Lemma gives ( ) N3m Λτ [Γτ∗τ − j(ε)Γτ∗τ −ε ] ; 1, κ∗τ τ ∈[0,δ)
1 1 c = N3m Λτ j ετ Λ ([τ − ετ , τ ]) j ετ Λcτ −ετ ; 1, κ∗τ −ετ 2 2 +
d−1
N3m Λτ j 2−1 ε Λ [τ − 2 ε, τ ] j 2−1 ε Λcτ −2 ε ; 1, κ∗τ −2 ε
=1
+N3m Λτ j(ε)Λcτ −ε ; 1, κ∗τ −ε For each 1 ≤ ≤ d − 1, we apply Lemma G.5((i)a), with d replaced by 3md, L1 = Λτ , L2 = Λ [τ − 2 ε, τ ] , L3 = Λcτ −2 ε , O1 = X, O2 = Λτ 3m m d, d˜ = 5md, κ = κ∗τ −2 ε , R = 4R(2ετ ) δ1 = 0, δ2 = d, δ = 3 2 and DL ≥ 32 m d(Λτ , Λcτ −2 ε ) ≥ 32 mc(2+1 ε), to get N3m Λτ j 2−1 ε Λ ([τ − ε , τ ]) j 2−1 ε Λcτ −ε ; 1, κ∗τ −ε c 3 ≤ 4R(2ετ ) e− 2 md(Λτ ,Λτ −ε ) |||j 2−1 ε ||| |||j 2−1 ε ||| +1 +1 3 1 5 ≤ 4eKj 2 ε R(2ετ ) e− 2 mc(2 ε) ≤ e− 4 mc(2 ε) 4 by (F.6b).
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For the last term, we again apply Lemma G.5(i(a)), this time with L1 = Λτ , δ1 = 0,
L2 = L3 = Λcτ −ε , O1 = X, O2 = Λτ 3m m d, d˜ = 5md, κ = κ∗τ −ε , δ2 = d, δ = 3 2
R = 4R(2ετ )
and DL ≥ 32 m d(Λτ , Λcτ −ε ) ≥ 32 mc(2ε), to get 3 1 5 N3m Λτ j(ε)Λcτ −ε ; 1, κ∗τ −ε ≤ 4R(2ετ )eKj ε e− 2 mc(2ε) ≤ e− 4 mc(2ε) 4 For the first term, the same Lemma gives 1 1 c c N3m Λτ j( ετ )Λ ([τ − ετ , τ ]) j( ετ )Λτ −ετ ; 1, κ∗τ −ετ 2 2 c 3 1 1 ≤ 4R(2ετ ) e− 2 md(Λτ ,Λ([τ −ετ ,τ ]) ) |||j( ετ )||| |||j( ετ )||| 2 2 3 ≤ 4eKj ετ R(2ετ ) e− 2 mc(ετ ) 1 5 ≤ e− 4 mc(ετ ) 4 Using (F.4d) to bound the sum over 1 < < d − 1 and adding the bounds on the first and last terms gives the desired bound for Λτ [Γτ∗τ − j(ε)Γτ∗τ −ε ]. The proof of the other bound is similar. (ii) We prove the first bound. If τ = 2 ε for all ∈ {0, . . . , n}, then 0 Γ∗τ − j(ε)Γ0∗τ −ε = 0 by Remark 3.5.iv. If τ = 2 ε for some ∈ {1, . . . , n − 1}, then Γ0∗τ − j(ε)Γ0∗τ −ε = Λ [0, 2+1 ε] j(2 ε) − j 2−1 ε Λ [0, 2 ε] j 2−1 ε c c = −Λ [0, 2+1 ε] j(2 ε) + j 2−1 ε Λ [0, 2 ε] j 2−1 ε Therefore, ( )
N2m Γ0∗τ − j(ε)Γ0∗τ −ε O ; e−m d(x,O) , λ ( ) c ≤ N2m Λ [0, 2+1 ε] j(2 ε) O ; e−m d(x,O) , λ ( ) c +N2m j 2−1 ε Λ [0, 2 ε] j 2−1 ε O ; e−m d(x,O) , λ +1 c c ≤ re−2md(O,Λ([0,2 ε]) ) |||j(2 ε)||| + re−2md(O,Λ([0,2 ε]) ) |||j 2−1 ε |||2 −2m[d(O,ΩcS )+c(2+1 ε)] if 1 ≤ ≤ n − 2 Kj 2 ε e ≤ 2re c if = n − 1 e−2md(O,ΩS ) In the second inequality we applied Lemma G.5.ii with d replaced by 2md, L1 = X, L3 = O1 = O2 = O, δ1 = md, δ2 = 0, δ = 2md, d˜ = 5md, κ = λ, R = r
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The Temporal Ultraviolet Limit
c Λ [0, 2+1 ε] c L2 = Λ [0, 2 ε]
Similarly, for τ = ε,
323
for the first summand for the second summand
(
)
Γ0∗ε − j(ε)Γ0∗0 O ; e−m d(x,O) , λ ( ) c = N2m Λ ([0, 2ε]) j(ε)O ; e−m d(x,O) , λ
N2m
≤ re−2m[d(O,ΩS )+c(2ε)] eKj ε c
and, for τ = δ,
( )
Γ0∗δ − j(ε)Γ0∗δ−ε O ; e−m d(x,O) , λ δ c δ −m d(x,O) ,λ = N2m j( )ΛS j( )O ; e 2 2
N2m
≤ re−2md(O,ΩS ) eKj δ c
Summing up the last three bounds, using (F.4d), gives ( )
N2m Γ0∗τ − j(ε)Γ0∗τ −ε O ; e−m d(x,O) , λ τ ∈(0,δ]
≤e
Kj
−2m d(O,ΩcS )
re
n−2
−2mc(2+1 ε)
2e
+ 2 + 1
=0
≤e
Kj
−2m d(O,ΩcS )
re
n−2 =0
1 2+1 ε
−2mc(2+1 ε)
e
+ 2 + 1
≤ 4eKj r e−2m d(O,ΩS ) c
which is the desired result.
Lemma E.16. Let τ ∈ εZ ∩ (0, δ). Recall that ε = 2−n δ with the integer n ≥ depthS. (i)
If d(τ ) = n (that is, τ ∈ εZ\2εZ), then Λτ (γτ − j(ε)γτ +ε ) = 0. If 0 < d(τ ) < n (that is, τ ∈ 2εZ), then
1 1 c Λτ (γτ − j(ε)γτ +ε ) = Λτ j ετ Λ ([τ, τ + ετ ]) j ετ Λcτ +ετ ατ +ετ 2 2 −
n−d(τ )−1 (
) ( ) ( ) 2 −1 ε Λ [τ, τ + 2 ε] j 2 −1 ε Λcτ +2 ε ατ +2 ε
=1
− j(ε)Λcτ +ε ατ +ε
(ii)
where, if τ = δ − ετ , then ατ +ετ = β. If d(τ ) = n (that is, τ ∈ εZ\2εZ), then Λτ (γ∗τ − j(ε)γ∗τ −ε ) = 0. If 0 < d(τ ) < n (that is, τ ∈ 2εZ), then
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1 1 c − j(ε)γ∗τ −ε ) = Λτ j ετ Λ ([τ − ετ , τ ]) j ετ Λcτ −ετ ατ∗ −ετ 2 2 n−d(τ )−1
−
( ) ( ) ( ) j 2 −1 ε Λ [τ −2 ε, τ ] j 2 −1 ε
=1
× Λcτ −2 ε ατ∗ −2 ε
− j(ε)Λcτ −ε ατ∗ −ε
where, if τ = ετ , then α∗τ −ετ = α∗ . Proof. We give the proof for part (ii). The proof of part (i) is similar. For τ ∈ (ε, δ) and τ ∈ [0, δ), directly from Definition 2.9, 0 if τ ∈ / (τ , τ + ετ ) Λτ Γτ∗τ = Λτ j(τ − τ − 2m−1 ε)Λ([τ , τ + 2m ε])j(2m−1 ε)Λcτ if τ ∈ [τ + 2m−1 ε, τ + 2m ε) with m ≥ 1, 2m ε ≤ ετ
(so that, in particular, Λτ Γτ∗τ = 0 only for ετ < ετ ) and ⎧ ⎪ if τ ∈ / (τ , τ + ετ ] ⎨0 j(ε)Γτ∗τ −ε = j(ε)Λcτ if τ = τ + ε ⎪ ⎩ j(τ − τ − 2m −1 ε)Λ([τ , τ + 2m ε])j(2m −1 ε)Λcτ
if τ ∈ (τ + 2m −1 ε, τ + 2m ε]
with m ≥ 1, 2m ε ≤ ετ
(so that, in particular, j(ε)Γτ∗τ −ε = 0 only for ετ < ετ or τ = τ + ετ ). ◦ If τ = τ + ε with τ ∈ εZ\2εZ so that ετ = ε, then " # Λτ Γτ∗τ − j(ε)Γτ∗τ −ε = −Λτ j(ε)Λcτ This gives the last term in the statement, for the case τ ∈ 2εZ. ◦ If τ = τ + ε with ετ > ε, then m = 1 and " # Λτ Γτ∗τ − j(ε)Γτ∗τ −ε = Λτ Λ ([τ , τ + 2ε]) j(ε)Λcτ − Λτ j(ε)Λcτ = 0 since Λτ = Λ([τ , τ +2ε]). This (together with the last ◦) gives the d(τ ) = n case in the statement. ◦ If τ = τ + 2k ε for some k > 0 with 2k ε < ετ , then m = k + 1 and m = k. As ετ = 2k ε, we have Λτ = Λ([τ , τ + 2k+1 ε]) and " # Λτ Γτ∗τ − j(ε)Γτ∗τ −ε
= Λ([τ , τ +2k+1 ε]) j(2k ε) − j(2k−1 ε)Λ([τ , τ +2k ε]) j(2k−1 ε) Λcτ = Λτ j(2k−1 ε)Λ([τ , τ + 2k ε])c j(2k−1 ε)Λcτ In this case ε < 2k ε = ετ . This gives the first term in the statement, for the case τ ∈ 2εZ.
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◦ If τ = τ + ετ , with ετ > ε, then Γτ∗τ = 0 and 2m ε = ετ so that " # 1 1 τ τ ετ Λ([τ , τ + ετ ]) j ετ Λcτ Λτ Γ∗τ − j(ε)Γ∗τ −ε = −Λτ j 2 2 In this case ετ > ετ . This gives the th term in the statement, with determined by 2 ε = ετ , for the case τ ∈ 2εZ. ◦ If τ ∈ (τ , τ + ετ ) but τ = τ + 2k ε for all k ≥ 0 with 2k ε ≤ ετ , then m = m ≥ 1 with 2m ε = 2m ε ≤ ετ and Γτ∗τ = j(ε)Γτ∗τ −ε . Finally, we consider τ = ε. Then γ∗ε = Γε∗ε αε∗ + Γ0∗ε α∗ = Λcε αε∗ + Λ ([0, 2ε]) j(ε)α∗ so that Λε γ∗ε = Λ ([0, 2ε]) j(ε)α∗
Λε j(ε)γ∗0 = Λ ([0, 2ε]) j(ε)α∗
and Λτ (γ∗τ − j(ε)γ∗τ −ε ) = 0 Lemma E.17. Assume that h ≡ 1. Then 1 |γ∗τ − γτ∗ , Λτ (γτ − j(ε)γτ +ε )| ≤ 4 τ ∈(0,δ)
∗ |γτ − γ∗τ , Λτ (γ∗τ − j(ε)γ∗τ −ε )| ≤
τ ∈(0,δ)
1 4
e−mc(ετ ) |Λcτ |
τ ∈(0,δ)
e−mc(ετ ) |Λcτ |
τ ∈(0,δ)
Proof. We prove the first bound. Write ετ = 2d ε. (That is, d = n − d(τ ).) By Lemma E.16(i), for d > 0, 1 1 c ετ Λ ([τ, τ + ετ ]) j ετ Λcτ +ετ ατ +ετ Λτ (γτ − j(ε)γτ +ε ) = Λτ j 2 2 d−1 − j 2−1 ε Λ [τ, τ + 2 ε] j 2−1 ε Λcτ +2 ε ατ +2 ε =1
− j(ε)Λcτ +ε ατ +ε and, if d = 0, then Λτ (γτ − j(ε)γτ +ε ) = 0. For most terms that result from inserting this into the left hand side of the first claim, we shall use the bound |γ∗τ − γτ∗ , Λτ AΛcτ ατ | ≤ sup |γ∗τ (y) − γτ (y)∗ | N0 (Λτ AΛcτ ; 1, κτ ) |Λcτ | y∈Λτ
≤ 80 eKj R(2ετ ) N0 (Λτ AΛcτ ; 1, κτ ) |Λcτ | by Proposition E.11(i). Using this bound, we have, for each τ ∈ (0, δ) with τ ∈ 2εZ and each 1 ≤ ≤ d − 1,
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& ' γ∗τ − γτ∗ , Λτ j 2−1 ε Λ [τ, τ + 2 ε] j 2−1 ε Λc ατ +2 ε τ +2 ε ≤ 80 eKj R(2ετ ) N0 (Λτ j 2−1 ε Λ [τ, τ + 2 ε] j 2−1 ε × Λcτ +2 ε ; 1, κτ +2 ε ) |Λcτ +2 ε | ≤ 320 eKj R(2ετ )2 eKj 2 ε e−4md(Λτ ,Λτ +2 ε ) |Λcτ +2 ε | c
≤ 320 R(ετ )2 e2Kj e−4mc(2 ε) |Λcτ +2 ε | 1 ≤ e−mc(2 ε) |Λcτ +2 ε | by (F.4a) and (F.6b) 8 +1
˜ by Lemma G.5(ii) with d replaced by 0, δ1 = 0, δ2 = m 3 d, δ = 4md, δ = 5md, κ = κτ +2 ε , R = 4R(2ετ ), O2 = Λτ = L1 , L2 = Λ([τ, τ +2 ε]) and L3 = Λcτ +2 ε . We also have, for each τ ∈ 2εZ ∩ (0, δ), & ' γ∗τ − γτ∗ , Λτ j(ε)Λcτ +ε ατ +ε ≤ 80 eKj R(2ετ ) N0 (Λτ j(ε)Λcτ +ε ; 1, κτ +ε ) |Λcτ +ε | ≤ 320 eKj R(2ετ )2 eKj ε e−4mc(2ε) |Λcτ +ε | 1 ≤ e−mc(ε) |Λcτ +ε | 8 ˜ by Lemma G.5(ii) with d replaced by 0, δ1 = 0, δ2 = m 3 d, δ = 4md, δ = 5md, c κ = κτ +ε , R = 4R(2ετ ), O2 = Λτ = L1 = L2 and L3 = Λτ +ε , followed by (F.4a) and (F.6b). We still have the “Λ([τ, τ +ετ ])c = Λcτ + 1 ετ ” terms to deal with. For these, 2 we use > ? γ∗τ − γτ∗ , Λτ j 1 ετ Λc 1 j 1 ετ Λcτ +ε ατ +ετ τ + 2 ετ τ 2 2 1 1 ∗ c ≤ ετ (x, y)Λτ + 1 ετ (y)j ετ (y, z) |γ∗τ (x)−γτ (x) | Λτ (x)j 2 2 2 x,y,z∈X
×Λcτ +ετ (z) |ατ +ετ (z)| 1 Kj 2 m d(x,z) 3 ετ (x, y) ≤ 320 e R(ετ ) e Λτ (x)j 2 x,y,z∈X 1 c ετ (y, z)Λcτ +ετ (z) ×Λτ + 1 ετ (y)j 2 2 by Proposition E.11(i), and the fact, from Lemma B.1(ii), that m
m
|ατ +ετ (z)| ≤ κτ +ετ (z) ≤ 4R(ετ )e 3 d(z,Λ([τ,τ +ετ ]) ≤ 4R(ετ )e 3 d(x,z) when x ∈ Λτ ⊂ Λ([τ, τ + ετ ]). For all x, y, z for which the summand does not vanish, m
−4md(Λτ ,Λcτ + 1 ε ) 4md(x,y) 2 τ
e 3 d(x,z)
−4md(Λτ ,Λcτ + 1 ε 2 τ
e
e 3 d(x,z) ≤ e ≤e
e
m
) 5md(x,y) 5md(y,z)
e
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and we have > ? γ∗τ − γτ∗ , Λτ j 1 ετ Λc 1 j 1 ετ Λcτ +ε ατ +ε τ τ + ε τ τ 2 2 2 −4md(Λτ ,Λcτ + 1 ε ) 1 1 2 τ ≤ ετ ||| Λcτ + 1 ετ (y) |||j ετ ||| 320 eKj R(ετ )2 e |||j 2 2 2 y∈X ≤ 320 eKj R(ετ )2 eKj ετ e−4mc(ετ ) Λcτ + 1 ετ 2 1 1 ≤ e−mc( 2 ετ ) Λcτ + 1 ετ 2 8 by (F.4a) and (F.6b). All together |γ∗τ − γτ∗ , Λτ (γτ − j(ε)γτ +ε )| τ ∈(0,δ)
⎫ n−d(τ )−1 ⎬ 1 1 −mc(2 ε) c Λτ +2 ε e−mc( 2 ετ ) Λcτ + 1 ετ + e ≤ 2 ⎭ ⎩8 8 =0 τ ∈2εZ∩(0,δ) 1 1 e−mc(ετ ) |Λcτ | # τ ∈ (0, δ) τ + ετ = τ ≤ 8 2 τ ∈(0,δ)
+ # (τ, ) τ ∈ (0, δ), 0 ≤ ≤ n − d(τ ) − 1, τ + 2 ε = τ
⎧ ⎨1
Here, we have used that τ + 2 ε ∈ (0, δ) for all τ ∈ (0, δ) and < n − d(τ ). On the other hand, given any τ ∈ (0, δ) there is at most one τ ∈ (0, δ) with τ + 12 ετ = τ (because it is necessary that 12 ετ = ετ ), and there is at most one pair (τ, ) with τ ∈ (0, δ), 0 ≤ ≤ n − d(τ ) − 1 and τ + 2 ε = τ (because it is necessary that 2 ε = ετ ). So we end up with 1 e−mc(ετ ) |Λcτ | |γ∗τ − γτ∗ , Λτ (γτ − j(ε)γτ +ε )| ≤ 4 τ ∈(0,δ)
τ ∈(0,δ)
E.5. Error Terms in the Recursive Construction Now assume that S is a hierarchy for scale 2δ, preceded by hierarchies (S1 , S2 ) for scale δ. Let ε = 2−n δ with n ≥ max{depth(S1 ), depth(S2 )}. For simplicity we again write Λ = ΛS . As in (5.19) we set ⎧ ⎪ if τ ∈ [0, δ) or τ = 2δ ⎨0 ∂c Γ∗τ = Λ jδ if τ = δ ⎪ ⎩ 0 ∂Γ∗τ + Γ∗τ −δ (S2 ) Λ jδ if τ ∈ (δ, 2δ) ⎧ δ ⎪ ⎨∂Γτ + Γ∗τ (S1 ) Λ jδ if τ ∈ (0, δ) ∂c Γτ = Λ jδ if τ = δ ⎪ ⎩ 0 if τ ∈ (δ, 2δ] or τ = 0
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with jδ = j(δ) − jc (δ) and, as in Proposition 3.6, c
∂Γ∗τ = j(τ −δ−2m−1 ε)Λ ([δ, δ+2m ε]) j(2m−1 ε)Λj(δ) for τ ∈ [δ+2m−1 ε, δ+2m ε) c
∂Γτ = j(δ−2m−1 ε−τ )Λ ([δ−2m ε, δ]) j(2m−1 ε)Λj(δ) for τ ∈ (δ−2m ε, δ−2m−1 ε] with 1 ≤ m ≤ n. Lemma E.18. ( ) ) ( 3 N2m ∂c Γ∗τ ; e− 2 m d(x,Λ) , κ∗0 ≤ 4e2Kj R(2δ) δe−m c + e−mc(δ) ( ) ) ( 3 N2m ∂c Γτ ; e− 2 m d(x,Λ) , κ2δ ≤ 4e2Kj R(2δ) δe−m c + e−mc(δ) Proof. We prove the first bound. In the case τ = δ we use Remark G.4(i), Lemma B.1(i) and Lemma 3.21(iii) to see that ( ) 3 N2m ∂c Γ∗δ ; e− 2 m d(x,Λ) , κ∗0 ( ) 3 = N2m Λjδ ; e− 2 m d(x,Λ) , κ∗0 = N2m (Λjδ ; 1, κ∗0 ) −md(x,y) κ∗0 (y) ≤ |||jδ ||| 2R(2δ) sup e κ∗0 (x) x,y∈X ≤ 4δ Kj R(2δ)eKj δ e−m c Now let τ ∈ (δ, 2δ). There is a unique 1 ≤ m ≤ n such that τ ∈ [δ + 3 2m−1 ε, δ+2m ε). We estimate N2m (∂Γ∗τ ; e− 2 m d(x,Λ) , κ∗0 ) and N2m (Γ0∗τ −δ (S2 ) 3 Λ jδ ; e− 2 m d(x,Λ) , κ∗0 ) separately. By [4, (IV.1)] ( ) 3 N2m ∂Γ∗τ ; e− 2 m d(x,Λ) , κ∗0 ( ) 3 c = N2m j(τ −δ− 2m−1 ε) Λ ([δ, δ+2m ε]) j(2m−1 ε)Λj(δ) ; e− 2 m d(x,Λ) , κ∗0 ( ) 3 3 ≤ N2m j(τ − δ − 2m−1 ε) ; e− 2 m d(x,Λ) , e− 2 m d(x,Λ) ( ) 3 c ×N2m Λ ([δ, δ + 2m ε]) j(2m−1 ε)Λj(δ) ; e− 2 m d(x,Λ) , κ∗0 (E.9) Using Remark G.4(i), we bound the first factor on the right hand side of (E.9) by ( ) 3 3 N2m j(τ − δ − 2m−1 ε) ; e− 2 m d(x,Λ) , e− 2 m d(x,Λ) e− 2 m d(y,Λ) 3
≤ |||j(τ − δ − 2m−1 ε)||| sup e− 2 md(x,y) 3
x,y∈X
≤ eKj (τ −δ−2
m−1
ε)
e− 2 m d(x,Λ) 3
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The second factor on the right hand side of (E.9) is bounded by ( ) 3 c N2m Λ ([δ, δ + 2m ε]) j(2m−1 ε)Λj(δ) ; e− 2 m d(x,Λ) , κ∗0 ≤ 4R(2δ) e−md(Λ,Λ([δ,δ+2
m
≤ 4eKj (δ+2
m−1
ε)
ε])c )
R(2δ) e−mc(2
m
|||j(2m−1 ε)||| |||j(δ)||| ε)
Here we used Lemma G.5(ii) with d replaced by 2md and c
L1 = Λ ([δ, δ + 2m ε]) , L2 = Λ, L3 = X, O1 = O2 = Λ 3m m d, δ2 = d, δ = md, d˜ = 5md, κ = κ∗0 , R = 4R(2δ) δ1 = 2 3 Putting the last two estimates together we get ( ) m 3 N2m ∂Γ∗τ ; e− 2 m d(x,Λ) , κ∗0 ≤ 4e2Kj δ R(2δ)e−mc(2 ε) ≤ 4eKj R(2δ)e−mc(δ) Similarly ( ) 3 N2m Γ0∗τ −δ (S2 )Λ jδ ; e− 2 m d(x,Λ) , κ∗0 ( ) 3 = N2m j(τ −δ−2m−1 ε) Λ ([δ, δ + 2m ε]) j(2m−1 ε) Λ jδ ; e− 2 m d(x,Λ) , κ∗0 ( ) 3 3 ≤ N2m j(τ − δ − 2m−1 ε) ; e− 2 m d(x,Λ) , e− 2 m d(x,Λ) ( ) 3 ×N2m Λ ([δ, δ + 2m ε]) j(2m−1 ε)Λjδ ; e− 2 m d(x,Λ) , κ∗0 ≤ eKj (τ −δ−2
m−1
ε)
4R(2δ) |||j(2m−1 ε)||| |||jδ ||| ≤ 4eKj τ R(2δ) δKj e−mc
≤ 4Kj eKj R(2δ) δe−m c Remark E.19. (a) Let τ ∈ (0, δ] and t > 0. If [τ − t, τ ) ∩ εZ = ∅, then ∂Γτ −t = j(t) ∂Γτ . (b) Let τ ∈ [0, δ) and t > 0. If (τ, τ + t] ∩ εZ = ∅, then ∂Γ∗τ +t = j(t) ∂Γ∗τ . Proof. For all τ ∈ (0, δ] ∂Γτ β = ΓS (τ ; α , β) − ΓS1 (τ ; α l , ΛS j(δ)β + ΛcS αδ )
Apply Lemma E.14. Lemma E.20. Write ∂j Γ∗τ = ∂c Γ∗τ − j(ε)∂c Γ∗τ −ε . (i)
∂j Γ∗τ = 0 if τ ∈ (0, 2δ]\{δ, δ + ε, δ + ε1 , . . . , δ + 2δ , 2δ} Furthermore ∂j Γ∗δ = Λjδ c
∂j Γ∗δ+ε = Λ ([δ, δ + 2ε]) j(ε)Λjc (δ) c ∂j Γ∗δ+2 ε = Λ [δ, δ + 2+1 ε] j(2 ε)Λjc (δ) − j 2−1 ε c ×Λ [δ, δ + 2 ε] j 2−1 ε Λjc (δ) for = 1, . . . , n − 1 δ δ c ∂j Γ∗2δ = −j(δ)Λjδ − j Λ ([δ, 2δ]) j Λjc (δ) 2 2
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(ii)
Ann. Henri Poincar´e
( ) m N3m ∂j Γ∗τ ; e− 2 d(x,Λ) , κ∗0
τ ∈(δ,2δ)
( ) m 1 +N3m ∂j Γ∗2δ + j(δ)Λjδ ; e− 2 d(x,Λ) , κ∗0 ≤ e− 2 mc(δ)
" # Proof. (i) If τ ∈ (δ, 2δ) and m = min m τ ∈ (δ, δ + 2m ε) then, by construction ∂c Γ∗τ = ∂Γ∗τ + Γ0∗τ −δ (S2 )Λ jδ c
= j(τ − δ − 2m−1 ε) Λ ([δ, δ + 2m ε]) j(2m−1 ε)Λj(δ) +j(τ − δ − 2m−1 ε) Λ ([δ, δ + 2m ε]) j(2m−1 ε) Λ jδ = j(τ − δ)Λj(δ) − j(τ − δ − 2m−1 ε) Λ ([δ, δ + 2m ε]) j(2m−1 ε)Λj(δ) +j(τ − δ − 2m−1 ε) Λ ([δ, δ + 2m ε]) j(2m−1 ε) Λ jδ = j(τ − δ)Λj(δ) − j(τ − δ − 2m−1 ε) Λ ([δ, δ + 2m ε]) j(2m−1 ε)Λjc (δ) Consequently, if τ ∈ (δ, 2δ) is not of the form δ + 2 ε for any = 0, . . . , n, then ∂c Γ∗τ − j(ε)∂c Γ∗τ −ε = 0. If τ = δ + 2 ε for some = 1, . . . , n − 1, then ∂c Γ∗τ − j(ε)∂c Γ∗τ −ε = −Λ [δ, δ + 2+1 ε] j(2 ε)Λjc (δ) +j 2−1 ε Λ [δ, δ + 2 ε] j 2−1 ε Λjc (δ) c = Λ [δ, δ + 2+1 ε] j(2 ε)Λjc (δ) c −j 2−1 ε Λ [δ, δ + 2 ε] j 2−1 ε Λjc (δ) If τ ∈ (0, δ), both ∂c Γ∗τ and j(ε)∂c Γ∗τ −ε are zero. For τ = δ, we have ∂c Γ∗δ − j(ε)∂c Γ∗δ−ε = Λjδ . For τ = δ + ε, ∂c Γ∗δ+ε − j(ε)∂c Γ∗δ = j(ε)Λj(δ) − Λ ([δ, δ + 2ε]) j(ε)Λjc (δ) − j(ε)Λjδ c
= Λ ([δ, δ + 2ε]) j(ε)Λjc (δ) For τ = 2δ, ∂c Γ∗2δ − j(ε)∂c Γ∗2δ−ε
δ δ = −j(δ)Λj(δ) + j Λ ([δ, 2δ]) j Λjc (δ) 2 2 δ δ c = −j(δ)Λjδ − j Λ ([δ, 2δ]) j Λjc (δ) 2 2
(ii) By part (i), ( ) 1 N3m ∂j Γ∗τ ; e− 2 m d(x,Λ) , κ∗0 τ ∈(δ,2δ)
( ) 1 +N3m ∂j Γ∗2δ + j(δ)Λjδ ; e− 2 m d(x,Λ) , κ∗0
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( ) c 1 N3m Λ [δ, δ + 2+1 ε] j(2 ε)Λjc (δ) ; e− 2 m d(x,Λ) , κ∗0
=0 n
+
( ) c 1 N3m j 2−1 ε Λ [δ, δ+2 ε] j 2−1 ε Λjc (δ) ; e− 2 m d(x,Λ) , κ∗0
=1
(E.10) To bound the terms of the first sum on the right hand side of (E.10), we use Lemma G.5.ii with d replaced by 3md and c L1 = Λ [δ, δ + 2+1 ε] , L2 = Λ, L3 = X, O1 = O2 = Λ m m δ1 = d, δ2 = d, δ = md, d˜ = 5md, κ = κ∗0 , R = 4R(2δ) 2 3 to get ( ) c 1 N3m Λ [δ, δ + 2+1 ε] j(2 ε)Λjc (δ) ; e− 2 m d(x,Λ) , κ∗0 ≤ 4R(2δ)e−md(Λ,Λ([δ,δ+2
+1
≤ 4eKj (δ+2
ε)
R(2δ)e−mc(2
ε])c )
+1
|||j(2 ε)||| |||jc (δ)|||
ε)
(E.11)
To bound the terms of the second sum on the right hand side of (E.10), we use [4, (IV.1)] to see that ( ) c 1 N3m j 2−1 ε Λ [δ, δ + 2 ε] j 2−1 ε Λ jc (δ) ; e− 2 m d(x,Λ) , κ∗0 ( ) 1 1 ≤ N3m j 2−1 ε ; e− 2 m d(x,Λ) , e− 2 m d(x,Λ) ( ) c 1 ×N3m Λ [δ, δ + 2 ε] j 2−1 ε Λjc (δ); e− 2 m d(x,Λ) , κ∗0 Remark G.4(i) gives the bound ( ) 1 1 N3m j 2−1 ε ; e− 2 m d(x,Λ) , e− 2 m d(x,Λ) 1 1 e− 2 m d(y,Λ) ≤ |||j 2−1 ε ||| sup e− 2 md(x,y) − 1 m d(x,Λ) e 2 x,y∈X
≤ eKj 2
−1
ε
for the first factor on the right hand side. (E.11), with replaced by − 1, −1 shows that the second factor is bounded by 4eKj (δ+2 ε) R(2δ)e−mc(2 ε) . Consequently ( ) c 1 N3m j 2−1 ε Λ [δ, δ + 2 ε] j 2−1 ε Λ jc (δ) ; e− 2 m d(x,Λ) , κ∗0 ≤ 4 eKj (δ+2
ε)
R(2δ)e−mc(2
ε)
(E.12)
Inserting (E.11) and (E.12), n−1 n +1 1 e−mc(2 ε) + e−mc(2 ε) ≤ e− 2 mc(δ) (E.10) ≤ 4eKj 2δ R(2δ) =0
by (F.4d) and (F.6b).
=1
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Appendix F. Properties of the Various Constants The model under consideration is determined by the kinetic energy h = ∇∗ H∇, the two-body potential 2v(x, y), the temperature T > 0, and the chemical potential μ. We are assuming that both H and v are exponentially decaying. That is, there is a “mass” m > 0 such that e6 md(x,0) |H (bi (0), bj (x))| DH = x∈X 1≤i,j≤d
|||v||| = sup
x∈X
e5m d(x,y) |v(x, y)|
y∈X
are finite. See (2.16) and (2.5). We have also assumed that there are constants 0 < cH < CH such that all of the eigenvalues of H lie between cH and CH . In Lemmas 3.21 and D.1, we introduced constants Kj , Kj (depending only on DH , m and μ) for bounds on the semigroup j(t) = e−t(h−μ) . Our bounds are uniform for two–body potentials lying in the annulus 14 v ≤ |||v||| ≤ 12 v and for which the lowest eigenvalue obeys v1 ≥ cv |||v|||. See Hypothesis 2.14. The constant v > 0 must be sufficiently small and the constant cv ∈ (0, 1) is arbitrary. The chemical potential μ is required to obey |μ| ≤ max{Kμ veμ , 1} with strictly positive Kμ and 12 < eμ ≤ 1. For the bounds and the construction of the large field/small field decomposition, we introduced, in (2.17)–(2.20), the cutoff functions er eR eR 1 1 1 r(t) = R(t) = r(t) R (t) = r(t) tv tv t e 1 1 1 c(t) = log2 (t) = (F.1) c = log2 v tv tv with strictly positive exponents er , eR , eR and e that obey 3eR + 4er < 1 eR + er < 1
1 ≤ 4eR + 2er 2(eR + er ) < eμ ≤ 1 1 ≤ eR e < 2er 2
(F.2)
Our main results Theorems 2.16 and 2.18 apply when v and the time interval length θ are sufficiently small. The precise restrictions are determined by a number of technical conditions that are specified in Hypothesis F.7 at the end of this appendix. Clearly R(2t) ≤ R(t) ≤ 2 R(2t)
(F.3a)
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r(2t) ≤ r(t) ≤ 2 r(2t) ∞
(F.3b)
R(2n t) ≤ 6 R(t)
(F.3c)
n=1
Lemma F.1. There is a constant tmax > 0, depending only on er , eR and eR such that, for all 0 < t ≤ tmax and 0 < v ≤ 1, R(2t) + r(t) < R(t) R (2t) + ∇ r(t) ≤ R (t) ≤ 2R (2t) r(t) 1 ≤ 10 R(t) 2
(F.3d) (F.3e) (F.3f)
Proof. These all follow directly from the definitions (F.1) and conditions (F.2). Lemma F.2. There is a constant 0 < vmax < 1, depending only on m, such that, for all 0 < t < 12 and 0 < v ≤ vmax , c < c(2t) < c(t) ≤ 2c(2t) −m c
e
− 14 m c(t)
e ∞ k=0
(F.4a)
≤v
(F.4b)
1 ≤ min tv, 32
(F.4c)
1 − 1 mc(2−k t) e 8 ≤1 2−k t
(F.4d)
Proof. Parts (F.5a–c) are obvious. For (F.4d), ∞ k=0
1 2−k t
e− 8 mc(2 1
−k
t)
= ≤
∞ k=0 ∞
elog
2k t
− 18 m log2
elog
2k t
−2 log
2k tv
2k tv
≤
k=0
∞ k=0
e− log
2k tv
=
∞ tv 2k
k=0
= 2tv For the first inequality we used 18 m log used v ≤ 1.
2 v
≥ 2 and for the second inequality we
Lemma F.3. There are constants tmax > 0 and 0 < vmax < 1, depending only on m, such that the following is true for all 0 < δ ≤ tmax and 0 < v ≤ vmax . Let S be a hierarchy for scale δ and let J J be decimation intervals for S, with lengths t and t, respectively. For all x ∈ Λ(J ) and y ∈ Λ(J )c we have m t (F.5) R ≤ e 3 d(x,y) R(t) 2 Proof. Recall from (F.3a) that R(t) ≤ 2 R(2t).So t 2t R ≤ R(t) 2 t
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and, by Definition 2.4, 1 d(x, y) ≥ c(t ) = log = tv
assuming 2 log
2
1 2tv
≥
2t 1 log + log t 2tv
Ann. Henri Poincar´e
2 ≥ 2 log
2t 2t 1 3 log ≥ log 2tv t m t
3 m.
The next Proposition involves the constant Kd = supy∈X x∈X e−d(x,y) > 1, which is a characteristic quantity of the spatial lattice X alone and was defined in the proof of Lemma 3.42. Lemma F.4. There are constants tmax > 0 and vmax > 0 (depending only on eR , eR , er , e , Kd , Kμ , Kj and Kj ), such that 1 40 1 96 eKj R(t)e− 4 mc(t) ≤ 1
1 r(t) e2Kj +Kj t R (t) + e−mc R(t) + |μ|R(t) ≤ 32 1 19 + 4D + 10 log R(t) ≤ (2t) 4 for all 0 < t ≤ tmax and 0 < v ≤ vmax . 248 Kd e6Kj t v r(t) R(t)3 <
(F.6a) (F.6b) (F.6c) (F.6d)
Proof. For (F.6a), just recall that tv r(t)R(t)3 = (tv)1−3eR−4er and 3eR+4er < 1. 1 By (F.4c), R(t)e− 4 mc(t) ≤ tv R(t) = (tv)1−eR −er tends to zero as tv → 0 and (F.6b) follows. Since (F.4b)
e2Kj +Kj t R (t) + e−mc R(t) + |μ|R(t)
1 r(t) ≤ e2Kj +Kj t1−eR + (tv)1−eR + Kμ t1−eR veμ −eR r(t) ≤ 32 and the constraint (F.6c) is satisfied if t is small enough. The inequality (F.6d) is satisfied since the ratio between log R(t) = (eR + 1 1 e and (2t) = ( 2tv ) converges to zero as tv tends to zero. er ) log tv As in the proof of Proposition 3.36, set KL2 =
1 32
min{cH e−4DCH ,
1 32 }.
Lemma F.5. There are constants17 CL and tmax such that, setting L(t) = CL r(t)2 , 1 2 KL2 t R (t) ≥ L(t) (F.7a) 2 (F.7b) KL2 r(t)2 ≥ L(t) cv tvR(t)4 ≥ L(t) (F.7c) 512
L(t) 1 ≤ (r(t) − r(2t)) (F.7d) 6e2Kj t |μ| + e−5mc R(2t) + tR (t) ≤ r(t) 3 17
The constant CL depends only on cH , CH , cv , eR and er . The constant tmax depends only on CL , Kj , Kμ , eμ , eR , er , e .
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n
⎛ (2p ε) ⎝
p=q
335
⎞3
p−1
c(2k ε)⎠ ≤ L(2q ε)
(F.7e)
k=q−1
for all 0 < t ≤ tmax , 0 < v ≤ vmax (with the vmax of Lemma F.2 and all ε > 0 and n ∈ N with 2n ε ≤ tmax . Proof. Using (F.4b), the constraint (F.7d) will be satisfied if 6e2Kj t1−eR [Kμ veμ −eR + v5−eR ] + t1−eR r(t) L(t) 1 1 ≤ = CL r(t) ≤ (1 − er )r(t) r(t) 3 2 As long as v is smaller than the vmax of Lemma F.2 and t is small enough, we may choose L(t) to be a small constant times r(t)2 . We may also choose cv . Then constraints (F.8a,b,c) this constant to be smaller than 12 KL2 and 512 are also satisfied. Observe that, for any 0 < 2tv ≤ 1 and real b ≥ 1, b+1 1 log tv b+1 b+1 b 1 1 1 = log 2 + log ≥ log + (b + 1) (log 2) log 2tv 2tv 2tv b+1 b 1 1 ≥ log + log 2tv 2tv Iterating, we have, for all ε > 0 and 0 ≤ m < n with 0 < 2n εv ≤ 1 and any real b ≥ 1, b+1 b+1 b n 1 1 1 ≥ log n + log j log m 2 εv 2 εv 2 εv j=m+1 b b+1 n 1 1 =⇒ ≤ 2 log m log j 2 εv 2 εv j=m and hence 211
n p=q
⎛ (2p ε) ⎝
p−1
⎞3 c(2k ε)⎠ ≤ 214
n
(2p ε)
p=q
k=q−1
≤ 223
log
1 2q εv
log
1 2q−1 εv
9 n
(2p ε)
p=q
Since, for any e > 0, e e e n n 1 1 1 1 1 = = 2p εv εv 2pe 1 − 2−e 2q εv p=q p=q
9
336
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the left hand side of (F.7e) ⎛ ⎞3 p−1 n 211 (2p ε) ⎝ c(2k ε)⎠ ≤ p=q
k=q−1
Ann. Henri Poincar´e
223 1 − 2−e
log
1 q 2 εv
9
1 q 2 εv
e
is indeed smaller that L(2q ε) = CL ( 2q1εv )2er if e < 2er and 2n εv is small enough. During the construction, we introduced the auxiliary constants ◦ ◦ ◦ ◦
KR = 212 Kj2 and KE = 223 in Theorem 3.24, KD = 235 e6Kj and KL = 248 e6Kj in Theorem 3.26, KΔ = 240 e10Kj in Theorem 3.27 and KQ = 29 e2Kj and KV = 225 e6Kj in Propositions 5.12 and 5.13, respectively.
Lemma F.6. There is a constant tmax > 0 (depending only on eR , er and Kj ), such that, for all 0 < t ≤ tmax and 0 < v ≤ 1, we have √ 2 Kj2 e2Kj t etKj KR t e−mc 25+er Kj eKj t 2er √ + (2tv) + 1−eR −2er + ≤1 21−2er KR 2 (tv)eR +2er KR (F.8a) e2tKj 21−(2eR +4er )
+
1 51 16Kj e8Kj t e−mc + 2 (2tv)r(2t)R(2t)3 KE r(2t)2 R(2t)2 v KE
210 e− 2 mc 220 + ≤1 2 KE vR(2t) KE 1
+
(F.8b)
2−2er e2Kj + 228 (tv)r(t)R(t)3 ≤ 1 Proof. (a)
Since er <
etKj + 21−2er
(F.8c)
1 10 ,
25+er Kj eKj t √ KR
2 ≤
1 21−2er
+
1 22−2er
e2Kj t ≤
7 2Kj t e 8
By (F.4b), all remaining contributions may be made arbitrarily small by choosing tmax small enough. (b) Since 1 − (2eR + 4er ) = 1 − (3eR + 4er ) + eR > eR ≥ 15 , by (2.17), we have e2tKj 21−(2eR +4er )
(c) are
+
1 51 220 1 2 (2tv)r(2t)R(2t)3 + ≤1− 8 KE KE 2
by (F.6a), if t is small enough. Using (F.4b) and (2.17), the remaining two terms can be made arbitrarily small. is obvious. The precise smallness assumptions on θ and v in Theorems 2.16 and 2.18
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Hypothesis F.7.
The Temporal Ultraviolet Limit
(i) θ is smaller than
cv 3) 248 e6Kj (1+Kμ
337
( −2(−3e −4e ))1−2e 1−4e r r R R , 1−2217 e10Kj
and each of the tmax ’s of Lemmas F.1 to F.6. 1 (ii) v is smaller than ( 212 e12Kj ) 1−2eR −4er and each of the vmax ’s of Lemmas F.1 to F.5. These are also all of the restrictions that we put on the constants Θ and v0 of Theorems 3.24, 3.26 and Proposition 3.29.
Appendix G. Changes of Variables and Estimates of Operator Norms The basic change of variables formula for the operator norms of Definition 3.18 is Proposition 3.19. It, and consequences of it, are proven in [4, §IV]. For the purposes of our construction, we need variants of these results for special situations, namely that the operators implementing the change of variables have restricted ranges or that the functions to which the change of variables is applied are polynomials. The first is treated in Proposition G.1 and the second in Lemma G.2 and Corollary G.3. Also, in Remark G.4 and Lemma G.5, we develop tools to bound the operator norms of Definition 3.18. We work in the same abstract environment as in [4, §IV]. We are given weight factors κ1 , . . . , κs on an abstract metric space X with metric d. We consider analytic functions f (α1 , . . . , αs ; h) of the complex fields α1 , . . . , αs and the additional “history” field h. Denote by w the weight system with metric d that associates the weight factor κj to the field αj , and the constant weight factor 1 to the history field h (see Definition 3.12). Proposition G.1. Let Γj , 1 ≤ j ≤ r, be h-operators on CX . Let Λ ⊂ X and, for each 1 ≤ j ≤ r, Λj be either Λ or Λc . Set ⎛ ⎞ r Λj Γj βj ; h⎠ f˜(α1 , . . . , αs−1 , β1 , . . . , βr ; h) = f ⎝α1 , . . . , αs−1 , j=1
Furthermore let κ ˜ i , 1 ≤ i ≤ s − 1 and λj , 1 ≤ j ≤ r be weight factors. Let w ˜ be the weight system with metric d that associates the weight factor κ ˜ i to the field αi , for 1 ≤ i ≤ s − 1, the weight factor λj to the field βj , for 1 ≤ j ≤ r, and the constant weight factor 1 to the history field h. Assume that κ ˜ i (x) ≤ κi (x) for all 1 ≤ i ≤ s − 1 and x ∈ X and that there is a ν > 1 such that 1 1 and Nd (Λj Γj ; κs , λj ) ≤ Nd (Λj Γj ; κs , λj ) ≤ ν ν 1≤j≤r Λj =Λ
1≤j≤r Λj =Λc
Then f˜ w˜ ≤ Cν f w
⎧ 4ν ⎪ ⎨ (e ln ν)2 with Cν =
4 ⎪ν
⎩ 1
if ν ≤ e if e ≤ ν < 4 if ν ≥ 4
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Proof. We introduce auxiliary fields βj 1≤j≤r and {γ1 , γ2 } and define f (α1 , . . . , αs−1 , γ1 , γ2 ; h) = f (α1 , . . . , αs−1 , Λγ1 + Λc γ2 ; h) ⎛
⎞
⎜ ⎟ f (α1 , . . . , αs−1 , β1 , . . . , βr ; h) = f ⎜ βj , βj ; h⎟ ⎝α1 , . . . , αs−1 , ⎠ 1≤j≤r Λj =Λ
1≤j≤r Λj =Λc
Then f˜(α1 , . . . , αs−1 , β1 , . . . , βr ; h) = f (α1 , . . . , αs−1 , Λ1 Γ1 β1 , . . . , Λr Γr βr ; h) Set, for each 1 ≤ j ≤ r, tj = Nd (Λj Γj ; κs , λj ) and introduce the auxiliary weight system w with metric d that associates the weight factor one to the history field, ◦ the weight factor κi to the field αi , for each 1 ≤ i ≤ s − 1, κs (x) to the field βj , for each 1 ≤ j ≤ r, and ◦ the weight factor λ j (x) = tj ◦ the weight factors κ1 (x) = 1≤j≤r tj κs (x) and κ 2 (x) = 1≤j≤r tj κs (x), Λj =Λc
Λj =Λ
respectively, to the fields γ1 and γ2 . By [4, Proposition IV.4], f˜(α1 , . . . , αs−1 , β1 , . . . , βr ; h) w˜ ≤ f (α1 , . . . , αs−1 , β1 , . . . , βr ; h) w since Nd (Λj Γj ; λ j , λj ) = Nd (Λj Γj ; tj κs , λj ) =
1 Nd (Λj Γj ; κs , λj ) = 1 tj
By [4, Lemma IV.5.i], applied twice, f (α1 , . . . , αs−1 , β1 , . . . , βr ; h) w ≤ f (α1 , . . . , αs−1 , γ1 , γ2 ; h) w since, 1≤j≤r Λj =Λ
sup x
λ j (x) λ j (x) = =1 sup κ1 (x) x κ2 (x) 1≤j≤r Λj =Λc
By part (iii) of [4, Lemma IV.5], with r replaced by 2, f (α1 , . . . , αs−1 , γ1 , γ2 ; h) w ≤ Cν f (α1 , . . . , αs ; h) w since Cν = Cν,2 and κ (x) 1 = tj ≤ sup 1 ν x κs (x) 1≤j≤r Λj =Λ
and
sup x
κ 1 (x) 1 = tj ≤ κs (x) ν 1≤j≤r Λj =Λc
In Sect. 5, we use a refinement of [4, Proposition IV.4] for functions of low degree in the fields that exploits exponential decay of operators away from a given subset L of X. This estimate is contained in
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Lemma G.2. Let A1 , . . . , As be h-operators on CX and let κ 1 , . . . κ s be weight factors. Furthermore, let f (α1 , . . . , αs ; h) be a polynomial in the fields α1 , . . . , αs and set f (α1 , . . . , αs ; h) = f (A1 α1 , . . . , As αs ; h) (a)
Assume that f (α1 , . . . , αs ; h) is homogeneous of degree dj in αj for each 1 ≤ j ≤ s. Then s
f w˜ ≤ f w
Nd (Aj ; κj , κ j )dj
j=1
where w ˜ is the weight system with metric d that associates the weight factor κ j to the field αj and the constant weight factor 1 to the history field h. (b) Let t1 , . . . , ts ∈ R with t1 + · · · + ts ≤ 0. Also, Let d and δ be metrics that obey ⎞ ⎛ s ⎟ ⎜ ti ⎠ δ d ≥ d + ⎝ i=1 ti ≥0
Denote by w the weight system with metric d that associates the weight factor κ j to the field αj and the constant weight factor 1 to the history field h. Assume that f (α1 , . . . , αs ; h) is homogeneous of degree one in each of the fields α1 , . . . , αs . Then, for any subset L of X, s ( ) Nd Aj ; κj etj δ(x,L) , κ j f w ≤ f w j=1
Proof. (a) follows from [4, Proposition IV.4] by scaling. (b) Denote by waux the weight system with metric d that associates the weight factor κj etj δ(x,L) to the field αj . We claim that f waux ≤ f w
(G.1)
The claim follows from (G.1), since, by part (a), s ( ) Nd Aj ; κj etj δ(x,L) , κ j f w ≤ f waux j=1
To prove (G.1) we may assume that r there is r s≥ 0 such that t1 , · · · , tr ≥ 0 and tr+1 , . . . , ts ≤ 0. By hypothesis, i=1 ti ≤ j=r+1 (−tj ). Therefore there are aij ≥ 0, i = 1, . . . , r, j = r + 1, . . . , s such that ti = (−tj ) ≥
s j=r+1 r
aij
aij
i=1
for i = 1, . . . , r
(G.2)
for j = r + 1, . . . , s
(G.3)
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Now fix x1 , . . . , xs ∈ X, ≥ 0 and z ∈ X . Let T be a tree whose set of vertices contains x1 , . . . , xs , z1 , . . . , z . Denote by lengthδ (T ) the length of T with respect to the metric δ. For each 1 ≤ i, j ≤ s, the points xi and xj can be connected by a path in T . Therefore δ(xi , L) ≤ δ(xj , L) + lengthδ (T ) Consequently s
ti δ(xi , L)
i=1
=
r s
s
aij δ(xi , L) +
i=1 j=r+1
≤
r s
aij δ(xj , L) +
i=1 j=r+1
=
s
4
tj +
j=r+1
4
≤
r
r
aij lengthδ (T ) +
i=1 j=r+1
5 aij
r s
δ(xj , L) +
i=1
5 ti
ti δ(xi , L)
i=r+1
r
s
tj δ(xj , L)
j=r+1
ti lengthδ (T )
i=1
lengthδ (T )
i=1
by (G.2). Therefore lengthd (T ) +
s
ti δ(xi , L) ≤ lengthd (T )
i=1
This holds for any tree T , whose set of vertices contains x1 , . . . , xs , z1 , . . . , z , so that τd (x1 , . . . , xs ,z) +
s
ti δ(xi , L) ≤ τd (x1 , . . . , xs ,z)
i=1
If we expand f (α1 , . . . , αs ; h) =
a (x1 , . . . , xs ;z) α1 (x1 ) · · · αs (xs ) h(z1 ) · · · h(z )
≥0 x1 ,...,x ∈X z∈X
and apply Definition 3.12, we get (G.1)
Corollary G.3. Let h(γ1 , . . . , γr ; h) be a multilinear form in the fields γ1 , . . . , γr . Furthermore, let ˜ j , Aj and A˜j Γji , Γ i
Vol. 11 (2010)
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341
(i = 1, . . . , r; j = 1, . . . , s) be h-operators on CX . Set ⎛ ⎞ s s j f1 (α1 , . . . , αs ; h) = h ⎝ Γ1 αj , . . . , Γjr αj ; h⎠ ⎛ f2 (α1 , . . . , αs ; h) = h ⎝
j=1 s
j=1
Γj1 αj , . . . ,
j=1
⎛ −h ⎝
s
s
⎞ Γjr αj ; h⎠
j=1
˜ j αj , · · · Γ 1
j=1
,
s
⎞ ˜ j αj ; h⎠ Γ r
j=1
⎛ ⎞ s s f3 (α1 , . . . , αs ; h) = h ⎝ A1 Γj1 αj , . . . , Ar Γjr αj ; h⎠ j=1
j=1
⎛ ⎞ s s −h ⎝ A˜1 Γj1 αj , . . . , A˜r Γjr αj ; h⎠ j=1
j=1
j=1
j=1
j=1
j=1
⎛ ⎞ s s ˜ j αj , . . . , ˜ j αj ; h⎠ −h ⎝ A1 Γ Ar Γ r 1 ⎛ ⎞ s s ˜ j αj , . . . , ˜ jr αj ; h⎠ +h ⎝ A˜1 Γ A˜r Γ 1 Let λ1 , . . . , λr be weight factors. Let w be the weight system with some metric d that associates the weight factor κj to the field αj ; and let wλ be the weight system with metric d that associates the weight factor λi to the field γi . Let δ either be 0 or a metric which obeys d ≥ d + (r − 1) δ and let L ⊂ X. Then (i)
(ii)
0 r s f1 w ≤ h wλ i=1 ( j=1 Nd (Γji ; λi eti δ(x,L) , κj )) where each ti , 1 ≤ i ≤ r, is either 1 or −(r − 1) and at least one of them is −(r − 1) f2 w ≤ r h wλ σδ σ r−1 where ⎧ ⎫ s s ⎨ ⎬ ˜ j ; λi eδ(x,L) , κj ) σ = max max Nd (Γji ; λi eδ(x,L) , κj ), Nd (Γ i i=1,...,r ⎩ ⎭ j=1
σδ = max
i=1,...,r
(iii)
s j=1
κj j ˜ j |||Γ − Γi ||| λi i
f3 w ≤ r2 h wλ σδ aδ (σa)r−1
j=1
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where a=
" # max max Nd (Ai ; λi etδ(x,L) , λi etδ(x,L) ), Nd (A˜i ; λi etδ(x,L) , λi etδ(x,L) )
i=1,...,r t=1,−(r−1)
aδ =
max
i=1,...,r t=1,−(r−1)
˜i ; λi etδ(x,L) , λi etδ(x,L) ) Nd (Ai − A
Proof. (i) It suffices to prove that, for each choice of 1 ≤ j1 , . . . , jr ≤ s, 0r the · w norm of h(Γj11 αj1 , . . . , Γjrr αjr ; h) is bounded by h wλ i=1 Nd (Γji i ; λi eti δ(x,L) , κji ). This follows from Lemma G.2(b) and [4, Lemma IV.5.ii] (when two or more of the ji ’s happen to be the same). (ii) Write the telescoping sum ⎞ ⎛ s s s ˜ j )αj , f2 (α1 , . . . , αs ) = h ⎝ (Γj1 − Γ Γj2 αj , . . . , Γjr αj ⎠ 1 j=1
⎛
+h⎝
j=1
s
˜ j αj , Γ 1
j=1
s
j=1
˜ j )αj , . . . , (Γj2 − Γ 2
j=1
s
⎞ Γjr αj ⎠
j=1
⎛ ⎞ s s s ˜ j αj , ˜ j αj , . . . , ˜ jr − Γ ˜ jr )αj ⎠ +··· + h⎝ Γ Γ (Γ 1 2 j=1
j=1
j=1
and apply part (i) to each of the summands. For term number i0 , which con˜ j , choose ti = −(r − 1) and ti = 1 for all i = i0 . (We have tains Γji0 − Γ 0 i0 suppressed the argument h and will do so for the rest of the proof.) (iii) Write the telescoping sum f3 (α1 , . . . , αs ) ⎞ ⎛ s s ˜ j )αj , . . . , Ar Γjr αj ⎠ = h ⎝ A1 (Γj1 − Γ 1 j=1
⎛ −h ⎝
s j=1
j=1
˜ j )αj , . . . , A˜1 (Γj1 − Γ 1 ⎛
+··· + h⎝ ⎛ −h ⎝
j=1
⎞ A˜r Γjr αj ⎠
j=1 s
˜ j αj , . . . , A1 Γ 1
j=1 s
s
˜ j αj , . . . , A˜1 Γ 1
s j=1
s
⎞
˜ jr − Γ ˜ jr )αj ⎠ Ar (Γ ⎞
˜ jr − Γ ˜ jr )αj ⎠ A˜r (Γ
j=1
We claim that the · w norm of each of the r lines is bounded by r h wλ σδ aδ (σa)r−1 . We prove this for the first line. The proof for the other lines is similar.
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343
We again write a telescoping sum ⎛ h⎝
s j=1
˜ j )αj , . . . , A1 (Γj1 − Γ 1 ⎛
−h ⎝
s
⎞ Ar Γjr αj ⎠
j=1 s
˜ j )αj , . . . , A˜1 (Γj1 − Γ 1
j=1
⎛
s
⎞ A˜r Γjr αj ⎠
j=1
⎞ s s s ˜ j )αj , = h ⎝ (A1 − A˜1 )(Γj1 − Γ A2 Γj2 αj , . . . , Ar Γjr αj ⎠ 1 j=1
⎛ +h ⎝
j=1
s j=1
j=1
⎞ s s ˜ j )αj , (A2 − A˜2 )Γj αj , . . . , A˜1 (Γj1 − Γ Ar Γjr αj ⎠ 1 2 j=1
⎛
j=1
⎞ s s s ˜ j )αj , A˜2 Γj2 αj , . . . , (Ar − A˜r )Γjr αj ⎠ + · · · + h ⎝ A˜1 (Γj1 − Γ 1 j=1
j=1
j=1
By the first bound, the · w norm of the first term is bounded by ⎞ ⎛ s ( ) ˜ j ); λ1 e−(r−1)δ(x,L) , κj ⎠ h wλ ⎝ Nd (A1 − A˜1 )(Γj1 − Γ 1 j=1
×
r
⎛ ⎝
i=2
s
( Nd
Ai Γji ; λi eδ(x,L) , κj
)
⎞ ⎠
j=1
≤ h wλ Nd (A1 − A˜1 ; λ1 e−(r−1)δ(x,L) , λ1 e−(r−1)δ(x,L) ) ⎞ ⎛ s ˜ j ; λ1 e−(r−1)δ(x,L) , κj )⎠ Nd (Γj1 − Γ ×⎝ 1 j=1
×
r i=2
⎛ ⎝Nd (Ai ; λi eδ(x,L) , λi eδ(x,L) ) ⎛
≤ h wλ ⎝aδ
×
r i=2
⎛ ⎝
s j=1
s
⎞ ˜ j ; λ1 e−(r−1)δ(x,L) , κj )⎠ Nd (Γj1 − Γ 1
j=1 s
⎞ Nd (Γji ; λi eδ(x,L) , κj )⎠
⎞ aNd (Γji ; λi eδ(x,L) , κj )⎠
j=1
≤ h wλ σδ aδ (σa)r−1
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by [4, Remark IV.3.ii]. The norm of the second term is bounded by hwλ
4 s
( Nd
˜1 (Γj1 − Γ ˜ j1 ); λ1 e−(r−1)δ(x,L) , κj A
j=1
×
4 s
( Nd
)
5
5 r 4 s 5 ) ( ) j δ(x,L) j δ(x,L) ˜ (A2 − A2 )Γ2 ; λ2 e , κj Nd Ai Γi ; λi e , κj
j=1
i=3
j=1
r−2
≤ hwλ (σδ a)(σaδ )(σa)
Similarly, one bounds the norms of each of the other r − 2 terms by h wλ σδ aδ (σa)r−1 . In order to apply Proposition 3.19, [4, Corollary IV.6] or Lemma G.2 we need techniques to estimate operator norms. They are given in Remark G.4 and Lemma G.5, below. Remark G.4. Let A be an h-linear map from CX to CX . Let d1 , d2 be metrics with d1 − d2 ≥ d. Furthermore let κ, κ be weight factors. (i)
⎞⎛
⎛ ⎜ Nd (A; κ, κ ) ≤ Nd1 (A; 1, 1) ⎝ sup ⎛
x,y∈X A(x,y)=0
⎜ Nd (A; κ, κ ) ≤ Nd1 (A; 1, 1) ⎝ sup
x,y∈X A(x,y)=0
e−d2 (x,y)
κ (y)⎟⎜ ⎠⎝ sup κ (x) x∈X
A(x, · )=0
⎞⎛ e−d2 (x,y)
κ(y) ⎟ ⎜ ⎠ ⎝ sup κ(x) y∈X
(ii) Let c > 0 and define the h-linear operator Ac by A(x;z; y) if d2 (x, y) ≤ c Ac (x;z; y) = 0 if d2 (x, y) > c Then Nd (Ac − A ; κ, κ ) ≤ e−c Nd1 (A; κ, κ ) (iii) Denote K=
sup
x,y∈X A(x; · ;y)=0
⎞
κ (y) −d2 (x,y) e κ(x)
Then Nd (A ; κ, κ ) ≤ KNd1 (A; 1, 1) (iv) Let J be a linear operator on CX . Then Nd (exph(J) ; κ, κ) ≤ eNd (J;κ,κ) Nd (exph(J) − h ; κ, κ) ≤ Nd (J; κ, κ) eNd (J;κ,κ)
A( ·,y)=0
κ (x)⎟ ⎠ κ(x) ⎞ κ (y)⎟ ⎠ κ(y)
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Proof. To prove parts (i) and (ii), observe that, for all x, y ∈ X and all z ∈ X (1) eτd (supp (x, z,y)) |A(x;z; y)|
κ (y) κ (y) ≤ e−d2 (x,y) eτd1 (supp (x, z,y)) |A(x;z; y)| κ(x) κ(x)
This immediately gives part (ii). The two inequalities of part (i) follow by writing κ (y) κ (x) κ(y) κ (y) κ (y) = = κ(x) κ (x) κ(x) κ(x) κ(y) Similarly, part (iii) follows from eτd (supp (x, z,y)) |A(x;z; y)|
κ (y) ≤ Ked2 (x,y) eτd (supp (x, z,y)) |A(x;z; y)| κ(x) ≤ Keτd1 (supp (x, z,y)) |A(x;z; y)|
Part (iv) follows from the expansion ¯
exph(J) = eJ = h +
∞ 1 ¯ J ! =1
and [4, (IV.1)]. In Appendix E, we use a more sophisticated Lemma G.5. Let ◦ L1 , L2 , L3 , O1 , O2 ⊂ X ◦ A1 , A2 be h–linear operators on CX ◦ δ1 , δ2 , δ, d˜ metrics on X ◦ R > 0 and κ a weight factor such that κ(x) ≤ R eδ2 (x,O2 ) for all x ∈ L3 . Set DL = max {δ(L1 , L2 ), δ(L2 , L3 ), δ(L3 , L1 )} DO = max {DL , δ(L1 , O1 ), δ(L2 , O1 ), δ(L3 , O1 )} (i)
˜ If Assume that d + δ2 + δ ≤ d. (a) Lj ⊂ O2 for at least one j ∈ {1, 2, 3} and D = DL or if (b) Lj ⊂ O2 for at least one j ∈ {1, 2, 3} , δ1 ≥ δ and D = DO or if (c) O1 ⊂ O2 , δ1 ≥ δ + δ2 and D = DO then ( ) Nd L1 A1 L2 A2 L3 ; eδ1 (x,O1 ) , κ ≤ R e−D Nd˜ (L1 A1 L2 ; 1, 1) Nd˜ (L2 A2 L3 ; 1, 1) (d) Assume that d + δ1 + δ2 + δ ≤ d˜ and that Li ⊂ O1 , Lj ⊂ O2 for some 1 ≤ i, j ≤ 3. Then ( ) Nd L1 A1 L2 A2 L3 ; e−δ1 (x,O1 ) , κ ≤ R e−DL Nd˜ (L1 A1 L2 ; 1, 1) Nd˜ (L2 A2 L3 ; 1, 1)
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Proof. We may assume that R = 1 and that κ(x) = eδ2 (x,O2 ) . We write the kernel of the operator L1 A1 L2 A2 L3 as (L1 A1 L2 A2 L3 )(x;z; y) = (L1 A1 L2 ) (x;z1 ; u) (L2 A2 L3 ) (u;z2 ; y) u∈L2 z2 ∈X (1) z1 , z2 = z z1 ◦(u)◦
(i) Fix x ∈ L1 , u ∈ L2 , y ∈ L3 , z1 ,z2 ∈ X (1) . Then −δ1 (x, O1 ) + τd (supp (x; z1 ◦ u ◦ z2 ; y)) + δ2 (y, O2 ) ≤ τd (supp (x;z1 ; u)) + τd (supp (u;z2 ; y)) + δ(x, u) + δ(u, y) + δ2 (y, O2 ) −δ1 (x, O1 ) − δ(x, u) − δ(u, y) ≤ τd˜ (supp (x;z1 ; u))+τd˜ (supp (u;z2 ; y))−δ2 (x, u)−δ2 (u, y)+δ2 (y, O2 ) −δ1 (x, O1 ) − δ(x, u) − δ(u, y) As x ∈ L1 , u ∈ L2 , y ∈ L3 δ(x, u) + δ(u, y) ≥ DL δ(x, O1 ) + δ(x, u) + δ(u, y) ≥ DO If Lj ⊂ O2 for at least one j ∈ {1, 2, 3} then −δ2 (x, u) − δ2 (u, y) + δ2 (y, O2 ) ≤ 0 Therefore −δ2 (x, u) − δ2 (u, y) + δ2 (y, O2 ) − δ1 (x, O1 ) − δ(x, u) − δ(u, y) −DL in case a) ≤ −DO in case b) In case (c), δ1 (x, O1 ) ≥ δ2 (x, O2 ) + δ(x, O1 ) so that −δ2 (x, u) − δ2 (u, y) + δ2 (y, O2 ) − δ1 (x, O1 ) − δ(x, u) − δ(u, y) ≤ −δ2 (x, u)−δ2 (u, y) + δ2 (y, O2 )−δ2 (x, O2 )−δ(x, O1 )−δ(x, u)−δ(u, y) ≤ −DO Consequently, in all three cases −δ1 (x, O1 ) + τd (supp (x; z1 ◦ u ◦ z2 ; y)) + δ2 (y, O2 ) ≤ −D + τd˜ (supp (x;z1 ; u)) + τd˜ (supp (u;z2 ; y)) so that e−δ1 (x,O1 ) eτd (supp (x; z1 ◦u◦ z2 ;y)) |(L1 A1 L2 ) (x;z1 ; u) (L2 A2 L3 ) (u;z2 ; y)| κ(y) ≤ e−D eτd˜(supp (x; z1 ;u)) |(L1 A1 L2 ) (x;z1 ; u)| eτd˜(supp (u; z2 ;y)) × |(L2 A2 L3 ) (u;z2 ; y)| If we define the auxiliary h-linear operators A˜1 and A˜2 by A˜1 (x;z; y) = eτd˜(supp (x; z;y)) |A1 (x;z; y)| A˜2 (x;z; y) = eτd˜(supp (x; z;y)) |A2 (x;z; y)|
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we now have, by [4, (IV.1)], ( ( ) ) Nd L1 A1 L2 A2 L3 ; eδ1 (x,O1 ) , κ ≤ R e−D N0 L1 A˜1 L2 A˜2 L3 ; 1, 1 ( ) ( ) ≤ R e−D N0 L1 A˜1 L2 ; 1, 1 N0 L2 A˜2 L3 ; 1, 1 = R e−D Nd˜ (L1 A1 L2 ; 1, 1)Nd˜ (L2 A2 L3 ; 1, 1) (ii) As in part (i), fix x ∈ L1 , u ∈ L2 , y ∈ L3 , z1 ,z2 ∈ X (1) and bound δ1 (x, O1 ) + τd (supp (x; z1 ◦ u ◦ z2 ; y)) + δ2 (y, O2 ) ≤ τd (supp (x;z1 ; u)) + τd (supp (u;z2 ; y)) + δ(x, u) + δ(u, y) + δ1 (x, O1 ) +δ2 (y, O2 ) − δ(x, u) − δ(u, y) ≤ τd˜ (supp (x;z1 ; u))+τd˜ (supp (u;z2 ; y))−(δ1 (x, u)+δ1 (u, y)−δ1 (x, O1 )) − (δ2 (x, u) + δ2 (u, y) − δ2 (y, O2 )) − (δ(x, u) + δ(u, y)) ≤ −DL + τd˜ (supp (x;z1 ; u)) + τd˜ (supp (u;z2 ; y))
Appendix H. Symbol Table Notation
Definition
Comments
∇
discrete gradient
|||v|||
(∇f ) ((x, y)) = f (y) − f (x) supx∈X e5m d(x,y) |v(x, y)|
|||A|||
N5m (A; 1, 1)
Definition 3.20
χθ (Ω; α, β)
Theorem 2.16
small field conditions
Definition 2.4
hierarchy
y∈X
S ∗
dμΩ,r (z , z)
0
dz(x)∗ ∧dz(x) x∈Ω 2πı −z(x)∗ z(x)
e DΩ;0 (ε; α∗ , β) DΩ;θ (α∗ , β)
(2.5)
before (2.1)
χ (|z(x)| < r)
before (2.2) −m
limm→∞ DΩ;m (2
part of effective action
θ; α∗ , β)
(2.2)
d(x, y)
standard metric on X
d(τ )
Notation 2.2(iii)
decimation index of τ
exph (J)
hJh
Definition 3.4(iii)
Γ∗S , ΓS Γτ∗τ (S),
Γττ (S)
e
h
Definition 2.9
background field
Definition 2.9
coefficients
∗
h
∇ H∇
kinetic energy
H
kernel in kinetic energy
see beginning of Appendix D
∗
Iθ (α , β)
−m
limm→∞ Im (2
∗
θ; α , β)
Theorem 2.16
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In (ε; α∗ , β)
(1.3), (1.10)
I(S;α∗ ,β)
Definition 2.8(ii)
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effective density large field integral operator
I(J ; α∗ ,β)
Definition 2.8(i)
large field integral operator
Jτ
Notation 2.2(iii)
decimation interval centred on τ
Jτ−
Notation 2.2(iii)
left half of Jτ
Jτ+
Notation 2.2(iii)
right half of Jτ
j(t)
exph (−t(h − μ)) 1 if d(x, y) ≤ c j(τ )(x, y) · 0 if d(x, y) > c −d(x,y) sup e
(3.1)
jc (τ )(x, y) Kd
y∈X x∈X 35 6Kj
KD
2 e
2 e
Theorem 3.27
23
KE
2
Theorem 3.24
48 6Kj
KL 1 32
KL2
2 e min cH e−4DCH ,
in proof of Lemma 3.42 Theorem 3.26
40 10Kj
KΔ
(3.2)
1 32
Theorem 3.26 in proof of Proposition 3.36
9 2Kj
KQ
2 e 12
KR
2
KV
Kj2
Nδ (A; κ, κ )
Theorem 3.24
25 6Kj
Proposition 5.13
Definitions 2.4, A.1, Sect. 5.1
small field set
Definition 3.18
weighted L1 –L∞
2 e
Λ(J )
Proposition 5.12
operator norm Ω(J )
Definitions 2.4, Sect. 5.3
small field set
Pα (J ), Pβ (J )
Definitions 2.4, A.1, Sect. 5.1
large field sets
Pα (J ), Pβ (J )
Definitions 2.4, A.1, Sect. 5.1
large field sets
Q(J )
Definitions 2.4, A.1, Sect. 5.1
large field set
Qε,δ (α∗ , β; γ∗ , γ )
(2.10)
dominant quadratic part
QS (α∗ , β; α
∗, α
)
Qε,δ α∗ , β; Γ∗S ( · ; α∗ , α
∗ ), ΓS ( · ; α
, β)
R(J )
Definitions 2.4, Sect. 5.3
(2.9), (3.3)
Stokes’ large field sets
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renormalization group operator
VΩ,δ (ε; α∗ , β) VΩ;θ (α∗ , β)
(2.1)
principal interaction
−m
limm→∞ VΩ;θ (2
θ; α∗ , β)
(2.4)
VS (α∗ , β; α
∗, α
)
(2.11)
dominant quartic part
v(x, y)
two-body potential
see Hypothesis 2.14
X Y
∗
Z /LZ D
D
bonds with at least
space just before Notation 2.2
one end in Y Y
points within
just before Notation 2.2
distance one of Y Zδ
Lemma 2.7
normalization constant
References [1] Balaban, T., Feldman, J., Kn¨ orrer, H., Trubowitz, E.: A functional integral representation for many boson systems. I. The partition function. Annales Henri Poincar´e 9, 1229–1273 (2008) [2] Balaban, T., Feldman, J., Kn¨ orrer, H., Trubowitz, E.: A functional integral representation for many boson systems. II. Correlation functions. Annales Henri Poincar´e 9, 1275–1307 (2008) [3] Balaban, T., Feldman, J., Kn¨ orrer, H., Trubowitz, E.: Power series representations for bosonic effective actions. J. Stat. Phys. 134, 839–857 (2009) [4] Balaban, T., Feldman, J., Kn¨ orrer, H., Trubowitz, E.: Power series representations for complex bosonic effective actions. I. A small field renormalization group step. J. Math. Phys. (to appear) [5] Balaban, T., Feldman, J., Kn¨ orrer, H., Trubowitz, E.: Power series representations for complex bosonic effective actions. II. A small field renormalization group flow. J. Math. Phys. (to appear) [6] Brydges, D., Federbush, P.: The cluster expansion in statistical physics. Commun. Math. Phys. 49, 233–246 (1976) [7] Brydges, D., Federbush, P.: The cluster expansion for potentials with exponential fall-off. Commun. Math. Phys. 53, 19–30 (1977) [8] Edwards, R.E.: Functional Analysis: Theory and Applications. Dover, New York (1995) [9] Ginibre, J.: Reduced density matrices of quantum gases. I. Limit of infinite volume. J. Math. Phys. 6, 238–251 (1965) [10] Ginibre, J.: Some applications of functional integration in statistical mechanics. In: DeWitt, C., Stora, R. (eds.) Statistical Mechanics and Quantum Field Theory, Proceedings of the Les Houches Summer School of Theoretical Physics, p. 327. Gordon and Breach, New York (1971)
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[11] Negele, J.W., Orland, H.: Quantum Many-Particle Systems. AddisonWesley, Reading (1988) Tadeusz Balaban Department of Mathematics Rutgers, The State University of New Jersey 110 Frelinghuysen Rd Piscataway, NJ 08854-8019, USA e-mail:
[email protected] Joel Feldman Department of Mathematics University of British Columbia Vancouver, BC V6T 1Z2, Canada e-mail:
[email protected] Horst Kn¨ orrer, Eugene Trubowitz Mathematik ETH-Zentrum 8092 Z¨ urich, Switzerland e-mail:
[email protected];
[email protected] Communicated by Vincent Rivasseau. Received: December 19, 2009. Accepted: January 12, 2010.
Ann. Henri Poincar´e 11 (2010), 351–431 c 2010 Springer Basel AG 1424-0637/10/030351-81 published online June 29, 2010 DOI 10.1007/s00023-010-0044-5
Annales Henri Poincar´ e
Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c = 1 Matrix Models Sara Pasquetti and Ricardo Schiappa Abstract. We address the nonperturbative structure of topological strings and c = 1 matrix models, focusing on understanding the nature of instanton effects alongside with exploring their relation to the large-order behavior of the 1/N expansion. We consider the Gaussian, Penner and Chern–Simons matrix models, together with their holographic duals, the c = 1 minimal string at self-dual radius and topological string theory on the resolved conifold. We employ Borel analysis to obtain the exact all-loop multi-instanton corrections to the free energies of the aforementioned models, and show that the leading poles in the Borel plane control the large-order behavior of perturbation theory. We understand the nonperturbative effects in terms of the Schwinger effect and provide a semiclassical picture in terms of eigenvalue tunneling between critical points of the multi-sheeted matrix model effective potentials. In particular, we relate instantons to Stokes phenomena via a hyperasymptotic analysis, providing a smoothing of the nonperturbative ambiguity. Our predictions for the multi-instanton expansions are confirmed within the trans-series set-up, which in the double-scaling limit describes nonperturbative corrections to the Toda equation. Finally, we provide a spacetime realization of our nonperturbative corrections in terms of toric D-brane instantons which, in the double-scaling limit, precisely match D-instanton contributions to c = 1 minimal strings.
1. Introduction and Summary The nonperturbative realm of quantum field and string theories has often been a source of many new results and surprises. Another recurrent topic of great interest over the past decades has been the large N approximation, lately relating gauge and string theories in a nonperturbative fashion.
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Of particular interest to us in this work is the case of the perturbative 1/N expansion of hermitian matrix models, whose nonperturbative corrections are exponentially suppressed as exp (−N ). In the double-scaling limit, these models describe noncritical or minimal (super)string theories, and the nonperturbative structure of the matrix model is related to that of the corresponding string theory [1]: the exp (−N ) contributions are instanton effects in the matrix model [2,3] and they are interpreted as D-brane configurations in the string theoretic description [4–6]. Of course the study of matrix models is not confined to the vicinity of their critical points and one may also study nonperturbative effects away from the double-scaling limit. The interesting point is that off-critical matrix models may be dual to topological string theories. For instance, this happens in the case first suggested by Dijkgraaf and Vafa [7], where some off-critical matrix models describe the topological string B-model on certain non-compact Calabi–Yau (CY) backgrounds, with the string genus expansion (in powers of the string coupling, gs ) being identified with the 1/N matrix model expansion; and it is also the case for topological strings with a Chern–Simons dual, first studied in [8,9]. This turns out to be a more general statement, as it was later shown in [10–12] that topological string theory on mirrors of toric manifolds also enjoys a dual holographic description in terms of off-critical matrix models. It is thus evident that fully understanding the nonperturbative structure of matrix models, both at and off criticality, will have many applications in both minimal and topological string theories. Recently [13–15] there has been significative progress in understanding and in quantitatively computing nonperturbative effects in matrix models away from criticality. In [13], and building upon double-scaled results [2,3,16,17], off-critical saddle-point techniques were developed in order to compute instanton amplitudes (up to two loops) in terms of spectral curve geometrical data. This work focused upon one-instanton contributions in one-cut models, and in [15] an extension to multi-instanton contributions, again in one-cut models, was obtained, starting from a two-cut analysis. Extensive checks of the nonperturbative proposals in these papers were also performed, by matching against the large-order behavior of the 1/N expansion. Another approach to multi-instanton amplitudes was developed in [14], this time around based on orthogonal polynomial methods, via the use of trans-series solutions to the string equations [18]. Further progress along these lines recently led to the proposal of [19], where a proper nonperturbative definition of a modularinvariant holomorphic partition function was presented, which was also shown to be manifestly background independent. Remarkably, many of the results uncovered in [13–15] appear to extend beyond the context of matrix models; e.g., in cases where the theory is controlled by a finite-difference equation, such as the string equation [18] for matrix models, it is possible to compute nonperturbative effects and relate them to the large-order behavior of the theory. This is the case of Hurwitz theory [13], which is controlled by a Toda-like equation, and also the case of topological strings on the background considered
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in [20].1 However, all models considered in the aforementioned articles lie in the universality class of 2D gravity, with c = 0, and methods that have been worked out in this case cannot be applied in a straightforward fashion to the case of topological strings in the universality class of c = 1. In view of this, it is necessary to develop new techniques in order to approach nonperturbative effects in models which belong to the universality class of the c = 1 string at the self-dual radius. Let us be a bit more specific about the nature of the string perturbative expansion and the type of nonperturbative contributions we shall be looking for. Topological strings, much like physical string theory, are perturbatively defined in terms of two couplings, α and gs , as2 F (gs ; {ti }) =
+∞
gs2g−2 Fg (ti ),
(1.1)
g=0
where F = log Z is the free energy and Z the partition function, and where the fixed genus free energies Fg (ti ) are themselves perturbatively expanded in α . In some sense, the α expansion is the milder one: it has finite convergence radius, with this radius given by the critical value of the K¨ ahler parameters where one reaches a conifold point in moduli space. As it turns out, the problem of finding a nonperturbative formulation of the A-model free energy, in α , may be reduced to that of solving the mirror B-model description, where topological string amplitudes become exact in α . In this way, the A-model solution is found by translating B-model amplitudes back to the A-model, by means of the mirror map. This topic has been extensively studied in the literature and we refer the reader to the recent developments [11,12] and references therein. The situation gets more complicated as one tries to go beyond perturbation theory in gs . In this case, one is immediately faced with the familiar string theoretic large-order behavior Fg ∼ (2g)! rendering (1.1) as an asymptotic expansion [4]. In this case, one expects nonperturbative corrections of order ∼ exp (−1/gs ), and an adequate nonperturbative formulation of the theory must encode all these corrections. As described above, there are certain cases, such as the backgrounds considered by Dijkgraaf and Vafa [7], or models with a dual Chern–Simons interpretation, where topological strings have a holographic matrix model description, with the matrix model large N expansion reproducing the topological string genus expansion. In these set of backgrounds, one would be tempted to use the finite N matrix model free energy as the gs nonperturbative definition of topological string theory.3 In order to establish this result, one must first understand how the finite N matrix model 1
A local CY threefold given by a bundle over a two-sphere, Xp = O(p − 2) ⊕ O(−p) → P1 , p ∈ Z, which may be regarded as a quantum group deformation of Hurwitz theory; see [20] for further details. 2 Recall that in the A-model the {t } are K¨ ahler parameters while in the B-model they are i complex parameters. 3 Other nonperturbative completions, provided by a holographic dual, have been proposed in [21].
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would encompass all nonperturbative contributions ∼ exp (−1/gs ). This situation is clear for minimal strings, realized in the double-scaling limit of hermitian matrix models: the nonperturbative effects associated with the asymptotic nature of the genus expansion are implemented via eigenvalue tunneling effects in the dual matrix model, and are interpreted in the continuum formulation in terms of Liouville branes in spacetime [1,6]. For topological strings, a similar understanding has been achieved in the case of the local curve [10,13], where a matrix model description is available [20]. In this case, the nonperturbative effects associated to the asymptotic behavior, or large-order behavior, have again been matched to instantons arising from matrix eigenvalue tunneling, and a spacetime interpretation in terms of domain walls has been provided [13]. However, there are several cases where this paradigm seems not to apply, at least not in a straightforward fashion. It is our goal to address such issues in the present work in the prototypical example of the resolved conifold, but also encompassing matrix models in the c = 1 universality class. Topological strings on the resolved conifold are holographically described by the Chern–Simons matrix model, but there are now no obvious instantons associated to eigenvalue tunneling as the Chern–Simons potential has no local maxima outside of the cut, where the eigenvalue instantons could tunnel to. This problem, which was not an issue in any of the previously mentioned examples, also appears in other matrix models, such as the Gaussian and Penner models; all of them in the c = 1 universality class. One may then ask where do nonperturbative corrections arise from, or what exactly controls the large-order behavior of the 1/N perturbative expansion in these models. We shall answer these questions in this paper. One way out is to directly compute the (would-be) instanton action that controls the large-order behavior of the perturbative expansion, by means of a standard Borel analysis (see, e.g., [22]). At first this may look like a formidable task, as one may expect the topological string genus expansion to be rather complicated, not amenable to a Borel transform. Happily, the free energies of all cases we consider enjoy a Gopakumar–Vafa (GV) integral representation [23,24] which allows for an exact location of the singularities in the Borel complex plane controlling the divergence of the asymptotic perturbative series, i.e., the instanton action [22]. This is the topological string generalization of a celebrated c = 1 string result [25]. Interestingly enough, this integral representation may also be regarded as an one-loop Schwinger integral [24], thus providing a spacetime interpretation of these nonperturbative effects; as already pointed out in [24] they control the pair-production rate of BPS bound states. As we shall later see, these results, which we further identify as Stokes phenomena of the finite N partition function, will also allow us to explain the nonperturbative contributions as one-eigenvalue effects in the matrix model picture. We find, from a saddle-point analysis, that the c = 1 nonperturbative effects arise due to the multi-valued structure of the effective potential (as preliminarily suggested in [26]); a different picture from that of matrix models in the universality class of 2D gravity plus matter, where the one-eigenvalue tunneling occurs from a metastable minimum to the most stable one.
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This paper is organized as follows. We begin in Sect. 2 by reviewing the main ideas behind our subsequent work. This includes the definition of the Borel transform and the relation between instantons and the large-order behavior of perturbation theory, both related to the existence of a nonperturbative ambiguity in the calculation of the free energy. In this section, we also discuss the Schwinger effect, where one actually has a physical prescription to define the inverse Borel transform, which will turn out to be the case for topological strings and c = 1 matrix models via the GV integral representation of the topological string free energy. In Sect. 3, we then move on to presenting the matrix models we shall be focusing upon. We review some of their properties, such as their spectral curves and their perturbative genus expansions, and also obtain expressions for their exact, finite N partition functions and holomorphic effective potentials, both of which play important roles in sections to come. In this section, we also discuss the double-scaling limit of these models and show how they relate to FZZT branes. Section 4 presents one of the main topics in this paper, the Borel analysis of the Gaussian, Penner and Chern–Simons matrix models. We show how to obtain Schwinger-like integral representations of the free energy, via Borel resummation, and how the correct identification of the leading poles in the complex Borel plane leads to the one-instanton action in all our examples. We further show in this section that while the all-loop multi-instanton amplitudes precisely reconstruct the perturbative series, the one-instanton results control the large-order behavior of perturbation theory. We then move on to another of our main topics in Sect. 5, namely the issue of Stokes phenomena. We recall how to obtain Stokes phenomena for integrals with saddles via hyperasymptotic analysis, and perform a detailed calculation for the Gamma function. This extends to the Barnes function and, in this way, allows us to identify instantons with Stokes phenomena as we reproduce the results we have previously found in Sect. 4, out of hyperasymptotic analysis. In Sect. 6, we provide a semiclassical interpretation of our instantons via eigenvalue tunneling, where this tunneling is now associated with the existence of a branched multi-sheeted structure in the relevant holomorphic effective potentials. Indeed, simple monodromy calculations reproduce our results for the multi-instanton action straight out of this interpretation. We further show in this section how to interpret our instantons in spacetime, from the point of view of ZZ branes. In Sect. 7, we discuss the trans-series approach to c = 1 matrix models and how it further validates our results. Finally, we conclude in Sect. 8 with an outlook and future prospects. We also include two appendices, one dedicated to the study of the monodromy structure of the polylogarithm, and the other dedicated to the Cauchy dispersion relation, in the case of more general topological string theories than the ones we address in this paper.
2. Asymptotic Series, Large Order and Topological Strings We start by reviewing some useful facts concerning asymptotic series, the relation of their large-order behavior to nonperturbative effects, as described by
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instantons or by the Schwinger effect, and put them in the context of topological string theory as we wish to study in the present work. For an introduction to these topics with applications in quantum mechanics and quantum field theory, we refer the reader to [22] and references therein. Let us consider the perturbative expansion of some function, F (z), with z the specific perturbative expansion parameter, F (z) ∼
+∞
Fn z n .
(2.1)
n=0
In many interesting examples one may infer that, at large n, the coefficients behave as Fn ∼ (βn)!, thus rendering the series divergent. As an approximation to the function F (z), the asymptotic series (2.1) must necessarily be truncated. As such, one is faced with an obvious problem: how to deal with the fact that the perturbative expansion has zero convergence radius? In particular, if we do not know the function F (z), but only its asymptotic series expansion, how do we associate a value to the divergent sum? The best framework to address issues related to asymptotic series is Borel analysis. One starts by introducing the Borel transform of the asymptotic series (2.1) as B[F ](ξ) =
+∞ Fn n ξ , (βn)! n=0
(2.2)
which removes the divergent part of the coefficients Fn and renders B[F ](ξ) with finite convergence radius. In particular, if F (z) originally had a finite radius of convergence (i.e., if it was not an asymptotic series), B[F ](ξ) would be an entire function in the Borel complex ξ-plane. In general, however, B[F ](ξ) will have singularities and it is crucial to locate them in the complex plane. The reason for this is simple to understand: if B[F ](ξ) has no singularities for real positive ξ one may analytically continue this function on R+ and thus define the inverse Borel transform by means of a Laplace transform as4 F(z) =
+∞ ds B[F ] zsβ e−s .
(2.3)
0
The function F(z) has, by construction, the same asymptotic expansion as F (z) and may thus provide a solution to our original question; it associates a value to the divergent sum (2.1). If, however, the function B[F ](ξ) has poles or branch cuts on the real axis, things get a bit more subtle: in order to perform the integral (2.3) one needs to choose a contour which avoids such singularities. This choice of contour naturally introduces an ambiguity (as we shall see next, a nonperturbative ambiguity) in the reconstruction of the original function, which renders F (z) non-Borel summable.5 As it turns out, different 4
For simplicity, we are assuming z ∈ R+ in this expression. Strictly speaking, the function is said not to be Borel summable if different integration contours yield different results. It may still be the case that, in spite of having singularities in the real axis, all alternative integration contours yield the same result. 5
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integration paths produce functions with the same asymptotic behavior, but differing by exponentially suppressed terms. For instance, in the presence of a singularity at a distance A from the origin, on the real axis, one may define the integral (2.3) on contours C± , either avoiding the singularity from above, and leading to F+ (z), or from below, and leading to F− (z). One finds that these two functions differ by a nonperturbative term [22] A
− F+ (z) − F− (z) ∼ i e z1/β .
(2.4)
In certain cases, e.g., when one has a Schwinger representation for the function [27,28], there is a natural and rigorous way to define the integral (2.3) on a contour which avoids the singularities, and which also allows for a physical interpretation of the nonperturbative contributions. So far our discussion has been rather general. However, it takes no effort to figure out the physical relevance of our discussion: divergent series are almost ubiquitous in physics and appear basically each time we approach an interesting problem in perturbation theory [22]. A typical and extensively studied case in quantum mechanics is the anharmonic oscillator (see, e.g., [22,29,30]). Herein, the ground state energy may be computed in perturbation theory—as a power series in the quartic coupling—and one finds that it is analytic in all the (coupling constant) complex plane except for a branch cut on the negative real axis, associated with the instability of the potential which becomes unbounded for negative values of the coupling. This instability is reflected by the fact that the series is, as expected, asymptotic. In particular, one can perform a Borel analysis as above and discover that the Borel transform of the ground state energy has singularities on the positive real axis, leading to an ambiguity of order ∼ i e−1/g , with g the quartic coupling constant. In this simple quantum mechanical example the nonperturbative ambiguity has a clear physical interpretation: it signals the presence—at negative g—of instantons mediating the decay from the unstable to the true vacuum, via tunneling under the local maximum of the potential. What these ideas illustrate is that by means of a purely perturbative analysis, i.e., finding the singularities of the Borel transform of the original perturbative series, it is possible to learn about nonperturbative effects—at least the intensity of the nonperturbative ambiguity (but we shall say more on this in the following). In some examples, it is possible to independently compute these nonperturbative terms directly, e.g., using WKB methods or computing the path integral around non-trivial (subdominant) saddle points [22]. In these examples one may then proceed in the opposite way from above and obtain information on the large-order behavior of the perturbative expansion out of the nonperturbative data. This is what we shall illustrate next. 2.1. From Instantons to Large-Order and Back In physical applications, the factor A appearing in (2.4) is the one-instanton action (see, e.g., [22]). Let us make this relation between instantons and the large-order behavior of perturbation theory a bit more precise, as it will play a crucial role in our later analysis. Consider a quantum system whose free
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energy is expressed as a perturbative expansion in g, the coupling constant,6 F (0) (g) =
+∞
(0)
fk g k .
(2.5)
k=0
The series (2.5) will generically be asymptotic, with zero radius of convergence. This is naturally associated to a branch cut of F (g) in the complex g-plane, located in the negative real axis and associated to instanton effects (just like in the anharmonic oscillator example above). The function F (g) is expected to be analytic otherwise. In fact this is saying that our quantum system should actually be thought of as anasymptotic formal power series in two expansion parameters, g and exp − g1 , see [14] for a discussion in the matrix model context. The appropriate expansion of the free energy is thus [14] F (g) =
+∞
C F () (g),
=0
F () (g) =
+∞ i − A () g e fk+1 g k . gb
(2.6)
k=0
Here, C is a parameter corresponding to the nonperturbative ambiguity. Also, () A is the one-instanton action, b a characteristic exponent and fk is the k-loop contribution around the -instanton configuration. Typically, the coefficients () fk are factorially divergent for any [22], in which case we may think about the ( + 1)-instanton sector as the nonperturbative contribution related to the asymptotic nature of the loop expansion around the -instanton sector. A standard procedure then relates the coefficients of the perturbative (0) expansion around the zero-instanton sector, fk , with the one-instanton free energy as follows. The discontinuity of the free energy across the branch cut (associated with the instability of the theory for negative g) is expressed, at first order, in terms of the leading instanton expansion (2.6) Disc F (g) ≡ lim F (g + i) − F (g − i) = 2i Im F (g) = F (1) (g) + · · · . (2.7) →0+
At the same time, we may use the Cauchy formula to write 1 F (g) = 2πi
0 dw −∞
Disc F (w) − w−g
dw F (w) . 2πi w − g
(2.8)
(∞)
In certain situations, e.g., in the aforementioned anharmonic oscillator example [29], it is possible to show by scaling arguments that the last integral in the expression above does not contribute. In such cases, (2.8) provides a remarkable connection between perturbative and nonperturbative expansions. Using the perturbative expansion (2.5) and the leading one-instanton contribution to the discontinuity Disc F (g) ∼ F (1) , one may obtain from the Cauchy formula 6
Here and below the index () labels the -instanton sector, so that (0) labels the perturbative expansion.
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(2.8) the following large order (or large k) relation (0) fk
+∞
= 0
+∞ dz F (1) (z) Γ (k + b) Γ (k + b − n) (1) n fn+1 A . ∼ 2πi z k+1 2πAk+b n=0 Γ (k + b)
(2.9)
This explicitly shows that the computation of the one-loop one-instanton partition function determines the leading order of the asymptotic expansion for the perturbative coefficients of the zero-instanton partition function. Higher loop corrections then yield the successive k1 corrections. Furthermore, instanton corrections with action A > A, where we have in mind multi-instanton corrections with action A, ≥ 2, will yield corrections to the asymptotics of (0) the fk coefficients which are exponentially suppressed in k. For the cases we shall consider in this work, namely matrix models and (0) string theory, one finds genus expansions as in (1.1), with Fg ∼ (2g)!, so that the relation (2.9) gets slightly re-written as follows (see, e.g., [13]). Begin with the free energy in the zero-instanton sector, gs2 F (0) (gs ). Setting z = gs2 , the one-instanton path integral then yields a series of the form zF (1) (z) =
i z
b 2
A −√ z
e
+∞
g
(1)
z 2 Fg+1 (t).
(2.10)
g=0
Following a procedure analogous to the one above, where one further assumes that the standard dispersion relation (2.8) still holds, it follows for the zeroinstanton sector perturbative coefficients +∞
Fg(0) (t)
= 0
+∞ dz zF (1) (z) Γ (2g + b) Γ (2g + b − h) (1) Fh+1 (t) Ah . ∼ 2πi z g+1 πA2g+b Γ (2g + b) h=0
(2.11) Again, the computation of the one-loop one-instanton free energy determines the leading order of the asymptotic expansion for the perturbative coefficients of the zero-instanton free energy. Higher loop corrections then yield the suc1 corrections. One should further notice that recently, in [13–15], the cessive 2g relation (2.11) has been tested in several models and rather conclusive numerical checks have confirmed its validity. 2.2. The Schwinger Effect and a Semiclassical Interpretation As discussed above, a nonperturbative ambiguity—typically associated to instantons, from a physical point of view—can arise when defining the integration contour for the inverse Borel transform. However, this is not always the case: we shall now review an example where a prescription to define the inverse Borel transform naturally arises, together with a physical interpretation for the nonperturbative contributions [28]. This is the Schwinger effect [27]. The one-loop effective Lagrangian describing a charged scalar particle, of charge e and mass m, in a constant electric field, E > 0, has an integral
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representation given by Schwinger [27] (see, e.g., [31] for a recent review) e2 E 2 L= 16π 2
+∞
ds s2
1 1 s − − sin s s 6
m2
e−s eE ,
(2.12)
0
which admits the weak coupling expansion 2n+4
+∞ B 2n+4 2eE m4 n L∼ (−1) . 16π 2 n=0 (2n + 4)(2n + 3)(2n + 2) m2
(2.13)
2n−1
In here we used the shorthand B 2n = 1−2 22n−1 B2n , with B2n the Bernoulli numbers. Since one may further relate the Bernoulli numbers to the Riemann zeta function via 2 ζ(2n) (2n)! (2.14) B2n = (−1)(n+1) (2π)2n it becomes evident that B2n diverges factorially fast. In this case, if one first writes the expansion (2.13) as L∼
+∞
a2n+4 x2n+4 ,
(2.15)
n=0
with x = 2eE m2 , it follows that at large n one has a2n+4 ∼ (2n + 1)! rendering this perturbative expansion asymptotic—and actually non-Borel summable as we shall see next. Indeed, computing the Borel transform it follows
+∞ 2 (ξ/2) a2n+4 m4 ξ/2 2n+4 ξ −1− , (2.16) B[L](ξ) = = (2n + 1)! 16π 2 sin (ξ/2) 6 n=0 from where one immediately notices that the Schwinger integral representation of the effective Lagrangian (2.12) is essentially the inverse Borel transform L(x) =
+∞
0
dt e2 E 2 −t B[L] (xt) e = t3 16π 2
+∞
ds s2
1 1 s − − sin s s 6
m2
e−s eE .
0
(2.17) Of course so far we still have a nonperturbative ambiguity to deal with: in order to perform the integration on the real axis one still needs to specify a prescription in order to avoid the poles at s = nπ, n ∈ N. This introduces the usual ambiguities leading to exponentially suppressed contributions to the effective Lagrangian. The novelty in this case is that there is now a natural way to address the integration avoiding the singularities in an unambiguous way [28]. As it turns out, the contour of integration needs to be deformed in such a way that the integral picks up the contributions of all the poles as if the real axis is approached from above, tantamount to a +i prescription; and this is the requirement which is dictated by unitarity [28]. As such, one has a physical principle behind the unambiguous choice of contour. Furthermore, the
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Lagrangian develops an imaginary part which is simple to compute by summing residues,7 and which cannot be seen to any finite order in perturbation theory,
2 +∞ (−1)n−1 πm2 1 eE exp −n Im L = , (2.18) 8π π n2 eE n=1 an expression with an evident multi-instanton flavor [32], as in (2.6). Besides the appropriate, physical prescription to perform the integration, and as such unambiguously compute the nonperturbative contributions to the Lagrangian, the Schwinger effect gives us something else: a physical interpretation of this imaginary part. Indeed, the imaginary part of the effective Lagrangian (2.18) is precisely the pair-production rate, or probability per unit volume for pair creation, for scalar electrodynamics in a constant electric field [27]. In other words, the above unitary +i prescription for the integration contour guarantees that this probability is a positive number between zero and one (which basically demands (2.18) to be real and positive). Another interesting illustration of the Schwinger effect, which will be of particular relevance in our subsequent discussion on topological strings and matrix models, is the case of a constant (Euclidean) self-dual electromagnetic background [33–35], satisfying 1 μνρλ F ρλ . (2.19) 2 Following [31], we introduce F 2 = 14 Fμν F μν and the natural dimensionless parameter γ = 2eF m2 . In this case, the one-loop effective Lagrangian describing a charged scalar particle is now given by Schwinger [27] and Dunne [31] Fμν = Fμν ≡
e2 F 2 L= 16π 2
+∞
0
ds s
1 1 1 2 − s2 + 3 sinh s
2s
e− γ ,
(2.20)
admitting the weak coupling expansion 2n+2
+∞ 2eF m4 B2n+2 . L∼− 16π 2 n=1 2n (2n + 2) m2
(2.21)
Notice that there are two possible self-dual backgrounds [33–35]: a magneticlike background with F real, in which case (2.21) has an alternating sign; and an electric-like background with F imaginary, in which case (2.21) is not alternating. If one carries through a Borel analysis similar to the previous one, where we studied the case of constant electric field, one may further notice that while the alternating (magnetic) series has Borel poles on the positive imaginary axis, the non-alternating (electric) series has the Borel poles on the positive real axis, making both situations rather distinct on what respects evaluating the inverse Borel transforms. For the non-alternating (electric) series it is also possible to see that the aforementioned unitarity prescription will pick 7
Notice that the residue at s = 0 precisely vanishes.
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a Borel contour leading to a nonperturbative imaginary contribution to the Lagrangian, unambiguously given by +∞
2πn γ em2 F 2π + Im L = (2.22) e− γ . 3 2 32π n=1 n n This expression can similarly be obtained by first considering the magnetic series, reflecting the integrand in (2.20) to the negative real axis in order to obtain an integral over the entire real line, and then deforming this integration contour such that it just incloses all the poles in the positive imaginary axis (a contour surrounding iR+ ). The resulting integral will then produce a sum over residues which, upon “Wick rotation” of the dimensionless coupling γ → i¯ γ , leads to the same expression as above, (2.22). As we shall see in the course of this work, this expression is also at the basis of the nonperturbative structure of topological strings and c = 1 matrix models. Another important feature of the Schwinger effect, that we shall further explore later on, is the fact that in the presence of a constant electric field the pair-production process can be given a semiclassical interpretation in terms of a tunneling process, where electrons of negative energy are extracted from the Dirac background by the application of the external field [36]. The motion under the potential barrier, classically forbidden, is considered for imaginary values of time, allowing for a computation of the tunneling probability corresponding to the pair-production rate as w ∼ e−2 Im S ,
(2.23)
where Im S is the imaginary part of the action developed during motion under the barrier. In here, a crucial point is that a particle in a sub-barrier trajectory satisfies the classical equations of motion. One may then use standard classical mechanics of a relativistic particle in order to describe this process. Indeed, energy conservation E = ± p2 + m2 − eEx, (2.24) together with the equation of motion ∂t p = eE, allow for an immediate re-writing of the action (for E = 0) as: p 2 m2 (2.25) log p + p2 + m2 . S(p) = p + m2 − 2eE 2eE Notice that, because of the logarithm, the action is a multi-valued function. The spectrum of possible values for the energy is displayed in Fig. 1. A potential barrier separates the lower continuum of negative-energy states (the minus sign of p2+ m2 ) from the upper continuum of positive-energy states (the plus sign of p2 + m2 ). Sub-barrier motion between points A1 and A2 will start at A1 , where t = 0 = p, and corresponds to the variation of the imaginary time/momentum along the path A1 BA2 , while the real part of the energy remains constant. Indeed, at the classical turning point B we will have t = im/eE and p = im, which corresponds to a square-root branch point of the function S(p). The motion ends back at t = 0 = p in point A2 , as shown
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Figure 1. On the left, the energy as a function of the position. The blue (orange) region corresponds to the lower (upper) continuum of particles with negative (positive) energy. The white region is classically forbidden. On the right, the variation of the imaginary momentum in the sub-barrier motion (color figure online) in Fig. 1. In this case, we see that the sub-barrier trajectory correspond to an increment of the imaginary part of the action as ΔA1 BA2 S = Im S,
(2.26)
which we may compute as the shift of the multi-valued function S(p) as we move in-between the sheets of the logarithm. In fact, it is rather simple to realize that the value of S(p) on a generic sheet differs from its value on the principal sheet, S ∗ (p), by πm2 , n ∈ Z. (2.27) eE In the illustration above we went (once) “half way” around the branch cut [−im, +im] in which case the shift in the action is given by S(p) = S ∗ (p) + in
πm2 . (2.28) 2eE For a generic sub-barrier motion, corresponding to a repeated wandering of the particle between the turning points A1 and A2 we may write πm2 , n ∈ Z, (2.29) 2 Im S = p 2 + m2 = n eE ΔA1 BA2 S =
γn
where γn is a contour encircling n-times the branch cut of the action. One may notice [36] that this result is in complete agreement with the Schwinger computation result (2.18). Naturally, this semiclassical argument may be refined in order to reproduce the pre-factors of the exponential term, and also so as to include the effect of a magnetic field. The point we wanted to make is that, from a semiclassical perspective, the instanton action describing the pair-production rate as a tunneling process may be computed via the branch cut discontinuities of the
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multi-valued function S(p). This is a technique that we shall deploy later on in order to provide for a semiclassical interpretation of nonperturbative effects in c = 1 matrix models in terms of eigenvalue tunneling. In this paper, we shall apply the techniques we have just described, Borel analysis, instanton calculus, and the Schwinger effect, in order to study the nonperturbative structure of topological strings and c = 1 matrix models. As such, we now turn to topological string theory with emphasis towards the integral representation of its free energy. 2.3. The Topological String Free Energy Asymptotic series and the Schwinger integral representation also appear in the context of topological string theory (see, e.g., [37] for an introduction). Let us start by describing the free energy of the A-model. The closed string sector of the A-model is a theory of maps φ : Σg → X from a genus-g Riemann surface, Σg , into a CY threefold X , which may be topologically classified by their homology class β = [φ∗ (Σg )] ∈ H2 (X , Z). One may expand dim H (X ,Z) ni [Si ] on a basis [Si ] of H2 (X , Z), with associated complexβ = i=1 2 ified K¨ ahler parameters ti . The topological string free energy has a standard genus expansion in powers of the string coupling gs , as in (1.1), which in the large-radius phase (i.e., for large values of the K¨ ahler parameters, in units of α ) becomes F (gs ; {ti }) =
+∞
gs2g−2 Fg (ti ),
Fg (ti ) =
g=0
Ng,β Qβ .
(2.30)
β≥0
Here, the sum over β is a sum over topological sectors or, equivalently, over world-sheet Qi = e−ti , with Qβ denot ni instantons. We have further introduced ing i Qi , and we have chosen units in which α = 2π. The coefficients Ng,β are the Gromov–Witten invariants of X , counting world-sheet instantons, i.e., the number of curves of genus g in the two-homology class β. The expansion in world-sheet instantons in (2.30), regarded as a power series in e−ti , generically has a finite convergence radius, tc , that can be estimated from the asymptotic large β behavior of Gromov–Witten invariants [38] Ng,β ∼ β (γ−2)(1−g)−1 eβtc ,
β → +∞.
(2.31)
In here, γ is a critical exponent. At the critical value of the K¨ ahler parameter, tc , the so-called conifold point, the geometric interpretation of the A-model large-radius phase breaks down and the topological string free energy undergoes a phase transition to a non-geometric phase, nonperturbative in α . One may characterize the theory by its critical behavior at the conifold point. In particular, one can consider the following double-scaling limit t → tc ,
gs → 0,
μ=
e−tc − e−t gs
fixed.
(2.32)
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In this case, the double-scaled free energy is universal, as first noticed in [39], and reads +∞ B2g 1 1 log μ + μ2−2g , (2.33) FDSL (μ) = Fc=1 (μ) = μ2 log μ − 2 12 2g(2g − 2) g=2 where Fc=1 (μ) is the all-genus free energy of the c = 1 string at the self-dual radius (for a review on these issues see, e.g., [40]). The critical behavior (2.33) has been checked in many examples, such as [41,42]. Furthermore, in [20], it has been shown that certain local CYs have a critical behavior which is in the universality class of 2d quantum gravity, i.e., they have γ = − 12 . Another feature to notice is that the above genus expansion depends on the alternating Bernoulli numbers and, thus, is alternating for real μ. Of particular interest to our present work is the fact that the free energy Fc=1 (μ) has a Schwinger-like nonperturbative integral formulation [25,43], given by +∞
1 ds 1 1 −sμ 1 + , (2.34) Fc=1 (μ) = − e 4 s sinh2 s s2 3 0
which coincides, after an appropriate identification of the parameters, with the one-loop effective Lagrangian for a charged particle in a constant selfdual background (2.20). This means that Fc=1 (μ) enjoys an asymptotic weak coupling expansion as in (2.21) and further develops a nonperturbative imaginary contribution akin to (2.22). In this line of thought, the exploration of Schwinger-like integral representations for the free energies of topological strings and c = 1 matrix models is one of the main topics in this paper. 2.4. A Schwinger–Gopakumar–Vafa Integral Representation As should be clear by now, Schwinger-like integral representations for the free energy are bound to play a critical role in our analysis. Happily, for topological string theory, such representations have been provided by Gopakumar and Vafa [23,24,44]. These works explored both the connection of topological strings to the physical IIA string, as well as the duality between type IIA compactified on a CY threefold, at strong coupling, and M-theory compactified on the same CY times a circle, in order to relate topological string amplitudes to the BPS structure of wrapped M2-branes and thus re-write the topological string free energy in terms of an integral representation. The final result in [44] for the all-genus topological string free energy, on a CY threefold X , is FX (gs ; {ti }) =
{di },r,m
+∞ i) n(d r
(X )
s 2r−2 ds 2πs 2 sin exp − (d · t + i m) . s 2 gs
0
(2.35) (d ) nr i
(X ) Let us explain the diverse quantities in this expression. The integers are the GV invariants of the threefold X . They depend on the K¨ ahler class di
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and on a spin label r. Later on we shall be focusing on the case where X is the (1) resolved conifold, for which there is only one non-vanishing integer, n0 = 1. The combination Z = d · t + i m represents the central charge of 4D BPS states obtained in the following fashion [44]. Start with M-theory compactified on X × S1 and consider the BPS spectrum of M2-branes wrapped on cycles dim H (X ,Z) of the CY threefold with fixed central charge A = d · t = i=1 2 di ti , ahler parameters. The mass of the with di as above and ti the complexified K¨ wrapped M2-branes is 2πA. Upon reduction on S1 each BPS state may have in addition an arbitrary (quantized) momentum m around the circle, leading to BPS states of central charge Z and mass 2πZ. Notice that these 4D BPS states contributing to the topological string free energy may be understood, from a IIA point of view, as bound states of D2 and D0-branes, and it is the physics of this system which can be related to a Schwinger-type computation and thus to the above integral representation [in fact, thanks to the N = 2 supersymmetry in the problem, the Schwinger calculation one has to perform in this context turns out to be equivalent to that of a vacuum amplitude for a charged scalar field in the presence of a self-dual electromagnetic field strength, as in (2.20)]. Furthermore, the integer m, associated to the winding around S1 , counts the number of D0 branes in the D2D0 BPS bound state. This should make (2.35) clear. One may also recover the perturbative genus expansion from this integral representation. Using the familiar identity
s s exp −2πim δ −n , (2.36) = gs gs m∈Z
n∈Z
with δ(x) the Dirac delta function, one may explicitly evaluate the sum over m in (2.35) and thus obtain, after the trivial integration over s, FX (gs ; {ti }) =
+∞ +∞
i) n(d r (X )
r=0 di =1
+∞ ngs 2r−2 −2πn d·t 1 2 sin e . n 2 n=1
(2.37)
This result expresses the topological string free energy, on a CY threefold X , in terms of the GV integer invariants [23,24,44]. To be completely precise, it is important to notice that in order to recover the full topological string free energy one still has to add to (2.37) the (alternating) constant map contribution [45,46] FK (gs ) =
+∞ g=0
gs2g−2 χK (X )
(−1)g |B2g B2g−2 | , 4g (2g − 2) (2g − 2)!
(2.38)
where χK (X ) = 2 h1,1 − h2,1 is the Euler characteristic of X . This term can also be given a Schwinger-like integral representation. From the point of view of the duality between type IIA and M-theory, this amounts to considering only the contribution arising from the D0-branes, or Kaluza–Klein modes.
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The result is [24] +∞ ds s 1 1 e−2πim gs FK (gs ) = χK (X ) 8 s sinh2 2s m∈Z 0
=
+∞
1 1 1 . χK (X ) 2 8 n sinh ng2 s n=1
(2.39)
In this paper we shall mainly consider the resolved conifold, a toric CY threefold for which dim H2 (X , Z) = 1 and thus the only non-vanishing integer (1) GV invariant is n0 = 1. In this case, the GV integral representation (2.35) immediately yields +∞ 2πs 1 ds 1 s e− gs (t+i m) , FX (gs ; t) = 2 4 s sin 2 m∈Z
(2.40)
0
an expression which carries a Schwinger flavor, as we have seen above. It is also important to point out that the case of r = 0 is the only one in which the integrand of the GV integral representation will have “interesting” poles, i.e., poles of the sine function on the real axis. When r > 0 the only poles of the integrand will be at zero and ∞ in the Borel complex plane. So, in particular, when studying more complicated CY threefolds where there is a sum over r ≥ 0, it will always be the contribution from GV invariants with r = 0 which will be the most relevant for the Schwinger analysis we shall carry through later in the paper and, as such, the case of the resolved conifold is a prototypical example for those situations. From the previous expression it is also simple to obtain the perturbative expansion, by summing over m as previously described, and one obtains +∞ 1 11 e−2πn t . FX (gs ; t) = 4 n=1 n sin2 ng2 s
(2.41)
By carrying through this sum, expanding in powers of gs , and adding the constant map contribution, one finally obtains the resolved conifold genus expansion as Fg (t) =
|B2g | (−1)g |B2g B2g−2 | + Li3−2g e−t . 2g (2g − 2) (2g − 2)! 2g (2g − 2)!
(2.42)
with Lip (x) the polylogarithm function. We shall later see how a Borel analysis allows for a nonperturbative completion of this expansion and moreover how to relate this nonperturbative completion to the large-order behavior of the above genus expansion. One final word pertains to the matrix model description of strings on the resolved conifold via a large N duality. It was shown in [47] that there is a duality between closed and open topological A-model string theory on, respectively, the resolved and the deformed conifold; two smooth manifolds related
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to the same singular geometry. In the resolved conifold case the conifold singularity is removed by blowing up a two-sphere around the singularity; while in the deformed conifold case the conifold singularity is removed by growing a three-sphere around it, which is also a Lagrangian sub-manifold thus providing boundary conditions for open strings. As it turns out, the full open topological string field theory in this latter background, T ∗ S3 , where we wrap N D-branes on the Lagrangian sub-manifold base, S3 , reduces to SU(N ) Chern–Simons gauge theory on S3 [48], whose partition function further admits a matrix model description [8]. The matrix model in question, which we shall review in the next section, has a potential with a single minimum and no local maxima. In this paper we refer to this type of matrix models (which will also include the Gaussian and Penner cases) as c = 1 matrix models since, as we shall see, they all admit a very natural double-scaling limit to the c = 1 string at self-dual radius. Notice that c = 1 matrix models do not belong to the class of matrix models for which the off-critical instanton analysis has been carried out so far. Because understanding nonperturbative corrections to the topological string free energy on the resolved conifold is undissociated from understanding nonperturbative corrections to c = 1 matrix models, we shall consider this latter case more broadly in order to shed full light on this class of instanton phenomena. As such, c = 1 matrix models is the subject we shall turn to next.
3. c = 1 Matrix Models and Topological String Theory We shall now introduce three distinct matrix models, all in the universality class of the c = 1 string, and which will be the main focus of our subsequent discussion. As mentioned in the previous section, one of these models is the one describing Chern–Simons gauge theory on S3 , known as the Stieltjes– Wigert matrix model. Another interesting, and rather elementary, matrix model is the Gaussian model. Yet, we shall find that it already displays many features that will also appear for the resolved conifold. Finally, we also address the Penner matrix model, first introduced to study the orbifold Euler characteristic of the moduli space of punctured Riemann surfaces. These three models have been extensively studied in the literature and in the present section we will mostly gather some general facts necessary to obtain their topological large N expansions and their holomorphic effective potentials. Then, in the following section, we shall analyze their large N asymptotic expansions from the point of view of Borel analysis. Let us begin by recalling some basic notions about matrix models (see, e.g., [1,18,49,50]). The hermitian N × N one-matrix model partition function is 1 1 (3.1) dM e− gs Tr V (M ) , Z= vol (U(N )) with vol (U(N )) the usual volume factor of the gauge group. In the eigenvalue diagonal gauge this becomes
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N
N 1 dλi 1 Z= (3.2) Δ2 (λ) e− gs i=1 V (λi ) , N ! i=1 2π where Δ(λ) = i<j (λi − λj ) is the Vandermonde determinant. The free energy of the matrix model is then defined as usual F = log Z and, in the large N limit, it has a perturbative genus expansion F =
+∞
gs2g−2 Fg (t),
(3.3)
g=0
with t = N gs the ’t Hooft coupling. Multi-trace correlation functions in the matrix model may be obtained from their generating functions, the connected correlation functions defined by 1 1 · · · Tr Wh (z1 , . . . , zh ) = Tr z1 − M zh − M (c) =
+∞
gs2g+h−2 Wg,h (z1 , . . . , zh ; t).
(3.4)
g=0
In particular, the generator of single-trace correlation functions is W1 (z) = N ω(z) where ω(z) is the resolvent, i.e., the Hilbert transform of the eigenvalue density ρ(λ) characterizing the saddle-point associated with the matrix model large N limit. In the most general case, this saddle-point is such that ρ(λ) has support C, with C a multi-cut region given by an union of s intervals Ci . At large N , the eigenvalues condense on these intervals Ci in the complex plane and one may interpret them geometrically as branch cuts of a spectral curve which, in the hermitian one-matrix model, would be a hyperelliptic Riemann surface corresponding to a double-sheet covering of the complex plane C, with the two sheets sewed together by the cuts Ci . The spectral curve, to be denoted by y(z), may be written in terms of the genus zero resolvent which, for a generic one-cut solution with C = [a, b], is given by the ans¨ atz dw V (w) (z − a)(z − b) 1 , (3.5) ω0 (z) = 2t 2πi z − w (w − a)(w − b) C
where one still has to impose that ω0 (z) ∼ z1 as z → +∞, in order to fix the position of the cut endpoints8 . The spectral curve is then defined as y(z) = V (z) − 2t ω0 (z) ≡ M (z) (z − a)(z − b). (3.6) For future reference, it is also useful to define the holomorphic effective potential, defined as the line integral of the one-form y(z) dz along the spectral 8
This boundary condition states that the eigenvalue density is normalized to one in the cut, C dλ ρ(λ) = 1.
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curve, λ Vh;eff (λ) =
dz y(z),
(3.7)
a
which appears at leading order in the large N expansion of the matrix integral as
N N 1 dλi exp − Vh;eff (λi ) + · · · . (3.8) Z∼ gs i=1 i=1 Because the real part of the spectral curve relates to the force exerted on a given eigenvalue, it turns out that the effective potential Veff (z) = Re Vh;eff (z) is constant inside the cut C, i.e., inside the cut the eigenvalues are free. The imaginary part of the spectral curve, on the other hand, relates to the eigenvalue density as Im y(z) = 2πt ρ(z), thus implying that the imaginary part of Vh;eff (z) is zero outside the cut and monotonic inside. These two conditions guarantee that the eigenvalue density is real with support on C. Furthermore, as should be clear from the expression above, if the matrix integral Z is to be convergent a careful choice of integration contour for the eigenvalues has to be made based also on the properties of the holomorphic effective potential [2,3]. In particular, this contour may be analytically continued to any contour which includes the cut C and does not cross any region where Veff (z) = Re Vh;eff (z) < 0, thus guaranteeing global stability of the saddle-point configuration and convergence of the matrix integral (as Re Vh;eff (λ) → +∞ at the endpoints of the integration contour). These properties of Vh;eff (z) ensure that, in the large N limit, the matrix integral can be evaluated with the steepest-descendant method [2,3]. There are many ways to solve matrix models. In particular, [51] proposed a recursive method for computing the connected correlation functions (3.4) and the genus-g free energies, Fg (t), entirely in terms of the spectral curve. This recursive method, sometimes denoted by the topological recursion, appears to be extremely general and applies beyond the context of matrix models; see [52] for a review. For our purposes of computing the genus expansion of the free energy one of the most efficient and simple methods is that of orthogonal polynomials [18], which we now briefly introduce. If one regards 1
dμ(z) ≡ e− gs V (z)
dz 2π
(3.9)
as a positive-definite measure in R, it is immediate to introduce orthogonal polynomials, {pn (z)}, with respect to this measure as dμ(z) pn (z)pm (z) = hn δnm , n ≥ 0, (3.10) R
where one further normalizes pn (z) such that pn (z) = z n + · · ·. Further noticing that the Vandermonde determinant is Δ(λ) = det pj−1 (λi ), the one-matrix
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model partition function may be computed as ZN =
N −1
hn = hN 0
n=0
N
rnN −n ,
(3.11)
n=1
n where we have defined rn = hhn−1 for n ≥ 1. These coefficients also appear in the recursion relations of the orthogonal polynomials,
pn+1 (z) = (z + sn ) pn (z) − rn pn−1 (z).
(3.12)
In the large N limit the recursion coefficients approach a continuous function n ∈ [0, 1], and one may proceed to compute the genus rn → R(x), with x = N expansion of log Z by making use of the Euler–MacLaurin formula; see [18,50] for details. 3.1. The Gaussian Matrix Model Let us first focus on the Gaussian matrix model, defined by the potential VG (z) = 12 z 2 . This case is rather simple as the matrix integral can be straightforwardly evaluated via gaussian integration, and the volume of the compact unitary group follows by a theorem of Macdonald [53] as 1
N (N +1)
(2π) 2 , vol (U(N )) = G2 (N + 1)
(3.13)
where G2 (z) is the Barnes function, G2 (z + 1) = Γ(z)G2 (z). The Gaussian partition function thus reads N2
ZG =
gs 2
(3.14) N G2 (N + 1). (2π) 2 The same result can be obtained with orthogonal polynomials. With respect to 2 the Gaussian measure dμ(x) = e−x dx one finds Hermite polynomials, Hn (x), and for the Gaussian matrix model it follows
g n2 z gs s G n , (3.15) pn (z) = Hn √ , hn = gs n! 2 2π 2gs indeed reproducing the expected result for the partition function as Z=
N −1 n=0
N2
hG n
=
gs 2
N −1
N
(2π) 2
n=0
n! = ZG ,
(3.16)
N −1 where we have also used that G2 (N + 1) = n=0 n!. The asymptotic genus expansion of the Gaussian free energy FG = log ZG simply follows from the asymptotic expansion of the logarithm of the Barnes function and one obtains
1 2 3 G F0 (t) = t log t − , (3.17) 2 2 1 (3.18) F1G (t) = − log t + ζ (−1), 12 B2g FgG (t) = t2−2g , g ≥ 2, (3.19) 2g(2g − 2)
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Figure 2. The Gaussian algebraic curve for values of t = −0.5, −0.1, 0, +0.1, +0.5, from left to right, respectively. Notice that the algebraic curve is singular for t = 0 where ζ(z) is the Riemann zeta function. One immediately notices that all free energies with g ≥ 1 diverge when t → 0. It is then quite obvious to consider the double-scaling limit, approaching the critical point tc = 0, as t → 0,
gs → 0,
μ=
t − tc gs
fixed,
(3.20)
in order to obtain the c = 1 string at self-dual radius behavior gs2g−2 FgG (t) →
B2g μ2−2g , 2g(2g − 2)
g ≥ 2.
(3.21)
Finally, it is very simple to compute the one-form on the spectral curve of the Gaussian model y(z) dz = z 2 − 4t dz, (3.22) as well as the holomorphic effective potential
√ z 2 − 4t √ , (3.23) 2 t √ G where we have normalized the result such that Vh;eff (b = 2 t) = 0. In Figs. 2 and 3 we plot the Gaussian algebraic curve for different values of t, as well as the real value of the holomorphic effective potential in the complex plane. We notice that, with an appropriate identification of parameters, the Gaussian holomorphic effective potential coincides with the action associated to the semiclassical Schwinger effect (2.25). In the following, we shall further comment about this interesting coincidence. G Vh;eff (z)
1 = z z 2 − 4t − 2t log 2
z+
3.2. The Penner Matrix Model The second example we wish to address is the Penner matrix model [54]. First introduced to study the orbifold Euler characteristic of the moduli space of Riemann surfaces at genus g, with n punctures, it turns out that in the double-scaling limit this model is actually related to the usual c = 1 noncritical string theory, its free energy being a Legendre transform of the free energy of the c = 1 string compactified at self-dual radius [55,56]. The Penner matrix model is defined by the potential VP (z) = z − log z and one may simply compute its partition function again using orthogonal polynomials. Indeed, one
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Figure 3. The real part of the Gaussian holomorphic effective potential in the complex z-plane, for t = 0.2. In G (z) < 0, in orange the region blue is the region Re Vh;eff G Re Vh;eff (z) > 0, and the black lines correspond to the Stokes G (z) = 0 (which also include the cut of the speclines Re Vh;eff tral curve). The white cut corresponds to the logarithmic branch cut (color figure online)
may write the Penner measure as 1
z
dμ(z) = z gs e− gs
dz , 2π
(3.24)
which is, up to normalization, the measure for the generalized, or associated, (α) 1 x −α dn Laguerre polynomials Ln (x) = n! e x dxn (e−x xn+α ). It thus follows for the Penner matrix model
z 1 2n+1+ g1s 1 P s) g = n! Γ n + + 1 . , h pn (z) = (−1)n gsn n! L(1/g s n n gs 2π gs (3.25) This immediately leads to the calculation of the partition function in this model as 1 N (N + g1s ) G (N + 1) G N −1 + 1 N + 2 2 gs gs ZP = hP , (3.26) n = N 1 (2π) G +1 n=0 2
gs
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where we made use of the Barnes function, satisfying N −1
Γ (n + α + 1) =
n=0
G2 (N + α + 1) . G2 (α + 1)
The normalized Penner free energy FP = FP − FG = log FP =
(3.27) ZP ZG
is given by
1 2 N 1 N log gs + log gs − N log 2π 2 gs 2
1 1 + log G2 N + + 1 − log G2 +1 , gs gs
(3.28)
and it admits the following genus expansion, obtained from the asymptotic expansion of the logarithm of the Barnes functions,
1 3 3 2 F0P (t) = (t + 1) log (t + 1) − (3.29) + , 2 2 4 1 (3.30) F1P (t) = − log (t + 1) , 12 B2g 2−2g FgP (t) = (t + 1) − 1 , g ≥ 2. (3.31) 2g(2g − 2) One immediately notices that all free energies with g ≥ 1 diverge when t → −1. It is then quite obvious to consider the double-scaling limit, approaching the critical point tc = −1, as t → −1,
gs → 0,
μ=
t − tc gs
fixed,
(3.32)
in order to obtain the c = 1 string at self-dual radius [55] gs2g−2 FgP (t) →
B2g μ2−2g , 2g(2g − 2)
g ≥ 2.
(3.33)
Next, let us address the large N expansion of the Penner matrix model by making use of saddle-point techniques [57,58]. This time around, the ans¨ atz for the large N , genus zero resolvent is [57]
1 1 ω0 (z) = (z − a)(z − b) , (3.34) V (z) − √ 2t z ab so that its large z asymptotics, ω0 (z) ∼ z1 + · · · as z → ∞, immediately determine the endpoints of the cut C = [a, b] to be (3.35) a = 1 + 2t − 2 t(t + 1), b = 1 + 2t + 2 t(t + 1). (3.36) It is now simple to obtain the one-form on the spectral curve of the Penner model 1 2 y(z) dz = z − 2 (2t + 1) z + 1 dz (3.37) z
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-2
-2
-2
-2
-2
-4
-4 -4 -2
0
2
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-4 -4 -2
0
2
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-4 -4 -2
0
2
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-4 -4 -2
0
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-4 -2
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Figure 4. The Penner algebraic curve for values of t = −0.2, −0.01, 0, +0.01, +0.2, from left to right, respectively. Notice that the algebraic curve is singular for t = 0 as well as the holomorphic effective potential P (z) Vh;eff = z 2 − 2 (2t + 1) z + 1 + log z − log 1 − (2t + 1) z + z 2 − 2 (2t + 1) z + 1 − (2t + 1) log z − (2t + 1) + z 2 − 2 (2t + 1) z + 1
+ (t + 1) log (t(t + 1)) + (t + 1) log 4 + iπ,
(3.38)
P (b) Vh;eff
= 0. In Figs. 4 and where we have normalized the result such that 5, we plot the Penner algebraic curve for different values of t, as well as the real value of the holomorphic effective potential in the complex plane. The structure of Stokes lines for this potential is now more complicated (see, e.g., [57,58]) than in the familiar polynomial cases (see, e.g., [2,3]). 3.3. The Chern–Simons Matrix Model We now turn to the Chern–Simons, or Stieltjes–Wigert, matrix model. As we previously stated this model is particularly interesting for its relation, via a large N duality, to topological string theory on the resolved conifold [47]. The SU(N ) Chern–Simons gauge theory on a generic three-manifold has been realized as a matrix model in [8]; see [59] for a review.. Here, we shall focus on the resolved conifold case, where the partition function of SU(N ) Chern– Simons gauge theory on S3 is, up to a factor, given by the Stieltjes–Wigert 2 matrix model [60] defined by the potential VSW (z) = 12 (log z) . To be precise, the Chern–Simons partition function relates tothe Stieltjes–Wigert partition 2 t 7N − 1 ZSW , so that function by the simple expression ZCS = exp − 12 the corresponding free energies equate as 7 t3 t + FSW . + (3.39) 2 12 gs 12 For a review of the main features of this matrix model, including saddle-point methods and orthogonal polynomial analysis, we refer the reader to, e.g., [50]. Let us start by computing the partition function ZCS using orthogonal polynomials—as we shall see one may regard the Stieltjes–Wigert matrix model as a q-deformation, in the quantum group sense, of the Gaussian matrix 2 1 dz is well-known in the model. The logarithmic measure dμ(z) = e− 2gs (log z) 2π FCS = −
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2
1
0
1
2 1
0
1
2
3
Figure 5. The real part of the Penner holomorphic effective potential in the complex z-plane, for t = 0.1. In P (z) < 0, in yellow the region blue is the region Re Vh;eff P Re Vh;eff (z) > 0, and the black lines correspond to the Stokes P (z) = 0 (which also include the cut of the speclines Re Vh;eff tral curve). The white cuts corresponds to the logarithmic branch cuts (color figure online) literature precisely because it leads to so-called Stieltjes–Wigert orthogonal polynomials, n k k(k−n) 2 2 n 1 n pn (z) = (−1)n q n + 2 q 2 −k −q − 2 z , k q k=0 (3.40) 7 1 gs SW n(n+1)+ 2 [n] ! hn = q 4 , q 2π where we have introduced gs
q=e ,
n 2
[n]q = q − q
−n 2
,
[n]q ! n . = m q [m]q ! [n − m]q !
(3.41)
With this information at hand, one may now explicitly compute the Stieltjes– Wigert partition function from definition ZSW =
N −1 n=0
hSW = n
g N2 s
2π
N
q 12 (7N
2
−1)
N −1 n=0
[n]q !.
(3.42)
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A simple glance at (3.16) immediately shows that, up to normalization, one may indeed regard the Stieltjes–Wigert matrix model as a q-deformation of the Gaussian matrix model, at least at the level of the partition functions. One may further define the q-deformed, or quantum Barnes function as Gq (N + 1) =
N −1
[n]q !,
(3.43)
n=0
gs N2 so that the Stieltjes–Wigert partition function is simply ZSW = 2π 2 N q 12 (7N −1) Gq (N + 1). These expressions may then be used to address the large N topological expansion of the Stieltjes–Wigert matrix model. Standard use of orthogonal polynomial techniques [18], as described, e.g., in [50], yield9 π2 t t3 − − Li3 e−t + ζ(3), (3.44) F0CS (t) = 12 6 1 t Li1 e−t + ζ (−1), F1CS (t) = − + (3.45) 24 12 B2g B2g B2g−2 FgCS (t) = + Li3−2g e−t , g ≥ 2, 2g (2g − 2) (2g − 2)! 2g (2g − 2)! (3.46) where Lip (z) is the polylogarithm of index p, Lip (z) =
+∞ n z . p n n=1
(3.47)
At genus g ≥ 2 of course the topological expansions of Chern–Simons and Stieltjes–Wigert perturbative free energies coincide, FgCS (t) = FgSW (t). Furgs , the free energies FgCS (t) thermore, after analytical continuation gs → i¯ coincide with the free energies of topological strings on the resolved conifold (2.42), once one identifies ’t Hooft coupling and K¨ ahler parameter. Finally, notice that all free energies with g ≥ 1 diverge when t → 0 which corresponds to e−t → 1; with this second variable the natural one as the divergences are associated to the singular point of the (negative index) polylogarithm, Li−p (1). It is then quite natural to consider the double-scaling limit, approaching the critical point e−tc = 1, as e−tc − e−t fixed, (3.48) gs in which case one again obtains the c = 1 string at self-dual radius B2g gs2g−2 FgCS (t) → μ2−2g , g ≥ 2. (3.49) 2g(2g − 2) Finally, we address the spectral curve and holomorphic effective potential for the Stieltjes–Wigert matrix model by making use of saddle-point tech2 (z) = z1 log z one must be niques. With potential VSW (z) = 12 (log z) and VSW e−t → 1,
9
gs → 0,
μ=
Notice that, unlike in the previous example of the Penner model, in here we have not normalized the Chern–Simons free energy by the Gaussian free energy.
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a bit careful in applying (3.5) to compute the resolvent: indeed, the deformation of the contour around the cut, C = [a, b], must now be done differently due to the logarithmic branch-cut. Instead of capturing the pole at z and the pole at ∞, this time around one captures the pole at z and the branch cut along the negative real axis (zero included); we refer the reader to [50] for further details. The endpoints of the cut are 3t √ (3.50) a, b = 2e2t − et ± 2e 2 et − 1, while the one-form on the spectral curve reads 2 −t z + (1 + e−t z) − 4z 1 + e 2 √ dz, y(z) dz = log z 2 z
(3.51)
which coincides with the one-form log Y (Z) dZ on the mirror curve Z H(Z, Y ) = 0 of the resolved conifold, written in terms of the C∗ variables Z = ez and Y = ey . One further computes the holomorphic effective potential as CS Vh;eff (z) 1 = − log2 z+log2 ξ −2 log ξ log(1 − e−t ξ) + Li2 (1 − ξ) + Li2 (e−t ξ) − V0 2
1 = − log2 z + log2 ξ − 2 log ξ log(1 − e−t ξ) + Li2 (e−t ξ) − Li2 (ξ) 2 π2 − log (1 − ξ) log ξ + (3.52) − V0 , 6
where equality holds due to the Euler’s reflection formula for dilogarithms Li2 (ξ) + Li2 (1 − ξ) = In here we have set ξ(z) =
1 + e−t z +
π2 − log (1 − ξ) log ξ. 6
(3.53)
2
(1 + e−t z) − 4z
(3.54)
2
to simplify notation10 and we have defined 3t √ t √ 1 V0 = − log2 2 e2t − et + 2e 2 et − 1 + log2 et + e 2 et − 1 2 t√ t√ t√ −2 log et + e 2 et − 1 log −e− 2 et − 1 −2Li2 1−et − e 2 et − 1 t√ (3.55) −2Li2 1 + e− 2 et − 1 , CS to ensure that the result is normalized such that Vh;eff (b) = 0. In Figs. 6 and 7, we plot the Stieltjes–Wigert algebraic curve for different values of t, as well as the real value of the holomorphic effective potential in the complex 10
Using this variable, the spectral curve is also compactly re-written as y(z) =
2 z
log
√ξ . z
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Figure 6. The Stieltjes–Wigert algebraic curve for values of t = −0.1, −0.01, 0, +0.01, +0.05, from left to right, respectively. Notice that the algebraic curve is singular for t = 0 1.5
1.0
0.5
0.0
0.5
1.0
1.5 0
1
2
3
Figure 7. The real part of the Stieltjes–Wigert holomorphic effective potential in the complex z-plane, for t = 0.1. The CS (z) = 0 black lines correspond to the Stokes lines Re Vh;eff (which also include the cut of the spectral curve). The white cuts and regions correspond to the logarithmic and dilogarithmic branch cuts. Because of the choice of principal sheets in Mathematica the colored regions are now not so clear. Akin to the Penner model, the region in yellow, to the right of the vertical black lines, and the region in blue, inside the “closed CS (z) > 0, in the principal sheet. The rest bubble”, have Re Vh;eff CS is Re Vh;eff (z) < 0 (color figure online) plane. Given that Li2 (z) is the standard dilogarithm function, with its intricate branch structure, it is not too hard to realize that the structure of Stokes lines of the present effective potential is now much more complicated than usual.
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3.4. Double-Scaling Limit and c = 1 Behavior We have just seen that the Gaussian, Penner and Chern–Simons free energies admit rather simple double-scaling limits to the c = 1 string at self-dual radius. In the Chern–Simons case, this relates to our earlier discussion in Sect. 2.3, where we pointed out that, at the conifold point of moduli space, the A-model may still be characterized by its critical behavior in the double-scaling limit (2.32). Free energies of the topological string reduce, in this situation, to free energies of the c = 1 string. Let us now briefly discuss, in the example of the Chern–Simons matrix model, how one may also study the destiny of the open string sector, i.e., of the matrix model correlators Wh (p1 , . . . , ph ) introduced in (3.4), in this c = 1 double-scaling limit. Introducing a parameter ζ as e−t ≡ 1 − ζ,
(3.56)
the conifold point of the Chern–Simons model is thus located at ζ = 0. The expansion of the branch points of the spectral curve (3.51) near the conifold point yields 1
a, b = 1 ± 2ζ 2 + · · · ,
(3.57)
in which case it is natural to scale also the z variable in the spectral curve as 1
z = 1+ζ2 s
(3.58)
in order to appropriately zoom into the critical region. In here, s is the doublescaled open coordinate. In these variables the Chern–Simons one-form y(z) dz, at criticality, scales to y(z) dz → y(s) ds = ζ s2 − 4 ds (3.59) which one immediately recognizes as the one-form of the Gaussian matrix model. The interesting point is that, in the same variables, also the two-point correlator W0 (p, q) reduces to the Gaussian one
1 1 st − 4 W0,2 (p, q) dp dq → − 1 ds dt. (3.60) 2 (s − t)2 (s2 − 4) (t2 − 4) In fact, there is a property of the topological recursion, proved in [51], which states that one may either first compute matrix model amplitudes and then take their double-scaling limits, or else recursively compute amplitudes directly from the double-scaled curve, the result being the same (i.e., the operations commute). As such, in the c = 1 double-scaling limit Chern–Simons open correlators will all reduce to Gaussian open correlators CS G (z1 , . . . , zh ) dz1 · · · dzh → ζ 2−2g−h Wg,h (s1 , . . . , sh ) ds1 · · · dsh . Wg,h
(3.61)
Now recall that open topological string amplitudes may be computed from the matrix model correlators Wg,h (z1 , . . . , zh ) as [10,11,61,62] (g) Ah (p1 , . . . , ph )
p1 =
ph ...
dz1 . . . dzh Wg,h (z1 , . . . , zh ),
(3.62)
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where the {pi } are the open string parameters which parametrize the moduli space of the brane. As such (3.61) shows how, near the conifold point, open amplitudes of topological strings on the resolved conifold reduce to Gaussian amplitudes. This is actually generic for topological string theory near the conifold point [12]. We shall now relate these Gaussian amplitudes with open amplitudes in the dual c = 1 model. We first need to recall some results in non-critical strings, holographically duals to matrix models. We are interested in minimal models obtained by coupling 2D gravity to minimal (p, q) matter models, with central charge cp,q = 1 − 6(p − q)2 /pq. The coupling to gravity leads to the appearance of the Liouville field, φ, with world-sheet action 2 d z√ 2 (3.63) g ∂φ + QRφ + 4πμL e2bφ , SL = 4π where μL is the bulk cosmological constant. The central charge of the Liouville sector is cL = 1 + 6Q2 , and the parameter b above relates to the background charge Q as Q = b + 1/b. The bosonic string requirement that the total central charge of Liouville theory plus minimal matter equals c = 26 eventually fixes b = pq . There are two distinct types of boundary conditions in Liouville theory [63,64]. There is a one-parameter family of Neumann boundary conditions, the so-called FZZT branes, parameterized by the boundary cosmological constant μB , usually expressed in terms of a parameter s as μL μB = cosh (πbs) . (3.64) sin (πb2 ) Besides FZZT branes, there are also ZZ branes, associated to Dirichlet boundary conditions. These correspond to a two-parameter family, parameterized by the pair of integers (m, n), and are localized at φ = ∞. At the quantum level FZZT and ZZ boundary conditions, or, respectively, the Bs | and B(m,n) | boundary states, are related as [65–67] m B(m,n) | = Bs(m,n) | − Bs(m,−n) |, with s(m, n) = i + bn . (3.65) b Both types of branes have been given a geometrical interpretation in terms of a complex curve, in [67]. This is accomplished by introducing the variables πs ∂ . (3.66) x = μB ∼ cosh (πbs) , y = Z FZZT ∼ sinh ∂μB b Considered as complex variables, the coordinates {x, y} define an algebraic curve F (x, y) = 0 embedded into C2 , which is identified with the spectral curve of the dual matrix model (i.e., a double-scaled hermitian one-matrix model) [67]. The FZZT brane disk partition function may be equivalently written as the line integral of the one-form y dx as μB FZZT Z (μB ) = dx y. (3.67)
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Analogously, the h-point matrix model correlators (3.4) are identified with open amplitudes with FZZT boundary conditions, e.g., the two-point function W0,2 (p, q) is identified as the annulus amplitude for FZZT branes and so on. The ZZ brane disk partition function is instead defined as the line integral of y dx over a closed contour ZZ dx y, (3.68) Z(m,n) = γm,n
where γm,n is a non-contractible contour conjugate to a “pinched cycle”, starting and ending at the singular point x(m,n) = x (s(m, n)) and y(m,n) = y (s(m, n)) [67]. The case of c = 1 is a bit more subtle since one has to consider the singular b → 1 limit. It is first necessary to introduce the renormalized couplings (3.69) μc=1 = lim π 1 − b2 μL , μB,c=1 = lim π 1 − b2 μB . b→1
b→1
Furthermore, an appropriate subtraction is required in order to define the FZZT disk partition function. This may be expressed in terms of the one-form w(s) dμB,c=1 (s) with
∂μB Z FZZT 4 w(s) ≡ lim + Z μ (3.70) D B , b→1 π (1 − b2 ) π where ZD is the disk partition function in the c = 1 CFT. The relevant c = 1 curve then reads √ x(s) = μB,c=1 (s) = μc=1 cosh (πs) , (3.71) √ y(s) = w(s) = −D μc=1 πs sinh (πs) , where D is some constant. The identification of the curve (3.71), arising from CFT considerations, with the curve of the dual matrix model is another delicate point. Here, the relevant matrix model is a double-scaled version of Matrix Quantum Mechanics (MQM) with a Sine–Liouville perturbation. It is know for quite some time [49] that the singlet sector of this MQM can be reduced to a system of free fermions, in an inverted harmonic oscillator. In the semiclassical limit, the ground state of this system is completely determined by the shape of the Fermi sea, which can be parameterized in terms of an uniformization parameter τ as [40] (3.72) x(τ ) = 2μ cosh(τ ), y(τ ) = 2μ sinh(τ ), where μ denotes the Fermi level. In analogy with the c < 1 case, one would like to identify the above MQM curve with the CFT curve (3.71). However, these two curves are clearly distinct. A solution to this puzzled has been offered in [68–70], where it was proposed that one should instead identify w(s) with the resolvent, rather than directly with the spectral curve of the dual matrix model. The spectral curve can then be extracted, following a very standard matrix model procedure, from the discontinuity of w(s), 1 √ (w(s + i) − w(s − i)) = −D μc=1 sinh (πs) . (3.73) ρ(s) ≡ − 2πi
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Clearly the new CFT curve, defined as x(s) = μB,c=1 (s),
y(s) = ρ(s),
(3.74)
agrees with the one-matrix model spectral curve (3.72) after an appropriate identification of parameters. The FZZT brane partition function can then be obtained from the line integral of the one-form y(τ ) ∂τ x(τ ) dτ , while the ZZ brane partition function can be defined by the following closed integral on the MQM curve [69] ZZ Z(n,1)
=i γn
iπn dx y = i
dτ ∂τ x(τ ) y(τ ) = 2πnμ,
n ∈ Z,
(3.75)
−iπn
corresponding to a (n, 1) ZZ brane partition function. Indeed, it has been shown that only an one-parameter set of the c = 1 ZZ branes may be identified in the dual MQM [71]. Let us further notice that the above matrix quantum mechanics spectral curve (3.72), with the uniformization parameter τ , is just an infinite covering of the hyperboloid x2 − y 2 = 2μ,
(3.76)
which is precisely the spectral curve of the Gaussian matrix model. In particular, this explains how open matrix model correlators of the Gaussian model, G , get identified with D-brane amplitudes with FZZT boundary conditions Wg,h in the c = 1 model at self-dual radius. Indeed, in [72] it was checked that the double-scaled Gaussian correlators are related to macroscopic loop operators in the c = 1 theory. Finally, the limit (3.61) shows that topological string amplitudes with toric-brane boundary conditions reduce to c = 1 amplitudes for FZZT branes. Hence, and as already pointed out in a related context in [10], toric branes reduce to FZZT branes in the double-scaling limit, at the conifold point.
4. Nonperturbative Effects, Large Order and the Borel Transform We may now turn to the study of the asymptotic perturbative expansions for the free energies of the matrix models and topological strings we are interested in. In particular, we shall perform a detailed Borel analysis of each case, and thus understand what type of nonperturbative effects control the large-order behavior of the distinct perturbative expansions. 4.1. The Gaussian Matrix Model and c = 1 Strings Let us begin with the Gaussian matrix model. The genus expansion of its free energy (3.19), is clearly an asymptotic expansion with FgG ∼ (2g − 3)!, given the growth of Bernoulli numbers as B2g ∼ (2g)!. Recalling our discussion in
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Sect. 2, we may then consider the Borel transform of the divergent Bernoulli sum, i.e., restricting to genus g ≥ 2, and obtain B[FG ](ξ) =
+∞ FgG (t) 2g−2 t2 1 1 1 . ξ =− + 2 − 2 ξ (2g − 3)! 12 ξ 4 sinh 2t g=2
(4.1)
This function has no poles on the positive real axis, for real argument (the genus expansion (3.19) is an alternating series). As such, one can define its inverse Borel transform11 1 FG (gs ) = − 4
+∞
ds s
0
1 − sinh2 g2ts s
2t gs
2
1 1 + s2 3
e−s ,
(4.2)
providing a nonperturbative completion for the asymptotic expansion of the free energy in the Gaussian matrix model. It is quite interesting to notice that, upon the trivial change of variables s → σ = g2ts s, this expression precisely coincides with the one-loop effective Lagrangian for a charged scalar particle in a constant self-dual electromagnetic field (of magnetic type) introduced in Sect. 2.2. Comparing with (2.20) we see that in here γ=
1 2eF = . 2 m N
(4.3)
If one instead considers imaginary string coupling, g¯s = igs , the asymptotic expansion (4.2) will coincide with the one-loop effective Lagrangian corresponding to a self-dual background of electric type, which is exactly the same as that for c = 1 strings at self-dual radius. This time around the perturbative series is not alternating in sign, and the Borel integral representation 1 Fc=1 (¯ gs ) = 4
+∞
dσ σ
1 1 1 − − 3 sin2 σ σ 2
2tσ
e− g¯s
(4.4)
0
has an integrand with poles on the positive real axis, in principle leading to ambiguities in the reconstruction of the function, as discussed in an earlier section. However, we may now use the analogy of this expression to the results in Sect. 2.2 in order to use the unitarity prescription to perform an unambiguous calculation, which basically yields an i prescription which reduces the imaginary part of the integral to a sum over the residues of its integrand. The nonperturbative imaginary contribution to the above free energy is thus simple
11
Notice that since FgG ∼ (2g − 3)! the inverse of the Borel transform will now have an extra factor of 1s with respect to the definition of Sect. 2, which dealt with asymptotic growths of the type ∼ (βn)!.
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to compute as12
+∞ 1 dσ 1 π 1 − 2tσ 1 Im Fc=1 (¯ gs ) = − − e g¯s 4 n=0 2πi σ sin2 σ σ 2 3 =−
nπ +∞
1 4π¯ gs
n=1
2πt g¯s + 2 n n
2πt n exp − . g¯s
(4.5)
As expected from the discussion above, this formula precisely matches with the Schwinger result in a self-dual background, expressed in (2.22). It can also be obtained from the “alternating” result (4.2) by analytic continuation and contour rotation. Furthermore, as discussed in Sect. 2.1, it follows that the discontinuity of the free energy across its branch cut consists of an instanton expansion given by +∞
n g¯s − 2πt i 2πt + 2 e g¯s gs ) = − Disc Fc=1 (¯ 2π¯ gs n=1 n n = F (1) (¯ gs ) + F (2) (¯ gs ) + · · · .
(4.6)
We may now relate this instanton series to the full c = 1 perturbative expansion (2.33), by means of the Cauchy formula (2.8). One first observes that the integral over the contour at infinity in (2.8) has, in here, no contribution, since the Barnes function is regular at infinity (see, e.g., [73]). As such, the dispersion relation (2.8) reads,13 after power series expansion of the integrand’s denominator,
0 +∞ +∞ 2k g ¯ dz 2πt n s √ Fc=1 (¯ gs ) = − 1 + 2 z k+1 z (2πn) n=1 k=0
√ n − 2πt z
e
−∞
+∞ +∞ 2 (2k + 1) g¯s 2k = 2k+2 t (2πn) n=1
(4.7)
k=1
Γ(2k) =
+∞ 2 (2g − 1) g=2
(2π)
2g
ζ(2g)Γ(2g − 2)
g¯ 2g−2 s
t
where we used the definition of the Riemann zeta function as +∞ 1 ζ(z) = . nz n=1
,
(4.8)
Notice that, from the first to the second line, we truncated k = 0 from the k sum. Indeed, for this particular value of k the integral would require regularization. However, this would only contribute to terms at genus zero and one, which we are not considering here in any case. As such we shall simply truncate the k = 0 contribution from the sum, without the need to regularize the divergence, and focus on the genus g ≥ 2 contributions. In this way, if in 12 13
Notice that the pole at σ = 0 has vanishing residue. Recall that for matrix models and strings one uses z = g¯s2 ; see Sect. 2.1.
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the above formula for Fc=1 (¯ gs ) we further relate the Riemann zeta function to the Bernoulli numbers via ζ(2n) = (−1)−(n+1)
(2π)2n B2n , 2 (2n)!
(4.9)
it immediately follows Fc=1 (¯ gs ) =
g¯ 2g−2 |B2g | s 2g (2g − 2) t g=2
+∞
(4.10)
which is indeed the c = 1 perturbative expansion, for genus g ≥ 2, with μ = g¯ts . In some sense, (4.10) takes us back to where we started the discussion, i.e., the Gaussian matrix model perturbative series. Indeed we have seen that the alternating Gaussian perturbative series admits a simple Borel transform, which may be inverted unambiguously to provide a nonperturbative completion of the theory. Upon “Wick rotation” of the coupling constant, this completion also describes the non-alternating c = 1 string theory alongside with its instanton effects (obtained in a fashion very similar to our earlier discussion of the Schwinger effect). Of course that a key aspect of this analysis is the fact that the integral representation of the free energy, provided by the inverse Borel transform, precisely coincides with the nonperturbative integral formulation of the c = 1 theory put forward in [25,43]. One may thus consistently pick either starting point and obtain the very same results. To end our analysis, we shall now address the large-order behavior of perturbation theory and see that it is controlled—as expected—by one-instanton contributions, i.e., by the closest pole to the origin in the complex Borel plane. If one considers the first term in the instanton expansion (4.6) and, following the discussion in Sect. 2.1, one sets g¯s − 2πt i t+ e g¯s , gs ) = − (4.11) F (1) (¯ g¯s 2π a comparison with (2.10) immediately yields A = 2πt,
b = −1,
(1)
F1
= t,
(1)
F2
=
1 , 2π
(4.12)
thus identifying the instanton action, the characteristic exponent, and the loop expansion around the one-instanton configuration. In fact, it is rather interesting to observe that in this situation the loop expansion around each -instanton sector is finite. This is quite unusual; typically the -instanton loop expansion is itself asymptotic, with its large-order behavior being controlled by the ( + 1)-instanton configuration. What we observe is that in this case only the zero-instanton sector displays non-trivial large-order behavior. Now, with the identifications (4.12) the large-order equation (2.11) implies Fg(0) (t) ∼
(2g − 1) (2g − 3)! 2g−2
2π 2 (2πt)
.
(4.13)
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Checking the expected large-order behavior in this case is straightforward, as one simply needs to use the standard relation 2 (2g)! 2 (2g)! −2g log 2 + e−2g log 3 + · · · (4.14) |B2g | = 2g ζ(2g) = 2g 1 + e (2π) (2π) in the asymptotic series (4.10) and the above (4.13) immediately follows; exponentially suppressed contributions in (4.14) are not contributing to the large order of the zero-instanton sector. If one is instead interested in the large-order behavior of the Gaussian matrix model, one essentially just needs to use the analytically continued instanton action A = 2πit and everything else follows in a similar fashion. 4.2. The Penner Matrix Model Having worked out the Borel analysis of the free energy in the Gaussian matrix model, which essentially reduces to the Borel analysis of the logarithm of the Barnes function G2 (z), we have all we need in order to write down the nonperturbative part of the free energy of the Penner model (3.31). In fact the whole procedure is essentially the same as before and we shall leave most calculations to the reader. The Borel transform is now +∞ FgP (t) 2g−2 t (t + 2) 1 1 1 1 − . ξ = + B[FP ](ξ) = 2 2 2 ξ ξ (2g − 3)! ξ 4 sinh 4 sinh g=2 2 2(t+1) (4.15) It should be simple to spot the similarities to the Gaussian case, as the free energy of the Penner model may be written in terms of Barnes functions as in (3.28). As such, the total discontinuity (for gs )is now given by the sum gs → i¯ t+1 1 of the discontinuities of log G2 gs + 1 and log G2 gs + 1 , which yields Disc FP (¯ gs )
+∞
+∞
2π(t + 1) 2π(t+1)n 2πm 2π g¯s g ¯ i s + 2 e− g¯s + 2 e− g¯s . =− − 2π¯ gs n=1 n n m m m=1 (4.16)
This is quite simple to obtain by following a procedure identical to what we used in the Gaussian case, but starting from the above Penner Borel-transform. Notice that in this case we have two sets of nonperturbative contributions, with instanton actions 2π (t + 1) and 2π, respectively. The large-order behavior of the theory is controlled, as usual, by the closest pole to the origin in the Borel plane. For t ≥ 0 the relevant pole is located at 2π; however, close to criticality t → −1, the first instanton tower in (4.16) is the relevant one. 4.3. The Chern–Simons Matrix Model and the Resolved Conifold We may now turn to the Chern–Simons matrix model, holographically describing topological strings on the resolved conifold. The genus expansion of its free energy (3.46), is asymptotic, and in here we wish to analyze this divergent
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series from the viewpoint of Borel analysis, as we did earlier with both the Gaussian and the Penner matrix models. Let us start with the Chern–Simons genus expansion (3.46) where, for the moment, we drop the constant map contribution and focus on genus g ≥ 2. This will allow us to better understand the divergence of the series arising from the term with both a Bernoulli number and a polylogarithm function contributions. One has in this case14 +∞ B2g Li3−2g e−t FCS (gs ) = gs2g−2 2g (2g − 2)! g=2 =
+∞ g=2
gs2g−2
B2g 1 2g−2 , 2g (2g − 2) (t + 2πim) m∈Z
(4.17)
where we have used an integral representation of the polylogarithm in terms of a Hankel contour, re-written as a sum over residues [74], to express 1 g ≥ 2. (4.18) Li3−2g e−t = Γ (2g − 2) 2g−2 , (t + 2πim) m∈Z Now, since the Bernoulli numbers grow as B2g ∼ (2g)! and the polylogarithm functions behave, in worse growth scenario,15 as lim|t|→0 Li3−2g (e−t ) ∼ Γ (2g − 2) t2−2g , this series is asymptotic and, like in Gaussian and Penner models, its coefficients grow factorially as (2g − 3)!. In this case one is led to the Borel transform +∞ +∞ FgCS (t) 2g−2 B2g ξ 2g−2 ξ = B[FCS ](ξ) = 2g−2 (2g − 3)! 2g (2g − 2)! (t + 2πim) g=2 g=2 m∈Z ⎛ ⎞ 2 (t + 2πim) 1 1 1 ⎝− + ⎠ . (4.19) = − ξ 12 ξ2 4 sinh2 m∈Z
2(t+2πim)
This function has no poles in the positive real axis, for real argument. This is expected since we started off with an alternating sign expansion and, in this case, we may define the free energy via the inverse Borel transform +∞ ds 1 FCS (gs ) = − 4 s m∈Z 0 ⎞ ⎛ 2
1 2 (t + 2πim) 1 1 − ×⎝ + ⎠ e−s . (4.20) 2 2 gs g s 3 s sinh 2(t+2πim) s 14
Notice that the m = 0 contribution, in the sum in the second expression, equals +∞ B2g gs 2g−2 , which is of course the Gaussian free energy at genus g ≥ 2. This is g=2 2g(2g−2) t a consequence of working with a Chern–Simons free energy which is not normalized against the Gaussian free energy; see Sect. 3.3. 15 At large t the polylogarithm’s growth is not factorial in genus, as one has limRe t→±∞ Li3−2g e−t = 0.
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This provides an unambiguous nonperturbative completion for the asymptotic expansion of the free energy in the Chern–Simons model. Interestingly enough, and like in previous examples, for each distinct m a trivial change of variables turns the corresponding expression into the one-loop effective Lagrangian for a charged scalar particle in a constant self-dual electromagnetic field (of magnetic type) introduced in Sect. 2.2, the sum over all integer m thus corresponding to a sum over an infinite number of Lagrangians of this type. If we further report to the discussion in Sect. 2.4, we see that our Borel resummation is essentially (up to analytic continuation, as we move between alternating and non-alternating perturbative series) equal to the GV integral representation of the free energy of topological strings on the resolved conifold. Let us thus consider the case of the resolved conifold in greater detail, gs , and correwhich is obtained by the simple analytic continuation gs → i¯ sponds to the electric version of the above result. In this case the free energy perturbative series is non-alternating and the Borel transform has poles on the positive real axis, making the reconstruction of the free energy possibly affected by nonperturbative ambiguities, which we may, however, understand in the computation of the imaginary part of the integral, +∞ dσ 1 1 1 − 2(t+2πim) 1 σ g ¯s Fconif (¯ gs ) = − . − e 4 σ sin2 σ σ 2 3 m∈Z
(4.21)
0
Moreover, since (4.21) agrees, after a simple change of variables, with the GV integral representation (2.35), at least for genus g ≥ 2, we have a physical interpretation for the nonperturbative terms we find: as observed earlier, in Sect. 2.4, the imaginary part of the integral will compute the BPS pair-production rate in the presence of a constant self-dual graviphoton background. The imaginary part of the integral (4.21) may be computed by the use of the unitarity +i prescription, yielding a sum over residues of the integrand. Equivalently it equals one-half of the integral over the whole real axis (the imaginary part is symmetric), which may be computed by closing the contour on the upper half of the complex plane, thus enclosing the poles of the hyperbolic sine; see Fig. 8. It follows gs ) Im Fconif (¯
+∞ 1 dσ 1 1 − 2(t+2πim) π 1 σ g ¯s − = − e 4 n=1 2πi σ sin2 σ σ 2 3 m∈Z nπ
+∞ 1 2π (t + 2πim) g¯s 2π (t+2πim) n + 2 exp − =− . (4.22) 4π¯ gs n=1 n n g¯s m∈Z
One observes without surprise that this formula matches (an infinite sum of) the Schwinger result in a self-dual background. The discontinuity of the free
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energy across its branch cut is thus given by the instanton expansion
+∞ 2π(t+2πim)n g¯s i 2π (t + 2πim) g ¯s Disc Fconif (¯ + 2 e− gs ) = − 2π¯ gs n=1 n n m∈Z
=F Making use of m∈Z
(1)
(¯ gs ) + F (2) (¯ gs ) + · · · .
2πn 2πn exp −2πim δ −k = gs gs
(4.23)
(4.24)
k∈Z
this discontinuity may also be written as i Disc Fconif (¯ gs ) = − 2π¯ gs n=1 k∈Z
2πt n 2πt 2πn 2πn g¯s g¯2 ∂ + 2 δ × − k + s2 δ − k e− g¯s . n n g¯s n ∂¯ gs g¯s +∞
(4.25)
As we did in the previous cases, we may now use the Cauchy formula to relate this instanton series to the perturbative expansion of the resolved conifold’s free energies. Once again, the integral over the contour at infinity in (2.8) has no contribution (see “Appendix”), and the dispersion relation thus reads, successively,
0 +∞ +∞ g¯s2m dz 2πn 2πt n Fconif (¯ gs ) = − 1+ √ δ √ −k 2 z m+1 z z n=1 k∈Z m=0 (2πn) −∞
√ ∂ 2πn √ n − 2πt z + z √ δ √ −k e ∂ z z +∞ +∞ +∞ k 2 (2m + 1) g¯s2m (e−t ) = 2m+2 k 1−2m (2πn) n=1 m=0 k=1
=
+∞
g¯s2g−2
g=1
=
+∞
g¯s2g−2
g=1
2 (2g − 1) (2π)
2g
ζ(2g) Li3−2g e−t
|B2g | Li3−2g e−t . 2g (2g − 2)!
where we made use of +∞ f (xi ) , dx f (x) δ (g(x)) = |g (xi )| i
(4.26)
(4.27)
−∞
with xi the real simple roots of g(x); of the definition of the polylogarithm of index p Lip (z) =
+∞ n z ; p n n=1
(4.28)
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Figure 8. Poles in the complex Borel plane, for the case of the resolved conifold free energy. The poles in red coincide with the spurious c = 1 string contribution, arising from the Gaussian normalization (color figure online)
and the definition of the Riemann zeta function alongside with its relation to the Bernoulli numbers. This result shows that the instanton expansion (4.23) indeed has enough information to completely rebuild the full free energy perturbative expansion for topological strings on the resolved conifold, at genus g ≥ 2 (at genus 0 and 1 one still needs to take into consideration the relation between Chern–Simons and Stieltjes–Wigert perturbative free energies, and additional regularizations may be needed as, e.g., in the Gaussian case). One may further show that, in particular, the closest pole to the origin in the Borel complex plane controls the large-order behavior of perturbation theory, corresponding to the familiar one-instanton contribution. A glance at Fig. 8 makes it clear that the closest pole to the origin corresponds to the oneinstanton contribution of the c = 1 string. This, of course, is due to the fact that the Chern–Simons free energy in (3.46) is not normalized by the Gaussian free energy. This is easily corrected by considering, in the following, the normalized free energy gs ) = Fconif (¯ gs ) − Fc=1 (¯ gs ). Fconif (¯
(4.29)
This will guarantee that our large-order tests will precisely look at the true “resolved conifold contribution”, without being plagued by ghost effects due to the Gaussian measure. In this normalized case, and as is simple to check by looking at Fig. 8 again, we have two complex conjugate poles equally distant from the origin, at ξ = 2π (t ± 2πi) /¯ gs , and we have to consider the contributions from them both. Let us consider the first terms in the instanton expansion (4.23) F (1) (¯ gs ) = −
i g¯s
m∈{±1}
t + 2πim +
g¯s − 2π(t+2πim) g ¯s , e 2π
(4.30)
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and thus define, following Sect. 2.1,
2π t2 + 4π 2 exp ±i arctan , t (4.31)
2π = t2 + 4π 2 exp ±i arctan , t (4.32)
A± = 2π (t ± 2πi) ⇒ A± = |A|e±iθA = 2π
F1± = t ± 2πi ⇒ F1± = |F1 |e±iθF (1)
(1)
F2
(1)
=
1 , 2π
(1)
b = −1.
(4.33)
Again, as in the previous examples, the loop expansion around each -instanton sector is finite. Thus, also for the resolved conifold only the zero-instanton sector displays non-trivial large-order behavior. In particular, the large-order equation (2.11) implies Fg(0) (t) ∼
Γ(2g − 1) π 2 |A|2g−2
1+
1 2g − 2
cos ((2g − 2) θA ) .
(4.34)
The check of this behavior is now harder than before, and we shall need to perform numerical tests to confirm its validity. In this sense, one constructs the test ratio
1 2π = cos (2g − 2) arctan 1 + O , (1) t g 2 |F1 | Γ(2g − 1) (0)
Rg ≡
π Fg |A|2g−1
(4.35)
where equality holds at large g due to (4.34). In Fig. 9, we can see that numerical analysis undoubtedly confirms our prediction. The analysis so far has focused only on the contribution to the resolved conifold free energy arising from the D2D0 bound states of branes. As is clear in (3.46), to this term one must still add the contribution from bound states of D0-branes, i.e., the contribution of constant maps. As it turns out, in this case the discontinuity is obtained from (4.23) by simply setting t = 0 in that expression, i.e.,
+∞ g¯s i 2πim + gs ) = − Disc FK (¯ g¯s n=1 n 2πn2 m∈Z
− 4π
e
2i m n g ¯s
=
(1) FK (¯ gs )
+
(2) FK (¯ gs )
(4.36)
+ ··· .
A procedure that should be familiar by now yields back the perturbative expansion (2.38) or (3.46) via the Cauchy formula and an integration of the above
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1.0 1.0 0.5 0.5 5
10
15
20
25
30
20
-0.5
-0.5
-1.0
-1.0
40
60
80
1.5 1.5 1.0
1.0
0.5
0.5 50
100
150
50
-0.5
-0.5
-1.0
-1.0
100
150
200
250
Figure 9. Test of large-order behavior for the resolved conifold. The plots represent the test ratio Rg (t), in red, versus the expected behavior cos (2g − 2) arctan 2π t , in blue, as a function of genus g, and for t = 20, 40, 60, 80, from left top to right bottom, respectively. The matching at high genera is evident (color figure online) discontinuity over the free energy branch cut: FK (¯ gs ) =−
=
g¯s2m
n=1 k∈Z m=0
(2πn)
2 −∞
dz z m+1
+∞ +∞ +∞ 2 (2m + 1) g¯s2m n=1 k=1 m=0
=
0
+∞ +∞
+∞ g=1
g¯s2g−2
(2πn)
2m+2
1
k
√ ∂ 2πn 2πn δ √ −k + z √ δ √ − k z ∂ z z
= 1−2m
+∞
g¯s2g−2
g=1
(−1)g B2g B2g−2 , 2g (2g − 2) (2g − 2)!
2 (2g − 1) (2π)
2g
ζ (2g) ζ (3−2g)
(4.37)
where we have used familiar properties of the zeta function and Bernoulli numbers, including ζ(−n) =
(−1)n Bn+1 . n+1
(4.38)
The last thing we want to show is that, as expected, it is the closest pole to the origin in the Borel complex plane that controls the large-order behavior of the theory. Repeating our earlier discussion we find the one-instanton contribution
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F (1) (¯ gs ) = −
i g¯s
2πim +
m∈{±1}
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g¯s − 4πg¯2 im s e , 2π
(4.39)
leading to A± = ±4π 2 i,
b = −1,
(4.40) 1 . 2π In this case the large-order equation (2.11) yields
2 Γ(2g − 1) 1 g−1 16π Fg(0) (t) = (2g − 3)! (2g − 1) . 1 + = (−1) 2g−2 4g 2g − 2 π 2 (4π 2 i) (2π) (4.41) (1)
F1± = ±2πi,
(1)
F2
=
To check that this is the right answer all one has to do is to use (−1)g B2g B2g−2 16π 2 = (−1)g+1 4g (2g)! (2g − 2)! ζ(2g) ζ(2g − 2) (2π) 16π 2 −6g log 2 = (−1)g−1 + 4 e−4g log 2 + · · · . 4g (2g)! (2g − 2)! 1 + 20 e (2π) (4.42) Notice that the above exponentially suppressed contributions do not contribute to the large order of the zero-instanton sector.
5. Stokes Phenomena and Instantons from Hyperasymptotics Having understood the Borel analysis of topological strings and c = 1 matrix models, benefiting in this course of the identification of instanton effects in these models, we shall now make a brief detour into the realm of hyperasymptotics, as first introduced in [75] (see, e.g., [76] for a review), i.e., a series of techniques to refine optimally truncated asymptotic expansions by the inclusion of exponentially small contributions. In particular, our focus will concern hyperasymptotic approximations for integrals with saddles [77,78], the prototypical example for problems dealing with the calculation of partition functions or free energies. In this case, the exponentially suppressed contributions arise from saddles other than the one chosen in the steepest–descent asymptotic approximation. The main interest of this analysis for the present work is that the Stokes phenomenon—certain “discontinuities” which we shall explain below and later relate to instanton effects—is automatically incorporated into the hyperasymptotic scheme. Suppose one wants to use the method of steepest descents in order to find an asymptotic expansion, as |κ| → ∞ with κ = |κ| eiθ , of the 1D “partition function” (5.1) Z(κ) = dz e−κW (z) , C
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where C is a contour we specify below. A typical calculation goes as follows: one begins with the calculation of the saddle points of the “potential function” W (z), computed as the set of points {zk }k=1,2,... such that W (zk ) = 0; then, chosen a reference saddle-point zn , the contour of integration C is deformed to the infinite oriented path of steepest descent through zn , which we shall denote by Cn (θ). This contour is defined as Im [κ (W (z) − W (zn ))] = 0
(5.2)
and with κ (W (z) − W (zn )) increasing away from zn . This immediately implies that the phase of W (z) − W (zn ) must equal −θ + 2πm, m ∈ Z. We then introduce the “partition function” Zn (κ) evaluated on the nth saddle 1 Zn (κ) ≡ √ e−κW (zn ) Zn (κ), κ √ dz e−κ(W (z)−W (zn )) . Zn (κ) = κ
(5.3) (5.4)
Cn (θ)
As we shall see, Zn (κ) will display Stokes phenomena in the form of a discontinuity associated to a jump in the steepest–descent path whenever it passes through one of the other saddles, k = n. The integral (5.4) can be evaluated via the steepest–descent method and one obtains a function of κ for each saddle, n, given by a series in negative powers of κ Zn (κ) ∼
+∞ ζg (n) g=0
κg
,
with (see [77,78] for details)
1 dz 1 ζg (n) = Γ g + . 2 2πi (W (z) − W (zn ))g+ 12 z
(5.5)
(5.6)
n
The series in (5.5) is asymptotic. The point of view of [77] is to understand this divergence as a consequence of the existence of other saddles {zk=n }, through which Cn does not pass. Because one is free to choose the reference saddle n at will, all possible asymptotic series are thus related by a requirement of mutual consistency, also known as the principle of resurgence: each divergent series will contain, in its late terms, and albeit in coded form due to their divergent nature, all the terms associated with the asymptotic series from all other saddles. Another important point concerning the asymptotic series (5.5) dwells with the fact that this expression only holds in a wedge of the complex κ-plane, i.e., for a restricted range of θ, a property which is associated to Stokes phenomena. Suppose that in the above set-up, and once (5.5) has been computed, we start varying θ in such a way that we always choose the contour of integration to be the steepest–descent through the saddle zn . As it turns out, this is a continuous process only for a finite range of θ: indeed, one faces a discontinuity if θ reaches a value such that the contour of integration passes
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Figure 10. On the left, the complex κ plane, showing a region −σnm1 < θ < −σnm2 . On the right, the equivalent region in z space, with saddle zn , adjacent saddles zm1 and zm2 , and respective steepest–descent contours through zn (red) hitting the adjacent saddles. In blue, we plot the adjacent contours (color figure online) through a second saddle, zm . This will happen when θ reaches the value −σnm , with − σnm = − arg (W (zm ) − W (zn )) .
(5.7)
For this value of θ the steepest–descent contour will change discontinuously and exponentially suppressed contributions to (5.4) will “suddenly” become of order one. In this case, and in order for the steepest–descent contour to go through a single saddle, one must restrict θ to an interval −σnm1 < θ < −σnm2 , where zm1 and zm2 are saddles adjacent to zn , i.e., saddles which may be reached from zn through steepest–descent paths.16 This is illustrated in Fig. 10. One of the goals of hyperasymptotics [75] is to deploy resurgence in order to better understand Stokes phenomena, and this is the aspect we shall be mostly interested in. Let us make these ideas more precise. In hyperasymptotics one begins with (optimal) truncation of the asymptotic series (5.5). In this case, Zn (κ) =
N −1 g=0
ζg (n) ) + R(N n (κ), κg
(5.8)
(N )
where Rn (κ) is the remainder associated to the finite truncation. The main contribution of the hyperasymptotic calculation in [77] was to produce an 16
A saddle zm is said to be adjacent to the saddle zn iff there is a path of steepest descent from zn to zm , i.e., zm will be adjacent to zn whenever θ = −σnm and thus arg (W (z) − W (zn )) = σnm (naturally this is also the condition that defines the Stokes lines for Zn (κ)). One similarly defines the adjacent contour through the adjacent saddle as the steepest–descent contour Cm (−σnm ), through zm .
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expression for the reminder which led to an exact resurgence formula for (5.5), and we shall now present these results. Let us first define the singulant, for every adjacent saddle zm , as Wnm ≡ W (zm ) − W (zn ) ≡ |Wnm | eiσnm .
(5.9)
In this case the remainder term (see [77,78] for details) can be expressed as a sum over integrals through all adjacent saddles to zn , {zm }: ) R(N n (κ)
1 1 = 2πi κN m
∞·e−iσnm
dη 0
η N −1 −ηWnm e Zm (η), 1 − κη
(5.10)
where the Zm (η) are defined on the adjacent contours Cm (−σnm ). There are several interesting points to this formula. First, it provides an exact and explicit expression for the reminder and one now explicitly sees that the divergence of the asymptotic series (5.5) is directly related to the existence of adjacent saddles. Second, inserting the above expression back in (5.8), one obtains the exact resurgence formula Zn (κ) =
N −1 g=0
ζg (n) 1 1 + g κ 2πi κN m
∞·e−iσnm
dη 0
η N −1 −ηWnm e Zm (η), 1 − κη
(5.11)
which is the basis for the hyperasymptotic analysis of [77]: indeed, each Zm (η) in the above integrands may itself be expanded as an asymptotic series leading, via iterations of the above formula, to exponentially improved asymptotic results for the original Zn (κ) (in the sense that the error associated to the approximation is reduced from polynomially small to exponentially small17 ). Third, as the resurgent formula holds for any N , one may write it down for N = 0, obtaining either the resurgent expression [78] 1 Zn (κ) = 2πi m
∞·e−iσnm
dη 0
e−ηWnm η−
η2 κ
Zm (η),
(5.12)
which leads to interesting functional relations in selected examples; or the (formal) resurgent relation [77] 1 (g − h − 1)! ζh (m), g−h 2πi m Wnm h=0 +∞
ζg (n) =
(5.13)
which expresses the late terms (g 1) of the asymptotic series at a given saddle as a sum over the early terms of the corresponding asymptotic series at the adjacent saddles. In particular, the leading contribution arises from the adjacent saddle m∗ with smallest singulant, i.e., to leading order one obtains ζg (n) ∼ 17
(g − 1)! ζ0 (m∗ ), g Wnm ∗
For a recent discussion in the field theoretic context see [79].
(5.14)
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Figure 11. Crossing the Stokes line, between the saddle zn and the adjacent saddle zm which makes manifest the characteristic factorial behavior of asymptotic series. Fourth, and finally, the resurgence formula (5.11) precisely incorporates Stokes phenomenon [77]; the appearance of suppressed exponential terms as the steepest–descent contour Cn (θ) sweeps through one of the adjacent saddles, m (see Fig. 11). This, as we mentioned, will happen as θ crosses the Stokes line Cn (−σnm ), in which case the asymptotic expansion (5.5) will have a discontinuity + − (5.15) Δ Zn (κ) ≡ Zn |κ| ei(−σnm +0 ) − Zn |κ| ei(−σnm +0 ) = 0. θ=−σnm
It is not too hard to compute the precise value of this discontinuity straight from the resurgence formula for the remainder (5.11). One obtains, without surprise, 1 dη 1 η N −1 e−ηWnm Zm (η) = N −1 Δ Zn (κ) κ 2πi η − κ θ=−σnm κ
= e−κWnm Zm (κ),
(5.16)
where, naturally, any further asymptotic expansion on the adjacent saddle, i.e., for Zm (κ), is to be evaluated precisely along the Stokes line θ = −σnm . This discontinuity is exponentially small as, on the Stokes line, κWnm is real and positive. The Stokes discontinuity is particularly relevant to us as we later wish to identify it with instanton effects and, as such, we shall dwell upon it later in this section. An important thing to notice is that this is actually not a discontinuity of the function Z(κ) but rather a discontinuity of the asymptotic
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approximation to Z(κ). A trivial example is the function sinh z1 . In the right + half-plane 1 Re z > 0 its asymptotic behavior as z → 0 is well described by 1 2 exp z , with the error being exponentially suppressed. However, should we try to rotate this asymptotic approximation to the left half-plane Re z < 0, it would no longer be valid as the subdominant error − 12 exp − z1 will no longer be small! This is clearly a discontinuity of the asymptotic approximation we chose, and not of the function itself. We shall next see how to apply this formalism within the case of the Gamma function, which will later lead to a hyperasymptotic understanding of the free energies in the Gaussian and Penner matrix models. In particular, we shall build up our analysis in order to see how to obtain the multiple instanton sectors of these models straight out of the above resurgence formulae. 5.1. Stokes Phenomena in the Gamma Function Consider the “Gamma partition-function” ZΓ (κ) ≡ Γ(κ).
(5.17)
One usually defines the Gamma function via Euler’s integral [80] +∞ Γ(κ) = dw wκ−1 e−w ,
Re (κ) > 0,
(5.18)
0
where the contour of integration is the positive real axis. The logarithm of the Gamma function, the “Gamma free-energy”, has a well-known representation [80]
1 1 (5.19) FΓ (κ) ≡ log Γ(κ) = κ − log κ − κ + log 2π + Ω(κ), 2 2 where Ω(κ) is meromorphic with simple poles at κ = −n, n ∈ N0 . One then obtains asymptotic expansions for the Gamma function by first obtaining asymptotic expansions for the function Ω(κ). One such familiar case is the Stirling series, which is the Poincar´e asymptotic expansion Ω(κ) ∼
+∞
1 B2g , 2g−1 2g (2g − 1) κ g=1
(5.20)
valid as |κ| → +∞, in the sector | arg(κ)| < π. Our goal in the following is to obtain an exponentially improved version of this asymptotic expansion, in the spirit of our previous discussion on hyperasymptotics and Stokes phenomena, along the guidelines in [80,81]. Let us begin with the “Gamma partition-function”, applying the hyperasymptotic analysis in the preceding section as in [81]. One first changes variables as w = κ ez and then re-writes Euler’s integral (5.18) as +∞ Γ(κ) = κ dz e−κW (z) , κ
−∞
(5.21)
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where W (z) = ez − z. The saddle points of this function are zk = 2πik, for k ∈ Z, with W (zk ) = 1−2πik. We may now apply the machinery we previously described, with the nuance that one now has an infinite number of saddles. We first select the reference saddle z0 = 0 and define √ 1 Γ0 (κ) ≡ 2π κκ− 2 e−κ G0 (κ), (5.22) κ G0 (κ) = dz e−κ(W (z)−1) , Re (κ) > 0, (5.23) 2π C0 (θ)
with log G0 (κ) = Ω(κ). It is not too hard [81] to identify all saddles {zm }m=0 as adjacent saddles to z0 : the singulants are now 1 π (5.24) W0m = −2πim = 2π |m| exp i(−1) 2 (1+sgn(m)) 2 and the steepest–descent contour from z0 to each zm is that for which θ = −σ0m , i.e., 1 π (5.25) arg (ez − z − 1) = (−1) 2 (1+sgn(m)) . 2 As such, for m > 0 the Stokes line is at θ = π2 and for m < 0 the Stokes line is at θ = − π2 , so that the imaginary axis is a Stokes line for the Gamma function. Parametrizing z = x+iy, this contour may also be written as cos y = (1 + x) e−x with e−x < siny y if m < 0 and greater than if m > 0 (see Fig. 12). As such, the remainder associated to the finite truncation of the asymptotic series for G0 (κ) follows from (5.11) as ⎧ +∞ ⎨ +i∞ 1 η N −1 2πiηm (N ) dη e Gm (η) R0 (κ) = 2πiκN m=1 ⎩ 1 − κη 0 ⎫ 0 ⎬ N −1 η −2πiηm dη e G (η) , (5.26) − −m ⎭ 1 − κη −i∞
where one should recall that the Gm (η) are to be evaluated over the adjacent steepest–descent contours Cm (−σ0m ), e.g., for m > 0, η Gm (η) = (5.27) dz e−η(W (z)−W (zm )) . 2π π Cm ( 2 ) In this integral, consider the shift w = z − 2πim, where we move the contour downwards in the complex z-plane by 2πim. It is simple to see that the shifted contour C¯m π2 will now go through z0 rather than zm and the integral becomes η (5.28) dw e−η(W (w)−1) = G0 (η). Gm (η) = 2π C¯m ( π 2)
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Figure 12. Paths of steepest descent for the Gamma function, in the (x, y) plane, starting at z0 = 0
The exact same reasoning applies if m < 0, in which case one just shifts the adjacent contours upwards. Thus ⎧ +∞ ⎨ +i∞ 1 η N −1 2πiηm (N ) R0 (κ) = dη e G0 (η) 2πiκN m=1 ⎩ 1 − κη 0 ⎫ 0 ⎬ η N −1 −2πiηm − dη e G (η) . (5.29) 0 ⎭ 1 − κη −i∞
One may now address Stokes phenomena for the “Gamma partitionfunction” and simply confirm that indeed the imaginary axis in the complex κ-plane is a Stokes line for G0 (κ): as κ becomes purely imaginary, one of the two integrals above will have a pole. As we have seen before, this leads to the discontinuities Δ G0 (κ)
θ=± π 2
=±
+∞ m=1
e±2πiκm G0 (κ) = ∓
1 G0 (κ). 1 − e∓2πiκ
(5.30)
Notice that the discontinuities are evaluated on the Stokes lines, κ = ±i|κ|, and are thus always exponentially suppressed. We will interpret these terms as nonperturbative “instanton” contributions to the “Gamma partitionfunction”. As we turn to the “Gamma free-energy”, the same nonperturbative corrections can be obtained very easily from the reflection formula
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Γ (κ) Γ (−κ) = −
Ann. Henri Poincar´e
π . κ sin πκ
(5.31)
Indeed, one may now obtain the final result from a two line calculation [82]. The idea is to directly find the (exponentially) improved version of the asymptotic expansion (5.20), in the sector π2 < θ < π, and which we name Ω+ (κ). Upon the analytic continuation −κ = e−iπ κ to the sector π2 < θ < π, where − π2 < arg (−κ) < 0, one may use the expansion (5.19) for Γ(−κ) in (5.31) to obtain, upon taking the logarithm of (5.31), Ω+ (κ) = Ω(κ) − log 1 − e2πiκ , (5.32) where the plus subscript refers to the sector past + π2 . In the sector −π < θ < − π2 one still obtains the above expression (5.32), but with e2πiκ replaced by e−2πiκ instead (and the plus subscript naturally gets replaced by a minus subscript). Thus, in both sectors the corrections to the expansion (5.19) are always exponentially suppressed, with the discontinuity across the Stokes lines θ = ± π2 given by Δ Ω(κ)
θ=± π 2
+∞ ±2πiκm e , = − log 1 − e±2πiκ = m m=1
(5.33)
which is, as expected, essentially related to the logarithm of the Stokes discontinuity for the “Gamma partition-function”, (5.30). It is rather tempting to understand these terms as instanton contributions, with “instanton (m) action” Sinst = W (zm ) − W (z0 ) = W0m = −2πim. For this identification to be valid, one expects that the instanton(s) with least action, S (−1) = 2πi and S (1) = −2πi, will yield the leading contributions controlling the large-order behavior of the perturbative expansion (5.20), as we have discussed before in Sect. 2.1. Setting as usual +∞ 1 (0) Ω (κ) ∼ κ−2g Ω(0) g , κ g=1
Ω(0) g ≡
B2g , 2g (2g − 1)
(5.34)
and +∞ 1 (1) (1) Ω (κ) ∼ iκb e−κA κ−g Ωg+1 , κ g=0
the standard large-order analysis yields
(1) Γ (2g + b) Ω2 A (1) (0) + ··· Ωg ∼ Ω1 + πA2g+b 2g + b − 1
(5.35)
(5.36)
and, consequently, (0)
Ωg+1 (0)
4g 2 Ωg
1 = 2 +O A
1 . g
(5.37)
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(0)
From the exact expression for the Ωg coefficients it immediately follows
(0) Ωg+1 1 1 B2g+2 2 1 = 2 +O 1− +O =− , (5.38) 2 2 (0) 2 4g B2g g g g (2π) 4g Ωg so that A = ±2πi just as expected from the instanton analysis. Thus, we see that indeed the one-instanton contributions control the large-order behavior of the perturbation theory, supporting the important identification of Stokes discontinuities with instanton contributions. An interesting property of our instanton actions is that S (m) = mS (1) so that the action of the mth one-instanton equals the mth multi-instanton action. In this case, one could interpret the result (5.33) as being exact, in the sense of including all multi-instanton corrections, and to all loop orders. For this identification to be valid, one expects to fully reconstruct the perturbative coefficients of the “Gamma free-energy” out of its complete multi-instanton series, as in (see, e.g., [13]) Ω(0) g
1 = 2πi
+i∞
dξ ξ 2g−2 Δ Ω(ξ) −i∞
⎛ +∞ ⎞ 0 +∞ (−1)g+1 1 ⎝ = dx x2g−2 e−2πmx + dx x2g−2 e2πmx ⎠ . 2π m m=1 −∞
0
(5.39) This is actually just one of the few examples on resurgent relations we have obtained in our earlier hyperasymptotic discussion. It is simple to see that the precise result is obtained, in full accordance with our multi-instanton expectations: Ω(0) g
=
+∞ 2(−1)g+1 (2g − 2)!
(2π)
2g
1 , 2g m m=1
(5.40)
where one just needs to use the representation of even Bernoulli numbers in terms of the zeta function, that we have used several times before, in order to check the result. We have thus seen very clearly, at the familiar free-energy level, that the contributions arising from the Stokes line discontinuities are precisely the usual nonperturbative instanton contributions. The exact matching within the matrix model examples we consider in this work will be made complete in the next section. 5.2. Instantons as Stokes Phenomena in Matrix Models In the previous section, we have identified the Stokes discontinuities of the Gamma function with instanton effects of the corresponding 1D integral, defining either a partition function or a free energy. We shall now see how this analysis carries through to the matrix models we are interested in. Let us start by
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considering the Gaussian and Penner matrix models, and how their instantonic sectors follow from a Stokes analysis of the corresponding free energies. As computed at an earlier stage, the exact free energy for the Gaussian matrix model is given by 1 2 1 N log gs − N log 2π + log G2 (N + 1), (5.41) 2 2 and the exact free energy for the Penner matrix model (normalized against the Gaussian measure) is FG =
FP =
N 1 1 log gs + N 2 log gs − N log 2π gs 2 2
1 1 + log G2 N + + 1 − log G2 +1 . gs gs
(5.42)
It is rather evident from these expressions that both cases have their asymptotic expansions associated to Poincar´e asymptotic expansions of the Barnes function. As such, the instanton sectors of these two matrix models will be dictated by the Stokes structure of the logarithm of the Barnes function. But due to the integral representation [73] 1 1 log G2 (N + 1) = N log 2π − N (N − 1) + N log Γ(N ) − 2 2
N dn log Γ(n) 0
(5.43) the Stokes structure of the Barnes free-energy is given in terms of the Stokes structure of the Gamma free-energy, which we have previously analyzed in greater detail. In particular, N Δ log G2 (N + 1)
θ=± π 2
= N Δ Ω(N )
=
θ=± π 2
−
dn Δ Ω(n) 0
θ=± π 2
+∞
N 1 iπ ∓ e±2πiN m ∓ . 2 m 2πim 12 m=1
(5.44)
At the Stokes lines N = ±i|N | and the discontinuities are exponentially suppressed as expected. It immediately follows18 +∞
2πt m g¯s i 2πt + 2 e− g¯s Δ FG = (5.45) 2π¯ gs m=1 m m iπ Notice that the factor 12 is a genus one artifact and we drop it in the following. Indeed, when computing the Borel transform of the Gaussian free energy (equivalently, of the logarithm of the Barnes function), we start the sum at genus g = 2 in order to avoid the problematic logarithmic terms at genus zero and one, which do not contribute to the largeorder behavior in any case. If one were to—incorrectly—start the sum at genus g = 1, while still making use of the Bernoulli expression (3.19) in order to compute this genus g = 1 term 1 contribution. in the sum, one would precisely find this spurious 12 18
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+∞
g¯s i 2π (t + 1) + 2 Δ FP = 2π¯ gs m=1 m m
+∞ 2π(t+1)m 2πm g¯s i 2π + 2 e− g¯s . − ×e− g¯s 2π¯ gs m=1 m m
405
(5.46)
These results, where we have used t = gs N and restricted to the Stokes line at θ = + π2 , and upon the identification of the Δ discontinuity with −Disc, precisely match our results for the full instanton sector of both Gaussian and Penner matrix models, obtained earlier in Sect. 4 via Borel summation (or to be later analyzed via the use of trans-series methods). We have not addressed the case of the Chern–Simons matrix model, as its free energy is given by the logarithm of the quantum Barnes function, for which we do not know of any appropriate hyperasymptotic framework which would allow for a derivation of its Stokes discontinuities. However, we believe it should be possible to study the hyperasymptotics of the quantum Barnes function in a similar fashion to the one above (see also [83]). Some interesting lessons may be drawn from our Stokes analysis. Because the appearance of the nonperturbative ambiguity of the matrix models’ free energies is related to Stokes phenomena, intimately associated to discontinuities of the asymptotic approximation, we see that this nonperturbative ambiguity is in fact an artifact of the semiclassical, large N analysis. Clearly, the exact free energies, (5.41) and (5.42), are given by the logarithm of entire functions in the complex plane, with no discontinuities. The Stokes lines, and thus the instanton corrections, appear only at the very moment we select a particular saddle and semiclassically evaluate the partition functions or free energies. On the other hand, it is also clear that it is just in this semiclassical limit that the notion of target space in the holographically dual theory emerges. For instance, we have seen in an earlier section that, in the c = 1 case, the geometry of the target space arises from the matrix model spectral curve, which gets identified with the derivative of the FZZT disk partition function. Even more manifestly, in the case of the Chern–Simons matrix model the spectral curve of the matrix model coincides with the mirror curve of the mirror CY to the resolved conifold. One is led to conclude that if one is to consider the exact free energies, (5.41) and (5.42), as the nonperturbative definitions of the holographically dual models, then it appears the “exact quantum” target spaces are very different from the semiclassical ones; basically at the nonperturbative level the notion of target space as a smooth geometry is lost. Interestingly enough, this discrepancy between “semiclassical” and “exact quantum” target spaces has also been advocated in [84], focusing on the example of non-critical strings with c < 1. In particular, Stokes phenomena was also identified therein as a source of instanton corrections. As we shall review in the upcoming Sect. 6.1, in the context of matrix models the nonperturbative partition function is obtained by summing over all saddles of the matrix integral. In particular, it is the averaging over all
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possible semiclassical geometries that leads to the background independence of the nonperturbative partition function, as described in [19,85]. 5.3. Smoothing the Nonperturbative Ambiguity As we have explained, the Stokes discontinuities are discontinuities of the asymptotic expansions and not of the functions under approximation, and we have made this clear in some examples. It so happens that in most cases one does not know the function we wish to approximate and, as such, one needs to devise methods to smooth the Stokes discontinuity [86]. The need for the smoothing should be clear: it maintains the validity of the asymptotic expansions even as we cross the Stokes lines. This is what we shall describe now: the universal smoothing proposed in [86], given by an uniform approximation involving the error function, and universally describing Stokes phenomena, which naturally makes the strength of the subdominant contribution grow smoothly from 0 to 1 across the Stokes line (upon where it equals 12 ). The idea goes as follows. Hoping to maintain the validity of our asymptotic expansions as we cross Stokes lines, let us write these as Zn (κ) ∼
+∞ ζg (n) g=0
κg
+i
Sm (κ) Δ Zn (κ)
m
θ=−σnm
,
(5.47)
where we have implicitly included the remainder associated to optimal truncation within the infinite sum, and where we have introduced the Stokes multiplier function Sm (κ) weighting the subdominant exponentials and which will smooth the transitions across the Stokes lines. The true power of the Stokes multiplier function arises from the fact that, if the appropriate variables are chosen to cross the Stokes lines, then this function is universal, within a wide class of problems.19 This appropriate “universal” variable involves the singulant Wnm , specifying contours through adjacent saddles, i.e., specifying the location of the Stokes lines. The Stokes multiplier function may thus be written as [86] 1 Sm (κ) = (1 + erf (snm (κ))) , (5.48) 2 with erf(x) the error function and where we defined the Stokes variable Im (κWnm ) snm (κ) ≡ . 2 Re (κWnm )
(5.49)
Observe that this is not a solution to the nonperturbative ambiguity problem as there is a choice of a real constant implicit in this result: the choice that before the Stokes line the exponentially suppressed contributions actually vanish. This is of course related to a choice of integration contour in the inverse Borel transform used in the calculation of the Stokes multiplier [86]. What the Stokes multiplier function does is to reduce the nonperturbative ambiguity to the choice of the real constant describing the intensity of the exponentially suppressed terms before the Stokes line, describing the crossing in an universal 19
But see [87] for a counter example.
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fashion. For a recent discussion of the nonperturbative ambiguity and choice of inverse Borel transform, within the matrix model context, see [14]. For the “Gamma partition-function” with singulant κW0m = −2πiκm, the Stokes variable becomes Im (−2πiκm) cos θ i π (1−sgn(m)) e 4 = − π|κ||m| √ (5.50) s0m (κ) = sin θ 2 Re (−2πiκm) and the smoothing becomes implemented by the Stokes multiplier
1 cos θ Sm>0 (κ) = 1 + erf − π|κ|m √ , 2 sin θ
(5.51)
where we have explicitly written the case where one crosses the upper imaginary axis. Noticethat in the immediate vicinity of the Stokes line one has θ = θ − π2 + · · ·, simplifying the argument of the error function in − √cos sin θ that region. At the level of the “Gamma free-energy”, completely analogous to the partition function analysis as the singulants are precisely the same, this result was interpreted in [82] as a distinct—but universal—Stokes smoothing for each mth small exponential. One may, in this light, separately understand the appearance of each exponential. For both Gaussian and Penner models, the discontinuity’s singulants are the same as for the Gamma function and, akin to the previous discussion, one is in the presence of infinitely many smoothings [82] with Stokes multipliers π 1 1 + erf θ − π|N |m , (5.52) Sm>0 (N ) = 2 2 near the Stokes line θ = π2 . This explains the appearance of each suppressed exponential, or each distinct instanton contribution, in a separate but universal and smooth manner. In this way, one may readily obtain formal expressions for the Stokes smoothing, also yielding formal expressions for the free energies of these models which allow, for instance, one to cross the Stokes lines and reach any nonperturbative point in the complex N -plane.
6. Semiclassical Interpretation of Instantons At this stage we have a very good understanding of nonperturbative phenomena in c = 1 matrix models and topological strings, with all the information we have gathered both from Borel and Stokes analysis. However, and because at the end of the day we are analyzing large N matrix models in a saddle-point approximation, we would like to understand these nonperturbative instanton corrections directly from a semiclassical large N point of view, i.e., directly in the matrix model language. This is what we shall do in this section, as we provide a semiclassical interpretation of instantons in terms of eigenvalue tunneling, across a multi-sheeted effective potential, and we also suggest a spacetime interpretation for the nonperturbative effects we have just obtained in the preceding sections.
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6.1. Instantons as Eigenvalue Tunneling We shall begin by recalling how instanton effects, which control the large-order behavior of the 1/N expansion, are interpreted as an eigenvalue tunneling effect, within the matrix model context. Recalling our discussion on matrix models, in diagonal gauge the one-matrix model partition functions is N
N 1 dλi 1 (6.1) Δ2 (λ) e− gs i=1 V (λi ) . Z= N ! i=1 2π As it stands this definition cannot be complete, as the integration contours for each eigenvalue λi still need to be specified. It may happen that the above integral is not well-defined as a convergent, real integral, in which case the model needs to be properly defined by analytic continuation, i.e., by the choice of an appropriate contour in the complex plane such that the integral becomes convergent. Indeed, in general, the various phases of matrix models are separated by singular domains (in the space of complex potentials), where no large N limit exists [2,3,88]. Removal of the “divergent regions” where Re V (z) → −∞ as |z| → ∞ corresponds to holes in the complex plane, in which case one is led to decompose a generic integration path γ on a homological basis of paths {γ1 , . . . , γs } as20 γ=
s
ζk γk ,
(6.2)
k=1
where we shall place Ni eigenvalues on the path γi , with arbitrary distribution s {Ni } but such that i=1 Ni = N . As we shall make clear in the following, the coefficients ζk may be regarded as theta-parameters leading to different theta-vacua [3]. We may then define, with the appropriate symmetrizations, ' 1 , . . . , Ns ) Z(N =
1 N 1 ! · · · Ns !
N1 Ns 1 dλi1 dλis 2 ... Δ (λ) e− gs 2π 2π i =1 i =1
γ1
1
γs
N
i=1
V (λi )
.
(6.3)
s
For particular choices of the integration contours and particular choices of the i ' filling fractions i = N N , with i = 1, . . . , s, the free energy log Z(N1 , . . . , Ns ) 21 may have a perturbative large N expansion ' 1 , . . . , Ns ) = F' (N1 , . . . , Ns ) = log Z(N
+∞
N 2−2g Fg (t),
(6.4)
g=0
where one finds the usual large-order behavior of Fg ∼ (2g)! rendering the topological 1/N expansion asymptotic. Nonperturbative effects associated to 20
For a polynomial potential of degree d there will be d holes in the complex plane, in which case the dimension of the homological basis will be s = d − 1. 21 While in general there is no topological large N expansion, there are of course some cases where this may be achieved: for instance in the case of degree d polynomial potential one may choose as homological basis the d − 1 steepest–descents paths which go through each of the d − 1 critical points of the potential [2, 3].
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singularities in the complex Borel plane are interpreted as instanton configurations as follows. ' 1 , . . . , Ns ) as the partition function associated to a specific Regarding Z(N topological sector, characterized by the filling fraction 1 , . . . , s , it becomes natural to consider the general partition function where one sums over all possible ways of distributing the N eigenvalues ' 1 , . . . , Ns ), Z(ζ1 , . . . , ζs ) = ζ1N1 · · · ζsNs Z(N (6.5) s
i=1
Ni =N
and where it is now clear that the ζk play the role of theta-parameters. This expression has been proposed by Marino [14], Eynard and Marino [19], and Eynard [85] to provide a nonperturbative partition function for the matrix model. That (6.5) realizes such a nonperturbative completion of the theory is made clear by understanding how it encodes all possible multi-instanton corrections. It was pointed out in [15] that if one is to consider the partition functions associated to two distinct topological sectors, with distinct fillings, {Ni } and {Ni }, one finds [15] ' , . . . , Ns ) Z(N − 1 1 ∼ e gs ' Z(N1 , . . . , Ns )
s
i=1
0 (Ni −Ni ) ∂F ∂ti ,
ti = gs Ni ,
(6.6)
implying that once one selects a reference background, {Ni } say, all other sectors are different instanton sectors of the matrix model. Let us consider the one-cut cubic matrix model in the following, in order to be a bit more concrete. The cubic potential V (z) = 12 z 2 + g3 z 3 has two critical points; the maximum located at z = z∗ and the metastable minimum z = 0, as illustrated in Fig. 13. There are two steepest–descent paths naturally associated to these critical points, γ0 through z = 0 and γ1 through z = z∗ . While the lowest energy configuration is associated to having all eigenvalues integrated along γ0 , this is an unstable configuration due to tunneling mediated by instanton configurations, which correspond to the integration of eigenvalues along γ1 [13]. In particular, the partition function in the one-instanton sector is given by Marino et al. [13] )(0) ( 1 1 (0) (1) ZN −1 dx det(x1 − M )2 N −1 e− gs V (x) . (6.7) ZN = 2π x∈γ1
Let us explain this expression. We removed one out of the N eigenvalues in the cut, x, and we are integrating it over the non-trivial saddle-point, γ1 . The remaining N − 1 eigenvalues are, of course, still integrated over the leading (0) saddle, associated to γ0 , and ZN is the zero-instanton partition function evaluated around this standard saddle-point. Finally, M is an (N − 1) × (N − 1) hermitian matrix, all of its eigenvalues still integrated around the standard saddle-point in the zero-instanton correlation function. We refer the reader to [13] for the details on the explicit computation of the quantity above. As it turns out this expression implies that, at leading order, the one-instanton
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20
10
0
10
20 15
10
5
0
5
Figure 13. The real part of the holomorphic effective potential for the one-cut cubic matrix model, in blue, and the simple cubic potential, in purple, for the values g = 0.1 and t = 3 (color figure online)
contribution to the free energy is given by (1)
F
(1)
=
ZN
(0)
ZN
(0)
1 ZN −1 = 2π Z (0) N
)(0) ( 1 A dx det(x1 − M )2 N −1 e− gs V (x) ∼ i e− gs ,
x∈γ1
(6.8) where the instanton action is [13] x0 A = Vh,eff (x0 ) − Vh,eff (b) =
dz y(z),
(6.9)
b
with b the endpoint of the single cut C = [a, b]. This formula has an obvious semiclassical interpretation, as the instanton action (6.9) is nothing but the height of the potential barrier under which instantons are tunneling. Furthermore, a configuration where N1 eigenvalues are integrated along the contour γ1 , and N0 = N − N1 along γ0 , can be naturally regarded either as a two-cut solution with filling fractions N0 and N1 , or as a N1 -instanton excitation above the reference one-cut solution [15]. It has thus become clear that the general partition function (6.5) provides the nonperturbative completion of the matrix model since, by summing over all the filling fractions, it naturally encompasses all the multi-instanton configurations of the theory. What we shall see in the following is that, for our class of c = 1 matrix models and topological strings, one also has to allow for generalized integration contours which contain copies of the leading saddle-point and which find themselves going through different sheets of the multi-valued holomorphic effective potential.
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411
1.0
1.0
0.15
0.8 0.8 0.6
0.10
0.6 0.4
0.4
0.05 0.2
0.2 0.0
0.00
0.0 -2
-1
0
1
2
0
1
2
3
4
0
1
2
3
4
Figure 14. The real part of holomorphic effective potentials for Gaussian, Penner and Chern–Simons matrix models, from left to right, respectively, when t = 0.2 6.2. Instantons and the Multi-Sheeted Effective Potential Let us now turn to the c = 1 matrix models we are considering in this work. A quick glance at the real part of the holomorphic effective potentials for the three models, plotted in Figure 14, is enough to realize that there are no critical points of these potentials outside their single cuts. While this may not seem surprising for real and positive gs , as the asymptotic expansions are Borel summable and there are no nonperturbative ambiguities in the reconstruction of the partition functions, we also know that when allowing for imaginary values of the string coupling our topological expansions become non-Borel summable, with instantons controlling large order. As such, and given that the discussion in the previous section cannot be applied, at least not in a straightforward fashion, in what follows we shall have to define the instanton actions for our models by exploiting the structure of the holomorphic effective potentials in the complex plane. Let us have a closer look at the holomorphic effective potentials for Gaussian, (3.23), Penner, (3.38), and Chern–Simons, (3.52), matrix models. Due to the presence of either logarithmic or dilogarithmic functions these holomorphic effective potentials have a multi-sheeted branch structure in all examples, a clear feature from the structure of their Stokes lines (recall Figures 3, 5 and 7). As one takes the derivative of the effective potential, in order to obtain the spectral curve, all logarithmic or dilogarithmic sheets artificially collapse on top of each other. As such, the spectral curve cannot explicitly see this structure. One way to lift this artificial degeneracy and make the multisheeted structure of the effective potentials manifest, is to repeat what we did in Sect. 5.1, in the context of hyperasymptotic analysis. Let us start with the Gaussian model. Akin to what we did for the “Gamma partition-function” it is useful to first change variables from z to (6.10) eu = z + z 2 − 4t, in which case the Gaussian holomorphic effective potential gets written as √ 1 G Vh;eff (u) = e2u − 2t2 e−2u − 2t u + 2t log 2 t. (6.11) 8
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In this new variable the critical points (or saddles) of the potential are located at22 √ (n) e2u = 4t ⇒ u± = log(±2 t) + iπn, n ∈ Z, (6.12) which is just the solution of a, b = eu . The interesting property of the u variable is that it lifts the aforementioned artificial degeneracy where all sheets have collapsed on top of each other, and insures that one identifies all possible critical points of the holomorphic effective potential in its fully unfolded multi-sheet domain. One furthermore selects n = 0 for the reference saddle and identifies all saddles m = 0 as adjacent saddles to n = 0. In this new variable the multi-sheeted structure of the holomorphic effective potential is much simpler to visualize: one finds an infinite number of copies of the original single cut (with each distinct degenerate sheet param(n) (n) G (u) is constant etrized by n), with endpoints u− and u+ , and where Vh;eff on each replica of the cut. This, of course, has dramatic implications: as we proceed to evaluate the instanton action via A = Vh;eff (x0 ) − Vh;eff (b) the sheets are no longer degenerate and we shall find non-zero values every time we place an eigenvalue in another sheet. The placing of the eigenvalue is simple and totally analogous to the previous discussion of eigenvalue tunneling: one removes the single eigenvalue from the endpoint of the cut and places it at the starting point of the “next” cut, in the following sheet (in such a way that the spectral curve cannot “see” any difference in the configuration, only the holomorphic effective potential can), an idea first suggested in [26]. We conclude that the multi-instanton action (the singulant, in the hyperasymptotic language we used earlier, e.g., in section 5.1) is given by the difference between the holomorphic effective potential evaluated on the principal sheet (corresponding to the choice n = 0 and denoted with a in the following) and its value on a generic sheet, 1 (n) G G = V (u ) − V (u ) = −2πint = dz y(z). (6.13) AG n h;eff h;eff + + 2 γn
In the last equality we compute the instanton action as the integral of y(z) dz along γn , a non-contractible contour encircling the eigenvalue cut n times23 . Analogously, the instanton action may be regarded as half the shift due to G (recall equation the additive monodromy of the logarithm present in Vh;eff(x) (3.23)). The action (6.13) we have just obtained from a semiclassical viewpoint coincides with the one previously obtained with either Borel or Stokes analysis. It is also important to notice that the definition we are now suggesting for the instanton action is in perfect analogy with the one that we have previously described when addressing the semiclassical derivation of the 22
We use the standard definition of the logarithm, log z ≡ log |z| + i arg z, and make explicit its multi-sheeted structure by defining arg z up to 2πi, i.e., arg z|n ≡ arg z+2πin with n ∈ Z. 23 Here, γ is the contour used to define the filling fraction in matrix models, i.e., the A-cycle. 1 The factor of 12 insures that the action describes moving an eigenvalue from the endpoint of the cut to the beginning of the “next” cut in the multi-sheeted structure, i.e., that we keep the standard eigenvalue tunneling picture.
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Schwinger pair-production effect, in Sect. 2.2. With an appropriate identification of the parameters, one may identify the semiclassical Schwinger effective action (2.25) with the matrix model effective potential (3.23). As such, in here one may still interpret the instantons as an eigenvalue tunneling process, with the sub-barrier motion consisting of multiple windings in the complex plane around the eigenvalue cut, as in the discussion of Sect. 2.2. As we move to the Penner model the holomorphic effective potential becomes more intricate, as displayed in (3.38), and we shall focus on obtaining the instanton action straight out of the monodromy shift of (3.38) around the cut C = [a, b]. There are now contributions from more than one logarithmic branch cut and one needs to properly consider them all. In this case, there are two possible ways to choose the contour: one may encircle the eigenvalue cut without crossing the log z cut in (3.38), in which case one is forced to cross the other two logarithmic branch cuts; or one may encircle the eigenvalue cut crossing the log z cut. In the first option, clockwise winding n times around the cut C = [a, b] one will cross two logarithmic branch cuts and obtain the global shift −(2t + 1)2πin − 2πin = −4πin (t + 1)24 . This immediately yield the multi-instanton action 1 AP = −2πi (t + 1) n = dz y(z), (6.14) n 2 γn
a result which agrees with the one we have previously obtained via Borel and Stokes analysis. If, instead, one chooses the second option, which involves crossing the log z cut, this will naturally yield an additional contribution of 2πim (a result we expected to find from Borel or Stokes analysis in any case), but it is not entirely clear to us how this relates to a cycle of the curve. Finally, we need to confirm the validity of our proposal within the Chern– Simons matrix model. In this case, the holomorphic effective potential (3.52) is rather complex, involving both logarithmic and dilogarithmic branch cuts. As such, we shall restrain to computing the instanton action from the monodromy around the cut C = [a, b]. If the cut C = [a, b] is placed on the positive real axis, we may choose our contour γn such that Re (z) ≥ 0 when contouring the cut, in which case one also finds Re (ξ) > 0 and both log z and log ξ will have no monodromy. We will then be left with the contributions of log (1 − ξ) and log (1 − e−t ξ). Since both e−t ξ and ξ will be bigger than one on the right hand side of the cut, on the real axis, then both these logarithms will produce the same shift of 2πi, at each winding. Turning to the dilogarithmic dependence, we find a more complicated monodromy structure and we refer the reader to, e.g., [89,90] for a more in depth analysis, or to “Appendix A” for a brief review of this topic. The dilogarithm Li2 (z) has a branch cut starting at z = 1, and 24
The reader * may be puzzled by the fact the integral of the curve around the cut, i.e., the A-cycle A dz y(z), yields 2(t + 1) instead of 2t (as one would have expected from the normalization of the genus-zero resolvent at infinity, ω0 (z) ∼ z1 + · · · as z → ∞). The reason for this is that, when deforming the contour back from infinity to the cut C, we still have to pick the residues at z = 0. This unusual feature is, of course, due to the fact that we are not dealing with a polynomial potential.
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we choose it to be on the positive real axis. However, as one moves from the principal sheet to a generic sheet, by crossing the principal branch cut, a new branch cut will emerge from z = 0 to the right. As such, we shall need two distinct integers in order to specify the value of Li2 (z) on a generic sheet, in terms of its value on the principal sheet, Li2 (z). This turns out to be [89,90] Li2 (z) = Li2 (z) + 2πin log(z) − 4π 2 kn,
n ∈ N+ ,
k ∈ Z.
(6.15)
In here, the integer n counts how many times we wound clockwise in the complex plane, by crossing the principal branch cut, while the integer k counts how many times we crossed the “hidden” branch cut. Putting it all together, it follows Vh;eff |CS = Vh;eff + 4πin log e−t ξ − 8π 2 nk1 − 4πin log ξ + 8π 2 nk2 CS = Vh;eff
CS
− 4πitn − 8π 2 n (k1 − k2 ) .
(6.16)
Setting k = k1 − k2 , with k1 and k2 being the two, a priori different, windings around the “hidden” branch cut of the two dilogarithms in (3.52), one may write dz y(z) = −4πitn − 8π 2 nk (6.17) γn,k
where γn,k is a contour winding around the cut C until it reaches the (n, k) sheet. Again, on each (n, k) sheet we find a copy of C, with the effective potential being constant on each replica of the cut. The instanton action is thus given by 1 CS 2 dz y(z). (6.18) An,k = −2πitn − 4π nk = 2 γn,k
Once again the instanton action we have obtained with our proposed semiclassical reasoning precisely matches the one we previously obtained from Borel or Stokes analysis. It is interesting to note that in all cases above the instanton action is essentially given by the integral of the spectral curve along the A-cycle of the single cut, while in the cases considered in [13,15] the instanton action was given by the integral of the spectral curve along the B-cycle of the cut(s). This complete our matrix model derivation of the instanton action. In the following we focus on the spacetime interpretation of the nonperturbative effects we have discussed do far. 6.3. Spacetime D-Instanton Interpretation Nonperturbative effects in hermitian matrix models have been realized in terms of D-brane instanton effects in the holographically dual minimal models with c < 1. In particular, in [6] matrix model instantons, as described by eigenvalue tunneling, have been shown to match the disk contribution of ZZ branes. An ZZ disk conanalogous situation takes place in the case of c = 1 where the Z(k,1) tribution (3.75) coincides with the instanton action obtained from the MQM nonperturbative integral formula [43,25,69]. In this case, the singularities of
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the curve leading to the ZZ brane disk amplitude are not pinched cycles, but instead are points where the curve has self-intersections, as described in [69]. As usual, let us first analyze the Gaussian matrix model, or, equivalently, the c = 1 string at self-dual radius. We have already observed that the MQM spectral curve is just an infinite covering of the hyperboloid defining the Gaussian spectral curve and identified correlators in the Gaussian matrix model with open amplitudes with FZZT boundary conditions. The natural next step is to identify the instanton action (6.13) with the ZZ brane partition function in the dual c = 1 theory. Moreover, considering the relation (3.65) between FZZT and ZZ boundary states, one may write the instanton action as the difference between two FZZT branes located at the branch point u+ , (n) on the principal sheet, and its replica u+ , on the n sheet,
AG (n) n c=1 An = = Z FZZT (u+ ) − Z FZZT (u+ ) i¯ gs 1 = Z ZZ = dz y(z) = 2πnμ. (6.19) 2i¯ gs γn
A very similar story goes through for the Penner model, thus completing the identification of the (double-scaled) Gaussian and Penner matrix model nonperturbative effects as ZZ brane D-instantons, in the dual c = 1 string theory. Let us now turn to the case of the Chern–Simons matrix model where, much as in the local curve backgrounds studied in [10], the instanton action may be interpreted, in the dual large N description on the resolved conifold, in terms of toric branes. In this case, the disk amplitude with (mirror of) toricbrane boundary conditions may be written, in terms of the mirror curve,25 as [61,62] (0) A1 (x)
x =
ds y(s).
(6.20)
It thus follows that one may write the instanton action as the difference of two disk amplitudes; simply set
CS A i (0) (n,k) i (0) (n,k) = A1 (u+ ) − A1 (u+ ) Aconif (n,k) = i¯ gs g¯s g¯s 2πn i dz y(z) = (t + 2πik) , (6.21) =− 2¯ gs g¯s γn,k
where the toric branes have been place at the endpoint u+ and at its copy, (n,k) u+ , on the sheet labelled by n and k (recall our previous discussion). The instanton action is in this way given by the tension of the domain wall interpolating in between the two branes (see as well [13]). 25
For backgrounds with a matrix model dual, this mirror curve precisely coincides with the spectral curve.
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At this point, two comments are in order. First, when we discussed the c = 1 double-scaling limit we also checked that the spectral curve—alongside with all the open correlators—of the Chern–Simons model collapse, at the conifold point, to their corresponding c = 1 values. In particular, this implies that in this limit also the instanton action reduces to the c = 1 one which indicates, as also observed in [10], that in the double-scaling limit nonperturbative effects due to toric branes reduce to the ones due to Liouville branes in minimal models. The second observation we want to make is that the instanton effects leading to the above action (6.21) involve vector multiplet moduli. This can be realized by simply noting that the instanton action (6.21) is given by the integral of the one-form y dz on the contour γ, which are, respectively, reductions of the complex structure three-form Ω and of the three-cycle Γ for a local CY, (n,k) = i Ω, Γ = C+ − C+ , (6.22) Aconif (n,k) Γ C+
(n,k) C+
where the two-cycles and are given by line-bundles, with base points (n,k) u+ and u+ on the spectral curve. The instanton action may in this way be identified with the tension of the domain wall interpolating in-between the two branes (see as well [13]). So far we provided a spacetime interpretation of nonperturbative effects in c = 1 matrix models and topological strings in terms of Liouville and toric branes. However, there is a further, obvious, spacetime interpretation of the instanton expansion as due to BPS particle production, via the Schwinger effect that we have discussed earlier in this paper. Furthermore, it is possible to relate these particle production effects to spacetime D-brane instantons via a compactification to three dimensions, followed by a T -duality [91]. Recently, in [92], this connection has been exploited in order to study the continuity, across walls of marginal stability, of nonperturbative effects in type II compactifications.
7. Trans-Series and the Toda Equation Another approach to the calculation of instanton corrections to the free energy of matrix models and topological strings, and first developed away from criticality in [14], deals with trans-series solutions in the orthogonal polynomial framework, rather than the spectral geometry as in the preceding section. In this section we wish to learn what this approach has to say on what considers Gaussian, Penner and Chern–Simons matrix models; in particular, we wish to confirm our instanton results in yet a novel setting. Let us begin by briefly reviewing the ideas behind the trans-series approach, and then apply it within our interests. Let us first recall that in the orthogonal polynomial formalism one may compute the partition function via (3.11) once one knows the recursion coefficients {rn }. Making use of the definition (3.11) it is not too hard to obtain
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ZN +1 ZN −1 = rN . 2 ZN
(7.1)
In the continuum N → ∞ limit the coefficients rn become a function rn → n ∈ [0, t], where the function R(x, gs ) can be R(x, gs ) of the variable x = t N determined by solving the so-called pre-string equation, a finite difference equation obtained from the continuum limit of the recursion for the coefficients {rn } [18]. Analogously, the continuum limit of equation (7.1) produces the following Toda-like equation for the free energy, FN = log ZN → F (t, gs ), exp (F (t + gs , gs ) − 2F (t, gs ) + F (t − gs , gs )) = R(t, gs ).
(7.2)
Given a solution to the recursion coefficients R(x, gs ), this equation then determines the free energy of our model as a solution to the Toda hierarchy. Let us now consider trans-series solutions (in the sense of exponential asymptotics) to the equations above.26 For that, consider the trans-series ans¨ atz for the recursion coefficients +∞ R(x, gs ) = C R() (x, gs ), (7.3) =0
with the zero-instanton contribution given by R(0) (x, gs ) =
+∞
(0)
gs2n R2n (x)
(7.4)
n=0
and the -instanton contributions given by
+∞ A(x) () () − R() (x, gs ) = R1 (x) e gs gsn Rn+1 (x) , 1+
≥ 1.
(7.5)
n=1
Plugging this ans¨ atz into the pre-string equations one may, in principle, deter() mine recursively both the instanton action A and all the loop terms Rn (x). Once this is done, one finally plugs the trans-series solution for R(t, gs ) on the right-hand side of equation (7.2) and solves it with a trans-series ans¨ atz for the free energy F (t, gs ) =
+∞
C F () (t, gs ),
(7.6)
=0
where F (0) (t, gs ) =
+∞
gs2g−2 Fg(0) (t),
(7.7)
g=0
and
F () (t, gs ) =
A(t) () F1 (t) e− gs
1+
+∞
() gsn Fn+1 (t)
,
≥ 1.
(7.8)
n=1 26 Notice that these trans-series solutions will only be valid in a specific region of the complex plane; as we change sectors and cross a Stokes line the asymptotics will change. This change will be given by a shift in the nonperturbative ambiguity parameter, C, as C → C +S with S the Stokes multiplier.
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()
Again, at least in principle, all the coefficients Fn (t) may be determined recursively. In [14], this formalism has been applied and checked against a large-order analysis in several matrix models. Remarkably, this method appears to work beyond the context of matrix models: for instance, in [14], the full instanton series has been obtained for the case of Hurwitz theory, which is also controlled by a Toda-like equation. 7.1. The Trans-Series Approach for c = 1 Matrix Models Let us now try to apply the trans-series method to solve our c = 1 models. As we have just learned, the first thing to do is to look for a trans-series solution to the pre-string equation. But it so happens that it is a common feature of all our c = 1 models that the recursion relations for the coefficients {rn } may be solved exactly, without a genus expansion. In particular, for Gaussian, Penner and Chern–Simons models we find rnP rnCS = q 3n (q n − 1)
rnG = gs n
→
= gs n (1 + gs n)
→
R (x, gs ) = x (1 + x) ,
q = egs
→
RCS (x, gs ) = e3x (ex − 1) . (7.11)
with
RG (x, gs ) = x, P
(7.9) (7.10)
Since we have exact solutions for the functions R(x, gs ) ≡ R(x), without a genus expansion, there is clearly no asymptotics and thus no trans-series expansion for R(x). Thus, for all three cases that we are considering, we only have to worry about the trans-series ans¨ atz (7.8) and plug into the homogeneous Toda equation F (t + gs ) − 2F (t) + F (t − gs ) = 0.
(7.12)
This is a rather interesting point, also implying that all Borel poles are controlled by the Toda equation (and not by the pre-string equations). In hindsight this is not so surprising, as at criticality all our examples are in the universality class of the c = 1 string, and it is precisely the case that the partition function of c = 1 string theory is a τ -function of the Toda hierarchy, satisfying the Toda equation [93]. In some sense, the above Toda equation plays a role very analogous to the one played by the Painlev´e I equation in c = 0 string theory, and which was also studied in connection to the large-order behavior of topological strings in [13,15]. Let us start with the one-instanton sector. By plugging into the homogeneous Toda equation the = 1 term in (7.8), at first order in gs one obtains
A (t) (1) − A(t) 2 g s 4F1 (t) e sinh = 0, (7.13) 2 (1)
and setting F1 (t) = 0 it follows
A (t) sinh =0 2
⇔
A (t) = iπk, 2
k ∈ Z.
(7.14)
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It is quite remarkable that the solution to the above equation already includes all k-instanton actions Ak (t) = 2πit k + α(k),
(7.15)
where we have also allowed the integration constant to depend on k. Making further use of the trans-series ans¨ atz we may compute
+∞ A (t) (k) (k) − k (k) n F (t ± gs , gs ) = F1 (t) e gs gs F(n+1) , 1+ (7.16) n=1
where, for example, (k)
(k)
(k)
F(2) = F2 (t) ±
∂t F 1
(k)
(k)
(k)
(k)
F(3) = F3 (t) ±
(t),
(7.17)
F1
(k)
F2 ∂ t F 1
(k)
(k)
(t) ± ∂t F2 (t) +
1 ∂t2 F1 (t). 2 F (k)
(7.18) (k) F1 1 We now insert these expressions into the Toda equation, and solve it perturbatively in gs . At second order in the string coupling it follows, (k)
⇒
∂t2 F1 (t) = 0
(k)
F1 (t) = Φ1 (k) + Φ2 (k) t.
(7.19)
In fact, solving the infinite chain of differential equations one perturbatively obtains, in a recursive fashion, one is always led to second order differential equations. All the higher order terms are then fixed to be (n+1)
(k)
Fn+1 (t) = −
Φ1
(k)
(k) Φ2 (k) F1 (t)
(n+1)
+ Φ2
(k),
n ≥ 1.
(7.20)
The k-instanton contribution in (7.8) is thus given by 2πit k+α(k)
gs F (k) (t, gs ) = (Φ1 (k) + Φ2 (k) t) e−
+∞ (n+1) (k) Φ1 (n+1) n + Φ2 × 1+ gs − (k) . Φ2 (k) (Φ1 (k) + Φ2 (k) t) n=1
(7.21) It is now simple to see that the instanton series of our three models, that we have computed either via Borel or Stokes analysis, (4.6), (4.16) and (4.23), are all solutions to the homogeneous Toda equation.27 This confirms, within the trans-series setting, the validity of our results. However, it is not equally clear how to flow in the other direction, i.e., how to obtain our results, (4.6), (4.16) and (4.23), starting from the trans-series formalism. In particular, it is not obvious to us how to provide enough boundary conditions in order to fix, for each distinct case, the integration constants of the general solution (7.21). A natural boundary condition is to impose matching to the c = 1 solution, in the double-scaling limit [94]. But this cannot be quite enough off-criticality: in 27 Because the Toda equation is linear, linear combinations of (7.21) are also a solution to the problem, in particular linear combinations where α also depends on an integer m and we sum over m ∈ Z as in the conifold.
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fact all our distinct examples are satisfying the same difference equation with the same aforementioned boundary condition. Thus, one is forced to demand, as a final boundary condition and in order to obtain a unique solution in each case, a comparison with the large-order behavior of perturbation theory, in each distinct model we wish to study. So, at least for these models, Borel or saddle-point methods seem more powerful and efficient ways to proceed. 7.2. A Comment on Parametric Resurgence In many cases it is not possible to resum the Borel transform of an asymptotic series, or even to explicitly compute this Borel transform, and thus one cannot locate the singularities in the Borel complex plane (some of which, in particular, control the large-order behavior of the perturbation theory). However, it may be the case that, even if we are not able to resum the Borel transform, we may know that the asymptotic series arises as a solution to a finite difference equation (where we are, of course, interested in the example of the Toda equation). In this case we may still obtain some interesting information, as shown in [95]. Let us quickly apply [95] to our problem. Consider the asymptotic series F (t, gs ) ∼
+∞
gs2n Fn (t),
(7.22)
n=0
which we take as a perturbative solution to the finite difference equation
gs ∂ F (t + gs )−2F (t)+F (t − gs ) = gs2 G(t) ⇔ 4 sinh2 F (t) = gs2 G(t). 2 ∂t (7.23) Then, it is possible to formally solve this differential equation [95], with the formal solution being expressed in terms of the power-series coefficients of the function +∞ x2 H(x) = −1 + Hn x2n+2 . (7.24) 2 x = 4 sinh 2 n=0 Furthermore, the poles of the Borel transform B[F ](ξ) may be related to the poles of the Borel transform of H(x). In particular [95] this implies that in this case the Borel poles are located at 2πint, with n ∈ Z. It is indeed the case that the instanton actions for the three matrix models we have studied in this paper are of this kind.
8. Conclusions and Outlook In this paper we have addressed the nonperturbative structure of topological strings and c = 1 matrix models, focusing on Gaussian, Penner and Chern– Simons matrix models together with their holographic duals, c = 1 minimal strings and topological strings on the resolved conifold. Making use of either Borel or Stokes analysis, we have uncovered the nature of instanton effects
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in these models, and have further explored the relation of these nonperturbative phenomena to the large-order behavior of the 1/N expansion. While this builds up on previous work along the same direction [13–15,19], clarifying the discussion for a big class of models, we believe there is still much work to be done. In particular, let us end by listing several issues raised in this paper which we believe deserve further and immediate investigation (in no particular order): • While our results were checked in the trans-series formalism, we have seen that it is not obvious how to obtain these results for the multi-instanton series starting straight from within the trans-series set up. In particular, the choice of boundary conditions is not completely clear. It would be interesting to further explore and better understand the trans-series ans¨ atz in this context, possibly solving the questions we have just mentioned. • In the Gaussian and Penner models, we have explicitly shown that instanton effects may be understood as Stokes phenomena for the logarithm of the G2 (z) Barnes function. However, we could not say much along these lines for the Chern–Simons model, as we do not know of any appropriate hyperasymptotic framework for the quantum Barnes function, Gq (z). It would be very interesting to study the hyperasymptotics of Gq (z) and show that Stokes phenomena in this case is also related to the instanton effects of the Chern–Simons model. • In our matrix model derivation of the instanton effects, in terms of eigenvalue tunneling, we have obtained the instanton action expressed in terms of the spectral curve (a cycle of the y dz one-form). It would be rather interesting to extend this calculation in order to contemplate loop corrections. Indeed, in [13,15], higher loop terms around the multi-instanton configurations were computed, in terms of matrix model open correlators. Extending that calculation to the present set up would be very interesting, since these correlators can be computed entirely in terms of the spectral curve and, as such, this formalism could be extended to other topological string scenarios where a dual matrix model description is not available. In particular, this could allow for a direct understanding—from a spectral geometry point of view—of why the loop expansion around an one/multi-instanton configuration truncates in all our examples. Furthermore, this would also provide an explanation of the instanton expansion in terms of open string amplitudes with either Liouville or toric boundary conditions, for the c = 1 string and the resolved conifold, respectively. • Much of our Borel analysis was very much related to the existence of a GV integral representation for the free energies of our models. Since also on a generic CY background the topological string free energy admits a GV integral representation, one is led to wonder if our approach may be applied to other, more general cases. Recall that on a general CY threefold X , the topological string free energy is given in terms of GV integer invariants by the expansion
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FX (gs ) =
+∞ +∞
i) n(d r (X )
r=0 di =1
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+∞ ds s 2r−2 − 2πs e gs (d·t+im) . (8.1) 2 sin s 2
m∈Z 0
When r = 0 the zeros of the sine will be poles of the integrand and this contribution to the total sum will, as in the case of the resolved conifold which we address in the paper, yield a nonperturbative contribution to the free energy. However, for generic CY backgrounds, one will also have to consider (d ) the summation over the K¨ ahler classes {di }, as weighted by n0 i . In this way, one is led to write (we neglect the pole at zero) +∞ +∞ i (di ) n0 (X ) 2π¯ gs n=1 m∈Z di =1
2 (2π)2 n g¯s (2π) (d · t + im) + 2 e− g¯s (d·t+im) . × n n
gs ) = − DiscFX (¯
(8.2)
Higher terms with r > 0 in (8.1) have no poles in the complex plane and will only contribute to the nonperturbative corrections through the residues at infinity (recall (2.8) and our discussion in the appendix). What role they might play is beyond the scope of our analysis. It seems likely that nonperturbative corrections obtained from the GV representation actually provide the full nonperturbative corrections to the topological string free energy, in those cases where the number of GV invari(d ) ants nr i (X ) is finite (as in our example of the resolved conifold). Indeed, in these cases (8.2) may provide the complete tower of nonperturbative corrections to the topological string free energy, as (8.1) is basically given by a (d ) finite sum, not a power series expansion. In particular, the number of n0 i invariants is finite for non-singular curves of any genus [96], for rational curves with nodal singularities [97], and for the configurations studied in [98] (among which are the CY threefolds which are Ak -type ALE spaces, times C). Further notice that in (8.2) the loop expansion around multi-instanton configurations truncates. In this case, and if indeed (8.2) turns out to be the full answer for backgrounds with a finite number of GV invariants, then it must also be the case that these backgrounds will display no nontrivial large-order behavior in their multi-instanton sectors. Finally, observe that the Ak -type ALE backgrounds have also been studied in the context of the OSV conjecture [21], in [99]. The OSV conjecture [21] relates the topological string partition function to the partition function of a configuration of branes in type II string theory, giving rise to a 4D BPS black hole. The brane partition function was further suggested to provide a nonperturbative completion of topological string theory. For Ak -type ALE spaces, times C, it would thus be interesting to compare OSV and Schwinger completions. (d ) • When the number of GV invariants nr i (X ) is infinite, it seems very unlikely that the GV integral representation can still provide the full set of nonperturbative corrections to the topological string free energy. Indeed, it is now
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(d)
the case that n0 ∼ ed , for large d, and it seems to be the case that (8.2) cannot provide the complete nonperturbative information. A particularly interesting example to further explore this issue is that of the local curve, which indeed has an infinite number of GV invariants. The good news is that this background has been extensively studied in [13], with instanton configurations identified and checked against the large-order behavior. As such, it would be extremely interesting to analyze the relation between the nonperturbative corrections arising due to the GV representation, with the ones derived in [13]. In particular, the analysis of [13,15] seems to indicate that, for the local curve, the loop expansion around multi-instanton configurations will not truncate, which is to say that (8.2) cannot be the full correct answer. On the other hand, and as we have already remarked, the local curve is in the universality class of 2D gravity, with c = 0, and as such it should better be understood as a first step to understand backgrounds with an infinite number of GV invariants. A second natural step in this direction would be looking at the case of local P2 , a c = 1 toric geometry with an infinite number of GV invariants. Indeed, the free energy of this model can be computed very efficiently to all genus, by means of direct integration of the holomorphic anomaly equations as shown in [100], and would thus provide for a natural testing ground for our aforementioned questions. • Besides an explicit check against the large-order behavior of the theory, another way to test the Schwinger completion of topological string theory, for generic backgrounds, would be to study the modular properties of its nonperturbative free energy. In fact, it is expected that modular invariance may be recovered at the nonperturbative level [19], an important issue also in the context of large N dualities. Clearly, for the case of the resolved conifold, which we studied in this paper, there are no constraints arising from modularity since the moduli space is trivial, with the mirror geometry having genus zero, but this will not be the case for, e.g., local P2 , whose mirror geometry has a spectral curve of genus one. Notice that backgrounds with a finite number of GV invariants seem not to give rise to mirror geometries with spectral curves of genus one and, as such, modularity should become an issue precisely when (8.2) ceases to be the full correct answer. In particular, modularity may play a key role in order to understand exactly what type of information is (8.2) missing in general backgrounds, and these issues should be addressed in future work.
Acknowledgements We would like to thank Jacopo Belfi, Andrea Brini, Luca Griguolo, Jos´e Mour˜ ao, Nicolas Orantin, Christoffer Petersson, Domenico Seminara, Jorge Drumond Silva, Angel Uranga, Marcel Vonk and, specially, Marcos Mari˜ no, for useful discussions, comments and/or correspondence. RS would like to thank CERN TH, Division for hospitality, where a part of this work was conducted.
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Appendix A. The Polylogarithm: Branch Points and Monodromy This appendix is devoted to the study of the polylogarithm, with emphasis towards its branch points and monodromy. This function is defined by Lip (z) =
+∞ n z , np n=1
(A.1)
and on its principal sheet it has a branch point at z = 1 leading, by convention, to a branch cut discontinuity in the complex z plane running from 1 to infinity. As one starts “exploring” the multi-sheeted structure of the polylogarithm and moves off its principal sheet, one finds that there exists another branch point, at z = 0. In this case, the resulting monodromy group will be generated by two elements, acting on the covering space of the bouquet S 1 ∨S 1 of homotopy classes of loops in C\{0, 1}, passing around the branch points z = 0 or z = 1. For further details, we refer the reader to the very thorough explanations that can be found in, e.g., [89,90]. For our purposes in this paper, a simple analysis in terms of explicit topological language will suffice. Let m1 represent the homotopy class of all loops based at some point z in C, which wind once, clockwise around the branch point at z = 1. The action of m1 on the polylogarithm has the effect of carrying this function from one sheet to the next. It was shown in [90] that one may write m1 · Lis (z) = Lis (z) − Δ1 ,
(A.2)
where Δ1 is a function, whose specific form is not important at the moment, but which includes a logarithm with a branch point at z = 0. This implies that, after acting once with m1 , one finds oneself on a sheet which has a branch cut discontinuity running from 0 to minus infinity. If we now let m0 represent the homotopy class of all loops based at some point z in C, which wind once, clockwise around this new branch point at z = 0, its action on the logarithm is the familiar one: m0 · log z = log z + 2πi.
(A.3)
Now, because the principal sheet of the polylogarithm has no branch point at z = 0, it simply follows m0 · Lis (z) = Lis (z).
(A.4)
If one now winds with m1 in the opposite direction, one is led to write instead m−1 1 · Lis (z) = Lis (z) − Δ−1 ,
(A.5)
where again Δ−1 is a function we shall leave unspecified; see [90] for details. If m1 is to be properly considered the group-theoretic inverse of m1 it better −1 be the case that m1 · m−1 1 = 1 = m1 · m1 , when acting on Lis (z). This immediately implies, e.g., m−1 1 Δ1 = −Δ−1 , where we recall that Δ1 includes the standard logarithmic branch cut discontinuity starting off at z = 0. This relation thus seems odd, as the logarithm has no branch point at z = 1 and there
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should be nothing to wind around. This is a subtle point, further explained in [90], and it should be stressed that it is the joining of polylogarithmic and logarithmic cuts that causes this effect. One way to capture the idea of there being no obstruction for the logarithm at z = 1 is [90] to define group elements g1 = m1 · m−1 0 and g0 = m0 , such that g1 · log z = log z and g0 · Lis (z) = Lis (z), g1 · Lis (z) = Lis (z) − Δ1 .
(A.6) (A.7)
We shall explore this monodromy group for the dilogarithm in the following. The free combinations of powers of the two generators g0 and g1 form the monodromy group of the polylogarithm. If s is a positive integer, this monodromy group has a finite-dimensional representation with dimension s + 1. A particularly well-known case is the dilogarithm s = 2, also further discussed in [89]. In this case, the monodromy group is the discrete Heisenberg group [90]. In particular, one finds Δn = 2πi (log z + 2πi (n − 1)) .
(A.8)
As such, repeated applications of g0 and g1 will only result in linear combinations of the dilogarithm Li2 (z), the logarithm log z, and the identity operator. Indeed, one could further take each of these three elements as a basis of a 3D vector space, e1 = 4π 2 , e2 = −2πi log z and e3 = Li2 (z), in which case the matrix representation of the monodromy group would become ⎡ ⎤ ⎡ ⎤ 1 0 0 1 1 0 (A.9) g1 = ⎣ 0 1 1 ⎦ and g0 = ⎣ 0 1 0 ⎦ . 0 0 1 0 0 1 These two matrices are in fact the generators of the discrete Heisenberg group H3 (Z), see [90] for full details on this discussion. In this paper, we are interested in the following action (see [90] for any missing details) g0k · g1n · Li2 (z) = g0k · (Li2 (z) − n Δ1 ) = Li2 (z) − n Δk+1 = Li2 (z) − 2πi n log z + 4π 2 k n, (A.10) and we make use of this result in the main body of the paper.
Appendix B. Dispersion Relation for Topological Strings In this appendix we address the Cauchy dispersion relation (2.8) for the case of general topological string theories on a CY threefold X , whose free energy is given in terms of GV integer invariants by the expansion (2.37), and which we recall in here as +∞ +∞ +∞ ngs 2r−2 −2πn d·t 1 i) 2 sin FX (gs ) = n(d (X ) e . (B.1) r n 2 r=0 n=1 di =1
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In particular, we wish to evaluate
Ann. Henri Poincar´e
dw FX (w) , 2πi w − g
(B.2)
(∞)
and show that this vanishes in the case of the resolved conifold, a result we have used in the main body of the paper. Notice that one cannot compute the pole at infinity straight: infinity is an essential singularity of the integrand in the GV representation and, as such, residue calculus does not apply. Nonetheless, for r = 0 one may decompose the contour C∞ into the sum of two contours in upper and lower hemispheres, C+ and C− , respectively, 1 dw 1 2πi w − g 2 sin nw 2 2 C∞ einw e−inw dw dw = + , (B.3) 2 2πi (w − g) (1 − einw ) 2πi (w − g) (1 − e−inw )2 C+
C−
and then use Jordan’s lemma—applicable as lim
max
R→+∞ θ∈[0,π]
(Reiθ
1 2 = 0 − g) 1 − einReiθ
(B.4)
in the upper hemisphere, and analogously in the lower—in order to find that this implies that the integral vanishes at infinity, and it thus follows that for (1) the resolved conifold, where only n0 = 1 is non-zero, the (B.2) contribution indeed vanishes. For more complicated CY threefolds with r ≥ 1, it seems rather likely that there will be a Cauchy contribution at infinity, and a complete analysis of this situation is beyond the scope of the present work.
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[71] Alexandrov, S.: (m, n) ZZ Branes and the c = 1 Matrix Model. Phys. Lett. B 604, 115 (2004). arXiv:hep-th/0310135 [72] Bertoldi, G., Hollowood, T.J.: Large N gauge theories and topological cigars. JHEP 0704, 078 (2007). arXiv:hep-th/0611016 [73] Adamchik, V.S.: Contributions to the theory of the Barnes function. arXiv: math/0308086[math.CA] [74] Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic Press, London (2007) [75] Berry, M.V., Howls, C.J.: Hyperasymptotics. Proc. R. Soc. Lond. A 430, 653 (1990) [76] Boyd, J.P.: The Devil’s invention: asymptotic, superasymptotic and hyperasymptotic series. Acta Appl. Math. 56, 1 (1999) [77] Berry, M.V., Howls, C.J.: Hyperasymptotics for integrals with saddles. Proc. R. Soc. Lond. A 434, 657 (1991) [78] Boyd, W.G.C.: Error bounds for the method of steepest descents. Proc. R. Soc. Lond. A 440, 493 (1993) [79] Friot, S., Greynat, D.: Non-perturbative asymptotic improvement of perturbation theory and Mellin–Barnes representation. arXiv:0907.5593[hep-th] [80] Paris, R.B., Kaminski, D.: Asymptotics and Mellin–Barnes Integrals. Cambridge University Press, Cambridge (2001) [81] Boyd, W.G.C.: Gamma function asymptotics by an extension of the method of steepest descents. Proc. R. Soc. Lond. A 447, 609 (1994) [82] Berry, M.V.: Infinitely many stokes smoothings in the gamma function. Proc. R. Soc. Lond. A 434, 465 (1991) [83] Koshkin, S.: Quantum Barnes function as the partition function of the resolved conifold. arXiv:0710.2929[math.AG] [84] Maldacena, J.M., Moore, G.W., Seiberg, N., Shih, D.: Exact vs. semiclassical target space of the minimal string. JHEP 0410, 020 (2004). arXiv:hepth/0408039 [85] Eynard, B.: Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence. JHEP 0903, 003 (2009). arXiv:0802.1788[math-ph] [86] Berry, M.V.: Uniform asymptotic smoothing of Stokes’ discontinuities. Proc. R. Soc. Lond. A 422, 7 (1989) [87] Chapman, S.J.: On the non-universality of the error function in the smoothing of stokes discontinuities. Proc. R. Soc. Lond. A 452, 2225 (1996) [88] Bonnet, G., David, F., Eynard, B.: Breakdown of universality in multicut matrix models. J. Phys. A 33, 6739 (2000). arXiv:cond-mat/0003324 [89] Maximon, L.C.: The dilogarithm function for complex argument. Proc. R. Soc. Lond. A 459, 2807 (2003) [90] Vepˇstas, L.: An Efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions. Numer. Algorithm 47, 211 (2008). arXiv:math/0702243 [91] Ooguri, H., Vafa, C.: Summing up D-Instantons. Phys. Rev. Lett. 77, 3296 (1996). arXiv:hep-th/9608079
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[92] Collinucci, A., Soler, P., Uranga, A.M.: Nonperturbative effects and wall-crossing from topological strings. arXiv:0904.1133[hep-th] [93] Dijkgraaf, R., Moore, G.W., Plesser, R.: The partition function of 2D string theory. Nucl. Phys. B 394, 356 (1993). arXiv:hep-th/9208031 [94] Kazakov, V., Kostov, I.K., Kutasov, D.: A matrix model for the two-dimensional black hole. Nucl. Phys. B 622, 141 (2002). arXiv:hep-th/0101011 [95] Sauzin, D.: Resurgent functions and splitting problems. arXiv:0706.0137 [math.DS] [96] Bryan, J., Pandharipande, R.: BPS states of curves in Calabi–Yau Threefolds. Geom. Topol. 5, 287 (2001). arXiv:math/0009025 [97] Bryan, J., Katz, S., Leung, N.C.: Multiple covers and the integrality conjecture for rational curves in Calabi–Yau threefolds. J. Algebraic Geom. 10, 549 (2001). arXiv:math/9911056 [98] Karp, D., Liu, C.-C.M., Mari˜ no, M.: The local Gromov–Witten invariants of configurations of rational curves. Geom. Topol. 10, 115 (2006). arXiv:math/0506488 [99] Aganagic, M., Jafferis, D., Saulina, N.: Branes, black holes and topological strings on toric Calabi–Yau manifolds. JHEP 0612, 018 (2006). arXiv:hepth/0512245 [100] Haghighat, B., Klemm, A., Rauch, M.: Integrability of the holomorphic anomaly equations. JHEP 0810, 097 (2008). arXiv:0809.1674[hep-th] Sara Pasquetti Theory Division Department of Physics CERN 1211 Geneva 23, Switzerland e-mail:
[email protected] Ricardo Schiappa Departamento de Matem´ atica Instituto Superior T´ecnico CAMGSD Av. Rovisco Pais 1 1049–001 Lisbon, Portugal e-mail:
[email protected] Communicated by Marcos Mari˜ no. Received: March 7, 2010. Accepted: April 8, 2010.
Ann. Henri Poincar´e 11 (2010), 433–497 c 2010 Springer Basel AG 1424-0637/10/030433-65 published online May 21, 2010 DOI 10.1007/s00023-010-0032-9
Annales Henri Poincar´ e
Non Linear Perturbations of Kerr Spacetime in External Regions and the Peeling Decay Giulio Caciotta and Francesco Nicol`o Abstract. We prove, outside the influence region of a ball of radius R0 centred in the origin of the initial data hypersurface, Σ0 , the existence of global solutions near to Kerr spacetime, provided that the initial data are sufficiently near to those of Kerr. This external region is the “far” part of the outer region of the perturbed Kerr spacetime. Moreover, if we assume that the corrections to the Kerr metric decay sufficiently fast, o(r−3 ), we prove that the various null components of the Riemann tensor decay in agreement with the “Peeling theorem”.
1. Introduction 1.1. The Problem and the Results The problem of the global stability for the Kerr spacetime is a very difficult and open problem. The more difficult issue is that of proving the existence of solutions of the vacuum Einstein equations with initial data “near to Kerr” in the whole outer region up to the event horizon.1 What is known up to now relative to the whole outer region are some relevant uniform boundedness results for solutions to the wave equation in the Kerr spacetime used as a background spacetime (see Dafermos and Rodnianski [7], Klainerman [9]).2 If we consider the existence problem in an external region sufficiently far from the Kerr event horizon for a slow rotating Kerr spacetime, the result is included in the version of Minkowski stability result proved by S. Klainerman and one of the present author, F. Nicol` o, see [10] and also [6]. In this case F. Nicol` o, recently proved [15], that the asymptotic behaviour of the Riemann components is in agreement with the “Peeling theorem” if the corrections to the Kerr initial data decay sufficiently fast. 1 2
Which is also an unknown of the problem. See also for the J = 0 case, [1] and references therein.
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In this paper, we prove the non-linear stability of the Kerr spacetime for any J ≤ M 2 and an appropriate class of initial data near to Kerr, in an external region3 where4 r ≥ R0 and M/R0 ≤ λ with λ sufficiently small. Moreover, if we restrict the class of initial data, once subtracted the Kerr part, to those which decay toward the spacelike infinity faster than r−3 , we prove that the null asymptotic decay of the Riemann tensor is in agreement with the “Peeling theorem”. Our main result is, in a somewhat preliminary version, the detailed version in Sect. 4, Theorem 1.1. Assume that initial data are given on Σ0 such that, outside of a ball centred in the origin of radius R0 , they are different from the “Kerr initial data of a Kerr spacetime with mass M satisfying M 1, J ≤ M 2 R0 for some metric corrections decaying faster than r−3 towards spacelike infinity together with its derivatives up to an order q ≥ 4, namely5 (Kerr)
gij = gij
γ
+ oq+1 (r−(3+ 2 ) ),
(Kerr)
kij = kij
γ
+ oq (r−(4+ 2 ) )
(1.1)
where γ > 0. Let us assume that the metric correction δgij and the second fundamental form correction δkij are sufficiently small, namely that we can define a function made by L2 norms on Σ0 of these quantities and require it to be small: J (Σ0 , R0 ; δ (3)g, δk) ≤ ε,
(1.2)
defined outside the then this initial data set has a unique development, M, can be foliated by a canonical doudomain of influence of BR0 . Moreover, M ble-null foliation {C(u), C(u)} whose outgoing leaves C(u) are complete6 and the various null components of the Riemann tensor relative to a null frame associated with this foliation decay as expected from the Peeling theorem. The proof of this result depends on many previous results: the proof of the stability of the Minkowski spacetime in the external region by Klainerman and Nicol` o, in [10], a result in turn based on the seminal work by Christodoulou and Klainerman [6], and concerning the peeling decay, important ideas of the proof come from the previous work by Klainerman and Nicol` o [11] and the recent [15]. We observe that the two results proved in this work, the global stability in the external region and the asymptotic decay in agreement with the peeling, are basically independent; relaxing the decay conditions on the initial data we can prove the stability with a worst null asymptotic decay. In this paper the 3
See the remark after the statement of Theorem 4.1 in Sect. 4 r is a radial coordinate which will be defined later on. 5 The components of the metric tensor written in dimensional coordinates. f = o (r −a ) q means that f asymptotically behaves as o(r −a ) and its partial derivatives ∂ k f , up to order q behave as o(r −a−k ). 6 By this we mean that the null geodesics generating C(u) can be indefinitely extended toward the future. 4
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two results are proved together, but it is easy to enlarge the class of initial data to prove only the first one, proceeding as in [11] where it has been shown in detail, for the perturbed Minkowski spacetime, how the spacelike decays of the initial data are connected to the null decay of the Riemann components. Therefore, as many steps to prove Theorem 1.1 have been discussed in previous works, we prove here the new part of this result, namely Theorem 3.1 which is the core result to obtain, via a bootstrap mechanism, the global existence and the decay satisfying the “Peeling theorem”. In the remaining part of the introduction we examine the difficulties one encounters to prove this result and how they have been solved. 1.2. The Global Existence in an External Region Around the Kerr Spacetime To understand the problem of perturbing around the Kerr spacetime it is appropriate to remember how the problem of perturbing around the Minkowski spacetime has been solved. The general strategy is a “bootstrap mechanism”: one proves (given a local existence result) that there is a finite region V , whose metric satisfies the Einstein equations, endowed with some specific properties, mainly that some norms associated with the metric components and its derivatives are bounded by a (small) constant, then assumes that the largest possible region, V∗ , where these “bootstrap bounds”, hold is finite and proves that, if the initial data are sufficiently small, the previous bounds can be improved. Therefore this region can be extended, and to avoid a contradiction, has to coincide with the whole spacetime. More precisely, we assume that V∗ is endowed with a foliation made by outgoing and incoming null cones, {C(λ)} and {C(ν)} and that the norms we assume bounded are those relative to the connection coefficients and to the components of the Riemann tensor. The central part of the proof is, therefore, to show that these norms can have better bounds. To do it we use in the manifold V∗ the structure equations and the Bianchi equations. The structure equations are transport equations for the connection coefficients along the incoming and outgoing cones and elliptic Hodge systems on the two-dimensional surface intersections of the incoming and an outgoing cones, S = C ∩ C. These equations are inhomogeneous equations whose inhomogeneous part depends on the Riemann components. The Bianchi equations, at their turn, can be written as transport equations for the Riemann components along the cones whose inhomogeneous part is made by products of the Riemann components and the connection coefficients. To use these equations a sort of “linearization” is done, namely we consider the Riemann components as external sources satisfying the “bootstrap bounds” and show, using the equations for the connection coefficients, that the bounds of the connection coefficients can be improved. Then we control the Riemann components; to do it, we use in a crucial way the fact that it is possible to define for the Riemann components a set of norms, globally denoted by Q, weighted L2 integrals along the outgoing and incoming cones, of the Bel-Robinson tensor, which play the role of the energy norms associated with the Bianchi equations. Assuming the “bootstrap bounds” for the connection
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coefficients, we prove that these energy norms are bounded by the corresponding initial data norms and using the Bianchi equations obtain finally, better norms for the Riemann components. If we ask ourselves how to transport this strategy to perturb non-linearly around the Kerr spacetime solution instead than around the Minkowski one, we realize that the main difference is, in broad terms, that now we are perturbing around a solution different from zero, while the Minkowski spacetime can be considered a “zero solution”. That is, in the Minkowski spacetime all the connection coefficients are identically zero with the exception of the second null fundamental forms χ and χ (which, due to the spherical symmetry, reduce to the two scalar functions trχ = 2r−1 and trχ = −2r−1 ). Moreover, all the Riemann components are identically zero. Therefore, to prove that the norms are bounded and small we do not have to subtract their Minkowski part (with the exception of trχ and trχ). The Kerr spacetime is not a “zero solution” and some kind of subtraction has to be done. This “subtraction” mechanism is delicate as we are not looking for a linearly perturbed solution,7 and it is realized through four different steps: (i)
We state the “bootstrap assumptions” in V∗ for the corrections of the connection coefficients and the Riemann tensor, that is for the various components to which we have subtracted their Kerr parts.8 Symbolically, δO = O − O(Kerr) ,
δR = R − R(Kerr)
(1.3)
and in some detail, see [10, Chapter 3], for all the definitions,
(ii)
δχ = χ − χ(Kerr) ,
δχ = χ − χ(Kerr) ,
δζ = ζ − ζ (Kerr) . . .
δα = α − α(Kerr) ,
δβ = β − β (Kerr) , . . .
Due to (i) we write the structure equation (in the V∗ region) instead that for the connection coefficients, χ, χ, ζ, ω, ω . . . . for their corrections.9 Recalling the proof for the Minkowski stability, these equations have inhomogeneous terms which depend on the Riemann tensor, and in this case, on the correction to the Riemann tensor δR = R − R(Kerr) . We use these modified structure equations to obtain better estimates for the norms of these connection coefficient corrections, δO, provided we have a control for the norms of the Riemann components corrections, δR.
7
Observe that also the linear perturbation around Kerr has some problems due again to the fact that the Riemann tensor in Kerr spacetime is different from zero, see [1] and reference therein. 8 More precisely the Kerr part “projected” on the V foliation, see the details in Sect. 2.2.2. ∗ 9 The technical details for this “Kerr decoupling” for the connection coefficients are discussed later on.
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(iii)
The third step consists in obtaining estimates for the Riemann components corrections. This requires to subtract the Kerr part of the Riemann tensor. This cannot be done in a direct way as the basic step to control the Riemann norms is to prove the boundedness of the energy type norms, Q. We would need analogous norms for the δR corrections again with a positive integrand,10 to get from them estimates for the (correction of the) null Riemann components, but to define them is a difficult and unsolved task. We proceed in a different way based on the fact that the Kerr spacetime is sta∂ is a Killing vector field. Therefore, if, instead of considering the tionary and ∂t Riemann components, we consider their time derivatives, they do not depend anymore on the Kerr part of the Riemann tensor, their initial data can have a better decay, and if we control their Q norms we can obtain a good control of the δR norms in V∗ and also a good asymptotic decay along the null direc∂ is tions. This argument is not rigorous as in the perturbed Kerr spacetime ∂t not anymore a Killing vector field, but it turns out that the basic idea can be implemented in the following sense: Instead of the time derivative of the Riemann tensor we define its (modˆ T R, where T0 , whose precise definition will be given ified11 ) Lie derivative L 0 ∂ later on and is equal to ∂t in the Kerr spacetime, is not anymore a Killing ˜ norms relative vector field, but only “nearly Killing”,12 then we define some Q ˆ to LT0 R with appropriate weights in the integrand and prove that they are bounded in terms of the corresponding quantities written in terms of the initial data; from it we can prove, after quite a few steps, that the δR norms are smaller than that assumed in the bootstrap assumptions and satisfy appropriate decays. In V∗ we can build a null frame, {e3 , e4 , e1 , e2 }, adapted to the foliation, where e1 , e2 are vector fields orthonormal and tangent to the two-dimensional surfaces S = C(λ) ∩ C(ν) while e3 , e4 are proportional to the null geodesics generating the cones and in the coordinates {u, u, ω 1 , ω 2 }13 have the following expressions: 1 ∂ 1 ∂ + X(Kerr) , + X ; 3 = (1.4) e4 = Ω ∂u Ω ∂u while if we were considering the Kerr metric (see Sect. 2 for greater details), ∂ 1 (Kerr) = + X(Kerr) ; e4 Ω(Kerr) ∂u (1.5) ∂ 1 (Kerr)
3 + X(Kerr) . = Ω(Kerr) ∂u 10
See the explicit expression of the Q norms in Minkowski case in [10, Chapter 3]. See later for its precise definition. 12 With “nearly Killing” we mean that its deformation tensor is small with respect to some Sobolev norms. 13 u and u are the affine parameters of the null geodesics generating the null cones, ω 1 , ω 2 are the angular variables. 11
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with ∂ ∂ , = ωB ∂ω 2 ∂φ are defined in Eq. 2.33, From this we define X(Kerr) = ωB
where ωB and Ω(Kerr)
X + X(Kerr) ∂ ∂ Ω (e3 + e4 ) − = + . (1.6) 2 2 ∂u ∂u ˜ norms have to It is important to point out that the integrands of the Q be a sum of non-negative terms (see for instance [10], Chapter 3 equations (3.5.1),. . . ,(3.5.3)), which requires that the Bel-Robinson tensor has to be saturated by appropriate vector fields, linear combinations of e3 , e4 with positive weights. Therefore, as in [10] we saturate the Bel-Robinson tensor with the following vector fields: 1 1 2 1 2 e4 + τ− e3 ) (1.7) T = (e4 + e3 ), S = (τ+ e4 + τ− e3 ), K = (τ+ 2 2 2 where τ + = 1 + u 2 , τ− = 1 + u 2 . T0 =
These vector fields, when we perturb Minkowski spacetime, are nearly Killing; here, they are not Killing vectors even in the Kerr spacetime.14 The relevance of the T, S, K vector fields in the present case is connected to the fact that they are non spacelike fields in the region outside the ergosphere and made by e3 and e4 , the null vectors of the frame adapted to the foliation which have the property that the fields N = Ωe4 , N = Ωe3 are equivariant vector fields. 1.3. The Decay of the Riemann Components, the Peeling Beside the proof of the Kerr stability in a region with r ≥ R0 M , we prove that the null Riemann components have a null asymptotic decay consistent with the “Peeling theorem”. This result has already been obtained in [15], if we restrict ourselves to the perturbation of a very slow rotating Kerr spacetime or to a “very external region” which was defined through the condition 1
˜ 2 M ≤ λR 0
(1.8) ˜ where λ is a small number depending on the smallness of the initial data. As discussed in [15], the advantage of restricting to this “much farther” region is that we do not have to prove again a global existence result as the Kerr part of the metric can itself be considered a perturbation of the Minkowski one satisfying the conditions of [10]. In this paper, once we have proved a global existence result, the way to prove again the “peeling decay”, now in a much larger region, is basically the same as the one discussed in [15]. We sketch now the main ideas involved and we refer to [15] and to next sections for a more detailed discussion. In [6,10], the null asymptotic behaviour of some of the null components of the Riemann tensor, specifically the α and the β components, see later for 14
Perturbing Minkowski spacetime we can choose T = T0 as in Minkowski spacetime ∂ T = ∂t .
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their definitions, is different from the one expected from the “Peeling Theorem”, [16], as the proved decay is slower. More precisely, the components α and 7 β 15 do not follow the “Peeling theorem” decaying both as r− 2 while we expect r−5 and r−4 , respectively.16 In a subsequent paper, [11], S. Klainerman and F. Nicol` o proved that the decay suggested from the “Peeling theorem” could be obtained assuming stronger spacelike decays for the initial data. Unfortunately, that result required an initial data decay too strong for proving the “peeling decay” in spacetimes near to Kerr. To show how this result can be improved let us first recall it in some detail. In [11] the following result was proved: Theorem. Let assume that on Σ0 /B the metric and the second fundamental form have the following asymptotic behaviour17 gij = gS ij + Oq+1 (r−(3+) )
(1.9)
kij = Oq (r−(4+) )
where gS denotes the restriction of the Schwarzschild metric on the initial hypersurface. Let us assume that a smallness condition for the initial data is satisfied.18 Then along the outgoing null hypersurfaces C(u) (of the external region) the following limits hold, with < and u and u the generalization of the Finkelstein variables u = t − r∗ , u = t + r∗ in the Schwarzschild spacetime19 : lim
r(1 + |u|)(4+) |α| = C0 ,
lim
r3 |ρ| = C0 ,
lim
r4 (1 + |u|)(1+) |β| = C0 ,
C(u);u→∞ C(u);u→∞ C(u);u→∞
lim
C(u);u→∞
lim
C(u);u→∞
r2 (1 + |u|)(3+) |β| = C0
r3 |σ| = C0
(1.10)
sup r5 (1 + |u|) |α| ≤ C0 .
(u,u)∈K
This result was obtained, basically, in two steps. First, we proved that ˜ of the same type as those used to prove a family of energy-type norms, Q the global existence near Minkowski in [10], but with a different weight in the integrand, were bounded in terms of the initial data. The new weights are the previous weights multiplied by a function |u|γ with appropriate γ > 0, and ˜ the central point is that the extra terms appearing in the “Error” of the Q norms have a definite sign and can be discarded. This allowed to prove that the various null components of the Riemann tensor, beside the decay in r, have a decay factor in the |u| variable. In the second crucial step it was proved that, 15
Components relative to a null frame adapted to the null outgoing and incoming cones which foliate the “external region”. 16 In principle some log powers can be present, see Kroon [13]. 17 Here, f = O (r −a ) means that f asymptotically behaves as O(r −a ) and its partial derivq atives ∂ k f , up to order q behave as O(r −a−k ). Here, with gij we mean the components written in Cartesian coordinates. 18 The details of the smallness condition are in [11]. 19 α, β, . . . are the null components of the Riemann tensor defined with respect to a null frame adapted to the double null foliation, see [10, Chapter 3], and later on. The norm | · | is, in this case, the sup norm relative to the S 2 -sections of C(u).
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integrating along the incoming cones, the extra decay in the |u| variable can be transformed in an extra decay in the r variable proving the final result. This result cannot be immediately translated to the present case as the required decay for the initial data does not admit initial data near Kerr. The way out, see [15], is the one used to prove global stability: instead of looking directly to the decay of the Riemann components, we look at the decay of their (modified) Lie derivative with respect to the nearly Killing vector field T0 . This basically subtracts the Kerr part, allows to prove the boundedness of the ˜ norms, proves that, after a “time” integration, δR satisfies bounds modified Q which allow to extend the region V∗ proving global existence and finally shows that the null components of the Riemann tensor satisfy the peeling decay. Summarizing the previous discussion, we can say that together with the proof of the “peeling decay” the relevant result of this paper is that we are able to extend the external region where the global solution of a non linear Kerr perturbation does exist, with respect to [10,15]. Nevertheless it has to be pointed out that, even technically improving the kind of estimates we are using here, this strategy does not allow us to cover the whole outer region, (that is from the event horizon on). This should be evident already looking at [1] for the Schwarzschild case where it is clear that the control of the “error”, crucial to control the “energy-norms”, is very problematic, even in the linear case, around the so-called “photosphere region”. Therefore, this is the more M 1 is required in Theorem 1.1, fundamental reason why the assumption R 0 and it is obvious that a different approach is required to cope with this region in the non-linear case.
2. The Bootstrap Assumptions The main difference with [10] and also [11] is that, in this case, the initial data we are considering are a perturbation of Kerr initial data (in Σ0 /BR0 ) and therefore, we do not assume the ADM mass small. We denote with O the connection coefficient norms defined as in [10] and we make specific assumptions on them. We denote by R the norms associated with the various null Riemann components, where ρ − ρ and σ − σ are in place of ρ and σ. The choice of norms as | · |∞ , | · |p,S or | · |L2 norms follows exactly the pattern of [10]; here we are interested in their weight factors and their smallness; therefore we denote all the norms with | · |. 2.1. The Null Canonical Foliation and the Metric in V∗ The Kerr connection coefficients satisfy transport equations similar to those in [10] with respect to a double null cone foliation of the Kerr spacetime. On the other side, the connection coefficients of the perturbed Kerr spacetime satisfy the transport equations with respect to a double null cone foliation of the perturbed Kerr spacetime. This implies that to subtract the Kerr part (to these transport equations) we have to control the difference between the double null cone foliation of the Kerr spacetime and the one associated with the perturbed Kerr and prove this difference “small”. This also has to be part
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of the bootstrap mechanism and requires that the bootstrap assumptions on the connection coefficient terms imply analogous estimates at the level of the corrections to the metric components, we denote hereafter globally δO(0) . The double null cone foliation: We assume that a, possibly finite, region V∗ exists whose boundary is made by the union of V∗ ∩ Σ0 , ∂V∗1 and ∂V∗2 , the first part being a spacelike hypersurface the second and the third two null hypersurfaces, the first incoming and the second outgoing; we also assume that in this region we have a metric whose components satisfy the Einstein vacuum equations. We can solve the eikonal equation g μν ∂μ w∂ν w = 0
(2.11)
20
choosing as initial data a function u0 on V∗ ∩ Σ0 or a function u0 on ∂V∗ 1 . Let us call u(p) and u(p) these solutions, respectively. The level hypersurfaces u(p) = u, u(p) = u define two family of null hypersurfaces we call null cones in analogy with the Minkowski case and denote C(u), C(u). These two families form the “double null cone foliation”. The null tangent vector fields of the geodesics generating the C and C null “cones” are ∂ ∂ (2.12) Lμ = −g μν ∂ν u ν ; Lμ = −g μν ∂ν u ν ∂x ∂x and the “lapse function” Ω is defined through the relation g(L, L) = −(2Ω2 )−1 . Associated with the double null cone foliation we define two null fields {e3 , e4 }: e4 = 2ΩL,
e3 = 2ΩL
(2.13)
such that g(e3 , e4 ) = −2. Given the double null foliation we define S(u, u) = C(u)∩C(u). On each S(u, u) we define two vector fields {ea }, a ∈ {1, 2} orthonormal to e3 , e4 , obtaining at each point p ∈ V∗ a null orthonormal frame. The foliation made by the two-dimensional surfaces {S(u, u)} is null outgoing and null incoming integrable. This means that the distributions and made by {e4 , e1 , e2 } and by {e3 , e1 , e2 }, respectively, are integrable. Moreover the integrability property of the S-foliation implies that the connection coefficients ξ and ξ, see for their definitions equations 2.44, are identically zero and that the second null fundamental forms are symmetric. The frame {e4 , e3 , e1 , e2 } is called the “adapted (to the double null foliation) frame”. One can have different double null foliations choosing different “initial data”, and among the initial data we choose some specific ones we will discuss later on, see Sects. 3.5, 3.6 and also [10, Chapter 3], and the associated foliation will be called “double null canonical foliation”. In conclusion, the first of the “bootstrap assumptions” we are stating is the following one: (Assumption I): V∗ is endowed with a double null canonical foliation. 20
u0 defines an appropriate radial foliation of Σ0 /BR0 , see Sect. 3.6.
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2.1.1. The “Adapted” Coordinates and the Adapted Null Frame. Theorem 2.1. let us assume V∗ be endowed with a double null cone foliation; then, in the “adapted” coordinates the metric tensor has the following form: g(·, ·) = −4Ω2 dudu + γab dω a − (X(Kerr) a du + X a du) × dω b − (X(Kerr) b du + X b du) , (2.14) where
∂ ∂ g(L, L) , X = X a a , X(Kerr) = ωB 2 (2.15) 2 ∂ω ∂ω Proof. Let us choose u(p) and u(p) as coordinates. As they satisfy the eikonal equation it follows that Ω=
−
g uu = g uu = 0.
(2.16)
Therefore, we have from (2.12), where the coordinates {xa } are still generic ones, ∂ ∂ g au ∂ g au ∂ uu uu + + L = −g ; L = −g . (2.17) ∂u g uu ∂xa ∂u g uu ∂xa The vector fields, see [10, Chapter 3], ∂ ∂ g au ∂ g au ∂ + + N= ; N= (2.18) ∂u g uu ∂xa ∂u g uu ∂xa are equivariant. This means that the diffeomorphism generated by them sends a surface S(u, u) = C(u) ∩ C(u) to another surface S on the same outgoing or incoming cone, respectively. To specify the “angular” coordinates {x1 , x2 }, we proceed in two steps. We consider the diffeomorphism generated by N , we denote it Φλ , which sends S(u, u) to S(u + λ, u). Let p ∈ S(u, u) there exist a point p0 ∈ C ∩ Σ0 such that p = Φ(u; p0 ). Let us denote the “angular” coordinates on Σ0 , ω01 , ω02 and make the following choice for the angular coordinates of p: x1 (p) = ω01 (p0 ),
x2 (p) = ω02 (p0 ).
Therefore, the integral curve of the vector field N , Φ(λ; p0 ), in these coordinates is Φu (λ; {u, ω0b }) = λ Φu (λ; {u, ω0b }) = u Φ
a
(λ; {u, ω0b })
=
(2.19)
ω0a
and ∂ . ∂u To have an expression for N and N as similar as possible to the one in Kerr spacetime, in the Pretorius–Israel coordinates, see [8], we perform a change of coordinates {xμ } ≡ {u, u, x1 , x2 } → {u, u, ω 1 , ω 2 } ≡ {y μ } where N=
ω a = xa + f a (λ, u, {ω b }).
(2.20)
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and from it ∂ N =N = ∂xμ μ
∂y ν N ∂xμ μ
∂ ∂f a ∂ ∂ + = . ∂y ν ∂u ∂λ ∂ω a
443
(2.21)
We choose f a (λ, u, {ω b }) such that ∂f a = δ2a ωB (λ, u, ω 1 ) (2.22) ∂λ where the explicit expression of the function ωB (λ, u, ω 1 ) will be given later on. Therefore, with this change of coordinates, ∂ ∂ ∂ + ωB 2 ≡ + X(Kerr) . (2.23) ∂u ∂ω ∂u Once N has been defined there is no more freedom in the expression of the equivariant vector field N ; an explicit calculation, see [10, Chapter 3], gives N=
[N, N ] = −4Ω2 ζ(ea )ea ,
(2.24)
with ζ, the “torsion” connection coefficient, which in the adapted frame is 1 ζ(ea ) = g(Dea e4 , e3 ). (2.25) 2 ∂ ∂ + δX = X(Kerr) + δX + X where X = ωB (2.26) Therefore N = ∂u ∂φ and δX satisfies the equation, which will be needed later on, ∂N X(Kerr) − ∂N (X(Kerr) + δX) = −4Ω2 ζ(ea )ea .
(2.27)
Associated with the double null cone foliation we define an adapted null orthonormal frame: ∂ ∂ 1 1 Le4 = 2ΩL = N, e3 = 2ΩL = N , e1 = e11 1 , e2 = e22 2 , Ω Ω ∂ω ∂ω (2.28) where e1 , e2 are S tangent vector fields orthonormal and orthogonal to e3 , e4 . At the generic point of V∗ , of the coordinates {u, u, θ, φ}, the inverse metric is g
μν
=
−2 (eμ4 eν3
+
eμ3 eν4 )
+
2
eμa eνa
a=1 2 2 μ ν μ ν (N N + N N ) + eμa eνa Ω2 a=1 1 μ ν = − 2 δu δu + δuμ δuν + X c (δcμ δuν + δcν δuμ ) + X(Kerr) d δdμ δuν + δdν δuμ 2Ω (2.29) + γ ab δaμ δbν .
=−
and the metric tensor is
g(·, ·) = −4Ω2 dudu + γab dω a − (X(Kerr) a du + X a du) × dω b − (X(Kerr) b du + X b du) .
(2.30)
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The previous result has still a certain arbitrariness as e3 , e4 depend on the foliation which in turn depends on the choice of the initial conditions on V∗ ∩ Σ0 and on ∂V∗1 . The natural choice is that of a foliation “near” to the Pretorius–Israel one used in the Kerr spacetime. Let us recall some aspects of this foliation. In [8] u=
t + r∗ ; 2
u=
t − r∗ , 2
(2.31)
where r∗ is the solution of the eikonal equation described there.21 With these definitions the Kerr null frame “adapted” to the P-I double null foliation is ∂ ∂ R 1 (Kerr) + ωB = 2Ω(Kerr) L = √ (∂t − ∂r∗ + ωB ∂φ ) = (Kerr) e3 ∂u ∂φ Ω Δ ∂ ∂ R 1 (Kerr) e4 = 2Ω(Kerr) L = √ (∂t + ∂r∗ + ωB ∂φ ) = (Kerr) + ωB ∂u ∂φ Ω Δ ∂ ∂ R R ∂ 1 (Kerr) (Kerr) , e2 , (2.32) e1 = = = Lsin 2θ∗ ∂θ∗ L ∂λ R sin θ ∂φ where
Δ 2marb , Ω = , λ = sin2 θ∗ . ΣR2 R2 Δ = rb2 + a2 − 2M r, Σ = rb2 + a2 cos2 θ
ωB =
(2.33)
ΣR2 = (rb2 + a2 )2 − Δa2 sin2 θ
and rb is the Boyer–Lindquist radial coordinate. L and λ are defined in [8] (see also equations 3.127). Denoting {θ∗ , φ}22 again {ω 1 , ω 2 }, a point of the Kerr spacetime is specified assigning {u, u, ω 1 , ω 2 }. The null hypersurfaces u = const, u = const define the double null foliation. As done before, starting from the null orthonormal frame, the Kerr metric in the {u, u, ω 1 , ω 2 } coordinates is g(Kerr) (·, ·) = −4Ω2(Kerr) dudu (Kerr) a b dω a − X(Kerr) + γab (du + du) dω b − X(Kerr) (du + du) . (2.34) where X(Kerr)
∂ , = ωB ∂φ
Ω(Kerr) =
Δ R2
2
γ11 21
L2 (sin 2θ∗ ) = , R2
γ12 = 0,
(2.35) 2
2
γ22 = R sin θ.
To specify it uniquely we still have to give the initial data, we discuss it later on. Observe that in the Pretorius–Israel frame the variable which stays near to the spherical coordinate θ of Minkowski spacetime is θ∗ and not θ.
22
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The equivariant vector fields in the Kerr spacetime ∂ ∂ ∂ ∂ + ωB + ωB N (Kerr) = , N (Kerr) = ∂u ∂φ ∂u ∂φ satisfy
(2.36)
[N (Kerr) , N (Kerr) ] = −4Ω2 ζ (Kerr) (ea )ea with
QR sin θ ∂rb ωB eφ . (ea )ea = − (2.37) ζ 2Σ It is now possible to compare the foliation associated with the Kerr spacetime and to the perturbed Kerr spacetime. If we endow V∗ with the metric (2.34), (V∗ , g(Kerr) ) is a submanifold of the Kerr spacetime; if the metric associated with V∗ is (2.30), then (V∗ , g) is a submanifold of the global perturbed Kerr spacetime whose existence we are proving. Comparing the two metrics (2.34) and (2.30) 23 we see that, at the metric level, the components which are modified due to the initial data perturbation of the “Kerr initial data” are (Kerr)
a δX a = X a − X(Kerr) ,
δΩ = Ω − Ω(Kerr) ,
(Kerr)
δγab = γab − γab
. (2.38)
We will have to prove that these corrections are “small” once the connection coefficients satisfy the “Bootstrap assumptions”. Therefore, we define the following norms, we denote globally δO(0) , which will be proved small later on: |r2 |u|1+δ δΩ|∞ ≡ sup |r|u|2+δ δΩ| V∗
2
2+δ
|r |u|
δX|∞ ≡ sup V∗
a
2
2+δ
|r |u|
1+δ
δX |, ||u| a
δγ|∞ ≡ sup V∗
(2.39) 1+δ
||u|
δγab |.
ab
2.2. The Connection Coefficients and Riemann Tensor Bootstrap Assumptions Once V∗ is endowed with a double null canonical foliation we have to specify the remaining bootstrap assumptions in V∗ . They are relative to the connection coefficients and to the Riemann components: more precisely to their difference from the analogous quantities in the Kerr spacetime. 2.2.1. Some Dimensional Remarks. All along this paper we use systematically, as a good help, “dimensional” arguments to obtain insight into the various decay factors of the Kerr connection coefficients and the Kerr Riemann tensor; moreover, the dimensionality of all the constants which appear in the theorems and in the various estimates is carefully specified. All the constants c which will appear are adimensional. The only constants with a natural dimension are M , R0 which both have a length dimension, L1 , and ε and 0 which define the smallness of the various energy type norms which have dimension: 23
In the perturbed Kerr spacetime we defined ωB as in the Kerr spacetime; this requires some care. In fact we define it as a function in the {u, u, ω 1 , ω 2 } coordinates while in Kerr spacetime ωB = ωB (rb , θ, φ) and rb , θ, φ are Boyer Lindquist coordinates. Therefore, using the Pretorius–Israel coordinates we can rewrite it as a function ωB = ωB (r∗ , θ∗ , φ), and finally following [8] we express r∗ as a function of u and u. This is the expression we use in (V∗ , g).
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L3 . Attention has, therefore, to be paid in considering the weight factors which appear in the various norms. They are of the following type: some integer powers of r times |u| factors with exponents 1 + δ, 2 + δ, . . . or 5 + γ (in the Q norms), or 1 + 2 , 2 + 2 , . . . (see later on). To keep consistent the dimension of the various terms in the various inequalities we have to interpret each term δ γ |u|k+δ with k integer as |u|k |u| , in the same way |u|5+γ as |u|5 |u| and also Rγ Rδ 2
0
0
− 2
|u|k+ 2 as |u|k |u|2 . We omit hereafter the factors R0−δ , . . . , R0−γ , . . . R0
R0
not to
burden the notations. Therefore, in all the expressions the terms |u|δ , |u|γ , |u| 2 have to be considered dimensionless. 2.2.2. The O Connection Coefficient Norms. We write any connection coefficient as the sum of the Kerr connection coefficient part plus a correction and make assumptions for the estimates of the correction parts. The O connection coefficients are tensor fields tangent to the S two-dimensional surfaces associated with the foliation of the perturbed Kerr spacetime, while the O(Kerr) connection coefficients are tangent to the S(Kerr) two-dimensional surfaces. Therefore, their difference would not be an S-tangent tensor. To avoid this problem we observe that O(Kerr) , assuming it, for instance, being a (0, 2) covariant tensor, can be written as ρ
σ
(Kerr) (Kerr) Oμν = Π(Kerr) μ Π(Kerr) ν Hρσ ,
(2.40)
where H (Kerr) is not S-tangent and Π(Kerr) projects it on the T S(Kerr) tangent ˆ space. Therefore, we substitute O(Kerr) with O (Kerr) ˆ μν = Πρμ Πσν Hρσ O .
(2.41)
where Π projects on the T S tangent space. Then we will have to prove that ˆ − O(Kerr) ) are small (see Sect. 3.4). Therefore, we write each the differences (O connection coefficient as ˆ + δO O=O and we make the bootstrap assumptions on the norms of the δO parts denoted globally as δO, while we control the sup norms of the Kerr part. These bootstrap norm estimates control the smallness and the decay along the outgoing or incoming cones. The decay is expressed in terms of a “radial variable” denoted by r defined, at a generic point p ∈ V∗ , as 1 1 r = r(u, u) = √ |S(u, u)| 2 . 4π
(2.42)
where |S(u, u)| is the area of the surface to which the point p belongs. Observe that this variable, although not far from, is different from the radial variable r∗ = u − u used by Israel and Pretorius, [8], in the Kerr spacetime and also with respect to the quantity u − u of the perturbed Kerr spacetime. All the decay are expressed in this variable and in the |u| variable, and we show, in Sect. 3.4.3, that all the various radial coordinates appearing in the Kerr spacetime and in the perturbed Kerr spacetime stay near to r(u, u), (of course in
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this external region). Therefore, hereafter with r, we always mean the quantity defined in (2.42), a well-defined function of u and u. Finally, the norms we are using for the connection coefficients are, as in [10], or sup norms, | · |∞ , or (p, S)-norms, | · |p,S , with p ∈ [2, 4], where ⎞ p1 ⎛ |f |p,S = ⎝ |f |p dμS ⎠ . (2.43) S
If f denotes a generic connection coefficient the pointwise norm of the integrand is made with the restriction of the metric on S, γab . Even if we are considering the norm of the Kerr part we use the γ metric associated with the perturbed metric, Eq. (2.30), which does not change, apart from a constant, the estimates for the Kerr part. In fact we will show in detail, from the boot(Kerr) , step IV strap assumptions it follows, that the metric γab stays near to γab of Sect. 3.2. 2.2.3. Decay of Kerr Connection Coefficients. We start recalling the general definitions of the connection coefficients, see [10, Chapter 3], for more details: 1 1 ξa = g(De4 e4 , ea ), ξ a = g(De3 e3 , ea ) 2 2 1 1 ηa = − g(De3 ea , e4 ), η a = − g(De4 ea , e3 ) (2.44) 2 2 1 1 ω = − g(De4 e3 , e4 ), ω = − g(De3 e4 , e3 ) 4 4 1 ζa = g(Dea e4 , e3 ). 2 The decay of the Kerr connection coefficients,24 is the following one25 : |rtrχ(Kerr) | ≤ κ; 2
|r ω
(Kerr)
3 (Kerr)
ˆ |r χ
| ≤ κM ;
|rtrχ(Kerr) | ≤ κ 2
|r |ω
(Kerr)
2
| ≤ κaM ≤ κM ;
|r4 ∇ / trχ(Kerr) | ≤ κaM ≤ κM 2 ;
(2.45)
| ≤ κM 3 (Kerr)
|r χ ˆ
(2.46) | ≤ κaM ≤ κM
2
|r4 ∇ / trχ(Kerr) | ≤ κaM ≤ κM 2 (2.47)
|r3 ζ (Kerr) | ≤ κaM ≤ κM 2 |r4 ∇ / ω (Kerr) | ≤ κaM ≤ κM 2 ,
|r4 ∇ / ω (Kerr) | ≤ κaM ≤ κM 2
where κ > 1 is a definite adimensional constant. The norm estimates in (2.45) refer to those connection coefficients which are different from zero in Minkowski, those in (2.46) refer to the coefficients 24 Those for the tangential derivatives follows in the standard way, getting an extra r for each tangential derivative. 25 As trχ and trχ are different from zero even in the Minkowski spacetime their norm estimates does not depend at the lowest order on M , in fact
trχ(Kerr) =
O(1) O(M 2 ) O(M ) + + ··· , + 2 r r r3
trχ(Kerr) =
O(1) O(M 2 ) O(M ) + + ··· . + 2 r r r3
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different from zero in Schwarzschild, but zero in Minkowski and the norm estimates in (2.47) refer to the connection coefficients or derivatives of connection coefficients different from zero in Kerr, but zero in Schwarzschild and in Minkowski. These “Kerr decays” can be easily obtained by dimensional arguments recalling that the Kerr metric, written in the Boyer–Lindquist coordinates, in the limit M → 0, a kept fixed and different from zero, reduces to the Minkowski metric written in the “oblate coordinates” and as we know that in Minkowski spacetime all connection coefficients, with the exception of trχ and trχ, are zero, it follows that performing an 1r expansion of the connection coefficients, all the coefficients of the terms in r1k which depend only on a and not on M must be identically zero. The dimensional argument goes as following: we assume the metric written in “cartesian” coordinates, namely coordinates with the dimension of a length: L1 . Then as the metric tensor has dimension L2 its components gμν has dimension L0 . Proceeding in the same way it follows that the connection coefficient components have dimension L−1 and the same happens for their norms. Therefore, as for instance, [χ] = L−1 , it follows that in Kerr it must be O(aM ) χ ˆ(Kerr) = . r3 Moreover, as in Kerr spacetime a = J/M and we consider those spacetimes where J ≤ M 2 , it follows that a ≤ M and the Kerr coefficient satisfy the condition |r3 χ ˆ(Kerr) | ≤ κM 2 . The bounds for ω (Kerr) and ω (Kerr) and for trχ(Kerr) and trχ(Kerr) are different as they reflect the fact that ω (Kerr) , ω (Kerr) are different from zero also in Schwarzschild and trχ(Kerr) , trχ(Kerr) even in Minkowski. 2.2.4. The δO Connection Coefficient Norms. The assumptions δO ≤ 0
(2.48)
summarize the conditions on the connection coefficients and their tangential l derivatives, ∇ / , with 0 ≤ l ≤ 4,26 |r2+l |u|2+δ ∇ / δtrχ| ≤ 0 ; |r1+l |u|3+δ ∇ / δtrχ| ≤ 0 l
l
|r2+l |u|2+δ ∇ / δ χ| ˆ ≤ 0 ; |r1+l |u|3+δ ∇ / δ χ| ˆ ≤ 0 l
l
|r2+l |u|2+δ ∇ / δζ| ≤ 0 l
|r
2+l
2+δ
|u|
l
∇ / δω| ≤ 0 , |r
(2.49) 1+l
3+δ
|u|
l
∇ / δω| ≤ 0 .
Remark. The choice of the δ coefficient is connected to the decay of the Riemann tensor. δ > 0 is required to prove the decay in agreement with the peeling. δ ≤ 0 could also be chosen, as discussed in the introduction, but, 26
The bounds foe η and η follow from these ones in the adapted frame, see for details [10, Chapter 3].
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in this case, a weaker decay for (some components of) the Riemann tensor follows.27 In conclusion the second of the “bootstrap assumptions” we are requiring is the following one: (Assumption II): In V∗ the connection coefficient norms satisfy the following bounds: δO ≤ 0 . 2.2.5. The δR Null Riemann Component Norms. We start recalling the general definitions, [10, Chapter 3], of the null components of the Riemann tensor with respect to the adapted null frame, where W is a generic Weyl tensor and X, Y are S-tangent vector fields, α(W )(X, Y ) = W (X, e4 , Y, e4 ),
β(W )(X) =
1 W (X, e4 , e3 , e4 ) 2
1 1 W (e3 , e4 , e3 , e4 ), σ(W ) = W (e3 , e4 , e3 , e4 ) (2.50) 4 4 1 β(W )(X) = W (X, e3 , e3 , e4 ), α(W )(X, Y ) = W (X, e3 , Y, e3 ). 2 As done for the connection coefficients, to prove our result we “subtract” the Kerr part of the Riemann tensor. Therefore, writing all the various components of the Riemann tensor as ρ(W ) =
R = R(Kerr) + δR we state bootstrap assumptions on the norms relative to the “correction part”, we denote globally δR. The norms bounded in the “bootstrap assumptions” are the sup norms of the various null components of δR and their tangential derivatives.28 In a compact way δR ≤ 0 ,
(2.51)
and in more detail, with l ≤ q − 1:
sup r5+l |u| 2 |∇ / α(δR)| ≤ 0 , l
V∗
sup r V∗
3+l
2+ 2
|u|
sup r4+l |u|1+ 2 |∇ / β(δR)| ≤ 0 l
V∗
l
|∇ / (ρ(δR) − ρ(δR))| ≤ 0 ,
l
l
(2.52)
sup r3+l |u|2+ 2 |∇ / (σ(δR) − σ(δR))| ≤ 0 sup r2+l |u|3+ 2 |∇ / β(δR)| ≤ 0 , V∗
sup r1+l |u|4+ 2 |∇ / α(δR)| ≤ 0 . l
V∗
27
The factor |u|δ could also describe, symbolically, a log factor, for instance |r 2 |u|2+δ δtrχ| ≤ 0 could mean |δtrχ| ≤ c 0
28
1 . (log |u|/R0 )δ r 2 |u|2
Once we control the tangential derivatives, via the Bianchi equations we control also the derivatives along e3 and e4 , see [10] for the more delicate control of De4 α and De3 α.
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Finally, we add a bootstrap assumption for ρ(δR),
sup |r3 |u|2+ 2 ρ(δR)| ≤ 0 ,
(2.53)
V∗
where ρ denotes the average over the S surface. Remarks. (i) The exponent 2 appearing here in the decay factors has to be assumed, at the end, equal to δ to complete the proof of the main theorem, Theorem 3.1. This has not to be confused with the ε denoting the smallness of the initial data in (1.2). (ii) The estimate for α(δR) is not yet the estimate for α(R) as we have to add α(R(Kerr) ), the same, of course, for all the remaining components. In fact, all the Riemann components contain also a Kerr part, even the components different from ρ and σ as we are not using the principal null direction frame, see [2] and later on for its precise definition. The precise estimates of the Kerr parts are given later on in Sect. 3.3.5. In conclusion the third “bootstrap assumption” is the following one: (Assumption III): In V∗ the norms of the components of the correction to the Kerr Riemann tensor, δR, satisfy the following bounds: δR ≤ 0 .
3. The Results 3.1. The Main Theorem Theorem 3.1. If the bootstrap assumptions, (2.48), (2.51), (2.53) hold with δ = 2 , the following smallness assumptions are satisfied,
20 M < ε < 0 , 1 R03 R0
(3.54)
and the smallness of the initial data is controlled by a function J explicitly defined in Sect. 3.6, Eq. (3.232), J (Σ0 , R0 ; δ (3)g, δ (3)k) ≤ ε, then in the V∗ region the following bounds are satisfied:
0
0
0 δO ≤ , δR ≤ , sup |r3 |u|2+ 2 ρ(δR)| ≤ . 2 2 2 V∗
(3.55)
(3.56)
This theorem is the main step to prove the global existence and the peeling properties in the external region defined by the condition |u| ≥ R0 (see also Sect. 3.7). 3.2. The Structure of the Proof of Theorem 3.1 The proof of Theorem 3.1 is divided into many steps we list in the following: ˜ norms I step: The definition of the Q ˜ ˜=L ˆ T R; their explicit expression We denote Q the Q norms associated with R 0 will be given in the sequel.
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˜ norms and of ρ(R) ˜ II step: the estimate of of the Q We prove that under the bootstrap assumptions (2.48), (2.51) and initial data such that on Σ0 : ˜ Σ ≤ ε2 ; Q 0 we have
˜ ≤ cε sup |r6+ 2 ρ(R)|
Σ0 /BR0
˜ ≤ c ε2 + M 20 ; Q R0
˜ ≤ 0 . sup |r3 |u|3+ 2 ρ(R)| 2 V∗
(3.57)
III step: Proceeding basically as in [15], we prove that from the results in the first two steps all the norms δR are bounded by 20 . The proof is divided into many consecutive lemmas; it requires also the control, based on the δO bootstrap assumptions, of the deformation tensor (T0 )π. IV step: We complete the proof of Theorem 3.1 showing that, using the results of the previous steps, we have
0 (3.58) δO ≤ . 2 This last step is divided in three parts to do in a precise order, namely: (a) Using the bootstrap assumptions for the δO norms, δO ≤ 0 , we prove that, under the initial data assumptions, the metric component corrections norms, defined in (2.39), satisfy, δO(0) ≤ c 0 .
(3.59)
(b) Using this result plus the initial data assumptions and the bootstrap assumptions, we obtain that the connection coefficient norms satisfy
0 (3.60) δO ≤ . 2 (c) Finally, we prove that under this result also the metric components corrections have better estimates, namely
0 (3.61) δO(0) ≤ . 2 This last result will be needed with all the other ones to perform step VI. V step: In this step we collect all the required initial data conditions and show how they can be satisfied imposing a global decay condition for the initial data and the smallness condition (3.55). VI step: In this step we recall how, by a partial local existence theorem, in view of the previous results we can extend the region V∗ showing that it has to coincide with the whole spacetime. 3.3. Proof of the Various Steps We call a constant “independent” if it does not depend on a, M, 0 , ε.
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˜ are ˜ Norms. These norms, we denote Q, 3.3.1. I Step: The Definition of the Q analogous to the Q norms defined in [10, Chapter 3]:
u) = Q 1 (u, u) + Q 2 (u, u) + Q(u,
q−1
(q) (u, u) Q
i=2
u) = Q (u, u) + Q (u, u) + Q(u, 1 2
q−1
(3.62) (u, u), Q (q)
i=2
where, with S(u, u) ⊂ V∗ , we denote V (u, u) = J (−) (S(u, u)) ∩ V∗ , ˜ K, ¯ K, ¯ K, ¯ e4 ) ˆ T R)( Q1 (u, u) ≡ |u|5+γ Q(L C(u)∩V (u,u)
˜ K, ¯ K, ¯ T, e4 ) ˆ O R)( |u|5+γ Q(L
+ C(u)∩V (u,u)
2 (u, u) ≡ Q
ˆ T R)( ˜ K, ¯ K, ¯ K, ¯ e4 ) ˆ OL |u|5+γ Q(L
C(u)∩V (u,u)
˜ K, ¯ K, ¯ T, e4 ) ˆ 2 R)( |u|5+γ Q(L O
+ C(u)∩V (u,u)
ˆ T R)( ˜ K, ¯ K, ¯ K, ¯ e4 ) ˆSL |u|5+γ Q(L
+
(3.63)
C(u)∩V (u,u)
⎧
(q) (u, u) ≡ Q
q−1 ⎪ ⎨ i=2
⎪ ⎩
C(u)∩V (u,u)
˜ ¯ ¯ ˆ i+1 |u|5+γ Q(L O R)(K, K, T, e4 )
+ C(u)∩V (u,u)
+ C(u)∩V (u,u)
(u, u) ≡ Q 1
ˆ iO R)( ˜ K, ¯ K, ¯ K, ¯ e4 ) ˆT L |u|5+γ Q(L
⎫ ⎪ ⎬ ˆT L ˆ i−1 R)( ˜ K, ¯ K, ¯ K, ¯ e4 ) ˆSL |u|5+γ Q(L O ⎪ ⎭
˜ K, ¯ K, ¯ K, ¯ e3 ) ˆ T R)( |u|5+γ Q(L
C(u)∩V (u,u)
+ C(u)∩V (u,u)
˜ K, ¯ K, ¯ T, e3 ), ˆ O R)( |u|5+γ Q(L
(3.64)
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453
ˆ T R)( ˜ K, ¯ K, ¯ K, ¯ e3 ) ˆ OL |u|5+γ Q(L
C(u)∩V (u,u)
2
˜ K, ¯ K, ¯ T, e3 ) ˆ R)( |u|5+γ Q(L O
+ C(u)∩V (u,u)
ˆ T R)( ˜ K, ¯ K, ¯ K, ¯ e3 ) ˆSL |u|5+γ Q(L
+
(3.65)
C(u)∩V (u,u)
(u, u) ≡ Q (q)
⎧ q−1 ⎪ ⎨ ⎪ i=2 ⎩
ˆ i R)( ˜ K, ¯ K, ¯ K, ¯ e3 ) ˆT L |u|5+γ Q(L O
C(λ)∩V (λ,ν)
˜ ¯ ¯ ˆ i+1 |u|5+γ Q(L O R)(K, K, T0 , e3 )
+ C(u)∩V (u,u)
+ C(u)∩V (u,u)
⎫ ⎪ ⎬ ˆT L ˆ i−1 R)( ˜ K, ¯ K, ¯ K, ¯ e3 ) . ˆSL |u|5+γ Q(L O 0 ⎪ ⎭
(3.66)
and 1 (Σ0 ) ≡ Q
˜ K, ¯ K, ¯ K, ¯ T) ˆ T R)( |u|5+γ Q(L
Σ0 ∩V∗
˜ K, ¯ K, ¯ T, T ) ˆ O R)( |u|5+γ Q(L
+
(3.67)
Σ0 ∩V∗
2 (Σ0 ) ≡ Q
ˆ T R)( ˜ K, ¯ K, ¯ K, ¯ T) ˆ OL |u|5+γ Q(L
Σ0 ∩V∗
+
2
˜ K, ¯ K, ¯ T, T ) ˆ O R)( |u|5+γ Q(L
Σ0 ∩V∗
+
ˆ T R)( ˜ K, ¯ K, ¯ K, ¯ T ). ˆSL |u|5+γ Q(L
(3.68)
Σ0 ∩V∗ q−1 i=2
⎧
(q) (Σ0 ) ≡ Q
q−1 ⎨
ˆ i R)( ˜ K, ¯ K, ¯ K, ¯ T) ˆT L |u|5+γ Q(L O ⎩ i=2 Σ ∩V 0 ∗ ˜ K, ¯ K, ¯ T, T ) ˆ i+1 R)( + |λ|5+γ Q(L O Σ0 ∩V∗
+ Σ0 ∩V∗
⎫ ⎬ ˆT L ˆ i−1 R)( ˜ K, ¯ K, ¯ K, ¯ T) ˆSL |λ|5+γ Q(L O ⎭
(3.69)
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We also introduce the following quantity: V ≡ Q ∗
sup {u,u|S(u,u)⊆V∗ }
u) + Q(u, u)}. {Q(u,
(3.70)
Q norms The initial conditions have to be chosen in such a way that the Q, are bounded on Σ0 : 1 (Σ0 ) + Q 2 (Σ0 ) + Q
q−1
(q) (Σ0 ) ≤ ε2 . Q
(3.71)
i=2
Let us look at the differences with the similar norms in [10, Chapter 3]: ˜ is not present, differently ˜ norms a term associated with ρ(R) In these Q from the definitions in [10]. ˜ = (ii) We are considering these norms associated with the Weyl tensor R ˆ LT0 R which can be interpreted as the “time derivative” of R; this is required, as discussed in the introduction, to “subtract” the Kerr part of the Riemann tensor. ˜ norms have an extra decay factor, |u|5+γ , with (iii) The weights of the Q γ > 0. It has the effect, in a complicated way shown in the following lemmas, of giving a better decay for the various null components ˆ T R. This improved decay is required, first, to guarantee that the of L 0 decay of Riemann tensor be in agreement with the “peeling”; second, to recover the estimates for the Riemann tensor by “time integration”.29 Once we prove that these norms are bounded in V∗ , the null comˆ T R, α, β, . . . decay with an exponent 5 + , 4 + , 3 + , . . . . ponents of L 0 2 2 2 depending on the null components we are considering, with = γ. ˜ norms, the Q| ˜ Σ norms are bounded Observe that, assigned γ in the Q 0 if the Riemann components on Σ0 decay with an ˆ > γ, implying that the correction of the metric components have to decay on Σ0 toward ˆ spacelike infinity as O(r−(3+ 2 ) ).30 Finally, the connection coefficients satisfy the structure equations which depend on the Riemann tensor so that the exponent factor δ must be equal to γ/2, see later on, and δ must coincide with the exponent /2, see Lemmas 3.2, 3.3, 3.5. In conclusion, in the bootstrap assumptions for the δO norms we have to choose (i)
γ = 2δ = ∈ (0, ˆ).
29 To perform the “time integration” the |u| exponent could be less the 5 + γ. To have the same decay as in [10], 2 + γ would be enough. 30 Therefore, as starting from the Q ˜ norm with a weight |u|5+γ it follows that the sup norm estimates for the Riemann components have the same factor γ; this implies that there is a loss of decay going from the initial data decay to the decay of the Riemann components in the whole V∗ and therefore, on the whole spacetime.
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˜=L ˆ T R the vector field T0 is In the definition of R 0 X(Kerr) + X Ω . (3.72) T0 = (e3 + e4 ) − 2 2 T0 is a “nearly” Killing vector field if the corrections to the Kerr metric ∂ ˜ norms are small and equal to ∂t in Kerr spacetime. Vice versa in the Q definition Ω T = (e3 + e4 ) (3.73) 2 is one of the vector fields saturating the Bel-Robinson tensor and also ˜ The reason for the use of appearing in the various Lie derivatives of R. ˜ both T and T0 is that to go back from R to the “time derivatives” of the components of R controlling both the norm smallness and the “peeling decay” we need to use T0 which is nearly a Killing vector field. Vice versa ˜ norms the control of the null comto obtain from the control of the Q ˜ ponents norms of R it is crucial to have their integrands positive which requires to use the vector field T = Ω2 (e3 + e4 ).
˜ and of Q ˜ in V∗ . 3.3.2. II Step: The Estimate of ρ(R) Theorem 3.2. Let the bootstrap assumptions 2.48 hold in V∗ ; assume that on Σ0 the initial data satisfy
˜ ≤ ε; sup |r6+ 2 ρ(R)|
Σ0 /BR0
˜ Σ ≤ ε2 Q 0
(3.74)
with = γ; then in V∗ the following estimates hold: 12 1 M M 2 3 3+ 2 2 2 ˜ ˜ ρ(R)| ≤ c1 ε +
0 ; Q ≤ c1 ε +
sup |r |u| (3.75) R0 R0 0 V∗ where c is an adimensional constant independent from M . Proof. The proof similar to the analogous proof in [15] uses a bootstrap inside the region V∗ . The rationale for it is the following: to prove the estimate for ˜ we need to control the other null components of R. ˜ These are controlled ρ(R) ˜ norms which at their turn require, to be controlled, an in terms of the Q ˜ estimate for ρ(R).31 Let V∗ = J (−) (S(u(R0 ), u∗ )), we consider V˜ = J (−) (S(u(R0 ), u ˜ )) ⊂ V∗ where u ˜ ≤ u∗ is the largest value of the variable u such that in V˜ the following estimates hold32 : 12 1 M M 2 3 3+ 2 2 2 ˜ ˜ sup |r |u| ρ(R)| ≤ c1 ε +
0 ; Q ≤ c1 ε +
(3.76) R0 R0 0 V˜ with c1 > 1. 31
The bootstrap we use here is different from the bootstrap we need to extend the region V∗ ; in fact the main bootstrap assumptions refer to the null components of δR = R −R(Kerr) ˜=L ˆ T R. and not to those of R 0 32 u(R ) is the value of u| 0 Σ0 = r∗ associated to r = R0 .
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˜ is,33 The transport equation for r3 ρ(R) d 3 1 (r ρ) = dλ |S(λ, ν)|
Ann. Henri Poincar´e
G
S(λ,ν)
where
(Ωtrχ − Ωtrχ) 3 1 3 G≡ r (ρ − ρ) + Ωr div ˆ·α+ζ ·β / Ω · β − 2η · β − χ . 2 2 (3.77)
Integrating along the incoming cones we obtain u |r3 ρ|(u, u) ≤ |r3 ρ|(u0 , u) + |r3 G|(u , u)
(3.78)
u0
Using the connection coefficients sup norm estimates 2.45, 2.46, 2.47, the bootstrap assumptions 2.48, recalling that in the external region |u| ≤ r, we have 2 M M2 2 M2 3 2 |r |r |r|u| |r3 G| ≤ c (ρ − ρ)| + |u|β| + α| p,S r2 |u| r2 |u| r2 |u| M2 M2 3 3+ 2 2 4+ 2 (ρ − ρ)| + β| ≤c |r |u| |r |u| 4+ r2 |u| 2 r2 |u|4+ 2 M2 5+ 2 + 2 4+ |r|u| α| (3.79) r |u| 2 ˜ null components in (3.79) can be bounded by the Q ˜ 21 The norms of the R quantity exactly as it was done in [10, Chapter 5], with the only difference ˜ and Q by Q. ˜ Therefore, using the second inequality that R is substituted by R in (3.76),
|r3 |u|3+ 2 ρ|(u, u) M2 ˜ 1 6+ 2 2 ρ|(u0 , u) + c 2 Q ≤ |r R0 1 M 2 M 2 12 ≤ ε + c 2 c1 ε +
0 R0 R0 52 1 M M 2 12 2
0 ≤ c 1 + 2 c1 ε + cc1 R0 R0 12 1 M 2 < c1 ε +
0 R0
(3.80)
provided ε and M/R0 have been chosen sufficiently small and c1 > c2 , so that 2 1 M M2 1 1 + c 2 c12 < c12 , c < 1. (3.81) R0 R0 33
The derivation of this equation is in Ref. [10, Chapter 5].
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˜ norms in V∗ . We are left to prove that in V˜ the second The control of the Q condition of (3.76) ˜ ≤ c1 ε2 + M 2 , Q R0 0 ˜−Q ˜ Σ . The can be improved. This requires the estimate of the error E˜ = Q 0 estimate of the error proceeds basically as in [10], the main differences being ˜ norms there are extra weight factors. Let us that in the definition of the Q study some error terms analogous to those discussed in [10, Chapter 6]. ˜ βγδ ˜ L ˆ O R) Let V ⊂ V˜ and consider one of the terms in V(u,u) |Div Q( (K β K γ T δ )|, namely the first term of
4 ˜ 444 = 1 τ+ |u|5+γ D(O, R) 2
V(u,u)
4 ˜ · Θ(O, R) ˜ ˆ O R) τ+ |u|5+γ α(L
V(u,u)
4 ˜ · Ξ(O, R) ˜ ˆ O R) τ+ |u|5+γ β(L
−
(3.82)
V(u,u)
We have
⎛ 4 ˜ · Θ(O, R) ˜ ≤⎜ ˆ O R) τ+ |u|5+γ α(L ⎝sup V
V(u,u)
u ×
⎛
1 2
≤ cQ V
u
u0
˜ 2⎟ u4 |u|5+γ |Θ(O, R)| ⎠
⎛ 3 ⎜ du ⎝ i=0
˜ 2⎟ ˆ O R)| u4 |u|5+γ |α(L ⎠
C(u ;[u0 ,u])
C(u ;[u0 ,u])
u0
⎞ 12
⎞ 12
⎜ du ⎝
⎞ 12
˜ 2⎟ u4 |u|5+γ |Θ(i) (O, R)| ⎠ .
(3.83)
C(u ;[u0 ,u])
The term with i = 0 is the only one not present in the [10] error estimates, the reason being that in that case the Riemann tensor Rμνρσ satisfied the Bianchi equation Dμ Rμνρσ = 0, while here Dμ LT0 Rμνρσ = 0. We show in detail how to control it; for all the remaining error terms, we refer to [5] where a detailed discussion on the other terms is done and to [10, Chapter 6], where for the Q norms, the control of the error terms is done in a very accurate way.
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The term in the error with i = 0, absent in [10],34 is proportional to ⎛ ⎞ 12 u 1 2 du ⎜ ˜ 2⎟ Q u4 |u|5+γ |Θ(0) (O, R)| (3.84) ⎝ ⎠ . V u0
C(u ;[u0 ,u])
This term arises from the first term of ˜ βγδ = Dα (L ˆ T R)βγδ = L ˆ O R) ˆ OL ˆ O Dα (L ˆ T R)βγδ + Dα (L 0 0
3
˜ βγδ . J i (O; R)
i=1
(3.85) As, see [10], Chapter 6, Eqs. (6.1.6) ˆ O Dα (L ˆ T R) = L ˆ O J(T0 , R) = L ˆ O (J 1 (T0 , R) + J 2 (T0 , R) + J 3 (T0 , R)) L 0 (3.86) the term we have to estimate is ⎛ u 1 ⎜ 2 QV du ⎝ u0
⎞ 12 ˆ O J(T0 , R)|2 ⎟ u4 |u|5+γ |L ⎠ .
(3.87)
C(u ;[u0 ,u])
This term has to be treated in a different way from the remaining ones as it ˜ in fact, it depends on R instead of on R. ˜ cannot be bounded in terms of the Q; It is easy to recognize that all the terms in which we can decompose J(T0 , R) can be estimated in the same way; therefore, we consider only J 1 (T0 , R) and again all the various terms which compose it can be treated in the same way, see [10, pp. 245–247]. They all have the structure R (3.88) r and recalling that T0 is nearly Killing, it follows that the generic term of ˆ O J 1 (T0 , R) considering, for simplicity, all these norms being sup norms, satL isfies the following bound, with σ > 0, (T0 )
πDR + (T0 )π
ˆ O J 1 (T0 , R)| ≤ |(T0 )π||DR| ≤ |L
0 2 r |u|2+δ
M M
0 ≤ c 5 2+δ+σ 1−σ , (3.89) 4 r r |u| R0
ˆ 0 does not improve the decay. Substituting this estimate remembering that L in 3.87 we obtain, ⎛ ⎞ 12 u 1 2 du ⎜ ˆ O J(T0 , R)|2 ⎟ Q u4 |u|5+γ |L ⎝ ⎠ V u0 34
C(u ;[u0 ,u])
There is an analogous term which is not present in [10] coming from the error term ˜ L ˆ T R) ˜ βγδ (K β K γ T δ )|, its estimate goes exactly in the same way and we do |Div Q(
V(u,u)
not report it here.
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1 2 0 M ≤ cQ V R01−σ
u
u0
1
2 0 ≤ cQ V
M R01−σ
u
u0
1 2
0 ≤ cQ V
M R01−σ
u
⎛ ⎜ du ⎝
⎞ 12
u4 |u|5+γ
C(u ;[u0 ,u])
⎛ ⎜ du ⎝
∞
⎛
γ 1 ⎜ du |u | 2 + 2 −δ ⎝
1
⎟
⎠ r10+2σ |u |4+2δ+2σ ⎞ 12
∞
|u|
u0
⎟
1
⎠ r10+2σ |u|4+2δ+2σ ⎞ 12
drr6 |u |5+γ
|u|
459
dr
1 ⎟ ⎠ r4+2σ
(3.90)
Let us choose35 γ ; 2 then the last integral satisfies the following inequality: ⎛ ⎞ 12 u 1 2 du ⎜ ˆ O J(T0 , R)|2 ⎟ Q u4 |u|5+γ |L ⎝ ⎠ V δ≥
(3.91)
C(u ;[u0 ,u])
u0
u 1 1 M 1 0 2 0 M ≤ cQ du |u | 2 ≤ cQ 3 V V 1−σ +σ R0 R0 |u | 2 u0 V M + 2 M . ≤c Q 0 R0 R0 If this were the only error term we could write M 2M ˜ + 0 |E| ≤ c QV R0 R0 and V + c M 2 V ≤ Q Σ + c M Q Q 0 R0 R0 0 implying 1 V ≤ Σ + c M 20 < c1 ε2 + M 20 Q Q 0 M 2 R0 1 − cR 1 − c M R0 1 2
0
(3.92)
(3.93)
(3.94)
(3.95)
R0
choosing c1 appropriately. Clearly the estimates discussed here and in [5] for ˜ norms, are far from some of the error terms to prove the boundedness of the Q giving a complete proof of the result; in fact, the error term is made by a great number of integrals, more than one hundred, and their complete estimates would take a huge number of pages. On the other side, these estimates have been done in a complete way for the Q norms in [10, Chapter 6], and what we want to point out here is that, apart some differences explicitly examined, 35
To complete the bootstrap procedure we need the equal sign, see Sect. 3.4.2.
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the way of controlling the error terms follows exactly the same pattern and therefore the present proof is just a consequence of the proof given there. 3.3.3. III Step: The Control of the δR Norms. In the previous step we have proved estimates (3.75). In this step using these estimates we prove estimates (3.56) for all the null components of the correction to the Riemann tensor δR = R − R(Kerr) . Theorem 3.3. Assume that in V∗ the estimates (2.48) and (2.51) hold; then, in V∗ , we have
0 (3.96) δR ≤ . 2 Proof. The proof goes basically as in [15]. We repeat here the main steps stating the various lemmas which prove the theorem. Remark. In the following Lemmas 3.1,. . . ,3.6 we prove the estimates for the sup norms of the null components of δR. To complete the bootstrap we need analogous estimates for their derivatives up to l = q − 1. Again, apart from notational complications, the more delicate part is the control of the nondifferentiated components. The remaining estimates are just a repetition and follow the pattern discussed in [10]. Lemma 3.1. Assuming the estimates (2.48) and the condition M 1 R0
(3.97)
˜=L ˆ T R satisfy the following it follows that the various null components of R 0 inequalities with an “independent” constant c2 > c1 and = γ: 7 5 ˆ T R)| ≤ c2 ε + 0 sup r 2 |u| 2 + 2 |α(L 0 N0 K 7 5 ˆ T R)| ≤ c2 ε + 0 sup r 2 |u| 2 + 2 |β(L 0 N0 K ˆ T R) − ρ(L ˆ T R)| ≤ c2 ε + 0 sup r3 |u|3+ 2 |ρ(L (3.98) 0 0 N0 K ˆ T R) − σ(L ˆ T R)| ≤ c2 ε + 0 sup r3 |u|3+ 2 |σ(L 0 0 N0 K ˆ T R)| ≤ c2 ε + 0 sup r2 |u|4+ 2 |β(L 0 N0 K ˆ T R)| ≤ c2 ε + 0 . sup r|u|5+ 2 |α(L 0 N0 K Proof. Under the estimates 2.48 and the condition M 1 < 2 1 R0 N0
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with N0 positive integer, we proved in Theorem 3.2 the inequalities 2 2 ˜ ≤ c1 ε2 + M 2 ≤ c1 ε2 + 0 ≤ c1 ε + 0 Q R0 0 N02 N0 12 1 1 M
0 3 3+ 2 2 2 ˜ ρ(R)| ≤ c1 ε +
0 ≤ c1 ε + sup |r |u| R0 N0 V∗
(3.99)
From it proceeding as in Chapter 5 of [10] the thesis follows. Next lemma shows that integrating along the incoming cones in V∗ , we can transform the decay in |u| proved in the previous lemma in a better decay in r. Lemma 3.2. From the results of Lemma 3.1, using assumptions (2.48) and also the condition κ
M ≤ 1, R0
the following estimates hold:
ˆ T R)| ≤ c˜4 ε + sup r5 |u|1+ 2 |α(L 0 V∗ 4 2+ 2 ˆ |β(LT0 R)| ≤ c˜3 ε + sup r |u| V∗ 3 3+ 2 ˆ ˆ |ρ(LT0 R) − ρ(LT0 R)| ≤ c3 ε + sup r |u| V∗
ˆ T R) − σ(L ˆ T R)| ≤ c3 ε + sup r3 |u|3+ 2 |σ(L 0 0 V∗ 2 4+ 2 ˆ β(LT0 R) ≤ c3 ε + sup r |u| V∗ 5+ 2 ˆ |α(LT0 R)| ≤ c3 ε + sup r|u| V∗
0 N0
0 N0
0 N0
0 N0
0 N0
(3.100)
0 . N0
with an “independent” constant c3 > c2 > c1 > c0 , and c˜3 , c˜4 satisfying c˜3 ≥ c(1 + c + c + c )
(3.101)
where c is a constant > 1, which can be different in different inequalities, where c , c satisfy 2 2 M c c ≥ cc2 κ M c ≥ cκ M + c (3.102) 2 0 R0 , R2 R2 0
and
0
c˜4 ≥ c(1 + c4 )
(3.103)
1 M2 M2 c4 ≥ c˜3 1 + κ 2 + κc12 2 . R0 R0
(3.104)
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Proof. This Lemma is basically Theorem 2.3 of [15] (see also [11] where the way of transforming the |u| decay in r decay was first introduced). As it is one of the central step of the whole result we repeat its proof for the β component. Proof of the second line of (3.100). From the Bianchi equations, see (3.2.8) ˆ T R)(ea ) satisfies, along the incoming null of [10], it follows that βa = β(L 0 hypersurface C(ν), the evolution equation36 $ # ∂βa + Ωtrχβa = 2Ωωβa + Ω ∇ / a ρ + ∇ / a σ + 2(χ ˆ · β)a + 3(ηρ + ησ)a ∂λ (3.105) From this equation, see Chapter 4 of [10], we obtain the following inequality: 2 d (2− p2 ) |r β|p,S ≤ ||2Ωω − (1 − 1/p)(Ωtrχ − Ωtrχ)||∞ |r(2− p ) β|p,S dλ 2 2 2 +Ω∞ |r(2− p ) ∇ / ρ|p,S + 3|r(2− p ) ηρ|p,S + |r(2− p ) F˜ |p,S ,
(3.106) where F˜ (·) = 2χ ˆ · β + ( ∇ / σ + 3 ησ).37 Integrating along C(ν), with λ1 = u|C(ν)∩Σ0 , we obtain 2
2
|r(2− p ) β|p,S (λ, ν) ≤ |r(2− p ) β|p,S (λ1 ) λ 2 + ||2Ωω − (1 − 1/p)(Ωtrχ − Ωtrχ)||∞ |r(2− p ) β|p,S (λ , ν) λ1
⎛ λ ⎞ λ λ 2 2 2 1 +Ω∞ ⎝ |r(2− p ) ∇ / ρ|p,S +3 |r(2− p ) ηρ|p,S + |r(2− p ) F˜ |p,S ⎠ . 2 λ1
λ1
λ1
In V∗ the previous assumptions imply the following estimates: r|λ|Ωω∞ ≤ κM
and
r|λ|Ω(trχ − Ωtrχ)∞ ≤ κM.
36 All the notations used in this paper without an explicit definition are those already introduced in [10]. The moving frame compatible with equation 3.105 is the Fermi transported one (see the detailed discussion in [10, Chapter 3]). 37 The term ∇ / σ + 3ησ behaves as the term ∇ / a ρ + 3ηa ρ and, therefore, we will not consider it explicitly.
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Therefore, we can apply the Gronwall’s Lemma obtaining:38 ⎛λ ⎡ 2 2 2 |r2− p β|p,S (λ, ν) ≤ c ⎣|r2− p β|p,S (λ1 ) + Ω∞ ⎝ |r2− p ∇ / ρ|p,S λ1
⎞⎤ λ λ 2 2 1 + 3 |r2− p ηρ|p,S + |r2− p F˜ |p,S ⎠⎦ 2 λ1
(3.107)
λ1
From inequality (3.59) and the explicit expression of Ω(Kerr) , Ω∞ ≤ c, mul tiplying both sides by r2 |λ|2+ 2 , with > > 0, remembering that r(λ, ν) < r(λ1 , ν) and |λ| < |λ1 |, we obtain ⎛
|r
2 4− p
|λ|
2+ 2
2
4− β|p,S (λ, ν) ≤ c ⎝|r p |λ|2+
λ |r
+
2 4− p
2
β|p,S (λ1 )
2+ 2
|λ |
λ ∇ / ρ|p,S +3
λ1
1 + 2
⎞
λ |r
2 4− p
|λ |
2+ 2
|r
2 4− p
|λ |2+
2
ηρ|p,S
λ1
F˜ |p,S⎠ .
(3.108)
λ1
We examine the integrals in (3.108). This first integral has the following estimate we prove in the Appendix: 2
0 ˜ sup r4− p |λ|3+ 2 ∇ / ρ(R) ≤ c ε + . (3.109) N0 p,S K Therefore, λ |r λ1
2 4− p
2+ 2
|λ |
λ 1
0 ˜ ∇ / ρ(R)|p,S ≤ c ε + − 1+ N0 2 |λ | λ1 1
0 ≤ c ε + . N0 |λ| − 2
(3.110)
To estimate the second integral, observe, recalling assumption (2.48), that η satisfies |r2−2/p |λ|η|p,S (λ, ν) ≤ κM 2 ,
p ∈ [2, ∞].
(3.111)
38 The constant c can be chosen as an “independent” constant for the following reason: in this application of the Gronwall Lemma the constant c has to bound the following exponent ⎧ ⎫ ⎪ ⎨∞ κM ⎪ ⎬ kM exp , ≤ exp 2 ⎪ ⎪ λ R0 ⎩ ⎭ λ1
therefore, under the assumption of the lemma we can choose c ≥ e.
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˜ in V∗ and the results of the previous Using the previous estimate for ρ(R) lemma we can conclude that
0 ˜ ≤ c2 + c0 M sup r3 |λ|3+ 2 ρ(R) ε+ ; R0 N0 V∗ it follows immediately λ |r
2 4− p
2+ 2
|λ |
ηρ|p,S
λ1
λ M 1
0 ≤ κM c2 + c0 ε+ · − 2+ R0 N0 2 r|λ | λ1 1 M M2
0 ≤ cκ 2 c2 + c0 ε+ R0 R0 N0 |λ| − 2 1
0 ≤ c ε + − N0 |λ| 2 2
with c ≥ cκ
M2 R02
c2 + c0
M R0
.
(3.112)
/ σ + 3 ησ + 2χ ˆ·β To estimate the third integral, from the expression F˜ (·) = ∇ and the previous remark concerning ∇ / σ + 3 ησ, we are left to prove that λ
2
|r4− p |λ |2+
2
χβ| ˆ p,S ≤ c
(3.113)
λ1
This is easy, as, from the estimates (3.98) and (2.48), we have
0 2 4+ 2 sup |r |λ| β| ≤ c2 ε + , N0 V∗ ˆ p,S ≤ κM 2 sup ||λ|r2−2/p χ| V∗
p ∈ [2, ∞].
(3.114)
Therefore, λ λ1
2
|r4− p |λ |2+
2
λ 1
0 F˜ |p,S ≤ c2 ε + κM 2 − 3+ N0 2 |λ | λ1 1
0 M2 ≤ cc2 ε + κ 2 N0 R0 |λ| − 2 1
0 ≤ c ε + − N0 |λ| 2
(3.115)
with c ≥ cc2 κ
M2 M2 ≥ cc2 κ 2 . 2 R0 R0
(3.116)
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Collecting all these estimates for the integrals in (3.108), we infer that
2
|r4− p |λ|2+ 2 β|p,S (λ, ν)
1
0 ≤ c |r |λ| β|p,S (λ1 ) + (c + c + c ) ε + N0 |λ| − 2 1
0 ≤ c(1 + c + c + c ) ε + N0 |λ| − 2 1
0 ≤ c˜3 ε + (3.117) − N0 |λ| 2 2+ 2
2 4− p
and finally,
2
0 |r4− p |λ|2+ 2 β|p,S (λ, ν) ≤ c˜3 ε + (3.118) N0 where we used the initial data assumptions QΣ0 ≤ ε which implies, assuming
≤ γ, 2
|r4− p |λ|2+ 2 β|p,S (λ1 , u = |λ1 |) ≤ cε. To prove the sup estimate in (3.100) we have to repeat for ∇ / β the previous estimate for β. This requires the transport equation for ∇ / β along the C “cones” which in turn requires the control of an extra derivative for ρ and σ. This is the reason why we need a greater regularity in the initial data which translate in the definition of Q norms with more Lie derivatives than in [10]. We do not write the proof as it goes, with the obvious changes, exactly as for the β estimate. Therefore, we have proved the following inequality: ˆ T R)| ≤ c˜3 ε + 0 (3.119) sup r4 |u|2+ 2 |β(L N0 K with c˜3 ≥ c(1 + c + c + c )
where c , c
(3.120)
satisfy
M M2 M2 + c (3.121) c , c ≥ cc2 κ 2 . 2 0 2 R0 R0 R0 The analogous estimate for α follows the same lines; we do not report it here and we refer to [5] where it is written in all the details. c ≥ cκ
˜ . . . to the ˆ T R), Next Lemma allows us to go from the estimates of α(L 0 39 ˜ estimates for α(LT0 R), . . .; it is (a simplified version of ) Theorem 2.4 of [15]. To prove it we have to use some norm estimates for the components of the (T0 )π deformation tensor based on the δO bootstrap assumptions. Moreover we also use the estimates of the norms of the Riemann tensor R = R(Kerr) + δR, which requires the control of the various components of the Kerr Riemann tensor. 39
The reason is that in V∗ the bootstrap assumptions are stronger than those which can be done “ab initio” on the whole spacetime in [15], see the initial discussion in the introduction there.
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3.3.4. The (T0 )π Deformation Tensor Estimates. The explicit expressions of the null components of the (T0 )π deformation tensor are (T0 )
π(e3 , e3 ) = (T0 )π(e4 , e4 ) = 0
(T0 )
π(e3 , e4 ) = 4(ω + ω) − (g(D3 X, e4 ) + g(D4 X, e3 ))
(T0 )
π(e3 , ea ) = 2Ωζ(ea ) − g(D3 X, ea ) − g(Da X, e3 )
(3.122)
(T0 )
π(e4 , ea ) = −2Ωζ(ea ) − g(D4 X, ea ) − g(Da X, e4 )
(T0 )
π(ea , eb ) = Ω(χ + χ)(ea , eb ) − (g(Da X, eb ) + g(Db X, ea ))
They are identically zero in Kerr spacetime. Under the bootstrap assumptions 2.48 we prove the following estimates: |r2 |u|2+δ (T0 )π(e3 , e4 )| ≤ c 0 |r2 |u|2+δ (T0 )π(e3 , ea )| ≤ c 0 |r2 |u|2+δ (T0 )π(e4 , ea )| ≤ c 0 2
(3.123)
2+δ (T0 )
|r |u|
π(ea , eb )| ≤ c 0
The proof follows immediately from the bootstrap assumptions (2.48) and from the estimates for the metric component corrections, (3.59), with the only exception of the estimate of the D4 δX part which requires a separate estimate proved in Lemma 6.1 of [5]. Observe that in the Kerr spacetime (g(D3 X, e4 ) + g(D4 X, e3 ))
(Kerr)
(g(Dλ X, eλ ) + g(Dλ X, eλ ))
(Kerr)
(g(Dφ X, eφ ) + g(Dφ X, eφ ))
(Kerr)
=0 =0
(3.124)
= 0.
This implies that the following combination of the connection coefficients are identically zero in Kerr and, therefore, in the perturbed Kerr they satisfy the following inequalities: |r2 |u|2+δ (ω + ω)| ≤ c 0 , R0−δ |r2 |u|2+δ (χ + χ)λλ | ≤ c 0 ,
|r2 |u|2+δ (trχ + trχ)| ≤ c 0 R0−δ |r2 |u|2+δ (χ + χ)φφ | ≤ c 0 .
(3.125)
Remark. The estimates of the (T0 )π components are required to move from the ˆ T R components to those of the LT R, later to the estimates estimates of the L 0 0 of the ∂T0 R components and finally to the δR ones. To prove that they have the right smallness and the appropriate decay we need to control the corrections δO. It is at this point that, to close the bootstrap mechanism, we need the transport equations for the δO parts which require the delicate subtraction of the Kerr part from these equations, we discuss in Sect. 3.4. 3.3.5. The Riemann Null Components in the Kerr Spacetime. The “principal null directions” frame {l, n, e˜θ , e˜φ }, see [2], is made by the following vector
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fields, where r is the Boyer–Lindquist radial coordinate rb : a ∂ ∂ r2 + a2 ∂ + + , Δ ∂t Δ ∂φ ∂r 2 Δ r + a2 ∂ a ∂ ∂ n= + − 2Σ Δ ∂t Δ ∂φ ∂r 1 ∂ , e˜θ = √ Σ ∂θ 1 ∂ 1 ∂ e˜φ = √ a sin θ + ∂t sin θ ∂φ Σ l=
where l, n are the principal null directions. As, in the Petrov classification, the Kerr spacetime is of type D, in this frame the only null Riemann component different from zero are ρ and σ or, in the Newman-Penrose notations, Ψ2 , 1 1 1 3a cos θ Ψ2 = ρ(R) + iσ(R) = = + i + O (3.126) 3 3 4 (r − i cos θ) r r r5 Beside the fact that our initial data are not exactly those of the Kerr spacetime due to corrections δ (3)g and δ (3)k, the null orthonormal frame we use is not the one associated with the principal null directions. Therefore, in the frame introduced in [8], all the Riemann components of the Kerr spacetime are different from zero. The relation between the frame adapted to the (Kerr) double null foliation of V∗ , see (2.32), and the “principal null directions” one is (Kerr)
e4
√
Δ = 2ΣR
)
RΣr r +a + Q 2
2
RΣr l+ r +a − Q 2
2
√ 2Σ n − 2 Σa sin θ˜ eφ Δ
*
(Kerr)
e3
√
=
Δ 2ΣR
)
* √ RΣr 2Σ n − 2 Σa sin θ˜ eφ l + r2 + a2 + Q Δ Q ΔP 2Σ (Kerr) eλ e˜θ − n , =√ l− 2ΣR Δ ΣR 2 2 2Σ r +a Δ (Kerr) eφ e˜φ − a sin θ l + n . = √ 2RΣ Δ ΣR
r2 + a2 −
RΣr Q
where40 Q2 = (r2 + a2 )2 − a2 λ, P 2 = a2 (λ − sin θ2 ), L = μP Q, λ = sin2 θ∗ . (3.127)
40
μ is an integrating factor defined in [8] equation (25) and θ∗ is in the M → 0 limit the spherical θ coordinate of the Minkowski spacetime.
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Assuming M/r ≤ M/R0 small the previous relations become approximately ) 2 * M M n + O e4 = l + O e˜φ , r2 r ) 2 * M M e3 = n + O l + O (3.128) e˜φ r2 r M M eλ = e˜θ + O (l − 2n) , eφ = e˜φ + O (l + 2n) . r r Denoting with the upperscript (P N ) the Riemann components in the “Principal null directions frame”, we have 2 2 M M (Kerr) (P N ) α (ea , eb ) = R(ea , e4 , eb , e4 ) = O +O ρ
ab σ (P N ) r2 r2 M M (Kerr) −1 (P N ) β (ea ) = 2 R(ea , e4 , e3 , e4 ) = O +O ρ
ab σ (P N ) r r (3.129) which implies the following estimates: r5 α(R(Kerr) )∞ ≤ cM 3 ,
r4 β(R(Kerr) )∞ ≤ cM 2 .
(3.130) 41
Estimates (3.130) and the “Bootstrap assumptions” for δR imply
0 r5 α(R)∞ ≤ cM 3 + c˜ 0 , r4 β(R)∞ ≤ cM 2 + c˜ . (3.131) R0 ˆ T R estimates the LT R ones. We are now in the position to obtain from the L 0 0 Lemma 3.3. Under the same assumptions as in Lemma 3.2, using the results proved there we have in the region V∗ the following inequalities:
0 M2 5 1+ 2 sup r |u| |α(LT0 R)| ≤ c˜6 ε + + c 2 0 N0 R0 K
M 0 sup r4 |u|2+ 2 |β(LT0 R)| ≤ c˜5 ε + + c 0 N0 R0 K
0 sup r3 |u|3+ 2 |ρ(LT0 R) − ρ(LT0 R)| ≤ c5 ε + N0 K
0 3 3+ 2 sup r |u| |σ(LT0 R) − σ(LT0 R)| ≤ c5 ε + N0 K
0 2 4+ 2 sup r |u| |β(LT0 R)| ≤ c5 ε + N0 K
0 5+ 2 sup r|u| |α(LT0 R)| ≤ c5 ε + . N0 K 41
It has to be pointed out that the estimates (3.130) refer to the various components of the (Kerr) Riemann tensor in the orthonormal frame associated with the Kerr spacetime, what in fact we have to consider here are the null Riemann components relative to the null orthonormal frame associated with the perturbed Kerr spacetime (see Eqs. (2.28)). It is easy to see, using the bootstrap assumptions for the metric components, that estimates (3.130) still hold possibly with a different c˜ constant.
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where, we assume δ ≥ 2 , c˜5 ≥ c˜3 + c˜;
c˜6 ≥ (˜ c4 + c˜ c) .
Proof. We start recalling the following expressions: ˆ T R + 1 (T0 ) [R] − 3 (tr(T0 ) π)R, LT0 R = L 0 2 8 where (T0 )
(3.132)
(3.133)
[R]αβγδ = (T0 ) παμ Rμβγδ + (T0 ) πβμ Rαμγδ + (T0 ) πγμ Rαβμδ + (T0 ) πδμ Rαβγμ . (3.134)
From these equations it follows 1 (T0 ) 3 [R]a4b4 − (tr(T0 ) π)α(R)ab 2 8 1 3 ˆ T R)a + (T0 ) [R]a434 − (tr(T0 ) π)β(R)a β(LT0 R)a = β(L 0 2 8 (T0 ) and, observing that π44 = 0 we easily obtain ˆ T R)ab + α(LT0 R)ab = α(L 0
(3.135)
(T0 )
[R]a4b4 1 3 = − (T0 ) π ˆa4 R34b4 + (T0 ) π ˆac Rc4b4 + (T0 ) π ˆbc Ra4c4 + (tr(T0 ) π)Ra4b4 2 4 1 (T0 ) 1 − π ˆ43 Ra4b4 + (T0 ) π ˆ4c (Racb4 + Ra4bc ) − (T0 ) π ˆb4 Ra434 . 2 2 Therefore estimating (T0 ) [R]a4b4 we obtain (T0 ) [R]a4b4 ∞ ≤ c (|(T0 )i|∞ + (T0 )j∞ )α(R)∞ + (T0 )m∞ β(R)∞ 1 ≤ c 2+5 2+δ (r2 |λ|2+δ (T0 )i∞ + r2 |λ|2+δ (T0 )j∞ )r5 α(R)∞ r |λ| 1 2 2+δ (T0 ) 4 + 2+4 2+δ r |λ| m∞ r β(R)∞ r |λ| 1 1 5 4 ≤ c 0 7 2+δ r α(R)∞ + 6 2+δ r β(R)∞ r |λ| r |λ| 2 c 0 (cM 3 + c˜ 0 ) (cM + c˜ R00 ) ≤ 5 1+ + (3.136) r |λ| 2 r2 |λ|1+δ− 2 r|λ|1+δ− 2 where in the last two lines we used estimates (3.131) so that, finally, 2 (cM 3 + c˜ 0 ) (cM + c˜ R00 ) 5 1+ 2 (T0 ) r |λ| [R]a4b4 ∞ ≤ c 0 + . (3.137) r2 |λ|1+δ− 2 r|λ|1+δ− 2 Moreover, (tr(T0 ) π)α(R)∞ ≤ ≤
c r7 |λ|2+δ c 0 r5 |λ|1+ 2
|r2 |λ|2+δ tr(T0 )π|∞ r5 α(R)∞ (cM 3 + c˜ 0 ) r2 |λ|1+δ− 2
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and
r5 |λ|1+ 2 (tr(T0 ) π)α(R)∞ ≤ c 0
(cM 3 + c˜ 0 ) . r2 |λ|1+δ− 2
(3.138)
Using estimates (3.137) and (3.138) it follows, using condition (3.54), (δ ≥ 2 ),
sup |r5 |u|1+ 2 α(LT R)| K (cM 3 + c˜ 0 )
0 ≤ c˜4 ε + + + c 0 N0 r2 |λ|1+δ− 2
0 M2 ≤ c˜6 ε + + c 2+δ− 0 ≤ c˜6 ε + 2 N0 R0
0 ) (cM 2 + c˜ |λ|
r|λ|1+δ− 2
0 M2 + c 2 0 . N0 R0
(3.139)
The estimates for the β term go in the same way and we do not report them here. For the other terms there is no need to repeat this computation. In fact ˆ T to LT the R null components already satisfy the peeling and going from L 0 0 42 the decay factors do not worsen. Next step is to obtain from the estimates for α(LT0 R), β(LT0 R) the estimates for ∂T0 α(R), ∂T0 β(R). This requires the control of [T0 , ea ], [T0 , e4 ] and [T0 , e3 ]. Lemma 3.4. Under the bootstrap assumptions in the region V∗ the following estimates hold:
0 M2 |g([T0 , ea ], ed )| ≤ c 1 + 2 2 R0 r |u|2+δ
0 M2 |g([T0 , e4 ], e3 )| ≤ c 1 + 2 R0 r2 |u|2+δ ) *
0 M2
0 |g([T0 , e4 ], ed )| ≤ c 1 + 2 1 + 3 R0 R0 r2 |u|2+δ
0 M2 |g([T0 , e3 ], ed )| ≤ c 1 + 2 R0 r2 |u|2+δ
0 M2 |g([T0 , e3 ], e4 )| ≤ c 1 + 2 R0 r2 |u|2+δ Proof. The proof consists in a long, but elementary set of estimates which together with the explicit expressions of the various commutators are in [5]. In the next Lemma 3.5 we prove the estimates we are looking for relative to ∂T0 (α(R)(ea , eb )) and ∂T0 (β(R)(ea )). Lemma 3.5. Under the same assumptions as in Lemma 3.2, using the results proved there and in Lemmas 3.3, 3.4, we have in the region V∗ the following 42
The corrections due to the change of null frames are inglobed in the choice of the c5 constant.
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inequalities: 5
1+ 2
sup r |u| K
4
|∂T0 (α(R)(ea , eb ))| ≤ c˜7
2+ 2
sup r |u| K
|∂T0 (β(R)(ea ))| ≤ c˜8
0 ε+ N0
0 ε+ N0
+c
471
M2
0 R02
M + c 0 , R0
(3.140)
with c˜7 ≥ (˜ c6 + c) ;
c˜8 ≥ (˜ c5 + c) .
(3.141)
Proof. It is a long, but elementary estimate obtained starting from the relations α(LT0 R)(ea , eb ) = (LT0 R)(ea , e4 , eb , e4 ) = ∂T0 (α(R)(ea , eb )) + R([T0 , ea ], e4 , eb , e4 ) + R(ea , [T0 , e4 ], eb , e4 ) β(LT0 R)(ea ) = (LT0 R)(ea , e4 , e3 , e4 ) = ∂T0 (β(R)(ea )) + R([T0 , ea ], e4 , e3 , e4 ) +R(ea , [T0 , e4 ], e3 , e4 ) + R(ea , e4 , [T0 , e3 ], e4 ).
(3.142)
We do not report it here, but for details see [5]. 3.3.6. The Estimate of δR. The final step consists in integrating along the integral curves of T0 . The |u| weight factors will allow to bound uniformly these integrals. Denoting by γ(s) the integral curve of T0 starting in Σ0 at a distance r0∗ from the origin and α(t) = α(t)(ea , eb ), α(t, r∗ ) = α
(Kerr)
s (0, r∗0 ) + δα(0, r ∗0 ) +
(∂T0 α)(γ(s))ds
(3.143)
0
where r∗ = r∗ (u, u) = u − u = γ r∗ (s), t = t(u, u) = u + u = γ 0 (s) r∗ r∗ u = u(t, r∗ ) = u(0, r∗1 ) = − 1 , u = u(t, r∗ ) = u(0, r∗2 ) = 2 2 2 t + r∗ − r∗ 1 r∗ 2 − r∗ 1 = . (3.144) t = t(u, u) = 2 2 As T0 =
∂ ∂ + ∂u ∂u
(3.145)
it follows that dγ μ = T0μ = δuμ + δuμ ds
(3.146)
therefore, with these definitions, in the coordinates {xμ } = {u, u, ω 1 , ω 2 }, d dγ μ ∂u dγ u u(γ(s)) = =1 = μ ds ds ∂x ds d dγ u dγ μ ∂u ∂u dγ u r∗ (γ(s)) = − =0 − = μ μ ds ds ∂x ∂x ds ds
(3.147)
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and, as ω 1 , ω 2 do not change along γ(s) d dr d r(γ(s)) = r∗ (γ(s)) = 0 ds dr∗ ds
(3.148)
and u(γ(s; r ∗ )) = u(γ(0; r ∗ )) + s, u(γ(s; r ∗ )) = u(γ(0; r ∗ )) + s t = u + u = (u(γ(0; r ∗ )) + u(γ(0; r ∗ ))) + 2s = 2s. (3.149) It follows δα(u, u) = α(u, u) − α
(Kerr)
s (u, u) = δα(0, r∗ ) +
(∂T0 α)(γ(s))ds 0
and s |δα(u, u)| ≤ |δα(0, r ∗ )| +
|(∂T0 α)(γ(s))|ds ≤ cˆ 0
+ c˜7 ε + c
M2
0 R02
s 0
ε r∗ 5
1 ds. r(γ(s))5 |u(γ(s))|1+ 2
(3.150)
Therefore, s 1 r5 (u, u) M2 |r δα(u, u)| ≤ cˆ 5 ε + c˜7 ε + c 2 0 ds r∗ (u, u) R0 |u(γ(s))|1+ 2 5
0
s
1 M2 r5 (u, u) ε + c ˜ ε + c
0 ds 7 r∗5 (u, u) R02 |u(γ(0; r ∗ )) + s|1+ 2 0 M2
0 , (3.151) ≤ cˆ1 ε + c c˜7 ε + c 2 0 ≤ c˜9 ε + R0 N0
≤ cˆ
where we have chosen cˆ1 such that cˆ
r5 (u, u) ≤ cˆ1 , r∗5 (u, u)
(3.152)
which is possible as we have proved that r∗ and r(u, u) stay near, and chosen c˜9 such that c˜9 ε ≥ (ˆ c1 + c˜7 ) ε.
(3.153)
The proof for δβ goes exactly in the same way and we do not repeat it. Therefore, we have proved the following lemma:
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Lemma 3.6. Under the same assumptions as in Lemma 3.2, using the results proved there and in Lemmas 3.3, 3.4, 3.5 we have in the region V∗ the following inequalities:
0 |r5 δα(R)| ≤ c˜9 ε + N0 (3.154)
0 4 |r δβ(R)| ≤ c˜10 ε + N0 where c˜9 ≥ cˆ1 + c˜7 ;
c˜10 ≥ cˆ2 + c˜8 .
(3.155)
3.4. IV Step: The Control of the δO Norms To prove better estimate for these norms in V∗ , we have to use the transport equations along the incoming and outgoing cones. The use of the outgoing cones is made, as in [10], obtaining estimates starting from “scri”, here the upper boundary of the region V∗ , a portion of an incoming cone. Therefore, first we have to control the not underlined connection coefficients43 on this last slice and to avoid a loss of derivatives we have to prove the existence, on it, of an appropriate foliation called the “last slice canonical foliation”, see [14,10].44 The decay factors for the bootstrap assumptions for the various (not underlined) δO norms have to be consistent with the weight factors of the norms we control on the “last slice”. The estimates for the underlined connection coefficients are made, vice versa, starting from the initial data hypersurface Σ0 . Also in this case an appropriate (canonical) foliation has to be introduced on Σ0 .45 Finally, as anticipated in Sect. 3.2 to control the δO norms we need estimates for the corrections to the Kerr metric. These estimates follow from the bootstrap assumptions on δO and are the content of the following lemma: Lemma 3.7. Assume that in V∗ the norms δO satisfy the bootstrap assumptions δO ≤ 0 (0)
then, assuming for the δO norms appropriate initial data conditions, see Sect. 3.6, the following estimates hold in V∗ : |r|u|2+δ δΩ|∞ ≤ c 0 ;
|r2 |u|2+δ δX|∞ ≤ c 0 ; 46
The proof of this lemma is given later on, tant to remark the order of the various proofs:
||u|1+δ δγ|∞ ≤ c 0 .
(3.156)
(see Sect. 3.4.4). It is impor-
43 Basically we denote as not underlined connection coefficients those coefficients whose transport equations we use are those along the outgoing cones, the opposite for the underlined ones. Remember, as discussed in detail in [10] that the choice of the transport equations to use is not arbitrary and is uniquely fixed by the request of avoiding any loss of derivatives which will make the bootstrap mechanism to fail. 44 See also [6] where the original idea was first stated. 45 This could be in principle avoided requiring more regularity for the initial data. 46 In fact we prove the equivalent Lemma 3.11.
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The bootstrap assumptions δO ≤ 0 imply, Lemma 3.7, the metric correction estimates δO(0) ≤ c 0 . (b) The metric correction estimates δO(0) ≤ c 0 , the initial data assumptions and the Riemann bootstrap assumptions imply better estimates for the δO connection coefficients δO ≤ 20 . (c) The improved estimates for the connection coefficients imply better estimates for the metric corrections (see Lemma 3.11, δO(0) ≤ 20 ). (a)
3.4.1. The Control of the δO Norms. The δO norms involve tangential derivatives up to fifth order to prove the bootstrap and the peeling, see [10,11]. The proof we sketch here is restricted to the zero and first derivatives as the control of the higher derivatives is simpler and is basically a repetition of what has been done in [10]. To control the δO norms we subtract from the connection coefficients their Kerr parts to obtain transport equations and Hodge equations for the δO corrections. This operation, we call “Kerr decoupling” is a central step of the whole procedure and we discuss it in some generality. Let us consider a two-covariant tensor connection coefficient O = Oμν dxμ ⊗dxν .
(3.157)
O is a tensor field tangent to the two-dimensional surfaces S(u, u) = C(u) ∩ C(u) intersections of the outgoing and incoming cones of the canonical null cone foliation assumed in V∗ ; Oμν can be written Oμν = Πρμ Πσν Hρσ
(3.158)
where H is a (0, 2) tensor in (V∗ , g), a priori not S-tangent, and Πμν projects from T V∗ to T S.47 As an example, look at the second null fundamental form χ, g(Dea e4 , eb ) = χ(ea , eb ) = χμν eμa eνb .
(3.159)
Denoting {θ (·)} the one forms dual to the T S orthonormal frame {ea }, χμν = χ(ea , eb )θμa θνb = g(Dea e4 , eb )θμa θνb b
a,b
= =
a,b
gρσ eτa (Dτ e4 )ρ eσb θμa θνb
= Πτμ Πσν (Dτ e4 )ρ gρσ
a,b Πτμ (Dτ e4 )σ Πσν
and finally, with {xμ } = {u, u, θ, φ}, χ = χμν dxμ ⊗dxν = Πρμ Πσν (Dρ e4 )σ dxμ ⊗dxν . 47
(3.160)
It is important to remark that it is not true that O = Oab dω a ⊗ dω b . This can be easily recognized looking at (3.160). In fact a a u b u θμ = g(ea , ·) = gμc eca = δμ γac + δμ (γbc X(Kerr) ) + δμ (γbc X b ) eca .
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The generic structure equations for the connection coefficients O are of two types: equations which are “transport equations” along the outgoing or incoming cones and “elliptic Hodge type” equations on the S surfaces. The transport equation has the following general structure, indicating with O a connection coefficient or a tangential derivative: D / 4 O + ktrχO = F,
(3.161)
where the integer k depends on the connection coefficient O we are considering, D / 4 is the projection on T S of the differential operator D4 = De4 , (D / 4 O)μν = Πρμ Πσν (D4 O)ρσ ,
(3.162)
F is a covariant “S-tangent” tensor whose components are quadratic or cubic functions of the (components of the) connection coefficients and possibly, if O is a tangential derivative of a connection coefficient, of the Riemann tensor, F = Fμν dxμ ⊗dxν ,
Fμν = Fμν ({O}, {R}).
The Hodge equations for the connection coefficients or their tangential derivatives have the form48 div / O =G+R
(3.163)
where div / is the divergence associated with ∇ / , the covariant derivative relative to the induced metric, γ (S) , on S, div / O = (div / O)σ dxσ ,
(div / O)σ = γ (S)
μν
(∇ / μ O)νσ ,
(3.164)
G is a covariant tensor and R denotes a null component of the Riemann tensor tangent to S of the same degree. To have a specific example of structure equations with this structure, we consider in [5] the transport equation for ∇ / trχ and the Hodge equation for χ ˆ (see also equations (4.3.6) and (4.3.13) of [10]). To obtain the structure equations for δO, we define O(Kerr) , the connection coefficient analogous to O but associated with the Kerr spacetime, (Kerr) O(Kerr) = Oμν dxμ ⊗dxν .
We would like to subtract the Kerr part and define δO ≡ O − O(Kerr) .
(3.165)
We need, nevertheless, a slight modification; in fact in the transport equations for the corrections, the δO terms have to be S-tangent to the S intersections of the outgoing and incoming cones of the foliation in the perturbed Kerr spacetime (V∗ , g). Nevertheless, the O(Kerr) tensor field is tangent to the S surfaces, intersections of the outgoing and incoming cones of the foliation in V∗ thought as a region of the Kerr spacetime. This means that, as in 3.158, we have ρ
σ
(Kerr) (Kerr) Oμν = Π(Kerr) μ Π(Kerr) ν Hρσ 48
Applied to a two S-tangent covariant tensor field.
(3.166)
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which is not S-tangent in (V∗ , g). Therefore, instead of (3.165) we define where
ˆ δO ≡ O − O.
(3.167)
(Kerr) ˆ μν = Πρμ Πσν Hρσ . O
(3.168)
To obtain a transport equation for δO we apply to it D / 4 obtaining, recalling that (Kerr)
O(Kerr) − ktrχ(Kerr) O(Kerr) + F (Kerr) = 0, ˆ O(Kerr) ) + δF (δO, O, ˆ O(Kerr) , δR) D / 4 δO + ktrχδO = H(δO(0) , O, −D /4
(3.169) where
(Kerr) ˆ (Kerr) ˆ ˆ O(Kerr) ) −(D H(δO(0) , O, /4 − D /4 )O − D /4 (O − O(Kerr) )
ˆ − ktrχ(Kerr) (O ˆ − O(Kerr) ) −kδtrχO δF (δO, O(Kerr) , δR) = F − F (Kerr) .
(3.170)
Now we proceed as in [10, Chapter 4], and we just sketch the proof. Let |δO| be a | · |p,S norm, applying Gronwall inequality and Lemma 4.1.5 of [10] we obtain the following estimate, with σ > 0: 2 2 ||u|2+δ r(2−σ)− p δO|p,S (u, u) ≤ c0 ||u|2+δ r(2−σ)− p δO|p,S (u, u∗ ) ⎞ u∗
2 2 ||u|2+δ r(2−σ)− p H|p,S + ||u|2+δ r(2−σ)− p δF |p,S (u, u )⎠ . (3.171) + u
To control the right-hand side of (3.171) we have to estimate the norm 2
||u|2+δ r3− p δO|p,S (u, u∗ ) on the last slice and the norms of H and δF .49 The norm on the last slice is discussed later on when we prove the existence of the “last slice canonical 2 2 foliation”. The bounds for the norms ||u|2+δ r3− p H|p,S and ||u|2+δ r3− p δF |p,S are proved in the following lemma: Lemma 3.8. Under the bootstrap assumptions, the following estimates hold in V∗ 2 2 2 2 M M 2+δ 3− p 2+δ 3− p ||u| r H|p,S ≤ c 0 , ||u| r δF |p,S ≤ c 0 . R0 R0 Once Lemma 3.8 is proved, assuming an analogous estimate on the last slice, 2 2 M 2+δ 2− p ||u| r δO|p,S (u, u∗ ) ≤ c 0 , (3.172) R0 49
It i to obtain these estimates that we consider the transport equations starting from the last slice which, therefore, has to be assigned as “initial data”, see Sect. 2.1.
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we obtain, integrating, 2
|u|2+δ r(2−σ)− p δO|p,S (u, u) ⎛ ≤ c0 ⎝||u|2+δ r
2 (2−σ)− p
δO|p,S (u, u∗ ) + c 0
M R0
2u∗ r
⎞ 1 ⎠ (3.173) 1+σ
u
2 M and, as |u| ≥ r, choosing R0 such that c R < 12 , the result 0 2
||u|2+δ r2− p δO|p,S (u, u) ≤ c 0
M R0
2 <
0 , 2
(3.174)
proving that the norms δO satisfy in V∗ better estimates than those in the “Bootstrap assumptions”. Proof of Lemma 3.8. We do not report it, see for details [5]. Although complicated by the need of subtracting the Kerr part, the strategy we have discussed follows the one described in [10]. There to complete the bootstrap mechanism, beside the transport equations, also the structure equations which are elliptic Hodge systems on S are used. Also for these equations the Kerr part has to be subtracted. Performing this subtraction there are no new ideas involved, different from those already described, therefore we do not discuss here their Kerr decoupling and the reader can find a detailed discussion in [5]. Here we estimate in detail the norm of the tangential derivative of a specific connection coefficient (its correction), δ(∇ / trχ). 2
/ trχ)|. We look at the 3.4.2. Detailed Estimate of the Norm |r 3− p |u|2+δ δ(∇ correction δO associated to the S-tangent vector field / trχ + trχζ) U / = Ω−1 (∇ and show in detail, for its transport equation, the structure we sketched in general. The transport equation U / satisfies is the following one: 3 / =F /. ΩD / 4U / + ΩtrχU 2
(3.175)
Proceeding as discussed in general, see Sect. 3.4.1, we write U / in the following way: U / =U /ˆ + δU / where
(3.176)
50
U /ˆ =
1 Ω(Kerr)
∇ / trχ(Kerr) + trχ(Kerr) ζˆ
(3.177)
−1 (Kerr) (Kerr) and Hν Πνμ Ω(Kerr) Dν trχ(Kerr) + trχ(Kerr) Hν (Kerr) (Kerr) (Kerr) μ (Kerr) ρ gμρ (Dν e4 ) e 3 .
50
In fact U /ˆ μ
=
=
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and δU / =−
δΩ 1 (Kerr) (∇ / trχ +trχζ) + δζ ∇ / δtrχ + δtrχζ + trχ ΩΩ(Kerr) Ω(Kerr) (3.178)
where ˆ δζ = ζ − ζ.
(3.179)
F / =F /ˆ + δF /.
(3.180)
Analogously we write
Therefore, Eq. (3.175) can be written as 3 (Kerr) ΩD / 4 δU / = δF / + (F /ˆ − F / / + ΩtrχδU ) 2
(Kerr) (Kerr) (Kerr) (Kerr) /ˆ − U / ) + Ω(δD / 4 )U / + δΩD /4 U / +L − ΩD / 4 (U ) 3 (Kerr) /ˆ − U / − Ωtrχ(U ) 2 * 3 3 (Kerr) (Kerr) (Kerr) / + δΩtrχ(Kerr) U / + (F /ˆ − F / ) + ΩδtrχU 2 2 ) * 3 (Kerr) (Kerr) (Kerr) (Kerr) + −Ω(Kerr) D /4 U / − Ω(Kerr) trχ(Kerr) U / +F / 2
(3.181)
where the last line is identically zero, being the structure equation satisfied in Kerr spacetime; therefore, we can rewrite the equation as 3 (Kerr) / = δF / + [(δU / ), U /ˆ , U / / + ΩtrχδU ] ΩD / 4 δU 2
(3.182)
where (Kerr) ] [(δU / ), U /ˆ , U /
(Kerr) ˆ (Kerr) (Kerr) (Kerr) = − Ω(δD / 4 )U /ˆ + ΩD /4 (U / −U / ) + +δΩD /4 U / ) * 3 3 3 (Kerr) (Kerr) (Kerr) (Kerr) ˆ / −U / / − Ωtrχ(U ) + ΩδtrχU + δΩtrχ U / 2 2 2
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Lemma 3.9. Under all the previous assumptions the following inequality holds in V∗ 2 2 (Kerr) r4− p |u|2+δ [(δU / ), U /ˆ , U / ] ≤ c |r4− p |u|2+δ (δD / 4 )U /ˆ |p,S p,S
≤
2 4− p
(Kerr) ˆ (Kerr) +|r |u| ΩD /4 (U / −U / )|p,S 2 (Kerr) (Kerr) 4− p 2+δ +|r|u| δΩ|∞ |r D /4 U / |p,S 2+δ
2 3 (Kerr) + |rΩtrχ|∞ |r3− p |u|2+δ (U /ˆ − U / )|p,S 2 2 (Kerr) +|r2 |u|2+δ δtrχ|∞ |r3− p U / |p,S 2
+|r|u|2+δ δΩ|∞ |rtrχ(Kerr) |∞ |r3− p U / 2 (Kerr) 4− p 2+δ ˆ +|r |u| ((F / −F / ))|p,S ≤ c 0
(Kerr)
|p,S
M2 . R02
(3.183)
Proof. We do not report here the easy part of the estimates which simply fol(Kerr) lows from the bootstrap assumptions and the explicit expression of U / . 2 4− p 2+δ ˆ |u| (δD / 4 )U / |p,S is done exactly as in the proof The bound of the norm |r of Lemma 3.8 obtaining 2
/ 4 )U / |r4− p |u|2+δ (δD
(Kerr)
|p,S ≤ c 0
M2 . r2
(3.184)
2 2 (Kerr) ˆ (Kerr) The term |r4− p |u|2+δ ΩD /4 (U / −U / )|p,S requires the control of |r3− p | (Kerr) (Kerr) u|2+δ (U /ˆ − U / )|p,S , the effect of D /4 is only that of adding a power of r in the decay. On the other side again as in the proof of Lemma 3.8, we have immediately
2
2 M (Kerr) /ˆ − U / )|p,S ≤ c 0 2 |r3− p |u|2+δ (U R0
(3.185)
so that finally we have 2
(Kerr)
r4− p |u|2+δ [(δU / ), U /ˆ , U /
]
p,S
≤ c 0
M2 . R02
(3.186)
(Kerr) (Kerr) The explicit expression of δF / + (F /ˆ − F / )=F / −F / is
δF / = −Ωχ ˆ · δU / − Ωδ χU ˆ/
(Kerr)
− δΩχ ˆ(Kerr) U /
(Kerr)
(Kerr) (Kerr)
− 2χ∇ ˆ / δχ ˆ − 2χ(δ∇ ˆ / )χ ˆ(Kerr) − 2δ χ ˆ∇ / (Kerr)
− ηχ ˆ · δχ ˆ − ηδ χ ˆ·χ ˆ
+ trχ χ ˆ · δη + trχ δ χ ˆ·η
− δη|χ ˆ (Kerr)
χ ˆ
(Kerr) 2
|
+ δtrχ χ ˆ(Kerr) · η (Kerr)
− trχδβ − δtrχβ (Kerr) .
(3.187)
The first term of δF / has the following estimate: 2
|r4− p |u|2+δ Ωχ ˆ · δU / |p,S ≤ c
M 2 4− p2 2+δ |r |u| δU /| r3
(3.188)
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The norms of all the other terms satisfy the following estimates, as proved in the Appendix: 2 2 M M4 |r4− p |u|2+δ (δF / − Ωχ ˆ · δU / )|p,S ≤ c 0 + . (3.189) R02 R04 Again proceeding as we did in general for δO, applying Gronwall inequality and Lemma 4.1.5 of [10] we obtain the following estimate, with σ > 0: 2 2 / |p,S (u, u) ≤ c0 ||u|2+δ r(3−σ)− p δU / |p,S (u, u∗ ) ||u|2+δ r(3−σ)− p δU ⎞ u∗
2 2 ||u|2+δ r(3−σ)− p [(δU )U (Kerr) ]|p,S + ||u|2+δ r(3−σ)− p δF + / |p,S (u, u )⎠ . u
(3.190) Integrating, provided M/R0 1 and provided we have the appropriate last slice canonical estimates, we obtain cε M
0 cε ≤ 3 2+δ . (3.191) |δU / | ≤ 3 2+δ + c r |u| R0 r3 |u|2+δ 2r |u| if c(ε +
M
0
0 ) ≤ . R0 2
(3.192)
Remark. Observe that when starting from the transport equation for δU /, Eq. (3.182), we write a transport equation for the | · |p,S norms we use a Fermi transported frame. This Fermi transported frame is used only at this stage, while the “Kerr decoupling” is performed at the level of the tensorial equations. Next step is to obtain from the knowledge of the bounds for the norms ˆ satisfies the following of U / the bounds for the norms of ∇ / trχ and for χ. ˆ 51 χ equation, see [10] Eq. (4.3.13), div / χ ˆ+ζ ·χ ˆ−
Ω U / + β = 0. 2
(3.193)
Therefore52 : δΩ ˆ Ω /+ U / − δβ (3.194) div / δχ ˆ = −(δdiv / )(+ χ) ˆ − ζ · δχ ˆ − δζ · (+ χ) ˆ + δU 2 2 where δβ = β − βˆ and βˆ is defined as the corresponding connection coefficients. As we have σμ / (+ + γ (Kerr) δ∇ / (+ (3.195) χ) ˆ χ) ˆ (δdiv / )(+ χ) ˆ = δγ σμ ∇ σ
μν
σ
μν
it follows that to use (3.194) to estimate ∇ / δχ ˆ we need to be able to estimate the right-hand side and for that we need to have only a norm estimate for 51
The way we prove this result here is slightly different from what has been done in [10] (see in particular remark 1 at page 132). 52 The small hat denotes the traceless part of χ the large hat is the one introduced in (3.168).
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the first derivatives of δγ which are present in δ∇ / σ . From (3.194) we obtain immediately the following estimate: 2
||u|2+δ r3− p ∇ / δ χ| ˆ p,S 2 2 ≤ c ||u|2+δ r3− p (δdiv / )(+ χ)| ˆ p,S + ||u|2+δ r3− p ζ · δ χ| ˆ p,S 2 2 Ω 2 δΩ / |p,S + ||u|2+δ r3− p U /ˆ |p,S +||u|2+δ r3− p δζ · (+ χ)| ˆ p,S + ||u|2+δ r3− p δU 2 2 2 +||u|2+δ r3− p δβ|p,S 2 2 / )(+ χ)| ˆ p,S + ||u|2+δ r2− p δ χ| ˆ p,S |rζ|∞ ≤ c ||u|2+δ r3− p (δdiv 2 2 +||u|2+δ r2− p δζ|p,S |r(+ χ)| ˆ ∞ + ||u|2+δ r3− p δU / |p,S |Ω|∞ 2 2+δ 1− p 2ˆ +||u| r δΩ|p,S |r U / |∞ 2 M M
0 ≤ c 0 2 + c˜8 ε + c 0 ≤ R0 R0 2 provided M M2
0 c 0 + 2 + c˜8 ε ≤ . R0 R0 2
(3.196)
(3.197)
Remarks. (a) The estimates of the quantities with the hat are the same as the one for the Kerr terms. In fact their difference is a small correction as it has been proved in [5]. (b) It is here, to close the bootstrap that we need to require δ = 2 ,53 In fact looking at the inequality (3.196) it follows that the term depending on δβ is bounded only if δ ≤ 2 . As, on the other side, to control the ˜ norms, we required that δ ≥ , (see the footnote boundedness of the Q 2 after (3.90)); the conclusion is that, in the bootstrap assumptions, we must choose
(3.198) δ= . 2 3.4.3. The Various Radial Coordinates. In this paper various quantities play the role of “radial coordinates”; let us compare them. In the Kerr metric written in the Pretorius Israel coordinates, (Kerr) a dω a − X(Kerr) (du + du) g(Kerr) = −4Ω2(Kerr) dudu + γab b (du + du) × dω b − X(Kerr) and in the perturbed Kerr metric,
a du + X a du) g(pert.Kerr) = −4Ω2 dudu + γab dω a − (X(Kerr) b du + X b du) , × dω b − (X(Kerr)
53
If, in the other instances where we required δ =
, 2
we had avoided it the only price to δ− 2
be paid would have been the presence of dimensional constants like R0
.
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we define, respectively,54 (Kerr)
r∗
=u−u
and r∗ = u − u.
(3.199)
Finally, a radial function r(u, u) was defined in (2.42) proportional to the square root of the corresponding surface S(u, u), both for the Kerr and for the perturbed Kerr case, √ 1 1 r = r(u, u) = ( 4π)− 2 |S(u, u)| 2 . We use all these radial functions in a interchangeable way; this is possible as we can control their norm differences. In fact it can be easily proved, see [5] for details, that the following inequalities hold:
0 (Kerr) |r∗ − r∗ (u, u)| ≤ c (3.200) r|u|1+δ M (Kerr) r∗ ≤ r(Kerr) 1 + c , (3.201) R0
0 . (3.202) |r(u, u) − r(Kerr) (u, u)| ≤ c r|u|1+δ Therefore, as expected, if (Kerr) r∗ , r∗ , rb .
M R0
1, we can identify r(u, u), r(Kerr) (u, u),
3.4.4. The Estimates of the δO (0) Norms. To obtain this result, we have to first find the transport equations for δΩ, δX a and δγab . This is the content of the following lemma: Lemma 3.10. The corrections to the Kerr components of the metric δΩ, δX a and δγab , satisfy the following equations: δΩ(Ω + Ω(Kerr) ) (3.203) ∂N (δΩ) = −2(Ω(Kerr) + δΩ)2 δω + ∂N Ω(Kerr) (Kerr) 2 Ω 2 QΔ ∂ a a )δX c = −Ω X δΩ + 4Ω2 δζ a ∂N δX a − Ω(∂c X(Kerr) Ω(Kerr) ΣR2 ∂rb (Kerr) (3.204) ) * c c ∂X(Kerr) ∂X(Kerr) ∂N (δγab ) − Ωtrχ(δγab ) = − (δγcb ) + (δγac ) ∂ω a ∂ω b
(Kerr) (Kerr) . + Ωγab δtrχ + δΩγab trχ(Kerr) + 2Ω(Kerr) δ χ ˆab + 2δΩχ ˆ(Kerr) ab (3.205) Proof. The equation for δX a : With the definition of the metric, (2.30), the commutation relation (2.24) has the following aspect: a − ∂e3 X a = −4Ω2 ζ(ec )eac (3.206) Ω ∂e4 X(Kerr) 54
(Kerr)
The {r∗ , θ∗ } Pretorius Israel coordinates (for Kerr spacetime) can be expressed in (Kerr) (Kerr) terms of the Boyer–Lindquist coordinates, r∗ = r∗ (θ, rb ), θ∗ = θ∗ (θ, rb ) where rb is the Boyer–Lindquist radial coordinate, see [8].
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which we can write as a a − ∂e3 X(Kerr) Ω∂e3 δX a = Ω ∂e4 X(Kerr) + 4Ω2 ζ(ec )eac a = Ω(δeμ4 − δeμ3 )∂μ X(Kerr) + 4Ω2 δζ a
a a a + 4Ω2 ζ (Kerr) + Ω ∂eˆ4 X(Kerr) − ∂eˆ3 X(Kerr) a = Ω(δeμ4 − δeμ3 )∂μ X(Kerr) + 4Ω2 δζ a
(3.207)
where δe4 = −
δΩ δX ; e4 + Ω Ω(Kerr)
δe3 = −
δΩ e3 . Ω(Kerr)
(3.208)
Therefore, Eq. (3.207) becomes a )δX c ∂e3 δX a = (∂c X(Kerr) Ω a a δΩ + 4Ω2 δζ a . (3.209) + (Kerr) ∂e3 X(Kerr) − ∂e4 X(Kerr) Ω
As
∂ a a a a − ∂e4 X(Kerr) − ∂u X(Kerr) Xa Ω ∂e3 X(Kerr) = ∂u X(Kerr) = −2 ∂r∗ (Kerr) ∂rb ∂ a QΔ ∂ a = −2 X = −2 X (3.210) ∂r∗ ∂rb (Kerr) ΣR2 ∂rb (Kerr)
the final expression is,55 ∂e3 δX − a
a (∂c X(Kerr) )δX c
=−
QΔ ∂ a X δΩ + 4Ω2 δζ a . Ω(Kerr) ΣR2 ∂rb (Kerr) (3.211) 2
The equation for δΩ: We start from the definition of ω in Eqs. (2.44) ω = − 12 D3 log Ω. From it 1 1 − D3 log Ω = − 2 ∂N (Ω(Kerr) + δΩ) 2 2Ω 1 =− ∂N Ω(Kerr) (Kerr) 2 2Ω 1 1 1 − − ∂N δΩ ∂N Ω(Kerr) − 2 2 (Kerr) 2Ω 2Ω2 2Ω and the final equation we have for δΩ is δΩ(Ω + Ω(Kerr) ) (Kerr) 2 ∂N δΩ = −2(Ω + δΩ) δω + ∂N Ω(Kerr) . (3.212) 2 Ω(Kerr) 55 Observe that these equations refer to the metric components in the {u, u, ω 1 , ω 2 } coordinates. We do not have here the previous problem of considering tensor fields tangent to S; ˆ quantities. therefore, we do not have to introduce the “auxiliary” O
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The equation for δγab : The definition of the induced metric on the generic (S) S(u, u), whose components we denote {γ ρσ }, is56 Πρμ Πσν gρσ = γ (S) μν
(3.213)
Πρμ = δμρ − (θμ3 eρ3 + θμ4 eρ4 ).
(3.214)
where
Observe that, as the spacetime is not static,57 γ (S) , the induced metric on the generic S, is not a 2 × 2 matrix (with the only γab components different from zero); Nevertheless the following holds58 ∂ ∂ (S) (S) γ ab = γab = g , , γ (S) (eA , eB ) = γ ab eaA ebB = γab eaA ebB . ∂ω a ∂ω b (3.215) Therefore,
∂ ∂ ∂ ∂ ∂ ∂ ∂N γab = DN g , , , D = g D + g N N ∂ω a ∂ω b ∂ω a ∂ω b ∂ω a ∂ω b ∂ ∂ ∂ ∂ ∂ ×g [N , ], , [N , ] + g D + g ∂ N, ∂ω a ∂ω a ∂ω b ∂ω a ∂ω b ∂ω b ∂ +g , D ∂b N ∂ω ∂ω a ∂ ∂ ∂ ∂ = 2Ωχab + g [N , ], , [N , ] . (3.216) + g ∂ω a ∂ω b ∂ω a ∂ω b
We have
)
* c ∂X(Kerr) ∂ ∂ ; = − N, a a ∂ω ∂ω ∂ω c
(3.217)
therefore, ∂N γab = 2Ωχab −
c ∂X(Kerr)
∂ω a
γcb −
c ∂X(Kerr)
∂ω b
γca .
(3.218)
56
We use here small latin letters a, b, . . . to indicate the θ, φ coordinates; the frame vector fields tangential to S will be indicated with eA , eB . . .. 57 In fact this is not possible even in the Kerr spacetime. 58 To prove it one has to compute explicitly θ 3 and θ 4 and from them the projection 3 = −1g components Πρμ , θ3 (·) = − 21 g(e4 , ·) therefore θμ eν 2 μν 4 1 1 1 1 3 c d θu = − guu eu Xc) = Ω (−2Ω2 + X · X(Kerr) ) − (−γcd X(Kerr) 4 − guc e4 = − 2 2 2Ω 2Ω c 1 1 1 1 3 c ˆd X ) = 0 X · X(Kerr) − (−γcd X θu = − guu eu 4 − guc e4 = − 2 2 2Ω 2 Ω 1 1 1 1 c (−γac X c ) − γac X c = 0. θa3 = − gau eu 4 − gac e4 = − 2 2 2Ω 2Ω 4 = Ω, θ 4 = θ 4 = 0. Analogously θu u c
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Analogously, (Kerr)
∂N γab
= 2Ω(Kerr) χ(Kerr) − ab
c ∂X(Kerr)
∂ω a
(Kerr)
γcb
−
c ∂X(Kerr)
∂ω b
(Kerr) γca
(3.219) and subtracting the result follows. Remark. The previous equations seem to imply a potentially dangerous loss of derivatives. In fact on the right-hand side of all the three equations there are terms of the order of first derivatives of the metric components, namely connection coefficients like ω, ζ, trχ. Nevertheless, this loss of derivatives is not harmful as it does not propagate when we estimate the connection coefficients and their tangential derivatives up to the order we need. A detailed discussion of this delicate aspect is in [5]. Next lemma is the stronger version of Lemma 3.7 and its proof includes the proof of this lemma (see also Step IV in Sect. 3.2). Lemma 3.11. Assume that in V∗ we have already proved that the norms δO satisfy better estimates than those of the bootstrap assumptions, namely
0 , δO ≤ N0 with N0 a large integer number; then, assuming for the O(0) norms appropriate initial data conditions, we prove that in V∗ better estimates hold:
0
0
0 (3.220) |r|u|2+δ δΩ| ≤ ; |r2 |u|2+δ δX| ≤ ; ||u|1+δ δγ| ≤ . 2 2 2 Proof. See the Appendix. 3.5. The Last Slice Canonical Foliation To control ∇ / δtrχ, we use transport equation 3.182, and this requires an estimate of ∇ / δtrχ on the last slice of V∗ we denote C ∗ ; this as discussed in [10] and in [14] requires a delicate choice of its foliation. Let us recall the main problem we have to cure: the equation for trχ along an incoming cone is 1 ˆ·χ ˆ − 2|ζ|2 − 4ζ · ∇ / log Ω − 2|∇ / log Ω|2 D3 trχ + trχtrχ + (D3 log Ω)trχ + χ 2 = 2( / log Ω + div / ζ + ρ). (3.221) Looking at this expression it is clear that there is a loss of derivatives due to the term in the right-hand side. To cure this problem in [10] we require that log Ω satisfies the equation / log Ω = −div / ζ − ρ + ρ.
(3.222)
To satisfy it we introduce a background foliation whose leaves are the level surfaces of the affine parameter v, associated with the null geodesic generators
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of C ∗ . Then we look for a new foliation u∗ = u∗ (v), expressed relatively to the background one, such that, relatively to it, Ω satisfies the equations / log Ω = −div / ζ − ρ + ρ;
log 2Ω = 0
(3.223) du∗ 2 −1 = (2Ω ) ; u∗ |S∗ (0) = λ1 dv where S∗ (0) = C ∗ ∩ Σ0 . Once these conditions are satisfied, the evolution equation for trχ becomes 1 D3 trχ + trχtrχ 2 ˆ·χ ˆ − 2|ζ|2 − 4ζ · δ∇ / log Ω − 2|∇ / log Ω|2 = 2ρ +(D3 log Ω)trχ + χ and the loss of derivatives disappears when we apply ∇ / . The proof of the existence of this foliation is in [14]. It is then clear that the last slice transport equation for ∇ / trχ has the following expression: 1 1 D3 ∇ / trχ)trχ / trχ + trχ∇ / trχ + (D3 log Ω)∇ / trχ + [∇ / , D3 ]trχ + (∇ 2 2 / log Ω)trχ + [∇ / , D3 ] log Ωtrχ + ∇ /χ ˆ·χ ˆ+χ ˆ·∇ /χ ˆ − 4ζ · ∇ /ζ +(D3 ∇ −4∇ /ζ · ∇ / log Ω − 4η · (−div / ζ − ρ + ρ) = 0. In the present case the problem has to be worked in a slight different way; let us consider again the transport equation (3.221), 1 D3 trχ + trχtrχ + (D3 log Ω)trχ + χ ˆ·χ ˆ − 2|ζ|2 − 4ζ · ∇ / log Ω − 2|∇ / log Ω|2 2 = 2( / log Ω + div / ζ + ρ). and subtract to it the “Kerr part” obtaining 1 1 D3 δtrχ + trχδtrχ + δtrχtrχ(Kerr) − 2ωδtrχ − 2δωtrχ(Kerr) 2 2 +χ ˆ · δχ ˆ + δχ ˆ·χ ˆ(Kerr) − 2ζ · δζ − 2δζ · ζ (Kerr) − 4ζ · δ∇ / log Ω / log Ω · δ∇ / log Ω − 4δ∇ / log Ω · (δ∇ / log Ω)(Kerr) − 4δζ · (∇ / log Ω)(Kerr) − 4∇ (Kerr)
= 2( / δ log Ω + ( / − /
)(log Ω)(Kerr)
+ div / δζ + (div / − div / (Kerr) )ζ (Kerr) + δρ). To avoid the loss of derivatives δ log Ω has to satisfy the following equation: / δ log Ω + div / δζ + δρ − δρ = 0,
(3.224)
and the important point is that the term (Kerr)
( / − /
)(log Ω)(Kerr) + (div / − div / (Kerr) )ζ (Kerr)
does not produce loss of derivatives as it contains second derivatives only of (log Ω)(Kerr) which is a given function. Using the transport equations for the not underlined connection coefficients on the last slice, C ∗ , allows to control their norms on the last slice in
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terms of the corresponding norms in the intersection C ∗ ∩ Σ0 and prove they are bounded again by59 M2 c ε + 0 2 . R0 3.6. V Step: The Initial Data The global existence and the peeling is proved here assuming a strong regularity for the initial data. We collect in the next subsections all the initial data conditions we have used in the various proofs. Moreover, all these conditions can be expressed in terms of quantities relative only to the initial data hypersurface, namely the three-dimensional metric (3) g and the second fundamental form (3)k, together with their covariant derivatives.60 This will allow to express the initial data smallness conditions requiring that a L2 integral on Σ0 /BR0 , whose integrand depends only on δ (3)k = (3)k − (3)k(Kerr) and on δ (3)Ricci = (3)Ricci − (3)Ricci(Kerr) be sufficiently small. 3.6.1. The Asymptotic Conditions on the Initial Data Metric. From equation 2.30 it follows g|Σ0 (·, ·)
1 1 = Ω2 dr∗2 + γab dω a + (X(Kerr) a − X a )dr∗ dω b + (X(Kerr) b −X b )dr∗ 2 2 1 1 = Ω2 + δX 2 dr∗2 − γab δX a dr∗ dω b + γab dω a dω b 4 2
(Kerr) = Ω2(Kerr) dr∗2 + γab dω a dω b 1 1 2 2 2 a b a b + (2Ω(Kerr) δΩ + δΩ + δX )dr∗ − γab δX dr∗ dω + δγab dω dω 4 2 (3.225)
Therefore, (g|Σ0 − g(Kerr) |Σ0 )(·, ·) 1 1 = 2Ω(Kerr) δΩ + δΩ2 + δX 2 dr∗2 − γab δX a dr∗ dω b + δγab dω a dω b 4 2 and the asymptotic conditions on the metric components are 1 1 1 a δΩ = o5 ; δX = o5 ; δγab = o5 , (3.226) r3+γ r4+γ r1+γ where r is defined as in equation 2.42, and the S’s are the two-dimensional surfaces associated with the canonical foliation of Σ0 we are going to define. We call these conditions “Kerr asymptotic flatness”. 59 The details of the “last slice problem”, first discussed and solved for a different foliation in [6], are discussed in detail in [10], Chapter 7 and in [14]. 60 Covariant with respect to the three-dimensional metric (3) g.
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3.6.2. The Smallness Conditions on the Initial Data Metric. The smallness conditions for the metric components are sup |r3+δ δΩ| ≤ ε;
Σ0 /BR0
sup |r4+δ δX| ≤ ε;
sup |r1+δ δγ| ≤ ε. (3.227)
Σ0 /BR0
Σ0 /BR0
3.6.3. The Smallness Conditions on the Initial Data Connection Coefficients. As the estimates of the connection coefficients in V∗ are obtained in terms of the initial data ones,61 the initial data have to be such that the following estimates hold, with l ≤ 4: / δtrχ| ≤ ε; |r4+l+δ ∇ / δtrχ| ≤ ε |r4+l+δ ∇ l
l
/ δ χ| ˆ ≤ ε; |r4+l+δ ∇ / δ χ| ˆ ≤ε |r4+l+δ ∇ l
l
/ δζ| ≤ ε |r4+l+δ ∇ l
|r
4+l+δ
(3.228)
l
∇ / δω| ≤ ε, |r
4+l+δ
l
∇ / δω| ≤ ε.
3.6.4. The Smallness Conditions on the Initial Data Riemann Components. Our initial data have to guarantee that the norms of the connection coefficients and their tangential derivatives are small and bounded on Σ0 , but, ˜ norms defined in terms of initial data have to be finite; more than that, the Q this implies that they have to be such that ˜ Σ /B ≤ ε2 Q (3.229) 0
R0
and also, on Σ0 /BR0 the following condition must hold: γ
˜ 2 ≤ ε2 . sup |r6+ 2 ρ(R)|
(3.230)
Σ0 /BR0
3.6.5. The Canonical Foliation on Σ0 . A foliation of Σ0 /BR0 in terms of twodimensional surfaces {S0 } is specified through a function u0 (p), the generic S0 is defined as S0 (u0 = ν) = {p ∈ Σ0 /BR0 |u0 (p) = ν}.
(3.231)
The solution of the eikonal equation u(p) with initial data u0 (p) on Σ0 /BR0 is such that the level hypersurfaces u(p) = ν defines the incoming cones of the double null foliation. As the norms of the “underlined” connection coefficients, see [10, Chapter 3] for all the detailed definitions, are estimated, using the transport equations along the incoming cones in terms of the same norms on the initial hypersurface, while the opposite happens for the not underlined coefficients estimated in terms of the norms on “Scri”, an analogous, although mild, problem appears in this case. To avoid a loss of derivatives we have to choose an appropriate (canonical) foliation of Σ0 /BR0 . We do not go here through the details as the way to obtain it is analogous to what was done in [10, Chapter 7]. Moreover, differently from the last slice case, here in principle we could even avoid the choice of the canonical foliation admitting a loss of 61 Remark: see the discussion about the last canonical slice, that the way in which the connection coefficients norms depend on the corresponding ones on the initial data, is different from the underlined and the not underlined ones.
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regularity going from the initial data to the solution, while this could not be allowed when we prove that V∗ can be extended. The previous discussion is not completely precise. In fact the intersections of the outgoing cones which are defined starting from “Scri” with the hypersurface Σ0 are not the S0 surfaces. This minor, although delicate, problem requires to define a spacelike hypersurface Σ0 near to Σ0 and control the various norms in the strip between these two hypersurfaces. This has been done in full detail in [10], subsection 4.1.3, and we do not repeat it here. 3.6.6. The Initial Data Condition in Terms of δ (3) g and δ (3) k. All the initial data conditions can be reexpressed requiring that the metric stays near to the Kerr metric in a definite way. This is obtained requiring a condition similar to the “Strong asymptotic flat condition” defined in [6] and, moreover, that an L2 integral, J , over Σ0 /BR0 (whose integrand is made by δ (3)k, δ (3)Ricci and their derivatives) is bounded by ε. Its explicit expression is # $ J (δ (3) g, δ (3) k) = R0−δ sup |r3+δ δΩ| + |r4+δ δX| + |r1+δ δγ| Σ0 /BR0
⎡
⎢ + R0−δ ⎣
4
Σ0 /BR0 l=0
+ Σ0 /BR0
3
5
(1 + d2 )(1+l)+ 2 +δ |∇ / δk|2 l
⎤ 12
5 ⎥ (1 + d2 )(3+l)+ 2 +δ |∇l δB|2 ⎦ . (3.232)
l=0
To express the tensor quantities of the four-dimensional spacetime restricted to Σ0 in terms of the three-dimensional quantities and their norm bounds in term of the corresponding three-dimensional ones, requires some work.This is a repetition, with some obvious modifications, of what has been done in [10], (mainly in Chapter 7) and in the original work [6], (mainly in Chapter 5) and we do not report it here; the interested reader can, nevertheless, look at the extended discussion in [5]. 3.7. VI Step: The Extension of the Region V∗ As previously said once, we have proved that in the region V∗ the estimates for the norms of the connection coefficients and for the Riemann tensor are better than those assumed in the “Bootstrap assumptions”; we have done the basic step to conclude that a region larger than V∗ does exist where the previous quantity satisfies again the “Bootstrap assumptions”. To obtain this result, nevertheless, something more is needed. In fact the bootstrap argument requires two conditions to be satisfied. First, that a region with the assumed properties of the V∗ region does exist, possibly a very small one and second it requires to prove that this region can be extended. The first condition can be implemented by a local existence result in a small strip above Σ0 /BR0 . In fact, of this solution we are interested only in the dependence region V associated with the annulus in Σ0 with r ∈ [R0 , R0 +δR],
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with δR arbitrary small, and it is clear that starting from appropriate initial data we can satisfy in V all the “bootstrap assumptions”. The second condition to be satisfied requires to solve again an existence problem starting from the upper boundary of V∗ . The strategy is not completely standard, but it has been already discussed in [10, Chapter 3, section 3.7.8] (see also [14]). Therefore, we sketch here only the more relevant points and the differences from the situation discussed there. The region whose existence we have to prove is a “strip” above the last slice of the region V∗ of arbitrary small width. Nevertheless, as there is no bound for the size of the region V∗ , this existence problem is not a local problem in both directions; it is local in the “outgoing cones” direction, but not local in the “incoming cones” directions. Moreover this is a characteristic problem as its initial data are on the intersection of the portions of two cones, one outgoing starting at the intersection of the last slice of V∗ with Σ0 , the other one being exactly the last slice. As the extension of the last slice cannot be controlled, to prove this result requires again a bootstrap argument. The difference is now that once we assume (again by a bootstrap assumption) that a portion of this strip does exist to prove that it can be extended, and that the whole strip does exist, requires to solve a characteristic local problem which is easy to manage. We do not give more details on it as this problem has already been treated as solved in two previous papers by the same authors, see [3,4]. Observe finally that the initial data in this case automatically satisfy the constraints for the characteristic problem as they are the restriction on these hypersurfaces of the Einstein solutions; the difference with respect to the discussion on [4] is that in this case the “initial data” are near to Kerr instead than to Minkowski, but this is not a problem. Finally, to complete the result we have to prove that, again, in the extended region, V∗ + δV , the estimates δR ≤ 0 , δO ≤ 0 hold; this requires some care. In fact from the “last slice initial data” one would be tempted to ˜ norms, to be finite avoiding possible logarithmic divergences, infer that the Q would require a weight |u|γ˜ with γ˜ < γ, the analog of the loss of decay from the initial data to the solutions. The problem is in fact not present, as once we have a solution in V∗ + δV and a new double null canonical foliation, we can, ˜ on the outgoing and incoming cones in terms exactly as before, estimate the Q of the initial norms on Σ0 and repeat all the previous lemmas to reobtain the correct decay for the δR norms. Once this is done, we repeat exactly the same procedure done in V∗ for the connection coefficients so that, finally, we have proved that in V∗ + δV all the bootstrap assumptions are still valid and, by contradiction, that V∗ has to coincide with the global (external) spacetime.
4. The Final Result We can state now with all the details our final result: Theorem 4.1. Assume that initial data are given on Σ0 such that, outside of a ball centred in the origin of radius R0 , they are different from the “Kerr initial
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data of a Kerr spacetime with mass M satisfying M 1, J ≤ M 2 R0 for some metric corrections decaying faster than r−3 toward spacelike infinity together with its derivatives up to an order q ≥ 4, namely62 (Kerr)
gij = gij
γ
+ oq+1 (r−(3+ 2 ) ),
(Kerr)
kij = kij
γ
+ oq (r−(4+ 2 ) )
(4.233)
where γ > 0. Let us assume that the metric correction δgij , the second fundamental form correction δkij are sufficiently small, namely the function J equation (3.232) made by L2 norms on Σ0 of these quantities is small,63 J (Σ0 , R0 ; δ (3)g, δ (3)k) ≤ ε,
(4.234)
defined outside the then this initial data set has a unique development, M, domain of influence of BR0 with the following properties: − where M + consists of the part of M which is in the =M + ∪ M (i) M − the one to the past. future of Σ/BR0 , M + (ii) (M , g) can be foliated by a canonical double null foliation {C(u), C(u)} whose outgoing leaves C(u) are complete64 for all |λ| ≥ |u0 | = R0 . The boundary of BR0 can be chosen to be the intersection of C(u0 ) with Σ0 . (iii) The various null components of the Riemann tensor relative to the null frame associated with the double null canonical foliation, decay along the outgoing “cones” in agreement with the “Peeling Theorem”. Remark. It is clear that, from the way this result has been obtained, the conM 1 has to be such that the development we prove is far from the dition R 0 event horizon we assume to exist in a spacetime, near to the Kerr spacetime, which is the boundary of the complete outer region. In fact, trying to go near to the event horizon or even to the “photosphere region”, see [1] and references ˜ therein, we would immediately find serious problems trying to control the Q norms in terms of the initial data ones.
5. Conclusions As mentioned in the introduction the global existence proof is separated from the “peeling result”. The global existence near Kerr spacetime required, in a broad sense, to subtract the Kerr part. This is done concerning the Riemann components looking for the estimates for the “time derivative” of the Riemann tensor, which eliminates the contribution of the Kerr spacetime, or more in general of any stationary spacetime. The subtraction for the connection coefficients is vice versa made in a more general way as the subtracted part has 62
The components of the metric tensor written in dimensional coordinates. This will also imply a slightly stronger condition on the decay of the metric and second fun damental form components, basically that R∞ drr 5+γ |δgij |2 < ∞, R∞ drr 7+γ |δkij |2 < ∞. 0 0 64 By this we mean that the null geodesics generating C(u) can be indefinitely extended toward the future. 63
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not to be time-independent.65 This is in some sense the more original part as, in a more external region, the peeling decay has been already proved in [15].
Acknowledgements The initial part of this work has been done during a visit of one of the authors, F. Nicol` o, to the Institut Mittag-Leffler (Djursholm, Sweden), where he was invited for the General Relativity semester and where he enjoyed many scientific discussions. Moreover, the same author is deeply indebted to S. Klainerman for pointing to him the importance of considering the Lie derivative, with respect to the “time” vector field T0 , of the Riemann tensor to obtain more detailed estimates for the various components of the Riemann tensor. Besides F. Nicol`o is also indebted for many illuminating discussions he had with him about this subject and many related ones. We also want to state clearly that the present result is deeply based on the previous works [10,11] and on the original fundamental work by Christodoulou and Klainerman [6]. Therefore, nothing has been “gracefully [an adverb sometimes very improperly used.] acknowledged”, but all the due credits have been explicitly given at the best to our knowledge.
6. Appendix 6.1. Proof of Various Equations, Inequalities and Lemmas In the proofs of various Lemmas, for the metric components we can use directly the difference between a quantity and its Kerr counterpart, Therefore, for the ˆ (0) = O(Kerr) , while their metric components withˆwe denote the Kerr part, O (0) meaning is different for the connection coefficients and for the null components of the Riemann tensor. Proof of inequality (3.109). This result is just one of the standard estimates used in Chapter 5 of [10] to show how from the control of the Q norms one can obtain the control of the sup norms of the null Riemann components. ˜ Weyl field instead The only difference is that here we are considering the R ˜ of the Riemann tensor R and that the Q norms are substituted by the Q norms. The detailed proof in this specific case is given in the Appendix of [5]. Proof of inequalities (3.123). Looking at the explicit expressions (3.122) and (3.125) it is immediate to realize that inequalities 3.125 are immediately derived by the bootstrap assumptions (2.48) while for the right-hand side terms in (3.122) we only need to control D3 δX, D4 δX and ∇ / δX. The control of the first two terms arises from the explicit expression of D3 δX (see (3.211), Lemma 3.11). The control of ∇ / δX requires to derive equation (3.211) and 65
In principle one could use the same strategy used for the Riemann components, namely to define “time derivatives” of the connection coefficients, write for them the structure equations, estimate their norms and recover by a time integration the connection coefficients, this method, although less general, should give the same result we have obtained.
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obtain from this equation an estimate for ∇ / δX proceeding as in Lemma 3.11. Observe that the loss of derivatives due to the presence on the right end side of ∇ / ζ is only apparent as in the estimate ∇ / ζ there is not any loss of derivatives (see the discussion in subsection 3.4.5 of [5]). Proof of Lemma 3.11. {δΩ}: δΩ satisfies the equation (Ω + Ω(Kerr) ) ∂N Ω(Kerr) . ∂N (δΩ) = −2(Ω)2 δω + (δΩ) 2 (Kerr) Ω It follows ∂N |δΩ| ≤ F |δΩ| + 2Ω2 |δω|
(6.236)
where F = (Ω + Ω(Kerr) )Ω(Kerr)
−2
∂N Ω(Kerr) = O
and integrating on C(u; [u0 , u]) we obtain ⎛ |δΩ|(u, u) ≤ c⎝|δΩ|C(u)∩Σ0 +
u
(6.235)
M r2
.
(6.237)
⎞ du |2Ω2 δω|(u, u )⎠ .
(6.238)
u0
As we have already proved that the estimates for δO(1) are better than the bootstrap assumptions, then for δω the following estimate holds:
0 |r2 |u|2+δ δω| ≤ N0 and we can obtain from the previous inequality the following one: u 2 3+δ |r |u|δΩ|(u, u) ≤ |r δΩ|C(u)∩Σ0 + du |2Ω2 δω|(u, u )r(u , u)2 |u | ⎛
u0
≤ c⎝|r3 δΩ|C(u)∩Σ0 + ≤ c |r3 δΩ|C(u)∩Σ0 + From it
0 N0
u u0
0 N0 |u|δ
⎞ du
1 ⎠ |u |1+δ
.
0
0 |r2 |u|1+δ δΩ|(u, u) ≤ c |r3 δΩ|C(u)∩Σ0 + , ≤ N0 N1
(6.239)
(6.240)
with 2−1 > N1−1 > N0−1 , choosing the initial data sufficiently small and N0 sufficiently large. {δX}: δX satisfies the following equation: 2 QΔ ∂ a a ∂e3 δX a − (∂c X(Kerr) )δX c = − X δΩ + 4Ω2 δζ a Ω(Kerr) ΣR2 ∂rb (Kerr) (6.241)
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where we have immediately, with σ > 0, M4 1 2M arb a a ≤ |∂c X(Kerr) | = |(∂c ωB )δφ | = ∂c ΣR2 R04−σ r1+σ and
2
QΔ ∂ a X (Kerr) ΣR2 ∂rb (Kerr) Ω
≤
(6.242)
M2 1 . R02 r2
(6.243)
Finally, using the improved estimates for the δO(1) norms we have |δζ a | ≤ c
1
0 . N0 r3 |u|2+δ
(6.244)
Using these estimates from the definition . |δX| = |δX a |2 a
we write, defining 2 QΔ ∂ a X G=− , Ω(Kerr) ΣR2 ∂rb (Kerr) δX a ∂e3 δX a ∂N |δX|2 = 2|δX|∂N |δX| = 2Ω = 2Ω
δX
a
a
≤ 4Ω
a a (∂c X(Kerr) )δX c
+ GδΩ + 4Ω2 δζ a
(6.245)
a |(∂c X(Kerr) )| |δX|2 + |G||δΩ||δX| + 4Ω2
a
|δζ a | |δX|
a
(6.246) and immediately ∂N |δX| ≤ 4Ω
a
a |(∂c X(Kerr) )| |δX| + |G||δΩ| + 4Ω2
) 2 * M M4 1
0
0 /N0 ≤ c 4−σ 1+σ |δX| + + R02 r4 |u|1+δ r3 |u|2+δ R0 r 1 M4 1
0 M2 ≤ c 4−σ 1+σ |δX| + 3 2+δ +c 2 . r |u| N0 R0 R0 r
|δζ a |
a
(6.247)
Applying the Gronwall Lemma we obtain ⎧ ⎫⎡ ⎤ u ⎨ M 4 u 1 ⎬ 2 1 1 M ⎣c 0 + 2 du ⎦ |δX| ≤ exp c 4−σ ⎩ R0 r1+σ ⎭ N0 R0 r3 |u |2+δ u0 u0 / 0 1 1 1 M2
0 M4 + 2 ≤ (6.248) ≤ exp c 4 c 0 3 1+δ 3 R0 N0 R0 r |u| 2 r |u|1+δ which proves the result.
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{δγ}: The Eq. (3.203) satisfied by δγab , )
c ∂X(Kerr)
c ∂X(Kerr)
* (δγac )
∂N (δγab ) − Ωtrχ(δγab ) = − (δγcb ) + ∂ω a ∂ω b
(Kerr) (Kerr) + Ωγab δtrχ + δΩγab trχ(Kerr) + 2Ω(Kerr) δ χ ˆab + 2δΩχ ˆ(Kerr) ab can be written as ∂N (δγab ) − Ωtrχ(δγab ) = (Ωtrχ − Ωtrχ)(δγab ) + (Gca (δγcb ) + Gcb (δγca )) +F (O(Kerr) , δΩ, δtrχ, δχ)
(6.249)
where F (O(Kerr) , δΩ, δtrχ, δχ)
(Kerr) (Kerr) . δtrχ + δΩγab trχ(Kerr) + 2Ω(Kerr) δ χ ˆab + 2δΩχ ˆ(Kerr) = Ωγab ab (6.250) As ∂N r(u, u) =
∂ r(u, u) r(u, u) = Ωtrχ ∂u 2
(6.251)
it follows ∂N
(δγab ) 1 2 r = 2 ∂N (δγab ) − 3 Ωtrχ(δγab ) r2 r r 2 $ 1 # = 2 ∂N (δγab ) − Ωtrχ(δγab ) r 1 # = 2 (Ωtrχ − Ωtrχ)(δγab ) + (Gca (δγcb ) + Gcb (δγca )) r
+F (O(Kerr) , δΩ, δtrχ, δχ)
Therefore, ∂N (r−2 δγab ) = (Ωtrχ − Ωtrχ)(r−2 δγab ) F + Gca (r−2 δγcb ) + Gcb (r−2 δγca ) + 2 r Defining |δγ| ≡
ab
|δγab |
(6.252)
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we have the following inequality: |δγ|2 |δγ| 2|δγ| 1 4 rΩtrχ |δγ| ∂N 4 = 2 ∂N 2 = 4 2|δγ|∂N |δγ| − r r r r r 2 ) * 1 = (δγab ) 4 −Ωtrχ(δγab ) + ∂N (δγab ) r ab (δγab ) (δγab ) c (δγcb ) c (δγca ) (Ωtrχ − Ωtrχ) 2 + Ga 2 + Gb = r2 r r r2 ab F (O(Kerr) , δΩ, δtrχ, δχ) + r2 ) * |δγ| |δγ| |F | |δγ| ≤ 2 |(Ωtrχ − Ωtrχ)| 2 + 2|G| 2 + 2 r r r r which implies ∂N
|δγ| r2
$ # ≤ |(Ωtrχ − Ωtrχ) + 2|G|
|δγ| r2
+
|F | . r2
(6.253)
As before we have for |G| and |(Ωtrχ − Ωtrχ)| the following estimates, with σ>0 M4 1 c |Gac | ≤ c 4−σ 1+σ , |(Ωtrχ − Ωtrχ)| ≤ 2 , (6.254) |G| = r r R0 ac therefore, applying Gronwall Lemma,
|δγ| r2
|δγ| (u, u) ≤ r2 u ≤c
⎧ ⎫ ⎨ M 4 u 1 ⎬ u |F | (u , u)du (u0 , u) + exp c 4−σ ⎩ R0 r1+σ ⎭ r2 u0
u0
|F | (u , u)du r2
(6.255)
u0
Observe now that the following inequality holds: |F | 1 (Kerr) (Kerr) ≤ 2 Ωγab δtrχ + δΩγab trχ(Kerr) + 2Ω(Kerr) δ χ ˆab + 2δΩχ ˆ(Kerr) ab r2 r
1 ≤ 2 |Ω|r2 |δtrχ| + |δΩ|r2 |trχ(Kerr) | + 2|Ω(Kerr) ||δ χ| ˆ + 2|δΩ||χ ˆ(Kerr) | r
0 ≤ c 2 2+δ . (6.256) r |u| Integrating the final result is 1+δ
||u| proving the result.
δγ| ≤ c 0
1 1 + N0 N1
≤
0 , 2
(6.257)
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References [1] Blue, P.: Decay of the Maxwell field on the Schwarzschild manifold. J. Hyperbolic Differ. Equ. 5(4), 807–856 (2008) [2] Chandrasekhar, S.: The Mathematical Theory of Black Holes. Oxford University Press, Oxford (1983) [3] Caciotta, G., Nicol` o, F.: Global characteristic problem for Einstein vacuum equations with small initial data. Part I: the initial data constraints. JHDE 2(1), 201–277 (2005) [4] Caciotta, G., Nicol` o, F.: Global characteristic problem for Einstein vacuum equations with small initial data II. arXiv-gr-qc/0608038 (2006) [5] Caciotta, G., Nicol` o, F.: The non linear perturbation of the Kerr spacetime in an external region. arXiv-gr-qc/0908.4330v1 (2009) [6] Christodoulou, D., Klainerman, S.: The global non linear stability of the Minkowski space. In: Princeton Mathematical Series, vol. 41 (1993) [7] Dafermos, M., Rodnianski, I.: A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds. arXiv-0805.4309v1 (2008) [8] Israel, W., Pretorius, F.: Quasi-spherical light cones of the Kerr geometry. Class. Quantum Gravity 15, 2289–2301 (1998) [9] Klainerman, S.: Linear stability of black holes following M. Dafermos and I. Rodnianski. Bourbaki Seminar (2009) [10] Klainerman, S., Nicol` o, F.: The evolution problem in general relativity. In: Progress in Mathematical Physics, vol. 25. Birkh¨ auser, Boston (2002) [11] Klainerman, S., Nicol` o, F.: Peeling proerties of asymptotically flat solutions to the Einstein vacuum equation. Class. Quantum Gravity 20, 3215–3257 (2003) [12] Kroon, J.A.V.: Logarithmic Newman-Penrose constants for arbitrary polyhomogeneous spacetime. Class. Quantum Gravity 16, 1653–1665 (1999) [13] Kroon, J.A.V.: Polyhomogeneity and zero rest mass fields with applications to Newman-Penrose constants. Class. Quantum Gravity 17, 605–621 (2000) [14] Nicol` o, F.: Canonical foliation on a null hypersurface. JHDE 1(3), 367–427 (2004) [15] Nicol` o, F.: The peeling in the “very external region” of non linear perturbations of the Kerr spacetime. ArXiv gr-qc:0901.3316 [16] Wald, R.M.: General Relativity. University of Chicago Press, Chicago (1984) Giulio Caciotta and Francesco Nicol` o Dipartimento di Matematica Universit` a degli Studi di Roma “Tor Vergata” Via della Ricerca Scientifica 00133 Rome, Italy e-mail:
[email protected];
[email protected] Communicated by Piotr T. Chrusciel. Received: December 14, 2009. Accepted: March 10, 2010.
Ann. Henri Poincar´e 11 (2010), 499–537 c 2010 Springer Basel AG 1424-0637/10/030499-39 published online July 27, 2010 DOI 10.1007/s00023-010-0036-5
Annales Henri Poincar´ e
Almost Exponential Decay of Quantum Resonance States and Paley–Wiener Type Estimates in Gevrey Spaces M. Klein and J. Rama Abstract. Let H0 be a self-adjoint operator in some Hilbert space H , and let λ0 be a (possibly degenerate) eigenvalue of H0 embedded in its essential spectrum σess (H0 ) with corresponding eigenprojection Π0 . For small |κ|, let H(κ) be a family of perturbed Hamiltonians, which is analytic in a generalized Balslev–Combes sense. Following Hunziker’s approach in (Commun Math Phys 132:177–188, 1990), we discuss the corrections to exponential decay in Π0 e−itH(κ) g(H(κ))Π0 = D(κ)e−ith(κ) D(κ) + R(κ, t), where D(κ) = Π0 + O(κ2 ) (κ → 0) and h(κ) is some family of in general non self-adjoint bounded operators with Ranh(κ) = RanΠ0 , leaving RanΠ0 invariant, and 0 ≤ g ≤ 1 is a cut-off function with g(λ0 ) = 1 and sufficiently small support. Our main result is a sharp estimate of the remainder R(κ, t) in terms of the Gevrey index a > 1, b > 0 of g ∈ Γa,b (R): 1 1 a t ≥ 0, C < ab− a , κ → 0 . R(κ, t) ≤ O(κ2 ) e−Ct
1. Introduction and Results If H is a semibounded self-adjoint operator in some Hilbert space H , it is well known (see, e.g., [24]) that the estimate |(ψ, e−itH ψ)|2 ≤ Ce−A|t|
(t ∈ R)
(1.1)
cannot hold for any ψ ∈ H \{0}, C > 0, A > 0: By the spectral theorem −itH F (t) := (ψ, e ψ) = e−itλ dμψ (λ) R
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is the Fourier transform of the spectral measure dμψ , and, by a Paley–Wiener argument, the estimate (1.1) implies that dμψ has a density analytic in some complex strip around the real axis. Since H is semibounded, this gives ψ = 0 (this is basically Exercise 16 in [21, Chapter 19]). Since, however, exponential decay in the context of quantum resonances (over certain finite time scales) is a much used concept in physics, it is a natural question to investigate, how close to the wrong estimate (1.1) one may come if the spectral measure dμψ for ψ = 0 is “as close to analytic as possible”. Somewhat surprisingly, this question has not been treated in the literature. In this paper we answer this question in the framework of the Gevrey spaces Γa and Γa,b (a > 1, b > 0); cf. Definition 1.1. For a = 1, Γa is a space of real analytic functions. We combine this use of Gevrey spaces with the approach of Hunziker in [10], which in an abstract Balslev–Combes setting derives corrections to the exponential decay of F (t) in the case of quantum resonances. In the original paper of Hunziker, these corrections are given for the case of spectral measures with a density in C0∞ . Our main result is a much better estimate on the remainder, if the density of the spectral measure is in Γa,b ∩C0∞ and a is close to 1; cf. Theorem 1.7 and Corollary 1.8. Not surprisingly, the crucial point to obtain such results are estimates of Paley–Wiener type for the spaces Γa,b . To explain our results in more detail, we shall introduce some notation. We recall that Gevrey spaces where introduced in [5]. A modern exposition from the point of PDE is in Rodino’s book [22]. In our context, we shall use a slightly refined definition following Jung’s thesis [12]. For any set M ⊂ C and any continuous function f on M , we write f ∞,M := supz∈M |f (z)|. Definition 1.1. Let a ∈ [1, ∞), b > 0. For Ω ⊂ R open, Γa,b (Ω) := f ∈ C ∞ (Ω, C)there exists a constant c0 (f ) > 0 such that for all m ∈ N ∂ m f ∞,Ω ≤ c0 (f ) (m + 1)c0 (f ) bm m!a
(1.2)
is called the Gevrey class (in Ω) with index (a, b). The global Gevrey class (in Ω) with index a is defined to be the set b>0 Γa,b (Ω) =: Γa (Ω). The small Gevrey class (in Ω) with index a is defined to be the set b>0 Γa,b (Ω) =: γ a (Ω). We introduce the spaces of Gevrey functions with compact support a,b Γa,b (Ω) ∩ C0∞ (Ω), 0 (Ω) := Γ
γ0a (Ω) := γ a (Ω) ∩ C0∞ (Ω)
Γa0 (Ω) := Γa (Ω) ∩ C0∞ (Ω), (a > 1).
a a It is well known that Γa,b 0 (Ω), Γ0 (Ω) and γ0 (Ω) (a > 1) contain elements different from zero. This is essentially a consequence of the Denjoy–Carleman Theorem; see, e.g., [8, Theorem 1.3.8] or [21, Theorem 19.10 and Theorem
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19.11]. For the sake of the reader we collect some basic facts on Gevrey spaces in Appendix A. Remark 1.2. The parameter a ≥ 1 is the main parameter characterizing the regularity of f ∈ Γa,b (Ω). To understand the role of b > 0, it might be helpful to consider the extreme case a = 1. Then estimate (1.2) implies that the radius of convergence of the Taylor series of f at any point t ∈ Ω satisfies ≥ b−1 . Thus f ∈ Γ1,b (Ω) has an analytic continuation into a complex set {z ∈ C | dist(z, ∂Ω) < b−1 }. If Ω = R, a Paley–Wiener theorem (see, e.g., [19, Theorem IX.13]) implies that, for f ∈ Γ1,b (R) with sup f (· + is)L2 (R) < ∞,
|s|≤β ∧
one has | f (τ )| ≤ C e−β|τ | (0 ≤ β < b−1 , τ ∈ R). Thus b is an additional parameter which, for a = 1, gives a finer characterization of the regularity of real analytic functions. For a > 1 and the space Γa,b (Ω), the parameter b plays a similar role; see Remark 4.5. In our context, b fixes the prefactor in the time scale T in (1.11), while the exponent is determined solely by a. This is dual to the above statement on regularity. Next we introduce our class of Hamiltonians, following Hunziker [10]. This class is defined by assumptions (A1)–(A5) below. (A1)
Abstract Balslev–Combes Setting: For κ ∈ K, K an open real neighborhood of 0, let H(κ) be a family of self-adjoint operators in a complex Hilbert space H . Let U (θ) (θ ∈ R) be a strongly continuous one parameter unitary group, such that for fixed κ ∈ K the relation H(κ, θ) := U (θ)H(κ)U (θ)−1
(θ ∈ R)
(1.3)
extends analytically (in the sense of Kato) into the strip Sβ := θ ∈ C |Imθ| < β for some β > 0. We assume that the spectrum σ(H(κ, θ)) for Imθ > 0 belongs to the closed lower half plane, i.e., σ(H(κ, θ)) ⊂ C− ,
C− := {z ∈ C | Imz < 0} (θ ∈ Sβ with Imθ > 0).
Analogously, σ(H(κ, θ)) ⊂ C+ ,
C+ := {z ∈ C | Imz > 0} (θ ∈ Sβ with Imθ < 0).
Remark 1.3. For θ ∈ Sβ the group U (θ) is defined by functional calculus, having the (formal) representation U (θ) := eiμθ dEμ (θ ∈ Sβ ), R
where Eμ is the spectral resolution associated to the self-adjoint infinitesimal generator of U (θ), θ ∈ R. It satisfies U (θ)∗ = U (θ)−1 and (U (θ)−1 )∗ = U (θ)(θ ∈ Sβ ) on some natural dense domain (depending on θ) in H .
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The following assumptions (A2) and (A3) are stability conditions on embedded eigenvalues. (A2) Let λ0 be an eigenvalue of H0 := H(κ, θ)κ=θ=0 embedded in its essential spectrum σess (H0 ) with corresponding eigenprojection Π0 , dim RanΠ0 < ∞. Assume λ0 ∈ σdisc (H0 (θ)) (θ ∈ Sβ \R), where H0 (θ) := H(κ, θ) κ=0
(A3)
for θ ∈ Sβ , and σdisc denotes the discrete spectrum. For fixed θ ∈ Sβ \R there exists a punctured neighborhood W (λ0 ; θ)• := {z ∈ C | 0 < |z − λ0 | < rθ } for some rθ > 0
(1.4)
of λ0 such that −1
R(κ, θ, z) := (H(κ, θ) − z)
(z ∈ W (λ0 ; θ)• , |κ| < κ0 (z))
(1.5)
•
exists and is bounded for each fixed z ∈ W (λ0 ; θ) , uniformly for |κ| < κ0 (z), where κ0 (·) : W (λ0 ; θ)• → R+ is some continuous map. Remark 1.4. (A3) ensures that for small |κ| only the λ0 -group of H(κ, θ) (and no other eigenvalue of H(κ, θ)) is contained in each compact subset of W (λ0 ; θ)• ∪ {λ0 }. (A4)
For fixed θ ∈ Sβ \R the perturbed total Riesz projection (for the λ0 group of H(κ, θ) in the sense of Kato [14, Chapter II §1.2, before Remark 1.3])
1 R(κ, θ, z) dz |κ| < inf κ0 (z) , (1.6) Π(κ, θ) := − z∈Γθ 2πi Γθ
with Γθ some loop around λ0 in W (λ0 ; θ)• , satisfies lim Π(κ, θ) − Π0 (θ) = 0, Π0 (θ) := Π(κ, θ)
κ=0
κ→0
.
(1.7)
Let θ ∈ R. Then, by (1.3), H0 (θ) is self-adjoint in H . The eigenprojection Π0 (θ) for the eigenvalue λ0 of H0 (θ) is given via functional calculus by Π0 (θ) = s -lim (∓iε) (H0 (θ) − λ0 ∓ iε) ε↓0
−1
(θ ∈ R).
(1.8)
Furthermore: Proposition 1.5. Assume (A1) and (A2). For θ ∈ R, let Π0 (θ) denote the eigenprojection for the eigenvalue λ0 of the self-adjoint operator H0 (θ). Then Π0 (θ) (θ ∈ R) extends analytically into the full strip Sβ . In particular, dim Ran Π0 (θ) = dim Ran Π0 (θ ∈ Sβ ); see [14, Chapter I §4.6, Lemma 4.10]. The statement of Proposition 1.5 is contained in [10], without proof. The statement goes back to [1]. But, in our opinion, the proof in [1] contains a serious gap (there is no explicit argument showing uniform boundedness of Π0 (θ) for Imθ near zero). The subsequent article [2] simply refers the reader to the proof in [1]. Simon in his form version [23] of Balslev–Combes theory
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(and in [20]) carefully avoids an analogous statement on the analyticity of Π0 (θ). We learnt a rigorous proof of Proposition 1.5 from Herbst, which we include in Appendix B. (A5) For fixed θ ∈ Sβ \R, the family H(κ, θ) is type (A) analytic for κ in some complex neighborhood K0 of zero in the sense of Kato; see [14, Chapter VII §2]. Remark 1.6. One could eliminate the hypothesis of “type (A)” in (A5) and consider general analytic families in the sense of Kato. But then the Rayleigh–Schr¨ odinger expansion in [10] changes, and one has to reprove the representation of Π(κ, θ) in Remark 3.1, the estimate (3.1) and the analysis of the singular part in the proof of Theorem 1.7. This is left to the interested reader. Now we are ready to formulate our main result. Theorem 1.7. Assume (A1)–(A5). Let I0 := [λ0 − ε, λ0 + ε] for some ε > 0 sufficiently small. Let a > 1, b > 0 and let gI0 ∈ Γa,b 0 (R) be a smoothed out version of the characteristic function 1I0 , i.e.,
1, x in some neighborhood I1 of I0 gI0 (x) = , 0 ≤ gI0 ≤ 1. 0, x outside some (other) neighborhood of I0 Assume that supp gI0 contains no eigenvalue of H0 different from λ0 . Then for κ ∈ K with |κ| sufficiently small and t ∈ R one has F (κ, t) := Π0 e−itH(κ) gI0 (H(κ))Π0 = D(κ) e−ith(κ) D(κ)Π0 + R(κ, t), (1.9) where h(κ), given by (3.4), belongs to the class A of operators, which are analytic families of (in general) non self-adjoint operators in B(H , RanΠ0 ), the space of bounded linear operators in H with range RanΠ0 , leaving RanΠ0 invariant. D(κ) and R(κ, t) belong to the same class A. The operator D(κ) = Π0 + O(κ2 ) (κ → 0) is explicitly given by (3.59). The remainder R(κ, t) is given by (3.81)and satisfies the estimate 1
|(u, R(κ, t)v)| ≤ O(κ2 ) u v e−Ct a (κ → 0), (1.10) × [0, ∞) × C , where K ⊂ K is uniformly for u, v, κ, t, C ∈ H × H × K compact with sufficiently small diameter and C is compact in {C ∈ R | 0 ≤ 1 C < ab− a }. Proof. See Section 3.
Our main result is the remainder estimate (1.10), which improves Hunziker’s upper bound cm κ2 (t + 1)−m (m ∈ N); see [10, (37)]. Hunziker’s result in [10] is optimal for g in the class C0∞ (R). It allows a control of F (κ, t) by the first term on the r.h.s. of (1.9) up to times only slightly larger than the expected physical lifetime τ ∼ |Imλ(κ)|−1 = O(κ−2 ) (where λ(κ) is a resonance associated to λ0 ), actually, for 0 ≤ t ≤ O(κ−2 | ln κ|) as κ → 0. The point of (1.10) is a control over much larger time intervals, which include arbitrary powers of τ , if the index a of Gevrey regularity is taken
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sufficiently close to 1. For simplicity, we give a precise statement in the case dim RanΠ0 = 1. Corollary 1.8. Assume (A1)–(A5) with dim RanΠ0 = 1 and let I0 and gI0 be as in Theorem 1.7. Then, for κ ∈ K with |κ| sufficiently small, one has h(κ) = λ(κ)Π0 for some λ(κ) ∈ C with |Imλ(κ)| ≤ μκ2 for some μ ≥ 0. Let ψ0 denote the normalized eigenfunction corresponding to the embedded eigenvalue λ0 of H0 . Set a 1 C a−1 , 0 ≤ C < ab− a . (1.11) T := 2 μκ Then one has for all 0 ≤ t ≤ T ψ0 , e−itH(κ) gI0 (H(κ))ψ0 = e−iλ(κ)t (1 + r(κ)), r(κ) = O(κ2 )
(1.12)
(κ → 0).
Proof. See Section 3.
Similar statements hold in the case 2 ≤ dim RanΠ0 < ∞. Because of this improved control over time scales very large with respect to τ , it is of interest to investigate more closely the asymptotic behavior of F (κ, t) as t → ∞. This can be done by analyzing the Jordan decomposition of h(κ) with the methods of finite-dimensional analytic perturbation theory (in the case of nonnormal families of matrices). For somewhat preliminary results in this direction, see [18]. Theorem 1.7 and Corollary 1.8 considerably sharpen the results in [15] (imposing the stronger assumptions (A1)–(A5)). The main technical tool for proving Theorem 1.7 are estimates of Paley–Wiener type for functions in the Gevrey class Γa,b (Ω). Our estimates are slightly more refined than the estimates which we found in the literature. This we shall briefly explain. Recall that for a compact set E ⊂ Rn (with n ∈ N) the supporting function HE is defined by HE (ξ) := sup x · ξ = sup x · ξ x∈E
(ξ ∈ Rn ),
(1.13)
x∈chE
where x · ξ denotes the inner product in Rn and chE denotes the closed convex hull of E; cf. [8, (4.3.1) and Definition 4.3.1]. Our main tool is Theorem 1.9. Let D be a domain in C meeting R and let f be meromorphic in D. Let Ω ⊂ R be an open set containing R\(D ∩ R) such that Ω contains no poles of f . Assume that f extends to a function in Γa,b (Ω) (which we shall also denote by f ) with f (t) = 0 for |t| > R for some (sufficiently large) R > 0. Let γ be a contour in (R ∪ D)\{poles of f } running from −∞ to +∞ (see Fig. 3). Define 1 e−iζz f (z) dz (ζ ∈ C). (1.14) f # (ζ) := √ 2π γ
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Then, if Imγ ≥ 0 (Imγ ≤ 0, respectively), for all ζ ∈ C with Reζ ≤ 0 1 (Reζ ≥ 0, respectively) and for all C < C(a, b) := ab− a one has 1
|f # (ζ)| ≤ C (c0 (f ) γ(f ) + f ∞,γ + β0 f ∞,U ) eHsupp f R (Imζ) e−C|Reζ| a (1.15) where C < ∞, 0 < β0 < ∞ and U is any sufficiently small complex neighborhood of some compact set I ⊂ (γ ∩ R ∩ D). If we assume that all poles z∗ of f above the contour γ in C+ (below the contour γ in C− , respectively) satisfy |Imz∗ | ≥ β0 > 0, then the constant C depends on f only through the constants a, b, β0 , R, the neighborhood U and the contour γ. c0 (f ) is the constant appearing in the Gevrey estimate (1.2) for the space Γa,b (Ω) and γ(f ) := γ1 (f ) + γ2 (f ), where c0 (f )+ 3a−1 2 c0 (f ) + 3a−1 2 , (1.16) γ1 (f ) := (a − 1)δ −c0 (f )
γ2 (f ) := e
c0 (f ) + a−1 2 3 a−1
c0 (f )+ a−1 2 (1.17)
with some δ ∈ (0, 1); cf. (4.3) and (4.4).
Proof. See Section 4. ∧
Remark 1.10. If γ = R, the function f # is the Fourier transform f of f . Theorem 1.9 is a generalization of part (1) of the following Paley–Wiener theorem. The main new points of Theorem 1.9 are the extension to possibly complex contours and the stated uniformity with respect to f . Theorem 1.11. (1)
∧
Let f ∈ Γa,b 0 (R) with a > 1, b > 0. Let E := supp f . Then f is an 1 entire function; for all ζ ∈ C, C < C(a, b) := ab− a one has ∧
1
| f (ζ)| ≤ C eHE (Imζ) e−C|Reζ| a
(1.18)
with some constant C < ∞, where C depends only on the constant c0 (f ) in the Gevrey estimate (1.2), f ∞ , a, b and E. (2)
∧
Let a > 1. Let f be an entire function with ∧
1
| f (ζ)| ≤ c1 eHE (Imζ) e−c2 |Reζ| a
(1.19)
for some compact interval E ⊂ R, some c1 , c2 > 0 and all ζ ∈ C. Then, ∧
(3)
0 a (R), b0 := (c−1 the inverse Fourier transform f of f is in Γa,b 2 a) , and 0 suppf ⊂ E. Let a > 1. Assume that f ∈ L∞ (R) satisfies
∧
1
| f (τ )| ≤ c1 e−c2 |τ | a Then f ∈ Γ
a,b0
(R) for b0 =
a (c−1 2 a) .
(τ ∈ R).
(1.20)
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Proof. See Section 5. A direct consequence of Theorem 1.11 (1) and (2) is the following Corollary 1.12. Let a > 1. Let E be a compact interval in R. Then: ∧
f ∈ Γa0 (R) with supp f ⊂ E ⇔ f is an entire function, which satisfies ∧
1
| f (ζ)| ≤ c1 eHE (Imζ) e−c2 |Re ζ| a for some c1 , c2 > 0 and all ζ ∈ C. Remark 1.13. Corollary 1.12 is stated in the book of Rodino [22, Theorem 1.6.7]. For a proof he refers to the original paper of Komatsu [16]. Komatsu proves a more general result (in the language of the spaces C{Mn } appearing in the Denjoy-Carleman Theorem, see, e.g., [21, Theorem 19.11] or [8, Theorem 1.3.8]), which in the context of Gevrey spaces specializes to the ormander’s book [9] (1983) there class Γa0 ; cf. [16, Theorem 9.1] (1973). In H¨ is a (short) proof of a version of Corollary 1.12 for the small Gevrey class γ0a ; cf. [9, Lemma 12.7.4]. Moreover, we found two Chinese papers ([3] (1988) and [4] (1992)) proving Corollary 1.12, without reference to neither [16] nor [9]. None of these works uses the concept of an almost analytic extension of f ∈ Γa0 . While the arguments given in [9] certainly could be adapted to prove Theorem 1.11, (1) and (2), for the spaces Γa,b 0 , this is less clear (and certainly much less intuitive) for Theorem 1.9 (and Theorem 1.11 (3)). In our context, Theorem 1.9 is the crucial result. The contour γ appears naturally via a contour deformation, and it is in that context that an almost analytic extension f of f becomes profitable. We recall that almost analytic extensions have been systematically introduced to analysis by Melin and Sj¨ ostrand in [17], and now play a prominent role in spectral theory in view of the Helffer–Sj¨ ostrand formula from [6]. For the sake of the reader, we give complete proofs of the Paley–Wiener type results Theorem 1.9 and Theorem 1.11 by use of almost analytic extensions and contour deformation. This might be of independent interest. The plan of the paper is as follows: Section 2 contains a preliminary result on the regularity of the spectral measure generated by g(H)ψ0 (which is explained in Proposition 2.1) and the asymptotic of its Fourier transform (see Theorem 2.2). Section 3 contains the proof of Theorem 1.7 and Corollary 1.8, granted the Paley–Wiener type results. For the sake of the reader, we have written down complete arguments, following the strategy of Hunziker in [10]. In Section 4 we prove Theorem 1.9. In Section 5 we prove Theorem 1.11. Our main tool in both sections is the characterization of functions f ∈ Γa,b by use of a suitable almost analytic extension f. Here we rely on results in the thesis of Jung [12], which are partially published in [11] and [13]. We refine these results by proving uniformity with respect to f . This uniformity is neither stated nor proved in [12]. Unfortunately, it was tacitly taken for granted and, implicitly, it has been used at crucial steps to
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prove the results in [12,13]. In the Appendix we have collected a few facts and proofs concerning Gevrey spaces and the proof of Proposition 1.5.
2. Spectral Measure and Time Behavior In this section we investigate the relation between the decay of the Fourier transform of the spectral measure of Φ = g(H)ψ with respect to H (where ψ is an analytic vector) and the regularity of g. The main new point of this section is the inverse result of Theorem 2.2 (2). We assume (C1) Let H be an (unbounded) self-adjoint operator in a complex Hilbert space H . Let U (θ) (θ ∈ R) be a strongly continuous one parameter unitary group, such that H(θ) := U (θ)HU (θ)−1
(θ ∈ R)
(2.1)
extends analytically (in the sense of Kato) into the strip Sβ := {θ ∈ C |Imθ| < β} for some β > 0. The spectrum σ(H(θ)) belongs to C− for θ ∈ Sβ with Imθ > 0 and to C+ for θ ∈ Sβ with Imθ < 0. (C2) For z0 ∈ R, θ ∈ Sβ \R fixed there exists W (z0 ; θ) := {z ∈ C | 0 ≤ |z − z0 | < rθ } for some rθ > 0, such that (H(θ) − z)−1 exists for all z ∈ W (z0 ; θ). Let {Eλ }λ∈R denote the resolution of the identity associated to H. For any φ ∈ H we write dμφ (λ) := d(Eλ φ, φ) for the spectral measure with respect to H. Proposition 2.1. Assume (C1) and (C2). Let g ∈ Γa,b 0 (R) for some a > 1 and b > 0 be real valued, with supp g ⊂ W (z0 ; θ)∩R. Let ψ ∈ H , such that U (θ)ψ (θ ∈ R) extends analytically into Sβ (i.e., ψ is an analytic vector for the selfadjoint infinitesimal generator of U (θ), θ ∈ R). Let Φ := g(H)ψ. Then the spectral measure μΦ has a density in Γa,b 0 (R), i.e., dμΦ (λ) = G(λ)dλ for (R) (which is explicitly given by (2.7)). some G ∈ Γa,b 0 Proof. Let f be any Borel function. The functional calculus applied to the operator H gives 2 f (λ) dμΦ (λ) = ψ, f (H)g (H)ψ = f (λ)g 2 (λ) dμψ (λ). (2.2) R
R
By Stone’s formula dμψ (λ) has the (formal) density 1 ψ, (H − (λ + iε))−1 − (H − (λ − iε))−1 ψ ρ(λ) := lim ε↓0 2πi
(λ ∈ R). (2.3)
Let θ ∈ Sβ \R with Imθ > 0. (The case Imθ < 0 is similar.) Then, since ψ is an analytic vector, one has for λ ∈ W (z0 ; θ) ∩ R (2.1) −1 −1 lim ψ, (H − (λ + iε)) ψ = U (θ)ψ, (H(θ) − λ) U (θ)ψ =: f (λ; θ) ε↓0
(2.4)
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and
−1 −1 U (θ)ψ = f (λ; θ). lim ψ, (H − (λ − iε)) ψ = U (θ)ψ, H(θ) − λ ε↓0
(2.5) Thus combining (2.5), (2.4) and (2.3) gives 1 f (λ; θ) − f (λ; θ) ρ(λ) = (λ ∈ W (z0 ; θ) ∩ R). (2.6) 2πi The functions f (λ; θ) and f (λ; θ) are analytic in the variable λ for all λ in some complex neighborhood of W (z0 ; θ) ∩ R and independent of θ (of θ, respectively) for θ ∈ Sβ with Imθ > 0, using the group property U (θ1 + θ2 ) = U (θ1 )U (θ2 ) and U (θ)∗ = U (θ)−1 in (2.5) and (2.4). Thus by (2.6) the density ρ(·) is analytic in some neighborhood of W (z0 ; θ) ∩ R and thus ρ ∈ Γ1 (W (z0 ; θ) ∩ R). Now, combining (2.3) and (2.2) yields dμΦ (λ) = G(λ) dλ, Then Proposition A.1 gives g 2 ρ ∈
G := g 2 ρ.
Γa,b 0 (W (z0 ; θ)
(2.7)
∩ R).
Theorem 2.2. Assume (C1) and (C2). Let ψ be as in Proposition 2.1. Let g ∈ C0 (R) with supp g ⊂ W (z0 ; θ) ∩ R. Define Φ := g(H)ψ. (1)
∧
μ Let a > 1, b > 0. If g ∈ Γa,b 0 (R), then the Fourier transform Φ of the spectral measure dμΦ (λ) satisfies ∧
1
| μΦ (t)| ≤ C e−C|t| a
(2.8)
1 −a
and some constant C < ∞. ∧ For all λ ∈ W (z0 ; θ) ∩ R define ρ(λ) as in (2.3). If μΦ satisfies
for all t ∈ R, C < C(a, b) := ab (2)
∧
1
| μΦ (t)| ≤ c1 e−c2 |t| a
(t ∈ R)
for some a > 1 and some constants c1 , c2 > 0, then g 2 ∈ Γa,b 0 (R) for a a) . all b ≥ (c−1 2 Proof of Theorem 2.2 (1). Combining the definition of a measure’s Fourier transform with Proposition 2.1 and (2.7) yields ∧ 1 μΦ (t) := √ e−itλ dμΦ (λ) 2π R ∧ 1 (2.7) = √ e−itλ G(λ) dλ = G (t) (t ∈ R). (2.9) 2π R
Then applying Theorem 1.11 (1) to G and taking into account Imt = 0 proves (2.8). Proof of Theorem 2.2 (2). As in the proof of Proposition 2.1 one checks that the density of dμΦ (λ) is G = g 2 ρ ∈ C0 (R). By Theorem 1.11 (3) and (2.9), 2 −1 G ∈ Γa,b G ∈ Γa,b 0 (R). We claim that g = ρ 0 (R). Actually, since ρ is analytic in some neighborhood of supp G ⊂ supp g, the zeros of ρ are discrete in
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supp G and of finite order. Thus, we may assume that z0 = 0, and we have −k to show that, for any k ∈ N\{0}, x−k G is in Γa,b G is in C0 (R). 0 (R), if x This follows from Taylor’s formula 1 k
G(x) = x h(x),
h(x) := 0
(1 − t)k−1 (k) G (tx) dt, (k − 1)!
since a straightforward calculation, using Proposition A.1, proves h ∈ Γa,b 0 (R).
3. Proof of Theorem 1.7 and Corollary 1.8 We assume (A1)–(A5) and recall the basic formulae of [10] in this simplified context. Let θ ∈ Sβ \R. From (A5) and (A4) it follows that Π(κ, θ) = Π0 (θ) + O(κ)
(κ → 0)
(3.1)
is analytic in κ ∈ K0 . (K0 is specified in (A5).) Thus dim RanΠ(κ, θ) = dim RanΠ0 (θ)
(3.2)
for |κ| sufficiently small. (For example, this can be seen by [14, Chapter I §4.6, Lemma 4.10].) So, by Proposition 1.5, λ0 is the limit as |κ| → 0 of a group of perturbed eigenvalues λ(κ) (of H(κ, θ)) having total algebraic multiplicity dim RanΠ0 . For |κ| sufficiently small these are the eigenvalues of the reduced operator H(κ, θ) := Π(κ, θ)H(κ, θ)Π(κ, θ) : RanΠ(κ, θ) → RanΠ(κ, θ).
(3.3)
More specifically, we shall now assume that Imθ > 0. (The case Imθ < 0 is similar.) Note that the λ(κ) are precisely the eigenvalues of H(κ, θ). By definition, they are the resonances corresponding to the unperturbed eigenvalue λ0 of H0 (θ); cf. (A2). Recall that in view of (A5) the domain of definition of λ(·) depends on Imθ. With the help of some auxiliary operators T (κ, θ) : RanΠ0 → RanΠ(κ, θ),
T (κ, θ)−1 : RanΠ(κ, θ) → RanΠ0
the reduced Hamiltonian H(κ, θ) is transformed into an endomorphism on RanΠ0 : θ)T (κ, θ) : RanΠ0 → RanΠ0 h(κ) := T (κ, θ)−1 H(κ,
(|κ| sufficiently small). (3.4)
We recall the explicit form of T (κ, θ) from [10]: Remark 3.1. Hunziker’s results use a Rayleigh–Schr¨ odinger expansion of Π(κ, θ) (i.e., a power series expansion in the perturbation parameter κ) Π(κ, θ) = P (N ) (κ, θ) + O(κN )
(κ → 0)
(3.5)
for θ ∈ Sβ \R being fixed and |κ| sufficiently small. P (N ) (κ, θ) denotes the perturbative expression up to order N − 1 in this expansion. For the case of
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embedded eigenvalues (of H0 ), which interests us in this paper, this expansion is of 0th order (i.e., N = 1). In the more general version of Hunziker [10] his operator P (1) (κ, θ) is our Π0 (θ). Then by [10, (17),(18),(19),(20)] one has for θ ∈ Sβ \R fixed with Imθ > 0 and all |κ| sufficiently small T (κ, θ) := Π(κ, θ)Π0 (θ)U (θ)D(1) (κ)−1/2 , −1/2 T (κ, θ)−1 = D(1) (κ) U (θ)−1 Π0 (θ)Π(κ, θ), D
(1)
−1
(κ) := U (θ)
Π0 (θ)Π(κ, θ)Π0 (θ)U (θ).
(3.6) (3.7) (3.8)
We set for Imθ > 0
∗ T (κ, θ) := Π(κ, θ)Π0 (θ)U (θ) D(1) (κ)−1/2 .
Then one finds that
T (κ, θ)−1
∗
= T (κ, θ).
(3.9)
(3.10)
The family h(κ), κ ∈ K0 , is an analytic family of operators in the class A. Even for κ real, h(κ) may be non self-adjoint. By (3.4), the resonances λ(κ) arising from λ0 are the eigenvalues of h(κ) for |κ| sufficiently small. Furthermore, one can show h(κ)∗ − h(κ) = O(κ2 ),
(A1)
0 ≥ Imλ(κ) = O(κ2 ) (|κ| → 0);
(3.11)
see [10, (24),(25)]. Proof of Theorem 1.7. Assume the conditions and notations of Theorem 1.7. We abbreviate g := gI0 . We may assume that for some fixed θ ∈ Sβ \R the support of g is contained in (W (λ0 ; θ)• ∪ {λ0 }) ∩ R; cf. (A3). For κ ∈ K, where K is specified in (A1), we define F (κ, t) := Π0 e−itH(κ) g(H(κ))Π0
(t ∈ R).
By functional calculus and Stone’s formula one gets 1 −1 F (κ, t) = lim e−itz g(z)Π0 (H(κ) − (z + iε)) ε↓0 2πi supp g −1 Π0 dz − (H(κ) − (z − iε))
(3.12)
for κ ∈ K, t ∈ R. Recall that g I0 ≡ 1, and set I0 = [α, β]. Let ω := supp g\I0 . Using (A3) and choosing supp g sufficiently small, we may assume that there are no eigenvalues of H(κ) in ω. Define Q± (κ, ε, z) := Π0 (H(κ) − (z ± iε))
−1
Π0 .
(3.13)
Then for ε > 0, κ ∈ K, z ∈ R the operators Q± (κ, ε, z) are bounded operators from H to RanΠ0 . Combining (3.13) and (3.12) gives ⎛ ⎞ β 1 ⎝ −itz −itz F (κ, t) = lim j e g(z)Qj (κ, ε, z) dz + e Qj (κ, ε, z) dz ⎠ ε↓0 2πi j=± ω
α
(3.14)
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for κ ∈ K, t ∈ R. Now, for θ ∈ Sβ \R with Imθ > 0 and for ε > 0, κ ∈ K, z ∈ R one has Q+ (κ, ε, z) = U (θ)−1 U (θ)Π0 U (θ)−1 U (θ) (H(κ) − (z + iε))
−1
U (θ)−1 U (θ)Π0 U (θ)−1 U (θ) = U (θ)−1 Π0 (θ) (H(κ, θ) − (z + iε)) (1.5)
−1
= U (θ)
−1
Π0 (θ)U (θ)
Π0 (θ)R(κ, θ, z + iε)Π0 (θ)U (θ),
(3.15)
Q− (κ, ε, z) = U (θ)−1 Π0 (θ)R(κ, θ, z − iε)Π0 (θ)U (θ).
(3.16)
By (A1) and the following Remark, the expressions on the r.h.s. of (3.15) and (3.16) are bounded operators from H to RanΠ0 . Remark 3.2. Analyticity of Π0 (θ) (θ ∈ Sβ ) implies that, for θ ∈ Sβ , U (θ) and U (θ)−1 act as bounded operators from RanΠ0 =: M0 to RanΠ0 (θ) =: M0 (θ) and vice versa; see [10, p.179]. In fact, for θ ∈ Sβ the relation U (θ)Π0 = Π0 (θ)U (θ) holds on the dense domain D(U (θ)) of U (θ). Since Π0 D(U (θ)) is dense in M0 , U (θ) is a priori well defined on a dense set in M0 with values in M0 (θ). Since dim M0 < ∞, U (θ) extends to a bounded operator from M0 to M0 (θ). By (3.15), for κ ∈ K, t ∈ R and ε > 0, the operator-valued function e−itz Q+ (κ, ε, z) is analytic in the variable z for z ∈ C+ . Analogously, by (3.16), e−itz Q− (κ, ε, z) is analytic in the variable z for z ∈ C− . Thus by Cauchy’s Theorem we have β
−itz
e α
Q± (κ, ε, z) dz =
e−itz Q± (κ, ε, z) dz
(κ ∈ K, t ∈ R, ε > 0),
γ±
(3.17) where we have fixed θ ∈ Sβ \R with Imθ > 0 and γ± is some contour in the closed upper half-plane (in the closed lower half-plane, respectively) running from α to β in the strip {z ∈ C | Rez ∈ I0 } and omitting the real poles of R(κ, θ, · ) (omitting the real poles of R(κ, θ, · ), respectively). See Fig. 1. Without loss of generality we assume that γ± ⊂ W (λ0 ; θ)• .
Figure 1. Example for a contour γ+
Figure 2. Deformation of Γ+ + into Γ
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Thus, for κ ∈ K, t ∈ R and ε > 0, F (κ, t) = lim (F+ (κ, ε, t) − F− (κ, ε, t)) , (3.18) ε↓0 ⎛ ⎞ 1 ⎝ e−itz g(z)Q± (κ, ε, z) dz + e−itz Q± (κ, ε, z) dz ⎠ . F± (κ, ε, t) := 2πi ω
γ±
(3.19) Note that for all ε ≥ 0 and κ ∈ K the operator-valued function Q± (κ, ε, z) is analytic in the variable z in some complex neighborhood of γ± ∪ ω. Thus, for fixed θ ∈ Sβ \R with Imθ > 0 and κ ∈ K, lim Q+ (κ, z, ε) = U (θ)−1 Π0 (θ)R(κ, θ, z)Π0 (θ)U (θ) (z ∈ γ+ ∪ ω), (3.20) ε↓0
lim Q− (κ, z, ε) = U (θ)−1 Π0 (θ)R(κ, θ, z)Π0 (θ)U (θ) (z ∈ γ− ∪ ω) ε↓0
(3.21)
are bounded operators in H , which map RanΠ0 to itself; cf. Remark 3.2. Now define Γ± := γ± ∪ R\I0 , running from −∞ to +∞. Extend g analytically (as the constant function g(·) = 1) to {z ∈ C | Rez ∈ I1 }. Then, for fixed θ ∈ Sβ \R with Imθ > 0, κ ∈ K and t ∈ R, we get by dominated convergence 1 e−itz g(z)U (θ)−1 Π0 (θ)R(κ, θ, z)Π0 (θ)U (θ) dz lim F+ (κ, ε, t) = ε↓0 2πi Γ+
=: f+ (κ, θ, t) and lim F− (κ, ε, t) = ε↓0
1 2πi
(3.22)
e−itz g(z)U (θ)−1 Π0 (θ)R(κ, θ, z)Π0 (θ)U (θ) dz
Γ−
=: f− (κ, θ, t).
(3.23)
For fixed θ ∈ Sβ \R with Imθ > 0 the operators f+ (κ, θ, t) and f− (κ, θ, t) are in B(H , RanΠ0 ); see Remark 3.2. Combining (3.23), (3.22) and (3.18) gives, for fixed θ ∈ Sβ \R with Imθ > 0, F (κ, t) = f+ (κ, θ, t) − f− (κ, θ, t)
(κ ∈ K, t ∈ R).
(3.24)
For any θ ∈ Sβ \R the resolvent R(κ, θ, z) can be written as the sum of ∧
a singular part Π(κ, θ)R(κ, θ, z) and a regular part R (κ, θ, z): ∧
R (κ, θ, z) := (1 − Π(κ, θ)) R(κ, θ, z), ∧
R(κ, θ, z) = Π(κ, θ)R(κ, θ, z)+ R (κ, θ, z).
(3.25) (3.26)
Inserting (3.26) into (3.22) and (3.23) splits f± into a regular part fr± and a singular part fs± . More precisely, for fixed θ ∈ Sβ \R with Imθ > 0, κ ∈ K
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and t ∈ R, f+ (κ, θ, t) = fr+ (κ, θ, t) + fs+ (κ, θ, t), 1 fr+ (κ, θ, t) := e−itz g(z)G1 (κ, θ, z) dz, 2πi
(3.27) (3.28)
Γ+
1 fs+ (κ, θ, t) := 2πi
e−itz g(z)G2 (κ, θ, z) dz
(3.29)
Γ+
and f− (κ, θ, t) = fr− (κ, θ, t) + fs− (κ, θ, t), 1 fr− (κ, θ, t) := e−itz g(z)G1 (κ, θ, z) dz, 2πi
(3.30) (3.31)
Γ−
1 fs− (κ, θ, t) := 2πi
e−itz g(z)G2 (κ, θ, z) dz,
(3.32)
Γ−
where ∧
G1 (κ, θ, z) := U (θ)−1 Π0 (θ) R (κ, θ, z)Π0 (θ)U (θ),
(3.33)
G2 (κ, θ, z) := U (θ)−1 Π0 (θ)Π(κ, θ)R(κ, θ, z)Π(κ, θ)Π0 (θ)U (θ)
(3.34)
:= {κ ∈ R | |κ| < for θ ∈ Sβ \R. Fixing θ ∈ Sβ \R with Imθ > 0, we set K minz∈γ± κ0 (z)}, where κ0 is the continuous function from (A3). We then and t ∈ R, get, for κ ∈ K F (κ, t) = fs+ (κ, θ, t) − fs− (κ, θ, t) + fr+ (κ, θ, t) − fr− (κ, θ, t) . (3.35) We shall now analyze the regular part fr+ (κ, θ, t) − fr− (κ, θ, t) and the singular part fs+ (κ, θ, t) − fs− (κ, θ, t) of F (κ, t), by estimating (u, f±s (κ, θ, t)v) and (u, f±r (κ, θ, t)v) for u, v ∈ H with u = v = 1. By sesquilinearity, this suffices to prove an estimate for all u, v ∈ H . Analysis of the regular part: For u, v ∈ H Fubini’s Theorem gives (u, fr+ (κ, θ, t)v)
(3.28)
=
1 2πi
e−itz g(z)Φuv (κ, θ, z) dz
t ∈ R), (3.36) (κ ∈ K,
Γ+
Φuv (κ, θ, z) := (u, G1 (κ, θ, z)v) .
(3.37)
Note that in view of (A3) and (A4) the function Φuv (κ, θ, ·) is analytic in W (λ0 ; θ)• ∪ {λ0 } =: W (λ0 ; θ). Thus g(·)Φuv (κ, θ, ·) is analytic in any open subset of {z ∈ C | Rez ∈ I0 }∩W (λ0 ; θ). In (3.36) we can thus deform the path Γ+ into Γ− without changing the integral. By Proposition A.1 and (A.1), the function g(·)Φuv (κ, θ, ·) belongs to Γa,b (R\I0 ). Applying Theorem 1.9 to
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g(·)Φuv (κ, θ, ·) and the contour Γ− gives 1 e−itz g(z)Φuv (κ, θ, z) dz ≤ C (c0 (gΦ) γ(gΦ) 2πi Γ−
1 + g(·) Φuv (κ, θ, ·)∞,Γ− + β0 g(·) Φuv (κ, θ, ·)∞,U e−Ct a
(3.38)
t ≥ 0, C < ab− a1 and some C < ∞. β0 > 0 for all u, v ∈ H , κ ∈ K, is any sufficiently small number (depending on W (λ0 ; θ)). U is any sufficiently small complex neighborhood (contained in the strip {z ∈ C | |Imz| < β0 }) of some compact set I ⊂ (Γ− ∩ I1 ). A priori, the constant c0 (gΦ) := c0 (g(·)Φuv (κ, θ, ·)) is the constant appearing in the Gevrey estimate (1.2) for the space Γa,b (R\I0 ), and γ(gΦ) := γ(g(·)Φuv (κ, θ, ·)) is defined in Theorem 1.9. To obtain a useful estimate on c0 (gΦ), we have to replace the space Γa,b by Γa,b , where b > b is chosen such that the constant C in (3.38) still satisfies C < ab −1 . Then, c0 (gΦ) in (3.38) may be taken as the constant associated to Γa,b , and Lemma A.2 gives the estimate c0 (gΦ) ≤ c Φuv (κ, θ, ·)∞,Ω
(3.39)
some complex neighborhood of ω = supp g\I0 and c < ∞ some conwith Ω and stant independent of Φuv (κ, θ, ·). We shall now prove that for κ ∈ K u, v ∈ H with u = v = 1 Φuv (κ, θ, ·)∞,Ω = O(κ2 ) (κ → 0).
(3.40)
To do this, by (3.37), it suffices to show G1 (κ, θ, z) = O(κ2 )
z ∈ Ω). (κ → 0, κ ∈ K,
(3.41)
∧
(possibly But, by analyticity, R (κ, θ, z) is uniformly bounded for κ ∈ K shrinking our original K) and z ∈ Ω, where without loss of generality we is contained in W (λ0 ; θ), and one has may assume that the closure of Ω ∧
∧
Π0 (θ) R (κ, θ, z)Π0 (θ) = Π0 (θ) (1 − Π(κ, θ)) R (κ, θ, z) (1 − Π(κ, θ)) Π0 (θ), (3.42) ∧
which follows from Π(κ, θ) R (κ, θ, z) (3.42) one has ∧
Π0 (θ) R (κ, θ, z)Π0 (θ) = O(κ2 )
(3.25)
=
0. Furthermore, by (3.1) and
κ ∈ K). (κ → 0, z ∈ Ω,
(3.43)
Then combining (3.43), (3.33) and Remark 3.2 proves (3.41), which implies (3.40). Similar arguments prove gΦu,v ∞,Γ− + gΦu,v ∞,U = O(κ2 )
(κ → 0).
(3.44)
Now observe that, combining (3.39) and (3.40), one obtains c0 (gΦ) = O(κ2 ),
γ(gΦ) = γ1 (gΦ) + γ2 (gΦ) = O(1)
(κ → 0), (3.45)
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uniformly for u, v ∈ H with u = v = 1, where we have used the definition of γ1 (·) and γ2 (·) in (1.16) and (1.17) to show the second estimate. Thus combining (3.45), (3.44), (3.38) and (3.36) gives 1
|(u, fr+ (κ, θ, t)v)| ≤ O(κ2 ) u v e−Ct a
(κ → 0),
(3.46)
× [0, ∞) × C , where K ⊂ K is uniformly for u, v, κ, t, C ∈ H × H × K compact with sufficiently small diameter and C is compact in {C ∈ R | 0 ≤ 1 C < ab− a }. A similar estimate for fr− (κ, θ, t) proves 1
|(u, fr− (κ, θ, t)v)| ≤ O(κ2 ) u v e−Ct a
(κ → 0),
(3.47)
× [0, ∞) × C , where K ⊂ K is uniformly for u, v, κ, t, C ∈ H × H × K compact with sufficiently small diameter and C is compact in {C ∈ R | 0 ≤ 1 C < ab− a }. Analysis of the singular part: By use of (3.3) and (3.4), we get for all and z ∈ Γ+ κ∈K −1 Π(κ, θ)R(κ, θ, z) = H(κ, θ) − z Π(κ, θ) −1 Π(κ, θ) = T (κ, θ) h(κ) T (κ, θ)−1 − z −1
= T (κ, θ) (h(κ) − z)
T (κ, θ)−1 .
(3.48)
Combining (3.48), (3.10) and (3.4) gives −1
Π(κ, θ)R(κ, θ, z) = T (κ, θ) (h(κ)∗ − z)
T (κ, θ)−1
(3.49)
and z ∈ Γ− . Inserting (3.48), respectively (3.49), into (3.34) yields for κ ∈ K −1
G2 (κ, θ, z) = U (θ)−1 Π0 (θ)T (κ, θ) (h(κ) − z)
T (κ, θ)−1 Π0 (θ)U (θ) (3.50)
z ∈ Γ+ and for κ ∈ K, −1
G2 (κ, θ, z) = U (θ)−1 Π0 (θ)T (κ, θ) (h(κ)∗ − z)
T (κ, θ)−1 Π0 (θ)U (θ) (3.51)
z ∈ Γ− . Combining (3.50), (3.29), (3.8), (3.7) and (3.6) gives for κ ∈ K, 1 −1 e−itz g(z) (h(κ) − z) dz D(1) (κ)1/2 fs+ (κ, θ, t) = D(1) (κ)1/2 2πi Γ+
(3.52) Analogously, combining (3.51), (3.32), (3.8) and (3.9) gives for t ∈ R, κ ∈ K. fs− (κ, θ, t) ∗ 1 ∗ −1 = D(1) (κ)1/2 e−itz g(z) (h(κ)∗ − z) dz D(1) (κ)1/2 2πi Γ−
(3.53)
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As in [10], we shall now calculate the integrals in (3.52) for t ∈ R, κ ∈ K. + in the closed and (3.53). To this end we deform the path Γ+ into a path Γ + = R\I0 . Moreover, lower half-plane, running from −∞ to +∞, with Γ+ ∩ Γ Γ+ −Γ+ encloses the spectrum of h(κ), i.e., all eigenvalues of h(κ); see Fig. 2. + − Γ+ the Gevrey function g can Furthermore, in the region enclosed by Γ be analytically continued to 1. By construction, eigenvalues of h(κ) lie in the closed lower half-plane; cf. (3.11). So by a symmetry argument eigenvalues of h(κ)∗ lie in the closed upper half-plane, and the real eigenvalues of h(κ) and t ∈ R, the Dunford–Taylor integral and h(κ)∗ coincide. Then, for κ ∈ K (see [14, Chapter I §5.6, (5.47)]) gives 1 2πi
−
−1
e−itz g(z) (h(κ) − z)
dz = e−ith(κ) .
(3.54)
Γ+
+ Γ
Thus by (3.54), Eq. (3.52) gives fs+ (κ, θ, t) = D(1) (κ)1/2 e−ith(κ) D(1) (κ)1/2 −1 (1) 1/2 1 + D (κ) e−itz g(z) (h(κ) − z) dz D(1) (κ)1/2 2πi + Γ
(3.55) t ∈ R. Analogously, we calculate the integral in (3.53): For for all κ ∈ K, κ ∈ K and t ∈ R we get 1 2πi
− + Γ
−1
e−itz g(z) (h(κ)∗ − z)
dz = 0
(3.56)
Γ−
by Cauchy’s Theorem, since there are no eigenvalues of h(κ)∗ with negative imaginary parts. Thus combining (3.56) and (3.53) gives ∗ 1 ∗ −1 fs− (κ, θ, t) = D(1) (κ)1/2 e−itz g(z) h(κ)∗ − z dz D(1) (κ)1/2 2πi + Γ
(3.57)
and t ∈ R. Then combining (3.55) and (3.57) gives, for κ ∈ K for κ ∈ K and t ∈ R, fs+ (κ, θ, t) − fs− (κ, θ, t) = D(κ)e−ith(κ) D(κ) + D(κ)I(κ, t)D(κ) − D(κ)∗ I ∗ (κ, t)D(κ)∗ , (3.58)
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and t ∈ R, where, for κ ∈ K D(κ) := D(1) (κ)1/2 , 1 −1 I(κ, t) := Π0 e−itz g(z) (h(κ) − z) dz Π0 , 2πi
(3.59) (3.60)
+ Γ
1 I (κ, t) := Π0 2πi ∗
−1
e−itz g(z) (h(κ)∗ − z)
dz Π0 .
(3.61)
+ Γ
We remark that I(κ, t) and I ∗ (κ, t) are in B(H , RanΠ0 ). An estimate of the term D(κ)I(κ, t)D(κ) − D(κ)∗ I ∗ (κ, t)D(κ)∗ in (3.58) needs an analysis of the operator D(1) (κ) given by (3.8). Therefore we use the identity Π0 (θ)Π(κ, θ)Π0 (θ) 2
= Π0 (θ) − Π0 (θ) (Π0 (θ) − Π(κ, θ)) Π0 (θ) (3.1)
(κ → 0, κ ∈ K).
= Π0 (θ) + Π0 (θ)O(κ2 )Π0 (θ)
(3.62)
Then inserting (3.62) into (3.8) and using Remark 3.2 yields D(1) (κ) = Π0 + O(κ2 )
(κ → 0, κ ∈ K).
one obtains Analogously, for κ ∈ K D(1) (κ)∗ = Π0 + O(κ2 )
(κ → 0).
(3.63)
By standard arguments, D(κ) and D(κ)∗ are in B(H , RanΠ0 ), and, for and t ∈ R, κ∈K D(κ)I(κ, t)D(κ) − D(κ)∗ I ∗ (κ, t)D(κ)∗ = I(κ, t) − I ∗ (κ, t) + r(κ, t), (3.64) where r(κ, t) = I(κ, t)O(κ2 ) + O(κ2 )I(κ, t) + O(κ2 )I(κ, t)O(κ2 ) − I ∗ (κ, t)O(κ2 ) − O(κ2 )I ∗ (κ, t) − O(κ2 )I ∗ (κ, t)O(κ2 )
(κ → 0)
(3.65)
and O(κ2 ) does not depend on t. We shall now estimate the operator I(κ, t)− I ∗ (κ, t) appearing in (3.64): By use of (3.60), (3.61) and the second resolvent equation we get for t∈R κ ∈ K, I(κ, t) − I ∗ (κ, t) 1 −1 −1 = Π0 e−itz g(z) (h(κ) − z) (h(κ)∗ − h(κ)) (h(κ)∗ − z) dz Π0 . 2πi + Γ
and t ∈ R Fubini’s Theorem yields For u, v ∈ H , κ ∈ K 1 ∗ (u, (I(κ, t) − I (κ, t))v) = e−itz g(z)Juv (κ, z) dz, 2πi + Γ
(3.66)
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where
−1 −1 Juv (κ, z) := u, Π0 (h(κ) − z) (h(κ)∗ − h(κ)) (h(κ)∗ − z) Π0 v . (3.67)
The function g(·)Juv (κ, ·) is in Γa,b (R\I0 ) with g(z)Juv (κ, z) = 0 for z ∈ + R\supp g. Then applying Theorem 1.9 to g(z)Juv (κ, z) and the contour Γ gives 1 e−itz g(z)Juv (κ, z) dz ≤ C c0 (gJ)γ(gJ) + g(·)Juv (κ, ·)∞,Γ + 2πi + Γ
1 +β0 g(·)Juv (κ, ·)∞,U e−Ct a
(3.68)
t ≥ 0, C < ab− a1 and some C < ∞. for all u, v ∈ H , κ ∈ K, c0 (gJ) := c0 (g(·)Juv (κ, ·)) is the constant appearing in (1.2) for the Gevrey space Γa,b (R\I0 ) for some b > b (b depending on C; here we argue as after (3.38)) and γ(gJ) := γ(g(·)Juv (κ, ·)) is defined in Theorem 1.9. β0 > 0 is any sufficiently small number (depending on W (λ0 ; θ)). U is any sufficiently small complex neighborhood (contained in the strip {z ∈ C | |Imz| < β0 }) + ∩ I1 ). For z in any compact subset of Γ + and of some compact set I ⊂ (Γ one gets, using Schwarz’ inequality, κ∈K (3.67)
−1
|Juv (κ, z)| ≤ (h(κ) − z)
−1
(h(κ)∗ − h(κ)) (h(κ)∗ − z)
u v
(3.11)
= O(κ2 ) u v (κ → 0).
(3.69)
This estimate also holds for z ∈ U, taking U sufficiently small, and z ∈ Ω, is some complex neighborhood of ω. Thus, by Lemma A.2 and the where Ω definition of γ(gJ), one gets c0 (gJ) ≤ O(κ2 ),
γ(gJ) = O(1)
(κ → 0),
(3.70)
and u, v ∈ H with u = v = 1. Then combining uniformly in κ ∈ K (3.70), (3.69), (3.68) and (3.66) leads to 1
|(u, (I(κ, t) − I ∗ (κ, t))v)| ≤ O(κ2 ) u v e−Ct a
(κ → 0),
(3.71)
× [0, ∞) × C , where K ⊂ K is uniformly for u, v, κ, t, C ∈ H × H × K compact with sufficiently small diameter and C is compact in {C ∈ R | 0 ≤ 1 C < ab− a }. To estimate the term r(κ, t) appearing in (3.64), we start with the first t ∈ R it follows from term on the r.h.s. of (3.65): For u, v ∈ H , κ ∈ K, (3.60) and Fubini’s Theorem that e−itz g(z)Yuv (κ, z) dz, (3.72) (u, I(κ, t)v) = + Γ
−1 Yuv (κ, z) := u, (h(κ) − z) v
(κ → 0).
(3.73)
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Again, the function g(·)Yuv (κ, ·) is in Γa,b (R\I0 ) with g(z)Yuv (κ, z) = 0 for + gives z ∈ R\supp g. Applying Theorem 1.9 to g(·)Yuv (κ, ·) and Γ 1 e−itz g(z)Yuv (κ, z) dz 2πi + Γ
1
≤ C (c0 (gY )γ(gY ) + β0 g(·)Yuv (κ, ·)∞,U ) e−Ct a
(3.74)
t ≥ 0, C < ab− a1 and some C < ∞. c0 (gY ) := for all u, v ∈ H , κ ∈ K, c0 (g(·)Yuv (κ, ·)) is the constant appearing in (1.2) for the space Γa,b (R\I0 ) for some b > b (b depending on C; we argue as after (3.38)) and γ(gY ) := γ(g(·)Yuv (κ, ·)) is defined in Theorem 1.9. β0 > 0 is any sufficiently small number (depending on W (λ0 ; θ)) and U is any sufficiently small complex neighborhood (contained in the strip {z ∈ C | |Imz| < β0 }) of some compact and u, v ∈ H with u = v = 1 + ∩I1 ). We claim that for κ ∈ K set I ⊂ (Γ Yuv (κ, ·)∞,Ω = O(1)
(κ → 0),
(3.75)
denotes some complex neighborhood of ω. Indeed, for z ∈ Ω and where Ω κ ∈ K, (3.73) −1 −1 |Yuv (κ, z)| = u, Π0 (h(κ) − z) v ≤ (h(κ) − z) u v
= O(1) u v as κ → 0, which proves (3.75). Lemma A.2 together with (3.75) and the definition of γ(gY ) gives c0 (gY ) ≤ O(1),
(κ → 0),
γ(gY ) = O(1)
(3.76)
and u, v ∈ H with u = v = 1. Then combining uniformly in κ ∈ K (3.76), (3.75), (3.74) and (3.72) leads to 1 u, Π0 I(κ, t)O(κ2 )v ≤ O(κ2 ) u v e−Ct a
(3.77)
× [0, ∞) × C , where as κ → 0, uniformly for u, v, κ, t, C ∈ H × H × K ⊂ K is compact with sufficiently small diameter and C is compact in K 1 {C ∈ R | 0 ≤ C < ab− a }. The remaining summands in (3.65) we estimate in a similar way, and we finally get 1
|(u, r(κ, t)v)| ≤ O(κ2 ) u v e−Ct a
(κ → 0),
(3.78)
× [0, ∞) × C , where K ⊂ K is uniformly for u, v, κ, t, C ∈ H × H × K compact with sufficiently small diameter and C is compact in {C ∈ R | 0 ≤ 1 C < ab− a }. Then combining (3.78), (3.71) and (3.64) gives |(u, D(κ)I(κ, t)D(κ)v) − (u, D(κ)∗ I ∗ (κ, t)D(κ)∗ v)| 1
≤ O(κ2 ) u v e−Ct a
(3.79)
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× [0, ∞) × C , where as κ → 0, uniformly for u, v, κ, t, C ∈ H × H × K K ⊂ K is compact with sufficiently small diameter and C is compact in 1 {C ∈ R | 0 ≤ C < ab− a }. Finishing the proof: Combining (3.35), (3.47), (3.46), (3.58) and (3.79) yields F (κ, t) = D(κ)e−ith(κ) D(κ)Π0 + R(κ, t)
t ∈ R), (κ ∈ K,
(3.80)
where R(κ, t) := D(κ)I(κ, t)D(κ) − D(κ)∗ I ∗ (κ, t)D(κ)∗ + fr+ (κ, θ, t) − fr− (κ, θ, t). (3.81)
(The operators D(κ), I(κ, t), I ∗ (κ, t), fr+ (κ, θ, t) and fr− (κ, θ, t) are defined in (3.59), (3.60), (3.61), (3.28) and (3.31).) Now (3.80) gives (1.9). Finally, combining (3.46), (3.47), (3.79) and (3.81) proves (1.10). Proof of Corollary 1.8. Assume the conditions and notation of the Corollary. By (3.11), we have |Imλ(κ)| ≤ μκ2 for |κ| sufficiently small and some 1 μ ≥ 0. Thus, by (1.10), for all 0 < C < ab− a and t ≥ 0, we have, as κ → 0, 1 iλ(κ)t (ψ0 , R(κ, t)ψ0 ) ≤ O(κ2 ) eϕ(κ,t) , ϕ(κ, t) := −Ct a + μκ2 t. e Then φ(κ, ·) has a unique zero at T given by (1.11), and φ(κ, t) ≤ 0 (t ≤ T ).
4. Proof of Theorem 1.9 We use the notation Rez =: t, Imz =: s (z ∈ C), ∂z := 12 (∂t + i∂s ), ∂z := 1 2 (∂t − i∂s ). We need the following results, which refine some estimates from [12]; cf. [12, Definition A.1 and Propositions A.1 & A.2 ]: Definition & Proposition 4.1. Let Ω ⊂ R be open. Let f ∈ Γa,b (Ω) with a ∈ (1, ∞), b > 0. For 0 < δ < 1 let χδ ∈ C0∞ (R) be a cut-off function with 0 ≤ χδ ≤ 1, supp χδ ⊂ [−1, 1] and χδ [−1 + δ, 1 − δ] = 1. For t ∈ Ω, s ∈ R (Fχa,b f )(t + is) := δ
∞ (is)k k=0
k!
∂tk f (t)χδ (k(b|s|)1/(a−1) )
(4.1)
f of f in the following sense: defines an almost analytic extension fδ := Fχa,b δ ∞ fδ ∈ C (Ω × R) and lims→0 fδ (t + is) = f (t), uniformly in t ∈ Ω. Furthermore, for all ε > 0 there is δ0 ∈ (0, 1), such that for all 0 < δ, δ < δ0 , s∈R B(a, b) − ε , (4.2) sup |∂z fδ (t + is)| ≤ c c0 (f ) γ1 (f ) exp − 1 t∈Ω |s| a−1
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where (1.16)
γ1 (f ) :=
c (f ) + 0
3a−1 2
c0 (f )+ 3a−1 2
(a − 1)δ
,
B(a, b) := (a − 1)b−1/(a−1) , (4.3)
c0 (f ) is the constant appearing in the Gevrey estimate (1.2) and c < ∞ (explicitly given by (A.27)) is some constant depending on f only through a, b, χ ∞ and ε. Furthermore, uniformly for s ∈ R, sup |fδ (t + is)| ≤ c c0 (f ) γ2 (f ), t∈Ω
(4.4)
c (f ) + a−1 c0 (f )+ a−1 (1.17) 2 0 2 γ2 (f ) := e−c0 (f ) 3 , a−1
where c < ∞ is some constant depending only on a.
Proof. See Appendix A.
Remark 4.2. Compared with [12, Proposition A.2], the main new point of estimate (4.2) is the uniformity with respect to f . Obviously, the only nontrivial point in estimate (4.4) is the stated uniformity in f . Both estimates, (4.2) and (4.4), are not quite optimal, but sufficient for our purpose. We shall apply them in case c0 (f ) = O(κ2 ) as κ → 0. The following Proposition shows that the estimate (4.2) for some almost analytic extension of f is “almost equivalent” to f ∈ Γa,b (Ω): Proposition 4.3 [12, Proposition A.5]. Let Ω ⊂ R be open. Let a ∈ (1, ∞), b > 0. Let B(a, b) be as in (4.3). Let f ∈ C ∞ (Ω × {|s| ≤ ω}, C), for some ω > 0, be bounded and satisfy B(a, b) sup ∂z f(t + is) = O exp − 1 t∈Ω |s| a−1
(s → 0).
Then f := f I ∈ Γa,b (I) for all I ⊂ Ω open with dist(∂I, ∂Ω) > 0. (In the case Ω = R, the latter condition is void and thus absent.) Proof. The proof is a slight modification of the arguments in [12, Anhang A, p. 102/103] for the case Ω = R. If ∂Ω = ∅, application of the inhomogeneous Cauchy formula ∂tn f(t) =
n! 2πi
∂G
∂z f(z) f(z) n! dz − dz ∧ dz n+1 (z − t) π (z − t)n+1
G := {z ∈ C | |z − t| ≤ r}
(n ∈ N),
G
for some r > 0,
requires to shrink Ω to some slightly smaller set I.
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For the proof of Theorem 1.9 we need the following estimate: Lemma 4.4. Let a > 1, b > 0. Let B(a, b) be as in (4.3). Let 0 < ε < B(a, b) and let τ, β ∈ R be given. Then
|β| B(a, b) − ε −|τ | s J(τ ) := e exp − ds 1 s a−1
(4.5)
0
satisfies 1
|J(τ )| ≤ |β| e−C(a,b,ε)|τ | a , where
1 C(a, b, ε) := b− a−1 −
(4.6)
1 ε − a1 − a−1 ε . a b − a−1 a−1
ε↓0
1
(4.7)
a↓1
0 < C(a, b, ε) −→ a b− a −→ b−1 ; cf. Remark 1.2.
Remark 4.5. Note that
Lemma 4.6. Let f ∈ Γa,b (J), J ⊂ R open. Let I ⊂ J be compact and let U be an open complex neighborhood of I, (U ∩ R) ⊂ J. Assume that f has an anaf lytic continuation from I to U, which we also denote by f . Let fδ := Fχa,b δ denote the almost analytic extension of f given by (4.1). Then, for each C > 0, C (t ∈ I), |(f − fδ )(t + is)| ≤ C1 f ∞,U exp − 1 |s| a−1 where |s| < dist(I, ∂U) e−C/(1−δ) and C1 < ∞ is some constant independent of f . We omit the straightforward proofs of Lemma 4.4 and Lemma 4.6. Proof of Theorem 1.9. Assume the conditions and notation of Theorem 1.9. We use the notation τ := Reζ, σ := Imζ (ζ ∈ C). We denote by Ω0 the (open) domain enclosed by γ and the real axis; cf. Fig. 3. Choose β ≷ 0 (depending on Imγ ≷ 0) such that, writing Sβ := {z ∈ C | 0 ≤ Imz ≤ β (0 ≥ Imz ≥ β)} , Sβ \Ω0 contains no poles of f . (In particular, |β| < β0 .) Denote by Ω1 the union of those components of Sβ \Ω0 which meet C\D. Thus f a priori is not well defined on all of Ω1 . (Note that Ω1 is closed.) For |β| > 0 sufficiently small (possibly decreasing |β|), we can choose a partition of unity χ1 + χ2 = 1 in a neighborhood of Ω1 ∩ R, with
χ1 (Rez) =
0, 1,
χ1 , χ2 ∈ C ∞ (R) (4.8)
z in a neighborhood of Ω1 \D . z in a neighborhood of (γ\R) ∩ Ω1
Denoting by fδ (where δ will be chosen later on) the almost analytic extension of f in the sense of Definition 4.1, we set (4.9) f(z) := χ2 (Rez)fδ (z) + χ1 (Rez)f (z)
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Figure 3. Example for a contour γ given in Theorem 1.9; Imγ ≤ 0, corresponding to Reζ ≥ 0
523
Figure 4. Deformation of γ into γ
for z in some neighborhood of Ω1 . Thus f is C ∞ (in the real sense) in some neighborhood of Ω1 . We define uζ (z) := e−izζ f(z) (ζ ∈ C, z in some neighborhood of Ω1 ).
(4.10)
Note that f(z) = f (z) for z ∈ (γ\R) ∩ Ω1 . Thus f defines an analytic continuation of f ((γ\R) ∩ Ω1 ) to a complex neighborhood of γ\R, which we shall also denote by f. With this notation, uζ (z) = e−izζ f(z) denotes the corresponding analytic continuation to this neighborhood. Clearly, uζ (z) = e−izζ f (z)
(z ∈ γ).
(4.11)
We now deform the path γ into a contour γ defined as follows: Set γ1 := {z ∈ C | Imz = β} ∩ Ω1 ,
γ2 := γ\Ω1 .
Then γ := γ1 + γ2 denotes a contour running from −∞ + iβ to +∞ + iβ, which encloses with the real line only poles of f in Ω0 ; cf. Fig. 4. Then Stoke’s Theorem gives duζ ∧ dz = uζ dz = uζ dz − uζ dz, (4.12) Ω1
∂Ω1
γ
γ
where duζ ∧ dz = ∂z uζ (z) dz ∧ dz = 2i∂z uζ (z) dt ∧ ds, (4.10)
∂z uζ (z) = e−izζ (∂z f)(z)
(z ∈ Ω1 ),
(4.13)
e−izζ = e−itτ etσ esτ eisσ . (4.14)
Use of (4.13) and (4.14) gives duζ ∧ dz = 2i e−itτ etσ esτ eisσ (∂z f)(t + is) dt ∧ ds. Ω1
Ω1
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Setting E := πt (Ω1 ∩ supp f), where πt : Rt × Rs → Rt is the projection onto the t-axis, one gets |β| duζ ∧ dz ≤ 2 etσ e−s|τ | |∂z f(t + is)| dt ds, (4.15) Ω1
0 E
where we have used |e−izζ | = etσ esτ and, by construction, sτ < 0. For z ∈ Ω1 a direct calculation shows (4.9)
∂z f(z) = χ2 (t)∂z fδ (z) + (∂z χ2 (t))fδ (z) + χ1 (t)∂z f + (∂z χ1 (t))f (z) 1 1 = χ2 (t)∂z fδ (z) + χ 2 (t)fδ (z) + χ 1 (t)f (z) 2 2 1 (4.8) = χ2 (t)∂z fδ (z) + χ 2 (t) fδ − f (z). (4.16) 2 Then applying (4.2) and Lemma 4.6 to (4.16) yields, possibly decreasing the value of |β| in the definition of Ω1 , for z ∈ Ω1 B(a, b) − ε (4.17) |∂z f(z)| ≤ c (c0 (f ) γ1 (f ) + f ∞,U ) exp − 1 |s| a−1 for any ε > 0, where δ in (4.9) is chosen sufficiently small and U is any sufficiently small complex neighborhood of I := supp χ 2 . The constants γ1 (f ) and B(a, b) are given by (4.3), c0 (f ) is the constant in (1.2) for the Gevrey space Γa,b (Ω) and c < ∞ is some constant independent of f (in the sense that c depends only on a, b, β0 and U). Then using (4.17) in (4.15) gives duζ ∧ dz ≤ c (c0 (f ) γ1 (f ) + f ∞,U ) |E| eHE (σ) Ω1
|β| B(a, b) − ε ds · e−s|τ | exp − 1 s a−1
(4.18)
0
for any ε > 0, if δ in (4.9) is chosen sufficiently small. c < ∞ denotes some constant independent of f . HE (σ) is the supporting function of E defined in (1.13) and |E| denotes the Lebesgue measure of E. Applying Lemma 4.4 to (4.18) gives 1 duζ ∧ dz ≤ c (c0 (f ) γ1 (f ) + f ∞,U ) |E| eHE (σ) |β| e−C(a,b,ε)|τ | a Ω1
(4.19) for any ε > 0. C(a, b, ε) is given by (4.7) and c < ∞ is some constant independent of f . We set α := γ ∩ supp uζ . Without loss of generality we may assume that |Imz| ≥ |β| for all z ∈ γ . (If γ (and thus γ ) meets the real line inside D, we may, without changing the integral in the definition (1.14) of
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525
f # , deform it to get nonzero imaginary part inside D.) Then, for z ∈ α, we have |uζ (z)| ≤ f∞,α eHE (σ) e−|τ | |β| ,
(4.20)
since |e−izζ | = etσ esτ and, by construction, sτ ≤ 0. Using (4.20) leads to (4.21) uζ dz = uζ dz ≤ |α| f∞,α eHE (σ) e−|τ | |β| , γ α where |α| denotes the length of the path α. Obviously, f∞,α ≤ f∞,γ . By (4.4), f∞,γ ≤ c γ2 (f ) c0 (f ) + f ∞,γ for some constant c < ∞ independent of f . Thus (4.21) turns into (4.22) uζ dz ≤ |α| (c γ2 (f ) c0 (f ) + f ∞,γ ) eHE (σ) e−|τ | |β| . γ By combining (4.22), (4.19) and (4.12) one gets, using |E| ≤ |α|, uζ dz ≤ (c |β| (c0 (f ) γ1 (f ) + f ∞,U ) + c c0 (f ) γ2 (f ) + f ∞,γ ) γ
· |α| eHE (σ) F (τ ; |β|, a, b)
(4.23)
with 1
F (τ ; |β|, a, b) := e−|τ | |β| + e−C(a,b,ε)|τ | a
(τ ∈ R),
where in (4.23) c , c < ∞ are some constants independent of f . Now, for all τ ∈ R there exists 0 < c < ∞, such that 1
F (τ ; |β|, a, b) ≤ c e−C(a,b,ε)|τ | a ,
(4.24)
where the constant c < ∞ depends only on |β|, a, b. Then combining (4.24), (4.23), (4.11) and (1.14) gives |f # (ζ)| ≤ C (c0 (f )(γ1 (f ) + γ2 (f )) + f ∞,γ + |β| f ∞,U ) 1
· eHE (σ) e−C(a,b,ε)|τ | a ,
(4.25)
where C < ∞ is some constant independent of f (in the sense that it depends only on a, b, β0 , U and γ). Since by construction suppf R ⊃ E, we have eHE (σ) ≤ eHsuppf R (σ)
(σ ∈ R).
(4.26)
Then inserting (4.26) into (4.25) yields |f # (ζ)| ≤ C (c0 (f )(γ1 (f ) + γ2 (f )) + f ∞,γ + |β| f ∞,U ) 1
· eHsuppf R (σ) e−C(a,b,ε)|τ | a ,
(4.27)
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where the constant C < ∞ is independent of f . Finally (4.27) proves (1.15), (4.7)
1
where we have taken into account C(a, b, ε) < C(a, b) := ab− a (ε > 0) and C(a, b, ε) → C(a, b) (ε ↓ 0); see Remark 4.5.
5. Proof of Theorem 1.11 We recall the following classical Paley–Wiener Theorem (see, e.g., [22, Chapter I.1.3, Eq. (1.3.25) ff.]): Theorem 5.1 (Paley–Wiener). f ∈C0∞ (Rn ) with suppf contained in some compact convex subset E of Rn ⇔
∧
f is an entire function, and for all N ∈ N there exists CN > 0 such that ∧
| f (ζ)| ≤ CN (1 + |ζ|)−N eHE (Imζ)
(ζ ∈ Cn ).
For the proof of Theorem 1.11 (2) and (3) we will need the following Lemma: Lemma 5.2. Let E ⊂ R be compact with supporting function HE . Let g ∈ L∞ (R) satisfy 1
|g(τ )| ≤ c1 e−c2 |τ | a
(τ ∈ R),
(5.1)
C0∞ (R)
be a smoothed out characteristic where a > 1 and c1 , c2 > 0. Let χ ∈ function satisfying supp χ ⊂ [−1, 1] and χ [− 12 , 12 ] = 1. Then 1 hα,β (z) := √ eizτ χ(β|Imz|α τ )g(τ ) dτ (5.2) 2π R α a a for α := , β := (5.3) a−1 c2 ∨
is an almost analytic extension of the inverse Fourier transform g in the sense of Definition & Proposition 4.1. More precisely: ∨
hα,β R = g R,
(5.4)
hα,β ∈ C ∞ (C)
(5.5)
(in the real sense), c α 1 C 2 |∂z hα,β (z)| ≤ exp −(a − 1) |Imz|− a−1 1+α |Imz| a
for z ∈ C, with C =
√1 c1 α 2π 2β
(5.6)
χ ∞ .
Proof. We recall the notation t := Rez, s := Imz for z ∈ C. Obviously, g ∈ L1 (R) and χ(β|s|α · ) → 1 as |s| ↓ 0 pointwise, which proves (5.4) by dominated convergence. Further, χ ∈ C0∞ (R) and supp χ ⊂ {x ∈ R | 12 ≤ |x| ≤ 1}. In particular
1 1 supp χ (β|s|α · ) ⊂ τ ∈ R ≤ |τ | ≤ . (5.7) 2β|s|α β|s|α
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We shall prove (5.5) by standard applications of the dominated convergence theorem. Therefore we calculate ∂t χ(β|s|α τ ) = 0 (t ∈ R), 1 = lim (χ(β|s|α τ ) − 1) = 0, ∂s χ(β|s|α τ ) s→0 s s=0
(5.8)
since χ(x) = 1 for |x| ≤ 12 . For s = 0 one obtains Fα,β (τ, s) := ∂s χ(β|s|α τ ) = χ (β|s|α τ )τ βα|s|α−1 sign(s) = χ (β|s|α τ )τ βα|s|α−2 s, |Fα,β (τ, s)| ≤ Cα,β |τ |
(5.9)
(τ ∈ R),
(5.10)
locally uniformly in |s| ≤ 1, since α > 1. We claim that, for |s| ≤ const., (5.9)
A(τ, z) := eizτ g(τ ) ∂z χ(β|s|α τ ) =
i i(t+is)τ e g(τ )Fα,β (τ, s) 2
(5.11)
is majorized by an integrable function v(τ ). Indeed, using (5.1) in (5.11) gives c1 |Fα,β (τ, s)| exp{|τ |1/a φα (τ, s)}, |A(τ, z)| ≤ 2 (5.12) −1 φα (τ, s) := |τ |α |s| − c2 . Note that by (5.9) and (5.11) supp A( ·, z)=supp Fα,β ( ·, s)=supp χ (β|s|α · ), −1 −1 which satisfies (5.7). On supp χ (β|s|α · ) one has |τ |α |s| ≤ β −α (which follows from (5.7)) and thus α −α−1 a 1 (5.3) −α−1 −1 <0 φα (τ, s) ≤ β − c2 = − c2 = c2 c2 a (5.13) for all s ∈ R. Thus combining (5.10), (5.12) and (5.13) leads to |A(τ, z)| ≤ v(τ ), where 1
1
v(τ ) := C|τ |e−c2 |τ | a (1− a )
(τ ∈ R)
is in L1 (R, dτ ). By combining (5.8) and (5.11), the function A(τ, z) is continuous in z. Thus, by dominated convergence, (5.2) 1 (5.11) 1 izτ α ∂z hα,β (z) = √ e g(τ )∂z χ(β|s| τ )dτ = √ A(τ, z)dτ (5.14) 2π 2π R
R
is continuous in z ∈ C. By similar arguments, the integrand of hα,β in (5.2) is majorized by 1
1
|eizτ χ(β|Imz|α τ )g(τ )| ≤ c1 e−c2 (1− a )|τ | a
(τ ∈ R, z ∈ C).
Thus, using dominated convergence again, hα,β is continuous, and so is ∂z hα,β . Inductively, such arguments give (5.5). Using (5.1) and (5.9) in (5.11)
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yields |A(τ, z)| ≤
1/a c1 αβ |χ (β|s|α τ )| |τ | |s|α−1 e−sτ e−c2 |τ | 2
(τ ∈ R, s = 0). (5.15)
To prove (5.6), combine (5.14) and (5.15) to get for s = 0 1/a 1 c1 αβ |∂z hα,β (z)| ≤ √ |χ (β|s|α τ )| |τ | |s|α−1 e−sτ e−c2 |τ | dτ 2π 2 supp χ (β|s|α · )
1 c1 αβ −1−α = √ |s| 2π 2
e−ϕ(x;s,α,c2 ) |x| |χ (βx)| dx,
supp χ (β · )
(5.16) where x := |s|α τ,
ϕ(x; s, α, c2 ) :=
s c2 x + 1/(a−1) |x|1/a . |s|α |s|
e−ϕ(x;s,α,c2 ) is maximal for βx = ∓1, corresponding to s ≷ 0. Thus, since (5.3)
1 , 1 − α = − a−1 1 1 1 c2 ϕ ∓ ; s, α, c2 = − |s|1−α + 1/(a−1) β − a β β |s| 1 c2 1 = |s|− a−1 − + 1/a (s ≷ 0). β β
(5.17)
Note that |x| ≤ β −1 on supp χ (β · ). Thus combining (5.17) and (5.16) gives, 1α χ ∞ , with C := √12π c2β
1 1 c2 |∂z hα,β (z)| ≤ C |s|−1−α exp |s|− a−1 − 1/a β β
a 1 c2 a−1 (5.3) = C |s|−1−α exp −(a − 1) |s|− a−1 . a Proof of Theorem 1.11 (1). Let f ∈ Γa,b 0 (R) with supp f ⊂ E. Then f is ∧
bounded and its Fourier transform f is an entire function by (the easy part of) Theorem 5.1. Applying Theorem 1.9 for γ = R and D = ∅ (thus U = ∅) to f proves (1.18). Proof of Theorem 1.11 (3). Because of Proposition 4.3 it suffices to show: There is an almost analytic extension f of f (in the sense of Proposition 4.3) satisfying B(a, b0 ) (|Imz| ↓ 0), (5.18) ∂z f(z) = O exp − 1 |Imz| a−1 where B(a, b0 ) is given by (4.3). We shall construct f by inverse Fourier ∧
transform of a cut-off version of f . More precisely: Choose χ ∈ C0∞ (R) with
Vol. 11 (2010)
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529 ∧
supp χ ⊂ [−1, 1] and χ [− 12 , 12 ] = 1. Then applying Lemma 5.2 to g =f a and β = ( ca2 )α yields that for α = a−1 ∧ 1 √ eizτ χ(β|Imz|α τ ) f (τ ) dτ f (z) := hα,β (z) := 2π R
a
is an almost analytic extension of f satisfying (5.6). Since (a − 1)( ca2 ) a−1 = B(a, b0 ) for b0 := ( ca2 )−a , (5.18) is proven. Proof of Theorem 1.11 (2). First, one observes that suppf ⊂ E by apply∧
ing the classical Paley–Wiener Theorem 5.1 to f . By Theorem 1.11 (3), f ∈ Γa,b0 (R).
Appendix A In this Appendix, we collect a few facts on Gevrey spaces. We remark that our Definition 1.1 for f ∈ Γa (Ω), Ω ⊂ R open, is equivalent to the usual definition |∂ n f (x)| ≤ Cf Anf n!a
(x ∈ Ω)
for some Cf , Af < ∞ and all n ∈ N (by use of Stirling’s formula, n!a might also be replaced by Γ(na + 1)). The point of including our additional factor (n + 1)c0 (f ) is the following Proposition A.1. Γa,b (Ω), defined in Definition 1.1, is an algebra which is stable under differentiation. If Ω = R, it is also stable under translation. For Ω = R, this is proven in [12, Proposition 3.1]. For general Ω ⊂ R, the same proof applies. We also remark that instead of the estimate (1.2), uniformly for all x ∈ Ω, where Ω ⊂ R is open, one sometimes requires the estimate only for all compact subsets K ⊂ Ω, letting all constants depend on K. This gives a Gevrey space of possibly unbounded functions and leads to a natural inductive limit topology. For our purpose, this is irrelevant. Next we mention the connection to the spaces C{Mn } appearing in the Denjoy– Carleman Theorem (see, e.g., [8, Theorem 1.3.8], [21, Theorem 19.11]): f is said to belong to the space C{Mn }, where Mn is an increasing sequence, if there exist Cf < ∞ and Af < ∞, such that |∂ n f (x)| ≤ Cf Anf Mn
(x ∈ R, n ∈ N).
Thus Γ (R) = C{Mn } with Mn := n! . (This sequence satisfies the usual convenient requirement that ln(Mn ) is convex.) The Denjoy–Carleman Theorem then implies that Γa (R) for a > 1 is a non quasi-analytic class. In particular, there exist cut-off functions g ∈ Γa0 (R) (and partitions of unity); see [21, Theorem 19.10]. A very explicit proof of this fact might be obtained by observing that, for β > 0, the function
−1 xβ , x>0 fβ (x) := e 0, x≤0 a
a
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belongs to Γa (R) for a ≥ 1 + β1 (and fβ ∈ Γa (R) for a < 1 + β1 ); see [11, Lemma 2]. Taking products of translates, fβ (x − x0 )fβ (x1 − x), one constructs functions with support on arbitrary intervals [x0 , x1 ] and, by a standard procedure (see [7, Chapter 2]), partitions of unity. Since Γa0 (Ω) ⊂ Γa1 ,b (Ω)
for all a0 < a1
and
b > 0,
(A.1)
Γa,b 0 (Ω)
this proves = {0} for all a > 1, b > 0, Ω open in R. Finally, we shortly discuss the role of the constant c0 (f ) in the definition (1.2) of the class Γa,b (Ω). Clearly, c0 (f ) ≥ f ∞,Ω = supx∈Ω |f (x)|. If f is ⊃ Ω with a uniform bound dist(∂ Ω, Ω) ≥ analytic in an open complex set Ω δ > 0, then the Cauchy estimates imply |∂ n f (x)| ≤ f ∞,Ω δ −n n!
(x ∈ Ω, n ∈ N).
(A.2)
−1
This estimate is slightly sharper than to say f ∈ Γa,δ (Ω) with c0 (f ) = f ∞,Ω (because of the factor (n + 1)c0 (f ) ). We also observe that the proof (given in [12]) of Proposition A.1 only gives ∂ n (f g)∞,Ω ≤ c0 (f ) c0 (g) (n + 1)c0 (f )+c0 (g)+1 bn n!a , which in general only yields c0 (f g) ≤ max {c0 (f ) c0 (g), c0 (f ) + c0 (g) + 1} . For our purpose, such a result is useless. We need a stronger result in the case where one of the factors f , g is analytic. Lemma A.2. Let g ∈ Γa,b (Ω) with a > 1, b > 0. Let f be analytic in the ⊂ C with Ω ⊂ Ω, dist(∂ Ω, Ω) ≥ δ > 0. Then f g ∈ Γa,b (Ω) and open set Ω the constant c0 (f g) in the Definition 1.1 of Γa,b (Ω), for any b > b, might be taken to satisfy c0 (f g) ≤ C f ∞,Ω , where the constant C is independent of f . Proof. Leibniz’ rule and Eq. (A.2) give n n n ∂ (f g)∞,Ω ≤ c0 (g) (k + 1)c0 (g) bk k!a f ∞ δ k−n (n − k)! k k=0 n b c0 (g) f ∞ a n n! b Ln , (A.3) ≤ c0 (g) f ∞ (n + 1) b where Ln :=
n
pk;n ,
c0 (g)
pk;n := (k + 1)
k=0
k−n
(b δ)
k!a (n − k)! n!a
n . (A.4) k
It suffices to prove: For any r := b/b < 1, rn Ln is uniformly bounded above, independent of Ln ≤ L (n ∈ N) with some L < ∞. It is easy to estimate pk;n = o(1) (n → ∞). 0≤k≤ n 2
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Thus it remains to estimate
n 2
531
pk;n . Here we may use Stirling’s for-
mula. Using (A.3), (A.4) and the boundedness of rn (n + 1)c0 (g)+N for any r < 1 and any N ∈ N, it suffices to prove that k−n n bδ qk;n with qk;n := e(a−1) (k ln(k)−n ln(n)) In := ea−1 n k> 2
1
is polynomially bounded in n. Setting η := a − 1 and B := e−1 (b δ) η , one finds qk;n = eln(B) η(k−n)+η(k ln(k)−n ln(n)) = eη (k ln(Bk)−n ln(Bn)) . Setting φ(k) := ηk ln(Bk) and using that φ(k) is increasing, we obtain −φ(n)
In = e
n
eφ(k) ≤
k> n 2
n . 2
(A.5)
This finishes the proof.
We close Appendix A with the proof of the estimates (4.2) and (4.4) contained in Definition & Proposition 4.1. Proof of Definition and Proposition 4.1, (4.2) and (4.4). A straightforward calculation proves 2∂z fδ (t + is) = (∂t + i∂s )fδ (t + is) ∞ (is)k k+1 ∂t f (t) χδ (k(b|s|)1/(a−1) ) − χδ ((k + 1)(b|s|)1/(a−1) ) = k! k=0
+i
∞ (is)k k=0
k!
1
1
∂tk f (t) χ (k|bs| a−1 ) k b a−1
1 1 |s| a−1 −1 a−1
for all t ∈ Ω and s ∈ R. Now observe that χδ (k(b|s|)1/(a−1) ) − χδ ((k + 1)(b|s|)1/(a−1) ) = 0 for k ∈ I := (kmin , kmax ] (i.e., for k ∈ N\I) with 1
kmin := (1 − δ) |bs|− a−1 − 1, Similarly, χ (k|bs|
1 a−1
1
kmax := |bs|− a−1 .
(A.6)
) = 0 for k ∈ I. Thus, for t ∈ Ω and s ∈ R,
2 |∂z fδ (t + is)| |s|k 1 kb |s|k k+1 ≤ |∂tk+1 f (t)| + |∂t f (t)| |bs| a−1 −1 χ ∞ . k! k! a−1
(A.7)
k∈I
Using (k + 2) ≤ 2(k + 1) for all k ∈ N gives (1.2)
∂tk+1 f ∞,Ω ≤ c0 (f ) (k + 2)c0 (f ) bk+1 (k + 1)!a ≤ c0 (f ) 2c0 (f ) (k + 1)c0 (f ) b bk (k!(k + 1))a = c1 (k + 1)c2 bk k!a
(k ∈ N),
(A.8)
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where c1 := b 2c0 (f ) c0 (f ),
c2 := c0 (f ) + a.
(A.9)
Combining (A.8) and (A.7) gives, for t ∈ Ω and s ∈ R, 1 kb χ ∞ |bs|k k!a−1 (k + 1)c2 1 + |bs| a−1 −1 2 |∂z fδ (t + is)| ≤ c1 a−1 k∈I 1 kmax b ≤ c1 1 + |bs| a−1 −1 χ ∞ |bs|k k!a−1 (k + 1)c2 a−1 k∈I −1 |s| (A.6) χ ∞ SI , = c1 1 + (A.10) a−1 where SI :=
|bs|k k!a−1 (k + 1)c2
(s ∈ R).
(A.11)
k∈I
Using Stirling’s formula, one finds some (universal) constant C, such that for all k ∈ N k(a−1) a−1 k a−1 a−1 ≤C (2π(k + 1)) 2 . (A.12) k! e Thus by (A.11) and (A.12) one obtains √ a−1 SI ≤ (C 2π)a−1 |bs|k (k + 1)c2 + 2 k k(a−1) e−k(a−1)
(s ∈ R).
k∈I
For k ≤ kmax we use (A.6) to get
k(a−1) 1 k(a−1) (A.6) |bs|k k k(a−1) ≤ |bs|k kmax = |bs|k |bs|− a−1 = 1,
yielding SI ≤ (C
√
2π)a−1
(k + 1)c2 +
k∈I
a−1 2
e−k(a−1) ≤ Csup
e−k(a−1)(1−δ) ,
k∈I
(A.13) where 0 < δ < 1 can be chosen arbitrarily small and √ 3a−1 Csup := (C 2π)a−1 sup E(k), E(k) := e−k(a−1)δ (k + 1)c0 (f )+ 2 . k∈I
(A.14) Using the geometric series in (A.13) leads to q SI ≤ Csup q kmin , q := e−(a−1)(1−δ) < 1. (A.15) q k ≤ Csup 1−q k∈I
Inserting (A.6) and (4.3) into (A.15) gives − δ) 1 B(a, b)(1 − δ)(1 SI = Csup exp − . 1 1−q |s| a−1
(A.16)
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Then combining (A.10) and (A.16) gives, for t ∈ Ω and s ∈ R, − δ) B(a, b)(1 − δ)(1 c1 Csup |s|−1 2|∂z fδ (t + is)| ≤ 1+ χ ∞ exp − . 1 1−q a−1 |s| a−1 (A.17)
Observe that for all ε > 0 there exists cε < ∞ and δ0 > 0, such that for all 0 < δ, δ < δ0 one has − δ) B(a, b)(1 − δ)(1 B(a, b) − ε −1 |s| exp − ≤ cε exp − (s ∈ R). 1 1 |s| a−1 |s| a−1 (A.18) To get an estimate for Csup we shall now determine the supremum in (A.14): Setting β := c0 (f ) + 3a − 1 , (A.19) α := (a − 1)δ, 2 one obtains E(k) = eψ(k) ,
ψ(k) := −kα + β ln(k + 1)
(k ∈ N).
(A.20)
Since ψ(0) = 0, limk→∞ ψ(k) = −∞ and ψ is concave, the only possible critical point (i.e., a zero of ψ ) in [0, ∞) is a local maximum, which is given by β (A.21) k∗ := − 1. α Using a > 1 leads to 3a 1 3a − a (A.19) β = c0 (f ) + − > c0 (f ) + = c0 (f ) + a (A.22) 2 2 2 and, in particular, β−α
(A.22) (A.19)
>
c0 (f ) + a − (a − 1)δ > c0 (f )
(0 < δ < 1).
(A.23)
Then inserting (A.21) into (A.20) gives sup E(k)
≤
k∈I (A.23)
≤
β E(k∗ ) = exp{−(β − α)} exp{β ln( )} α β β β e−c0 (f ) exp{β ln( )} = e−c0 (f ) α α
(A.19)
=
e−c0 (f )
c0 (f ) +
3a−1 2
(a − 1)δ
c0 (f )+ 3a−1 2 .
Combining (A.24), (A.20) and (A.14) yields c0 (f )+ 3a−1 2 √ a−1 −c (f ) c0 (f ) + 3a−1 2 0 e . Csup ≤ (C 2π) (a − 1)δ
(A.24)
(A.25)
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Combining (A.9) and (A.25) gives c1 Csup cε χ ∞ 1+ 1−q a−1 √ a−1 c0 (f )+ 3a−1 cε 2 1 + a−1 b(C 2π) χ ∞ c0 (f ) + 3a−1 2 c0 (f ) ≤ . 1−q (a − 1)δ (A.26) Finally combining (A.15), (A.17), (A.18) and (A.26) proves (4.2) with √ a−1 cε b 1 + (C 2π) χ ∞ 2 a−1 . (A.27) c = c(a, b, ε, χ ) := 1 − e−(a−1)(1−δ) The estimate (4.4) follows from similar (but simpler) arguments.
Appendix B Proof of Proposition 1.5. [I. Herbst; private communication] By D we denote the set of entire vectors for the self-adjoint infinitesimal generator A of the unitary group U (θ) = eiθA (θ ∈ R); cf. Remark 1.3. Then, for φ, ψ ∈ D, Im z > 0 and θ ∈ R, −1 (B.1) φ, (H0 − z)−1 ψ = φ(θ), (H0 (θ) − z) ψ(θ) , φ(θ) := U (θ)φ,
ψ(θ) := U (θ)ψ.
By assumption (A1), (H0 (·) − z)−1 extends to an analytic function in Sβ+ := {θ ∈ Sβ | Im θ > 0}. Thus (B.1) also holds for θ ∈ Sβ+ . Setting θ = 0 in (1.8) gives Π0 = s -lim (−iε) (H0 − (λ0 + iε)) ε↓0
−1
.
(B.2)
By (A2), for θ ∈ Sβ+ , (H0 (θ) − z)−1 has a Laurent series expansion around λ0 . Thus, combining (B.2) and (B.1), we get by Cauchy’s formula (φ, Π0 ψ) = φ(θ), Π0 (θ)ψ(θ) , (B.3)
−1 −1 (H0 (θ) − z) dz (θ ∈ Sβ+ , φ, ψ ∈ D), (B.4) Π0 (θ) = 2πi |z−λ0 |=ε
where ε > 0 is sufficiently small, such that |z − λ0 | = ε encloses no point of σ(H0 (θ)) different from λ0 . By 1M we denote the characteristic function and ψ = ψ(−θ) for φ = 1[−n,m] (A)u and of M ⊂ R. Setting φ = φ(−θ) ψ = 1[−n,m] (A)v with m, n > 0 sufficiently large and u, v ∈ H , and fixing θ = iα for some 0 < α < β, we get u, 1[−n,m] (A)e−αA Π0 eαA 1[−n,m] (A)v (B.3) = u, 1[−n,m] (A)Π0 (iα)1[−n,m] (A)v (u, v ∈ H ). (B.5)
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Since Π0 (iα) ≤ Cα < ∞ by (B.4) and assumption (A2), one obtains (using (B.5) and Tα,m,n = sup v = u =1 |(u, Tα,m,n v)| ) Tα,m,n ≤ Cα ,
Tα,m,n := 1[−n,m] (A)e−αA Π0 eαA 1[−n,m] (A),
(B.6)
uniformly in m, n > 0. Choosing an orthonormal basis {ej | j ∈ {1, . . . , N }} N of Ran Π0 , we insert Π0 = j=1 (ej , · ) ej into (B.6) to get αA
Tα,m,n e
1[−n,m] (A)ek =
N
(n,m)
cjk
e−αA 1[−n,m] (A)ej ,
(B.7)
j=1
(n,m)
cjk
:= eαA 1[−n,m] (A)ej , eαA 1[−n,m] (A)ek
for m, n ∈ N and j, k ∈ {1, . . . , N }. Note that (n,m) (m) = eαA 1(−∞,m] ej , eαA 1(−∞,m] ek =: cjk lim cjk
n→∞
(j, k ∈ {1, . . . , N }).
Choosing m > 0 sufficiently large and using the spectral theorem, the vectors ej (m) := 1(−∞,m] (A)ej and ej (m, n) := 1[−n,m] (A)ej (j ∈ {1, . . . , N }) are linearly independent for all sufficiently large m and n. By the spectral theorem again, eαA ej (m) and eαA ej (m, n) are linearly independent (j ∈ {1, . . . , N }). Thus, by a standard theorem on Gram matrices, the (n,m) (m) matrices C (n,m) := (cjk )1≤j,k≤N and C (m) := (cjk )1≤j,k≤N are invertible for m and n sufficiently large, and one has a bound M on the inverse (C (n,m) )−1 ≤ M < ∞ (since C (n,m) → C (m) , n → ∞), which is uniform with respect to n sufficiently large. Combining this with (B.7) and the estimate (B.6), we get e−αA ej (m, n) ≤ M sup Tα,m,n eαA ek (m, n) 1≤k≤N
≤ M Cα sup eαA ek (m, n) ≤ Cα,m < ∞ 1≤k≤N
(B.8)
for all j ∈ {1, . . . , N }, uniformly in n ∈ N. Letting n → ∞ on the l.h.s. of (B.8), the spectral theorem yields that ej is in D(e−αA ), the domain of e−αA . Analogously, ej ∈ D(eαA ) (j ∈ {1, . . . , N }). Writing e±sA ej = e±sA (1{μ≥0} (A)+1{μ<0} (A)) ej , by functional calculus one gets e±sA ej ≤ eαA ej + e−αA ej (|s| ≤ α), which proves ej ∈ D(e±sA )(|s| ≤ α). Thus the eigenfunctions ej (j ∈ {1, . . . , N }) are analytic vectors for A, and Π0 (θ) = N j=1 (ej (θ), · ) ej (θ) is analytic in θ ∈ Sβ .
Acknowledgements The authors thank Rainer W¨ ust for initiating their cooperation on the subject of this paper, Ira Herbst for the proof in Appendix B and an anonymous referee for helpful remarks to improve this article.
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References [1] Aguilar, J., Combes, J.M.: A class of analytic perturbations for one-body Schr¨ odinger Hamiltonians. Commun. Math. Phys. 22, 269–279 (1971) [2] Balslev, E., Combes, J.M.: Spectral properties of many-body Schr¨ odinger operators with dilation-analytic interactions. Commun. Math. Phys. 22, 280–294 (1971) [3] Chen, H.: Paley–Wiener type theorems and their application in Gevrey classes and ultradistributions. J. Math. Res. Exposition 8(2), 245–254 (1988) ` [4] Cui, S.B.: A generalization of the Paley–Wiener–Schwartz and Eskin theorems. Acta Math. Sci. 12(3), 332–337 (1992) [5] Gevrey, M.: Sur la Nature Analytique des Solutions des Equations aux ´ D´eriv´ees Partielles (Premier M´emoire). Annales scientifiques de l’E.N.S. 3e s´erie, tome 35, 129–190 (1918) [6] Helffer, B., Sj¨ ostrand, J.: Equation de Schr¨ odinger avec champ magn´etique et ´equation de Harper. In: Holden, H., Jensen, A. (eds.) Lecture Notes in Physics, vol. 345. Schr¨ odinger Operators. Springer, Berlin (1989) [7] Hirsch, M.W.: Differential Topology. Graduate Texts in Mathematics, vol. 33. Springer, Berlin (1994) [8] H¨ ormander, L.: The Analysis of Linear Partial Differential Operators I, 2nd edn. Springer Study Edition. Springer, Berlin (1990) [9] H¨ ormander, L.: The Analysis of Linear Partial Differential Operators II. Grundlehren der mathematischen Wissenschaften, vol. 257. Springer, Berlin (1983) [10] Hunziker, W.: Resonances, metastable states and exponential decay laws in perturbation theory. Commun. Math. Phys. 132, 177–188 (1990) [11] Jung, K.: Adiabatic invariance and the regularity of perturbations. Nonlinearity 8, 891–900 (1995) [12] Jung, K.: Adiabatik und Semiklassik bei Regularit¨ at vom Gevrey-Typ. Dissertation (Thesis), Technische Universit¨ at Berlin, Fachbereich Mathematik, D 83, Berlin (1997) [13] Jung, K.: Phase space tunneling for operators with symbols in a Gevrey class. J. Math. Phys. 41(7), 4478–4496 (2000) [14] Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1980) [15] Klein, M., Rama, J., W¨ ust, R.: Time evolution of quantum resonance states. Asymptot. Anal. 51(1), 1–16 (2007) [16] Komatsu, H.: Ultradistributions. I. Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20, 25–105 (1973) [17] Melin, A., Sj¨ ostrand, J.: Fourier integral operators with complex-valued phase functions. Fourier integral operators and partial differential equations. (Colloq. Internat., Univ. Nice, 1974), pp. 120–223. Lecture Notes in Math., vol. 459. Springer, Berlin (1975) [18] Rama, J.: Time evolution of quantum resonance states. Dissertation (Thesis), Technische Universit¨ at Berlin, Fakult¨ at II, Institut f¨ ur Mathematik, D 83, Berlin (2007)
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[19] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. II. Fourier Analysis, Self-Adjointness. Academic Press, New York (1975) [20] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. IV. Analysis of Operators. Academic Press, New York (1978) [21] Rudin, W.: Real and Complex Analysis, International Edition. McGraw-Hill, New York (1987) [22] Rodino, L.: Linear Partial Differential Operators in Gevrey Spaces. World Scientific, Singapore (1993) [23] Simon, B.: Quadratic form techniques and the Balslev–Combes theorem. Commun. Math. Phys. 27, 1–9 (1972) [24] Simon, B.: Resonances and complex scaling: a rigorous overview. Int. J. Quant. Chem. 14, 529–542 (1978) M. Klein and J. Rama Institut f¨ ur Mathematik Universit¨ at Potsdam Am Neuen Palais 10 14415 Potsdam, Germany e-mail:
[email protected] Communicated by Christian Gerard. Received: April 27, 2009. Accepted: February 24, 2010.
Ann. Henri Poincar´e 11 (2010), 539–564 c 2010 Springer Basel AG 1424-0637/10/030539-26 published online June 3, 2010 DOI 10.1007/s00023-010-0037-4
Annales Henri Poincar´ e
A Time-Dependent Perturbative Analysis for a Quantum Particle in a Cloud Chamber Gianfausto Dell’Antonio, Rodolfo Figari and Alessandro Teta Abstract. We consider a simple model of a cloud chamber consisting of a test particle (the α-particle) interacting with two quantum systems (the atoms of the vapor) initially confined around a1 , a2 ∈ R3 . At time zero, the α-particle is described by an outgoing spherical wave centered in the origin and the atoms are in their ground state. We show that, under suitable assumptions on the physical parameters of the system and up to second order in perturbation theory, the probability that both atoms are ionized is negligible unless a2 lies on the line joining the origin with a1 . The work is a fully time-dependent version of the original analysis proposed by Mott in 1929.
1. Introduction The classical limit of quantum mechanics is a widely studied subject in mathematical physics and many detailed results on the asymptotic regime → 0 are available (see e.g., [20] and references therein). In most cases the limit → 0 for the time-dependent Schr¨ odinger equation relative to a given quantum system is studied only for suitable chosen, “almost classical” states, namely WKB or coherent states. Roughly speaking, these results guarantee that if one chooses an almost classical initial state then for → 0 its propagation remains close to a classical propagation at any (not too long) later time. The problem arises when one considers a situation in which the initial state of the system is genuinely non-classical , e.g., a superposition state, and nevertheless the system exhibits, at later times, a classical behavior. Examples of this situation are the localization effect in chiral molecules or the suppression of the interference fringes for a heavy particle in a two-slit experiment. It is clear that the emergence of such classical behavior cannot be understood if one insists to consider the limit → 0 for the isolated quantum system. It is worth mentioning that the problem has some relevance from the conceptual point of view. In fact it was already raised in the earliest debate on the foundation of
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Quantum Mechanics (see e.g., [5]). The accepted explanation in these cases is based on the consideration of the interaction of the quantum system with an environment, and in particular on the decoherence effect produced by the environment. It is important to stress that the decoherence effect must be proved in each situation, starting from specific models of system plus environment and introducing suitable assumptions on the parameters of the model. Many results in this direction have been obtained in the physical literature (see e.g., the reviews [13,17] and references therein) but only few mathematical results are available. Here, we want to focus on a problem of a different kind, raised by Mott [18] in 1929, concerning the explanation of the straight tracks left by an α-particle in a cloud chamber. According to quantum mechanics [8,12] the α-particle, isotropically emitted by a radioactive source, is initially described by a spherical wave function and then interacts with the atoms of the vapor surrounding the radioactive source. The observed tracks are the macroscopic manifestation of ionizations of the atoms induced by the α-particle and “it is a little difficult to picture how it is that an outgoing spherical wave can produce a straight track; we think intuitively that it should ionise atoms at random throughout space” [18]. The explanation proposed by Mott was based on a simple model describing the α-particle in interaction with only two atoms. Exploiting time-independent perturbation arguments, he concluded that the probability that both atoms are ionized is negligible unless the two atoms and the center of the spherical wave lie on the same line. In such rather indirect sense, this would explain why we see straight tracks in the experiments. We will remark on this aspect in Sect. 3. Notice that one typically speaks of tracks due to the decay of an α-particle. A standard theoretical examination of the meaning of tracks left by an α-particle should rely on the reduction postulate where the cloud of vapor is assumed to act as a macroscopic apparatus measuring positions (or better tracks) instantaneously localizing the particle at the place where the ionization takes place. We mention that the problem is also discussed in [15] and later in [2], and some further elaborations on the subject can be found in [4,6,7,9,14,21]. In this paper, we reconsider the three-particle model of a cloud chamber. Under suitable assumptions on the parameters of the model, which will be specified later, we give a proof of Mott’s result through a fully time-dependent analysis and up to second order in perturbation theory. We remark that in our model a crucial assumption is the choice of a semiclassical initial state for the α-particle, i.e., a spherical wave in the short wavelength limit. In this sense our result should be considered as an example of semiclassical analysis in presence of an environment. The method of the proof is rather elementary and it basically relies on stationary and non stationary phase arguments for the estimate of the oscillatory integrals appearing in the perturbative expansion. The work extends to the three-dimensional case the result obtained in [10] for the simpler one-dimensional case, where the spherical wave reduces to the coherent superposition of two wave packets with opposite average momentum.
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2. Description of the Model In this section, we describe the model, i.e., the Hamiltonian, the initial state, the assumptions on the physical parameters, and introduce some notation. Let us first introduce the Hamiltonian. We consider a three-particle non relativistic, spinless quantum system in dimension three, made of a particle with mass M (the α-particle) and two other particles with mass m which play the role of electrons in two model-atoms with fixed nuclei. More precisely we describe such electrons as particles subject to an attractive point interaction placed at fixed positions a1 , a2 ∈ R3 , with a1 = 0, a2 = 0, and a1 = a2 . Moreover we assume that the interaction between the α-particle and each atom is given by a smooth two-body potential V . We denote by R the position coordinate of the α-particle and by r1 , r2 the position coordinates of the two electrons. The Hamiltonian of the system in L2 (R9 ) is formally written as H = H0 + λH1 H0 = K 0 + K 1 + K 2
(2.1) (2.2)
H1 = V (γ −1 (R − r1 )) + V (γ −1 (R − r2 ))
(2.3)
where K0 denotes the free Hamiltonian for the α-particle 2 ΔR , (2.4) 2M odinger operator λ > 0 is a coupling constant and Kj , j = 1, 2, is the Schr¨ in L2 (R3 ) with an attractive point interaction of strength −(4πγ)−1 , γ > 0, placed at aj . We recall that the operator Kj is by definition a non trivial selfadjoint extension of the free Hamiltonian restricted on C0∞ (R3 \{aj }). In the Appendix I, (Sect. 7) we collect some basic facts on this kind of Hamiltonians while for a complete treatment we refer to [1]. Here we only specify the spectrum K0 = −
σp (Kj ) = {E0 },
E0 = −
2 , 2mγ 2
σc (Kj ) = σac (Kj ) = [0, ∞)
(2.5)
and the proper and generalized eigenfunctions, respectively given by 1 e−|x| ζ 0 (x) = √ γ 2π |x| ik·aj 0 −1 φ (γ (r − aj ), γk), φj (r, k) = e e−i|y||x| 1 1 0 iy·x φ (x, y) = − e 1 − i|y| |x| (2π)3/2 ζj (r) =
1
ζ 0 (γ −1 (r − aj )), 3/2
(2.6)
(2.7)
The parameter γ has the physical meaning of a scattering length and it characterizes the effective range of the point interaction. From (2.6) it is also clear that γ is a measure of the linear spread of the ground state, i.e., of the atoms. The unperturbed Hamiltonian H0 is obviously selfadjoint and bounded from below in L2 (R9 ) and moreover the smoothness assumption on the interaction potential V (see Theorems 1, 2) guarantees that the perturbed Hamiltonian H is also selfadjoint and bounded from below on the same domain
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of H0 (see e.g., [19]). In particular this implies that the evolution problem associated with the Hamiltonian H is well posed. We choose the initial state in the product form Ψ0 (R, r1 , r2 ) = ψ(R)ζ1 (r1 )ζ2 (r2 )
(2.8)
where ζj is defined in (2.6) and ψ(R) is a spherical wave defined as follows. Let us consider a gaussian wave packet localized in space around the origin, with standard deviation γ and mean momentum P0 > 0 along the direction u ˆ ∈ S 2 . Integrating over the unit sphere S 2 , one obtains |x|2 i R Nε ψ(R) = 3/2 f (γ −1 R) dˆ u e ε uˆ· γ , f (x) = e− 2 (2.9) εγ S2
where ε > 0 is the dimensionless parameter P0 γ
ε≡
(2.10)
and Nε is the normalization factor chosen to guarantee that ψL2 (R3 ) = 1, i.e., Nε =
1 1
4π 7/4 1 − e− ε2
1/2
(2.11)
Notice that Nε → (4π 7/4 )−1 for ε → 0. By an elementary integration we also get −
R2
4πNε e 2γ 2 sin(ε−1 γ −1 |R|) ψ(R) = 1/2 |R| γ
(2.12)
We remark that the characteristic length γ appears in the definition of the Hamiltonian as well as in the initial state in such a way that the range of the interaction between the α-particle and the atoms, the linear dimension of the atoms and the localization in space of the spherical wave are all of order γ. As it will become clear further on, this is a crucial ingredient for the proof of our result. We also notice that the initial state (2.8) coincides with the one considered by Mott. Namely the α-particle is emitted as an outgoing spherical wave and the atoms are in their ground state. Let us describe the hypotheses on the physical parameters of the model. We assume ε 1
(2.13)
Moreover there are two positive constants c < C < ∞, independent of ε, such that γ < Cε, j = 1, 2 (2.14) cε < |aj | m < Cε (2.15) cε < M
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Condition (2.13) means that the wavelength P0−1 associated to the initial state of the α-particle is much smaller than the linear dimension of the atoms and the range of the interaction, which means that we are in a semi-classical regime for the α-particle. In (2.14) we assume that |a1 |, |a2 | are macroscopic distances with respect to the characteristic length γ and in (2.15) we require that the mass ratio is small. Finally we tacitly assume λ < Cε M v02
cε < λ0 ≡
(2.16)
where v0 = P0 M −1 . Condition (2.16) is necessary in order to make reasonable the application of our perturbative techniques, even if it is not strictly required for the proof of our results. The above assumptions (2.13), (2.14), (2.15) have some relevant physical implications. In particular from (2.13), (2.15) one sees that the binding energy of the atoms is small compared to the kinetic energy of the α-particle M 2|E0 | = 2 M v0 m
P0 γ
2 ≡
M 2 ε m
(2.17)
Furthermore the assumptions (2.13) and (2.14) imply two relations among the characteristic times of the system which will be relevant in what follows. In particular, we define the flight times to the atoms of the α-particle τj =
|aj | , v0
j = 1, 2
(2.18)
the characteristic “period” of the atoms Ta = 2π
mγ 2 = 4π |E0 |
(2.19)
and the transit time of the α-particle in the region where the atom are localized Tt =
γ v0
(2.20)
Then one has Tt γ , j = 1, 2 = τj |aj | Tt 1 M 1 M ≡ ε = Ta 4π m P0 γ 4π m
(2.21) (2.22)
i.e., the transit time Tt is small with respect to the flight times τj but it is comparable with the characteristic period of the atoms Ta . This means that the α-particle can “see” the internal structure of the atoms.
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Let us introduce some notation to streamline the presentation. a ˆj =
aj , |aj |
j = 1, 2
(2.23)
1 (1 + y 2 ), y ∈ R3 2 t τj 2 t a= , b = , c = ω(yj ), yj ∈ R3 , j = 1, 2 j j M γ2 M γ2 P0 γmγ 2 1 dx e−iξ·x φ0 (x, y)ζ 0 (x), ξ, y ∈ R3 h(ξ, y) = (2π)3/2 g(ξ, y) = V (ξ)h(ξ, y) ω(y) =
(2.24) (2.25) (2.26) (2.27)
where F˜ (q) =
1 (2π)3/2
dx e−iq·x F (x)
(2.28)
denotes the Fourier transform of F . Moreover, for n, m ∈ N, we denote 1,n = uWm <x>m Dα uL1 (R3 ) (2.29) 0≤|α|≤n
where <x>2 = 1 + x2 , x = (x1 , x2 , x3 ) ∈ R3 , α = (α1 , α2 , α3 ) ∈ N3 , |α| = α1 + α2 + α3 and Dα u =
∂ α1 ∂ α2 ∂ α3 α2 α3 u 1 ∂xα 1 ∂x2 ∂x3
(2.30)
Finally Ck denotes a positive numerical constant, depending on k ∈ N and, possibly, on the dimensionless parameters (2.25). It is important to notice that, for any fixed t > τ2 and yj ∈ R3 , the quantities a, bj , cj are all of order one. In fact it is sufficient to notice that a=
|a2 | t , P0 γ γ τ2
bj =
|aj | , P0 γ γ
cj =
M m
P0 γ
2
|a2 | t ω(yj ) γ τ2
(2.31)
and to use (2.13), (2.14), and (2.15).
3. Result We are now in a position to formulate our result. We are interested in the computation of the probability that both atoms are ionized at time t > 0. An exact computation obviously requires the complete knowledge of the state Ψ(t) of the system, which is not available. Following the original strategy of Mott we shall limit to consider the second order approximation Ψ2 (t) of the
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state Ψ(t) which, iterating twice Duhamel’s formula, is given by i
ˆ 2 (t) Ψ2 (t) = e− tH0 Ψ t i i λ ˆ Ψ2 (t) = Ψ0 − i dt1 e t1 H0 H1 e− t1 H0 Ψ0
(3.1)
0
λ2 − 2
t dt1 e
i t1 H0
− i t1 H0
t1
H1 e
0
i
i
dt2 e t2 H0 H1 e− t2 H0 Ψ0 (3.2)
0
Therefore we shall study the probability that both atoms are ionized up to second order in perturbation theory, i.e. 2 ˆ 2 (R, r1 , r2 , t) (3.3) P(t) = dRdk1 dk2 dr1 dr2 φ1 (r1 , k1 )φ2 (r2 , k2 )Ψ Our main result is the characterization of the ionization probability P(t) for a fixed time t > τ2 and it is summarized in Theorems 1 and 2 below. In Theorem 1 we consider the case in which a2 is not aligned with a1 and the origin and we show that the ionization probability decays faster than any power of ε. ˆ1 · a ˆ2 < 1 and let us assume (2.13), Theorem 1. Let us fix t > τ2 , |a1 | < |a2 |, a (2.14), and (2.15), V ∈ S(R3 ). Then for any k ∈ N there exists Ck > 0 such that
4 −2k λt |a1 | −k 2 4 P(t) ≤ Nε 2Ck V W 1,k + (1 − a ˆ1 · a ˆ2 ) 1− ε2k−2 k |a2 | (3.4) We notice that the estimate (3.4) is still meaningful (for some values of k) if the angle between a1 and a2 is of order εp with 0 < p < 1, while it gives no information for p ≥ 1. In the latter case the second atom lies inside a small cone of aperture proportional to ε, apex in the position a1 of the first atom and axis on the line joining the origin and a1 . This situation is considered in Theorem 2 where we compute the leading term of the asymptotic expansion for ε → 0 of the ionization probability. We explicitly mention that we do not consider some intermediate situations, like the case of an angle proportional to ε| log ε|. ˆ1 · a ˆ2 = cos χε , where χε ∈ Theorem 2. Let us fix t > τ2 , |a1 | < |a2 |, a [0, χ0 ε], χ0 > 0, and let us assume (2.13), (2.14), and (2.15), V ∈ S(R3 ). Then, at the leading order for ε → 0, we have 2 4 2 −1 4 −1 γ 2 P(t) ∼ ε (ε λ0 ) ε Nε dxdydz dη1 dη2 F (η1 , η2 ; x, y, z) |a1 | R9
R2
(3.5) where the function F is independent of ε and will be specified during the proof.
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In the following remarks we briefly comment on the above results. Remark 3.1. The estimate (3.4) is valid for t larger but of the same order of τ2 , while it loses its meaning for t → ∞. This is only due to the method we use for the proof, based on second order perturbation theory. A non-perturbative approach or a more detailed perturbative analysis should provide an estimate which is uniform in time. We also remark that in Theorem 2 we limit ourselves to the computation of the leading term, without making any attempt to estimate the remainder. Such leading term is small for ε → 0 being proportional, as expected, to the solid angle that the atoms subtend at the origin. It would be interesting to extend the results given here at any order in perturbation theory and in particular to verify that the leading term has the same behavior for ε → 0 at all orders. Remark 3.2. We recall that the normalization factor Nε goes to (4π 7/4 )−1 for ε → 0 (see (2.11)). Moreover from (2.10), (2.14), and (2.16) one has that 4 4 λ0 |a2 | t λt = (3.6) ε γ τ2 is proportional to ε−4 for any t larger but of the same order of τ2 . This means that it is sufficient to take any integer k > 4 in (3.4) to conclude that the ionization probability estimated in Theorem 1 is much smaller than the one computed in Theorem 2. We underline this point since it allows to understand the results in Theorems 1, and 2 on the basis of the original physical argument given by Mott, which can be described as follows. At time zero the spherical wave starts to propagate in the chamber and at time τ1 it interacts with the atom in a1 . If, as result of the interaction, such atom is ionized then a localized wave packet emerges from a1 with momentum along the direction Oa1 . In order to obtain also ionization of the atom in a2 the localized wave packet must hit the atom in a2 (at time τ2 ) and this can happen only if a2 approximately lies on the line Oa1 . It should be stressed that such physical behavior is far from being universal and it strongly depends on our assumptions on the physical parameters of the model. Remark 3.3. Finally we observe that our result states that one can only observe straight tracks in a cloud chamber. With this we do not mean that there is any focusing of the support of the wave packet of the α-particle along a classical straight trajectory, corresponding to the observed track. In fact the solution of the Schr¨ odinger equation with Hamiltonian (2.1) and initial datum (2.8) has the form Ψ(R, r1 , r2 , t) = F00 (R, t)ζ1 (r1 )ζ2 (r2 ) + dk1 Fc0 (R, k1 , t)φ1 (r1 , k1 ) · ζ2 (r2 ) + dk2 F0c (R, k2 , t)φ2 (r2 , k2 ) · ζ1 (r1 ) + dk1 dk2 Fcc (R, k1 , k2 , t)φ1 (r1 , k1 )φ2 (r2 , k2 ) (3.7)
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where the four probability amplitudes F00 , Fc0 , F0c , Fcc are localized in different regions of the configuration space of the whole system and therefore describe not interfering “quantum histories”. If one interprets double ionization as the only case of macroscopic ionization, giving then rise to an observable track, one can say, in a pictorial language, that is along the track the expected value of the position of the α particle in the states in which the track (as an observable) has expectation one. Let us outline the strategy of the proof of Theorems 1 and 2. The starting point is the following, more convenient, representation formula for the ionization probability λ4 t4 Nε2 2 ε ε (x, y1 , y2 , t) + G21 (x, y1 , y2 , t)| (3.8) dxdy1 dy2 |G12 P(t) = 4 2 ε where for l, j = 1, 2, j = l one has ε Glj (x, y1 , y2 , t)
=
1 dˆ u
S2
0
α dα
dβ
i
dηdξGlj (α, β, η, ξ; x, y1 , y2 , t)e ε Θlj (ˆu,α,β,η,ξ;x,y1 ,y2 ,t)
0
(3.9) and dropping the parametric dependence on x, y1 , y2 , t Θlj (ˆ u, α, β, η, ξ) = u ˆ · (x + a(αη + βξ)) − bj a ˆ j · η − bl a ˆ l · ξ + cj α + cl β (3.10) Glj (α, β, η, ξ) = g(η, yj )g(ξ, yl )f (x + a(αη + βξ))eiφ(α,β,η,ξ) a φ(α, β, η, ξ) = x · (η + ξ) + (αη 2 + βξ 2 + 2αη · ξ) 2
(3.11) (3.12)
In (3.8) and (3.9), we have denoted by x, y1 , y2 the rescaled position of the α-particle and the rescaled momenta of the electrons respectively, while α, β play the role of rescaled time variables. Moreover we recall that f is the gaussian defined in (2.9). The proof of (3.8) is a long but straightforward computation and it is postponed to the Appendix II (Section 8). Due to formula (3.8), we are reduced to the analysis of the two oscillatory ε corresponding to the possible graphs in the second order perturintegrals Glj ε bative expansion. In particular G12 describes the graph in which the atom in ε the opposite case. Since we always a1 is ionized before the atom in a2 and G21 ε is negligible. In assume |a1 | < |a2 |, we expect that the contribution of G21 fact, in Sect. 4 we shall see that the phase Θ21 has no critical points and then, by standard integration by parts, we shall prove that the contribution of the ε is bounded by a constant times εk , for any k ∈ N. oscillatory integral G21 ε is more delicate and we have to distinguish The estimate of the term G12 the non aligned and the aligned case. It turns out that the phase Θ12 has no ε is bounded by critical points in the first case and then the contribution of G12
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a constant times εk , for any k ∈ N. This will be proved in Sect. 5, concluding also the proof of Theorem 1. In Sect. 6, we consider the aligned case, where the phase Θ12 has a manifold of critical points parametrized by a vector in R2 . By a careful applicaε , we compute the leading term of tion of the stationary phase method to G12 the asymptotic expansion for ε → 0 and then we also conclude the proof of theorem 2. ε 4. Estimate of G21
In this section, we shall prove that the the contribution of the oscillatory inteε is negligible for any orientation of the unit vectors a ˆ1 , a ˆ2 . gral G21 Proposition 4.1. Let us fix t > τ2 , |a1 | < |a2 | and let us assume (2.13), (2.14), (2.15), V ∈ S(R3 ). Then for any k ∈ N there exists Ck > 0 such that −2k |a1 | 2 ε (x, y1 , y2 )| ≤ Ck V 4W 1,k 1 − ε2k (4.1) dxdy1 dy2 |G21 k |a2 | Proof. The crucial point is that the gradient of the phase Θ21 = u ˆ · (x + a(αη + βξ)) − b1 a ˆ 1 · η − b2 a ˆ 2 · ξ + c1 α + c2 β
(4.2)
does not vanish in the integration region. To see this it is sufficient to compute
2 2 3 ∂Θ21 ∂Θ21 + ˆ1 )2 + (aβ u ˆ − b2 a ˆ2 )2 = (aαˆ u − b1 a ∂ηk ∂ξk k=1
≥ (aα − b1 )2 + (aβ − b2 )2
τ 1 2 τ 2 2 ≡ a2 α − + β− t t
(4.3)
In the region {(α, β) ∈ R2 |0 ≤ α ≤ 1, 0 ≤ β ≤ α} the r.h.s. of (4.3) takes its +τ2 , then minimum in (α0 , β0 ), with α0 = β0 = τ12t
2 2 3 ∂Θ21 ∂Θ21 + (4.4) ≥ Δ221 ∂ηk ∂ξk k=1
where Δ21
|a2 | =√ (τ2 − τ1 ) ≡ √ 2 2M γ 2P0 γ γ
|a1 | 1− |a2 |
(4.5)
Notice that, under the assumptions (2.13), (2.14), (2.15) and |a1 | < |a2 |, there exists a positive constant c0 , independent of ε, such that Δ21 > c0 for any ε > 0. The estimate (4.4) allows to control G21 using standard non stationary phase methods [3,11,16]. In fact, recalling the identity ∇b ∇b ib a + ie div a (4.6) aeib = −idiv eib |∇b|2 |∇b|2
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and performing k integration by parts we have i i Θ21 k ε = (iε) dηdξ(Lk G21 )e ε Θ21 dηdξG21 e
(4.7)
where the operator L acts on the variables ζ = (ζ1 , . . . , ζ6 ) ≡ (η1 , η2 , η3 , ξ1 , ξ2 , ξ3 ) as follows LG21 =
6
uj
j=1
∂G21 , ∂ζj
∂Θ21 1 |∇ζ Θ21 |2 ∂ζj
(4.8)
uj1 , . . . , ujk Dζkj1 ,...,ζj G21
(4.9)
uj =
and moreover 6
Lk G21 =
j1 ,...,jk =1
k
In (4.9) we have denoted by Dζkj ,...,ζj the derivative of order k with respect 1 k to ζj1 , . . . , ζjk . From (4.9), (4.8), (4.4), (4.5) we easily get the estimate ε |G21 |
≤ε
1
k
u S 2 dˆ
dα 0
≤ 4π
k
ε Δk21
α dβ 0
1
α dα
0
dβ 0
dηdξ Lk G21
6 dηdξ Dζkj1 ,...,ζj G21 k j1 ,...,jk =1
(4.10)
If we square (4.10), integrate w.r.t. the variables x, y1 , y2 and use Schwartz inequality we find ε 2 | dxdy1 dy2 |G21
≤ 4π 2
ε2k Δ2k 21
⎡ ⎤ ⎛ 2 ⎞1/2 2 6 ⎟ ⎥ ⎢ ⎜ k ⎠ ⎥ sup ⎢ dy D G dηdξ dxdy ⎝ 1 2 21 ,...,ζ ζ j j ⎣ ⎦ 1 k α,β j1 ,...,jk =1 (4.11)
From the definition of G21 (see (3.11)), we have 6 j1 ,...,jk =1
≤ Ck
|Dζkj1 ,...,ζj G21 | k
k i1 =1
·
k i3 =1
|Dηi1 g(η, y1 )|
k i2 =1
|Dξi2 g(ξ, y2 )|
k i Dx3 f (x + a(αη + βξ)) (|x| + a|η| + a|ξ|)i4 i4 =1
(4.12)
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The last term in (4.12) can be easily estimated as follows k
(|x| + a|η| + a|ξ|)i4
i4 =1
≤
k
(|x + a(αη + βξ)| + 2a|η| + 2a|ξ|)i4
i4 =1
≤
k √
i4 24a2 <x + a(αη + βξ)><η><ξ>
i4 =1
≤
k
a2i4 25i4 /2
<x + a(αη + βξ)>k <η>k <ξ>k
(4.13)
i4 =1
where <x>2 = 1 + x2 , x ∈ R3 . Hence 6 j1 ,...,jk =1
·
k i2 =1
k i k Dη1 g(η, y1 ) <ξ>k Dζj1 ,...,ζjk G21 ≤ Ck <η>k i1 =1
k i i2 Dx3 f (x + a(αη + βξ)) (4.14) Dξ g(ξ, y2 ) <x + a(αη + βξ)>k i3 =1
Moreover, recalling the definition (2.27), we get 6 j1 ,...,jk =1
k k k i1 i2 k k <ξ> ≤ C V (η) G <η> Dζj1 ,...,ζjk 21 Dη Dξ V (ξ) k i1 =1
i2 =1
· <x + a(αη + βξ)>k
k
i Dx3 f (x + a(αη + βξ))
i3 =1
×
k
i Dη4 h(η, y1 )
i4 =1
k i5 =1
i5 Dξ h(ξ, y2 )
Using (4.15) in estimate (4.11) we find ε 2 dxdy1 dy2 |G21 | k ε2k i1 ≤ 2k Ck dη<η>k Dη V (η) dξ<ξ>k Δ21 i1 =1 ⎡ 2 k k i2 i3 ⎣ Dη h(η, y1 ) dy1 · |Dξ V (ξ)| i2 =1
i3 =1
×
dy2
k
i4 =1
i4 Dξ h(ξ, y2 )
⎫ 2 ⎤1/2 ⎪2 ⎬ ⎦ ⎪ ⎭
(4.15)
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⎡ 2 ⎤2 k ε2k ≤ 2k Ck V 4W 1,k ⎣sup dy |Dηm h(η, y)| ⎦ k Δ21 η m=1
k 1/2 4 ε2k 2 ≤ 2k Ck V 4W 1,k sup dy Dηm h(η, y) k Δ21 η m=1
(4.16)
It remains to show that the last term in the r.h.s. of (4.16) is finite. From the definition (2.26) we have (−i)m 1 m2 m3 0 (4.17) dxe−iη·x xm Dηm h(η, y) = 1 x2 x3 ζ0 (x)φ (x, y) (2π)3/2 where m1 + m2 + m3 = m. The integral kernel φ0 (x, y) defines a bounded operator in L2 (R3 ), with norm less or equal to one. This fact directly follows from (7.9), in Appendix I. Hence m 2 1 2 1 m2 m3 dy Dη h(η, y) ≤ (4.18) dx |xm 1 x2 x3 ζ0 (x)| < ∞ (2π)3 Taking into account inequality (4.18)) in (4.16), we conclude the proof.
ε 5. Estimate of G12 in the Case a ˆ1 · a ˆ2 < 1
Following the same line of the previous section, we prove that the contribuε to the ionization probability is negligible provided that the two tion of G12 ˆ2 are not parallel. Of course, the estimate shall crucially unit vectors a ˆ1 and a depend on the angle between the two unit vectors. Proposition 5.1. Let us fix t > τ2 , a ˆ1 · a ˆ2 < 1 and let us assume (2.13), (2.14), (2.15), V ∈ S(R3 ). Then for any k ∈ N there exists a strictly positive constant Ck such that 2 −k ε (x, y1 , y2 )| ≤ Ck V 4W 1,k (1 − a ˆ1 · a ˆ2 ) ε2k (5.1) dxdy1 dy2 |G12 k
Proof. As in the case of Proposition 4.1, we consider the phase Θ12 = u ˆ · (x + a(αη + βξ)) − b2 a ˆ 2 · η − b1 a ˆ 1 · ξ + c2 α + c1 β
(5.2)
and we show that its gradient is strictly different from zero in the integration region. In fact
2 2 3 ∂Θ12 ∂Θ12 + ˆ2 )2 + (aβ u ˆ − b1 a ˆ1 )2 (5.3) = (aαˆ u − b2 a ∂ηk ∂ξk k=1
The r.h.s. of (5.3), considered as a function of the variables (α ≥ 0, β ≥ 0), b2 b1 ˆ·a ˆ2 , u ˆ·a ˆ1 ) when u takes its minimum in (α1 , β1 ) = ( u ˆ·a ˆ2 ≥ 0, u ˆ·a ˆ1 ≥ 0 a a and in α = 0 and/or β = 0 when u ˆ·a ˆ2 < 0, u ˆ·a ˆ1 < 0, respectively. Hence if
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u ˆ·a ˆi < 0 for at least one i then the r.h.s. of (5.3) is larger than min{b21 , b22 }. ˆ·a ˆ2 ≥ 0. We have Let us consider the case u ˆ·a ˆ1 ≥ 0, u
2 2 3 ∂Θ12 ∂Θ12 + u·a ˆ2 u ˆ−a ˆ2 )2 + b21 (ˆ u·a ˆ1 u ˆ−a ˆ1 )2 ≥ b22 (ˆ ∂ηk ∂ξk k=1 ⎧ % $ 1 1 ⎪ if u ˆ·a ˆ2 ≥ √ , u ˆ·a ˆ1 ≥ √ u·a ˆ2 )2 − (ˆ u·a ˆ1 )2 ⎨ ≥ min{b21 , b22 } 2 − (ˆ 2 2 1 1 ⎪ ⎩ ≥ min{b21 , b22 } √ if u ˆ·a ˆi < for at least one i 2 2 (5.4) ˆ·a ˆi . From the convexity of cos2 x for x ∈ [0, π/4] Let us denote θi = arccos u it follows that 1 1 θ1 + θ2 1 1 1 cos2 θ1 + cos2 θ2 ≤ cos2 = + cos(θ1 + θ2 ) ≤ (1 + a ˆ1 · a ˆ2 ) 2 2 2 2 2 2 (5.5) 1 1 ˆ·a ˆ1 ≥ √ we have Then for u ˆ·a ˆ2 ≥ √ , u 2 2 u·a ˆ1 )2 ≥ 1 − a ˆ1 · a ˆ2 2 − (ˆ u·a ˆ2 )2 − (ˆ
(5.6)
which implies
2 2 3 ∂Θ12 ∂Θ12 + ≥ Δ212 ∂ηk ∂ξk
(5.7)
k=1
where & 1 min{|a1 |, |a2 |} & Δ12 = √ min{b1 , b2 } 1 − a ˆ1 · a ˆ2 = √ 1−a ˆ1 · a ˆ2 γ 2 2P0 γ (5.8) ˆ2 < 1, We notice that, under the assumptions (2.13), (2.14), (2.15) and a ˆ1 · a the square modulus of the phase gradient remains strictly larger than zero. From now on the proof proceeds exactly in the same way as in the previous Proposition 4.1 and we omit the details. Proof of Theorem 1. From (3.8) and taking into account Propositions 4.1, 5.1 we immediately get the proof. Remark 5.1. The same kind of estimate proved in Proposition 5.1 is also valid ε which means that Theorem 1 could be proved without the assumption for G21 |a1 | < |a2 |. On the other hand the estimate given in Proposition 4.1 is crucial for the analysis of the stationary case of the next section.
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6. The Stationary Case Here, we consider the case a ˆ1 · a ˆ2 = 1−O(εq ), with q ≥ 2. We shall see that the ε has stationary points when exactly this phase of the oscillatory integral G12 case occurs. This implies that the ionization probability P(t) is not negligible for ε small as in the previous situation and the leading term of its asymptotic expansion in powers of ε can be computed. ˆ2 as follows Throughout this section we shall fix the unit vectors a ˆ1 , a a ˆ1 = (0, 0, 1),
a ˆ2 = (sin χε , 0, cos χε )
(6.1)
where χε ∈ [0, χ0 ε], χ0 > 0. Moreover, in order to characterize the asymptotic ε , we introduce a convenient decomposition of the unit sphere S 2 . behavior of G12 More precisely we define Γθ¯ as the portion of S 2 inside a cone with apex in the ¯ 0 < θ¯ < π , and we denote Γ ¯ θ¯ = S 2 \Γθ¯. origin, axis parallel to a ˆ1 , aperture θ, 2 ε ˆ2 as in (6.1) the corresponding decomposition of G12 in a For any choice of a ˆ1 , a non-stationary part (denoted with the label n) and a stationary part (denoted with the label s) is ε,n ε,s ε = G12 + G12 G12 1 α i ε,n = dˆ u dα dβ dηdξGε12 e ε Θ G12 ¯¯ Γ θ
0
ε,s G12 = Γθ¯
α dα
0
(6.3)
0
1 dˆ u
(6.2)
dβ
i
dηdξGε12 e ε Θ
(6.4)
0
where Gε12 = G12 eiδε sin χε 1 − cos χε δε = − b2 η1 + b2 η3 ε ε Θ=u ˆ · (x + a(αη + βξ)) − b1 ξ3 − b2 η3 + c2 α + c1 β
(6.5) (6.6) (6.7)
ε,n We shall analyze the asymptotic behavior of the two oscillatory integrals G12 ε,s and G12 separately. We first show that the phase Θ has no stationary points ¯ θ¯ and then the contribution of G ε,n is negligible. in Γ 12
Proposition 6.1. Let us fix t > τ2 , a ˆ1 , a ˆ2 as in (6.1) and let us assume (2.13), (2.14), (2.15), V ∈ S(R3 ). Then for any k ∈ N there exists Ck > 0 such that ε 2k 2 ε,n (x, y1 , y2 )| ≤ Ck V 4W 1,k (6.8) dxdy1 dy2 |G12 k sin θ¯ ¯ θ¯, we have Proof. If we denote u ˆ = (sin θ cos φ, sin θ sin φ, cos θ) ∈ Γ
2 2 3 ∂Θ ∂Θ + = a2 (α2 + β 2 ) + b21 + b22 − 2a(b1 β + b2 α) cos θ ∂ηk ∂ξk k=1
≥ a2 (α2 + β 2 ) + b21 + b22 −2a(b1 β + b2 α) cos θ¯
(6.9)
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¯ b1 cos θ), ¯ then The r.h.s. of (6.9) takes its minimum in (α2 , β2 ) = ( ba2 cos θ, a
2 2 3 ∂Θ ∂Θ + (6.10) ≥ Δ2 ∂ηk ∂ξk k=1
where Δ=
√ √ min{|a1 |, |a2 |} sin θ¯ 2 min{b1 , b2 } sin θ¯ = 2 P0 γ γ
(6.11)
Exploiting estimate (6.10), (6.11) it is now straightforward to obtain (6.8) proceeding exactly as in the proof of Proposition 4.1. Remark 6.1. We notice that the estimate (6.8) is still meaningful if we choose the angle θ¯ proportional to εd , with 0 < d < 1. This in particular means that only a small fraction of the unit sphere, of area proportional to ε2 around the direction a ˆ1 , can give a non trivial contribution to the ionization probability. ε,s Let us consider the oscillatory integral G12 . It turns out that the phase Θ has a manifold of critical points in the integration region, parametrized by a vector in R2 . Therefore we fix the variables (η1 , η2 ) ∈ R2 as parameters and ε,s in the form we write G12 ε,s G12 = dη1 dη2 I ε (η1 , η2 ) (6.12) i I ε (η1 , η2 ) = dqGε12 (q; η1 , η2 )e ε Θ(q;η1 ,η2 ) (6.13) Ω
where ( ' Ω = q ≡ (ˆ u, α, β, η3 , ξ) |ˆ u ∈ Γθ¯, α ∈ [0, 1], β ∈ [0, α], η3 ∈ R, ξ ∈ R3
(6.14)
In the next lemma we show that for each value of the parameters (η1 , η2 ) the phase in (6.13) has one, non degenerate stationary point. It is relevant that the value of the phase and of the Hessian of the phase at the critical point do not depend on (η1 , η2 ). Lemma 6.2. For each (η1 , η2 ) ∈ R2 the phase Θ(q; η1 , η2 ), q ∈ Ω, has exactly one critical point ) 0 0 0 0 0 0 0* (6.15) ˆ , α , β , η3 , ξ1 , ξ2 , ξ3 q0 ≡ u where u ˆ0 = (0, 0, 1), ξ10 = −
b1 c2 , η30 = − , a a + b η x c1 2 2 2 ξ20 = − , ξ30 = − b1 a
α0 =
x1 + b2 η1 , b1
b2 , a
β0 =
(6.16) (6.17)
and x = (x1 , x2 , x3 ) ∈ R3 . Moreover b2 c 2 b1 c 1 + a a |D2 Θ0 | ≡ |Dq2 Θ(q0 ; η1 , η2 )| = a4 b41 Θ0 ≡ Θ(q0 ; η1 , η2 ) = x3 +
(6.18) (6.19)
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Proof. In order to compute the critical points & of the phase (6.7) as a function of q ∈ Ω it is convenient to write u ˆ = (μ, ν, 1 − μ2 − ν 2 ), where (μ, ν) ∈ R2 ¯ Therefore with μ2 + ν 2 ≤ sin2 θ. & Θ(q; η1 , η2 ) = μw1 + νw2 + 1 − μ2 − ν 2 w3 − b2 η3 − b1 ξ3 + c2 α + c1 β (6.20) where we have introduced the short hand notation w = (w1 , w2 , w3 ),
wj = xj + a(αηj + βξj )
(6.21)
By an explicit computation, one finds that the critical points are solutions of the system ∂Θ μw3 = w1 − & =0 ∂μ 1 − μ2 − ν 2 ∂Θ νw3 = w2 − & =0 ∂ν 1 − μ2 − ν 2 & ∂Θ = aμη1 + aνη2 + a 1 − μ2 − ν 2 η3 + c2 = 0 ∂α & ∂Θ = aμξ1 + aνξ2 + a 1 − μ2 − ν 2 ξ3 + c1 = 0 ∂β & ∂Θ = a 1 − μ2 − ν 2 α − b2 = 0 ∂η3 ∂Θ = aμβ = 0 ∂ξ1 ∂Θ = aνβ = 0 ∂ξ2 & ∂Θ = a 1 − μ2 − ν 2 β − b1 = 0 ∂ξ3
(6.22) (6.23) (6.24) (6.25) (6.26) (6.27) (6.28) (6.29)
First we notice that α and β cannot be zero, otherwise from (6.26), (6.29) one would have b2 = b1 = 0. Then from (6.27), (6.28) we have μ = ν = 0 and from (6.26), (6.27) we have α = ba2 , β = ba1 . Exploiting the remaining equations it is now trivial to find the unique solution (6.16), (6.17). Furthermore the value of the phase at the critical point (6.18) is easily obtained. For the proof of (6.19) we need the second derivatives of the phase evaluated at the critical point ∂2Θ ∂μ2 ∂2Θ ∂μ∂ξ1 ∂2Θ ∂ν 2 ∂2Θ ∂ν∂ξ1
∂2Θ ∂2Θ ∂2Θ = 0, = aη1 , = aξ1 , ∂μ∂ν ∂μ∂α ∂μ∂β ∂2Θ ∂2Θ = b1 , = 0, =0 ∂μ∂ξ2 ∂μ∂ξ3 ∂2Θ ∂2Θ ∂2Θ = aη2 , = aξ2 , = −w3 , =0 ∂ν∂α ∂ν∂β ∂ν∂η3 ∂2Θ ∂2Θ = 0, = b1 , =0 ∂ν∂ξ2 ∂ν∂ξ3 = −w3 ,
∂2Θ =0 ∂μ∂η3
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∂2Θ = 0, ∂α2 ∂2Θ = 0, ∂β 2 ∂2Θ = 0, ∂η32
∂2Θ ∂2Θ ∂2Θ = 0, = a, =0 ∂α∂β ∂α∂η3 ∂α∂ξj ∂2Θ ∂2Θ ∂2Θ = 0, = 0, = 0, ∂β∂η3 ∂β∂ξ1 ∂β∂ξ2 ∂2Θ ∂2Θ = 0, =0 ∂η3 ∂ξj ∂ξj ∂ξk
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∂2Θ =a ∂β∂ξ3 (6.30)
The computation of the Hessian is now a tedious but straightforward exercise and it is omitted for the sake of brevity. We are now ready to conclude the proof of theorem 2. Proof of Theorem 2. Exploiting the stationary phase theorem [3,11,16] and the previous lemma, we find the leading term of the asymptotic expansion of (6.13) for ε → 0 I ε (η1 , η2 ) ∼
π (2πε)4 i Θ0 e ε G12 (q0 ; η1 , η2 )eiδ0 ei 4 μ0 a2 b21
(6.31)
where μ0 denotes the signature of the Hessian matrix at the critical point and moreover sin χε b2 η1 (6.32) δ0 = − lim ε→0 ε In particular δ0 = 0 if χε = O(ε) and δ0 = 0 if χε = O(εq ), q > 1, or χε = 0. From (6.12) we also obtain π (2πε)4 i Θ02 ε,s ε (6.33) dη1 dη2 G12 (q0 ; η1 , η2 )eiδ0 ei 4 μ0 G12 ∼ 2 2 e a b1 We notice that the integrand in (6.33) is a function of x (position of the α-particle), yj (momentum of the jth ionized atom) and η1 , η2 . Hence we denote π
F (η1 , η2 ; x, y1 , y2 ) ≡ (2π)4 G12 (q0 ; η1 , η2 )eiδ0 ei 4 μ0
(6.34)
and, taking into account (3.8), Proposition 4.1, (6.2), Lemma 6.2 and (6.33), we find 2 4 λt Nε2 6 dη1 dη2 F (η1 , η2 ; x, y, z) P(t) ∼ ε (6.35) dxdydz 4 4 a b1 Using the definition of a, b1 , τ1 in (6.35), we easily get formula (3.5). It remains to show that the integral in (6.35) is finite. From the definitions (3.11), (2.27), (2.26) and the boundedness of h(ξ, y) we have ) * ) * (6.36) |F (η1 , η2 ; x, y, z)| ≤ c V η1 , η2 , η30 V ξ10 , ξ20 , ξ30 |f (w0 )| where γ γ M τ2 τ2 ω(z), ξ10 = −ε−1 x1 − η1 , ξ20 = −ε−1 x2 − η2 , m |a1 | τ1 |a1 | τ1 M |a1 | M |a2 | M ω(y) − ε2 ω(z) (6.37) ξ30 = −ε ω(y), w0 = x3 − ε2 m m γ m γ
η30 = −ε
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We recall that f and ω(y) are defined in (2.9), (2.24), respectively, and x = (x1 , x2 , x3 ) ∈ R3 . Then we write 2 dxdydz dη1 dη2 F (η1 , η2 ; x, y, z)
≤c
dx1 dx2 dydz
* 2 ) dx3 |f (w0 )|2 dη1 dη2 |V (η1 , η2 , η30 )||V ξ10 , ξ20 , ξ30 |
(6.38) Using the Schwartz inequality in the integral with respect to (η1 , η2 ) we have 2 dxdydz dη1 dη2 F (η1 , η2 ; x, y, z) ≤ c dz dη1 dη2 <η1 >4 <η2 >4 ) 0 0 0 * 2 0 2 ×|V (η1 , η2 , η3 )| (6.39) dy dx1 dx2 V ξ1 , ξ2 , ξ3 dx3 |f (w0 )|2 The last integral is finite due to the assumptions on V and this concludes the proof of the theorem. As we already pointed out in Remark 3.1, an explicit estimate of the remainder in the expansion of (6.13) goes beyond the scope of this paper. We notice that a complete asymptotic expansion, with an estimate of the rest, for ε → 0 of I ε (η1 , η2 ) can be obtained following the general analysis of [11] where, in Theorem 2.4, the case of oscillatory integrals depending on several parameters is discussed in detail. Here, we want just to outline an alternative possible strategy to evaluate the rest based on an elementary manipulation of the integral and exploiting the specific form of the phase. We define p = (μ, ν, α, β),
k = (η3 , ξ1 , ξ2 , ξ3 )
B(p; η, η2 ) = μ(x1 + aαη1 ) + ν(x2 + aαη2 ) +
&
(6.40) 1−
μ2
−
ν2
x3
+ c2 α + c1 β (6.41) & & (6.42) A(p) = 1 − μ2 − ν 2 aα − b2 , μaβ, νaβ, 1 − μ− ν 2 aβ − b1 Then, dropping the dependence on the parameters η1 , η2 , we have i i I ε = dpe ε B(p) dkGε12 (p, k)e ε A(p)·k
(6.43)
D
where D is the domain of integration corresponding to the variables p. We introduce the following linear change of coordinates p = (μ, ν, α, β) → z = (z1 , z2 , z3 , z4 ) ε ε ε b2 b1 ε + z1 , β = + z4 z2 , ν = z3 , α = b1 b1 a a a a and denote p = Lε z. Hence i i ε4 Iε = 2 2 dze ε B(Lε z) dkGε12 (Lε z, k)e ε A(Lε z)·k a b1 μ=
Dε
(6.44)
(6.45)
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where Dε is the domain of integration corresponding to the variables z. We notice that B(Lε z) = Θ0 − εk0 · z + Λε (z), A(Lε z) = εz + Γε (z) where k0 = −
c2 x1 + b2 η1 x2 + b2 η2 c1 , , , a b1 b1 a
(6.46) (6.47) (6.48)
and Λε (z), Γε (z) are explicitly known functions of order ε2 for ε → 0. We also notice that Dε reduces to R4 for ε → 0. Taking into account (6.46), (6.47), we have i ε4 i Θ 0 ε −ik0 ·z εi Λε (z) ε dze e dkGε12 (Lε z, k)eiz·k e ε Γε (z)·k (6.49) I = 2 2e a b1 Dε
From the above formula one sees that the expansion of I ε for ε → 0, with an explicit remainder, is reduced to the Taylor expansion of the integrand in the r.h.s. of (6.49) and then to the estimate of the error done if the domain Dε is replaced by R4 . Finally, by suitable integration by parts, one can show that each term of the expansion and the remainder are summable w.r.t. the other variables (η1 , η2 and then x, y, z). We plan to follow this strategy in further work where the analysis of the model will be carried out at any order in perturbation theory.
7. Appendix I We recall here the definition and the main properties of the Schr¨ odinger operator with an attractive point interaction, which is used as model Hamiltonian for the two “atoms” (for further details and for the proofs we refer to [1]). We fix for simplicity = 1, m = 1/2 and define domain and action of the Schr¨ odinger operator Kα in L2 (R3 ) with a point interaction placed at the origin and strength α < 0 as follows D(Kα ) =
√ , + λ u ∈ L2 (R3 )|u = wλ + qGλ , wλ ∈ D(−Δ), q ∈ C, wλ (0) = α + q 4π (7.1)
Kα u = −Δwλ − λqGλ
(7.2) 2
λ
where D(−Δ) is the domain of the free Hamiltonian, λ > (4πα) and G (x) = √ e− λ|x| 4π|x| .
The operator defined in (7.1), (7.2) is selfadjoint and bounded from below, with spectrum explicitly given by σp (Kα ) = {−(4πα)2 },
σc (Kα ) = σac (Kα ) = [0, ∞)
(7.3)
The unique proper eigenfunction corresponding to the negative eigenvalue and a set of generalized eigenfunctions corresponding to the absolutely continuous
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spectrum are respectively given by ζα (x) = (−α)1/2 φα (x, k) =
1 (2π)3/2
e4πα|x| |x| eik·x +
(7.4) e−i|k||x| 1 4πα+ i|k| |x|
(7.5)
We notice that for α = −(4π)−1 the expressions (7.4), (7.5) reduce to ζ 0 (x), φ0 (x, k) (defined in (2.6), (2.7)). Exploiting ζα and φα one obtains an eigenfunction expansion theorem for the operator Kα . More precisely one defines the map u → {u0 , u ˆ} u0 = dxu(x)ζα (x),
(7.6)
u ˆ(k) =
dxu(x)φα (x, k)
(7.7)
for u ∈ L2 (R3 ) sufficiently smooth and then one can prove the following statements. (1) The map (7.6), (7.7) extends to a unitary operator ˆ} ∈ C ⊕ L2 (R3 ) Uα : u ∈ L2 (R3 ) → {u0 , u
(7.8)
where the last integral in (7.7) should be now intended in the L2 -sense, i.e., ˆ is by definition the limit in L2 (R3 ) for N → ∞ of the sequence - u dxu(x)φα (x, ·). In particular one has |x|
(7.9)
3
Moreover for any u ∈ L (R ) the following inversion formula holds u(x) = u0 ζα (x) + dkˆ u(k)φα (x, k) (7.10) (2)
where the two terms of the sum in r.h.s. of (7.10) are orthogonal. Uα Kα Uα−1 is a diagonal operator , i.e. for u ∈ D(Kα ) 2 . ˆ(k) (Kα u)0 = −(4πα)2 u0 , (K α u)(k) = k u
(7.11)
which in particular means that the pure point and the absolutely continuous subspaces associated to Kα are explicitly characterized as follows L2pp (R3 ) = {u ∈ L2 (R3 )|ˆ u = 0},
L2ac (R3 ) = {u ∈ L2 (R3 )|u0 = 0} (7.12)
(3)
2
3
For any u ∈ L (R ) the unitary propagator reads ) −itKα * 2 −i(4πα)2 t e u (x) = e u0 ζα (x) + dke−itk u ˆ(k)φα (x, k)
(7.13)
where again the two terms of the sum in the r.h.s. of (7.13) are orthogonal for any t > 0. In principle the above statements can be derived exploiting general results about eigenfunction expansions for selfadjoint operators. On the other hand in the case of the operator Kα the eigenfunctions ζα , φα are explicitly known
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in terms of elementary functions and therefore all the above properties can be directly checked. In particular it is a nice exercise to verify that Uα is unitary and (7.10) holds, by simply mimicking the standard proof of Plancherel’s theorem for the Fourier transform in L2 . Moreover a direct computation shows that the two terms of the sum in the r.h.s. of (7.10) are orthogonal. Formulae (7.11), (7.12) can be directly checked exploiting the definition of Kα and the representation (7.13) for the unitary propagator follows from (7.11) and the fact that Uα is unitary.
8. Appendix II Here, we give a proof of the representation formula (3.8). The relevant object to compute is the probability amplitude in (3.3) ˆ 2 (R, r1 , r2 , t) (8.1) F(R, k1 , k2 , t) = dr1 dr2 φ1 (r1 , k1 )φ2 (r2 , k2 )Ψ As we already pointed out in Appendix I, formula (8.1) -defines F(R, - ·, ·, t) as the limit in L2 (R6 ) for N, M → ∞ of the sequence |r1 |
i
e tH0 H1 e− tH0 = W1 (t) + W2 (t) Wj (t) = e
i tK0
e
i tKj
(8.2) − i tK0 − i tKj
Vj e
e
(8.3)
where Vj denotes the multiplication operator by Vj (R, rj ) = V (γ −1 (R − rj ))
(8.4)
Using (8.2) we rewrite the r.h.s. of (8.1) in the more convenient form F(R, k1 , k2 , t) = dr1 dr2 φ1 (r1 , k1 )φ2 (r2 , k2 )Ψ0 (R, r1 , r2 ) λ −i
t
dt1
dr1 dr2 φ1 (r1 , k1 )φ2 (r2 , k2 )
0
× [(W1 (t1 ) + W2 (t1 )) Ψ0 ] (R, r1 , r2 ) t t1 λ2 dt1 dt2 dr1 dr2 φ1 (r1 , k1 )φ2 (r2 , k2 ) − 2 0
0
× [(W1 (t1 ) + W2 (t1 )) (W1 (t2 ) + W2 (t2 )) Ψ0 ] (R, r1 , r2 ) (8.5) We observe that the operator Wj (t) acts non trivially only on the variable R and rj . Exploiting this fact and the orthogonality relation (see Appendix I) drφj (r, k)ζj (r) = 0, j = 1, 2 (8.6)
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we obtain (8.7) F = F12 + F21 Flj (R, k1 , k2 , t) t t1 λ2 = 2 dt1 dt2 dr1 dr2 φ1 (r1 , k1 )φ2 (r2 , k2 ) (Wj (t1 )Wl (t2 )Ψ0 ) (R, r1 , r2 ) 0
0
(8.8) where l, j = 1, 2, j = l. Due to the specific factorized form of the initial state of the system, we have i
i
i
i
i
Wj (t1 )Wl (t2 )Ψ0 = e (t1 +t2 )|E0 | e t1 K0 e t1 Kj Vj ζj e t2 Kl Vl ζl e− t2 K0 ψ (8.9) Moreover, according to the eigenfunction expansion theorem recalled in Appendix I, the propagator generated by Kj is i i i tKj tE0 0 e (Vj (R, ·)ζj ) (r) = e cj (R)ζj (r) + dke tE(k) Vˆj (R, k)φj (r, k) (8.10) where c0j (R)
=
Vˆj (R, k) = E(k) = Therefore
drVj (R, r)ζj2 (r)
(8.11)
drφj (r, k)V (γ −1 (R − r))ζj (r)
(8.12)
2 k 2 2m
i i drφj (r, k) e tKj (Vj (R, ·)ζj ) (r) = e tE(k) Vˆj (R, k)
(8.13)
(8.14)
Exploiting (8.9) and (8.14), formula (8.8) reduces to Flj (R, k1 , k2 , t) =
t t1 i i λ2 dt dt2 e t1 (E(kj )+|E0 |) e t2 (E(kl )+|E0 |) 1 2 0 0 i i i i t K 1 0 Vˆj (·, kj )e− t1 K0 e t2 K0 Vˆl (·, kl )e− t2 K0 ψ (R) × e (8.15)
We notice that the r.h.s. of (8.12) can be more conveniently written in terms of the Fourier transform V of the interaction potential as follows −1 Vˆj (R, kj ) = e−ik·aj γ 3/2 dξeiγ (R−aj )·ξ g(ξ, γkj ) (8.16)
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where g(ξ, y) has been defined in (2.27). From (8.16) and the explicit expression of the free propagator we have i i e t2 K0 Vˆl (·, kl )e− t2 K0 ψ (R) al t2 2 R t2 i 2M γ −ikl ·al 3/2 2 ξ +i γ ·ξ−i γ ·ξ =e ξ (8.17) γ ψ R+ dξg(ξ, γkl )e Mγ and
i i i i e t1 K0 Vˆj (·, kj )e− t1 K0 e t2 K0 Vˆl (·, kl )e− t2 K0 ψ (R) = e−ikl ·al −ikj ·aj γ 3 dξdηg(η, γkj )g(ξ, γkl ) t1 t2 t1 a 2 2 a R R i 2M γ i(− γj ·η− γl ·ξ ) 2 η + γ ·η+ 2M γ 2 ξ + γ ·ξ+ M γ 2 η·ξ
·e
t2 t1 η+ ξ × R+ Mγ Mγ
e
ψ (8.18)
Finally we consider the time-dependent phase factor in (8.15). We notice that t1 t2 (E(kj ) + |E0 |) + (E(kl ) + |E0 |) = (t1 w(γkj ) + t2 w(γkl )) (8.19) mγ 2 Taking into account (8.18), (8.19), (2.9), and rescaling the time variables according to t1 = tα, t2 = tβ, we can rewrite (8.15) as follows Flj (R, k1 , k2 , t) =
λ2 t2 Nε 3/2 −ikl ·al −ikj ·aj γ e 2 ε 1 α × dˆ u dα dβ dξdηg(η, γkj )g(ξ, γkl )f 0
S2
× i
0
R t + (αη + βξ) γ M γ2
2 t R t αη 2 + R γ ·η+ 2M γ 2 βξ + γ ·ξ+ M γ 2 αη·ξ a 0 al j t t i 1ε u ˆ· R γ + M γ 2 (αη+βξ) − γ ·η− γ ·ξ+ mγ 2 (ω(γkj )α+ω(γkl )β) t
· e / 2M γ2 ·e
(8.20) We observe that t =a M γ2
(8.21)
is of order one for ε → 0 and this means that the first exponential in the integral in (8.20) has a slowly oscillating phase for ε 1. On other hand t M a = 2 mγ m
(8.22)
is proportional to ε−1 and therefore the last exponential in the integral in (8.20) has a rapidly oscillating phase for ε 1. Denoting R = γx, kj = γ −1 yj and using the notation (2.25), we find
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λ2 t2 Nε 3/2 −ikl ·al −ikj ·aj γ e 2 ε 1 α × dˆ u dα dβ dξdηg(η, yj )g(ξ, yl )f (x + a(αη + βξ))
Flj (γx, γ −1 y1 , γ −1 y2 ) =
S2
0
0
2 2 i[x·(η+ξ)+ a 2 (αη +βξ +2αη·ξ)]
i
e ε [ˆu·x+ˆu·a(αη+βξ)−bj aˆj ·η−bl aˆl ·ξ+cj α+cl β] ×e 2 2 λ t Nε 3/2 −ikl ·al −ikj ·aj ε γ e ≡ 2 Glj (x, y1 , y2 , t) (8.23) ε where in the last line we have used (3.9), (3.10), (3.11), (3.12). From (3.3), (8.1) and (8.23) we obtain 2 −3 P(t) = γ dxdy1 dy2 F12 (γx, γ −1 y1 , γ −1 y2 ) + F21 (γx, γ −1 y1 , γ −1 y2 ) λ 4 t4 N 2 2 ε ε = 4 2ε (x, y1 , y2 ) + G21 (x, y1 , y2 )| (8.24) dxdy1 dy2 |G12 ε and this concludes the proof of (3.8).
References [1] Albeverio, S., Gesztesy, F., Hogh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics. Springer, New York (1988) [2] Bell, J.: Quantum mechanics for cosmologists. In: Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press (1988) [3] Bleinstein, N., Handelsman, R.A.: Asymptotic Expansions of Integrals. Dover Publication, New York (1975) [4] Blasi, R., Pascazio, S., Takagi, S.: Particle tracks and the mechanism of decoherence in a model bubble chamber. Phys. Lett. A 250, 230–240 (1998) [5] Born, M., Einstein, A.: Born–Einstein Letters, 1916–1955. Macmillan Science Publ., (2004) [6] Broyles, A.A.: Wave mechanics of particle detectors. Phys. Rev. A 48(2), 1055– 1065 (1993) [7] Cacciapuoti, C., Carlone, R., Figari, R.: A solvable model of a tracking chamber. Rep. Math. Phys. 59 (2007) [8] Condon, E., Gurney, R.: Nature 122, 439 (1928) [9] Castagnino, M., Laura, R.: Functional approach to quantum decoherence and the classical final limit: the Mott and cosmological problems. Int. J. Theo. Phys. 39(7), 1737–1765 (2000) [10] Dell’Antonio, G., Figari, R., Teta, A.: Joint excitation probability for two harmonic oscillators in dimension one and the Mott problem. J. Math. Phys. 49(4), 042105 (2008) [11] Fedoryuk, M.V.: The stationary phase method and pseudodifferential operators. Usp. Mat. Nauk 26(1), 67–112 (1971) [12] Gamow, G.: Zur Quantentheorie des Atomkernes. Zeit. F. Phys. 51, 204–212 (1928)
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[13] Giulini, D., Joos, E., Kiefer, C., Kupsch, J., Stamatescu, I.-O., Zeh, H.D.: Decoherence and the Appearance of a Classical World in Quantum Theory. Springer, Berlin (1996) [14] Halliwell, J.J.: Trajectories for the wave function of the universe from a simple detector model. Phys. Rev. D 64, 044008 (2001) [15] Heisenberg, W.: The Physical Principles of Quantum Theory. Dover Publication, New York (1951) [16] H¨ ormander, L.: The Analysis of Linear Partial Differential Operators. Springer, Berlin (1983) [17] Hornberger, K.: Introduction to decoherence theory. arXiv:quant-ph/0612118v3, 5 Nov 2008 [18] Mott, N.F.: The wave mechanics of α-ray tracks. Proc. R. Soc. Lond. A 126, 79–84 (1929) [19] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness. Academic Press, London (1975) [20] Robert, D.: Semi-classical approximation in quantum mechanics. A survey of old and recent mathematical results. Helv. Phys. Acta 71, 44–116 (1998) [21] Steinmann, O.: Particle localization in field theory. Comm. Math. Phys. 7, 112–137 (1968) Gianfausto Dell’Antonio Dipartimento di Matematica, Universit´ a di Roma “La Sapienza” P.le A. Moro, 2 00185 Rome, Italy and S.I.S.S.A. via Beirut, 2-4 34151 Trieste, Italy e-mail:
[email protected] Rodolfo Figari Dipartimento di Scienze Fisiche Sezione I.N.F.N. di Napoli, Universit` a “Federico II” Via Cinthia, 45 80126 Naples, Italy e-mail:
[email protected] Alessandro Teta Dipartimento di Matematica Pura ed Applicata Universit` a di L’Aquila Via Vetoio loc. Coppito 67010 L’Aquila, Italy e-mail:
[email protected] Communicated by Claude Alain Pillet. Received: September 24, 2009. Accepted: March 28, 2010.
Ann. Henri Poincar´e 11 (2010), 565–584 c 2010 Springer Basel AG 1424-0637/10/040565-20 published online June 8, 2010 DOI 10.1007/s00023-010-0035-6
Annales Henri Poincar´ e
Topological Graph Polynomials in Colored Group Field Theory Razvan Gurau Abstract. In this paper, we analyze the open Feynman graphs of the Colored Group Field Theory introduced in Gurau (Colored group field theory, arXiv:0907.2582 [hep-th]). We define the boundary graph G∂ of an open graph G and prove it is a cellular complex. Using this structure we generalize the topological (Bollob´ as–Riordan) Tutte polynomials associated to (ribbon) graphs to topological polynomials adapted to Colored Group Field Theory graphs in arbitrary dimension.
1. Introduction Discrete structures over finite sets, in particular graphs, are paramount to our present understanding of physics. Since Feynman realized that the perturbation series of quantum field theory is indexed by subclasses of graphs, the best experimentally tested physical predictions we have to this date rely solely on them. Different quantum field theories generate different classes of graphs. The scalar Φ4 field theory generates graphs formed of four valent vertices and lines. More involved quantum field theories, like Yang-Mills gauge theories [2,3], require further structure to be added (new particles, space-time indices, etc.). Random matrix models [4–7] and non commutative quantum field theories [8,9] generate ribbon graphs. A striking feature of the random matrix models and non commutative quantum field theories [10–14] is that the graphs are organized hierarchically. That is, the dominant contribution to the partition function is given by planar graphs, first order corrections are given by genus one graphs, second order corrections by genus two graphs, etc. Random matrix models are relevant to very diverse physical and mathematical questions ranging from two dimensional quantum gravity to knot theory and quark confinement [15]. In the context of non commutative quantum field theory the topological power counting of the ribbon graphs has been shown in a series of papers to lead to a non trivial fixed point of the
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renormalization group flow [16–20]. One can therefore expect that an appropriate generalization of such models to higher dimensions should also pose non trivial renormalization fixed points. The study of such generalizations holds essential clues for problems ranging from the quantization of gravity in higher dimensions to condensed matter. Random matrix models generalize in higher dimensions to random tensor models, or group field theories (GFT) [21–23]. The perturbative development of such theories generates “stranded graphs [24].”. The connection between GFTs and quantum gravity has been largely investigated [25]. Different models have been considered [26–29], and their semiclassical limit analyzed [30,31]. The study of the renormalization properties of such models has been started [32–34]. However, classical GFT models generate many singular graphs (that is graphs whose dual topological spaces have extended singularities). In a previous paper [1] we proposed a solution to this problem in the form of the “colored group field theory” (CGFT). The singular graphs are absent in this context in any dimension and the surviving graphs possess a cellular complex structure. In this paper, we extend the study started in [1] of the Feynman graphs of the CGFT to open graphs (that is graphs with external half lines). For every such graph G we define its boundary graph G∂ . We prove that G∂ has a cellular structure inherited from the graph G. Extending the definition of the boundary operator of [1], we introduce the homology of G∂ and explore some of its properties. Our model has been further studied in [35]. A simple and yet powerful way to encode information about a graph is through topological polynomials. Introduced first by Kirchhoff [36] they were studied (much) later by Tutte [37] as the solution of an inductive contraction deletion equation. The topological polynomials appear naturally in the dimensional regularization of quantum field theories [38] or in the study of statistical physics models [39–41]. The Tutte polynomials have been generalized by Bollob´ as and Riordan [42–45] to ribbon graphs. Further generalizations of these polynomials, respecting more involved induction equations, have been put in relation with the Feynman amplitudes of random matrix models and non commutative quantum field theories [46–49]. Relying on the cellular complex structure of G and G∂ we propose a generalization of the classical topological polynomials adapted to CGFT graphs. These polynomials respect a contraction deletion equation and encode information about the cellular homology of the CGFT graph. This paper is organized as follows. In Sect. 2 we briefly review the classical Tutte and Bollob´ as–Riordan polynomials. In Sect. 3 we detail the GFT graphs and define the boundary cellular complex and cellular homology for open graphs. In Sect. 4 we define the topological polynomials of CGFT graphs and show that they obey a contraction deletion relation. Section 6 draws the conclusions of our work. The mathematics and physics nomenclature for graphs is very different and sometimes quite confusing. The reader is strongly encouraged to consult [49] for a dictionary. Also, some familiarity with ribbon graphs is assumed.
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Again [49] (specifically Sections 4.1 and 4.3) provides a very good and concise introduction to this topic.
2. Tutte and Bollob´as–Riordan Polynomials This section is a short introduction to topological graph polynomials, see [49] and references therein for more detailed presentations. A graph G is defined by the sets of its vertices V(G) and lines L(G). A line, connecting the vertices v1 , v2 ∈ V(G) is denoted lv1 v2 ∈ L(G). For any line lv1 v2 of G one can define two additional graphs1 • •
The graph with the line lv1 v2 deleted, denoted G − lv1 v2 , with set of lines L(G − lv1 v2 ) = L(G)\{lv1 v2 } and set of vertices V(G − lv1 v2 ) = V(G). The graph with the line lv1 v2 contracted, denoted G/lv1 v2 , is the graph obtained from G by deleting lv1 v2 and identifying the two end vertices v1 and v2 . That is L(G/lv1 v2 ) = [L(G)\{lv1 v2 }]/(v1 ∼ v2 ), V(G/lv1 v2 ) = V(G)/(v1 ∼ v2 ). Note that if v1 = v2 the G/lv1 v2 = G − lv1 v2 .
Given a graph G one can consider the family of its subgraphs. H is a subgraph of G (denoted H ⊂ G) if V(H) = V(G) and L(H) ⊂ L(G). Thus G − lv1 v2 is a subgraph of G, whereas G/lv1 v2 is not. The multivariate Tutte polynomial ZG (q, {β}) of the graph G depends on one variable βlv1 v2 associated to each line lv1 v2 and an unique variable q counting the connected components of G Definition 1 (Sum over subgraphs). ZG (q, {β}) = q |k(H)| H⊂G
βlv1 v2 ,
(1)
lv1 v2 ∈L(H)
where k(H) is the number of connected components of the subgraph H. This polynomial obeys a contraction deletion equation Lemma 1. For any line lv1 v2 ∈ L(G), ZG (q, {β}) = βlv1 v2 ZG/lv1 v2 (q, {β}\{βlv1 v2 })+ZG−lv1 v2 (q, {β}\{βlv1 v2 }). (2) For a graphs with no lines but with v vertices ZG (q, ∅) = q v . In quantum field theory one deals with graphs whose vertices are furthermore decorated with “half lines”, or external legs.2 We use halflines to encode information about the graph G in a subgraph H. We will always replace a line belonging to G but not to H by two halflines on its end vertices. 1 2
The two end vertices might coincide, v1 = v2 . Or flags in the mathematical literature.
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F2
v1
F1
v2
l v1v 2
Figure 1. Ribbon vertices, ribbon lines and strands The Tutte polynomial can be generalized to ribbon graphs. A typical ribbon graph with half lines is presented in Fig. 1. It is made of ribbon vertices (v1 and v2 in Fig. 1) and ribbon lines (lv1 v2 in Fig. 1). The lines and half lines in a ribbon graph have two sides, also called strands, represented by solid lines in Fig. 1. The strands of a graph encode an extra structure. Tracing a strand one encounters one of the two cases • Either one does not encounter a half line (F1 in Fig. 1). In this case the closed strand defines an internal face. • Or one does encounter a half line (F2 in Fig. 1). In this case one continues on the second strand of this external half line (one “pinches” the external half line). The strands thus traced define an external face. This “pinching” is represented by the dotted curves in Fig. 1. A ribbon subgraph H ⊂ G of the ribbon graph G has the same set of vertices V(H) = V(G), but only a subset of the lines L(H) ⊂ L(G). Again, for a subgraph H all lines lv1 v2 ∈ L(G)\L(H) are replaced by pinched external half lines. Thus, all internal faces of H are internal faces of G, but there might exist external faces of H consisting of the union of pieces belonging to several internal faces of G. We are now in position to generalize the Definition 1 to ribbon graphs. We introduce an extra variable z counting all the faces (internal or external) of the graph, and define Definition 2. The multivariate Bollob´ as–Riordan polynomial of a ribbon graph, analog to the multivariate polynomial of Eq. (1), is: ⎛ ⎞ k(H) ⎝ q βlv1 v2 ⎠ z F (H) , (3) VG (q, {βl }, z) = H⊂G
lv1 v2 ∈L(H)
where k(H) is again the number of connected components of H, and F (H) the total number of faces. The deletion of a ribbon line lv1 v2 consists in replacing it by two pinched halflines on its end vertices v1 and v2 . It is well defined for all the lines of a graph. On the contrary, the contraction must respect the strand structure and is well defined only for lines lv1 v2 connecting two different vertices v1 = v2 . The polynomial define by Eq. (3) respects the contraction deletion equation (2)
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Figure 2. GFT vertex and a GFT line in four dimensions
only for such lines. The end graphs (those which cannot be contracted further) consist of connected components with only one vertex, but possibly many lines and faces. The polynomial of such end graphs can be read from Eq. (3). The crucial property of the topological polynomials is that the definitions in term of subgraphs and the contraction deletion properties can be exchanged. That is, the polynomials of Definitions 1 and 2 are the unique solutions of the deletion contraction equation (2) respecting the appropriate forms for the end graphs. Although, given just Eq. (2), one might think that its solution depends on the order in which the lines are contracted (deleted), Eqs. (1) and (3) show that it does not.
3. Colored Group Field Theory Graphs Ribbon graphs generalize in higher dimensions to group field theory graphs [22–24]. The GFT graphs are generated by a path integral and are built by the following rules. The GFT vertex in n dimension has coordination n + 1. Each halfline (and consequently line) has exactly n strands. Inside a vertex, the strands connect two half lines. In n dimensions, if we label the strands of a halfline 1 to n turning anticlockwise, the strand p connects to the p’th successor halfline when turning clockwise around the vertex. Every GFT line connects two half lines with an arbitrary permutation of the strands. Figure 2 presents the GFT vertex and a typical GFT line in four dimensions. The reader can check that a GFT graph in two dimensions is a ribbon graph with vertices of coordination three. As such, it is dual to a triangulation of a two dimensional surface. Considering the ribbon vertices of the graph as 0-cells, its lines as 1-cells and its faces as 2-cells, a ribbon graph becomes a two dimensional cellular complex. One would expect that the GFT graphs in higher dimension also have a cellular complex structure. This is not true in general because the permutations of strands on the lines prevent one from defining cells of dimension higher that two! A solution is to consider only the colored group field theory graphs introduced in [1]. In fact, to our knowledge, this is the only category of graphs generated by a path integral which has an associated complex structure in
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Figure 3. Two vertices and a line in a colored graph arbitrary dimension3 ! The graphs obtained by the perturbative development of the color group field theory action of [1] obey Definition 3. A CGFT graph in n dimensions is a GFT graph such that • The CGFT vertices are stranded vertices. The set of vertices V(G) = {v1 , . . . vn } is the disjoint union of two sets V(G) = V + (G) ∪ V − (G). V + (G) is the set of positive vertices and V − (G) is the set of negative vertices. • The lines lvi 1 v2 ∈ L(G) connect a positive and a negative vertex (v1 ∈ V + (G) and v2 ∈ V − (G)) and posses a color index i ∈ {0, . . . n}. The n strands of all CGFT lines are parallel. Halflines also possess a color index. • Each color appears exactly once among the lines or halflines touching a vertex. The colors are encountered in the order 0, . . . , n when turning clockwise around a positive vertex and anticlockwise around a negative one. A CGFT graph admits two equivalent representations, either as a stranded graph, or simply as an edge colored graph, obtained by collapsing all the strands belonging to all lines. As the connectivity of strands inside the CGFT vertex and lines are fixed the two representations are in one to one correspondence. A colored graph is made of colored lines connecting positive and negative vertices. In Fig. 3, the line of color 3 connects the positive vertex on the left with the negative one on the right. Figure 4 gives the two representations for the same graph. 3.1. Bubbles and Cellular Structure In the definition of the Bollob´ as–Riordan polynomial the faces (internal and external) played a crucial role. In higher dimensions the faces generalize to higher dimensional cells, called bubbles. First consider G a CGFT graph with no external half lines. In [1] we defined the p-cells of G as Definition 4. A “p-bubble” with colors i1 < · · · < ip of a graph with n + 1 colors G with no external halflines is a maximal connected components made of lines of colors i1 , . . . , ip . We denote it BVC , where C = {i1 , . . . , ip } is the ordered set of colors of the lines in the bubbles and V is the set of vertices. 3
In three dimensions one also has the alternative to use the orientable model of [32], but this cannot be generalized to higher dimensions.
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0 0 1
v
1
3
1
v
2
2
2 3
Figure 4. A closed colored graph in three dimensions Note that, unlike the subgraphs of Sect. 2, the connected components do not have half lines. For example, for the graph in Fig. 4 we have the 3-bubbles , Bv013 , Bv023 and Bv123 , the 2-bubbles (that is faces) Bv011 v2 , Bv021 v2 , Bv031 v2 , Bv012 1 v2 1 v2 1 v2 1 v2 12 13 23 Bv1 v2 , Bv1 v2 , Bv1 v2 , the one bubbles (that is lines) Bv01 v2 , Bv11 v2 , Bv21 v2 , Bv31 v2 , and finally the 0-bubbles (that is vertices) Bv1 , Bv2 . Like the graph G, the p-bubbles themselves admit graphical representations either as stranded graphs or as edge colored graphs. For instance in Fig. 4, is obtained by deleting all strands the stranded graph of the 3-bubble Bv012 1 v2 belonging to the line lv31 v2 . Similarly the stranded graph of the 2-bubble Bv011 v2 is obtained by deleting all strands belonging to the lines lv21 v2 and lv31 v2 . Considering the representation of bubbles as stranded graphs it is easy to see that in any dimension, the strands themselves always correspond to 2-bubbles. This remark is crucial for the next section. As proved in [1], the p-bubbles define a cellular complex and a cellular homology induced by the boundary operator Definition 5. The p’th boundary operator dp acting on a p-bubble BVC with colors C = {i1 , . . . ip } is • for p ≥ 2, C (−)q+1 B V , (4) dp (BVC ) = q
B C ∈Bp−1 V
V ⊂V C =C\iq
•
which associates to a p-bubble the alternating sum of all (p − 1)-bubbles formed by subsets of its vertices. for p = 1, as the lines Bvi 1 v2 connect a positive vertex (v1 ∈ V + (G)) to a negative one, v2 ∈ V − (G) d1 Bvi 1 v2 = Bv1 − Bv2 .
•
(5)
for p = 0, d0 Bv = 0.
3.2. External Half Lines and the Boundary Complex A graph G with external half lines is dual to a topological space with boundary. We will first associate to G a “boundary graph” G∂ , dual to a triangulation of the boundary of the topological space and then identify a cellular complex structure for G∂ .
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1
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01 0
012
Figure 5. Tetrahedron dual to a CGFT vertex To understand the construction of G∂ one needs to consider the topological space dual to G (see [1] and [32] for details). The dual of a colored graph is essentially a simplicial complex.4 Each CGFT vertex is dual to a n-simplex Δn . The half lines of a vertex are dual to the “sides” of Δn , that is the (n − 1)simplices Δn−1 bounding it. A boundary simplex Δn−1 inherits the color of the halfline to which it corresponds. The lines (which are identifications of halflines) correspond to the gluing of the two Δn simplices along a common Δn−1 boundary simplex. Higher dimensional p-bubbles are dual to (n − p)simplices, in particular the 2-bubbles are dual to Δn−2 simplices. In particular, in the stranded representation of a CGFT graph, the Δn−2 simplices are dual to the strands. In three dimensions this is represented in Fig. 5. The vertex 0123 is dual to the tetrahedron 0123, the halfline 0 is dual to the triangle 0, the 2-bubble 01 is dual to the edge common to the triangles 0 and 1, and the 3-bubble 012 is dual to the the vertex of the tetrahedron common to the triangles 0, 1, and 2. If a vertex in a CGFT graph has no half lines then its dual simplex Δn sits in the interior of the simplicial complex (in the bulk). On the contrary, if a vertex has half lines, then its dual simplex sits on the boundary of the simplicial complex, and contributes to the triangulation of this boundary with the Δn−1 simplex dual to the half line. The triangulation of the boundary of the simplicial complex is therefore made of all the Δn−1 simplices dual to the halflines of the graph. These Δn−1 simplices are glued along there boundary Δn−2 . The boundary Δn−2 simplices are dual, in the stranded representation of a CGFT graph to the open strands. To obtain the graph G∂ dual to the boundary of the simplicial complex one must draw a vertex for each external halfline of G and a line for each open strand of G. This can be achieved starting with the stranded representation of the graph G (see Fig. 6), delete all closed strands, and “pinch” the external strands into a vertex for each external half line. The graph thus obtained is the edge colored representation of G∂ . We call G∂ the “boundary graph” of G. The vertices of G∂ inherit the color of the halfline and the lines of G∂ inherit the couple of colors of the strand to which they correspond. In the example of Fig. 6, the graph G∂ (represented on the right) has one connected 4
It is in fact a slightly more general gluing of simplices along there faces.
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03 1 3
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3 w1
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Figure 6. A CGFT graph G and it boundary graph G∂ component with two vertices, w1 and w2 , both of color 3 and three lines of colors 03, 13 and 23. Note that G∂ is a graph of vertices with one color and lines colored by couples of colors: a priori it is very different from a CGFT graph. Nevertheless G∂ has a cellular complex structure, strongly reminiscent of the one of G. We denote the set of vertices of G∂ (obtained after pinching) by V∂ . They are the 0-bubbles of the cellular complex of G∂ . For p ≥ 1, we have Definition 6. Let a graph G and its boundary graph G∂ obtained after pinch ing. For p ≥ 1 the “boundary p-bubbles” (B∂ )CV are the maximally connected ∂ components of G∂ formed by boundary vertices V∂ ⊂ V∂ and boundary lines of colors ia ib , with {ia , ib } ⊂ C ⊂ {0, . . . n} and |C | = p + 1. For example G∂ in Fig. 6 has • • •
0 bubbles (B∂ )3w1 , (B∂ )3w2 , which are the vertices of G∂ . 13 23 1 bubbles (B∂ )03 w1 w2 , (B∂ )w1 w2 , (B∂ )w1 w2 , which are the lines of G∂ . 013 023 123 2 bubbles (B∂ )w1 w2 , (B∂ )w1 w2 , (B∂ )w1 w2 , which are the connected components with lines (03, 13), (03, 23) and (13, 23) respectively.
We denote Bp∂ the set of all boundary p-bubbles, and following [1] we define the operator Definition 7. The p’th boundary operator d∂p of the boundary complex, acting on a boundary p-bubble (B∂ )CV∂ with colors C = {i1 , . . . ip+1 } is •
for p ≥ 1, d∂p [(B∂ )CV∂ ] =
•
(−)q+1
q
p−1 (B )C ∈B ∂ V ∂ ∂ V∂ ⊂V∂ C =C\iq
(B∂ )CV∂ ,
(6)
for p = 0, d∂0 [(B∂ )iw1 ] = 0. For G∂ of Fig. 6 for instance, 13 03 d∂2 [(B∂ )013 w1 w2 ] = (B∂ )w1 w2 − (B∂ )w1 w2 3 3 3 3 d∂1 [d∂2 [(B∂ )013 w1 w2 ]] = (B∂ )w1 + (B∂ )w2 − (B∂ )w1 − (B∂ )w2 = 0.
The operator d∂p is a boundary operator in the sense
(7)
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Lemma 2. d∂p−1 ◦ d∂p = 0.
(8)
Proof. The proof goes much like its counterpart presented in [1]. Consider the application of two consecutive boundary operators on a boundary p-bubble d∂p−1 d∂p [(B∂ )CV∂ ] = (−)q+1 d∂p−1 [(B∂ )CV∂ ] (9) q
=
p−1 (B )C ∈B ∂ V ∂ ∂ V∂ ⊂V∂ C =C\iq
(−)q+1
q
+
p−1 (B )C ∈B ∂ V ∂ ∂ V∂ ⊂V∂ C =C\iq
(−)r+1
r
(−)r
r>q
p−2 (B )C ∈B ∂ V ∂ ∂ V∂ ⊂V∂ C =C\iq \ir
p−2 (B )C ∈B ∂ V ∂ ∂ V∂ ⊂V∂ C =C\iq \ir
(B∂ )CV∂ ,
(B∂ )CV∂
(10)
as ir is the r − 1’th color of C \iq if q < r. The two terms cancel by exchanging q and r in the second term. The boundary bubbles define a cellular complex with attaching maps induced by the boundary operator of Definition 7. With the appropriate substitutions, one reproduces the main results of [1] for the cellular homology of G∂ defined by d∂p . Lemma 3. Let G∂ a connected boundary CGFT graph with n + 1 colors. The operator d∂p has the following properties •
The d∂0 operator respects ker(d∂0 ) =
Z.
(11)
|B0∂ |
•
The d∂1 operator respects ker(d∂1 ) =
Z,
Im(d∂1 )
|B1∂ |−|B0∂ |+1
•
Z.
(12)
|B0∂ |−1
The d∂n−1 operator respects ker(d∂n−1 ) = Z ,
Im(d∂n−1 )
Z.
(13)
n−1 |B∂ |−1
In consequence, for all graphs, denoting the homology groups of G∂ as Hq∂ , we have H0∂ = Z,
Hn∂ = Z.
(14)
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And if G is moreover a three dimensional graph (that is it has four colors), then for each connected component of G∂ we have Z, H2∂ = Z, (15) H0∂ = Z, H1∂ = 2g
that is G∂ is a union of tori.
4. Topological Polynomials of GFT Graphs Having at our disposal a good definition of bubbles in arbitrary colored graphs we proceed to generalize the topological polynomials to higher dimensional graphs. However one encounters a problem. There is an incompatibility between the contraction of lines of Sect. 2 and the colored graphs of Definition 3. If G is a colored graph and l one of its lines, G − l is still a colored graph, but G/l is not. The vertex obtained by identifying the endvertices of l does not respect the conditions of Definition 3. But the p-bubbles are defined only for colored graphs. It is therefore needed to modify the contraction move to ensure that G/l remains a colored graph. This is achieved by slightly enlarging the class of graphs we consider to graphs with active and passive lines. Definition 8. A colored graph with active and passive lines is a colored graph G and a partition of the lines L(G) into two disjoint sets, L(G) = L1 (G) ∪ L2 (G), such that L2 (G) is a forest.5 The lines in the first set, L1 (G) are called active lines whereas the lines in the second set L2 (G) are called passive. Note that a colored graph with no passive lines is just a colored graph in the sense of Definition 3. For a colored graph with active and passive lines, we define the deletion and contraction only for the active lines l ∈ L1 (G) as follows Definition 9. For all active lines l ∈ L1 (G) we define • The graph with the line l deleted, G − l with V(G − l) = V(G), L1 (G − l) = L1 (G)\{l} and L2 (G − l) = L2 (G). • The graph with the line l contracted G/l with V(G/l) = V(G), L1 (G/l) = L1 (G)\{l} and L2 (G/l) = L2 (G) ∪ {l}. That is the contraction is reinterpreted as transforming an active lines into a passive one, instead of the identification of the end vertices. The contraction is defined therefore only if {l} ∪ L2 (G) is still a forest (that is it has no loops). Note that one can use the new definitions of G − l and G/l also for the graphs of Sect. 2. Then Eq. (2) holds for all active lines and Definition 1 holds if L2 (G) = ∅. Let G be a CGFT graph with n + 1 colors, and G∂ its boundary graph. As before, let Bp , 0 ≤ p ≤ n be the set of all bulk p-cells (defined by Definition 4), and Bp∂ , 0 ≤ p ≤ n − 1 the set of boundary p-cells (defined by Definition 6). 5
That is the lines in L2 (G) do not form loops.
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Denote Bn+1 the set of the connected components of G and Bn∂ the set of connected components of G∂ . To define the topological polynomial associated to G, we introduce a variable xp counting all the bulk p-cells and a variable yp counting all the boundary p-cells. Furthermore, we associate a variable βl to all active lines in G. Definition 10. The topological polynomial PG ({βl }, {xp }, {yp }) is n
n+1 |Bp | |Bp | PG ({βl }, {xp }, {yp }) = βl xp yp ∂ . H⊂G;L2 (H)=L2 (G)
l∈L1 (H)
p=0
p=0
(16) Note that the variables x0 and x1 are redundant: the number of vertices of |B0 | any subgraph is equal to the number of vertices of the initial graph, thus x0 is just an overall multiplicative factor and x1 contributes just with a global L (G) x1 2 multiplicative factor after a uniform rescaling of the line parameters βl . An explicit example is detailed at length in the Appendix. The polynomial of Eq. (16) has the following behavior under various rescalings p
PG ({βl }, {ρ(−) xp }, {ρ(−)
p+1
yp }) = ρχ(G) PG ({βl }, {xp }, {yp })
p
PG ({βl }, {xp }, {ρ(−) yp }) = ρχ(G∂ ) PG ({βl }, {xp }, {yp }),
(17)
with χ(G) and χ(G∂ ) the Euler characteristics of G and G∂ respectively. Moreover it respects the contraction deletion relation Lemma 4. For all active lines l ∈ L1 (G) such that {l} ∪ L2 (G) is a forest P ({β}, {xp }, {yp }) = βl PG/l ({β}\{βl }, {xp }, {yp }) +PG−l ({β}\{βl }, {xp }, {yp }),
(18)
Proof. Note that any active line l divides the subgraphs indexing the sum in (16), H ⊂ G with L2 (H) = L2 (G), into two families, namely Fl∈ (G) = {H|l ∈ L1 (H)}
Fl∈/ (G) = {H|l ∈ / L1 (H)} .
(19)
We split (16) into two terms corresponding to these two families. All the subgraphs in the first family contain l, thus we can factor βl in front of the first term, and reinterpret the line l as a passive line in the graph H/l. The set of graphs Fl∈ (G) is in one to one correspondence to the set of all the subgraphs H/l ⊂ G/l with L2 (H/l) = L2 (G/l) = L2 (G) ∪ {l}, therefore the first term on the rhs of Eq. (18) is recovered. The graphs in the second family Fl∈/ (G) coincide with the subgraphs of G − l, and one recovers the second term in (18). The classical Tutte and Bollob´ as–Riordan polynomials are recovered as limit cases of the higher dimensional polynomial defined here. For the CGFT graphs with three colors (which are trivalent ribbon graphs) Eqs. (16) and (3) imply P ({β}, {1, 1, z, q}, {1, 1, z}) = VG (q, {β}, z) ,
(20)
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and for an arbitrary CGFT with L2 (G) = ∅ P ({β}, {1, q}, {1}) = ZG (q, {β}).
(21)
5. A Discussion of Universality Perhaps the most appealing trait of the Bollob´ as Riordan and Tutte polynomials is their universality, namely the fact that any graph invariant respecting the deletion-contraction equations can be computed starting from them. In the classical case the proof of such a result is lengthy and technical (see [42,43]) and becomes considerably more difficult for multivariate polynomials [50]. The classical proofs of universality [42,43] rely on a classification of the end graphs of the deletion contraction, chord diagrams in [42,43] (referred to as “rosettes” following [46–49] in the sequel ). The first core result established in [42,43] is that all chord diagrams can be reduced by topological “rotation about chords” to canonical rosettes. The proof of universality proceeds then by a multi layer analysis over increasingly complex graphs starting with trivial chord diagrams, and proceeding step by step to canonical chord diagrams, arbitrary chord diagrams and finally arbitrary graphs. In this section we will introduce the first notions needed for such an analysis for the polynomials presented in this paper. The question is much more subtle and difficult than in the ribbon graphs case, and for the moment out of reach. Identifying canonical colored rosettes with, say, four colors is equivalent to a full classification of three dimensional piecewise linear manifolds and pseudomanifolds, an extremely difficult open question in algebraic topology. Even supposing that one would obtain such canonical rosettes (and we emphasize again that we do not expect such a result any time soon), it would still be a highly non trivial problem to generalize the proof of universality of [42]. 5.1. Colored Rosettes In the classical case the rosettes are the end graphs with one vertex (obtained after the contraction of a tree in the graph) decorated by some loop lines (which cannot be contracted further as they start and end on the same vertex). Similarly, for colored graphs the end graphs are made of forests of passive lines L2 (G) decorated by active lines l, such that ∀{l}, {l} ∪ L2 (G) is not a forest. A connected end graph G (that is L2 (G) contains exactly one tree) admits a representation as a colored rosette obtained by splitting all lines in L2 (G) longitudinally into two pieces (leaving the external half lines of the tree on the appropriate sides) inheriting the color of the line, and deforming the closed circuit of the pieces to a circle. A self explanatory graphical representation of this procedure is given in Fig. 7, where the passive lines are dotted and the active ones are solid. Note that the colored rosettes have a natural counterclockwise orientation. The inverse procedure is also well defined. To obtain the passive tree starting from the rosette one turns around the circle and identifies adjacent
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Figure 7. Obtaining a colored rosette pieces having the same color into a tree line. Note that in order to obtain the full passive tree one might need to go around the circle more than once. For the example of Fig. 7 the first pass reconstitutes the lines of colors 0, 1 and 3 in the passive tree, but one needs a second pass to reconstruct the line of color 2. At first sight the colored rosettes are very similar to the ribbon graphs rosettes, but as the boundary data along the colored rosettes is richer they are noticeably harder to deal with. The classical proofs of [42,43] proceed by classifying topologically related rosettes. This is achieved by two topological operations, the “rotation about lines” and the “sum of rosettes”. We note that there is a slight inconsistency in the definition of the “rotation” in the literature, namely the definition used in [43], is different from the one in [42]. It is not clear to us if these two are equivalent, as the definition of [43] imposes some restrictions on “rotations” and, at least at first sight, a “rotation” in the sense of [42] cannot be obtained by a sequence of “rotations” in the sense of [43]. The second topological operation in [42,43], the “sum of rosettes” allows to freely move parts of a rosette with respect to each other, and disentangle complicated rosettes. In the rest of this section, we will give an appropriate generalization of the “rotation about lines” to colored rosettes. In some cases this operation will allow us to disentangle rosettes, but we will show through counterexamples that this is not generic. Moreover, up to now we have not been able to find an appropriate generalization of the “sum of rosettes” to colored rosettes. 5.2. The R Relation Following [43] we consider now graphs with two effective vertices separated by at least two lines (that is L2 (G) has exactly two trees connected by at least two active lines). Contraction of either one or the other of the active lines leads to two distinct rosettes as in Fig. 8 and called R related. From the point of view of the underlying tree of passive lines, the R relation can be seen as applying the inverse of a contraction move along a line followed by a contraction along another tree line. In the rosette amounts to choose the two pieces on the rosette coming from a passive line (1 in Fig. 8) and identifying them to reconstruct it followed by the split of a newly formed tree line (2 in Fig. 8).
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Figure 9. Three lines disentangled by an R move Any line on a rosette separates two different vertices in the colored graph. Call the unique (nonempty) path in the passive tree connecting these two vertices P. On the rosettes, the line lets on the same side (both on the interior or both on the exterior) the two pieces on the rosette coming from the lines in the passive tree not belonging to P, and separates on its two sides (one on the interior and one on the exterior of the line) the two pieces on the rosette coming form the lines belonging to P. Thus one can perform the R move and exchange the line on the rosette with any one of the lines in P. As the two vertices it connects are of opposing orientation, any line on the rosette encompasses an odd number of pieces, at least one of which comes form P. In the classical case the rotations allow one to always simplify rosettes with three lines such that at most two intersect. In some cases this holds also for colored rosettes, as is apparent from Fig. 9 (remember that the pieces α, β, γ, δ are all oriented counterclockwise and note that the R relation preserves these orientations). However this is not generic. For the graph of Fig. 10 no R move can disentangle the three lines in the corresponding rosette. It is thus very difficult to give a full characterization of colored rosettes and define some canonical rosettes to implement the usual proofs of universality. However the language developed here and the notion of colored rosettes should provide clues to a partial characterization of three dimensional manifolds and pseudomanifolds.
6. Conclusion In this paper, we introduced topological polynomials adapted to CGFT graphs, obeying a deletion contraction equation. To each CGFT graph we associated
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Figure 10. Three lines which cannot be disentangled a boundary graph, and defined and studied its homology. Moreover, in an attempt to address the question of universality of our polynomials we were led to introduce colored rosettes and adapted R moves. The generalized polynomials reproduce the classical ones for certain values of the parameters. Although the polynomials we define are not the unique generalization one can consider, they already encode nontrivial topological information as seen by the behavior under rescaling of their arguments. One can for instance consider generalizations, in which instead of associating a unique variable xp which counts all the p-cells, one associates a different variable to each p-cell. Such a polynomial would presumably obey a generalized deletion-contraction for p-cells instead of lines.
Acknowledgements The author would like to thank Vincent Rivasseau for very useful discussions at an early stage of this work, and an anonymous referee for suggesting the inclusion of the discussion on universality. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.
Appendix In this appendix, we detail the topological polynomial and check the contraction deletion relation for the graph in Fig. 4. The subgraphs of this graph are: the total graph formed by the lines 0123, subgraphs with three lines 123, 023, 013, 012, sub graphs with two lines 01, 02, 03, 12, 13, 23, those with one line 0, 1, 2, 3 and the subgraph with zero lines. The polynomial of the complete graph is then
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Figure 11. The graphs G − l and G/l PG = β0 β1 β2 β3 x20 x41 x62 x43 x4
+ (β1 β2 β3 + β0 β2 β3 + β0 β1 β3 + β0 β1 β2 ) x20 x31 x32 x3 x4 y02 y13 y23 y3
+ (β0 β1 + β0 β2 + β0 β3 + β1 β2 + β1 β3 + β2 β3 ) x20 x21 x2 x4 y04 y16 y24 y3 + (β0 + β1 + β2 + β3 ) x20 x1 x4 y06 y19 y25 y3 + x20 x24 y08 y112 y28 y32 .
(22)
Consider for instance the contributions of the subgraph 012, represented in Fig. 6. It has two vertices, three lines 0, 1 and 2, three internal faces 01, 02 and 12, one internal bubble 012 and one connected component. This yields a factor x20 x31 x32 x3 x4 . Its boundary graph is represented on the right hand side of Fig. 6. It has two vertices (both colored 3), three lines colored 03, 13 and 23, three faces, one formed by the lines 01 and 02, another one formed by the lines 01, 03 and the third one formed by the lines 02 and 03, and one connected component, yielding a factor y02 y13 y23 y3 . Multiplying the two factors reproduces the coefficient of β0 β1 β2 in Eq. (22) Chose a line, say 0. The graphs G − l and G/l are represented in Fig. 11 where the passive line l0 of G/l is represented as a dotted line. The graph G − l has subgraphs made of lines 123, 12, 23, 13, 1, 2, 3 and the subgraph with zero lines. Thus PG−l = β1 β2 β3 x20 x31 x32 x3 x4 y02 y13 y23 y3
+ (β1 β2 + β1 β3 + β2 β3 ) x20 x21 x2 x4 y04 y16 y24 y3 + (β1 + β2 + β3 ) x20 x1 x4 y06 y19 y25 y3 + x20 x24 y08 y112 y28 y32 .
(23)
All the subgraphs of G/l will have l0 ∈ L2 as a passive line. They are formed by the active lines 123, 12, 23, 13, 1, 2, 3 and the graph with no active line. Therefore PG/l = β1 β2 β3 x20 x41 x62 x43 x4 + (β2 β3 + β1 β3 + β1 β2 ) x20 x31 x32 x3 x4 y02 y13 y23 y3 + (β1 + β2 + β3 ) x20 x21 x2 x4 y04 y16 y24 y3 + x20 x1 x4 y06 y19 y25 y3 ,
(24)
and direct inspection shows that PG = β0 PG/l + PG−l .
(25)
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References [1] Gurau, R.: Colored group field theory. arXiv:0907.2582 [hep-th] [2] Nakanishi, N.: Graph Theory and Feynman Integrals. Gordon and Breach, New York (1970) [3] Itzykson, C., Zuber, J.-B.: Quantum Field Theory. McGraw and Hill, New York (1980) [4] David, F.: A model of random surfaces with nontrivial critical behavior. Nucl. Phys. B 257, 543 (1985) [5] Ginsparg, P.: Matrix models of 2-d gravity. arXiv:hep-th/9112013 [6] Gross, M.: Tensor models and simplicial quantum gravity in >2-D. Nucl. Phys. Proc. Suppl. 25A, 144–149 (1992) [7] Sasakura, N.: Tensor model for gravity and orientability of manifold. Mod. Phys. Lett. A 6, 2613 (1991) [8] Connes, A.: Noncommutative Geometry. Academic Press Inc., San Diego (1994) [9] Douglas, M.R., Nekrasov, N.A.: Noncommutative field theory. Rev. Mod. Phys. 73, 977 (2001). arXiv:hep-th/0106048 [10] Grosse, H., Wulkenhaar, R.: Renormalization of φ4 -theory on noncommutative R4 in the matrix base. Commun. Math. Phys. 256(2), 305 (2005). arXiv:hepth/0401128 [11] Grosse, H., Wulkenhaar, R.: Power-counting theorem for non-local matrix models and renormalization. Commun. Math. Phys. 254(1), 91 (2005). arXiv:hepth/0305066 [12] Rivasseau, V., Vignes-Tourneret, F., Wulkenhaar, R.: Renormalization of noncommutative φ4 -theory by multi-scale analysis. Commun. Math. Phys. 262, 565 (2006). arXiv:hep-th/0501036 [13] Gurau, R., Magnen, J., Rivasseau, V., Vignes-Tourneret, F.: Renormalization of non-commutative φ44 field theory in x space. Commun. Math. Phys. 267(2), 515 (2006). arXiv:hep-th/0512271 [14] Gurau, R., Magnen, J., Rivasseau, V., Tanasa, A.: A translation-invariant renormalizable non-commutative scalar model. Commun. Math. Phys. 287, 275 (2009). arXiv:0802.0791 [math-ph] [15] ’t Hooft, G.: A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461 (1974) [16] Grosse, H., Wulkenhaar, R.: The beta-function in duality-covariant noncommutative φ4 -theory. Eur. Phys. J. C35, 277 (2004). arXiv:hep-th/0402093 [17] Disertori, M., Rivasseau, V.: Two and three loops beta function of non commutative phi(4)**4 theory. Eur. Phys. J. C 50, 661 (2007). arXiv:hep-th/0610224 [18] Disertori, M., Gurau, R., Magnen, J., Rivasseau, V.: Vanishing of beta function of non commutative phi(4)**4 theory to all orders. Phys. Lett. B 649, 95 (2007). arXiv:hep-th/0612251 [19] Gurau, R., Rosten, O.J.: Wilsonian renormalization of noncommutative scalar field theory. JHEP 0907, 064 (2009). arXiv:0902.4888 [hep-th] [20] Geloun, J.B., Gurau, R., Rivasseau, V.: Vanishing beta function for Grosse-Wulkenhaar model in a magnetic field. Phys. Lett. B 671, 284 (2009). arXiv:0805.4362 [hep-th]
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[21] Boulatov, D.: A model of three-dimensional lattice gravity. Mod. Phys. Lett. A 7, 1629–1646 (1992). arXiv:hep-th/9202074 [22] Freidel, L.: Group field theory: an overview. Int. J. Phys. 44, 1769–1783 (2005). arXiv:hep-th/0505016 [23] Oriti, D.: Quantum Gravity. In: Fauser, B., Tolksdorf, J., Zeidler, E. (eds.) Birkhauser, Basel (2007). arXiv:gr-qc/0512103 [24] De Pietri, R., Petronio, C.: Feynman diagrams of generalized matrix models and the associated manifolds in dimension 4. J. Math. Phys. 41, 6671–6688 (2000). arXiv:gr-qc/0004045 [25] Barrett, J., Nash-Guzman, I.: arXiv:0803.3319 (gr-qc) [26] Engle, J., Pereira, R., Rovelli, C.: The loop-quantum-gravity vertex-amplitude. Phys. Rev. Lett. 99, 161301 (2007). arXiv:0705.2388 [27] Engle, J., Pereira, R., Rovelli, C.: Flipped spinfoam vertex and loop gravity. Nucl. Phys. B 798, 251 (2008). arXiv:0708.1236 [gr-qc] [28] Livine, E.R., Speziale, S.: A new spinfoam vertex for quantum gravity. Phys. Rev. D 76, 084028 (2007). arXiv:0705.0674 [gr-qc] [29] Freidel, L., Krasnov, K.: A new spin foam model for 4d gravity. Class. Quant. Grav. 25, 125018 (2008). arXiv:0708.1595 [gr-qc] [30] Conrady, F., Freidel, L.: On the semiclassical limit of 4d spin foam models. Phys. Rev. D 78, 104023 (2008). arXiv:0809.2280 [gr-qc] [31] Bonzom, V., Livine, E.R., Smerlak, M., Speziale, S.: Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model. Nucl. Phys. B 804, 507 (2008). arXiv:0802.3983 [gr-qc] [32] Freidel, L., Gurau, R., Oriti, D.: Group field theory renormalization—the 3d case: power counting of divergences. Phys. Rev. D 80, 044007 (2009). arXiv: 0905.3772 [hep-th] [33] Magnen, J., Noui, K., Rivasseau, V., Smerlak, M.: arXiv:0906.5477 [hep-th] [34] Adbesselam, A.: On the volume conjecture for classical spin networks. arXiv: 0904.1734[math.GT] [35] Geloun, J.B., Magnen, J., Rivasseau, V.: Bosonic Colored Group Field Theory. arXiv:0911.1719 [hep-th] [36] Kirchhoff, G.: Uber die Aufl¨ osung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Str¨ ome gef¨ urht wird. Ann. Phys. Chem. 72, 497–508 (1847) [37] Tutte, W.T.: Graph Theory. Addison-Wesley, Reading (1984) [38] ‘t Hooft, G., Veltman, M.: Regularization and renormalization of gauge fields. Nucl. Phys. B44(1), 189–213 (1972) [39] Crapo, H.H.: The Tutte polynomial. Aequationes Mathematicae 3, 211–229 (1969) [40] Sokal, A.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids, Surveys in combinatorics 2005, pp. 173–226. London Math. Soc. Lecture Note Ser., vol. 327. Cambridge University Press, Cambridge (2005). arXiv:math/0503607 [41] Jackson, B., Procacci, A., Sokal, A.D.: Complex zero-free regions at large |q| for multivariate Tutte polynomials (alias Potts-model partition functions) with general complex edge weights. arXiv:0810.4703v1 [math.CO]
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[42] Bollob´ as, B., Riordan, O.: A polynomial invariant of graphs on orientable surfaces. Proc. Lond. Math. Soc. 83, 513–531 (2001) [43] Bollob´ as, B., Riordan, O.: A polynomial of graphs on surfaces. Math. Ann. 323, 81–96 (2002) [44] Ellis-Monaghan, J., Merino, C.: Graph polynomials and their applications. I. The Tutte polynomial. arXiv:0803.3079 [45] Ellis-Monaghan, J., Merino, C.: Graph polynomials and their applications. II. Interrelations and interpretations. arXiv:0806.4699 [46] Gurau, R., Rivasseau, V.: Parametric representation of noncommutative field theory. Commun. Math. Phys. 272, 811 (2007). arXiv:math-ph/0606030 [47] Rivasseau, V., Tanasa, A.: Parametric representation of ‘critical’ noncommutative QFT models. Commun. Math. Phys. 279, 355 (2008). arXiv:mathph/0701034 [48] Tanasa, A.: Parametric representation of a translation-invariant renormalizable noncommutative model. arXiv:0807.2779 [math-ph] [49] Krajewski, T., Rivasseau, V., Tanasa, A., Wang, Z.: Topological Graph Polynomials and Quantum Field Theory. Part I. Heat Kernel Theories. arXiv:0811.0186 [math-ph] [50] Bollob´ as, B., Riordan, O.: A Tutte polynomial for coloured graphs. Combin. Probab. Comput. 8, 45–93 (1999) Razvan Gurau Perimeter Institute for Theoretical Physics Waterloo, ON N2L 2Y5, Canada e-mail:
[email protected] Communicated by Carlo Rovelli. Received: November 16, 2009. Accepted: April 6, 2010.
Ann. Henri Poincar´e 11 (2010), 585–609 c 2010 Springer Basel AG 1424-0637/10/040585-25 published online June 8, 2010 DOI 10.1007/s00023-010-0038-3
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A Uniqueness Theorem for Degenerate Kerr–Newman Black Holes Piotr T. Chru´sciel and Luc Nguyen Abstract. We show that the domains of dependence of stationary, I + -regular, analytic, electro-vacuum space–times with a connected, non-empty, rotating, degenerate event horizon arise from Kerr–Newman space–times.
1. Introduction A classical problem in general relativity is that of classification of domains of outer communication of suitably regular black hole space–times. A complete solution for stationary, I + -regular, analytic, vacuum, connected nondegenerate black holes has been given in [12], building on the fundamental work in [7,24,34,37,38] and others; see [1,2] for some progress towards removing the hypothesis of analyticity. The analysis in [12] has been extended to the electro-vacuum case in [18,19] (see [6,8,29] for previous results). The aim of this work is to remove the condition of non-degeneracy in the rotating case (here Mext denotes the domain of outer communications; the reader is referred to [12] for terminology and further notation): Theorem 1.1. Let (M , g) be a stationary, I + -regular, analytic, electro-vacuum space–time with connected, non-empty, rotating, degenerate future event horizon I + (Mext ) ∩ ∂Mext . Then (Mext , g) is isometrically diffeomorphic to the domain of outer communications of a Kerr–Newman space–time. Non-rotating, degenerate, vacuum and suitably well-behaved solutions are expected not to exist; here one should keep in mind that while the usual staticity argument for non-rotating configurations applies both for non-degenerate [37] (compare [12, end of Section 7]) and degenerate [14, Section 5] configurations, it requires existence of a maximal surface, which has only been proved in the non-degenerate case so far [17]. An alternative proof of nonexistence of non-rotating, vacuum, degenerate solutions could proceed by showing that all associated near-horizon geometries are axisymmetric, but this remains to be seen. Static electrovacuum solutions with degenerate components have been classified in [15], see also [14].
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We note two recent papers [3,21] where it is explicitly shown that the horizon of the near-horizon metric corresponds to a point on the ρ = 0 axis, where ρ is a Weyl coordinate defined for the near-horizon metric. The remaining uniqueness claims, in the spirit of our Theorem 1.1 of [3,21] do not appear to be properly justified. Indeed, the first element needed to prove Theorem 1.1, and missing in the existing literature when degenerate horizons are present, is the global reduction to a harmonic map problem; equivalently, one needs to prove that the area density of the orbits of the isometry group can be used as one of global coordinates on the domain of outer communications; this is established below in Theorem 3.3. The other element missing in the existing literature is the proof that the harmonic map associated to (M , g) lies a finite distance to a Kerr–Newman one when degenerate horizons are present; we do this below in Theorem 3.4 in vacuum and Theorem 3.5 in electrovacuum. The remaining arguments of the proof of Theorem 1.1 are as in [12,19]; for the convenience of the reader we describe a few key steps in Sect. 4. A different approach to the uniqueness problem can be found in [30,33]. The results proved here are similarly needed to justify the approach there. Our analysis below can be used to provide a uniqueness theorem for stationary and axisymmetric space–times with several black hole components, along the lines of Corollary 6.3 of [12]; note that many such vacuum configurations are excluded by the analysis in [32].
2. Adapted Coordinates Assuming I + -regularity and analyticity, it follows from the Structure Theorem 4.5 in [12] that Hawking’s rigidity theorem [12, Theorem 4.13] applies, and so for each rotating connected component of the future event horizon I + (Mext ) ∩ ∂Mext there exists on Mext a Killing vector field ξ tangent to the generators, without zeros on I + (Mext ) ∩ ∂Mext , as well as a second Killing vector field η, commuting with ξ, and generating a U (1) action on M . Introducing null Gaussian coordinates [31] near a connected degenerate component of I + (Mext ) ∩ ∂Mext , the metric there takes the form r, x ˜) dv 2 − 2 dv d˜ g = −˜ r2 F˜ (˜ r + 2˜ r ha (˜ r, x ˜) dv d˜ xa + hab (˜ r, x ˜) d˜ xa d˜ xb , (2.1) ˜ = (˜ xa ), and the x ˜a ’s are where ξ = ∂v , the horizon is at r˜ = 0, we write x coordinates on a two dimensional cross-section of the horizon, which is spherical by the topology theorem [16]. All functions are smooth functions of their arguments near r˜ = 0. It has been shown in [23], and rediscovered in [28] (see also [27]), that, for axisymmetric stationary vacuum metrics, the leading order behaviour of the functions above coincides with that of the extreme Kerr metric. We choose the coordinates x ˜a at r˜ = 0 to coincide with the spherical Boyer–Lindquist ˜ coordinates (θ, ϕ) of the Kerr metric [compare (3.3)–(3.8) below]. A similar procedure applies to the electro-vacuum situation, using [28]. We will return to the details of those constructions in Sects. 3.1 and 3.2. The coordinates on the horizon are then propagated away from the horizon so as to obtain
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the form (2.1) of the metric. Then η = ∂ϕ , and since the commutator [ξ, η] vanishes, the construction leading to (2.1) can be carried out so that all metric functions are independent of both v and ϕ. It turns out to be convenient to rewrite (2.1) as ˜ dv 2 − 2 dv (d˜ ˜ dθ) ˜ + hϕϕ (dϕ + r˜ α(˜ ˜ dv)2 g = −˜ r2 F (˜ r, θ) r + r˜λ(˜ r, θ) r, θ) ˜ dv) dθ˜ + h ˜˜ dθ˜2 . + 2h ˜ (dϕ + r˜ α(˜ r, θ) (2.2) ϕθ
θθ
To obtain this form of the metric one defines α as α :=
g(∂ϕ , ∂v ) gϕv , ≡ r˜gϕϕ r˜g(∂ϕ , ∂ϕ )
(2.3)
and the other functions in (2.2) are then obtained by redefinitions: F = F˜ − gϕϕ α2 ,
λ = −hθ˜ + gϕθ˜α,
(2.4)
with hϕϕ = gϕϕ , etc. Since gϕϕ vanishes at zeros of ϕ, smoothness of α at the zero-set of ∂ϕ requires justification; this proceeds as follows: Since ∂ϕ and ∂v are Killing vector fields, both g(∂ϕ , ∂v ) and g(∂ϕ , ∂ϕ ), and hence their ratio, are scalar functions on space–time. So smoothness of the ratio is obvious away from zeros of g(∂ϕ , ∂ϕ ). To proceed further, we need to understand the nature of the zero-set of ∂ϕ . In the current coordinate system the Killing vector field η coincides with ∂ϕ . It is well known that a periodic Killing vector field cannot be null on a causal domain of outer communications Mext . It further follows from [12, Theorem 4.5] that in I + -regular space–times the Killing vector field η cannot be null on I + (Mext )∩∂Mext . So, under the hypothesis of I + -regularity, on this last region the function g(∂ϕ , ∂ϕ ) vanishes only at zeros of η. Consider then a point p at which a Killing vector field η vanishes. It is also well known (see, e.g., [11, Proposition 7.1]) that, in four-dimensional space–times, there exists a normal coordinate system {xμ } centred at p such that: 1.
either there exist constants βμ ∈ R, μ = 0, 1, not both zero, such that η = β0 (x0 ∂1 + x1 ∂0 ) + β1 (x3 ∂2 − x2 ∂3 );
2.
or there exists a constant a ∈ R∗ such that η = a (x0 − x2 )∂1 + x1 (∂0 + ∂2 ) .
(2.5)
(2.6)
Exponentiating, in normal coordinates near p the action of the isometry group φt generated by η is linear and takes the form ⎛ ⎞⎛ 0⎞ 0 0 cosh(β0 t) sinh(β0 t) x ⎜ sinh(β0 t) cosh(β0 t) ⎟ ⎜ x1 ⎟ 0 0 ⎜ ⎟⎜ ⎟ (2.7) ⎝ 0 0 cos(β1 t) − sin(β1 t) ⎠ ⎝ x2 ⎠ x3 0 0 sin(β1 t) cos(β1 t)
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in case 1., while in case 2. the matrix B ν μ := ∇μ η ν is nilpotent, with B 3 = 0, so that the matrix Λt associated with the action of φt is ⎛ ⎞ 0 a 0 0 ⎜a 0 a 0⎟ 2 ⎟ Λt = Id + tB + t2 B 2 , with B = ⎜ ⎝ 0 −a 0 0 ⎠ . 0 0 0 0 This shows that periodic orbits are not possible in the second case, while in the first they are possible if and only if β0 = 0. Smoothness of α can now be established by adapting the analysis of the proof of [9, Proposition 3.1] to the current setting; we provide the details to exhibit some key factorizations needed in the arguments that follow: Consider a covering of A := {η = 0} by domains of definition O of smooth coordinate systems xA , A = 0, 1, and for q ∈ O let xa , a = 2, 3, denote normal coordinates on expq {(Tq A )⊥ }. Note that the coordinates (x2 , x3 ) here are not identical with the ones in (2.5)–(2.6), but the xa |expp {(Tp A )⊥ } ’s coincide, where p is as in the analysis leading to (2.5)– (2.6). We have just seen that A is a smooth timelike submanifold of M ; and A is totally geodesic (in the sense of having vanishing second fundamental form) by standard arguments. Set (xμ ) = (xA , xa ), and
(2.8) ρ˜ = (x2 )2 + (x3 )2 . We have the following local form of the metric g =˚ g dxA dxB + AB =:˚ g
+
A,a
3
(dxa )2
a=2
O(˜ ρ) dxA dxa +
O(˜ ρ2 ) dxA dxB +
A,B
O(˜ ρ2 ) dxa dxb ,
(2.9)
a,b
with ˚ g the (Lorentzian) metric induced by g on A . The O(˜ ρ2 ) character of a b ρ) character the dx dx error terms is standard in normal coordinates; the O(˜ ρ2 ) character of of the dxa dxA error terms comes from orthogonality; the O(˜ the dxA dxB error terms follows from the totally geodesic character of A . The Killing vector field η takes the form η = x3 ∂2 − x2 ∂3 = ∂ϕ , where ρ cos ϕ, ρ˜ sin ϕ). (x2 , x3 ) = (˜
(2.10)
When expressed in terms of ρ˜ and ϕ, the functions gμν := g(∂xμ , ∂xν ) are smooth functions of the xμ ’s. Let Rπ denote a rotation by π in the (xa )planes; Rπ is obtained by flowing along η a parameter-time π and is therefore an isometry, leading to gab (xA , −x2 , −x3 ) = gab (xA , x2 , x3 ), gAB (xA , −x2 , −x3 ) = gAB (xA , x2 , x3 ), gAa (xA , −x2 , −x3 ) = −gAa (xA , x2 , x3 ).
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In particular all odd-order derivatives of gab with respect to the xa ’s vanish at {xa = 0}, etc. Those symmetry properties together with Borel’s summation Lemma imply that there exist smooth fields bAB (xC , s), γA (xC , s), and γ(xC , s) such that
gAB (xC , x2 , x3 ) = bAB (xC , ρ˜2 ), (2.11) gAb η b (xB , x2 , x3 ) = ρ˜2 γA (xB , ρ˜2 ), u(xA , x2 , x3 ) := (g(η, η)) (xA , x2 , x3 ) = ρ˜ 1+ ρ˜2 γ(xA , ρ˜2 ) .
Similarly, let n = xa ∂a , then gab η a nb and gab na nb are smooth funcρ4 ), tions invariant under the flow of η, with gab η a nb = (gab − δab )η a nb = O(˜ ρ4 ), hence there exist smooth functions ζ(xA , s) and σ(xA , s) gab na nb = ρ˜2 +O(˜ such that gab η a nb (xA , x2 , x3 ) = ρ˜4 ζ(xA , ρ˜2 ), gab na nb (xA , x2 , x3 ) = ρ˜2 (1 + ρ˜2 σ(xA , ρ˜2 )). We note similar formulae for the Maxwell two-form F and its Hodge-dual ∗F: ˆ A , ρ˜2 ), ∗Fab η a nb (xA , x2 , x3 ) Fab η a nb (xA , x2 , x3 ) = ρ˜2 ζ(x ˜ A , ρ˜2 ), (2.12) = ρ˜2 ζ(x B 2 3 b B 2 3 2 B 2 b FAb η (x , x , x ) = ρ˜ γˆA (x , ρ˜ ), ∗FAb η (x , x , x ) = ρ˜2 γ˜A (xB , ρ˜2 ), FAb nb (xB , x2 , x3 ) = ρ˜2 γˇA (xB , ρ˜2 ), ∗FAb nb (xB , x2 , x3 )
(2.13)
= ρ˜2 γ˙ A (xB , ρ˜2 ),
(2.14)
ˆ ζ, ˜ γˆA , γ˜A , γˇA for some smooth “sphere functions” (to be defined shortly) ζ, and γ˙ A . In the same fashion one finds existence of a smooth one-form λA (s, xb )dxA such that ρ2 , xb ). (gAa na ) (x1 , x2 , xb ) = ρ˜2 λA (˜ In polar coordinates (2.10) one therefore obtains g(η, ·) = ρ˜2 (1 + ρ˜2 γ)2 dϕ + ζ ρ˜ d˜ ρ + γA dxA . Writing g in the form g = u2 (dϕ + χj dy j )2 + γjk dy j dy k ,
(2.15)
=:χ
with y = (x , ρ˜), one has g(η, ·) = u2 (dϕ + χj dy j ) leading to j
A
χ=
γA ρ˜ζ d˜ ρ+ dxA , (1 + ρ˜2 γ)2 (1 + ρ˜2 ψ)2
ρ2 + bAB dxA dxB + 2λA ρ˜ d˜ ρ dxA γjk dy j dy k = (1 + ρ˜2 σ) d˜ 2
− u χi χj dy dy , i
j
(2.16)
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in particular the functions γρ˜ρ˜, γAB , and γAρ˜/˜ ρ are smooth functions of ρ˜2 A and x . We have proved: Proposition 2.1. The one-form χ defined in (2.15) extends smoothly to the rotation axis A = {η = 0}. In particular gvϕ = χ(∂v ) gϕϕ is a smooth function on space–time. Since it vanishes at r˜ = 0, the quotient rgϕϕ ) is also a smooth function on space–time by Taylor’s theorem. Hence gvϕ /(˜ the function α defined in (2.3) is smooth. Smoothness of A and λ as in (2.4) follows. In the coordinate system adapted to the horizon as in (2.2), the intersection of the axis A and of the Killing horizon corresponds to sin θ˜ = 0. To see that this remains true in a neighbourhood of the horizon, recall that the construction of the Gauss normal coordinates in (2.1) involves the family of null geodesics normal to the section S := {v = 0} of the connected component ˜ ϕ) of the future event horizon under consideration: the local coordinates (θ, − on S are first Lie-propagated to J˙ (S) along the normal null geodesics, and then to a neighbourhood of the Killing horizon along the flow of ∂v . Since S is invariant under the action of U (1), so is its normal bundle. It follows from (2.7) that, at the north and south poles of S, which are fixed points of the rotational Killing vector η, those normal geodesics are initially tangent to A . But A is totally geodesic, so in fact those geodesics remain on A : one of them is the generator of the event horizon, the second one is the one which is used to ˜ ϕ) away from the horizon. Now, A is also invaripropagate the coordinates (θ, ant under the flow of ∂v , which is tangent on A to that null normal geodesic to S which coincides with the generator of the horizon. Thus ∂v is transversal to the other null geodesic on A , so flowing this other geodesic along ∂v fills out a neighbourhood of this geodesic within A . Since θ˜ is constant along the flow of ∂v , we conclude that sin θ˜ = 0 on A . Finally, e.g. by dimension considerations, we obtain that {sin θ˜ = 0} coincides with A = {˜ ρ = 0} in a collar neighbourhood of S. ˜ which is equivaSo, near θ˜ = 0 the function ρ˜ of (2.8) is equivalent to θ, ˜ ˜ lent to sin θ, and by the arguments above for small θ we have ρ˜ ˜ = f˜(˜ r, θ), (2.17) sin θ˜ for some function f˜, smooth in its arguments, bounded away from zero, and which can be smoothly extended to an even function of θ˜ across zero. Similarly ˜ which is again equivalent to near θ˜ = π the function ρ˜ is equivalent to π − θ, ˜ ˜ sin θ near θ = π, and so the function f in (2.17) extends smoothly across θ = π to a function which is bounded away from zero and even in π − θ for θ close ˜ we conclude that (2.17) to π. Since (2.17) is trivial away from the zeros of sin θ, holds everywhere.
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Functions of θ˜ ∈ [0, π] with the smooth even extension properties near zero and π, as just described in the last paragraph, will be called sphere functions: indeed, a function of θ˜ defines a smooth function on a sphere if and only if it is a sphere-function in the sense just defined. Equations (2.11) and (2.18) lead us to hϕϕ ˜ = f (˜ r, θ), sin2 θ˜
(2.18)
for some sphere function f , smooth in its arguments, and bounded away from zero. It also follows from what has been said so far that the functions α, λ, ˜ and λ/ sin θ˜ are smooth sphere functions of r˜ ˜ h ˜˜, h ˜/ sin θ, F , hϕϕ / sin2 θ, ϕθ θθ ˜ and θ. As the next step, we modify the coordinate r˜ to a new coordinate rˆ by setting ˜ r + r˜λ(˜ ˜ dθ), ˜ r, θ)(d˜ r, θ) dˆ r = eχ˜ (˜
(2.19)
normalized so that rˆ(˜ r = 0, θ˜ = 0) = 0. Equivalently, ∂r˜rˆ = eχ˜ ,
∂θ˜rˆ = r˜eχ˜ λ.
(2.20)
˜ − r˜λ∂r˜χ ˜ = ∂r˜(˜ rλ), ∂θ˜χ
(2.21)
The integrability conditions for rˆ give
which can be solved by shooting characteristics from the north pole θ˜ = 0, where we impose χ ˜ = 0. Again smoothness of χ ˜ and of rˆ at the north and south poles requires justification: Since λ/ sin θ˜ is a smooth sphere function of ˜ by matching powers in a power-series expansion of χ r˜ and θ, ˜ in (2.21) one ˜ finds that χ ˜ is a smooth sphere function of r˜ and θ. In other words, for each r˜, χ ˜ defines naturally a smooth function on S 2 . A similar argument applies to (2.21). Since ∂θ˜rˆ = 0 at r˜ = 0 from (2.19), we have ˜ =0 rˆ(˜ r = 0, θ) ˜ Since χ for all θ. ˜ is a smooth function on Ir˜ × S 2 , where Ir˜ is the interval of definition of r˜, (2.20) implies that both rˆ r˜
and
r˜ rˆ
are smooth functions on Ir˜ × S 2 near {ˆ r = 0}.
(2.22)
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To summarise, we have shown: Proposition 2.2. Near a spherical degenerate Killing horizon in an axially symmetric spacetime the metric can be written in the form ˜ dv2 + 2 ψ(ˆr, θ) ˜ dv dˆr + hϕϕ (ˆr, θ)(dϕ ˜ ˜ dv)2 g = −ˆ r2 F (ˆ r, θ) + ˆr α(ˆr, θ) ˜ (dϕ + ˆr α(ˆr, θ) ˜ dv)dθ˜ + h ˜˜(ˆr, θ) ˜ dθ˜2 , + h ˜ (ˆ r, θ) (2.23) θϕ
θθ
where ∂v is the Killing field defining the Killing horizon, ∂ϕ is the axial Killing ˜ parameterize a two-dimensional spherifield, the horizon is at rˆ = 0, (ϕ, θ) ˜ h ˜˜, h ˜/ sin θ˜ (and cal cross-section of the horizon, and F , α, ψ, hϕϕ / sin2 θ, ϕθ θθ 2 ˜ hence also det hab / sin θ) are smooth sphere functions in a neighbourhood of rˆ = 0. ˜ Similarly, for any anti-symmetric tensor F the functions Fvϕ / sin2 θ, 2 ˜ ˜ ˜ Frϕ / sin θ, Fvθ˜/ sin θ, Fθϕ ˜ / sin θ and Fvˆ r are smooth sphere functions.
3. Geometric Analysis Near a Degenerate Horizon In this section, we study the geometric implications, near a degenerate horizon E0 in an axially symmetric and stationary electro-vacuum space–time (M , g), of the metric form (2.23). Recall that it has been shown in [23] in vacuum, and in [28] in electrovacuum, that the near-horizon geometry is determined uniquely by the area A0 of a cross-section S0 of the horizon E0 , the electric charge qe and the magnetic charge qb of the horizon. For convenience of notation we introduce the area radius of the horizon: A0 . (3.1) r0 = 4π Note that, by the near-horizon analysis in [28], r02 ≥ qe2 + qb2 .
(3.2)
3.1. The Near-Horizon Limit in Vacuum Assume that (M , g) is a vacuum space–time, so qe = qb = 0. The near-horizon geometry of the extreme Kerr solution which has horizon area A0 is given by (see, e.g., [4]): 2 2 1 + cos2 θ r02 2 2r02 sin2 θ rˆ rˆ 2 2 2 gNHK = r + r0 dθ + − 2 dt + 2 dˆ dφ + 2 dt , 2 r0 rˆ 1 + cos2 θ r0 √ where (t, rˆ+r0 / 2, θ, φ) is the Boyer–Lindquist coordinate system for the Kerr solution. By the change of variables rˆ r2 v = t − 0 , ϕ = φ − log , rˆ r0
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the above metric can be rewritten as 2 2 1 + cos2 θ 2r2 sin2 θ rˆ rˆ r + r02 dθ2 + 0 dv . gNHK = − 2 dv 2 + 2 dv dˆ dϕ + 2 r0 1 + cos2 θ r02 We then use the results in [23] and the analysis in Sect. 2 to obtain, in a ˜ such that neighbourhood of E0 , a null Gaussian coordinate system (v, ϕ, rˆ, θ) ˜ ϕ) agree with the the metric g takes the form (2.23) and the coordinates (θ, coordinate (θ, ϕ) of the above Kerr metric at rˆ = 0. Furthermore, ˜ = 1 (1 + cos2 θ), ˜ F (0, θ) (3.3) 2r02 ˜ = 1 (1 + cos2 θ), ˜ ψ(0, θ) (3.4) 2 2 ˜ 2 ˜ = 2r0 sin θ , (3.5) hϕϕ (0, θ) 1 + cos2 θ˜ ˜ = 0, (3.6) h ˜(0, θ) ϕθ
˜ = 1 r2 (1 + cos2 θ), ˜ hθ˜θ˜(0, θ) 2 0 ˜ = 1. α(0, θ) r02
(3.7) (3.8)
Observe that Eqs. (3.5)–(3.7) together with Proposition 2.2 allow us to write hϕϕ =
2r02 sin2 θ˜ βϕϕ , 1 + cos2 θ˜
det h = r04 sin2 θ˜ β,
(3.9)
˜ which satisfy βϕϕ (0, θ) ˜ ≡ for some smooth sphere functions βϕϕ and β of (ˆ r, θ), ˜ ≡ 1. β(0, θ) 3.2. The Near-Horizon Limit in Electrovacuum In the general case where (M , g) is electrovacuum, by [28], the near-horizon geometry is characterized by that of the Kerr–Newman solution which has the same horizon area parameter A0 and charge parameters qe and qb . In the Kerr–Newman case, the near-horizon fields can be obtained by first applying a duality rotation to FKN (to account for the magnetic charge), and then calculating the near-horizon limit. Using, e.g., [25, pp.79–80] one finds (compare [4]): 2 m20 + a20 cos2 θ r02 2 rˆ 2 2 2 r + r0 dθ − 2 dt + 2 dˆ gNHKN = r02 r0 rˆ 2 r4 sin2 θ 2a0 m0 rˆ + 2 0 2 dt , dφ + m0 + a0 cos2 θ r04 (m2 − a2 cos2 θ) 2 a0 m0 r02 sin θ cos θ FNHKN = qe − dˆ r ∧ dt dφ ∧ dθ + 2 0 2 0 2 2 2 2 2 (m0 + a0 cos θ) r0 (m0 + a0 cos2 θ) 4 a2 m2 rˆ sin θ cos θ + 2 0 20 2 dθ ∧ dt r0 (m0 + a0 cos2 θ)2
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r02 (m20 − a20 cos2 θ) sin θ 2a0 m0 cos θ dˆ r ∧ dt dφ ∧ dθ+ 2 2 2 2 2 2 2 (m0 +a0 cos θ) r0 (m0 +a0 cos2 θ) 2 a0 m0 (m20 − a20 cos2 θ) rˆ sin θ − dθ ∧ dt . r02 (m20 + a20 cos2 θ)2
Here r0 is as in (3.1), a0 = (r02 − qb2 − qb2 )/2 and m0 = a20 + qe2 + qb2 . Note that the sign of a0 is not determined in [28], but we can always make it positive using the transformation φ → −φ. Introducing the change of variables + qb
v =t−
r02 , rˆ
ϕ=φ−
2a0 m0 log rˆ, r02
we obtain
2 m20 + a20 cos2 θ rˆ 2 2 2 2 dv + 2 dv dˆ r + r dθ − 0 r02 r02 2 r4 sin2 θ 2a0 m0 rˆ + 2 0 2 dv , (3.10) dϕ + 4 2 m + a0 cos θ r0 0 (m20 − a20 cos2 θ) 2 a0 m0 r02 sin θ cos θ dˆ r ∧ dv FNHKN = qe − dϕ ∧ dθ+ 2 2 (m0 + a0 cos2 θ)2 r02 (m20 +a20 cos2 θ) 4 a2 m2 rˆ sin θ cos θ + 2 0 20 2 dθ ∧ dv r0 (m0 + a0 cos2 θ)2 2 2 2 r0 (m0 −a0 cos2 θ) sin θ 2a0 m0 cos θ +qb dˆ r ∧ dv dϕ ∧ dθ + 2 2 2 (m20 +a20 cos2 θ)2 r0 (m0 +a0 cos2 θ) 2 a0 m0 (m20 − a20 cos2 θ) rˆ sin θ − dθ ∧ dv . (3.11) r02 (m20 + a20 cos2 θ)2 gNHKN =
Thus, as already explained, we can select a null Gaussian coordinate sys˜ ϕ) in a neighbourhood of E0 in M such that g takes the form tem (v, rˆ, θ, ˜ ϕ) coincide with the coordinates (θ, ϕ) as in (2.23) there, the coordinates (θ, (3.10) on E0 , and 2 2 2 ˜ ˜ = m0 + a0 cos θ , F (0, θ) r04 2 2 2 ˜ ˜ = m0 + a0 cos θ , ψ(0, θ) 2 r0 4 r0 sin2 θ˜ ˜ = , hϕϕ (0, θ) m2 + a2 cos2 θ˜ 0
0
m20
a20
˜ = 0, hϕθ˜(0, θ) ˜ = hθ˜θ˜(0, θ)
(3.12) (3.13) (3.14) (3.15)
2
˜ + cos θ, ˜ = 2a0 m0 . α(0, θ) r04
(3.16) (3.17)
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Moreover, by Proposition 2.2 we have r4 sin2 θ˜ hϕϕ = 2 0 2 βϕϕ , det h = r04 sin2 θ˜ β, m0 + a0 cos2 θ˜
595
(3.18)
˜ which satisfy βϕϕ (0, θ) ˜ ≡ for some smooth sphere functions βϕϕ and β of (ˆ r, θ) ˜ β(0, θ) ≡ 1. 3.3. The Orbit-Space Metric In the following, we use xA as the dummy variable for r and θ˜ and xa as the dummy variable for v and ϕ. This should not be confused with the coordinates (xA , xa ) of the proof of Proposition 2.1. The Killing part of the metric g is defined as ˜ dv 2 + hϕϕ (ˆ ˜ r2 F (ˆ r, θ) r, θ)(dϕ + rˆ α(r) dv)2 . (3.19) g = −ˆ Note that r2 F hϕϕ . det g = −ˆ In particular, g is Lorentzian for rˆ = 0 if and only if Ahϕϕ is non-negative. In electro-vacuum this follows from (2.18) and from the analysis of the nearhorizon geometry in [23,28] [compare (3.3) and (3.12)]. Alternatively, one can simply assume that this is true and carry-on the analysis from there. The orbit-space metric q is defined as qAB = gAB − gab gAa gBb , where gab is the matrix inverse to g (∂a , ∂b ). In matrix notation (3.19) reads 2 rˆ (−F + α2 hϕϕ ) rˆ α hϕϕ g = , rˆ α hϕϕ hϕϕ and so its inverse reads g−1 = −
1 hϕϕ r α hϕϕ rˆ2 F hϕϕ −ˆ
−ˆ r α hϕϕ . rˆ2 (−F + α2 hϕϕ )
The orbit-space metric is then dˆ r2 det h ˜2 r2 + qθ˜θ˜ dθ˜2 = F −1 ψ 2 2 + dθ . q = qrˆrˆ dˆ rˆ hϕϕ By (3.12)–(3.17) we have 1 qrˆrˆ = 2 m20 + a20 cos2 θ˜ + O(ˆ r) , qθ˜θ˜ = m20 + a20 cos2 θ˜ + O(ˆ r), rˆ with the error terms meant for small rˆ. Now, define the function ρ by
ρ = − det g = rˆ F hϕϕ .
(3.20)
(3.21)
(3.22)
By (3.12) and (3.18) we have ρ = rˆ β˜ρ sin θ˜
(3.23)
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˜ such that for some smooth sphere function β˜ρ = β˜ρ (ˆ r, θ) ˜ = 1 + O(ˆ r, θ) r). β˜ρ (ˆ By the Einstein–Maxwell electro-vacuum equation, ρ is harmonic with respect to q (see, e.g., [40, Section 2]). Let z be minus the harmonic conjugate of ρ, i.e. z is defined up to a constant by
hϕϕ ψ θ˜θ˜ ρ ˜, z,ˆr = q det q ρ,θ˜ = F det h rˆ ,θ
F det h rˆ ρ,ˆr . z,θ˜ = −q rˆrˆ det q ρ,ˆr = − hϕϕ ψ By (3.12)–(3.16) and (3.23), we have ˜ z,ˆr = γ β˜ρ cos θ,
1 ˜ r β˜ρ ),ˆr sin θ, z,θ˜ = − rˆ (ˆ γ
˜ such that γ(0, θ) ˜ ≡ 1. Thus, where γ is some smooth positive function of (ˆ r, θ) up to a shift by a constant, ˜ z = rˆ β˜z cos θ,
(3.24)
where ˜ =1 r, θ) β˜z (ˆ rˆ
rˆ ˜ β˜ρ (s, θ) ˜ ds = 1 + O(ˆ γ(s, θ) r). 0
Altogether (3.23) and (3.24) imply that r2 := ρ2 + z 2 = rˆ2 + O(ˆ r3 )
as rˆ → 0.
We have thus proved: Proposition 3.1. In I + –regular, axisymmetric, stationary and electro-vacuum space–times, every degenerate component of the event horizon corresponds to a point lying on the axis ρ = 0 in the (ρ, z) plane. Remark 3.2. For the sake of simplicity we have stated the result under the hypotheses of Theorem 1.1. However, the analysis above only uses the following: the horizon is degenerate, and has a spherical cross-section S on which the Killing vector ∂v has no zeros; the Killing vector field ∂ϕ is spacelike wherever non-zero; the function ρ is harmonic with respect to the orbit-space metric q; and finally lim
rˆ→0
F det h F hϕϕ . = 1 = lim ˜ rˆ→0 sin2 θ ψ 2 hϕϕ
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3.4. Global Isothermal Coordinates We wish to show that the functions ρ and z provide global coordinates on the quotient manifold Mext /(R × U (1)), where Mext is the domain of outer communications in (M , g). For this we adopt the strategy in [12], which in turn draws on [10]; the arguments there need to be extended in a non-trivial way to cover the current setting. Let B be the manifold obtained from the orbit space Mext /(R×U (1)) by doubling along the axis, as in [10]. The metric q extends smoothly to a smooth metric on B, which we will also denote by q. In this section, we show that the functions ρ and z, defined in the previous section and appropriately extended to the double, provide global isothermal coordinates for (B, q) and hence, by restriction, for Mext /(R × U (1)). As shown in Proposition 3.1, in the connected case the horizon corresponds to a point p in a one-point completion B := B ∪ {p} of B. The point p will be denoted by 0; the reason for this slight abuse of notation will be clear momentarily. For configurations with Nd degenerate components of the horizon and Nr non-degenerate ones, each degenerate horizon will correspond to a point pi in a completion Nr 1 d B := B ∪N i=1 {pi } ∪a=1 Da
where the Da1 ’s are disks corresponding to smooth boundary components for B; see [12] for a detailed description of the non-degenerate components of the event horizon. It should be noted that the point 0 in the former case, and the pi ’s in the latter case, are genuinely not points in B. In a B–neighbourhood of each pi we parameterize B by a small punc˜ with rˆ ∈ (0, 4). x, yˆ) → (ˆ r, θ), tured disc D4 \{0} ⊂ R2 via the polar map (ˆ By (3.21) in this region, q is conformal to ˜ dθ˜2 , r, θ) qˆ := dˆ r2 + r2 f (ˆ ˜ ≡ 1. This can be rewritwhere f is a smooth sphere function such that f (0, θ) ten as ˜ (dˆ r, θ) x2 + dˆ y 2 − dˆ r2 ) qˆ = dˆ r2 + f (ˆ ˜ −1 f (ˆ r, θ) ˜ (dˆ = f (ˆ r, θ) x2 + dˆ y2 ) + (ˆ x dˆ x + yˆ dˆ y )2 . rˆ2 So qˆ will extend smoothly across x ˆ = yˆ = 0 if and only if f − 1 equals rˆ2 times a smooth function of x ˆ and yˆ. If this happens to be the case, we can apply [10, Theorem 2.9] to reach the desired conclusion, Theorem 3.3 below. However, it is not clear that f will take this form in general, so the above strategy needs to be revised to allow general metrics qˆ as above. For this we need to provide first some preliminary analysis. Let Rqˆ denote the Gaussian curvature of qˆ. It is evident that Rqˆ is smooth in all sufficiently small punctured discs D4 \{0}, 0 < < 0 for some 0 , with Rqˆ = O(ˆ r−1 )
and
|DRqˆ| = O(r−2 ) for small rˆ > 0.
(3.25)
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Moreover, the usual formula for the scalar curvature in a frame formalism in dimension two shows that there are functions (3.26) fˆx , fˆy ∈ C ∞ (D4 \{0}) ∩ L∞ (D4 ) such that Rqˆ = ∂x fˆx + ∂y fˆy . ˆ ∈ H01 (D4 ) be the solution to (see e.g. In particular, Rqˆ ∈ H −1 (D4 ). Let u [22, Theorem 8.3]) R −Δqˆu ˆ = 2qˆ in D4 , u ˆ=0
on ∂D4 ,
−2ˆ u
qˆ is flat in D2 \{0}. By (3.26) and standard ellipso that the metric e tic estimates (see e.g. [22, Theorem 8.24]), u ˆ is smooth in D4 \{0}, μ-H¨older continuous in D4 for some μ ∈ (0, 1) with ˆ uC μ (D2 ) ≤ C(, f L∞ (D4 ) , fˆx L∞ (D4 ) , fˆy L∞ (D4 ) ). ˆ∗ ≡ u ˆ in the region which Now, pick any u ˆ∗ ∈ Cc∞ (B)∩C μ (B) such that u is parameterized by D . Define q˜ = e−2ˆu∗ qˆ. It is readily seen that q˜ is flat near 0. Since u ˆ∗ is continuous, q˜ has no conical singularity at the origin, and so the metric q˜ is smooth across 0 in an appropriate differentiable structure (which might, or might not, coincide with the one defined by the coordinates x ˆ and yˆ, but this turns out to be irrelevant for what follows). We can now apply [10, Theorem 2.19] to find a function u ˜ ∈ C ∞ (B) such that q := e−2˜u q˜ is a smooth flat metric on the complete simply connected manifold B. Since the relevant equations are conformally invariant, one can ignore the possible singularities at the pi ’s of the conformal factor relating q and q, and proceed as in [12] (see in particular the argument leading from Equation (6.8) to Equation (6.11) there) to show that ρ and z provide a global coordinate system on Mext /(R × U (1)). We conclude that Theorem 3.3. Let (M , g) be a stationary, axisymmetric, I + -regular, electrovacuum space–time. Then the area function ρ and its harmonic conjugate −z form a global manifestly asymptotically flat coordinate system on Mext /(R × U (1)). For the sake of completeness, we note some regularity properties of u ˆ. Fix some point p ∈ D \{0}. Applying [22, Theorem 8.32] to u − u(0) in D|p|/2 (p) and recalling (3.25), we have |Dˆ u(p)| ≤ C |p|−1 ˆ u − u(0)L∞ (D|p|/2 (p)) + |p| RqˆL∞ (D|p|/2 (p)) ≤ C |p|μ−1 + 1 . It follows that |Dˆ u(p)| ≤ |p|μ−1
for any p ∈ D \{0}.
(3.27)
Similarly, applying [22, Theorem 6.2] to u − u(0) in D|p|/2 (p) and noting that, by (3.25)
RqˆC μ (D|p|/2 (|p|)) ≤ C |p|−1−μ
for any μ ∈ (0, 1],
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we get
|D2 u ˆ(p)| ≤ C |p|−2 ˆ u − u(0)L∞ (D|p|/2 (p)) + |p|μ RqˆC μ (D|p|/2 (p)) ≤ C |p|μ−2 + |p|μ −1 .
We thus have ˆ(p)| ≤ |p|μ−2 |D2 u
for any p ∈ D \{0}.
(3.28)
3.5. Hypersurface-Orthogonality Recall that ξ = ∂v , η = ∂ϕ . It is well-known that, in electro-vacuum (see, e.g., [26]), the plane distribution (Span{ξ, η})⊥ is integrable; equivalently dξ ∧ ξ ∧ η = dη ∧ ξ ∧ η = 0.
(3.29)
By direct computations, we find ! dξ ∧ ξ ∧ η = rˆ2 hϕϕ −α hϕϕ [α ψ],θ˜ + F,θ˜ ψ − ψ,θ˜ F " ˜ dv ∧ dˆ r ∧ dϕ ∧ dθ, + rˆ3 α F −hϕϕ,ˆr hϕθ˜ + hϕϕ hϕθ,ˆ ˜r ! " ˜ dη ∧ ξ ∧ η = −ˆ r ψ h2ϕϕ α,θ˜rˆ2 F −hϕϕ,ˆr hϕθ˜ +hϕϕ hϕθ,ˆ dv∧dˆ r ∧dϕ∧dθ. ˜r Thus the hypersurface orthogonality condition (3.29) reads − ψ h2ϕϕ α,θ˜ + rˆ F −hϕϕ,ˆr hϕθ˜ + hϕϕ hϕθ,ˆ ˜ r = 0,
(3.30)
F,θ˜ ψ + (−F + α2 hϕϕ ) ψ,θ˜ = 0.
(3.31)
3.6. The Ernst Potential of ∂ϕ in Vacuum We now turn our attention to the second missing ingredient required for the uniqueness argument. Namely, we will show that, in a neighbourhood of the horizon E0 , the harmonic map associated to (M , g) lies a finite distance from that associated to the Kerr–Newman solution which has the same parameters A0 , qe and qb . We start with the special case where (M , g) is vacuum. The electro-vacuum case will be considered in Sect. 3.7. The (complex) Ernst potential associated with the Killing vector η = ∂ϕ is defined as X + i Y where X = g(η, η),
dY = ∗(η ∧ dη),
where ∗ is the Hodge operator of g. Here, by a common abuse of notation, we use the same symbol η for the vector η and its metric dual g(η, ·). The existence of the twist potential Y is a consequence of the Einstein vacuum equations; see, e.g., [38, Section 2]. The reference Kerr metric has been chosen to have the same area radius r0 as the metric under consideration, and so from (3.9), we have X=
2r02 sin2 θ˜ (1 + O(ˆ r)). 1 + cos2 θ˜
(3.32)
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To obtain the twist potential Y , a computation gives η ∧ dη = −h2ϕϕ [ˆ r α],ˆr dv ∧ dˆ r ∧ dϕ + rˆ h2ϕϕ α,θ˜ dv ∧ dϕ ∧ dθ˜ r ∧ dθ˜ r α hϕϕ ],ˆr dv ∧ dˆ + rˆ α hϕϕ hϕθ,ˆ ˜ r − hϕθ˜ [ˆ ˜ + −hϕϕ h ˜ + h ˜ hϕϕ,ˆr dˆ r ∧ dϕ ∧ dθ. ϕθ,ˆ r
ϕθ
Using det gμν = −ψ 2 det h, we are led to
" dv ! ∗(η ∧ dη) = √ −ˆ r h2ϕϕ α,θ˜ + rˆ2 F ψ −1 −hϕϕ,ˆr hϕθ˜ + hϕϕ hϕθ,ˆ ˜r det h dˆ r −√ −hϕϕ,ˆr hϕθ˜ + hϕϕ hϕθ,ˆ ˜r det h √ r α],ˆr . + dθ˜ hϕϕ det h ψ −1 [ˆ
In the above formula, the dv component must vanish. This is a consequence of one of the hypersurface orthogonality conditions, namely that dη ∧ ξ ∧ η = 0 (see (3.30)). Thus, 1 r −hϕϕ,ˆr hϕθ˜ + hϕϕ hϕθ,ˆ ∗ (η ∧ dη) = − √ ˜ r dˆ det h √ ˜ + hϕϕ det h ψ −1 [ˆ r α],ˆr dθ. (3.33) Now, as dY = ∗(η ∧ dη), the relations (3.4), (3.8) and (3.9) imply ˜ Y,ˆr = γrˆ sin θ,
Y,θ˜ = 4r02 γθ˜
sin3 θ˜ , ˜2 (1 + cos2 θ)
(3.34)
˜ with γ ˜(0, θ) ˜ ≡ 1. r, θ) where γrˆ and γθ˜ are smooth sphere function of (ˆ θ By [12, Section 6], in a sufficiently regular black hole space–time, in a collar neighbourhood of every component of the Killing horizon the axis of rotation A has exactly two connected components, each of which meets a cross-section of the horizon at exactly one point. Now, by Proposition 2.2, in a neighbourhood of the horizon, ∂ϕ vanishes along {θ˜ = 0} and {θ˜ = π}. Evidently these two sets correspond to different component of A . Denote by A+ and A− the components of A that contain {θ˜ = 0} and {θ˜ = π}, respectively. It is well-known that Y is constant on each component of A . In a neighbourhood of the horizon, this can be seen readily from the first equation in (3.34). Away from the horizon, see e.g. [13, Eq. (2.6)] or [38]. By (3.34), we have π π # # sin3 θ˜ 2 ˜ dθ˜ = 4 r dθ˜ = 4 r02 . Y #A− − Y #A+ = Y,θ˜(0, θ) 0 ˜2 (1 + cos2 θ) 0
0
Hence, shifting Y by a constant if necessary, we can assume that # # Y #A− = 2r02 , Y #A+ = −2r02 .
(3.35)
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Then, by integrating (3.34), Y =−
4r02 cos θ˜ ˜ + δY (ˆ r, θ), 1 + cos2 θ˜
where δY is given by ˜ = 4r2 δY (ˆ r, θ) 0
θ˜ (γθ˜(ˆ r, τ ) − 1) 0
=
−4r02
sin3 τ dτ (1 + cos2 τ )2
π (γθ˜(ˆ r, τ ) − 1) θ˜
sin3 τ dτ. (1 + cos2 τ )2
It thus follows that Y =−
4r02 cos θ˜ ˜ + O(ˆ r sin4 θ). 1 + cos2 θ˜
(3.36)
To proceed, we recall that the distance db between two points (X1 , Y1 ) and (X2 , Y2 ) in the (real) hyperbolic plane is implicitly given by the formula [5, Theorem 7.2.1]: ⎛$ ⎞ %2 X1 X2 1⎝ (Y1 − Y2 )2 ⎠ . − + cosh db − 1 = 2 X2 X1 X1 X2 Also, recall that we have shown that the functions z and ρ defined in Sect. 3.3 provide global isothermal coordinates on the orbit space. Define (r, θ) by (z, ρ) = (r cos θ, r sin θ). Now consider a reference Ernst potential XKerr + i YKerr as given in [20]: % $ √ r03 (r 2 + r0 ) sin2 θ 1 √ r02 2 XKerr (r, θ) = (r 2 + r0 ) + + √ sin2 θ, (3.37) 2 2 (r 2 + r0 )2 + r02 cos2 θ YKerr (r, θ) = r02 (cos3 θ − 3 cos θ) −
r04 cos θ sin4 θ . (r 2 + r0 )2 + r02 cos2 θ √
(3.38)
Here r and θ are polar coordinates associated to Kerr’s own (z, ρ) coordinates. It is convenient to rewrite XKerr and YKerr as 2r02 sin2 θ + O(r sin2 θ), (3.39) 1 + cos2 θ 4r2 cos θ YKerr (r, θ) = − 0 + O(r sin4 θ). (3.40) 1 + cos2 θ The leading order term near r = 0 for YKerr can also be rewritten in the following form XKerr (r, θ) =
−
r02 sin4 θ 4r02 cos θ = −2r02 + , 2 1 + cos θ 2(1 + cos2 θ) cos4 (θ/2)
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useful away from θ = π, or as −
4r02 cos θ r02 sin4 θ 2 = 2r − , 0 1 + cos2 θ 2(1 + cos2 θ) sin4 (θ/2)
which is useful away from θ = 0. This shows that in either case the deviation from the constant terms ±2r02 in YKerr factors out through sin4 θ. In the remainder of this section we derive a bound for the hyperbolic distance between (X, Y ) and (XKerr , YKerr ), which are compared after identifying the (z, ρ) coordinates of the solution under consideration with the (z, ρ) coordinates of the reference Kerr solution. This leads to relations between (r, θ) ˜ which we analyse now. By (3.23) and (3.24) we have and (ˆ r, θ), ˜ r sin θ = ρ = rˆ β˜ρ sin θ,
and
˜ r cos θ = z = rˆ β˜z cos θ.
Thus, for small rˆ, r2 = ρ2 + z 2 = rˆ2 + O(ˆ r3 ), ˜ sin θ = (1 + O(ˆ r)) sin θ,
(3.41)
˜ cos θ = (1 + O(ˆ r)) cos θ.
(3.43)
(3.42)
Substituting (3.41)–(3.43) into (3.39)–(3.40) we get 2r02 sin2 θ˜ (1 + O(ˆ r)), 1 + cos2 θ˜ 4r2 cos θ˜ ˜ + O(ˆ r sin4 θ). YKerr (r, θ) = − 0 1 + cos2 θ˜
XKerr (r, θ) =
(3.44) (3.45)
From (3.32), (3.36), (3.44) and (3.45) we arrive at db ((X, Y ), (XKerr , YKerr )) = O(1)
for small rˆ.
(3.46)
We have therefore proved (for terminology, see [12]): Theorem 3.4. Let (M , g) be a vacuum, I + regular, stationary and axisymmetric asymptotically flat black hole space–time. Let E0 be a degenerate component of the event horizon I + (Mext ) ∩ ∂Mext with cross-section area A0 . There exists a neighbourhood of E0 on which the hyperbolic-plane distance between the complex Ernst potential of (M , g) and that of the extreme Kerr space–time with the same area of the cross-sections of the horizon is bounded. 3.7. The Ernst Potential of ∂ϕ in Electrovacuum We continue with the extension of the analysis in Sect. 3.6 to the electrovacuum case. Let F be the electro-magnetic two-form in a stationary axisymmetric space–time (M , g) satisfying the sourceless Einstein–Maxwell equations: thus F is invariant under both ξ and η and satisfies the Maxwell equations. In particular, F = dA,
d ∗ F = 0.
(3.47)
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To fix terminology, the vectorial Ernst potential (U, V, χe , χm ) of the rotational Killing field η, is defined as follows. First, U is defined as 1 1 U = − log X = − log g(η, η). 2 2 Next, the electric and magnetic potentials χe and χm of η are defined by dχe = iη ∗ F,
dχm = iη F.
To dispel confusion, we emphasize that these are not the same as the standard electric and magnetic potentials which are defined using the stationary Killing field. The existence of χe and χm is a consequence of (3.47). Note that η vanishes on the axis A , which implies that χe and χm are constant on each connected component of A . It further follows from the Einstein-Maxwell equations that the 1-form ∗(η ∧ dη) − 2χm dχe + 2χe dχm is closed (see e.g. [40]), and so we can define V by 2 d ≡ dY := ∗(η ∧ dη) − 2χm dχe + 2χe dχm . Similarly to the vacuum case, V is constant on each connected component of A . In the sequel, we analyse the asymptotic behaviour of (U, V, χe , χm ) as rˆ → 0. It is desired to relate this potential to that of the reference degenerate Kerr–Newman solution which has the same horizon area A0 and charge parameters qe and qb . To this end, we introduce the variable (r, θ) as in Sect. 3.6 by (z, ρ) = (r cos θ, r sin θ), where the functions z and ρ are defined in Sect. 3.3. The Ernst potential of the reference Kerr–Newman solution then takes the following form in terms of r and θ) (see, e.g., [25]): sin2 θ((r + m0 )2 + a20 )2 − r2 a20 sin2 θ) 1 UKN = − log , 2 (r + m0 )2 + a2 cos2 θ VKN = −a0 m0 (3 cos θ − cos3 θ) a20 sin2 θ − q02 m−1 0 (r + m0 ) , (r + m0 )2 + a20 cos2 θ (r + m0 )2 + a20 r + m0 χeKN = −qe cos θ +qb a0 sin2 θ , (r + m0 )2 + a20 cos2 θ (r + m0 )2 + a20 cos2 θ (r + m0 )2 + a20 r + m0 χm −qe a0 sin2 θ . KN = −qb cos θ (r + m0 )2 + a20 cos2 θ (r + m0 )2 + a20 cos2 θ − a0 m0 cos θ sin2 θ
Here q0 is the total charge,
q0 :=
qe2 + qb2 .
Note that in [25], only the case of vanishing magnetic charge is considered. The general magnetically charged solution can be obtained from this by a duality rotation, F → cos λF + sin λ ∗ F, where λ is a real constant. Under this transformation the new potentials χeKN and χm KN are obtained from the old ones
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by a constant rotation in the (χeKN , χm KN ) plane, while UKN and VKN remain unchanged, whence the above formulae. As shown in Sect. 3.6, we have the relations ˜ sin θ = (1 + O(ˆ r)) sin θ,
and
cos θ = (1 + O(ˆ r)) cos θ˜ for small rˆ,
(3.48)
where the error terms are smooth sphere functions. Thus, by a simple calculation, in a neighbourhood of the horizon we have UKN = − log sin θ˜ + O(1), ˜ + O(sin2 θ), ˜ VKN = 2a0 m0 s(θ)
(3.49)
˜ + O(sin2 θ), ˜ = qe s(θ) ˜ + O(sin2 θ), ˜ = qb s(θ)
(3.51)
χeKN χm KN
(3.50) (3.52)
˜ ≡ −1 in [0, π/6] and where s is some smooth function of θ˜ such that s(θ) ˜ s(θ) ≡ 1 in [5π/6, π]. We are ready for the analysis of (U, V, χe , χm ) near the horizon. From the flux formulae for the total electric and magnetic charges of the horizon we have π 1 ˜ χm dθ, (3.53) qb = ,θ˜ 2 0
qe =
1 2
π
˜ χe,θ˜ dθ.
(3.54)
0
In view of the above two identities, we can assume without loss of generality that # # (3.55) χm #A± = ∓qb , χe #A± = ∓qe , where A± are the connected components of the axis A as defined in Sect. 3.6. By Proposition 2.2 we have1 ˜ = 2F ˜(ˆ ˜ ˜ χm (ˆ r, θ) ϕθ r , θ) = O(sin θ) ,θ˜
for small rˆ,
(3.56)
and so, by (3.55), again for small rˆ, m
θ˜
˜ = −qb + χ (ˆ r, θ) 0
˜ χm (ˆ r, θ)dτ ,θ˜
π = qb −
2 ˜ ˜ = qb s(θ)+O(sin ˜ χm (ˆ r, θ) θ). ,θ˜
(3.57)
θ˜
Similarly, we have ˜ = 2 ∗ F ˜(ˆ ˜ ˜ r, θ) χe,θ˜(ˆ ϕθ r , θ) = O(sin θ)
for small rˆ,
(3.58)
leading to ˜ = qe s(θ) ˜ + O(sin2 θ) ˜ χe (ˆ r, θ) 1
for small rˆ.
We use the convention F = Fμν dxμ ∧ dxν for the coefficients Fμν of a two-form.
(3.59)
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Next, applying the result of [28] to (M , g, F) and to (M , g, ∗F) we find ˜ =F ˜ ˜ = ∗F ˜ Fϕθ˜(0, θ) and ∗ Fϕθ˜(0, θ) KNϕθ˜(0, θ), KNϕθ˜(0, θ), where FKN is the electro-magnetic two-form associated to the reference Kerr– Newman solution. Note that here we have used θ ≡ θ˜ on the horizon. It thus follows from the definition of χe and χm that ˜ ≡ χe (0, θ), ˜ ˜ ≡ χm (0, θ). ˜ χe (0, θ) and χm (0, θ) KN
KN
By a direct computation, we then get π q 2 r4 a0 q04 −χm χe,θ˜ + χe χm dθ˜ = 20 02 arctan − . ,θ˜ rˆ=0 a0 m0 m0 a0 m0
(3.60)
0
To continue, we recall (3.33) which gives ∗(η ∧ dη)θ˜ = hϕϕ ψ −1
√ det h [ˆ r α],ˆr .
Hence, by (3.13), (3.17) and (3.18), 2a0 m0 r04 sin3 θ˜ γ , (3.61) ˜ 2 θ˜ (m20 + a20 cos2 θ) ˜ ≡ 1. It follows that where γθ˜ is a smooth sphere function satisfying γθ˜(0, θ) π 2 q 2 r4 a0 r4 ∗(η ∧ dη)θ˜ |rˆ=0 dθ˜ = − 2 0 20 arctan +2 0 . (3.62) a0 m0 m0 a0 m0 ∗ (η ∧ dη)θ˜ =
0
Using (3.60), (3.62) and recalling the definition of V we get π 4 4 # # ˜ dθ˜ = r0 − q0 = 4 a0 m0 . # # V A− − V A+ = V,θ˜(0, θ) a0 m0 a0 m0
(3.63)
0
Thus, we can assume without loss of generality that # V #A± = ∓2 a0 m0 . Now, taking (3.56), (3.57), (3.58), (3.59) and (3.61) into account we get ˜ = O(sin θ) ˜ for small rˆ. V,θ˜(ˆ r, θ) Thus ˜ = 2a0 m0 s(θ) ˜ + O(sin2 θ) ˜ for small rˆ. V (ˆ r, θ)
(3.64)
Finally, by (3.18), ˜ = − log sin θ˜ + O(1) U (ˆ r, θ)
for small rˆ.
(3.65) e
m 1 , χ1 )
Recall that the distance db between two points p1 = (U1 , V1 , χ and p2 = (U2 , V2 , χe 2 , χm 2 ) in the complex hyperbolic plane is given by (see, e.g., [39, Equation (55), p. 26]) & ' m 2 2 cosh2 db (p1 , p2 ) = cosh(U1 − U2 ) + eU1 +U2 [(χe1 − χe2 )2 + (χm 1 − χ2 ) ] 2
e e e m m + e2(U1 +U2 ) {V1 − V2 − χm 2 (χ1 − χ2 ) + χ2 (χ1 − χ2 )} .
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We conclude from (3.65), (3.64), (3.59), (3.57) and (3.49)–(3.52) that db ((U, V, χe , χm ), (UKN , VKN , χeKN , χm KN )) = O(1)
for small rˆ.
(3.66)
We have therefore proved (for terminology, see [12]): Theorem 3.5. Let (M , g, F) be an electrovacuum, I + regular, stationary and axisymmetric asymptotically flat black hole space–time. Let E0 be a degenerate component of the event horizon I + (Mext ) ∩ ∂Mext with cross-section area A0 , electric charge qe and magnetic charge qb . There exists a neighbourhood of E0 on which the complex-hyperbolic-plane distance between the vectorial Ernst potential of the rotational Killing vector field of (M , g) and that of the extreme Kerr–Newman space–time with the same horizon cross-section area, electric charge and magnetic charge is bounded. Remark 3.6. A more careful analysis as in Sect. 3.6 shows the following asymptotic behaviour for small rˆ: ˜ |χe − χeKN | + |χm − χm ˆ sin2 θ, KN | + |V − VKN | ≤ C r 4 ˜ m e e e m m |V − VKN − χKN (χ − χKN ) + χKN (χ − χKN )| ≤ C rˆ sin θ.
4. Proof of Theorem 1.1 Let (M , g) be a stationary, I + -regular, analytic electro-vacuum space–time with connected, non-empty, rotating future event horizon E0 . As justified in detail in [12], we only need to consider the case where the metric is axisymmetric. By Theorem 3.3, the area function ρ and its harmonic conjugate −z form a global manifestly asymptotically flat coordinate system on Mext /(R×U (1)) where, by Proposition 3.1, E0 corresponds to the point ρ = z = 0. It is well known that the vectorial Ernst potential (U, V, χe , χm ) is a harmonic map from R3 \{ρ = 0} = {(ρ, z, ϕ : ρ > 0, z ∈ R, ϕ ∈ [0, 2π]} into the complex hyperbolic plane. Define a reference vectorial Ernst potential (UKN , VKN , χeKN , χm KN ) as in Sect. 3.7. By the asymptotic analysis of [35,36] (compare [19]) and Theorem 3.5, the hyperbolic distance db between the two Ernst potentials is finite and goes to zero as one recedes to infinity. Using the subharmonicity of db and [13, Proposition C.4], we conclude that db ≡ 0 and so (U, V, χe , χm ) ≡ (UKN , VKN , χeKN , χm KN ). It is then customary to show that (Mext , g) is diffeomorphic to the corresponding domain of outer communications in that Kerr– Newman space–time to which the reference Ernst potential is associated.
Acknowledgements Both authors were supported in part by the EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1). PTC was further supported in part by the Polish Ministry of Science and Higher Education grant Nr N N201 372736. LN would like to thank Dr. Willie W.Y. Wong for drawing his attention to the problem studied in the present paper.
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[20] Dain, S.: A variational principle for stationary, axisymmetric solutions of Einstein’s equations. Class. Quantum Gravity 23, 6857–6871 (2006). arXiv:grqc/0508061 [21] Figueras, P., Lucietti, J.: On the uniqueness of extremal vacuum black holes. Class. Quantum Gravity 27, 095001 (2010). arXiv:0906.5565 [hep-th] [22] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983) [23] H´ aj´ıˇcek, P.: Three remarks on axisymmetric stationary horizons. Commun. Math. Phys. 36, 305–320 (1974) [24] Hawking, S.W.: Black holes in general relativity. Commun. Math. Phys. 25, 152– 166 (1972) [25] Heusler, M.: Black Hole Uniqueness Theorems. Cambridge University Press, Cambridge (1996) [26] Kundt, W., Tr¨ umper, M.: Orthogonal decomposition of axi-symmetric stationary space–times. Z. Physik 192, 419–422 (1966) [27] Kunduri, H.K., Lucietti, J.: A classification of near-horizon geometries of extremal vacuum black holes. J. Math. Phys. 50, 082502 (2009). arXiv:0806.2051 [hep-th] [28] Lewandowski, J., Pawlowski, T.: Extremal isolated horizons: A local uniqueness theorem. Class. Quantum Gravity 20, 587–606. arXiv:gr-qc/0208032 (2003) [29] Mazur, P.: Proof of uniqueness of the Kerr–Newman black hole solution. J. Phys. A Math. Gen. 15, 3173–3180 (1982) [30] Meinel, R., Ansorg, R.M., Kleinw¨ achter, A., Neugebauer, G., Petroff, D.: Relativistic Figures of Equilibrium. Cambridge University Press, Cambridge (2008) [31] Moncrief, V., Isenberg, J.: Symmetries of cosmological Cauchy horizons. Commun. Math. Phys. 89, 387–413 (1983) [32] Neugebauer, G., Hennig, J.: Non-existence of stationary two-black-hole configurations. Gen. Relativ. Gravit. 41(9), 2113–2130 (2009) [33] Neugebauer, G., Meinel, R.: Progress in relativistic gravitational theory using the inverse scattering method. J. Math. Phys. 44, 3407–3429 (2003). arXiv:grqc/0304086 [34] Robinson, D.C.: Uniqueness of the Kerr black hole. Phys. Rev. Lett. 34, 905–906 (1975) [35] Simon, W.: Radiative Einstein–Maxwell spacetimes and ‘no-hair’ theorems. Class. Quantum Gravity 9, 241–256 (1992) [36] Simon, W., Beig, R.: The multipole structure of stationary space–times. J. Math. Phys. 24, 1163–1171 (1983) [37] Sudarsky, D., Wald, R.M.: Extrema of mass, stationarity and staticity, and solutions to the Einstein–Yang–Mills equations. Phys. Rev. D46, 1453–1474 (1993) [38] Weinstein, G.: On rotating black-holes in equilibrium in general relativity. Commun. Pure Appl. Math. XLIII, 903–948 (1990) [39] Weinstein, G.: On the Dirichlet problem for harmonic maps with prescribed singularities. Duke Math. J. 77, 135–165 (1995) [40] Weinstein, G.: N -black hole stationary and axially symmetric solutions of the Einstein/Maxwell equations. Commun. Partial Differ. Equ. 21, 1389–1430 (1996)
Vol. 11 (2010)
A Uniqueness Theorem for Degenerate Kerr–Newman
Piotr T. Chru´sciel LMPT, F´ed´eration Denis Poisson Tours, France and Mathematical Institute and Hertford College Oxford, UK e-mail:
[email protected] Luc Nguyen OxPDE, Mathematical Institute Oxford, UK e-mail:
[email protected] Communicated by Marcos Marino. Received: February 10, 2010. Accepted: March 24, 2010.
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Ann. Henri Poincar´e 11 (2010), 611–658 c 2010 Springer Basel AG 1424-0637/10/040611-48 published online June 23, 2010 DOI 10.1007/s00023-010-0042-7
Annales Henri Poincar´ e
Particle Decays and Stability on the de Sitter Universe Jacques Bros, Henri Epstein and Ugo Moschella Abstract. We study particle decay in de Sitter space–time as given by first-order perturbation theory in a Lagrangian interacting quantum field theory. We study in detail the adiabatic limit of the perturbative amplitude and compute the “phase space” coefficient exactly in the case of two equal particles produced in the disintegration. We show that for fields with masses above a critical mass mc there is no such thing as particle stability, so that decays forbidden in flat space–time do occur here. The lifetime of such a particle also turns out to be independent of its velocity when that lifetime is comparable with de Sitter radius. Particles with mass lower than critical have a completely different behavior: the masses of their decay products must obey quantification rules, and their lifetime is zero.
1. Introduction Some important progress in the astronomical observations of the last 10 years [1,2] have led in a progressively convincing way to the surprising conclusion that the recent universe is dominated by an almost spatially homogeneous exotic form of energy density to which there corresponds an effective negative pressure. Such negative pressure acts repulsively at large scales, opposing itself to the gravitational attraction. It has become customary to characterize such energy density by the term “dark”. The simplest and best known candidate for the “dark energy” is the cosmological constant. As of today, the ΛCDM (Cold Dark Matter) model, which is obtained by adding a cosmological constant to the standard model, is the one which is in better agreement with the cosmological observations. Recent data show that dark energy behaves as a cosmological constant within a few percent error. In addition, if the description provided by the ΛCDM model is correct, Friedmann’s equation shows that the remaining energy components must in the future progressively thin out and eventually vanish thus letting the cosmological constant term alone survive.
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In the above scenario the de Sitter geometry [3,4], which is the homogeneous and isotropic solution of the vacuum Einstein equations with cosmological term, appears to take the double role of reference geometry of the universe, namely the geometry of space–time deprived of its matter and radiation content and of geometry that the universe approaches asymptotically. On the other hand, it seems reasonable to imagine that the presence of a small cosmological constant, while having a huge impact on our understanding of the universe as a whole, would not influence microphysics in its quantum aspects. However, this conclusion may have to be reassessed, because in the presence of a cosmological constant, however small, it is the notion of elementary particle itself which has to be reconsidered. The particle concept arises in Minkowski quantum field theory in two different but related contexts: on the one hand, it is a crucial ingredient in scattering theory and in the construction of an S-matrix and, on the other hand, it emerges as an output of Wigner’s classification of the irreducible unitary representations of the Poincar´e group. In both cases, the mathematical construction and the physical interpretation are based on the Fourier representation of Minkowski space–time, a tool which is closely related to the global structure of the Minkowski space–time, in particular its translation invariance. When considering quantum field theory in a curved background the situation is radically different and much of what is learnt in standard QFT gets lost, in particular, the idea of what a particle is. This fact has been known for a long time (see, e.g., [5]), and much work has been done to try to find a replacement for that notion. In this respect, a useful quantity that has been studied is the vacuum expectation value of the renormalized stress–energy tensor. Unfortunately, such a calculation is already very difficult for linear field theories, and even when the geometry of the background is explicitly given, not to mention a certain degree of arbitrariness in the construction. The technology seems to have attained its limits, and an application to the interacting case seems hopeless. Even if the notion of particle is generally speaking ill-defined in curved space–times, and if there is no S-matrix, one can still study perturbative expansions of the transition amplitudes. General schemes to address the question of how to define interacting quantum field theories at the perturbative level on curved backgrounds have been recently introduced and developed in [6–9]. The adiabatic problem however has not been treated by the above authors who only address the problem of ultraviolet divergences (sufficient from a local algebraic viewpoint), and it seems out of reach of such a general framework and left to a case-by-case study. Unfortunately, even if one may think that interactions between elementary particles happen in a “laboratory” so that “infinity” is a distance of the order of meters, the calculation of perturbative amplitudes involves integrations over the whole space–time manifold and requires the removal of the infrared cutoffs; it should be expected that different topological global structures result in different physical properties in the “small”.
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In this paper, we provide an example of such a phenomenon by giving a full description of how to solve the problem of calculating the mean lifetime of unstable scalar particles on de Sitter space–time at first order in perturbation theory. This interesting physical problem provides also an example of a concrete perturbative calculation in presence of the cosmological constant. The task already presents considerable mathematical difficulties. To our knowledge this calculation was first taken up by Nachtmann [10]. He showed, in a very special case, that while a Minkowskian particle can never decay into heavier products, a dS-particle can, although this effect is exponentially small in the dS-radius. The subject has acquired a greater physical interest with the advent of inflationary cosmology and more recently of the dark energy era. In particular, a similar question has been addressed by Polyakov who recently studied the vacuum decay and de Sitter instability in [15,16]. The idea that particle decays during the (quasi-)de Sitter phase may have important consequences on the physics of the early universe has been suggested recently [11–13]. The mathematical and physical difficulties related to the lack of time-translation symmetry of the de Sitter universe, and more generally of non-static cosmological backgrounds, have been tackled [11–13] by using the Schwinger–Keldysh formalism, which is suitable for studying certain aspects of the quantum dynamics of systems out of equilibrium. An important ingredient of this approach is the so called Dynamic Renormalization Group [14] which allows a kind of resummation of an infinite series of infrared diverging quantities. That method is, however, based on the introduction of a practical notion of lifetime of an unstable particle which is quite different from the definition commonly used in quantum physics. Also, the hard technical difficulties of the concrete calculation involved in solving a complicated integro-differential equation have only been faced in the favorably special conformal and minimally coupled massless cases although in principle the method can be used to deal with particles of generic mass [11–13]. In this paper, we perform a computation which is similar to the one outlined by Nachtmann and follows the conventional quantum field theoretical perturbative approach for computing probability amplitudes. Our work gives significantly wider results with respect to [10], e.g., regarding the so-called adiabatic limit, complementary-series-particles, and explicit expressions of the relevant K¨ all´en–Lehmann weights. On the other side, comparing our result with those of [11–13] is not easy because of the non standard (but interesting) definition of lifetime chosen in [11–13]. These findings have been summarized in a recent short communication [17] and developed in a related paper [18]. The results exhibit significant differences compared to the Minkowski case, and decay processes which are normally forbidden become possible and, vice-versa, processes that are normally possible are now forbidden (see also [15,16]). The maximal symmetry of the de Sitter universe implies the existence of a global square-mass operator, one of the two Casimir operators of the de Sitter group SO0 (1, d) (see, e.g., [19]); this quantity is conserved for de Sitter invariant field theories. However, in
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contrast with the Poincar´e group case, the tensor product of two unitary irreducible representations of masses m1 and m2 decomposes into a direct integral of representations whose masses m do not satisfy the ‘subadditivity condition’ m ≥ m1 + m2 : all representations of mass larger than a certain critical value (principal series) appear in the decomposition. This fact was shown in [10] for the two-dimensional case and will be established here in general. This means that the de Sitter symmetry does not prevent a particle with mass in the principal series from decaying into, e.g., pairs of heavier particles. This phenomenon also implies that there can be nothing like a mass gap in that range. This is a major obstruction to attempts at constructing a de Sitter S-matrix; the Minkowskian asymptotic theory makes essential use of an isolated point in the spectrum of the mass operator, and this will generally not occur in the de Sitter case. We will also show that the tensor product of two representations of sufficiently small mass below the critical value (complementary series) contains an additional finite sum of discrete terms in the complementary series itself (at most one term in dimension 4). This implies a form of particle stability, but the new phenomenon is that a particle of this kind cannot disintegrate unless the masses of the decay products have certain quantized values. Stability for the same range of masses has also been recently found [8] in a completely different context. Other remarks about the physical meaning and applicability of our results will be presented in the concluding section. 1.1. Notation We denote C+ = −C− the open upper complex half-plane. Let Δ = C\[−1, 1], Δ1 = C\(−∞, 1]. The function log is defined as holomorphic on C\(−∞, 0] and real on (0, +∞) and ζ → ζ μ as exp(μ log(ζ)). It is entire in μ. If ζ ∈ Δ1 and ρ > 0, then (ρζ)μ = ρμ ζ μ . If ζ ∈ C+ and s ∈ C− , then (sζ)μ = sμ ζ μ . We define z → (z 2 − 1)1/2 as holomorphic on Δ and asymptotic to z at large |z|. It is Herglotz, negative on (−∞, −1) and positive on (1, +∞).
2. Free and Generalized Free Fields in Minkowski and de Sitter Space–times In this section, we give a short summary of the theory of free and generalized free quantum fields on de Sitter space–time. Since there are infinitely many inequivalent representations of the field algebra, a (mathematical) choice has to be made on physical grounds. Ours is based on the analyticity properties of the vacuum expectation values: see the condition (W2) below. In the Minkowski space, this is equivalent to the positivity of the energy. In the de Sitter case, it admits a thermal interpretation [20–23]. The reader can find in [21–23] a general approach to de Sitter QFT based on such analytic properties. It includes the so called Bunch–Davies, also called Euclidean vacuum of de Sitter scalar Klein–Gordon fields as a basic example. The real (resp. complex) d-dimensional Minkowski space–time Md (resp. (c) Md ) is Rd (resp. Cd ) equipped with the Lorentzian inner product
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x · x = x0 x − x1 x − · · · − xd−1 x 0
1
d−1
615
= x0 x − x · x 0
(1)
with respect to an arbitrarily chosen Lorentz frame {eσ , σ = 0, . . . , d − 1}. def
When no ambiguity arises, x2 = x · x. The real (resp. complex) de Sitter (c) space–time Xd (resp. Xd ) with radius R > 0 are the hyperboloids Xd = {x ∈ Md+1 : x · x + R2 = 0}, (c)
(2)
(c)
Xd = {x ∈ Md+1 : x · x + R2 = 0},
equipped with the pseudo-Riemannian metric induced by (1). The connected (c) real (resp. complex) Lorentz group acting on Md (resp. Md ) is L↑+ (d) = SO0 (1, d − 1; R) (resp. L+ (C; d) = SO0 (1, d − 1; C)). The connected group (c) of displacements on Xd (resp. Xd ) is L↑+ (d+1) (resp. L+ (C; d+1)), sometimes (c) denoted G0 (resp. G0 ). These groups act transitively. Note that our definition of Md etc. arbitrarily selects a particular orthonormal basis (e0 , . . . , ed−1 ) in Md or (e0 , . . . , ed ) in Md+1 . These particular Lorentz frames will be useful in the sequel. In Md the future and past open cones V± and the future and past light-cones C± are given by V+ = {x ∈ Md : x · x > 0,
x0 > 0} = −V− ,
C+ = {x ∈ Md : x · x = 0,
x0 ≥ 0} = −C− .
(3) (c)
The future and past tubes in the complex Minkowski space–time Md given by: T± = Rd + iV± .
are (4)
(c)
The future and past tuboids in Xd are the intersections of the future and (c) (c) past tubes in Md+1 with the complex de Sitter manifold Xd : (c)
T± = T± ∩ Xd .
(5)
We will use the letter X to denote either Md or Xd when the same discussion applies to both, X (c) denoting the complexified object. dx will denote the standard invariant measure on X , i.e., using the frame (e0 , . . . , en ), dx = dx0 . . . dxd−1 in the case of Md , and dx = 2δ(x2 + R2 ) dx0 . . . dxd for Xd . A (neutral scalar) generalized free field φ on X is entirely specified by its 2-point function. This is a tempered distribution W on X × X (we denote W (x, x ) = W(x , x)), which we require to have the following properties: . (W1) Hermiticity: W(x, x ) = W(x , x).
(6)
. (W2) Analyticity and invariance: there is a function w of one complex variable, holomorphic in the cut plane C\R+ , with tempered behavior at infinity and at the boundaries, such that, in the sense of tempered distributions, W(x, x ) =
lim
z∈T− ,z ∈T+ z→x,z →x
w((z − z )2 ),
(7)
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hence W (x, x ) = W(x , x) =
lim
z∈T+ ,z ∈T− z→x,z →x
w((z − z )2 ).
(8)
For complex z, z ∈ Xd such that (z − z )2 ∈ C\R+ we will denote W (z, z ) = w((z − z )2 ). Note that this implies (c)
W (z, z ) = W (z , z) = W (−z , −z),
(9)
W(x, x ) = W(−x , −x).
(10)
and (W1) and (W2) also imply W (z, z ) = W (z, z ).
(11)
. (W3) Positivity: For every f ∈ S(X ), W, f ⊗ f = f (x)W(x, x )f (x ) dx dx ≥ 0.
(12)
X ×X
Conversely, given W and w having these properties, after having identified the kernel N1 = {f ∈ S(X ) : W, f ⊗ f = 0} we can construct a Hilbert space F1 by completing S(X )/N1 equipped with the scalar product (f, g) = W, f ⊗ g , and then exponentiate F1 into a Fock space F F=
∞
Fn ,
F0 = C,
Fn = SF1⊗n
for n ≥ 1.
(13)
n=0
The vacuum Ω is the unit vector 1 ∈ F0 = C. There is a continuous unitary representation U of the Poincar´e or de Sitter group acting on F and preserving the Fn , with U Ω = Ω. The generalized free field φ is defined on a dense domain in F and (Ω, φ(x)φ(x )Ω) = W(x, x ). As a result of the analyticity property (W2), the Wick powers of a generalized free field are well-defined local fields operating in the same Fock space. Their vacuum expectation values are obtained by the standard Wick formulae as sums of products of W. We note that a function W on Xd × Xd possessing the properties (W1) and (W2) automatically extends [through (7)] to a function with the same properties on Md+1 × Md+1 , so that a generalized free field on Xd has an extension as a generalized free field on Md+1 . However the extension of W need not satisfy (W3) on Md+1 × Md+1 even if it does on Xd × Xd . A free field φ of mass m > 0 on X is a generalized free field such that W is a solution of the Klein–Gordon equation with mass m in both arguments, and is normalized so as to obey the canonical commutation relations. In that case W is uniquely determined by m and will be denoted Wm . In the Minkowskian case, the representation U |F1 is irreducible and equivalent to the representation [m, 0] of the Poincar´e group. As usual, the representation U provides a representation of the Lie algebra of the (Poincar´e or de Sitter) group and its enveloping algebra by
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self-adjoint (or i× self-adjoint) operators on F. In particular, the squaremass operator M 2 is given by M 2 = P μ Pμ in the Minkowskian case, and by M 2 = M μν Mμν /2R2 in the de Sitter case. In both cases, M 2 Ψ = m2 Ψ for every Ψ ∈ F1 (see, e.g., [19]). 2.1. Special Features Of Free Fields in de Sitter Space–Time In the de Sitter case the mass m can be related to a dimensionless parameter ν as follows 2
2
m R =
ν=± m R − 2
2
d−1 2
2 1/2
d−1 2
2 + ν2,
= ±R(m2 − m2c )1/2 ,
(14) mc =
d−1 . 2R
(15)
In this case, if no ambiguity arises, we shall often denote Wν = W−ν to mean (c) Wm , and similarly Wν and wν . Explicitly, if z, z ∈ Xd , (z − z )2 ∈ / R+ , and hence ζ = z · z /R2 does not belong to the real interval (−∞, −1],
Wν (z, z ) =
Γ
d−1 2
+ iν Γ d−1 2 − iν
(ζ 2 − 1)−
d−2 4
− d−2
2 P− 1 +iν (ζ)
(16) 2 2(2π) Rd−2 d−1 Γ d−1 d−1 d−1 d 1−ζ 2 + iν Γ 2 − iν + iν, − iν; ; F = . d 2 2 2 2 (4π) 2 Rd−2 Γ d2 (17) d 2
Since Γ(c)−1 F (a, b; c; z) is entire in a, b, and c the RHS of (17) is meromorphic in ν with simple poles at ν = ±i((d − 1)/2 + n), n ≥ 0 an integer. In other words, Wν (z, z ) extends to a holomorphic function of ν, z and z in the (c) (c) domain {ν ∈ C, z ∈ Xd , z ∈ Xd : ν ∈ / ±i((d − 1)/2 + Z+ ), (z − z )2 ∈ / R+ }. However, wν possesses the positivity property (W3) [see (12)] only if either . (1) ν is real, i.e., m ≥ mc = (d − 1)/2R. In this case, U |F1 is an irreducible unitary representation of the “principal series”. or . (2) ν is pure imaginary with iν ∈ (−(d − 1)/2, (d − 1)/2), i.e., 0 < m ≤ (d − 1)/2R. In this case U |F1 is an irreducible unitary representation of the “complementary series”. We shall need a small part of the harmonic analysis on the de Sitter (c) space–time as developed in [22]. If z ∈ T± ⊂ Xd and ξ ∈ C+ \{0} ⊂ Md+1 , then ± Im(z · ξ) > 0, so that (z · ξ)λ is well-defined and holomorphic in (z, λ) in (T+ ∪ T− ) × C. The role of plane waves on Xd is played by the distributions ψλ± (x, ξ) =
lim
y∈V+ ,y→0
((x ± iy) · ξ)λ = ψλ∓ ¯ (x, ξ).
(18)
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An important formula expressing the de Sitter case two-point Wν as a Fourier superposition of plane-waves is the following (see [22]): − d−1 −iν − d−1 +iν (ξ · z ) 2 α(ξ), (19) Wν (z, z ) = Rcd,ν (z · ξ) 2 γ
where z ∈ T− , z ∈ T+ , and cd,ν =
d−1 −πν Γ( d−1 2 + iν)Γ( 2 − iν)e . d+1 d 2 π
(20)
In (19), γ is a (d − 1)-cycle in C+ \{0} homologous to the sphere S0 = C+ ∩ {ξ : ξ 0 = 1}. The (d − 1)-form α is given, in the standard coordinates, by α = (ξ 0 )−1
d
j . . . dξ d . (−1)j+1 ξ j dξ 1 . . . dξ
(21)
j=1
If a smooth function f on C+ \{0} is homogeneous of degree (1 − d), the form f α is closed, so that the linear functional (22) f → I0 (f ) = f (ξ)α(ξ) γ
is independent of γ. This implies that it is Lorentz-invariant. We often denote dμγ the measure defined on γ by the restriction of α. In particular, the restriction of α to the (d − 1)-sphere S0 is the standard volume form on that sphere, normalized by S0 dμS0 (ξ) = 2π d/2 /Γ(d/2). It is possible to take the limit of (19), in the sense of distributions, when z and z tend to the reals: − + (x, ξ)ψ− (x , ξ) dμγ (ξ). (23) Wν (x, x ) = Rcd,ν ψ− d−1 d−1 +iν −iν 2
2
γ
Comparing (17) with (19) and (20) gives − d−1 −iν − d−1 +iν (z · ξ) 2 (ξ · z ) 2 dμγ (ξ) γ
eπν 2π d/2 = d−1 d F R Γ 2
d−1 d−1 d 1−ζ + iν, − iν; ; 2 2 2 2
.
(24)
Both sides of this equation are holomorphic in z, z , ν in the domain T− × T+ × C, hence (24) holds in this domain. Remark 2.1. If T is a homogeneous distribution of degree β on C+ \{0}, it can be restricted to any C ∞ submanifold of dimension d − 1 which is transversal to the generators of C+ , in particular to hyperplanar sections such as S0 = {ξ ∈ C+ : ξ 0 = 1} and V0 = {ξ ∈ C+ : ξ 0 + ξ d = 1}. If γ is of this type and compact, γ T (ξ)α(ξ) is well-defined and, if β = 1 − d, it is independent of γ.
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Remark 2.2. For any complex α, (z, ξ) → (z · ξ)α is C ∞ in ξ and holomorphic in z on T± ×(C+ \{0}) and it is an entire function of α. For each ξ it has a limit in the sense of tempered distributions on Xd as z tends to the reals, and this has been denoted ψα± (x, ξ). It is an entire function in α. Furthermore its invariance under G0 implies that, if ϕ ∈ S(C+ \{0}), C+ ψα± (x, ξ)ϕ(ξ) dξ is C ∞ in x. Indeed any small displacement of x can be effected by a group transformation close to the identity, which can be transferred to ξ and thence to ϕ. In the same way, ψα± (x, ξ) is C ∞ in ξ (as well as homogeneous) when integrated with a smooth test-function in x. This explains the meaning of formulae such as (23). Note that the integral in this formula is entire in ν. For similar reasons, for any ϕ ∈ S(Xd ), Xd ϕ(x)Wν (x, x ) dx is C ∞ in x and meromorphic in ν. 2.2. More Features Common to Minkowski and de Sitter Space–Time An important formula, which holds in Minkowski as well as in de Sitter space– time (but in this case only if m, m ≥ mc ), is the projector identity: Wm (z, x)Wm (x, y) dx = C1 (m, d)δ(m2 − m2 )Wm (z, y). (25) X
Here C1 (m, d) = 2π for Minkowski space–time, C1 (m, d) = C0 (ν) = 2π| coth(πν)| for de Sitter space–time.
(26) (27)
The proof of the above identity is trivial in the Minkowskian case. For the de Sitter case it will be provided in Appendix D. Note that C0 (mR) tends to 2π as R → +∞ for a fixed m > 0. The K¨ all´en–Lehmann decomposition theorem exists in both Md and Xd . In the case of Md , (see [24, p. 360]), it asserts that, for every W having the properties (W1) and (W2) there is a tempered ρ such that ρ(m2 )Wm (z, z ) dm2 . (28) W (z, z ) = R+
If W satisfies (W3), then ρ is a tempered positive measure. The same holds in the dS case provided W satisfies some decrease property. In this case, the integral runs on masses of the principal series, i.e., m > mc = (d − 1)/2R. For proofs and details, see [22,25]. In particular, if mj ≥ 0 and, in the dS case, mj > mc for 1 ≤ j ≤ N , ∞ N
Wmj (x, x ) = ρ(a2 ; m1 , . . . , mN )Wa (x, x ) da2 . (29) j=1
a≥b
Here b = mc in the de Sitter case, b =
j
mj in the Minkowski case.
3. Particle Decays: General Formalism There is at the moment nothing like the Haag–Ruelle asymptotic theory (HRT) (see [24,26,27]) for the de Sitter universe. Indeed all the ingredients of that
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theory are missing in the de Sitter case. For example, as it will be shown in this paper, even in a free field theory of mass m > mc , the mass m is not an isolated point in the mass spectrum. Moreover, the solutions of the Klein–Gordon equation do not have the kind of localization at infinity which plays an essential role in the HRT. The concept of a particle is, therefore, not obvious in de Sitter space–time, except for localized observations. Here we adopt Wigner’s point of view: a one-particle vector state is a state belonging to an invariant subspace of the Hilbert space in which the representation of the invariance group reduces to an irreducible representation. In the dS case, we also require that this irreducible representation belong to the principal or complementary series, i.e., it should be equivalent to one of the representations which occur in the F1 of a free field. We shall study the decay of a particle using first-order perturbation theory. The initial framework and calculations are the same for the Minkowski and de Sitter cases: its ingredients are the projector identity and the K¨ all´en– Lehmann representation. (It can also be extended to the Minkowskian thermal case ([28]) although there is no K¨ all´en–Lehmann representation there). Let φ0 , φ1 , . . . , φN
(30)
be 1 + N independent free scalar fields with masses mj > 0, j = 0, . . . , N , acting in a common Fock space H, the tensor product of the individual Fock spaces for the φk : H=
N
F (k) ,
(31)
k=0
(Ω, φj (x)φk (y)Ω) = δjk Wmj (x, y).
(32)
We denote (0)
(N )
Hj0 ,...,jN = Fj0 ⊗ · · · ⊗ FjN .
(33)
This is the subspace of states in H containing jk k-particles. Ej0 ,...,jN denotes the Hermitian projector onto this subspace. We now switch on an interaction term γg(x)L(x) dx, L(x) =: φ0 (x)φ1 (x)q1 . . . φN (x)qN : . (34) X
N Here the qj are non-negative integers, and we denote q! = j=1 qj !. γ is a small constant. g is a smooth, rapidly decreasing function over X . In the end, g should be made to tend to 1 (adiabatic limit). According to perturbation theory, the transition amplitude between two normalized states ψ0 and ψ1 in H is given by (ψ0 , S(γg)ψ1 ), where S(γg) is the formal series in γg ∞ n n
i γ S(γg) = g(x1 ) dx1 · · · g(xn ) dxn T (L(x1 ) . . . L(xn )) (35) n! n=0 Xn
In (35), T (L(x1 ) . . . L(xn )) denotes the (renormalized) time-ordered product of L(x1 ), . . . L(xn ). In the first order in γg, the transition amplitude between
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two orthogonal states ψ0 and ψ1 is (ψ0 , iT1 (γg)ψ1 ),
621
T1 (γg) =
γg(x)L(x) dx.
(36)
X
We take
ψ0 =
f0 (x)φ0 (x)Ω dx,
ψ1 =
f1 (x11 , . . . , x1q1 , . . . , xN 1 , . . . , xN qN ) :
qj N
(37) φj (xjk ) dxjk : Ω,
j=1 k=1
(38) where f0 and f1 are smooth rapidly decreasing functions. The states of the form (37) generate H1,0...,0 and the states of the form (38) generate H0,q1 ,...,qN . The probability of transition from ψ0 to any state in H0,q1 ,...,qN is: q!γ 2 (ψ0 , T1 (γg)E0,q1 ,...,qN T1 (γg)∗ ψ0 ) = f0 (x)f0 (y)g(u)g(v) Γ= (ψ0 , ψ0 ) (ψ0 , ψ0 ) ⎧ ⎫ N ⎨ ⎬ × Wm0 (x, u) Wmj (u, v)qj Wm0 (v, y) dx du dv dy. (39) ⎩ ⎭ j=1
From now on, we suppose, in the dS case, that mk > mc , 0 ≤ k ≤ N , i.e all particles belong to the principal series. We may then replace the central two-point function in u and v by its K¨ all´en–Lehmann decomposition: N
Wmj (u, v)qj = ρ(a2 ; m1 , . . . , m1 , . . . , mN , . . . , mN )Wa (u, v) da2 . (40) j=1
Here mj occurs qj times as an argument of ρ. This gives q!γ 2 f0 (x)f0 (y)g(u)g(v)ρ(a2 ; m1 , . . . , m1 , . . . , mN , . . . , mN ) Γ= (Ψ0 , Ψ0 ) × Wm0 (x, u)Wa (u, v)Wm0 (v, y) dx du dv dy da2 .
(41)
The next step would be the so-called adiabatic limit, and should consist in letting the cut-off g tend to 1 in this formula. It is however easier to set first only one of the g’s equal to 1, say g(u) = 1 in (41). It then becomes possible to perform the integration over u by using the projector identity (25) and we find for the transition probability: Γ = L1 (f0 , g) × q!ρ(m20 ; m1 , . . . , m1 , . . . , mN , . . . , mN ), where L1 (f0 , g) =
γ 2 C1 (m0 , d)
g(v)f0 (x)Wm0 (x, v)Wm0 (v, y)f0 (y)dx dy dv . f0 (x)Wm0 (x, y)f0 (y) dx dy
(42)
(43)
This formula exhibits an interesting factorization: the first factor depends only on the wavepacket f0 , the mass m0 of the incoming particle and the
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switching-off factor γ 2 g; the adiabatic limit still remains to be done there; the second factor contains all the information about the decay products. If we now attempt to set g(v) = 1 in (43) and to integrate over v using again (25), the result is proportional to δ(m20 − m20 ), i.e., the integral diverges. This difficulty was resolved in the 1930’s by aiming at the average transition probability per unit time (see, e.g., [29], pp. 60–62). We first review the well-known Minkowski case, in a form which can serve as a model for the de Sitter case. In fact even this famous old case deserves some re-examination on its own right and it is possible, in this case, to allow the two g in (41) to tend to 1 simultaneously, or even at different rates. This is done in Appendix A. It is found that, if both g are taken as in (46), the result is the same as found above. But this is not necessarily the case for other g. Nevertheless the procedure announced above (i.e., setting the first g in (41) be equal to 1, then discussing the time average of the limit as the second g tends to 1) will be used in the de Sitter case, since it gives good results in the Minkowski case, and since calculations in the dS case would become much more difficult otherwise. Note that in the de Sitter case (42) and (43) are applicable only when m0 > mc and the range of integration over a2 in (41) contains only values a2 > m2c (mc = (d − 1)/2/R). In the case of the decay into two particles below that this includes the case m1 > mc , but of mass m1 , it will be seen √ also the case mc > m1 > mc 3/2.
4. Minkowski Case 4.1. Adiabatic Limit: The Fermi Golden Rule The simplicity of the Minkowskian case arises from being able to use of the Fourier representations: f0 (x) = e−ipx f0 (p) dp, g(x) = e−ipx g(p) dp, (44) wm (x, y) = (2π)1−d eip(y−x) δ(p2 − m2 )θ(p0 ) dp. Then the factor in (43) becomes L1 (f0 , g) =
(2π)2 γ 2
g (p − q) dp dq f0 (p)δ(p2 − m20 )θ(p0 )f0 (q)δ(q 2 − m20 )θ(q 0 ) . (45) 2 |f0 (p)|2 δ(p2 − m )θ(p0 ) dp 0
We now specialize the cut-off g to depend only on the time coordinate of the chosen frame g(v) = h(v 0 ) = h(t), i.e., we think of the interaction as smoothly switched on and then turned off. The Fourier representation is then p) and (45) becomes g(p) = h(p0 )δ( (2π)2 γ 2 h(0) (2p0 )−1 |f0 (p)|2 δ(p2 − m20 )θ(p0 ) dp L1 (f0 , g) = . (46) |f0 (p)|2 δ(p2 − m2 )θ(p0 ) dp 0
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If we choose for g the indicator function of a time-slice of thickness T , i.e., h(t) = θ(t + T /2)θ(T /2 − t), h(0) = T /2π, we get (2π)γ 2 (2p0 )−1 |f0 (p)|2 δ(p2 − m20 )θ(p0 ) dp . (47) L1 (f0 , g) = T × |f0 (p)|2 δ(p2 − m2 )θ(p0 ) dp 0
Therefore, as noted above, removing the cut-off produces infinity. However, according to the Fermi golden rule, what is physically meaningful is not the amplitude but the amplitude per unit time. Therefore, dividing this by T and taking the limit as T → ∞ (a particularly trivial operation in this case) we finally get the following expression for the transition probability per unit time: (2π)γ 2 (2p0 )−1 |f0 (p)|2 δ(p2 − m20 )θ(p0 ) dp 1 = τ (f0 ) |f0 (p)|2 δ(p2 − m2 )θ(p0 ) dp 0
× q!ρ(m20 ; m1 , . . . , m1 , . . . , mN , . . . , mN ).
(48)
The reciprocal of this expression is the lifetime of the 0-particle in the state f0 . The dependence on the wavepacket f0 is a crucial feature of the special relativistic Minkowski case as it will be readily recognized. For instance to compute the lifetime τ0 of a particle at rest in the chosen frame we may let p), e.g., by taking |f0 (p)|2 tend to δ( f0 (p) = ε(1−d)/2 ϕ( p/ε),
f0 (x) = 2πδ(x0 )ε(d−1)/2 ϕ(εx),
ε > 0,
(49)
with ϕ ∈ S(Rd−1 ), and letting ε → 0. Then (48) tends to 1 πγ 2 = q!ρ(m20 ; m1 , . . . , m1 , . . . , mN , . . . , mN ). τ0 m0
(50)
We may act with a Lorentz boost on the same particle by replacing in (48) the wavepacket f0 by f0Λ (x) = f0 (Λ−1 x), Λ ∈ L↑+ ; the amplitude is modified as follows (2π)γ 2 (2(Λp)0 )−1 |f0 (p)|2 δ(p2 − m20 )θ(p0 ) dp 1 = τ (f0Λ ) |f0 (p)|2 δ(p2 − m2 )θ(p0 ) dp 0
× q!ρ(m20 ; m1 , . . . , m1 , . . . , mN , . . . , mN ).
(51)
If again |f0 (p)|2 → δ( p), the final result is the expression in (50) multiplied by 1/(Λ)00 . If Λ = exp(sM10 ), i.e., the particle is moving with velocity v = th s, (Λ)00 = cosh s = (1 − v2 )−1/2 gives the usual correction to the lifetime: τ v = τ0 / 1 − v 2 . (52) Remark 4.1. It is worthwhile to stress once more that this effect, which expresses the behavior of the life-time of a moving particle in special relativity, crucially depends on the peculiar way in which the wavepacket enters in the transition amplitude per unit time (48).
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4.2. K¨all´en–Lehmann Weights The weight ρ can be explicitly computed only in the case of one particle decaying into two particles. For a particle of mass m0 > 0 decaying into two identical particles of mass m1 > 0, i.e., the case N = 1, q1 = 2, the well-known formula is 2 d−3 m0 − 4m21 2 2 ρ(m0 ; m1 , m1 ) = θ(m20 − 4m21 ), (53) d−1 m (4π) 2 2d−2 Γ d−1 0 2 and (50) becomes d−3 πγ 2 m20 − 4m21 2 1 −1 2 2 = (lifetime of m0 ) = d−1 2 θ(m0 − 4m1 ). d−1 τ0 d−3 2 (4π) 2 Γ 2 m0 For d = 4 this is
1 γ 2 m20 − 4m21 2 1 = θ(m20 − 4m21 ), τ0 8π m20
(54)
(55)
in agreement with the computation in, e.g., [29].
5. de Sitter Case 5.1. Adiabatic Limit in the de Sitter Case The discussion of the adiabatic limit is more complicated in the de Sitter case. Taking the adiabatic limit is of course technically much more involved than in the Minkowski case (and we will relegate all the technical details to the appendices). But the really intricate and maybe perplexing issue is the physical interpretation of the whole procedure and, even more, of the somewhat surprising results. Having in mind the Minkowskian case that we have just discussed, the first question that should be asked is what is “time” in the de Sitter universe and what does it means that an interaction lasts for a certain time. In the Minkowski case we have the solid foundation of special relativity and a privileged class of frames, the inertial frames, each of them having an inherent precise notion of time. In the de Sitter case (and the situation is even worse in a general curved space–time) we have no such thing. Instead we have many possible coordinate systems, that may or may not cover the whole manifold, and many possible choices of temporal coordinates that have no special relation to each other. For example, the de Sitter universe is the only known space–time manifold admitting three different inequivalent choices of cosmic time so that the de Sitter metric takes the appearance of a, respectively, closed, flat, or open Friedmann–Robertson–Walker universe. But there are also other possibilities. The choice of time coordinate made in 1917 by de Sitter in his original papers [3,4] describes a wedge-like region of the de Sitter manifold as a static space– time with bifurcate Killing horizons [30].
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Figure 1. Time-slices of the de Sitter space–time in the closed and in the flat coordinate systems
We choose to proceed heuristically in analogy with the Minkowskian case. Concretely, we will work out the adiabatic limit using two of the three possible cosmological coordinate systems, namely the closed and the flat systems. Starting again from (43) we take the cutoff g appearing in there as the indicator (or characteristic) function of some “cosmic time-slice” of thickness T with respect to the relevant choice of cosmic time. We will see that in both the closed and flat case the amplitude diverges linearly in T precisely as in the Minkowskian case. Therefore, to extract a finite limit we are entitled (and have no other choice than) to use the Fermi golden rule and compute in the above two frames the probability per unit time by dividing by T ; there is at this point a small difference with respect to the flat case: the amplitude per unit time at finite T depends on T . However, letting T → ∞ gives a well-defined limit which exhibits a much more disturbing difference with the Minkowskian case. Closed FRW Model: The relevant coordinate system is the following: x0 = R sinh(t/R), x(t, u) = xi = R cosh(t/R)(−u),
(56)
u ∈ S d−1 ,
(the minus sign at RHS is for further convenience). In this coordinate system, the constant time slices are hyperspheres. These coordinates have the advantage to globally cover the de Sitter manifold (Fig. 1); they gives to the metric the form of a closed FRW model with scale factor a(t) = cosh(t/R): ds2 = dt2 − R2 cosh2 (t/R)dσ 2 (u)
(57)
= dt − R cosh (t/R)(dχ + sin χ(dθ + sin θ dφ )) 2
2
2
2
2
2
2
2
(d = 4).
(58)
In (57), dσ 2 (u) is the square line element on S d−1 at u, and (58) includes its expression in Euler angles. In these coordinates, we choose g(x) = gT (x) = θ(t + T /2)θ(T /2 − t).
(59)
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Flat FRW Model: These are the coordinates currently used in the context of inflationary models. Hypersurfaces of constant time are flat: ⎧ t t 1 0 2 ⎪ ⎨x = R sinh R + 2R e R y , t x(t, y) = xj = e R yj , (1 ≤ j ≤ d − 1), y ∈ Rd−1 , (60) ⎪ t ⎩ d 1 R e y2 , x = R cosh Rt − 2R 2 ds2 = dt2 − e2t/R (dy12 + · · · + dyd−1 ),
(61)
In these coordinates we choose g(x) = gT (x) = θ(t + T /2)θ(T /2 − t).
(62) 0
But the coordinates (60) only cover one half of Xd , the region where x + xd > 0, and the adiabatic limit will have to include the contribution of the other half not covered by the coordinate system. It turns out that the limit L1 (f0 , g) 1 = lim (63) × q!ρ(m20 ; m1 , . . . , m1 , . . . , mN , . . . , mN ) T →∞ τ T exists and is the same for both kinds of slices. The calculations are tedious and not quite straightforward, and will be given in Appendices B, C. For the spherical slices of the closed FRW system, only the calculations for d = 2, 3, 4 have been carried out. The method for the flat FRW coordinates works for all d. The inverse lifetime that results is γ 2 π coth(πκ)2 R × q!ρ(m20 ; m1 , . . . , m1 , . . . , mN , . . . , mN ) Γ1,q1 ,...,qN (f0 ) = |κ| (64) where we denoted κ = R(m20 − m2c )1/2 . Note the similarity of this formula with (50) and in fact the first factor in (64) tends to the corresponding factor in (50) when R → ∞ at fixed m0 . However, there is a most striking difference with (48), the RHS of (64) does not depend on f0 , the initial wave function of the decaying particle. Therefore, in particular, the lifetime of a particle does not depend on its velocity. We will comment on this feature, at first sight embarrassing, in the conclusions.
6. K¨all´en–Lehmann Weights As in the flat case, an explicit computation of the K¨ all´en–Lehmann weight is only possible for decays of one particle into two. Here the discussion will be restricted to the case of a particle of mass m0 > 0 decaying into two particles of equal masses m1 = m2 > 0 and we suppose at the beginning m1 > mc . The more difficult case m1 = m2 will be treated in a paper in preparation [31]. We shall find an explicit ρ(a2 ; m1 , m1 ) such that Wm (z, z )2 = ρ(a2 ; m1 , m1 )Wa (z, z ) da2 . (65)
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We change to variables κ = [a2 R2 − (d − 1)2 /4]1/2 , ν = [m21 R2 − (d − 1)2 /4]1/2 , and (by abuse of notation) seek a function ρ(κ; ν, ν) (mostly abbreviated as ρ(κ)) such that ∞ (66) Wν (z, z )2 = 2κρ(κ; ν, ν)Wκ (z, z ) dκ. 0
By (16), this is equivalent to 2 (x2 Cd,ν
− d−2 4
− 1)
− d−2 2 P− 1 +iν (x)2 2
∞ =
− d−2
2 2Cd,κ κρ(κ)P− 1 +iκ (x) dκ, 2
(67)
0
with Cd,ν
=
Γ
d−1 2
+ iν Γ d−1 2 − iν d
2(2π) 2 Rd−2
.
(68)
The generalized Mehler–Fock theorem [32, p. 398] asserts that ∞ g(x) = P−σ 1 +iκ (x)f (κ)dκ ⇐⇒ 2
0
κ f (κ) = sinh(πκ)Γ π
∞ 1 1 − σ + iκ Γ − σ − iκ P−σ 1 +iκ (x)g(x) dx. 2 2 2 1
(69) Therefore, (67) implies 2 Cd,ν κ sinh(πκ)Γ κρ(κ) = 2Cd,κ π
d−1 d−1 + iκ Γ − iκ × hd (κ, ν, ν), 2 2 (70)
∞ 2 d−2 − d−2 − d−2 def 2 2 (x) P− 1 +iκ (x) dx. hd (κ, ν, ν) = (x2 − 1)− 4 P− 1 +iν 2
2
(71)
1
It is possible to obtain an explicit expression of hd (κ, ν, ν) by using Mellin transform techniques (see [33]) and a lemma of Barnes (see [34]). Recall that if ϕ ∈ D((0, ∞)), its Mellin transform ϕ is given by ∞ (72) ϕ(s) = ζ s−1 ϕ(ζ) dζ, 0
It is entire in s = σ + iτ , decreasing faster than any negative power of τ for fixed σ, and σ+i∞ 1 ϕ(ζ) = ζ −s ϕ(s) ds ∀σ ∈ R, (73) 2iπ σ−i∞
ψ(ζ) = ϕ(1/ζ)
⇐⇒
= ϕ(−s). ψ(s)
(74)
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If ϕ, ϕ1 , ϕ2 are in D((0, ∞)), ∞ du ϕ(ζ) = ϕ1 (ζ/u)ϕ2 (u) u
Ann. Henri Poincar´e
⇐⇒
ϕ(s) =ϕ 1 (s)ϕ 2 (s).
(75)
0
In particular (Mellin–Plancherel identity) ∞
σ+i∞
1 du = ϕ1 (u)ϕ2 (u) u 2iπ
0
ϕ 1 (−s)ϕ 2 (s) ds.
(76)
σ−i∞
These properties can be extended to other functions and generalized functions (see [33]), and, in many interesting cases, although the Mellin transforms are no longer entire, the above formulae survive provided the integration in (73) or (76) is performed on a suitable contour. √ By the change of variable x = 1 + ζ in (71) we find ∞ dζ (77) hd (κ, ν, ν) = G1 (ζ)G2 (ζ) , ζ 0
with
√ Pαμ ( 1 + ζ) √ G1 (ζ) = , 2 1+ζ
µ G2 (ζ) = ζ 2 +1 [Pβμ ( 1 + ζ)]2 .
(78)
and 1 d 1 (79) α = − + iκ, β = − + iν, μ = 1 − . 2 2 2 The Mellin transforms of G1 and G2 are known (see [33, 17(1) p. 257, and 28(1) p. 263.]) ! 2μ−1 s − μ2 , 1 + α2 − s, 1−α 2 −s 1 (s) = G Γ (80) 1 − μ2 − s Γ 1 + α−μ Γ 1−μ−α 2
2
provided Re μ < 2 Re s < min{2 + Re α, 1 − Re α}, 1 2 (s) = 1 G π 2 Γ(1 + β − μ)Γ(−β − μ) ×Γ
1 − μ2 + s, β − μ2 − s, −β − 1 − μ − 3μ 2 − s, − 2 − s
μ 2
− s, − 1+μ 2 −s
! (81)
/ N. provided Re μ < Re(s + 1 − μ/2) < min{− Re β, 12 }, and μ ∈ Remark 6.1. We have actually checked the above formulae using the methods described in [33]. However, some other formulae appearing in that extremely useful reference have misprints. By the Mellin–Plancherel theorem hd (κ, ν, ν) =
1 2iπ
σ+i∞
1 (s)G 2 (−s) ds, G
σ−i∞
(82)
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and the preceding formulae give 1 (s)G 2 (−s) = G d−4 ×Γ
4
√ " 2 πΓ d+1 4 + d 2
+ iν + s, d−4 4 − iν
3d−6 4
iκ d+1 2 , 4 + s, d−4 4
1 − +
#
iκ d−1 d−1 2 , 2 + iν, 2 − iν 3 iκ s, 34 + iκ 2 − s, 4 − 2 −
s
. (83)
+s
It is now possible to use Barnes’ Second Lemma [34, p. 112]: Lemma 6.1 (Barnes). 1 2iπ
i∞
! a1 + s, a2 + s, a3 + s, b1 − s, b2 − s Γ ds c+s
−i∞
! a1 + b1 , a2 + b1 , a3 + b1 , a1 + b2 , a2 + b2 , a3 + b2 , =Γ c − a1 , c − a2 , c − a3
(84)
provided a1 + a2 + a3 + b1 + b2 − c = 0
(85)
and that the contour of integration in (84) separates the increasing and decreasing series of poles. 2 (−s), as given by (83), is, up to a factor, of 1 (s)G As a function of s, G the form of the integrand of (84) if we take d−4 d−4 d−4 + iν, a2 = − iν, a3 = , 4 4 4 3d − 6 3 iκ 3 iκ . b1 = + , b2 = − , c = 4 2 4 2 4 This choice satisfies the condition (85). Therefore, " # iκ d−1 iκ Γ d−1 4 + 2 , 4 − 2 hd (κ, ν, ν) = d √ " d+1 iκ d+1 iκ d−1 # d−1 2 2 πΓ 4 + 2 , 4 − 2 , 2 + iν, d−1 2 − iν, 2 d−1 iκ d−1 iκ d−1 iκ d−1 iκ 4 + 2 + iν, 4 + 2 − iν, 4 − 2 + iν, 4 − 2 − iν, . ×Γ d−1 d−1 2 − iν, 2 + iν a1 =
(86)
(87)
From here on, we will use the notation μ = (d − 1)/4. We can recast the above expression for hd (κ, ν, ν) using Legendre’s duplication formula: 1 hd (κ, ν, ν) = 3−d/2 3/2 2 π Γ(2μ + iκ)Γ(2μ − iκ)Γ(2μ + iν)2 Γ(2μ − iν)2 Γ(2μ) 2 iκ iκ iκ + iν Γ μ + − iν Γ μ + ×Γ μ + 2 2 2 2 iκ iκ iκ d−1 + iν Γ μ − − iν Γ μ − . ×Γ μ − , μ= 2 2 2 4 (88)
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In this form the formula is a special case of the formula for two unequal masses which will appear in [31]. We note that in the derivation of (87) or (88) with hd defined in (71), d is not restricted to be an integer. These formulae hold wherever both sides are defined. Eqs. (87) and (70) give κρ(κ; ν, ν) =
κ sinh(πκ) 1 iκ 2d+2 π Rd−2 Γ 2 + μ + iκ 2 Γ 2 + μ − 2 Γ(2μ) iκ iκ iκ ×Γ μ + + iν Γ μ + − iν Γ μ + 2 2 2 iκ iκ iκ + iν Γ μ − − iν Γ μ − ×Γ μ − , 2 2 2 1
d+3 2
(89)
or, using κ sinh(πκ) = π[Γ(iκ)Γ(−iκ)]−1 , κρ(κ; ν, ν) =
1
1 iκ + μ + iκ 2 Γ 2 + μ − 2 Γ(2μ)
1 iκ iκ × + i ν Γ μ+ Γ μ+ Γ(iκ)Γ(−iκ) =±1 2 2 2d+2 π
d+1 2
Rd−2 Γ
1 2
=±1
(90) This obviously extends to an even analytic function of κ, hence 2
∞
wν (z, z ) =
κρ(κ; ν, ν)wκ (z, z ) dκ.
(91)
−∞
Moreover, for real ν and κ = 0, κρ(κ; ν, ν) is strictly positive. This shows that, in the presence of a suitable interaction term [see (34)], any “principal” particle can decay into any pair of equal-mass “principal” particles. 6.1. Minkowskian Limit Setting κ = M R > 0 and ν = mR > 0 in (89) gives $2 $ 3 $ $ sinh(πM R) $ j=1 Γ(xj + iRuj ) $ ρ(M R; mR, mR) = $ , $ d+3 2d+2 π 2 Rd−2 Γ d−1 $ Γ(x4 + iRu4 ) $
(92)
2
with x1 = x2 = x3 = u1 =
M + m, 2
d−1 d+1 , x4 = , 4 4 M M u2 = − m, u3 = , 2 2
u4 =
M . 2
(93)
Recall Stirling’s formula [35, p. 47]: 1 1 Γ(z) = (2π) 2 e−z+(z− 2 ) log z 1 + a1 z −1 + a2 z −2 + O(z −3 ) , a1 = 1/12,
a2 = 1/288,
(94)
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valid for z ∈ / R− . By a straightforward calculation it follows that if z = x + iy and |x| remains bounded while |y| → +∞, (x − 12 )x2 + 2(a1 x − a2 ) + a21 |Γ(x + iy)|2 ∼ 2πe−π|y| |y|2x−1 1 + . (95) y2 Using this in (92), we find that as R → ∞, R2 ρ(RM ; Rm, Rm) $M $ d−3 $ $ exp πR M M 2 − 4m2 2 2 −m− 2 −m ∼ (1 + AR−2 ), (96) d−1 4 M 2d π 2 Γ d−1 2 where 3
(xj − 12 )x2j + 2(a1 xj − a2 ) + a21 (x4 − 12 )x24 + 2(a1 x4 − a2 ) + a21 A= − 2 uj u24 j=1 1 17 1 107 = + for d = 4. (97) − 16 (M + 2m)2 (M − 2m)2 24M 2
Note that the argument of the exponential in (96) is 0 if M −2m ≥ 0, otherwise −πR(2m − M ) and, in this case, R2 ρ(RM ; Rm, Rm) tends rapidly to 0. In all cases, (96) shows that R2 ρ(RM ; Rm, Rm) tends to ρMink. (M 2 ; m, m) (see (53)). 6.2. Complementary Particles One benefit of having the explicit formula (89) is being able to examine the case of “complementary” particles. The integrand of (91) is meromorphic in κ and ν. We can rewrite (91) as κ sinh(πκ) wν (z, z )2 = 5 d+ d+5 2 R2d−4 2 π Γ d2 Γ d−1 2 R 2 2 d−1 d−1 d 1−ζ iκ iκ + iκ, −iκ; ; ×F Γ μ− Γ μ+ 2 2 2 2 2 2
iκ i ν d−1 + . (98) × Γ μ+ dκ, μ = 2 2 4
, =±1
The integrand is meromorphic in κ and ν. It has no singularity when both are real. The LHS is holomorphic in {ν : ν ∈ / ±i((d − 1)/2 + Z+ )}. We analytically continue the integral in the variable ν: choose ν complex with Re ν > 0 and α = Im ν > 0. Recall that, for integer n ≥ 0, z+n∼0
⇒
Γ(z) ∼
(−1)n . n!(z + n)
(99)
The poles of the functions κ → Γ(μ ± iκ/2) are at κ = ±2i(μ + n) (n ≥ 0 integer), and are independent of ν. The other poles of the integrand are as
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follows (n ≥ 0 integer): iκ + μ ± iν + n ∼ 0 ⇒ Γ 2
iκ + μ ± iν 2
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∼
(−1)n , i 2 n!(κ − 2i(μ ± iν + n))
(100) iκ iκ (−1)n − + μ ± iν + n ∼ 0 ⇒ Γ − + μ ± iν ∼ i . 2 2 − 2 n!(κ + 2i(μ ± iν + n)) (101) The poles κ−2i(μ+iν+n) = 0 [see (100)] and the poles κ+2i(μ−iν+n) = 0 [see (101)] are on the line −2 Re ν + iR. Their mutual distances do not change as ν varies, and they all move down as Im ν increases. The poles κ+2i(μ+iν+n) = 0 [see (101)] and κ − 2i(μ − iν + n) = 0 [see (100)] are the opposites of those described before. They lie on 2 Re ν + iR and move up as Im ν increases. If Im ν increases from 0 but 0 < Im ν < μ, no pole reaches the real axis and the formula (91) continues to hold. This is true in particular √ if μ = iα with 0 < α < (d − 1)/4 = mc /2, corresponding to mc > m1 > mc 3/2. If this condition is satisfied and m0 > mc , Eqs. (42) and (43) hold and the adiabatic limit exists just as in the case m1 > mc . When Im ν reaches μ we have (102) wν (z, z )2 = κρ(κ, ν, ν)wκ (z, z ) dκ, C
where the contour C is obtained from R by a small downward excursion to avoid the pole at −2 Re ν, and another small upward excursion to avoid the pole at 2 Re ν. Once μ < Im ν < μ + 1, we can extract the residues of the poles at κ = ±2i(μ + iν). A similar situation occurs when the successive poles κ = ±2i(μ+iν+n) cross the real axis, so that, for Re ν > 0, Im ν ≥ 0, Im ν−μ ∈ / Z, N = max {j ∈ Z : j < Im ν − μ}, wν (z, z )2 = κρ(κ, ν, ν)wκ (z, z ) dκ R
! N
An (ν) An (ν) + w2i(μ+iν+n) (z, z ) + w−2i(μ+iν+n) (z, z ) . 2 2 n=0 (103)
Note that if Im ν < μ(N < 0), the discrete sum is not present. It turns out that An (ν) = An (ν), which is consistent with κρ being even in κ. Recall that wτ = w−τ for any τ . Thus, for N = max {j ∈ Z : j < Im ν − μ}, (always supposing Re ν > 0), 2
wν (z, z ) = R
κρ(κ, ν, ν)wκ (z, z ) dκ +
N
n=0
An (ν)w2i(μ+iν+n) (z, z ).
(104)
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We find, for integer n ≥ 0, An (ν) =
(−1)n d−1
n!2d−1 π 2 Rd−2 Γ(2μ) Γ(2μ + 2iν + n)Γ(−2iν − n)Γ(2μ + n)Γ(−iν − n)Γ(2μ + iν + n) × . Γ(−2μ − 2iν − 2n)Γ(2μ + 2iν + 2n)Γ( 12 − iν − n)Γ( 12 + 2μ + iν + n) (105)
If now we let ν tend to iα (Re ν tends to 0), (104) will continue to hold provided both parts of the RHS remain meaningful. Therefore, if 0 < α < (d − 1)/2, α − μ ∈ / Z, and N = max {j ∈ Z : j < α − μ} , μ = (d − 1)/4, N
wiα (z, z )2 = κρ(κ; iα, iα)wκ (z, z ) dκ + An (iα)w2i(α−μ−n) (z, z ). n=0
R
(106) κρ(κ; iα, iα) =
iκ Γ(μ + iκ 2 )Γ(μ − 2 ) 1+d 2d+2 π 2 Rd−2 Γ(2μ)
×
Γ(μ +
iκ iκ − α)Γ(μ − iκ 2 − α)Γ(μ + 2 + α)Γ(μ − 2 + α) 1 iκ 1 iκ Γ(iκ)Γ(−iκ)Γ(μ + 2 + 2 )Γ(μ + 2 − 2 ) (107)
iκ 2
This is obviously positive. For An (iα) we find An (iα) =
1 n!2d−1 π
d−1 2
Rd−2 Γ(2μ) Γ(2α − n)Γ(2μ + n)Γ(α − n)Γ(2μ − α + n) × Γ(2α − 2μ − 2n)Γ( 12 + α − n)Γ( 12 + 2μ − α + n) Γ(2μ − 2α + n) ×(−1)n Γ(2μ − 2α + 2n)
(108)
The first two factors in this expression are positive since the arguments of all Γ functions are positive due to α − μ − n > 0 and 2μ − α > 0. The last factor is of the form 2n−1
(−1)n Γ(n + x) = (−1)n (q + x)−1 . Γ(2n + x) q=n
(109)
The last product contains n negative factors and the result is positive, so that An (iα) ≥ 0. Thus the Hilbert space with scalar product given by the LHS of (106) appears as a direct integral of Hilbert spaces associated with unitary irreducible representations of G0 . We conclude that . 1. Any particle from the principal series can decay into two particles (of equal masses) of any series. . 2. A particle of the complementary series with parameter κ = iβ, with 0 < β < 2μ can decay into two particles with parameter iα, α = 12 β + μ + n,
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where n is any integer such that 0 ≤ n and α < 2μ, i.e., n < μ − β/2. This relation can also be written as (2μ − β) = 2(2μ − α) + 2n < 2μ.
(110)
This implies a form of particle stability, but the new phenomenon is that a particle of this kind cannot disintegrate unless the masses of the decay products have certain quantized values. Stability for the same range of masses has also been recently found [36] in a completely different context.
7. Concluding Remarks In trying to interpret the results concerning the lack of mass subadditivity in the de Sitter universe, one can wonder whether they might be due to the thermodynamical properties [20–23] of the fundamental state we have been using. We have tested this possibility against a similar computation in flat thermal field theory that however does not exhibit this phenomenon in two-particle decays. Another issue has to do with energy conservation and the relation mass/energy. dS invariant field theories admit ten conserved quantities (in d = 4). The identification of a conserved energy among these quantities has proven to be useful in classical field theory [37]. The same quantity remains exactly conserved also at the quantum level although it becomes an operator whose spectrum is not positive [21–23] even when restricted to the region where the corresponding classical expression is positive [37]; the thermodynamical properties of dS fields arise precisely in this restriction [20–23]. Energy is conserved also in the decay processes that violate mass subadditivity, once the adiabatic limit has been performed. The breakdown of the subadditivity property of masses in dS space–time just reflects the nonexistence of an Abelian translation group and thereby of a linear energy–momentum space. When we consider the adiabatic limit problem and its meaning in the de Sitter context a first complication is the existence of several choices of cosmic time, having different physical implications and the result might depend on one’s preferred choice. We have studied the closed and the flat cosmological and found that in both models the first factor in (42) diverges like T ; thus it has to be divided by T to extract a finite result which is the same in both models. Here the second (unforeseen) result comes in: in contrast to the Minkowskian case the limiting probability per unit of time does not depend on the wave packet! This result seems to contradict what we see everyday in laboratory experiments, a well known effect of special relativity (52). Furthermore, in contrast with the violation of particle stability that is exponentially small in the de Sitter radius, this phenomenon does not depend on how small is the cosmological constant. How can we solve this paradox and reconcile the result with everyday experience? The point is that the idea of probability per unit time (Fermi’s golden rule) has no scale-invariant meaning in de Sitter: if we use the limiting probability to evaluate amplitudes of processes that take place in a short time we get a grossly wrong result. This is in strong disagreement with
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what happens in the Minkowski case where the limiting probability is attained almost immediately (i.e., already for finite T ). Therefore, to describe what we are really doing in a laboratory we should not take the limit T → ∞ and rather use the probability per unit of time relative to a laboratory consistent scale of time. In that case we will recover all the standard wisdom even in presence of a cosmological constant. But, if an unstable particle lives a very long time (>>R) and we can accumulate observations then a nonvanishing cosmological constant would radically modify the Minkowski result and de Sitter invariant result will emerge. This result should not be shocking: after all erasing any inhomogeneity is precisely what the quasi de Sitter phase is supposed to do at the epoch of inflation; in the same way, from the viewpoint of an accelerating universe all the long-lived particles look as if they were at rest and so their lifetime would not depend on their peculiar motion.
Acknowledgements We thank T. Damour, H. de Vega, M. Gaudin, G. Gibbons, D. Marolf, M. Milgram and V. Pasquier for enlightening discussions. U. M. thanks the SPhT and the IHES for hospitality and support.
Appendix A. More Details in the Minkowski Case In this appendix, we study in more detail the adiabatic limit in the Minkowski case: it is possible to let the two occurrences of g in (41) tend to 1 together, or even at different rates. Let U(f0 , ϕ1 , ϕ2 , ρ) = f0 (x)f0 (y)ϕ1 (u)ϕ2 (v)ρ(σ 2 ) × wm0 (x, u)wa (u, v)wm0 (v, y) dx du dv dy dσ 2 .
(111)
We will assume that ρ is C ∞ and has support in c2 + R+ , with 0 < c < m0 , and that, for each integer n ≥ 0, there are constants Cn ≥ 0 and Ln ≥ 0, such that, for all real t ≥ c2 , |ρ(n) (t)| ≤ Cn (1 + |t|)Ln . We take
ϕj (x) =
e−ipx ϕ j (p) dp,
(112)
0 )δ( ϕ j (p) = ε−1 j (p0 /εj )ψ(p p ), j g
Rd
1 gj (εj (x0 − t))ψ(t) dt, (j = 1, 2), 2π R −itw g j (w) dw, ψ(t) = e−itw ψ(w) gj (t) = e dw.
ϕj (x) =
R
(113) (114)
R
Here εj = Tj−1 > 0. The function ψ belongs to S(R) with ψ(0) = 1. The ∞ function gj is L with compact support. (The cases of real interest are
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gj (t) = θ(1/2 − |t|) or gj (t) = θ(t)θ(1 − t).) We find, after using the various delta-functions, −1 ε (2p0 )−1 |f0 (p)|2 U(f0 , ϕ1 , ϕ2 , ρ) = (2π)d+3 ε−1 1 2 × g 1
p0 − h 0 ε1
g 2
h 0 − p0 ε2
p∈Rd h0 ∈R
0 − p0 ) 0 − h0 )ψ(h ψ(p
× δ(p2 − m20 )θ(p0 )ρ(h20 − p20 + m20 )θ(h0 ) dp dh0 . 0
(115) 0
0
We now change from the variable h to the variable w such that h = p + w: w w −1 H(w) g1 − ε2 U(f0 , ϕ1 , ϕ2 , ρ) = ε1 g 2 dw ε1 ε2 ε1 r 1 r) g1 (−r) g2 dr. (116) = H(ε ε2 Here 2 H(w) = (2π)d+3 |ψ(w)|
(2p0 )−1 |f0 (p)|2 δ(p2 − m20 )θ(p0 )
× ρ(m20 + w(2p0 + w))θ(p0 + w) dp, and we set
H(t) =
dw. e−itw H(w)
(117)
(118)
R
Then ε2 U(f0 , ϕ1 , ϕ2 , ρ) = (2π)−2
H(x)g1 (ε1 x +
R2
ε1 y)g2 (y) dx dy. ε2
(119)
With our assumptions on ρ, H ∈ S(R). Since gj is L∞ with compact support and H ∈ S(R), the above integral (119) is absolutely convergent, uniformly in ε1 and ε2 . Hence −2 ε2 U(f0 , ϕ1 , ϕ2 , ρ) = (2π) H(x)G(x, ε1 , ε2 ) dx, (120) G(x, ε1 , ε2 ) =
R
g1 ε1 x + R
ε1 ε1 y g2 (y) dy = g1 y g2 (y − ε2 x) dy. ε2 ε2 R
(121) We assume from now on 0 < ε1 ≤ ε2 ≤ 1. Since gj ∈ L∞ ∩ L1 and translation is continuous on L1 , G is continuous in x. Example 1. We suppose that g1 , g2 are C ∞ with compact support. In this case the limits when εj tend to 0 can be taken under the integral sign in (121). . (1.1) if ε1 tends to 0 at fixed ε2 , G tends to the constant g1 (0) g2 (y) dy, independent of ε2 .
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Figure 2. Graph of G(x, ε1 , ε2 ) when g1 (x) = g2 (x) = θ(x)θ(1 − x)
Figure 3. Graph of G(x, ε1 , ε2 ) when g1 (x) = g2 (x) = θ(1/2 − |x|) . (1.2) if both ε1 and ε2 tend to 0 and ε1 /ε2 → 0, G also tends to g1 (0) g2 (y) dy. . (1.3) if both ε1 and ε2 tend to 0 and ε1 /ε2 → λ ∈ (0, 1], then G tends to the constant g1 (λy)g2 (y) dy, and −2 ε2 U(f0 , ϕ1 , ϕ2 , ρ) → (2π) (122) H(x) dx g1 (λy)g2 (y) dy. This holds in particular if ε1 and ε2 are kept equal so that λ = 1. The constant g1 (λy)g2 (y) dy may be equal to the preceding constant g1 (0) g2 (y) dy, for example if g1 (λy) = g1 (0) on the support of g2 . Example 2. We consider the case when gj (x) = θ(x)θ(1 − x), i.e., gj is the indicator function of [0, 1]. Then (see Fig. 2) −1 G(x, ε1 , ε2 ) = (1 + ε2 x)θ(1 + ε2 x)θ(−x) + θ(x)θ(ε−1 1 − ε2 − x) −1 −1 −1 + ε2 (ε−1 1 − x)θ(x − ε1 + ε2 )θ(ε1 − x).
(123)
G(x, ε1 , ε2 ) tends to 1 when both εj → 0 (with ε1 ≤ ε2 ). G also tends to 1 if ε1 → 0 at fixed ε2 and then ε2 → 0. Since H ∈ S(R), the integral (120) tends to (2π)−2 R H(x) dx = (2π)−1 H(0). Example 3. Consider now the case gj (x) = θ(1/2 − |x|), i.e., gj is the indicator function of [−1/2, 1/2]. In that case G(x, ε1 , ε2 ) is even in x (see Fig. 3): G(x, ε1 , ε2 ) = θ(ε−1 − ε−1 − 2|x|) 1 2 1 ε2 −1 −1 −1 + −ε2 |x| θ(2|x| − ε−1 1+ 1 +ε2 )θ(ε1 + ε2 −2|x|). 2 ε1 (124) G(x, ε1 , ε2 ) tends to 1 either if ε1 tends to 0 at fixed ε 2 , or if both εj → 0 (with ε1 ≤ ε2 ), and the integral (120) tends to (2π)−2 R H(x) dx = (2π)−1 H(0).
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Conclusion. With the two last choices of gj just described, d+2 2 ε2 U(f0 , ϕ1 , ϕ2 , ρ) → (2π) ρ(m0 ) (2p0 )−1 |f0 (p)|2 δ(p2 − m20 )θ(p0 ) dp. (125) For other choices of gj , the limit as ε1 → 0, then ε2 → 0 need not be the same as when ε1 = ε2 → 0. If we relax the conditions set on ρ, the same conclusions hold if, e.g., ρ(s) = ρ(s; m1 , m2 ) and d ≥ 4.
Appendix B. Adiabatic Limit (dS): Horizontal Slices Horizontal slices have been described in subsect. 5.1. In this appendix, we study limT →+∞ T −1 L1 (f0 , g) where g(x) is given by (59) in the coordinates (56). We denote κ = [m20 − (d − 1)2 /4]1/2 and recall that m0 > (d − 1)/2 (hence κ > 0), and C1 (mo , d) = C0 (κ) [see (27)]. Inserting the representation (23) for the three occurrences of wm0 + (denoted also wκ+ ) in the formula (43) for L1 (f0 , g) gives L1 (f0 , g) =
γ 2 C0 (κ)cd,κ
h0 (ξ) =
γ×γ
h0 (ξ)Kκ (ξ, ξ , g)h0 (ξ ) dμγ (ξ) dμγ (ξ ) , (126) h (ξ)h0 (ξ) dμγ (ξ) γ 0
+ ψ− (x, ξ)f0 (x) dx, d−1 −iκ
(127)
+ − ψ− (x, ξ)ψ− (x, ξ )g(x) dx. d−1 d−1 −iκ +iκ
(128)
2
Xd
Kκ (ξ, ξ , g) =
2
2
Xd
We take γ = S0 = {ξ ∈ C+ : ξ 0 = 1} S d−1 , the unit sphere in Rd . In this appendix, we also set R = 1: a general R can be reinstated in the results by homogeneity. Note that, for any ϕ ∈ C ∞ (S0 × S0 ), + − ϕ(ξ, ξ )ψ− (x, ξ)ψ− (x, ξ ) dμS0 (ξ) dμS0 (ξ ) d−1 d−1 −iκ +iκ 2
2
is C ∞ in x. Hence, for any bounded g with bounded support, Kκ is a distribution on S0 × S0 in the variables (ξ, ξ ). We will take g invariant under the rotation group in d dimensions (leaving e0 invariant), hence Kκ (ξ, ξ , g) = Kκ (Lξ, Lξ , g) for every such rotation L. Hence Kκ is C ∞ in ξ when smeared with a test-function in ξ . Studying the limit of T −1 L1 (f0 , g) for g as in (59), is, therefore, equivalent to studying the limit of T −1 Kκ (ξ, ξ , g) as a distribution in ξ for fixed ξ. In this appendix, the cases d = 2 and d = 4 will be treated. The case d = 3, more straightforward than d = 4 (no need to use d = 2), will be omitted. The result in these three cases is the same [see (157) and (218)].
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B.1. Case d = 2 We use the following parametrizations ⎧ ⎧ 0 0 ⎪ ⎪ ⎨x = sinh t ⎨ξ = 1 1 x = cosh t sin θ ξ1 = 0 ⎪ ⎪ ⎩ 2 ⎩ 2 x = cosh t cos θ ξ = −1
⎧ 0 ⎪ ⎨ξ = 1 ξ 1 = − sin φ ⎪ ⎩ 2 ξ = − cos φ
639
(129)
with t ∈ R, −π < θ < π, (x ∈ X2 ), −π < φ < π, (ξ, ξ ∈ ∂V+ ).
(130)
In these variables, the measure dx takes the form cosh t dt dθ. For small ε > 0, changing t into t ± iε pushes x into T± . If g(x) = g0 (t), 1 Kκ (ξ, ξ , g) = g0 (t)[sinh(t + i0) + cosh(t + i0) cos(θ)]− 2 −iκ t∈R,−π≤θ≤π 1
×[sinh(t − i0) + cosh(t − i0) cos(θ − φ)]− 2 +iκ cosh t dt dθ. (131) For real s with 0 < |s| < π/2 and real α, sinh(t + is) + cosh(t + is) cos(α) = cos(s)(sinh t + cosh t cos(α)) + i sin(s)(cosh t + sinh t cos(α))
(132)
has a non-zero imaginary part of the same sign as s, so its power μ can be taken for any complex μ and remains analytic in t for all real t, smooth and periodic with period 2π in α. The integral over θ in (131) will be performed, using Plancherel’s formula, by first computing the discrete Fourier transform, in the variable θ, of the two last factors in the integrand, i.e., 1
eimφ Fm (t + i0)Fm (t + i0)g0 (t) cosh(t) dt, (133) Kκ (ξ, ξ , g) = 2π m∈Zt∈R
with π Fm (t + is) =
1
[sinh(t + is) + cosh(t + is) cos(θ)]− 2 −iκ eimθ dθ,
(134)
−π
where 0 < |s| < π/2. By changing θ to −θ in the integration, we get: Fm (t + is) = F−m (t + is).
(135)
We use the formula (see [35], (15) p. 157 and (7) p. 140) Pμm (z)
Γ(μ + m + 1) = 2πΓ(μ + 1)
π [z + (z 2 − 1)1/2 cos φ]μ eimφ dφ, −π
(z ∈ Δ1 , Re z > 0),
(136)
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valid for all m ∈ Z. This gives, for 0 < s < π/2, 2πΓ( 12 − iκ) m P 1 (−i sinh(t + is)) Γ(m + 12 − iκ) − 2 −iκ for t > 0, (137) 1 2πΓ( 2 − iκ) m P 1 (−i sinh(t + is)) Fm (t + is) = (−1)m e−iπ/4+πκ/2 Γ(m + 12 − iκ) − 2 −iκ for t < 0, (138) 1 2πΓ( + iκ) 2 P −m Fm (t + is) = eiπ/4+πκ/2 (i sinh(t − is)) 1 Γ(−m + 12 + iκ) − 2 +iκ for t > 0, (139) 1 2πΓ( 2 + iκ) P −m (i sinh(t − is)) = (−1)m eiπ/4+πκ/2 1 Γ(−m + 12 + iκ) − 2 +iκ for t < 0. (140)
Fm (t + is) = e−iπ/4+πκ/2
Taking g0 (t) = θ(T /2 − t)θ(t + T /2), we rewrite (133) as
+ − Kκ (ξ, ξ , g) = eimφ (Im + Im ),
(141)
m∈Z
where + Im
T /2 Fm (t + i0)Fm (t + i0) cosh(t) dt
1 = 2π
(142)
0 sinh(T /2) πκ
= 2πe
P−m1 −iκ (−iu + ε)P−−m (iu + ε) du. (143) 1 +iκ
m
(−1)
2
2
0
In the last expression we have used Γ( 12 +z)Γ( 12 −z) = π/ cos(πz), and changed to the variable u = sinh t. Similarly (with now u = − sinh t), − Im
1 = 2π
0 Fm (t + i0)Fm (t + i0) cosh(t) dt
(144)
−T /2 sinh(T /2) πκ
= 2πe
P−m1 −iκ (iu + ε)P−−m (−iu + ε) du. (145) 1 +iκ
m
(−1)
2
2
0
Remark B.1. Using (ah.10) and Γ( 12 + z)Γ( 12 − z) = π/ cos(πz) shows that the RHS of (145) can be obtained from the RHS of (143) by changing in the ± ± integrand (but not outside the integral) κ to −κ. Note also that Im = I−m by (135). The meaning of (141) is that Kκ (ξ, ξ , g) is a distribution in ξ as + − + Im is its discrete Fouexpressed in the coordinate φ, and that m → Im + − rier transform. It is tempered, i.e., |Im + Im | does not increase faster than a power of |m| as |m| → ∞. To prove that T −1 Kκ tends to a limit (also a distribution in φ) as T → ∞ is equivalent to proving that
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+ − . (1) For each m, T −1 (Im + Im ) tends to a limit Um as T → ∞, + − + Im | ≤ . (2) there are two positive constants P and Q such that T −1 |Im Q P (1 + |m| ) for all m and T .
If both conditions are satisfied, Um is the mth Fourier coefficient of the limit, i.e., lim T −1 Kκ = m Um eimφ . B.1.1. Condition (1). We need the asymptotic behavior of Pλm (z) as |z| → ∞, as described in [35, pp. 123, 124, 126, and 164]. For z ∈ Δ1 and ζ = z −2 , Pλμ (z) =
2−λ−1 π −1/2 Γ(− 12 − λ)z −λ−1+μ (z 2 − 1)−μ/2 Γ(−λ − μ) ×F ( 12 + λ/2 − μ/2, 1 + λ/2 − μ/2; λ + 3/2; ζ) 2λ π −1/2 Γ( 12 + λ)z λ+μ (z 2 − 1)−μ/2 Γ(1 + λ − μ) ×F (−λ/2 − μ/2, 12 − λ/2 − μ/2; 12 − λ; ζ).
+
(146)
If λ+ 12 ∈ / Z, the two hypergeometric functions can be expanded into convergent power series for |ζ| < 1. For z ∈ Δ1 and |z| → ∞, we find Pλμ (z) ∼
2λ π −1/2 Γ( 12 + λ) z λ 2−λ−1 π −1/2 Γ(− 12 − λ)z −λ−1 + Γ(−λ − μ) Γ(1 + λ − μ)
(147)
Hence, as u → +∞, P−m1 −iκ (−iu)P−−m (iu) 1 2 2 +iκ 1 1 2− 2 +iκ π −1/2 Γ(iκ)eiπ/4+πκ/2 u− 2 +iκ ∼ Γ( 12 + iκ − m)
1 1 2− 2 −iκ π −1/2 Γ(−iκ)eiπ/4−πκ/2 u− 2 −iκ + Γ( 12 − iκ − m) 1 1 2− 2 −iκ π −1/2 Γ(−iκ)e−iπ/4+πκ/2 u− 2 −iκ × Γ( 12 − iκ + m) 1 1 2− 2 +iκ π −1/2 Γ(iκ)e−iπ/4−πκ/2 u− 2 +iκ + . Γ( 12 + iκ + m)
(148)
We first consider the “off-diagonal terms” of this product: 2−1+2iκ π −1 Γ(iκ)2 u−1+2iκ 2−1−2iκ π −1 Γ(−iκ)2 u−1−2iκ + 1 . 1 1 Γ( 2 + iκ − m)Γ( 2 + iκ + m) Γ( 2 − iκ − m)Γ( 12 − iκ + m)
(149)
These two terms are exchanged by changing κ to −κ. The contribution of the + first to Im /T is of the form 1 Const. T
sinh(T /2)
u−1+2iκ du = Const.
1
1 (sinh(T /2)2iκ − 1). 2iκT
(150)
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This tends to zero as T → +∞. The same happens for the second term. The “diagonal terms” are 2−1 π −1 Γ(iκ)Γ(−iκ)eπκ u−1 2−1 π −1 Γ(−iκ)Γ(+iκ)e−πκ u−1 + . 1 1 Γ( 2 + iκ − m)Γ( 2 − iκ + m) Γ( 12 − iκ − m)Γ( 12 + iκ + m)
(151)
Again these two terms are exchanged by changing κ to −κ. Their sum can be reexpressed as (−1)m cosh(πκ)2 u−1 . πκ sinh(πκ)
(152)
u−1 du = log(sinh(T /2)) ∼ T /2,
(153)
1 + eπκ cosh(πκ)2 Im ∼ . T πκ sinh(πκ)
(154)
Since sinh(T /2)
1
− + Because of Remark B.1, Im /T has the same limit as Im /T and
1 + 2eπκ cosh(πκ)2 − (Im + Im . )= T →+∞ T κ sinh(πκ)
Um = lim
(155)
Um is independent of m, so that if Condition (2) is satisfied, 1 4πeπκ cosh(πκ)2 Kκ (ξ, ξ , gT ) = δ(φ) T →+∞ T κ sinh(πκ) 4πeπκ cosh(πκ)2 δS 1 (ξ, ξ ), = κ sinh(πκ) lim
(156)
and (see (126)) lim T −1 L1 (f0 , gT ) = γ 2 C0 (κ)c2,κ
T →+∞
γ 2 π coth(πκ)2 4πeπκ cosh(πκ)2 = . κ sinh(πκ) |κ| (157)
We note that, owing to the delta function in (156) the dependence on h0 (i.e., on f0 ) has completely disappeared from the limit. This result agrees with (64). B.1.2. Condition (2). In this section, λ always denotes − 21 − iκ with κ ∈ R and κ = 0. We first return to the first step of the preceding subsection in the case m = 0. From the identity (146) and the analyticity of ζ → F (a, b, c; ζ) in the unit disk, it follows that there is a M0 (κ) > 0 such that $ $ 1 1 1 1 $ 2− 2 −iκ π −1/2 Γ(−iκ)z − 2 −iκ $$ 2− 2 +iκ π −1/2 Γ(iκ)z − 2 +iκ $ − $ $Pλ (z) − $ $ Γ( 12 + iκ) Γ( 12 − iκ) < M0 (κ)|z|−5/2
z ∈ Δ1 ,
|z| > 2.
(158)
By (137)–(138), there is also an M1 (κ) > 0 such that, for 0 < s < π/2, |F0 (t + is)| ≤ M1 (κ)| sinh(t + is)|−1/2 .
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We now obtain crude bounds for |Fm |. For t ≥ 0 and 0 < |s| < π/2, changing θ to θ + π in (134), we get π m
λ
Fm (t + is) = (−1) cosh(t + is)
λ
((1 − cos θ) − (1 − th(t + is))) eimθ dθ.
−π
(160) Changing to the variable ϕ = θ/2, Fm (t + is) = (−1)m 2(2 cosh(t + is))λ Am (z),
(161)
π/2 (sin2 (ϕ) − z 2 )λ e2imϕ dϕ,
Am (z) =
(162)
−π/2
z2 =
1 (1 − th t)(1 − i tg s) 1 (1 − th(t + is)) = . 2 2 1 + i th t tg s
We now suppose 0 < tg s < 1/4. It follows, after some calculations: $ $ $ $ $ Im z $ $ Im z 2 $ $ $ $ ≤ tg s, $ ≤ tg(2s), tg(s/2) ≤ $ tg s ≤ $ Re z 2 $ Re z $
(163)
(164)
We define z = x − iy with x > 0. Then 0 < x < |z| < 3/4, 0 < y ≤ x tg s < 3/16, % e−t √ ≤ |z| ≤ x 1 + tg2 s ≤ x 17/16. 2
(165)
Recall that for ρ > 0, −π < θ < π, ζ ∈ C, |(ρeiθ )ζ | = ρRe ζ e−θ Im ζ ≤ ρRe ζ eπ| Im ζ| .
(166)
Thus −π|κ|
e
π/2 |Am (z)| ≤ H(z) =
| sin2 ϕ − z 2 |−1/2 dϕ
−π/2
1 √
=2 0
1−
t2
dt . |t2 − z 2 |
(167)
√ √ After splitting the integration interval as [0, 1] = [0, x] ∪ [x, 3/2] ∪ [ 3/2, 1], straightforward estimates give √ 4π e−π|κ| Am (z) ≤ H(z) ≤ √ + 8 + 4 log( 3/x) ≤ 27 + 4t. 3 3
(168)
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We now consider π/2 A0 (z) − Am (z) =
(sin2 (ϕ) − z 2 )λ (1 − e2imϕ ) dϕ −π/2
π/2 (sin2 (ϕ) − z 2 )λ (1 − e2imϕ + 2imϕ) dϕ.
=
(169)
−π/2
Using |1 − e2imϕ + 2imϕ| ≤ 2m2 ϕ2 ≤
m2 π 2 sin2 (ϕ) 2
∀ϕ ∈ [−π/2, π/2] (170)
we get −π|κ|
e
π/2 |A0 (z) − Am (z)| ≤ m π | sin2 (ϕ) − z 2 |−1/2 sin2 (ϕ) dϕ 2 2
0
m2 π 2 z 2 H(z) + m2 π 2 = 2
π/2 | sin2 (ϕ) − z 2 |1/2 dϕ 0
5π 3 m2 m2 π 2 z 2 H(z) + (171) ≤ 2 8 Since |z 2 | ≤ 17x2 /16 and 2x2 log(1/x) < 1/e, there is a constant M2 > 0 such that, for all m, |A0 (z) − Am (z)| ≤ eπ|κ| M2 m2 , and hence |F0 (t + is) − (−1)m Fm (t + is)| ≤
√
(172)
2e2π|κ| | cosh(t + is)−1/2−iκ |M2 m2 . (173)
With 0 < tg s < 1/4, as we have chosen, | cosh(t + is)−1/2−iκ | ≤ (cosh t)−1/2 (17/16)1/4 e|κ|/4 , so that
√ |F0 (t + is) − (−1)m Fm (t+is)| ≤ 2e(2π+1/4)|κ| (17/16)1/4 | cosh(t)|−1/2 M2 m2 , (174)
and, by (159), there is an M3 (κ) > 0 such that |Fm (t + is)| ≤ M3 (κ)(1 + m2 )| sinh(t)|−1/2 .
(175)
Therefore, using the bound (167), independent of m, for 0 ≤ t ≤ t1 , and the bound (175) for t1 ≤ t ≤ T /2, we find that there is a constant M4 (κ) > 0 such that + |T −1 Im | ≤ M42 (κ)(1 + m2 )2
∀m ∈ Z.
− The same holds for |T −1 Im |. This proves that Condition (2) is satisfied.
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B.2. Other Dimensions In this subsection the dimension of the de Sitter space–time X is n = d > 2, i.e., the ambient Minkowski space–time is Rd+1 . The notation n = d is used to stay close to [38], Chap. IX, p 448 ff, which is constantly used in this section. As in Sect. B.1.1, we wish to compute (177) Kν (ξ, ξ , g) = ψλ+ (x, ξ)g(x)ψλ− ¯ (x, ξ ) dx X
where ξ, ξ ∈ ∂V+ ⊂ Rd+1 and ξ 0 = ξ = 1, λ = −(n − 1)/2 − iν, ψλ± (x, ξ) are as defined in (18). We use the following parametrization for x = (x0 , x) ∈ X, ξ = ∈ ∂V+ , ξ = (1, ξ ) ∈ ∂V+ (see [38], p. 448]). (1, ξ) 0
⎧ 0 x = sinh t, x = − cosh t u ⎪ ⎪ ⎪ ⎪ ⎪u1 = sin θn−1 . . . sin θ2 sin θ1 ⎪ ⎪ ⎪ ⎪ ⎪ u2 = sin θn−1 . . . sin θ2 cos θ1 ⎪ ⎨ u3 = sin θn−1 . . . cos θ2 ⎪ . ⎪ ⎪ ⎪.. ⎪ ⎪ ⎪ ⎪un−1 = sin θ ⎪ n−1 cos θn−2 ⎪ ⎪ ⎩ n u = cos θn−1 ξ 0 = ξ n = 1,
⎧ 0 ξ =1 ⎪ ⎪ ⎪ ⎪ ⎪ξ 1 = sin φn−1 . . . sin φ2 sin φ1 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ξ = sin φn−1 . . . sin φ2 cos φ1 ⎪ ⎨ 3 ξ = sin φn−1 . . . cos φ2 ⎪ . ⎪ ⎪ ⎪.. ⎪ ⎪ ⎪ ⎪ξ n−1 = sin φ ⎪ n−1 cos φn−2 ⎪ ⎪ ⎩ n ξ = cos φn−1 for 1 ≤ j < n.
ξj = 0
(178)
(179)
Here t ∈ R, 0 ≤ θ1 < 2π, 0 ≤ φ1 < 2π, 0 ≤ θk < π for k > 1, 0 ≤ φk < π for k > 1. With these notations dx = coshn−1 t dt du,
du = sinn−2 θn−1 dθn−1 . . . sin θ2 dθ2 dθ1 .
(180)
We also use the normalized measure dσ(u) on S n−1 , dσ(u) =
Ω−1 n
du,
Ωn =
du =
2π n/2 . Γ(n/2)
(181)
S n−1
We restrict g to be of the form g(x) = gT (x) = g0 (t) = θ(T /2 − t)θ(t + T /2), T > 0. The integral (177) takes the form (182) Kν (ξ, ξ , g) = g0 (t)(cosh t)n−1 dt F (t, u)G(t, u) du, R
F (t, u) = (sinh(t + i0) + cosh(t + i0) cos(θn−1 ))λ .
(183)
For G, we have G(t, u) = (x+ · ξ )λ . Note that ξ = Rξ, where R is the rotation in Rn R = eφ1 M21 . . . eφn−1 Mnn−1 ,
Mjk = ej ∧ ek .
(184)
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For example
eφn−1 Mnn−1
⎛ 1 ⎜ .. ⎜. ⎜ = ⎜0 ⎜ ⎝0 0
Ann. Henri Poincar´e
...
0 .. .
0
0
... ... ...
1 0 0
0 cos φn−1 − sin φn−1
⎞
⎟ ⎟ ⎟ ⎟ 0 ⎟ sin φn−1 ⎠ cos φn−1
(185)
Therefore, G(t, u) = F (t, R−1 u).
(186)
As in the case d = 2, we reexpress the integral over u in (182) using harmonic analysis on the sphere. Harmonic analysis on S n−1 uses an orthonormal basis {ΞK } of functions on the sphere ( = 0, 1, 2, . . . , K is a multiindex). This is fully described in [38, Chap IX]): ΞK (u)ΞK (u) dσ(u) = δ δKK . (187) S n−1
For fixed the functions {ΞK } generate a finite-dimensional subspace Hn of L2 (S n−1 ) in which the regular representation of SO(n) reduces to an irreducible unitary representation, characterized by its matrix elements in the basis {ΞK }: for any g ∈ SO(n),
ΞK (g −1 u) = tM K (g)ΞM (u). (188) M
{ΞK }
{tKM }
The functions and are analytic on S n−1 and SO(n) respectively. 2 Given two arbitrary L functions h1 , h2 on S n−1 we have (for j = 1, 2)
hj K ΞK (u), (189) hj (u) = ,K
hj K =
hj (u)ΞK (u) dσ(u),
(190)
S n−1
h1 (u)h2 (u) dσ(u) =
h1 K h2 K .
(191)
,K
S n−1
These formulae imply that
ΞK (u)ΞK (v ) = Ωn δS n−1 (u, v )
(192)
,K
where δS n−1 (u, v ) denotes the distribution (actually measure) on S n−1 × S n−1 defined by δS n−1 (u, v )ϕ(u, v ) du dv = ϕ(u, u) du. (193) S n−1 ×S n−1
S n−1
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Actually, as is the case for all invariant distributions on S n−1 ×S n−1 , smearing δS n−1 (u, v ) only in v with a C ∞ function produces a C ∞ function of u: δS n−1 (u, v )ψ(v ) dv = ψ(u). (194) S n−1
Choosing in particular u = en , we can use the formula ([38, IX 4.1 (1-4)] and text there) , Γ( + n − 2)(2 + n − 2) . (195) ΞK (en ) = δK0 !Γ(n − 1) Inserting this in (192) gives δS n−1 (en , v ) =
Ω−1 n
,
Γ( + n − 2)(2 + n − 2) Ξ0 (v ). !Γ(n − 1)
Taking v = ξ with ξ given by (178) and using [38] IX 4.1 (3-4), , Γ( + n − 2)(2 + n − 2) tM 0 (g) ΞM (g en ) = !Γ(n − 1)
(196)
(197)
we get δS n−1 (en , ξ ) = Ω−1 n
Γ( + n − 2)(2 + n − 2) !Γ(n − 1)
t00 (R),
(198)
with R given by (184). Harmonic analysis extends to distributions on the sphere, as it does on S 1 . We apply (189)-(191) to the case h1 (u) = F (t, u), h2 (u) = G(t, u). Because F (t, u) depends only on cos θn−1 , (t) = F (t, u)ΞK (u) dσ(u) = δK0 f0 (t). (199) fK S n−1
Note that t can be complexified in (199), i.e., t can be replaced by t + is with (t), writing simply 0 < |s| < π/2. In the sequel we omit the t-dependence of fK fK unless the t-dependence becomes significant. We have , n−2 !Γ(n − 2)(2 + n − 2) . (200) Ξ0 (u) = A0 C 2 (cos θn−1 ), A0 = Γ( + n − 2)(n − 2)
f0 Ξ0 (R−1 u) = (f0 tK0 (R))ΞK (u). (201) G(t, u) =
,K
Therefore,
F (t, u)G(t, u) du = Ωn t00 (R)|f0 |2 ,
(202)
S n−1
Kν (ξ, ξ , gT ) = Ωn
T /2
t00 (R) −T /2
|f0 (t)|2 (cosh t)n−1 dt
(203)
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Also t00 (R) =
!Γ(n − 2) n−2 C 2 (cos φn−1 ), Γ( + n − 2)
(204)
For (200) see [38] IX 3.6 (6, 7) p. 480. For (204) see [38], IX 4.2 (8) p. 484. We thus have n−2 f0 = Ω−1 A [sinh(t + i0) + cosh(t + i0) cos θn−1 ]λ C 2 (cos θn−1 ) n 0 × sinn−2 θn−1 dθn−1 . . . sin θ2 dθ2 dθ1 π −1 = Ωn Ωn−1 A0 [sinh(t + i0) + cosh(t + i0) cos θ]λ sinn−2 θ 0 n−2 2
× C
(cos θ) dθ
(205)
Cμ
In these formulae is a Gegenbauer polynomial: see [35] p. 175 for the definition. The formulae [35] p. 176 (9), and [38], IX 3.1 (3), giving the explicit coefficients of Cμ coincide, so we are dealing with the same objects. B.3. The Case d = n = 4 We now restrict our attention to the case d = 4, keeping the notations of the preceding subsection. In this case λ = −3/2 − iκ, Ω4 = 2π 2 , A0 = 1. We exclude the case κ = 0. Since (n − 2)/2 = 1, the formula (205) gives: π 2 [sinh(t + i0) + cosh(t + i0) cos θ]λ C1 (cos θ) sin2 θ dθ. (206) f0 = π 0
We have ([35], 3.15.1 (15) p. 177) C1 (cos θ) =
sin( + 1)θ . sin θ
(207)
Therefore, for sufficiently small s > 0, π 2 f0 (t + is) = [sinh(t + is) + cosh(t + is) cos θ]λ sin( + 1)θ sin θ dθ (208) π 0
1 = cosh(t + is)λ π
π [th(t + is) + cos θ]λ sin( + 1)θ sin θ dθ
−π
( + 1) cosh(t + is)λ = π(λ + 1)
(209)
π [th(t + is) + cos θ]λ+1 cos( + 1)θ dθ
−π
(210) =
( + 1) cosh(t + is)−1 π(λ + 1)
π [sinh(t + is) + cosh(t + is) cos θ]λ+1
−π
× cos( + 1)θ dθ.
(211)
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Recall that λ + 1 = −1/2 − iκ. Therefore, comparing (211) with (134), we find, for 0 < |s| < π/2, ( + 1) F+1 (t + is) + F−(+1) (t + is) 2π(− − iκ) cosh(t + is) ( + 1) F+1 (t + is), = (212) π(− 12 − iκ) cosh(t + is)
f0 (t + is) =
1 2
using (135). Hence Kκ (ξ, ξ , gT ) = Ω4
t00 (R)
( + 1)2 + 1/4)
π 2 (κ2
T /2
×
F+1 (t + i0)F+1 (t + i0) cosh t dt
(213)
−T /2
=
t00 (R)
4π( + 1)2 + − (I + I+1 ) (κ2 + 1/4) +1
(214)
with the notations of (142). Therefore, by (155) and the proof of Condition (2) for d = 2 (Sect. B.1.2), 1 8πeπκ cosh(πκ)2 Kκ (ξ, ξ , gT ) = 2 t00 (R)( + 1)2 . (215) T →+∞ T (κ + 1/4)κ sinh(πκ) lim
In the case n = 4, (198) becomes
ξ ). (l + 1)2 t00 (R) = Ω4 δS 3 (e4 , ξ ) = 2π 2 δS 3 (ξ,
(216)
Thus 1 16π 3 eπκ cosh(πκ)2 ξ ). Kκ (ξ, ξ , gT ) = 2 δS 3 (ξ, T →+∞ T (κ + 1/4)κ sinh(πκ) lim
(217)
It follows [see (126)] that lim T −1 L1 (f0 , gT ) =
T →+∞
γ 2 π coth(πκ)2 . |κ|
(218)
This is the same as in the case d = 2.
Appendix C. Adiabatic Limit (dS): Parabolic Slices We again take R = 1. We again start from the formulae (126)–(128) of Appendix B, but we only require κ ∈ R\{0}. The function g will be chosen as announced in Sect. 5.1. The map (t, y) → x(t, y) defined in (60) is a diffeomorphism of Rd onto the “upper half” Xdup = {x ∈ Xd : x0 +xd > 0}, and (t, y) → −x(t, y) is a diffeomorphism of Rd onto the “lower half” Xddown = −Xdup . The
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cycle γ appearing in (126) will be chosen as V0 = C+ ∩{ξ ∈ Md+1 : ξ 0 +ξ d = 1}. It can be parametrized by the diffeomorphism η → ξ(η) of Rd−1 onto V0 : ⎧ 0 d−1 ⎨ ξ = 12 (1 + η 2 ),
ηj2 . (219) ξ(η) = ξ j = ηj , (1 ≤ j ≤ d − 1), η 2 = ⎩ d 2 j=1 ξ = 12 (1 − η ), Thus V0 is a Euclidean space with (dξ · dξ) = −dη 2 on V0 . The stability group of the vector e0 − ed in G0 leaves V0 invariant and acts as the group of Euclidean displacements there. As noted in Remarks 2.1 and 2.2, the G0 invariance and homogeneity of ψλ± (x, ξ) imply that it can be regarded as a distribution in ξ on V0 , C ∞ in x on Xd . For a real g ∈ S(Xd ), if we denote gˇ(x) = g(−x), we find Kκ (ξ, ξ , gˇ) = e2πκ K−κ (ξ, ξ , g).
(220)
It will turn out that g can be chosen invariant under the stability group of e0 − ed . Then Kκ is an invariant distribution on V0 × V0 . For our purposes it will suffice (and be possible) to study the limit of T −1 Kκ (ξ, ξ , gT ) with gT (x) = θ(t + T /2)θ(T /2 − t),
t = log(x0 + xd ),
(221)
and to add in the end the limit of T −1 Kκ (ξ, ξ , gˇT ) obtained from (220). With x parametrized as in (60) and ξ as in (219), we have s (y − η)2 1 t [e (y − η)2 − e−t ] = − , 2 2s 2
x(t, y) · ξ =
For k ∈ Rd−1 , we find
± (k, s, η) = ψ− d−1 +iν
def
=2
d−1 2 −iν
d−1 2 −iν
2
Rd−1
!− d−1 2 +iν y2 − (s ∓ i) eiky dy s ∓ i
eikη Rd−1 ∞
eikη 0
= 2d−1−iν π
d−1 2
k
!− d−1 2 +iν a2 − (s ∓ i) da2 s ∓ i
3−d 2
∞ eikη 0
×y
d−1 2
(222)
± ψ− (x(t, y), ξ)eiky dy d−1 +iν
2
=2
s = e−t .
δ(y2 − a2 )eiky dy
Rd−1
y2 s ∓ i
!− d−1 2 +iν − (s ∓ i)
J d−3 (ky) dy, 2
(223)
where k = |k| and s = e−t . We use the following formula [39, (51) p. 95] with some notational changes) λ−iα−1 iα ∞ k z K−iα (kz) 2 2 −λ+iα λ . (y + z ) y Jλ−1 (ky) dy = 2 Γ(λ − iα) 0
(224)
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This is valid provided k > 0, Re z > 0, Re λ > 0, and Re(λ − 2iα + 1/2) > 0. Note that none of these parameters except k needs to be real. In our application, λ = (d − 1)/2 and α = ν ∈ R\{0}. We take z = −is + ε, s > 0, ε > 0 arbitrarily small, α = ν. Since (y 2 + z 2 ) then has a small negative imaginary − in (223). In (224) K denotes part, this will correspond to the case of ψ −iα
−λ+iν
the McDonald function. This will introduce no lasting ambiguity since we will use the identities [39, (5), (6) p. 4, (15) p. 5] K−iν (−iks) =
iπ πν/2 (1) iπeπν/2 e Jiν (ks) − e−πν J−iν (ks) . H−iν (ks) = 2 2 sinh(πν) (225)
This yields λ λ+1 −iν λ k s − ikη i2 π (eπν Jiν (ks) − J−iν (ks)) , ψ −λ+iν (k, s, η) = e Γ(λ − iν) sinh(πν) d−1 . (226) λ= 2 + ψ −λ−iν (k, s, η) can be obtained from this since, for real ν, it is the complex − (−k, s, η): conjugate of ψ −λ+iν
λ λ+1 iν λ k s + ikη (−i)2 π (eπν J−iν (ks) − Jiν (ks)) , ψ −λ−iν (k, s, η) = e Γ(λ + iν) sinh(πν) d−1 λ= . (227) 2
Remark C.1. By the preceding remarks, if ξ ∈ V0 is expressed in terms of η ∈ Rd−1 as in (219), ψα± (x, ξ) is a tempered distribution in η, a C ∞ function of x, and an entire function in α. If x is expressed as in (60), its Fourier transform with respect to the variable y is also a tempered distribution in the variable k conjugated to y and in η, C ∞ in s and holomorphic in α, and, in this sense, the formulae (226) and (227) can be continued to all ν. If ν is taken real in these formulae, their RHS becomes locally bounded in k, in particular locally L2 . Supposing g(x) = G((x0 +xd )−1 ) (for example if g(x) = gT (x) = GT (s) = θ(s − e−T /2 )θ(eT /2 − s)), Plancherel’s formula gives Kκ (ξ, ξ , g) = (2π)1−d
s−d G(s)
s>0,k∈Rd−1 + (k, s, η)ψ − (−k, s, η ) ds dk. ×ψ− d−1 −iκ − d−1 +iκ 2
2
(228)
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Inserting (226), we find Kκ (ξ, ξ , g) = (2π)1−d
-κ (k, g) = K
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-κ (k, g) dk, eik·(η−η ) K
(229)
Rd−1
" # 2 2 s−1 G(s) A Jiκ (ks)J−iκ (ks) + BJiκ (ks) + CJ−iκ (ks) ds,
0
(230) where A=
Γ
2d π d+1 eπκ cosh(πκ) , 2 + iκ Γ d−1 2 − iκ sinh (πκ)
d−1
B=C=
2
Γ
d−1 2
−2d−1 π d+1 eπκ . 2 + iκ Γ d−1 2 − iκ sinh (πκ)
(231) (232)
Going back to (230), we divide the integration range into the intervals [0, 1] and [1, ∞]. After dividing by T , the contribution of the second interval is bounded by ∞ 1 1 Const. . (233) k −1 s−2 ds = Const. T kT 1
This is because |Jα (x)| < Const.x−1/2 as x → +∞ (see [39], p. 85). Hence the contribution of the second interval tends to 0 as T tends to +∞. The function Jα can be written as ∞
1 (−1)m (z/2)2m α α 2 = (z/2) + O(z ) . (234) Jα (z) = (z/2) m!Γ(m + α + 1) Γ(1 + α) m=0 Thus as T tends to +∞, 1 Kκ (k, gT ) T 1 " # 1 2 2 ∼ s−1 AJiκ (ks)J−iκ (ks) + BJiκ (ks) + CJ−iκ (ks) ds T e− T /2
1 ∼ T
1 e− T /2
s−1
A B(ks/2)2iκ + Γ(1 + iκ)Γ(1 − iκ) Γ(1 + iκ)2 +
=
! C(ks/2)−2iκ 2 2 + Const. k s ds Γ(1 − iκ)2
Bk 2iκ (1 − e−iT κ ) A + 2Γ(1 + iκ)Γ(1 − iκ) 2iT κΓ(1 + iκ)2 Ck −2iκ (1 − eiT κ ) Const. k 2 (1 − e−T ) . + + −2iT κΓ(1 − iκ)2 2T
(235)
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Hence lim
T →+∞
A sinh(πκ) 1 A Kκ (k, gT ) = = . T 2Γ(1 + iκ)Γ(1 − iκ) 2πκ
(236)
This gives 1 A sinh(πκ) Kκ (ξ, ξ , gT ) = δ(η − η ), T →+∞ T 2πκ
(237)
γ 2 π coth2 (πκ) 1 γ 2 C0 (κ)cd,κ A sinh(πκ) L1 (f0 , gT ) = = . T →+∞ T 2πκ 2|κ|
(238)
lim
and (see (126)) lim
This is half of the result in (157) or (218), but it is doubled by the addition of the contribution of gˇT .
Appendix D. Proof of the Projector Identity In this appendix, we give a proof of the formula (25) in the de Sitter case, with masses m and m in the principal series, i.e., m2 = μ2 + (d − 1)2 /4, m2 = ν 2 + (d − 1)2 /4, with real μ = 0 and ν = 0. We set R equal to 1. The meaning of (25) is Wm (z, x)Wm (x, y)g(x) dx lim g∈S(Xd ),g→1
X
= C1 (m, d)δ(m2 − m2 )Wm (z, y).
(239)
For g ∈ S(Xd ) the integral in this formula is well defined (see Remark 2.2). The same method as in Appendix C will be used. Using (23) reduces the problem to the study, as g tends to 1, of + − ψ− (x, ξ)ψ− (x, ξ )g(x) dx. (240) Kμ,ν (ξ, ξ , g) = d−1 d−1 −iμ +iν 2
2
Xd
Recalling Remarks 2.1, 2.2 and C.1, and parametrizing ξ and ξ in terms of η and η as in (219), we see that, for a general smooth fast decreasing g, this is well defined as a distribution in η and η , and an entire function in μ and ν, and, denoting gˇ(x) = g(−x), it satisfies Kμ,ν (ξ, ξ , gˇ) = eπ(μ+ν) K−¯μ,−¯ν (ξ, ξ , g).
(241)
(It is sufficient to verify this formula for real μ and ν). We will use the same coordinates (60) and many of the formulae of Appendix C. We wish to take g as gu (x) = θ(x0 + xd ), or gd = gˇu . Thus gu (u stands for “upper”) is the indicator function of the domain covered by the coordinates (60), We u d (ξ, ξ ) = Kμ,ν (ξ, ξ , gu ) and Kμ,ν (ξ, ξ ) = Kμ,ν (ξ, ξ , gˇu ). To make denote Kμ,ν
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the integral converge, we first replace gu by a better behaved guε of the form guε (x(t, y)) = Gε (e−t )gu (x) which will tend to gu (x) as ε → 0. We thus consider u,ε Kμ,ν (ξ, ξ ) = Kμ,ν (ξ, ξ , guε ) + − = ψ− (x, ξ)ψ− (x, ξ )guε (x) dx, d−1 d−1 −i(μ) +iν 2
(242)
2
Xdu u,ε d,ε (ξ, ξ ) = Kμ,ν (ξ, ξ , gˇuε ) = eπ(μ+ν) K−¯ Kμ,ν μ,−¯ ν (ξ, ξ ).
(243)
We now take μ and ν real and furthermore require μν > 0. Using the coordinates (60) and parametrizing ξ and ξ as in Appendix C [see (219)], we may use the Plancherel formula as was done there. We obtain u,ε u,ε Kμ,ν (ξ, ξ ) = (2π)1−d eik·(η−η ) K (244) μ,ν (k) dk, Rd−1 d,ε Kμ,ν (ξ, ξ )
1−d
= (2π)
d,ε eik·(η−η ) Kμ,ν (k) dk,
Rd−1
d,ε u,ε (k) = eπ(μ+ν) K Kμ,ν −μ,−ν (−k), u,ε K μ,ν (k) =
(245)
d−1 d+1 i(μ−ν)
k 2 π d−1 sinh(πμ) sinh(πν)Γ d−1 2 − iμ Γ 2 + iν ∞ ds Gε (s) [eπμ J−iμ (sk) − Jiμ (sk)] [eπν Jiν (sk) − J−iν (sk)] , × s 0
(246) where k = |k|. We can use the following formula [39, 7.7.4 (30), p. 51]: ∞
Jα (as)Jβ (as)s−ρ ds
0
=
(a/2)ρ−1 Γ(ρ)Γ((α + β + 1 − ρ)/2) , 2Γ((1 + α + β + ρ)/2)Γ((1 − α + β + ρ)/2)Γ((1 + α − β + ρ)/2) Re(α + β + 1) > Re ρ > 0, a > 0. (247)
d,ε Choosing Gε (s) = sε with 0 < ε < 1, and using (245) to obtain Kμ,ν (k) from u,ε Kμ,ν (k), we obtain u,ε d,ε K μ,ν (k) + Kμ,ν (k) = ×
2d−2 π d+1 k i(μ−ν) (k/2)−ε Γ(1 − ε) d−1 sinh(πμ) sinh(πν)Γ d−1 2 − iμ Γ 2 + iν
(eπ(μ+ν) + 1)Γ((−iμ + iν + ε)/2) Γ((2 − iμ + iν − ε)/2)Γ((2 + iμ + iν − ε)/2)Γ((2 − iμ − iν − ε)/2) (eπμ + eπν )Γ((−iμ − iν + ε)/2) − Γ((2 − iμ − iν − ε)/2)Γ((2 + iμ − iν − ε)/2)Γ((2 − iμ + iν − ε)/2)
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(eπμ + eπν )Γ((iμ + iν + ε)/2) Γ((2 + iμ + iν − ε)/2)Γ((2 − iμ + iν − ε)/2)Γ((2 + iμ − iν − ε)/2)
! (eπ(μ+ν) + 1)Γ((iμ − iν + ε)/2) + . Γ((2 + iμ − iν − ε)/2)Γ((2 − iμ − iν − ε)/2)Γ((2 + iμ + iν − ε)/2) (248) These expressions have well-defined limits in the sense of distributions in μ and ν. In the numerator of each term inside the square brackets we make the substitution Γ(z) = Γ(1 + z)/z. As ε → 0, we find 2d−1 π d+1 k i(μ−ν) d−1 sinh(πμ) sinh(πν)Γ d−1 2 − iμ Γ 2 + iν 1 1 eπ(μ+ν) + 1 + × i(μ − ν) + ε −i(μ − ν) + ε Γ 1 − i μ+ν Γ 1 + i μ+ν 2 2 1 1 eπμ + eπν + − . (249) i(μ + ν) + ε −i(μ + ν) + ε Γ 1 − i μ−ν Γ 1 + i μ−ν 2 2
u,ε d,ε K μ,ν (k) + Kμ,ν (k) ∼
Using (it + ε)−1 + (−it + ε)−1 ∼ 2πδ(t), and Γ(1 + iz)Γ(1 − iz) = πz/ sinh(πz), this gives 2d π d+1 k i(μ−ν) d−1 sinh(πμ)Γ d−1 2 − iμ Γ 2 + iν ! e2πμ + 1 eπμ + e−πμ δ(μ − ν) + δ(μ + ν) . × μ μ
u,ε d,ε K μ,ν (k) + Kμ,ν (k) ∼
(250)
Recall that we are interested in the case when μ = 0 and ν = 0 have the same sign. In this case δ(μ + ν) = 0, and |μ|−1 δ(μ − ν) = 2δ(μ2 − ν 2 ). Thus, in this case, u,ε d,ε K μ,ν (k) + Kμ,ν (k) ∼
2d+2 π d+1 eπμ | coth(μ)| d−1 δ(μ2 − ν 2 ). Γ 2 − iμ Γ d−1 + iμ 2
(251)
Therefore, by (244) and (245), u d Kμ,ν (ξ, ξ , g = 1) = Kμ,ν (ξ, ξ ) + Kμ,ν (ξ, ξ )
=
2d+2 π d+1 eπμ | coth(μ)| d−1 δ(μ2 − ν 2 )δ(η − η ), Γ 2 − iμ Γ d−1 2 + iμ
(252)
Recall that this holds when μ and ν are both non-zero and have the same sign. Still in the same case, using (23) we have Wμ (x, y)Wν (y, x ) dy = cd,μ cd,ν ψ− d−1 +iμ (x, ξ) 2
γ×γ
Xd
× Kμ,ν (ξ, ξ , 1)ψ− d−1 −iν (x , ξ ) dμγ (ξ) dμγ (ξ ). 2
(253)
We choose γ = V0 as described at the beginning of Appendix C and of this appendix. With the parametrization (219), this is a (d − 1)-Euclidean space
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and dμγ (ξ) = dd−1 η. Therefore, by (252), the RHS of (253) is given by 2d+2 π d+1 eπμ | coth(μ)| (cd,μ )2 δ(μ2 − ν 2 ) d−1 Γ 2 − iμ Γ d−1 2 + iμ × ψ− d−1 +iμ (x, ξ)ψ− d−1 −iμ (x , ξ) dμγ (ξ), 2
2
(254)
γ
and finally
Wμ (x, y)Wν (y, x ) dy = 2π| coth(μ)|δ(μ2 − ν 2 )Wμ (x, x ).
(255)
Xd
Although we have assumed μ and ν to have the same sign in the derivation, it follows from Wμ = W−μ and the form of the formula above that it holds for all possible relative signs, provided μ = 0 and ν = 0.
References [1] Riess, A.G., et al.: Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009–1038 (1998) [2] Perlmutter, S., et al.: Measurements of omega and lambda from 42 high-redshift supernovae. Astrophys. J. 517, 565–586 (1999) [3] De Sitter, W.: On the relativity of inertia: remarks concerning Einstein’s latest hypothesis. Proc. Kon. Ned. Acad. Wet. 19, 1217–1225 (1917) [4] De Sitter, W.: On the curvature of space. Proc. Kon. Ned. Acad. Wet. 20, 229– 243 (1917) [5] Birrell, N.D., Davies, P.C.W.: Quantum fields in curved space. In: Cambridge Monographs on Mathematical Physics, vol. 7. Cambridge University Press, New York (1982) [6] Brunetti, R., Fredenhagen, K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623 (2000). arXiv:math-ph/9903028 [7] Hollands, S., Wald, R.M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223, 289 (2001). arXiv:gr-qc/0103074 [8] Hollands, S., Wald, R.M.: Existence of local covariant time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309 (2002). arXiv:gr-qc/0111108 [9] Hollands, S.: Renormalized quantum Yang–Mills fields in curved space–time. Rev. Math. Phys. 20, 1033 (2008). arXiv:0705.3340 [gr-qc] [10] Nachtmann, O.: Dynamische Stabilit¨ at im de-Sitter-raum. Osterr. Akad. Wiss. Math.-Naturw. Kl. Abt. II 176, 363–379 (1968) [11] Boyanovsky, D., Holman, R., Prem Kumar, S.: Inflaton decay in De Sitter spacetime. Phys. Rev. D 56, 1958–1972 (1997) [12] Boyanovsky, D., de Vega, H.J.: Particle decay in inflationary cosmology. Phys. Rev. D 70, 063508 (2004)
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[13] Boyanovsky, D., de Vega, H.J., Sanchez, N.G.: Particle decay during inflation: self-decay of inflaton quantum fluctuations during slow roll. Phys. Rev. D 71, 023509 (2005) [14] Boyanovsky, D., de Vega, H.J.: Dynamical renormalization group approach to relaxation in quantum field theory. Ann. Phys. 307, 335–371 (2003) [15] Polyakov, A.M.: De Sitter space and eternity. Nucl. Phys. B 797, 199 (2008). arXiv:0709.2899 [hep-th] [16] Polyakov, A.M.: Decay of Vacuum Energy. arXiv:0912.5503 [hep-th] [17] Bros, J., Epstein, H., Moschella, U.: Lifetime of a massive particle in a de Sitter universe. JCAP 0802, 003 (2008) [18] Bros, J., Epstein, H., Gaudin, M., Moschella, U., Pasquier, V.: Triangular invariants, three-point functions and particle stability on the de Sitter universe. Commun. Math. Phys. 295, 261 (2010) [19] G¨ ursey, F.: Introduction to the de Sitter group. In: Group Theoretical Concepts and Methods in Elementary Particle Physics, pp. 365–389. Gordon and Breach, New York (1964) [20] Gibbons, G.W., Hawking, S.W.: Cosmological event horizons, thermodynamics, and particle creation. Phys. Rev. D 15, 2738–2751 (1977) [21] Bros, J., Moschella, U., Gazeau, J.P.: Quantum field theory in the de Sitter universe. Phys. Rev. Lett. 73, 1746–1749 (1994) [22] Bros, J., Moschella, U.: Two-point functions and quantum fields in de Sitter universe. Rev. Math. Phys. 8, 327–392 (1996) [23] Bros, J., Epstein, H., Moschella, U.: Analyticity properties and thermal effects for general quantum field theory on de Sitter space–time. Commun. Math. Phys. 196, 535–570 (1998) [24] Bogolubov, N.N., Logunov, A.A., Oksak, A.I., Todorov, I.T.: General Principles of Quantum Field Theory. Springer, Berlin (1990) [25] Bros, J., Viano, G.A.: Forum Math. 8, 621 (1996) [26] Jost, R.: The General Theory of Quantized Fields. American Mathematical Society, Providence (1965) [27] Araki, H.: Mathematical Theory of Quantum Fields. Oxford University Press, Oxford (1999) [28] Bros, J., Buchholz, D.: Axiomatic analyticity properties and representations of particles in thermal quantum field theory. Ann. Poincar´e 64, 495–522 (1996) [29] Veltman, M.: Diagrammatica. vol. I. Cambridge University Press, Cambridge (1994) [30] Kay, B.S., Wald, R.M.: Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on space–times with a bifurcate killing horizon. Phys. Rept. 207, 49–136 (1991) [31] Bros, J., Epstein, H., Gaudin, M., Moschella, U., Pasquier, V.: Triangular invariants, three-point functions and particle stability on the de Sitter universe. (2010, in preparation) [32] Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics. Springer, Berlin (1966) [33] Marichev, O.I.: Handbook of Integral Transforms of Higher Transcendental Functions. Ellis Horwood Limited, Chichester (1982)
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[34] Slater, L.J.: Generalized Hypergeometric Functions. Cambridge University Press, Cambridge (1966) [35] Erd´elyi, A.: The Bateman manuscript project. In: Higher Transcendental Functions, vol. I. McGraw-Hill, New York (1953) [36] Skenderis, K., Townsend, P.K.: Pseudo-supersymmetry and the domain-wall/ cosmology correspondence. J. Phys. A 40, 6733 (2007) [37] Abbott, L.F., Deser, S.: Stability of gravity with a cosmological constant. Nucl. Phys. B 195, 76 (1982) [38] Vilenkin, N.J.: Special Functions and the Theory of Group Representations. Nauka, Moscow (1968) [39] Erd´elyi, A.: The Bateman manuscript project. In: Higher Transcendental Functions, vol. II. McGraw-Hill, New York (1953) Jacques Bros Service de Physique th´eorique CEA Saclay 91191 Gif-sur Yvette France e-mail:
[email protected] Henri Epstein ´ Institut des Hautes Etudes Scientifiques 91440 Bures-sur-Yvette France e-mail:
[email protected] Ugo Moschella Universit` a dell’Insubria Como, Italy e-mail:
[email protected] and INFN Milano Milan, Italy Communicated by Klaus Fredenhagen. Received: October 22, 2009. Accepted: April 12, 2010.
Ann. Henri Poincar´e 11 (2010), 659–764 c 2010 Springer Basel AG 1424-0637/10/040659-106 published online July 10, 2010 DOI 10.1007/s00023-010-0041-8
Annales Henri Poincar´ e
Nonlocal Potentials and Complex Angular Momentum Theory Jacques Bros, Enrico De Micheli and Giovanni Alberto Viano Abstract. The purpose of this paper is to establish meromorphy properties of the partial scattering amplitude T (λ, k) associated with physically relevant classes Nwγ (ε),α of nonlocal potentials in corresponding domains (δ)
Dγ,α of the space C2 of the complex angular momentum λ and of the complex momentum k (namely, the square root of the energy). The general expression of T as a quotient Θ(λ, k)/σ(λ, k) of two holomorphic (δ) functions in Dγ,α is obtained by using the Fredholm–Smithies theory for complex k, at first for λ = integer, and in a second step for λ complex (Re λ > −1/2). Finally, we justify the “Watson resummation” of the partial wave amplitudes in an angular sector of the λ-plane in terms of the various components of the polar manifold of T with equation σ(λ, k) = 0. While integrating the basic Regge notion of interpolation of resonances in the upper half-plane of λ, this unified representation of the singularities of T also provides an attractive possible description of echoes in the lower half-plane of λ. Such a possibility, which is forbidden in the usual theory of local potentials, represents an enriching alternative to the standard Breit–Wigner hard-sphere picture of echoes.
1. Introduction In the standard Breit–Wigner theory of scattering the notions of “time delay” and “time advance”, corresponding respectively to the increasing or decreasing character of a given phase-shift as a function of the energy (see, e.g., [1, pp. 110 and 111]) are not described in a symmetric way: while the former is described by a pole singularity of the scattering amplitude, the latter relies on the model of scattering by an impenetrable sphere. A question then arises: can the time advance still be evaluated in terms of hard-sphere scattering in collisions between composite particles, when Pauli exclusion principle comes into play? We recall, indeed, that when two composite particles collide, the fermionic character of the components emerges and the antisymmetrization
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of the whole system generates repulsive exchange-forces, which can produce echoes, and are thus responsible for the occurrence of time advance. A theoretical characterization of the phenomenon of echo has been given in [2] (see Section II-B of the latter) who introduced this notion in parallel with the notion of resonance: echoes manifest themselves as bumps in the energy-dependance plot of the cross-section, like resonances, but in contrast with the latter, they are associated with the downward (instead of upward) passage through π2 of a given phase-shift. However, in spite of this parallel, the status of the echoes has remained rather obscure in terms of the analyticity properties in energy of the scattering functions, in comparison with the success of the standard pole-resonance conceptual correspondence. The reason for this discrepancy lies in the fact that the Breit–Wigner poles associated with resonances necessarily occur in the “second-sheet” of the energy variable √ E, namely in the lower half-plane of the usual momentum variable k = E, near the positive real axis; this is a situation which readily accounts for the upward passage of the phase-shift through π2 . But then the natural candidacy of poles in the upper half-plane for accounting the downward passage characterizing the echoes turns out to be strictly forbidden in any formalism of scattering theory (as well in nonrelativistic potential theory as in relativistic quantum field theory) as a constraint imposed by the causality principle. It is however to be noted that from both experimental and theoretical viewpoints, one knows that for any given phase-shift the echoes are present together with the resonances. In fact, the physically observed occurrence of resonance–echo pairs finds its firm theoretical basis in the Levinson theorem (see [1, p. 206]) which prescribes the equality of each phase-shift at zero and infinite energies in the absence of bound states. Considering the present sum of knowledge that is available in the general theory of quantum scattering processes, it seems to us that there remains a problem of global understanding of the organization of sequences of resonances and associated echoes for successive values of the angular momentum and of the corresponding relationships between the energy variable E (or k) and . Let us recall the historical situation from both theoretical and experimental aspects. At first, the standard theory of scattering does not attempt to group resonances in families: each resonance is simply described by a fixed pole singularity in the energy variable. As for echoes, they are individually parametrized in a very rough way and from the outset by the hard-sphere picture. However, phenomenological data clearly show that the resonances often appear in ordered sequences, such as rotational bands. Typical examples can be observed in α−α, α−40 Ca,12 C−12 C, 28 Si−28 Si, and other heavy-ion collisions (see [3] and references quoted therein). In these examples, the resonances are ordered in rotational bands of levels whose energy spectrum can be fitted by an expression of the form E = A + B( + 1), where is the angular momentum of the level, and A and B are constants. Furthermore, the widths of the resonances increase as a function of the energy. Since 1960, a wide opening on the previous problem was given by Regge’s basic formalism of the holomorphic interpolation of scattering partial waves
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in the complex angular momentum (CAM) variable λ (see [4]). In this formalism, the concept of pole reached his full fecundity through the association of sequences of resonances with polar manifolds λ = λ(k) in the complex space C2 of the variables (λ, k). Each such polar manifold gives rise to a “trajectory” on which a sequence of complex momentum values k with Im k < 0 satisfying the equation λ(k ) = corresponds to a sequence of resonances. In view of the complex geometrical framework, a new possibility offered by that description was the study of the trajectories λ = λ(k) for k real and positive, realized as curves in the complex upper half-plane of λ. Coming back to the general problem that we have raised above, it now seems interesting to inquire whether this enlarged framework of complex geometry in the CAM and momentum variables might give some new insight on the description of echoes. In other words, are there some polar manifolds λ = λ(k) which have something to do with the description of echoes? A tentative answer to this question was proposed by various authors (see [5]) by considering in particular the case of scattering by Yukawa-type potentials. In such cases each pole trajectory in the upper half-plane of λ behaves as follows. After having traveled forward (i.e., with d Re λ/dk > 0) and described thereby a sequence of resonances at Re λ(k) = , it starts going up and turning backwards, and then passes again but in decreasing order (and with larger values of Im λ) through points whose real parts Re λ = had been previously visited. It was then proposed that this second part of such trajectories be associated with echoes. However it turns out that such a global ordering is inconsistent with most of the experimental data, which exhibit the alternance of resonances and echoes for successive values of when one follows the increase of the variable k. From a formal viewpoint, a scenario which might appear to be more appropriate would be the possible coupling of a pure resonance trajectory in the upper half-plane with one or several echo trajectories in the lower half-plane Im λ < 0. However, this latter possibility has been excluded from the whole theory of local potentials by a theorem proved by Regge (see [4]). It is at this point of our considerations that we wish to advocate for the necessity of enlarging the framework of Schr¨ odinger theory so as to include the occurrence of nonlocal potentials V (R, R ) and thereby invalidate the application of Regge’s no-go theorem. As a matter of fact, whenever the echoes are generated by exchange forces due to Pauli’s exclusion principle, the manybody structure of the colliding particles is involved and leads one to introduce nonlocal potentials. This class of potentials has been indeed considered long time ago, mainly in connection with the theory of nuclear matter [6,7]. The procedures commonly used for treating the many-body dynamics, like the resonating group method [8], the complex generator coordinate technique [9], the cluster coordinate method [10], all lead to an extension of the standard Schr¨ odinger equation, which now becomes an integro-differential equation of the same form as that obtained in the study of the nonlocal potentials. In this paper, we intend to study the singularities of the partial scattering amplitude for appropriate classes of nonlocal potentials, both in the complex momentum
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k-plane and in the complex angular momentum λ-plane. Concerning the possible location of these singularities, one of the results that we have obtained is a modified extension of the previous Regge theorem to the case of nonlocal potentials, which now opens the way to singularities in the lower half-plane of λ (at k real and positive). For local potentials (notably, Yukawian potentials) all the singularities of the partial scattering function, such as those which manifest themselves as resonances, must be located at k > 0 in the region Im λ > 0. Here we have considered as an example a class of nonlocal potentials V (λ; R, R ) = V∗ (R, R )F˜ (λ), where F˜ (λ) is a function holomorphic, bounded, . and of Hermitian-type in the half-plane C+ = {λ ∈ C : Re λ > − 12 }. For such −1 2
a class of potentials, we denote by Lj (j ∈ Z) the set of lines in C+ where − 12 ˜ Im F (λ) = 0. All the points of a curve Lj with j > 0 (resp., j < 0) belong to the region Im λ > 0 (resp., Im λ < 0), L0 being along the real positive axis. We prove that no singular pairs (λ, k) can occur with λ in any line Lj in the lower half-plane (j ≤ 0), and with Im k > 0 and Re k > 0. But there is a possible occurrence of singular manifolds containing branches in the region Im k ≥ 0, Re k ≥ 0 and Re λ > 0, Im λ < 0, always located in strips of the fourth quadrant of the λ-plane, well-separated from one another by the set of lines Lj (j ≤ 0). These strips therefore set the ground for possible echo trajectories. Although it is still premature to conclude on the existence of the latter before some numerical exploration be performed, we think that this promising result may suggest how to fill a gap between phenomenological and theoretical analysis of scattering data. Indeed, in refs. [11,12] two of us have performed an extensive phenomenological analysis, fitting the scattering data of α − α and π + -p elastic scattering by using two pole trajectories in the CAM-plane. The resonances have been described by poles in the first quadrant, the echoes by poles in the fourth quadrant. As a further possible motivation to the investigation of the theory of nonlocal potentials, we remind the reader that in a parallel presentation of the two-particle scattering theory in Schr¨ odinger wave-mechanics and in Quantum Field Theory (QFT), it is a suitable Bethe–Salpeter-type kernel which plays the role of the potential [13], provided the latter be understood as a generalized potential of nonlocal-type (and also energy-dependent). Recently, the CAM formalism has been introduced in QFT [14], and it has been used to obtain a corresponding CAM-diagonalization of the Bethe–Salpeter equation [15]. Therefore, also from this viewpoint, exploring the CAM-singularities generated by appropriate classes of nonlocal potentials deserves some interest. As far as we know, the connection between nonlocal potentials and complex angular momentum theory is totally missing in the literature. Let us quote what is written in the classical textbook by De Alfaro and Regge [4] on this question: “Our philosophy in regard to these potentials (i.e., nonlocal potentials) is the following: maybe they are there, but if they are there we do not know what to do with them. They will not be discussed in this book anymore.” Even in excellent texts on scattering theory, like the one by Reed and Simon
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[16], this problem has not been considered at all. Long time ago, one of the authors, in collaboration with others, treated the scattering theory for large classes of nonlocal potentials in a series of papers [17–20], where, however, the analysis in the CAM-plane was not considered. The present paper can be regarded as a continuation and completion of these works, in particular of [20] (whose main results are recalled in Sect. 2). So, before considering the interesting problem of describing resonances and echoes, which will be outlined in the last section of this paper, one needs to settle on a firm basis the whole theory of scattering functions with respect to the complex variables (λ, k) in the framework of appropriate classes of nonlocal potentials. The paper is organized as follows: in Sect. 2 we recall the spectral properties associated with the Schr¨ odinger two-particle operators for a large class of potentials U , which include a local part called V0 and a nonlocal part called V . All these properties concern bound states and scattering solutions, and are treated in the complex half-plane Im k ≥ 0. In Sect. 3 we study a class Nw,α of rotationally invariant nonlocal potentials V , which is characterized by a positive parameter α; for such a class the previous treatment is extended to the half-plane Im k ≥ −α, so as to include the analysis of resonances. In this section we use Smithies’ theory [21] of Fredholm-type integral equations for studying various properties of the resolvent when k belongs to the half-plane {k ∈ C : Im k ≥ −α} and the angular momentum is a non-negative integer. The Smithies formalism produces modified Fredholm formulae, whose advantage is to fit rigorously with the convenient framework of Hilbert–Schmidt-type kernels. It is therefore in this formalism that we investigate in detail bound states, spurious bound states (or bound states embedded in the continuum), antibound states, resonances, scattering solutions, and partial scattering amplitudes in the strip Ωα = {k ∈ C : | Im k| < α}. In Sect. 4 we study the interpolation of the so-called partial potentials (i.e., the coefficients of the Fourier–Legendre expansion of the potentials) in the plane of the complexified angular momentum variable λ. The potentials which admit this interpolation will and satisfy an exponential decrease of the form e−γ Re λ (γ > 0) in C+ − 12 be called “Carlsonian potentials with CAM-interpolation V (λ; R, R ) and rate of decrease γ”, in view of the fact that the uniqueness of the interpolation is guaranteed by Carlson’s theorem. The class of such potentials which belong to γ . Then, by using the bounds on the complex anguNw,α will be denoted Nw,α lar momentum Green function (i.e., the extension of the Green function from integral non-negative values of the angular momentum to complex values λ) one can extend the Fredholm–Smithies formalism of the previous section in terms of vector-valued and operator-valued holomorphic functions of the two complex variables λ and k. Correspondingly, one then studies the properties of the partial scattering amplitude T (λ, k) as a meromorphic function of λ and k (δ) in appropriate domains Dγ,α of the space C2 . This sets the basis for analyzing the notions of resonances and of echoes in terms of the polar singularities of T (δ) in either part Im λ ≷ 0 of the domain Dγ,α . The first part of Sect. 5 is devoted to the Watson resummation of the partial wave amplitudes in an angular sector
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of the complex λ-plane for all positive values of k (and also in some domain of the complex k-plane). In the second part of this section, the notion of a dominant pole in the Watson representation of the scattering amplitude is introduced and a tentative parallel study of resonances and echoes is proposed in that framework. Finally, in order to make simpler the presentation of our results, we have reserved two appendices for mathematical ingredients to be used in the text: while Appendix A is devoted to the derivation of bounds on the complex angular momentum Green function and on the Bessel and Hankel functions, Appendix B deals with continuity and holomorphy properties of vector-valued and operator-valued functions.
2. A Review of Spectral Properties for a Class of Schr¨ odinger Two-Particle Operators Including Local and Nonlocal Potentials We consider Schr¨ odinger operators of the following form: (Hψ)(x) = −Δψ(x)+(U ψ)(x) = −Δψ(x)+V0 (x)ψ(x)+ V (x, y)ψ(y) dy, (2.1) 3 where x = (x1 , x2 , x3 ) is a three-dimensional real vector, |x|2 = k=1 x2k , Δ = 3 2 2 3 k=1 ∂ /∂xk is the Laplace operator, and integration is on the whole R space. The integro-differential operator H represents, in the center of mass system, the energy operator of two particles interacting through a local plus a nonlocal potential. Our assumptions on the potential functions V0 and V are the following: (a) V0 andV are real-valued; V is symmetric: i.e., V (x, y) = V (y, x). p 1/p (b) Ap = [ (1 + |x|)p |V (p = 1, 2). 0 (x)| dx] p< +∞ p (c) Bp = [ (1 + |x|) ( |V (x, y)| dy) dx]1/p < +∞ (p = 1, 2). Note that condition (a) guarantees that the operator H is a time-reversal invariant and formally Hermitian operator. We then have (see [20]): Proposition 1. If conditions (a), (b) and (c) hold, then the operator H defined by (2.1) is self-adjoint in L2 with domain DH = W 2,2 . In this section L2 and W 2,2 will indicate L2 (R3 ) and W 2,2 (R3 ), respectively; moreover, we recall that W 2,2 is the Sobolev space of all the square integrable functions which have square integrable distributional derivatives up to the second order. We now consider the following problems. Problem 1 (Bound state problem). Suppose that the potential functions V0 and V satisfy conditions (a), (b), and (c), then we study the solutions ψ with ψ ∈ W 2,2
and ψL2 = 1,
(2.2)
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of the Schr¨ odinger equation (Hψ)(x) = k 2 ψ(x), namely: Δψ(x) + k 2 ψ(x) = (U ψ)(x) = V0 (x)ψ(x) + V (x, y)ψ(y) dy,
(2.3)
where k is a complex number with Im k ≥ 0. The following theorem can then be proved. Theorem 2. Let B be the set of values of k (Im k ≥ 0) such that a non trivial solution of Problem 1 exists; then: (i) for any k ∈ B the number of linearly independent solutions of Problem 1 (i.e., the multiplicity of k) is finite; (ii) B is contained in the union of the real axis and the positive imaginary axis; (iii) B is bounded; (iv) B is countable with no limit points except k = 0; (v) if a real value of k belongs to B, then (−k) ∈ B and Problem 1 has precisely the same solutions associated with k and −k.
Proof. See [20] (proof of Theorem 2.1).
We introduce the index n (n = 1, 2, 3, . . .) to label the imaginary and the positive real values of k which belong to B, and we understand to count any such value of k as many times as its multiplicity. Then the numbers En = kn2 (n = 1, 2, 3, . . .) are the eigenvalues of the operator H = −Δ + U ; with each eigenvalue En we can associate one and only one solution ψn (x) of Problem 1, i.e., an eigenfunction of H. Since the operator H is self-adjoint (see Proposition 1), the functions ψn (x) can be regarded as forming an orthonormal system. odinger Problem 2 (Scattering problem). We study solutions Ψξ (x) of the Schr¨ equation (HΨξ )(x) = |ξ|2 Ψξ (x), which are of the following form: Ψξ (x) = eiξ,x + Φ(ξ, x), where ξ = (ξ1 , ξ2 , ξ3 ) is a given three-dimensional vector, ξ, x = and Φ(ξ, x) satisfies the following properties:
(2.4) 3
j=1 ξj xj
2,2 (i) Φ(ξ, ·) ∈ Wloc ; 1 (|x| → +∞); (ii) Φ(ξ, x) = O |x| 2 ∂Φ (iii) |x|=r ∂|x| (ξ, x) − i|ξ|Φ(ξ, x) μ(dx) −−−−−→ 0, μ denoting the Lebesgue r→+∞
measure on the sphere; (iv) ΔΦ(ξ, x)+|ξ|2 Φ(ξ, x) = V0 (x)Φ(ξ, x)+ V (x, y)Φ(ξ, y) dy+V0 (x)eiξ,x + V (x, y)eiξ,y dy, (Schr¨ odinger equation written in terms of Φ).
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2,2 Note that Φ ∈ Wloc means that f Φ ∈ W 2,2 for any f ∈ C0∞ (R3 ). Besides, we remark that the third condition involves the notion of trace in the sense of Sobolev of the function Φ and of its first derivatives. To this purpose, let us note that if u ∈ W 2,2 and Σ is a smooth surface, then the trace of u on Σ can be defined as the restriction of u on Σ. This definition is meaningful because u is a continuous function, but in the case of the first derivatives, the previous definition breaks down. However, the linear operator relating every Φ ∈ C0∞ (R3 ) to ∂Φ/∂xk , restricted on the sphere of radius r, is a continuous mapping of a subspace of W 2,2 into the space of the functions which are square integrable on the sphere of radius r. Since C0∞ (R3 ) is dense in W 2,2 , this operator can be extended in a unique way to the whole W 2,2 (see the Appendix of 2,2 is continuous (Lemma A.3 of [20]), [20]). Furthermore, any function Φ ∈ Wloc so that from the second condition of Problem 2 it follows that Ψξ is a bounded and continuous function of x in R3 . Then the following theorem holds.
Theorem 3. For any ξ ∈ R3 , ξ = 0, a solution of Problem 2 exists. The solution is unique if and only if |ξ| ∈ B (the set defined in Theorem 2). Any solution of Problem 2 has the following asymptotic behavior: 1 x ei|ξ||x| F |ξ| , ξ + o Φ(ξ, x) = (|x| → +∞), |x| |x| |x| 1 e−ix,y (U Ψξ ) (y) dy. F (x, ξ) = − 4π x The quantity F |ξ| |x| , ξ , the so-called scattering amplitude, is uniquely defined for any ξ ∈ R3 . Proof. See [20] (proof of Theorem 2.2).
Besides, let ΩR be the set of vectors ξ ∈ R3 such that |ξ| ∈ B, ξ = 0; then the functions ΩR ξ → Φ(ξ, x) are equicontinuous, i.e.: sup |Φ(ξ, x) − Φ(η, x)| −→ 0,
x∈R3
(2.5)
where ξ ∈ ΩR is fixed, and |ξ − η| → 0. We can rapidly sketch the approach to Problems 1 and 2. One first introduces the following operator L(k): eik|x−y| f (y) dy L(k)f (x) = − V0 (x) 4π|x − y| eik|z−y| − V (x, z) dz f (y) dy, (2.6) 4π|z − y| acting on the Hilbert space X 2 defined by
1/2 . 2 2 2 2 (1 + |x|) |f (x)| dx < +∞ . X = f ∈ L : f X 2 = Note that the function eik|x−y| /4π|x − y| [in definition (2.6) of the operator L(k)], is the Green function associated with the classical radiation problem.
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Next, one applies the Riesz–Schauder theory [22] to the resolvent R(k) = [1 − L(k)]−1 , and then the following alternative holds: (a) either R(k) = [1 − L(k)]−1 exists and is a bounded operator on X 2 ; (b) or the space spanned by the eigenfunctions of L(k) has dimension n ≥ 1. In case (a) the integral equation v(ξ, ·) = v0 (ξ, ·) + L(|ξ|)v(ξ, ·), where
(2.7)
v0 (ξ, x) = V0 (x) eiξ,x +
V (x, y)eiξ,y dy,
has a unique solution in X 2 , i.e.: v(ξ, ·) = R(|ξ|)v0 (ξ, ·). In case (b), Eq. (2.7) has a solution v(ξ, ·) ∈ X 2 if and only if v0 (ξ, ·) is orthogonal in X 2 to n linearly independent eigenfunctions of the adjoint operator L† (|ξ|). For this purpose the following theorem can be proved. Theorem 4.
(i) If the function f is a solution of the problem
f ∈ X 2,
f = L(|k|)f,
then the function
ψ(x) = −
f ≡ 0
(k = 0, Im k ≥ 0),
(2.8)
eik|x−y| f (y) dy 4π|x − y|
is a solution of the problem ψ ∈ W 2,2 ,
Δψ + k 2 ψ = U ψ
(ψ ≡ 0).
(2.9)
(ii) Conversely, if ψ(x) is a solution of problem (2.9) with Im k ≥ 0, then the function f (x) = Δψ(x) + k 2 ψ(x) is a solution of problem (2.8). (iii) If ψ(x) is a solution of problem (2.9) with Im k = 0, then the function f ∗ (x) = (1 + |x|)−2 ψ(x) is an eigenfunction in X 2 of the adjoint operator L† (k). Proof. See [20] (proof of Theorem 4.2).
Let now ψ1 , . . . , ψn be n linearly independent solutions of problem (2.8); by statement (iii) of the theorem above it follows that the functions fj∗ (x) = (1+|x|)−2 ψj (x) are linearly independent eigenfunctions of the adjoint operator L† (|ξ|). Since the following equality holds [see [20], Eq. (6.5)]: ∗
fj , v0 (ξ, ·) X 2 = 0, where ·, · X 2 denotes the scalar product in X 2 , then a solution to Eq. (2.7) in X 2 exists but it is no longer unique. Nevertheless, the scattering amplitude x , ξ) is unique (see [20]). F (|ξ| |x|
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Once the solutions of the time-independent Schr¨ odinger equation have been obtained one can write, following a procedure which goes back to T. Ikebe1 [23,24], an expansion formula for an arbitrary function f ∈ L2 in terms of the eigenfunctions ψn (x) and of the scattering solutions Ψξ (x) (see [20], Theorem 2.3). By the use of this expansion one obtains a spectral representation of the operator H: ∂u(x, t) , (Hu(·, t)) (x) = −Δu(x, t) + V0 (x)u(x, t) + V (x, y)u(y, t) dy = i ∂t and of functions of H. In particular, a representation of the evolution operator exp(−itH) is obtained, and then the solution of the time-dependent Schr¨ odinger equation can be studied. In the case of the nonlocal potentials being considered, we face an additional problem: the possible existence of positive energy eigenvalues and, correspondingly, the non-uniqueness of the scattering solutions. Nevertheless the scattering amplitude exists and is unique for any ξ ∈ R3 (see Theorem 3) even if the scattering solution is not unique. This allows us to define uniquely the scattering operator S = W+† W− , where W± = s−limt→±∞ exp(itH) exp(itΔ) (s−lim ≡ strong limit), and prove the unitarity of S in a very general setting (see [20]).
3. Rotationally Invariant Nonlocal Potentials: Analyticity in the k-Plane of the Resolvents and of the Partial Scattering Amplitudes at Fixed Angular Momentum Hereafter we shall be concerned with a class of nonlocal potentials, which represents a natural generalization of the central character of the local interaction. These potentials, denoted by V (R, R ), are assumed to depend only on the lengths R, R of the vectors R and R , and on the angle η between them, or equivalently, on the shape and dimension of the triangle (O, R, R ), but not on its orientation. We then rewrite Eq. (2.3) in the following form: (3.1) (Hψ)(R) = −Δψ(R) + g V (R, R )ψ(R ) dR = k 2 ψ(R). R3
This equation can be seen to represent the two-body Schr¨ odinger equation in its reduced form with respect to the coordinate R of the relative motion between the two interacting particles; accordingly, ψ(R) represents the relative motion wavefunction, and g is the coupling constant of the interacting particles. The Planck constant and the reduced mass μ do not appear in Eq. (3.1) corresponding to a simple choice of units ( = 2μ = 1). In view of the assumptions on V (R, R ), we can write the following formal expansion: ∞ 1 . V (R, R ) = V (R, R ; cos η) = (2 + 1)V (R, R )P (cos η), (3.2) 4πRR =0
1 It must be mentioned that there was a subtle error in the original paper by Ikebe [23], which was subsequently corrected by Simon [25]; the interested reader is referred to Simon’s monograph [25].
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where cos η = (R · R )/(RR ), and the P (·) are the Legendre polynomials. The Fourier–Legendre coefficients V (R, R ) of V (R, R ) are given by: +1 V (R, R ) = 2πRR V (R, R ; cos η)P (cos η) d(cos η).
(3.3)
−1
Next, from the current conservation law it follows that V (R, R ) is a real and symmetric function: V (R, R ) = V (R, R ) = V (R , R). We can thus conclude that, provided the coupling constant g is restricted to real values, the Hamiltonian H is a formally Hermitian and rotationally invariant operator. The relative motion wavefunction ψ(R) can now be expanded in the form: ∞
ψ(R) =
1 ψ (R)P (cos θ), R
(3.4)
=0
where is now the relative angular momentum between the interacting particles. Representing the unit vectors (R/R) and (R /R ) respectively by the angles (θ, ϕ) and (θ , ϕ ), we have: cos η = cos θ cos θ + sin θ sin θ cos(ϕ − ϕ ). Then, using the following addition formula for the Legendre polynomials: π 2π 0
Ps (cos η)P (cos θ ) sin θ dθ dϕ =
4π P (cos θ)δs , 2 + 1
(3.5)
0
one readily obtains from formulae (3.1)–(3.5) the following nonlocal Schr¨ odinger-type integro-differential equation at fixed angular momentum: ( + 1) . D,k ψ (R) = ψ (R)+k 2 ψ (R)− ψ (R) = g R2
+∞ V (R, R )ψ (R ) dR , 0
(3.6) where k 2 = E is the relative kinetic energy of the two particles in the center of mass system, and (for all integer ): V (R, R ) = V (R, R ) = V (R , R).
(3.7)
We study two types of solutions of the Schr¨ odinger-type equation (3.6): (S-a) Bound state solutions, which satisfy the following conditions: (i) ψ (R) is absolutely continuous; (ii) ψ (0) = 0; +∞ (iii) 0 |ψ (R)|2 dR < +∞. (S-b) Scattering solutions, denoted by Ψ (k; R), which satisfy the following conditions: (i ) Ψ (k; R) is absolutely continuous;
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(ii ) Ψ (k; R) can be written as Ψ (k; R) = kRj (kR) + Φ (k; R),
(3.8)
where j (·) denotes the spherical Bessel function (note that Rj (kR) is a solution of the differential equation D,k [Rj (kR)] = 0 which vanishes at R = 0), and Φ (k; R) satisfies the following conditions: d Φ (k; R) − ikΦ (k; R) = 0 (k ∈ R+ ). lim Φ (k; 0) = 0, R→+∞ dR (3.9) The second equality in (3.9) is the so-called Sommerfeld radiation condition. It will be shown below in Theorem 18 that (in accordance with the study given in refs. [17–19]) the research of solutions of type (S-b) of the Schr¨ odinger-type equation (3.6) reduces to the problem of solving a Lippmann–Schwinger-type linear integral equation, namely the following inhomogeneous Fredholm equation: +∞ v (k, g; R) = v,0 (k; R) + g L (k; R, R )v (k, g; R ) dR , (3.10a) 0
where: +∞ v,0 (k; R) = V (R, R )kR j (kR ) dR ,
L (k; R, R ) =
(3.10b)
0 +∞
V (R, R )G (k; R , R ) dR .
(3.10c)
0
In the latter, G (k; R, R ) satisfies the “Green function” distributional identity D,k (R) G (k; R, R ) = δ(R − R ),
(3.10d)
and is explicitly given by the following formula: G (k; R, R ) = G (k; R , R) = −ikRR j [k min(R, R )] h [k max(R, R )] , (3.10e) (1)
(1)
where h (·) denotes the spherical Hankel function. Similarly, as it will be shown in Theorem 16, the solutions of type (S-a) of Eq. (3.6) are associated with solutions of the corresponding homogeneous Fredholm equation [obtained by replacing v,0 by 0 in Eq. (3.10a)]. Symmetry properties in the complex k-plane. In view of the parity and complex conjugation properties satisfied by the spherical Bessel and Hankel functions, namely j (z) = j (z) = (−1) j (−z), h (z) = (−1) h (−z) (see [26, formulae (9.1.35), (9.1.39), (9.1.40)]), the following symmetry properties (1)
(1)
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readily follow from the reality condition (3.7) on the potentials V and from Eqs. (3.10b), (3.10c), (3.10e): G (k; R, R ) = G (−k; R, R ), L
(k; R, R )
= L (−k; R, R ),
v,0 (k; R) = v,0 (k; R).
(3.11) (3.12) (3.13)
The present section is organised as follows. After having defined an appropriate class of nonlocal potentials together with the corresponding Hilbert-space framework, we give a complete study of the Fredholm-resolvent integral equation at fixed angular momentum , associated with Eqs. (3.10); this is done in Sect. 3.1. Then the general meromorphy properties of this resolvent with respect to the complex momentum variable k are presented in Sect. 3.2, where we also outline the algebraic correspondence between the pole structure of the latter and bound-state-type solutions of the nonlocal Schr¨ odinger equation. Complete results concerning the relationship between the integral equation formalism and the Schr¨ odinger-type formalism are then given in Sect. 3.3, including the introduction and analyticity properties in k of the partial scattering amplitudes T (k; g). Finally, a short Sect. 3.4 is devoted to the partial wave expansion of the total scattering amplitude F (k, cos θ; g) and to its general analyticity properties in k and cos θ. 3.1. Classes Nw,α of Nonlocal Potentials: Properties of the Functions v,0 (k; ·), of the Operators L (k), and of the Resolvents R (k; g) in the k-Plane Definitions. In what follows, all the functions of the real positive variable R are considered as defined for almost every (a.e.) R with respect to an appropriate measure on R+ . For each positive number α, we introduce: (1) The Hilbert space ⎧ ⎫ ⎡ +∞ ⎤1/2 ⎪ ⎪ ⎨ ⎬ . Xw,α = x(R) : xw,α = ⎣ w(R)e2αR |x(R)|2 dR⎦ < +∞ . ⎪ ⎪ ⎩ ⎭ 0
(3.14) In (3.14) w denotes a given continuous and strictly positive weight-function on the interval [0, +∞). For any bounded operator A on Xw,α , the corresponding norm will be denoted by AXw,α , or simply A. A subspace of bounded operators equipped with an appropriate Hilbert–Schmidt (HS) norm AHSw,α , or simply AHS (such that AHS ≥ A), will be introduced below. (2) The class Nw,α of rotationally invariant nonlocal potentials V (R, R ) (defined for a.e. R, R ), which satisfy the conditions V (R, R ) = V (R, R ) = V (R , R) (or equivalently conditions (3.7)), together with
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the following condition: ⎡ ⎤1/2 . C(V ) = ⎣ w(R)e2αR dR w(R )e2αR V 2 (R, R ) dR ⎦ < +∞. R3
R3
(3.15) In view of Parseval’s equality, we also have: ⎤1/2 ⎡ +∞ +∞ +∞ w(R)e2αR dR w(R )e2αR (2 + 1)V2 (R, R ) dR ⎦ , C(V ) = ⎣ 0
=0
0
(3.16) so that the partial potentials V (R, R ) satisfy (for all ≥ 0) the condition ⎤1/2 ⎡ +∞ +∞ . w(R)e2αR dR w(R )e2αR V2 (R, R ) dR ⎦ < +∞, C(V ) = ⎣ 0
0
(3.17) or, in terms of the function (defined for a.e. R) ⎞1/2 ⎛ +∞ . (w) V (R) = ⎝ w(R )e2αR V2 (R, R ) dR ⎠ ,
(3.18)
0
which belongs to Xw,α , (w)
C(V ) = V
w,α
C(V ) ≤√ . 2 + 1
(3.19)
Our aim is to consider the integral equation (3.10a) as a linear equation in Xw,α depending on the complex parameters k and g and on the integer ( ≥ 0), which we rewrite in operator form as follows: [I − gL (k)] v (k, g; ·) = v,0 (k; ·).
(3.20)
In the latter, I denotes the identity operator in Xw,α ; v,0 (k; ·) is the function defined by (3.10b), and L (k) denotes the integral operator with kernel L (k; R, R ) [see (3.10c)]. In the complex plane of k, we consider the strip . . Ωα = {k ∈ C : | Im k| < α} and the half-plane Πα = {k ∈ C : Im k > −α}.Πα and Ωα will denote the closures of Πα and Ωα , respectively. We shall then specify classes of nonlocal potentials Nw,α , with appropriate conditions on the weight-function w in such a way that the following properties can be established: (i) the functions v,0 (k; ·) belong to Xw,α for all k ∈ Ωα , ( = 0, 1, 2, . . .); (ii) the operators L (k) are compact operators of Hilbert–Schmidt-type in Xw,α for all k ∈ Πα , ( = 0, 1, 2, . . .).
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In fact, for all such classes of potentials, the kernel L (k; R, R ) will be majorized by an appropriate kernel of rank one, and this will then allow us to apply Smithies’ refined version of the Fredholm theory [21] for describing and discussing the solutions of Eq. (3.20). Note that a similar study, which was more based on the results of the Riesz–Schauder theory [22], had been performed for the particular case = 0 (s-wave) and a slightly different class of potentials in [17]. 3.1.1. Properties of the Vector-Valued Functions k → v ,0 (k; ·). We shall rely on the fact that the spherical Bessel functions j (z) are entire functions for all integers ( ≥ 0), which satisfy bounds of the form (A.42), valid for all integers ( ≥ 0) and for all k ∈ C (see [27]). For k ∈ Ωα we shall use the ∗ of Xw,α , namely norm of the function kRj (kR) in the dual space Xw,α ⎛ +∞ ⎞1/2 |kRj (kR)|2 . ⎝ ∗ dR⎠ . (3.21) k·j (k·)w,α = w(R)e2αR 0
In fact, in view of (A.42), we have for all k ∈ Ωα and ≥ 0: ⎤1/2 ⎡ ∞ 2 |k|R dR 1 ∗ ⎦ k·j (k·)w,α ≤ ⎣ ≤ Aw (|k|), c 1 + |k|R w(R)
(3.22)
0
where the last inequality expresses a requirement on the weight-function w, namely the existence of a positive and non-decreasing function |k| → Aw (|k|) to be defined on R+ . We shall then prove the following lemma. Lemma 5. For every potential V in a class Nw,α such that w satisfies a condition of the type (3.22), the corresponding functions k → v,0 (k; ·) [formally defined in (3.10b)] are well-defined for all integers , ≥ 0, as functions on Ωα with values in Xw,α ; for each the corresponding norm v,0 (k; ·)w,α admits the following bound for k varying in Ωα : v,0 (k; ·)w,α ≤
C(V ) C(V ) ∗ k·j (k·)w,α ≤ c Aw (|k|). (2 + 1)1/2 (2 + 1)1/2
(3.23)
Moreover, this vector-valued function is continuous in Ωα and holomorphic in Ωα . Proof. Starting from Eq. (3.10b), using the Schwarz inequality and taking into account Eqs. (3.18), (3.21) and condition (3.22), we can write for a.e. R: +∞ (w) ∗ |v,0 (k; R)| = V (R, R )kR j (kR ) dR ≤ V (R) k·j (k·)w,α . 0
(3.24) It then follows from (3.19) that the function v,0 (k; ·) belongs to Xw,α and the bounds (3.23) readily follow from (3.19), (3.24), and (3.22) for k ∈ Ωα .
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Proof of the last statement: For k varying in any bounded domain of the form (K) (K) . Ωα = {k ∈ Ωα : |k| < K} (or in its closure Ωα ), Eq. (3.24) yields the fol(w) lowing bound |v,0 (k; R)| ≤ c Aw (K)V (R). One concludes that v,0 (k; R) (K)
(K)
belongs to a class C(D, μ, p) (see Lemma B.8), with D = Ωα (or Ωα ), μ(R) = w(R) e2αR , and p = 2. One moreover checks that the function v,0 (k; R) (K)
(K)
is continuous in Ωα , holomorphic in Ωα for a.e. R in view of Lemma B.9, since the integrand of (3.10b), holomorphic with respect to k in C, can be (K)
uniformly bounded in Ωα
[in view of (A.42)] by the integrable function: KR 1/2 αR −1/2 R −
→ c V (R, R )w (R )e (R ) w 1 + KR
[the integrability property of the latter being obtained by the Schwarz inequality in view of (3.18) and (3.22)]. Lemma B.8 is thus applicable and allows one to state that the vector-valued function k → v,0 (k; ·) ∈ Xw,α is continuous in (K)
(K)
Ωα , holomorphic in Ωα . Since the argument is valid for any value of K, the last statement of the lemma is thus established. Choice of the weight-function w. Our choice of relevant weight-functions w satisfying a condition of the type (3.22) will obey the following criteria: (a) w(R) should be chosen as small as possible near R = 0 in order to include in the class Nw,α potentials V (R, R ), whose behaviour is as much singular as possible near R = 0 and R = 0. Note that the behaviour of w(R) for R tending to infinity is not so relevant, since the dominant behaviour of V (R, R ) at large R and R is in fact dictated by the factors eαR and eαR , which make the classes Nw,α look like nonlocal versions of Yukawa-type potentials. The convergence of the integral (3.22) at R → ∞ will only serve to ensure the validity of the boundedness of v,0 (k; ·)w,α for k lying on the boundary of the strip Ωα . (b) the behaviour of the majorant Aw (|k|) may keep some flexibility, according to whether one is interested in improving the behaviour of v,0 (k; ·)w,α at k → 0 or at k → ∞. In view of these considerations, we can propose the following three specifications of the weight-function w, which will correspond respectively to the following convenient majorizations of the integrand of (3.22): +∞ dR |k|R (i) 1+|k|R < 1 leads one to a choice w = w0 with A2w0 (|k|) = 0 w0 (R) . An (ε) . 1−ε 2ε appropriate choice is: w0 = w0 = R (1 + R) , so that A2w0 (ii)
√
=
A2ε
. =
+∞
dR . R1−ε (1 + R)2ε
(3.25)
0
+∞ RdR leads one to a choice w = w1 with A2w1 (|k|) = |k| 4 w1 (R) . 0 (ε) . 2−ε 2ε An appropriate √ choice is: w1 = w1 = R (1 + R) , so that one has: |k| Aw1 (|k|) = 2 Aε [with Aε given by formula (3.25)]. |k|R 1+|k|R
≤
|k|R 2
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+∞ R2 dR ≤ |k|R leads one to a choice w = w2 with A2w2 (|k|) = |k|2 0 w2 (R) . (ε) . 3−ε 2ε An appropriate choice is: w2 = w2 = R (1 + R) , so that one has: Aw2 (|k|) = |k|Aε . |k|R 1+|k|R
As a consequence of the previous analysis and of majorization (3.23), we can now give the following Complement to Lemma 5. The following bounds hold for each vector-valued function k → v,0 (k; ·): (ε)
(i) for w = w0 = R1−ε (1 + R)2ε , v,0 (k; ·)w,α ≤ c√ C(V )Aε ; |k| (ε) 2−ε 2ε (ii) for w = w1 = R (1 + R) , v,0 (k; ·)w,α ≤ 2 c C(V )Aε ; (ε) (iii) for w = w2 = R3−ε (1 + R)2ε , v,0 (k; ·)w,α ≤ |k|c C(V )Aε . 3.1.2. Properties of the Operator-Valued Functions k → L (k). In Appendix A we have derived bounds on the angular-momentum Green functions G (k; R, R ) which imply global majorizations of the following form for k varying in Πα :
|G (k; R, R )| ≤ hM (, |k|)eα(R+R ) M (R)M (R ),
(3.26)
with the following three specifications: ! √ π (i) M (R) = R, hM (, |k|) = 12 2+1 [implied by (A.5)]; [implied by (A.11)]; (ii) M (R) = 1, hM (, |k|) = 1+π |k| ! √ 1 π (iii) M (R) = 1 + R, hM (, |k|) = min 1+π [implied by (A.12)]. |k| , 2 2+1 We then introduce a condition of the following type on the weight-function w: ⎡ . B(M, w) = ⎣
⎤1/2
∞ M 2 (R)
dR ⎦ w(R)
< ∞,
(3.27)
0
which allows one to prove Theorem 6. For every potential V in a class Nw,α such that w satisfies a condition of the type (3.27), the corresponding kernels L (k; R, R ) (formally defined in (3.10c)) are well-defined as compact operators L (k) of Hilbert–Schmidt-type acting in the Hilbert space Xw,α for all non-negative integral value of and for all k ∈ Πα \{0}. More precisely, |L (k; R, R )| is bounded (for each and k) by a kernel of rank one, and the corresponding Hilbert–Schmidt norm L (k)HS "w,α admits the following majorization for of L (k) in a Hilbert space called X k ∈ Πα \{0}: C(V ) L (k)HS ≤ B 2 (M, w) √ hM (, |k|). 2 + 1
(3.28)
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Moreover, for each , the HS-operator-valued function k → L (k), taking its "w,α , is continuous in Πα \{0} and holomorphic in Πα \{0}. values in X Proof. Assuming that L (k) exists as an operator in Xw,α , let L† (k) be its adjoint, given by the standard definition L† (k)x, y w,α = x, L (k)y w,α ,
(x, y ∈ Xw,α ), where ·, · w,α denotes the scalar product in Xw,α .L† (k) is the integral operator with kernel: L† (k; R, R ) =
w(R ) 2α(R −R) e L (k; R , R). w(R)
"w,α , whose finiteness has Therefore, the Hilbert–Schmidt norm of L (k) in X to be proven, is given by the following double-integral: . 2 L (k)HS = Tr[L† (k)L (k)] +∞ +∞ e−2αR = dR w(R )e2αR |L (k; R , R)|2 dR . w(R) 0
(3.29)
0
Let us first show that the integral on the r.h.s. of (3.10c) is absolutely convergent, and therefore defines L (k; R, R ) for a.e. R. In view of (3.26), this integral is bounded in modulus by ⎤ +∞ hM (, |k|) ⎣ |V (R, R )| eαR M (R ) dR ⎦ eαR M (R ), ⎡
(3.30)
0
and thereby, in view of the Schwarz inequality and of (3.18) and (3.27), one (w) obtains for a.e. R (since V ∈ Xw,α ) the following majorization by a kernel of rank one: |L (k; R, R )| ≤ hM (, |k|)B(M, w)V
(w)
(R)eαR M (R ).
(3.31)
In view of (3.31), we now obtain the following majorization for the expression (3.29) of L (k)2HS : +∞
L (k)2HS
≤
h2M (, |k|)B 2 (M, w)
$ e−2αR # 2αR 2 e M (R) dR w(R)
0 +∞ 2 (w) × w(R )e2αR V (R ) dR , 0
(3.32)
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which yields [in view of (3.27)]: (w) 2 w,α ,
2
L (k)HS ≤ B 4 (M, w)h2M (, |k|)V
(3.33)
and therefore [in view of (3.19)] the majorization (3.28). Proof of the last statement: We note that the space Xw,α , defined in (3.14), is a Hilbert space of the type Xμ introduced in Appendix B [see formula (B.6)], with μ(R) = w(R)e2αR . For each k ∈ Πα \{0}, L (k) is an element of the corre"w,α introduced in (B.9), with the coincidence of nota"μ = X sponding space X tions L (k)HS = L (k)(μ) . It can be checked that for k varying in any given (K) . domain Πα = {k ∈ Πα : |k| > K}, and for each , the function (k; R, R ) → L (k; R, R ) satisfies all the assumptions of Lemma B.10 with [in view of (3.10c) and (3.26)]: ζ = k, F1 (k; R, R ) = G1 (R, R ) = |V (R, R )|, F2 (k; R, R ) = G (k; R, R ), and G2 (R, R ) = hM (, K)eα(R+R ) M (R)M (R ). The fact that the majorizing kernel ⎤ ⎡ +∞ |V (R, R )|eαR M (R ) dR ⎦ eαR M (R ) (3.34) G(R, R ) = hM (, K) ⎣ 0
coincides with the expression (3.30), taken for |k| = K, implies that GHS = G(μ) is finite in view of the previous HS-norm majorization that yielded the r.h.s. of (3.32). Since F1 and F2 are continuous (resp., holomorphic) with respect to k in Πα (resp., Πα ), Lemma B.10 implies that the HS-operator(K)
valued function k → L (k) is continuous (resp., holomorphic) in Πα (resp., (K) Πα ), and therefore in Πα \{0} (resp., Πα \{0}), since the argument is valid for any K > 0. Complement to the choice of the weight-function w. For the three given specifications of M (R) in majorization (3.26), one can always obtain the equality B(M, w) = Aε , with Aε given by Eq. (3.25), provided one chooses respectively (ε) (ε) the weight-functions w1 , w0 (see the complement to Lemma 5), and the weight-function . (3.35) w(ε) (R) = R1−ε (1 + R)1+2ε , (ε)
(ε)
which is such that w(ε) ≥ max(w0 , w1 ). In view of this remark, the majorization (3.28) can thus be specified as follows: Complement to Theorem 6. The following bounds hold for each operator-valued function k → L (k): (ε)
(i) for w = w0 = R1−ε (1 + R)2ε , L (k)HS ≤ (ε)
(ii) for w = w1 = R2−ε (1 + R)2ε , L (k)HS ≤
C(V ) 1+π √ A2 ; |k| ! 2+1 ε C(V ) 1 π √ 2 2 2+1 2+1 Aε ;
(iii) for w = w(ε) = R1−ε (1 + R)1+2ε , % 1 + π 1 C(V ) 2 π √ , L (k)HS ≤ min Aε . |k| 2 2 + 1 2 + 1
(3.36)
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It will appear in the following that the choice w = w(ε) [see Eq. (3.35)], for any positive ε, allows one to obtain the most interesting results, in view of the following corollary of the “Complements to Lemma 5 and to Theorem 6”. Corollary 5–6. For any weight-function w(ε) (R), one has: √ |k| (a) v,0 (k; ·)w(ε),α ≤ min(1, 2 )c C(V )Aε , for k ∈ Ωα ; (b) for each , the HS-operator-valued function k → L (k), taking its values "w(ε),α , is also holomorphic at the origin and therefore in the whole in X . " half-plane Πα . Moreover, in view of (3.36), the function L (r) = HS
supmax(|k|,)≥r L (k)HS is uniformly bounded for all r ≥ 0, and tends to zero when r tends to +∞. 3.1.3. Smithies’ Formalism for the Resolvents R (k;g). Let us introduce the resolvent associated with equation (3.20), i.e., R (k; g) = [I − gL (k)]
−1
.
(3.37)
In this formalism, g is treated as a general complex parameter, keeping in mind that each “physical” theory is obtained by fixing g at a real value interpreted as a coupling constant. The fact that L (k) is a Hilbert–Schmidt operator on the Hilbert space Xw,α allows us to use Smithies’ formulae and bounds [21], which all make sense in terms of Hilbert–Schmidt kernels. According to Theorem 5.6 of Ref. [21], we have (with the identification of notations g ↔ λ, L ↔ K, N ↔ H, σ ↔ δ, R ↔ Δ δ ): R (k; g) = I + g
N (k; g) σ (k; g)
( = 0, 1, 2, . . .),
(3.38)
where: (i) σ (k; g) =
+∞
(σ )n (k)g n ,
(3.39)
n=0
(−1)n (Q )n (k) (n ≥ 1), (σ )0 (k) = 1, (σ )n (k) = n! 0 n−1 0 ··· 0 0 (ρ )2 (k) 0 n − 2 ··· 0 0 (ρ )3 (k) (ρ )2 (k) 0 ··· 0 0 (Q )n (k) = ··· ··· ··· ··· ··· ··· (ρ )n−1 (k) (ρ )n−2 (k) (ρ )n−3 (k) · · · (ρ )2 (k) 0 (ρ )n (k) (ρ )n−1 (k) (ρ )n−2 (k) · · · (ρ )3 (k) (ρ )2 (k)
(3.40) 0 0 0 , · · · 1 0 (3.41)
with: (ρ )n (k) = Tr [Ln (k)] (Note that (Q )1 (k) = (σ )1 (k) = 0).
(n ≥ 2).
(3.42)
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(ii) N (k; g) =
+∞
(N )n (k) g n ,
(3.43)
n=0
with: (N )n (k) = L (k)(Δ )n (k) = (Δ )n (k)L (k), I n ··· 0 ··· 0 L (k) (−1)n L2 (k) (Δ )0 = I, (Δ )n (k) = (Q )n (k) n! · · · ··· n L (k)
(3.44)
(n ≥ 1).
(3.45) Note that there holds for all n ≥ 1 the following recurrence relation between the bounded operators (Δ )n (k) and (Δ )n−1 (k) [see formula (5.4.1) of [21]]: (Δ )n (k) = (σ )n (k)I + L (k)(Δ )n−1 (k) = (σ )n (k)I + (Δ )n−1 (k)L (k). (3.46) The latter directly yields the following identity in the sense of power series of g: +∞ . (Δ )n (k)g n = σ (k; g)I + gΔ (k; g)L (k), Δ (k; g) =
(3.47)
n=0
which then yields: Δ (k; g)[I − gL (k)] = σ (k; g)I,
i.e.,
Δ (k; g) = σ (k; g)R (k; g),
(3.48)
and also, in view of (3.44): Δ (k; g) = σ (k; g)I + gN (k; g).
(3.49)
By putting Eqs. (3.48) and (3.49) together, one concludes that Eq. (3.38) is then satisfied in the sense of formal series by the functionals σ (k; g) and N (k; g), defined as functionals of L (k) by Eqs. (3.39)–(3.45). Since for each the function k → L (k) is holomorphic in Πα and con"w,α (see Theorem 6 and tinuous in Πα as a function taking its values in X Corollary 5–6), it follows from Lemma B.6 that the same property holds for all the corresponding power functions k → Ln (k). Moreover one also has: (i) In view of Lemma B.5 (applied to K[ζ] = L (k) and Kt [ζ] = L (k)), the function k → Tr[L2 (k)], and similarly all functions k → (ρ )n (k) = Tr[Ln−1 (k)L (k)] [see Eq. (3.42)] are holomorphic in Πα and continuous in Πα . Since [in view of (3.40), (3.41)] (Q )n (k) and (σ )n (k) are polynomials of the variables (ρ )p (k)(p ≤ n), all these functions are also holomorphic in Πα and continuous in Πα .
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(ii) In view of (3.44) and (3.45), the kernels (N )n (k) are polynomials of L (k) n+1 of the form (N )n (k) = j=1 aj (k)Lj (k), with complex coefficients aj (k), which are themselves polynomials of the variables (ρ )p (k). Then, in view of Lemma B.7, each function k → (N )n (k) is (like k → L (k)) holomorphic in Πα and continuous in Πα as a function taking its values in "w,α . X From Smithies’ theory it follows that, for any k in Πα , and any non-negative integral value of , σ (k; g) is an entire function of g, and g → N (k; g) is an entire Hilbert–Schmidt operator-valued function [the series (3.43) being "w,α ]. More precisely, we can prove the following convergent in the norm of X theorems. Theorem 7. For every potential V in a class Nw(ε),α , the functions σ (k; g) satisfy the following properties, for all integers ≥ 0: (a) σ (k; g) is defined and uniformly bounded in modulus by a function Φ (|g|) in Πα × C; it is continuous in Πα × C and holomorphic in Πα × C; (b) σ (k; g) = σ (−k; g); (c) for any fixed value of g, there holds: supmax(|k|,)≥r |σ (k; g) − 1| −−−−−→ 0. r→+∞
Proof. By combining the basic inequalities of Smithies’ theory (see [21, Lemma 5.4]) with the uniform bounds (3.36) on L (k)HS , one obtains the following majorizations, valid for all integers n ≥ 1, ≥ 0 and for all k in Πα : e n/2 e n/2 # $n n |(σ )n (k)| ≤ A2ε C(V )h(, |k|) (2+1)−n/2 , L (k)HS ≤ n n (3.50) where . h(, |k|) = min
π + 1 1 , |k| 2
%
π 2 + 1
.
(3.51)
In view of (3.50), the series (3.39) defining σ (k; g) − 1 is dominated for all k ∈ Πα by a convergent series with positive terms. It is convenient to associate with this series the entire function ∞ . e n/2 n Φ(z) = z , (3.52) n n=1 which is a positive and increasing function of z for z > 0, such that Φ(0) = 0. From (3.50), one then concludes that σ (k; g) is for all k ∈ Πα an entire function of g, which satisfies the following uniform majorization: 2 C(V ) |σ (k; g) − 1| ≤ Φ (|g|L (k)HS ) ≤ Φ |g|Aε √ h(, |k|) . (3.53) 2 + 1 Moreover, in view of the holomorphy (resp., continuity) property of the functions (σ )n (k) in Πα (resp., Πα ), Lemma B.1 can be applied to the sequence of functions {(k, g) → (σ )n (k)g n ; n ∈ N}; it follows that the sum of the series (3.39) defines σ (k; g) as a holomorphic function of (k, g) in Πα × C, which is moreover continuous in Πα × C.
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The symmetry relation (3.12) implies analogous relations for the quantities (ρ )n (k), (Q )n (k), (σ )n (k), and therefore property (b). In view of the behaviour of h(, |k|) [given by (3.51)], one checks that . ) h(, |k|)) is finite and tends the quantity " h(r, g) = supmax(|k|,)≥r (|g|A2ε √C(V 2+1 to zero for r tending to infinity (for each fixed g). Correspondingly, one has: C(V ) lim sup Φ |g| A2ε √ h(r, g)) = Φ(0) = 0. h(, |k|) = lim Φ(" r→+∞ max(|k|,)≥r r→+∞ 2 + 1 (3.54) Property (c) is then readily implied by inequality (3.53).
Theorem 8. For every potential V in a class Nw(ε),α , the operators N (k; g) exist as Hilbert–Schmidt operators acting on Xw(ε),α , for all integers ≥ 0 and for all (k, g) in Πα × C; in this set, the function (k, g) → N (k; g)HS is uniformly bounded in k by a function Ψ (|g|). Moreover, the following properties hold: (a) The HS-operator-valued function (k, g) → N (k; g), taking its values in "w(ε),α , is continuous in Πα × C and holomorphic in Πα × C; X (b) N (k; g; R, R ) = N (−k; g; R, R ); (c) supmax(|k|,)≥r N (k; g)HS −−−−−→ 0. r→+∞
Proof. In view of Smithies’ theory, there hold the following inequalities (see Lemmas 2.6 and 5.4 and the proof of Theorem 5.6 of [21]) for all integers n ≥ 1: e(n+1)/2 L (k)n+1 (3.55) HS . nn/2 In view of the latter, the series (3.43) is dominated term-by-term in the HSnorm by a convergent series; the sum of the latter is therefore well-defined as a HS-operator N (k; g) for all values of (k, g) in Πα × C. The entire function . Ψ(z) = z[1 + e1/2 Φ(z)], (3.56) (N )n (k)HS ≤ (Δ )n (k)L (k)HS ≤
[with Φ given by Eq. (3.52)], is like Φ an increasing function of z, for z ≥ 0. It follows from (3.43), (3.55) and from the bound (3.36) on L (k)HS (used as "w(ε),α satisfies the in the proof of Theorem 7) that the norm of N (k; g) in X bound: C(V ) 1 1 Ψ(|g|L (k)HS ) ≤ Ψ |g|A2ε √ h(, |k|) . N (k; g)HS ≤ |g| |g| 2 + 1 (3.57) "w(ε),α Moreover, since the functions (k, g) → (N )n (k), taking their values in X are continuous in Πα × C and holomorphic in Πα × C, Lemma B.1 can be applied to the sequence of functions {(k, g) → (N )n (k)g n ; n ∈ N}; it follows that for each g ∈ C the sum of the series (3.43) defines (k, g) → N (k; g) as a function satisfying property (a).
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Property (b) follows from the symmetry relation (3.12), through all the analogous symmetry relations satisfied by the quantities (Q )n (k), (Δ )n (k), (N )n (k). By using the fact that limz→0 Ψ(z) = 0, and taking into account expression (3.51) of h(, |k|), one obtains property (c) as a by-product of inequality (3.57). 3.2. Meromorphy Properties of the Resolvent and Their Physical Interpretation It is convenient to rewrite Eq. (3.37) in terms of the Fredholm resolvent kernel . (tr) or truncated2 resolvent R (k; g) = g1 [R (k; g) − I], as the following Fredholm resolvent equation: (tr)
(tr)
R (k; g) = L (k) + gL (k)R (k; g),
(3.58)
whose solution is given in view of (3.38) by N (k; g) . (3.59) σ (k; g) We now give an analysis of the meromorphy properties of the operator-valued (tr) function (k, g) → R (k; g), which follow from Theorems 7 and 8 and formula (3.59). (tr)
R (k; g) =
3.2.1. Meromorphy in (k, g) and Meromorphy in k at Each Fixed g. A singu(tr) larity (more precisely, a pole) of the function (k, g) → R (k; g) is generated by a zero of the modified Fredholm determinant σ (k; g), namely a connected component in Πα × C of the complex manifold with equation σ (k; g) = 0. An essential property of this manifold to be checked at first is the fact that it cannot contain components of the form g − g0 = 0. In fact, this would imply σ (k; g0 ) = 0 for all k, which (for |k| → ∞) would contradict property (c) of Theorem 7. So, for each fixed value of g ∈ C, the corresponding restriction of the function σ (k; g) is a non-zero holomorphic function of k in Πα . Then, in (tr) view of (3.59) and of Theorem 8, we can conclude that R (k; g) is defined for each ( = 0, 1, 2, . . .) and for each g ∈ C as a meromorphic HS-opera"w(ε),α . At fixed g, tor-valued function of k in Πα , which takes its values in X (tr)
all the possible poles of the function k → R (k; g) can thus be generically defined as solutions k = k (j) (, g) of the implicit equation σ (k; g) = 0 (at points where ∂σ /∂k = 0, considering the generic case). The complement of this discrete set, namely the set of all points k in Πα (resp., Πα ) such that σ (k, g) = 0 will be denoted by Πα, (g) (resp., Πα, (g)). We therefore have (in view of Theorems 7 and 8): Theorem 9. For every potential V in a class Nw(ε),α , the function (k, g) → (tr)
R (k; g) is meromorphic in Πα × C as a HS-operator-valued function. More 2
We use here the same terminology as in relativistic quantum field theory, in which the truncated four-point function plays the same role as the truncated resolvent in the present nonrelativistic framework.
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(tr)
precisely, the operators R (k; g) exist for all (k, g) such that k ∈ Πα, (g), g ∈ C, as Hilbert–Schmidt operators acting on Xw(ε),α , and for any fixed g ∈ C, (tr) "w(ε),α , the HS-operator-valued function k → R (k; g), taking its values in X is holomorphic in Πα, (g). As a corollary, we also have: Theorem 10. The function (k, g) → R (k; g) is meromorphic in Πα × C, and for any fixed g in C the function k → R (k; g) is holomorphic in Πα, (g), as operator-valued functions taking their values in the space of bounded operators in Xw(ε),α . (tr)
In fact, the holomorphy properties of R (k; g) as a HS-operator-valued function imply the same holomorphy properties as a bounded operator-val(tr) (tr) ued function (since R (k; g) ≤ R (k; g)HS ). By adding the constant operator I (holomorphic as a bounded operator), one thus concludes (tr) that R (k; g) = I + R (k; g) has the same holomorphy (and meromorphy) (tr) properties in k as R (k; g), but in the sense of a bounded (not Hilbert– Schmidt)-operator-valued function. 3.2.2. Poles of the Resolvent and Solutions of the Schr¨ odinger-Type Equation. All the possible poles k = k (j) (, g) correspond to situations in which there exists a non-zero solution x = x(R) of the homogeneous equation gL (k)x = x. In fact, Eqs. (3.58) and (3.59) imply the following identity between HS-operator-valued functions, which is valid for all (k, g) ∈ Πα × C: N (k; g) = σ (k; g)L (k) + gL (k)N (k; g).
(3.60) (tr)
A value k = k (j) (, g) corresponds to a pole of the function k → R (k; g) iff the previous equation reduces to the homogeneous equation N (k; g) = gL (k)N (k; g),
with N (k (j) (, g); g) = 0.
(3.61)
Then it follows from Fredholm’s theory that the latter kernel is of the form N (k (j) (, g); g)(R, R ) = i∈I xi (R)yi (R ), with xi ∈ Xw(ε),α and yi ∈ Xw∗ (ε),α , I denoting a finite set. The existence of a pole of the function (tr)
k → R (k; g) is therefore equivalent to the existence of at least one (nonzero) solution x = x(R) in Xw(ε),α of the equation gL (k)x = x. As shown below (see Lemma 13), one can associate with any function x(R) in Xw(ε),α , for every , and for k ∈ Πα , the function +∞ ψ(R) = g G (k; R, R )x(R ) dR ,
(3.62)
0
which satisfies the equation D,k ψ = gx, since G (k; R, R ) is a Green function of the differential operator D,k . Now, in view of Eq. (3.62), the definition
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bound states
k − plane
spurious bound states resonances antibound states
−α
Figure 1. Representation of bound states and resonances in the complex k-plane (3.10c) of L (k) implies the following equality: +∞ g[L (k)x](R) = V (R, R )ψ(R ) dR .
(3.63)
0
So, if x is a non-zero solution of the homogeneous Fredholm equation gL (k)x = x (associated with a value of k which is a pole of the function k → R (k; g)), then the function ψ defined by Eq. (3.62) satisfies the following relations: +∞ V (R, R )ψ(R ) dR , (3.64) D,k ψ(R) = gx = g 0
and therefore ψ is a non-zero solution of the Schr¨ odinger-type equation (3.6). 3.2.3. Some Results on the Location of the Poles and Their Physical Interpretation (see Fig. 1). For g real (interpreted physically as a coupling constant) and for k such that Im k ≥ 0, Eq. (3.62) defines a one-to-one correspondence between the solutions of the homogeneous equation gL (k)x = x and a class of square-integrable solutions of the Schr¨ odinger-type equation (3.6), which will be fully described in Theorem 16 below. According to the latter, all the possible zeros k = k (j) (, g) of σ (k; g) which lie in Im k ≥ 0 correspond to “bound state solutions” of (3.6): these solutions are the contributions with a given angular momentum to the set of solutions of Problem 1, whose general properties have been listed in Theorem 2 (see Sect. 2). In particular, as a general by-product of Theorem 2, it follows that for each real value of g, all the possible zeros k = k (j) (, g) in the closed half-plane Im k ≥ 0
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can be located only either on the imaginary axis or on the real axis. These two situations, which will be analyzed in detail below (see Theorem 16), correspond respectively to bound states and to spurious bound states (i.e., “bound states embedded in the continuum”). Furthermore, the zeros on the real axis are distributed in pairs symmetric with respect to k = 0 (in view of statement (tr) (b) of Theorem 7). Concerning the possible poles of R (k; g) in the strip −α < Im k < 0, we can also say that (for the same reason) they occur either on the imaginary axis (anti-bound states) or in pairs symmetric with respect to the imaginary axis (resonances). Since (at fixed g) σ (k; g) is holomorphic in Πα ( = 0, 1, 2, . . .), and since the function (k, ) → σ (k; g) tends uniformly to 1 for max(|k|, ) → +∞ (see statement (c) of Theorem 7), there holds a finiteness property of the set of zeros of all the functions σ in the domain Πα of the k-plane, which can be stated as follows in terms of the corresponding physical interpretation: Proposition 11. For any nonlocal potential V in a class Nw(ε),α , and for each fixed real value of the coupling constant g: (a) there exists an integer L = L(V, g) such that, for every ≥ L, there occur no bound states, anti-bound states and resonances of angular momentum ; (b) for each integer such that 0 ≤ < L, there occur at most a finite number n of (normal or spurious) bound states and a finite number of resonances and anti-bound states of angular momentum in any strip . Ω− α−ε = {k ∈ C : −(α − ε) ≤ Im k < 0}, all of them being localized in a finite disk of the form |k| < kV,g . 3.3. Correspondence Between the Solutions of the Fredholm Equation with Kernel L (k) and Those of the Nonlocal Schr¨ odinger-Type Equation 3.3.1. Preliminary Properties. We need to state four lemmas. Lemma 12. For every function x in Xw(ε),α , there hold the following properties: (a) the function R → kRj (kR)x(R) is integrable on [0, +∞) for all k in Ωα , and satisfies the following majorization: +∞ c kR j (kR )x(R ) dR ≤ e−(α−| Im k|)R xw(ε),α , 1/2 [2(α − | Im k|)] R
(3.65) which is valid for all k in Ωα and R ≥ 1; (1) (b) the function R → kRh (kR)x(R) is integrable on any interval [R, +∞) (R > 0) for all k in Πα , and satisfies the following majorization:
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+∞ (1) c (1 + |k|−1 ) −(α+Im k)R (1) ≤ kR h (kR )x(R ) dR xw(ε),α , [2(α + Im k)]1/2 e R
(3.66) which is valid for all k in Πα and R ≥ 1. Proof. We make use of the Schwarz inequality in the space Xw(ε),α : (ε)
(a) By using bound (3.22), with w = w(ε) > w0 [see (3.35)], which allows us to take Aw(ε) (k) = Aw(ε) = Aε [given by (3.25)], we obtain for all k ∈ Ωα : 0
+∞
∗
|kRj (kR)||x(R)| dR ≤ k·j (k·)w(ε),α xw(ε),α ≤ c Aε xw(ε),α . 0
(3.67) The majorization (3.65) of the remainder of this integral from R to +∞ is obtained similarly (for k ∈ Ωα and R ≥ 1) by using the (e| Im k|R )-dependence of the bound (A.42) on [kRj (kR)], and the inequality [w(ε) (R)]−1 < 1. (1) (b) By using the bound (A.43) on [kRh (kR)], one obtains a Schwarz inequality similar to (3.67), except for the replacement of the integration interval [0, +∞) by [R, +∞) (for all R > 0), which proves the corresponding integrability property for all k in Πα . The bound (3.66) is obtained by using again the inequality [w(ε) (R)]−1 < 1 together with a majorant of bound (A.43) for R ≥ 1. Lemma 13. For every function x in Xw(ε),α , the corresponding function +∞ ψx;,k (R) = g G (k; R, R )x(R ) dR
(3.68)
0
is well-defined for all k with Im k ≥ −α, and enjoys the following properties: (a) there hold majorizations of the following form: (i) for Im k ≥ 0, c,ε xw(ε),α Re− min(α,Im k)R ; |ψx;,k (R)| ≤ |g| "
(3.69)
(ii) for −α ≤ Im k < 0, |ψx;,k (R)| ≤ |g| " c,ε xw(ε),α Re| Im k| R ;
(3.70)
− α2
(b) for every k such that < Im k < α, k = 0, there holds the following limit: (1) (3.71) ψx;,k (R) + ig bx;,k Rh (kR) −−−−−→ 0, R→+∞
where: bx;,k
. =
+∞ kRj (kR)x(R) dR, 0
(3.72)
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with the specifications listed below. There exists a function c (k) such that, for R ≥ 1, there hold the following inequalities: (i) if 0 ≤ Im k < α, (1) ψx;,k (R) + igbx;,k Rh (kR) ≤ c (k)|g|xw(ε),α e−αR ; (3.73) (ii) if − α2 ≤ Im k < 0, (1) ψx;,k (R) + igbx;,k Rh (kR) ≤ c (k)|g|xw(ε),α e−(α−2| Im k|)R ;
(3.74)
(R) of ψx;,k (R) is (c) for all k with Im k ≥ −α, k = 0, the derivative ψx;,k well-defined, absolutely continuous and bounded for R tending to zero. For Im k ≥ 0, there holds a majorization of the following form: |ψx;,k (R)| ≤ |g|" c ,ε xw(ε),α e− min(α,Im k)R ,
(3.75)
and the following limit is valid for all k such that 0 ≤ Im k < α, k = 0: eR Im k ψx;,k (R) − ikψx;,k (R) −−−−−→ 0. (3.76) R→+∞
Proof. In view of expression (3.10e) of the Green function G , we can rewrite Eq. (3.68) under the following form, whose validity is established below: ⎡ (1) ψx;,k (R) = −ig ⎣Rh (kR)
R
kR j (kR )x(R ) dR
0 +∞
⎤
kR h (kR )x(R ) dR ⎦ .
+ Rj (kR)
(1)
(3.77)
R
In fact, for every k such that Im k ≥ −α, the convergence of the integrals in (3.77) is obtained, together with appropriate majorizations on the latter (1) by using the bounds (A.42) and (A.43) on the functions j and h , in the following way: |ψx;,k (R)|
⎡ +1 R (1) c c ⎣ 1 + |k|R |k|R −R Im k e eR | Im k| |x(R )| dR ≤ |g| |k| |k|R 1 + |k|R 0 ⎤ +∞ +1 |k|R 1 + |k|R + eR| Im k| e−R Im k |x(R )| dR ⎦ , 1 + |k|R |k|R R
(3.78)
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which yields (by using the increase property of the function
|k|R 1+|k|R ):
|ψx;,k (R)| (1)
|g|c c R −R Im k
R
≤e
R | Im k|
e
+∞ e−R Im k |x(R )| dR .
R| Im k|
|x(R )| dR + e
R
0
(3.79) By using the assumption that x belongs to Xw(ε),α , the two integrals on the r.h.s. of the latter can be seen to be convergent for all k in Πα in view of the Schwarz inequality, which therefore implies the existence (and analyticity in k for every R ≥ 0) of ψx;,k (R) in this domain of the k-plane. We now exhibit these convergence properties together with the majorizations listed under (a). For k in the half-plane Im k ≥ 0, majorization (3.79) also implies the following one: |ψx;,k (R)| (1)
|g|c c R
≤ e−R min(α,Im k)
R
eαR |x(R )| dR + e−αR
+∞ eαR |x(R )| dR , R
0
(3.80) (1)
which then yields (3.69) (with " c,ε = c c Aε ) by directly using the Schwarz inequality. For k in the strip −α ≤ Im k < 0, the majorization (3.79) readily implies +∞ (1) that |ψx;,k (R)| ≤ |g|c c Re| Im k|R 0 eαR |x(R )| dR , which then yields (3.70). (b) Let bx;,k be given by the integral in (3.72), whose convergence has been established in Lemma 12 (a), provided | Im k| ≤ α. Equation (3.77) can then be rewritten as follows: ⎡ +∞ (1) (1) ψx;,k (R) + igbx;,k Rh (kR) = ig ⎣Rh (kR) kR j (kR )x(R ) dR R
⎤ +∞ (1) −Rj (kR) kR h (kR )x(R ) dR ⎦ . R
(3.81) Let us show that for − α2 < Im k < α (and k = 0), each term in the bracket on the r.h.s. of the latter tends to zero in the limit R → +∞, with the specifications (i), (ii) listed under (b). In view of Lemma 12 (a) and of bound (A.43), the first term on the r.h.s. of (3.81) can be majorized by |g|c,1 (k)e−R Im k e−(α−| Im k|)R xw(ε),α , where . (1) we have put: c,1 (k) = c c (1 + |k|−1 ) [2(α − | Im k|)]−1/2 . This bound correspond to the following two regimes:
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(i) |g|c,1 (k)e−αR xw(ε),α , if 0 ≤ Im k < α, (ii) |g|c,1 (k)e−(α−2| Im k|)R xw(ε),α , if − α2 < Im k < 0. Similarly, in view of Lemma 12 (b) and of bound (A.42), the second term on the r.h.s. of (3.81) can be majorized by |g|c,2 (k)e| Im k|R e−(α+Im k)R . (1) xw(ε),α , where we have put: c,2 (k) = c c (1+|k|−1 ) [2(α+Im k)]−1/2 . Here again, this bound corresponds to the previous two regimes (i) and (ii), except for the substitution c,1 (k) → c,2 (k). The inequalities (3.73) and (3.74) are . therefore established with c (k) = c,1 (k) + c,2 (k). of ψx;,k , which can be obtained (c) We now consider the derivative ψx;,k for every positive R by a direct computation from the r.h.s. of (3.77) (since (1) for all k ∈ C, k = 0, and R > 0, j (kR) and h (kR) define analytic functions + of R on R ). This yields: ψx;,k (R)
' R d & (1) Rh (kR) = −igk R j (kR )x(R ) dR dR 0 +∞ d (1) [Rj (kR)] − igk R h (kR )x(R ) dR . dR
(3.82)
R
ψx;,k
The fact that is absolutely continuous is then an immediate consequence of (3.82), in view of the convergence of the integral factors established above for Im k ≥ −α. The fact that ψx;,k (R) remains a bounded and absolutely continuous function in the limit R → 0 is also implied by Eq. (3.82) by taking (1) into account the fact that the holomorphic functions j and h respectively admit a zero of order and a pole of order + 1 at the origin. The majorization (3.75) is then deduced from (3.82) in the same way as (3.69) is deduced from (3.77); this is because, in view of bounds (A.47) and (A.50) (respectively similar to (A.42) and (A.43)), a majorization of the form . (3.79) is equally valid for ψx;,k Let us finally establish limit (3.76). We note that the second term on the r.h.s. of (3.82) tends to zero as a constant times e−αR , like the second term on the r.h.s. of (3.77) divided by R [in view of the majorization of the latterused (R) − ikψx;,k (R) and in (3.80)]. If we now form the expression eR Im k × ψx;,k consider it as given by the corresponding difference of the r.h.s. of Eqs. (3.82) and (3.77), we conclude that its limit for R → +∞ is the same as the limit of the expression ( ) R ' d & (1) (1) R Im k e × (−igk) R j (kR )x(R ) dR . Rh (kR) − ikRh (kR) dR 0
(3.83) Now, in view of (A.51), the latter can be bounded by a quantity of the form R O( R12 ) × 0 eR | Im k| |x(R )| dR ≤ O( R12 ) × Aε xw(ε),α , which proves the limit (3.76) for all k such that 0 ≤ Im k < α, k = 0.
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Lemma 14. For every function ψ on [0, +∞[ such that |ψ(R)| ≤ c(ψ)R, or more generally for every ψ in the dual space Xw∗ (ε),α of Xw(ε),α , the following properties hold: (a) the function R → ψ(R)v,0 (k; R), where v,0 (k; R) is the function defined by Eq. (3.10b) and Lemma 5, is integrable on [0, +∞) for all k in Ωα ; . +∞ (b) the function xψ (R) = 0 V (R, R )ψ(R ) dR is well-defined as an element of Xw(ε),α , and one has (for all k in Ωα ): +∞ +∞ kRj (kR)xψ (R) dR = ψ(R)v,0 (k; R) dR; 0
(3.84)
0
(c) the following double integral is well-defined: +∞
+∞
dR 0
dR ψ(R)V (R, R )ψ(R ) < +∞.
(3.85)
0
Proof. (a) The assumption |ψ(R)| ≤ c(ψ)R implies: +∞ |ψ(R)||v,0 (k; R)| dR ≤
c(ψ) v,0 (k; ·)w(ε),α . (2α)1/2
(3.86)
0
(b) Similarly, formula (3.18) implies: |xψ (R)| ≤
+∞ |V (R, R )| |ψ(R )| dR ≤
c(ψ) (w(ε) ) V (R), (2α)1/2
(3.87)
0
and therefore, in view of (3.19): xψ w(ε),α ≤
c(ψ) C(V ). (2α)1/2
(3.88)
Then, in view of (3.10b) and of the symmetry relation V (R, R ) = V (R , R), one readily obtains equality (3.84), together with the following inequality [in view of (3.67) and (3.88)]: +∞ c(ψ) kRj (kR) xψ (R) dR ≤ c C(V )Aε . (3.89) 1/2 (2α) 0
(c) Finally, the double integral in (3.85) can be rewritten and majorized as follows: ∞ 2 ψ(R)xψ (R) dR ≤ c(ψ) xψ w(ε),α ≤ c (ψ) C(V ). (3.90) (2α)1/2 2α 0
Note that inequalities similar to (3.86)–(3.90) are obtained by using the ∗ c(ψ) more general assumption ψ ∈ Xw∗ (ε),α and replacing (2α) 1/2 by ψw (ε),α .
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Lemma 15 (Wronskian Lemma). Let ψ(R) be a function on R+ which enjoys the following conditions: (i) its derivative is absolutely continuous and bounded for R tending to zero; (ii) it satisfies a bound of the form |ψ(R)| ≤ cR; (iii) for given values of and k ∈ Πα , it is a solution of the integro-differential +∞ equation: D,k ψ(R) = g 0 V (R, R )ψ(R ) dR , where g is real. Then one has: ⎡ lim ⎣ψ(R)ψ (R) − ψ (R)ψ(R) − (k − k 2 ) 2
R
R→+∞
⎤ ψ(R )ψ(R ) dR ⎦ = 0.
0
(3.91) Proof. Equation (3.6) directly implies the following equalities for g real: 2
ψ(R)ψ (R) − ψ (R)ψ(R) + (k 2 − k )ψ(R)ψ(R) = ψ(R)[D,k ψ](R) − [D,k ψ](R) ψ(R) +∞ # $ V (R, R ) ψ(R)ψ(R ) − ψ(R )ψ(R) dR . =g
(3.92)
0
We note that, in view of Lemma 14 (b) and (c) (and by taking condition (ii) into account), the r.h.s. of the latter is well-defined as an integrable function I(R) on [0, +∞), and that, in view of the symmetry condition on the potential R" V (R, R ), one then has limR→+∞ I(R) dR = 0. Therefore, by integrating " 0 " and taking into account the Eq. (3.92) side by side over R between 0 and R fact that ψ(0) = 0 with ψ (0) bounded, one readily obtains Eq. (3.91). 3.3.2. Homogeneous Integral Equation and Bound State Solutions. We shall now focus on the basic relationship between (i) the solutions of the homogeneous Fredholm equation gL (k)x = x associated with the zeros k (j) (, g) of σ in the closed upper half-plane of k, and (ii) the bound state solutions of the corresponding nonlocal Schr¨ odinger equation. It is worthwhile to study this relationship specially for real values of the coupling g, which correspond to the Hermitian character of the Hamiltonian (3.1). The results are described by the following Theorem 16, whose proof relies basically on Lemmas 12, 13, 14, and 15. Theorem 16. (a) Let x,k ∈ Xw(ε),α be a non-zero solution of the homogeneous integral equation: gL (k)x,k = x,k ,
(3.93)
for fixed values of (non-negative integer), g real, and k such that Im k ≥ 0, k = 0; then there exists a corresponding non-zero solution ψ,k (R) of the Schr¨ odinger-type integro-differential equation (3.6), which is defined
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by +∞ G (k; R, R )x,k (R ) dR , ψ,k (R) = g
(3.94)
0
and enjoys the following properties: (i) ψ,k (R) is absolutely continuous and bounded in [0, +∞); (ii) ψ,k (0) = 0 and there exists a constant c(ψ) such that |ψ(R)| ≤ c(ψ)R; & '1/2 . +∞ 2 (iii) ψ,k = |ψ (R)| dR < +∞. ,k 0 Moreover, there holds the following inversion formula: +∞ V (R, R )ψ,k (R ) dR . x,k (R) =
(3.95)
0
(b) If k is such that 0 ≤ Im k ≤ α, there is a finite constant b,k such that: b,k
+∞ +∞ = kRj (kR)x,k (R) dR = ψ,k (R)v,0 (k; R) dR. 0
(3.96)
0
(c) The respective cases Im k > 0 (ordinary bound states) and k real, k = 0 (bound states embedded in the continuum or “spurious bound states”) are distinguished from each other by the following additional properties: (i) if Im k > 0, one necessarily has Re k = 0; moreover, the solution |ψ,k | satisfies the following global majorization: |ψ,k (R)| ≤ |g|c c Aε x,k w(ε),α Re− min(α,Im k)R , (1)
(3.97)
|ψ,k (R)|
and also tends to zero for R tending to +∞; (ii) if Im k = 0, k = 0, there holds the following majorization: |ψ,k (R)| ≤ |g|" c,k,ε x,k w(ε),α Re−αR ,
(3.98)
where " c,k,ε denotes a suitable constant, and the previous relations (3.96) become orthogonality relations, namely there holds the implication: Im k = 0
=⇒
b,k = 0.
(3.99)
(d) Conversely, if for fixed values of and k ( ≥ 0, Im k ≥ 0, k = 0), there exists a solution ψ,k (R) of the integro-differential equation (3.6) which satisfies properties (i), (ii), and (iii) listed in (a), then Eq. (3.95) defines a corresponding solution x,k (R) in Xw(ε),α of the homogeneous equation (3.93); moreover, ψ,k is reconstructed from x,k by Eq. (3.94) and all the properties described under (b) and (c) are valid. Proof. As proved in Lemma 13, the fact that the function ψ,k is well-defined by formula (3.94) and satisfies properties (i) and (ii) listed in (a) is simply ensured by the assumption that x,k ∈ Xw(ε),α . It follows that the action on
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ψ,k of the second-order differential operator D,k is well-defined and since D,k (R)G (k; R, R ) = δ(R − R ), Eq. (3.94) yields: D,k ψ,k = gx,k .
(3.100)
In particular, since x,k is non-zero, ψ,k is also non-zero. On the other hand, property (ii) of ψ,k implies that this function satisfies the assumptions of Lemma 14, and therefore [in view of Lemma 14 (b)] the integral +∞ V (R, R )ψ,k (R ) dR is convergent (for a.e. R) and defines an element of 0 Xw(ε),α . Now, by plugging the expression (3.94) of ψ,k in this integral, applying the Fubini theorem, and recognizing the definition (3.10c) of L (k; R, R ), one obtains the following equality: +∞ g[L (k)x,k ](R) = V (R, R )ψ,k (R ) dR .
(3.101)
0
Then, in view of Eqs. (3.100) and (3.101), the assumption (3.93) of (a) implies the two equalities (3.64) (for ψ = ψ,k ); in other words, the integro-differential equation (3.6) and the inversion formula (3.95) are satisfied respectively by ψ,k and x,k . Another result of Lemma 14 (b) [namely, Eq. (3.84)] implies that ψ,k and x,k satisfy our statement (b). The proof of the statements listed in (c) will rely crucially on the Wronskian lemma (Lemma 15). We distinguish the two cases: (i) Im k > 0: The bound (3.97) coincides with (3.69), which has been established in Lemma 13 under the same assumption for x,k . As a by-product, property (iii) of ψ,k is established for the case Im k > 0. Besides, (R) also tends to zero for R Eq. (3.75) implies that the function ψ,k tending to infinity. Moreover, the uniform boundedness of ψ,k together with the bound . (R) − (3.97) on ψ,k imply that the Wronskian W (R) = [ψ,k (R) ψ,k (R)ψ ψ,k (R)] tends to zero for R tending to infinity. Now, since ψ,k ,k is a solution of (3.6), we can apply Lemma 15; Eq. (3.91) then entails +∞ 2 that (k − k 2 ) 0 ψ ,k (R)ψ,k (R) dR = 0, which is only possible (for Im k > 0) if Re k = 0, since ψ,k is non-zero. (ii) Im k = 0: The following steps can be taken. (1) |ψ,k (R)| is uniformly bounded. This results from the inequalities (3.69) and (3.73) of Lemma 13 together with the bound (A.43) on (1) |kRh (kR)|. (2) limR→∞ |ψ,k (R)| = 0. This is implied (for k = 0) by the following two statements. At first, Eq. (3.91) of Lemma 15 now implies (since k = k) that limR→∞ W (R) = 0. Secondly, in view of (1) and of Eq. (3.76) of Lemma 13 (for Im k = 0), it follows that the expression [2ikψ ,k (R) ψ,k (R)−W (R)] = ψ ,k (R)[ikψ,k −ψ,k (R)]− ψ,k (R)[ikψ,k − ψ,k (R)] tends itself to zero for R tending to infinity.
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(3) there holds the orthogonality relation (3.99), i.e. b,k = 0. In fact, it also follows from Eq. (3.71) of Lemma 13 that the difference (1) |ψ,k (R)| − |g||b,k ||Rh (kR)| tends to zero for R tending to infinity. But we know from (A.54) that in the limit R → ∞, the function (1) kRh (kR) behaves like exp{i[kR−(+1) π2 ]}. Therefore the validity of (2) (|ψ,k (R)| → 0) necessitates that b,k = 0. Let us now consider again Eq. (3.73) of Lemma 13. Since b,k = 0, this bound reduces to the following one: |ψ,k (R)| ≤ c (k)|g|xw(ε),α e−αR
for R ≥ 1.
(3.102)
But, in view of Eq. (3.69) of Lemma 13 (for Im k = 0), ψ,k (R) also satisfies (for all R ≥ 0) the bound |ψ.k (R)| ≤ |g|" c,ε Rx,k w(ε),α .
(3.103)
Then, it is clear that the two bounds (3.102) and (3.103) can be replaced by a unique bound of the form (3.98), valid for all R ≥ 0. The latter also implies property (iii) of ψ,k for the case Im k = 0. Proof of (d). Let ψ(R) ≡ ψ,k (R) be a solution of Eq. (3.6) satisfying +∞ properties (i), (ii), (iii) of (a); one then has: D,k ψ(R) = g 0 V (R, R )ψ . (R ) dR = gxψ (R), where xψ belongs to Xw(ε),α in view of Lemma 14 (b). If . ∞ " we now introduce the function ψ(R) = g 0 G (k; R, R )xψ (R ) dR , to which the previous study of (3.94) can be applied, we can assert that this function " satisfies equation (3.100), namely, D,k ψ(R) = gxψ (R), together with properties (i), (ii) of (a), property (iii) being only obtained for the case Im k > 0. . " Therefore the function y(R) = [ψ(R) − ψ(R)], which is such that D,k y(R) = 0 and satisfies properties (i) and (ii) of (a), is a constant multiple of kRj (kR); this multiple vanishes if property (iii) of (a) is also satisfied and therefore, for the case Im k > 0, we readily obtain that ∞ " (3.104) ψ(R) = ψ(R) = g G (k; R, R )xψ (R ) dR . 0
It requires a further argument to obtain the corresponding result for the case Im k = 0, since we can only write a priori that: ∞ ψ(R) = g G (k; R, R )xψ (R ) dR + ρkRj (kR), (3.105) 0
where ρ denotes a constant factor. Now, we can again deduce from Lemma 13 " (applied to the function ψ(R)) that ψ(R) admits an asymptotic behaviour of the following form [given by (3.71)]: (1) (3.106) lim ψ(R) − ρ kRj (kR) + igb Rh (kR) = 0. R→∞
The proof that both constants ρ and b also vanish in this case relies on the fact (1) that ψ(R) is assumed to satisfy property (iii). In fact, the functions kRh (kR)
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and kRj (kR) behave at infinity as exp{i[kR −(+1) π2 ]}, respectively, [in view π of (A.54)] and π2 cos[kR ∞− ( + 21) 2 ] (see, e.g., [28]), and therefore the finiteness of the quantity 0 |ψ(R)| dR is consistent with (3.106) if and only if ρ = b = 0. To conclude, we have obtained that Eq. (3.104) is valid for all cases (Im k ≥ 0). First, this implies that the function xψ is non-zero and, moreover, by applying to both sides of Eq. (3.104) the integral operator associated with V (R, R ), one gets in view of (3.10c) (in operator form): xψ = V ψ = gV G (k)xψ = gL (k)xψ .
(3.107)
So, by starting from the assumptions of point (d), we have been able to derive Eqs. (3.104) and (3.107), namely, we have reproduced the basic assumptions (3.93) and (3.94) of (a) for all cases Im k ≥ 0, which ends the proof of the theorem. Remark 1. Had property (iii) not been imposed on ψ in the assumptions of point (d), one would have obtained from the general form (3.105) (by applying the operator V to both sides of the latter and also accounting for (3.10b)): ∞ xψ (R) = g L (k; R, R ) xψ (R ) dR + ρv,0 (k; R). (3.108) 0
In particular, the latter form is relevant for ρ = 1, since it coincides with . Eq. (3.10a), whose solution xψ (R) = v (k, g; R) will be used below for describing the scattering solution of Eq. (3.6) (see Theorem 18 below). 3.3.3. Inhomogeneous Integral Equation and Scattering Solutions: the Partial Scattering Amplitude T (k; g). Theorem 17. Being given any potential V in a class Nw(ε),α and any given complex number g, let Ωα, (g) be the set of all points k ∈ Ωα such that the corresponding Fredholm–Smithies denominator σ (k, g) of the resolvent R (k; g) does not vanish. Then the inhomogeneous integral equation [1 − gL (k)]v (k, g; ·) = v,0 (k; ·)
( = 0, 1, 2, . . .),
(3.109)
admits for every g ∈ C and k ∈ Ωα, (g) a unique solution v (k, g; R) in Xw(ε),α , which is well-defined by the formula v (k, g; ·) = R (k; g)v,0 (k; ·).
(3.110)
Furthermore, the function (k, g) → v (k, g; ·) is meromorphic in Ωα × C, and for any g in C, the function k → v (k, g; ·) is holomorphic in Ωα, (g), in the sense of the vector-valued functions taking their values in Xw(ε),α . Proof. The solution (3.110) of (3.109), which follows from (3.37), defines v (k, g; ·) as an element of Xw(ε),α in view of the fact that v,0 (k; ·) belongs to Xw(ε),α (see Lemma 5, more precisely Corollary 5–6) and that R (k; g) is a bounded operator in Xw(ε),α (see Theorem 10). Moreover, the meromorphy properties in (k, g) and the holomorphy properties in k at fixed g of R (k; g) and v,0 (k; ·), established respectively in Theorem 10 and Lemma 5, imply the corresponding properties for v (k, g; ·) in view of Lemma B.3 (ii).
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Let us now define for every g ∈ C and k in Ωα, (g) the following functions: Ψ (k, g; R) = kRj (kR) + Φ (k, g; R), +∞ Φ (k, g; R) = g G (k; R, R )v (k, g; R ) dR .
(3.111) (3.112)
0
Since v (k, g; ·) ∈ Xw(ε),α , the function Φ (k, g; ·) is well-defined in view of Lemma 13, and we have: D,k Ψ (k, g; R) = D,k Φ (k, g; R) = gv (k, g; R) = g 2 [L (k)v (k, g; ·)](R) + gv,0 (k; R).
(3.113)
But, by taking Eqs. (3.10c) and (3.112) into account (and interchanging convergent integrals) along with Eq. (3.10b) (and Lemma 5), the r.h.s. of (3.113) can be rewritten as follows: ∞ ∞ g V (R, R )Φ (k, g; R ) dR + g V (R, R )kR j (kR ) dR , 0
0
so that [in view of Eqs. (3.111), (3.113)], one has: ∞ D,k Ψ (k, g; R) = g
V (R, R )Ψ (k, g; R ) dR .
(3.114)
0
We then conclude that Eq. (3.111) reproduces the form (3.8) of the scattering solution of the Schr¨ odinger-type equation (3.6). We shall now see that Lemma 13 not only allows one to obtain the properties of this solution for k and g real, which have been listed under (S-b) and include in particular the Sommerfeld radiation condition [given in (3.9)], but also implies extensions of these properties to the complex domain Ωα, (g) of the k-plane (for each g ∈ C). Moreover, one will show that there also hold meromorphy properties of this solution with respect to (k, g) in the domain Ωα × C. This will be the scope of the following two theorems. Theorem 18. For every g ∈ C, k ∈ Ωα, (g) and for any non-negative integer , there exists a solution of Eq. (3.6) which is of the form Ψ (k, g; R) = kRj (kR) + Φ (k, g; R), and satisfies the following properties: d dR Ψ (k, g; R)
is absolutely continuous for all R and bounded for R tending to zero; (ii) the function Φ (k, g; ·) is expressed in terms of the solution v (k, g; ·) of the integral equation (3.109) by the formula (3.112), and satisfies a bound of the following form: (i)
c,ε v (k, g; ·)w(ε),α Re−R Im k , |Φ (k, g; R)| ≤ |g|"
(3.115)
in particular one has: Φ (k, g; 0) = 0;
(3.116)
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(iii) for 0 ≤ Im k < α, there holds the following limit: d R Im k Φ (k, g; R) − ikΦ (k, g; R) = 0; lim e R→+∞ dR
697
(3.117)
(iv) for − α2 < Im k < α, there holds the following majorization (containing a suitable constant cˆ (k, g)): (1) Φ (k, g; R) − T (k; g)ikRh (kR) ≤ cˆ (k, g)e−αR max 1, e−2(Im k)R , (3.118) where: +∞ T (k; g) = −g Rj (kR)v (k, g; R) dR,
(3.119)
0
is well-defined as an analytic function of k in the domain Ωα, (g); (v) for k ∈ R+ and g ∈ R, the function Ψ (k, g; R) satisfies all the properties listed under (S-b) of the scattering solution of Eq. (3.6), and the corresponding function T (k; g), which is the physical partial scattering amplitude, can also be defined as Φ (k, g; R) (3.120) T (k; g) = lim . R→+∞ ikRh(1) (kR)
Proof. By applying Lemma 13 with x(R) = v (k, g; R) ∈ Xw(ε),α , ψx;,k (R) = Φ (k, g; R) and bx;,k = − kg T (k; g), one readily obtains property (i) (given by (c)) and properties (ii), (iii), (iv) [since formulae (3.115), (3.117), (3.118), and (3.119), correspond respectively to [(3.69), (3.70)], (3.76), [(3.73), (3.74)] and (3.72)]. Property (v) is directly obtained by inspection, the limit (3.120) being also directly implied by (3.118) in view of the asymptotic behaviour (A.54) (1) of Rh (kR) for R tending to infinity. The proof of the analyticity property of the function k → T (k; g) in the domain Ωα, (g) is left to the following theorem. Now, by exploiting the holomorphy properties of σ (k; g) and N (k; g), obtained respectively in Theorems 7 and 8, we can derive meromorphy properties in the complex variables (k, g) of the scattering solution Ψ (k, g; R) and of the partial scattering amplitude T (k; g). For this purpose, by taking into account Eq. (3.38), we can now re-express v (k, g; R) as follows, for all k ∈ Ωα, and g ∈ C: v (k, g; ·) =
u (k, g; ·) , σ (k; g)
(3.121)
where: . u (k, g; ·) = [σ (k; g) + gN (k; g)] v,0 (k; ·).
(3.122)
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We then have: Theorem 19.
(i) T (k; g) can be written as follows:
T (k; g) = −
g σ (k; g)
+∞ Θ (k; g) , (3.123) R j (kR )u (k, g; R ) dR = σ (k; g) 0
where: +∞ Θ (k; g) = −gσ (k; g) R j (kR )v,0 (k; R ) dR 0 +∞ +∞ − g2 R j (kR ) dR N (k; g; R , R)v,0 (k; R) dR. 0
(3.124)
0
The following properties hold for any non-negative integral value of : (i.a) the function k → σ (k; g) is defined and uniformly bounded in Πα , holomorphic in Πα ; (i.b) the function k → Θ (k; g) is defined and uniformly bounded in Ωα , holomorphic in Ωα ; (i.c) k → T (k; g) is a meromorphic function in Ωα . (ii) The function k → S (k; g), given by S (k; g) = 1 + 2iT (k; g),
(3.125)
is meromorphic in Ωα . It satisfies the condition of elastic unitarity for k real and can be written as follows: S (k; g) = e2iδ (k;g)
( = 0, 1, 2, . . .),
(3.126)
where δ (k; g) is a real-valued function of k on R+ (which is defined modulo π). Accordingly, the following representation of T (k; g) holds: T (k; g) = eiδ (k;g) sin δ (k; g)
(k ∈ R+ ).
(3.127)
Proof. (i) Formulae (3.123), (3.124) obviously follow from (3.119), (3.121), and (3.122). Statement (i.a) has been proved in Theorem 7(a). Proof of (i.b). For each k ∈ Πα , N (k; g) acts on Xw(ε),α as a bounded (and even Hilbert–Schmidt) operator and, in view of Theorem 8(a), the function k → N (k; g) is holomorphic in Πα as a bounded-operator-valued function (since it is holomorphic as a HS-operator-valued function). Since v,0 (k; ·) is holomorphic in Ωα as a function with values in Xw(ε),α (see Lemma 5 and Corol(N )
lary 5–6), it then follows from Lemma B.3(ii) that the function k → u defined by (N ) u (k, g; R)
+∞ = N (k; g; R, R )v,0 (k; R ) dR, 0
(k; ·),
(3.128)
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is also holomorphic in Ωα as a function with values in Xw(ε),α . In view of Eq. (3.124) and of (i.a), proving the analyticity of Θ (k; g) in Ωα amounts to proving that the functions of k, defined by the integrals +∞ Rj (kR)v,0 (k; R) dR,
and
0
+∞ (N ) Rj (kR)u (k, g; R) dR,
(3.129)
0
are holomorphic in Ωα . But this follows from Lemma B.4(ii) by noting that: (a)
the function R → Rj (kR) takes its values in the dual space Xw∗ (ε),α of Xw(ε),α , for all k ∈ Ωα ; in fact [in view of (A.42) and (3.25)] the following bound holds: '2 ∞ |Rj (kR)|2 e−2αR & ∗ ·j (k·)w(ε),α = dR Fork ∈ Ωα , R1−ε (1 + R)1+2ε ≤
(b)
c2 |k|2
∞
0
|k|R c2 dR ≤ A2ε ; 1+2ε + R) |k|
R1−ε (1 0
(3.130)
the holomorphic function k → Rj (kR) satisfies (for k ∈ Ωα ) the conditions of Lemma B.8(ii), and therefore defines a holomorphic vector-valued function of k in Ωα taking its values in Xw∗ (ε),α . |v
(k,·)|
is uniformly bounded in Xw(ε),α for k ∈ Ωα (see Moreover, since ,0 |k|1/2 Corollary 5–6), and since N (k; g)HS is uniformly bounded (at fixed g) in (N )
Πα (see Theorem 8), it follows from (3.128) that
|u
(k,g;·)| |k|1/2
is also uniformly
bounded in Xw(ε),α for k ∈ Ωα . Then, it results from the uniform bound (3.130) in Xw∗ (ε),α that the functions defined by the two integrals in (3.129), which are ∗ ∗ respectively bounded by ·j (k·)w(ε),α × v,0 (k, ·)w(ε),α and ·j (k·)w(ε),α × (N )
u
(k; ·)
w(ε),α
, are uniformly bounded for k ∈ Ωα . In view of (3.124) and
(3.128) (and of (i.a)), this implies that the functions Θ (k; g) are uniformly bounded for k ∈ Ωα , which ends the proof of (i.b). Statement (i.c) obviously follows from (3.123) and statements (i.a) and (i.b). (ii) From (3.125) and (i.c) it follows that S (k; g) is a meromorphic function of k for k ∈ Ωα . The condition of elastic unitarity S (k; g)S (k, g) = 1 follows from the unitarity of the scattering operator S proved in [20] in a very general setting (see Sect. 2). This implies the representations (3.126) and (3.127). 3.3.4. Complement on Spurious Bound States: the Corresponding Properties of Scattering Solutions, Partial Scattering Amplitudes and Phase-Shifts. The phenomenon of the “bound states embedded in the continuum” (or “positiveenergy bound states” or “spurious bound states”) traces back to a classical paper of Wigner and von Neumann [29]. It can be qualitatively explained as follows: bumps in a potential will reflect a wave, and well-arranged bumps can
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act constructively and prevent a wave from reaching infinity: i.e., stationary waves with falloff can be formed [25]. For example, a bound state embedded in the continuum seems to appear in the negative helium ion. The level 4 P5/2 of this system lies in fact in the continuum, and it is not liable to auto-ionization. An analogous state seems to be present in the helium atom (both of these examples have been indicated by Wigner to the author of Ref. [30], as written in a footnote of that paper). For the class of nonlocal potentials considered here, we can say that the partial scattering amplitude T (k; g), and therefore also the phase-shift δ (k; g), remain well-defined at all the values of k corresponding to spurious bound states. This is a consequence of the previous theorem, namely of the meromorphy property of T (k; g) in Ωα and of its boundedness for all real k, which is implied by Eq. (3.127) (expressing the unitarity condition). Putting these two properties together entails that T (k; g) is finite and holomorphic at all real values of k, and therefore in particular at those for which σ (k; g) = 0, corresponding to the occurrence of spurious bound states (note that it is thus necessary that any such value of k be also a zero of the holomorphic function Θ (k; g) introduced in Theorem 19). We are now going to show that not only the partial scattering amplitude T (k; g) but also the scattering solution Ψ (k, g; R) remains finite at all pairs (k, g) which correspond to spurious bound states. More precisely we can state: Theorem 20. Let g = g"(k) be any solution of the equation σ (k; g) = 0 considered as an analytic curve C in a complex neighborhood N0 in C2 of a certain real point (k0 , g0 ), such that g0 = g"(k0 ) with k0 ∈ R+ . Let us also assume that the curve C is associated with a “simple pole” of the Fredholm resolvent kernel (tr) R (k; g). Then the following properties are valid: (i) For (k, g) ∈ N0 , there exists a decomposition of the following form of (tr) R (k; g): R (k; g; R, R ) = (tr)
p (k; R, R ) "(tr) (k; g; R, R ), +R g"(k) − g
(3.131)
"w(ε),α , where p (k; R, R ) is (for each k) a kernel of finite rank r in X which depends holomorphically on k in a suitable neighborhood V0 of k0 , and satisfies the functional relation +∞ p (k; R, R )p (k; R , R ) dR = p (k; R, R ),
(3.132)
0
"(tr) (k; g) is uniformly bounded in X "w(ε),α , for (k, g) ∈ while the kernel R N0 ; in this open set, it defines a HS-operator-valued holomorphic function "w(ε),α . of (k, g) taking its values in X
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(ii) The following equalities hold for all k in V0 : +∞ Rj (kR)p (k; R, R ) dR = 0,
+∞ p (k; R, R )v,0 (k; R ) dR = 0.
0
0
(3.133) (iii) The solution v (k, g; R) of the inhomogeneous equation (3.109), which is well-defined as an element of Xw(ε),α by the formula v (k, g; ·) = R (k; g)v,0 (k; ·) for σ(k; g) = 0, tends to a finite limit v (k, g"(k); ·) in Xw(ε),α , when g tends to g"(k) and the vector-valued function (k, g) → v (k, g; ·) is holomorphic in N0 . (iv) The function Φ (k, g; R) defined in Eq. (3.112), the scattering solution Ψ (k, g; R) = kRj (kR) + Φ (k, g; R), and the partial scattering amplitude T (k; g), defined in (3.119), remain finite and satisfy all the properties listed in Theorem 18 at all points (k, g) ∈ N0 . . Proof. (i) For convenience, we will choose N0 = {(k, g) : k ∈ V0 ; |g − g"(k)| < a}, for a suitable choice of V0 containing k0 > 0, and of a > 0. For any k fixed in V0 , the existence of a decomposition of the form (3.131) for the Fredholm resolvent kernel of L(k) is a standard result (see, e.g., [31]). A simple presentation given in Subsection III-3 of [32] (see, in particular, formulae (77) through (83), Lemma 1 of the latter, which we here consider for the simple-pole case n = 1) allows one to write the following formula for p (k; R, R ): 1 (tr) p (k; R, R ) = − R (k; g; R, R ) dg, (3.134) 2πi γ
where γ denotes a closed (anticlockwise) contour surrounding the point g"(k) inside the disk |g − g"(k)| < a of the g-plane (note that for V0 sufficiently small, γ may be considered as independent of k). This integral "w(ε),α , since the function is meaningful in the sense of HS-kernels in X
(k;g) g → R (k; g) = N σ (k;g) is holomorphic in the complement of C as a HS-operator-valued function (see Theorem 8 and the last page of Appendix B). Moreover, in view of the fact that the function k → (tr) R (k; g; R, R ) is holomorphic in V0 for all g ∈ γ (as a result of Theorem 9), it follows that k → p (k; R, R ) is itself a HS-operator-valued holomorphic function (of finite rank r) in V0 . The projector formula (3.132) (true for all k ∈ V0 ) refers to formula (80) of [32]. Finally, in view of (3.131) and (3.134), one also has (for k ∈ V0 ): "(tr) (k; g; R, R ) dg = 0, R (3.135)
(tr)
γ
which proves (also in view of Theorem 9) that the function (k, g) → "(tr) (k; g; R, R ) is holomorphic in N0 as a HS-operator-valued function. R
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(ii) Since (see [31,32]) the kernel p (k0 ) of rank r is such that p (k0 ) = g0 L (k0 )p (k0 ) = g0 p (k0 )L (k0 ),
(3.136)
"w(ε),α , there exist r linearly independent solutions and since p (k0 ) ∈ X (j) x (R) ∈ Xw(ε),α of the homogeneous Fredholm equation g0 L (k0 )x = x, and r linearly independent solutions ψ ∗(j) (R) ∈ Xw∗ (ε),α of the corresponding equation g0 ψ ∗ L (k0 ) = ψ ∗ such that: p (k0 ; R, R ) =
r
x(j) (R)ψ ∗(j) (R ).
(3.137)
j=1
But we are in the case when formula (3.99) of Theorem 16 applies, namely +∞ we have 0 Rj (k0 R) x(j) (R) dR = 0, for 1 ≤ j ≤ r, which entails the +∞ orthogonality relation 0 Rj (k0 R)p (k0 ; R, R ) dR = 0. . Let us now associate with ψ ∗(j) (1 ≤ j ≤ r) the function x∗(j) = V ψ ∗(j) , which belongs to Xw(ε),α since ψ ∗(j) ∈ Xw∗ (ε),α [see Lemma 12 (c)]. We then have: ψ ∗(j) = g0 ψ ∗(j) L (k0 ) = g0 ψ ∗(j) V G (k0 ) [in view of (3.10c)], which also yields: V ψ ∗(j) = g0 V [G (k0 )V ψ ∗(j) ] = g0 L (k0 )[V ψ ∗(j) ], (by using the symmetry relations (3.10e) and (3.7)), namely x∗(j) = g0 L (k0 )x∗(j) . It then follows from Lemma 12 (c) and Eq. (3.99) (applied to the eigenfunction x∗(j) of L (k0 )) that: +∞ ∗(j) +∞ ψ (R)v,0 (k; R) dR = 0 Rj (k0 R) x∗(j) (R) dR = 0, for 1 ≤ j ≤ 0 +∞ r, which entails therefore the orthogonality relation 0 p (k0 ; R, R )v,0 (k; R) dR = 0. The previous argument can of course be applied to any real neighbouring point (k, g) of (k0 , g0 ) on the curve C, thus implying that relations (3.133) hold for all k in V0 ∩ R+ . Therefore, they also hold in V0 by the principle of analytic continuation. (iii) In view of Eqs. (3.131) and (3.133), one has for all (k, g) ∈ N0 \C: (tr)
v (k, g; ·) = v,0 (k; ·) + R (k; g)v,0 (k; ·) "(tr) (k; g)v,0 (k; ·). = v,0 (k; ·) + R
(3.138)
"(tr) (k; g) is holomorphic at fixed g for But, since the function k → R "w(ε),α , (k, g) ∈ N0 as a HS-operator-valued function taking its values in X it follows from Lemmas 5 and B.3(ii) (as in the proof of Theorem 17) that the r.h.s. of Eq. (3.138) defines the analytic continuation of the function k → v (k, g; ·) at fixed g from Ω+ α, (g) to the set {k : (k, g) ∈ N0 } (as a vector-valued function taking its values in Xw(ε),α ). (iv) In view of the properties of v (k, g; ·) established in (iii), the finiteness and holomorphy properties of Φ (k, g), Ψ (k, g), and T (k; g), and all the bounds and asymptotic limits of the latter, which have been established in Theorem 18, are directly extended to the set N0 , since the arguments given in the proof of that theorem remain valid there without modification.
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Remark 2. In view of Theorem 16, one can say that at each pair (k, g = g"(k)), k > 0, corresponding to a spurious bound state, there exists a finitedimensional affine subspace of functions [v ]μ (k, g; ·) ∈ Xw(ε),α of the form r . [v ]μ (k, g; R) = v (k, g; R)+ j=1 μj x(j) (R), (μ = (μ1 , . . . , μr ) ∈ Rr ), which all are solutions of the inhomogeneous equation (3.109), while the corresponding affine subspace of functions [Ψ ]μ (k, g; R) = kRj (kR) + [Φ ]μ (k, g; R), where +∞ we have put [Φ ]μ (k, g; R) = g 0 G (k; R, R )[v ]μ (k, g; R ) dR , satisfy all the properties of scattering solutions of the Schr¨ odinger-type equation (3.6). However, in view of Eqs. (3.119) and (3.99), all these solutions give rise to the +∞ (unique) scattering amplitude T (k; g) = −g 0 R j (kR )[v ]μ (k, g; R ) dR . Moreover, we can say that the particular solutions v (k, g"(k); ·) and Ψ (k, g"(k); ·) are distinguished from all the others by their property of being the restrictions to the curve C(g = g"(k)) of the (respective) solutions v (k, g; ·) and Ψ (k, g; ·), which depend holomorphically on g and k in a complex neighborhood of C in C2 . We shall now complete this study of bound states embedded in the continuum by recalling the following two interesting properties, which concern the behavior of the phase-shift δ (k; g) for k varying between zero and infinity. (a) The following proposition was proved by Gourdin, Martin and Chadan and quoted in [33] for the case of separable nonlocal potentials. Proposition 21 (G.M.C. [33]). (i) Positive energy bound states correspond to those energies at which the phase-shift crosses a value of nπ, n = 0, ±1, . . ., downward, (i.e., with a negative slope) and conversely. (ii) The phase-shift never crosses nπ upward. (iii) The phase-shift may become tangent to nπ either from below or from above. (b) An extension of Levinson’s theorem has been proved, in the case of nonlocal separable potentials, in Ref. [34]; see also Ref. [18], where Levinson’s theorem has been proved, in the case = 0, for a class of potentials very close to that considered here. The extension to any integral value of is straightforward. One can then state that the total variation of the phase-shift in the interval 0 ≤ k < +∞ is given by: δ (0) − δ (∞) = π(N + N ) ( integer),
(3.139) N
is the where N is the number of negative energy bound states and number of positive energy bound states; for simplicity, one assumes that all the bound states are represented by simple poles, and that there are no bound states at k = 0. 3.4. Partial Wave Expansion of the Scattering Amplitude The absence of bound states for sufficiently large was already noted earlier as being a corollary of property (c) of σ (k; g) stated in Theorem 7. It can also
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be presented in a simpler and more precise way as follows. In view of the complement of Theorem 6, the norm L (k) of L (k), considered as a bounded operator on Xw(ε),α , satisfies the majorization (3.36) for all k in Πα : √ π 2 C(V ) A . (3.140) L (k) ≤ L (k)HS ≤ 2 ε 2 + 1 Let = 0 (g) be the smallest integer such that the r.h.s. of (3.140) is major√ . ) ized by 1/|g| and let κg = |g| 2π A2ε 2C(V < 1. Then for all integers such 0 (g)+1 that ≥ 0 (g), the operator gL (k) is a contraction in Xw(ε),α and therefore there exist no bound state or spurious bound state and even no resonance in the strip −α ≤ Im k < 0. Next we prove the following proposition. Proposition 22. For k real and positive, the asymptotic behavior of the partial scattering amplitude T (k; g), for large values of , is governed by the following ˆ g)): majorization (including a suitable constant C(k, |T (k; g)| ≤
ˆ g) Γ( + 1) C(k, −β(k) = O −1 e−β(k) , 1 3 e (2 + 1) 2 Γ( + 2 ) with
cosh β(k) = 1 +
2α2 . k2
(3.141)
Proof. In view of Theorem 17, we know that, for k ∈ Ωα, (g), v (k, g; ·) belongs to Xw(ε),α , and we therefore have in view of (3.119): +∞ |T (k; g)| = Rj (kR)v (k, g; R) dR |g| 0
v (k, g; ·)w(ε),α ∗ ≤ k·j (k·)w(ε),α . |k|
(3.142)
But, for ≥ 0 (g), Ωα, (g) contains the whole real set k > 0, and from Eq. (3.10a) and bound (3.140) we can also write: √ π |g|A2ε C(V ) v (k, g; ·)w(ε),α , v (k, g; ·)w(ε),α ≤ v,0 (k; ·)w(ε),α + 2 2 + 1 (3.143) which yields: v (k, g; ·)w(ε),α ≤ (1 − κg )−1 v,0 (k; ·)w(ε),α .
(3.144)
So, for all k > 0 and ≥ 0 (g), we have [in view of (3.142) and (3.144)]: |T (k; g)| ≤
|g| ∗ (1 − κg )−1 v,0 (k; ·)w(ε),α k·j (k·)w(ε),α , k
and therefore, by applying Lemma 5 [formula (3.23)] for w = w(ε) : '2 C(V ) & |g| ∗ k·j |T (k; g)| ≤ (1 − κg )−1 (k·) . w(ε),α k (2 + 1)1/2
(3.145)
(3.146)
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∗
We shall now introduce a majorization of k·j (k·)w(ε),α , which is uniform with respect to all values of ; for this purpose, we shall use the following equality (valid for all and k > 0): +∞ 1 2α2 −2αR 2 Q 1 + 2 , e J+1/2 (kR) dR = (3.147) πk k 0
Q (·) denoting the second kind Legendre function (see [35]). In fact, we can write, in view of Eq. (A.45): &
k·
∗ j (k·)w(ε),α
'2
. =
+∞ e−2αR
|kRj (kR)|2 1−ε R (1 + R)1+2ε
dR
0 +∞ πk Rε ≤ e−2αR |J+1/2 (kR)|2 dR, 2 (1 + R)1+2ε
(3.148)
0
and therefore, by making use of (3.147) (since J+1/2 (kR) is real-valued for k > 0): & '2 1 2α2 ∗ k·j (k·)w(ε),α ≤ Q 1 + 2 . (3.149) 2 k In view of the latter, majorization (3.146) then yields: C(V ) |g| 2α2 (1 − κg )−1 Q |T (k; g)| ≤ 1 + . (3.150) 2k k2 (2 + 1)1/2 We now use the following uniform bound on the second-kind Legendre functions [19] and [36]: √ Γ( + 1) −β(+1) |Q (cosh β)| ≤ π (1 − e−2β )−1/2 (β > 0). (3.151) e Γ( + 32 ) By taking the latter into account at the r.h.s. of Eq. (3.150), one then readily obtains the√majorization (3.141), with the following definition of the constant: e−β(k) ˆ g) = π|g| (1 − κg )−1 C(V ) C(k, 1 . 2k 1−e ( −2β(k) ) 2 We can now introduce the so-called partial waves: e2iδ (k;g) − 1 T (k; g) = . (3.152) k 2ik Then the total scattering amplitude F (k, cos θ; g) can be expanded as a Fourier-Legendre series in terms of the a (k; g), usually called the partial wave expansion: ∞ (2 + 1)a (k; g)P (cos θ), (3.153) F (k, cos θ; g) = a (k; g) =
=0
where the P (·) are the Legendre polynomials, and θ is the scattering angle in the center of mass system. We have thus obtained results very similar to those which hold for the class of local Yukawian potentials. In particular, in view of
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(3.141), expansion (3.153) converges in an ellipse Eβ(k) with foci ±1 and half major axis cosh β(k) = (1 + 2α2 /k 2 ). This convergent expansion defines the total scattering amplitude F (k, cos θ; g) as a holomorphic function of cos θ in the ellipse Eβ(k) for each positive value of k. Finally, we study the behaviour of the partial scattering amplitude T (k; g) for small and large values of k. We prove the following proposition. Proposition 23. (i) For any real g and any integer ( ≥ 0), the asymptotic behaviour of the partial scattering amplitude T (k; g) for k tending to infinity in Ωα is such that 1 T (k; g) = T" (k; g), (3.154) k where T" is bounded in k at infinity. (ii) Considering the generic case of values of g and such that σ (0; g) = 0, the corresponding threshold behaviour (for k → 0) of the partial scattering amplitude T (k; g) is such that T (k; g) = k 2+1 T"
(loc)
where
(loc) T"
(k; g),
(3.155)
is bounded in k in a complex neighbourhood of k = 0.
Proof. In view of Eqs. (3.123), (3.124), and of the corresponding norm inequalities that have been used in the proof of Theorem 19, we can write: |g| |g| ∗ N (k; g)HS k· j (k·)w(ε),α v,0 (k; ·)w(ε),α , |T (k; g)| ≤ 1+ |k| |σ (k; g)| (3.156) and therefore, in view of Lemma 5 [Eq. (3.23)] and of (3.57): '2 C(V ) & |g| Ψ(|g|L (k)HS ) ∗ k·j |T (k; g)| ≤ (k·) . 1+ (ε) w ,α |k| |σ (k; g)| (2 + 1)1/2 (3.157) (i) For all k ∈ Ωα , we can apply the global bounds (3.36) and (3.22) with Aw(ε) (k) = Aε [see Eq. (3.25)], which allow one to replace the majorization (3.157) by ⎤ ⎡ 1+π −1 2 Ψ |k| |g| C(V )A 1/2 ε |g| ⎣ (2+1) ⎦ C(V ) c2 A2ε . 1+ |T (k; g)| ≤ |k| |σ (k; g)| (2 + 1)1/2 (3.158) Now, we know that, for k tending to infinity, one has Ψ(c/|k|) = O(1/|k|) [in view of (3.56)] and, moreover, |σ (k; g)| tends uniformly to 1 [see Theorem 7 (c)]. Then, majorization (3.158) implies: 1 C(V ) |g| c2 A2 , (3.159) |T (k; g)| ≤ 1+O |k| |k| (2 + 1)1/2 ε which directly yields the asymptotic behaviour (3.154).
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(ii) We now consider a (real) value g0 of g such that σ (0; g0 ) = 0; moreover, we know from Theorem 7 that σ (k; g0 ) is holomorphic as a function of k in Πα . It follows that there exists a neighborhood of k = 0, say Sη = {k : |k| ≤ η}, and a constant M such that |σ (k; g0 )|−1 ≤ M for all k in Sη . Here we shall make use of a majorization on the norm of the function kRj (kR) in Xw∗ (ε),α , which is different from those obtained in (3.22) and (3.148). For this purpose, we shall use the bound (A.42), which yields for all R ≥ 0 and k ∈ Sη : |kRj (kR)| ≤ c |k|+1 R+1 eηR . If η is chosen such that η < α, the latter implies the following majorization: ⎡ +∞ ⎤1/2 2+1+ε −2(α−η)R R e ∗ ⎦ k·j (k·)w(ε),α ≤ c |k|+1 ⎣ =" c (α, η, ε)|k|+1 , (1 + R)1+2ε 0
(3.160) where " c is a new constant (also depending of α, η, ε). By taking (3.160) into account and also the k-independent option of the global bound (3.36) on L (k)HS , the majorization (3.157) can then be replaced by the following one, valid for all k in Sη : C(V ) |g| π 1/2 C(V ) 2 Aε " c2 |k|2+2 , |T (k; g)| ≤ 1 + M Ψ |g| |k| 2 2 + 1 (2 + 1)1/2 (3.161) which directly yields the threshold behaviour (3.155).
Remark 3. In view of formula (3.125), it follows from the previous propositions that S (k) → 1 both for k tending to zero and for |k| tending to infinity. Condition (ii) also implies that all the partial waves a (k) = T (k, g)/k are finite at k = 0. Finally, we can also re-express the statement (i) of Proposition 23 in terms of the physically interesting quantities δ [namely, the phase-shifts defined by Eq. (3.126)] as the following Corollary 24. For any real g and any integer ( ≥ 0), the limit of the phaseshift δ (k; g) for k tending to infinity in Ωα exists and is equal to zero (mod. π).
4. Nonlocal Potentials with Interpolation in the CAM-Plane and Analyticity Properties of the Partial Scattering Amplitude 4.1. Interpolation of the Partial Potentials V in the CAM-Plane: Classes γ Nw (ε),α of Nonlocal Potentials Let us recall that the complex angular momentum (CAM) theory in potential scattering has been rigorously stated for potentials which do not depend on the angular momentum, namely for a large class of local potentials, including the
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Yukawian potentials. For this class of potentials one can show the existence of a distinguished meromorphic interpolation T (λ, k) of the sequence of partial waves T (k) in the half-plane C+ (Re λ > − 12 ), which allows for performing a − 12 Watson resummation of the partial wave expansion. Concerning the notion of + analytic interpolation of a sequence {f }∞ =0 in the half-plane C− 1 , namely the 2
existence of a function F (λ), holomorphic in C+ and such that F () = f for − 12 all integers ≥ 0, it is worthwhile to recall the important notion of Carlsonian interpolation. An analytic interpolation F is said to be Carlsonian if it satisfies (which of a global bound of the form |F (λ)| ≤ Ce(π−ε)|λ| , with ε > 0, in C+ − 12 course requires that the given sequence {f } itself satisfies such inequalities). Carlson’s theorem [37] then asserts that F (λ) is the unique Carlsonian interpolation of the sequence {f }; in fact, any other analytic interpolation of this sequence, such as the one obtained by adding to F the non-Carlsonian function sin πλ, is a non-Carlsonian interpolation. It is a remarkable feature of the theory of Yukawian potentials (and of a more general class of local potentials V (R) enjoying suitable analyticity properties in R) that all the relevant CAM analytic interpolations of the angular momentum formalism can be performed in the sense of Carlsonian interpolations. In this connection, let us also mention that the introduction of so-called “exchange potentials” leads one to split the set of partial waves into two separate subsets, namely the even- partial waves, and the odd- partial waves. When it is attempted to perform the CAM interpolation of the partial waves, one is faced with the problem of handling the factor (−1) = cos π; indeed, the function cos πλ, whose restriction to integers gives (−1) , is not a Carlsonian function. Therefore, in that case, one is led to perform Carlsonian analytic interpolations of the two sets (even- and odd-) separately. Let us also note that such a separation was also found to occur at a more fundamental level in our approach of complex angular momentum analysis in the general framework of relativistic Quantum Field Theory, with the notion of Bethe–Salpeter kernel playing the role of a generalized nonlocal potential (see [15]). This separation is also exhibited in a striking way when one considers the scattering of identical particles, since in view of the symmetrization (resp., antisymmetrization) properties of the boson (resp., fermion) quantum description, one can only obtain an analytic interpolation of even- waves with all the odd- waves identically equal to zero for the case of bosons, and the converse for the case of fermions. Here we wish to exhibit the mechanism of analytic interpolation of the partial waves in the framework of nonlocal rotationally invariant potentials belonging to appropriate subclasses of the classes considered previously in Sect. 3: in view of their expansion in partial potentials V , they can also be called (with a small abuse of language) “angular-momentum-dependent potentials”. The first step in order to introduce such a relevant subclass of nonlocal potentials consists in finding suitable conditions which allow one to perform the analytic interpolation of the partial potentials V (R, R ) [see Eqs. (3.2), (3.3)] from integral -values to complex λ-values in a specific domain of the CAM-plane, containing the real positive semi-axis. For the sake of simplicity
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and without loss of generality, we only consider the case of an interpolation with respect to the set of all integers ; the case of separate interpolations with respect to the subsets of even integers and of odd integers can be treated similarly. As we shall now explain it below, the existence of a Carlsonian analytic interpolation in λ for the sequence {V (R, R )}∞ =0 can be established either from appropriate bounds to be satisfied by multiple differences of the sequential elements V themselves, or from appropriate analyticity and increase properties of the complete potential V (R, R ) = V (R, R ; cos η) with respect to the complexified angular variable cos η. (i) Hausdorff-type bounds on the V : Let us suppose that the partial potentials V (R, R ) ( = 0, 1, 2, . . .) form, for arbitrary values of R and R , a sequence of real numbers which are constrained as follows. Denote by Δ the difference operator: ΔV (·, ·) = V+1 (·, ·) − V (·, ·); we thus have: k k Δk V (·, ·) = Δ × Δ × · · · × Δ V (·, ·) = (−1)m V+k−m (·, ·), +, * m m=0 k times
(for every k ≥ 0); Δ0 is the identity operator, by definition. Now, let us suppose that the sequence {V (·, ·)}∞ =0 is constrained by the following Hausdorff-type bound: (2+) i (1+) Δ V(−i) (·, ·)(2+) < M ( + 1) i i=0 ( = 0, 1, 2, . . . ; > 0),
(4.1)
where M and ε are given positive constants (ε being as small as wanted). It can be proved [38] that condition (4.1) is necessary and sufficient to represent the sequence {V (·, ·)}∞ =0 as: 1 V (·, ·) =
x u(x; ·, ·) dx
( = 0, 1, 2, . . .),
(4.2)
0
where u(x; ·, ·) belongs to L(2+) (0, 1). Let us now put x = e−v into the integral on the r.h.s of (4.2); we thus obtain: +∞ e−v e−v u(e−v ; ·, ·) dv V (·, ·) =
( = 0, 1, 2, . . .).
0
The numbers V (·, ·) can be regarded as the restriction to the integers of the following Laplace transform: +∞ e−(λ+1/2)v e−v/2 u(e−v ; ·, ·) dv V (λ; ·, ·) =
1 λ ∈ C, Re λ ≥ − 2
.
0
(4.3)
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. In the latter, the function v → U (v; ·, ·) = e−v/2 u(e−v ; ·, ·) can be shown to be in L1 (0, +∞) as a consequence of the inclusion u(x; ·, ·) ∈ L(2+) (0, 1) (whatever small ε is). Therefore, for all R, R , the Laplace transform V (λ; R, R ) of U (v, R, R ), defined by (4.3), is holomorphic in the halfplane Re λ > − 12 . In view of the Riemann-Lebesgue theorem, completed by the analysis of [39, p. 125] (and also by taking the Carlson theorem into account), we can state Proposition 25. Let the partial potentials V (R, R ) ( = 0, 1, 2, . . .) of a given rotationally invariant nonlocal potential V (R, R ) satisfy Hausdorff-type conditions of the form (4.1). Then there exists a unique Carlsonian interpolation + V (λ; R, R ) of the sequence {V (R, R )}∞ =0 , which is analytic in C− 1 , contin2
uous at Re λ = − 12 and tends uniformly to zero for |λ| → ∞ inside any fixed , with δ > 0. half-plane C+ − 1 +δ 2
(ii) Analyticity and behaviour at infinity of V (R, R ; cos η) in the complex cos η-plane: Since every rotationally-invariant nonlocal potential V (R, R ) defines for every (R, R ) an invariant kernel on the sphere, R . namely a function of the scalar product R R · R = cos η, whose FourierLegendre coefficients V (R, R ) are given by (3.3), we can apply the results of [40] (see Theorem 1 of the latter) concerning the Fourier-Laplace-type transformation of holomorphic invariant kernels on cut-domains of the complexified sphere. These results can be summarized in the following Proposition 26. Let V (R, R ; ζ), initially defined for −1 ≤ ζ ≡ cos η ≤ +1, satisfy (for each (R, R ) fixed) the following conditions: (a) it admits an analytic continuation V" (R, R ; ζ) with respect to ζ in a given cut-plane Πγ = C\{ζ ∈ R : ζ ≥ γ}, where γ ≥ 0; (b) there exists a real number m, with m > −1, and a function g(v) in L1 (R+ ) such that for ζ = cos(u + iv) varying in the closure of the cut-plane Πγ (represented by the set 0 ≤ u ≤ 2π, v ≥ 0), there holds a uniform bound of the following form |V" (R, R ; cos(u + iv))| ≤ (RR )−1 V∗ (R, R )g(v)emv . Then there exists a function V (λ; R, R ), holomorphic in the complex half plane C+ m such that for all integers > m one has: V (; R, R ) = V (R, R ). ∞ Moreover, this interpolation of the sequence {V (R, R )}=0 is Carlsonian: in fact, it satisfies a global bound of the following form in C+ m (with a suitable constant K): |V (λ; R, R )| ≤ KV∗ (R, R )e−γ Re λ .
(4.4)
It also results from Theorem 3 of [40] that, conversely, if the sequence + {V (R, R )}∞ =0 admits a Carlsonian interpolation V (λ; R, R ) in Cm satisfying a global bound of the previous type, then the potential V (R, R ; cos η) admits (for each R, R ) an analytic continuation in the cut-plane Πγ of the complex variable ζ = cos(u + iv), which behaves at infinity like emv (up to a power of v).
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Remark 4. In the following, we shall consider a framework in which the majorizing potential V∗ (R, R ) satisfies an appropriate L2 -norm. The previous conditions of analyticity of V" (R, R ; cos η) in cos η and of V (λ; R, R ) in λ must then be understood to hold for a.e. (R, R ), with respect to the chosen L2 -norm. In view of these results, one can exhibit examples of such nonlocal potentials, namely those of the form V" (R, R ; cos η) = (RR )−1 V∗ (R, R )f(cos(u + iv)), where f denotes a holomorphic function in the cut-plane Πγ , which behaves at infinity like emv . As a basic explicit example of this type, for which m = −1, one can take f(cos η) of the following form: f(cos η) = (eγ − cos η)−1 , (γ > 0). Then, in view of formula (3.3), one has: V (R, R ) = 2πV∗ (R, R )f , where f is expressed in terms of the second-kind Legendre function Q (see [41, Vol. 2, p. 316, formula (17)]) by the formula: +1 f = −1
P (cos η) d(cos η) = 2Q (eγ ). eγ − cos η
(4.5)
It is known that the function λ → Qλ (eγ ) is holomorphic in the half-plane Re λ > −1, and tends to zero uniformly as e−γ(λ+1) for |λ| → ∞ in the halfplane Re λ ≥ − 12 . It therefore represents the unique Carlsonian interpolation of the sequence {Q (eγ )}∞ =0 . This example therefore illustrates in a typical way the previous proposition [including a bound of the form (4.4)]. Another example of a function V (λ; R, R ) which satisfies (4.4) is obtained in the previous approach (i) (Hausdorff-type bounds) by imposing the support condition u(e−v ; ·, ·) = 0 for v ∈ [0, γ) into the Laplace representation (4.3) of V (λ; R, R ). γ Classes Nw (ε),α of nonlocal potentials. For the purpose of the present Sect. 4, we shall introduce subclasses Nwγ(ε),α , of the previously considered classes Nw(ε),α of rotationally-invariant nonlocal potentials V (R, R ) (see Sect. 3.1), by imposing the following additional
Assumption. The Fourier–Legendre coefficients V (R, R ) of V (R, R ) admit a Carlsonian interpolation V (λ; R, R ) in the half-plane Re λ > − 21 satisfying a global bound of the form (4.4); the given number γ is supposed to be positive and the analyticity property of V with respect to λ is supposed to hold for a.e. (R, R ), according to the choice of the majorizing potential V∗ (R, R ) (see our previous remark and the Hilbertian requirement on V∗ specified below). For brevity, we shall call these potentials Carlsonian potentials with CAM-interpolation V (λ; R, R ) and rate of decrease γ. Regarding the Hilbert space on which the potentials V (λ; R, R ) are acting, we thus consider the space Xw,α defined in (3.14) with the choice (3.35) for the weight-function w, namely: w(ε) (R) = R(1−ε) (1 + R)(1+2ε) (ε > 0), which already played the main role in Sect. 3. Next, we wish to ensure the following condition on the potential (note that we use the same notation as in Sect. 3, formula (3.17) for the constant
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C(·)): for all λ with Re λ > − 21 : ⎧ +∞ ⎨ . C(V (λ; ..)) = R(1−ε) (1 + R)(1+2ε) e2αR dR ⎩ 0
⎫1/2 +∞ ⎬ 2 (R )(1−ε) (1 + R )(1+2ε) e2αR |V (λ; R, R )| dR < ∞. × ⎭ 0
(4.6) Since V (λ; R, R ) is assumed to satisfy a bound of the form (4.4), it is natural to impose the following condition on the kernel V∗ (R, R ): ⎧ +∞ ⎨ . R(1−ε) (1 + R)(1+2ε) e2αR dR C(V∗ ) = ⎩ 0
⎫1/2 +∞ ⎬ 2 (R )(1−ε) (1 + R )(1+2ε) e2αR |V∗ (R, R )| dR < ∞, × ⎭ 0
(4.7) or [as in (3.18) and (3.19)] in terms of the function (w(ε) )
V∗
⎞1/2 ⎛ +∞ . ⎝ (R) = w(ε) (R )e2αR V∗2 (R, R ) dR ⎠ ,
(4.8)
0
which belongs to Xw(ε),α , (w(ε) ) C(V∗ ) = V∗
w(ε),α
< ∞.
In view of (4.4), one therefore has the following global majorization: 1 Re λ > − C(V (λ; ..)) ≤ Ke−γ Re λ C(V∗ ) . 2
(4.9)
(4.10)
The set of conditions (4.4), (4.6), (4.7), and (4.10) characterize the class Nwγ(ε),α of nonlocal potentials for which we are going to study the analyticity properties of the partial scattering amplitudes. γ Simple examples of potentials in Nw (ε),α : The simplest examples which can be seen to satisfy the previous conditions are obtained by choosing V∗ as a kernel of rank one V∗ = v ⊗ v and V (λ; R, R ) = v(R)v(R )e−γλ (or v(R)v(R )Qλ (eγ )). Concerning v, one saturates bound (4.7) with a 3 “Yukawa-type” function such as v(R) = (1 + R)− 2 −ε e−αR .
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4.2. Analyticity and Boundedness Properties in Complex (λ, k)-Space of the Functions v0 (λ, k; ·), the Operators L(λ, k), and the Resolvent R(λ, k; g) Our assumptions on the CAM interpolation V (λ; R, R ) of the potentials V will allow us to introduce corresponding CAM interpolations v0 (λ, k; ·), L(λ, k), and R(tr) (λ, k; g) for the respective sequences {v,0 (k; ·)}, {L (k)}, and (tr) {R (k; g)}, defined earlier in (3.10b), (3.10c), (3.58), and (3.59). The derivation of the properties of all these CAM interpolations relies not only on the assumptions on V (λ; R, R ), but also on the properties of the spherical functions jλ (kR) and of the CAM Green function G(λ, k; R, R ) for (λ, k) ∈ C+ × C(cut) , where C(cut) = C\(−∞, 0]. These properties have been − 12 established in Appendix A (Sects. A.II, A.III) and, in particular, it has been shown there that for nonintegral values of , these analytic functions of the two complex variables λ, k are holomorphic with respect to k in a ramified domain with branch point at k = 0, from which we only retain here a distinguished sheet C(cut) . For all k in this domain, the functions jλ (kR) and G(λ, k; R, R ) are CAM interpolations of the corresponding sequences of func ∞ tions {j (kR)}∞ =0 and {G (k; R, R )}=0 ; however, it is to be noted that these interpolations are non-Carlsonian for general values of k. It is only when k = iκ, κ > 0, that [in view of bound (A.19)] the Green function G(λ, iκ; R, R ) appears to be the (unique) Carlsonian interpolation of the corresponding sequence {G (iκ; R, R )}∞ =0 , thus implying similar Carlsonian properties for the functions L(λ, iκ) and R(tr) (λ, iκ; g) (κ > 0). The (non-Carlsonian) interpolations obtained for general values of k in C(cut) are the analytic continuations of the latter with respect to k. The previous considerations concerning the occurrence of a branch-point at k = 0 and of non-Carlsonian bounds in the half-plane Re λ > − 21 lead us to introduce the following domains: (i) In the complex λ-plane: for each pair of positive numbers (γ, δ), the (truncated ) angular sector . Λγ(δ) =
) ( γ 1 1 λ ∈ C : | Im λ| < Re λ + ; Re λ > − + δ , 3π 2 2
(4.11)
(δ)
whose closure is denoted by Λγ . (ii) In the complex k-plane: (cut) (cut) . the cut-strip Ωα = Ωα \(−∞, 0], and its closure Ωα ; (cut) (cut) . the cut half-plane Πα = Πα \(−∞, 0], and its closure Πα . . (δ) (δ) (cut) (iii) In the complex (λ, k)-space C2 : the domain Dγ,α = Λγ × Ωα , whose (δ) . (δ) (cut) closure is Dγ,α = Λγ × Ωα . 4.2.1. The Vector-Valued Function (λ, k) → v0 (λ, k; R). The CAM interpolation of the sequence of functions {v,0 (k; R)}∞ =0 [see Eq. (3.10b)] is formally
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defined by the following expression, whose analyticity and boundedness properties are stated below: +∞ v0 (λ, k; R) = V (λ; R, R )kR jλ (kR ) dR .
(4.12)
0
In order to obtain a majorization of the integrand on the r.h.s. of the latter, we shall use bound (A.63) for the spherical Bessel function jλ (kR), which holds for (λ, k) ∈ C+ × C(cut) , and yields [compare to (3.21) and (3.22)]: −1 2
∗
k· jλ (k·)w(ε),α
⎛ +∞ ⎞1/2 2 |kRj (kR)| . λ =⎝ dR⎠ w(ε) (R)e2αR 0
% ≤
1 3π π |k| 2 e 2 | Im λ| 2
3 1 + 2 π(Re λ + 12 )
⎞1/2 ⎛ +∞ −2(α−| Im k|)R Re ⎝ dR⎠ . w(ε) (R) 0
(4.13) By taking Eqs. (3.35) and (3.25) into account for majorizing the latter integral, (cut)
we then obtain, for (λ, k) ∈ C+ × Ωα −1 ∗ k·jλ (k·)w(ε),α
%
≤
2
:
1 3π π Aε |k| 2 e 2 | Im λ| 2
3 1 + 2 π(Re λ + 12 )
.
(4.14)
We then have: Theorem 27. For every nonlocal potential V in Nwγ(ε),α , the corresponding function (λ, k) → v0 (λ, k; ·), is well-defined by the integral (4.12) as a vector-valued (cut)
(cut)
function in the set C+ × Ωα , holomorphic in C+ × Ωα , taking its val− 12 − 12 ues in Xw(ε),α ; the corresponding norm v0 (λ, k; ·)w(ε),α admits the following (cut)
× Ωα majorization in C+ −1 2
%
v0 (λ, k; ·)w(ε),α ≤
:
1 3π π Aε KC(V∗ )|k| 2 e−γ Re λ e 2 | Im λ| 2
3 1 + , 2 π(Re λ + 12 ) (4.15)
[the constants on the r.h.s. being defined by Eqs. (3.25), (4.4), (4.8), and (4.9)]. 1 Moreover, the function k − 2 v0 (λ, k; ·) is defined as a continuous and uniformly (δ)
bounded vector-valued function of (λ, k) in any closed set D2γ,α (for any δ > 0). Proof. The proof is quite similar to the one of Lemma 5. By applying the Schwarz inequality to integral (4.12) and taking Eqs. (4.4) and (4.8) into account, one obtains (for a.e. R): (w(ε) ) ∗ (R) k·jλ (k·)w(ε),α . |v0 (λ, k; R)| ≤ Ke−γ Re λ V
(4.16)
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It then follows from (4.9) and (4.14) that the function R → v0 (λ, k; R) belongs (cut)
to Xw(ε),α for all (λ, k) ∈ C+ × Ωα −1 2
, and satisfies bound (4.15). Then, it (δ)
follows from the latter that, for (λ, k) varying in any set D2γ,α [defined for any 1 δ > 0 via (4.11)], the function (λ, k) → |k|− 2 v0 (λ, k; ·)w(ε),α is uniformly .π 1 ). bounded by the constant 2 Aε KC(V∗ )eγ/2 ( 32 + πδ Finally, the holomorphy and continuity properties of the vector-valued function (λ, k) → v0 (λ, k; ·) are obtained as in Lemma 5 by a direct application of Lemma B.9 (giving the holomorphy and continuity properties in (λ, k) of the integral (4.12), for a.e. R) and of Lemma B.8, by noting that the function (λ, k, R) → v0 (λ, k; R) belongs to a relevant class C(D, μ, p), with D = (cut)
C+ × Ωα − 12 and p = 2.
(δ)
(resp., D2γ,α for the continuity property), μ(R) = w(ε) (R)e2αR
4.2.2. The Operator-Valued Function L(λ, k). The CAM interpolation of the sequence of kernels {L (k; R, R )}∞ =0 [see Eq. (3.10c)] is formally defined by the following expression, whose analyticity and boundedness properties are stated below: +∞ V (λ; R, R )G(λ, k; R , R )dR . (4.17) L(λ, k; R, R ) = 0
As for the case of the kernels L (k; R, R ) (see Sect. 3), we are going to show that L(λ, k; R, R ) is bounded by a kernel of rank one. To this effect, we shall use bounds (A.34) and (A.41) for the complex angular momentum Green function, which hold respectively for Im k ≥ 0 and Im k < 0 in the domain (λ, k) ∈ C+ × C(cut) . These bounds imply the following global majorization, which −1 2
(cut)
holds for (λ, k) ∈ C+ × Πα −1 2
:
√ α(R+R ) 3π| Im λ| e |G(λ, k; R, R )| ≤ c RR e 1+
1 2 Re λ + 1
.
(4.18)
Now, if the potential V (λ; R, R ) belongs to the class Nwγ(ε),α , it follows from (4.4) and (4.18) that the following majorization holds for a.e. (R, R , R ) and (cut)
× Πα (λ, k) ∈ C+ −1 2
:
√ |V (λ; R, R )G(λ, k; R , R )| ≤ Mγ (λ)V∗ (R, R ) R R eα(R +R ) , (4.19)
where:
. Mγ (λ) = cKe3π| Im λ| e−γ Re λ 1 +
1 2 Re λ + 1
.
(4.20)
In view of the latter, we obtain the following bound for the integral (4.17): +∞ √ √ αR V∗ (R, R ) R eαR dR , |L(λ, k; R, R )| ≤ Mγ (λ) R e
0
(4.21)
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which yields, by taking (4.8) into account, using the Schwarz inequality and +∞ R the bound 0 dR ≤ A2ε [with Aε given by (3.25)]: w(ε) (R ) √ (w(ε) ) |L(λ, k; R, R )| ≤ Mγ (λ)Aε V∗ (R) R eαR . (4.22) As in Sect. 3 (see Theorem 6), it is then appropriate to introduce the Hilbert "w(ε),α of Hilbert-Schmidt kernels K(R, R ) with respect to the measure space X μ(R) dR = w(ε) (R)e2αR dR = R1−ε (1 + R)1+2ε e2αR dR [see Appendix B.I, formula (B.9)], whose norm is given by . 2 2 KHS = KX /
w(ε),α
+∞
=
e−2αR dR w(ε) (R)
0
+∞ 2 w(ε) (R )e2αR |K(R , R)| dR . 0
(4.23) In fact, the kernel of rank one on the r.h.s. of Eq. (4.22) belongs to this space, its norm being expressed and majorized as follows: ⎤1/2 ⎡ +∞ (ε) . ∗ R w ( ) V∗ (·)e(α ·) (ε) = C(V∗ ) ⎣ dR ⎦ w ,α w(ε) (R ) w(ε),α 0
≤ C(V∗ )Aε .
(4.24) (cut) Πα ,
It then follows from (4.22) that, for each (λ, k) ∈ C+ × the kernel − 12 " L(λ, k; R, R ) belongs to Xw(ε),α and satisfies the following norm inequality: L(λ, k)HS ≤ Mγ (λ) C(V∗ ) A2ε .
(4.25)
We can then state the following theorem. Theorem 28. For every nonlocal potential V ∈ Nwγ(ε),α , the corresponding kernels L(λ, k; R, R ) [formally defined by (4.17)] are well-defined as compact operators L(λ, k) of Hilbert-Schmidt-type acting in the Hilbert space Xw(ε),α (cut)
for all (λ, k) in C+ × Πα , and the corresponding Hilbert–Schmidt norm − 12 "w(ε),α admits the following global majorization: L(λ, k)HS of L(λ, k) in X 1 L(λ, k)HS ≤ c K C(V∗ ) A2ε e3π| Im λ| e−γ Re λ 1 + . (4.26) Re λ + 12 Moreover, the HS-operator-valued function (λ, k) → L(λ, k), taking its values "w(ε),α , is holomorphic in C+ 1 × Π(cut) , and is continuous and uniformly in X α − 2
(δ)
(cut)
bounded in any set of the form Λγ × Πα
.
Proof. The main part of it has been given in the previous argument; in particular, the global majorization (4.26) [which is a rewriting of (4.25) and (4.20)] is (δ)
(cut)
seen to give a uniform bound for L(λ, k)HS in any set of the form Λγ ×Πα [see Eq. (4.11)]. As in Theorem 6, the holomorphy and continuity properties of the HS-operator-valued function (λ, k) → L(λ, k) are directly obtained by applying Lemma B.10 (now with ζ = (λ, k)) to integral (4.17).
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4.2.3. Smithies’ Formalism for the Resolvent R(λ, k; g). We can now formally write the following expression for the resolvent: R(λ, k; g) = [I − gL(λ, k)]
−1
.
(4.27)
The fact that L(λ, k) is a Hilbert-Schmidt operator on the Hilbert space Xw(ε),α allows us to use Smithies’ formulae and bounds. Accordingly, we can write: R(λ, k; g) = I + g
N (λ, k; g) , σ(λ, k; g)
(4.28)
where the operators N (λ, k; g) and the functions σ(λ, k; g) are defined by extending formally all the formulae (3.39) through (3.45) of Smithies’ formalism from non-negative integral values of to complex values of λ in C+ . − 12 More precisely, we can now prove the following theorems. Theorem 29. For every nonlocal potential V ∈ Nwγ(ε),α , the function (λ, k, g) → (cut)
(cut)
×Πα ×C and continuous in C+ ×Πα × σ(λ, k; g) is holomorphic in C+ − 12 − 12 C. Moreover, at fixed g, it is uniformly bounded in any closed set of the form (δ) (cut) Λγ × Πα , and the function σ(λ, k; g) − 1 tends uniformly to zero for |λ| (δ)
(cut)
tending to infinity in any subset Λγ × Πα
with γ < γ.
Proof. In view of the holomorphy and continuity properties of L(λ, k) stated in Theorem 28, one can then follow the argument given in the proof of Theorem 7 for justifying successively (and for every n ≥ 2) the corresponding holomorphy and continuity properties of the functions ρn (λ, k) = Tr[Ln (λ, k)], Qn (λ, k) and σn (λ, k) [defined as in Eqs. (3.41) and (3.40)]. Now, by combining the basic inequalities of Smithies’ theory with the bound (4.25) on L(λ, k)HS , one obtains the following majorizations, similar to (3.50): e n/2 e n/2 # $n n C(V∗ )A2ε Mγ (λ) . (4.29) L(λ, k)HS ≤ |σn (λ, k)| ≤ n n ∞ It follows that the series σ(λ, k; g) = n=0 σn (λ, k)g n (with σ0 = 1) is domi(cut)
nated, for all (λ, k, g) in C+ × Πα × C by a convergent series with positive − 12 terms. By associating with the latter the entire function Φ(z) as in the proof of Theorem 7 [see Eq. (3.52)], one then concludes from inequality (4.29), written for all values of n, that the sum of the series σ(λ, k; g) is well-defined and satisfies the following global majorization: 0 1 |σ(λ, k; g) − 1| ≤ Φ (|g|L(λ, k)HS ) ≤ Φ |g|C(V∗ )A2ε Mγ (λ) . (4.30) Inequalities (4.29) and (4.30) entail (by applying Lemma B.1 to the sequence of functions {(λ, k, g) → σn (λ, k)g n ; n ∈ N}) that the function σ(λ, k; g) is (cut) holomorphic in the domain C+ × Πα × C of C3 , and also defined and con−1 (cut)
tinuous for k ∈ Πα
2
. Moreover, since the function Mγ (λ) [see Eq. (4.20)] is
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(δ)
uniformly bounded in any set Λγ and tends uniformly to zero for |λ| tending (δ)
to infinity in any set Λγ , the last statement of the theorem directly follows from majorization (4.30). Theorem 30. For every nonlocal potential V ∈ Nwγ(ε),α , the operators N (λ, k; g) exist as Hilbert-Schmidt operators acting on Xw(ε),α for all (λ, k, g) in the (cut)
× Πα subset C+ −1 2
× C of C3 . The HS-operator-valued function (λ, k, g) →
"w(ε),α , is holomorphic in C+ 1 × Π(cut) × C and N (λ, k; g), taking its values in X α − (cut)
continuous in C+ × Πα −1 2
2
× C. Moreover, at fixed g, N (λ, k; g)HS is uni(δ)
(cut)
formly bounded in any closed set of the form Λγ × Πα to zero for |λ| tending to infinity in any subset
(δ) Λγ
×
and tends uniformly
(cut) Πα
with γ < γ.
Proof. By defining successively (and for every n ≥ 1) the bounded-operatorvalued functions Δn (λ, k) and the HS-operator-valued functions Nn (λ, k) in terms of L(λ, k) as in Eqs. (3.45) and (3.44), one deduces from Lemma B.6 that all these functions satisfy the same holomorphy and continuity properties as those of L(λ, k) specified in Theorem 28. Moreover, in view of Smithies’ theory, there hold the following inequalities, similar to (3.55): Nn (λ, k)HS ≤ Δn (λ, k) L(λ, k)HS ≤
e(n+1)/2 n+1 L(λ, k)HS . (4.31) nn/2
∞ In view of the latter, the series n=0 Nn (λ, k)g n is dominated term by term in the HS-norm by a convergent series; the sum of this operator-valued entire series is therefore well-defined as a HS-operator N (λ, k; g) for all (λ, k, g) in (cut)
C+ × Πα −1 2
L(λ, k)HS
× C. Now, it follows from (4.31) and from the bound (4.25) on "w(ε),α satisfies the bound: that the norm of N (λ, k; g) in X
N (λ, k; g)HS ≤
1 1 1 0 Ψ (|g|L(λ, k)HS ) ≤ Ψ |g|C(V∗ ) A2ε Mγ (λ) , |g| |g|
(4.32)
where Ψ(z) is the entire function introduced in the proof of Theorem 8 [see Eq. (3.56)]. The fact that Ψ(z) is an increasing function of z for z ≥ 0 has been used for obtaining the second inequality in (4.32). Inequalities (4.31) and (4.32) entail (by applying Lemma B.1 to the sequence of functions {(λ, k, g) → Nn (λ, k)g n ; n ∈ N}) that the HS-opera(cut) tor-valued function N (λ, k; g) is holomorphic in the domain C+ × Πα × C −1 (cut)
2
of C3 , and also defined and continuous for k ∈ Πα . Finally, the last statement of the theorem directly follows from (4.32) and from the expression (4.20) of Mγ (λ) (as for the last statement of Theorem 29).
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4.3. Meromorphy Properties of the Resolvent and Their Physical Interpretation 4.3.1. General Structure. Let us introduce, as in Sect. 3, the truncated Fredholm resolvent R(tr) (λ, k; g) as follows: R(tr) (λ, k; g) =
N (λ, k; g) . σ(λ, k; g)
(4.33)
It follows from Theorems 29 and 30 that R(tr) (λ, k; g) is a HS-operator-valued meromorphic function of (λ, k, g), whose “poles” are localized on the various possible connected components of the complex analytic set defined by the equation σ(λ, k; g) = 0. Now, in view of the last property stated in Theorem 29, this set cannot contain any component of the form g = g0 , and therefore for each fixed g (in C) the subset 2 3 . × Π(cut) : σ(λ, k; g) = 0 (4.34) Dα (V ; g) = (λ, k) ∈ C+ α −1 2
(cut)
× Πα is a domain, which is the complement in C+ − 12 analytic set. We can then state: Theorem 31. For every nonlocal potential V
of a one-dimensional
∈ Nwγ(ε),α , the operators
R(tr) (λ, k; g) exist as Hilbert-Schmidt operators acting on Xw(ε),α for all (cut)
× Πα (λ, k, g) in the dense subdomain of C+ −1 2
× C where σ(λ, k; g) = 0.
Moreover, the HS-operator-valued function (λ, k, g) → R(tr) (λ, k; g), taking its "w(ε),α , is a meromorphic function whose restriction to each fixed values in X value of g is holomorphic in Dα (V ; g). As in Sect. 3 (see Theorem 10), one can also state the following property of the “complete resolvent” R(λ, k; g) = [I − gL(λ, k)]−1 = I + gR(tr) (λ, k; g): Theorem 32. For any fixed g, the function R(λ, k; g) is holomorphic in the domain Dα (V ; g) as an operator-valued function, taking its values in the space of bounded operators in Xw(ε),α . 4.3.2. Symmetry Properties in the Complex Variables (λ, k, g). In Sect. 3 we have shown that the basic symmetry properties (3.7), (3.11), and (3.12) imply the corresponding invariance of the quantities σ (k; g), N (k; g), and R (k; g) under the transformation (k → −k, g → g) in their respective analyticity domains of the complex (k, g)-space [see Theorems 7 (b), 8 (b)]. Here one can similarly justify the invariance of the corresponding interpolated quantities in the CAM-plane, namely σ(λ, k; g), N (λ, k; g), and R(λ, k; g), under the transformation (λ → λ, k → −k, g → g). Note however that some additional specifications must be given in view of the occurrence of the branch point at k = 0 and of the affiliated cut on the real negative k-axis, which , but already for noninare effective not only for complex values of λ in C+ −1 2
tegral real values of λ (λ > − 12 ). As a matter of fact, it can be seen that for the quantities previously mentioned, their invariance under the transformation (λ → λ, k → −k, g → g) can be first established for k varying in the upper
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half-plane; the extension of this invariance property to the cut-plane Πα then follows by analytic continuation, provided one introduces a k → −kinvariant ramified analyticity domain over Πα \{k = 0} (thus including also a cut-domain with a cut along the positive real k-axis, which is the symmetric (cut) domain of Πα ). Here again, these properties are based on: (a) the symmetry properties of the potential, namely V (λ; R, R ) = V (λ; R, R ), and V (λ; R, R ) = V (λ; R , R) for all λ ∈ C+ , which cor− 12 respond to the Carlsonian interpolation of (3.7); (b) the relation G(λ, −k; R, R ) = G(λ, k; R, R ) for all (λ, k) such that λ ∈ C+ and Im k > 0. This relation, which extrapolates (3.11), is easily − 12 derived from the integral representation (A.15) of G(λ, iκ; R, R ) and the analytic continuation of the latter to complex values of λ and κ. From (a) and (b), one then derives the analog of (3.12), namely: (c) L(λ, −k; R, R ) = L(λ, k; R, R ), and subsequently (by arguments similar to those given in the proofs of Theorems 7 (b) and 8 (b)), (d) σ(λ, −k; g) = σ(λ, k; g), N (λ, −k; g) = N (λ, k; g), R(λ, −k; g) = R(λ, k; g). 4.3.3. Poles of the Resolvent and Solutions of the Schr¨ odinger-Type Equation. By the same analysis as in Sect. 3.2 for the case integer, we can say that the existence of a pole k = k(λ, g) of the meromorphic function k → R(λ, k; g), namely a value of k such that σ(λ, k(λ, g); g) = 0 with N (λ, k; g) = 0, is equivalent to the existence of at least one non-zero solution x = x(R) in Xw(ε),α of the homogeneous Fredholm equation gL(λ, k)x = x. Concerning the terminology, we prefer to say that such a solution x is associated with a singular pair (λ, k) (i.e., a pair satisfying the equation σ(λ, k; g) = 0 for a fixed value of g) rather than with the variable-dependent notion of “pole”. At a singular pair (λ, k) indeed, it can be advantageous as well to consider the pole of the meromorphic function λ → R(λ, k; g), at the value λ = λ(k, g) such that σ(λ(k, g), k; g) = 0 (instead of the pole in the variable k, as always considered before). As shown below in Lemma 33, one can associate with any function x(R) (cut)
in Xw(ε),α , and for every λ ∈ C+ and k ∈ Πα −1 2
, the function
+∞ ψ(R) = g G(λ, k; R, R )x(R ) dR ,
(4.35)
0
which satisfies the equation Dλ,k ψ = gx, since G(λ, k; R, R ) is (for every complex pair (λ, k)) the Green function of the corresponding differential operator Dλ,k [defined as D,k by complexification of in Eq. (3.6)]. Now, in view of Eq. (4.35), the definition (4.17) of L(λ, k) implies the following equality: +∞ g[L(λ, k)x](R) = V (λ; R, R )ψ(R ) dR . 0
(4.36)
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So, if x is a non-zero solution of the homogeneous Fredholm equation gL(λ, k)x = x associated with a given singular pair (λ, k) of the resolvent R(tr) (λ, k; g) (g being fixed), then one has: +∞ V (λ; R, R )ψ(R ) dR . Dλ,k ψ(R) = gx(R) = g
(4.37)
0
Thus ψ appears as a non-zero solution of the extension to complex λ of the Schr¨ odinger-type equation (3.6). As in Sect. 3 (see Theorem 16), some specific properties of this type of solution ψ(R) will be given below. 4.3.4. Some Results on the Locations of the Poles of R(tr) (λ, k; g). First of all, we shall take into account the fact that each class of nonlocal potentials Nwγ(ε),α (for any γ > 0) is contained in the corresponding class Nw(ε),α introduced in Sect. 3. It follows that all the properties of the poles in k at fixed integer proved above hold true for the potentials in Nwγ(ε),α . By incorporating the results of this previous analysis and taking g real, we are led to distinguish between two situations, whose specifications and interest will be justified below. (cut)
(a) λ real and larger than − 12 , k complex in Πα . In this case we shall see that one obtains a “natural” extension of the results obtained in Sect. 3 for λ = integer (which leads us to use the same terminology): (a.1) bound states: zeros of σ sitting on the positive imaginary axis; no other zeros of σ can occur in the upper half-plane Im k > 0; (a.2) spurious bound states: zeros of σ sitting on the positive real axis Im k = 0, Re k > 0; (a.3) anti-bound states: zeros of σ sitting on the negative imaginary axis −α < Im k < 0, Re k = 0; (a.4) resonances: zeros of σ in the half-strip −α < Im k < 0, Re k > 0. and k real in R+ . In this case we may have: (b) λ complex in C+ −1 2
(b.1) zeros of σ in the first quadrant of the λ-plane (Im λ > 0, Re λ > − 12 ), which correspond to an alternative description of resonances; (b.2) zeros of σ in the fourth quadrant of the λ-plane (Im λ < 0, Re λ > − 12 ), corresponding to echoes (see the end of Sects. 4.3.4, 5.2).
Note that in this description a dissymmetric role is played by the singularities in the first and in the fourth quadrant of the λ-plane, since we have to keep in mind that here the range of k has been restricted to k ≥ 0. So this dissymetric role does not enter in conflict with the symmetry λ → λ accompanied by k → −k which governs the global set of singularities of R(λ, k; g) (as explained in the previous section), but does not carry any physical meaning. We shall first prove the following variant of Lemma 13.
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Lemma 33. For every function x in Xw(ε),α and for all (λ, k) such that Re λ > − 12 and Im k ≥ 0, the corresponding function +∞ ψx;λ,k (R) = g G(λ, k; R, R )x(R ) dR
(4.38)
0
is well-defined as a locally bounded function, contained in the space Xw∗ (ε),α . Moreover, it enjoys the following properties: (i) If Re λ > 0, there holds a global majorization of the following form for R varying on the whole half-line {R ∈ R+ }: 1 0 1 |ψx;λ,k (R)| ≤ |g| xw(ε),α " c0 (λ) [Φ(k)]−(Re λ+ 2 ) min R, β(λ, α)R−Re λ , (4.39) where: " c0 (λ) = ceπ| Im λ| 1 +
1 2 Re λ + 1
1 1 + 2(Re λ + 1) 2 Re λ
1/2 ,
(4.40)
Φ(k) = 1 if | Arg(−ik)| ≤ π4 , while Φ(k) = sin 2|φ(k)| if π4 ≤ |φ(k)| = | Arg(−ik)| < π2 , and β denotes a suitable positive function of λ and α; (ii) the derivative ψx;λ,k (R) of ψx;λ,k is well-defined on R+ and satisfies a global majorization of the form: |ψx;λ,k (R)| ≤ |g|
" c1 (λ, k) √ xw(ε),α R−1/2 ; 2α
(4.41)
(iii) for any potential V in a class Nwγ(ε),α , the corresponding double integral +∞
+∞
dR 0
dR ψ(R)V (λ; R, R )ψ(R )
(4.42)
0
is absolutely convergent. Proof. By using the assumption that x belongs to Xw(ε),α and the Schwarz inequality, we obtain the following majorization of the expression (4.38) of ψx;λ,k (R): ⎡
⎤1/2 +∞ −2αR e |G(λ, k; R, R )|2 (ε) dR ⎦ , (4.43) |ψx;λ,k (R)| ≤ |g|xw(ε),α ⎣ w (R ) 0
with w(ε) defined by Eq. (3.35). In the latter integral, we can now plug the bound (A.39) on |G(λ, k; R, R )|, which is valid for Im k ≥ 0 and λ ∈ C+ . −1 2
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A straightforward majorization then shows that the integral in (4.43) is convergent and bounded by R [c(λ, k)]2 /(2α), which implies that ψx;λ,k (R) is well√ √ defined and bounded by |g|xw(ε),α c(λ,k) R. It then also follows that 2α ∗
ψx;λ,k w(ε),α
⎡ +∞ ⎤1/2 −2αR Re c(λ, k) |g|xw(ε),α ⎣ ≤ √ < +∞. dR⎦ w(ε) (R) 2α
(4.44)
0
Proof of (i). Using (A.36) (along with inequality Rε (1 + R)−1−2ε < 1) allows one to majorize the integral in (4.43) by the expression c2 (λ, k)[Φ(k)]−(2 Re λ+1) ⎤ ⎡ R +∞ 2 Re λ+1 2 Re λ+1 R R ×R ⎣ e−2αR dR + e−2αR dR ⎦ , R R R
0
(4.45) which can itself be majorized by either one of the following two expressions (by respectively majorizing e−2αR by one or not): 1 1 2 −(2 Re λ+1) 2 + R (a) c (λ, k)[Φ(k)] , (4.46) 2(Re λ + 1) 2 Re λ which holds under the additional condition Re λ > 0. 4 5 2 −(2 Re λ+1) −2 Re λ Γ(2 Re λ + 2) −1 −2αR + (2α) Re R . (b) c (λ, k)[Φ(k)] 2 Re λ+2 (2α) (4.47) 2
Here, the expression inside the bracket can itself be majorized by β (λ, α) R−2 Re λ [in terms of a suitable positive constant β(λ, α)]. Therefore the inequalities (4.46) and (4.47) imply a global majorization of the r.h.s. of (4.43) of the form (4.39), by noting that, in view of (A.35), the expression 1 1/2 " c0 (λ), as defined by Eq. (4.40), is such that " c0 (λ) = [ 2(Re1λ+1) + 2 Re × λ] sup{k:Im k≥0} c(λ, k). Proof of (ii): In view of Eq. (4.38), one is led to and ∞prove the convergence boundedness properties of the integral J (R) = 0 g ∂G ∂R (λ, k; R, R )x(R ) dR , which will then define the function ψx;λ,k (R). Similarly to formula (4.43), one can now write, in view of the Schwarz inequality and of bound (A.40), the following successive majorizations: ⎡ +∞ ⎤1/2 2 −2αR ∂G e ⎦ |J (R)| ≤ |g| xw(ε),α ⎣ ∂R (λ, k; R, R ) w(ε) (R ) dR 0
⎡ +∞ ⎤1/2 −2αR Re " c1 (λ, k) |g|xw(ε),α ⎣ ≤ √ < +∞, dR⎦ w(ε) (R) R 0
(4.48)
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which therefore proves that ψx;λ,k (R) is well-defined on R+ and satisfies the global majorization (4.41).
Proof of (iii): In view of the conditions (4.6) and (4.10) on the potential V and of the fact that ψ belongs to the space Xw∗ (ε),α , it results from the Schwarz inequality that the integral (4.42) is absolutely convergent and bounded by the constant C(V (λ; ..))(ψ∗w(ε),α )2 . We can now give an appropriate variant of the Wronskian Lemma 15. Lemma 34 (Wronskian Lemma). For given values of λ, k, g such that Re λ > 0, Im k > 0, and g ∈ R, let ψ(R) be a solution of the integro-differential equa +∞ tion: Dλ,k ψ(R) = g 0 V (λ; R, R )ψ(R ) dR , associated with a solution x ∈ Xw(ε),α of the corresponding equation x = gL(λ, k)x via Eq. (4.35). Then there holds the following identity, in which all the integrals are absolutely convergent: +∞ +∞ ψ(R) ψ(R) ψ(R)ψ(R) dR − Im λ (2 Re λ + 1) dR 2 Im k Re k R2 0
0
+∞ +∞ [V (λ; R, R ) − V (λ; R, R )] =g dR dR ψ(R)ψ(R ). 2i 0
(4.49)
0
Proof. The following equation [analogous to Eq. (3.92)] results from the extension of Eq. (3.6) to complex values of (λ, k): 2
ψ(R)ψ (R) − ψ (R)ψ(R) + (k 2 − k )ψ(R)ψ(R) ψ(R)ψ(R) R2 = ψ(R)[Dλ,k ψ](R) − [Dλ,k ψ](R)ψ(R) +∞ # $ V (λ; R, R )ψ(R)ψ(R ) − V (λ; R, R )ψ(R )ψ(R) dR . =g −[λ(λ + 1) − λ(λ + 1)]
(4.50)
0
Then, since ψ(R) satisfies all the properties described in Lemma 33, it is legitimate to integrate over R from 0 to +∞ both sides of the latter equation; more precisely, one deduces from (i) and (iii) of Lemma 33 that, under the conditions Re λ > 0 and Im k > 0, the r.h.s. of (4.50), as well as the function ψ(R)ψ(R)/R2 are integrable over R from 0 to +∞. Moreover, the majorizations (4.39) and (4.41) imply that the function [ψ(R) ψ (R) − ψ (R) ψ(R)] tends to zero at both ends of the half-line [0, +∞). It therefore follows that the R" remaining integrated term of Eq. (4.50), namely, 4i Im k Re k 0 ψ(R)ψ(R) dR, " tends to +∞. Finally, by taking into account the has a finite limit when R symmetry property of V (λ; R, R ), the integral in Eq. (4.50) can be rewritten under the form of Eq. (4.49), in which all terms are well-defined.
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Application: regions of (λ, k)-space free of singularities In Theorem 16 (c), we had proved (as an application of the Wronskian Lemma) that no singularity of the resolvent R (k; g) can occur in the upper half-plane of k, except on the imaginary axis. In other words, there are no singular pairs (λ, k) with λ = integer ( ≥ 0) and Im k > 0, Re k = 0. Now, there are some extensions of that result to the location of the singularity manifolds in complex (λ, k)-space of the “interpolated resolvent” R(λ, k; g), which similarly follow from Lemma 34. (a) For λ real, Eq. (4.49) reduces to: +∞ ψ(R)ψ(R) dR = 0, Im k Re k
(4.51)
0
which entails that there are no singular pairs (λ, k) with λ real positive and Im k > 0, Re k = 0, since Eq. (4.51) excludes the possibility of a non-zero function ψ in these situations. (b) If the potential V (λ; R, R ) is constant with respect to λ, the r.h.s. of Eq. (4.49) still vanishes. Then one sees that no singular pairs (λ, k) can occur such that Im λ < 0 with Im k > 0 and Re k > 0 (and in the symmetric region obtained by (λ, k) → (λ, −k)). In all such situations indeed, the l.h.s. of Eq. (4.49) would have to be strictly positive for any non-zero function ψ. Note that, via an argument of continuity, this result also implies that there is no singular pair (λ, k) with Im λ < 0 and k > 0. We notice that an analog of this situation is always encountered in the complex angular momentum formalism of the theory of local potentials, since the correspondence with the present case is simply given formally by the relation V (λ; R, R ) = δ(R − R )V (R): in other words, the complex angular momentum interpolation of the potential is always constant in λ. As a matter of fact, in that framework, the previous result appeared as a basic theorem, which was proved by Regge [4]: according to the latter, all the singularity manifolds of R(λ, k; g), such as those which manifest themselves as resonances (i.e., containing pairs (λ, k) with λ = integer and Im k < 0), can only manifest themselves at k > 0 in the region Im λ > 0. (c) An interesting simple class of nonlocal potentials (already mentioned in Sect. 4.1) are the potentials of the form V (λ; R, R ) = V∗ (R, R )F6(λ), where the function F6 is holomorphic, bounded and of Hermitian-type in the half-plane C+ . As typical simple examples, we mentioned F6(λ) = − 12 e−γλ or F6(λ) = Qλ (eγ ). For such a class, let {Lj ; j ∈ Z} be the set of lines in C+ 1 on which one has Im F6(λ) = 0, Lj and L−j being complex −2
conjugate of each other with the following property: all the points of a curve Lj with j > 0 (resp., j < 0) belong to the region Im λ > 0 (resp., Im λ < 0), L0 being along the real positive axis. Then, at any point λ ∈ Lj we have V (λ; R, R ) = V (λ; R, R ), and therefore the r.h.s. of Eq. (4.49) vanishes at such points. Therefore, on the basis of Eq. (4.49) as in (b),
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no singular pairs (λ, k) can occur with λ in any line Lj , j ≤ 0, and with Im k ≥ 0 and Re k ≥ 0. This property indicates the possible occurrence of singularity manifolds containing branches in the region Im k ≥ 0, Re k ≥ 0 and Re λ > 0, Im λ < 0, but always located in “strips” of this fourth quadrant of the λ-plane, well-separated from one another by the set of lines Lj , j ≤ 0. These possible singularities, which did not exist for the case of local potentials, can be seen to enjoy properties which are related to the notion of echo (see Sect. 5). 4.4. Analyticity and Boundedness Properties in Complex (λ, k)-Space of the Partial Scattering Amplitude T (λ, k; g) We now extend Eq. (3.20) from non-negative integral values to complex λ by considering the integral equation [I − gL(λ, k)]v(λ, k; g; ·) = v0 (λ, k; ·),
(4.52)
where, according to Theorems 27 and 28, the vector-valued and operator-valued functions (λ, k) → v0 (λ, k; ·) and (λ, k) → L(λ, k) are well-defined in the (cut)
(cut)
set C+ × Ωα , and holomorphic in the domain C+ × Ωα . Then there − 12 − 12 holds the following analog of Theorem 17, in which we have put Dα (V ; g) = (cut) Dα (V ; g) ∩ Ωα [see Eq. (4.34)]. Theorem 35. For any nonlocal potential V in a class Nwγ(ε),α , the inhomogeneous equation (4.52) admits for any g ∈ C and (k, λ) ∈ Dα (V ; g) a unique solution v(λ, k; g; ·) in Xw(ε),α , which is well-defined by the formula: v(λ, k; g; ·) = R(λ, k; g)v0 (λ, k; ·).
(4.53)
Furthermore, for any g, the function v(λ, k; g; ·) is holomorphic in Dα (V ; g) as a vector-valued function, taking its values in Xw(ε),α . Proof. The solution (4.53) of equation (4.52), which follows from (4.27), defines v(λ, k; g; ·) as an element of Xw(ε),α , in view of the fact that v0 (λ, k; ·) belongs to Xw(ε),α (see Theorem 27) and that R(λ, k; g) is a bounded operator in Xw(ε),α (see Theorem 32) for all g ∈ C and (k, λ) ∈ Dα (V ; g). Moreover, the holomorphy properties of R(λ, k; g) and v0 (λ, k; ·), established in Theorems 32 and 27 respectively, imply the corresponding property for v(λ, k; g; ·) in view of Lemma B.3(ii). Now, by taking into account the expression (4.28) of R(λ, k; g) in Eq. (4.53), we can re-express v(λ, k; g; ·) as follows, for all (λ, k) ∈ Dα (V ; g): v(λ, k; g; ·) =
u(λ, k; g; ·) , σ(λ, k; g)
(4.54)
where: . u(λ, k; g; ·) = [σ(λ, k; g) + gN (λ, k; g)] v0 (λ, k; ·).
(4.55) (cut)
In fact, in view of Theorems 29 and 30, for every (λ, k) ∈ C+ × Ωα , one − 12 can act with the multiplier σ(λ, k; g) and with the Hilbert-Schmidt operator
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N (λ, k; g) on the vector v0 (λ, k; ·) ∈ Xw(ε),α . One can then state (also in view of Lemma B.3(ii)) the following theorem. Theorem 36. For any nonlocal potential V in a class Nwγ(ε),α , the quantity (cut)
u(λ, k; g; ·) is well-defined for every g ∈ C and every (λ, k) ∈ C+ × Ωα − 12 a vector in Xw(ε),α , depending holomorphically on (λ, k), such that
, as
u(λ, k; g; ·)w(ε),α ≤ {|σ(λ, k; g)| + |g| N (λ, k; g)HS } v0 (λ, k; g; ·)w(ε),α . (4.56) Remark 5. It is clear that the various functions (λ, k) → σ(λ, k; g), N (λ, k; g), u(λ, k; g; ·) are respectively CAM interpolations of the corresponding sequences of functions k → σ (k; g), N (k; g), u (k; g; ·) [see Eq. (3.122)], since all the integral relations of the present section reduce legitimately to the corresponding equations of Sect. 3 for λ = ∈ N (any potential in Nwγ(ε),α being contained in Nw(ε),α ). We also stress the fact that while the functions σ and N are holo(cut)
(cut)
morphic in C+ × Πα , the domain of the function u is only C+ × Ωα , − 12 − 12 which is the maximal domain in which the function v0 can be proved to be holomorphic (see Theorem 27). Next, by substituting the complex variable λ to the integer in formulae (3.121), (3.122), (3.123), and (3.124), we shall introduce an analytic interpolation T (λ, k; g) of the sequence of partial scattering amplitudes {T (k; g)}∞ =0 by the following formula: +∞ Θ(λ, k; g) , T (λ, k; g) = −g R jλ (kR )v(λ, k; g; R ) dR = σ(λ, k; g)
(4.57)
0
where: +∞ Θ(λ, k; g) = −g R jλ (kR )u(λ, k; g; R ) dR .
(4.58)
0
T (λ, k; g) will be called here the CAM-partial-scattering-amplitude. We can in fact prove the following Theorem 37. For every nonlocal potential V belonging to the class Nwγ(ε),α , the following properties hold: (i) the function (λ, k, g) → Θ(λ, k; g) is defined and holomorphic in C+ × −1 (δ) Dγ,α
(cut) Ωα
2
× C; at fixed g, it is uniformly bounded in any sector (for any δ > 0); (cut) (ii) for every g the function T (λ, k; g) is meromorphic in C+ × Ωα and − 12 holomorphic in Dα (V ; g). Moreover, for every γ , with γ < γ, there exists a number δ0 (depending on γ and g) such that T (λ, k; g) is holomorphic (δ0 )
in the corresponding truncated sector Dγ ,α and satisfies an exponentially
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decreasing bound of the following form: |T (λ, k; g)| ≤ cγ ,g e−(γ−γ
) Re λ
.
(4.59)
(cut)
× Ωα the function Rjλ (kR) is a vector Proof. Since for every (λ, k) ∈ C+ − 12 in the dual space Xw∗ (ε),α , the quantity Θ(λ, k; g) is well-defined by Eq. (4.58) and such that ∗
|Θ(λ, k; g)| ≤ |g| ·jλ (k·)w(ε),α u(λ, k; g; ·)w(ε),α ∗
≤ |g| ·jλ (k·)w(ε),α v0 (λ, k; g; ·)w(ε),α {|σ(λ, k; g)| + |g|N (λ, k; g)HS } . (4.60) × The fact that the function (λ, k, g) → Θ(λ, k; g) is holomorphic in C+ −1 2
(cut)
Ωα × C is then directly implied by Lemma B.4. By now taking the majorizations (4.15), (4.14), (4.30), and (4.32) into account, we derive from (4.60) the following global bound: 2 3 1 π + |Θ(λ, k; g)| ≤ |g|A2ε KC(V∗ )e−γ Re λ e3π| Im λ| 2 2 π(Re λ + 12 ) 7 0 18 × 1 + [Φ + Ψ] |g|C(V∗ )A2ε Mγ (λ) , (4.61) with Mγ (λ) given by Eq. (4.20). In view of Eq. (4.11), one then easily checks (δ)
that for (λ, k) ∈ Dγ,α (for any δ > 0), the r.h.s. of (4.61) is uniformly major1 2 1 ized by π2 |g|A2ε KC(V∗ )eγ/2 ( 32 + πδ ) ×{1+[Φ+Ψ](|g|C(V∗ )A2ε cKeγ/2 (1+ 2δ ))}, which is independent of λ and k [here one takes into account the regularity properties of the entire functions Φ and Ψ, defined by Eqs. (3.52) and (3.56)]. This ends the proof of (i). The first part of (ii) is a straightforward consequence of Eq. (4.57). The holomorphy property of T (λ, k; g) and its majorization of the form (4.59) in (δ0 )
the truncated sectors Dγ ,α is implied by the fact that |σ(λ, k; g) − 1| tends uniformly to zero for λ going to infinity (see Theorem 29), and that |Θ(λ, k; g)| satisfies the global bound (4.61). When using the latter, one now takes into account Eq. (4.11) with γ replaced by γ (γ < γ), which finally yields the exponential factor on the r.h.s. of (4.59).
5. Watson Resummation of the Partial Wave Amplitudes: Resonances and Echoes Since each class of nonlocal potentials Nwγ(ε),α is a subclass of Nw(ε),α , one can apply all the results of scattering theory obtained in Sects. 3.4 and 4.4 to the case of potentials V in any given class Nwγ(ε),α . In particular, we shall rely on the expansion (3.153) of the scattering amplitude F (k, cos θ; g) in terms of the partial waves a (k; g) = T (k; g)/k [see Eq. (3.152)], whose finiteness at k = 0 is a consequence of the threshold behavior (3.155) (see Proposition 23). A physically important related function,
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introduced in (3.152), is the phase-shift δ (k; g) [see also Eqs. (3.125), (3.126), and (3.127)]. Then, in Sect. 4.3 an analytic interpolation T (λ, k; g) of the sequence (cut) + × C, {T (k; g)}∞ =0 has been defined as a meromorphic function in C− 1 × Ωα 2 whose various properties have been listed in Theorem 37. In view of the exponential decrease properties of a (k; g) (resp., a(λ, k; g)) for (resp., Re λ) tending to infinity, specified in formulae (3.141) and (3.152) (resp., (4.59)), we can safely apply the Watson resummation method to expansion (3.153), written for any fixed values of k in R+ , g real and θ in the interval 0 < θ ≤ π. It yields: ∞ (2λ + 1)a(λ, k; g)Pλ (− cos θ) i dλ, (5.1) (2 + 1)a (k; g)P (cos θ) = 2 sin πλ =0
C
where the path C encircles the positive real semi-axis in the λ-plane (see Fig. 2a). This path must be chosen with some care in order to include only the singularities of the integrand on the r.h.s. of formula (5.1) which are the poles generated by the zeros of sin πλ: other singularities in the first and in the fourth quadrants of the λ-plane, but close to the real semi-axis, must be avoided. We now introduce for every (γ , δ) such that 0 < γ < γ and 0 < δ < 12 , (δ) the path Γ = Γγ , whose support is the boundary of the truncated angular sec(δ)
tor Λγ [see Eq. (4.11)] and whose orientation is given by continuous distortion from C to Γ in the λ-plane (see Fig. 2b). According to Theorem 37, the number (δ) N of poles λ = λn (k, g) of T (λ, k; g) which are contained in Λγ is finite, since (δ)
(δ )
all these poles must be confined in the bounded region Λγ \Λγ 0 . Then, in view of the exponentially decreasing majorization (4.59) on a(λ, k; g) = T (λ, k; g)/k and of the following type of bound on the Legendre function (see [42, p. 709, formula II.107]): " θ)eπ| Im λ| (0 ≤ θ < π), (5.2) |Pλ (cos θ)| ≤ C(cos which is compensated by | sin πλ|−1 , the integration contour C in Eq. (5.1) can be legitimately replaced by Γ, provided the contributions of the poles λn (k; g) be taken into account via the residue theorem. We thus obtain: (2λ + 1) a(λ, k; g)Pλ (− cos θ) i dλ 2 sin πλ C
i = 2
Γ
N (2λ + 1)a(λ, k; g) Pλ (− cos θ) ρn (k; g) Pλn (− cos θ) dλ − π . sin πλ sin πλn (k; g) n=1
(5.3) According to the analysis of Sect. 4.3.4, the poles λn (k; g) may lie either in the upper or in the lower half-plane. In the generic case these poles can be considered as first order poles (see also the considerations on the spectrum of the resolvent and the physical arguments given by R.G. Newton in [43];
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see Sections 9.1, p. 240 and 9.3, p. 257). In (5.3), the factors ρn (k; g) are the corresponding residues of the function [(2λ + 1)a(λ, k; g)]. The previous analysis can thus be summarized in the following Theorem 38. For every nonlocal potential V ∈ Nwγ(ε),α the following representation of the total scattering amplitude holds: F (k, cos θ; g) N i (2λ + 1)a(λ, k; g)Pλ (− cos θ) ρn (k; g)Pλn (− cos θ) = dλ − π , 2 sin πλ sin πλn (k; g) n=1 Γ
(5.4) for 0 < θ ≤ π, k ∈ R+ , and Γ denotes any choice Γ = Γγ such that γ < γ and 0 < δ < 12 . (δ)
5.1. Extension of Representation (5.4) in the Two Complex Variables k and cos θ Formula (II.107) of [42] provides us with the following majorization on : Pλ (cos θ), which is valid for all cos θ in the cut-plane C\]−∞, −1] and λ ∈ C+ −1 2
π| Im λ| | Im θ| Re λ
|Pλ (cos θ)| ≤ C(cos θ)e
e
,
(5.5)
where the function C(cos θ) is bounded at infinity. Then, in view of the latter and of bound (4.59), the integrability condition of the background integral of (δ) (5.4) on a given path Γγ is: | Im θ| < γ − γ ,
(5.6)
which means that the corresponding integral representation of F (k, cos θ; g) is valid and defines F as an analytic function of the two variables (k, cos θ) in (cut) the domain Ωα × {Eγ−γ \[1, +∞[} (Eγ−γ denoting the ellipse with foci +1 and −1 and major semi-axis cosh(γ − γ )). Note that the maximal ellipse Eγ of analyticity is obtained for a choice of γ arbitrarily small, namely for the choice of the original path Γ = C. Finally, by using this path C, it can be seen that F (k, cos θ; g) can be (cut) analytically continued in the product domain Ωα × Eγ . This is based on the following argument: (i) As noticed in [5, p. 7], the discontinuity of the integral of (5.1) across the cut cos θ ∈ [1, cosh γ[ is proportional to C (2λ + 1)a(λ, k; g)Pλ (cos θ) dλ, which vanishes in view of the Cauchy theorem. (ii) For cos θ tending to 1, the limit of the integral (5.1) is infinite since lim Pλ (−z) = ∞ for z tending to one. However, since Pλ (−z) is bounded by a multiple of ln(z − 1), a similar bound holds for the integral in the neighborhood of z = 1; so this point cannot be an isolated singularity for the holomorphic function F (k, z; g).
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Remark 6. In the applications of the CAM method to high energy physics, it is essential that the path Γ of the so-called background integral of the WatsonRegge representation (5.4) can be taken along the imaginary axis at Re λ = − 12 . Of course, this requires that the region of the λ-plane in which a(λ, k; g) ; this is decreasing uniformly with respect to Re λ is the full half-plane C+ − 12 condition is fulfilled if F (k, cos θ; g) is analytic in a cut-plane of cos θ for each k fixed (i.e., in particular if the Mandelstam representation is satisfied; for example, this is the case for the scattering amplitude of the theory of Yukawa-type local potentials). Indeed, if such a property is valid, use can be made of the following asymptotic behaviour of the Legendre function Pλ (z)(z = cos θ) as |z| → +∞, for Re λ ≥ − 12 (see [5, p. 6, formula 2.5]): Pλ (z) π −1/2 2λ z λ
Γ(λ + 12 ) Γ(λ + 1)
(z ∈ C\(−∞, −1]),
(5.7)
which implies that the background integral is of the order z −1/2 as |z| → +∞. Assuming that the number of poles of the scattering amplitude is finite (as in the case of Yukawian potentials), one obtains the leading term in the asymptotic behaviour of the scattering amplitude as |z| → +∞ from the pole term with the largest real part. However, in the present framework, we are working in the physical region of cos θ(−1 ≤ cos θ ≤ 1), and we are not interested in the asymptotic behaviour of the scattering amplitude for large momentum transfer; accordingly, we shall (δ) fully exploit representation (5.4) with its path integral along Γγ , as shown in Fig. 2b.
5.2. Analysis of Resonances and Echoes as Contributions of Poles in the λ-Plane We shall now apply the previous Watson-type representation (5.4) of the total scattering amplitude F (k, cos θ; g) to the computation of the set of partial waves a (k; g), ( = 0, 1, . . .), which are defined by the standard inversion formula of expansion (3.153): 1 a (k; g) = 2
+1 F (k, cos θ; g)P (cos θ) d(cos θ).
(5.8)
−1
For this purpose, we shall use the basic projection formula (see [36, Vol. 1, p. 170, Eq. (7)]): 1 2
1 P (cos θ)Pλ (− cos θ) d(cos θ) = −1
sin πλ . π(λ − )(λ + + 1)
(5.9)
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Im λ
(a)
0
/
− 1 2
1
2
Re λ
2
Re λ
C
Im λ
(b)
Γγ ’
(δ)
Λ(δγ ’) 0
λ n+ /
− 1 2
C
+δ − 1 2
/
0
1
λ n−
Figure 2. a the integration path C in formula (5.1). b the (δ) integration path Γ = Γγ in formulae (5.3) and (5.4). The grey (δ )
sector Λγ 0 represents a region without poles for the function a(λ, k; g). Typical poles, namely λn+ and λn− , are indicated in the first and fourth quadrants, respectively
By plugging the expression (5.4) of F in (5.8) and applying (5.9) to the various terms, one obtains: i a (k; g) = 2π
Γ
N (2λ + 1) a(λ, k; g) ρn (k; g) dλ − . (λ − )(λ + + 1) (λ (k; g) − )(λn (k; g) + + 1) n n=1
(5.10) In the r.h.s. of the latter, we distinguish the so-called “background integral” over Γ [which is always convergent in view of the exponential decrease property (4.59)] from the individual contributions of the poles λ = λn (k; g) of a(λ, k; g).
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We will show that under certain simple assumptions these poles can be seen to induce properties of the partial waves a (k; g), which are characteristic of resonances and echoes; these properties are: (i) the rapid variation in the momentum variable k (or the energy E = k 2 ), including the passage through π2 (mod.π), of the phase-shift function k → δ (k; g), which satisfies [in view of (3.127)]: a (k; g) =
eiδ (k;g) sin δ (k; g) k
(k ∈ R+ ),
(5.11)
and therefore: δ (k; g) = Arg a (k; g)mod.π.
(5.12)
Resonances (resp., echoes) are characterized by the upward (resp., downward) passage of the phase-shift through ± π2 at a certain value k = kr (resp., k = ke ) at which [in view of (5.11)] a (k; g) = i/k. In the basic literature on the subject (see, e.g., [1, Chapter 2, Subsection 2.11 (c)]), the quantity 2∂δ /∂E(= k1 ×∂δ /∂k), whose positive or negative sign plays a role in the previous description of resonances or echoes, has been interpreted in terms of the time-delay or time-advance that the incident wave-packet undergoes in the scattering process, in a sense which has been introduced by Eisenbud. (ii) The production of a “bump” around k = kr or ke in the plot of the function k → |a (k; g)| and therefore of the cross-section σc (k; g), since one can write σc (k; g) = 4π(2 + 1)|a (k; g)|2 + 4π (2 + 1)|a (k; g)|2 , (5.13) =
the sum at the r.h.s. of the latter being subdominant near k = kr or ke . Assumption. We shall concentrate on the function a (k; g) for given fixed values of and g, and assume that among the various poles λ = λn (k; g) of the meromorphic function a(λ, k; g) which contribute to the sum on the r.h.s. of Eq. (5.10), there exists a distinguished pole, denoted simply by λ(k; g), with the corresponding residue ρ(k; g) = |ρ(k; g)|eiϕ(k;g) . We assume that the corresponding analytic function k → λ(k; g) satisfies the following properties: There exists a finite interval I = {k : 0 < kmin < k < kmax } such that for k ∈ I: (a) The function k → α(k; g) = Re λ(k; g) is an increasing function such that for a certain value k = k ∈ I one has: α(k ; g) = . (b) The function k → β(k; g) = Im λ(k; g) is such that 0 < |β(k; g)| 1 and |(∂β/∂k)(k; g)| (∂α/∂k)(k; g). a (k; g)| 1, where we have put: (c) “One-pole dominance”: |a (k; g) − " " a (k; g) =
−ρ(k; g) . [λ(k; g) − ][λ(k; g) + + 1]
(5.14)
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We shall then consider that the unitarity relation k|a (k; g)|2 − Im a (k; g) = 0,
(5.15)
[implemented by the parametrization (5.11)] is approximately satisfied by this one-pole dominant contribution " a (k; g) itself. Applying this approximation yields the following relation between the modulus and the argument ϕ of the residue ρ: # $ 8 17 (sin ϕ) −(α − )(α + + 1) + β 2 + (cos ϕ)β(2α + 1) , (5.16) |ρ| = k which, in particular, yields for k = k [in view of (a)]: 1 0 (cos ϕ)β(k )(2 + 1) |ρ(k ; g)| = (5.17) + O [β(k )]2 , k and therefore, in view of assumption (c): |a (k ; g)| ≈ |" a (k ; g)| ≈
| cos ϕ(k ; g)| . k
(5.18)
Variation of the phase-shift near k = k . In view of assumption (c) and of (5.12), and by taking the arguments of both sides of Eq. (5.14), one then obtains: δ (k; g) ≈ ϕ(k; g) ± π − Arg(λ(k; g) − ) − Arg(λ(k; g) + + 1).
(5.19)
A simple geometrical analysis, making an essential use of assumption (a), shows that: (1) If β(k; g) is positive, the function k → δ (k; g) admits an upward variation from a value of the form (ϕ(k; g) + ε) to (ϕ(k; g) + π − ε) for k varying on a short interval [k− , k+ ] centered at k (and depending on ε). In view of the assumption β(k; g) 1, the size of this interval can always be assumed to be such that: − 1 < α(k− ) < < α(k+ ) < + 1. We thus conclude that in a “generic way” (i.e., except if ϕ(k; g) = ± π2 ), the interval [k− , k+ ] will contain a value k = kr at which δ (kr ; g) = ± π2 , which therefore exhibits the typical behaviour of a resonance at k = kr . (2) Similarly, if β(k; g) is negative, the function k → δ (k; g) admits a downward variation from (ϕ(k; g)−ε) to (ϕ(k; g)−π+ε) for k varying on a short interval [k− , k+ ] centered at k . The latter also contains (in a generic way) a value k = ke at which δ (ke ; g) = ± π2 , which then exhibits the typical behaviour of an echo at k = ke . Bump for the cross-section near k = k . In both cases (1) and (2) of the previous analysis, one has, in view of (5.11): 1 |a (kr,e )| = , (5.20) kr,e (where kr,e = kr or ke ), and therefore the plot of the partial wave a (k; g) exhibits a bump around k = kr,e which is tangent to the upper limiting curve (k; g)| = 1/k. One can also notice the difference between (5.20) and the |amax value (5.18) of |" a (k ; g)|, which (if | cos ϕ(k ; g)| = 1) expresses the rapid variation of the phase-shift between k and kr,e . Finally, in view of (5.13), the
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plot of the cross-section σc (k; g) will also present a bump in an interval of the momentum variable containing the values k and kr,e . 5.3. Connection Between Descriptions of Phenomena in the λ-Plane and in the k-Plane The description of resonances and echoes of a given partial wave a (k; g), which has been given above, appeared to be completely symmetric. In both cases indeed, it was based on the assumption of a dominant one-pole approximation of the partial scattering function a(λ, k; g) in the complex λ-plane, such that the dominant pole λ = λ(k; g) be located at a very small distance |β(k; g)| from the real axis; the two cases are distinguished from each other by the sign of the function k → β(k; g). We are now going to show that, in spite of the previous apparent symmetry, these two cases necessarily correspond to completely different types of analyticity properties of the dominant function k → " a (k; g) in the complex k-plane. (1) The case of resonances. In the r.h.s. of (5.14), it is the factor λ(k; g) − = α(k; g) − + iβ(k; g) at the denominator which is responsible for the rapid variation of δ (k; g) near k = k , and whose vanishing in the complex k-plane must be analysed. Since the function (k real) → λ(k; g) is holomorphic in a small complex neighbourhood V of k , one can postulate the validity (in V) of the following first-order Taylor approximation λ(k; g) ≈ +iβ(k ; g)+ ∂λ ∂k (k ; g)(k −k ), which therefore yields: λ(k; g) − ≈
∂λ (k ; g)[(k − k ) + iγ], ∂k
(5.21)
where we have put: γ=
β(k ; g) . ∂λ ∂k (k ; g)
(5.22)
∂α ∂λ ∂α Since it was supposed that ∂β ∂k ∂k , we can say that ∂k (k ; g) ≈ ∂k , which is real and positive, and [in view of (5.22)] this positivity property is also true for γ. In view of (5.21), the expression (5.14) of " a (k; g) therefore factorizes a pole located at k = k − iγ, and γ can thus be related to the standard width parameter of the Breit-Wigner one-pole approximation. The shape of the bump for the cross-section: If one first considers the case when the residue ρ(k; g) is real (for k real), so > 0, we can see that the expression (5.14) that, in view of (5.16): ρ = β(2α+1) k of " a (k; g) can be approximated by −β −γ ≈ . k(α − + iβ) k(k − k + iγ)
(5.23)
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This approximation, which is valid for k varying in a suitable interval centered at k , gives the standard Lorentzian contribution to the cross-section, namely: σ " (k; g) ≈
k 2 [(k
4πγ 2 . − k )2 + γ 2 ]
(5.24)
In the general case, the residue is complex and satisfies Eq. (5.16). Then, by taking Eqs. (5.21) and (5.22) into account, expression (5.14) now yields a contribution to the cross-section which is of the following form: 4π[sin ϕ(k; g)(k − k ) − γ cos ϕ(k; g)]2 . (5.25) k 2 [(k − k )2 + γ 2 ] This asymmetric contribution corresponds to the generalized form of a BreitWigner one-pole model, when the unitary partial wave S (k) includes an additional phase function ϕ(k), namely: k − k − iγ . (5.26) S (k) = e2iϕ(k) k − k + iγ σ " (k; g) ≈
(2) The case of echoes. In view of the apparent symmetric treatment of resonances and echoes that we have given above, one would be tempted to apply again the previous Taylor expansion argument to the analysis of the factor [λ(k; g) − ] when β(k; g) is negative. However, the analog of formula (5.21) would exhibit a pole in the upper half-plane at k = k − iγ, with γ < 0, corresponding to the real value λ = . But such a result is contradictory with the constraint imposed by the Wronskian Lemma (see, after Lemma 34, the paragraph (a) in “Application: regions of (λ, k)-space free of singularities”). This impossibility of having singular pairs (λ, k) at λ real and Re k > 0, Im k > 0, which was known to hold in general for local potentials, has indeed been extended here to the large classes Nwγ(ε),α of nonlocal potentials. In order to have a full account of echoes associated with dominant poles of a(λ, k; g) that may be produced by the theory of nonlocal potentials, one is thus faced to imagine the following type of mathematical model (the parameter g being not considered here): construct a holomorphic function k → λ(k) satisfying for each value of ( ≥ 0) the three previous assumptions (a), (b), and (c) with β(k) = Im λ(k) < 0 for k ∈ I, and such that, in addition, Im λ(k) remains strictly negative when k varies in the upper half-plane, near I. As a tutorial model, one can propose the following function, in which c0 and θ0 denote positive constants, with θ0 1: 2θ0 1 λ(k) = − + c0 e−iθ0 k π . (5.27) 2 One easily checks that (i) the real values k = k ; Re λ(k ) = are such that the corresponding points λ(k ) belong to the half-line Arg(λ + 12 ) = −θ0 and that the assumptions (a) and (b) (see Sect. 5.2) are satisfied for each . (ii) For Re k > 0, Im k > 0, the corresponding point λ(k) remains in the lower half-plane (namely in the sector −θ0 < Arg(λ + 12 ) < 0).
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Note that real values λ(k) = are produced for imaginary values k = iκ (possibly interpreted as bound states) and that the holomorphic function k → λ(k) admits an analytic continuation in the second quadrant of the k−plane, such that Im λ(k) > 0, in agreement with the symmetry property: λ(−k) = λ(k). So, if we assume that for each , the function " a (k) associated with λ(k) by formula (5.14) is a dominant-pole contribution to the partial-wave a (k), we have seen in Sect. 5.2 that the latter enjoys the characteristic phase-shift properties of an echo near k = k and that its contribution to the crosssection produces a bump near k = k . We have thus checked that this tutorial mathematical model produces what can be called “an echo trajectory”, which is consistent with the general requirements that have been obtained above for the locations of the singularities of the partial scattering amplitude T (λ, k; g) in the complex (λ, k)-space. In conclusion, we can say that the theory of nonlocal potentials presented in this paper provides a theoretical ground for analyzing the echoes in terms of poles of the partial scattering amplitude localized for k real in the fourth quadrant of the λ-plane, a possibility which is excluded in the usual theory of local potentials. From the conceptual viewpoint, such a scenario based on a unified description of analytic singularities of the partial scattering amplitudes in the variables (λ, k) would be more satisfactory than the standard one based on the ad hoc adjunction of an impenetrable sphere to a given local potential. It would indeed be attractive to show that certain nonlocal potentials can produce at the same time a “Regge trajectory” λ = λ(k) with β > 0 describing a sequence of resonances, whose angular momentum increases with k, and a similar “image-trajectory” λ = λ(k) with β < 0 describing a corresponding sequence of echoes, whose angular momentum would also increase with k, in agreement with the phenomenological study of various nuclear scattering processes (see [11,12]).
Appendix A. Bounds on the “Angular-Momentum Green Function” A.I. Discrete Angular Momentum Analysis We shall use the partial wave expansion of the (free-Hamiltonian) Green function:
2
2
1/2
eik(R +R −2RR cos θ) 1 . 1 eik|R−R | = , g(cos θ, k; R, R ) = 4π |R − R | 4π (R2 + R 2 − 2RR cos θ)1/2
(A.1)
namely: g(cos θ, k; R, R ) = −
∞ =0
(2 + 1)
G (k; R, R ) P (cos θ), 4πRR
(A.2)
in which the set of coefficients G (k; R, R ), called “angular-momentum Green function” are given by:
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+1 g(cos θ, k; R, R )P (cos θ) d cos θ.
(A.3)
−1 Relations (3.10e): i.e., G (k; R, R ) = −ikRR j [k min(R, R )]h(1) [k max(R, R )], (implied by the fact that G satisfies Bessel equations with appropriate boundary conditions separately with respect to R and R ) will not be used here, since we shall obtain relevant bounds on G by direct use of formula (A.3). In formula (A.3), k may be real or complex, namely the functions G are defined for all R > 0, R > 0 as entire functions of k, and one has: 2
4π|g(cos θ, k; R, R )| =
e− Im k[(R−R )
1
+2RR (1−cos θ)] 2 1
[(R − R )2 + 2RR (1 − cos θ)] 2
.
(A.4) 1
For Im k ≥ 0, the latter is uniformly bounded by [2RR (1 − cos θ)]− 2 , while for Im k < 0 it is uniformly bounded by
e
| Im k|(R+R )
1
[2RR (1−cos θ)] 2
. From (A.3) one thus
obtains the following global majorization: |G (k; R, R )| ≤
1 1 (RR ) 2 max 1, e− Im k(R+R ) 2
+1
−1
≤
1 max 1, e− Im k(R+R ) 2
|P (t)| . dt 2(1 − t)
πRR 12 2 + 1
.
(A.5)
For writing the rightmost inequality of (A.5), we1have used the Martin inequal0 ity (see [44]) |P (cos θ)| < min 1, 2[π sin θ]−1/2 . (Note that for k real, bound (A.5) itself lies in [44]). We shall now derive an alternative bound on G , which exhibits a decrease property with respect to |k| (k ∈ C). Let us rewrite Eq. (A.3) as follows, by 1 2 introducing the change of integration variable u = [R2 + R − 2RR cos θ] 2 , 2 2 −u2 i.e., cos θ(u) = R +R : 2RR 1 G (k; R, R ) = 2
R+R
eiku P (cos θ(u)) du.
(A.6)
|R−R |
We now have (by using integration by parts): 1 ikG (k; R, R ) = 2
=
1 2RR
R+R
|R−R |
R+R
|R−R |
d iku [e ] P (cos θ(u)) du du
eiku P (cos θ(u)) u du +
' 1 & ik(R+R ) − (−1) eik|R−R | , e 2 (A.7)
(where we have used the fact that P (1) = 1, P (−1) = (−1) ).
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From the integral representation P (cos θ) = one readily obtains the following bound |P (cos θ)| ≤
1 π
π 0
739
(cos θ+i sin θ cos α) dα,
. | sin θ|
(A.8)
The latter allows one to give a majorization for the r.h.s. of Eq. (A.7), which yields: ⎡ ⎤ R+R u du ⎢ ⎥ |k||G (k; R, R )| ≤ max 1, e− Im k(R+R ) ⎣1 + ⎦. 2 RR | sin θ(u)| |R−R |
(A.9) 1
Since 2RR sin θ(u) = [(R + R + u)(R + R − u)(u + R − R )(u + R − R)] 2 , one then gets the following majorization: 2
R+R
|R−R |
u du ≤ RR | sin θ|
R+R
|R−R |
du 1
1
(R + R − u) 2 (u − |R − R |) 2
= π,
(A.10)
.
(A.11)
and therefore, from (A.9):
− Im k(R+R )
|G (k; R, R )| ≤ max 1, e
1 + π |k|
As a result of (A.5) and (A.11), we can thus write the following global uniform bound, which exhibits decrease properties with respect to both variables and |k| when they go to infinity: |G (k; R, R )| % 1 1 π + 1 1 π ≤ max 1, e− Im k(R+R ) (1 + R) 2 (1 + R ) 2 min , . |k| 2 2 + 1 (A.12) A.II. Complex Angular Momentum Analysis We shall now introduce a function G(λ, k; R, R ), called the complex-angularmomentum Green function, defined for all complex λ in the half-plane C+ = − 12 7 8 1 λ : Re λ > − 2 , such that for all positive integers , one has: G (k; R, R ) = G(, k; R, R ). For every R, R (R > 0, R > 0), the function G will be uniquely " where C " defined as a holomorphic function of (λ, k) in the product C+ 1 × C, −2
denotes the universal covering of C\{0}. The uniqueness of this interpolation of G will be ensured by the fact that for k = iκ, κ > 0, G(λ, iκ; R, R ) is a Carlsonian interpolation, which will thus allow us to specify the “basic” first. sheet C(cut) = C\(−∞, 0] of G. Our purpose now is the derivation of uniform bounds for |G(λ, k; R, R )| in {(λ, k) ∈ C+ × C(cut) }. −1 2
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(1) Analysis for k = iκ, κ > 0. Let z0 ≡ z0 (R, R ) = 12 ( RR + RR ). For k = iκ, κ > 0, the Green function g [see (A.1)] can be conveniently rewritten as follows in terms of the complex variable z = cos θ: 1
1
1 e−κ(2RR ) 2 (z0 −z) 2 g(z, iκ; R, R ) = . 4π (2RR ) 12 (z0 − z) 12
(A.13)
1 1 . Since the function u(R, R ; z) = (2RR ) 2 (z0 − z) 2 (specified as being positive for z real, z < z0 ) is such that Re u(R, R ; z) ≥ 0 for z varying in the (closed) (cut) . cut-plane Cz0 = C\[z0 , +∞[, the following uniform bound holds:
|g(z, iκ; R, R )| ≤
For z ∈ C(cut) , z0
1 1 . 4π (2RR ) 12 |z0 − z| 12
(A.14)
It follows that, as a holomorphic function of z, g(z, iκ; R, R ) satisfies the conditions of the Froissart–Gribov theorem (or Laplace-transformation on the one-sheeted hyperboloid in the sense of [40]). Therefore, there exists a function G(λ, iκ; R, R ), holomorphic in the half-plane {λ ∈ C+ } such that for all − 12 integers ( ≥ 0), one has G(, iκ; R, R ) = G (iκ; R, R ). Moreover, this function G is given by the following integral in terms of the discontinuity Δg of 'g & 1
across the cut z ∈ [z0 , +∞[, namely Δg(z, iκ; R, R ) = RR G(λ, iκ; R, R ) = − 2
1
1
z0
1
(2RR ) 2 (z−z0 ) 2
1 1 +∞ cos κ(2RR ) 2 (z − z0 ) 2 1
(2RR ) 2 (z − z0 ) 2
(R,R )
1
cos κ(2RR ) 2 (z−z0 ) 2
1 4π
:
Qλ (z) dz. (A.15)
In this equation, Qλ denotes the second-kind Legendre function, and we note . that the integral is convergent for all λ in C+ −1 2
Bounds on G(λ, iκ; R, R ): (a) In view of (A.15), we have: +∞
1
(RR ) 2 √ |G(λ, iκ; R, R )| ≤ 2 2
|Qλ (z)|
dz.
1
z0 (R,R )
[z − z0 (R, R )] 2
(A.16)
Then, by using the following integral representation of Qλ (see, e.g., [40, formula (III-11)]): 1 Qλ (z) = π
+∞
. v =cosh−1 z
1
1
e−(λ+ 2 )w [2(cosh w − cosh v)]− 2 dw,
(A.17)
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and the relation cosh−1 z0 (R, R ) = ln RR , we obtain: +∞
|Qλ (z)| 1
z0 (R,R )
≤
1 π
[z − z0 (R, R )] 2 +∞
dz .
1
e−(Re λ+ 2 )w dw
|ln RR |
ζ =cosh w
dz 1
z0 (R,R )
{2(ζ − z)[z − z0 (R, R )]} 2
.
(A.18)
But, since the subintegral in the r.h.s of (A.18) is equal to the constant π √ , we obtain, in view of (A.16) and (A.17): 2 |G(λ, iκ; R, R )| ≤
(Re λ+ 12 ) 1 R R (RR ) 2 , . min 2(2 Re λ + 1) R R
(A.19)
(b) We shall now derive an alternative bound on G, which exhibits a decrease property with respect to κ. By making use of the integration variable u "= 1 1 u "2 u) = z0 + 2RR u "(z) = (2RR ) 2 (z −z0 ) 2 and of the inverse mapping z = z(" , we rewrite Eq. (A.15) as follows: 1 G(λ, iκ; R, R ) = − 2
+∞ cos κ" u Qλ (z(" u)) d" u.
(A.20)
0
We then have: 1 κG(λ, iκ; R, R ) = − 2
+∞
d [sin κ" u]Qλ (z(" u)) d" u d" u
0 +∞ 1 u " = sin κ" uQλ (z(" u)) d" u. 2 RR
(A.21)
0
From (A.17) (and making use of a partial integration procedure) we can deduce the following integral representation for Qλ : λ+ 12 Qλ (z) = − π
+∞
z
1
e−(λ+ 2 )w dζ
1 − 1 2 π 2 [2(ζ − z)] (ζ − 1)
+∞
1
e−(λ+ 2 )w ζ dζ 1
z
3
[2(ζ − z)] 2 (ζ 2 − 1) 2
, (A.22)
in which w = cosh−1 ζ. By taking the latter into account in Eq. (A.21) and inverting the integrations over z and ζ, one obtains:
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ζ sin κ(2RR ) 12 (z − z0 ) 12 λ+ e dζ κG(λ, iκ; R, R ) = − dz 1 2π ζ2 − 1 [2(ζ − z)] 2 z0 z0 +∞ ζ sin κ(2RR ) 12 (z − z0 ) 12 −(λ+ 12 )w e 1 − dz. 3 ζ dζ 1 2π (ζ 2 − 1) 2 [2(ζ − z)] 2 1 2
+∞
z0
−(λ+ 12 )w
z0
(A.23) A uniform bound for the first term on the r.h.s. of Eq. (A.23) is obtained by simply majorizing the sine-function by one. In fact, this term is majorized in } by: the whole half-plane {λ ∈ C+ −1 2
Re λ + 2π
+∞
1 2 z0
(R,R )
1
{2[ζ − z0 (R, R )]} 2 dζ ≤ A1 × (2 Re λ + 1), (ζ 2 − 1)
(A.24)
where A1 is a numerical constant (independent of k, R and R ). For obtaining a uniform bound for the second term on the r.h.s. of Eq. (A.23) it is necessary to . majorize the sine-function: (a) by one in the range z ≥ z1 = z0 + (2RR κ2 )−1 , 1 12 and (b) by κ(2RR ) (z−z0 ) 2 in the range z0 ≤ z ≤ z1 . One is then led to introduce a partition of the integration region into three subregions of the (z, ζ)plane, namely R1 = {ζ ≥ z1 , z0 ≤ z ≤ z1 }, R2 = {z0 ≤ ζ ≤ z1 , z0 ≤ z ≤ ζ}, and R3 = {ζ ≥ z1 , z1 ≤ z ≤ ζ}. While the integrals in R1 and R2 yield uniform majorizations by numerical constants, 0 the integral in1R3 is majorized by an expression of the form [a1 + a2 max ln (RR κ2 )−1 , 0 ] (a1 and a2 being numerical constants). By taking all these estimates into account for the r.h.s. of (A.23), one obtains a majorization of the following form in the half-plane {λ ∈ C+ }: − 12 1 0 A1 (2 Re λ + 1) + A2 + A3 ln 1 + (RR κ2 )−1 |G(λ, iκ; R, R )| ≤ 0 κ 1 2(Re λ + 1) + ln 1 + κ−2 1 ≤C 1 + ln 1 + κ R 1 × 1 + ln 1 + , (A.25) R where A1 , A2 , A3 , and C are suitable numerical constants. As a result of (A.19) and (A.25), we can thus write the following global uniform bound, which exhibits decrease properties with respect to both variables λ and κ when they go to infinity: √ √ 1 1 |G(λ, iκ; R, R )| ≤ 1 + ln 1 + + R 1 + ln 1 + + R R R 4 0 15 2(Re λ+1)+ln 1+κ−2 1 ,C . (A.26) × min 2(2 Re λ + 1) κ Bounds on
∂ ∂R G(λ, iκ; R, R ):
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We shall obtain a relevant expression for this derivative of G by computing the derivative of the double integral on the r.h.s. of Eq. (A.23) with respect to R. Using the fact that the successive integrands of the latter in the variables ζ and z vanish at their common threshold z0 (R, R ), we obtain the following integral representation (in which w = cosh−1 ζ): ∂ G(λ, iκ; R, R ) ∂R +∞ 12 λ + 12 R ζ −(λ+ 12 )w + dζe = 2R (ζ 2 − 1) (ζ 2 − 1) 32 z0 (R,R ) 1 1 ζ cos κ(2RR ) 2 (z − z0 ) 2 × dz 1 [2(ζ − z)] 2 z0 (R,R ) & ' ⎧ ⎫ R R ⎨ ⎬ R − R 1 2 × . 1 − [z − z0 (R, R )] ⎩ 2[z − z0 (R, R )] 2 ⎭
2π
(A.27)
By proceeding as for the bound (A.19) on G, we now deduce the following bound from (A.27): ∂ G(λ, iκ; R, R ) ∂R 1 ≤ √ 4 2
R 2R
12 R R − R R
12
1 + √ 4 2
+∞
e
z0
R 2R
+∞
−(Re λ+ 12 )w
(R,R )
−(Re λ+ 12 )w
e z0
(R,R )
|λ+ 12 | ζ + 2 2 (ζ − 1) (ζ − 1) 32
|λ + 12 | ζ + (ζ 2 − 1) (ζ 2 − 1) 32
× [ζ − z0 (R, R )] dζ.
dζ
(A.28)
By using the inequality ζ −z0 ≤ ζ −1 one readily obtains that the latter integral in (A.28) is convergent and bounded by a (λ-dependent) constant in the . whole . By now making the change of variable u = sinh w = ζ2 − 1 half-plane C+ − 12 in the former integral of0 (A.28), this integral can be major one1 also ∞sees that −2 1 + 1 × ized (for λ ∈ C+ ) by λ + du. As a result, one can u 1 R R 2 −1 | − | 2
2
R
R
replace the inequality (A.28) by a simple majorization of the following form: 12 ∂ R ≤ c G(λ, iκ; R, R ) (λ) × , 1 ∂R R which is valid for all k = iκ(κ > 0) and λ ∈ C+ . −1 2
(A.29)
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(2) Analytic continuation in k. We start from the definition (A.15) of G, in which we insert the integral representation (A.17) of Qλ and then invert the integrations over z and ζ = cosh w. We thus obtain: G(λ, iκ; R, R ) =−
+∞
1 2
(RR ) √ 2 2π
−(λ+ 12 )w
e
1
z0 (R,R )
(ζ 2 − 1) 2
dζ
ζ cos κ(2RR ) 12 (z − z0 ) 12
z0
1
{2(ζ − z)[z − z0 (R, R )]} 2
dz. (A.30)
We now wish to define the analytic continuation of this double integral with respect to the complex variable k by putting k = iκe−iφ ; φ will be taken in the (cut) of interval − π2 ≤ φ ≤ 3π 2 so that G be defined in the “basic first sheet” C the k-plane. (a) For |φ| ≤ π2 , this analytic continuation of G(λ, k; R, R ) = G(λ, iκe−iφ ; R, R ) is well-defined by shifting in C2 the integration region from its initial . situation at k = iκ, namely Γ0 = {(ζ, z) : z0 = z0 (R, R ) ≤ z ≤ ζ < +∞} . to the set Γφ = {(ζ, z) : ζ −z0 = |ζ −z0 |e2iφ , z −z0 = |z −z0 |e2iφ ; z0 ≤ |z| ≤ 1 |ζ| < +∞}. The corresponding rotation of angle φ of (z − z0 ) 2 will then cancel the rotation of angle −φ of κ in the cosine-factor under the integral on the r.h.s. of (A.30), so that this factor can always be bounded by one. It follows that one obtains a majorization for the analytic continuation at k = iκe−iφ of the r.h.s. of (A.30) which involves the same subintegral over z as in (A.18) (equal to the constant √π2 ), namely: 1
(RR ) 2 |G(λ, k; R, R )| ≤ 4
−(λ+ 12 )(w+iϕ) e |d(w + iϕ)|;
γφ (z0 (R,R ))
(A.31) . in (A.31), γφ (z0 (R, R )) is the image of γ "φ = {ζ = z0 (R, R ) + ρe2iφ , ρ ∈ . [0, +∞)} by the mapping ζ → w " = w + iϕ = cosh−1 ζ. One can check that in the path γφ (z0 (R, R )) the variables |ϕ| and w vary respectively in the intervals |ϕ| ∈ [0, 2φ] and w ∈ [w0 (R, R ), +∞), where w0 (R, R ) is positive and such that: (i) if |φ| ≤ π4 , cosh w0 = z0 (R, R ) = 12 RR + RR , i.e.: R R −w0 (R,R ) e = min , ; (A.32) R R π 4
π 2,
2 2 2 cosh2 w0 (R, R ) = z0 (R, R ) sin 2φ + cos 2φ, which yields: sinh w0 (R, R ) = 12 RR − RR sin 2|φ|, and thereby: R R e−w0 (R,R ) ≤ (sin 2|φ|)−1 min , . (A.33) R R
(ii) if
≤ |φ| ≤
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Note that in case (i), γφ (z0 (R, R )) defines |ϕ| as an increasing function of w in the interval [w0 (R, R ), +∞), while in case (ii), γφ (z0 (R, R )) is tangent at the line w = w0 (R, R ) at some point ϕ0 (with 0 ≤ ϕ0 ≤ π2 ). By taking these geometrical facts into account, one then deduces from (A.31) a majorization of the following form, which is valid for all k in the closed upper half-plane: 1 1 (RR ) 2 e−(Re λ+ 2 )w+(Im λ)ϕ | d(w + iϕ)| |G(λ, k; R, R )| ≤ 4 γφ (z0 )
1
1
≤ c(λ, k)(RR ) 2 e−(Re λ+ 2 )w0 (R,R ) , where:
2 Im λφ(k)
c(λ, k) = c max e
,1 1+
1 2 Re λ + 1
(A.34) .
(A.35)
In the latter, c is a numerical constant and φ(k) = − Arg(−ik); more precisely, k = iκe−iφ , with φ = φ(k) such that |φ| ≤ π2 . Moreover, by taking Eqs. (A.32) and (A.33) into account, we see that (A.34) implies the following majorization, which is valid globally in the set {(λ, k) : λ ∈ C+ ; Im k > 0}: −1 2
−(Re λ+ 12 )
|G(λ, k; R, R )| ≤ c(λ, k)[Φ(k)]
Re λ+ 12 R R (RR ) min , , R R (A.36)
1 2
in which we have put: Φ(k) = 1 if |φ(k)| ≤
π , 4
(A.37)
and π π ≤ |φ(k)| < . (A.38) 4 2 We also notice that, since w0 (R, R ) ≥ 0, majorization (A.34) also yields for all k in the closed upper half-plane Im k ≥ 0: 1 1 12 π| Im λ| |G(λ, k; R, R )| ≤ c(λ, k)(RR ) ≤ c e 1+ (RR ) 2 . 2 Re λ + 1 (A.39) Φ(k) = (sin 2|φ(k)|)
if
By performing the same contour distortion argument on the integral (A.27) for defining the analytic continuation in k of the function ∂G ∂R (λ, iκ; R, R ), and by proceeding as for the derivation of bound (A.29), we obtain an extension of the latter to the full half-plane Im k > 0, which is of the following form: 12 ∂G R ≤ " c (λ, k; R, R ) (λ, k) . (A.40) 1 ∂R R
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−iφ (b) For π2 < φ ≤ 3π ; R, R ) may be 2 , the analytic continuation of G(λ, iκe pursued, but the “rotated cycle” Γφ now acquires an additional part whose . support is the real set Γr = {(ζ, z) : −1 ≤ ζ ≤ z ≤ z0 (R, R )}; in fact, " since for φ ≥ π2 , the the inclusion of Γr is more easily seen in the w-plane, contour γφ (z0 (R, R )) may be distorted so as to contain the broken line [v0 , 0]∪[0, −2iπ]∪[−2iπ, −2iπ +v0 ], completed by an infinite branch whose asymptote is the line ϕ = 2φ (i.e., the image of γ "φ from a second sheet). In this new situation, the bound that one obtains for the analytic continuation at k = iκe−iφ of the r.h.s. of (A.30) still contains the constant subintegral over z (equal to √π2 ), but the latter is now obtained after a majorization of the cosine-factor by cosh[κ(R+R ) cos φ] = cosh[Im k(R+R )]. Moreover, since the range of values of ϕ in this analytic continuation of (A.30) admits 3π as its maximal value, the latter majorization (A.37) must now be replaced by
|G(λ, k; R, R )|
0 1 1 ≤ c(RR ) 2 cosh[Im k(R + R )] max e3π Im λ , 1 1 +
1 2 Re λ + 1
, (A.41)
which is valid for all k in the lower half-plane (of the basic first sheet) and λ ∈ C+ . (For simplicity, we have used the same constant c in (A.41) as − 12 in (A.34) and (A.37), being not concerned with the best values of these constants). A.III. Complements on Bessel and Hankel Functions (1)
(a) Bounds on the spherical Bessel and Hankel functions j and h for integer ( ≥ 0): The following inequalities have been established in [27,26]; for all k ∈ C and R ≥ 0, there hold: (+1) |k|R eR| Im k| , (A.42) |kRj (kR)| ≤ c 1 + |k|R 1 + |k|R (1) (1) e−R Im k , (A.43) kRh (kR) ≤ c |k|R (1)
where c and c are constants whose dependence on is not exploited here. (b) The derivatives of the spherical Bessel and Hankel functions: Starting from the following relation (see [36, Vol. 2, p. 11, formula (50)]): d ν [z Jν (z)] = z ν Jν−1 (z) (ν ∈ C), dz and recalling that % π J+1/2 (z), j (z) = 2z
(A.44)
(A.45)
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one obtains: d [zj (z)] = −j (z) + zj−1 (z), dz which yields, in view of (A.42): d |k|R R| Im k| , dR [Rj (kR)] ≤ c 1 + |k|R e
(A.46)
(A.47)
where c = c + c−1 . This bound is valid for all k ∈ C and R ≥ 0. (1) By using a similar formula for the Hankel functions Hν (z), namely (see, e.g., [26, p. 361, Eq. (9.1.30)]): d & ν (1) ' (1) z Hν (z) = z ν Hν−1 (z) (ν ∈ C), (A.48) dz . π (1) (1) H+1/2 (z), one also obtains: together with the relation h (z) = 2z d & (1) ' (1) (1) zh (z) = −h (z) + zh−1 (z). dz
(A.49)
In view of (A.43), the latter equality yields the following bound, which is valid for all k ∈ C and R ≥ 0: +1 ' d & (1) 1 + |k|R (1) e−R Im k , (A.50) dR Rh (kR) ≤ c |k|R where c = c + c−1 . (c) The Sommerfeld condition for the spherical Hankel functions: We now want to prove the following property: For all k such that k = 0, there holds the following behaviour in the limit R → +∞: & ' 1 1 (1) (1) R Im k d e (A.51) dR Rh (kR) − ikRh (kR) = |k|3 O R2 . (1)
(1)
(1)
Formula (A.51) derives from the following representation of the Hankel functions [45, p. 117] by an asymptotic series (in the sense of Poincar´e): % 2 i[ρ−(ν+ 1 ) π ] (ν, m) (1) 2 2 e , (A.52) Hν (ρ) = πρ (−2iρ)m m=0,1,2,... where: (ν, m) =
(4ν 2 − 1)(4ν 2 − 9) · · · (4ν 2 − {2m − 1}2 ) , 22m m!
(ν, 0) = 1. (A.53)
This series reduces exceptionally to a finite sum whenever the subscript ν takes a half-integral value. In fact, one can easily verify that if ν = + 12 the symbols (ν, m) are zero for all integers m such that m > . In these cases, the series (A.52) represents exactly the Hankel functions. In view
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. ! (1) (1) of the relation Rh (kR) = π2 R k H+1/2 (kR), we deduce from (A.52) the following representation: 0 1 % % π R (1) 1 i[kR−(+1) π ] + 12 , m (1) 2 H (kR) = e . Rh (kR) = 2 k +1/2 k (−2ikR)m 0≤m≤
(A.54) Then we get by a direct computation ⎧ 0 1⎫ ⎨1 ' + 1, m ⎬ π d & (1) 2 ei[kR−(+1) 2 ] Rh (kR) = ik ⎩k dR (−2ikR)m ⎭ 0≤m≤ 0 1 m + 12 , m π 1 − ei[kR−(+1) 2 ] k (−2ikR)m+1 1≤m≤
(1)
= ikRh (kR) +
1 i[kR−(+1) π ] P (−1) (kR) 2 e , k (kR)+1
(A.55)
in which P (−1) denotes a polynomial of degree − 1 whose all coefficients are different from zero. One readily checks that the latter yields limit (A.51) (for all k such that k = 0). (d) Bounds on the spherical Bessel functions jλ (z) for λ ∈ C+ and z ∈ −1 2
C(cut) : .π We recall that jλ (z) = 2z Jλ+1/2 (z), and start from the following integral representation of the Bessel function Jλ (z) [28], which is valid for z ∈ R+ and Re λ > 0: +∞ π 1 sin πλ −iz sin ϕ+iλϕ Jλ (z) = e dϕ − e−z sinh ξ−λξ dξ. (A.56) 2π π −π
0
This representation defines Jλ (z) as the sum of two holomorphic functions J (1) and J (2) of λ and z. As an integral on the interval [−π, π], the first term on the r.h.s. of (A.56) defines J (1) as an entire function of (λ, z) satisfying the following global bound in C2 : (1) (A.57) J (λ, z) ≤ e| Im z| eπ| Im λ| . Consider now the function J (2) (λ, z) to be defined by the second term on the r.h.s. of (A.56). For (λ, z) ∈ C+ × C+ , the integral in that term is easily majorized by a convergent integral (thanks to the minoration sinh ξ > ξ in the exponential under the integral), which shows that J (2) is well-defined and analytic in this set and such that: eπ| Im λ| eπ| Im λ| (2) ≤ . (A.58) J (λ, z) ≤ π(Re λ + Re z) π Re λ We now obtain an analytic continuation of J (2) and an extension of the previous bound for (λ, z) ∈ C+ × C(cut) by distorting the integration path
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γ0 = [0, +∞[ of the second integral of (A.56) into any path γφ whose support is the following set: {ξ ∈ [0, −iφ]} ∪ {ξ = −iφ + β : β ≥ 0}, with |φ| ≤ π2 . The corresponding integral can then be replaced by φ
iz sin u+iλu
e
iφ
du + e
0
+∞ e−z(cos φ sinh β−i sin φ cosh β) e−λβ dβ,
(A.59)
0
which can be bounded in modulus by ⎡ ⎤ ∞ e| Im λ||φ| ⎣|φ|e| Im z|| sin φ| + e−(Re z cos φ+Im z sin φ) sinh β e−(Re λ)β dβ ⎦ , 0
(A.60) in the half-plane ε(φ) Im z > 0 (ε(φ) denoting the sign of φ). In the sector with equation Re z + Im z tan φ > 0 of this half-plane, the previous integral can then be majorized by (Re λ)−1 (for any value of φ). For z varying in C(cut) , one then obtains a global bound for the expression $ (A.60) with the choice π # φ = ± π2 , which is equal to e| Im λ| 2 π2 e| Im z| + Re1 λ . It then yields: 1 | Im z| 1 (2) | Im λ| 3π 2 ≤ e e (λ, z) + . (A.61) J 2 π Re λ By now putting together the bounds (A.57), (A.61), we obtain the √ (A.58), and. following majorizations for the function kRjλ (kR) = π2 Jλ+1/2 (kR): for (λ, k) ∈ C+ × C+ , − 12 %π √ 1 e| Im k|R eπ| Im λ| 1 + ; (A.62) kRjλ (kR) ≤ 2 π(Re λ + 12 ) × C(cut) , for (λ, k) ∈ C+ − 12 %π √ 3π 3 1 e| Im k|R e 2 | Im λ| + . kRjλ (kR) ≤ 2 2 π(Re λ + 12 )
(A.63)
Appendix B. Continuity and Holomorphy Properties of Vector-Valued and Operator-Valued Functions We recall some general facts about continuous and holomorphic functions taking their values in complete normed spaces on the field of complex numbers, denoted by A (resp., B or C) in the following. The norm of an element a of A is denoted by aA , or simply a if there is no ambiguity. D will denote a given domain either in Rm or in Cm . The real or complex variables whose range is D are called ζ = (ζ1 , . . . , ζm ), and we shall consider vector-valued functions ζ → a[ζ] such that for all ζ ∈ D, a[ζ] belongs to A and a[ζ] is uniformly bounded on every compact subset of D. By definition, the function ζ → a[ζ] is continuous in D if limh→0 a[ζ + h] − a[ζ] = 0 for every ζ ∈ D and h varying in a neighborhood Vζ of zero such that ζ + Vζ ⊂ D. When ζ is complex, the function ζ → a[ζ] is holomorphic in D if there exist
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m functions ζ → a˙ j [ζ] defined and continuous in D with values in A, called the partial derivatives of a[ζ], such that: limh→0
a[ζ+hj ]−a[ζ] h th
− a˙ j [ζ] = 0 for
every ζ ∈ D and hj = (0, . . . , 0, h, 0, . . . , 0), h being the j -component of hj ; the increments hj , (1 ≤ j ≤ m), are supposed to vary in a neighborhood Vζ of zero, which is chosen such that ζ + Vζ ⊂ D. These definitions of “continuity” and “holomorphy” (or “analyticity”) for vector-valued functions, (also called more precisely “strong continuity” and “strong holomorphy or analyticity”) will be used directly in the following survey of various product operations given in Sect. B.I and of the vector-valued functional interpretation of functions depending on real or complex parameters that we give in Sect. B.II. For completeness, we shall also summarize in Sect. B.III the various (equivalent) “weak” and “strong” characterizations of “vector-valued holomorphy”, by presenting them directly in the several-variable case. From the previous definitions and by using the norm inequality in A, one readily checks that any sum of continuous (resp., holomorphic) vector-valued functions is a continuous (resp., holomorphic) vector-valued function. This property extends to uniformly convergent series of vector-valued functions: Lemma B.1. Let {ζ → an [ζ]; n ∈ N} be a sequence of vector-valued functions in D taking their values in the complete normed space A and such that for all ζ in D there holds a majorization ∞of the form: an [ζ] ≤ un , where the sequence is such that M = {un }∞ n=0 n=0 un < ∞. Then there exists a vector-valued function ζ →
s[ζ] taking its values in A, such that for all ζ in D one has ∞ s[ζ] = n=0 an [ζ], as the sum of a convergent series in A, with s[ζ] ≤ M . Moreover, (i) if the functions an [ζ] are continuous in D, then s[ζ] is continuous in D; (ii) if ζ is complex and if the functions an [ζ] are holomorphic in D, then s[ζ] is holomorphic in D. Proof. For all ζ, the series with general term an [ζ] is dominated by the series with general term un and therefore convergent. Then the norm inequality N2 N2 an [ζ] ≤ n=N an [ζ] and the completeness property of A imply n=N 1 1 . N the convergence in A of the sequence {sN [ζ] = n=0 an [ζ]; N ∈ N} to a vector s[ζ] such that s[ζ] ≤ M . Moreover, ∞ (i) For any given ε, let Nε be such that p=Nε up ≤ 3ε . If the functions an [ζ] are all continuous, for any given ζ in D there exists a neighborhood Vζ,ε of zero such that sNε [ζ + h] − sNε [ζ] ≤ 3ε for all h ∈ Vζ,ε . Then by writing the norm inequality s[ζ + h] − s[ζ] ≤ s[ζ + h] − sNε [ζ + h] + sNε [ζ + h] − sNε [ζ] + sNε [ζ] − s[ζ], one sees that each term on the r.h.s. of this inequality is bounded by 3ε , and therefore the l.h.s. is majorized by ε, which proves the continuity of the function ζ → s[ζ]. (ii) If ζ is complex and if the functions an [ζ] are all holomorphic, one uses s [ζ+hj ]−sN [ζ] − a similar Nε -argument with the holomorphic functions N h (s˙ N )j [ζ] (instead of sN [ζ +h]−sN [ζ]), after having proved that the series of . a [ζ+hj ]−an [ζ] and vector-valued functions with general terms (an )hj [ζ] = n h
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(a˙ n )j [ζ]are uniformly majorized (term by term) by cun , where c is some constant independent of h. The proof of the latter relies on Cauchy-type inequalities for vector-valued holomorphic functions, which are given in Sect. B.III [see our argument after formula (B.28)]. B.I. Products Which Preserve Continuity and Holomorphy Being given three complete normed spaces A, B, C, we shall denote by Π any mapping (a, b) → c = Π(a, b) from the direct product A × B into C, which is bilinear with respect to the two variables a and b and bicontinuous in the following sense; for all pairs (a, b), there holds: Π(a, b)C ≤ aA × bB .
(B.1)
(A general constant factor, different from one, could be inserted on the r.h.s. of the latter, but it would be of no use in the applications and can always be avoided by a suitable rescaling of the norms). Then we have: Lemma B.2. Being given any bilinear and bicontinuous mapping Π from A×B into C: (i) if a[ζ] and b[ζ] (are continuous functions in D, respectively vector-valued in A and B, then the function ζ → Π(a[ζ], b[ζ]) is continuous in D, as a vector-valued function with values in C. (ii) If ζ is complex and if a[ζ] and b[ζ] are holomorphic functions in D, respectively vector-valued in A and B, then the function ζ → Π(a[ζ], b[ζ]) is holomorphic in D, as a vector-valued function with values in C. Proof. (i) In view of the bilinearity of Π, of the norm inequality in C, and of (B.1), we have: Π(a[ζ + h], b[ζ + h]) − Π(a[ζ], b[ζ])C ≤ Π(a[ζ + h], (b[ζ + h] − b[ζ]))C + Π((a[ζ + h] − a[ζ]), b[ζ])C ≤ a[ζ + h]A × b[ζ + h] − b[ζ]B + a[ζ + h] − a[ζ]A × b[ζ]B . (B.2) For every ζ ∈ D, h is submitted to vary in such a sufficiently small neighborhood of zero that ζ + h remains in a compact subset of D, so that a[ζ + h]A remains uniformly bounded. Then in view of the continuity of a[ζ] and b[ζ], the last member of (B.2) tends to zero with h, which implies the continuity of Π(a[ζ], b[ζ]) in D. (ii) Let a˙ j [ζ] ∈ A and b˙ j [ζ] ∈ B denote respectively the partial derivatives of a[ζ] and b[ζ] with respect to ζj in the complex domain D. We shall then show that the function Π(a[ζ], b[ζ]) admits a partial derivative with respect to ζj in D, which is equal to Π(a[ζ], b˙ j [ζ]) + Π(a˙ j [ζ], b[ζ]). In fact, in view of the bilinearity of Π and of the norm inequality in C, we can write:
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Π(a[ζ + hj ], b[ζ + hj ])−Π(a[ζ], b[ζ]) −(Π(a[ζ], b˙ j [ζ])+Π(a˙ j [ζ], b[ζ])) h C Π(a[ζ + hj ], b[ζ]) − Π(a[ζ], b[ζ]) − Π(a˙ j [ζ], b[ζ]) ≤ h C Π(a[ζ], b[ζ + hj ]) − Π(a[ζ], b[ζ]) − Π(a[ζ], b˙ j [ζ]) + h C Π((a[ζ + hj ] − a[ζ]), (b[ζ + hj ] − b[ζ])) + . (B.3) h C By using again the bilinearity of Π and applying the bicontinuity inequality (B.1) to each term of the r.h.s. of (B.3), we can majorize the latter by a[ζ + hj ] − a[ζ] b[ζ + hj ] − b[ζ] ˙ − a˙ j [ζ] × b[ζ]B + a[ζ]A × − bj [ζ] h h A + |h| ×
a[ζ + hj ] − a[ζ] h
A
×
b[ζ + hj ] − b[ζ] h
B
.
B
(B.4)
Then, for any given ζ in D, each of the three terms of (B.4) tends to zero with h, in view of the hypothesis that a[ζ] and b[ζ] are holomorphic vector-valued functions in D whose values are bounded in the norm in every compact subset of D; this implies that for each j, (1 ≤ j ≤ m), the l.h.s. of (B.3) tends to zero with h, and therefore that the function Π(a[ζ], b[ζ]) is holomorphic in D. An immediate corollary of the previous lemma is obtained by taking C = A, and defining B as the space L(A) of bounded linear operators {L : a → A L(a); a ∈ A} on A equipped with the usual norm L = supa∈A L(a)
a A . The inequality L(a)A ≤ L × a
(B.5)
plays the role of (B.1) and there holds Lemma B.3. Let a[ζ] and L[ζ] denote functions in D which are respectively vector-valued in A and L(A), and let ζ → L(a)[ζ] = L[ζ](a[ζ]) denote the image function which is vector-valued in A. Then: (i) if a[ζ] and L[ζ] are continuous in D, L(a)[ζ] is continuous in D; (ii) if ζ is complex and if a[ζ] and L[ζ] are holomorphic in D, L(a)[ζ] is holomorphic in D. We shall now give applications of Lemma B.2 to particular structures which are relevant at several places of this paper. . (1) Let A be the Hilbert space Xμ = L2 ([0, ∞), μ(R) dR), where μ denotes a given continuous and strictly positive function on the interval [0, ∞). For
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any element x = x(R) in Xμ , we put: ⎡ ∞ ⎤ 12 . xμ = ⎣ |x(R)|2 μ(R) dR⎦ .
753
(B.6)
0
Let B be the dual space X μ1 of Xμ , C = C, and Π the bilinear form which associates with each pair (x, y) ∈ Xμ × X μ1 the “quasi-scalar product” ∞
. y, x =
y(R)x(R) dR.
(B.7)
0
Note that it differs from the usual scalar product which is sesquilinear, but since Xμ is stable under the operation x → x, where x denotes the complex conjugate function R → x(R) of R → x(R), there still holds the Schwarz inequality: |y, x | ≤ xμ × y μ1 ,
(B.8)
which appears as a bicontinuity inequality of the type (B.1). We can then state as a special case of Lemma B.2: Lemma B.4. (i) If x[ζ] and y[ζ] are continuous functions in D, respectively vector-valued in Xμ and X μ1 , the quasi-scalar-product-function ζ → y[ζ], x[ζ] is continuous in D. (ii) If ζ is complex and if x[ζ] and y[ζ] are holomorphic functions in D, respectively vector-valued in Xμ and X μ1 , the function ζ → y[ζ], x[ζ] is holomorphic in D. "μ (called “HS-kernel space”) of kernels (2) We take for A the Hilbert space X K(R, R ) on Xμ equipped with the Hilbert-Schmidt-type norm ⎡ +∞ ⎤ 12 +∞ μ(R) . ⎣ K(μ) = |K(R, R )|2 ⎦ . dR dR (B.9) μ(R ) 0
0
As it can be seen by applying Schwarz’s inequality, this definition of the HS-norm of K ensures that the linear operator defined by the formula +∞ (Kx)(R) = K(R, R )x(R ) dR , 0
associates with every element x of Xμ an element Kx of Xμ . As in (1), we " 1 = L2 ([0, ∞) × [0, ∞), μ−1 (R)μ(R ) dR dR ) take for B the dual space X μ "μ , C = C, and we choose for Π the corresponding quasi-scalarof X "μ × X "1: product of pairs (K, K ) ∈ X μ
. ≺ K , K =
∞
+∞
dR 0
0
dR K (R, R )K(R, R ),
(B.10)
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which also satisfies Schwarz’s inequality: | ≺ K , K | ≤ K(μ) × K ( 1 ) . μ
(B.11)
Note that by introducing the transposed kernel Kt of K , which is such "μ , one can rewrite the previous formulae (B.10) and (B.11) that Kt ∈ X in terms of the trace formalism, namely: ≺ K , K = Tr [KKt ] ,
| Tr [KKt ] | ≤ K(μ) × Kt (μ) .
(B.12)
Then, by specializing Lemma B.2 to the present case, we obtain Lemma B.5. (i) If K[ζ] and K [ζ] are continuous HS-operator-valued " 1 , the quasi"μ and X functions in D, taking their values respectively in X μ scalar-product-function ζ →≺ K [ζ], K[ζ] = Tr [K[ζ]Kt [ζ]] is continuous in D. (ii) If ζ is complex and if K[ζ] and K [ζ] are holomorphic HS-operator-valued " 1 , then the "μ and X functions in D, taking their values respectively in X μ function ζ →≺ K [ζ], K[ζ] = Tr [K[ζ]Kt [ζ]] is holomorphic in D. "μ , and Π denote (3) Let A = B = C denote the Hilbert space of HS-kernels X " "μ , the kernel the composition of kernels: for any pair (K1 , K2 ) in Xμ × X . K = K1 K2 = K1 ◦ K2 , defined by . K(R, R ) =
+∞ K1 (R, R )K2 (R , R ) dR ,
(B.13)
0
"μ . In fact, the proof of the standard HS-norm inequality, which belongs to X corresponds to the choice μ = 1 (see, e.g., [21] and references therein) can "μ , with an arbitrary function be directly reproduced for the case of X μ(μ > 0), namely; K(μ) ≤ K1 (μ) × K2 (μ) .
(B.14)
(To check it, one ! just has to introduce the “renormalized” kernels μ(R) (Kj )ren (R, R ) = μ(R ) Kj (R, R ), j = 1, 2, which are such (Kj )ren (1) = Kj (μ) ). Since (B.14) is a bicontinuity inequality of the type (B.1), Lemma B.2 applies and yields Lemma B.6. (i) If K1 [ζ] and K2 [ζ] are continuous HS-operator-valued func"μ , then the composition-product-function tions in D, with values in X K[ζ] = K1 [ζ] ◦ K2 [ζ] is continuous in D as an operator-valued function "μ . with values in X (ii) If ζ is complex, and if K1 [ζ] and K2 [ζ] are holomorphic HS-operator"μ , then the composition-productvalued functions in D, with values in X function K[ζ] = K1 [ζ] ◦ K2 [ζ] is holomorphic in D as an operator-valued "μ . function with values in X
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"μ , B = C, C = X "μ , and the product Π(K, λ) = λK, (4) Taking A = X " (K ∈ Xμ , λ ∈ C), which is such that λK = |λ|K, we immediately obtain from Lemma B.2 the continuity (resp., holomorphy) property of any product λ[ζ]K[ζ] of continuous (resp., holomorphic) functions "μ , ζ → λ[ζ] ∈ C. By combining this property with the result ζ → K[ζ] ∈ X . of Lemma B.6 applied iteratively to any power K n [ζ] = K[ζ] ◦ · · · ◦ K[ζ] (n factors), we obtain Lemma B.7. (i) If ζ → K[ζ] is a continuous HS-operator-valued function " in D, taking its values n in Xμ , then any polynomial function of the form ζ → Pn (K)[ζ] = j=1 aj [ζ]K j [ζ], where the aj’s are complex-valued continuous functions in D, is a continuous HS-operator-valued function in "μ . D, with values in X (ii) If ζ is complex, and if ζ → K[ζ] is a holomorphic HS-operator-valued " function in D, taking its values n in Xμ , jthen any polynomial function of the form ζ → Pn (K)[ζ] = j=1 aj [ζ]K [ζ], where the aj’s are holomorphic functions in D, is a holomorphic HS-operator-valued function in D, "μ . with values in X B.II. Passage from Functions Depending Continuously or Holomorphically on Parameters ζ to Continuous or Holomorphic Vector-Valued Functions of ζ We now introduce for each strictly positive function μ and each positive num(p) . ber p, the functional space Xμ = Lp ([0, ∞), μ(R) dR), of all functions f (R) (defined almost everywhere on [0, +∞)) with norm ⎡ +∞ ⎤ p1 . ⎣ f μ,p = |f (R)|p μ(R) dR⎦ , (B.15) 0
Spaces C(D, μ, p): Keeping the same notations as in B.I, we introduce C(D, μ, p) as the space of all functions (ζ, R) → f (ζ; R) which are defined on D × [0, +∞) for almost every (a.e.) R, namely up to a subset of measure zero in {R ∈ [0, +∞)}, and which enjoy the following property. For each func(p) tion f , there exists a positive function M (R) in Xμ such that the following uniform majorization holds, for all ζ ∈ D and a.e. R ∈ [0, +∞): |f (ζ; R)| ≤ M (R).
(B.16)
It follows from this definition that every function f in C(D, μ, p) defines a vector-valued function ζ → f [ζ](·) = f (ζ; ·) in D, which takes its values in (p) Xμ , since [in view of (B.16)], one has for all ζ ∈ D : f [ζ]μ,p ≤ M μ,p . We shall now prove: Lemma B.8. (i) Let (ζ, R) → f (ζ; R) be a function in C(D, μ, p) such that for a.e. R, f (·; R) is a continuous function of ζ in D. Then there exists a continuous vector-valued function ζ → f [ζ] in D which takes its values (p) in Xμ and such that f [ζ](R) = f (ζ; R).
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(ii) Let D be a domain of the space Cm of the complex variables ζ = (ζ1 , . . . , ζm ), and let (ζ, R) → f (ζ; R) be a function in C(D, μ, p) such that for a.e. R, f (·; R) is a holomorphic function of ζ in D. Then there exists a holomorphic vector-valued function ζ → f [ζ] in D which takes its (p) values in Xμ and such that f [ζ](R) = f (ζ; R). Proof. (i) ζ being fixed in D, one considers the family of functions R → . fζ,h (R) = [f (ζ + h; R) − f (ζ; R)] which, in view of the uniform majorization (B.16) (true for a.e. R), satisfy for all h such that ζ + h ∈ D the following uniform bound: fζ,h pμ,p =
+∞ +∞ |fζ,h (R)|p μ(R) dR ≤ 2p M (R)p μ(R) dR = 2p M pμ,p . 0
0
(B.17) Since, by the continuity assumption, one has limh→0 fζ,h (R) = 0 for a.e. R, it then follows from Lebesgue-Fatou’s theorem that the integral on the l.h.s. of (B.17), and therefore fζ,h μ,p = f [ζ + h] − f [ζ]μ,p , tends to zero with h. Since this holds for all ζ ∈ D, this proves the continuity in D of the function f [ζ](R) = f (ζ; R) as a vector-valued function with values (p) in Xμ . (ii) Let ζ be fixed in the complex domain D at a “j-distance” rj (ζ) from the boundary of D (by j-distance, we mean the distance of ζ from the boundary of the section of D by the complex one-dimensional submanifold passing at ζ and parallel to the ζj -plane). We then introduce the following family of functions (labeled by hj = (0, . . . , 0, h, 0, . . . , 0); 1 ≤ j ≤ m, with the condition ζ + hj ∈ D): . R −→ ghj [ζ](R) =
f (ζ + hj ; R) − f (ζ; R) ∂f − (ζ; R) . h ∂ζj
∂f In the latter, the derivative ∂ζ of the holomorphic function f satisfies for a.e. j R a Cauchy integral representation of the form: ∂f f (ζ ; R) 1 (ζ; R) = dζ , (B.18) ∂ζj 2πi (ζj − ζj )2 j γr
where γr denotes the circle centered at ζj with radius r, and where ζ has all its components ζk , k = j, respectively equal to ζk . In view of the uniform upper bound (B.16), and since r can be chosen arbitrarily close to rj (ζ), there holds the following bound (for a.e. R): ∂f M (R) (B.19) ∂ζj (ζ; R) ≤ rj (ζ) . Now, by taking hj such that |h| < r, we can write a Cauchy integral representation on γr for the holomorphic function ζ → ghj [ζ](R) (for a.e. value of R);
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by combining Eq. (B.18) with the usual Cauchy representation for f (ζ) and f (ζ + hj ), one obtains: h ghj [ζ](R) = 2πi
γr
f (ζ ; R) dζ . (ζj − ζj )2 (ζj − ζj − h) j
(B.20)
By restricting h to vary in a neighborhood of zero such as, e.g., {h ∈ C : |h| ≤ r 2 }, one obtains a uniform majorization for the r.h.s. of Eq. (B.20) which yields for a.e. R (since r can be chosen arbitrarily close to rj (ζ)): |ghj [ζ](R)| ≤
2 |h|M (R) . [rj (ζ)]2
(B.21)
From (B.19) and (B.21) one deduces that: . ∂f (ζ; R) (a) there exists (for each j) a vector-valued function ζ → f˙j [ζ](R) = ∂ζ j (p)
M
μ,p taking its values in Xμ and such that (for all ζ ∈ D): f˙j [ζ]μ,p ≤ . rj (ζ)
(b) There holds: ghj [ζ]μ,p ≤
2|h| [rj (ζ)]2 M μ,p ,
which proves that the vector. f [ζ+hj ]−f [ζ] − f˙j [ζ]μ,p valued function ζ → f [ζ](R) = f (ζ; R) is such that h tends to zero with h for all ζ in D. We have thus proved that the function ζ → f [ζ](R) is holomorphic in D as a vector-valued function taking its (p) values in Xμ .
In the text, we shall have to apply directly Lemma B.8 for the case p = 2, and with a weight-function of the form μ(R) = w(R)e2αR , w being specified in Sect. 3.1. We also need to apply the previous result to a more involved situation, which is described below in Lemma B.10. For this purpose, we shall first state the following property, which appears as a variant of Lemma B.8 for the case p = 1 (we also need this result only for μ(R) = 1). Lemma B.9. (i) Let (ζ, R) → f (ζ; R) be a function in C(D, 1, 1) such that for a.e. R, f (·; R) is a continuous function of ζ in D. Then the integral . +∞ I(ζ) = 0 f (ζ; R) dR is continuous in D. (ii) Let ζ be complex, D a domain of Cm and let (ζ, R) → f (ζ; R) be a function in C(D, 1, 1) such that for a.e. R, f (·; R) is a holomorphic func. +∞ tion of ζ in D. Then the integral I(ζ) = 0 f (ζ; R) dR is holomorphic in D. Proof. (i) One just has to check that |I(ζ)| ≤ f (ζ; ·)1,1 ≤ M 1,1 and that |I(ζ + h) − I(ζ)| ≤ fζ,h 1,1 , which tends to zero with h as in Lemma B.8 (i).
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(ii) As in the proof of Lemma B.8 (ii), one considers the function ghj [ζ](R) and its majorization (B.21), which allows one to check that: I(ζ + hj ) − I(ζ) ∂I − (ζ) h ∂ζj +∞ 2|h|M 1,1 = ghj [ζ](R) dR ≤ ghj [ζ] 1,1 ≤ . (B.22) [rj (ζ)]2 0
It follows that the l.h.s. of (B.22) tends to zero with h, which proves the holomorphy property of I(z) in D. Lemma B.10. (i) Let (ζ, R, R ) → F (ζ; R, R ) be a function defined for a.e. (R, R ) on D × [0, +∞) × [0, +∞) by a convergent integral of the following form: +∞ F1 (ζ; R, R )F2 (ζ; R , R ) dR , (B.23) F (ζ; R, R ) = 0
under the following assumptions: (a) the functions Fj (ζ; R, R )(j = 1, 2), are defined for a.e. (R, R ) on D × [0, +∞) × [0, +∞) and satisfy uniform bounds |Fj (ζ; R, R )| ≤ Gj (R, R ) on this set, such that the integral G(R, R ) = +∞ G1 (R, R ) G2 (R , R ) dR is convergent for almost every 0 value of (R, R ) and the function (R, R ) → G(R, R ) belongs to "μ , [see paragraph B.I-(2) of this Appendix]. X (b) For a.e. (R, R ), the functions F1 (·; R, R ) and F2 (·; R, R ) are continuous functions of ζ in D. Then there exists a continuous HS-operator-valued function ζ → "μ and such that K[ζ](R, R ) = K[ζ] in D which takes its values in X F (ζ; R, R ). (ii) Let ζ be complex, D a domain of Cm , and (ζ, R, R ) → F (ζ; R, R ) a function of the form (B.23) satisfying the previous conditions (a) together with the following additional condition: (b ) for a.e. (R, R ), the functions F1 (·; R, R ) and F2 (·; R, R ) are holomorphic functions of ζ in D. Then there exists a holomorphic HS-operator-valued function ζ → K[ζ] in D "μ and such that K[ζ](R, R ) = F (ζ; R, R ). which takes its values in X . Proof. The function R → FR,R (ζ; R ) = F1 (ζ; R, R )F2 (ζ; R , R ), which is defined for a.e. (R, R ), continuous in ζ in case (i), holomorphic in ζ in . case (ii), is uniformly bounded by the function in L1 : R → GR,R (R ) = G1 (R, R )G2 (R , R ). Then Lemma B.9 entails that, for a.e. (R, R ), . +∞ F (ζ; R, R ) = 0 FR,R (ζ; R ) dR is continuous (resp., holomorphic) with +∞ respect to ζ and uniformly bounded by G(R, R ) = 0 GR,R (R ) dR . Since "μ , it follows that F (ζ; R, R ) the function (R, R ) → G(R, R ) belongs to X satisfies conditions which are similar to those of Lemma B.8 for p = 2 (with
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either continuity or holomorphy properties in ζ according to the respective cases (i) or (ii)), up to the replacement of the integration space {R ∈ R+ } by "μ . The results (i) and {(R, R ) ∈ R+ × R+ } and of the Hilbert space Xμ by X (ii) then correspond directly to those of Lemma B.8. B.III. Complement on the Various Criteria of Vector-Valued Analyticity in Several Complex Variables We shall first recall the equivalence between several criteria of analyticity for the case of numerical functions of several complex variables defined in a domain of Cm . These criteria are: (1) “differentiability criterium”: existence of partial derivatives with respect to the complex variables ζj (1 ≤ j ≤ m) at all points of D; (2) solution of the system of Cauchy-Riemann equations in D; (3) “Cauchy integral criterium” in D and the associated Cauchy integral representations for the function and all its successive (partial) derivatives, implying corresponding Cauchy inequalities; (4) convergence of the Taylor series in an appropriate complex neighborhood of each point of D. For each of these characteristic properties, there is a corresponding “weak criterium of analyticity” for the vector-valued functions ζ → a[ζ] of several complex variables taking their values in the complete normed space A; it consists in stating that for every element ϕ of the dual space A of A, the numerical “scalar-product” function ζ → ϕ, a[ζ] satisfies the corresponding analyticity criterium. Then, it turns out that each weak criterium is equivalent to a “strong criterium of analyticity”, which involves either the notion of limit or that of integral in the sense of the norm in A. In particular, all the results that have been derived in this Appendix have made use of the differentiability criterium, which postulates the existence of partial derivatives ζ → a˙ j [ζ] of ζ → a[ζ] as vector-valued functions obtained in the sense of strong limits in A. The fact that it is implied by the corresponding weak criterium (1) for the numerical functions ζ → ϕ, a[ζ] (for all ϕ ∈ A ) is obtained by a direct adaptation of the Dunford theorem (see, e.g., [22, p. 128]) to the several variable case. The main ingredient of this “weak to strong passage” consists in the use of the “maximum boundedness theorem” (through its corollary called “Resonance Theorem” in [22]) and of the completeness property of A. The importance of another “weak to strong passage” concerns the Cauchy integral criterium (3), since in particular the latter allows one to give a direct proof of Cauchy-type inequalities, which majorize the norms a˙ j [ζ] of the partial derivatives at ζ in terms of the maximum of a[ζ ] in a neighborhood of ζ, and which also imply the strong convergence of the Taylor series in a complex neighborhood of ζ.
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Before giving a further description of the “strong Cauchy integral criterium”, and in order to make clear how the latter results from the “strong differentiability criterion” through the corresponding implications for the weak criteria, let us recall how the implication (1) =⇒ (3) is obtained for numerical functions of several complex variables. (1) =⇒ (2): choosing the increment h either real or purely imaginary in the following definition, one checks that the existence of continuous partial derivaf (ζ+hj )−f (ζ)] ∂f (ζ) for f (ζ), namely lim{h→0inC} − f˙j (ζ) = 0 for every tives ∂ζ h j . ζ ∈ D and hj = (0, . . . , 0, h, 0, . . . , 0), implies that f"(x, y) = f (ζ) (with ζj = xj + iyj ; 1 ≤ j ≤ m) satisfies the system of Cauchy–Riemann equations in D:
∂ f" ∂xj (x, y)
"
∂f = −i ∂y (x, y) or, by passing formally to the variables (ζj , ζ j = j
xj −iyj ): ∂ f"/∂ζ j (x, y) = 0.
(2) =⇒ (3): considering f"(x, y) as a 0-form in a domain of R2m ≡ Cm , ∂ f" and introducing the differential 1-form df"(x, y) = 1≤j≤m ∂xj (x, y) dxj + ∂ f" ∂ f" ∂ f" " 1≤j≤m ∂ζj dζj + ∂ζ (x, y) dζ j , the ∂yj (x, y) dyj or, equivalently, df = j
Cauchy-Riemann system can be equivalently written d(f"(x, y) dζ1 ∧ · · · ∧ dζm ) = 0. Then, in view of Stokes’ theorem, the latter is equivalent to the fact that the following Cauchy-type integral formula: (B.24) f"(x, y) dζ1 ∧ · · · ∧ dζm = 0, Γ
holds for every m-real-dimensional integration cycle Γ of the form Γ = ∂Δ, where Δ can be any (m + 1)-cycle whose support is contained in D. An usual and convenient choice for Γ is the “distinguished boundary” of a polydisk-type domain, namely Γ = γ1 × · · · × γm , where each γj (1 ≤ j ≤ m) is the boundary of a domain δj homeomorphic to a disk and such that the polydisk-type . " = domain Δ δ1 × · · · × δm be contained in D. As in the case of one complex variable, Eq. (B.24) implies the corresponding integral representation m 1 f (ζ ) dζ1 ∧ · · · ∧ dζm , (B.25) f (ζ) = −ζ ) 2πi (ζ1 − ζ1 ) · · · (ζm m γ1 ×···×γm
" Moreover, there also holds which is valid for every point ζ = (ζ1 , . . . , ζm ) in Δ. integral representations “of partial type”, namely with respect to any subset J(J ⊂ {1, 2, . . . , m}) of variables ζj integrated on the 1-cycle γj enclosing ζj , as those used in Eqs. (B.18) and (B.20). (This corresponds to exploiting the Cauchy–Riemann system under the following form: d(f"(x, y) ∧{j∈J} dζj )|{ζk =0,∀k∈J} = 0, i.e., by Stokes’ theorem: {; γj } f"(x, y)∧{j∈J} dζj = 0.) / j∈J
Coming back to the case of vector-valued functions ζ → a[ζ], the “weak to strong passage” for integral relations such as (B.24) and (B.25) can be presented as follows. Considering, e.g., the case of (B.24), the fact that for all ϕ ∈ A , the numerical functions ϕ, a[ζ] satisfy the equation Γ ϕ, a[ζ] dζ = 0
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. (with dζ = dζ1 ∧· · ·∧ dζm ) implies the corresponding vector-valued equation in A, namely Γ a[ζ] dζ = 0. This implication is based on the following argument: (a) The strong differentiability of a[ζ] implies its strong continuity inζ, which allows one to define (simple or multiple) integrals of the form Γ a[ζ] dζ with values in A (namely, as strong limits of Riemann sums in A). (b) In view of the continuity and the linearity of each ϕ ∈ A , and by applying the weak criterium, one has: ϕ, Γ a[ζ] dζ = Γ ϕ, a[ζ] dζ = 0. (c) If a vector I ∈ A (such as I = Γ a[ζ] dz) satisfies ϕ, I = 0 for all ϕ ∈ A , I is necessarily the zero-vector in A. This results from the following Lemma B.11 (see p. 108 of [22]). For any given I in A such as I = 0, there exists a continuous linear form ϕ0 such that ϕ0 , I = IA . Now, being given a function a[ζ] holomorphic in D with values in A, and satisfying a uniform bound a[ζ] ≤ M , one can write Cauchy-type vectorequations similar to Eqs. (B.18) and (B.20), namely (by putting ahj [ζ] = a[ζ+hj ]−a[ζ] ): h a[ζ ] 1 dζ , (B.26) a˙ j [ζ] = 2πi (ζj − ζj )2 j γr a[ζ ] h . dζ . (B.27) ghj [ζ] = ahj [ζ] − a˙ j [ζ] = 2 2πi (ζj − ζj ) (ζj − ζj − h) j γr
By using norm inequalities under the integration signs in the latter, one then obtains majorizations for these integrals which are similar to (B.19) and (B.21) and yield the Cauchy-type inequalities: 2|h|M M ghj [ζ] ≤ , . (B.28) a˙ j [ζ] ≤ rj (ζ) [rj (ζ)]2 As an application, we notice that the end of the proof of Lemma B.1 can . be obtained by applying (B.28) to the functions (a˙ n )j [ζ] and (gn )hj [ζ] = (an )hj [ζ] − (a˙ n )j [ζ], the positive constant M being then replaced by un . In view of these inequalities, the uniform majorization (by 3ε ) of the remainders of the series with general terms (a˙ n )j [ζ] and (an )hj [ζ] is then ensured (for all hj ∈ Vζ,ε ) by the convergence of the majorizing series with general term un .
References [1] Nussenzveig, H.M.: Causality and Dispersion Relations. Academic Press, New York (1972) [2] McVoy, K.W.: Giant Resonances and Neutron-Nucleus Total Cross Sections. Ann. Phys. 43, 91–125 (1967) [3] Cindro, N., Poˇcaniˇc, D.: Resonances in heavy-ion reactions—structural vs. diffractional models. In: Resonances Models and Phenomena. Lecture Notes in Physics, vol. 211, pp. 158–181. Springer, Berlin (1993) [4] De Alfaro, V., Regge, T.: Potential Scattering. North-Holland, Amsterdam (1965)
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[5] Newton, R.G.: The Complex j-plane. W. A. Benjamin, New York [6] Bethe, H.A.: Nuclear many-body problem. Phys. Rev. 103, 1353–1390 (1956) [7] Wildermuth, K., Tang, Y.C.: A Unified Theory of the Nucleus. Vieweg, Braunschweig (1977) [8] Tang, Y.C., LeMere, M., Thompson, D.R.: Resonating-group method for nuclear many-body problems. Phys. Rep. 47, 167–223 (1978) [9] LeMere, M., Tang, Y.C., Thompson, D.R.: Study of the α+16 O system with the resonating-group method. Phys. Rev. C 14, 23–27 (1976) [10] Wildermuth, K., McClure, W.: Cluster representation of nuclei. Springer Tracts in Modern Physics, vol. 41. Springer, Berlin (1966) [11] De Micheli, E., Viano, G.A.: Unified scheme for describing time delay and time advance in the interpolation of rotational bands of resonances. Phys. Rev. C 68, 064606 (2003) [12] De Micheli, E., Viano, G.A.: Time delay and time advance in resonance theory. Nucl. Phys. A 735, 515–539 (2004) [13] Bros, J.: On the notions of scattering state, potential and wave-function in quantum field theory: an analytic-viewpoint. In: Kashiwara, M., Kawai, T. (eds.) Prospect of Algebraic Analysis, vol. 1, pp. 49–74. Academic Press, New York (1988) [14] Bros, J., Viano, G.A.: Complex angular momentum in general quantum field theory. Ann. Henri Poincar´e 1, 101–172 (2000) [15] Bros, J., Viano, G.A.: Complex angular momentum diagonalization of the BetheSalpeter structure in general quantum field theory. Ann. Henri Poincar´e 4, 85–126 (2003) [16] Reed, M., Simon, B.: Methods of Modern Mathematical Physics—Scattering Theory, vol. 3. Academic Press, New York (1979) [17] Bertero, M., Talenti, G., Viano, G.A.: Scattering and bound states solutions for a class of non-local potentials (s-wave). Commun. Math. Phys. 6, 128–150 (1967) [18] Bertero, M., Talenti, G., Viano, G.A.: Bound states and Levinson’s theorem for a class of non-local potentials (s-wave). Nucl. Phys. A 113, 625–640 (1968) [19] Bertero, M., Talenti, G., Viano, G.A.: A note on non-local potentials. Nucl. Phys. A 115, 395–404 (1968) [20] Bertero, M., Talenti, G., Viano, G.A.: Eigenfunction expansions associated with Schr¨ odinger two-particle operators. Nuovo Cimento 62A(10), 27–87 (1969) [21] Smithies, F.: The Fredholm theory of integral equations. Duke Math. J. 8, 107– 130 (1941) [22] Yosida, K.: Functional Analysis. Springer, Berlin (1965) [23] Ikebe, T.: Eigenfunction expansions associated with the Schr¨ odinger operators and their applications to scattering theory. Arch. Ration. Mech. Anal. 5, 1–34 (1960) [24] Ikebe, T.: On the phase-shift formula for the scattering operator. Pac. J. Math. 15, 511–523 (1965) [25] Simon, B.: Quantum Mechanics for Hamiltonians Defined as Quadratic Forms. Princeton Univ. Press, Princeton (1971) (see also the references quoted therein)
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[26] Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965) [27] Newton, R.G.: Analytic properties of radial wave functions. J. Math. Phys. 1, 319–347 (1960) [28] Tichonov, A.N., Samarskij, A.A.: Uravnenija Matematiˇceskoi Fiziki. Mir, Moscow (1977) ¨ [29] Neumann, J.von , Wigner, E.P.: Uber Merkwurdige Diskrete Eigenwerte. Z. Physik 30, 465–467 (1929) [30] Fonda, L.: Bound states embedded in the continuum and the formal theory of scattering. Ann. Phys. 22, 123–132 (1963) [31] Goursat, E.: Cours d’Analyse Math´ematique, Tome III. Gauthier-Villars, Paris (1956) [32] Bros, J., Pesenti, D.: Fredholm resolvents of meromorphic kernels with complex parameters. J. Math. Pures et Appl. 62, 215–252 (1983) [33] Chadan, K., Sabatier, P.C.: Inverse Problems in Quantum Scattering Theory, p. 110. Springer, New York (1977) [34] Martin, A.: On the validity of Levinson’s theorem for non-local interactions. Nuovo Cimento 7, 607–627 (1958) [35] Watson, G.N.: Theory of Bessel Functions, p. 389. Cambridge University Press, Cambridge (1952) [36] Erdelyi, A.: Bateman Manuscript Project—Higher Trascendental Functions. McGraw-Hill, New York (1953) [37] Boas, R.P.: Entire Functions. Academic Press, New York (1954) [38] Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1972) [39] Hoffman, K.: Banach spaces of analytic functions, p. 125, Prentice-Hall, Englewood Cliffs (1962) [40] Bros, J., Viano, G.A.: Connection between the harmonic analysis on the sphere and the harmonic analysis on the one-sheeted hyperboloid-III. Forum Math. 9, 165–191 (1997) [41] Erdelyi, A.: Bateman Manuscript Project—Tables of Integral Transforms. McGraw-Hill, New York (1954) [42] Bros, J., Viano, G.A.: Connection between the harmonic analysis on the sphere and the harmonic analysis on the one-sheeted hyperboloid-II. Forum Math. 8, 659–722 (1996) [43] Newton, R.G.: Scattering Theory of Waves and Particles. McGraw-Hill, New York (1966) [44] Martin, A.: Some simple inequalities in scattering by complex potentials. Nuovo Cimento 23, 641–654 (1962) [45] Sommerfeld, A.: Partial Differential Equations in Physics, vol. 6. Academic Press, New York (1964)
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Jacques Bros Institut de Physique Th´eorique CEA-Saclay 91191 Gif-sur-Yvette Cedex France e-mail:
[email protected] Enrico De Micheli IBF, Consiglio Nazionale delle Ricerche Via De Marini, 6 16149 Genoa Italy e-mail:
[email protected] Giovanni Alberto Viano Dipartimento di Fisica, Universit` a di Genova Via Dodecaneso, 33 16146 Genoa Italy and Sezione di Genova, Istituto Nazionale di Fisica Nucleare Via Dodecaneso, 33 16146 Genoa Italy e-mail:
[email protected] Communicated by Klaus Fredenhagen. Received: October 21, 2009. Accepted: April 26, 2010.
Ann. Henri Poincar´e
Ann. Henri Poincar´e 11 (2010), 765–780 c 2010 Springer Basel AG 1424-0637/10/040765-16 published online June 22, 2010 DOI 10.1007/s00023-010-0045-4
Annales Henri Poincar´ e
Rotation Numbers of Linear Schr¨ odinger Equations with Almost Periodic Potentials and Phase Transmissions Meirong Zhang and Zhe Zhou Abstract. In this paper we study the linear Schr¨ odinger equation with an almost periodic potential and phase transmission. Based on the extended unique ergodic theorem by Johnson and Moser, we will show for such an equation the existence of the rotation number. This extends the work of Johnson and Moser (in Commun Math Phys 84:403–438, 1982; Erratum Commun Math Phys 90:317–318, 1983) where no phase transmission is considered. The continuous dependence of rotation numbers on potentials and transmissions will be proved.
1. Introduction In this paper we study the following generalized linear Schr¨ odinger equations with almost periodic potentials and phase transmissions: ⎧ ⎨ y + Q(t)y = 0 for t ∈ R\Γ y(ti −) y(ti +) (1.1) = Ai for ti ∈ Γ, ⎩ y (ti +) y (ti −) where Q(t) is an almost periodic function, Γ is a periodic lattice, and Ai ∈ 0 −1 SL(R2 ) are symplectic matrices, i.e., ATi JAi = J, where J = , and 1 0 T Ai denotes the transpose of Ai . These equations are introduced by Kronig and Penney [9] to describe the quantum effect. A typical spectrum on (1.1) is y + β δ(t − 2iπ) y = λy, i∈Z
M. Zhang is supported by the National Basic Research Program of China (Grant no. 2006CB805903), the National Natural Science Foundation of China (Grant no. 10531010), and the 111 Project of China (2007).
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where δ(t) is the unit Dirac measure located at t = 0, and the parameter β describes the tense of quantum effects. This corresponds to Eq. (1.1) with 1 0 . Q(t) ≡ −λ, Γ = 2πZ, Ai = β 1 In recent years, Niikuni [13,14] has established the existence of the rotation number for Eq. (1.1) when the potential Q(t), the lattice Γ, and the transmissions {Ai } are periodic of the same period. By the rotation number approach, the spectrum of (1.1) has been established, extending the works of [6,17]. For explanation to differential equations with general measures, see [11]. The main purpose of this paper is to initiate a study for Eq. (1.1) with almost periodic potentials Q(t), following the classical work [7] where no phase transmission is considered. Though it is desired that Eq. (1.1) will admit a well-defined rotation number for general almost periodic lattice Γ and general almost periodic transmission {Ai }, as an initial step, we will show this is true for a periodic lattice Γ and a choice of transmissions as rigid rotations cos αi sin αi Ai = , − sin αi cos αi where the sequence S := {αi } is almost periodic. For the corresponding definitions, see Sect. 2. With such a choice for Eq. (1.1), its dynamics can be described mainly by the argument equation with phase transmissions
θ = cos2 θ + Q(t) sin2 θ for t ∈ R\Γ (1.2) θ(ti +) = θ(ti −) + αi for ti ∈ Γ, which results from Eq. (1.1) in the polar coordinates y = r sin θ, y = r cos θ. For definiteness, we always understand solutions θ(t) of Eq. (1.2) to be rightcontinuous, i.e., θ(t+) = θ(t). Let us use θ(t) = θ(t; Q, Γ, S, ϑ), t ∈ R, to denote the unique solution of Eq. (1.2) satisfying the initial value θ(0) = ϑ ∈ R. The main result of this paper is as follows: Theorem 1.1. Let Q(t) and S = {αi } be almost periodic and Γ be periodic. Then for any ϑ ∈ R, the following limit lim
t→+∞
θ(t; Q, Γ, S, ϑ) − ϑ t
(1.3)
does exist. Moreover, the limit of (1.3) is independent of the choice of the initial value ϑ ∈ R. The complete proof of Theorem 1.1 is given in Sect. 4. The limit of (1.3) is called the rotation number of Eq. (1.2) and is denoted by Γ (Q, S). It can be used to describe the dynamics of the corresponding Eq. (1.1). For more precise statements of the results, see Theorem 4.2. It can be expected that such a rotation number is also useful in studying the spectrum structure of the operators from Eq. (1.1), as in [7,13,14] for the case of no phase transmission.
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When no phase transmission is considered, i.e., Ai = I in (1.1), or S = {0} in (1.2), Theorem 1.1 has been established by Johnson and Moser in a classical paper [7]. One extension to the so-called p-Laplacian with almost periodic potentials is given in [4]. For linear Hamiltonian systems of many degrees of freedom, see also Novo et al. [15] for the ergodic representation of rotation numbers. For the extension to random dynamical systems, see [1,10]. However, when transmissions are taken into account, the usual approaches in these works do not work since the arguments are no longer continuous in time t ∈ R. Therefore, in Sect. 3, inspired by the classical Poincar´e maps for periodic systems, we will construct for Eq. (1.1) a discrete skew-product dynamical system without discontinuity. See formula (3.5). Moreover, limits (1.3) will be reduced some Birkhoff sums of the skew-product dynamical system. See formula (3.13). In Sect. 4, based on the extended unique ergodic theorem by Johnson and Moser [7], we will show that limits (1.3) do exist and are independent of ϑ. This yields the rotation number of Eq. (1.1). By making use of the ergodic representation of rotation numbers, we will prove that rotation numbers are continuous in potentials and phase transmissions. See Theorem 4.4.
2. Basics on Almost Periodicity Let us first recall some facts on almost periodicity. For more details, see [5]. We say that a function Q = Q(t) : R → R is almost periodic, if Q is continuous and for any ε > 0, the set of the ε-periods T (Q, ε) := {τ ∈ R : |Q(t + τ ) − Q(t)| < ε ∀ t ∈ R} is relatively dense in R. That is, there exists some lε > 0 such that [t, t + lε ) ∩ T (Q, ε) = ∅
∀ t ∈ R.
Denote by APF the space of almost periodic functions, endowed with the uniform norm Q ∞ := sup |Q(t)|, t∈R
Q ∈ APF.
Then we have the following results: Lemma 2.1. [5] Any Q ∈ APF is bounded and uniformly continuous in R. Moreover, (APF, · ∞ ) is a Banach space. Naturally, one has the following flow in APF defined by translations APF × R (Q, τ ) → Q · τ ∈ APF,
Q · τ := Q(· + τ ).
In order to describe the phase transmissions, let us introduce APF 2 := {(Q, W ) : (Q, W ) : R → R2 is such that Q, W ∈ APF}, which is equipped with the uniform topology (Q, W ) ∞ := max{ Q ∞ , W ∞ }.
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The translations in APF 2 are simply (Q, W ) → (Q · τ, W · τ ). Given (Q, W ) ∈ APF 2 . The hull of (Q, W ) ∈ APF 2 is defined by E(Q, W ) := closure(APF 2 ,·∞ ) {(Q · τ, W · τ ) : τ ∈ R}. It is well known that E(Q, W ) is compact. For our purpose, we need to introduce the discrete hull as follows: Definition 2.2. Given T > 0. The T -hull of (Q, W ) ∈ APF 2 is defined by ET (Q, W ) := closure(APF 2 ,·∞ ) {(Q · nT, W · nT ) : n ∈ Z}. Then ET (Q, W ) ⊂ E(Q, W ) is also compact in (APF 2 , · ∞ ). Like the case of continuous time, we can equip ET (Q, W ) with a group structure by (Q1 , W1 ) · (Q2 , W2 ) := lim (Q(· + n1m T + n2m T ), W (· + n1m T + n2m T )), m→∞
(2.1) whenever (Qi , Wi ) = lim (Q(· + nim T ), W (· + nim T )) ∈ ET (Q, W ), m→∞
i = 1, 2. (2.2)
Note that the definition of product of ET (Q, W ) does make sense. That is, the limit of (2.1) does exist and is independent of the choice of times nim so long as (2.2) hold. Thus, ET (Q, W ) with the metric induced by · ∞ is a compact Abelian topological group. In the space APF 2 , the following time-T translation is continuous: ϕT (Q, W ) := (Q · T, W · T ),
(Q, W ) ∈ APF 2 .
(2.3)
By the definition of T -hulls, one sees that ET (Q, W ) is ϕT -invariant. We have the following results: Lemma 2.3. Given (Q, W ) ∈ APF 2 . (i) The transformation ϕT is minimal on ET (Q, W ). That is, ET (q, w) = ET (Q, W ) for any (q, w) ∈ ET (Q, W ). (ii) The transformation ϕT on ET (Q, W ) is uniquely ergodic with the unique invariant Borel probability measure being the Haar measure ν of the compact Abelian group ET (Q, W ). Proof. (i) Let (q, w) ∈ ET (Q, W ). Since ET (Q, W ) is ϕT -invariant, we have (q · nT, w · nT ) ∈ ET (Q, W )
∀ n ∈ Z.
By the compactness of ET (Q, W ), we have ET (q, w) ⊆ ET (Q, W ). On the other hand, suppose that (Q · nm T, W · nm T ) → (q, w) ∈ ET (Q, W ),
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where nm ∈ Z. Then by reversing times, we have (q · (−nm T ), w · (−nm T )) → (Q, W ). Thus (Q, W ) ∈ ET (q, w). One has then ET (Q, W ) ⊆ ET (q, w). (ii) By (2.1) and (2.3), we have ϕT (q, w) = (Q · T, W · T ) · (q, w)
∀ (q, w) ∈ ET (Q, W ).
Thus, ϕT is a rotation on the group ET (Q, W ). Then combining (i) with [16, Theorem 6.20], we can obtain the desired result. We say that a sequence S = {αi }i∈Z ⊂ R is almost periodic, if for any ε > 0, the set T (S, ε) := {k ∈ Z : |αi+k − αi | < ε ∀i ∈ Z}
(2.4)
is relatively dense (in Z) in the sense that there exists some kε ∈ N such that T (S, ε) ∩ [i, i + kε ) = ∅
∀ i ∈ Z.
Such a kε is called an ε-period of the sequence S. The set of all almost periodic sequences is denoted by APS. By a lattice Γ, it means that Γ = {· · · < t−2 < t−1 < t0 < t1 < t2 < · · ·} ⊂ R is a set of discrete points. We say that Γ is periodic if there exist T > 0 and n ∈ N such that ∀ i ∈ Z.
ti+n = ti + T
(2.5)
The numbers T and n are called the period and the length of the periodic lattice Γ, respectively. In the following, we always assume that (T, n) in (2.5) is minimal. In this case, Γ is called a (T, n)-periodic lattice. The set of all (T, n)-lattices is denoted by PLT,n . Note that any Γ ∈ PLT,n can be represented as Γ = {t0 , t1 , . . . , tn−1 } + T Z,
t0 < t1 < · · · < tn−1 (< t0 + T ).
Lemma 2.4. Given Γ ∈ PLT,n . (i) Let W ∈ APF. Then, the sequence {W (ti ) : ti ∈ Γ} ∈ APS. (ii) Conversely, let S = {αi }i∈Z ∈ APS. There exists W ∈ APF such that W (ti ) = αi for all ti ∈ Γ. Proof. (i) This follows from the definitions of two kinds of almost periodicity. For the detailed proof, one can see [5, p. 163]. (ii) Since S ∈ APS, we know that Sj := {βi := αin+j }i∈Z ∈ APS,
j = 0, 1, . . . , n − 1.
(2.6)
In fact, for any ε > 0, if k ∈ T (S, ε/n), then for any i ∈ Z, by (2.4) we have |αi+k − αi | < ε/n.
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Furthermore, |βi+k − βi | = |αin+kn+j − αin+j | ≤ |αin+kn+j − αin+k(n−1)+j | + · · · + |αin+k+j − αin+j | < ε. This implies that T (S, ε/n) ⊂ T (Sj , ε). Thus, T (Sj , ε) is also a relatively dense set. Define auxiliary functions for j = 0, 1, . . . , n − 1, ⎧ βi ⎪ ⎪ (t − tin+j−1 ) when t ∈ [tin+j−1 , tin+j ], ⎪ ⎪ t − ⎪ ⎨ in+j tin+j−1 −βi (2.7) Wj (t) = (t − tin+j ) when t ∈ [tin+j , tin+j+1 ], ⎪ ⎪ t − t ⎪ in+j+1 in+j ⎪ ⎪ ⎩ 0 otherwise, where i runs over Z, we know that Wj (t) is almost periodic. In fact,it follows from (2.5) and (2.7) that Wj · kT − Wj ∞ ≤ sup |βi+k − βi |. i∈Z
Combining this with (2.6), we conclude that {kT : k ∈ T (Sj , ε)} ⊂ T (Wj , ε), completing the assertion. Moreover, we observe that Wj (tin+j ) = βi = αin+j , Wj (tk ) = 0 for k = in + j. n−1 Lemma 2.1 implies that W (t) := j=0 Wj (t) is a desired representation for (Γ, S).
3. Reduction to Discrete Skew-Product Flows Given Γ ∈ PLT,n , S ∈ APS and Q ∈ APF. By Lemma 2.4, we can rewrite Eq. (1.2) as
θ = cos2 θ + Q(t) sin2 θ for t ∈ R\Γ (3.1) θ(ti ) = θ(ti −) + W (t) for ti ∈ Γ, where W (t) = WΓ,S (t) is determined by the periodic lattice Γ and the almost periodic sequence S. From now on we fix a Γ ∈ PLT,n . Without loss of generality, we assume that t0 = 0. Denote solutions of Eq. (3.1) by θ(t) = θ(t; Q, Γ, W, ϑ) = θΓ (t; Q, W, ϑ). Lemma 3.1. Given ϑ ∈ R. There holds the following relation: θΓ (t; Q, W, ϑ) − ϑ θΓ (kT ; Q, W, ϑ) − ϑ = lim . (3.2) lim t→+∞ k→+∞ t kT That is, if one of the limits exists, then another exists as well and they are equal.
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Proof. Let t ∈ [kT, (k + 1)T ), k ∈ N. Then tkn+j ≤ t < tkn+j+1 for some j, 0 ≤ j ≤ n − 1. Therefore, |θ(t) − θ(kT )| ⎫ ⎧ ⎪ kT+t1 kT +tj t ⎪ ⎨ ⎬ (cos2 θ(s) + Q(s) sin2 θ(s)) ds +··· + + = ⎪ ⎪ ⎭ ⎩ kT kT +tj−1 kT +tj kn+j (by (2.5) and (3.1)) + W (ti ) i=kn+1 ≤ T (1 + Q ∞ ) + n W ∞ . It implies that θ(t) − ϑ kT θ(kT ) − ϑ θ(kT ) − ϑ = lim = lim . k→+∞ t k→+∞ t kT kT The proof is complete. lim
t→+∞
In order to prove the existence of these limits, we will embed Eq. (3.1) into the following family of equations:
θ = cos2 θ + q(t) sin2 θ for t ∈ R\Γ (3.3) θ(ti +) = θ(ti −) + w(t) for ti ∈ Γ, where (q, w) runs over the hull ET (Q, W ), as it did in [7]. Due to the 2π-periodic in θ of the vector field of (3.3) and the uniqueness of solutions of ODEs, we have the following result: Lemma 3.2. For any (q, w) ∈ ET (Q, W ), ϑ ∈ R, t ∈ R, k ∈ Z, there holds θΓ (t; q, w, ϑ + 2kπ) − (ϑ + 2kπ) = θΓ (t; q, w, ϑ) − ϑ.
(3.4)
Let us introduce the product space Z := ET (Q, W ) × S2π with the distance d((q1 , w1 , ϑ1 ), (q2 , w2 , ϑ2 )) := max{ q1 − q2 ∞ , w1 − w2 ∞ , |ϑ1 − ϑ2 |S2π }. Note that Eq. (3.3) can yield a ‘dynamical system’ on ET (Q, W )×S2π , although θ(t) has discontinuity in time t because of the transmissions on the lattice. In order to overcome this discontinuity, by noticing that Γ is a periodic lattice, we can introduce ΦΓ := {ΦiΓ }i∈Z by ΦiΓ (q, w, ϑ) := (ϕiT (q, w), θΓ (iT ; q, w, ϑ)),
(q, w, ϑ) ∈ Z, i ∈ Z,
(3.5)
where ϕT is defined by (2.3), and both ϑ and θΓ (iT ; q, w, ϑ) are taken modulo 2π. Due to (3.4), it is well defined. In some sense, {ΦiΓ } is the Poincar´e map of (3.3) (of the period T ). Some observations on {ΦiΓ } are as follows: Lemma 3.3. (i) For any given i ∈ Z, θΓ (iT ; q, w, ϑ) is Lipschitz continuous on Z. (ii) For any given i ∈ Z, ΦiΓ (q, w, ϑ) is continuous on Z. Moreover, ΦnΓ1 +n2 = ΦnΓ1 ◦ ΦnΓ2
for n1 , n2 ∈ Z.
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Proof. (i) Without loss of generality, we can assume that i = 1 and n = 2. That is, Γ = {0, t1 } + T Z,
0 < t1 < T.
Let θi (t) = θΓ (t; qi , wi , ϑi ), i = 1, 2, be two solutions of (3.3). Due to (3.4), we only consider the case 0 ≤ ϑ2 − ϑ1 ≤ π. Then, |ϑ2 − ϑ1 | = |ϑ2 − ϑ1 |S2π . For t ∈ [0, t1 ), we have
θi (t) = cos2 θi (t) + qi (t) sin2 θi (t), t ∈ [0, t1 ) θi (0) = ϑi . Denote D(t) := θ2 (t) − θ1 (t). Then we have D (t) = cos2 θ2 (t) + q2 (t) sin2 θ2 (t) − (cos2 θ1 (t) + q1 (t) sin2 θ1 (t)) = (q2 (t) − q1 (t)) sin2 θ2 (t) + (cos θ2 (t) − cos θ1 (t))(cos θ2 (t) + cos θ1 (t)) + q1 (t)(sin θ2 (t) − sin θ1 (t))(sin θ2 (t) + sin θ1 (t)). One has some ζ(t), η(t) which are between θ1 (t) and θ2 (t) and satisfy cos θ2 (t) − cos θ1 (t) = −D(t) sin ζ(t),
sin θ2 (t) − sin θ1 (t) = D(t) cos η(t).
Thus, D (t) = (q1 (t)(sin θ2 (t) + sin θ1 (t)) cos η(t) − (cos θ2 (t) + cos θ1 (t)) × sin ζ(t)D(t) + (q2 (t) − q1 (t)) sin2 θ2 (t) := A(t)D(t) + B(t).
(3.6)
Note that (q1 , w1 ) ∈ ET (Q, W ) and q1 ∞ = Q ∞ . Therefore, A ∞ ≤ 2 Q ∞ + 2 =: M
B ∞ ≤ q1 − q2 ∞ .
The solution of (3.6) with the initial value D(0) = ϑ2 − ϑ1 is ⎞⎛ ⎞ ⎛ t t s D(t) = exp ⎝ A(s) ds⎠ ⎝ϑ2 − ϑ1 + B(s)e− 0 A(τ ) dτ ds⎠ , 0
(3.7)
t ∈ [0, t1 ).
0
Then for any t ∈ [0, t1 ), we have t ⎞ ⎞⎛ ⎛ t s |D(t)| ≤ exp ⎝ |A(s)| ds⎠ ⎝|ϑ2 − ϑ1 | + B(s)e− 0 A(τ ) dτ ds⎠ 0 0 ⎛ ⎞ t1 M t1 ⎝ ≤e |ϑ2 − ϑ1 | + q1 − q2 ∞ eM s ds⎠ ≤ C1 d, (3.8) 0
where (3.7) is used and d := d((q1 , w1 , ϑ1 ), (q2 , w2 , ϑ2 )).
(3.9)
By letting t → t1 −, we conclude that |θ2 (t1 −) − θ1 (t1 −)| ≤ C1 d.
(3.10)
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Now we consider t ∈ [t1 , T ). Then the solutions θi (t) satisfy
θi (t) = cos2 θi (t) + qi (t) sin2 θi (t), t ∈ [t1 , T ), θi (t1 ) = θi (t1 −) + wi (t1 ). Again by the same argument, we consider D(t) on [t1 , T ), and similarly as (3.8), we have |D(t)| ≤ eM (T −t1 ) ⎛ ×⎝|θ2 (t1 −) + w2 (t1 ) − θ1 (t1 −) − w1 (t1 )| + q1 −q2 ∞
T
⎞ eM (s−t1 ) ds⎠
t1
≤ eM (T −t1 ) ⎛ ×⎝|θ2 (t1 −) − θ1 (t1 −)| + w1 − w2 ∞ + q1 − q2 ∞
T
⎞ eM (s−t1 ) ds⎠
t1
≤ C2 d.
(by (3.9) and (3.10))
Then by letting t → T −, we have |θ2 (T −) − θ1 (T −)| = |D(T −)| ≤ C2 d for some C2 . Finally, |θΓ (T ; q2 , w2 , ϑ2 ) − θΓ (T ; q1 , w1 , ϑ1 )|S2π = |D(T −) + w2 (T ) − w1 (T )|S2π ≤ (C2 + 1)d, proving the Lipschitz continuity. (ii) We need only to prove that θΓ (n1 T ; q · n2 T, w · n2 T, θΓ (n2 T ; q, w, ϑ)) ≡ θΓ (n1 T + n2 T ; q, w, ϑ).
(3.11)
Let us denote θ1 (t) := θΓ (t + n2 T ; q, w, ϑ),
θ2 (t) := θΓ (t; q · n2 T, w · n2 T, θΓ (n2 T ; q, w, ϑ)).
Note that Γ is T -periodic. Then both θi (t) satisfy the following function:
θ = cos2 θ + q(t + n2 T ) sin2 θ, t ∈ R\Γ θ(t) = θ(t−) + w(t + n2 T ), t ∈ Γ. Since θ1 (0) = θ2 (0) = θΓ (n2 T ; q, w, ϑ), we conclude that θ1 (t) ≡ θ2 (t). By setting t = n1 T , we have the desired equality. Combining (i) with the definition (3.5), we have the continuity of ΦiΓ (q, w, ϑ). Lemma 3.3 (ii) shows that ΦΓ is a discrete skew-product flow on Z with the base flow {ϕiT }. Let us introduce the following observation function: θΓ (T ; q, w, ϑ) − ϑ , (3.12) T which can be considered as a function on ET (Q, W ) × S2π . It follows from Lemma 3.3 (i) that FΓ (q, w, ϑ) is Lipschitz continuous on Z. FΓ (q, w, ϑ) :=
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Lemma 3.4. There holds the following relation: k−1 θΓ (t; q, w, ϑ) − ϑ 1 = lim FΓ ◦ ΦiΓ (q, w, ϑ). t→+∞ k→+∞ k t i=0
lim
(3.13)
That is, if one of the limits exists, then another exists as well and they are equal. Proof. For any k ∈ N, we have θΓ (kT ; q, w, ϑ) − ϑ kT =
k−1 1 θΓ ((i + 1)T ; q, w, ϑ) − θΓ (iT ; q, w, ϑ) k i=0 T
=
k−1 1 θΓ (T ; q · iT, w · iT, θΓ (iT ; q, w, ϑ)) − θΓ (iT ; q, w, ϑ) k i=0 T
=
k−1 1 FΓ ◦ ΦiΓ (q, w, ϑ) k i=0
(by (3.11))
(by (3.5), (3.4) and (3.12)).
Now result (3.13) follows from (3.2).
4. Proof of the Main Result Define k−1 1 FΓ ◦ ΦiΓ (q, w, ϑ) k→+∞ k i=0
FΓ∗ (q, w, ϑ) := lim
(4.1)
whenever the limit exists. Due to uniqueness of solutions of ODEs, we know that θΓ (t; q, w, ϑ0 ) < θΓ (t; q, w, ϑ) < θΓ (t; q, w, ϑ0 + 2π)
when ϑ0 < ϑ < ϑ0 + 2π.
Thus, by (3.13) and (4.1), if FΓ∗ (q, w, ϑ0 ) exists for some ϑ0 , then FΓ∗ (q, w, ϑ) exists for any ϑ ∈ S2π and it is independent of ϑ. The extension of the unique ergodic theorem in [7] will be used to establish the existence of rotation numbers. For our purpose, we choose the discrete case. Theorem 4.1. [7] Let {ϕi }i∈Z be a continuous dynamical system of discrete time on a compact metric space X. Then, for any f ∈ C(X) := C(X, R) satisfying f dμ = 0 X
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for all invariant Borel probability measures μ of {ϕi }, one has k−1 1 lim f ◦ ϕi (x) = 0 k→+∞ k i=0
uniformly in x ∈ X. Now we concentrate on the ergodic limits (4.1). For simplicity, denote E := ET (Q, W ) and Z = ET (Q, W ) × S2π . The proof is along the line of [7]. For more details, see also [4]. Due to Lemma 3.3 (ii), {ΦiΓ } is a continuous dynamical system of discrete time which admits invariant Borel probability measures μ, see [16, Corollary 6.9.1] or [8,12]. Theorem 4.2. Given (Γ, Q, W ) ∈ PLT,n × APF 2 . We assert that Γ = Γ (Q, W ) := FΓ dμ
(4.2)
Z
is independent of invariant Borel probability measures μ of {ΦiΓ }. Moreover, one has the convergence k−1 1 FΓ ◦ ΦiΓ (q, w, ϑ) = Γ , k→+∞ k i=0
lim
(4.3)
which is uniform in the whole space (q, w, ϑ) ∈ Z. Proof. Let μ be any invariant Borel probability measure of {ΦiΓ }. By the Birkhoff ergodic theorem, there exists a Borel set Zμ ⊂ Z, depending on the measure μ, such that μ(Zμ ) = 1 and the limits FΓ∗ (q, w, ϑ) of (4.1) exist for all (q, w, ϑ) ∈ Zμ . Furthermore, FΓ∗ ∈ L1 (Z, μ) is integrable and satisfies FΓ∗ dμ = FΓ dμ =: Γ,μ . (4.4) Z
Z
FΓ∗ (q, w, ϑ)
Due to the independence of of ϑ, Zμ can be taken as the form Zμ = Eμ × S2π , where Eμ is a Borel set of E. Assertion 1. ν(Eμ ) = 1 where ν is the Haar measure on E = ET (Q, W ). To this end, denote the projection Z = E × S2π → E by π1 . Then the push (π1 )∗ μ = μ ◦ π1−1 of μ on Z to E is an invariant Borel probability measure of ϕT . Hence, (π1 )∗ μ must be the Haar measure ν on E. Thus, ν(Eμ ) = μ(π1−1 (Eμ )) = μ(Eμ × S2π ) = 1. ˆμ ) = 1 and ˆμ ⊂ Eμ such that ν(E Assertion 2. There exists E FΓ∗ (q, w, ϑ) ≡ Γ,μ , FΓ∗ (q, w, ϑ)
FΓ∗ (q, w)
ˆμ × S2π . (q, w, ϑ) ∈ E
(4.5)
Since = is independent of ϑ ∈ S2π for (q, w) ∈ Eμ , FΓ∗ (q, w) can be considered as an integrable function in L1 (E, ν) which is ϕT ˆ μ ⊂ Eμ invariant. As ν is the unique ergodic measure of ϕT , one has some E ∗ ˆ ˆ such that ν(Eμ ) = 1 and FΓ (q, w) is constant on Eμ . By (4.4), this constant must be Γ,μ . This gives (4.5).
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ˆμ × S2π . These two assertions show that (4.3) has the constant limit on E We need to extend this result to the whole space Z. (i) Let us prove that the constant Γ,μ of (4.4) is independent of invariant measures μ of ΦΓ . In fact, for another measure μ , arguing as above, one has ˆμ , Γ,μ . Since ν(E ˆμ ) = ν(E ˆμ ) = 1, E ˆμ ∩ E ˆμ = ∅. the corresponding objects E ˆ ˆ Taking a point (q, w) ∈ Eμ ∩ Eμ , we have k−1 1 FΓ ◦ ΦiΓ (q, w, ϑ) = Γ,μ k→+∞ k i=0
Γ,μ = lim
∀ ϑ ∈ S2π .
See (4.1) and (4.5). Thus, Γ,μ is independent of μ, which is denoted by Γ as in (4.3). (ii) Let us consider the function f := FΓ − Γ . By Lemma 3.3 (i), f is continuous on Z. From (4.4), f fulfills the requirement of Theorem 4.1. Thus, as k → +∞, k−1 k−1 1 1 i f ◦ ΦΓ (q, w, ϑ) = FΓ ◦ ΦiΓ (q, w, ϑ) − Γ → 0 k i=0 k i=0
uniformly in (q, w, ϑ) ∈ Z. This gives (4.3).
Definition 4.3. Given (Γ, Q, W ) ∈ PLT,n × APF 2 . We call the number Γ = Γ (Q, W ) of (4.2) the rotation number of (3.1) or that of (1.1). Proof of Theorem 1.1. In the present terminology, the solutions θ(t; Q, Γ, S, ϑ) of (1.3) are the same as θΓ (t; Q, W, ϑ). Now the existence of the limit of (1.3) follows simply from (3.13), (4.2) and (4.3). In fact, one has θΓ (t; q, w, ϑ) − ϑ = Γ (Q, W ) t→+∞ t lim
(4.6)
uniformly in (q, w, ϑ) ∈ ET (Q, W )×S2π . That is, the rotation number Γ (Q, W ) describes the dynamics for not only Eq. (3.1), but also the family of Eqs. (3.3). Note that for any (q, w) ∈ ET (Q, W ), one has (Γ, q, w) ∈ PLT,n × APF 2 with the same lattice Γ. The rotation number Γ (q, w) of Eq. (3.3) is defined as well. By (4.6), one has actually Γ (q, w) = Γ (Q, W )
∀(q, w) ∈ ET (Q, W ).
In fact, one has the ergodic representation (4.2) for rotation number Γ (Q, W ). Given a periodic lattice Γ ∈ PLT,n . Let us consider rotation number Γ (Q, W ) as a nonlinear functional of (Q, W ) ∈ (APF 2 , · ∞ ). We will show that Γ (Q, W ) is continuous in (Q, W ). Note that in the ergodic representation (4.2), even when Γ ∈ PLT,n is fixed, all of these objects depend on the pair (Q, W ) ∈ APF 2 . These include the space Z, the dynamical system {ΦiΓ } and its invariant measure μ, and the observation FΓ . In the following, we will apply the monotonicity of rotation numbers to give a considerable reduction for the continuity problem.
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Theorem 4.4. Given a periodic lattice Γ ∈ PLT,n . Then rotation number Γ (Q, W ) is continuous in (Q, W ) ∈ APF 2 with the uniform topology · ∞ . Proof. At first, let us prove the following monotonicity for rotation numbers: Suppose that Q0 ≤ Q1 and W0 ≤ W1 . By the comparison theorem of ODEs with respect to the vector field, one has θΓ (t; Q0 , W0 , ϑ) ≤ θΓ (t; Q1 , W1 , ϑ) for all t ≥ 0 and therefore, Γ (Q0 , W0 ) ≤ Γ (Q1 , W1 )
when Q0 ≤ Q1 , W0 ≤ W1 .
In case (Q1 , W1 ) − (Q0 , W0 ) ∞ = max{ Q1 − Q0 ∞ , W1 − W0 ∞ } < δ, we have Q0 − δ < Q1 < Q0 + δ,
W0 − δ < W1 < W0 + δ.
Hence, Γ (Q0 − δ, W0 − δ) ≤ Γ (Q1 , W1 ) ≤ Γ (Q0 + δ, W0 + δ). In order to show the continuity of Γ (Q, W ) in (Q, W ), it suffices to prove that for any fixed (Q, W ) ∈ APF 2 , Γ (Q + δ, W + δ) is continuous in δ at δ = 0. More generally, for any sequence δk → δ0 ∈ R, by denoting Qk := Q + δk ,
Wk := W + δk ,
k ∈ N ∪ {0},
(4.7)
we will show that Γ (Qk , Wk ) → Γ (Q0 , W0 )
as k → +∞.
(4.8)
For k ∈ N ∪ {0}, denote Ek = ET (Qk , Wk ),
Zk = Ek × S2π .
Note that Ek and Zk depend on k. However, by denoting E := ET (Q, W ),
Z = E × S2π ,
we know from (4.7) that Ek = E + δk (1, 1) ⊂ APF 2 are translations of E. Accordingly, one can introduce homeomorphisms σk : Z = E × S2π → Zk = (E + δk (1, 1)) × S2π by σk (q, w, ϑ) = (q + δk , w + δk , ϑ). Now, the flows ΦΓ = {ΦiΓ } on Zk induce ΦiΓ,k (q, w, ϑ) := σk−1 ◦ ΦiΓ ◦ σk (q, w, ϑ), One sees that these ΦΓ,k := one has
{ΦiΓ,k }
i ∈ Z, (q, w, ϑ) ∈ Z.
are flows on the same space Z. Explicitly,
ΦiΓ,k (q, w, ϑ) = (q · iT, w · iT, θΓ (iT ; q + δk , w + δk , ϑ))
(4.9)
for i ∈ Z and (q, w, ϑ) ∈ Z. Similarly, the corresponding observation functions FΓ on Zk are transformed into FΓ,k (q, w, ϑ) := FΓ ◦ σk (q, w, ϑ),
(q, w, ϑ) ∈ Z.
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By (3.12), one has θΓ (T ; q + δk , w + δk , ϑ) − ϑ , (q, w, ϑ) ∈ Z. (4.10) T For any k ∈ N ∪ {0}, let μ ˆk be any ΦΓ -invariant probability measure on Zk . ˆk ◦ σk is a ΦΓ,k -invariant measure on Z. Now Then, for each k ∈ N, μk := μ the ergodic representation (4.2) for Γ (Qk , Wk ) can be rewritten as Γ (Qk , Wk ) = FΓ dˆ μk = FΓ,k dμk . (4.11) FΓ,k (q, w, ϑ) =
Zk
Z
The last representations are now on the same space Z. According to [16, Theorem 6.5], we may assume that μk μ∞ in the weak topology for some measure μ∞ . That is, g dμk → g dμ∞ ∀ g ∈ C(Z) = C(Z, R). Z
Z
We have actually
gk → g∞ in (C(Z), · ∞ ) =⇒
gk dμk → Z
g∞ dμ∞ ,
(4.12)
Z
because ⎞ ⎛ gk dμk − g∞ dμ∞ = (gk − g∞ ) dμk + ⎝ g∞ dμk − g∞ dμ∞ ⎠ Z
Z
Z
Z
Z
Z
Z
⎞ ⎛ = O( gk − g∞ ∞ ) + ⎝ g∞ dμk − g∞ dμ∞ ⎠,
where the two terms tend to 0, following the definitions of gk → g∞ and the weak convergence μk μ∞ . At first, we assert that μ∞ is a ΦΓ,0 -invariant probability measure. Indeed, let f ∈ C(Z). It follows from Lemma 3.3 (i) and (4.9) that gk := f ◦ Φ1Γ,k → g∞ := f ◦ Φ1Γ,0 in (C(Z), · ∞ ). As μk is ΦΓ,k -invariant, we have f dμk = f ◦ Φ1Γ,k dμk = gk dμk . Z
Z
Z
By letting k → +∞, we obtain f dμ∞ = g∞ dμ∞ = f ◦ Φ1Γ,0 dμ∞ . Z
Z
(4.13)
Z
See (4.12). As f ∈ C(Z) is arbitrary, by [16, Theorem 6.8], (4.13) means that μ∞ is ΦΓ,0 -invariant. Next, it follows from Lemma 3.3 (i) and (4.10) that FΓ,k → FΓ,0 in (C(Z), · ∞ ).
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Now it follows from the ergodic representations (4.11) that Γ (Qk , Wk ) = FΓ,k dμk → FΓ,0 dμ∞ . Z
779
(4.14)
Z
See also (4.12). Since μ∞ is ΦΓ,0 -invariant, we know from Theorem 4.2 that FΓ,0 dμ∞ = Γ (Q0 , W0 ). Z
Hence, (4.14) proves the desired result (4.8).
We end the paper with some remarks. (i) When S = {0}, i.e., W = 0, the results of this paper reduce to those for classical Schr¨ odinger equations with almost periodic potentials [7,4]. (ii) For general almost periodic, symplectic transmissions A = {Ai }, it is also possible to deduce the existence of rotation numbers by considering the hull of transmissions. (iii) Following the reduction of Eq. (1.1) to multiplication of matrices as in [2,3], it is possible to prove the existence of Lyapunov exponents for Eq. (1.1) under the assumptions on (Q, Γ, S) above.
Acknowledgements The authors would like to thank Professor R. Krikorian for helpful discussions.
References [1] Arnold, L.: Random Dynamical Systems. In: Springer Monographs Mathematics. Springer-Verlag, Berlin (1998) [2] Fayad, B., Krikorian, R.: Exponential growth of product of matrices in SL(2, R). Nonlinearity 21, 319–323 (2008) [3] Feng, D.-J.: Lyapunov exponents for products of matrices and multifractal analysis, I. Positive matrices. Isr. J. Math. 138, 353–376 (2003) [4] Feng, H., Zhang, M.: Optimal estimates on rotation number of almost periodic systems. Z. Angew. Math. Phys. 57, 183–204 (2006) [5] Fink, A.: Almost Periodic Differential Equations. Springer, New York (1974) [6] Gan, S., Zhang, M.: Resonance pockets of Hill’s equations with two-step potentials. SIAM J. Math. Anal. 32, 651–664 (2000) [7] Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 84, 403–438 (1982). Erratum, Commun. Math. Phys. 90, 317–318 (1983) [8] Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995) [9] Kronig, R., Penney, W.: Quantum mechanics in crystal lattices. Proc. R. Soc. Lond. 130, 499–513 (1931)
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[10] Li, W., Lu, K.: Rotation numbers for random dynamical systems on the circle. Trans. Am. Math. Soc. 360, 5509–5528 (2008) [11] Meng, G., Zhang, M.: Measure differential equations, I. Continuity of solutions in measures with weak∗ topology. Preprint. http://faculty.math.tsinghua.edu. cn/∼mzhang (2009) [12] Nemytskii, V.V., Stepanov, V.V.: Qualitative Theory of Differential Equations. Princeton University Press, Princeton (1960) [13] Niikuni, H.: Absent spectral gaps of generalized Kronig-Penney Hamiltonians. Tsukuba J. Math. 31, 39–65 (2007) [14] Niikuni, H.: The rotation number for the generalized Kronig-Penney Hamiltonians. Ann. Henri Poincar´e 8, 1279–1301 (2007) [15] Novo, S., N´ un ˜ez, C., Obaya, R.: Ergodic properties and rotation number for linear Hamiltonian systems. J. Differ. Equ. 148, 148–185 (1998) [16] Walters, P.: An Introduction to Ergodic Theory. Springer-Verlag, New York (1982) [17] Zhang, M.: The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials. J. Lond. Math. Soc. 64(2), 125–143 (2001) Meirong Zhang and Zhe Zhou Department of Mathematical Sciences Tsinghua University Beijing 100084, China e-mail:
[email protected];
[email protected] Meirong Zhang Zhou Pei-Yuan Center for Applied Mathematics Tsinghua University Beijing 100084, China Communicated by Rafael D. Benguria. Received: May 26, 2009. Accepted: February 16, 2010.
Ann. Henri Poincar´e 11 (2010), 781–803 c 2010 Springer Basel AG 1424-0637/10/050781-23 published online October 15, 2010 DOI 10.1007/s00023-010-0039-2
Annales Henri Poincar´ e
Regularity Results for the Spherically Symmetric Einstein–Vlasov System H˚ akan Andr´easson Abstract. The spherically symmetric Einstein–Vlasov system is considered in Schwarzschild coordinates and in maximal-isotropic coordinates. An open problem is the issue of global existence for initial data without size restrictions. The main purpose of the present work is to propose a method of approach for general initial data, which improves the regularity of the terms that need to be estimated compared to previous methods. We prove that global existence holds outside the center in both these coordinate systems. In the Schwarzschild case we improve the bound on the momentum support obtained in Rein et al. (Commun Math Phys 168:467–478, 1995) for compact initial data. The improvement implies that we can admit non-compact data with both ingoing and outgoing matter. This extends one of the results in Andr´easson and Rein (Math Proc Camb Phil Soc 149:173–188, 2010). In particular our method avoids the difficult task of treating the pointwise matter terms. Furthermore, we show that singularities never form in Schwarzschild time for ingoing matter as long as 3m ≤ r. This removes an additional assumption made in Andr´easson (Indiana Univ Math J 56:523–552, 2007). Our result in maximal-isotropic coordinates is analogous to the result in Rendall (Banach Center Publ 41:35–68, 1997), but our method is different and it improves the regularity of the terms that need to be estimated for proving global existence in general.
1. Introduction In the present work we investigate the issue of global existence for the spherically symmetric Einstein–Vlasov system when the initial data is unrestricted in size. The system is analyzed in Schwarzschild coordinates, i.e. in a polar time gauge, and in maximal-isotropic coordinates where a maximal time gauge is imposed. These coordinate systems are often, in the literature, conjectured to be singularity avoiding. However, there is to our knowledge no proof of this
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statement for any matter model and it would be very satisfying to provide an answer to this conjecture for the Einstein–Vlasov system. Moreover, a proof of global existence would be of great importance due to its relation to the weak cosmic censorship conjecture, cf. [1,4,5,9,11,13]. A third motivation for our interest in this problem is the fact that global existence for general data have been obtained for the Vlasov–Poisson system, i.e. Newtonian gravity coupled to Vlasov matter. Batt [8] showed global existence in the spherically symmetric case 1977, and the general case was settled independently by Pfaffelmoser [16], and Lions and Perthame [14] in 1991. It is thus natural to ask if similar results can be obtained when Newtonian gravity is replaced by general relativity. One should of course bear in mind that the situations are fundamentally different since in the latter case there exist data which lead to spacetime singularities, cf. [4,7,21]. Nevertheless, as mentioned above, global existence may hold for polar or maximal time slicing. The issue of global existence for the spherically symmetric Einstein– Vlasov system has previously been investigated in several papers, cf. [1,3,4,6, 12,18–20]. Global existence for small initial data has been proved in [18] and [12] for massive and massless particles respectively. In [4] initial data which guarantee formation of black holes are constructed, and it is proved that for a particular class of such initial data, with a steady state in the interior of the surrounding matter, global existence holds in Schwarzschild time. In [3] global existence is shown to hold in a maximal time gauge for rapidly outgoing matter. The methods of proofs in these cases are all tailored to treat special initial data and they will likely not apply in more general situations. The investigations [1,6,19,20] are conditional in the sense that assumptions are made on the solutions, and not only on the initial data. However, the methods are general and cover large classes of initial data and can thus be thought of as possible approaches for treating the general case. The main purpose of the present work is to propose a method of approach for general initial data, which improves the regularity of the terms that need to be estimated compared to the methods in [1,6,19,20]. Furthermore, the method improves and simplifies some of the previous results. To be more precise, let us discuss the relation between these studies. An important quantity in the study of the Einstein–Vlasov system and the Vlasov-Poisson system is Q(t) := sup{1 + |v| : ∃(s, x) ∈ [0, t] × R3 such that f (s, x, v) = 0}. (1.1) Here f is the density function on phase, and v ∈ R3 is the momentum. Q(t) measures the support of the momenta, and the content of the continuation criterion for these systems, cf. [8,18,20], is that solutions can be extended as long as Q(t) remains bounded. The definition of Q only applies in the case when the initial data have compact support in the momentum variables. For non-compact initial data, it was shown in [6] that solutions can be extended ˜ as long as Q(t) is bounded where ◦ ˜ := sup 1 + |V (s, 0, r, v)| | 0 ≤ s ≤ t, (r, v) ∈ supp f , Q(t) (1.2) 1 + |v|
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Here V is a solution of the characteristic system, cf. (2.18). The analysis in [1,6,19] is carried out in Schwarzschild coordinates. The main result in [19] shows that as long as there is no matter in the ball {x ∈ R3 : |x| ≤ }, the estimate C()t
Q(t) ≤ elog Q(0)e
,
(1.3)
holds. Here C() is a constant which depends on . In view of the continuation criterion this can thus be viewed as a global existence result outside the center of symmetry for initial data with compact support. The bound is obtained by estimating each term individually in the characteristic equation (2.18) for the radial momentum. This involves a particular difficulty. Let us consider the term involving μr in (2.18). The Einstein equations imply that m μr = 2 e2λ + 4πrpe2λ =: T1 + T2 , r where m is the quasi local mass and p is the pressure. There is a distinct difference between the terms T1 and T2 due to the fact that m can be regarded as an average, since it is given as a space integral of the energy density ρ, whereas p is a pointwise term. Also the term involving λt in (2.18) is a pointwise term in this sense. The method in [19] is able to estimate the pointwise terms outside the center but generally it seems very unpleasant to have to treat these terms. In the present work we give an alternative and simplified proof of the result in [19]. In particular our method avoids the pointwise terms by using the fact that the characteristic system can be written in a form such that Green’s formula in the plane can be applied. This results in a combination of terms involving second order derivatives which can be substituted for by one of the Einstein equations. This method was first introduced in [1] but here the set up is different and the application of Green’s formula becomes very natural. In addition the bound of Q is improved compared to (1.3) and reads Ct Q(t) ≤ Q(0) + 2 eC(1+t)/ . ˜ This bound implies that also Q(t) is bounded and therefore a consequence of the method is that global existence outside the center also holds for noncompact initial data. This improves the result in [6] where only non-compact data for which the matter is ingoing, and such that the matter keeps on going inwards for all times, are admitted. Another consequence of our method is that global existence holds for ingoing matter as long as 3m(t, r) ≤ r. Note that in Schwarzschild coordinates 2m(t, r) ≤ r always, and that there are closed null geodesics if 3m = r in the Schwarzschild spacetime. This result was already proved in [1] but an additional assumption was imposed which now has been dropped. We now turn to the case of maximal-isotropic coordinates. Rendall shows in [20] global existence outside the center in maximal-isotropic coordinates.
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The bound on Q(t) is again obtained by estimating each term in the characteristic equation, cf. Eq. (6.17). In this case there are no pointwise terms in contrast to the case with Schwarzschild coordinates. The terms are however, in analogy with the Schwarzschild case, strongly singular at the center. The method that we use in Schwarzschild coordinates also applies in this case. The improvement lies in the fact that there is some gain in regularity, i.e. the terms that need to be estimated are less singular compared to the corresponding terms in [20]. Indeed, these terms can schematically, in both the Schwarzschild and maximal-isotropic case, be written as ∞ g(t, η)dη,
(1.4)
0
and the known a priori bounds read r ηg(t, η)dη.
(1.5)
0
Hence, the degree of the singularity is of order one in the radial variable. Roughly, for the methods in [19] and [20] the singularity is of second order. An advantage of our method is clearly that it applies in both cases, and it also turns out that the principal term to be estimated in the two cases is the spacetime integral of the Gauss curvature of the two-dimensional quotient manifold M/SO(3), where M is the four-dimensional spacetime manifold. This is interesting since the Gauss curvature is a coordinate independent quantity. Moreover, the principal term in [10, p. 1172], has the same form. Finally we mention that our method also applies in the case of maximal-areal coordinates which were used in the study [3]. The analysis in this case is completely analogous and is left out. The outline of the paper is as follows. The Einstein–Vlasov system in Schwarzschild coordinates is treated in Sects. 2–5. In Sect. 2, the system is formulated and the a priori bounds are given in Sect. 3. Section 4 is devoted to the proof of global existence outside the center and in Sect. 5 global existence is shown for ingoing matter which satisfies m/r ≤ 1/3. Sections 6–8 concern the system in maximal-isotropic coordinates. The system is given in Sect. 6, and in Sect. 7 the necessary a priori bounds are derived. Section 8 is devoted to the proof of the global existence theorem outside the center.
2. The Einstein–Vlasov System For an introduction to the Einstein–Vlasov system and kinetic theory we refer to [2,20], and for a careful derivation of the system given below we refer to [17]. In Schwarzschild coordinates the spherically symmetric metric takes the form ds2 = −e2μ(t,r) dt2 + e2λ(t,r) dr2 + r2 (dθ2 + sin2 θdϕ2 ).
(2.1)
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The Einstein equations read
e−2λ
e−2λ (2rλr − 1) + 1 = 8πr2 ρ,
(2.2)
e−2λ (2rμr + 1) − 1 = 8πr2 p,
(2.3)
(2.4) λt = −4πreλ+μ j, 1 μrr + (μr − λr ) μr + − e−2μ (λtt + λt (λt − μt )) = 8πpT . r (2.5)
The indices t and r denote derivatives. The Vlasov equation for the density distribution function f = f (t, r, w, L) is given by w L ∂t f + eμ−λ ∂r f − λt w + eμ−λ μr E − eμ−λ 3 (2.6) ∂w f = 0, E r E where E = E(r, w, L) =
1 + w2 + L/r2 .
(2.7)
Here w ∈ (−∞, ∞) can be thought of as the radial component of the momentum variables, and L ∈ [0, ∞) is the square of the angular momentum. The matter quantities are defined by π ρ(t, r) = 2 r p(t, r) =
j(t, r) =
pT (t, r) =
π r2 π r2
∞ ∞ −∞ 0 ∞ ∞
−∞ 0 ∞ ∞
Ef (t, r, w, L) dw dL,
(2.8)
w2 f (t, r, w, L) dw dL, E
(2.9)
wf (t, r, w, L), dw dL,
(2.10)
−∞ 0 ∞ ∞
π 2r4
−∞ 0
L f (t, r, w, L) dw dL. E
(2.11)
Here ρ, p, j and pT are the energy density, the radial pressure, the current and the tangential pressure respectively. The following boundary conditions are imposed to ensure asymptotic flatness lim λ(t, r) = lim μ(t, r) = 0,
r→∞
r→∞
(2.12)
and a regular center requires λ(t, 0) = 0,
t ≥ 0.
(2.13)
We point out that the Einstein equations are not independent and that e.g. the equations (2.4) and (2.5) follow by (2.2), (2.3) and (2.6).
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As initial data it is sufficient to prescribe a distribution function f = f(r, w, L) ≥ 0 such that ◦
r
◦
4πη 2 ρ(η) dη <
r . 2
(2.14)
0 ◦
Here we denote by ρ the energy density induced by the initial distribution ◦ function f. This condition ensures that no trapped surfaces are present ini◦ tially. Given f, Eqs. (2.2) and (2.3) can be solved to give λ and μ at t = 0. We ◦ will only consider initial data such that f = 0 if L > L2 , for some L2 > 0, and ◦ such that f = 0 if r > R2 , for some R2 > 0. If in addition the initial data is C 1 we say that it is regular. In most of the previous investigations the condition of compact support on the momentum variable w has been included in the definition of regular data. The exception is [6] where non-compact initial data is studied and a decay condition replaces the assumption of compact support. In this study we also include data with non-compact support and we impose the decay condition from [6] sup (r,w,L)∈R3
◦
|w|5 f(r, w, L) < ∞.
(2.15)
We distinguish between the two cases; regular initial data with compact support and regular initial data which satisfy the decay condition (2.15). We denote these classes of initial data by I C and I D respectively. The main results below concern subclasses of I C and I D . Given R1 > 0, we define the subclass I C (R1 ) with radial cut-off by ◦ ◦ I C (R1 ) = f ∈ I C : f = 0 for r ≤ R1 . The subclass I D (R1 ) is defined analogously. Let us now write down a couple of facts about the system (2.2)–(2.13). A solution to the Vlasov equation can be written f (t, r, w, L) = f0 (R(0, t, r, w, L), W (0, t, r, w, L), L),
(2.16)
where R and W are solutions of the characteristic system W dR = e(μ−λ)(s,R) , ds E(R, W, L) dW = −λt (s, R)W − e(μ−λ)(s,R) μr (s, R)E(R, W, L) ds L + e(μ−λ)(s,R) 3 , R E(R, W, L)
(2.17)
(2.18)
such that (R(s, t, r, w, L), W (s, t, r, w, L), L) = (r, w, L) when s = t. This representation shows that f is nonnegative for all t ≥ 0, f ∞ = f0 ∞ , and that
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f (t, r, w, L) = 0 if L > L2 . The quasi local mass m is defined by r m(t, r) = 4π
η 2 ρ(t, η)dη,
(2.19)
2m(t, r) . r
(2.20)
0
and by integrating (2.2) we find e−2λ(t,r) = 1 − A fact that we will need is that μ + λ ≤ 0. This is easily seen by adding Eqs. (2.2) and (2.3), which gives λr + μr ≥ 0, and then using the boundary conditions on λ and μ. Furthermore, from (2.20) we get that λ ≥ 0, and it follows that μ ≤ 0. We also introduce the notations μ ˆ and μ ˇ. From Eq. (2.3) we have ∞ μ(t, r) = − r
m(t, η) 2λ e − η2
∞ 4πηpe2λ dη =: μ ˆ+μ ˇ.
(2.21)
r
Finally, we note that in [18] and [6] local existence theorems are proved for compact and non-compact initial data respectively, and it will be used below that solutions exist on some time interval [0, T [.
3. A Priori Bounds in Schwarzschild Coordinates In this section we collect the a priori bounds that we need in the proofs below. There are two known conserved quantities for the Einstein–Vlasov system, the number of particles and the ADM mass M . Here, we will only need the latter which is given by ∞ r2 ρ(t, r)dr.
M = 4π
(3.1)
0
The conservation of the ADM mass follows from general arguments but it can easily be obtained by simply taking the time derivative of the integral expression and use of the Vlasov equation and the Einstein equations. The following results are given in [1] but since the proofs are very short we have included them here for completeness. By a regular solution we mean a solution which is launched by regular initial data.
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Lemma 1. Let (f, μ, λ) be a regular solution to the Einstein–Vlasov system. Then ∞ 4πr(ρ + p)e2λ eμ+λ dr ≤ 1, (3.2) 0
∞
m + 4πrp e2λ eμ dr ≤ 1. r2
(3.3)
0
Proof. Using the boundary condition (2.12) we get μ+λ
1≥1−e
∞ (t, 0) =
d μ+λ e dr dr
0
∞ = (μr + λr )eμ+λ dr. 0
The right-hand side equals (3.2) by Eqs. (2.2) and (2.3) which completes the first part of the lemma. The second part follows by studying eμ instead of eμ+λ . Next we show that not only ρ(t, ·) ∈ L1 , which follows from the conservation of the ADM mass, but that also e2λ ρ(t, ·) ∈ L1 . Lemma 2. Let (f, μ, λ) be a regular solution to the Einstein–Vlasov system. Then ∞ ∞ t 2 2λ . (3.4) r e ρ(t, r)dr ≤ r2 e2λ ρ(0, r)dr + 8π 0
0
Proof. Using the Vlasov equation we obtain
m ∂t r2 e2λ ρ(t, r) = −∂r r2 eμ+λ j − reμ+λ 2je2λ r 2 μ+λ 1 μ+λ ≤ −∂r r e j + re (ρ + p)e2λ . 2 Here we used that m/r ≤ 1/2 together with the elementary inequality 2|j| ≤ ρ + p, which follows from the expressions (2.8)–(2.10). In view of (2.16) and (2.17) we see that limr→∞ r2 j(t, r) = 0, since the initial data has compact support in r. Since the solution is regular and hence bounded the boundary term at r = 0 also vanishes (as a matter of fact spherical symmetry even implies that j(t, 0) = 0). Thus, by lemma 1 we get d dt
∞ r2 e2λ ρ(t, r)dr ≤
1 , 8π
0
which completes the proof of the lemma.
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4. A Regularity Result in Schwarzschild Coordinates Theorem 1. Let 0 < < R1 . Consider a solution of the spherically symmetric Einstein–Vlasov system, launched by initial data in I D (R1 ), on its maximal time interval [0, T [ of existence. If f (s, r, ·, ·) = 0, for (s, r) ∈ [0, t[×[0, ], then C(1+t) ˜ ≤ 1 + Ct e . Q(t) (4.1) 2 In particular, if f (t, r, ·, ·) = 0 for (t, r) ∈ [0, T [×[0, ], then T = ∞. The last statement in the theorem is a consequence of the continuation criterion derived in [6]. The theorem thus improves the result in [6] which was restricted to a special class of initial data where all the matter is ingoing. The result holds in particular for compactly supported data, and the bound (4.1) improves the bound in [19], cf. inequality (1.3). Proof. We consider the quantities G = E + W and H = E − W, which satisfy G > 0, H > 0. Along a characteristic (R(s), W (s), L) we have by (2.17) and (2.18). W Leμ−λ dG = − λt + μr eμ−λ G + 3 , (4.2) ds E R E and
dH W Leμ−λ μ−λ = λt + μr e H− 3 . ds E R E
Let us first consider the quantity H. From (4.3) we have dH W μ−λ ≤ λt + μr e H. ds E
(4.3)
(4.4)
It follows that
W (s) t (μ−λ)(s,R(s)) ] ds . H(t) ≤ H(0)e 0 [λt (s,R(s)) E(s) +μr (s,R(s))e
(4.5)
Let us denote the curve (s, R(s)), 0 ≤ s ≤ t, by γ. By using that dR W = eμ−λ , dt E the integral above can be written as the curve integral e(−μ+λ)(t,r) λt (t, r)dr + e(μ−λ)(t,r) μr (t, r)dt.
(4.6)
γ
Let Γ denote the closed curve Γ := γ + Ct + C∞ + C0 , oriented clockwise, where Ct = {(t, r) : R(t) ≤ r ≤ R∞ }, C∞ = {(s, R∞ ) : t ≥ s ≥ 0},
(4.7) (4.8)
C0 = {(0, r) : R∞ ≥ r ≥ R(0)}.
(4.9)
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Here R∞ ≥ R2 +t, so that f = 0 when r ≥ R∞ . We now apply Green’s formula in the plane and use Eq. (2.5) to obtain e−μ+λ λt dr + eμ−λ μr dt Γ
=
∂t e−μ+λ λt − ∂r eμ−λ μr dt dr
Ω
=
(λtt + (λt − μt ) λt − (μrr + (μr − λr )μr )eμ−λ dt dr
Ω
eμ−λ − 8πpT eμ+λ dt dr r Ω 2m = eμ+λ − 4π(ρ − p) − 8πp dt dr. T r3 (μr − λr )
=
(4.10)
Ω
Using the hypothesis that matter stays away from the region r ≤ , we have R(s) ≥ , 0 ≤ s ≤ t, and we get in view of (3.3), together with the facts that λ ≥ 0, ρ − p ≥ 0 and q ≥ 0, 2m 2t μ+λ (4.11) e − 4π(ρ − p) − 8πpT dt dr ≤ . r3 Ω
Remark 1. The integrand above has a geometrical meaning. It is the scalar curvature of the quotient manifold M/SO(3) with metric ds2 = −e2μ dt2 +e2λ dr2 . It is interesting to note that the principal term in [10, p. 1172] has the same form. Now we wish to estimate the curve integral (4.6). We have M · · · = · · · − 4πrje2λ dr − dt − 4πrje2λ dr. 2 R∞ γ
Γ
Ct
C∞
(4.12)
C0
Here we used that λt = 4πrjeμ+λ and that μr (t, R∞ ) = along C0 is given in terms of the initial data, 4πrje2λ dr ≤ C.
M 2 . R∞
The integral
(4.13)
C0
By letting R∞ → ∞ the integral along C∞ vanishes and it remains to estimate the contribution from the integral along Ct . In view of (3.4) we get ∞ 4πrje2λ dr ≤ 1 4πr2 ρ(t, r)e2λ(t,r) dr ≤ C (1 + t). (4.14) Ct
0
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The estimate of G is very similar. Indeed, by the hypothesis of the theorem we have R(t) ≥ , thus √ √ L L2 Leμ−λ ≤ 2 ≤ 2 . (4.15) 3 R E We derive in view of (4.2)
W (s) t (μ−λ)(s,R(s)) ]ds G(t) ≤ G(0)e 0 [−λt (s,R(s)) E(s) −μr (s,R(s))e t √ (s) (μ−λ)(s,R(s)) L2 τt [−λt (s,R(s)) W ]ds dτ. E(s) −μr (s,R(s))e + e 2
(4.16)
0
Note that the integrals in the exponent above are identical to the integral in (4.5) except for the sign. We can accordingly use the same arguments as above, the only difference is that we use (4.14) twice, for Ct and Cτ , and that the integral in (4.10) has opposite sign. Since 2pT ≤ ρ − p it is sufficient to estimate the integral 8π(ρ − p)eμ+λ dt dr. (4.17) Ω
We use the bound (3.2) to obtain the estimate (4.11) also in this case. The estimates of the boundary terms follow by the argument above for H. Hence, we have the following bounds C(1+t) Ct (4.18) G(t) ≤ G(0) + 2 e , and H(t) ≤ H(0)e
C(1+t)
.
(4.19)
Since E = G + H this implies that C(1+t) Ct ˜ Q(t) ≤ 1 + 2 e .
(4.20)
This completes the proof of Theorem 1.
5. Global Existence for Ingoing Matter with 3m ≤ r ◦
In this section we consider compactly supported initial data f ∈ I C (R1 ) such that for given 0 < L1 < L2 and P > 0, it holds that L ≥ L1 , and w < −P for all ◦ (w, L) ∈ supp f. Furthermore, the initial data have the property that 3m < r everywhere. Note that in Schwarzschild spacetime there are closed null geodesics when 3m = r. Let us denote this class of initial data by I C (R1 , L1 , P, 3) ⊂ I C (R1 ). By continuity there is a T1 > 0 such that w ≤ 0 for w ∈ supp f (t), and 3m(t, r) ≤ r everywhere, for t ≤ T1 . We will show that on the time interval [0, T1 ] singularities do not form in the evolution. This can be phrased as a global existence result for ingoing matter satisfying 3m ≤ r. A similar result
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is proved in [1] but an additional assumption on the solution is imposed and the present result is thus an improvement. Theorem 2. Consider a solution to the spherically symmetric Einstein–Vlasov ◦ system launched by initial data f ∈ I C (R1 , L1 , P, 3). Let T > 0 be the maximal time interval on which the solution exists and let T1 be as above. If T < ∞ then T1 < T. Proof. Let us consider the quantity Heμˆ along a given characteristic (t, R(t), W (t), L), with W ≤ 0. In view of (4.3) and (2.21) we have d W W Leμ−λ (Heμˆ ) = λt + μr eμ−λ + μ ˆt + μ ˆr eμ−λ Heμˆ − 3 eμˆ . (5.1) ds E E R E Since μ = μ ˆ+μ ˇ we have μr eμ−λ + μˆr
W μ−λ W e =μ ˇr eμ−λ + μ ˆr eμ−λ 1 + E E 1 + L/R2 . ˆr eμ−λ =μ ˇr eμ−λ + μ E(E − W )
(5.2)
Since H = E − W and μ ˆr =
me2λ , r2
we obtain
2λ 2 me (R + L) d L W (Heμˆ ) = λt + μ ˇr eμ−λ + μ − ˆt Heμˆ +eμ−λ eμˆ . ds E R R3 E R3 E (5.3)
We note that if m(t, R(t)) 1 ≤ , R(t) 3
(5.4)
m(t, R(t)) 2λ(t,R(t)) m(t, R(t)) e ≤ 1. = R(t) R(t)(1 − 2m(t,R(t)) ) R(t)
(5.5)
we have
Hence, as long as (5.4) holds true we get in view of (5.3) the inequality d W 1 μˆ μ ˆ μ−λ (He ) ≤ λt +μ ˇr e e . +μ ˆt Heμˆ + eμ−λ (5.6) ds E RE √ L1 L1 , the last term Since W ≤ 0, we have that H ≥ 1 + R 2 , and since ER ≥ 2
is bounded by CHeμˆ , and we get for t ≤ T1 ,
t
H(t)eμˆ(t,R(t)) ≤ H(0)eμˆ(0,R(0)) eCt+ 0 μˆt (s,R(s))ds t (μ−λ)(s,R(s)) W ]ds . × e 0 [λt (s,R(s)) E +ˇμr (s,R(s))e
(5.7)
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Let us denote by γ the curve (t, R(t)), 0 ≤ t ≤ T1 . The last integral in (5.7) can be written as e(−μ+λ)(t,r) λt (t, r)dr + e(μ−λ)(t,r) μ ˇr (t, r)dt. (5.8) γ
We will apply the Green formula to this curve integral and we introduce as above the closed curve Γ = γ + CT1 + C∞ + C0 where CT1 , C∞ and C0 are defined as in (8.7). We have e−μ+λ λt dr + eμ−λ μ ˇr ds Γ
= Ω
=
∂t e−μ+λ λt − ∂r eμ−λ μ ˇr ds dr
∂t e−μ+λ λt − ∂r eμ−λ μr ds dr +
Ω
m ∂r eμ+λ 2 ds dr. r
Ω
(5.9) For the first integral above we use 4.10 and we obtain the identity e−μ+λ λt dr + eμ−λ μ ˇr ds Γ
2m − 4π(ρ − p) − 8πp dt dr T r3 Ω m 2m μ+λ + e (μr + λr ) 2 + 4πρ − 3 dt dr r r Ω m 4πeμ+λ (ρ + p)e2λ + 2p − ρ dr ds = r
=
eμ+λ
Ω
∞ ∞ + Ω −∞ 0
4π 2 eμ+λ f (t, r, w, F )dF dw dr ds. r2 E
(5.10)
Here we used that μr + λr = 4πr(ρ + p)e2λ and that ∞ ∞ 2pT = ρ − p − −∞ 0
4π 2 eμ+λ f (t, r, w, F )dF dw. r2 E
(5.11)
Recall from (5.7) that the integral involving μ ˆ should also be taken into account. Since ∞ μ ˆt (t, r) = 4πje2λ eμ+λ dη, (5.12) r
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we get
t
4πje2λ eμ+λ ds dr.
μ ˆt (s, R(s))ds = 0
(5.13)
Ω
Hence we obtain with CB := CT1 + C∞ + C0 ,
−μ+λ
e γ
μ−λ
λt dr + e
=
T μ ˇr ds +
μ ˆt (s, r)ds 0
m 4πeμ+λ (ρ + p)e2λ + 2p − ρ + je2λ dr ds r
Ω
∞ ∞ + −
Ω −∞ 0
4π 2 eμ+λ f (t, r, w, F )dF dw dr ds r2 E
e−μ+λ λt dr + eμ−λ μ ˇr ds.
(5.14)
CB
The integral over CB equals −
−μ+λ
e
μ−λ
λt dr + e
CB
∞ μ ˇr ds =
4πrj(T1 , r)e2λ(T1 ,r) dr
R(T1 )
T1 4πR∞ p(s, R∞ )e(μ+λ)(s,R∞ ) ds
+ 0
∞ −
4πrj(0, r)e2λ(0,r) dr.
(5.15)
R(0)
The first integral on the right-hand side is negative since matter is ingoing, the second vanishes when R∞ is large since p has compact support and the third integral depends only on the initial data. For the second integral in (5.14) we note that ∞ ∞ 2 μ+λ 4π 2 4π e f (t, r, w, F )dF dw ≤ r ρ, (5.16) r2 E L1 −∞ 0
which in view of (3.1) implies that it is bounded. It remains to show that the 2λ = e2λ we find that the first integral in (5.14) is bounded. Since 1 + 2 m r e integrand is non-positive m (ρ + p)e2λ + 2p − ρ + je2λ r
m + (p + j)e2λ ≤ 0. = −(ρ − p) 1 − e2λ r
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Here we used that p ≤ ρ, e2λ m/r ≤ 1, and that p + j ≤ 0 in view of (2.9) and (2.10) since matter is ingoing. Hence Heμˆ is bounded on [0, T1 ]. Now, from the characteristic equation (2.17) we have that d −1 −W −W Heμˆ R = 2 eμ−λ ≤ √ eμˆ ≤ √ ≤ C(T1 )R−1 . ds R E R L1 R L1
(5.17)
It follows that for any characteristic (t, R(t), W (t)), R(T1 ) ≥ , for some > 0, and Theorem 1 thus applies which shows that the solution can be extended beyond t = T1 . Hence, if T < ∞, then T1 < T, and the proof of Theorem 2 is complete.
6. The Einstein–Vlasov System in Maximal-Isotropic Coordinates In [20], the Einstein–Vlasov system is studied in maximal-isotropic coordinates where the metric reads ds2 = −α(t, R)dt2 + A2 (t, R)[(dR + β(t, R)dt)2 + R2 (dθ2 + sin2 θdφ2 )]. The condition that the hypersurfaces of constant time are maximal implies, cf. [20], that the field equations take the following form: 1 5/2 2 3 2 2 AR K + 16πρ (6.1) ∂R R √ =− A R 8 2 2 A 3 2 2 1 K + 4π(ρ + p + 2pT ) (6.2) αRR + αR + AR αR = αA2 R A 2 1 1 AR + KR + 3 K = 8πj (6.3) A R 3 1 (6.4) βR − β = αK R 2 At = −αKA + ∂R (βA) (6.5) 1 1 Kt = − 2 αRR + 3 AR αR + βKR + 4πα(2pT − p − ρ) A A 2 2 2 A + α − 3 ARR + 4 (AR )2 − (6.6) R A A RA3 The indices t and R denote derivatives but sometimes we also use ∂t and ∂R . In [20] the Vlasov equation is written in different coordinates than we use here. The relation is as follows. Let xi be the coordinates (R sin θ cos φ, R sin θ cos φ, R cos θ) and define an orthonormal frame by ei = A−1 ∂/∂xi . In [20] the mass shell is coordinated by (t, xa , v i ), where v i denote the components of a vector in the orthonormal frame, and the Vlasov equation in [20] is accordingly written in these variables. Now, let w = (v · x)/R, and let L = A2 R2 (v 2 − (x · v)2 /R2 ), then the Vlasov equation for the density
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distribution function f = f (t, r, w, L) takes the form
α w 1 αL AR EαR ∂t f + − β ∂R f + − + αKw + + ∂w f = 0, AE A EA3 R2 R A (6.7) where E = E(R, w, L) =
1 + w2 +
L . A2 R2
(6.8)
The matter quantities are defined by π ρ(t, R) = 2 2 A R p(t, R) =
j(t, R) =
pT (t, R) =
π A2 R2 π A2 R2
∞ ∞ Ef (t, r, w, L) dw dL, −∞ 0 ∞ ∞
−∞ 0 ∞ ∞
w2 f (t, r, w, L) dw dL, E
(6.10)
wf (t, r, w, L), dw dL,
(6.11)
−∞ 0 ∞ ∞
π 2A4 R4
−∞ 0
(6.9)
L f (t, r, w, L) dw dL. E
(6.12)
To ensure asymptotical flatness we impose the boundary conditions lim A(t, R) = 1,
R→∞
lim α(t, R) = 1,
R→∞
lim β(t, R) = 0,
R→∞
(6.13)
and to ensure a regular center we require that A(t, 0) = 1
∀t ≥ 0.
(6.14)
In the case of Schwarzschild coordinates the constraint equations are easily ◦ solved for any given f ∈ I D . This is not the case in maximal-isotropic coordinates. Here we assume that initial data are given, such that the constraint equa◦ ◦ tions are satisfied, and such that 0 ≤ f ∈ C 1 has compact support, f (R, w, L) = 0 whenever R < R1 , R > R2 or if L > L2 , for given 0 < R1 < R2 < ∞, and L2 > 0. Hence, we assume properties which are similar to the initial data class I C (R1 ), with the notable difference that the condition (2.14) is not imposed since in these coordinates trapped surfaces are admitted. Let us denote this class of initial data by J C (R1 ). For a given metric the solution of the Vlasov equation is given by f (t, R, w, L) = f0 (R(0, t, R, w, L), W(0, t, R, w, L)),
(6.15)
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where R and W are solutions of the characteristic system αW dR = −β , ds AE dW 1 EαR αL AR =− + αKW + + . ds A EA3 R2 R A
797
(6.16) (6.17)
Here R(t, t, R, w, L) = R, and W(t, t, R, w, L) = w.
7. A Priori Bounds in Maximal-Isotropic Coordinates In this section we collect the bounds needed in the next section. The following estimates are derived in [20] by using the results of [15], |AR | 2 2 , ≤ . R A R Moreover, it follows from Eq. (6.4), cf. [20], that A|K| ≤
|β| ≤ 3.
(7.1)
(7.2)
Another useful bound follows from conservation of the ADM mass M, which is given by, cf. [20], ∞ 1 3 2 M= K + 16πρ dR. A5/2 R2 (7.3) 4 2 0
The following estimate plays the role of Lemma 1 in Schwarzschild coordinates. Lemma 3. Let (f, α, A, β) be a regular solution of the Einstein–Vlasov system (6.1)–(6.14). Then ∞ 3 2 2 K + 4π(ρ + p + 2pT ) dη ≤ 1. αA η (7.4) 2 0
Proof. Consider the second order equation (6.2) for α. The left-hand side of (6.2) equals 1 ∂R (R2 AαR ), R2 A which implies that 1 αR (R) = 2 R A
R αA3 η 2
3 2 K + 4π(ρ + p + 2pT ) dη. 2
0
By using the boundary condition α(t, ∞) = 1 we get ⎛ R ⎞ ∞ 3 1 ⎝ K 2 + 4π(ρ + p + 2pT ) dη ⎠ dR. 1 ≥ 1 − α(0) = αA3 η 2 R2 A 2 0
0
(7.5)
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From Eq. (6.1) we get that A is monotonic and decreasing. Changing the order of integration and using the monotonicity of A we get ⎞ ⎛ ∞ ∞ 3 2 1 3 2 ⎠ ⎝ dR αA η K + 4π(ρ + p + 2pT ) dη 1≥ R2 A(η) 2 0
η
∞ 2
αA η
=
3 2 K + 4π(ρ + p + 2pT ) dη. 2
(7.6)
0
We also need to establish a decay result for large R. Lemma 4. A solution of the Einstein–Vlasov system (6.1)–(6.14) satisfies t
αR (s, R) ds → 0 A(s, R)
as R → ∞.
(7.7)
0
Proof. Using (7.3) together with Eq. (6.1) we obtain A M √R ≥ − 2 , R A which in view of the boundary condition A(∞) = 1 implies that 2 M A(t, R) ≤ 1 + . 2R
(7.8)
From (6.16) and (7.2) we obtain that all characteristics originating from the support of the matter have R(t) ≤ R2 + 4t =: R3 (t). From Eq. (6.3) we have for R ≥ R3 (t), in view of (7.1) R R (t) 3 R3 |(A3 K)(t, R)| = 8πη 3 A4 j dη = 8πη 3 A4 j dη 0 0 = R33 (t)|(A3 K)(t, R3 (t))| ≤ 2R32 (t)A2 (t, R3 (t)). From (7.8) we have
A(t, R3 (t)) ≤
1+
M 2R2
(7.9)
2 ≤ C.
Thus, since A ≥ 1, CR32 (t)2 (R2 + 4t)2 =C . (7.10) 3 R R3 Now we use Eqs. (6.6) and (6.5) to derive A2 7 2αR 2αAR AR + R2 + , ∂t (AKR) = − RK 2 A + RKAβ + 1+ 4 A A 2A A2 (7.11) K(t, R) ≤
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where we used (6.1) and (6.2) to substitute the second order derivatives of A and α. Since AR √ ≤ M , A R2 we have for sufficiently large R that 0≥
AR A2 1 + R2 ≥ − . A 2A 2
Thus we get 7 2αAR αR ≤ ∂t (AKR) + RK 2 A − RKAβ − . (7.12) A 4 A2 In view of the decay estimates for A, AR and K and the bound of β, we obtain t αR (s, R) ds → 0 as R → ∞. A(s, R) 0≤
0
This completes the proof of the lemma.
8. A Regularity Result in Maximal-Isotropic Coordinates Theorem 3. Let 0 < < R1 . Consider a solution of the spherically symmetric Einstein–Vlasov system, launched by initial data in J C (R1 ), on its maximal time interval [0, T [ of existence. If f (s, R, ·, ·) = 0, for (s, R) ∈ [0, t[×[0, ], then C(1+t2 ) Ct (8.1) Q(t) ≤ Q(0) + 2 e . In particular, if f (t, R, ·, ·) = 0 for (t, R) ∈ [0, T [×[0, ], then T = ∞. The last statement in the theorem is a consequence of the continuation criterion derived in [20]. Indeed, the requirements are that Q(t) and A(t, 0) are bounded. Now, since there is no matter in the domain R ≤ , a bound on A(t, 0) follows from Eqs. (6.1) and (7.3), cf. the bound (7.8). The result of the theorem is not new, it is included in [20], but as was mentioned in the introduction the method is different and the terms that need to be estimated, in order to obtain global existence without assuming a lower bound of R, are more regular in our approach. Proof. The method of proof is to a large extent analogous to the proof in Schwarzschild coordinates. Consider the quantities G = E(t, R, W) + W > 0 and H = E(t, R, W) − W > 0. Along a characteristic (R(t), W(t), L) we have by (6.16) and (6.17) αR αKW 1 AR αL dG LαK =− − + + 3 2 G− , (8.2) ds A E 2EA2 R2 A R E R A and αR αKW 1 AR αL dH LαK = − + − H− . (8.3) ds A E 2EA2 R2 A3 R2 E R A
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First we consider the quantity H. Using equation (7.1) we conclude that √ 1 AR C L C − LαK − αL + ≤ ≤ 2. 2EA2 R2 3 2 2 A R E R A R Thus t
H(t) ≤ H(0)e
αR 0 A
− αKW ds E
t +
C t eτ 2
αR A
− αKW ds E
dτ.
(8.4)
0
Let us consider the first of the integrals in the exponent, the second is analogous. Since αW dR = −β , ds AE we can write the integral as a curve integral αR (t, R) − (βKA)(t, R) dt, −(KA)(t, R)dR + A(t, R) γ
where γ is the curve (s, R(s)), 0 ≤ s ≤ t. Let Γ denote the closed curve Γ := γ + Ct + C∞ + C0 , oriented clockwise, where Ct = {(t, r) : R(t) ≤ r ≤ R∞ }, C∞ = {(s, R∞ ) : t ≥ s ≥ 0},
(8.5) (8.6)
C0 = {(0, r) : R∞ ≥ r ≥ R(0)}.
(8.7)
Here R∞ ≥ R2 + 4t, so that f = 0 when r ≥ R∞ . By applying Green’s formula in the plane we get αR (t, R) − (βKA)(t, R) dt −(KA)(t, R)dR + A(t, R) Γ αR (t, R) = − (βKA)(t, R) dt dR =: IΩ . ∂t (−(KA)(t, R)) − ∂R A(t, R) Ω
Now we use Eq. (6.6) to substitute for ∂t K above. We obtain 2ARR 2(AR )2 2AR αA − + + 4π(ρ + p − 2pT ) + αK 2 A dR dt. IΩ = A3 A4 RA3 Ω
(8.8) From (6.1) we derive ARR (AR )2 AR 3 = − − K 2 − 8πρ. 3 4 3 A A RA 4 We thus get
IΩ =
αA Ω
K2 2AR R2 A2R − − − 4π(ρ − p + 2pT ) . 4 RA3 A4
(8.9)
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Remark 2. The quasi local mass m is given by r m = (1 − |∇r|2 ). 2 Here r is the area radius, and ∇r is the gradient of r. In our case r = AR, which implies that R3 A3 K 2 R3 (AR )2 − R2 AR − . 8 2A It thus follows that IΩ can be written as 2m αA − 4π(ρ − p + 2pT ) dR dt. IΩ = A3 R3 m=
(8.10)
(8.11)
Ω
This is thus identical to the structure of the corresponding term in Schwarzschild coordinates, cf. Remark 1. Here we will however stick to the form (8.9) for the estimates. To summarize, we have in view of (8.4) obtained the estimate H(t) ≤ H(0)eIΩ −ICt −IC∞ −I0 ,
(8.12)
R∞ =− (KA)(t, R) dR,
(8.13)
where ICt
R(t)
R∞ IC0 =
(KA)(0, R) dR,
(8.14)
αR (s, R∞ ) − (βKA)(s, R∞ )) ds. A(s, R∞ )
(8.15)
R(0)
and t IC∞ = 0
We now invoke the a priori bounds derived in the previous section. Since R(s) ≥ , 0 ≤ s ≤ t, we have in view of Lemma 3, 2m 2t αA 3 3 dR dt ≤ . IΩ ≤ A R Ω
Using the bounds (7.1), (7.8) and (7.10) we obtain C(1 + t2 ) . The integral IC∞ vanishes in view of Lemma 4 and IC0 depends only on the initial data. Since the second term in (8.4) can be estimated in the same way we obtain C(1+t2 ) Ct H(t) ≤ H(0) + 2 e . ICt ≤
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The estimate for G is analogous. Thus C(1+t2 ) Ct G(t) ≤ G(0) + 2 e , and we get
Q(t) ≤
Q(0) +
This completes the proof of Theorem 3.
Ct 2
e
C(1+t2 )
.
References [1] Andr´easson, H.: On global existence for the spherically symmetric Einstein– Vlasov system in Schwarzschild coordinates. Indiana Univ. Math. J. 56, 523–552 (2007) [2] Andr´easson, H.: The Einstein–Vlasov system/kinetic theory. Living Rev. Relativ. 8 (2005) [3] Andr´easson, H., Kunze, M., Rein, G.: Global existence for the spherically symmetric Einstein–Vlasov system with outgoing matter. Commun. Partial Differ. Equ. 33, 656–668 (2008) [4] Andr´easson, H., Kunze, M., Rein, G.: The formation of black holes in spherically symmetric gravitational collapse. arXiv:0706.3787 [5] Andr´easson, H., Kunze, M., Rein, G.: Gravitational collapse and the formation of black holes for the spherically symmetric Einstein–Vlasov system. Q. Appl. Math. 68, 17–42 (2010) [6] Andr´easson, H., Rein, G.: The asymptotic behaviour in Schwarzschild time of Vlasov matter in spherically symmetric gravitational collapse. Math. Proc. Camb. Phil. Soc. 149:173–188 (2010) [7] Andr´easson, H., Rein, G.: Formation of trapped surfaces for the spherically symmetric Einstein–Vlasov system. J. Hyperbolic Differ. Equ. (to appear). arXiv:0910.1254 [8] Batt, J.: Global symmetric solutions of the initial value problem of stellar dynamics. J. Differ. Equ. 25, 342–364 (1977) [9] Christodoulou, D.: On the global initial value problem and the issue of singularities. Class. Quantum Grav. 16, A23–A35 (1999) [10] Christodoulou, D.: Bounded variation solutions of the spherically symmetric Einstein-scalar field equations. Commun. Pure Appl. Math. 46, 1131–1220 (1993) [11] Dafermos, M.: Spherically symmetric spacetimes with a trapped surface. Class. Quantum Grav. 22, 2221–2232 (2005) [12] Dafermos, M.: A note on the collapse of small data self-gravitating massless collisionless matter. J. Hyperbolic Differ. Equ. 3, 589–598 (2006) [13] Dafermos, M., Rendall, A.D.: An extension principle for the Einstein–Vlasov system in spherical symmetry. Ann. Henri Poincar´e 6, 1137–1155 (2005) [14] Lions, P.L., Perthame, B.: Propagation of moments and regularity for the 3-dimensional Vlasov–Poisson system. Invent. Math. 105, 415–430 (1991) ´ Optical scalars and singularity avoidance in spher[15] Malec, E., Murchada, N.O.: ical spacetimes. Phys. Rev. 50, 6033–6036 (1994)
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[16] Pfaffelmoser, K.: Global classical solutions of the Vlasov–Poisson system in three dimensions for general initial data. J. Differ. Equ. 95, 281–303 (1992) [17] Rein, G.: The Vlasov–Einstein System with Surface Symmetry. Habilitationsschrift, Munich (1995) [18] Rein, G., Rendall, A.D.: Global existence of solutions of the spherically symmetric Vlasov–Einstein system with small initial data. Commun. Math. Phys. 150, 561–583 (1992). Erratum: Commun. Math. Phys. 176, 475–478 (1996) [19] Rein, G., Rendall, A.D., Schaeffer, J.: A regularity theorem for solutions of the spherically symmetric Vlasov–Einstein system. Commun. Math. Phys. 168, 467– 478 (1995) [20] Rendall, A.D.: An introduction to the Einstein–Vlasov system. Banach Center Publ. 41, 35–68 (1997) [21] Rendall, A.D.: Cosmic censorship and the Vlasov equation. Class. Quantum Grav. 9, L99–L104 (1992) H˚ akan Andr´easson Mathematical Sciences University of Gothenburg G¨ oteborg, Sweden and Mathematical Sciences Chalmers University of Technology 41296 G¨ oteborg, Sweden e-mail:
[email protected] Communicated by Piotr T. Chrusciel. Received: April 7, 2010. Accepted: May 10, 2010.
Ann. Henri Poincar´e 11 (2010), 805–880 c 2010 Springer Basel AG 1424-0637/10/050805-76 published online October 17, 2010 DOI 10.1007/s00023-010-0043-6
Annales Henri Poincar´ e
Improved Decay for Solutions to the Linear Wave Equation on a Schwarzschild Black Hole Jonathan Luk Abstract. We prove that sufficiently regular solutions to the wave equation g φ = 0 on the exterior of the Schwarzschild black hole obey the − 3 +δ
−2+δ and |∂t φ| ≤ Cδ v+ on a compact region of r, estimates |φ| ≤ Cδ v+ 2 including inside the black hole region. This is proved with the help of a new vector field commutator that is analogous to the scaling vector field on Minkowski spacetime. This result improves the known decay rates in the region of finite r and along the event horizon.
1. Introduction A major open problem in general relativity is that of the nonlinear stability of Kerr spacetimes. These spacetimes are stationary axisymmetric asymptotically flat black hole solutions to the vacuum Einstein equations Rμν = 0 in 3 + 1 dimensions. They are parametrized by two parameters (M, a), representing, respectively, the mass and the angular momentum of a black hole. It is conjectured that Kerr spacetimes are stable. In the framework of the initial value problem, the stability of Kerr would mean that for any solution to the vacuum Einstein equations with initial data close to the initial data of a Kerr spacetime, its maximal Cauchy development has an exterior region that approaches a nearby, but possibly different, Kerr spacetime. Kerr spacetimes have a one-parameter subfamily of spacetimes known as Schwarzschild spacetimes for which a = 0. The Schwarzschild metric in the so-called exterior region can be expressed as −1 2M 2M dr2 + r2 dσS2 , dt2 + 1 − g =− 1− r r
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where dσS2 denotes the standard metric on the unit sphere. In view of the nonlinear problem, it is conjectured that a spacetime that is close to Schwarzschild initially will approach a Kerr spacetime that is also close to Schwarzschild, i.e., a M . In other words, we can consider the stability of Schwarzschild spacetimes within Kerr spacetimes. (Notice that the Schwarzschild family itself is not asymptotically stable since a Kerr spacetime with small a can be considered as a small perturbation of a Schwarzschild spacetime.) To tackle the nonlinear stability of Schwarzschild spacetimes within the Kerr family, it is important to first understand the linear waves g φ = 0 on the exterior region of Schwarzschild spacetimes. This can be compared with the nonlinear stability of Minkowski spacetime whose proof requires a robust understanding of the quantitative decay of the solutions to the linear wave equation [5,20]. The pointwise decay of the solutions to the linear wave equation on Schwarzschild background is proved in [4,11]. In particular, Dafermos– −1 Rodnianski proved a decay rate of |φ| ≤ C (max{1, v}) everywhere in the exterior region, including along the event horizon [11]. The subject of this paper is to improve this decay rate. In particular, we will prove that for arbi− 3 +δ trarily small δ > 0, |φ| ≤ Cδ,R (max{1, v}) 2 in the region {rb ≤ r ≤ R} for any rb > 0 and R > 2M . This includes the decay rate along the event horizon and inside the black hole region. Our proof applies a new vector field commutator S that is analogous to the scaling vector field in Minkowski spacetime. We will show that for solutions to g φ = 0, g (Sφ) decays sufficiently towards spatial infinity and only grows mildly towards event horizon. We then prove energy estimates for Sφ with the help of (a slightly modified version of) the energy estimates of φ in [11]. This will enable us to prove the decay of Sφ. With this decay, we follow Klainerman and Sideris [16] to improve the decay rate for ∂t φ. We also introduce a novel method to improve the decay rates for φ and its spatial derivatives. We hope that this improved decay will be relevant for nonlinear problems. We recall for example the wave map equation from R3,1 to S2 given by: 2 m φ = φ (∂t φ) − |∇φ|2 . To prove the global existence for small data for this equation, it is insuf−1 ficient to have |∂φ| ≤ C (1 + |t|) . One needs an improved decay |∂φ| ≤ −1 −δ C (1 + |t + r|) (1 + |t − r|) . Moreover, one needs the nonlinearity to satisfy the so-called null condition (see [14]). In a future work, we will use the improved decay rate we prove in this paper and study the global well-posedness of small data for a nonlinear wave equation satisfying a null condition on a fixed Schwarzschild background. In Sects. 1.1 and 1.2, we will introduce the Schwarzschild spacetime and the class of solutions that we consider. This will introduce the terminologies necessary to state the main theorem in Sect. 1.3. We will motivate our proof
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Figure 1. Schwarzschild spacetime with a comparison with the linear waves on Minkowski spacetime (Sect. 1.4). We then mention some known results on linear waves on Schwarzschild spacetime (Sect. 1.5). We especially discuss the work [11] whose techniques are important for this paper. We will then provide some heuristics for our proof of the main theorem in the final subsection of the introduction (Sect. 1.6). 1.1. Schwarzschild Spacetime Schwarzschild spacetime is the spherically symmetric asymptotically flat solution to the vacuum Einstein equations. The Schwarzschild metric in the exterior region is −1 2M 2M 2 dr2 + r2 dσS2 , g =− 1− dt + 1 − r r where dσS2 denotes the standard metric on the unit sphere. It is easy to observe from the metric that the vector field ∂t is Killing and it is orthogonal to the hypersurfaces t = constant. Spacetimes with this property are called static. It is also manifestly spherically symmetry and, therefore, has a basis of Killing vector fields Ωi generating the symmetry. Moreover, Schwarzschild spacetimes are asymptotically flat. This means that the metric approaches the flat metric as we go to spatial infinity (r → ∞). Synge [24] and Kruskal [18] showed that the Schwarzschild metric can be extended past r = 2M as a solution to the vacuum Einstein equations. Its maximal development is usually described by a Penrose diagram, which depicts a conformal compactification of the 4D manifold quotiented out by spherical symmetry (Fig. 1). In this diagram, the coordinate system t, r > 2M, ω ∈ S2 with the metric described above represents the region I, which we will call from now on the exterior region. In the nonlinear stability problem, it is this region that is conjectured to be stable. Extended beyond r = 2M , the Schwarzschild spacetime contains a black hole (region II in the diagram). Physically, an observer outside the black hole region cannot receive signals emitted inside the black hole. The null hypersurface r = 2M separating the exterior region I and the black hole is known as the event horizon H+ .
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We return to the discussion of the exterior region of the Schwarzschild black hole. For notational convenience, we let 2M . r We denote as r∗ the Regge–Wheeler tortoise coordinate μ=
r∗ = r + 2M log (r − 2M ) − 3M − 2M log M. In these coordinates, the Schwarzschild metric in the exterior region is given by g = − (1 − μ) dt2 + (1 − μ) dr∗2 + r2 dσS2 . Notice that in the above equation we have used both r∗ and r. Here, and below, we think of r∗ as the coordinate and r as a function on Q, with r (q) = Area(q) 4π ,
i.e., the physical radius of the 2-sphere under which the metric is symmetric. The coordinate r∗ is +∞ at spatial and null infinity; −∞ at the event horizon and 0 at r = 3M . The set {r = 3M } is known as the photon sphere. On this set trapping occurs: there exist null geodesics that lie in this set. In particular, these geodesics neither cross the event horizon nor approach null infinity. This suggests, via geometrical optics considerations, that one has to lose derivatives while proving energy estimates. We will return to this point when we discuss the vector field X. We notice that as in the coordinates (t, r, ω), ∂t and Ω are Killing in the (t, r∗ , ω) coordinates. We also define the retarded and advanced Eddington–Finkelstein coordinates u and v by t = v + u,
r∗ = v − u.
At the event horizon H+ , u = +∞. At future null infinity I + , v = +∞. Notice that in these coordinates, the metric is given by −4 (1 − μ) du dv + r2 dσS2 . In particular, this shows that ∂u , ∂v in this coordinate system are null. In the following, we are also going to consider the coordinate system (v, r, ω), where v and r are defined as above. We will only use this coordinate system when considering the region near the event horizon or inside the black hole. When there is no confusion, we will not specify the coordinate system when we use the notation ∂v . In the (v, r, ω) coordinate system the metric is g = −(1 − μ) dv 2 + 2 dr dv + r2 dσS2 . Notice that the coordinate v is originally defined only for r > 2M . However, as the metric in the (r, v) coordinate system is non-singularity, we can extend the v coordinate to r > 0. We will also refer to u inside the black hole region with u = v −2r −4M log (r − 2M ) defined as it is in the exterior region. Notice that u is not defined on the event horizon.
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1.2. Wave Equation and Class of Solutions We would like to study the solutions to g φ = 0 in Schwarzschild spacetimes. Written in the local coordinates of the exterior region, −1 −1 g φ = − (1 − μ) ∂t2 φ + (1 − μ) r−2 ∂r∗ r2 ∂r∗ φ + Δ / φ, where Δ / denotes the Laplace–Beltrami operator on the standard 2-sphere with radius r. We notice that g commutes with Killing vector fields. In particular, g φ = 0 implies g ∂t φ = 0 and g Ωφ = 0. The decay result that we prove apply to solutions to the wave equation that is in some energy class initially. We define the energy classes using currents of vector fields. We will briefly introduce the relevant concepts here in order to present the energy classes. A more detailed description of the vector fields will be presented in the next section. Define the energy–momentum tensor 1 Tμν = ∂μ φ∂ν φ − gμν ∂ α φ∂α φ. 2 Given a vector field V μ , we define the associated current JμV (φ) = V ν Tμν (φ) and the modified current JμV,w (φ) = JμV (φ) +
1 w∂μ φ2 − ∂μ wφ2 . 8
To define the energy classes we need two vector fields: y1 (r) ∂u + y2 (r) ∂v , 1−μ Z = u 2 ∂u + v 2 ∂v ,
N = ∂t +
where y1 , y2 > 0 are supported near the event horizon with y1 = 1, y2 = 0 at the event horizon. The precise form of y1 , y2 will be defined later. Notice that we can also write N in the (v, r, ω) coordinates as 1 + y2 (r) ∂v − (y1 (r) − y2 (r)(1 − μ)) ∂r . N= 2 This indicates that N is regular and can be defined across the event horizon. We will define it inside the black hole so that it is smooth and timelike, future-directed. We also define a modifying function for the associated current of Z: wZ =
2tr∗ (1 − μ) . r
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We note here that ∞
2 φ) (∂ u 2 μ + (∂v φ) + |∇ / φ|2 r2 dA dr∗ , JμN (φ) nt0 dVolt0 ∼ 1−μ
JμZ,w
Z
(φ) nμt0
−∞ S2 ∞
2
2
u2 (∂u φ) + v 2 (∂v φ) + (1 − μ)
dVolt0 ∼ −∞ S2
×
2
u +v
2
2
t2 + (r∗ ) 2 |∇ / φ| + φ r2 2
r2 dA dr∗ ,
where dVolt0 is the volume form of the slice {t = t0 } (see Sect. 2). Let S = t∂t + r∗ ∂r∗ . Define
3 2 k μ k μ N Z,wZ E0 (φ) = Ω φ nt0 dVolt0 Jμ Ω φ nt0 + Jμ k=0
{r≥2M }
3
+ {rb ≤r≤r0 }
E1 (φ) = E0 (Sφ) +
k=0
JμN Ωk φ nμv0 dVolv0 ,
where r0 > 2M is to be picked,
k=0 1 4−m
E0 ∂tm Ωk φ ,
m=0 k=0
E2 (φ) =
2 2−m
E1 ∂tm Ωk φ ,
m=0 k=0
E3 (φ) =
1 1
E0 ∂tm Ωk φ + E1 (φ),
m=0 k=0
E4 (φ) =
2 4−m m=0 k=0
2 2−m E0 ∂tm Ωk φ + E1 ∂t Ωk φ . m=0 k=0
We notice that the boundedness of these quantities should be thought of as requirements of regularity and decay. In the above, E0 , E1 , E2 , E3 and E4 requires 4, 8, 10, 8 and 10 derivatives respectively. In terms of spatial decay, all the energy classes require decay of φ at spatial infinity. However, we note that φ is not required to decay toward the bifurcate sphere (H+ ∩H− in Fig. 1). In the following, we will work as if φ is smooth and supported away from spatial infinity. This assumption can be removed by a standard approximation argument. 1.3. Statement of the Main Theorem We prove both pointwise decay and energy decay for solutions of g φ = 0. From this point onwards, we assume t∗ > 1, v∗ > 1.
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Main Theorem 1. Suppose φ is a solution to the wave equation on the Schwarzschild spacetime, i.e., g φ = 0. Then for any δ > 0 and any 0 ≤ rb ≤ 2M ≤ R < ∞ 1.
Pointwise decay of φ − 32 +δ
|φ (v∗ , r) | ≤ Cδ,rb ,R v∗ 2.
1
E22 (φ)
for rb ≤ r ≤ R.
Pointwise decay of derivatives of φ 1 − 3 +δ E22 ∂tm Ωk φ |∇φ (v∗ , r) | ≤ Cδ,rb ,R v∗ 2
for rb ≤ r ≤ R,
k+m≤1
3.
where ∇ denotes any derivatives. Decay of nondegenerate energy in the region 2M ≤ r ≤ R R
2
φ(t∗ )2 + (∇φ (t∗ ))
r1∗
+ {−∞
dVolt∗
1 2 (∂u φ) + (1 − μ) |∇ / φ|2 dA du{v=v∗ } 1−μ
−3+δ
≤ Cδ,r1∗ ,R min {t∗ , v∗ }
E1 ∂tm Ωk φ ,
k+m≤1
4.
for any r1 satisfying r1∗ ≤ R∗ . Decay of nondegenerate energy in the region rb ≤ r ≤ 2M in (v, r, ω) coordinates v ∗ +1
φ2 + (∇φ)
2
dA dvr ≤ Cδ,rb v∗−3+δ
E1 ∂tm Ωk φ ,
k+m≤1
v∗
for any rb ≤ r ≤ 2M . Remark 1. The integral in statement 3 represents the part of nondegenerate energy restricted to the region r ≤ R (See Sect. 3.5). It should be compared with the corresponding part of the (degenerate) energy generated by the vector ∂ field T = ∂t R
∗
2
(∇φ (t∗ )) dVolt∗ + r1∗
2 (∂u φ) + (1 − μ) |∇ / φ|2 dA du{v=v∗ } .
{2M ≤r ∗ ≤r1∗ }
The nondegeneracy is more apparent if we write the second integral in the (v, r, ω) coordinates, which up to some constant is: 2 (∂r φ) + |∇ / φ|2 dA dr{v=v∗ } . {2M ≤r ∗ ≤r1∗ }
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For the time derivatives, we have better decay estimates both in the sense that we have a better decay rate and have a larger region of spacetime on which the estimates hold. Main Theorem 2. Suppose φ is a solution to the wave equation on the Schwarzschild spacetime, i.e., g φ = 0. Then for any δ > 0 and rb > 0 1.
Pointwise decay of ∂t φ 1
|∂t φ (v ∗ ) | ≤ Cδ,rb v∗−2+δ E42 (φ) 2.
for rb ≤ r ≤ 2M or r∗ ≤
Decay of nondegenerate energy of ∂t φ in the Region r∗ ≤ t∗
2
2
(∂t φ(t∗ )) + (∇∂t φ (t∗ )) r1 ∗
+ {−∞
2
t∗ . 2
t∗ 2
dVolt∗
1 2 (∂u ∂t φ) + (1 − μ) |∇ / ∂t φ|2 dAdu{v=v∗ } 1−μ
≤ Cδ,r∗ min{t∗ , v∗ }−4+δ E3 (φ) . 3.
for any r1∗ . Decay of nondegenerate energy of ∂t φ in the region rb ≤ r ≤ 2M in (v, r, ω) coordinates v ∗ +1
2
2
(∂t φ) + (∇∂t φ)
dA dvr ≤ Cδ,rb v∗−4+δ E3 (φ),
v∗
for any rb ≤ r ≤ 2M . We would like to point out that the pointwise decay rates in both theorems apply to region of finite r including inside the black hole. 1.4. The Case of Minkowski Spacetime At this point, we would like to discuss some decay results for the linear wave equation on Minkowski spacetimes. We would like to especially highlight techniques that are relevant to our result. In Minkowski space R3,1 , the solutions to the wave equation with initial conditions φ (t = 0, x) = φ0 and ∂t φ (t = 0, x) = φ1 can be written as ⎛ ⎞ 1 ⎝ ∂t tφ0 (x + ty) dA (y) + tφ1 (x + ty) dA (y)⎠ . (1) φ (t, x) = 4πt2 S2
S2
This formula implies immediately that |φ (t, x) | ≤ tC+ , where t+ = max{t, 1}. This decay is optimal in the variable t. However, improved decay canbe seen 3 in the null coordinates v = 12 (t + r) and u = 12 (t − r), where r2 = i=0 x2i . In particular, (1) implies the strong Huygens’ Principle, asserting that φ with
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compactly supported initial data is compactly supported in the variable u. Therefore, denoting v+ = max{v, 1}, u+ = max{u, 1}, we have in particular |φ| ≤
CN , v+ u N +
∀N ≥ 0.
If we just focus on the region {r ≤ 2t }, where t ∼ v ∼ u, the decay can be written as CN |φ| ≤ N , ∀N ≥ 0. t+ However, the use of the representation formula (1) is not available on perturbations of the Minkowski spacetime. In [5,20], a more robust understanding of the decay of the linear waves was necessary. This was achieved by the vector field method. Let φ be a solution to the linear wave equation on Minkowski spacetime, m φ = 0. Define the energy–momentum tensor 1 Tμν = ∂μ φ∂ν φ − mμν ∂ α φ∂α φ. 2 Notice that the wave equation implies that the energy–momentum tensor is divergence free, i.e., ∇μ Tμν = 0. Given a vector field V μ , we define the associated currents JμV (φ) = V ν Tμν (φ) , 1 K V (φ) = Tμν (∇μ V ν + ∇ν V μ ) ; 2 and the modified currents V 1 V w ∂μ φ2 − ∂μ w V φ2 , JμV,w (φ) = JμV (φ) + 8 1 1 V,wV V K (φ) = K (φ) + wV ∂ ν φ∂ν φ − g wV φ2 , 4 8 where wV is some scalar function associated to the vector field V . Since the energy–momentum tensor is divergence free, it is easy to check that ∇μ JμV (φ) = K V (φ), V
V
∇μ JμV,w (φ) = K V,w (φ). Notice that K V (φ) = 0 whenever V is Killing. In this case JμV (φ) is divergence free. Therefore, for any solution φ and Killing vector field V , there is a conservation law J0V (φ) dxt1 = J0V (φ) dxt0 . t=t1
t=t0
This is a manifestation of Noether’s Theorem, which states that a differentiable one-parameter family of symmetries gives rise to a conservation law. We call the vector field V in this application a multiplier because we “multiply”
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it to the energy–momentum tensor. An example of this is to take the Killing vector field ∂t and derive the energy conservation law
3 3 2 2 2 (∂t φ) + (∂xi φ) dxt = (∂t φ) + (∂xi φ) (2) dxt0 . i=1
i=1
Besides being multipliers, vector fields can also be used as commutators. This means that we commute the vector fields with m . For example, since ∂ ∈ {∂t , ∂xi } is Killing, [m , ∂] = 0 and, therefore, m (∂φ) = 0. Then the energy conservation law (2) can be applied to ∂φ and we can control the L2 norm of the derivatives of φ of orders 1 and 2. Then using a Sobolev-type inequal1 1 2 2 ity ||φ||L∞ (R3 ) ≤ C||φ||H ˙ 1 (R3 ) ||φ||H ˙ 2 (R3 ) (which holds for compactly supported functions), uniform boundedness of the solutions to the wave equation can be proved. The Killing vector fields Ωi generating the spherical symmetry can also be used as commutators. This is especially useful because compared to / . This allows the angular derivatives, Ωi has an extra factor of r, i.e., Ω ∼ r∇ one to prove in [15] that for φ decaying sufficiently fast at spatial infinity: |φ| ≤
2 2 C m k ||∂r Ω φ||L2 (R3 ) , r m=1 k=0
which implies a decay in the region {r > 2t }: |φ| ≤
C C ≤ r v+
after applying (2) to Ωk φ. t To achieve decay 2 of2 φ in {r ≤ 2 }, one can use the conformally Killing vector field Z = t + r ∂t + 2tr∂r introduced by Morawetz [23]. In this Z case, K Z (φ) = 0. Nevertheless, by defining wZ = 2t, K Z,w (φ) = 0 and, Z,wZ therefore, J0 (φ) dxt is a conserved quantity. Moreover, some algebraic manipulation would show Z J0Z,w (φ) dxt φ2 2 2 2 ≥c v 2 (∂v φ) + u2 (∂u φ) + v 2 + u2 + |∇ / φ| dx, r2 where ∇ / denotes the angular derivatives. The conserved nonnegative quantity Z,wZ J0 (φ) dxt is known as the conformal energy. For the region {r ≤ 2t }, notice that the boundedness of the conformal energy implies a local energy decay 2 φ C 2 2 + (∂ φ) + (∂ φ) + |∇ / φ| . dxt ≤ v u r2 t+ {r≤ 2t }
After considering the equations m ∂ k φ = 0, Sobolev embedding would imply the pointwise decay |φ| ≤ tC+ , for r ≤ 2t . Notice that in this region t+ is
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comparable to v+ . Therefore, we have in the whole of Minkowski spacetime |φ| ≤
C . v+
Klainerman and Sideris [16] showed that more decay can be achieved in the interior region {r ≤ 2t } for the derivatives of φ. They used the scaling vector field S = t∂t + r∂r as a commutator. Notice that S is conformally Killing and [m , S] = 2m . In particular, if one has m φ = 0, then m (Sφ) = Sm φ + 2m φ = 0. Therefore, any decay results that hold for φ also hold for Sφ. Klainerman and Sideris [16] showed that ||u+ ∂∂t φ||L2 (R3 ) ∂∈{∂t ,∂xi }
≤C
||∂Sφ||L2 (R3 ) + ||∂φ||L2 (R3 ) + ||∂ 2 φ||L2 (R3 ) + ||∂Ωφ||L2 (R3 ) .
∂∈{∂t ,∂xi }
By cutting off appropriately and using the local energy decay estimates, t C ||∂∂t φ||L2 ({r≤ t }) ≤ 2 in r ≤ 2 t+ 2 since u1+ ≤ tC+ in this region. Again, using the Sobolev-type inequality above, one shows that |∂t φ| ≤ tC2 in {r ≤ 2t }. The other derivatives can also be + estimated first by elliptic estimates and then the Sobolev inequality, since ||u+ ∂t2 φ||L2 (R2 ) = ||u+ Δφ||L2 (R2 ) by the linear wave equation. Therefore, t C |∂φ| ≤ 2 in r ≤ . t+ 2 We remark that in [16], the improved decay in {r ≤ 2t } can also be proved for the function φ itself by inverting the Laplacian. As we proceed to prove the analogous decay on Schwarzschild spacetimes, we will avoid doing so. This is because on Schwarzschild spacetimes, it is impossible to invert the Laplacian for functions that do not vanish on the bifurcate sphere (H+ ∩ H− in Fig. 1). 1.5. Some Known Results on the Wave Equation on Schwarzschild Spacetimes We now turn to the corresponding problem for linear waves on Schwarzschild spacetimes. The problem of the uniform boundedness of solutions to g φ = 0 on the exterior of Schwarzschild occupied the physics community for some time. The first mathematically rigorous result was obtained by Wald [25] for solutions vanishing on the bifurcate sphere (H+ ∩H− in Fig. 1). Kay and Wald [13] later removed this restriction and proved the uniform boundedness of a more general class of solutions. They used the energy conservation law given by using ∂t as a multiplier as well as the Killing fields {∂t , Ωi } as commutators. The decay rates −1 |φ| ≤ Cv+ ,
|rφ| ≤
−1 CR u+ 2 ,
∀r ≥ 2M, ∀r ≥ R,
(3)
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where v+ = max{v, 1}, u+ = max{u, 1} and CR depends only on an appropriate norm of the initial data, for sufficiently regular solutions to g φ = 0, were proved by Dafermos and Rodnianski [11]. We note that the decay rate (3) holds in the entire exterior region of Schwarzschild spacetimes, including along the event horizon. In addition to the vector fields in [13,25], their approach employed several other (non-Killing!) vector fields. One is an analog of the Morawetz vector field Z in Minkowski spacetime. It has an associated nonnegative quantity which we will call the conformal energy. It has weights similar to that of the conformal energy on Minkowski spacetime so that its boundedness would imply a local energy decay. Another is a vector field of the form X = f (r∗ ) ∂r∗ . The construction of this vector field was motivated by Laba and Soffer [19]. Unlike other multipliers, X is conX X structed so that K X,w (φ) (instead of J X,w (φ)) can be controlled. This is used to estimate some energy quantity integrated over spacetime, in particular error terms from the “conservation law” of the conformal energy. The estimates of X are iterated together with that of Z to achieve the boundedness of the conformal energy. This then implies the decay of φ away from the event horizon. The estimate associated to X can be thought of as an integrated in time local energy decay. It was extensively studied in [1,2,4,8,11,22]. In addition, [11] introduced a new, red shift vector field, which takes advantage of the geometry of the event horizon and is used crucially in proving the decay rate close to and along the event horizon. This vector field is one of the few stable features of the Schwarzschild spacetime. In particular, it can be used to give a more robust proof of boundedness of the solutions to the linear wave equation on Schwarzschild spacetimes. It also plays key roles in the boundedness results for the linear wave equation on small axisymmetric stationary perturbations of Schwarzschild spacetimes and in the decay result for the linear wave equation on slowly rotating Kerr spacetimes [9,10]. As we will see later, it will make a crucial appearance in this article to achieve the improved decay rate along the event horizon. The study of pointwise decay was carried out independently by Blue and Sterbenz [4]. They showed a similar quantitative decay result for initial data vanishing on the bifurcate sphere, with a decay rate that is weaker than [11] along the event horizon. In the proof they used analogues of the vector fields Z and X but not the vector field Y . Strichartz estimates for solutions of the wave equation on Schwarzschild background were shown in [22]. We refer the readers to Sects. 3 and 4 in [10] for further references on this problem. Considerable attention has also been given to the problem of decay of solutions of the wave equations on the Schwarzschild spacetime restricted to φ arising in the decomposition φ(t, r, ω) = a fixed spherical harmonic 2 φ (r, t)Y (ω), ω ∈ S . Such results for a fixed spherical harmonic have been obtained in [7,12,17,21]. We refer the readers to Sect. 4.6 in [10] for a more detailed discussion.
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1.6. Outline of the Proof Our proof uses ideas from Dafermos and Rodnianski [11] and Klainerman and Sideris [16]. In addition to the arguments used in [11], we introduce a vector field S = t∂t +r∗ ∂r∗ which is analogous to the scaling vector field in Minkowski spacetime. Since Schwarzschild spacetimes are asymptotically flat, S is still an “asymptotic conformal symmetry” generating an “asymptotic almost conservation law”. However, the error terms away from spacelike infinity are in general large. To see this more concretely, we recall that on Minkowski spacetimes, m φ = 0 implies m (Sφ) = 0. This does not hold in Schwarzschild spacetimes. Nevertheless, for g φ = 0, we still have a (schematic) equation g (Sφ) = h(r) ∇φ + ∇2 φ with h → 0 as r → ∞. The strategy is then to go through the argument in Dafermos and Rodnianski [11] and control the error terms that arise from g (Sφ) = 0. To do so, we use a slightly modified version of the energy estimates that are available from the proof in [11]. As in later parts of the paper, we define ψ = Sφ. We would like to prove energy estimates for ψ similar to those for φ that are established in [11], except for a loss of an arbitrarily small power of t. A key estimate that will be used to prove the main theorem is t
t2 2
ψ 2 + (∇ψ)
2
χ (r∗ ) dVol ≤ Cδ t−2+δ , 1
(4)
t1 − t S2 2
where χ is some weight and t1 ≤ t2 ≤ (1.1) t1 . A similar estimate is available with ψ replaced by φ from [11] using the X vector field. In order to prove this, we argue in a similar fashion. We want to show, using the vector field X, that for t1 , t2 as above t
t2 2
2
ψ 2 + (∇ψ)
χ (r∗ ) dVol ≤ Cδ t−2+δ {conf. energy(ψ)}, 1
t1 − t S2 2
where the conformal energy is the current of the vector field Z on the boundary {t = ti }. We then hope to show {conf. energy(ψ) at t2 } ⎛ ⎜ ≤ C ⎝{conf. energy(ψ) at t1 } +
t
t2 2
ψ 2 + (∇ψ)
2
⎞ ⎟ χ (r∗ ) dVol⎠ .
t1 − t S2 2
We then iterate two inequalities to obtain (4) as in [11]. The main difficulty in actually carrying out the above procedure is that each step is only true modulo some error terms that need not be small. These are error terms arising from the fact that ψ does not satisfy the homogeneous wave equation, but only satisfies an inhomogeneous wave equation, which sche matically can be thought of as g ψ = h(r∗ ) ∇φ + ∇2 φ . If one applies the vector field method to this equation, one would generate an error term of the
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form t2 ∞
V μ ∂μ ψh(r∗ ) ∇φ + ∇2 φ dVol,
(5)
t1 −∞ S2
2 for the vector fields V ∈ {∂t , X = f (r)∂r∗ , Z = t2 + (r∗ ) ∂t + 2tr∗ ∂r∗ }. (In practice there is still another error term if one uses the modified current, but since it can be controlled similarly, we omit the technicalities here.) Applying Cauchy–Schwarz, we can control (5) by ⎛t ∞ ⎞ 12 ⎛ t ∞ ⎞ 12 2 2 2 ˜ ∗ )((∇φ)2 + (∇2 φ)2 ) dVol⎠ (6) ⎝ (∇ψ) dVol⎠ ⎝ h(r t1 −∞ S2
t1 −∞ S2
We control the first factor by some energy quantities of ψ which we are in the process of proving. They are set up so that we can estimate them with a bootstrap argument. In order that the bootstrap can close, we would need to show that the second factor decays or does not grow as t1 , t2 → ∞ (for example with t2 = (1.1)t1 ). The precise rate of decay that is necessary depends on the vector field V under consideration and is ultimately dictated by what the bootstrap argument requires. To achieve this, we recall the energy estimates derived from the X vector field in [11]. In particular, we have t2 ∞
2
(∇φ) χ (r∗ ) dVol ≤ C,
(7)
t1 −∞ S2 t
t2 2
2
(∇φ) χ (r∗ ) dVol ≤ Ct−2 1 ,
(8)
t1 − t S2 2
where χ is a weight that decays at spatial infinity, (8) gives good control for the ˜ and χ behaves second factor in (6) for the region {− 2t ≤ r∗ ≤ 2t } as long as h appropriately. We will slightly improve the weight χ from [11] so that we have, ˜ (r∗ ) ≤ C (1 + |r∗ |)−2 χ (r∗ ). This would give control for the loosely speaking, h second factor in (6) for the region {− 2t ≤ r∗ ≤ 2t }. For the regions {r∗ ≤ − 2t } ˜ (r∗ ) ≤ C (1 + |r∗ |)−2 χ (r∗ ) ≤ C (1 + t)−2 χ (r∗ ). Then we can and {r∗ ≥ 2t }, h control the second factor in (6) in this region with (7) and the extra factor of (1 + t)−2 . The reader should keep in mind that these are only heuristics and are not true if directly applied. The actual estimates for these error terms are slightly more involved considering first that V μ might grow t; and second that we do not have energy estimates that control every derivatives of ψ; and thirdly that some error terms would tend to infinity as r approaches the event horizon. The relevant estimates will be proved in Sect. 5. In [16], the estimates for ψ are used to prove the decay for ∂t φ in Minkowski spacetime. We show that it is possible to argue similarly to prove the decay for ∂t φ in Schwarzschild spacetimes (Sect. 7.2). Recall that in [16], one then
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proceeds with elliptic estimates to prove the decay for other derivatives. However, on Schwarzschild spacetimes, if we are to prove an L2 elliptic estimate, we are bound to have some lower order terms involving only one derivative of φ. These terms cannot be controlled by the estimates of ψ and, therefore, we are unable to use a similar method to prove the decay of the spatial derivatives of φ. Therefore, we introduce in this paper a new method, based on a novel application of S,to prove the decay for the function φ as well as its derivatives in spatial directions (Sect. 7.1). We notice that by (8), ∗
t2 r2
2
φ2 + (∂r∗ φ)
dVol ≤ Ct−2 1 ,
t1 r1∗
for t1 ≤ t2 ≤ (1.1) t1 . Therefore, there exists a time t˜ ∈ [t1 , t2 ] such that ∗
r2
2
φ2 + (∂r∗ φ)
dVolt˜ ≤ C t˜−3 .
r1∗
In order to show that the same holds for any t, we note that S is strictly timelike on a compact set of r∗ . Therefore, we can integrate in the direction of S from the slice t˜ to a generic slice t. This integration would not give an extra factor of t precisely because we already have the estimates for ψ = Sφ. After controlling the spacetime terms by (4) and (8), we show that for any t, ∗
r2
2
φ2 + (∂r∗ φ)
dVolt ≤ Cδ t−3+δ .
r1∗
We use Sobolev Embedding to get the pointwise decay estimate for φ and its derivatives (for r1∗ ≤ r∗ ≤ r2∗ ) after commuting with an appropriate number of Killing vector fields. We note in particular that in this proof, it is unnecessary to invert the Laplacian on Schwarzschild spacetime to prove the decay of φ. The argument above gives the decay of φ and its derivatives in a compact region of r∗ , i.e., a compact region of space that is also away from the event horizon. (Recall r∗ is defined so that r∗ = −∞ at the event horizon.) In order to prove that φ also decays along the event horizon, we use the red-shift vector field introduced in [11]. This vector field was used in [11] to show that in some (explicitly identified) neighborhood of the event horizon, some energy quantity on an initial slice can control some similar energy quantity in a spacetime slab provided that the error terms that are supported in a compact region of r∗ can be controlled. It is then used to propagate the decay of φ from a compact region of r∗ to the event horizon. In this article, we show along these lines that any decay estimate proved on a compact region of r∗ can be propagated to the event horizon, giving rise to a decay estimate of the same rate. This will be carried out in Sect. 5. Moreover, using an identical argument with the red-shift vector field, which we also include for the sake of completeness, the
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decay estimate can be propagated to slightly beyond the event horizon into the black hole region. Once we have an estimate slightly beyond the event horizon, we can easily prove the same decay estimate anywhere inside the black hole region by taking advantage of the geometry of the region. This strategy for controlling the scalar field inside the black hole region was first used in [6] and [7] in the nonlinear setting, see also subsequent [22] in the linear setting. This will be carried out in Sect. 6 and will give the full improved decay result.
2. Notations Before proceeding, we would like to first define the notations used for the coordinates and volume form. For the r, r∗ coordinates, we always use ∗ to denote the Regge–Wheeler tortoise coordinate of the same point. For the t coordinates: t0 denotes the time slice on which the initial data is posed. t∗ denotes the time slice on which we would like to control the solution. ti denotes dyadic time slices (which will be defined in Sect. 4). t denotes a generic time slice. We assume t0 , t∗ , ti , t > 0. For volume forms: μ) dAdr∗ dt. dV ol denotes the spacetime volume form, dV ol = r2 (1 − √ 2 ∗ dVolt denotes the volume form on a time slice, dVolt = √r 1 − μ dA1dr . 2 dVolv denotes a volume form on a v slice, dVolv = r 1 − μ dA du. dA denotes the volume form on the standard sphere of radius 1. Whenever we write without integration limits, it denotes the integration over “whole space” that is appropriate for the volume form.
3. Vector Fields 3.1. Conservation Laws We consider the conservation laws for φ satisfying g φ = 0. Define the energy– momentum tensor 1 Tμν = ∂μ φ∂ν φ − gμν ∂ α φ∂α φ. 2 We note that Tμν is symmetric and the wave equation implies that ∇μ Tμν = 0. 1 Most of the time it is clear from context whether we are integrating over a t or v slice. We will specify in the case of possible ambiguity, for example dVol{t=2v−r∗ } is the volume form on a t slice, where t has the specified value.
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Given a vector field V μ , we define the associated currents JμV (φ) = V ν Tμν (φ) , V T μν (φ) , K V (φ) = πμν V is the deformation tensor defined by where πμν
1 (∇μ Vν + ∇ν Vμ ) . 2 = 0 if V is Killing. Since the energy-momentum
V = πμν V In particular, K V (φ) = πμν tensor is divergence-free,
∇μ JμV (φ) = K V (φ) . We also define the modified current
1 w∂μ φ2 − ∂μ wφ2 . 8 1 V,w V ν Define K (φ) = K (φ) + 4 w∂ φ∂ν φ − 18 g wφ2 . Then JμV,w (φ) = JμV (φ) +
∇μ JμV,w (φ) = K V,w (φ) . We integrate by parts with this in a region B bounded to the future by Σ1 and to the past by Σ0 . The region B should have no other boundary. Denoting the future-directed normal to Σ0 and Σ1 by nμΣ0 and nμΣ1 , respectively, we have Proposition 1. JμV (φ) nμΣ1 dVolΣ1 + K V (φ) dVol = JμV (φ) nμΣ0 dVolΣ0 .
Σ1
JμV,w
(φ) nμΣ1
dVolΣ1 +
Σ1
B
K
Σ0 V,w
(φ) dVol =
B
JμV,w (φ) nμΣ0 dVolΣ0 .
Σ0
In this paper, there are two choices of Σi that we will use. The first is to choose Σi to be t = constant slices. The second choice is for estimates near the event horizon. In this case, B = {v0 ≤ v ≤ v1 , t ≥ t0 }, Σ0 = {v = v0 , t ≥ t0 } ∪ {v0 ≤ v ≤ v1 , t = t0 } and Σ1 = {v = v1 , t ≥ t0 } ∪ {v0 ≤ v ≤ v1 , u = ∞} (See Fig. 2). One can similarly define the above quantities for the inhomogeneous wave equation g ψ = F . In this case, the energy-momentum is no longer divergence free. Instead, we have ∇μ Tμν = F ∂ν ψ. In this case, ∇μ JμV (ψ) = K V (ψ) + F V ν ∂ν ψ. For the modified current, 1 ∇μ JμV,w (ψ) = K V,w (ψ) − F wψ + F V ν ∂ν ψ. 4
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Figure 2. Regions of integration Proposition 2. JμV (ψ) nμΣ1 dVolΣ1 + K V (ψ) dVol + F V ν ∂ν ψ Σ1
=
B
JμV (ψ) nμΣ0
dVolΣ0 .
Σ0
JμV,w (ψ) nμΣ1 dVolΣ1 +
Σ1
+ B
B
K V,w (ψ) dVol
B
1 − F wψ + F V ν ∂ν ψ 4
dVol =
JμV,w (ψ) nμΣ0 dVolΣ0 .
Σ0
In the case of wave equation on Schwarzschild background, we can compute the energy–momentum tensor explicitly in local coordinates (t, r∗ , xA , xB ) or equivalently (u, v, xA , xB ), where xA , xB is an orthonormal basis on S2 . 2
Tuu (φ) = (∂u φ) , 2
Tvv (φ) = (∂v φ) , / φ|2 , Tuv (φ) = (1 − μ) |∇ / φ|2 − ∂ α φ∂α φ. TAA (φ) + TBB (φ) = |∇ As a result, K V (φ) =
1 2 −1 2 −1 + (∂v φ) ∂u Vu (1 − μ) (∂u φ) ∂v Vv (1 − μ) 4 (1 − μ) 1 (Vu − Vv ) |∇ / φ|2 − ∂ α φ∂α φ . +|∇ / φ|2 (∂u Vv + ∂v Vu ) − 2r
3.2. Vector Field Multiplier T Define T = ∂t . Recall that T is Killing. Therefore, K T (φ) = 0.
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In the following, we will consider this current on a constant t-slice. One computes that in local coordinates 1 2 2 (∂t φ) + (∂r∗ φ) + (1 − μ) |∇ JμT (φ) nμt = √ / φ|2 , 2 1−μ where nμt is the normal to a t-slice. 3.3. Vector Field Multiplier X Define X = f (r∗ ) ∂r∗ . In the following we will use different functions f . One computes that 2 2 − 3μ 1 f (r∗ ) (∂r∗ φ) X 2 + |∇ / φ| K (φ) = f (r∗ ) 1−μ 2 r 4 (1 − μ) 1 ∗ ∗ f (r ) ∂ α φ∂α φ. − 2f (r ) + 4 r We consider the modified current using wX = 2f (r∗ ) + K
X,wX
2
1 f (r∗ ) (∂r∗ φ) + |∇ / φ|2 = 1−μ 2
2 − 3μ r
4(1−μ) f r
(r∗ ). Then
1 f (r∗ ) − g wX φ2 8
2 2 − 3μ 1 f (r∗ ) (∂r∗ φ) + |∇ / φ|2 f (r∗ ) 1−μ 2 r 1 ∗ 4 ∗ μ 2μ 1 f (r )+ f (r )+ 2 f (r∗ )− 3 (3−4μ) f (r∗ ) φ2 , − 4 1−μ r r r X μ 1 JμX,w nt = √ f (r∗ ) ∂t φ∂r∗ φ 1−μ 2 (1 − μ) 1 ∗ ∗ f (r ) (∂t φ) φ, (9) + √ f (r ) + r 2 1−μ =
where nμt is the normal to a t-slice. The vector field X is constructed to control a spacetime integral by the X boundary terms, i.e., one hopes to control the integral of K X,w (φ) by the X integral of JμX,w (φ) nμt . In order for this to be useful, we need K X (φ) to be everywhere positive. Such vector fields are constructed in [11] using spherical harmonic decomposition. In particular, it was shown in [11] that there exists a family of vector fields Xl = fl (r∗ ) ∂r∗ for l ≥ 0 such that for any function φ (not necessarily satisfying the wave equation), if we write out the spherical ∞ X harmonic decomposition φ = l=0 φl , K Xl ,w l (φl ) ≥ 0. Moreover, one has
2 Xl φ2l (∂r∗ φl ) + dA ≤ C K Xl ,w (φl ) dA 2 4 ∗ ∗ (1 + |r |) (1 − μ) (1 + |r |) (1 − μ) S2
S2
(10)
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for l ≥ 1, where C can be picked to be independent of l, and S2
(∂r∗ φ0 ) (1 +
1+δ |r∗ |)
2
r2
(1 − μ)
dA ≤ C
K Xl ,w
Xl
(φ0 ) dA.
S2
Moreover for this choice of Xl , the boundary terms are also controllable as shown in [11]: Xl K Xl ,w (φl ) dVol ≤ C JμT (φ) nμt0 dVolt0 . X
Remark 2. We note that although K Xl ,w l (φ) is shown to be nonnegative everywhere, it has a weight in front of |∇ / φ|2 that degenerates at r = 3M . Therefore, we cannot directly estimate the integral of |∇ / φ|2 by that of Xl K Xl ,w (φ). Instead, we will consider the equation g (Ωφ) = 0 and esti X mate the relevant quantities with K Xl ,w l (Ωφ) dVol. This loss of derivative is related to the trapping phenomenon that we mentioned in Sect. 1.1. In Sect. 3, we will construct two more vector fields of this form. One will be a modified X0 to control a weighted L2 -norm of the zeroth spherical harmonic and the other will be used to control the behavior at infinity. 3.4. Vector Field Multiplier Z Define Z = u2 ∂u + v 2 ∂v . This is the analogue of the conformal vector field in Minkowski spacetime. Like the case in Minkowski spacetime, it is used to show decay for the solution to the wave equation. One computes that 1 μr∗ r∗ (1 − μ) 1 2tr∗ (1 − μ) α Z 2 + − ∂ φ∂α φ. / φ| K = −t|∇ − 2 4r 2r 4 r ∗
We consider the modified current using wZ = 2tr (1−μ) . Then r Z 1 μr∗ r∗ (1 − μ) 1 K Z,w = −t|∇ + − / φ|2 − g wZ φ2 2 4r 2r 8 1 μr∗ r∗ (1 − μ) r∗ (4μ − 3) t + − = −t|∇ / φ|2 − μr−2 φ2 2 + , 2 4r 2r 4 r Z 1 2 2 / φ|2 u2 (∂u φ) + v 2 (∂v φ) + (1 − μ) u2 + v 2 |∇ JμZ,w nμt = √ 4 1−μ 2tr∗ (1 − μ) r∗ (1 − μ) 2 φ∂t φ − φ . + r r where nμt is the normal to a t-slice. It is shown in [11] that there exist r1∗ , r2∗ such that for r∗ ≤ r1∗ or r∗ ≥ r2∗ , Z K Z,w ≥ 0. Z Moreover, it is shown that JμZ,w nμt dVolt is everywhere non-negative.
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More specifically, if we define S = u∂u + v∂v and S = −u∂u + v∂v , Z
JμZ,w nμ t dVolt
2 2 1 t r∗ 2 2 √ μ (Sφ) + (Sφ) + (1 − μ) = Sφ + φ + Sφ + φ r r 8 1−μ / φ|2 dVolt . + 2(1 − μ)(u2 + v 2 )|∇
3.5. Vector Field Multipliers Y , Y and N ∗
1 (r ) ∂u +y2 (r∗ ) ∂v , where y1 , y2 > 0 are supported in r ≤ (1.2) r0 , Define Y = y1−μ with y1 = 1, y2 = 0 at the event horizon and y1 (r∗ ) ∼ y2 (r∗ ) ∼ C (1 − μ) for 2M ≤ r ≤ r0 . Here we want to choose r0 small enough so that
1. 2. 3.
Y is supported on r < 3M (i.e., (1.2) r0 < 3M ), K Y (φ) ≥ 0 on 2M ≤ r ≤ r0 , 1 CK Y (φ) ≥ √1−μ JμY (φ) nμ{v=const.} on 2M ≤ r ≤ r0 .
The vector field Y is designed to capture the red-shift effect at the event hori zon [11]. Using the current J Y , we will not only produce estimates on constant t-slices, but also on constant v-slices. We will, therefore, record here all the relevant computations. We have 2 2 (∂v φ) (∂u φ) y1 μ 1 + − y y2 + |∇ / φ|2 KY = 1 2 r 2 (1 − μ) 2 2 (1 − μ) y1 (y2 (1 − μ)) y1 1 − − y2 ∂u φ∂v φ, × − 1−μ 1−μ r 1−μ y1 1 2 μ Y 2 √ (∂u φ) + (1 − μ) y2 |∇ / φ| , Jμ (φ) n{v=const.} = 2 1−μ 1−μ 1 JμY (φ) nμ{t=const.} = √ 2 1−μ y1 2 2 (∂u φ) + y2 (∂v φ) + (y1 + (1 − μ) y2 ) |∇ / φ|2 . × 1−μ From this we see that if r0 is chosen to be close enough to 2M , requirements 2 and 3 can be satisfied. We modify this vector field so that it has better bounds on constant t- slices. Define Y = Y + χ (r) T , where χ (r) is a cutoff function with χ (r) = 1 r ≤ r0 . Y has the following properties: 0 r ≥ (1.2) r0 1. 2. 3.
Y is supported on r < (1.2) r0 , K Y = K Y on r < r0 , 1 Y JμY (φ) nμ{v=const.} on 2M ≤ r ≤ r0 . CK (φ) ≥ √1−μ
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On the region 2M ≤ r ≤ r0 , we have JμY (φ) nμ{v=const.} y1 1 1 2 2 + = √ / φ| , (∂u φ) + (1 − μ) (y2 + 1) |∇ 1−μ 2 2 1−μ y1 1 1 2 + JμY (φ) nμ{t=const.} = √ (∂u φ) 1−μ 2 2 1−μ 1 2 2 + y2 + / φ| . (∂v φ) + (y1 + (1 − μ) (y2 + 1)) |∇ 2 We argue without computation that for r0 ≤ r ≤ (1.2) r0 , 1 J T (φ) nμ{t=const.} , |K Y | ≤ C √ 1−μ μ JμY (φ) nμ{v=const.} ≤ CJμT (φ) nμ{v=const.} , JμY (φ) nμ{t=const.} ≤ CJμT (φ) nμ{t=const.} . This is true because JμT (φ) nμ{t=const.} controls every derivative of φ while the
terms in JμT (φ) nμ{v=const.} and JμY (φ) nμ{t=const.} contain only derivatives ∂u and ∇ / . Thus the only difference is the weights, which are functions of r and are harmless since r is bounded on this region. Extend Y to inside the black hole smoothly by the requirement that N is future directed and causal. Define N = T + Y for r ≥ 2M . N is future directed and causal everywhere, thus JμN (φ) nμ{t=const.} ≥ 0. Away from the horizon, namely when r ≥ 1.2r0 , JμN (φ) nμ{t=const.} = JμT (φ) nμ{t=const.} . However, as we approach the horizon, JμN (φ) nμ{t=const.} ∼ JμY (φ) nμ{t=const.} and thus JμN (φ) nμ{t=const.} gives a much stronger bound. We assume for our energy classes that the integral of JμN (φ) nμ{t=const.} is bounded initially and this clearly implies the boundedness for the corresponding integrals for J T and J Y initially. The flux corresponding to J N should be thought of as a nondegenerate energy, which does not degenerate at the event horizon. This allows us to prove decay results along the event horizon. Before introducing the vector field commutator S, we end this part on vector field multipliers by explicitly noting what each of the positive quantities bounds. Most of these are direct consequences of the expressions of the curZ rents, except that for J Z,w , which requires some manipulation and is proved in [11]. 2 2 1 (∂r∗ φ) + (∂t φ) + (1 − μ) |∇ / φ|2 ≤ CJμT (φ) nμt , Proposition 3. 1. √1−μ ∞ 2 2 1 u2 (∂u φ) + v 2 (∂v φ) + (1 − μ) u2 + v 2 |∇ / φ|2 dVolt ≤ 2. −∞ S2 √1−μ ∞ Z C −∞ S2 JμZ,w (φ) nμt dVolt , ∞ ∞ ∗ 2 2 Z 1 (1 − μ) (r )r2+t φ2 dVolt ≤ C −∞ S2 JμZ,w (φ) nμt dVolt , 3. −∞ S2 √1−μ
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X (∂r∗ φ)2 dA ≤ C S2 l K Xl ,w l (φ) dA, S2 (1+|r ∗ |)2 r 1+δ (1−μ) X |∇ / φ|2 dA ≤ C S2 l K Xl ,w l (Ωφ) dA. S2 (1+|r ∗ |)4 (1−μ)
3.6. Vector Field Commutator S Define S = t∂t + r∗ ∂r∗ = v∂v + u∂u . This vector field, together with the usual Killing fields, will be commuted with g . We note that the vector field t∂t + r∂r is conformally Killing on Minkowski with [m , t∂t + r∂r ] = 2m . Therefore, the commutator [g , S] is expected to approach 2g towards spatial infinity, where the spacetime approaches Minkowski. We set ψ = Sφ and derive an equation for ψ. ∗ ∗ ∗ 2 + r rμ g + 2r rr − 1 − 2rr μ ∂r∗ + Proposition 4. 1. [g , S] = ∗ ∗ /. 2 rr − 1 − 3r2rμ Δ 2.
g ψ
g1 (r∗ ) ∂r∗ φ + g2 (r∗ ) Δ / φ,
=
(log r)+ r2 ∗
|g1 (r∗ ) |, |g2 (r r
where
∗
)|
r >> 2M , (log r) = max{log r, 1}. + r ∼ 2M
|r |
Remark 3. Equivalently, we write |g1 (r∗ ) |, |g2 (r r
∗
)|
∼
(1+|r ∗ |)(log r)+ . r3
Proof. −1
[− (1 − μ)
−1
∂t2 + r∗ ∂r∗ (1 − μ)
−1
∂t2 −
∂t2 , S] = −2 (1 − μ) = −2 (1 − μ)
−1
(1 − μ)
∂t2
r∗ μ ∂2, r (1 − μ) t
−1 −1 ∂r2∗ , S = 2 (1 − μ) ∂r2∗ − r∗ ∂r∗ (1 − μ) ∂r2∗ −1
= 2 (1 − μ)
−1
∂r2∗ +
r∗ μ ∂ 2∗ , r (1 − μ) r
2 2 2r∗ (1 − μ) ∗ ∂r , S = ∂r ∗ + ∂r ∗ r r r2 ∗ 2r (1 − μ) 2 4 = ∂r ∗ + − ∂r ∗ , r r2 r 2r∗ (1 − μ) Δ /, r ∗ 2r (1 − μ) 2 2r∗ μ r∗ μ [g , S] = 2 + − − 2 ∂r ∗ g + r r2 r r ∗ 2r (1 − μ) r∗ μ −2− + Δ / r r [Δ / , S] =
∼
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Ann. Henri Poincar´e
2r∗ μ r∗ μ 2 r∗ −1− = 2+ g + ∂r ∗ r r r r ∗ r 3r∗ μ −1 − +2 Δ /. r 2r ∗ ∗ 2. is immediate from 1. if we let g1 (r∗ ) = 2r rr − 1 − 2rr μ and g2 (r∗ ) = ∗ ∗ 2 rr − 1 − 3r2rμ .
4. Estimates for φ The following has been proved in [11] and is collected for later use. Theorem 5 (Dafermos–Rodnianski). T 1. Jμ (φ) nμt∗ dVolt∗ = JμT (φ) nμt0 dVolt0 , X ,wXl K l (φ) dVol ≤ C JμT (φ) nμt0 dVolt0 , 2. Zl 3. Jμ (φ) nμt∗ dVolt∗ ≤ CE0 (φ), t2∗ T 4. − t∗ Jμ (φ) nμt∗ dVolt∗ ≤ CE0 (φ) t−2 ∗ , t2 2 2t X ,wXl (φ) dVol ≤ CE0 (φ) t−2 5. t1 − t l K l 1 , where t1 ≤ t2 ≤ (1.1) t1 . 2
The following Hardy type inequality is also proved in [11] and will be used throughout this paper. Lemma 6. − 12 2 −1 ∗ −2 2 dVolt0 ≤ C (∂r∗ φ) (1 − μ) 2 dVolt0 . (1 + |r |) φ (1 − μ) Remark 4. This can be written equivalently in local coordinates as ∞ −∞ S2
φ2 (1 + |r∗ |)
2r
2
∞
∗
dA dr ≤ C
2
(∂r∗ φ) r2 dA dr∗ .
−∞ S2
We construct a vector field X0 to control the spacetime integral of φ2 itself. Proposition 7.
φ2 (1 + |r∗ |)
4
dVol ≤ C
JμT (φ) nμt0 dVolt0 .
Proof. We first notice that we already have control of a weighted L2 -norm of the non-zeroth spherical harmonics. This is because by (10), Xl φ2l dA ≤ C K Xl ,w (φl ) dA 4 ∗ (1 + |r |) (1 − μ) S2
S2
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for l ≥ 1. This together with Theorem 5.2 would give φ2l dVol ≤ C JμT (φ) nμt0 dVolt0 4 ∗ (1 + |r |) (1 − μ) for l ≥ 1. So it suffices to consider the zeroth spherical harmonic. μ3 r 3 M3 Define X0 = f0 ∂r∗ , with f0 (r∗ ) = f0 (r) = − (1+4μ −2 ) = − 8(1+4μ−2 ) . Suppose we act with X0 on the zeroth spherical harmonic of φ0 . Using (9), 2 X0 2 − 3μ 1 f (r∗ ) (∂r∗ φ0 ) + |∇ / φ0 |2 K X0 ,w (φ0 ) = 0 f0 (r∗ ) 1−μ 2 r 1 4 μ 1 f (r∗ ) + f0 (r∗ ) + 2 f0 (r∗ ) − 4 1−μ 0 r r 2μ − 3 (3 − 4μ) f0 (r∗ ) φ20 r 2 1 1 f0 (r∗ ) (∂r∗ φ0 ) 4 μ − f (r∗ ) + f0 (r∗ ) + 2 f0 (r∗ ) = 1−μ 4 1−μ 0 r r 2μ ∗ − 3 (3 − 4μ) f0 (r ) φ20 , r X0 1 f0 (r∗ ) ∂t φ0 ∂r∗ φ0 JμX0 ,w (φ0 ) nμt = √ 1−μ 1 2 (1 − μ) f0 (r∗ ) (∂t φ0 ) φ0 , + √ f0 (r∗ ) + r 2 1−μ where we have used ∇ / φ0 = 0. X0 We would have to show first that K X0 ,w (φ0 ) ≥ 0 and controls φ2 , and X0 second that JμX0 ,w (φ0 ) nμ is controllable by JμT (φ0 ) nμ . We first compute the derivatives of f0 : f0 (r∗ ) = (1 − μ) ∂r f0 (r) μr2 (1 − μ) = 2 ≥0 (1 + 4μ−2 ) μ (1 − μ) 2 ∂r f0 f0 (r∗ ) = (1 − μ) ∂r2 f0 + r
2μr 16r μ2 r (1 − μ) 2 = (1 − μ) − + + 3 2 2 μ (1 + 4μ−2 ) (1 + 4μ−2 ) (1 + 4μ−2 ) 2
2
3μ (1 − μ) 2 μ2 (1 − μ) 2μ (1 − μ) ∂r f0 + ∂r f0 − ∂r f0 2 r r r2 2 48 384 3μ (1 − μ) 3 = (1 − μ) − + 4 3 r μ3 (1 + 4μ−2 ) μ (1 + 4μ−2 )
16r μr μ2 (1 − μ) (3μ − 2) × − + . + 3 2 2 μ (1 + 4μ−2 ) (1 + 4μ−2 ) (1 + 4μ−2 ) 3
f0 (r∗ ) = (1−μ) ∂r3 f0 +
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A computation shows that 1 4 μ 2μ f + f0 + 2 f0 − 3 (3 − 4μ)f0 1−μ 0 r r r 6 μ (192 + μ(128 + μ(−784 + μ(464 + μ(−28 + μ(52 + μ(−3 + 4μ))))))) =− . 4(4 + μ2 )4 We need to show that 192 + μ(128 + μ(−784 + μ(464 + μ(−28 + μ(52 + μ(−3 + 4μ)))))) ≥ 0 for 0 ≤ μ ≤ 1. Case 1 11 20 ≤ μ ≤ 1 192 + 128μ − 784μ2 + 464μ3 = 16(−12 − 20μ + 29μ2 )(μ − 1) ≥ 0. 52 − 3μ + 4μ2 reaches its minimum at 38 . Hence, 52 − 3μ + 4μ2 ≥ 823 −28 + μ(52 − 3μ + 4μ2 ) ≥ −28 + 11 20 16 ≥ 0.
823 16 .
Case 2 0 ≤ μ ≤ 11 20 2 464 − 28μ + 823 μ has negative discriminant, hence ≥ 0. 16 Also, for this range of μ, 192 + 128μ − 784μ2 ≥ 0. X0 X0 Therefore, K X0 ,w (φ0 ) ≥ 0. Moreover, φ20 ≤ CK X0 ,w (φ0 ). It now remains only to control the boundary terms. Using Lemma 6 and Cauchy–Schwarz, X0 JμX0 ,w (φ0 ) nμ dVolt 1 √ = f0 r∗ ∂t φ0 ∂r∗ φ0 1−μ 2 (1 − μ) ∗ 1 f0 r∗ + + √ f0 r (∂t φ0 ) φ0 dVolt0 r 2 1−μ 1 1 2 √ (∂t φ0 )2 + (∂r∗ φ0 )2 + dVolt0 ≤C φ 0 1−μ (1 + |r∗ |)2 1 √ ≤C ((∂t φ0 )2 + ∂r∗ φ0 )2 dVolt0 1−μ ≤ C JμT (φ0 ) nμ t0 dVolt0 . ˜ = f˜ (r∗ ) ∂r∗ so as to improve We would like to construct a vector field X the weights in r of the spacetime integral that can be controlled. More precisely, we have the following: Proposition 8. t∗ ∞ r−1−δ (∂r∗ φ)2 + r−3−δ φ2 dVol ≤ C JμT (φ) nμ t0 dVolt0 , t0 1 S2
t∗ ∞ t0 1 S2
for 0 < δ < 12 .
r−1 |∇ / φ|2 dVol ≤ C
1 k=0
JμT Ωk φ nμ t0 dVolt0 ,
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Remark 5. The loss of derivative above is unnecessary because we are considering only a subregion of {r∗ > 0}. One can construct yet another variant of the vector field X to achieve the above estimate without any loss of derivatives. However, since this would not improve the regularity in our final result, it is not pursued here. 1 ˜ = f˜(r∗ )∂r∗ , where f˜ = χ(r∗ )(1 − ) and χ is a cutoff Proof. Let X (1+r ∗ )δ function satisfying 0 r∗ ≤ 1 χ= . 1 r∗ ≥ max{100, 100M }
We recall (9): ˜
2 2 − 3μ ˜ ∗ 1 f˜ (r∗ ) (∂r∗ φ) + |∇ / φ|2 f (r ) 1−μ 2 r 1 ˜ ∗ 4 μ 1 f (r ) + f˜ (r∗ ) + 2 f˜ (r∗ ) − 4 1−μ r r 2μ − 3 (3 − 4μ) f˜ (r∗ ) φ2 , r 1 (φ) nμt = √ f˜ (r∗ ) ∂t φ∂r∗ φ 1−μ 2 (1 − μ) ˜ ∗ 1 ∗ ˜ + √ f (r ) + f (r ) (∂t φ) φ, r 2 1−μ ˜ X
K X,w (φ) =
˜
JμX,w
˜ X
Since we already have control of the spacetime integrals on a compact set ˜ ˜ X using Theorem 5 and Proposition 7, we only have to show that K X,w (φ) ≥ 0 for r∗ ≥ max{100, 100M }. For r∗ ≥ max{100, 100M }, we have f˜ (r∗ ) =
δ 1+δ
(1 + r∗ ) δ (1 + δ) f˜ (r∗ ) = − 2+δ (1 + r∗ ) δ (1 + δ) (2 + δ) . f˜ (r∗ ) = 3+δ (1 + r∗ ) 2
˜
˜ X
/ φ|2 in K X,w (φ) is positive for r∗ ≥ Clearly, the coefficient of (∂r∗ φ) and |∇ ˜ ˜ X max{100, 100M }. We now study the coefficient of φ2 in K X,w (φ) for r∗ ≥ max{100, 100M }: μ 2μ 1 ˜ 4 ˜ f + f + 2 f˜ − 3 (3 − 4μ) f˜ 1−μ r r r 1 δ (1 + δ) (2 + δ) 4δ (1 + δ) 2M δ = − + 2+δ 1+δ 1 − μ (1 + r∗ )3+δ r (1 + r∗ ) r2 (1 + r∗ ) 12M 32M 2 2μ − + − 3 (3 − 4μ) δ δ 3 ∗ 5 ∗ r r (1 + r ) r (1 + r )
832
J. Luk
≤
3δ (1 + δ) (2 + δ) 3+δ r∗ )
−
2 (1 + 12M
−
4δ (1 + δ)
r (1 + 32M 2
2+δ r∗ )
Ann. Henri Poincar´e
+
2M δ r2
1+δ
(1 + r∗ )
+ δ δ r3 (1 + r∗ ) r5 (1 + r∗ ) 3δ M δ (1 + δ) 32 −1 + ≤ 2δ − 12 + 2+δ δ 2 100 r (1 + r∗ ) r3 (1 + r∗ ) < 0. ˜ X
˜
100M }. Hence K X,w (φ) ≥ 0 for r∗ ≥ max{100, 2 Moreover, on this region of r∗ , r−1−δ (∂r∗ φ) + r−3−δ φ2 + r−1 |∇ / φ|2 ≤ ˜
˜ X
CK X,w (φ). ˜ X˜ Finally, we have JμX,w (φ) nμ dV olt ≤ CJμT (φ) nμ dV olt using Lemma 6 and Cauchy–Schwarz exactly as in Proposition 7. Remark 6. The weights in the Proposition are the same as those for Minkowski space. Since Schwarzschild is asymptotically flat, they are the expected weights. Corollary 9. In local coordinates, Theorem 5, Propositions 7 and 8 imply via Proposition 3 the following bounds: 1 √ ((∂ ∗ φ)2 + (∂t φ)2 + (1 − μ)|∇ / φ|2 ) dVolt∗ ≤ C JμT (φ)nμt0 dVolt0 , 1. 1−μ r 1 2 2 √ u2 (∂u φ) + v 2 (∂v φ) + (1 − μ) |∇ 2. / φ|2 dVolt∗ ≤ CE0 (φ), 1−μ √ ∗ 2 2 3. 1 − μ (r )r2+t φ2 dVolt∗ ≤ CE0 (φ) t−2 ∗ , t2 2t r1−δ (∂r∗ φ)2 r 1−δ φ2 4. t1 − t (1+|r∗ |)2 (1−μ) + (1+|r∗ |)4 dVol ≤ CE0 (φ) t−2 1 , where t1 ≤ t2 ≤ (1.1) t1 , k −2 t2 2t 2 1 r 3 |∇ / φ|2 5. t1 − t (1+|r∗ |)4 (1−μ) dVol ≤ C k=0 E0 Ω φ t1 , where t1 ≤ t2 ≤ (1.1) t1 . 2
5. Estimates for ψ In this section, we would like to imitate [11] and prove an analogue of Theorem 5. For technical reasons, however, we will need to lose an arbitrarily small power of t. T T Theorem 10. 1. Jμ (ψ) nμt∗ dVolt∗ ≤ C Jμ (ψ) nμt0 dVolt0 + E1 (φ) , Z 2. J (ψ) nμt∗ dVolt∗ ≤ Ctδ∗ E1 (φ), μ , 3. {− t∗ ≤r∗ ≤ t∗ } JμT (ψ) nμt∗ dVolt∗ ≤ CE1 (φ) t−2+δ ∗ t2 2 2t 2 X ,wXl 4. t1 − t l K l (ψ) dVol ≤ CE1 (φ) t−2+δ , where t1 ≤ t2 ≤ (1.1) t1 , 1 2
The general strategy is as follows. We follow the argument in [11] but now in the conservation law for each of the vector fields, t∗ μthere is an extra error term which is a spacetime integral that looks like t0 V ∂μ ψg ψdVol (as well t as an extra term − 41 t0∗ wψg ψdVol for the modified currents). Very often, we need to show that this integral decays (or does not grow) with t∗ , thus
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
833
we need to “produce” some decay in t. We do this by splitting the domain of integration into three regions and estimating them separately: For the region { 2t ≤ r∗ ≤ ∞}, we use the fact that g ψ contains negative powers of r∗ , (which is a consequence of the asymptotic flatness of Schwarzschild). In this region, negative powers of r∗ can be estimated by negative powers of t. 2. For the region {− 2t ≤ r∗ ≤ 2t }, we note that we have decay in the spacetime integral of φ for each dyadic slab by Corollary 9.4 and 9.5. We, therefore, estimate the integral on this region by that of K X (φ). Here, it is essential that we use the improved X estimates given by Proposition 8. 3. For the region {−∞ ≤ r∗ ≤ − 2t }, we make use of the fact that there is 1.
1
an extra factor of (1 − μ) 2 in the spacetime volume form compared to the volume form on a time-slice (see Sect. 1.5). From the definition of r∗ , we 1 ∗ have (1 − μ) ≤ Cecr , thus the factor of (1 − μ) 2 gives exponential decay in r∗ , which translates to exponential decay in t in this region. Therefore, on this region, we first estimate on each time slice, and then carry out the integration in t. Since we will often perform integration dyadically, we first set up the notation. We define a dyadic partition of [t0 , t∗ ] by t0 ≤ t1 ≤ · · · ≤ tn = t∗ , where ti ≤ (1.1) ti−1 and n is the minimal integer such that this can be done. In particular, log (t∗ − t0 ) ∼ n. We begin with the T estimate. Proposition 11.
JμT (ψ) nμt∗ dVolt∗ ≤ C
JμT (ψ) nμt0 dVolt0 + C
2
E0 Ωk φ .
k=0
Proof. The conservation law gives
JμT (ψ) nμt∗ dVolt∗ =
JμT (ψ) nμt0 dVolt0 +
∂t ψg ψ dVol.
We split the error term into three parts and estimate them separately. By Corollary 9.1, ⎛ ⎞ 12 ∞ t∗ ∞ JμT (ψ) nμt dVolt ⎠ ∂t ψg ψ dVol ≤ C ⎝ t t0 −∞ S2 2 ⎞ 12 ⎛ ∞ 2 (log r)+ ⎟ ⎜ 2 (∂r∗ φ) + |∇ ×⎝ / Ωφ|2 dVolt ⎠ dt r4 t 2
S2
834
J. Luk
⎛ ≤ C sup ⎝ t0 ≤t≤t∗
⎛ ×⎝
Ann. Henri Poincar´e
⎞ 12
∞
JμT (ψ) nμt dVolt ⎠
−∞ S2
1 ∞
JμT
k
Ω φ
k=0−∞ S2
⎛
≤ C sup ⎝ t0 ≤t≤t∗
⎛ ×⎝
1
∞
nμt0
dVolt0 ⎠ ⎞ 12
t∗
3
t− 2 dt
t0
JμT (ψ) nμt dVolt ⎠
−∞ S2
∞
⎞ 12
⎞ 12
JμT Ωk φ nμt0 dVolt0 ⎠
k=0−∞ S2
For the middle region, we observe that by Corollaries 9.4 and 9.5, t 2 ∂t ψg ψ dVol − t 2 ⎛ ⎞ 12 t∗ ∞ ≤C ⎝ JμT (ψ) nμt dVolt ⎠ t0
⎛
⎜ ×⎝
−∞ S2 t 2
− 2t S2
(1 + |r∗ |) ⎛
≤ C sup ⎝ t0 ≤t≤t∗
⎛ ⎜ ×⎜ ⎝
(1 − μ) 3 2
⎞ 12
∞
JμT (ψ) nμt dVolt ⎠
⎞ 12 ⎞ t 2 ⎟ Xl ⎜ ⎟ Ωk φ (1 − μ) dVolt ⎠ dt⎟ K Xl ,w ⎝ ⎠
t i+1
k=0 i=0 t i
t0 ≤t≤t∗
− 2t
∞
l
⎞ 12
JμT (ψ) nμt dVolt ⎠
−∞ S2
⎛
2 n−1 k=0 i=0
1 ⎜ ti2 ⎝
t i+1 2t
ti − t 2
l
K Xl ,w
Xl
⎞ 12
⎟ 2 / Ωφ|2 dVolt ⎠ dt (∂r∗ φ) + |∇
⎛
2 n−1
≤ C sup ⎝
⎜ ×⎜ ⎝
2 (log r)+ r6
−∞ S2
⎛
⎛
2
⎞ 12 ⎞ k ⎟ ⎟ Ω φ dVol⎠ ⎟ ⎠
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
⎛ ≤ C sup ⎝ t0 ≤t≤t∗
∞
−∞
⎛ ≤ C sup ⎝ t0 ≤t≤t∗
⎞ 12
JμT
(ψ) nμt
dVolt ⎠
k
E0 Ω φ
12 n−1
⎞ 12
JμT
(ψ) nμt
dVolt ⎠
2
−1 ti 2
i=0
k=0
S2
∞
2
835
12
k
E0 Ω φ
k=0
−∞ S2
By Corollary 9.1, t ⎛ ⎞ 12 − 2 t∗ ∞ ∂t ψg ψ dVol ≤ C ⎝ JμT (ψ) nμt dVolt ⎠ −∞ t0 −∞ S2 ⎞ 12 ⎛ t − 2 3 ⎟ ⎜ 2 2 ×⎝ (1 + |r∗ |) (1 − μ) 2 (∂r∗ φ) + |∇ / Ωφ|2 dVolt ⎠ dt −∞ S2
⎛
≤ C sup ⎝ t0 ≤t≤t∗
t∗ ×
∞
⎞ 12 ⎛ JμT (ψ) nμt dVolt ⎠ ⎝
1 ∞
⎞ 12
JμT Ωk ψ nμt dVolt ⎠
k=0−∞ S2
−∞ S2
e−ct dt
t0
⎛
≤ C sup ⎝ t0 ≤t≤t∗
∞
⎞ 12 ⎛ JμT (ψ) nμt dVolt ⎠ ⎝
1 ∞
⎞ 12
JμT Ωk ψ nμt dVolt ⎠
k=0−∞ S2
−∞ S2
These together show that JμT (ψ)nμt∗ dVolt∗ ≤
JμT (ψ)nμt0 dVolt0 +C sup
t0 ≤t≤t∗
JμT (ψ)nμt dVolt
12 2
12 E0 (Ωk φ)
,
k=0
which implies the proposition with the following lemma, taking h1 (t) = 0 and 2 h2 (t) = k=0 E0 Ωk φ . Lemma 12. Suppose f (t) is continuous, h1 (t), h2 (t) are increasing and we have 1 1 f (t∗ ) ≤ C f (t0 ) + h1 (t∗ ) + sup f (t) 2 h2 (t∗ ) 2 , t0 ≤t≤t∗
for all t∗ ≥ t0 . Then f (t∗ ) ≤ C(f (t0 ) + h1 (t∗ ) + h2 (t∗ )) .
836
J. Luk
Ann. Henri Poincar´e
Proof. Suppose supt0 ≤t≤t∗ f (t) is achieved by f t˜ for some t0 ≤ t˜ ≤ t∗ . Then 1 1 f t˜ ≤ C f (t0 ) + h1 t˜ + f t˜ 2 h2 t˜ 2 . h1 (t) , h2 (t) increasing implies, 1 1 f t˜ ≤ C f (t0 ) + h1 (t∗ ) + f t˜ 2 h2 (t∗ ) 2 . Using Cauchy–Schwarz and subtracting 12 f t˜ from both sides, f t˜ ≤ C(f (t0 ) + h1 (t∗ ) + h2 (t∗ )) . Clearly, f (t∗ ) ≤ supt0 ≤t≤t∗ f (t) = f t˜ . Hence we have the lemma.
We then derive an X estimate. Here unlike in the case for φ, in which ˜ estimate was used to improve the already known estimate from Xl , we the X need to consider both of them at the same time. Proposition 13. t∗
˜
˜ X
|K X,w (ψ) | +
K Xl ,w
Xl
(ψl ) dVol
l
t0
≤C
JμT (ψ) nμt∗ dVolt∗ +
JμT (ψ) nμt0 dVolt0
+ Ct−2+δ 0
2
E0 Ωk φ
k=0
Remark 7. The reader may ask why this Proposition gives decay for the error term while the statement of Proposition 11 does not. In fact, the proof of Proposition 11 is sufficient to show that the error term decays. However, we do not pursue this as it is unnecessary for later use. Proof. Decompose ψ = l ψl into spherical harmonics. Since Schwarzschild spacetimes are spherically symmetric, g ψl = g1 (r∗ ) ∗ / (Ωφl ). ∂r φl + g2 (r∗ ) ∇ ˜ ˜ X Notice that K X,w (ψ) is not everywhere positive. It is identically zero ˜ ˜ X for r∗ ≤ 1 and as we have shown in the proof of Proposition 8, K X,w ≥ 0 for r∗ ≥ max{100, 100M }. On the remaining (not necessarily positive) region ˜ X ˜ X 1 ≤ r∗ ≤ max{100, 100M }, we have |K X,w (ψl ) | ≤ CK Xl ,w l (ψl ). (Notice that we have avoided the region around r = 3M where this inequality is potentially problematic.)
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
837
In particular, applying Proposition 2 for the vector field X, we have |K
˜ X X,w
≤
K
(ψl ) | dVol +
=
Xl
(ψl ) dVol
˜ ˜ X X,w
˜ ˜ X JμX,w
K Xl ,w
(ψl ) dVol + (C + 1)
(ψl ) nμt∗
+ (C + 1)
dVolt∗ −
Xl JμXl ,w
˜
K Xl ,w
Xl
(ψl ) dVol
˜ X
JμX,w (ψl ) nμt0 dVolt0
(ψl ) nμt∗
dVolt∗ −
Xl JμXl ,w
(ψl ) nμt0
dVolt0
2 (1 − μ) ˜ f˜ + f˜∂r∗ ψl ψl dVol f ψl ψl dVol − r 1 2 (1 − μ) fl ψl ψl dVol − (fl ∂r∗ ψl ) ψl dVol + (C + 1) fl + 4 r ≤C JμT (ψl ) nμt∗ dVolt∗ + JμT (ψl ) nμt0 dVolt0 +
1 4
+
| r−1 ψl + ∂r∗ ψl ψl | dVol .
We split the last term into three integrals and estimate them separately. By Theorem 5.2, t∗ ∞ t0
t 2
| r−1 ψl + ∂r∗ ψl ψl | dVol
S2
t∗ ∞ ≤C t0
t 2
3 δ δ 1 δ r−1+ 2 (log r)+ r− 2 − 4 |ψl | + r− 2 − 4 |∂r∗ ψl |
S2
1 δ / Ωφ|) dVol × r− 2 − 4 (|∂r∗ φ| + |∇ −1+ δ2
≤ Ct0
⎛ ⎞ 12 t∗ ∞ ˜ ˜ X ⎜ ⎟ |K X,w (ψl ) | dVol⎠ ⎝ t0
t 2
S2
⎞ 12 1 t∗ ∞ ˜ ˜ X ⎟ ⎜ ×⎝ |K X,w Ωk φl | dVol⎠ ⎛
k=0 t
0
t 2
S2
838
J. Luk
−1+ δ2
≤ Ct0
⎛ ×⎝
⎛t ∞ ⎞ 12 ∗ ˜ X ˜ ⎝ |K X,w (ψl ) | dVol⎠ t0 −∞ S2
1 t∗ ∞ k=0 t
1 4
t∗ ∞ t0 −∞ S2
+ Ct−2+δ 0
˜
|K X,w
˜ X
⎞ 12
Ωk φl | dVol⎠
−∞ S2
0
≤
Ann. Henri Poincar´e
˜ X
˜
|K X,w (ψl ) | dVol 1 ∞
JμT Ωk φl nμ0 dVolt0 .
k=0−∞ S2
By Theorem 5.5, t
t∗ 2
| r−1 ψl + ∂r∗ ψl ψl | dVol
t0 − t S2 2 t
≤C
t∗ 2
3 δ 1 δ r− 2 − 4 |ψl | + r− 2 − 4 |∂r∗ ψl |
t0 − t S2 2
1 δ / Ωφ|) dVol × r− 2 − 4 (|∂r∗ φ| + |∇ ⎛ ⎞ 12 t t∗ 2 ˜ Xl ˜ X ⎜ ⎟ ≤C⎝ |K X,w (ψl ) | + K Xl ,w (ψl ) dVol⎠ t0 − t S2 2
⎞ 12 t 2 t∗ 2 ˜ ˜ X ⎟ ⎜ ×⎝ |K X,w Ωk φl | dVol⎠ ⎛
k=0 t
0
1 ≤ 4
− 2t S2
t∗ ∞
˜
˜ X
|K X,w (ψl ) | + K Xl ,w
Xl
(ψl ) dVol
Xl
(ψl ) dVol
t0 −∞ S2
+C
n−1 2
k t−2 i E0 Ω φ l
i=0 k=0
≤
t∗ ∞ 1 4
˜
˜ X
|K X,w (ψl ) | + K Xl ,w
t0 −∞ S2
+ Ct−2 0
1 k=0
E0 Ωk φl .
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
839
By Theorem 5.1, Proposition 11 and Lemma 6, t
t∗ − 2
| r−1 ψl + ∂r∗ ψl ψl | dVol
t0 −∞ S2 t
t∗ − 2
|r∗ | (|ψl | + |∂r∗ ψl |) (|∂r∗ φl | + |∇ / Ωφl |) dVol
≤C t0 −∞ S2
t∗ ≤C t0
⎛
∞
⎝
(1 + |r∗ |)
−∞
⎛
t
− 2
⎜ ×⎝
2 2 ψl
≤C t0
⎛ ⎝
∞
×⎝
∞
⎞ 12
⎞ 12 (∂r∗ ψl )2 dA dr∗ ⎠ ⎞ 12
(∂r∗ φl )2 + (1 − μ)|∇ / Ωφl |2 dA dr∗ ⎠ e−ct dt ⎛
≤ C sup ⎝ t0 ≤t≤t∗
dA dr∗ ⎠
⎟ 4 2 (r∗ ) (1 − μ) (∂r∗ φl ) + (1 − μ) |∇ / Ωφl |2 dA dr∗ ⎠ dt
−∞
×⎝
2
−∞
⎛
⎛
+ (∂r∗ ψl )
−∞
t∗
⎞ 12
1
2 ∞
∞
⎞ 12 JμT (ψl ) nμt dVolt ⎠
−∞
JμT
⎞ 12
Ωk φl nμt
dVolt0 ⎠ e−ct0
k=0−∞
≤C
JμT (ψl ) nμt0 dVolt0 + Ct−2 0
2
E0 Ωk φl ,
k=0
Subtract the terms with K from both sides and get t∗
˜
˜ X
X
|K X,w (ψl ) | + K Xl ,w (ψl ) dVol
t0
≤C
JμT
(ψl ) nμt∗
dVolt∗ +
JμT
(ψl ) nμt0
dVolt0 +Ct−2+δ 0
2
E0 Ωk φl .
k=0
Sum over l ≥ 0 to get the Proposition.
We localize the estimates in the above Proposition to obtain decay as in [11].
840
J. Luk
Ann. Henri Poincar´e
Proposition 14. Let t0 ≤ t1 ≤ (1.1) t0 , |r1∗ | + |r2∗ | ≤ ˜ Xl ˜ X |K X,w (ψ) | + K Xl ,w (ψl ) dVol P
Then
l
≤C
t0 2.
t−2 0
JμZ
(ψ) nμ0 dVolt0
+
t−2+δ 0
2 1
E0 ∂tm Ωk φ
,
k=0 m=0
where P = {t0 ≤ t ≤ t1 , − 2t ≤ r∗ ≤ 2t } or P = {t0 ≤ t ≤ t1 , r1∗ − (t1 − t) ≤ r∗ ≤ r2∗ + (t1 − t)}. ∗ ∗ 1 |x| ≤ 1 r r ˜ φ, ∂ φ = χ Proof. Let χ = . On t = t0 , let φ˜ = χ 0.65t t 0.65t0 ∂t φ 0 0 |x| ≥ 1.1 and solve for g φ˜ = 0 for t ≥ t0 . Following [11], we have 0.715t 0
√ −0.715t0
1 φ2 dVolt0 ≤ 1−μ
JμZ (φ) nμt0 dVolt0 .
This is true because of Propositions 3.2, 3.3 and an elementary one-dimensional estimate: ⎛ a ⎞ a 1 |f (x) |2 dx ≤ Ca2 ⎝ |∂x f (x) |2 + |f (x) |2 dx⎠ , −a
−a
−1
for a ≥ 1.2 ˜ Using this, we can estimate the current of φ: JμT φ˜ nμt0 dVolt0 0.715t 0
≤
JμT
(φ) nμt0
JμZ
(φ) nμt0
dVolt0 +
−0.715t0
≤
Ct−2 0
Ct−2 0
0.715t 0
−0.715t0
1 √ φ2 dVolt0 1−μ
dVolt0 .
Similarly,
JμT Ωφ˜ nμt0 dVolt0 ≤ Ct−2 JμZ (Ωφ) nμt0 dVolt0 . 0
One can prove this one-dimensional estimate by first considering g = 0 on [− 12 , 12 ] and a a |g (x) |dx ≤ Ca −a |∂x g (x) |dx. Then one sets g (x) = f (x)2 and use the trivial bound −a a a 2 Cauchy–Schwarz to get −a |f (x) | dx ≤ Ca2 −a |∂x f (x) |2 dx.. Finally, one cuts off f (x) 1 1 to be identically zero in [− 2 , 2 ].
2
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
841
˜ By Proposition 13, Define ψ˜ = S φ. ˜ Xl ˜ X X,w K Xl ,w ψ˜ + ψ˜l dVol K l
P
≤C
2 JμT ψ˜ nμt1 dVolt1 + JμT ψ˜ nμt0 dVolt0 +Ct−2+δ E0 Ωk φ˜ . 0 k=0
˜ X ˜ X X,w (ψ) | + l K Xl ,w l (ψl ) dVol by finite The left-hand side equals P |K speed of propagation. JμT ψ˜ nμt0 dVolt0 can be estimated in a similar way as JμT φ˜ nμt0 dVolt0 . More specifically, we claim that μ Z (ψ) n dVol + E (φ) . J JμT ψ˜ nμt0 dVolt0 ≤ Ct−2 t 0 t0 μ 0 0 To see this, we first note that ∗ ∗ r r r∗ ˜ χ ψ+ φ. ψ=χ 0.65t0 0.65t0 0.65t0 ˜ Also note that on the support of ψ, JμT ψ˜ nμt0 dVolt0 0.715t 0
≤
|r ∗ | t0
≤ C. Therefore,
JμT (ψ) nμt0 dVolt0 + Ct−2 0
−0.715t0 0.715t 0
0.715t 0
√
−0.715t0
1 ψ 2 dVolt0 1−μ
0.715t 0
1 √ φ2 dVolt0 1−μ −0.715t0 −0.715t0 μ μ −2 Z Z ≤ Ct0 Jμ (ψ) nt0 dVolt0 + Jμ (φ) nt0 dVolt0 μ −2 Z ≤ Ct0 Jμ (ψ) nt0 dVolt0 + E0 (φ) . +C
JμT
(φ) nμt0
dVolt0 +
Ct−2 0
We would now want to control JμT ψ˜ nμt1 dVolt1 . Using the conservation law for T and an integration by parts in t, JμT ψ˜ nμt1 dVolt1 μ T ˜ g ψdVol ˜ ˜ = Jμ ψ nt0 dVolt0 − ∂t ψ μ T ˜ ˜ ˜ ˜ ˜ ≤ Jμ ψ nt0 dVolt0 + ψg ∂t ψ dVol + ψg ψ 1 − μ dVolt0 ˜ g ψ˜ 1 − μ dVolt . + ψ 1
842
J. Luk
Ann. Henri Poincar´e
We first estimate the spacetime error term in this expression. Using Proposition 3, 7, and 8, ˜ ˜ ˜ ˜ ˜ ˜ ψg ∂t ψ dVol ≤ ψg S ∂t φ dVol + ψ g ∂t φ dVol (1 + |r∗ |) (log r)+ ˜ ∂r∗ ∂t φ˜ + ∇ / ∂t Ωφ˜ dVol ≤C |ψ| 3 r
12 1− δ4 ˜2 r ψ ≤C rδ 4 dVol (1 + |r∗ |) ⎛ ⎞ 12 2 2 ˜ ˜ ∗ (1 + |r∗ |)6 ∇ / Ω∂ ∂ ∂ + φ φ t r t ⎜ ⎟ ×⎜ dVol⎟ δ ⎝ ⎠ 7+ r 4 δ 2
≤ Ct0
12 ˜ ˜ X X,w X K l ψ˜l dVol ψ˜ + K l
12 2 ˜ X X ˜ X,w l ∂t Ωk φ˜ + ∂t Ωk φ˜l dVol × K Xl ,w K
k=0 δ 2
≤ Ct0
+ t−2+δ 0 ≤
1 4
l
μ T ˜ Jμ ψ nt1 dVolt1 + JμT ψ˜ nμt0 dVolt0
12 ×
E0 Ω φ
2
JμT
k˜
∂t Ω φ
nμt0
12 dVolt0
k=0
2 k˜ Ω JμT ψ˜ nμt1 dVolt1 +C JμT ψ˜ nμt0 dVolt0 + Ct−2+δ E φ 0 0
k˜
k=0
+ Ct−2+δ 0 1 ≤ 4
2
2
JμZ,w
Z
∂t Ωk φ˜ nμt0 dVolt0
k=0
k=0
μ T ˜ Jμ ψ nt1 dVolt1 + C JμT ψ˜ nμt0 dVolt0
+ Ct−2+δ 0
1 2
E0 ∂tm Ωk φ˜
m=0 k=0
where at the third to last step we again used Proposition 13. We estimate the boundary terms using Lemma 6 and Corollary 9.1 ˜ ˜ ψg ψ 1 − μ dVolt0 ˜ |∂r∗ φ| ˜ + |∇ ˜ dVolt ≤ C (1 + |r∗ |) r−3 (log r) 1 − μ|ψ| / Ωφ| +
0
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
≤C
∗
(1 + |r |)
1
−2
− ψ˜2 (1 − μ) 2 dVolt0
843
12
12 2 ˜2 ∂r∗ φ˜ + |∇ / Ωφ| 1 − μ dVolt0 ≤ C JμT ψ˜ nμt0 dVolt0 + C JμT φ˜ + JμT Ωφ˜ nμt0 dVolt0 . The term for t = t1 is done analogously, but with a more careful choice of constant. ˜ g ψ˜ 1 − μ dVolt ψ 1 ≤C
(1 + |r∗ |) r−3 (log r)+
∗
≤C × ≤
1 4
=
1 4
(1 + |r |)
˜ |∂r∗ φ| ˜ + |∇ ˜ dVolt 1 − μ|ψ| / Ωφ| 1
− ψ˜2 (1 − μ) 2 dVolt1
12 2 2 ˜ ˜ ∂r∗ φ + |∇ / Ωφ| 1 − μ dVolt1
JμT φ˜ + JμT Ωφ˜ nμt1 dVolt1 JμT ψ˜ nμt1 dVolt1 + C
JμT φ˜ + JμT Ωφ˜ nμt0 dVolt0 . JμT ψ˜ nμt1 dVolt1 + C
Combining these estimates and subtracting sides, we get
1 2
JμT ψ˜ nμt1 dVolt1 on both
JμT ψ˜ nμt1 dVolt1 ≤ Ct−2 0
JμZ (ψ) nμt0 dVolt0 + Ct−2+δ 0
It remains to control 1 2
12
1
−2
1
m=0
2
k=0
1 2
E0 ∂tm Ωk φ˜ .
m=0 k=0
E0 Ωk ∂tm φ˜ .
E0 ∂tm Ωk φ˜
m=0 k=0 0.715t 0
≤C −0.715t0
1 5
m=0 k=0
JμN
∂tm Ωk φ˜
nμ t0
+
1 4 m=0 k=0
JμZ
∂tm Ωk φ˜
nμ t0
dVol{t=t0 }
844
J. Luk 0.715t 0
≤C
5 1
0.715t 0
∂tm Ωk φ
2 1
5 1
nμ t0 2
∂tm Ωk φ
+
4 1
JμZ
∂tm Ωk φ
m=0 k=0
+
m=0 k=0
−0.715t0
≤C
m=0 k=0
−0.715t0
+ Ct−2 0
JμN
Ann. Henri Poincar´e
4 1
t20
2
∂tm Ωk φ
nμ t0
dVol{t=t0 }
r2 dA dr∗
m=0 k=0
E0 ∂tm Ωk φ .
m=0 k=0
After establishing the X estimates, we turn to the Z estimates for ψ. Proposition 15.
Z
JμZ,w (ψ) nμt∗ dVolt∗ ∗
JμZ,w
≤C
Z
r 1 t∗ 2 Xl μ Ωk ψl dVol (ψ) nt0 dVolt0 + C t K Xl ,w k=0 t
0
r1∗
l
⎛ ⎞ 12 t
t∗ 2 ˜ Xl ˜ X ⎜ ⎟ +C ⎝ t2+2δ |K X,w (ψ) | + K Xl ,w (ψl ) dVol⎠
×
l
t0 − t 2 2 1
E0 ∂tm Ωk φ
12
m=0 k=0
+
Ctδ∗
2
E0 Ωk φ .
k=0
Proof. By Proposition 2 applied to the vector field Z,
Z
JμZ,w (ψ) nμt∗ dVol{t=t∗ } =
Z JμZ,w
+
(ψ) nμt0
tr∗ (1 − μ) Z,wZ g ψ dVol dVol{t=t0 } + K (ψ) dVol − 2r
2 u ∂u ψ + v 2 ∂v ψ g ψ dVol. Z
As remarked before, there exists r1∗ , r2∗ with r1∗ < r2∗ such that K Z,w (ψ) is non-positive for r∗ ≤ r1∗ or r∗ ≥ r2∗ . Therefore,
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
845
∗
Z
K Z,w (ψ) dVol ≤
r2
Z
K Z,w (ψ) dVol
r1∗ ∗
t∗ r2 ≤C t ψ 2 + |∇ / ψ|2 dVol t0 r1∗ ∗
r 1 t∗ 2 Xl ≤C Ωk ψl dVol. t K Xl ,w k=0 t
0
l
r1∗
For the first error term, we again estimate by looking at three separate regions. By Proposition 3.3 and Corollary 9.1, t∗ ∞ ∗ tr (1 − μ) ψg ψ dVol 2r t0 t 2 ⎛ ⎞1 2 t∗ ∞ 2 t 2 2 ⎜ ∗⎟ ψ r (1 − μ) dA dr ⎠ ≤C ⎝ r2 t0
⎛ ⎜ ×⎝
t 2
S2
⎞1 2 (log r)2+ 2 2 2 ∗⎟ (∂r∗ φ) + |∇ / Ωφ| r dA dr ⎠ dt r2
∞ t 2
S2
≤C
Z JμZ,w
sup
t0 ≤t≤t∗
t∗
(ψ) nμ t dVolt
1 1 2 k=0
δ
t0
δ 2
sup
t0 ≤t≤t∗
Z JμZ,w
(ψ) nμ t
dVolt
1 1 2
E0
k=0
By Proposition 3.3, Corollary 9.4 and Corollary 9.5, t∗ 2t tr∗ (1 − μ) ψg ψ dVol 2r t0 − t 2
t∗ ≤C t0
Ω k φ nμ t0 dVol{t=t0 }
t−1+ 2 dt
×
≤ Ct∗
JμT
⎛ ⎜ ⎝
t
2
− 2t S2
2
⎞ 12
t 2 2 ⎟ ψ r (1 − μ) dA dr∗ ⎠ 2 r
Ω φ k
12 .
12
846
J. Luk
⎛ ⎜ ×⎝
∗
t0
⎜ ⎝
2
(1 + |r |)
(log r)2+ (1 r4
Z
− μ)
JμZ,w (ψ)nμt dVolt
≤ C sup t∗
⎞ 12
t
2
− 2t S2
⎛
Ann. Henri Poincar´e
t0 ≤t≤t∗
2
⎟ (∂r∗ φ)2 + |∇ / Ωφ|2 r2 dA dr∗ ⎠ dt
12 ⎞ 12
t
2
− 2t S2
(log r)2+ r2
≤ C sup
t0 ≤t≤t∗
⎛ ⎜ ×⎜ ⎝
n−1
⎛
1 ⎜ ti2 ⎝
i=0
⎟ (∂r∗ φ)2 + |∇ / Ωφ|2 r2 (1 − μ) dA dr∗ ⎠ dt
Z JμZ,w
t i+1 2t
ti − t 2
≤ C sup
t0 ≤t≤t∗
(ψ) nμt
12 dVolt
⎞12 ⎞ ⎟ 2 (log r)2+ 2 2 ∗ ⎟ ⎟ ∗ φ) + |∇ (∂ r / Ωφ| (1 − μ) dA dr dt ⎠ r ⎠ r2
Z JμZ,w
(ψ) nμt
dVolt
12 n−1 2
−1 t i 2 E0
i=0 k=0
≤ C sup
t0 ≤t≤t∗
Z
JμZ,w (ψ) nμt dVolt
12 2
12
k
Ω φ
12
E0 Ωk φ
.
k=0 ∗
By Proposition 3.3 and Corollary 9.1 and using the fact that (1 − μ) ≤ Cecr , t∗ − 2t tr∗ (1 − μ) ψg ψ dVol 2r t0 −∞ ⎛ t ⎞ 12 t∗ − 2 2 t 2 2 ⎜ ⎟ ≤C ⎝ ψ r (1 − μ) dA dr∗ ⎠ r2 t0
⎛
⎜ ×⎝
−∞ S2
⎞ 12
t
− 2
⎟ (r∗ )4 (∂r∗ φ)2 + (1 − μ)|∇ / Ωφ|2 r2 (1 − μ) dA dr∗ ⎠ dt
−∞ S2
≤ C sup
t0 ≤t≤t∗
t0 ≤t≤t∗
1 12 12 t∗ (ψ)nμt dVolt e−ct JμT (Ωk φ)nμt dVolt dt k=0
t0
≤ C sup
JμZ,w
Z
Z JμZ,w
(ψ) nμt
dVolt
12 1 k=0
12
k
E0 Ω φ
.
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
847
The estimation of the second error term is slightly more involved because there is a factor of t2 in the integrand. In particular, even near spacelike infinity, one needs to use estimates for the spacetime integral for φ. We intend to estimate this term separately in three regions as above. However, for technical reasons, we will divide the regions slightly differently. Divide as usual the = t∗ . We then set the three regions interval n−1to be n−1 into t0 ≤ t1 ∗≤ · ·ti· ≤ tnn−1 ti ti ∗ i=0 {ti ≤ t ≤ ti+1 , r > 2 }, i=0 {ti ≤ t ≤ ti+1 , − 2 ≤ r ≤ 2 }, i=0 {ti ≤ t ≤ ti+1 , r∗ < − t2i }. n−1 In the region i=0 {ti ≤ t ≤ ti+1 , r∗ > t2i }, we estimate one power of t Z by that in J Z,w (φ) and the other is canceled with the decay in r. To achieve this we use Proposition 3.3, Theorem 5.1, 5.2 and Proposition 8, n−1 t i+1 ∞ 2 2 u ∂u ψ + v ∂v ψ g ψ dVol i=0 ti ti 2
≤C
n−1
⎛
t i+1
⎜ ⎝
i=0 t i
⎛ ⎜ ×⎝
ti 2
∞ t 2
∞
≤ C sup
t0 ≤t≤t∗
⎛ ⎜ ×⎜ ⎝
n−1
t0 ≤t≤t∗
⎛
1 2
Z JμZ,w
ti 2
ti
≤ C sup
t0 ≤t≤t∗
12 dVolt
Z
JμZ,w (ψ) nμt dVolt
ti 2
1 2
S2
Z JμZ,w
⎛
i=0
(ψ) nμt
S2
t i+1 ∞
⎜ ×⎝
⎞ 12
⎟ 2 (∂r∗ φ) + |∇ / Ωφ|2 r2 dA dr∗ ⎠ dt
ti
i=0
n−1
⎛ ⎞ 12 ⎞ ∞ 2 ⎟ (log r)+ ⎜ ⎟ 2 (∂r∗ φ) + |∇ / Ωφ|2 r2 dA dr∗ ⎠ dt⎟ ⎝ ⎠ 2 r
≤ C sup ⎜ ×⎝
⎟ r2 dA dr∗ ⎠
t i+1
i=0 t i
n−1
2
⎞ 12
S2
2 (log r)+ r2
S2
2
u2 (∂u ψ) + v 2 (∂v ψ)
− 12 + δ2
1 δ r2− 2
(ψ) nμt
t i+1 ∞
ti ti
ti
ti 2
S2
12 ⎞ 12
⎟ 2 (∂r∗ φ) + |∇ / Ωφ|2 r2 dA dr∗ dt⎠ 12 dVolt 1
δ r1+ 2
⎞ 12
⎟ 2 / Ωφ|2 r2 dA dr∗ dt⎠ (∂r∗ φ) + |∇
848
J. Luk
≤ C sup
t0 ≤t≤t∗
×
n−1
δ 2
≤ C sup
t0 ≤t≤t∗
≤ Ct∗
JμT
ti
i=0
δ 2
Z JμZ,w
(φ) nμti
Z JμZ,w
sup
t0 ≤t≤t∗
(ψ) nμt
Z JμZ,w
12 dVolt
dVolti +
(ψ) nμt
Ann. Henri Poincar´e
JμT
12 dVolt
(ψ) nμt
(φ) nμti+1 JμT
12 dVolt
12 dVolti+1
(φ) nμt0
JμT
dVolt0
(φ) nμt0
12 n−1 12
dVolt0
δ 2
ti
i=0
.
n−1 For the region i=0 {ti ≤ t ≤ ti+1 , − t2i ≤ r∗ ≤ t2i }, we first rewrite into (t, r∗ )-coordinates and then perform an integration by parts in t. It is to avoid extra boundary terms during this integration by parts that we have divided our regions differently from before. The reason that we perform this integration by parts is that instead of a spacetime integral term with ∂t ψ, we would prefer a term with ψ, which can then be controlled by the integral of ˜ X ˜ X |K X,w | + l K Xl ,w l . t t i+1 2i n−1 2 u ∂u ψ + v 2 ∂v ψ g ψ dVol i=0 ti − ti 2
≤C
n−1
t i+1
ti
2
|tr∗ ∂r∗ ψg ψ|r2 (1 − μ) dA dr∗ dt
i=0 t i − ti S2 2
n−1 t i+1 t2i 2 ∗ 2 2 ∗ t + (r ) ∂t ψg ψr (1 − μ) dA dr dt +C i=0 ti − ti S2 2
≤C
n−1
t i+1
ti 2
|tr∗ ∂r∗ ψg ψ|r2 (1 − μ) dA dr∗ dt
i=0 t i − ti S2 2
+C
n−1
t i+1
ti
2
t2 |ψg (∂t ψ) |r2 (1 − μ) dA dr∗ dt
i=0 t i − ti S2 2
+C
n−1
t i+1
ti
2
i=0 t i − ti S2 2
t|ψg ψ|r2 (1 − μ) dA dr∗ dt
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
+C
n−1 i=0
+C
t2 |ψg ψ| 1 − μ dVolti
ti 2
−
n−1 i=0
ti
2
849
ti
2
t2 |ψg ψ| 1 − μ dVolti+1
ti 2
−
t
t∗ 2
|tr∗ ∂r∗ ψg ψ|r2 (1 − μ) dA dr∗ dt
≤C t0 − t S2 2 t
t∗ 2 +C
t2 |ψg (∂t ψ) |r2 (1 − μ) dA dr∗ dt
t0 − t S2 2 t
t∗ 2 +C
t|ψg ψ|r2 (1 − μ) dA dr∗ dt
t0 − t S2 2 ti
+C
n 2 i=0
−
t2i |ψg ψ| 1 − μ dVolti .
ti 2
We now group this into three parts: first, the spacetime term that grows like t2 ; second, the spacetime terms that grow like t; and finally, the boundary terms. By Proposition 7, 8, Proposition 3.4, 3.5, Theorem 5.1 and 5.2, t
t∗ 2
t2 |ψg (∂t ψ) |r2 (1 − μ) dA dr∗ dt
t0 − t S2 2 t
t∗ 2 ≤C t0 − t 2
⎛
⎜ ≤C⎝
t2 (log r)+ (1 + |r∗ |) |ψ| (|∂r∗ φt | + |∇ / Ωφt |) 3 r
t2+δ rδ
t0 − t 2
⎛ ⎜ ×⎜ ⎝
⎞ 12
t
t∗ 2
n−1 i=0
1− δ2
ti
⎛ ⎜ ⎝
r
1− δ4
(1 +
t i+1 2t
ψ
2
4 |r∗ |)
⎟ dVol⎠
∗
(1 + |r |)
6
2
(∂r∗ φt ) + |∇ / Ωφt | δ
ti − t 2
2
r7+ 4
⎞ 12 ⎞ ⎟ ⎟ dVol⎠ ⎟ ⎠
850
J. Luk
⎛ ⎜ ≤C⎝
t
t∗ 2
×
⎜ ≤C⎝
×
K Xl ,w
Xl
⎞ 12
(ψl )
⎟ dVol⎠
⎞ 12
l
−δ ti 2
1 2
12
E0 ∂tm Ωk φ
m=0 k=0
i=0
⎛
˜ X
˜
|K X,w (ψ) | +
t2+δ rδ
t0 − t 2
n−1
Ann. Henri Poincar´e
t 2
t∗
˜ X
˜
t2+2δ
|K X,w (ψ) | +
K Xl ,w
Xl
(ψl )
⎟ dVol⎠
l
t0 − t 2 2 1
12
E0 ∂tm Ωk φ
.
m=0 k=0
By Proposition 7, 8, Proposition 3.4, 3.5, Corollaries 9.4 and 9.5, t
t∗ 2
|tr∗ ∂r∗ ψg ψ|r2 (1 − μ) dAdr∗ dt
t0 − t S2 2 t
t∗ 2 +
t|ψg ψ|r2 (1 − μ) dA dr∗ dt
t0 − t 2
⎛
⎜ ≤C⎝
t0 − t 2
⎛ ⎜ ≤C⎝
⎞
t
t∗ 2
∗
t (log r)+ (1 + |r |) (|r ∂r∗ ψ| + |ψ|) (|∂r∗ φ| + |∇ / Ωφ|) ⎟ dVol⎠ r3
t
t∗ 2
tδ r δ
t0 − t 2
⎛ ⎜ ×⎜ ⎝
n−1
⎛ ⎜ ≤C⎝
×
⎛
1− δ2
ti
r
⎜ ⎝
t i+1 2t
2
ψ2
4 |r∗ |)
+
(∂r∗ ψ)
(1 + |r∗ |)
6
˜
˜ X
|K X,w (ψ) | +
2
(∂r∗ φ) + |∇ / Ωφ|2
l
t0 − t 2 k
⎟ dVol⎠
δ
tδ r δ
⎞ 12
r7+ 4
t
δ r1+ 4
ti − t 2
t∗ 2
k=0
1− δ4
(1 +
i=0
2
∗
E0 Ω φ
12 n−1
i=0
−δ ti 2
K Xl ,w
Xl
(ψl )
⎞ 12 ⎞ ⎟ ⎟ dVol⎠ ⎟ ⎠ ⎞ 12
⎟ dVol⎠
Vol. 11 (2010)
⎛ ⎜ ≤C⎝
×
Improved Decay for Solutions to the Linear Waves
t
t∗ 2
⎞ 12
K Xl ,w
Xl
⎟ dVol⎠
(ψl )
l
t0 − t 2 2
˜ X
˜
|K X,w (ψ) | +
t2δ
851
12
k
E0 Ω φ
.
k=0
By Propositions 3.1, 3.3 and Theorem 5.4, ti
n 2 i=0
−
ti 2
t2i |ψg ψ| 1 − μ dVolti ⎛
n ⎜ ≤C⎜ ti ⎝ i=0
Z JμZ,w
≤ C sup
t0 ≤t≤t∗
(ψ) nμt
dVolt
12 1
12
k
E0 Ω φ
k=0
δ 2
⎞ 12 ⎞ ti 2 1 ⎜ ⎟ ⎟ JμT Ωk φ nμti dVolti ⎠ ⎟ ⎝ ⎠ ⎛
k=0 ti −2
≤ Ct∗
Z
JμZ,w (ψ) nμti dVolti
12
sup
t0 ≤t≤t∗
Z JμZ,w
(ψ) nμt
dVolt
12 1
12
k
E0 Ω φ
n
1
i=0
.
k=0
These together give t∗ 2t 2 2 u ∂u ψ + v ∂v ψ ψ dVol t0 − t 2
⎛t ⎞ 12
∗ 2 1 ˜ X X ˜ l ≤ C ⎝ t2+2δ |K X,w (ψ) | + K Xl ,w (ψl ) dVol⎠ m=0 k=0
l
t0
×E0 ∂tm Ωk φ
12
δ 2
+ Ct∗
sup
t0 ≤t≤t∗
JμZ,w
Z
(ψ) nμt
12 1
dVolt
k=0
E0 Ωk φ
12
.
n−1 We finally look at the third region, i=0 {ti ≤ t ≤ ti+1 , r∗ < − t2i }, for the second error term. By Propositions 3.1, 3.3 and Theorem 5.1. t∗ − 2t 2 2 u ∂u ψ + v ∂v ψ ψ dVol t0 −∞
t∗ ≤C t0
⎛ ⎜ ⎝
t
− 2
−∞
2
2
u2 (∂u ψ) + v 2 (∂v ψ)
⎞ (1 − μ)
− 12
⎟ dVolt ⎠
852
J. Luk
⎛ ⎜ ×⎝
t
− 2
Ann. Henri Poincar´e
⎞
3 ⎟ 2 2 (r∗ ) (1 − μ) 2 (∂r∗ φ) + |∇ / Ωφ|2 dVolt ⎠ dt
−∞
t∗ ≤C
⎛ ⎜ e−ct ⎝
t0
⎞⎛
t
− 2
⎟⎜ JμZ (ψ) nμt dVolt ⎠ ⎝
Z JμZ,w
≤ C sup
t0 ≤t≤t∗
⎟ JμT Ωk φ nμt dVolt ⎠ dt
k=0−∞
−∞
⎞
t
− 2
1
(ψ) nμt
12
1
(E0 (φ)) 2 .
dVolt
Therefore, Z JμZ,w (ψ) nμt∗ dVol{t=t∗ } ∗
JμZ,w
≤C
r 1 t∗ 2 Xl μ Ωk ψl dVol (ψ) nt0 dVolt0 + t K Xl ,w
Z
k=0 t
⎛ ⎜ +C ⎝
×
0
t
t∗ 2
˜ X
˜
t2+2δ
l
r1∗
|K X,w (ψ) | +
K Xl ,w
Xl
(ψl )
⎞ 12 ⎟ dVol⎠
l
t0 − t 2 2 1
12
E0 ∂tm Ωk φ
m=0 k=0 δ 2
+ C sup t∗ t0 ≤t≤t∗
Z
JμZ,w (ψ) nμt dVolt
12 2
12
E0 Ωk φ
.
k=0
The proof concludes with Lemma 12, taking ∗
r 1 t∗ 2 Xl h1 (t∗ ) = Ωk ψl dVol t K Xl ,w k=0 t
⎛ ⎜ +⎝
0
l
r1∗
t
t∗ 2
˜
2 1
h2 (t∗ ) =
2
K Xl ,w
Xl
(ψl )
⎟ dVol⎠
12
E0 ∂tm Ωk φ
m=0 k=0
⎞ 12
l
t0 − t 2
× tδ∗
˜ X
|K X,w (ψ) | +
t2+2δ
,
k
E0 Ω φ
.
k=0
We notice that h1 (t) and h2 (t) are increasing.
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
853
We now combine Propositions 11, 13, 14 and 15 to prove Theorem 10.2. This will then imply the other parts of Theorem 10. Proposition 16.
Z
JμZ,w (ψ) nμt∗ dVolt∗ ≤ CE1 (φ) tδ∗
Proof. We first show that Propositions 13 and 11, t∗
˜ X
˜
|K X,w (ψ) | +
JμZ,w (ψ) nμt∗ dVolt∗ grows only like t1+δ ∗ . Using Z
X
K Xl ,w (ψl ) dVol
l
t0
≤C ≤C
JμT (ψ) nμt∗ dVolt∗ +
JμT (ψ) nμt0 dVolt0
+ Ct−2+δ 0
2
E0 Ωk φ
k=0
JμT (ψ) nμt0 dVolt0 + C
2
E0 Ωk φ .
k=0
Similarly, t∗
˜ X
˜
|K X,w (Ωψ) | +
X
K Xl ,w (Ωψl ) dVol
l
t0
≤C
JμT
(Ωψ) nμt0
dVolt0 + C
3
E0 Ωk φ .
k=0
Apply Proposition 15 to get Z JμZ,w (ψ) nμt∗ dVolt∗ ≤C
∗
Z
JμZ,w (ψ) nμt0 dVolt0 + Ct∗
r 1 t∗ 2 k=0 t
0
r1∗
K Xl ,w
Xl
Ωk ψl dVol
l
⎛ ⎞ 12 t t∗ 2 ˜ Xl ˜ X ⎜ ⎟ + Ct1+δ |K X,w (ψ) | + K Xl ,w (ψl ) dVol⎠ ⎝ ∗ l
t0 − t 2
× ≤C
2 1
E0 ∂tm Ωk φ
12
m=0 k=0 Z
JμZ,w (ψ) nμt0 dVolt0
+ Ctδ∗
2 k=0
E0 Ωk φ
854
J. Luk 1
+ Ct∗
Ann. Henri Poincar´e
3 JμT Ωk ψ nμt0 dVolt0 + Ct∗ E0 Ωk φ
k=0
+ Ct1+δ ∗
+ Ct1+δ ∗
k=0
JμT (ψ) nμt0
2 1
dVolt0
12 2 1
12
E0 ∂tm Ωk φ
m=0 k=0
E0 ∂tm Ωk φ
m=0 k=0
Z
JμZ,w (ψ) nμt0 dVolt0 + t1+δ ∗
≤C
×
1
JμT
k
Ω ψ
nμt0
dVolt0 +
1 3−m
E0 ∂tm Ωk φ
.
m=0 k=0
k=0
Z We now have some control over JμZ,w (ψ) nμt∗ dVolt∗ and we will use Proposition 14 to estimate the spacetime integral terms by integrating dyadically. By Proposition 14 and 11, t i+1 r2∗
Xl
(ψl ) dVol
l
r1∗
ti
K Xl ,w
t−2 i
≤C
t−2 i
≤C
×
JμZ
(ψ) nμti
dVolti +
t−2+δ i
2 1
E0 ∂tm Ωk φ
m=0 k=0
JμZ (ψ) nμt0 dVolt0 + t−1+δ i
1
JμT
k
Ω ψ
nμt0
dVolt0 +
1 3−m
E0 ∂tm Ωk φ
,
m=0 k=0
k=0
where here we have not kept track of the constant factor in front of δ, but just note that it can be chosen to be arbitrarily small. t r∗ X We can apply the same argument to tii+1 r∗2 l K Xl ,w l (Ωψl ) dVol to 1 get t i+1 r2∗
ti
r1∗
≤C
Xl
(Ωψl ) dVol
l
t−2 i
×
K Xl ,w
2 k=0
JμZ (Ωψ) nμt0 dVolt0 + t−1+δ i JμT
k
Ω ψ
nμt0
dVolt0 +
1 4−m m=0 k=0
E0 ∂tm Ωk φ
.
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
This in turn provides more control on
855
JμZ,w (ψ)nμt∗ dV olt∗ by Proposition 15: Z
Z
JμZ,w (ψ)nμ t∗ dVolt∗
Z JμZ,w (ψ)nμ t0 dVolt0
≤C
⎛ n−1
+C
×
⎜ t1+δ ⎝ i
ti − t 2
2 1
˜ X
˜
|K X,w (ψ)| +
≤C
JμZ,w
k=0
⎛
n−1
+C
Z
⎜ t1+δ ⎝ i
i=0
×
∂tm Ωk φ k
Ω ψ
12
nμ t0
+ Ctδ∗
n−1
×
2 1
E0
K Xl ,w
Xl
Ωk ψl dVol
l
⎞ 12
Xl ⎟ K Xl ,w (ψl ) dVol⎠
2
E0 Ωk φ
k=0
Ctδ∗
E0 (ψ) +
1 4−m m=0 k=0
˜ X
K Xl ,w
Xl
E0
∂tm Ωk φ
⎞ 12
⎟ (ψl ) dVol⎠
l
⎜ t1+δ ⎝ i
i=0
dVolt0 +
˜
⎛
≤ CE1 (φ)tδ∗ + C
|K X,w (ψ) | +
E0 ∂tm Ωk φ
m=0 k=0
r1∗
ti
t
t i+1 2
ti − t 2
2 1
ti
l
m=0 k=0
1
t i+1 r2∗
k=0 i=0
t
t i+1 2
i=0
E0
+C
1 n−1
12
t
t i+1 2
˜
˜ X
|K X,w (ψ)| +
Xl ⎟ K Xl ,w (ψl ) dVol⎠
l
ti − t 2
∂tm Ωk φ
⎞ 12
12
.
m=0 k=0
m k Here, we recall that we have defined E1 (φ) = E0 (ψ)+ 1m=0 4−m k=0 E0 ∂t Ω φ in Sect. 1.3. Clearly, we can replace δ by with a different constant C which depends only on : Z JμZ,w (ψ) nμ t∗ dVolt∗ ≤ CE1 (φ) t∗ ⎛ +C
n−1
⎜ t1+ ⎝ i
i=0
×
2 1 m=0 k=0
t
t i+1 2
E0
K Xl ,w
Xl
⎟ (ψl ) dVol⎠
l
ti − t 2
˜ X
˜
|K X,w (ψ) | +
⎞ 12
∂tm Ωk φ
1 2
.
(11)
856
J. Luk
Ann. Henri Poincar´e
Notice that at this point, the only term that exhibits more growth than expected is ⎛ ⎞ 12 t i+1 2t n−1 ˜ Xl ˜ X ⎜ ⎟ t1+ |K X,w (ψ) | + K Xl ,w (ψl ) dVol⎠ ⎝ i i=0
l
ti − t 2
2 1
×
12
E0 ∂tm Ωk φ
.
m=0 k=0
We will close the argument with a bootstrap. For notational purposes, we define Z It∗ = JμZ,w (ψ) nμt∗ dVolt∗ , t i+1 2t
IIti =
˜
˜ X
|K X,w (ψ) | +
t ∗ E1
It∗ ≤ C
K Xl ,w
Xl
(ψl ) dVol,
l
ti − t 2
(11) is equivalent to
(φ) +
n−1
t1+ i
1 2
1 2
(IIti ) E1 (φ) .
(12)
i=0
On the other hand, Proposition 14 gives −2+δ IIti ≤ C t−2 E1 (φ) . i Iti + ti δ
Assume It ≤ At E1 (φ), where A ≥ 4C. We want to show that It ≤
δ4 1 1 for all t ≥
400C 2 A 2 +1
(13) A δ 2 t E1
(φ)
. From the assumption and (13) we have
A
E1 (φ) + t−2+δ E1 (φ) . IIti ≤ C At−2+δ i i Hence, by picking =
δ 4
in (12),
1 3δ 1 It ≤ Ct E1 (φ) + 100C 2 A 2 + 1 t 4 E1 (φ) 3δ 1 1 ≤ Ctδ E1 (φ) + 100C 2 A 2 + 1 t 4 E1 (φ) δ 4
≤
A δ t E1 (φ) , 2
1 1
since A ≥ 4C and t ≥
400C 2 A 2 +1 A
δ4
.
Remark 8. We would like to note that the number of derivatives used in the above argument is highly wasteful (we used a total of 8 derivatives!). Blue and Soffer [3] constructed a vector field to control trapping with only derivatives. Therefore, we can, at least in principle, repeat the above argument noting the
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
857
unnecessary loss of derivatives. The details, however, have not been pursued. It is likely that with this vector field, Theorem 5 holds with E0 (φ) only having 1 + derivatives. Moreover, in Propositions 11–15, instead of having two Ω derivatives on φ, one only needs 1 + of them. One can then go to the proof of Proposition 16 and reprove it assuming only that φ is in H 2+ initially with suitable decay. Now Theorem 10 follows directly from Propositions 16 and 14.
6. Estimates near the Event Horizon In this section, we will use the vector field Y to prove that any decay estimates that can be proved on a suitable compact set holds also along the horizon. We will also show that these estimates control enough derivatives to give pointwise decay estimate. Proposition 17. Suppose ∗ ∗ 2((1.2)r 0 ) −r0
JμT (φ) nμt dVolt ≤ Bt−α
for all t, for some α ≥ 0.
r0∗
Then
JμY (φ) nμ1 (t 2
∗ ∗ +r0 )
dVol{v= 12 (t∗ +r0∗ )}
{r≤r0 }
JμY (φ) nμ∞ dVol{u=∞}
+ { 12 (t∗ +r0∗ )≤v≤ 12 (t∗ +1+r0∗ )}
≤ C B + JμN (φ) nμt0 dVolt0 t−α ∗ . Remark 9. The reader should think of B as some energy quantity of the initial data. For example, the hypothesis of this proposition show later, as we will holds for B = C m+k≤1 E1 ∂tm Ωk φ . Proof. Apply Proposition 1 for Y , on the region R = { 12 (t1 + r0∗ ) ≤ v ≤ 1 ∗ 2 (t∗ + r0 ) , t ≥ t1 } as in the Fig. 3, we get JμY (φ) nμ1 (t +r∗ ) dVol{v= 12 (t∗ +r0∗ )} 2
∗
0
{t≥t1 }
+ { 12 (t1 +r0∗ )≤v≤ 12 (t∗ +r0∗ )}
JμY (φ) nμ∞ dVol{u=∞} +
R
K Y (φ) dVol
858
J. Luk
Ann. Henri Poincar´e
v = 12 (t1 + r0∗ ) v = 12 (t∗ + r0∗ ) t = t1 r = 1.2r0 r = r0
Figure 3. The region R
JμY (φ) nμ1 (t
=
2
∗ 1 +r0 )
dVol{v= 12 (t1 +r0∗ )}
{t≥t1 }
JμY (φ) nμt1 dVolt1 .
+
(14)
{ 12 (t1 +r0∗ )≤v≤ 12 (t∗ +r0∗ )}
We split up the integrals into r ≤ r0 and r > r0 parts. Notice that the domain of integration of {t≥t1 } JμY (φ)nμ1 (t1 +r∗ ) dVol{v= 1 (t1+r0∗ )} 2 0 2 lies inside {r ≤ r0 }. Moreover, we note that { 1 (t1 +r∗ )≤v≤ 1 (t∗ +r∗ )} JμY (φ)nμ∞ 0 0 2 2 dVol{u=∞} ≥ 0. Hence
JμY
{r≤r0 }
μ
(φ)n 1 (t 2
∗ ∗ +r0 )
dVol{v= 12 (t∗ +r0∗ )} +
K Y (φ) dVol
R∩{r≤r0 }
μ
JμY (φ)n 1 (t
≤
2
{r≤r0 }
∗ 1 +r0 )
dVol{v= 12 (t1 +r0∗ )}
JμY (φ)nμt1 dVolt1
+ { 12 (t1 +r0∗ )≤v≤ 12 (t∗ +r0∗ )}
JμY (φ)nμ1 (t
+
2
∗ ∗ +r0 )
dVol{v= 12 (t∗ +r0∗ )}
{r≥r0 }∩{t≥t1 }
+
|K Y (φ)| dVol.
R∩{r≥r0 }
We estimate three terms on the right hand side. Notice that Y is constructed to be supported in {r ≤ (1.2) r0 }.
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
859
JμY (φ) nμt1 dVolt1
{ 12 (t1 +r0∗ )≤v≤ 12 (t∗ +r0∗ )} ∗ ((1.2)r 0)
JμT (φ) nμt1 dVolt1
≤C r0∗
≤ CBt−α 1 , For the second and third term, we first use the compact support of Y and then apply the conservation law associated to the Killing vector T . JμY (φ) nμ1 (t +r∗ ) dVol{v= 12 (t∗ +r0∗ )} 2
{r≥r0 }∩{u≥ 12 (t1 −r0∗ )}
∗
0
JμT (φ) nμ1 (t
≤C
∗ ∗ +r0 )
2
dVol{v= 12 (t∗ +r0∗ )}
{r0 ≤r≤(1.2)r0 }∩{u≥ 12 (t1 −r0∗ )}
≤C
JμT (φ) nμt1 dVolt∗
{r0∗ ≤r ∗ ≤2((1.2)r0 )∗ −r0∗ }
≤ CBt−α ∗
≤ CBt−α 1 ,
t∗
Y
((1.2)r 0 )∗
|K Y (φ) | dVol
|K (φ) | dVol ≤ C r0∗
t1
R∩{r≥r0 }
t∗
((1.2)r 0 )∗
JμT (φ) nμt dVolt dt
≤C r0∗
t1
t∗ ≤ CB
t−α dt
t1
≤ CB (t∗ − t1 ) t−α 1 , since α ≥ 0. Write f (t) = {r≤r0 } JμY (φ) nμ1 (t+r∗ ) dVol{v= 12 (t+r0∗ )} . Then we have 2
t∗ f (t∗ ) +
0
. f (τ ) dτ ≤ C f (t1 ) + B max{t∗ − t1 , 1}t−α 1
(15)
t1
We take C to be fixed from this point on. We clearly can assume the C > 1. From this, we will prove the Proposition by a bootstrap argument. Assume f (t) ≤ At−α for some large A that is to be determined. We want to show that −α . f (t) ≤ A 2t
860
J. Luk
Ann. Henri Poincar´e
Let t1 = t∗ − 8C 2 . Since we are only concerned with t∗ large, we assume −1 1 1 C 2 so that t∗ < 2 α t1 . Then without loss of generality that t∗ > 8 1 − 2− α t∗ f (t∗ ) +
−α 2 f (τ ) dτ ≤ C At−α 1 + 8C Bt1
t∗ −8C 2
≤ 2C A + 8C 2 B t−α ∗ .
There exists t˜ with t∗ − 8C 2 ≤ t˜ ≤ t∗ such that f t˜ ≤
1 8C 2
t∗ f (τ ) dτ t∗
−8C 2
A + 8C 2 B −α t∗ . ≤ 4C 1
Now we let t1 = t˜. Notice that t∗ < 2 α t˜. Then t∗ f (t∗ ) +
f (τ ) dτ ≤ C f t˜ + 8C 2 B t˜−α
t˜
A −α 3 −α t + 2C 2 Bt−α ∗ + 16C Bt∗ 4 ∗ A ≤ t−α , 2 ∗ ≤
if A ≥ 72C 3 B. Of course to have f (t) ≤ At−α for all t, we also need it to hold initially, i.e., A ≥ f (t0 ). Therefore, we have μ μ Y N Jμ (φ)n 1 (t +r∗ ) dVol{v= 12 (t∗ +r0∗ )} ≤ C B + Jμ (φ)nt0 dVolt0 t−α ∗ , 2
∗
0
{r≤r0 }
where C is a universal constant different from the one above. Finally, we can also get the decay estimate along the event horizon (i.e., on u = −∞) by plugging in the decay rate of f into (14) and replacing the t interval by [t∗ , t∗ + 1]. Using Proposition 17, we claim that a similar estimate holds on t-slices. 2((1.2)r0 )∗ −r0∗ T Proposition 18. Suppose r∗ Jμ (φ) nμt dVolt ≤ Bt−α for all t, for 0 some α > 1. Then μ Y μ N Jμ (φ) nτ dVolτ ≤ C B + Jμ (φ) nt0 dVolt0 v∗−α , {v∗ ≤v≤v∗ +1}
for v∗ ≥ 1.
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
861
Proof. We prove this using the conservation law for Y on the region R = {v∗ ≤ v ≤ v∗ + 1, 2v∗ − r0∗ ≤ t ≤ τ }.
JμY (φ)nμv∗ +1 dVol{v=v∗ +1}
{t≥2v∗ −r0∗ }
JμY (φ)nμτ dVolτ +
+
R
{v∗ ≤v≤v∗ +1}
JμY (φ)nμv∗ dVolv∗ +
=
{t≥2v∗ −r0∗ }
K Y (φ) dVol
JμY (φ)nμ2v∗ −r0∗ dVol{t=2v∗ −r0∗ } .
{v∗ ≤v≤v∗ +1}
We split up the integrals into r ≤ r0 and r >r0 parts. Notice that the domain of integration of {t≥2v∗ −r∗ } JμY (φ) nμv∗ dVolv∗ lies 0 inside {r ≤ r0 }. Notice also that
JμY
(φ)nμv∗ +1 dVol{v=v∗ +1}
K Y (φ)dVol ≥ 0.
+
{t≥2v∗ −r0∗ }∩{r≤r0 }
R∩{r≤r0 }
Hence
JμY (φ)nμτ dVolτ
{v∗ ≤v≤v∗ +1}
JμY
≤
(φ)nμv∗ dVolv∗
{t≥2v∗ −r0∗ }
+
JμY (φ)nμ2v∗ −r0∗ dVol{t=2v∗ −r0∗ }
{v∗ ≤v≤v∗ +1}
JμY
+
(φ)nμv∗ +1 dVol{v=v∗ +1}
{t≥2v∗ −r0∗ }∩{r≥r0 }
+
|K Y (φ)| dVol.
R∩{r≥r0 }
We show that each term has the correct bound. The first term is bounded using Proposition 17,
JμY (φ) nμv∗ dVolv∗
{t≥2v∗ −r0∗ }
= {r≤r0 }
JμY (φ) nμv∗ dVolv∗
−α ≤ C B + JμN (φ) nμt0 dVolt0 (2v∗ − r0∗ ) ≤ C B + JμN (φ) nμt0 dVolt0 v∗−α .
862
J. Luk
Ann. Henri Poincar´e
The second term is controlled by assumption JμY (φ) nμ2v∗ −r0∗ dVol{t=2v∗ −r0∗ } {v∗ ≤v≤v∗ +1}
JμT (φ) nμ2v∗ −r0∗ dVol{t=2v∗ −r0∗ }
≤C {v∗ ≤v≤v∗ +1}
−α
≤ CB (2v∗ − r0∗ ) ≤ CBv∗−α .
The last two terms are bounded by noting that Y is supported in r ≤ (1.2) r0 . The details are identical to the proof of Proposition 17. Therefore, JμY (φ) nμτ dVolτ C B + JμN (φ) nμt0 dVolt0 v∗−α . {v∗ ≤v≤v∗ +1}
This, and Sobolev embedding, is sufficient to show pointwise decay of the derivatives of φ along the horizon. We show further that if on a compact set, we have both energy decay and L2 decay, then we have pointwise decay along the event horizon. More precisely, we have Proposition 19. There exist r˜ very close to 2M such that if 1 3−m
∗ ∗ 2((1.2)r 0 ) −r0
m=0 k=0
JμT ∂tm Ωk φ nμt + φ2 dVolt ≤ Bt−α
r˜∗
for all t, for some α ≥ 0, then
|φ (v∗ , r) |2 ≤ C
B+
|∂r∗ φ (v∗ , r) | ≤ C
N
Jμ Ωk φ nμt0 dVolt0
v∗−α ,
k=0
2
2
B+
1 3−m
m=0 k=0
JμN
∂tm Ωk φ
nμt0
dVolt0
v∗−α ,
for v∗ ≥ 1, r ≤ r˜. Proof. We first take r˜ to be small enough to apply Y , i.e., r˜ < r0 . The exact condition on r˜ will be determined later. For decay of φ (v∗ , r), we want to show that on any time-slice, say t = τ , 2 k 2 |∇ / φ| + |∇ / k ∂r∗ φ|2 dA drτ∗ ≤ Cv∗−α . k=0 {v∗ ≤v≤v∗ +1}
For decay of ∂r∗ φ (v∗ , r), we want to show that on any time-slice, say t = τ , 2 k |∇ / ∂r∗ φ|2 + |∇ / k ∂r2∗ φ|2 dA drτ∗ ≤ Cv∗−α . k=0{v ≤v≤v +1} ∗ ∗
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
863
Proposition 18 gives 2 μ 2 ∗ N (∂r∗ φ) + |∇ / φ| dA drτ ≤ C B + Jμ (φ) nt0 dVolt0 v∗−α . {v∗ ≤v≤v∗ +1}
After commuting with an appropriate number of Ω, Proposition 18 gives |∇ / ∂r∗ φ|2 + |∇ /∇ / ∂r∗ φ|2 dAdrτ∗ {v∗ ≤v≤v∗ +1}
≤C
2
B+
N
Jμ Ωk φ nμt0 dVolt0
v∗−α .
k=1
After commuting with ∂t and using the equation, Proposition 18 gives 2 ∂r2∗ φ + |∇ / ∂r2∗ φ|2 + |∇ / 2 ∂r2∗ φ|2 dA drτ∗ {v∗ ≤v≤v∗ +1}
≤C
1 3−m
B+
JμN
∂tm Ωk φ
m=0 k=0
Therefore, it remains to show
nμt0
dVolt0
v∗−α .
φ2 dA drτ∗ ≤ Cv∗−α .
{v∗ ≤v≤v∗ +1}
We rewrite φ
2
2v∗ −τ +2
dA drτ∗
φ2 dA drτ∗ .
= 2v∗ −τ
{v∗ ≤v≤v∗ +1}
To achieve decay, we integrate in the u-direction and use the estimates we have on the compact set. 2v∗ −τ +2
φ2 dA drτ∗
2v∗ −τ ∗ r˜ +2
≤
φ
2
∗ dA dr{t=2v r∗ } ∗ −˜
r˜∗
τ 2v∗ −˜ r∗
∗ φ2 dA dr{t=2v r∗ } + ∗ −˜
≤ r˜∗
τ
2v∗ −˜ r∗
τ
2v∗ −t+2
2v∗ −t 2v∗ −t+2
2v∗ −t
φ2 (1 − μ) dA dr∗ dt
+ 2v∗ −˜ r∗
φ (∂u φ) dA dr∗ dt
+
∗
r˜ +2
2v∗ −t+2
2v∗ −t
2
(∂u φ) dA dr∗ dt 1−μ
864
J. Luk
−α
≤ B (2v∗ − r˜∗ )
τ
Ann. Henri Poincar´e
2v∗ −t+2
K Y (φ) dVol
+ 2v∗ −˜ r 2v∗ −t
τ
2v∗ −t+2
φ2 (1 − μ) dA dr∗ dt.
+ 2v∗ −˜ r∗
2v∗ −t
Using the conservation law for Y , and controlling all the terms on the region {r ≥ r˜} with the assumption, we have τ
2v∗ −t+2
K Y (φ) dV ol
2v∗ −˜ r∗
2v∗ −t
≤
JμN (φ) nμv∗ dVolv∗ + CBv∗−α
{r≤˜ r}
−α μ N ≤ C B + Jμ (φ) nt0 dVolt0 (v∗ − r˜∗ ) + CBv∗−α , where in the last step we have used Proposition 17. Therefore, φ2 dA drτ∗ ≤ C B + JμN (φ) nμt0 dVolt0 v∗−α
2v∗ −τ +2
2v∗ −τ
τ
2v∗ −t+2
φ2 (1 − μ) dA dr∗ dt.
+ 2v∗ −˜ r∗
2v∗ −t
The decay from the last term comes from the exponentially decaying (towards r∗ = −∞) factor (1 − μ). To use this decay, we use a bootstrap argument. v −t+1 ∗ ≤ Av∗−α , independent of t (Note that Assume the decay v∗∗−t φ2 dA dr{t=t} we can do this initially (in v) independent of t because after we fix v, the region of integration is a bounded set of the manifold. The apparent infiniteness is just an artifact of the choice of coordinates). We want to show that v∗ −t+1 2 ∗ −α φ dA dr{t=t} ≤A 2 v∗ . v∗ −t ∗ From the above, and using that (1 − μ) ≤ Cecr we have 2v∗ −τ +2
φ2 dA drτ∗
2v∗ −τ
∞ μ N −α ≤ C B + Jμ (φ) nt0 dVolt0 v∗ + ACec(2v∗ −t+2) v∗−α dt v∗ −˜ r∗
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
865
∗ μ N ≤ C B + Jμ (φ) nt0 dVolt0 v∗−α + c−1 ACec(˜r +2) v∗−α A −α v , 2 ∗ if we choose A ≥ 4C B + JμN (φ) nμt0 dVolt0 and r˜∗ ≤ −2 + ≤
1 c
log
c 4C .
The above is already sufficient to prove the full decay rate in the exterior region. While Proposition 17 implies the L2 control of the derivatives of φ along the event horizon, it is useful to prove the L2 control of φ along the event horizon. Proposition 20. Suppose ∗ ∗ 2((1.2)r 0 ) −r0
JμT (φ) nμt + φ2 dVolt ≤ Bt−α
r0∗
for all t, for some α ≥ 0,, then along the event horizon r = 2M , ⎞ ⎛ v ∗ +1 ⎟ ⎜ φ(2M )2 dA dv ≤ C ⎝B + JμN (φ) nμv0 dVolv0 ⎠ v∗−α , v∗
S2
{2M ≤r≤r0 }
for v∗ ≥ 1. Proof. We note that when changing from the (v, u) coordinates to the (v, r) coordinates, we have 1 ∂u = −∂r . 1−μ Hence, written in the (v, r) coordinates, we have r0 Y μ Jμ (φ) nv dVolv ≥ c (∂r φ)2 dA drv . 2M S2
{2M ≤r≤r0 }
By Proposition 17,
JμY (φ) nμv∗ dVolv∗ ≤ Cv∗−α .
{2M ≤r≤r0 }
Therefore, we can take the L2 control from the compact region of r∗ as in the assumption and integrate it to the horizon using the control of ∂r φ, giving the conclusion.
7. Estimates in the Black Hole Region In this section, we will prove that any decay rate that can be proved along the event horizon can then be proved inside the black hole region. Combining this with the result of the previous section, we can show that any decay rate that is obtained in a compact region of r∗ can also be proved in the black hole region.
866
J. Luk
Ann. Henri Poincar´e
We will achieve this in two steps: First, we show that any decay rate along the event horizon can be propagated slightly inside the black hole region, say, to the timelike slice r = 2M − for some > 0. This step uses an argument that is identical to that in the previous section, which we include here for the sake of completeness. The proof uses the fact that the deformation of the redshift vector field K Y has a favorable sign (similar to the case 2M ≤ r ≤ r0 ) if is chosen to be small enough. Then, we show that a decay estimate on r = 2M − for any > 0 would imply the same decay rate further inside the black hole. This part is considerably easier because of the geometry of the region in question. In this region we take advantage of the spacelike character of the Killing vector field ∂t , the nondegeneracy of ∂r and the finiteness of r, which varies between 2M − and rb > 0, as well as the fact that along constant u slices the r and the v distances are comparable. In the context of the nonlinear spherically symmetric Einstein–Maxwell-scalar field model, such black hole interior estimates were obtained in [6]. All the computations in this section will be done in the (r, v, ω) coordinates, as it is the most convenient inside the black hole region. Proposition 21. Suppose on the event horizon that v+1 JμY (φ) nμr dVolr=2M ≤ Bv −α
for all v, for some α ≥ 0.
v
Then, for some > 0, and for all r ∈ [2M − , 2M ], ⎛
v ∗ +1
⎜ JμY (φ) nμr dVolr ≤ C ⎝B +
v∗
⎞
⎟ JμN (φ) nμv0 dVolv0 ⎠ v∗−α .
{2M − ≤r≤r0 }
Remark 10. As in Proposition 17, the reader should think of B as some energy quantity of the initial data. For example, as we will show later, the hypothesis of this Proposition holds for B = C m+k≤1 E1 ∂tm Ωk φ . Moreover, note that if the conclusion of Proposition 17 implies the hypothesis of this Proposition. Proof. Applying Proposition 1 for Y , on the region R = {v1 ≤ v ≤ v∗ , 2M − ≤ r ≤ 2M } , we get Y μ Jμ (φ) nv∗ dVolv∗ + JμY (φ) nμr dVol{r=2M − } {2M − ≤r≤2M }
+ R
{v1 ≤v≤v∗ }
K Y (φ) dVol
= {2M − ≤r≤2M }
JμY (φ) nμv1 dVolv1 +
{v1 ≤v≤v∗ }
JμY (φ) nμr dVolV ol{r=2M } .
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
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Notice that {v1 ≤v≤v∗ } JμY (φ)nμr dVol{r=2M − } ≥ 0 since {r = 2M − } is timelike for any > 0 and Y is timelike for sufficiently small. Hence Y μ Jμ (φ) nv∗ dVolv∗ + K Y (φ) dVol R
{2M − ≤r≤2M }
JμY
≤
(φ) nμv1
{2M − ≤r≤2M }
dVolv1 +
JμY (φ) nμr dVol{r=2M } .
{v1 ≤v≤v∗ }
(16) First, notice that by the assumption of the Proposition JμY (φ) nμr dVol{r=2M } ≤ B max{v∗ − v1 , 1}v1−α . {v1 ≤v≤v∗ }
Second, since is small we have K Y (φ) ≥ cJμY (φ) nμv∗ . Let f (v) =
{2M − ≤r≤2M }
v∗ f (v) +
JμY (φ) nμv dVolv . Then we have
f (ν) dν ≤ C f (v1 ) + B max{v∗ − v1 , 1}v1−α .
v1
This identity resembles (15) and an identical bootstrap would lead to expected decay rate of f . The conclusion of the proposition then follows from plugging this decay rate back into (16). As before, we would then like to prove the L2 decay of φ. We will show that any decay rate in L2 along the event horizon also holds inside the black hole region. More precisely, we have the following: Proposition 22. Suppose on the event horizon that v+1
JμY (φ) nμr + φ2 dVolr=2M ≤ B1 v −α
for all v, for some α ≥ 0.
v
and v+1 1 3−m
JμY ∂tm Ωk φ nμr + φ2 dVolr=2M ≤ Bt−α
m=0 k=0 v
for all v, for some α ≥ 0.
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Then, for some > 0, and for all r ∈ [2M − , 2M ], ⎞ ⎛ v ∗ +1 ⎟ ⎜ φ(r)2 dA dvr ≤ C ⎝B1 + JμN (φ) nμv0 dVolv0 ⎠ v∗−α , v∗
S2 (r)
{2M − ≤r≤r0 }
⎛
⎞
2
⎜ |φ (v∗ , r) |2 ≤ C ⎝B +
(17)
⎟ JμN Ωk φ nμv0 dVolv0⎠ v∗−α ,
k=0{2M − ≤r≤r } 0
(18) and ⎛ ⎜ |∂r∗ φ(v∗ , r)|2 ≤ C ⎝B +
1 3−m m=0 k=0
⎞
⎟ JμN (∂tm Ωk φ)nμv0 dVolv0 ⎠v∗−α ,
{2M − ≤r≤r0 }
(19) for v∗ ≥ 1, r ≤ r˜. Remark 11. The conclusions of Propositions 17 and 20 together imply the hypothesis of this Proposition. Proof. By Proposition 21 and Sobolev Embedding, (17) would imply (18) and (19). We note that when changing from the (v, u) coordinates to the (v, r) coordinates, we have 1 ∂u = −∂r . 1−μ Hence, written in the (v, r) coordinates, we have
JμY
(φ) nμv
r0 dVolv ≥ c
(∂r φ)2 dA drv .
2M − S2
{2M − ≤r≤r0 }
By Proposition 21, the assumption of this proposition implies that JμY (φ) nμv dVolv {2M − ≤r≤r0 }
⎛
⎜ ≤ C ⎝B1 +
{2M − ≤r≤r0 }
⎞ ⎟ JμN (φ) nμv0 dVolv0 ⎠ v −α .
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
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By integrating in the ∂r direction, we have that for any 2M − ≤ r ≤ 2M , v ∗ +1
v∗
φ(r)2 dA dv
S2 (r)
⎛ v +1 ⎞ v ∗ +1 2M ∗ 2 ≤C⎝ φ(2M )2 dA dvr=2M + (∂r φ) dA dr dv ⎠ v∗
S2
v∗
⎛
⎜ ≤ CB1 v∗−α + C ⎝B1 +
2M − S2
⎞
⎟ JμN (φ) nμv0 dVolv0 ⎠ v∗−α .
{2M − ≤r≤r0 }
Once we have the improved decay estimates slightly into the black hole region, we can propagate them to anywhere in the black hole region. As noted before, the problem becomes considerably easier due to the geometry inside the black hole. We will use the facts that −∂r is timelike and nondegenerate and that the energy associated to −∂r would control its deformation K −∂r . The fact that ∂r controls the deformation is in turn implied by the fact that ∂t is Killing. Since we would like to prove an estimate in a finite region of r, these facts would allow us to prove estimates in the whole black hole region (with a constant that degenerates as rb → 0) easily with Gronwall inequality. Proposition 23. Fix rb > 0. Suppose for some r˜ with rb ≤ r˜ < 2M we have v+1
JμY (φ) nμr˜ + φ(˜ r)2 dA dvr˜ ≤ B1 v −α ,
v S2 (˜ r)
and v+1 1 3−m
JμY ∂tm Ωk φ nμr˜ + φ(˜ r)2 dA dvr˜ ≤ Bv −α
m=0 k=0 v S2 (˜ r)
Then we have v ∗ +1
φ2 + (∇φ)
2
dAdvr ≤ Crb B1 v∗−α ,
v∗
and |φ|2 ≤ Crb Bv∗−α
for rb ≤ r ≤ r˜
with ∇ understood as the coordinate derivatives in the regular coordinates (r, v, ω). Remark 12. The hypothesis is implied by the conclusion of Propositions 21 and 22 with the appropriate B and B1 and with tilder = 2M − . Note that
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the following argument requires r < 2M strictly and, therefore, we need Propositions 21 and 22 to push the estimates at the horizon to slightly inside. Proof. Consider the region F = {(rb , v) : v∗ ≤ v ≤ v∗ + 1} which is the part of the r = rb spacelike hypersurface between v∗ and v∗ +1. We would like to determine the domain of dependence D(v∗ ) of this region. To do so, we notice that the hypersurfaces v = constant and u = v − 2r − 4M log (r − 2M ) = constant are the boundary of the past of this region. Hence D(v∗ ) ∩ {rb ≤ r ≤ r˜} ⊂ {|v − v∗ | ≤ C} ∩ {rb ≤ r ≤ r˜}. Now consider the energy with respect to the timelike vector field −∂r together with an L2 term (which is different from the energy we consider above) 2 φ + Jμ−∂r (φ)nμr dVolr , Eb (φ; r, v∗ ) = D(v∗ )∩{r}
where −∂r is taken in the (r, v) coordinate and is future-directed and time like in this region. Thus Eb (φ; r, v∗ ) ≥ D(v∗ ) (φ2 + (∇φ)2 )dVolr ≥ 0, with ∇ understood as the coordinate derivatives in the regular coordinates (r, v, ω). We have, −∂r K (φ) dVolr ≤ C Jμ−∂r (φ)nμr dVolr ≤ CEb (φ; r, v∗ ) D(v∗ )∩{r}
D(v∗ )∩{r}
with the constant C = C(rb ) is independent of v∗ and r as long as rb ≤ r ≤ r˜. This is, first, because ∂r is invariant under the spacelike Killing vector field ∂t and hence the coefficients in both K −∂r (φ) and Jμ−∂r (φ)nμr are independent of t. The constant can also be chosen to be uniform in r because we are in a finite region of r. Moreover, −∂r φ2 dVolr ≤ CEb (φ; r). D
Hence,
⎛ Eb (φ; r) ≤ C ⎝Eb (φ; r˜) +
r˜
⎞ Eb (φ; r ) dr ⎠ .
r
Gronwall inequality implies, since we are in a finite range of r, that Eb (φ; r) ≤ CEb (φ; r˜) ≤ CB1 v∗−α . Therefore, for every rb ≤ r ≤ r˜, v ∗ +1
φ2 + (∇φ)
2
dAdvr ≤ CB1 v∗−α
v∗
since {(r, v) : rb ≤ r ≤ r˜, v∗ ≤ v ≤ v∗ + 1} ⊂ D. The pointwise decay follows from commuting the equation with ∂t and Ω and Sobolev Embedding.
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
871
Figure 4. The region P
8. Proof of the Main Theorems 8.1. Improved Decay for φ (Proof of Main Theorem 1) To prove Main Theorem 1, we proceed in two steps. First, we show that for every t∗ , there exist t1 < t∗ , t ∼ t∗ such that a weighted L2 -norm of φ on the . We then use the estimates for ψ slice {t = t1 } has the desired decay of t−3+δ ∗ to upgrade this to decay estimates for a weighted L2 -norm of φ on the slice {t = t∗ }. We first set up some notation. Fix r1∗ , r2∗ . These are the r1∗ and r2∗ in the statement of Main Theorem 1. In other words, we would like to prove a decay estimate on the fixed compact region r1∗ ≤ r ≤ r2∗ . Let t∗ ≥ 2 (|r1∗ | + |r2∗ |) be −1 the time slice on which we want to show the decay estimate. Let t˜ = (1.1) t∗ ∗ ∗ ∗ and P = {t˜ ≤ t ≤ t∗ , r1 − t∗ + t ≤ r ≤ r2 + t∗ − t} (Fig. 4). Proposition 24. There exist a t1 with t˜ ≤ t1 ≤ t∗ such that 2 2 φ2 + (∂r∗ φ) r−2 (1 − μ) dA dr∗ ≤ Ct−3 ∗ E0 (φ) . P∩{t=t1 }
Proof. By Theorem 5, 2 2 φ2 + (∂r∗ φ) r−2 (1 − μ) dAdr∗ dt P
≤C
˜
˜ X
|K X,w (φ) | +
K Xl ,w
Xl
(φl ) dVol
l
P
≤ C t˜−2 E0 (φ) ≤ Ct−2 ∗ E0 (φ) . Now take t1 such that 2 2 φ2 + (∂r∗ φ) r−2 (1 − μ) dA dr∗ P∩{t=t1 }
= inf
t˜≤t≤t∗ P∩{t=t}
2
φ2 + (∂r∗ φ)
2
r−2 (1 − μ) dA dr∗ ,
872
J. Luk
Ann. Henri Poincar´e
which exists since we are taking the infimum over a compact interval, and note that 2 2 φ2 + (∂r∗ φ) r−2 (1 − μ) dA dr∗ inf t˜≤t≤t∗ P∩{t=t}
−1 ≤ t∗ − t˜ ≤ Ct−1 ∗ ≤
P −3 Ct∗ E0
2
φ2 + (∂r∗ φ)
P 2
φ2 + (∂r∗ φ)
2
r−2 (1 − μ) dA dr∗ dt 2
r−2 (1 − μ) dAdr∗ dt
(φ) .
To upgrade this to an estimate for a generic t, we make two observations about S. First, S is timelike away from the event horizon. Second, S has a weight ∼ t. We can, therefore, integrate from the “good slice” t = t1 to the slice t = t∗ and get the same decay estimate. This is done using integration by parts in the following Proposition. We prove a more general form but the 2 2 reader should keep in mind that we will use f = φ2 +(∂r∗ φ) , g = r−2 (1 − μ) . Proposition 25. Let f = f r∗ , t, ω ∈ S2 , g = g (r∗ ) , P = {t1 ≤ t ≤ t∗ , r1∗ − t∗ + t ≤ r∗ ≤ r2∗ + t∗ − t}. Then f g dA dr∗ + vf g dA dr∗ t∗ P∩{t=t∗ }
P∩{v= 12 (t∗ +r2∗ )}
uf g dA dr∗
+ P∩{u= 12 (t∗ −r1∗ )}
= t1
∗
f g dA dr + 2
+
f g dA dt dr + P
P∩{t=t1 }
∗
r∗ f g dA dt dr∗
P
(Sf ) g dA dt dr∗ .
P
Proof. We change to the variables u, v and integrate by parts, ∗ 1 2 (t∗ −r1 ) t∗ −u
∗
∗ 1 2 (t∗ −r2 )
v (∂v f ) g dA dt dr =
v (∂v f ) g dA dv du + 1 ∗ t1 −u 2 (t∗ −r2 )
P
1 ∗ 2 (2t1 −t∗ −r2 )
∗ 1 2 (t∗ +r2 )
v (∂v f ) g dA dv du t1 −u
=− P
f g dA dt dr∗ −
P
vf ∂r∗ g dA dt dr∗ +
P∩{t=t∗ }
vf g dA dr∗
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
vf g dA dr∗ ,
vf g dA dr −
+ P∩{v=t∗ +r2∗ }
∗
873
P∩{t=t1 } ∗ 1 2 (t∗ +r1 )
u (∂u f ) g dA dt dr∗ =
P
∗ 1 2 (t∗ −r1 )
u (∂u f ) g dA du dv t1 −v
1 ∗ 2 (2t1 −t∗ +r1 ) ∗ 1 2 (t∗ +r2 ) t∗ −v
+
u (∂u f ) g dA du dv 1 ∗ t1 −v 2 (t∗ +r1 )
=−
f g dA dt dr∗ +
P
P
P∩{u=t∗ −r1∗ }
uf g dA dr∗
P∩{t=t∗ }
uf g dA dr∗ −
+
uf ∂r∗ g dA dt dr∗ +
uf g dA dr∗ .
P∩{t=t1 }
The proposition is proved by adding these two equations.
To prove the main theorem, we use the above identity using f = φ2 + 2 2 (∂r∗ φ) , g = r−2 (1 − μ) . We notice that since f, g ≥ 0 by definition and u, v ≥ 0 in P = {t1 ≤ t ≤ t∗ , r1∗ − t∗ + t ≤ r∗ ≤ r2∗ + t∗ − t}. Therefore, vf g dA dr∗ + uf g dA dr∗ ≥ 0. P∩{v= 12 (t∗ +r2∗ )}
P∩{u= 12 (t∗ −r1∗ )}
Thus Proposition 23 would imply 2 2 φ2 + (∂r∗ φ) r−2 (1 − μ) dA dr∗ t∗ P∩{t=t∗ }
≤ t1
2
φ2 + (∂r∗ φ)
P∩{t=t1 }
+2 P
+2
2
φ2 + (∂r∗ φ)
2
r−2 (1 − μ) dA dr∗ 2
r−2 (1 − μ) dA dr∗ dt
2 2 r−2 (1 − μ) | dt dr∗ |r∗ φ2 + (∂r∗ φ)
P
+
2
(|ψφ| + |∂r∗ ψ∂r∗ φ|) r−2 (1 − μ) dA dt dr∗
P
≤
Ct−2 ∗ E0
(φ) + P
ψ 2 + (∂r∗ ψ)
2
2
r−2 (1 − μ) dA dt dr∗
874
J. Luk
+ ≤
2
φ2 + (∂r∗ φ)
P Ct−2+δ E1 ∗
Ann. Henri Poincar´e 2
r−2 (1 − μ) dA dt dr∗
(φ) ,
where we have used Proposition 22 at the second to last step and Theorems 5 and 10 at the last step. Therefore, ∗
r2
2
φ (t∗ ) + (∂r∗ φ (t∗ ))
2
dA dr∗ ≤ Ct−3+δ E1 (φ) . ∗
r1∗
Since ∂t , Ω are Killing, it follows immediately that r∗
1 2
2 ∂tm ∂rl ∗ φ (t∗ ) + |∇ / k ∂rl ∗ φ (t∗ ) |2 dA dr∗ ≤ Ct−3+δ E1 ∂tm Ωk φ , ∗
l=0 r ∗ 1
for any k, m. Using the equation g φ = 0, we get ∗
r2
∇l φ (t∗ )
2
dVol ≤ Ct−3+δ ∗
E1 ∂tm Ωk φ .
k+m≤l
r1∗
We have thus established the decay of the nondegenerate energy in the exterior region away from the event horizon. The full decay of nondegenerate energy part of Main Theorem 1 follows from Propositions 17 and 20–23. The pointwise decay part of Main Theorem 1 follows form the standard Sobolev Embedding Theorem for the part of the exterior region away from the event horizon and from Propositions 17, 20–23 for the region near the event horizon and inside the black hole. 8.2. Improved Decay for ∂t φ (Proof of Main Theorem 2) To estimate the time derivatives of φ, we follow an idea of Klainerman and Sideris [16]. The key observation is that the first derivatives of ∂t φ are con1 trolled with a weight of t−r ∗ by a linear combination of first derivatives of φ and ψ. This extra weight would give extra decay to ∂t φ. ∗
Proposition 26. Suppose t + r∗ ≥ max{ 2t , |r2 | }. (This is true for example when r∗ is bounded below and t is sufficiently large.) 1. |(t − r∗ )∂t2 φ| ≤ C(|∂t ψ| + |∂r∗ ψ| + |∂t φ| + |∂r∗ φ| + (1 − μ)|r∗ ||Δ / φ|), / φ|), 2. |(t − r∗ )∂r∗ ∂t φ| ≤ C(|∂t ψ| + |∂r∗ ψ| + |∂t φ| + |∂r∗ φ| + (1 − μ)|r∗ ||Δ / ψ| + |∂r∗ Ωφ|. 3. |t(1 − μ)∇ / ∂t φ| ≤ C((1 − μ)|∇ Proof. Define Δg φ = (1 − μ) (1 − μ) ∂t2 φ = Δg φ. Recall that
−1
∂r∗ φ + 2r ∂r∗ φ + Δ / φ. Then g φ = 0 reads
ψ = t∂t φ + r∗ ∂r∗ φ.
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Therefore, ∂t ψ − ∂t φ = t∂t2 φ + r∗ ∂r∗ ∂t φ, ∂r∗ ψ − ∂r∗ φ = r∗ ∂r2∗ φ + t∂r∗ ∂t φ. Hence, t (∂t ψ − ∂t φ) − r∗ (∂r∗ ψ − ∂r∗ φ) 2
= t2 ∂t2 φ − (r∗ ) ∂r2∗ φ 2 2 = t2 − (r∗ ) ∂t2 φ + (r∗ ) (1 − μ) Δg φ − ∂r2∗ φ 2 (1 − μ) 2 2 ∂r∗ φ + (1 − μ) Δ /φ . = t2 − (r∗ ) ∂t2 φ + (r∗ ) r Therefore, by re-arranging and dividing by (t + r∗ ), | (t − r∗ ) ∂t2 φ| 1 (t (∂t ψ − ∂t φ) − r∗ (∂r∗ ψ − ∂r∗ φ) = t + r∗ 2 (1 − μ) ∗ 2 ∂r∗ φ + (1 − μ) Δ − (r ) / φ) r ≤ C (|∂t ψ| + |∂r∗ ψ| + |∂t φ| + |∂r∗ φ| + (1 − μ) |r∗ ||Δ / φ|) . We have thus proved 1. On the other hand, using again the above equality, we also have (t − r∗ ) ∂r∗ ∂t φ = −∂t ψ + ∂r∗ ψ + ∂t φ − ∂r∗ φ + t∂t2 φ − r∗ ∂r2∗ φ
= −∂t ψ + ∂r∗ ψ + ∂t φ − ∂r∗ φ + (t − r∗ ) ∂t2 φ + r∗ (1 − μ)Δg φ − ∂r2∗ φ
= −∂t ψ + ∂r∗ ψ + ∂t φ − ∂r∗ φ + (t − r∗ )∂t2 φ 2r∗ (1 − μ) ∂r∗ φ + (1 − μ)|r∗ |Δ / φ. + r This, together with 1, implies 2. The proof of 3 is more direct. Using the definition of S, and that Ω is independent of t and r∗ , Ωψ = r∗ ∂r∗ Ωφ + t∂t Ωφ. Thus, by noting that Ω and r∇ / differ only by constant, |t (1 − μ) ∇ / ∂t φ|
r∗ ∂r∗ Ωφ| r ≤ C ((1 − μ) |∇ / ψ| + |∂r∗ Ωφ|) .
≤ | (1 − μ) ∇ / ψ| + | (1 − μ)
This would easily imply Main Theorem 2.2:
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Corollary 27.
1 1 ct∗ m k μ T −4+δ Jμ (∂t φ) nt∗ dVolt∗ ≤ Ct∗ E0 ∂t Ω φ + E1 (φ) , m=0 k=0
r˜∗
for all c < 1 and r˜. In particular, r˜ can be chosen as that given by Proposition 19. Proof. We can consider t∗ large enough so that first, the assumption of Proposition 24 holds and second, on the domain of integration, (t∗ − r∗ ) ∼ t∗ . ct∗ JμT (∂t φ) nμt∗ dVolt∗ r˜∗
ct∗ =
2 2 (∂r∗ ∂t φ) + ∂t2 φ + (1 − μ) |∇ / ∂t φ|2 dVolt∗
r˜∗
≤
Ct−2 ∗
ct∗
2
2
2
r˜∗ 2
+ (1 − μ) |∇ / Ω∂t φ|2 + (∂r∗ Ωφ) ≤
Ct−2 ∗
2
/ ψ|2 + (∂t φ) + (∂r∗ φ) (∂t ψ) + (∂r∗ ψ) + (1 − μ) |∇ dVolt∗
ct∗ J T (ψ) + J T (φ) + J T (Ωφ) + J T (∂t Ωφ) dVolt∗ . r˜∗
The corollary follows from Theorems 5 and 10. Finally, we proceed to the proof of Main Theorems 2.1 and 2.3: Corollary 28. We have v ∗ +1
2
2
(∂t φ) + (∇∂t φ)
dA dvr ≤ Cδ,rb v∗−4+δ E3 (φ) ,
for any r ≥ rb .
v∗
and
2
|∂t φ (v∗ ) | ≤
Cv∗−4+δ
2 4−m
E0 ∂tm Ωk φ
+
m=0 k=0
2 2−m
E1 ∂tm Ωk φ
,
m=0 k=0
if rb ≤ r ≤ 2M or r∗ ≤
t∗ . 2
Proof. We provea Sobolev-type inequality. We first work on R3 . We claim ∞ 3 that for u ∈ Cc R , 1
1
2 2 ||u||L∞ (R3 ) ≤ C||u||H ˙ 1 (R3 ) ||u||H ˙ 2 (R3 ) .
We give a simple proof using Littlewood–Paley theory. Let N ∈ 2Z be a dyadic number, χ (ξ) be a radial cutoff function which is supported in {|ξ| < 2} and
Vol. 11 (2010)
Improved Decay for Solutions to the Linear Waves
877
is identically 1 in {|ξ| < 1}. Define the Littlewood–Paley operators PN by ξ 2ξ P u = χ − χ u ˆ. N N
N
Since the inequality claimed is invariant under scaling u (x) → λu (x) and u (x) → u (λx), we can assume that ||u||H˙ 1 (R3 ) = ||u||H˙ 2 (R3 ) = 1. Then, by Bernstein inequality, ||PN u||L2 (R3 ) ≤ min{CN −1 , CN −2 }. Therefore, by Bernstein inequality again,
||u||L∞ (R3 ) ≤ C
⎛
N ||PN u||L2 (R3 ) ≤ C ⎝ 3 2
N
N
− 12
+
⎞ N ⎠ ≤ C. 1 2
N
N ≥N0
We note that a variant of this is true. We have for u ∈ Cc∞ R3 , 1
2 ||u||L∞ (R3 \Br (0)) ≤ C||u||H ˙ 1 (R3 \B
1
r
2 ||u||H ˙ 2 (R3 \B (0))
r (0))
.
This is true because one can extend u into Br (0) without increasing the H˙ 1 or H˙ 2 norm. We now apply this to a cutoff version of ∂t φ. ∗ 1 |x| ≤ 1 r φ for r ≥ r˜, where r˜ Let χ = . On t = t∗ , let φ˜ = χ 0.5t ∗ 0 |x| ≥ 1.1 is as in Proposition 19. The H˙ 1 norm is controlled with Corollary 23 and Theorem 5.4. ˜ ˙1 3 ||∂t φ|| H (R \Br˜ (0)) 0.55t∗ 2
0.55t∗ 2
JμT (∂t φ) nμt∗ dV olt∗ + Ct−2 ∗
≤C r˜∗
≤ Ct−4+δ ∗
r˜∗ 1
1
2
(∂t φ) r2 (1 − μ) dA dr∗
E0 ∂tm Ωk φ + E1 (φ) .
m=0 k=0
The H˙ 2 norm can be controlled similarly once we note that using the equation, we have for r ≥ r˜, 1 |∂r2∗ ∂t φ| ≤ C |∂t3 φ| + |∂r∗ ∂t φ| + |∇ / 2 ∂t φ| r 3 ≤ C |∂t φ| + |∂r∗ ∂t φ| + |∇ / Ω∂t φ| .
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Therefore, for t = t∗ , ˜ ˙2 3 ||∂t φ||
H (R \Br˜ (0)) 0.55t∗ 2
≤C
JμT ∂t2 φ + JμT (∂t Ωφ) + JμT (∂t φ) nμt∗ dVolt∗
r˜∗ 0.55t∗ 2
+ Ct−2 ∗
∂t2 φ
2
2
2
+ (∂t Ωφ) + (∂t φ)
r2 (1 − μ) dA dr∗
r˜∗
≤ Ct−4+δ ∗
2 2
E0 ∂tm Ωk φ +
m=0 k=0
1 1
E1 ∂tm Ωk φ
.
m=0 k=0
Therefore, ||∂t φ||L∞ ({˜r∗ ≤r∗ ≤ t∗ }) 2
˜ L∞ (R3 \B (0)) ≤ ||∂t φ|| r ˜ 1
1
˜ 2 ˜ 2 ≤ C||∂t φ|| ˙ 1 (R3 \Br˜ (0)) ||∂t φ||H ˙ 2 (R3 \Br˜ (0)) H
2 2 1 1 m k m k −4+δ ≤ Ct∗ E0 ∂ t Ω φ + E1 ∂ t Ω φ . m=0 k=0
m=0 k=0
In particular, for t sufficiently large, this L∞ estimate holds on sets of compact r∗ . Noting that the L∞ norm controls the L2 norm on compact sets, we have ∗ ((1.2)r 0)
2
(∂t φ) dVolt∗ r˜
≤
Ct−4+δ ∗
2 2
E0 ∂tm Ωk φ
+
m=0 k=0
1 1
E1 ∂tm Ωk φ
.
m=0 k=0
We also have, by Corollary 23, 1 3−m m=0 k=0
∗ ((1.2)r 0)
JμT ∂tm Ωk (∂t φ) nμt dVolt∗
r˜
≤ Ct−4+δ ∗
2 4−m
m=0 k=0
E0 ∂tm Ωk φ +
1 4−m
E1 ∂tm Ωk φ
.
m=0 k=0
The corollary then follows from the Sobolev Embedding Theorem and Propositions 17, 19, 20, 21, 22 and 23.
Acknowledgements The author thanks his advisor Igor Rodnianski for suggesting the problem and for sharing numerous insights. He thanks Mihalis Dafermos, Gustav Holzegel
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and Igor Rodnianski for very helpful comments on preliminary versions of the manuscript.
References [1] Alinhac, S.: Energy multipliers for perturbations of Schwarzschild metric. Commun. Math. Phys. 288(1), 199–224 (2009) [2] Blue, P., Soffer, A.: Semilinear wave equations on the Schwarzschild manifold. I. Local decay estimates (see also Errata for“Global existence and scattering for the nonlinear Schrodinger equation on schwarzschild manifolds”, “Semilinear wave equations on the Schwarzschild manifold i: Local decay estimates”, and “the wave equation on the Schwarzschild metric ii: local decay for the spin 2 Regge Wheeler equation”, arxiv:gr-qc/0608073). Adv. Differ. Equ. 8(5), 595–614 (2003). arXiv:gr-qc/0310091 [3] Blue, P., Soffer, A.: Improved decay rates with small regularity loss for the wave equation about a Schwarzschild black hole (2006). arXiv:math.AP/0612168 [4] Blue, P., Sterbenz, J.: Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space. Commun. Math. Phys. 268(2), 481–504 (2006). arXiv:math.AP/0510315 [5] Christodoulou, D., Klainerman, S.: The global nonlinear stability of the Minkowski space. Princeton University Press, New Jersey (1993) [6] Dafermos, M.: The interior of charged black holes and the problem of uniqueness in general relativity. Commun. Pure Appl. Math. 58(4), 445–504 (2005) [7] Dafermos, M., Rodnianski, I.: A proof of Price’s law for the collapse of a selfgravitating scalar field. Invent. Math. 162, 381–457 (2005). gr-qc/0309115 [8] Dafermos, M., Rodnianski, I.: A note on energy currents and decay for the wave equation on a Schwarzschild background (2007). arXiv:math.AP/0710.0171 [9] Dafermos, M., Rodnianski, I.: A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds (2008). arXiv:grqc/0805.4309 [10] Dafermos, M., Rodnianski, I.: Lectures on black holes and linear waves (2008). arXiv:gr-qc/0811.0354 [11] Dafermos, M., Rodnianski, I.: The red-shift effect and radiation decay on black hole spacetimes. Commun. Pure Appl. Math. (2009). arXiv:gr-qc/0512119 [12] Donninger, R., Schlag, W., Soffer, A.: A proof of Price’s Law on Schwarzschild black hole manifolds for all angular momenta (2009). gr-qc/0908.4292 [13] Kay, B.S., Wald, R.M.: Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation 2-sphere. Class. Quantum Gravity 4(4), 893–898 (1987) [14] Klainerman, S.: The null condition and global existence to nonlinear wave equations. In: Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984). Lectures in Applied Mathematics, vol. 23, pp. 293–326. American Mathematical Society, Providence (1986) [15] Klainerman, S.: Remarks on the global Sobolev inequalities in the Minkowski space Rn+1 . Commun. Pure Appl. Math. 40(1), 111–117 (1987) [16] Klainerman, S., Sideris, T.: On almost global existence for nonrelativistic wave equations in 3D. Commun. Pure Appl. Math. 49(3), 307–321 (1996)
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[17] Kronthaler, J.: Decay rates for spherical scalar waves in the schwarzschild geometry (2007). gr-qc/0709.3703 [18] Kruskal, M.D.: Maximal extension of Schwarzschild metric. Phys. Rev. 119(2), 1743–1745 (1960) [19] Laba, I., Soffer, A.: Global existence and scattering for the nonlinear Schrodinger equation on Schwarzschild manifolds. Helv. Phys. Acta 72(4), 272–294 (1999) [20] Lindblad, H., Rodnianski, I.: Global existence for the Einstein vacuum equations in wave coordinates. Commun. Math. Phys. 256(1), 43–110 (2005). arXiv:math.AP/0312479v2 [21] Machedon, M., Stalker, J.: Decay of solutions to the wave equation on a spherically symmetric background (2010, preprint) [22] Marzuola, J., Metcalfe, J., Tataru, D., Tohaneanu, M.: Strichartz estimates on Schwarzschild black hole backgrounds (2008). arXiv:math.AP/0802.3942 [23] Morawetz, C.S.: Notes on time decay and scattering for some hyperbolic problems. In: Society for Industrial and Applied Mathematics, Philadelphia. Regional Conference Series in Applied Mathematics, vol. 19 (1975) [24] Synge, J.L.: The gravitational field of a particle. Proc. R. Irish Acad. Sect. A 53, 83–114 (1950) [25] Wald, R.M.: Note on the stability of the Schwarzschild metric. J. Math. Phys. 20(6), 1056–1058 (1979) Jonathan Luk Department of Mathematics Princeton University Princeton, NJ 08544 USA e-mail:
[email protected] Communicated by Piotr T. Chrusciel. Received: October 12, 2009. Accepted: March 13, 2010.
Ann. Henri Poincar´e 11 (2010), 881–927 c 2010 Springer Basel AG 1424-0637/10/050881-47 published online October 17, 2010 DOI 10.1007/s00023-010-0049-0
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Asymptotic Gluing of Asymptotically Hyperbolic Solutions to the Einstein Constraint Equations James Isenberg, John M. Lee and Iva Stavrov Allen Abstract. We show that asymptotically hyperbolic solutions of the Einstein constraint equations with constant mean curvature can be glued in such a way that their asymptotic regions are connected.
1. Introduction One of the most useful ways to produce new solutions of the Einstein constraint equations is via gluing techniques. The standard gluing construction is the following: we presume that (M, g, K) is an Einstein initial data set, with M a smooth n-dimensional manifold, g a Riemannian metric on M , and K a symmetric tensor field on M . We further assume that this set of data satisfies the (vacuum) Einstein constraint equations (with a possibly nonzero cosmological constant Λ): divg K − ∇ Trg K = 0, R(g) −
|K|2g
2
+ (Trg K) = 2Λ,
(1) (2)
which are the necessary and sufficient conditions for (M, g, K) to generate a spacetime solution of the (vacuum) Einstein gravitational field equations via the Cauchy problem [5]. (Here divg is the divergence operator defined as the trace of the covariant derivative on the last two indices, Trg is the trace operator, R(g) is the scalar curvature, and | · |g is the tensor norm, all computed with respect to the metric g.) Choosing a pair of points p1 , p2 ∈ M , one shows that there is a family of new solutions (Mε , gε , Kε ) of the constraint equations indexed by a small positive parameter ε in which (i) Mε is obtained from M by (connected sum) surgery joining p1 and p2 and (ii) outside of a neighborhood of the connected Research supported in part by NSF grant DMS-0406060 at Washington and NSF grant PHY-0652903 at Oregon.
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sum bridge in Mε , the data (gε , Kε ) can be made as close as desired to (g, K) (in a sense to be made precise later) by taking ε sufficiently small. We note that this gluing construction allows for the possibility that the manifold M consists of two disconnected components; then if p1 is chosen to lie in one of the components and p2 in the other, the new glued solution effectively connects two disconnected solutions of the constraint equations. The mathematics and the utility of the gluing of solutions of the Einstein constraint equations are discussed in a series of papers [10,13–15], which show that gluing can be carried out for a wide variety of initial data sets: they can be compact, asymptotically Euclidean (“AE”), or asymptotically hyperbolic (“AH”), and they can be vacuum solutions or non-vacuum solutions with various coupled matter fields. This past work shows that in some cases the gluing can be done so that the glued solution exactly matches the original one outside the gluing region, so long as certain nondegeneracy conditions (“no KIDS”) hold at the points of gluing. When this can be done, the gluing is said to be localized. More generally, the glued solutions may not exactly match the original ones outside the gluing region, but can be constructed so that the data set is arbitrarily close to the original solution away from this region; this is called non-localized gluing. Say one chooses a pair of (disjoint) asymptotically hyperbolic solutions of the constraints and glues them at a pair of points satisfying the necessary conditions, as described in either [14] or [10]. If each of the original AH data sets has a single (connected) asymptotic region (as described below in Sect. 2.1), then the glued data set, which is also asymptotically hyperbolic, necessarily has two disjoint asymptotic regions. If we are working with AH initial data sets which are viewed as data on partial Cauchy surfaces that intersect null infinity in an asymptotically simple spacetime [20], then the existence of multiple asymptotic regions is problematic for physical modeling. In the present paper, we show that one can glue asymptotically hyperbolic solutions of the constraint equations in such a way that in fact the asymptotic region of the glued data set is connected. The idea, which is modeled after the studies of Mazzeo and Pacard on gluing asymptotically hyperbolic Einstein manifolds [19], is to use the conformally compactified representation of asymptotically hyperbolic geometries, which models a complete, asymptotically hyperbolic manifold as the interior of a compact manifold with boundary (see Sect. 2.1 below). The boundary of this manifold, which we call the ideal boundary, is not part of the physical initial manifold, but represents asymptotic directions at infinity. The gluing is done using points p1 and p2 lying on the ideal boundary. In this context, we will use the term asymptotic region to refer to any open collar neighborhood of the boundary, with the boundary itself deleted. If the original manifold has a single connected asymptotic region, then so does the glued manifold. (See Fig. 1.) If the original manifold has two disjoint connected asymptotic regions with one boundary point chosen on the ideal boundary of each region, then the glued manifold will have a single connected asymptotic region (Fig. 2). (Although the ideal boundary in Fig. 1 appears to be disconnected, in the cases of interest the boundary of
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Figure 1. Two points in the same ideal boundary component
Figure 2. Two points in different ideal boundary components the glued manifold will be a connected sum of connected 2-manifolds, which is always connected.) The results we present here do not hold for general AH initial data sets. We require that the data have constant mean curvature (“CMC”), in the sense that Trg K is constant on M . Also, our results thus far provide sufficient conditions for (asymptotic) non-localized gluing. In a future paper, we hope to both eliminate the CMC restriction, and find conditions which are sufficient for the gluing to be localized. Our results here do hold for initial data sets of any spatial dimension n ≥ 3. Note, however, that for the convenience of the exposition, many of our equations and calculations are written in the form appropriate for 3 dimensions rather than general dimensions. In or near the statements of our main propositions and theorems, we shall make clear what modifications of the hypotheses, if any, are needed so that the results apply for general n; we also indicate
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where appropriate modifications are needed to generalize certain statements and calculations to the n dimensional case. To set up our work here, we start in Sect. 2 with a definition and discussion of asymptotically hyperbolic initial data and their polyhomogeneous behavior in the asymptotic region. The section continues with a brief description of the conformal method for generating solutions of the constraint equations and the simplifications of the method which occur for constant mean curvature data. The conformal method is discussed here because it plays an important role both in constructing basic examples of AH initial data [1] and in carrying out our gluing procedure. We conclude Sect. 2 with an overview of those aspects of [1] on which our work relies. We state our main theorem in Sect. 3. We then carry out the first part of the gluing construction (“splicing”) in Sect. 4, producing a family of initial data sets depending on a small parameter ε, which satisfy the constraint equations approximately. Our splicing construction is modeled after the construction of asymptotically hyperbolic Einstein metrics in [19]. The proof of the main theorem relies on the use of weighted H¨ older spaces. We define and discuss these spaces in Sect. 5, following [16]. In the rest of Sect. 5, we study certain elliptic operators which act on weighted H¨ older spaces, and prove that they are invertible with norms of their inverses bounded uniformly in ε. One of the key steps here is a blow-up analysis argument similar to that of [19]. Our analytical results are applied in Sect. 6 to correcting the traceless part of the spliced second fundamental form. Finally, in Sect. 7 we use results of [12] and a contraction mapping argument to solve the Lichnerowicz equation and complete the proof of the main theorem. Section 8 contains concluding remarks.
2. Preliminaries In this section we define and discuss examples of asymptotically hyperbolic initial data sets. We also review the conformal method for creating solutions of the constraint equations. 2.1. Asymptotically Hyperbolic Initial Data Sets There are two principal models for asymptotically hyperbolic initial data sets. One is the hyperboloid in (n + 1)-dimensional Minkowski space, ˘ = {(x0 , x1 , . . . , xn ) : x0 > 0 M
and
(x0 )2 = (x1 )2 + · · · + (xn )2 + 1},
together with its induced Riemannian metric g˘ (which is a model for the hyperbolic metric of constant sectional curvature −1) and its extrinsic curvature ˘ ≡ g˘; it is straightforward to check that (M ˘ , g˘, K) ˘ satisfies (1) and (2) K with Λ = 0. The other is a totally geodesic spacelike hypersurface M in the (n + 1)-dimensional anti-de Sitter spacetime with constant sectional curvature −1, together with its induced Riemannian metric g and its extrinsic curvature K ≡ 0; another straightforward computation shows that (M , g , K ) satisfies (1) and (2) with Λ = −n(n − 1)/2.
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Roughly speaking, an asymptotically hyperbolic initial data set (M, g, K) is one in which the Riemannian metric g is complete and approaches constant negative curvature as one approaches the ends of the manifold, and the extrinsic curvature K asymptotically approaches a pure trace tensor field that is a constant (possibly zero) multiple of the metric. A more precise definition of asymptotic hyperbolicity is motivated by the Poincar´e disk model of hyperbolic space. In that model, hyperbolic space is given by the (smooth) metric g=
4 ((dx1 )2 + · · · + (dxn )2 ), (1 − |x|2 )2
(3)
on the open unit ball. If we define the function ρ = 12 (1 − |x|2 ), then g = ρ2 g is the Euclidean metric and thus extends smoothly to the closure of the open ball. Note that the function ρ is smooth on the closed ball, it picks out the boundary of the ball since ρ−1 (0) is equal to this boundary, and it satisfies the derivative condition |dρ|g = 12 d(1 − |x|2 )g = 1 on the boundary of the ball. With this example in mind, we make the following definitions: We suppose throughout this paper that M is the interior of a smooth, compact manifold with boundary M . A defining function for M is a nonnegative real-valued function ρ : M → R of class at least C 1 such that ρ−1 (0) = ∂M and dρ does not vanish on ∂M . Given a nonnegative integer k and a real number α ∈ [0, 1], a smooth Riemannian metric g on M is said to be conformally compact of class C k (or C k,α , or C ∞ ) if there exists a smooth defining function ρ and a Riemannian metric g on M of class C k , C k,α , or C ∞ , respectively, with g = ρ−2 g|M . Smoothly conformally compact means the same as conformally compact of class C ∞ . If g is conformally compact and if in addition |dρ|g = 1 on ∂M , then g is said to be asymptotically hyperbolic (of class C k , C k,α , or C ∞ , as appropriate). One verifies easily that any asymptotically hyperbolic metric of class at least C 2 has sectional curvatures approaching −1 near ∂M (see [18]), and so indeed has the intuitive properties mentioned above. The boundary ∂M is called the ideal boundary, and ∂M together with its induced metric ι∗ g (where ι : ∂M → M is inclusion), is called conformal infinity. Note that for a given AH metric, the choice of the defining function is not unique, so the geometry of the conformal infinity is only defined up to a conformal factor. Naively, one might hope to work with initial data sets (M, g, K) in which (M, g) is a smoothly asymptotically hyperbolic Riemannian manifold, and K is a symmetric 2-tensor field such that ρ2 K has a smooth extension to M and is equal to a constant multiple of the metric on the ideal boundary. Unfortunately, however, it is shown in [2] that there are obstructions to finding solutions with this degree of smoothness, marked by the presence of log terms in the asymptotic expansions of g and K near the ideal boundary. For this reason, instead of smoothness we have to settle for a slightly weaker notion called polyhomogeneity, which we now define. A function f : M → R is said to be polyhomogeneous (cf. [18]) if it is smooth in M , and there exist a sequence of real numbers si +∞, a sequence
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of nonnegative integers {qi }, and functions f ij ∈ C ∞ (M ) such that f∼
qi ∞
ρsi (log ρ)j f ij
(4)
i=1 j=0
in the sense that for any positive integer K, there exists a positive integer N such that the difference f−
qi N
ρsi (log ρ)j f ij
i=1 j=0
is O(ρK ) as ρ → 0, and remains O(ρK ) after being differentiated any number of times by smooth vector fields on M that are tangent to ∂M . It is easy to check that sums and products of polyhomogeneous functions are polyhomogeneous, as are quotients of polyhomogeneous functions provided that the denominator has no log terms with its lowest power of ρ (i.e., q1 = 0) and provided that its leading term f 10 does not vanish on the ideal boundary. A tensor field on M is said to be polyhomogeneous if it is smooth on M and its component functions are polyhomogeneous in some smooth coordinate chart in a neighborhood of every ideal boundary point. We define a polyhomogeneous asymptotically hyperbolic Riemannian metric on M to be a polyhomogeneous Riemannian metric g which is also conformally compact of class at least C 2 . Now we come to the main definition of this section. Definition 1. A polyhomogeneous asymptotically hyperbolic initial data set for the vacuum Einstein equations with cosmological constant Λ (sometimes called a hyperboloidal initial data set) is a triple (M, g, K), in which (i) (ii) (iii)
(iv)
M is the interior of a smooth, compact manifold with boundary M ; g is a polyhomogeneous AH Riemannian metric on M ; K is a polyhomogeneous symmetric covariant 2-tensor field on M with the property that for any smooth defining function ρ, ρ2 K has a C 2 extension to M whose restriction to ∂M is a constant multiple of (the extension of) ρ2 g there; the Einstein constraint equations (1) and (2) are satisfied.
For Λ = 0, AH initial data sets are also called asymptotically (anti-)de Sitter initial data sets. Our definition implies that g and K can be written in the form (5) g = ρ−2 g, τ −1 (6) K = g + ρ μ, n where g is a polyhomogeneous Riemannian metric on M that is of class at least C 2 (and thus has log terms, if any, only with powers of ρ greater than 2); τ is a polyhomogeneous C 2 scalar function on M whose restriction to ∂M is a constant; and μ is a polyhomogeneous symmetric 2-tensor field on M that is trace-free with respect to g and has log terms only with powers of ρ greater
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than 1. A polyhomogeneous AH initial data set is defined to be CMC if the mean curvature function τ = Trg K is constant. We note that for asymptotically hyperbolic CMC data, the value of the constant τ (up to sign) is determined by the cosmological constant. Indeed, the definition of CMC AH data guarantees that the scalar curvature approaches the constant −n(n − 1) at the ideal boundary. Since the g-norm of ρ−1 μ approaches zero at the ideal boundary, it follows from (6) that we must have |K|2g → τ 2 /n and (Trg K)2 → τ 2 . Inserting these relations into (2) and evaluating in the limit at the ideal boundary, we see that the mean curvature τ must satisfy 2nΛ . (7) τ 2 = n2 + n−1 Note that this relation imposes a restriction on Λ: n(n − 1) . 2 There are two special cases of interest: The case Λ = 0, τ = n is modeled by the hyperboloid in Minkowski space; and the case Λ = −n(n−1)/2, τ = 0 is modeled by a totally geodesic spacelike hypersurface in anti-de Sitter space. It follows from the work of Andersson and Chru´sciel in [1] that polyhomogeneous asymptotically hyperbolic initial data sets (for arbitrary Λ ≥ −n(n − 1)/2) exist in abundance; we review this work in Sect. 2.3. Λ≥−
2.2. The Conformal Method for Finding Solutions of the Constraint Equations Both the construction of AH initial data sets in [1] and our gluing construction here are based on the conformal method, which (along with the closely related conformal thin sandwich method) is the most widely used procedure for producing solutions of the constraint equation. We proceed by reviewing this method here. Note that, in this section especially, while the stated constructions and results work in general dimensions, it is convenient to write the numerical exponents and coefficients in the form appropriate for n = 3 dimensions. The modifications needed for the general case appear, for example, in [6] and [13]. We start by introducing two auxiliary differential operators which are involved in the conformal method. The first of the two is the conformal Killing operator Dλ which depends on a Riemannian metric λ and acts on vector fields X as follows: 1 1 1 1 (Dλ X)cd := LX λcd − (divλ X)λcd = (∇c Xd + ∇d Xc ) − ∇a X a λcd . 2 3 2 3 (8) The image of Dλ is contained in the space of symmetric 2-tensors which are traceless with respect to λ. The formal adjoint of Dλ is Dλ∗ T = − (divλ T ) ;
(9)
here (and throughout the paper) the symbol refers to the raising of an index. Another auxiliary operator we use is the elliptic, formally self-adjoint vector
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Laplacian operator Lλ X = (Dλ∗ ◦ Dλ )X = − (divλ (Dλ X)) .
The conformal method for 3-dimensional manifolds is based on the Lichnerowicz–York decomposition of data [7] gab = ψ 4 λab ,
(10) 1 (11) Kcd = ψ −2 (νcd + 2(Dλ W )cd ) + ψ 4 τ λcd , 3 where λab is a Riemannian metric, νcd is a symmetric tensor field that is divergence-free and trace-free with respect to λab , τ is a scalar function, ψ is a positive definite scalar function, and W c is a vector field. Substituting the field decompositions (10)–(11) into the vacuum constraint equations (1)–(2) and using standard conformal transformation formulas for the scalar curvature and for divergences, we obtain 1 6 ψ ∇c τ, 3 1 1 τ 2 − 3Λ 5 Δλ ψ = R(λ)ψ − |νcd + 2(Dλ W )cd |2λ ψ −7 + ψ . 8 8 12
(Lλ W )c =
(12) (13)
(Here Δλ denotes the scalar Laplace operator with respect to the metric λ; our convention is Δλ = divλ ◦ gradλ . See Eqs. (6), (7), (12), and (13) of [6] for the analogous formulas in general dimensions. Note, however, that in Eq. (12) of that reference, “4V ” should be replaced by “2V ”.) The idea of the conformal method is to choose any conformal data (M, λ, ν, τ ) in which ν is traceless and divergence-free with respect to λ, and then use the coupled PDE system (12)–(13) to solve for the determined data (ψ, W ). If, for a given set of conformal data, one can solve the system (12)–(13), then the initial data fields obtained by recomposing the fields as in (10)–(11) provide a solution to the constraint equations. To execute such a construction of initial data, one needs to have a symmetric, traceless, and divergence-free tensor ν. There is a standard method for finding such a tensor [21]. The idea behind the method is to start with an arbitrary traceless symmetric 2-tensor field μ and then find a vector field X which satisfies Lλ X = (divλ μ) .
(14)
Using (9) one easily verifies that ν := μ + Dλ X is symmetric, traceless and divergence-free. There have been extensive studies to determine which sets of conformal data lead to solutions, and which do not. (See [4] for a recent review.) This issue is best understood for conformal data with constant τ , which leads to initial data with constant mean curvature (“CMC”). In this case, the constraint equations (12)–(13) effectively decouple—the (unique) solution to (12) is Dλ W = 0—and one need only analyze the solvability of the (remaining) Lichnerowicz equation (13). In light of (7), which in dimension 3 reduces to
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τ 2 = 9 + 3Λ, the Lichnerowicz equation becomes 1 1 3 Δλ ψ − R(λ)ψ + |νcd |2λ ψ −7 − ψ 5 = 0. (15) 8 8 4 A similar cancellation occurs in general dimensions: one uses (7) to eliminate τ and Λ from the Lichnerowicz equation, making the coefficients and the exponents of the equation dependent only on the dimension n, the metric λ, and the tensor ν. The issue is also fairly well understood for “near CMC” conformal data sets, which are characterized by |∇τ |λ being sufficiently small compared with τ . 2.3. The Conformal Method and Andersson–Chru´sciel Initial Data Sets The Andersson-Chru´sciel construction [1] of polyhomogeneous AH CMC initial data on a smoothly conformally compact 3-dimensional manifold M starts by choosing a smoothly conformally compact metric λ on M and a traceless, ¯ for which μ ¯ ∈ C ∞ (M ). The construction consymmetric 2-tensor μ = ρ−1 μ tinues by finding a solution X of the vector Laplacian equation (14) and by considering the symmetric, traceless, divergence-free tensor ν := μ + Dλ X. The polyhomogeneity of X, and consequently of ν, arises naturally here as a consequence of the indicial roots of the vector Laplacian (for details on indicial roots see [16]). More precisely, we have the asymptotic expansions X ∼ ρ2 X 0 + (ρ4 log ρ)X 1 −1
ν∼ρ
with X 0 , X 1 ∈ C ∞ (M ),
(16)
with ν 0 , ν 1 ∈ C ∞ (M ).
(17)
ν 0 + (ρ log ρ)ν 1
The exponent 4 in ρ4 log ρ corresponds to the indicial root of the vector Laplacian in dimension 3; in dimension n, 4 should be replaced by n + 1. Note that the description in [1] treats ν as a contravariant 2-tensor, which accounts for the difference between our powers of ρ and the powers of Ω and x in [1]. It is useful that Andersson–Chru´sciel [1] also prove a sequence of existence and uniqueness results regarding solutions of equations such as (14) in the context of polyhomogeneous tensor fields; we rely on these results when we conclude that the perturbations we make are polyhomogenous. The construction in [1] proceeds with the analysis of the Lichnerowicz equation. It is shown that the boundary value problem consisting of the Lichnerowicz equation (15) with the boundary condition ψ ∂M = 1 has a polyhomogeneous solution of the form ψ ∼ 1 + ρψ 0 +
∞
(ρ3 log ρ)i ψ i ,
with ψ i ∈ C ∞ (M ).
(18)
i=1
The exponent 3 in ρ3 log ρ corresponds to the indicial root of the linearization of the Lichenrowicz operator (cf. the left-hand side of (15)) in the neighborhood of the constant function ψ0 ≡ 1. (In dimension n, this term should be ρn log ρ.) We point out that the analysis in [1] also includes an existence and uniqueness result for the Lichnerowicz boundary value problem with polyhomogeneous data. We need this result when we show that our solution of the Lichnerowicz equation is polyhomogeneous (see Theorem 25).
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Although technically the work in [1] addresses the case of Λ = 0 only, the fact that the Lichnerowicz equation (15) does not depend on the specific choices of τ and Λ allows us to extend the results of [1] to the asymptotically (anti-)de Sitter setting. Indeed, for a given cosmological constant Λ ≥ −n(n − 1)/2 one can determine τ (up to sign) from (7) and then assemble the initial data set using the Lichnerowicz–York prescription. For convenience, we say that a polyhomogeneous AH initial data set (M, g, K) is of Andersson–Chru´sciel (A–C) type if it can be written (for dimension 3) in the form g = ψ 4 λ and K = (τ /3)g + ψ −2 ν, in which λ is a smoothly conformally compact metric, ν is a symmetric 2-tensor field that is divergencefree and trace-free with respect to λ and has an asymptotic expansion of the form (17), and ψ is a positive function with an asymptotic expansion of the form (18). The discussion in [1, Appendix A] shows that generically, initial data of A–C type are the “smoothest possible” AH initial data. Throughout this paper, we assume only that the asymptotically hyperbolic initial data sets we work with are polyhomogenous in the sense of Definition 1 (which includes data of A–C type as a special case). Our gluing procedure then produces new data of the same type. We note that if our starting data set is of A–C type, it will not generally follow from our main theorem (Theorem 2 below) that the solution we obtain after gluing is also of A–C type; our results will only guarantee that this solution is polyhomogeneous in the sense of Definition 1. The difficulty is that while A–C data sets are obtained by solving the conformal constraints with smooth conformal data, in carrying out the gluing we must solve these equations for conformal data sets which include log terms.
3. Main Gluing Theorem With the conventions established above, we are ready to state our main theorem. Theorem 2. Let (M, g, K) be a polyhomogeneous asymptotically hyperbolic CMC initial data set that satisfies the Einstein (vacuum) constraint equations for some cosmological constant Λ ≥ −n(n − 1)/2, and let p1 , p2 be distinct points in the ideal boundary ∂M . Then for sufficiently small ε > 0 there exists a polyhomogeneous AH initial data set (Mε , gε , Kε ) such that (i)
(ii) (iii)
Mε is diffeomorphic to the interior of a boundary connected sum, obtained from M by excising small half-balls B1 around p1 and B2 around p2 , and identifying their boundaries. (Mε , gε , Kε ) is a solution to the vacuum constraints with cosmological constant Λ. On the complement of any fixed small half-balls surrounding p1 and p2 in M , and away from the corresponding neck region in Mε , the pullback to M of the data (gε , Kε ) converges uniformly (as ε → 0) in C 2,α × C 1,α to (g, K), for some α ∈ (0, 1).
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On any fixed compact subset of M , the pullback of (gε , Kε ) converges uniformly with all derivatives to (g, K).
In fact, the global convergence of Kε is a little better than C 1,α : away from the fixed half-balls, it actually converges in a weighted C 1,α space. See Theorem 19 for the precise statement. Note that, as is the case for the non-localized gluing of AH data sets at interior points (see [14]), there is no need to impose any nondegeneracy conditions on the data in the neighborhood of the gluing points p1 and p2 . The gluing construction is a step by step procedure, which we explain in detail (primarily for the 3-dimensional case) in the rest of the paper. Here, we briefly summarize the main steps: The first step, which we call splicing, involves the construction of a one-parameter family of manifolds and initial data sets that are CMC and polyhomogeneous AH, but that only approximately solve the constraint equations. (In most of the literature discussing gluing constructions, both the first step leading to approximate solutions and the complete construction leading to exact solutions are called “gluing.” In this paper, to distinguish the two, we call the procedure leading to the approximate solutions “splicing.”) The new manifolds Mε are obtained by a connected sum construction which is executed in the preferred background coordinates. The parameter ε labels the coordinate “size” of the “bridge”, or gluing region. Next we use cutoff functions tied to the parametrized gluing region to construct a parametrized set of metrics gε on Mε . To verify that these spliced metrics are all asymptotically hyperbolic, we also construct a parametrized set of spliced defining functions ρε . Using a different cutoff procedure, we produce a family of (spliced) symmetric 2-tensors με that, by construction, are trace-free with respect to the corresponding gε , but are generally not divergence-free with respect to gε . The next two steps involve deformations of the spliced data sets (Mε , gε , Kε = (τ /3)gε + με ) to produce the glued data sets which satisfy the constraints and have the desired limit properties. To deform με , we first estimate its divergence, and then (following the standard York prescription) solve a linear elliptic system (based on the vector Laplacian Lgε ) whose solution tensor deforms με to a new family of tensors νε that are divergence-free. To deform the metric, we treat (Mε , gε , νε ) as a set of CMC conformal data, and proceed to solve the Lichnerowicz equation (15) for a family of conformal factors ψε . The ε-parametrized data sets (Mε , ψε4 gε , ψε−2 νε +(τ /3)ψε4 gε ), which we call the glued data, then solve the constraint equations, and are verified to approach arbitrarily close (as ε → 0) to the original data away from the gluing region.
4. Splicing Construction We presume that we are given a (3-dimensional) CMC polyhomogeneous asymptotically hyperbolic initial data set (M, g, K), which is not assumed to be connected. We let ρ denote a chosen smooth defining function for M . We
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may write τ τ g + μ = ρ−2 g + ρ−1 μ, 3 3 where g and μ are polyhomogeneous, g ∈ C 2 (M ), μ ∈ C 1 (M ), and μ is tracefree with respect to g (or, equivalently, g). Note that the assumption that g is polyhomogeneous and of class C 2 means that the first log term in the expansion of g must occur with a power of ρ strictly greater than 2, and thus g is actually in C 2,α (M ) for some α ∈ (0, 1); and similarly μ ∈ C 1,α (M ). In the rest of this section, we detail the splicing constructions. We detail the deformation steps, which complete the gluing construction, in subsequent sections. K=
4.1. Preferred Background Coordinates We focus first on the given polyhomogeneous asymptotically hyperbolic geometry (M, g). Let ρ be a smooth defining function, and define g = ρ2 g, which is a C 2,α polyhomogenous Riemannian metric on M . For each point p ∈ ∂M , we can choose smooth functions θ1 , θ2 such that θ1 (p) = θ2 (p) = 0 and (ρ, θ1 , θ2 ) form smooth coordinates in a neighborhood U ⊂ M , which we call background coordinates. (For the n-dimensional case, of course, we would choose θ1 , θ2 , . . . , θn−1 .) Sometimes for reasons of notational symmetry we also set θ0 = ρ. Throughout this paper, we will index such background coordinates with indices named a, b, c, . . ., which we understand to run from 0 to 2; and we will use indices j, k, . . ., running from 1 to 2, to refer to coordinates on ∂M . We will use the Einstein summation convention when convenient. It is shown in [12] that when g is an asymptotically hyperbolic metric that is smoothly conformally compact, there is a smooth defining function ρ such that |dρ|2g /ρ2 ≡ 1 in a neighborhood of the ideal boundary ∂M , so the where h(ρ) is a metric can be written in the form g = ρ−2 dρ2 + h(ρ) there, smoothly varying family of metrics on ∂M with h(0) = g T ∂M . Unfortunately, that result does not apply in the present circumstances because we are not assuming that g has a smooth conformal compactification. As a substitute, however, we have the following lemma (the generalization of this lemma and its proof to n dimensions should be readily evident): Lemma 3. If (M, g) is a polyhomogeneous asymptotically hyperbolic Riemannian geometry, then there exists a smooth defining function ρ such that |dρ|2g = 1 + O(ρ2 ). ρ2
(19)
Also, for each p ∈ ∂M there exist smooth background coordinates (ρ, θ1 , θ2 ) on an open neighborhood U of p in M in which g can be written in the form g = ρ−2 (dρ2 + (dθ1 )2 + (dθ2 )2 + mab (ρ, θ)dθa dθb ),
(20)
where the “error terms” mab are uniformly bounded in U and satisfy m00 (ρ, θ) = mj0 (ρ, θ) = m0j (ρ, θ) = O(ρ2 ), j ∈ {1, 2}, j, k ∈ {1, 2}. mjk (ρ, θ) = O ρ + (θ1 )2 + (θ2 )2 ,
(21) (22)
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Moreover, g is uniformly equivalent in U to the metric ρ−2 (dρ2 + (dθ1 )2 + (dθ2 )2 ). Proof. Let ρ0 be any smooth defining function for M , and write g 0 = ρ20 g. The hypothesis implies that there is a smooth metric g 1 on M such that g 0 = g 1 + O(ρ2 ). (Just take g 1 locally to be equal to the leading smooth terms in an asymptotic expansion for g 0 , and then patch together with a partition of unity.) Let g1 = ρ−2 0 g 1 . Because g1 is asymptotically hyperbolic and smoothly conformally compact, the argument of [12] shows that there is a smooth defining function ρ such that |dρ|2g1 /ρ2 ≡ 1. It follows that |dρ|2g /ρ2 = 1 + O(ρ2 ). Let g = ι∗ g 0 = ι∗ g 1 be the metric induced on ∂M by inclusion. Given p ∈ ∂M , let (θ1 , θ2 ) be Riemannian normal coordinates for g on some neighborhood of p in ∂M . Extend (θ1 , θ2 ) to a neighborhood of p in M by declaring them to be constant along the integral curves of the smooth vector field gradg1 ρ. It follows that (ρ, θ1 , θ2 ) are smooth coordinates in a neighborhood U of p, in which g 1 has an expression of the form g 1 = dρ2 + (dθ1 )2 + (dθ2 )2 + mab (ρ, θ)dθa dθb , with m00 , mj0 , and m0j identically zero, and mjk satisfying (22). Because g = ρ−2 (g 1 + O(ρ2 )), this implies that g has the expansion claimed in the statement of the lemma. Since mab (0, 0) = 0, by shrinking U we may also ensure that the coefficients mab are uniformly small in U, and thus g is uni formly equivalent to ρ−2 (dρ2 + (dθ1 )2 + (dθ2 )2 ) there. From now on, we assume that ρ is a smooth defining function satisfying (19). We now argue that we can choose ρ so that it also satisfies Δg ρ ≤ 0 on M.
(23)
For asymptotically hyperbolic Riemannian manifolds of dimension n, an easy computation (see, for example, [12, p. 199]) shows that Δg ρ → −(n − 2) ρ
as ρ → 0.
Thus, there is some δ > 0 such that Δg ρ ≤ 0 on the set where ρ < δ. Let σ : [0, +∞) → 0, 34 δ be any smooth, increasing, concave-down function for which 3 σ(x) = x if x ≤ δ/2, σ(x) = δ if x ≥ δ. 4 Define ρ˜ := σ ◦ ρ. Note that the conclusions of the previous lemma still hold if the function ρ is replaced by ρ˜. Furthermore, we compute Δg ρ˜ = divg ((σ ◦ ρ)dρ) = (σ ◦ ρ) Δg ρ + (σ ◦ ρ) |dρ|2g ≤ 0, where the last inequality follows from the facts that σ ≥ 0, σ ≤ 0, and Δg ρ < 0 on the support of σ ◦ ρ. From now on, we replace ρ by ρ˜, and assume that (23) holds. We call any coordinates (ρ, θ1 , θ2 ) that satisfy the conclusions of the previous lemma and (23) preferred background coordinates centered at p.
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4.2. Splicing the Manifolds and the Metrics We now focus on the topological aspect of our gluing construction. Let p1 , p2 ∈ ∂M be two distinct points on the ideal boundary, and for i = 1, 2 let θi = (ρ, θi1 , θi2 ) = (θi0 , θi1 , θi2 ) be preferred background coordinates on a neighborhood Ui ⊂ M centered at pi . There is a positive constant c such that these preferred coordinates are defined and (20)–(22) hold for |θi | ≤ c; after multiplying ρ and each of the coordinate functions θij by 1/c (which does not affect (19), (20) or (23)), we may assume that these two preferred coordinate charts are defined for |θi | ≤ 1. We now let ε be a small positive parameter, and consider two “semiannular” regions Aε,1 , Aε,2 ⊂ M characterized by
Aε,i := θi ∈ Ui : ε2 < |θi | < 1 . We let Aε,i = Aε,i ∩ M . For each choice of ε, the two regions can be identified using an inversion map with respect to a sphere of radius ε, given explicitly in coordinates by θ2 = Iε (θ1 ), where Iε : Aε,1 → Aε,2 is the following diffeomorphism: ε2 Iε (θ1 ) = θ1 . |θ1 |2
(24)
Based on this map, we define an equivalence relation on M by saying θ1 ∼ θ2 when θ1 ∈ Aε,1 , θ2 ∈ Aε,2 , and θ2 = Iε (θ1 ). This produces the connected sum manifold M ε , defined as follows: Definition 4. For i = 1, 2 and a > 0, let B a,i be the closed subset of M that corresponds in coordinates to the ball |θi | ≤ a. We define Ωε ⊂ M to be the open subset Ωε := M (B ε2 ,1 ∪ B ε2 ,2 ), and define the spliced manifold M ε by M ε := Ωε /∼. We let Ωε = Ωε ∩ M , and let Mε denote the subset of M ε consisting of points whose representatives are in Ωε . Let πε : Ωε → M ε be the natural quotient map, and define the neck of M ε to be the open subset N ε := πε (Aε,1 ) = πε (Aε,2 ). We let Nε denote N ε ∩ Mε . We will parametrize the neck by an expanding family of half-annuli in the upper half-space. Let H3 denote the closed upper half-space, defined by H3 := {(y, x1 , x2 ) ∈ R3 : y ≥ 0}, and let H3 ⊂ H3 be the subset where y > 0. Let r = (y 2 + (x1 )2 + (x2 )2 )1/2 , and define Aε ⊂ H3 to be the half-annulus defined by 1 Aε := (y, x1 , x2 ) ∈ H3 : ε < r < , ε
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and let Aε = Aε ∩ H3 . Analogous to the case of background coordinates, on H3 we use the notations x0 = y
and x = (y, x) = (y, x1 , x2 ) = (x0 , x1 , x2 ).
To define the parametrization of the neck, we first define diffeomorphisms αε,i : Aε → Aε,i for i = 1, 2 by θi = αε,i (y, x), where αε,i (y, x) = (εy, εx). Then we define βε : Aε → Aε,2 by βε = Iε ◦ αε,1 = αε,2 ◦ I, where I : Aε → Aε is the inversion in the unit circle: I(y, x) = (y/r2 , x/r2 ). Our preferred parametrization of Nε is Ψε := πε ◦ αε,1 = πε ◦ βε = πε ◦ αε,2 ◦ I : Aε → Nε . The various diffeomorphisms are summarized in the following commutative diagram: I Aε Aε βε
αε,1 ? Aε,1
Iε
αε,2 - ? - Aε,2
πε
-
Nε . The topology of Mε does not change with (sufficiently small) ε. The Riemannian geometry on Mε (which we define next) does depend on ε; this is one of the reasons that we keep track of the parameter ε. To obtain a suitable Riemannian metric gε on Mε , we blend the metrics coming from the original annuli with the use of a cutoff function. Lemma 5. There exists a nonnegative and monotonically increasing smooth cutoff function ϕ : R → R that is identically 1 on [2, ∞), is supported in 12 , ∞ , and satisfies the condition
1 ϕ(r) + ϕ ≡ 1. (25) r Proof. Let ϕ0 be a nonnegative and decreasing smooth cutoff function such that ϕ0 (r) = 12 for r ≤ 12 and ϕ0 (r) = 0 for r ≥ 2, and set
1 1 ϕ(r) := − ϕ0 (r) + ϕ0 . 2 r An easy computation shows that ϕ satisfies the conclusions of the lemma.
Using this cutoff function and the maps αε,1 , βε , and Ψε defined above, we define the metric on Mε as follows:
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Definition 6. We define gε to be the metric on Mε that agrees with (πε )∗ g away from the neck Nε , while on Nε it satisfies
1 Ψ∗ε gε = ϕ(r)(αε,1 )∗ g + ϕ (26) (βε )∗ g. r Note the following: ∗ • On the set where r ≥ 2, Ψ∗ε gε agrees with αε,1 g; 1 ∗ • On the set where r ≤ 2 , Ψε gε agrees with βε∗ g. It is obvious from the definition (which holds for any dimension) that gε is polyhomogeneous and conformally compact of class C 2 . 4.3. Splicing the Defining Functions Next we construct a family of defining functions for the manifolds Mε that are specially adapted to the metrics gε , and that agree with the original defining function ρ away from Nε . To define them, we need the following auxiliary function: Lemma 7. There exist a constant Λ ≥ 2 and (0, ∞) that satisfies
1 F = r2 F (r), r F (r) = 1, 1 F (r) = 2 , r |3rF (r) + r2 F (r)| 1 ≤ , F (r) 2
a C ∞ function F : (0, ∞) → r ∈ (0, ∞),
(27)
r ≥ Λ,
(28)
r ≤ 1/Λ,
(29)
r ∈ (0, +∞).
(30)
Proof. We use the function 1 + 1/r2 to interpolate between 1/r2 for small r and the constant function 1 for large r. More specifically, let ψ0 be any smooth cut-off function such that ψ0 (r) = 1 for r ≤ 1,
0 ≤ ψ0 (r) ≤ 1 for 1 ≤ r ≤ 2,
ψ0 (r) = 0 for r ≥ 2,
let C=
sup |3rψ0 (r) + r2 ψ0 (r)|,
r∈(0,∞)
and choose Λ ≥ 2 large enough that
2 2 1 . ≤ Λ 2C
(31)
The function ψ(r) := ψ0 (Λr) satisfies ψ(r) = 1 for r ≤ 1/Λ, ψ(r) = 0 for r ≥ 2/Λ and |3rψ (r) + r2 ψ (r)| ≤ C
for all r.
Define F (r) := 1 +
1 1 − ψ(r) − 2 ψ r2 r
1 r
(32)
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and compute F
897
1 1 = 1 + r2 − ψ − r2 ψ(r) = r2 F (r). r r
One readily verifies (28), (29) and ⎧ −2 ⎪ if r ∈ [1/Λ, 2/Λ] ⎨1 + r − ψ(r) −2 F (r) = 1 + r if r ∈ [2/Λ, Λ/2] ⎪ 1 ⎩ 1 −2 if r ∈ [Λ/2, Λ]. 1 + r − r2 ψ r
(33)
Note that 1 if r ≤ 1, F (r) ≥ 1 if r ≥ 1. (34) r2 Furthermore, we see that 3rF (r) + r2 F (r) is nonzero only on the intervals [1/Λ, 2/Λ] and [Λ/2, Λ]. A straightforward computation using (33) and (34) shows r2 |3rψ (r) + r2 ψ (r)| if r ∈ [1/Λ, 2/Λ] |3rF (r) + r2 F (r)| ≤ 1 −2 1 F (r) 3 r ψ (1/r) + r2 ψ (1/r) if r ∈ [Λ/2, Λ]. r F (r) ≥
Property (30) is now immediate from (31) and (32).
Note that the function F of the preceding lemma is bounded below by 1, and satisfies an estimate of the form
1 |F (r)| =O . (35) F (r) r We now let ξ : H3 → R be the positive function defined by ξ(y, x) = yF (r), where F is the function of the preceding lemma and r = |x|. A computation using (27) shows that ξ ◦ I = ξ, and εξ agrees with εy = (αε,1 )∗ ρ where r ≥ 2 and with εy ◦ I = (βε )∗ ρ where r ≤ 1/2. Define ρε : Mε → R to be the defining function that is equal to ρ away from the neck, and on the neck satisfies Ψ∗ε ρε = εξ = εyF. We wish to show that gε is asymptotically hyperbolic. To do so, we first need to find an expression for gε along the lines of (20). Recall that g˘ := y −2 (dy 2 + (dx1 )2 + (dx2 )2 ) is the metric of the upper half-space model of hyperbolic space. The inversion I(y, x) = (y/r2 , x/r2 ) is an isometry for this model: I ∗ g˘ = g˘. For i = 1, 2, we can write (αε,i )∗ g = g˘ + mab,i (εy, εx)
dxa dxb , y y
(36)
where mab,i are the error terms from the expression (20) for g in preferred background coordinates centered at pi . This metric is uniformly equivalent on Aε to g˘; and because mab,i ∈ C 2,α (U ∩ M ) and mab,i (0, 0) = 0, it is immediate that as ε → 0, the C 2,α norm of the functions mab,i (εy, εx) is O(ε) on any subset of Aε where r is bounded above, including in particular the set where
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ϕ = 1. (The C 2,α norm in use here is the ordinary Euclidean one inherited from R3 .) To analyze (βε )∗ g = I ∗ (αε,2 )∗ g, we compute, for a = 0, 1, 2,
2 ac
a r δ − 2xa xc dx dxc ∗ ac ac , where Q (y, x) = Q (y, x) I , = y y r2 c and therefore (βε )∗ g = I ∗ (αε,2 )∗ g εy εx dxc dxd = g˘ + . mab,2 2 , 2 Qac (y, x)Qbd (y, x) r r y y
(37)
a,b,c,d
ac
Because Q is a rational function with nonvanishing denominator, and is homogeneous of degree zero, it is uniformly bounded on Aε , and thus (βε )∗ g is uniformly equivalent on Aε to g˘. Moreover, all of the derivatives of Qac are uniformly bounded on any subset of Aε where r is bounded below by a positive constant, such as the set where ϕ = 0. Also, on any such subset, an easy argument shows that the C 2,α -norm of mab,2 (εy/r2 , εx/r2 ) is O(ε) as ε → 0. Combining these observations with formula (36) for (αε,2 )∗ g, we conclude that the pullback of gε to Aε has the form (Ψε )∗ gε = g˘ + kab,ε (y, x)
dxa dxb , y y
(38)
where for each ε, the function kab,ε is bounded and in C 2,α (Aε ), (Ψε )∗ gε is uniformly equivalent to g˘ independently of ε, and kab,ε C 2,α (Ac ) = O(ε)
(39)
as ε → 0 for any fixed c ∈ (0, 1). To obtain estimates for the behavior of kab,ε at the ideal boundary analogous to (21) and (22), we need to explicitly expand the various terms in (37). Note that in addition to the functions Qab being uniformly bounded, we have y 2yxj Qj0 (y, x) = Q0j (y, x) = 2 = O . r r Therefore, using (21)–(22) for mab,i , we have
εy εx 1 mjk,2 2 , 2 Q0j (y, x)Q0k (y, x) k00,ε (y, x) = ϕ(r)m00,1 (εy, εx) + ϕ r r r εy εx + 2mj0,2 2 , 2 Q0j (y, x)Q00 (y, x) r r εy εx + m00,2 2 , 2 Q00 (y, x)Q00 (y, x) r r 1 εy 2 2 2 , (40) = ϕ(r)O(ε y ) + ϕ O r r4 where the implied constants on the right are uniform in ε on all of Aε . Note that, in view of the definition of ρε , the right-hand side of this equation is O(ρε ), uniformly in ε. Applying similar computations to the other terms, and
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using the notation gab,ε to denote the components of Ψ∗ε gε in standard coordinates on Aε , we conclude that g00,ε (y, x) = y −2 (1 + O(ρε )); g0j,ε (y, x) = y gjk,ε (y, x) = y
−2 −2
(41) j ∈ {1, 2};
(42)
j, k ∈ {1, 2}.
(43)
O(ρε ), (δjk + O(ε)),
It follows that the inverse matrix satisfies gε00 (y, x) = y 2 (1 + O(ρε ));
(44) j ∈ {1, 2};
(45)
j, k ∈ {1, 2}.
(46)
gε0j (y, x) = y 2 O(ρε ), gεjk (y, x)
2
= y (δ
jk
+ O(ε)),
Lemma 8. There exists a constant C > 0 independent of ε such that sufficiently close to the ideal boundary ∂M ε of Mε we have C |dρ |2 ε gε − 1 ≤ ρε . ρ2ε ε Proof. Away from the neck, ρε and gε match the original ρ and g on M , and the result follows there from Lemma 3. We compute on the neck by identifying it with Aε by means of the diffeomorphism Ψε . We obtain |εF (r)dy + εyF (r) dr|gε |dρε |2gε = ρ2ε (εyF (r))2 2 dy dy F (r) , dr = + 2 y y F (r) 2
gε
gε
F (r) 2 dr . + F (r) gε
(47)
Using (44), we see that the first term on the right-hand side of (47) is 2 dy = 1 + O(ρε ). y gε To estimate the other terms, note that ρε and gε agree on the neck with (pullbacks of) ρ and g except on the subset A1/2 ⊂ Aε . So in the computation below, we may assume that 1/2 ≤ r ≤ 2, which means that ρε is bounded above and below by constant multiples of εy, and Ψ∗ε gε is uniformly equivalent to g˘. Thus, up to a constant multiple, the second term in (47) is bounded on A1/2 by dy F (r) |F (r)| y 2 C , dr = ≤ ρε , 2 y F (r) F (r) r ε g ˘ and the third by
The result follows.
F (r) 2 F (r)2 2 C dr F (r) = F (r)2 y ≤ ε ρε . g ˘
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The previous lemma implies that |dρε |ρ2ε gε = 1 on the ideal boundary ∂M ε . Thus, the manifold (Mε , gε ) is asymptotically hyperbolic. A consequence of this property is that for each value of ε, R(gε ) approaches −6 (or, in general dimensions, it approaches −n(n − 1)) as ρε → 0. We need to show that the convergence is uniform in ε. It follows from [12] that any AH metric g satisfies g , g¯−1 , ∂¯ g ) + ρ2 · P2 (¯ g , g¯−1 , ∂¯ g , ∂ 2 g¯), R(g) = −6|dρ|2g¯ + ρ · P1 (¯
(48)
where (in accord with the definition of asymptotic hyperbolicity) g¯ = ρ2 g and where Pm , m = 1, 2, are certain universal polynomials whose terms involve mth -order derivatives of the components of g¯. Since (by Lemma 3) we have that |dρ|2g¯ = 1 + O(ρ2 ), it follows that R(g) + 6 = O(ρ). Clearly then, for each individual ε the function ρ−1 ε (R(gε ) + 6) is bounded on Mε . The uniformity question now is whether the C 0,α -norms of the functions are bounded uniformly in ε. The following lemma (stated for dimension 3, with 6 replaced by n(n − 1) for general dimensions) shows that they are: Lemma 9. If c > 0 is fixed, then (ρ−1 ε (R(gε ) + 6)) ◦ Ψε C 0,α (Ac ) is bounded uniformly in ε. Proof. First observe that −1 ρε (R(gε ) + 6) ◦ Ψε = (εyF )−1 (R(Ψ∗ε gε ) + 6) . ¯ ε := y 2 Ψ∗ gε . We then have To estimate the scalar curvature R(Ψ∗ε gε ) let h ε 2 ∗ ¯ ε, h ¯ −1 , ∂ h ¯ε) R (Ψε gε ) + 6 = 6 − 6 |dy|h¯ ε + y · P1 (h ε ¯ ε, h ¯ −1 , ∂ h ¯ ε, ∂2h ¯ ε ). + y 2 · P2 (h ε
(49)
¯ ε can be expressed as The components of the metric h ¯ ab,ε (y, x1 , x2 ) = δab + kab,ε (y, x) h ¯ ε, where kab,ε C 2,α (Ac ) = O(ε) as ε → 0. Consequently, the components of h −1 −1 ¯ −1 2 ¯ 0,α ¯ hε , ε ∂ hε and ε ∂ hε have uniformly bounded C (Ac )-norms. It follows that the terms in (εyF )−1 (R(Ψ∗ε gε ) + 6) coming from the last two terms of the right-hand side of (49) have uniformly bounded C 0,α (Ac )-norms. A careful consideration of the expansion (40) yields k00,ε (y, x) = O(εy 2 ) and ∂k00,ε (y, x) = O(εy) on Ac . Therefore, −1 −1 6 − 6|dy|2h¯ ε ε y
C 0,α (Ac )
= ε−1 y −1 k00,ε (y, x)C 0,α (A
is also uniformly bounded. This observation completes our proof.
c)
We conclude the discussion of the AH geometries (Mε , gε ) and their defining functions ρε by proving that each ρε is superharmonic. Lemma 10. If ε > 0 is small enough then Δgε ρε ≤ 0.
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Proof. Away from the gluing region Ψε (A1/Λ ), the quotient map πε is a diffeomorphism satisfying πε∗ gε = g and πε∗ ρε = ρ. Thus, away from the gluing region the inequality we need to show is an immediate consequence of (23). To prove the inequality on the gluing region we utilize the transformation law for the conformal Laplacian (see, e.g., [17, Eq. (2.7)]) to write the Laplace operator for (Ψε )∗ gε in terms of that of the conformally related met¯ ε := y 2 (Ψε )∗ gε : ric h
Δ(Ψε )∗ gε (yF ) Δgε ρε ∗ Ψε = ρε yF 1 y 3/2 1 ¯ 1/2 ∗ 1/2 = R ((Ψε ) gε ) + Δh¯ ε (y F ) − R(hε )y F . (50) 8 F 8 ¯ ε and the Euclidean metric It follows from (39) that the difference between h ¯ ε − δC 2,α (A ) → 0. Thus we have δ approaches zero, in the sense that h 1/Λ ¯ ε ) → 0 and R(h y 3/2 Δh¯ ε (y 1/2 F ) − Δδ (y 1/2 F ) → 0 as ε → 0, with both convergences uniform on A1/Λ . A straightforward computation shows that 1 y 1/2 F + y 1/2 F . Δδ (y 1/2 F ) = − y −3/2 F + 3 4 r Since Lemma 9 shows that R((Ψε )∗ gε ) = −6 + O(Ψ∗ε ρε ), which is equal to −6 + O(ε) on A1/Λ , it follows that
Δgε ρε y 2 3rF + r2 F →0 (51) Ψ∗ε +1− 2 · ρε r F uniformly on A1/Λ as ε → 0. Our result is now an immediate consequence of (30) and the fact that y 2 ≤ r2 everywhere. Note that while the statement and the validity of this lemma do not depend on dimension, a number of constants and exponents appearing in the proof do. In particular, we note that for n dimensions, the right-hand side of (50) becomes n−2 y n/2 n−2 ∗ (4−n)/2 (4−n)/2 ¯ R ((Ψε ) gε ) + R(hε )y F) − F ; Δh¯ ε (y 4(n − 1) F 4(n − 1) and in (51), the +1 term needs to replaced by +(n − 2) while the final term remains the same. 4.4. Splicing the Traceless Part of the Second Fundamental Form Recall that our given second fundamental form on M can be written K = μ + (τ /3)g, where τ is a constant and μ is a traceless, divergence-free symmetric 2-tensor field of the form μ = ρ−1 μ for some μ ∈ C 1,α (M ). Our goal in this section is to create on each Mε a traceless symmetric 2-tensor με that is “approximately divergence-free,” and such that πε∗ με is equal to μ away from the neck. Later, we will correct it so that it is divergence-free.
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Let χ : R → R be a smooth nonnegative function such that χ(r) = 1 for r ≥ 3 and supp(χ) ⊂ (2, ∞). For each ε > 0, define χε : M → R by ⎞ ⎛ 2 (θj )2 ρ 1 ⎠ χε (ρ, θ1a ) = χ ⎝ 2 + , on B 1,1 , ε ε j ⎞ ⎛ (52) 2 (θj )2 ρ 2 ⎠ χε (ρ, θ2a ) = χ ⎝ 2 + , on B 1,2 , ε ε j on M \ B √3ε,1 ∪ B √3ε,2 . χε ≡ 1, Then let μ ε = χε μ on M . (The level sets of√χε are half-ellipsoids with radii proportional to ε in the ρ direction and to ε in the θ-directions. We have designed these unusual cutoff functions so that the divergence of μ ε will be uniformly small, despite the fact that the tangential and normal components of μ vanish at different rates near the ideal boundary; see Lemma 18 for details.) ε is supported in the set M \(B 2ε,1 ∪ It follows from our choice of χε that μ B 2ε,2 ). Because πε restricts to a diffeomorphism from this set to an open subset of Mε , we can define a symmetric 2-tensor με on Mε by ε , με = πε∗ μ
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understood to be zero on the neck. Because gε = πε∗ g on the support of με , and με is a scalar multiple of πε∗ μ, it follows that με is traceless with respect to gε . Although it is generally not divergence-free, we will show below that its divergence is not too large (see Lemma 18).
5. Analysis on the Spliced Manifolds In this section, we develop the results we need concerning linear elliptic operators on our spliced manifolds. 5.1. Weighted H¨ older Spaces and Linear Differential Operators To carry out the needed analysis on AH geometries with AH data, and also to provide a convenient framework for specifying the rate at which various quantities like the trace-free part of K approach their requisite asymptotic values, it is convenient to work with weighted H¨ older spaces. Here we recall the definition of these spaces, using the conventions of [16]. Suppose (M, g) is an asymptotically hyperbolic Riemannian geometry of class C ,β and ρ is a smooth defining function. (Our polyhomogeneous metrics, for example, are automatically asymptotically hyperbolic of class C 2,α for every α ∈ (0, 1).) Continuing with the 3-dimensional case for notational convenience, let (ρ, θ1 , θ2 ) be background coordinates on an open subset U ⊂ M , which we may assume extend to a neighborhood of the closure of U in M . Let ˘ ⊂ H3 be a fixed precompact ball containing (1, 0, 0). A M¨ B obius chart for M
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˘→U (or more accurately a M¨ obius parametrization) is a diffeomorphism Φ : B whose coordinate representation has the form (ρ, θ1 , θ2 ) = Φ(y, x) = (ay, ax1 + b1 , ax2 + b2 ) for some constants (a, b1 , b2 ). There is a neighborhood W of ∂M in M covered by finitely many background charts, and then the resulting family of M¨ obius charts covers W ∩ M . We extend this cover to all of M by choosing finitely many interior charts, which we also call M¨ obius charts for uniformity, to cover M W. Let E be a tensor bundle over M . For any nonnegative integer k and real number α ∈ [0, 1] such that k + α ≤ + β, we define the intrinsic H¨ older space C k,α (M ; E) as the set of sections u of E whose coefficients are locally of class C k,α , and for which the following norm is finite: uk,α = sup Φ∗ uC k,α (B) ˘ , Φ
where the supremum is over our collection of M¨ obius charts, and the norm on the right-hand side is the usual Euclidean H¨ older norm of the components of a ˘ For any real number δ, we define the corresponding weighted tensor field on B. H¨ older space by Cδk,α (M ; E) = {ρδ u : u ∈ C k,α (M ; E)}, with norm uk,α,δ = ρ−δ uk,α . When the tensor bundle is clear from the context, we will usually abbreviate the notation by writing Cδk,α (M ) instead of Cδk,α (M ; E). The index δ labels the rate of asymptotic decay of a given quantity, measured in terms of the intrinsic (asymptotically hyperbolic) Riemannian metric g. In particular, we note that larger positive values of δ imply more rapid decay. It is shown in [16, Lemma 3.7] that if η is any covariant r-tensor field on M with coefficients in background coordinates that are C k,α up to the ideal boundary, then η ∈ Crk,α (M ). Similarly, any vector field with coefficients that are C k,α up to k,α the ideal boundary lies in C−1 (M ). A linear partial differential operator P of order m between tensor bundles is said to be geometric if the components of P u in any coordinates can be expressed as linear functions of the components of u and their covariant derivatives of order at most m, with coefficients that are constant-coefficient polynomials in the dimension, the components of g, their partial derivatives, ! and 1/ det gij , such that the coefficient of each jth derivative of u involves at most the first m − j derivatives of g. The operators Δg (the Laplace– Beltrami operator), divg (the divergence), Dg (the conformal Killing operator), Dg∗ (the adjoint of Dg ), and Lg (the vector Laplacian) introduced above are all examples of geometric operators. It is shown by Mazzeo [18] (see also [16]) that every geometric operator P of order m on an asymptotically hyperbolic Riemannian geometry of class C l,β defines a bounded linear map from
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Cδk,α (M ) to Cδk−m,α (M ): P uk−m,α,δ ≤ Cuk,α,δ ,
(54)
whenever m ≤ k + α ≤ + β; and moreover, if P is also elliptic, then it satisfies the following elliptic estimate for 0 < α < 1 and m < k + α ≤ + β: uk,α,δ ≤ C(P uk−m,α,δ + u0,0,δ ).
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Because our spliced manifolds (Mε , gε ) are polyhomogeneous and asymptotically hyperbolic of class C 2 , they are also asymptotically hyperbolic of class C 2,α for some α ∈ (0, 1), and thus the results we have just discussed hold on Mε for each ε, with (, β) = (2, α). However, for our subsequent analysis, we need to check that the constants in (54) and (55) can be chosen independently of ε when ε is sufficiently small. Threading through the arguments of [16], we see that for a given geometric operator P , the constants depend only on uniform bounds of the following type as Φ ranges over a collection of M¨ obius charts covering M : (56) sup (Φ∗ g)−1 g˘ ≤ C. Φ∗ g − g˘C 2,α (B) ˘ ≤ C, ˘ B
(In [16], attention is restricted to a countable, uniformly locally finite family of M¨ obius charts, but that additional restriction is used only for Sobolev estimates, which do not concern us here.) In fact, it is not necessary to use M¨ obius charts per se, in which the first background coordinate is exactly equal to ρ; the arguments of [16] show that it is sufficient to use any family of paramet˘0 of B ˘ rizations Φ satisfying (56), as long as there is a precompact subset B such that the images of the restrictions Φ|B˘0 still cover M , and the following uniform estimates hold in addition to (56): Φ∗ ρC 2,α (B) ˘ ≤ Cρ0 ,
1 ρ0 ≤ |Φ∗ ρ| ≤ Cρ0 , C
(57)
where ρ0 = Φ∗ ρ(1, 0, 0). Thus to obtain our uniform estimates, we need only exhibit a family of charts for each ε such that the corresponding estimates hold for g = gε and ρ = ρε , with constants independent of ε. Start with the family of all M¨ obius charts for M . On the portion of Mε away from the neck, these same charts (composed with πε ; see Definition 4) serve as charts for Mε , which satisfy (56) and (57) uniformly in ε. Recall that we use the diffeomorphism Ψε : Aε → Nε to parametrize the neck. Because Ψ∗ε gε is isometric to g except on a subset of A1/2 ⊂ Aε , we need only show how to construct appropriate charts covering points in Ψε (A1/2 ). On this set, we will use standard coordinates on A1/2 as a substitute for background coordinates. Given p = Ψε (y0 , x10 , x20 ) ∈ Ψε (A1/2 ), we define ˘ → Nε , where ϕp : B ˘ → Aε is the map Φp = Ψε ◦ ϕp : B ϕp (x, y) = (y0 y, y0 x1 + x10 , y0 x2 + x20 ).
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Note that the Jacobian of ϕp is y0 times the identity. Under this map, (38) shows that gε pulls back to Φ∗p gε = g˘ + kab,ε (y0 y, y0 x1 + x10 , y0 x2 + x20 )
dxa dxb , y y
where we recall that g˘ is the metric of the upper half space model of hyperbolic ˘ ⊂ A1/4 , these metrics space. If we assume that ε is small enough that ϕp (B) satisfy the estimates in (56) uniformly in ε because the functions kab,ε are uniformly small in C 2,α norm on A1/4 . The defining function ρε pulls back to Φ∗p ρε (x) = εy0 yF (|ϕp (x)|) , and Φ∗p ρε (1, 0, 0) = εy0 F (|(y0 , x10 , x20 )|). Because F is uniformly bounded above and below on A1/4 by positive constants, and all of its derivatives are uniformly bounded there, it follows that the functions Φ∗p ρε satisfy the estimates in (57) uniformly in ε. Summarizing the discussion above, we have proved Proposition 11. Suppose P is a geometric operator of order m ≤ 2 acting on sections of a tensor bundle E → M , and for each ε > 0, Pε is the corresponding operator on Mε . There exists a constant C independent of ε such that for all C 2,α sections u of E, all integers k such that m ≤ k ≤ 2, and all real numbers δ, Pε uk−m,α,δ ≤ Cuk,α,δ . If in addition P is elliptic, then uk,α,δ ≤ C (Pε uk−m,α,δ + u0,0,δ ) . This proposition holds for any dimension. 5.2. The Vector Laplacian on Hyperbolic Space In this section, we study the kernel of the vector Laplacian Lg˘ = Dg∗˘ ◦ Dg˘ on hyperbolic space (H3 , g˘). We denote the standard coordinates by (y, x) = (y, x1 , x2 ) = (x0 , x1 , x2 ) on H3 , and we use the notations |x| = ((x1 )2 + (x2 )2 )1/2 and r = |x| = (|x|2 + y 2 )1/2 . As a global defining function on H3 , we use 2y ρ˘(y, x1 , x2 ) = . 2 |x| + (y + 1)2 The function ρ˘ is the pullback to the upper half-space of the usual defining function 12 (1 − |x|2 ) on the unit ball. It is well known (see, for example, [14] or [16]) that the vector Laplacian Lg˘ : Cδ2,α (H3 ) → Cδ0,α (H3 ) is invertible for −1 < δ < 3 and 0 < α < 1. (On Hn , the analogous statement is true for −1 < δ < n, and all of the results and proofs of this section go through with appropriate substitutions of constants.) This leads to the following lemma:
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Lemma 12. If −1 < δ < 3, then there is no nonzero global vector field X on H3 satisfying both Lg˘ X = 0 and the estimate |X|g˘ ≤ C ρ˘δ . Proof. The hypothesis implies that X ∈ Cδ0,0 (H3 ), and then Lemma 4.8(b) of [16] implies that X ∈ Cδ2,α (H3 ) for 0 < α < 1. The result then follows from the injectivity of Lg˘ on the latter space. We need some variations on this result, in which the defining function ρ˘ is replaced by other weight functions. As in Sect. 4.3, we let ξ : H3 → R be the function ξ(y, x) = yF (r), where F is the function of Lemma 7. We have noted above that ξ ◦ I = ξ, where I : H3 → H3 is the g˘-isometry given by inversion with respect to the unit hemisphere. Away from 0 and ∞, ξ is a defining function for ∂H3 , but it blows up at both 0 and ∞. With these features in mind, we use ξ as our replacement for ρ˘ (again noting that this result generalizes to n dimensions if we replace the condition −1 < δ < 3 by the condition −1 < δ < n), and we prove the following: Proposition 13. If −1 < δ < 3, then there is no nonzero global vector field X on H3 satisfying both Lg˘ X = 0 and the estimate |X|g˘ ≤ Cξ δ . Proof. Suppose X is a nonzero vector field on H3 satisfying Lg˘ X = 0 and |X|g˘ ≤ Cξ δ for some −1 < δ < 3. As a consequence of the behavior of the metric g˘ near the ideal boundary, the components of X in standard coordinates on H3 satisfy the condition |X a | ≤ Cyξ δ . Let ϕ : R2 → R be a smooth bump function supported in the set where |x| ≤ 12 and satisfying 0 ≤ ϕ(x) ≤ 1, and define a smooth vector field Y = (Y 0 , Y 1 , Y 2 ) on H3 by " a (58) Y (y, x) = X a (y, x − u)ϕ(u) du. R2
Differentiation under the integral sign shows that Lg˘ Y = 0. We show first that on the set where r ≤ 1, Y satisfies an estimate of the form |Y |g˘ ≤ Cy s for some s with −1 < s < 3. Observe that by definition ξ(y, x) = O(y/r2 ) for r ≤ 1, and consequently
δ y a for r = |(y, x)| ≤ 1. |X (y, x − u)| ≤ Cy |x − u|2 + y 2 Making the substitution u = x − yv (thereby defining v), we have
δ " y y du |Y a | ≤ C |x − u|2 + y 2 |u|≤ 12
"
=C
y |x−yv|≤ 12
= Cy
"
3−δ |x−yv|≤ 12
y
δ
y 2 |v|2 + y 2 dv . (|v|2 + 1)δ
y 2 dv
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Because |(y, x)| ≤ 1, the triangle inequality implies that {v : |x − yv| ≤ 12 } is contained in {v : |v| ≤ 2/y}. We now distinguish three cases. Case 1: If δ > 1, then the integrand above has finite integral over all of R2 . Therefore, |Y a | ≤ Cy 3−δ , from which it follows that |Y |g˘ ≤ Cy 2−δ . Case 2: If δ < 1, then we let (t, ω) denote polar coordinates in the (v 1 , v 2 ) plane, and we compute "2π "2/y |Y | ≤ y a
3−δ 0
t dt dω (t2 + 1)δ
0
(t2 + 1)−δ+1 ≤ C y 3−δ 1 + y 2δ−2 = Cy
3−δ
t=2/y t=0
≤ C y 1+δ . It follows that |Y |g˘ ≤ C y δ . Case 3: If δ = 1, then computing in polar coordinates as before, we get t=2/y |Y a | = Cy 2 log(t2 + 1) t=0 ≤ C y 2 | log y| ≤ C y 1+s , for any s such that 0 < s < 1. It follows that |Y |g˘ ≤ Cy s . In the three cases above, on the set where r ≤ 1, we have obtained an estimate of the form |Y |g˘ ≤ Cy s for some s such that −1 < s ≤ min{δ, 2 − δ}. On the other hand, if r ≥ 1 and |u| ≤ 12 , then we have |(y, x−u)| ∼ |(y, x)| and ξ(y, x − u) ∼ ξ(y, x), where ∼ means “bounded above and below by constant multiples of.” It follows easily that |Y a | ≤ Cyξ δ , and therefore |Y |g˘ ≤ Cξ δ on this set. Now let Y# be the vector field Y# = I∗ Y . Because I is an isometry and ξ is I-invariant, the argument above implies that Cξ δ , r ≤ 1, # |Y |g˘ ≤ s C(y ◦ I) , r ≥ 1. Defining a new vector field Z on H3 by " Z a (y, x) = Y# a (y, x − u)ϕ(u) du, R2
we find that Lg˘ Z = 0, and consequently the same argument as above shows that Z satisfies the estimate Cy s , r ≤ 1, |Z|g˘ ≤ s C(y ◦ I) , r ≥ 1 ≤ C ρ˘s . As a consequence of Lemma 12, this implies that Z ≡ 0.
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If X ≡ 0, choose a point (y0 , x0 ) ∈ H3 at which some coordinate component X a (y0 , x0 ) is nonzero. After a translation in the x-variables (which is an isometry of H3 ), we may assume that x0 = 0. There is some ball Br (0) ⊂ R2 such that X a (y0 , x) does not change sign for x ∈ Br (0). If ϕ is chosen to be supported in this ball, it follows from (58) that Y a (y0 , 0) = 0. Repeating this argument with Y in place of X shows that there is a point at which Z = 0. This is a contradiction, so we conclude that X ≡ 0 as claimed. To adapt this argument to the general n-dimensional case, we need to make the following changes: The three cases are δ > (n − 1)/2, in which case we obtain |Y |g˘ ≤ Cy n−1−δ ; δ < (n − 1)/2, in which case |Y |g˘ ≤ Cy δ ; and δ = (n − 1)/2 in which case |Y |g˘ ≤ Cy s for any s such that 0 < s < (n − 1)/2. In all three cases we end up with an estimate on the set where r ≤ 1 of the form |Y |g˘ ≤ Cy s for some s such that −1 < s ≤ min{δ, n − 1 − δ}. The rest of the argument proceeds as in the 3-dimensional case. We also need the following consequence of this result, in which the weight function is taken to be the vertical coordinate y. (For the n-dimensional version of this corollary, we replace 3 in the statement by n.) Corollary 14. If −1 < δ < 3, then there is no nonzero global vector field X on H3 satisfying both Lg˘ X = 0 and the estimate |X|g˘ ≤ Cy δ . Proof. If δ ≥ 0, this follows from the previous proposition and the fact that y ≤ Cξ. If δ < 0, then it follows from Lemma 12 and the fact that y ≥ C ρ˘. 5.3. The Vector Laplacian on the Spliced Manifolds We now consider the vector Laplacians on our spliced manifolds. For each ε > 0, let (Mε , gε ) be the asymptotically hyperbolic spliced manifold defined in Definitions 4 and 6, and let Lε := Lgε be its corresponding vector Laplacian. Since gε is asymptotically hyperbolic of class C 2,α for some α ∈ (0, 1), the analysis in [14] or [16] shows that Lε : Cδ2,α (Mε ) → Cδ0,α (Mε ) is invertible so long as ε > 0 and −1 < δ < 3. We need to show that the norm of its inverse is bounded uniformly in ε. The main goal of this section is to understand this uniformity. Fix α as above and δ ∈ (−1, 3). We start with the uniform Schauder estimate (see Proposition 11) X2,α,δ ≤ C(Lε X0,α,δ + X0,0,δ ),
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where C is some constant independent of ε. We will show that there is a uniform constant D such that for sufficiently small ε X0,0,δ ≤ DLε X0,0,δ .
(60)
This last estimate implies that X2,α,δ ≤ C(D + 1)Lε X0,α,δ ; i.e., that the norm of the inverse (Lε )−1 is bounded above by 1/(C(D + 1)). We use blow-up analysis to prove (60). The main ingredient in the analysis is the following lemma: Lemma 15. Let (Σ, γ) be an asymptotically hyperbolic manifold, and let {Nj } be a sequence of open subsets of Σ such that every compact subset of Σ is
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contained in Nj for all but finitely many j. Suppose that for each j we are given a Riemannian metric gj on Nj such that gj → γ uniformly with two derivatives on every compact subset of Σ. Assume furthermore that there exist vector $ fields Yj on Nj , a positive real-valued function ζ on Σ, a compact subset K0 ⊂ j Nj , and positive constants C1 , C2 such that (a) inf K0 (ζ −δ |Yj |gj ) ≥ C2 ; (b) supNj (ζ −δ |Yj |gj ) ≤ C1 ; (c) supNj (ζ −δ |Lgj Yj |gj ) → 0. Then there exist a C ∞ vector field Y on Σ and a constant C3 for which Lγ Y = 0,
|Y |γ ≤ C3 ζ δ ,
Y ≡ 0.
be a slightly larger Proof. Let K ⊂ Σ be a precompact open set, and let K precompact open set containing K. Since the metrics gj converge uniformly (with two derivatives) to γ, we may assume that the following estimates on K when j is sufficiently large: hold on K |Yj |γ ≤ 2C1 ζ δ ,
|Lγ Yj |γ ≤ Cζ δ ,
for some constant C independent of j. The function ζ is bounded above so it follows that Yj 0,p % and and below by positive constants on K, H (K,γ) are bounded uniformly in j. Sobolev estimates now imply Lγ Yj H 0,p (K,γ) % that Yj H 2,p (K,γ) ≤ CK , for some new constant CK depending on K but independent of j. By the Rellich Lemma there exists a subsequence Yjn ,K of Yj that converges in H 1,p (K, γ). For sufficiently large p, we have a Sobolev embedding H 1,p (K, γ) → C 0,0 (K, γ). This means that there exists a pointwise limit YK := lim Yjn ,K jn →∞
where YK ∈ C 0,0 (K, γ). Note that by construction |YK |γ ≤ 2C1 ζ δ on K. Consider a nested sequence of precompact open sets whose union is Σ: K1 ⊂ K 1 ⊂ K2 ⊂ K 2 ⊂ K3 ⊂ K 3 · · · . We may use the process outlined above to inductively construct sequences Yjn ,Km for each Km , such that the sequence Yjn ,Km is a subsequence of Yjn ,Km−1 that converges uniformly on Km . The diagonal sequence Yjn ,Kn converges uniformly on every compact subset of Σ to a continuous limit Y on Σ that satisfies |Y |γ ≤ 2C1 ζ δ . The assumption (a) ensures that Y ≡ 0. So, it remains to show that Lγ Y = 0.
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Let X be a compactly supported test vector field on Σ. Since gjn converges to γ uniformly on supp X with two derivatives, we have " " & ' Lgjn X, Yjn ,Kn γ dVγ Lγ X, Y γ dVγ = lim jn →∞
Σ
Σ
"
= lim
jn →∞
&
X, Lgjn Yjn ,Kn
' γ
dVγ = 0.
Σ
Thus Y is a weak solution to Lγ Y = 0, and it follows from elliptic regularity that Y ∈ C ∞ (Σ) and Lγ Y = 0. We now focus on verifying inequality (60). (As usual, this lemma holds in dimension n with the hypothesis −1 < δ < n.) Lemma 16. If −1 < δ < 3, then there exists a constant D such that for sufficiently small ε and for all vector fields X ∈ Cδ2,0 (Mε ) we have X0,0,δ ≤ DLε X0,0,δ .
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Proof. Suppose not: Then there exist positive numbers εj → 0 and vector fields Xj ∈ Cδ2,0 (Mεj ) such that Xj 0,0,δ = 1 and Lεj Xj 0,0,δ → 0. The fact that Xj 0,0,δ = 1 means in particular that for each j there exists a point qεj ∈ Mεj such that 1 ρε (qε )δ . 2 j j Since πε maps Ωε/3 = M (B ε/3,1 ∪ B ε/3,2 ) surjectively onto Mε , for each j we may choose a representative qj for qεj such that qj ∈ Ωε/3 . Passing to a subsequence if necessary, we may assume that qj → q ∈ M . Our proof now splits into several cases depending on the location of q. Each case culminates in a contradiction. |Xj (qεj )|gεj ≥
Case 1: q ∈ M . This is the easiest of the cases as it allows immediate use of Lemma 15. Indeed, let (Σ, γ) = (M, g), Nj = Ωεj /3 , gj ≡ g, ζ ≡ ρ, and let K0 be a compact set containing a small neighborhood of q. Vector fields Yj on Nj for which (πεj )∗ Yj = Xj necessarily satisfy the hypotheses of Lemma 15. Thus there exists a C ∞ nonzero vector field Y on M with Lg Y = 0 and |Y |g = O(ρδ ). However, for δ ∈ (−1, 3), the vector Laplacian has no kernel in Cδ0,0 (M ), so this is a contradiction. Case 2: q ∈ ∂M {p1 , p2 }. Let θ := (ρ, θ1 , θ2 ) be a set of preferred background coordinates centered at q, which we may assume to be defined on a half-disk D ⊂ M whose coordinate radius is R. Let (ρj , θj1 , θj2 ) be the coordinates of qj , j 0. Note that (ρj , θj1 , θj2 ) → (0, 0, 0) as j → ∞. Let (Σ, γ) be the hyperbolic space (H3 , g˘). We define Nj to be the halfball in H3 centered at (0, −θj1 /ρj , −θj2 /ρj ) of (Euclidean) radius R/ρj . As soon as j is large enough that |θj | < R/2, the triangle inequality shows that the
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half-ball of radius R/2ρj centered at (0, 0, 0) is contained in Nj , so we see that ( j Nj = Σ and that each compact subset of M is contained in Nj for all but finitely many j. Consider the transformations Tj : Nj → M whose coordinate representations are given by Tj (y, x1 , x2 ) = (ρj y, ρj x1 + θj1 , ρj x2 + θj2 ).
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These transformations are chosen so that Tj (1, 0, 0) = qj . We will now construct metrics and vector fields on Nj satisfying the hypothesis of Lemma 15. First, let gj := Tj∗ g. It follows easily from Lemma 3 that gj → g˘ uniformly on compact sets together with two derivatives. ∗ Now define Yj := ρ−δ j Tj Xj . We compute: δ δ x))|g ≤ ρ−δ |Yj (x)|gj = ρ−δ j |Xj (Tj ( j (ρj y) = y , 1 −δ δ 1 |Yj (1, 0, 0)|gj = ρ−δ j |Xj (qj )|g ≥ ρj ρj = , 2 2 y −δ |Lgj Yj |gj = (ρj y)−δ (|Lg Xj |g ◦ Tj ) ≤ Lεj Xj 0,0,δ → 0.
In particular, the hypotheses of Lemma 15 are fulfilled for K0 = {(1, 0, 0)} and ζ ≡ y. It follows that there is a nonzero vector field Y on H3 for which Lg˘ Y = 0 and |Y |g˘ = O(y δ ). For δ ∈ (−1, 3) this is impossible by Corollary 14. Case 3: q ∈ {p1 , p2 }; without loss of generality we may assume that q = p1 . For sufficiently large j, the point qεj is contained in the neck Nεj ; let xj := (yj , x1j , x2j ) be the point in Aεj such that Ψεj (xj ) = qεj , and let rj := |xj |. It follows from the fact that qj ∈ Ωε/3 that rj > 13 . There are several different ways in which qj can converge to p1 . We consider now three subcases and use Lemma 15 in each subcase. Case 3a: There are uniform upper and lower bounds on rj and yj /rj ; i.e., for some d > 0, 1 < rj ≤ d 3
and
1≥
yj 1 ≥ > 0. rj d
(63)
In this case we take (Σ, γ) = (H3 , g˘), Nj := Aεj and gj := Ψ∗εj gεj . It is imme( diate that j Nj = Σ, and that every compact set in Σ is contained in almost all Nj . It follows from (38) that gj → g˘ uniformly on compact sets together with two derivatives. Consider the function ξ(y, x) = yF (r) and vector fields Yj such that ∗ (Ψεj )∗ Yj = ε−δ j Xj . Because Ψεj ρεj = εj ξ, we have |Yj |gj ≤ ξ δ and sup ξ −δ |Lgj Yj |gj → 0, Nj
so conditions (b) and (c) of Lemma 15 are satisfied with ζ ≡ ξ. The compact set K0 characterized by (63) contains the points (xj ), where we have |Yj (xj )|gj = ε−δ j |Xj (qεj )|gεj ≥
1 −δ 1 εj ρεj (qεj )δ = ξ(xj )δ . 2 2
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This means that the condition (a) of Lemma 15 also holds. Therefore, there exists a nonzero C ∞ vector field on Y on H3 that satisfies Lg˘ Y = 0 and |Y |g˘ = O(ξ δ ). However, this contradicts Proposition 13. Case 3b: There is a uniform positive lower bound on yj /rj , but rj are unbounded. Passing to a subsequence, we may assume that yj 1 rj → ∞, ≥ > 0. rj d We again take (Σ, γ) = (H3 , g˘). Consider the transformation Tj : H3 → H3 given by Tj (x) = rj x. This transformation is chosen so that the points aj := Tj−1 (xj ) = (yj /rj , x1j /rj , x2j /rj ) lie in a compact region K0 of the upper hemisphere {r = 1, y > 0}. The set Nj ⊆ H3 characterized by 1 1 Nj := x :
= F (rj r)δ ≤ C; = (εj rj y)−δ |Lgεj Xj |gεj ◦ Ψεj ◦ Tj ≤ F (rj r)δ Lεj Xj 0,0,δ → 0.
Moreover, since Ψεj ◦ Tj (aj ) = qεj and ρεj (qεj ) = Ψ∗εj ρεj (xj ) = εj yj F (rj ), we have 1 |Yj (aj )|gj = (εj rj )−δ |Xj (qεj )|gεj ≥ (εj rj )−δ ρεj (qεj )δ 2 1 δ δ = (yj /rj ) F (rj ) ≥ Cy(aj )δ . 2 Consequently, the conditions of Lemma 15 are fulfilled. This means that we now have a C ∞ nonzero vector field Y on H3 for which Lg˘ Y = 0 and |Y |g˘ = O(y δ ). This is a contradiction to Corollary 14. Case 3c: There is no positive lower bound on yj /rj . Passing to a subsequence, we may assume that one of the following holds:
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rj → ∞, or yj → 0 and rj is bounded.
Note that in both cases (i) and (ii), |(x1j , x2j )|/yj → ∞. In either case, we consider transformations Tj : H3 → H3 defined by Tj (y, x1 , x2 ) = (yj y, yj x1 + x1j , yj x2 + x2j ), + chosen so that Ψεj ◦ Tj (1, 0, 0) = qεj . Let Nj := Tj−1 (A+ εj ), where Aεj := {r > 16 } ∩ Aεj . The region Nj ⊆ H3 is the semiannular region centered at (0, −x1j /yj , −x2j /yj ) of inner radius 1/6yj and outer radius 1/εj yj . In case (i), we have |(x1j , x2j )| → ∞ and ) * −x1 −x2 |(x1j , x2j )| − 16 |(x1j , x2j )| 1 j j , = ≥ → ∞. − yj yj 6yj yj 2yj
In case (ii), once j is big enough, we have |(x1j , x2j )| − 16 ≥ 17 , as a consequence of rj > 13 . It follows that ) * −x1 −x2 |(x1j , x2j )| − 16 1 1 j j , = ≥ → ∞. − yj yj 6yj yj 7yj In particular, Nj contains the half-ball of radius ) * ) * + −x1 −x2 −x1 −x2 1 1 j j j j − , , Rj = min , − εj y j y j yj yj yj 6yj first expression in the minimum centered at the origin. Since εj xj → 0, the ( also converges to ∞. Therefore, Rj → +∞, j Nj = H3 , and every compact subset of H3 is contained in almost all Nj . As in the previous case, we take (Σ, γ) = (H3 , g˘) and gj := Tj∗ Ψ∗εj gεj . Note that (38) shows that gj can be expressed as gj = g˘ + kab,εj (Tj (y, x))
dxa dxb , y y
where kab,ε are functions defined on Aε such that kab,ε C 2,α (Ac ) → 0 for any fixed c > 0. We need to show that kab,ε ◦Tj C 2,α (K) → 0 for any fixed compact set K ⊂ H3 . We will consider cases (i) and (ii) separately. Let K ⊂ H3 be a compact set, and let R be the supremum of r|K . First assume we are in case (i). For any point x = (y, x) ∈ K, as soon as j is large enough that R < 12 |(x1j , x2j )|/yj , the reverse triangle inequality gives ) * * ) x1j x2j |(x1j , x2j )| −R + x ≥ yj r ◦ Tj (x) = yj 0, , yj yj yj ≥
1 |(x1j , x2j )| yj = |(x1j , x2j )| → ∞. 2 yj
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Eventually, therefore, the set Tj (K) lies in the portion of Aε where r > 2, and thus for x ∈ K we have kab,εj (Tj (x)) = mab,1 (εj yj y, εj yj x1 + εj x1j , εj yj x2 + εj x2j ), and it follows easily that kab,εj ◦ Tj C 2,α (K) → 0 because (εj yj , εj x1j , εj x2j ) → 0. On the other hand, if we are in case (ii), then for any x ∈ K, as soon as R < |(x1j , x2j )|/yj , we have * ) |(x1j , x2j )| + R ≤ 2|(x1j , x2j )| ≤ 2rj ≤ C, r ◦ Tj (x) ≤ yj yj since {rj } is bounded. Therefore, Tj (K) is contained in the fixed annulus {x : 16 < r < C}, on which kab,ε converges to zero in C 2,α norm. Because yj → 0, the transformations Tj are affine transformations with uniformly bounded Jacobians, and so again we conclude that kab,ε ◦ Tj C 2,α (K) → 0. In both cases, therefore, gj → g˘ in C 2,α (K). This time, we let 1 T ∗ Ψ∗ ρε and Yj := (εj yj )−δ Tj∗ Ψ∗εj Xj . ζ=y= εj yj F (r ◦ Tj ) j εj j Reasoning as in Case 3b, since F is bounded above on [ 16 , ∞) and bounded below everywhere, we find that δ y −δ |Yj |gj = (εj yj y)−δ |Xj |gεj ◦ Ψεj ◦ Tj ≤ (εj yj y)−δ Tj∗ Ψ∗εj ρεj y −δ |Lgj Yj |gj
= F (r ◦ Tj )δ ≤ C; = (εj yj y)−δ |Lgεj Xj |gεj ◦ Ψεj ◦ Tj
≤ F (r ◦ Tj )δ Lεj Xj 0,0,δ → 0; 1 |Yj (1, 0, 0)|gj = (εj yj )−δ |Xj (qεj )|gεj ≥ (εj yj )−δ ρεj (qεj )δ 2 1 −δ δ = yj (yj F (yj )) ≥ c. 2 These estimates show that the vector fields Yj satisfy the conditions of Lemma 15 for the choice of K0 = {(1, 0, 0)}. We now see that there exists a nonzero C ∞ vector field Y on H3 such that Lg˘ Y = 0,
Y = O(y δ )
for δ ∈ (−1, 3).
However, this is impossible by Corollary 14. We note that this argument goes through with very little modification for the case of general dimensions. Now we are ready for our main theorem concerning the vector Laplacian. Theorem 17. If ε is sufficiently small and if −1 < δ < 3, and 0 < α < 1, then the vector Laplacian Lε : Cδ2,α (Mε ) → Cδ0,α (Mε ) is invertible and the norm of its inverse is bounded uniformly in ε.
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In dimension n, this theorem holds with −1 < δ < n. Proof. The invertibility follows from the analysis in [14] or [16], and the uniform estimate follows by combining (59) with (61).
6. Correcting the Traceless Part of the Second Fundamental Form In this section, we use the elliptic PDE theory and analysis of the previous section to add a correction to our spliced tensor με to make it divergence-free. First we show that its divergence is not too large. Lemma 18. With με √ defined by (53), divgε με ∈ C10,α (Mε ) with norm equal to divgε με 0,α,1 = O( ε). Proof. Recall that we have defined μ ε = χε μ, where μ is the given traceless second fundamental form and χε is defined by (52). Restricted to the support ε to με , so it of μ ε , the projection πε is a diffeomorphism taking g to gε and μ √ ε ∈ C10,α (M ) with O( ε) norm. suffices to show that divg μ For a vector field Y and a symmetric 2-tensor η, let us use the notation Y η to denote the 1-form η(Y, ·). It is easy to check (by doing the computation in M¨ obius coordinates) that the map (Y, η) → Y η is a continuous bilinear map from Cδk,α (M ) × Cδk,α (M ) to Cδk,α (M ) for any δ1 , δ2 ∈ R. 1 2 1 +δ2 It follows easily from the definition of the divergence operator and the fact that μ is divergence-free that ε = χε divg μ + (gradg χε ) μ = (gradg χε ) μ. divg μ The support of gradg (χε ) is contained in the union of the two half-balls B 1,1 ∪ B 1,2 . Letting θj denote either θ1j or θ2j depending on which half-ball we are in, we compute ⎞⎛ ⎞ ⎛ 2 (θj )2 2θj gradg θj 2ρ grad ρ ρ g ⎠⎝ ⎠, gradg χε = χ ⎝ 2 + + 2 ε ε ε ε j j and therefore, ⎛
⎞⎛ ⎞ 2 (θj )2 2θj 2ρ ρ ⎠ ⎝ (gradg ρ) μ + divg μ (gradg θj ) μ⎠ . ε = χ ⎝ 2 + 2 ε ε ε ε j j (64)
Using the formula for the change in Christoffel symbols under a conformal change in metric (see, for example, [16, Eq. (3.10)]), we find that 0 = divg μ = ρ2 divg μ − ρ(gradg ρ) μ. After substituting μ = ρ−1 μ, this becomes 0 = ρ2 divg (ρ−1 μ) − ρ(gradg ρ) (ρ−1 μ) = ρ divg μ − 2(gradg ρ) μ.
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It follows that (gradg ρ) μ = 12 ρ divg μ, and thus (gradg ρ) μ = (ρ2 gradg ρ) (ρ−1 μ) 1 = ρ2 divg μ. 2 Since divg μ is a 1-form whose coefficients in background coordinates are in C 0,α (M ), divg μ is contained in C10,α (M ), and thus (gradg ρ) μ ∈ C30,α (M ). On the other hand, a straightforward computation shows that gradg θj ∈ C10,α (M ), and therefore (gradg θj ) μ ∈ C20,α (M ). We conclude that the following quantities are finite: ρ−2 (gradg ρ) μ0,α,1 ,
ρ−1 (gradg θj ) μ0,α,1
(j = 1, 2).
With this in mind, we rewrite (64) as ⎛ ⎞ 3 j 2θ ρ −1 2ρ divg μ ρ (gradg θj ) μ⎠ , ε = fε (ρ, θ) ⎝ 2 ρ−2 (gradg ρ) μ + ε ε j (65) where
⎞ 2 (θj )2 ρ ⎠. fε (ρ, θ) = χ ⎝ 2 + ε ε j ⎛
Note √ that fε is bounded √ independently of ε, and is supported in a region where ρ ≤ ε 3 and |θj | ≤ 3ε. Its differential satisfies ⎞⎛ ⎞ ⎛ 2 (θj )2 2θj dθj 2ρdρ ρ ⎠⎝ ⎠. dfε = χ ⎝ 2 + + ε ε ε2 ε j j Because |dρ|g and |dθj |g are both bounded by multiples of ρ, it follows that |dfε |g is bounded uniformly in ε. Therefore, fε is uniformly bounded in C 1 (M ) and thus also in C00,α (M ). Inserting these estimates into (65), we find that ⎛ √ 3 3) 2(ε ρ−2 (gradg ρ) μ ε 0,α,1 ≤ fε 0,α,0 ⎝ divg μ 2 0,α,1 ε ⎞ √ √ 2( 3ε)(ε 3) ρ−1 (gradg θj ) μ ⎠ + 0,α,1 ε j √ ≤ C (ε + ε) √ ≤ C ε. This completes the proof.
Using the preceding result and Theorem 17, the idea now is to make a small perturbation of με , which we denote by νε , for which divgε νε = 0. We have the following theorem for any dimension:
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Theorem 19. For each sufficiently small ε > 0, there is a polyhomogenous symmetric 2-tensor field νε , which is traceless and divergence-free with respect to gε , such that ρ2 νε has a C 2 extension to M ε that vanishes on ∂M ε , and νε satisfies √ (66) νε − με 1,α,1 = O( ε). In particular, away from the neck, νε converges uniformly in C11,α (and therefore also in C 1,α ) to (the projection of) μ. On any fixed compact subset of M , πε∗ νε converges to μ uniformly with all derivatives. Proof. We use the standard technique for finding a divergence-free perturbation of με discussed in Sect. 2 (see (14)). Relying on Theorem 17, for each small ε > 0, we let Xε be the unique vector field in C12,α (Mε ) that satisfies Lε Xε = (divgε με ) ,
(67)
and we set νε := με + Dε Xε . By definition of the conformal Killing operator, is traceless; and by construction it is divergence-free. Since divgε με 0,α,1 = νε √ O( ε), it follows from the uniform estimate of Theorem 17 that √ (68) Xε 2,α,1 = O( ε). The arguments of Sect. 5.1 show that Dε is bounded from Cδ2,α (Mε ) to Cδ1,α (Mε ) uniformly in ε, and therefore (66) is satisfied. On any fixed compact subset G ⊂ M , we have πε∗ με = μ and πε∗ gε = g for ε small enough, and therefore (67) shows that Lg (πε∗ Xε ) = 0 there. It follows from (68) that πε∗ Xε converges uniformly to zero on G, so the usual bootstrap argument shows that πε∗ Xε → 0 with all derivatives, and the last statement of the theorem follows. It now remains to show that νε is polyhomogeneous and that ρ2ε νε has 2 a C extension to M ε which vanishes on the ideal boundary. We start by observing that the right-hand side of (67) is polyhomogeneous and that, by Theorem 6.3.10 of [1], there exists a polyhomogeneous solution Xεphg of (67). Since μ has an asymptotic expansion beginning with ρ−1 and the first log term (if any) appearing with ρs , s > 0, a computation shows that the vector field on the right-hand side of (67) has an expansion beginning with a term, ρ2ε term, and with the first log terms (if any) appearing in the ρs+3 ε s > 0. Inserting the general asymptotic expansion for Xεphg into (67) and matching like terms inductively, we conclude that Xεphg has an asymptotic , expansion beginning with ρ2ε and the first log terms appearing with ρs+3 ε s > 0. (Note that the first log terms which arise from the indicial roots of Lε appear with ρ4ε .) On the other hand, Xε ∈ C12,α (Mε ) implies that its component functions in background coordinates are also O(ρ2ε ). The uniqueness part of Theorem 6.3.10 in [1] implies that Xε = Xεphg . It now follows easily that νε is polyhomogeneous and that it has an asymptotic expansion and the first log term appearing with ρsε , s > 0. Thus the starting with ρ−1 ε 2 extension of ρε νε to M ε is actually of class C 2 and vanishes on the ideal boundary, which is just what is needed for νε to be the traceless part of
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the second fundamental form for a polyhomogeneous AH initial data set (see Definition 1).
7. The Lichnerowicz Equation and Conformal Deformation to the Spliced Solutions of the Constraint Equations Thus far, starting with a set of asymptotically hyperbolic, polyhomogeoneous, constant mean curvature initial data (M, g, K) satisfying the Einstein constraint equations, together with a pair of points {p1 , p2 } both contained in the ideal boundary ∂M , we have first produced a one-parameter family of spliced data sets (Mε , gε , με ) which are asymptotically hyperbolic, polyhomogeneous, CMC, and not solutions of the constraints, and we have then corrected με to a new family of symmetric tensors νε which are all divergence-free as well as trace-free with respect to gε . We have verified that outside of the gluing region, νε approaches the original trace-free part of K in an appropriate sense. To complete our gluing construction, we now carry out a one-parameter family of conformal deformations that transform the data (Mε , gε , νε , τ ) to a family of data sets (Mε , ψε4 gε , ψε−2 νε + (τ /3)ψε4 gε ) satisfying the desired properties of the gluing construction (including the constraint equations) for all ε. Following the principles of the conformal method outlined in Sect. 2.2, if we want the conformally transformed data sets to satisfy the constraints, then the conformal functions ψε must solve the Lichnerowicz equation (15), which for the data (Mε , gε , νε , τ ) takes the form Lε (ψε ) = 0, where 1 1 3 Lε (u) := Δgε u − R(gε )u + |νε |2gε u−7 − u5 . 8 8 4
(69)
Hence, we need to do the following: prove that for each ε the Lichnerowicz equation (69) does admit a positive solution ψε which is polyhomogeneous and C 2 up to the ideal boundary, prove that ψε approaches 1 at the ideal boundary (so that the resulting Riemannian manifold is AH) and prove that as ε → 0 the solutions ψε approach 1 away from the gluing region. We carry out these proofs here, noting that with small modifications, these arguments hold for all dimensions n ≥ 3. The first step in our proof that, for the data sets (Mε , gε , νε , τ ), the Lichnerowicz equation admits solutions with the desired asymptotic properties, is to estimate the extent to which the constant function ψ0 ≡ 1 fails to be a solution of (69). While it is relatively straightforward to show that Lε (ψ0 ) is an element of the weighted H¨ older space C10,α (Mε ), we have been unsuccessful in proving that the corresponding norm of Lε (ψ0 ) is “small”. Consequently, we are able to find a solution ψε of the Lichnerowicz equation such that ψε − ψ0 “vanishes” on ∂M ε , but we are only able to obtain good estimates on ψε − ψ0 in C 2,α (Mε ). This, however, is sufficient to prove our main result. √ Lemma 20. We have Lε (ψ0 ) ∈ C10,α (Mε ) and Lε (ψ0 )0,α = O( ε) as ε → 0.
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Proof. Note that ψ0 ≡ 1 yields 1 Lε (ψ0 ) = − (R(gε ) + 6 − |νε |2gε ). 8 The fact that Lε (ψ0 ) ∈ C10,α (Mε ) is immediate from |νε |2gε ∈ C10,α (Mε ) which is true by construction, and (R(gε ) + 6) ∈ C10,α (Mε ) which is true by virtue of (48) and Lemma 9. To estimate the unweighted norm of Lε (ψ0 ), let Φ be one of our preferred charts for Mε (see Sect. 5.1). If the image of Φ is away from ˘ ∩ Ψε (A√ε/3 ) = ∅, then the gluing region, i.e., if Φ(B) √ 1 ∗ Φ (|νε |2gε − |με |2gε )C 0,α (B) ˘ = O( ε) 8 as a consequence of the second constraint equation 0 = R(g)−|K|2g +τ 2 −2Λ = R(g) − |μ|2g + 6 and (66). Thus, it remains to study the charts Φ for which Φ∗ Lε (ψ0 )C 0,α (B) ˘ =
˘ ⊆ Ψε (Ac√ε ), Φ(B) where c > 0 is some sufficiently small fixed number. We start with the inequality 1 ∗ Φ∗ Lε (ψ0 )C 0,α (B) ˘ ≤ Φ (R(gε ) + 6)C 0,α (B) ˘ 8 √ 1 + O( ε) + Φ∗ |με |2gε C 0,α (B) ˘ . 8 It follows from (48) and Lemma 9 that R(gε ) + 60,α,1 is bounded uniformly √ in ε. The uniformity properties (57) and the fact that ρε ≤ ε/c on Ψε (Ac√ε ) imply that √ (70) Φ∗ (R(gε ) + 6)C 0,α (B) ˘ = O( ε). To understand the με -term, note that πε is a diffeomorphism on the support of μ ε , where we also have 2 2 2 μ ε g ≤ μg = ρ2 μg¯ . It then follows that supB˘ |Φ∗ |με |2gε | = O(ε). Likewise, d(| με |2g )g = ρd(ρ2 χ2 |μ|2g¯ )g¯ ≤ ρ(2ρ|dρ|g¯ |μ|2g¯ + 2ρ2 |dχ|g¯ |μ|2g¯ + ρ2 |d(|μ|2g¯ )|g¯ ) = O(ρ2 ) as a consequence of the boundedness of |dρ|g¯ and the fact that ρ2 |dχ|g¯ = ρO(ε)O( 1ε ) = O(ρ) on the support of dχ. Overall, we see that Φ∗ |με |2gε C 0,α (B) ˘ = O(ε) √ and therefore Φ∗ Lε (ψ0 )C 0,α (B) ˘ = O( ε) as ε → 0. that
It should also be pointed out that |νε |2gε = R(gε ) + 6 + 8Lε (ψ0 ) implies
for some C > 0.
2 |νε |g ≤ C ε 0,α
(71)
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The main ingredient in our study of the solvability and the solutions of the Lichnerowicz equation is the uniform invertibility of the linearizations 1 R(gε ) + 7|νε |2gε + 30 Pε := Δgε − 8 of Lε at ψ0 = 1. In what follows we rely heavily on maximum principle(s). Part of the reason why this approach is successful is that the function 1 R(gε ) + 7|νε |2gε + 30 fε := 8 has a positive lower bound. Lemma 21. Let C < 3 be a positive constant and let ε be sufficiently small. We have fε ≥ C, pointwise. Proof. It is enough to show sup fε − |νε |2gε − 3 → 0 Mε
as ε → 0
(72)
or, equivalently, that the sup-norms of |R(gε ) − |νε |2gε + 6| both over the gluing region Ψε (A√ε/3 ) and over its complement Mε Ψε (A√ε/3 ) converge to 0. To prove this convergence on Ψε (A√ε/3 ), note that πε maps diffeomorphically onto the support of με , and that 2 2 |με |gε ≤ ρ2 |μ|g¯ ◦ πε−1 . 2
Thus, there is a constant c > 0 such that |με |gε ≤ cε on Ψε (A√ε/3 ). In light of (66) this means that 2
sup |νε |gε → 0 as ε → 0. Ψε (A√ε/3 ) The convergence result (72) on the gluing region now follows from Lemma 9 or, rather, estimate (70). To prove convergence away from the gluing region, note that Λ = 13 τ 2 − 3 (from (7)), so the second constraint equation yields R(g) = |K|2g − τ 2 + 2Λ = |μ|2g − 6.
(73) Mε Ψε (A√ε/3 )
Therefore, using (66), we conclude that the restriction of fε to satisfies 1 2 2 fε − |νε |2gε ◦ πε = R(g) + 7 |(πε )∗ νε |g + 30 − |(πε )∗ νε |g 8 √ 1 2 2 |μ|g − |(πε )∗ νε |g = 3 + O( ε), = 3+ 8 which completes the proof.
Strictly speaking, the operators Pε are not “geometric” due to the presence of the |νε |2gε term, so the analysis of [16] does not apply directly. There are many ways to circumvent this; for convenience, we base our argument on Proposition 3.7 of [12]. First we need the following uniform estimate (called
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the “basic estimate” in [12]). This estimate is analogous to the estimate of Lemma 16 above for the vector Laplacian. The proofs of the two lemmas, however, are quite different: Lemma 16 is proved using blow-up analysis, while the proof of the next lemma is direct and constructive. Consequently, the next lemma features a more optimal result on C (as opposed to the vector Laplacian case where we are only able to prove the existence of C). Lemma 22. Let C > 1.
1 3
be fixed, and assume ε > 0 is sufficiently small.
−1 If u is a C function on Mε with both ρ−1 ε u and ρε Pε u bounded then 2
−1 sup |ρ−1 ε u| ≤ C sup |ρε Pε u|. Mε
2.
(74)
Mε
If u is a bounded C 2 function on Mε with Pε u bounded, then sup |u| ≤ C sup |Pε u|. Mε
(75)
Mε
Similarly, if Ω is a precompact subset of Mε , and u is a continuous function on Ω that is C 2 in Ω and vanishes on ∂Ω, then −1 sup |ρ−1 ε u| ≤ C sup |ρε Pε u|, Ω
sup |u| ≤ C sup |Pε u|.
and
Ω
Ω
(76)
Ω
Proof. We start by proving (74). Note that it suffices to consider functions u for which −1 sup |ρ−1 ε u| = sup(ρε u). Mε
Mε
Given a fixed ε > 0 and a C 2 -function u ∈ C10,0 (Mε ), Yau’s Generalized Maximum Principle [12] implies that there is a sequence of points {xk } of Mε such that (i) (ii) (iii)
−1 limk→∞ [ρ u](xk ) = supMε [ρ−1 ε u] ε −1 limk→∞ d(ρε u)g (xk ) = 0 ε lim supk→∞ Δgε [ρ−1 ε u](xk ) ≤ 0.
Note that the condition (ii) can be re-written as −1 dρε lim ρ−1 du − ρ u (xk ) = 0. ε ε k→∞ ρε gε
(77)
A short computation shows that dρε −1 Δgε ρε −1 −1 dρε Δgε ρ−1 u = ρ Δ u − 2 , ρ du − ρ u . (78) − ρ−1 gε ε ε ε ε u ρε ε ρε ρε ε is bounded for each fixed ε > 0. Recall that, by Lemma 8, the quantity dρ ρε gε
Therefore, the identity (77) implies −1 Δgε ρε lim sup ρ−1 Δ u − ρ u (xk ) = lim sup Δgε ρ−1 gε ε ε ε u (xk ) ≤ 0. (79) ρε k→∞ k→∞
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Since the defining functions ρε for small ε > 0 are superharmonic (Lemma 10) and since fε ≥ C1 by Lemma 21, we see that Δgε ρε 1 −1 −1 −1 Δgε ρε −1 −1 ρ u. = ρε Pε u + ρε fε − ρε Δgε u − ρε u ≥ ρ−1 ε Pε u + ρε ρε C ε Conditions (i) and (79) now imply lim sup[ρ−1 ε Pε u](xk ) + k→+∞
1 sup[ρ−1 u] ≤ 0 C Mε ε
for small enough ε. Consequently, we have −1 −1 sup[ρ−1 ε u] ≤ C lim inf [−ρε Pε u](xk ) ≤ C sup |ρε Pε u|, Mε
k→+∞
Mε
as claimed. The proofs of the remaining three estimates are similar but considerably easier. Indeed, to prove (75) we use 1 u(xk ) C in place of (78), while (76) is proved using the ordinary maximum principle. Δgε u(xk ) = Pε u(xk ) + fε (xk )u(xk ) ≥ Pε u(xk ) +
Theorem 23. The operators Pε : C12,α (Mε ) → C10,α (Mε ) and Pε : C 2,α (Mε ) → C 0,α (Mε ) are invertible for sufficiently small ε > 0. The norm of the inverse of Pε : C 2,α (Mε ) → C 0,α (Mε ) is bounded uniformly in ε. Proof. It follows from Proposition 3.7 in [12] (together with Lemma 22) that Pε is invertible when ε is small enough, so it remains only to prove uniformity of the norm. We start by establishing a uniform elliptic estimate u2,α ≤ C (Pε u0,α + u0,0 )
(80)
in which C is independent of (sufficiently small) ε > 0. Let Φ be one of our preferred charts for Mε (see Sect. 5.1). Consider the elliptic operator PΦ,ε : ˘ → C 0,α (B) ˘ defined by C 2,α (B) PΦ,ε := ΔΦ∗ gε − Φ∗ (3 + |νε |2gε − Lε (ψ0 )); this operator is of interest since Φ∗ (Pε u) = PΦ,ε Φ∗ u. Recall that the metric Φ∗ gε is uniformly equivalent to the hyperbolic met˘ ric g˘. Furthermore, we see from Lemma 20 and (71) that the C 0,α (B)-norms ∗ 2 ∗ of Φ |νε |gε and Φ Lε (ψ0 ) are uniformly bounded. Thus, the eigenvalues of the principal symbol of PΦ,ε are uniformly bounded from below, while the ˘ of the coefficients of PΦ,ε are uniformly bounded from above. C 0,α (B)-norms ˘ such that the restrictions of our pre˘ Let B0 be a fixed precompact subset of B ˘ ferred charts to B0 still cover Mε . It follows from the standard elliptic theory [11] that there is a constant C (independent of ε, Φ and u) such that ∗ Φ∗ uC 2,α (B˘0 ) ≤ C(PΦ,ε Φ∗ uC 2,α (B) ˘ + Φ uC 0,0 (B) ˘ ).
Taking the supremum with respect to Φ now yields (80).
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Next, we combine Lemma 22 and the elliptic estimate (80). We conclude that there is a constant C (independent of ε) such that u2,α ≤ CPε u0,α for all u ∈ C (Mε ). This shows that the norm of Pε−1 : C 0,α (Mε ) → C 2,α (Mε ) is bounded independently of ε. 2,α
We solve the Lichnerowicz equation by interpreting it as a fixed point problem. More precisely, consider the quadratic error term Qε (η) := Lε (ψ0 + η) − Lε (ψ0 ) − Pε η 3 1 (1 + η)5 − 1 − 5η = |νε |2 (1 + η)−7 − 1 + 7η − 8 4 and the corresponding map Gε : η → −(Pε )−1 (Lε (ψ0 ) + Qε (η)). Note that, by Lemma 20 and Theorem 23, Gε : C 2,α (Mε ) → C 2,α (Mε )
and Gε : C12,α (Mε ) → C12,α (Mε ).
It is easy to see that the solutions ψε = ψ0 + ηε of the Lichnerowicz equation correspond to the fixed points ηε of Gε . In what follows we argue that Gε is a contraction mapping from a small ball in C 2,α (Mε ) to itself. Lemma 24. For sufficiently large C and √ sufficiently small ε, the map Gε is a contraction of the closed ball of radius C ε around 0 in C 2,α (Mε ). √ Proof. Let η1 , η2 ∈ C 2,α (Mε ) be of norm O( ε). Assuming in addition that |η1 | < 1 and |η2 | < 1, using the bound on |νε |2gε expressed in (71), and using the binomial expansion formulae, we find that for sufficiently small ε > 0, 1 2 −7 −7 Qε (η2 ) − Qε (η1 )0,α = 8 |νε | (1 + η1 ) − (1 + η2 ) + 7(η1 − η2 ) 3 5 5 − (1 + η1 ) − (1 + η2 ) − 5(η1 − η2 ) 4 0,α √ √ ≤ O( ε)η2 − η1 0,α ≤ O( ε)η2 − η1 2,α . √ A similar calculation shows that if η2,α = O( ε), then Qε (η) √ 0,α = O(ε). As a consequence of Lemma 20, functions η with η2,α = O( ε) also satisfy √ Lε (ψ0 ) + Qε (η)0,α = O( ε). Combining this with Theorem 23 we have that there is a sufficiently large constant C > 0 such that for sufficiently small ε > 0 √ Gε (η)2,α = (Pε )−1 (Lε (1) + Qε (η)) ≤ C ε. 2,α
→ Thus, we have determined that Gε : To see that this map is a contraction, we compute √ Gε (η1 ) − Gε (η2 )2,α = (Pε )−1 (Qε (η1 ) − Qε (η2 )) = O( ε)η1 − η2 2,α . B C √ε
B C √ε .
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It thus follows that if ε > 0 is small enough, the map Gε : B C √ε → B C √ε is a contraction. We are now ready to state and prove our main result regarding solutions of the Lichnerowicz equation for the parametrized sets of conformal data (Mε , gε , νε , τ ). This result holds, with appropriate modification of the coefficients and exponents in the Lichnerowicz equation, for arbitrary dimension n ≥ 3: Theorem 25. If ε is sufficiently small, there exists a polyhomogeneous function ψε on Mε which has a C 2 extension to M ε that is equal to 1 on ∂M ε , and satisfies 1 1 3 (81) Δgε ψε − R(gε )ψε + |νε |2gε ψε−7 − ψε5 = 0. 8 8 4 The function ψε is a small perturbation of the constant function ψ0 ≡ 1 in the sense that √ ψε − ψ0 2,α = O( ε) as ε → 0. (82) On any fixed compact subset of M , πε∗ ψε converges to 1 uniformly with all derivatives. Proof. By the Banach Fixed Point Theorem, the sequence η0,ε := 0, η1,ε := Gε (η0,ε ), . . . , ηn,ε := Gε (ηn−1,ε ), . . . converges in B C √ε ⊆ C 2,α (Mε ). Thus, there exists a function ηε on Mε such √ that ηε 2,α ≤ C ε and such that the function ψε := ψ0 + ηε solves the Lichnerowicz equation and satisfies (82). On a fixed compact subset G ⊂ M , (81) shows that Δg (πε∗ ψε ) is equal to a smooth function that converges in C 2,α (G) as ε → 0 to 18 (R(g) − |μ|2g + 6), which is identically zero by (73). By standard elliptic estimates, πε∗ ψε → 1 in C 4,α (G), and then iteration of this argument shows that the convergence is uniform with all derivatives. To address the regularity of ψε , note that ηn,ε ∈ C12,α (Mε ) for all n, ε. Consequently, each ηn,ε has a continuous extension to M ε that vanishes on ∂M ε . Because convergence in C 2,α (Mε ) implies uniform convergence, it follows that, for each fixed ε, the limit ηε := limn→∞ ηn,ε also has a continuous extension to M ε and vanishes on ∂M ε . We now conclude that ψε = ψ0 + ηε approaches 1 at the ideal boundary. Therefore, Corollary 7.4.2 of [1] applies and we see that ψε is polyhomogeneous. Inserting the asymptotic expansion for ψε into (81) and comparing like terms inductively, we find that the first log terms in ψε appear with ρ3ε . (These terms arise as a consequence of the indicial roots of the linearized Lichnerowicz operator.) It follows that ψε has a C 2 extension to M ε . With these solutions ψε to the Lichnerowicz equation in hand, we readily verify that the one-parameter family of initial data sets (Mε , ψε4 gε , ψε−2 νε + (τ /3)ψε4 gε ) satisfies the list of properties outlined in Theorem 2. Hence, we
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have constructed the desired asymptotic gluing of AH initial data satisfying the Einstein constraint equations.
8. Conclusions The gluing construction which we have discussed and verified here allows one to take a pair (or more) of CMC initial data sets for isolated systems with unique asymptotic regions—either asymptotically null data sets in asymptotically flat spacetimes, or data sets in asymptotically anti-de Sitter spacetimes—and glue them together in such a way that the spacetime which develops from this glued data has a single asymptotic region. In the case that the original data sets are asymptotically null, one may wonder how the Bondi mass [20] for the glued data compares with the Bondi masses for the original data sets. We will study this issue in future work. There are a number of ways in which the results proven here might be extended. It should be straightforward to be able to handle solutions of the Einstein–Maxwell or Einstein-fluid constraints, rather than the Einstein vacuum constraint equations. A more challenging generalization we plan to consider is to allow for initial data sets which do not have constant mean curvature. We have done this in earlier gluing work [10] using localized deformations of the original data sets so that, in small neighborhoods of the gluing points, the mildly perturbed original initial data sets do have constant mean curvature. The work of Bartnik [3] shows that this sort of deformation can always be done. A key first step in generalizing our results here to non CMC initial data sets is to generalize Bartnik’s local CMC deformation results to neighborhoods of asymptotic points in AH initial data sets. This issue is under consideration. One further generalization of some interest is to attempt to carry out localized gluing at asymptotic points in AH initial data sets. To do this, it would likely be necessary to determine if the work of Chru´sciel and Delay [8] generalizes so that it holds in asymptotic neighborhoods in AH initial data sets. While this may prove to be difficult, we do believe that we will be able to localize the gluing to the extent that in regions bounded away from the ideal boundary, the glued data is unchanged from the original data.
Acknowledgements We would like to thank the referee for several useful suggestions that have resulted in improvements to the paper.
References [1] Andersson, L., Chru´sciel, P. T.: Solutions of the constraint equations in general relativity satisfying “hyperboloidal boundary conditions”. Dissertationes Math. (Rozprawy Mat.) 355, 100 pp (1996)
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[2] Andersson, L., Chru´sciel, P.T.: On “hyperboloidal” Cauchy data for vacuum Einstein equations and obstructions to smoothness of scri. Commun. Math. Phys. 161, 533–568 (1994) [3] Bartnik, R.: Regularity of variational maximal surfaces. Acta Math. 161, 145–181 (1988) [4] Bartnik, R., Isenberg, J.: The constraint equations. In: Chrusciel, P.T., Friedrich, H. (eds.) The Einstein Equations and the Large Scale Behavior of Gravitational Fields, pp. 1–38. Birkh¨ auser, Basel (2004) [5] Choquet-Bruhat, Y.: Th´eor`eme d’existence pour certains syst`emes d’´equations aux d´eriv´ees partialles non lin´eaires. Acta Math. 88, 141–225 (1952) [6] Choquet-Bruhat, Y., Isenberg, J., Pollack, D.: The constraint equations for the Einstein-scalar field system on compact manifolds. Class. Quantum Gravit. 24, 809–828 (2007) [7] Choquet-Bruhat, Y., York, J.W. Jr.: The Cauchy problem. In: Held, A. (ed.) General Relativity and Gravitation, vol. 1, pp. 99–172. Plenum, New York (1980) [8] Chru´sciel, P.T., Delay, E.: On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications. M´em. Soc. Math. Fr. (N.S.) 94, vi+103 pp (2003) [9] Chru´sciel, P.T., Delay, E., Lee, J.M., Skinner, D.N.: Boundary regularity of conformally compact Einstein metrics. J. Differential Geom. 69, 111–136 (2005) [10] Chru´sciel, P.T., Isenberg, J., Pollack, D.: Initial data engineering. Commun. Math. Phys. 257, 29–42 (2005) [11] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of the Second Order. Springer, Berlin (1983) [12] Graham, C.R., Lee, J.M.: Einstein metrics with prescribed conformal infinity on the ball. Adv. Math. 87, 186–225 (1991) [13] Isenberg, J., Maxwell, D., Pollack, D.: A gluing construction for non-vacuum solutions of the Einstein-constraint equations. Adv. Theor. Math. Phys. 9, 129– 172 (2005) [14] Isenberg, J., Mazzeo, R., Pollack, D.: Gluing and wormholes for the Einstein constraint equations. Commun. Math. Phys. 231, 529–568 (2002) [15] Isenberg, J., Mazzeo, R., Pollack, D.: On the topology of vacuum spacetimes. Ann. Henri Poincar´e 4, 369–383 (2003) [16] Lee, J. M.: Fredholm operators and Einstein metrics on conformally compact manifolds. Mem. Am. Math. Soc. 183, vi+83 pp (2006) [17] Lee, J.M., Parker, T.H.: The Yamabe Problem. Bull. Am. Math. Soc. 17, 37–91 (1987) [18] Mazzeo, R.: Elliptic theory of differential edge operators I. Commun. Partial Differ. Equ. 16, 1615–1664 (1991) [19] Mazzeo, R., Pacard, F.: Maskit combinations of Poincar´e-Einstein metrics. Adv. Math. 204(2), 379–412 (2006) [20] Wald, R.M.: General Relativity. University of Chicago Press, Chicago (1984) [21] York, J.W. Jr.: Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial-value problem of general relativity. J. Math. Phys. 14, 456–464 (1973)
Vol. 11 (2010)
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James Isenberg Department of Mathematics University of Oregon Eugene Oregon 97403-5203 USA e-mail:
[email protected] John M. Lee University of Washington Mathematics Department Box 354350 Seattle WA 98195-4350 USA e-mail:
[email protected] Iva Stavrov Allen Department of Mathematical Sciences Lewis & Clark College Portland OR 97219 USA e-mail:
[email protected] Communicated by Piotr T. Chrusciel. Received: October 9, 2009. Accepted: June 19, 2010.
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Large Normally Hyperbolic Cylinders in a priori Stable Hamiltonian Systems Patrick Bernard Abstract. We prove the existence of normally hyperbolic cylinders in a priori stable Hamiltonian systems the size of which is bounded from below independently of the size of the perturbation. This result should have applications to the study of Arnold’s diffusion.
A major problem in dynamical systems consists in studying the Hamiltonian systems on Tn × Rn of the form H(q, p) = h(p) − 2 G(t, q, p),
(t, q, p) ∈ T × Tn × Rn .
(H)
Here should be considered as a small perturbation parameter, we put a square because the sign of the perturbation will play a role in our discussion. In the unperturbed system ( = 0) the momentum variable p is constant. We want to study the dynamics of the perturbed system in the neighborhood of a torus {p = p0 }, corresponding to a resonant frequency. There is no loss of generality in assuming that the frequency is of the form ∂h(p0 ) = (ω, 0) ∈ Rm × Rr . If the restricted frequency ω is non-resonant in Rm , then it is expected that the averaged system Ha (q, p) = Ha (q1 , q2 , p1 , p2 ) = h(p) − 2 V (q2 ) should locally approximate the dynamics of (H) near p = p0 = q = (q1 , q2 ) ∈ Tm × Tr and p = (p1 , p2 ) ∈ Rm × Rr , and where V (q2 ) = G(t, q1 , q2 , p0 )dt dq1 .
(Ha ) (p01 , p02 ),
where
We make the following hypothesis on the averaged system: Hypothesis 1. The function h is convex with positive definite Hessian and the averaged potential V has a non-degenerate local maximum at q2 = 0. membre de l’IUF.
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Under Hypothesis 1, the averaged system has an invariant manifold of equations (∂p2 h = 0, q2 = 0) ∈ Tn × Rn . Because h has positive definite Hessian, the equation ∂p2 h(p1 , p2 ) = 0 is nonsingular and it defines a smooth m-dimensional manifold in Rn which can also be described parametrically by the relation p2 = P2 (p1 ) for some function P2 : Rm −→ Rr . Therefore, the corresponding invariant manifold can be written in a parametric form as {(q1 , 0, p1 , P2 (p1 )); (q1 , p1 ) ∈ Tm × Rm }, it is a cylinder. Moreover, this manifold is normally hyperbolic in the sense of [12]. It is necessary at this point to precise the terminology. An open manifold will be called weakly invariant for a flow if the vector field is tangent at each point to this manifold. It will be called strongly invariant if it contains the full orbit of each of its points. A compact strongly invariant manifold is called normally hyperbolic if it is eventually absolutely 1-normally hyperbolic for the time-one flow in the sense of [12], Definition 4. Definition 1. A weakly invariant open manifold N (for some vector field X) is called normally hyperbolic if there exists: • A vector field Y on a compact manifold M . • An embedding i : U −→ M from a neighborhood U of N into M which conjugates X|U and Y|i(U ) . ˜ in M (for • A normally hyperbolic strongly invariant compact manifold N ˜. the vector field Y ) such that i(N ) ⊂ N Returning to the invariant cylinder of the averaged system, we observe that the open sub-cylinder {(q1 , 0, p1 , P2 (p1 )); (q1 , p1 ) ∈ Tm × Rm , p1 < δ} ,
δ>0
is a normally hyperbolic weakly (and even strongly) invariant open submanifold for the averaged system in the sense of Definition 1. From this observation, and from the fact that the full system can be considered locally (near p = p0 ) as a perturbation of the averaged system, one can prove the existence of a small normally hyperbolic weakly invariant cylinder in the full system, this is well understood. This cylinders can also be seen as the center manifold of a “whiskered” (or partially hyperbolic) torus, which is the continuation in the full systems of the invariant torus {(t, q1 , 0, p0 ),
(t, q1 ) ∈ T × Tm }
which exists in the averaged system. The name whiskered comes from the fact that this torus has hyperbolic normal directions, this name (as well as the corresponding object) was introduced by Arnold in [1]. The existence of a whiskered torus in the original system was proved in [18], following earlier works on the persistence of partially hyperbolic KAM tori. It is well understood, see for example [4] that such a torus must be contained in an invariant cylinder which is normally hyperbolic. Proving the existence of whiskered tori
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involves KAM theory, which is quite demanding in terms of regularity, while the existence of the invariant cylinder relies on the softer theory of normal hyperbolicity. The idea of embedding whiskered tori into a normally hyperbolic cylinder and to use the theory of normal hyperbolicity in the context of Arnold diffusion is more recent than the paper of Arnold. To the best of our knowledge, it appears first in Moeckel [16]. It was then progressively understood that normally hyperbolic invariant cylinders can be used to produce diffusion even in the absence of whiskered tori. We described two well-known methods allowing to prove the existence of small normally hyperbolic weakly invariant cylinders in the full system for > 0. However, the size of the invariant cylinder that has been obtained in the literature is small, meaning that it converges to 0 with . Our point in the present paper is that a large normally hyperbolic weakly invariant cylinder actually exists: Theorem 1. Assume that H is smooth (or at least C r for a sufficiently large r) and satisfies Hypothesis 1. Assume that ω is Diophantine, and fix κ > 0. Then, there exists an open ball B ⊂ Rm containing p01 , a neighborhood U of 0 in Tr , a positive number 0 and, for < 0 two C 1 functions Q2 : T × Tm × B −→ U ⊂ Tr
and
P2 : T × Tm × B −→ Rr
such that the annulus A = {(t, q1 , Q2 (t, q1 , p1 ), p1 , P2 (t, q1 , p1 )),
(t, q1 , p1 ) ∈ T × Tm × B}
is weakly invariant for (H) (in the sense that the Hamiltonian vector field is tangent to it). We have P2 −→ P20 uniformly as −→ 0, where P20 is the function (t, q1 , p1 ) −→ P2 (p1 ). Moreover, we have P2 − P20 C 1 κ, and Q2 C 1 κ/. Each strongly invariant set of (H) (in the sense that it contains the full orbit of each of its points, for example, a whiskered torus) contained in the domain D := T × Tm × U × B × {p2 ∈ Rr : p2 } is contained in A for < 0 . The cylinder A is normally hyperbolic and symplectic. The novelty here is that the ball B does not depend on . Easy examples show that we cannot expect a fine control of the asymptotic behavior of Q2 in terms of the averaged system only except if we restrict to smaller domains depending on . This asymptotic behavior also depends on the averaged systems at other frequencies. However, the very weak estimates we have are sufficient to describe the restricted dynamics. Let A0 ⊂ Tn × Rn be the restriction of the invariant annulus to the section {t = 0}, A0 = {(q1 , Q2 (0, q1 , p1 ), p1 , P2 (0, q1 , p1 )),
(q1 , p1 ) ∈ Tm × B},
and let φ : A0 −→ Tn × Rn be the time-one flow of H (which is well-defined on A0 when is small enough). Then, A0 is somewhat invariant for φ (although there are some difficulties near the boundary) in a sense that will be given
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more precisely below. We define the map Φ : Tm × B −→ Tm × Rm as the restriction of φ to A0 seen in coordinates (q1 , p1 ), more precisely Φ(q1 , p1 ) = (q1 , p1 ) ◦ φ (q1 , Q2 (0, q1 , p1 ), p1 , P2 (0, q1 , p1 )). Note that this map is well-defined on Tm × B. Let us finally consider an open ball B0 ⊂ Rm which contains p0 and whose closure is contained in B, and set A00 = {(q1 , Q2 (0, q1 , p1 ), p1 , P2 (0, q1 , p1 )),
(q1 , p1 ) ∈ Tm × B0 }.
Proposition 2. The map Φ is converging uniformly (when −→ 0) on Tm × B0 to the map q q + ∂p1 h (p1 , P2 (p1 )) Φ0 : 1 −→ 1 , p1 p1 which gives the unperturbed dynamics on the invariant cylinder of the averaged system. Moreover, we have φ(A00 ) ⊂ A0 when is small enough. Finally, given η > 0, we can choose the ball B0 small enough so that the inequality dΦ − dΦ0 C 0 η m
holds on T × B0 when is small enough. The frequency map p1 −→ Ω0 (p1 ) := ∂p1 h (p1 , P2 (p1 )) has positive torsion in the sense that ∂p1 Ω0 = ∂p21 h(p1 , P2 (p1 )) is a positive definite symmetric matrix for all p1 ∈ Rm . As a consequence, when is small enough, the restricted map Φ has positive torsion in a neighborhood (independent of ) of Tm × {p01 }, in the sense that ∂p1 (q1 ◦ Φ)(q1 ,p1 ) ρ1 · ρ1 > 0 ∀ρ1 ∈ Rm for all q1 ∈ Tm and p1 ∈ B0 provided that B0 has been chosen small enough. The map Φ is symplectic with respect to the symplectic form obtained by restriction of the ambient symplectic form to A0 . It is part of the statement of Theorem 1 that this form is non-degenerate on A0 . Note that this symplectic form is not dq1 ∧ dp1 in general. In the case m = 1, (but for any dimension n) one can combine these results with the existing techniques on the a priori unstable situation, like the variational methods coming from Mather Theory (see [2,14]), developed for the a priori unstable situation in [3,6,7] or more geometric methods such as [11] (the papers [8,18] also treat the a priori unstable situation, but it seems to me at first sight that they require too strong information on the restricted dynamics to be applicable here). One can then hope to obtain, under additional non-degeneracy assumptions, the existence of restricted Arnold diffusion in the following sense: there exists δ > 0 and 0 such that, for each ∈ ]0, 0 [ there exists an orbit (q (t), p (t)) with the following property: the image p (R) is not contained in any ball of radius δ in Rn . Once again, the key point here is that δ can be chosen independent of . Specifying the needed “non-degeneracy
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assumptions” will require some further work, but I believe it will not require any method beyond those which are already available. Of course, finding “global” Arnold diffusion, as announced in [15], that is, orbits wondering in the whole phase space along different resonant lines (or far away along a given resonant line) requires a specific study of relative resonances (when the restricted frequency ω is resonant), where the existence of normally hyperbolic invariant cylinders cannot be obtained by the method used in the present paper. Let us close this introduction with a remark on uniqueness. In general, there is no uniqueness statement for the normally invariant cylinder we obtain. + However, in the case m = 1, we can obtain a stronger result: let [p− 1 , p1 ] ⊂ + B ⊂ R be an interval such that both Ω0 (p− 1 ) and Ω0 (p1 ) are Diophantine. Then, there exists whiskered tori T− and T+ of dimension 2 in T × Tn × Rn which are close to the unperturbed tori − (t, q1 ) ∈ T × T T−0 = t, q1 , 0, p− 1 , P2 (p1 ) : and T+0 =
+ t, q1 , 0, p+ 1 , P2 (p1 :
(t, q1 ) ∈ T × T .
The whiskered tori T± are contained in the annulus A . They bound a compact part A= of A which is then strongly invariant in the sense that it contains the full orbit of each of its points. The annulus A= is then unique in the sense that if A˜ is another normally hyperbolic cylinder given by Theorem 1 (with the same domain B), then it must contain A= . The cylinder A= is a normally hyperbolic invariant cylinder in the genuine sense. If the interval [p− , p+ ] has been chosen small enough, then the restricted map Φ : A= −→ A= is a C 1 area preserving twist map (for the appropriate area form). When m > 1 one should not expect the same kind of properties, since Arnold diffusion may occur inside the invariant cylinder.
1. Averaging To apply averaging methods, it is easier to consider the extended phase space (t, e, q, p) ∈ T × R × Tn × Rn where the Hamiltonian flow can be seen as the Hamiltonian flow of the autonomous Hamiltonian function ˜ e, q, p) = h(p) + e − 2 G(t, q, p) H(t, ˜ = 0. Then, we consider a smooth on one of its energy surfaces, for example H solution f (t, q) of the Homological equation ∂t f + ∂q f · (ω, 0) = G(t, q, p0 ) − V (q2 ). Such a solution exists because ω is Diophantine, as can be checked easily by power series expansion. It is unique up to an additive constant. We consider the smooth symplectic diffeomorphism ψ : (t, e, q, p) −→ (t, e + 2 ∂t f (t, q), q, p + 2 ∂q f (t, q))
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and use the same notation for the diffeomorphism (t, q, p) −→ (t, q, p + 2 ∂q f (t, q)). We have ˜ ◦ ψ = h(p) + e − 2 V (q2 ) − 2 R(t, q, p) + O(4 ), H where R(t, q, p) = G(t, q, p) − G(t, q, p0 ). In other words, by the timedependent symplectic change of coordinates ψ , we have reduced the study of H to the study of the time-dependent Hamiltonian H1 (t, q, p) = h(p) − 2 V (q2 ) − 2 R(t, q, p) + O(4 ) where R = O(p − p0 ). As a consequence, Theorem 1 holds for H if it holds for H1 . More precisely, assume that there exists an invariant cylinder A = (t, q1 , Q2 (t, q1 , p1 ), p1 , P2 (t, q1 , p1 )) for H1 , with Q2 C 1 κ/2 and P2 − P20 C 1 κ/2. Then, the annulus A := ψ (A ) is invariant for H. Since ψ is 2 -close to the identity, while Q2 C 1 κ/2, the annulus A has the form A = (t, q1 , Q2 (t, q1 , p1 ), p1 , P2 (t, q1 , p1 )) for C 1 functions Q2 , P2 which satisfy Q2 C 1 κ/ and P2 − P20 C 1 κ. We will prove that Theorem 1 holds for H1 in Sect. 4. We first expose some useful tools.
2. Normally Hyperbolic Manifolds We shall now present a version of the classical theory of normally hyperbolic manifolds adapted for our purpose. On Rnz × Rnx × Rny , let us consider the time-dependent vector field z˙ = Z(t, z, x, y) x˙ = A(z)x y˙ = −B(z)y. We assume that the function Z : R × Rnz × Rnx × Rny −→ Rnz is C 1 -bounded in the domain R × Rnz × {x ∈ Rnx : x < 1} × {y ∈ Rny : y < 1},
(D)
and that the matrices A and B are C 1 -bounded functions of z. Moreover, we assume that there exists constants a > b > 0 such that A(z)x · x ax2 ,
B(z)y · y ay2
for all x, y, z, and such that ∂(t,z) Z(t, z, x, y) b
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for all (t, z, x, y) belonging to (D). We consider the perturbed vector field z˙ = Z(t, z, x, y) + Rz (t, z, x, y) x˙ = A(z)x + Rx (t, z, x, y) y˙ = −B(z)y + Rx (t, z, x, y). where R = (Rz , Rx , Ry ) is seen as a small perturbation. Theorem 2. There exists > 0 such that, when RC 1 < , the maximal invariant set of the perturbed vector field contained in the domain (D) is a graph of the form {(t, z, X(t, z), Y (t, z)),
(t, z) ∈ R × Rnz }
where X and Y are C 1 maps. This graph is normally hyperbolic, and it is contained in the domain Rnz × {x ∈ Rnx : x (2/a)RC 0 } × {y ∈ Rny : y (2/a)RC 0 }. In other words, we have (X, Y )C 0 (2/a)RC 0 . The C 1 norm of (X, Y ) is converging to zero when the C 1 norm of the perturbation converges to zero. Proof. The invariant space Rnz is normally hyperbolic in the sense of [10,12]. As a consequence, the standard theory applies and implies the existence of functions X and Y such that the graph (t, z, X(t, z), Y (t, z)) is invariant, normally hyperbolic, and contained in (D). Note that we are slightly outside of the hypotheses of the statements in [12] because our unperturbed manifold is not compact. However, the results actually depend on uniform estimates rather than on compactness (see [9, Appendix B], for example, see also [5]), and we assumed such uniform estimates. Let us now prove the estimate on (X, Y ). We have the inequality x˙ · x ax2 + x · Rx ax(x − Rx C 0 /a) which implies that x˙ · x xRx C 0 if 2Rx C 0 /a x 1, hence this domain cannot intersect the invariant graph. Similar considerations show that the domain 2Ry C 0 /a y 1 cannot intersect the graph.
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3. Hyperbolic Linear System Let us consider the linear Hamiltonian system on Rn × Rn generated by the Hamiltonian 1 1 H(q, p) = Bp, p − Aq, q , 2 2 where both A and B are positive definite symmetric matrices. We recall that this system can be reduced to G(x, y) = Dx, y , where D is a positive definite symmetric matrix, by a linear symplectic change of variables (q, p) −→ (x, y). To do so, we consider the symmetric positive definite matrix 1/2 L := A−1/2 (A1/2 BA1/2 )1/2 A−1/2 , which is the only symmetric and positive definite solution of the equation L2 AL2 = B. Considering the change of variables 1 1 x = √ (Lp + L−1 q); y = √ (Lp − L−1 q) 2 2 or equivalently 1 1 q = √ L(x − y); p = √ L−1 (x + y), 2 2 an elementary calculation shows that we obtain the desired form for the Hamiltonian in coordinates (x, y), with D = LAL = L−1 BL−1 . As a consequence, the equations of motions in the new variables take the block-diagonal form x˙ = Dx;
y˙ = −Dy.
In the original coordinates (q, p) the stable space (which is the space x = 0) is the space {(q, −L2 q), q ∈ Rn } while the unstable space is {(q, L2 q), q ∈ Rn }.
4. Proof of Theorem 1 We now prove Theorem 1 for the Hamiltonian H1 (t, q, p) = h(p) − 2 V (q2 ) − 2 R(t, q, p) + O(2+γ ), where R = O(p − p0 ) and γ > 0 (γ = 2 in our situation). We assume that Hypothesis 1 holds. We lift all the angular variables to the universal covering, and see H1 as a Hamiltonian of the variables (t, q, p) = (t, q1 , q2 , p1 , p2 ) ∈ R × Rm × Rr × Rm × Rr which is one-periodic in t, q. We assume that p0 = 0. We will need some notations. We set A := ∂ 2 V (0), it is a symmetric positive definite matrix. We will denote by B(p1 ) a matrix which depends
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smoothly on p1 , is uniformly positive definite, is constant outside of a neighborhood of p1 = 0 in Rm , and coincides with ∂p22 h(p1 , P2 (p1 )) in a neighborhood of p1 = 0. We will denote by P˜2 (p1 ) a compactly supported smooth function P˜2 : Rm −→ Rr which coincides with P2 around p1 = 0. Finally, we will denote by h0 (p1 ) a smooth compactly supported function which is equal to h(p1 , P2 (p1 )) around p1 = 0. It is useful to introduce two new positive parameters α and δ. We always assume that 0 < < δ < α < 1. In the sequel, we shall chose α small, then δ small with respect to α, and work with small enough with respect to α and δ. The parameter δ represents the size of the normally hyperbolic cylinder we intend to find. We will denote by χ a smooth function of its arguments which may depend (in an unexplicited way) on the parameters , δ, but which is C 2 -bounded, uniformly in , δ. The notation χ will be used in a similar way when only C 1 bounds are assumed. Lemma 3. There exists a smooth Hamiltonian function H2 (t, q, p) (which depends on the parameters , δ) of the form 1 2 H2 = h0 (p1 ) + B(p1 ) · (p2 − P˜2 (p1 ))2 − A · q22 2 2
√ 3 2 3/2 ˜ + χ p1 , (p2 − P2 (p1 ))/ + δ χ(q2 / δ) + 2 δχ(t, q, p/δ) + 2+γ χ(t, q, p) which coincides with H1 on the domain √ q2 δ, p1 δ, p2 − P2 (p1 ) . Proof. Let us expand the function h with respect to p2 at the point P2 (p1 ): h(p1 , p2 ) 1 = h(p1 , P2 (p1 ))+ ∂p22 h(p1 , P2 (p1 )) · (p2 −P2 (p1 ))2 +S(p) · (p1 −P2 (p1 ))3 2 where S(p) is a 3-linear form on Rr depending smoothly on p. We consider a 3-form S(p) which depends smoothly on p, is compactly supported, and is equal to S(p) near p = 0. Let i : Rk −→ Rk (for any k) be a compactly supported smooth map which is equal to the identity on the unit ball. Then the function 3 1 h0 (p1 ) + B(p1 ) · (p1 − P˜2 (p1 ))2 + 3 S(p) · i (p2 − P˜2 (p1 ))/ 2
1 = h0 (p1 ) + B(p1 ) · (p1 − P˜2 (p1 ))2 + 3 χ p1 , (p2 − P˜2 (p1 ))/ 2 is equal to h if p belongs to a given neighborhood of 0 (independent of , δ) and satisfies p2 − P˜2 (p1 ) . Similarly, we write 1 V (q2 ) = A · q22 + W (q2 ) · q23 2
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for some 3-linear form W (q2 ). It is equal to √ √ 3 1 1 A · q22 + δ 3/2 W (q2 ) · i q2 / δ = A · q22 + δ 3/2 χ q2 / δ 2 2 √ on {q2 δ}. Finally, we observe that the function R(t, q, p) can be written in the form R(t, q, p) = L(t, q, p) · p and is equal to the function δL(t, q, p) · i(p/δ) = δχ(t, q, p/δ) on {p δ}. Collecting all terms proves the Lemma.
We will now prove the existence of a normally hyperbolic invariant graph for H2 contained in the region √ {q2 δ, p2 − P˜2 (p1 ) } Its intersection with {p1 < δ} will give a weakly invariant manifold for H1 (meaning that the Hamiltonian vector field of H1 is tangent to it). To simplify the following equations, we set 1 h2 (p) := h0 (p1 ) + B(p1 ) · (p2 − P˜2 (p1 ))2 . 2 The Hamiltonian vector field of H2 can be written
q˙1 = ∂p1 h2 (p) + 2 χ p1 , (p2 − P˜2 (p1 ))/ + 2 χ(t, q, p/δ) p˙1 = 0 + 2 δχ(t, q, p/δ) q˙2 = B(p1 )(p2 − P˜2 (p1 )) + 2 χ(p1 , (p2 − P˜2 (p1 ))/) + 2 χ(t, q, p) √ p˙2 = 2 Aq2 + 2 δχ(q2 / δ) + 2 δχ(t, q1 , q2 , p/δ) recalling the convention that χ(.) always denotes a C 1 function of its arguments, depending on and δ, but bounded in C 1 independently of δ and . Motivated by Sect. 2, we set
1/2 , L(p1 ) = A−1/2 (A1/2 B(p1 )A1/2 )1/2 A−1/2 and perform the change of variables (t, q1 , p1 , q2 , p2 ) −→ (τ, θ, r, x, y) given by: τ = t,
θ = αq1 , r = p1 , x = L(p1 )(p2 − P˜2 (p1 ))+L−1 (p1 )q2 , y = L(p1 )(p2 − P˜2 (p1 ))−L−1 (p1 )q2 ,
recalling that α is a fixed positive parameter. Equivalently, this can be written t = τ /,
q1 = θ/α,
q2 = L(r)(x − y)/2,
p1 = r, p2 = P˜2 (r) + L−1 (r)(x + y)/2.
In the new coordinates, the principal part of the vector field takes the form (denoting f´ for df /dτ ) θ´ = αΩ(r, x, y),
r´ = 0,
x ´ = D(r)x,
y´ = −D(r)y,
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Ω(r, x, y) := ∂p1 h2 r, P˜2 (r) + L−1 (r)(x + y)/2
and D(r) := L(r)AL(r) = L−1 (r)B(r)L−1 (r). The equality above holds because L(r) solves the equation L2 (r)AL2 (r) = B(r). Let us write in details the calculations leading to the expressions of x ´ := dx/dτ (the calculation for y´ is similar):
´ x = x˙ = L(p1 ) p˙2 − ∂p1 P˜2 · p˙1 + L−1 (p1 )q˙2 + (∂p1 L · p˙1 ) p2 − P˜2 (p1 ) + ∂p1 (L−1 ) · p˙1 q2 = 2 L(p1 )Aq2 + L−1 (p1 )B(p1 )(p2 − P˜2 (p1 )) + 2 δχ(t, q, p/δ, x, y) + 3 χ(p1 , (x + y)/) √ + 2 δχ(q2 / δ) + 2+γ χ(t, q, p, x, y) = L(r)AL(r)(x − y)/2 + L−1 (r)B(r)L−1 (r)(x + y)/2 √ √ √ + 2 δχ(τ /, θ/, r/δ, x/δ, y/δ, x/, y/)+2 δχ(r/ δ, x/ δ, y/ δ) = D(r)x + 2 δχ(τ /, θ/, r/δ, x/δ, y/δ, x/, y/) √ √ √ + 2 δχ(r/ δ, x/ δ, y/ δ). The function Ω(r, x, y) is C 1 -bounded on {(r, x, y),
x 1, y 1} .
We can choose α < 1 once and for all in order that the principal part of the vector field satisfies the hypotheses of Theorem 2. The full vector field can be written in the new coordinates, (with the notation f´ := df /dτ ): θ´ = αΩ(r, x, y) + 2 χ(τ /, θ/α, r, x/, y/) r´ = 0 + δχ(τ /, θ/α, r/δ, x/δ, y/δ, x/, y/) x ´ = D(r)x + δχ(τ /, θ/α, r/δ, x/δ, y/δ, x/, y/) √ √ √ + δχ(r/ δ, x/ δ, y/ δ) y´ = −D(r)y + δχ(τ /, θ/α, r/δ, x/δ, y/δ, x/, y/) √ √ √ + δχ(r/ δ, x/ δ, y/ δ). In this expression, we observe √ that the uniform norm of the perturbation is O(δ) while the C 1 norm is O( δ) (recall that 0 < < δ < 1). We can apply Theorem 2 and find a unique bounded normally hyperbolic invariant graph (τ, θ, X(τ, θ, r), r, Y (τ, θ, r)). Moreover Theorem 2 also implies that (X, Y )C 0 Cδ. Because the invariant graph we have obtained is the maximal invariant set contained in the domain {x 1, y 1}, and since the vector field is -periodic in t and α-periodic in q1 , we conclude that the functions X and Y
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are -periodic in t and α-periodic in q1 . In the initial coordinates, we have an invariant graph (t, q1 , Q2 (t, q1 , p1 ), p1 , P2 (t, q1 , p1 )) with Q2 (t, q1 , p1 ) = L(p1 ) (X(t, q1 , p1 ) − Y (t, q1 , p1 )) /2 and P2 (t, q1 , p1 ) = P˜2 (p1 ) + L−1 (p1 ) (X(t, q1 , p1 ) + Y (t, q1 , p1 )) /2. The functions Q2 and P2 are 1-periodic in (t, q1 ). The invariant graph we have obtained is normally hyperbolic for the flow of H2 , and its strong stable and strong unstable directions have the same dimension r. It follows from general results on partial hyperbolicity in a symplectic context (see e.g. [13, Proposition 1.8.3]1 ) that it is a symplectic manifold. This means that the restriction to the invariant graph of the ambient symplectic form is a symplectic form. Observing that Q2 C 0 Cδ,
P2 C 0 Cδ,
we infer that the annulus {(t, q1 , Q2 (t, q1 , p1 ), p1 , P2 (t, q1 , p1 )) :
t ∈ T, q1 ∈ Tm , p1 ∈ Rm , p1 < δ} ⊂ T × Tn × Rn
is contained in the domain √ {q2 δ, p1 δ, p2 − P˜2 (p1 ) } where H2 = H1 provided δ has been chosen small enough. It is thus a weakly invariant cylinder for H1 i.e. the extended Hamiltonian vector field of H1 on T × Tn × Rn is tangent to this annulus at each point. Orbits may still exit from the cylinder through its boundary. We finish with √ the estimates on the C 1 norms. Since the C 1 size of the perturbation is O( δ), we can make it as small as we want by choosing δ small. We can thus assume that (X, Y )C 1 is small, and this implies the desired C 1 estimates on P2 and Q2 . We have proved Theorem 1 for H1 , we conclude from Sect. 1 that Theorem 1 holds for H.
5. Proof of Proposition 2 Let (q1 , p1 ) be given in Tm × B, and let (q1 (t), q2 (t), p1 (t), p2 (t)) be the orbit (under H) of the point (q1 , Q2 (0, q1 , p1 ), p1 , P2 (0, q2 , p2 )) .
1
In this text, the equality of the dimensions of the stable and unstable directions (that obviously holds here) is stated as a conclusion, although it should be taken as an assumption.
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We have the Hamilton equations q˙1 (t) = ∂p1 H(t, q1 (t), q2 (t), p1 (t), p2 (t)) p˙1 (t) = −∂q1 H(t, q1 (t), q2 (t), p1 (t), p2 (t)). They imply that p˙1 = O(2 ), and we conclude that p1 (t) ∈ B for all t ∈ [0, 1] if p1 ∈ B0 , provided is small enough. This implies the inclusion φ(A00 ) ⊂ A0 , and it also implies that (q1 (t), q2 (t), p1 (t), p2 (t)) = (q1 (t), Q2 (t, q1 (t), p1 (t)), p1 (t), P2 (t, q1 (t), p1 (t))) for each t ∈ [0, 1]. The Hamilton equations then take the form q˙1 (t) = ∂p1 h (p1 (t), P2 (t, q1 (t), p1 (t)))
− 2 ∂p1 G (t, q1 (t), Q2 (t, q1 (t), p1 (t)), P2 (t, q1 (t), p1 (t)))
p˙1 (t) = +2 ∂q1 G (t, q1 (t), Q2 (t, q1 (t), p1 (t)), P2 (t, q1 (t), p1 (t))) . The map Φ is thus the time-one flow of the vector field ∂p1 h (p1 , P2 (t, q1 , p1 )) − 2 ∂p1 G (t, q1 , Q2 (t, q1 , p1 ), P2 (t, q1 , p1 )) q1 −→ p1 2 ∂q1 G (t, q1 , Q2 (t, q1 , p1 ), P2 (t, q1 , p1 )) which converges uniformly to the vector field q1 ∂p1 h (p1 , P2 (p1 )) −→ p1 0 when −→ 0 on Tm × B. We conclude that Φ is converging uniformly to Φ0 (as defined in Proposition 2). Moreover, we see that the C 1 distance between these two vector fields is O(κ) (κ is a parameter introduced in the statement of Theorem 1), so it can be made arbitrarily small by taking B0 small enough. The same statement then holds for the time-one flows Φ and Φ0 .
References [1] Arnold, V.I.: Instability of dynamical systems with several degrees of freedom. Sov. Math. Doklady 5, 581–585 (1964) [2] Bernard, P.: Connecting orbits of time dependent Lagrangian systems. Ann. Inst. Fourier 52, 1533–1568 (2002) [3] Bernard, P.: The dynamics of pseudographs in convex Hamiltonian systems. J. AMS 21(3), 625–669 (2008) [4] Bolotin, S.V., Treschev, D.V.: Remarks on the definition of hyperbolic tori of Hamiltonian systems. Regul. Chaotic Dyn. 5(4), 401–412 (2000) [5] Chaperon, M.: Stable manifolds and the Perron–Irwin method. Ergod. Theory Dyn. Syst. 24, 1359–1394 (2004) [6] Cheng, C.-Q., Yan, J.: Existence of diffusion orbits in a priori unstable Hamiltonian systems. J. Differ. Geom. 67(3), 457–517 (2004) [7] Cheng, C.-Q., Yan, J.: Arnold diffusion in Hamiltonian systems: the a priori unstable case. J. Differ. Geom. 82(2), 229–277 (2009)
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[8] Delshams, A., de la Llave, R., Seara, T.M.: A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model. Mem. AMS 179(844) (2006) [9] Delshams, A., de la Llave, R., Seara, T.M.: Orbits of unbounded energy in quasiperiodic perturbations of geodesic flows. Adv. Math. 202, 64–188 (2006) [10] Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226 (1971) [11] Gidea, M., Robinson, C.: Obstruction argument for transition chains of Tori interspersed with gaps, preprint [12] Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant Manifolds. Lecture Notes in Mathematics. Springer, New York (1977) [13] Lochack, P., Marco, J.P., Sauzin, D.: On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian Systems. Mem. AMS 163(775) (2003) [14] Mather, J.N.: Variational construction of connecting orbits. Ann. Inst. Fourier 43, 1349–1368 (1993) [15] Mather, J.N.: Arnold diffusion: announcement of results. J. Math. Sci. (N.Y.) 124(5), 5275–5289 (2004) [16] Moeckel, R.: Transition tori in the five-body problem. JDE 129, 290–314 (1996) [17] Treschev, D.: Hyperbolic tori and asymptotic surfaces in Hamiltonian systems. Russ. J. Math. Phys. 2(1), 93–110 (1994) [18] Treschev, D.: Evolution of slow variables in a priori unstable Hamiltonian systems. Nonlinearity 17(5), 1803–1841 (2004) Patrick Bernard CEREMADE UMR CNRS 7534 Pl. du Mar´echal de Lattre de Tassigny 75775 Paris Cedex 16, France e-mail:
[email protected] Communicated by Viviane Baladi. Received: December 17, 2009. Accepted: March 24, 2010.
Ann. Henri Poincar´e 11 (2010), 943–971 c 2010 Springer Basel AG 1424-0637/10/050943-29 published online October 17, 2010 DOI 10.1007/s00023-010-0050-7
Annales Henri Poincar´ e
Exponential Renormalization Kurusch Ebrahimi-Fard and Fr´ed´eric Patras Abstract. Moving beyond the classical additive and multiplicative approaches, we present an “exponential” method for perturbative renormalization. Using Dyson’s identity for Green’s functions as well as the link between the Fa` a di Bruno Hopf algebra and the Hopf algebras of Feynman graphs, its relation to the composition of formal power series is analyzed. Eventually, we argue that the new method has several attractive features and encompasses the BPHZ method. The latter can be seen as a special case of the new procedure for renormalization scheme maps with the Rota–Baxter property. To our best knowledge, although very natural from group-theoretical and physical points of view, several ideas introduced in the present paper seem to be new (besides the exponential method, let us mention the notions of counter-factors and of order n bare coupling constants).
1. Introduction Renormalization theory [5,6,11,18] plays a major role in the perturbative approach to quantum field theory (QFT). Since its inception in the late 1930s [3] it has evolved from a highly technical and difficult set of tools, mainly used in precision calculations in high energy particle physics, into a fundamental physical principle encoded by the modern notion of the renormalization group. Recently, Alain Connes, Dirk Kreimer, Matilde Marcolli and collaborators developed a compelling mathematical setting capturing essential parts of the algebraic and combinatorial structure underlying the so-called BPHZ renormalization procedure in perturbative QFT [7–10,21]. The essential notion appearing in this approach is the one of combinatorial Hopf algebras. The latter typically consists of a graded vector space where the homogeneous components are spanned by finite sets of combinatorial objects, such as planar or K. Ebrahimi-Fard is on leave from Univ. de Haute Alsace, Mulhouse, France.
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non-planar rooted trees, or Feynman graphs, and the Hopf algebraic structures are given by particular constructions on those objects. For a particular QFT the set of Feynman rules corresponds to a multiplicative map from such a combinatorial Hopf algebra, generated, say, by one-particle irreducible (1PI) ultraviolet (UV) superficially divergent diagrams, into a commutative unital target algebra. This target algebra essentially reflects the regularization scheme. The process of renormalization in perturbative QFT can be performed in many different ways [6,18]. A convenient framework is provided by dimensional regularization (DR). It implies a target algebra of regularized probability amplitudes equipped with a natural Rota–Baxter (RB) algebra structure. The latter encodes nothing but minimal subtraction (MS). Introducing a combinatorial Hopf algebra of Feynman graphs in the context of φ3 -theory (in 6 dimensions) allows for example to reformulate the BPHZ renormalization method for Feynman graphs, in terms of a Birkhoff–Wiener–Hopf (BWH) decomposition inside the group of dimensionally regularized characters [8,21]. As it turns out, Bogoliubov’s recursive renormalization process is then best encoded by Atkinson’s recursion for noncommutative Rota–Baxter algebras, the solution of which was obtained in the form of a closed formula in [15]. Following Kreimer [20], Walter van Suijlekom extended the Hopf algebra approach to perturbative renormalization of gauge theories [24,25]. The Connes–Kreimer approach focused originally on DR+MS but can actually be extended to other regularization schemes, provided the subtraction method corresponds to a Rota–Baxter algebra structure. It applies for example to zero momentum subtraction as shown in [13]. However, essential parts of this algebraic machinery are not anymore available once the RB property is lost. More precisely, the remarkable result that Bogoliubov’s classical renormalization formulae give birth to Hopf algebra characters and are essentially equivalent to the BWH decomposition of Hopf algebra characters is lost if the renormalization scheme map is not RB [7,8]. Two remarks are in order. First, more insights from an algebraic point of view are needed in this particular direction. As a contribution to the subject, we propose and study in the last section of the present paper a non-MS scheme within DR which is not of Rota–Baxter type. Second, the characterization of the BPHZ method in terms of BWH decomposition might be too restrictive, as it excludes possible subtraction schemes that do not fall into the class of Rota–Baxter type ones. In this paper we present an exponential algorithm to perform perturbative renormalization (the term “exponential” refers to the way the algorithm is constructed and was also chosen for its similarity with the classical “additive” and “multiplicative” terminologies). One advantage of this method, besides its group-theoretical naturality, is that it does not rely on the Rota–Baxter property. Indeed, the exponential method is less restrictive than the BPHZ method in the Hopf algebraic picture. It only requires a projector P− (used to isolate the divergences of regularized amplitudes) such that the image of the associated orthogonal projector, P+ := id − P− , forms a subalgebra. This constraint on the image of P+ reflects the natural
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assumption that products of finite regularized amplitudes are supposed to be finite. Let us mention that the very process of exponential renormalization leads to the introduction of new objects and ideas in the algebro-combinatorial approach to perturbative QFT. Particularly promising are the ones of counter-factors and order n bare coupling constants, that fit particularly well some widespread ideas that do not always come with a rigorous mathematical foundation such as the one that “in the end everything boils down in perturbative QFT to power series substitutions”. The notion of order n bare coupling constants makes such a statement very precise from the algebraic point of view. Let us also mention that the exponential method is a further development of ideas sketched in our earlier paper [16] that pointed at a natural link between renormalization techniques and fine properties of Lie idempotents, with a particular emphasis on the family of Zassenhaus Lie idempotents. Here we do not further develop such aspects from the theory of free Lie algebras [23], and refer to the aforementioned article for details on the subject. The paper is organized as follows. The next section briefly recalls some general properties of graded Hopf algebras including the BWH decomposition of regularized Feynman rules viewed as Hopf algebra characters. We also dwell on the Fa` a di Bruno Hopf algebra and prove an elementary but useful Lemma that allows the translation of the Dyson formula (relating bare and renormalized Green’s functions) into the language of combinatorial Hopf algebras. In Sect. 3 we introduce the notion of n-regular characters and present an exponential recursion used to construct m-regular characters from m − 1-regular ones. We conclude the article by introducing and studying a toy-model nonRota–Baxter renormalization scheme on which the exponential recursion can be performed. We prove in particular that locality properties are preserved by this renormalization process.
2. From Dyson to Fa`a di Bruno 2.1. Preliminaries In this section we introduce some mathematical structures to be used in the sequel. We also recall the BWH decomposition of Hopf algebra characters. Complementary details can be found, e.g. in [13,17,22]. algebra H = Let us consider a graded, connected and commutative Hopf H over the field k, or its pro-unipotent completion n≥0 n n≥0 Hn . Recall that since the pioneering work of Pierre Cartier on formal groups [4], it is well-known that the two types of Hopf algebras behave identically, allowing to deal similarly with finite sums n≤N hn , hn ∈ Hn and formal series n∈N hn , hn ∈ Hn . The unit in H is denoted by 1. Natural candidates are the Hopf algebras of rooted trees and Feynman graphs [7,8] related to non-commutative geometry and pQFT, respectively.
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We remark here that graduation phenomena are essential for all our forthcoming computations, since in the examples of physical interest they incorporate information such as the number of loops (or vertices) in Feynman graphs, relevant for perturbative renormalization. The action of the grading operator Y : H → H is given by: nhn for h = hn ∈ Hn . Y (h) = n∈N
n∈N
n∈N
We write for the augmentation map from H to H0 = k ⊂ H and ∞ H + := n=1 Hn for the augmentation ideal of H. The identity map of H is denoted id. The product in H is written mH and its action on elements simply by concatenation. The co-product is written Δ; we use Sweedler’s notation n n (1) (2) and write h(1) ⊗h(2) or j=0 hj ⊗hn−j for Δ(h) ∈ j=0 Hj ⊗Hn−j , h ∈ Hn . := The space of k-linear maps from H to k, Lin(H, k) n∈N Lin(Hn , k), is naturally endowed with an associative unital algebra structure by the convolution product: f ∗ g := mk ◦ (f ⊗ g) ◦ Δ :
Δ
f ⊗g
m
k H −→ H ⊗ H −−−→ k ⊗ k −−→ k.
The unit for the convolution product is precisely the co-unit : H → k. Recall that a character is a linear map γ of unital algebras from H to the base field k: γ(hh ) = γ(h)γ(h ). The group of characters is denoted by G. With πn , n ∈ N, denoting the projection from H to Hn we write γ(n) = γ ◦ πn . An infinitesimal character is a linear map α from H to k such that: α(hh ) = α(h)(h ) + (h)α(h ). As for characters, we write α(h) = n∈N α(n) (hn ). We remark that by the definitions of characters and infinitesimal characters γ0 (1) = 1, that is γ0 = , whereas α0 (1) = 0, respectively. Recall that the graded vector space g of infinitesimal characters is a Lie subalgebra of Lin(H, k) for the Lie bracket induced on the latter by the convolution product. Let A be a commutative k-algebra, with unit 1A = ηA (1), ηA : k → A and with product mA , which we sometimes denote by a dot, i.e. mA (u ⊗ v) =: u · v or simply by concatenation. The main examples we have in mind are A = C, A = C[[ε, ε−1 ] and A = H. We extend now the definition of characters and call an (A-valued) character of H any algebra map from H to A. In particular H-valued characters are simply algebra endomorphisms of H. We extend as well the notion of infinitesimal characters to maps from H to the commutative k-algebra A, that is: α(hh ) = α(h) · e(h ) + e(h) · α(h ), where e := ηA ◦ is now the unit in the convolution algebra Lin(H, A). Observe that infinitesimal characters can be alternatively defined as k-linear maps from H to A with α ◦ π0 = 0 that vanish on the square of the augmentation ideal of H. The group (Lie algebra) of A-valued characters (infinitesimal characters)
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is denoted G(A) (g(A)) or GH (A) when we want to emphasize the underlying Hopf algebra. 2.2. Birkhoff–Wiener–Hopf Decomposition of G(A) In the introduction we already mentioned one of Connes–Kreimer’s seminal insights into the algebro-combinatorial structure underlying the process of perturbative renormalization in QFT. In the context of DR+MS, they reformulated the BPHZ-method as a Birkhoff–Wiener–Hopf decomposition of regularized Feynman rules, where the latter are seen as an element in the group G(A). Of pivotal role in this approach is a Rota–Baxter algebra structure on the target algebra A = C[[ε, ε−1 ]. In general, let us assume that the commutative algebra A = A+ ⊕ A− splits directly into the subalgebras A± = T± (A) with 1A ∈ A+ , defined in terms of the projectors T− and T+ := id − T− . The pair (A, T− ) is a special case of a (weight one) Rota–Baxter algebra [14] since T− , and similarly T+ , satisfies the (weight one RB) relation: T− (x) · T− (y) + T− (x · y) = T− (T− (x) · y + x · T− (y)) ,
x, y ∈ A.
(1)
One easily shows that Lin(H, A) with an idempotent operator T− defined by T− (f ) = T− ◦ f , for f ∈ Lin(H, A), is a (in general non-commutative) unital Rota–Baxter algebra (of weight one). The Rota–Baxter property (1) implies that G(A) decomposes as a set as the product of two subgroups: G(A) = G− (A) ∗ G+ (A),
where
G± (A) = exp∗ (T± (g(A))).
Corollary 1. [8,13] For any γ ∈ G(A) the unique characters γ+ ∈ G+ (A) −1 ∈ G− (A) in the decomposition of G(A) = G− (A) ∗ G+ (A) solve the and γ− equations: γ± = e ± T± (γ− ∗ (γ − e)).
(2)
That is, we have Connes–Kreimer’s Birkhoff–Wiener–Hopf decomposition: −1 ∗ γ+ . γ = γ−
(3)
Note that this corollary is true if and only if the operator T− on A is of Rota–Baxter type. That is, uniqueness of the decomposition follows from the idempotence of the map T− . In fact, in the sequel we will show that this result is a special case of a more general decomposition of characters. 2.3. The Fa`a di Bruno Hopf Algebra and a Key Lemma Another example of combinatorial Hopf algebra, i.e. a graded, connected, commutative bialgebra with a basis indexed by combinatorial objects, which we will see to be acutely important in the sequel, is the famous Fa` a di Bruno Hopf algebra F , for details see e.g. [2,17,19]. Recall that for series, say of a real variable x: f (x) =
∞ n=0
an (f ) xn+1 , h(x) =
∞ n=0
an (h) xn+1 , with a0 (f ) = a0 (h) = 1,
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the composition is given by: f (h(x)) =
∞
an (f ◦ h) xn+1 =
n=0
∞
an (f )(h(x))n+1 .
n=0
It defines the group structure on: ∞ GF := f (x) = an (f ) xn+1 | an (f ) ∈ C, a0 (f ) = 1 . n=0
One may interpret the functions an as a derivation evaluated at x = 0: 1 dn+1 f (0). n + 1! dxn+1 The coefficients an (f ◦ h) are given by: an (f ) =
an (f ◦ h) =
n
ak (f )
k=0
al0 (h) · · · alk (h).
l0 +···+lk =n−k li ≥0,i=0,...,k
For instance, with an obvious notation, the coefficient of x4 in the composed series is given by f3 + 3f2 h1 + f1 (h21 + 2h2 ) + h3 . The action of these coefficient functions on the elements of the group GF implies a pairing: an , f := an (f ). The group structure on GF allows to define the structure of a commutative Hopf algebra on the polynomial ring spanned by the an , denoted by F , with co-product: ΔF (an ) =
n
al0 · · · alk ⊗ ak .
k=0 l0 +···+lk =n−k li ≥0,i=0,...,k
Notice that, using the pairing, an element f of GF can be viewed as the R[x]valued character fˆ on F characterized by: fˆ(an ) := an (f )xn . The composition of formal power series translates then into the convolution product of characˆ ∗ fˆ. ters: f (h) = h Let us condense this into what we call the Fa`a di Bruno formula, that is, define a := n≥0 an . Then a satisfies: ΔF (a) = an+1 ⊗ an . (4) n≥0
Note that subindices indicate the graduation degree. We prove now a technical lemma, important in view of applications to perturbative renormalization. As we will see further below, it allows to translate the Dyson formulas for renormalized and bare 1PI Green’s functions into the language of Hopf algebras.
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Lemma 2. Let H = n≥0 Hn be a complete graded commutative Hopf algebra, which is an algebra of formal power series containing the free variables f1 , . . . , fn , . . .. We assume that fi has degree i andwrite f = 1 + k>0 fk n (so that, in particular, f is invertible). If Δ(f ) = n f α ⊗ fn , where α = n≥0 αn and the αn , n > 0, are algebraically independent as well as algebraa di Bruno formula: ically independent from the fi , then α satisfies the Fa` Δ(α) = αn+1 ⊗ αn . n≥0
Proof. Indeed, let us make explicit the associativity of the co-product, (Δ ⊗ id) ◦ Δ = (id ⊗ Δ) ◦ Δ. First: Δ(f αn ) ⊗ fn = f αn ⊗ Δ(fn ) = f αn ⊗ (f αn−p )p ⊗ fn−p . n≥0
n≥0
n,p≤n
Now we look at the component of this identity that lies in the subspace H ⊗ H ⊗ H1 and get: Δ(f α) = f αn ⊗ (f α)n−1 , n≥0
that is:
f αn α(1) ⊗ fn α(2) =
f αn ⊗ fp αn−p−1 .
n,p
n≥0
Since f is invertible: αn α(1) ⊗ fn α(2) = αn ⊗ fp αn−p−1 . n,p
n≥0
From the assumption of algebraic independence among the αi and fj , we get, looking at the component associated to f0 = 1 on the right hand side of the above tensor product: Δ(α) = α(1) ⊗ α(2) = αn+1 ⊗ αn . n≥0
Corollary 3. With the hypothesis of the Lemma, the map χ from F to H, an −→ αn is a Hopf algebra map. In particular, if f and g are in GH (R), f ◦ χ and g ◦ χ belong to GF (R) and: g ∗ f (αn )xn+1 = (g ◦ χ) ∗ (f ◦ χ)(an )xn+1 = f ◦ χ(g ◦ χ), n≥0
n≥0
where in the last equality we used the identification of GF (R) with x + xR[x] to view f ◦ χ and g ◦ χ as formal power series. In other terms, properties of H can be translated into the language of formal power series and their compositions.
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3. The Exponential Method
Let H = n≥0 Hn be an arbitrary graded connected commutative Hopf algebra and A a commutative k-algebra with unit 1A = ηA (1). Recall that πn stands for the projection on Hn orthogonally to the other graded components of H. As before, the group of characters with image in A is denoted by G(A), with unit e := ηA ◦ . We assume in this section that the target algebra A contains a subalgebra A+ , and that there is a linear projection map P+ from A onto A+ . We write P− := id − P+ . The purpose of the present section is to construct a map from G(A) to G(A+ ). In the particular case of a multiplicative renormalizable perturbative QFT, where H is a Hopf algebra of Feynman diagrams and A the target algebra of regularized Feynman rules, this map should send the corresponding Feynman rule character ψ ∈ G(A) to a renormalized, but still regularized, Feynman rule character R. The particular claim of A+ ⊂ A being a subalgebra implies G(A+ ) being a subgroup. This reflects the natural assumption, motivated by physics, that the resulting-renormalized-character R ∈ G(A+ ) maps products of graphs into A+ , i.e. R(Γ1 Γ2 ) = R(Γ1 )R(Γ2 ) ∈ A+ . Or, to say the same, products of finite and regularized amplitudes are still finite. In the case where the target algebra has the Rota–Baxter property, the map from G(A) to G(A+ ) should be induced by the BWH decomposition of characters. 3.1. An Algorithm for Constructing Regular Characters We first introduce the notion of n-regular characters. Later we identify them with characters renormalized up to degree n. Definition 1. A character ϕ ∈ G(A) is said to be regular up to order n, or n-regular, if P+ ◦ ϕ(l) = ϕ(l) for all l ≤ n. A character is called regular if it is n-regular for all n. In the next proposition we outline an iterative method to construct a regular character in G(A+ ) starting with an arbitrary one in G(A). The iteration proceeds in terms of the grading of H. Proposition 4. Let ϕ ∈ G(A) be regular up to order n. Define μϕ n+1 to be the linear map which is zero on Hi for i = n + 1 and: μϕ n+1 := P− ◦ ϕ ◦ πn+1 = P− ◦ ϕ(n+1) . Then 1. μϕ n+1 is an infinitesimal character. ϕ ∗ 2. The convolution exponential Υ− n+1 := exp (−μn+1 ) is therefore a character. + − 3. The product ϕn+1 := Υn+1 ∗ ϕ is a regular character up to order n + 1. Note that we use the same notation for the projectors P± on A and the ones defined on Lin(H, A).
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Remark. Let us emphasize two crucial points. First, we see the algebraic naturalness of the particular assumption on A+ being a subalgebra. Indeed, it allows for a simple construction of infinitesimal characters from characters in terms of the projector P− . Second, at each order in the presented process we stay strictly inside the group G(A). This property does not hold for other recursive renormalization algorithms. For example, in the BPHZ case, the recursion takes place in the larger algebra Lin(H, A), see e.g. [15]. Proof. Let us start by showing that μϕ n+1 is an infinitesimal character. That is, its value is zero on any non trivial product of elements in H. In fact, for / H0 , y = xz ∈ Hn+1 , x, z ∈ μϕ n+1 (y) = P− (ϕ(y)) = P− (ϕ(x)ϕ(z)) = P− (P+ ϕ(x)P+ φ(y)), since ϕ is n-regular by assumption. This implies that μϕ n+1 (y) = P− ◦ P+ (P+ ϕ(x)P+ ϕ(z)) = 0 as the image of P+ is a subalgebra in A. The second assertion is true for any infinitesimal character, see e.g. [13]. The third one follows from the next observations: ϕ + ∗ • For degree reasons (since μϕ n+1 = 0 on Hk , k ≤ n), ϕn+1 = exp (−μn+1 ) ∗ ϕ ∗ ϕ = ϕ on Hk , k ≤ n, so that exp (−μn+1 ) ∗ ϕ is regular up to order n. • In degree n+1: let y ∈ Hn+1 . With a Sweedler-type notation for the reduced ⊗ y(2) , we get: co-product Δ(y) − y ⊗ 1 − 1 ⊗ y = y(1) exp∗ (−μϕ n+1 ) ∗ ϕ(y) ϕ ∗ = exp∗ (−μϕ n+1 )(y) + ϕ(y) + exp (−μn+1 )(y(1) )ϕ(y(2) )
= −μϕ n+1 (y) + ϕ(y) = P+ ϕ(y), ∗
(5)
(−μϕ n+1 )
being zero on Hi , 1 ≤ i ≤ n and which follows from exp ϕ exp∗ (−μϕ ) = −μ on H . Hence, this implies immediately: n+1 n+1 n+1 P+ ((exp∗ (−μϕ n+1 ) ∗ ϕ)(y)) = P+ (P+ ϕ(y)) = P+ ϕ(y) = (exp∗ (−μϕ n+1 ) ∗ ϕ)(y). Note the following particular fact. When iterating the above construction of regular characters, say, by going from a n − 1-regular character ϕ+ n−1 to the , the n − 1-regular character is by construction almost n-regular character ϕ+ n (H ) is given by applying P+ to regular at order n. By this we mean that ϕ+ n n ϕ+ (H ), see (5). This amounts to a simple subtraction, i.e. for y ∈ H n n: n−1 + + + ϕ+ n (y) = P+ (ϕn−1 (y)) = ϕn−1 (y) − P− (ϕn−1 (y)).
Observe that by construction for y ∈ Hn : (1)
(2)
+ + + (6) ϕ+ n−1 (y) = ϕn−2 (y) − P− (ϕn−2 (yn−1 ))ϕn−2 (y1 ), n (1) (2) where the reader should recall the notation Δ(y) = i=0 yi ⊗ yn−i making the grading explicit in the co-product. Further below we will interpret these results in the context of perturbative renormalization of Feynman graphs: for example, when Γ is a UV divergent 1PI diagram of loop order n, the order one
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Ann. Henri Poincar´e
(2)
graph Γ1 on the right-hand side of the formula consists of the unique one loop (2) primitive co-graph. That is, Γ1 follows from Γ with all its 1PI UV divergent (2) subgraphs reduced to points. In the literature this is denoted as res(Γ) = Γ1 . The following propositions capture the basic construction of a character regular to all orders from an arbitrary character. We call it the exponential method. Proposition 5 (Exponential method). We consider the recursion: Υ− 0 := e, := ϕ, and: ϕ+ 0 − + ϕ+ n+1 := Υn+1 ∗ ϕn , ∗ + + + + where Υ− n+1 := exp (−P− ◦ϕn ◦πn+1 ). Then, we have that ϕ := ϕ∞ := lim ϕn + is regular to all orders. Moreover, Υ− ∞ ∗ ϕ = ϕ , where:
→
Υ− ∞ := lim Υ(n) →
and: − Υ(n) := Υ− n ∗ · · · ∗ Υ1 .
Remark. In the light of the application of the exponential method to perturbative renormalization in QFT, we introduce some useful terminology. We call + ∗ Υ− l := exp (−P− ◦ ϕl−1 ◦ πl ) the counter-factor of order l and the product − − Υ(n) := Υ− n ∗ · · · ∗ Υ1 = Υn ∗ Υ(n − 1) the counter-term of order n. 3.2. On the Construction of Bare Coupling Constants The following two propositions will be of interest in the sequel when we dwell on the physical interpretation of the exponential method. Let A be as in Proposition 4. We introduce a formal parameter g which commutes with all elements in A, which we extend to the filtered complete algebra A[[g]] (think of g as the renormalized -i.e. finite- coupling constant of a QFT). The character ϕ ∈ G(A) is extended to ϕ˜ ∈ G(A[[g]]) so as to map f = 1 + k>0 fk ∈ H to: ϕ(fk )g k ∈ A[[g]]. ϕ(f ˜ )(g) = 1 + k>0
Notice that we emphasize the functional dependency of ϕ(f ˜ ) on g for reasons that will become clear in our forthcoming developments. Recall Lemma 2. We assume that Δ(f ) = n≥0 f αn ⊗ fn where α = a di Bruno formula: n≥0 αn ∈ H satisfies the Fa` αn+1 ⊗ αn . Δ(α) = n≥0
˜ − ∗ ϕ˜+ ˜ Let := Υ ˜ ∈ G(A[[g]]) be the n + 1-regular characn = Υ(n + 1) ∗ ϕ n+1 ter constructed via the exponential method from ϕ˜ ∈ G(A[[g]]). Now we define ˜ − , l ≥ 0 a formal power series in g, which we call the for each counter-factor Υ l order l bare coupling constant: ˜ − (gα) = g + an(l) g n+1 , g(l) (g) := Υ l ϕ˜+ n+1
n>0
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(l) (l) ˜− an := Υ− l (αn ). Observe that Υ0 (gα) = gε(1) = g and by construction an = 0 for n < l.
Proposition 6 (Exponential counter-term and composition). With the afore˜ mentioned assumptions, we find that applying the order n counter-term Υ(n) to the series gα ∈ H equals the n-fold composition of the bare coupling constants g(1) (g), . . . , g(n) (g): ˜ ˜− ∗ ··· ∗ Υ ˜ − (gα)(g) Υ(n)(gα)(g) =Υ n 1 = g(1) ◦ · · · ◦ g(n) (g). Proof. The proof follows by induction together with the Fa` a di Bruno formula. ˜ ˜− ∗ Υ ˜ − (gα)(g) Υ(2)(gα)(g) =Υ 2 1 − n+1 ˜ = (Υ2 (α)(g))n+1 a(1) n g n≥0
=
n+1 ˜− a(1) = g(1) ◦ g(2) (g). n (Υ2 (gα)(g))
n≥0
Similarly: ˜ Υ(m)(gα)(g) ˜− ∗ ··· ∗ Υ ˜− ∗ Υ ˜ − (gα)(g) =Υ m 2 1 − − n+1 ˜ ˜ = (Υm ∗ · · · ∗ Υ2 (α)(g))n+1 a(1) n g n≥0
=
n+1 a(1) = g(1) ◦ (g(2) ◦ · · · ◦ g(m) )(g). n (g(2) ◦ · · · ◦ g(m) )(g))
n≥0
Proposition 7 (Exponential method and composition). With the assumption of the foregoing proposition we find that: ˜ ϕ˜+ ˜ ) ◦ g(1) ◦ · · · ◦ g(n) (g) n (f )(g) = Υ(n)(f )(g) · ϕ(f Proof. The proof follows from the co-product Δ(f ) = n f αn ⊗fn by a simple calculation. ˜ ˜ )(g) ϕ˜+ n (f )(g) = (Υ(n) ∗ ϕ)(f ˜ Υ(n)(f αm )(g)ϕ(fm )g m = m≥0
˜ = Υ(n)(f )(g)
m ˜ ϕ(fm )(Υ(n)(gα)(g))
m≥0
from which we derive the above formula using Proposition 6.
The reader may recognize in this formula a familiar structure. This identity is indeed an elaboration on the Dyson formula: we shall return to this point later.
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3.3. The BWH-Decomposition as a Special Case + in Proposition 5 may be interpreted as a The decomposition Υ− ∞ ∗ϕ = ϕ generalized BWH decomposition. Indeed, under the Rota–Baxter assumption, that is if P− is a proper idempotent Rota–Baxter map (i.e. if the image of P− is a subalgebra, denoted A− ), G(A) = G(A− ) ∗ G(A+ ) and the decomposition of a character ϕ into the convolution product of an element in G(A− ) and in G(A+ ) is necessarily unique (see [14,15] to which we refer for details on the Bogoliubov recursion in the context of Rota–Baxter algebras). In particular, Υ− ∞ identifies with the counter-term ϕ− of the BWH decomposition. Let us detail briefly this link with the BPHZ method under the Rota– Baxter assumption for the projection maps P− and P+ . Proposition 5 in the foregoing subsection leads to the following important remark (that holds independently of the RB assumption). Observe that by construction it is clear that for y ∈ Hk , k < n + 1: Υ(n + 1)(y) = Υ(k)(y). Using ϕ+ k−1 = Υ(k − 1) ∗ ϕ we see with y ∈ Hk that: − Υ(k)(y) = Υ− k ∗ · · · ∗ Υ1 (y)
= −P− (ϕ+ k−1 (y)) + Υ(k − 1)(y) = −P− (ϕ(y)) − P− (Υ(k − 1)(y)) −P− (Υ(k − 1)(y(1) )ϕ(y(2) )) + Υ(k − 1)(y) )ϕ(y(2) )) + P+ (Υ(k − 1)(y)) = −P− (ϕ(y) + Υ(k − 1)(y(1)
= −P− (Υ(k − 1) ∗ (ϕ − e)(y)) + P+ (Υ(k − 1)(y)).
(7)
Now, note that for all n > 0, the RB property implies that Υ(n)(y) is in A− for y ∈ H + . Hence, going to (7) we see that P+ (Υ(k − 1)(y)) = 0. n Proposition 8. For n > 0 the characters ϕ+ := n and Υ(n) restricted to H n H solve Bogoliubov’s renormalization recursion. i i=0
Proof. Let x ∈ H n . From our previous discussion: e(x) − P− ◦ (Υ(n) ∗ (ϕ − e))(x) = Υ(n)(x). Similarly: e(x) + P+ ◦ (Υ(n) ∗ (ϕ − e))(x) = e(x) + P+ ◦ (Υ(n) ∗ ϕ − Υ(n))(x) = e(x) + P+ ◦ (ϕ+ n − Υ(n))(x) = ϕ+ n (x). When going to the last line we used P+ ◦ P− = P− ◦ P+ = 0 as well as the Rota–Baxter property of P− and P+ . This implies that, on H n , ϕ+ n = e + P+ ◦ (Υ(n) ∗ (ϕ − e)) and Υ(n) = e − P− ◦ (Υ(n) ∗ (ϕ − e)) which are Bogoliubov’s renormalization equations for the counter-term and the renormalized character, respectively, see e.g. [14,15].
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3.4. On Counter-Terms in the BWH Decomposition Recall briefly how these results translate in the language of renormalization in perturbative QFT. This section also introduces several notations that will be useful later on. The reader is referred to the textbooks [6,18] and the articles [8,9] for more details. As often in the literature, the massless φ4 Lagrangian L = L(∂μ φ, φ, g) in four space-time dimensions shall serve as a paradigm: g 1 ∂μ φ∂ μ φ − φ4 . (8) 2 4! This is certainly a too simple Lagrangian to account for all the combinatorial subtleties of perturbative QFT, but its basic properties are quite enough for our present purpose. The quadratic part is called the free Lagrangian, denoted by L0 . The rest is called the interaction part, and is denoted by Li . The parameter g appearing in L = L0 + Li is the so-called renormalized, that is, finite coupling constant. Perturbation theory is most effectively expressed using Feynman graphs. Recall that from the above Lagrangian we can derive Feynman rules. Then any Feynman graph Γ corresponds by these Feynman rules to a Feynman amplitude. By |Γ| we denote the number of loops in the diagram. Recall that in any given theory exists a rigid relation between the numbers of loops and vertices, for each given m-point function. In φ4 theory, for graphs associated to the 2-point function the number of vertices equals the number of loops. For graphs associated to the 4-point function the number of vertices is equal to the number of loops plus one. A Feynman amplitude consists of the Feynman integral, i.e. a multiple d(= 4)-dimensional momentum space integral: ⎤ ⎡ |Γ| dd kl ⎦ IΓ (p, k), (9) Γ → ⎣ L :=
l=1
multiplied by a proper power of the coupling constant, i.e. g |Γ|+1 for 4-point graphs and g |Γ| for 2-point graphs. Here, k = (k1 , . . . , k|Γ| ) are the |Γ| independent internal (loop) momenta, that is, each independent loop yields one N integration, and p = (p1 , . . . , pN ), with k=1 pk = 0, denotes the N external momenta. Feynman integrals are most often divergent and require to be properly regularized and renormalized to acquire physical meaning. A regularization method is a prescription that parameterizes the divergencies appearing in Feynman amplitudes upon introducing non-physical parameters, denoted ε, ˜ ε) = g |Γ|+1 ψ(Γ; ε) thereby rendering them formally finite. Let us write g ψ(Γ; for the regularized Feynman amplitude (for example in DR; the notation ψ˜ is introduced for later use). Of pivotal interest are Green’s functions, in particular 1PI n-point (regularized) Green’s functions, denoted G(n) (g, ε) := G(n) (p1 , . . . , pn ; g, ε). In the following we will ignore the external momenta and omit the regularization parameter. Recall that for the renormalization of the Lagrangian (8), the 4- and 2-legs 1PI Feynman graphs, respectively the corresponding amplitudes,
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beyond tree level are of particular interest. As guiding examples we use therefore from now on the regularized momentum space 1PI 4- and 2-point Green’s function. These are power series in the coupling g with Feynman amplitudes as coefficients: ˜ g ) and G(2) (g) = ψ(z ˜ φ ), G(4) (g) = ψ(gz where zg and zφ stand for the formal coupling constant z-factors in the corresponding Hopf algebra of Feynman graphs H: (4) (2) zg = 1 + Γk and zφ = 1 − Γk . (10) k>0
k>1
Here, 1 is the empty graph in H and: (4)
(4) Γk
:=
(2)
Γk,m
m=1
sym(Γk,m )
Nk
and
(4)
(4)
(2) Γk
Γk,n
n=1
sym(Γk,n )
Nk
(4)
:=
(2)
(2)
(2)
denote the sums of the Nk 1PI 4-point and Nk 2-point graphs of loop order k, divided by their symmetry factors, respectively. To deal with the polynomial dependency of the Green’s functions on the coupling constant g, we write: G(4) (g) = g +
∞
(4)
g k+1 Gk
and
G(2) (g) = 1 −
k=1 (r)
∞
(2)
g k Gk ,
k=1
(r)
so that Gk = ψ(Γk ), for r = 2, 4. Hence, as perturbative 1PI Green’s functions are power series with individual–UV divergent–1PI Feynman amplitudes as coefficients, one way to render them finite is to renormalize graph by graph. This is the purpose of the Bogoliubov recursion, which, in the context of DR + MS, was nicely encoded in the group-theoretical language by Connes and Kreimer [8]. Indeed, let H be the graded connected commutative Hopf algebra of 1PI Feynman graphs associated to the Lagrangian (8) and let us choose the RB algebra of Laurent series A = C[ε−1 , ε]] as a target algebra for the regularized amplitudes (the natural ˜ ε) extends uniquely to a choice in DR). Then, the correspondence Γ → ψ(Γ; ˜ can be interpreted character on H. That is, the regularized Feynman rules, ψ, as an element of G(A[[g]]). Recall now that in the case of DR the underlying RB structure, i.e. the MS −1 ∗ ψ˜+ . This allows scheme, implies the unique BWH decomposition ψ˜ = ψ˜− to recover Bogoliubov’s classical counter-term map C and the renormalized Feynman rules map R. Indeed, for an arbitrary 1PI graph Γ ∈ H, one gets: C(Γ) = ψ˜− (Γ) and R(Γ) = ψ˜+ (Γ). (4)
The linearity of R then leads to renormalized 1PI Green’s functions: GR (g) = (2) R(gzg ), GR (g) = R(zφ ). We refer to [8] for further details.
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3.5. The Lagrangian Picture The counter-terms C(Γ) figure in the renormalization of the Lagrangian L. Indeed, for a multiplicative renormalizable QFT, it can be shown that the BPHZ method is equivalent to the method of additive, and hence multiplicative renormalization. Therefore, let us remind ourselves briefly of the additive method, characterized by adding order-by-order counter-terms to the Lagrangian L. Eventually, this amounts to multiplying each term in the Lagrangian by particular renormalization factors. Details can be found in standard textbooks on perturbative QFT, such as [6,18]. In general the additive renormalization prescription is defined as follows. The Lagrangian L is modified by adding the so-called counter-term Lagrangian, Lct , resulting in the renormalized Lagrangian: where Lct :=
s>0
Lren := L + Lct , (s)
Lct is defined by: 1 g Lct := C1 (g) ∂μ φ∂ μ φ − C2 (g) φ4 , 2 4!
(11)
(s) (s) with Cn (g) := s>0 g s Cn , n = 1, 2 being power series in g. The Cn , n = 1, 2, s > 0 are functions of the regularization parameter ε to be defined iteratively as follows. (1) To obtain the 1-loop counter-term Lct one starts with L = L0 + Li , computes the propagators and vertices, and generates all one-loop diagrams, that is, graphs of order g 2 . Among those one isolates the UV divergent 1PI (1) Feynman diagrams and chooses the 1-loop counter-term part Lct , that is, (1) Cn , n = 1, 2, so as to cancel these divergences. (1) (1) Now, use the 1-loop renormalized Lagrangian Lren := L + Lct + (s) s>1 Lct to generate all graphs up to 2-loops, that is, all graphs of order 3 g . Note that this includes for instance graphs with one loop where one of the (1) vertices is multiplied by g 2 C2 and the other one by g, leading to an order g 3 contribution. Again, as before, isolate the UV divergent 1PI ones and choose (2) the 2-loop counter-term part Lct , which is now of order g 3 , again so as to cancel these divergencies. Proceed with the 2-loop renormalized Lagrangian (2) (1) (2) (s) Lren := L + Lct + Lct + s>2 Lct , and so on. The 2-point graphs contribute to the wave function counter-term, whereas 4-point graphs contribute to the coupling constant counter-term (see e.g. [6, Chap. 5]). Note that after j steps in the iterative prescription one obtains the resulting jth-loop renormalized Lagrangian: (s) (1) (j) (j) := L0 + Li + Lct + · · · + Lct + Lct (12) Lren s>j (s) Cn ,
with counter-terms n = 1, 2 fixed up to order j, such that it gives finite (s) expressions up to loop order j. The part Lct , s > j, remains undetermined. In fact, later we will see that, in our terminology, some associated Feynman rules are j-regular.
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The multiplicative renormalizability of L implies that we may absorb the counter-terms into the coupling constant and wave function Z-factors:
where Cn (g) =
Zg := 1 + C2 (g), s (s) s=1 g Cn ,
Zφ := 1 + C1 (g),
n = 1, 2. We get:
1 1 Zφ ∂μ φ∂ μ φ − gZg φ4 . (13) 2 4! As it turns out, Bogoliubov’s counter-term map seen as C ∈ G(A[[g]]) gives: (j) = Lren
Zg (g) = C(zg )(g)
and Zφ (g) = C(zφ )(g),
where we made the g dependence explicit. Now we define the bare, or unrenormalized, field φ(0) := Zφ φ as well as the bare coupling constant: g B (g) :=
gZg (g) , Zφ2 (g)
and as C ∈ G(A[[g]]): g B (g) = gC(zB )(g) where zB := zg /zφ2 ∈ H is the formal bare coupling. Up to the rescaling of the wave functions, the locality of the counter-terms allows for the following renormalized Lagrangian: 1 1 Lren = ∂μ φ(0) ∂ μ φ(0) − g B (g)φ4(0) . (14) 2 4! 3.6. Dyson’s Formula Revisited Let us denote once again by H and F the Hopf algebra of 1PI Feynman graphs a Di Bruno of the massless φ4 theory in four space-time dimensions and the Fa` Hopf algebra, respectively. The purpose of the present section is to show how Dyson’s formula, relating renormalized and (regularized) bare Green’s functions, allows for a refined interpretation of the exponential method for constructing regular characters in the context of renormalization. We write R and C for the regularized renormalized Feynman rules and counter-term character, respectively. Recall the universal bare coupling constant: zB := zg zφ−2 . It can be expanded as a formal series in H: Γk ∈ H, zB = 1 +
(15)
k>0
where Γk ∈ Hk is a homogeneous polynomial of loop order k in 1PI 2- and (4) (2) 4-point graphs with a linear part Γk +2Γk . Notice that, as H is a polynomial (4) (2) algebra over Feynman graphs and since the family of the Γk and of the Γk (r) are algebraically independent in H, also the families of Γk and Γk , r = 2, 4, are algebraically independent in H.
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Coming back to 1PI Green’s functions. Dyson back then in the 1940s [12] showed-in the context of QED, but the result holds in general [18, Chap. 8]– that the bare and renormalized 1PI n-point Green’s functions satisfy the following simple identity: (n)
n/2
GR (g) = Zφ G(n) (g B ).
(16)
Recall that the renormalized as well as the bare 2- and 4-point Green’s functions and the Z-factors, Zφ and Zg , are obtained by applying respectively the renormalized Feynman rules map R, the Feynman rules ψ˜ and the counter-term C to the formal z-factors introduced in (10), respectively. When translated into the language of Hopf algebras, the Dyson equation reads, say, in the case of the 4-point function: (4)
GR (g) = R(gzg ) = C(zφ2 )
∞
(4)
˜ C(zB )j+1 ψ(gΓ j )
j=0
=
∞
(4) ˜ C(zB )j C(zg )ψ(gΓ j ).
j=0
This can be rewritten: ˜ R(zg ) = mA (C ⊗ ψ)
∞
(4)
j zB z g ⊗ Γj ,
(17)
j=0
where we recognize the convolution expression R = C ∗ ψ˜ of the BWH decomposition, with: (4) k zB z g ⊗ Γk . (18) Δ(zg ) = k≥0
Similarly, the study of the 2-point function yields: (2) k Δ(zφ ) = zφ ⊗ 1 − zB z φ ⊗ Γk . k>0
The equivalence between the two formulas (17) and (18) follow from the observation that the BWH decomposition of characters holds for arbitrary counter-terms and renormalized characters, ψ˜− and ψ˜+ , respectively. Choosing, e.g. ψ˜− = C and ψ˜+ = R in such a way that their values on Feynman diagrams form a family of algebraically independent elements (over the rationals) in C shows that (17) implies (18) (the converse being obvious). Notice that the co-product formulas can also be obtained directly from the combinatorics of Feynman graphs. We refer to [1,9,24,25] for complementary approaches and a self-contained study of co-product formulas for the various formal z-factors. Now, Lemma 2 implies immediately the Fa`a di Bruno formula for zB : Proposition 9. Δ(zB ) =
k≥0
k+1 zB ⊗ Γk
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Corollary 10. There exists a natural Hopf algebra homomorphism Φ from F to H: an → Φ(an ) := Γn .
(19)
Equivalently, there exists a natural group homomorphism ρ from G(A), the A-valued character group of H, to the A-valued character group GF of F : G(A) ϕ → ρ(ϕ) := ϕ ◦ Φ : F → A. 3.7. Dyson’s Formula and the Exponential Method Let us briefly make explicit the exponential method for perturbative renormalization in the particular context of the Hopf algebras of renormalization. We denote by H := n≥0 Hn the Connes–Kreimer Hopf algebra of 1PI–UVdivergent–Feynman graphs and by G(A[[g]]) the group of regularized characters from H to the commutative unital algebra A over C to be equipped with a C-linear projector P− such that the image of P+ := id − P− is a subalgebra. The algebra A and projector P− reflect the regularization method respectively the renormalization scheme. The unit in G(A[[g]]) is denoted by e. The corresponding graded Lie algebra of infinitesimal characters is denoted by g(A[[g]]) = n>0 gn (A[[g]]). Let ψ˜ ∈ G(A[[g]]) be the character corresponding to the regularized Feynman rules, derived from a Lagrangian of a—multiplicative renormalizable—perturbative quantum field theory, say, for instance φ4 in four space-time dimensions. Hence any l-loop graph Γ ∈ Hl is mapped to: ˜
ψ ˜ Γ− → ψ(Γ) := g |Γ| ψ(Γ) = g l ψ(Γ).
(20)
Note that the character ψ associates with a Feynman graph the corresponding Feynman integral whereas the character ψ˜ maps any graph with |Γ| loops to its regularized Feynman integral multiplied by the |Γ|th power of the coupling constant. Recall that the exponential method of renormalization proceeds orderby-order in the number of loops. At one-loop order, one starts by considering the infinitesimal character of order one from H to A[[g]]: τ˜1 := P− ◦ ψ˜ ◦ π1 ∈ g1 (A[[g]]), The corresponding exponential counter-factor from H to A[[g]] is given by: ˜ − := exp∗ (−˜ Υ τ1 ). 1 From the definition of the Feynman rules character (20) we get: ˜ − (Γk ) = exp∗ (−P− ◦ ψ˜ ◦ π1 )(Γk ) Υ 1
= g k exp∗ (−P− ◦ ψ ◦ π1 )(Γk )
= g k Υ− 1 (Γk ). The character: ˜ − ∗ ψ˜ ψ˜1+ := Υ 1
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is 1-regular, i.e. it maps H1 to A+ [[g]]. Indeed, as h ∈ H1 is primitive we find ˜ ˜ ˜ ˜ ˜ − (h) = ψ(h) +Υ − P− (ψ(h)) = P+ (ψ(h)). In general, by mulψ˜1+ (h) = ψ(h) 1 ˜ − we obtain tiplying the order n − 1-regular character by the counter-factor Υ n the n-regular character: ˜+ ˜ ˜− ˜ ψ˜n+ := Υ n ∗ ψn−1 = Υ(n) ∗ ψ, ˜ ˜− ∗ ··· ∗ Υ ˜ − . Hence, with the exponential order n counter-term Υ(n) := Υ n 1 in the Hopf algebra context the exponential method of iterative renormalization consists of a successive multiplicative construction of higher order regular characters from lower order regular characters, obtained by multiplication with counter-factors. Next, we define the nth-order bare coupling constant: ˜ − (gzB )(g) = g + g k+1 Υ− gn (g) = Υ n n (Γk ) ∈ gA[[g]]. k≥0
Recall that Υ− n (Γk ) = 0 for k < n. We denote the m-fold iteration: ◦ (g), g1 ◦ · · · ◦ gm (g) =: gm
where by Proposition 9 and from the general properties of Fa` a di Bruno formu◦ ˜ las, we have: gm (g) = Υ(n)(gz B ). We also introduce the nth-order Z-factors: ˜ Zg(n) (g) := Υ(n)(z g )(g)
(n) ˜ and Zφ (g) := Υ(n)(z φ )(g),
so that the nth-order renormalized 2- and 4-point 1PI Green’s functions are: (4) (4) l l+1 ˜ ˜ Υ(n)(z G (g) := g ψ˜n+ (zg )(g) = g Υ(n) ∗ ψ˜g (zg )(g) = ψ(Γ ) B zg )g R,n
l
2 ˜ = Υ(n)(z φ)
l≥0 (4)
l+1 ˜ (Υ(n)(gz ψ(Γl ) B )(g))
l≥0
=
(n) (Zφ (g))2
◦ gm (g)
l+1
(4)
ψ(Γl ),
l≥0 (n) ◦ or, = (Zφ (g))2 G(4) (gm (g)). (2) (2) l ˜ ( Υ(n)(gz )(g)) ψ(Γ ) and G B R,n (g) l≥0 l (4) GR,n (g)
(n) Similarly, ψ˜n+ (zφ )(g) = Zφ (g)
◦ = Zφ (g)G(2) (gm (g)). This corresponds to a Lagrangian multiplicatively renormalized up to order n: (n)
(n)
1 (n) gZg (g) 4 Zφ (g)∂μ φ∂ μ φ − φ . 2 4! However, using Propositions 6 and 7, we may also rescale the wave function and write: 1 g ◦ (g) (n) Lren := ∂μ φn,0 ∂ μ φn,0 − n φ4n,0 . 2 4! (n) where φn,0 := Zφ (g)φ. Physically, on the level of the Lagrangian, the exponential renormalization method corresponds therefore to successive reparametrizations of the bare coupling constant. (n) Lren :=
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4. On Locality and Non Rota–Baxter Type Subtraction Schemes In this last section we present a class of non-Rota–Baxter type subtraction schemes combining the idea of fixing the values of Feynman rules at given values of the parameters and the minimal subtraction scheme in dimensional regularization. The latter is known to be local [5,6] and we will use this fact to prove that the new class of non-Rota–Baxter type schemes is local as well. We first introduce some terminology. Let ψ denote a dimensionally regularized Feynman rules character corresponding to a perturbatively renormalizable (massless, for greater tractability) quantum field theory. It maps the graded connected Hopf algebra H = n≥0 Hn of 1PI Feynman graphs into the algebra A of Laurent series with finite pole part. In fact, to be more precise, the coefficients of such a Laurent series are functions of the external parameters. In this setting, (9) specializes to (see e.g. [6]): H Γ → ψ(Γ; μ, g, s) =
∞
aμn (Γ; g, s)εn .
n=−N
Here, μ denotes ’tHooft’s mass, ε the dimensional regularization parameter and s the set of external parameters others than the coupling constant g. The algebra A is equipped with a natural Rota–Baxter projector T− mapping any Laurent series to its pole part: T− (ψ(Γ; μ, g, s)) :=
−1
aμn (Γ, g, s)εn .
n=−N
This is equivalent to a direct decomposition of A into the subalgebras A− := T− (A) and A+ := T+ (A). In this setting, recall that the BWH decomposition gives rise to a unique −1 ∗ ψ+ into a counter-term map ψ− and the renormalized factorization: ψ = ψ− Feynman rules map ψ+ . Both maps are characterized by Bogoliubov’s renormalization recursions: ψ± = e ± T± ◦ (ψ− ∗ (ψ − e)). The Rota–Baxter property of T− ensures that both, ψ− and ψ+ , are characters. Recall the notion of locality [5,6]. We call a character ψ (and, more generally, a linear form on H) strongly local if the coefficients in the Laurent series which it associates to graphs are polynomials in the external parameter. Notice that the convolution product of two strongly local characters is strongly local: strongly local characters form a subgroup of the group of characters. On the other hand a character ψ is local if its counter-term ψ− is strongly local. Notice that strong locality implies locality. Indeed, since, by the Bogoliubov formula ψ− = e − T− (ψ− ◦ (ψ − e)), ψ− is strongly local if ψ is strongly local due to the recursive nature of the formula. It is well-known that for a multiplicatively renormalizable perturbative QFT with dimensionally regularized Feynman rules character ψ, the counterterm ψ− following from Bogoliubov’s recursion is strongly local. Moreover, as
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the Birkhoff decomposition is unique, recall that comparing with the exponen− − tial method we get: ψ− = Υ− ∞ := lim Υ(n) with: Υ(n) := Υn ∗ · · · ∗ Υ1 . Hence, → in the particular case of a Rota–Baxter type subtraction scheme the exponential method provides a decomposition of Bogoliubov’s counter-term character with respect to the grading of the Hopf algebra. The following Proposition shows that the exponential counter-factors inherit the strong locality property of the Bogoliubov’s counter-term character. Proposition 11. In the context of minimal subtraction, the exponential counterfactors Υ− i and hence the exponential counter-terms Υ(n) are strongly local iff ψ− = Υ− ∞ is strongly local. Proof. One direction is evident as strong locality of the counter-factors implies strong locality of Υ− ∞ . The proof of the opposite direction follows by induction. For any Γ ∈ H1 we find: ψ− (Γ) = −T− ◦ ψ ◦ π1 (Γ), which implies that −T− ◦ ψ ◦ π1 is strongly local. The strong locality of Υ− 1 := exp∗ (−T− ◦ ψ ◦ π1 ) follows from the usual properties of the exponential map in a graded algebra. − Let us assume that strong locality holds for Υ− 1 , . . . , Υn . For Γ ∈ Hn+1 we find (for degree reasons): − + ψ− ∗ Υ−1 (n)(Γ) = · · · ∗ Υn+2 ∗ Υ− n+1 (Γ) = Υn+1 (Γ) = −T− ◦ ψn ◦ πn+1 (Γ).
Strong locality of −T− ◦ ψn+ ◦ πn+1 follows, as well as strong locality of Υ− n+1 = exp∗ (−T− ◦ ψn+ ◦ πn+1 ). The next result with be useful later. Lemma 12. For a strongly local character φ in the context of a proper projector + P− on A, the exponential method leads to a decomposition φ = Υ− ∞ ∗ φ into a as well as a strongly local regular character φ+ . strongly local counter-term Υ− ∞ Proof. The proof follows once again from the definition of the recursion. Indeed, the first order counter-factor in the exponential method is: ∗ Υ− 1 = exp (−T− ◦ φ ◦ π1 ), − which is clearly strongly local, since φ is strongly local. Then φ+ 1 = Υ1 ∗ φ is strongly local as a product of strongly local characters. The same reasoning then applies at each order.
4.1. A Non-Rota–Baxter Subtraction Scheme We introduce now another projection, denoted T−q . It is a projector defined on A in terms of the RB map T− : n T−q := T− + δε,q ,
(21)
n where the linear map δε,q is the Taylor jet operator up to nth-order with respect to the variable ε at zero, which evaluates the coefficient functions at
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all orders between 1 and n at the fixed value q: ∞ n n m δε,q am (x)ε ai (q)εi . := i=1
m=−N
Note the condensed notation, where q stands for a fixed set of values of parameters. The choice of the projection amounts, from the point of view of the renormalized quantities, to fix the coefficient functions at 0 for given values of parameters (e.g. external momenta). One verifies that T−q defines a linear projection. Moreover, the image of T+q := id − T−q forms a subalgebra in A (the algebra of formal power series in ε whose coefficient functions of order less than n vanish at the chosen particular values q of parameters), but the image of T−q does not. This implies immediately that the projector T−q is not of Rota–Baxter type. Hence, we have in general: T−q (ψ(Γ; μ, g, s)) =
−1
aμl (Γ, g, s)εl +
l=−N
n
aμi (Γ, g, q)εi
i=1
and T+q (ψ(Γ; μ, g, s)) =
∞
aμl (Γ, g, s)εl −
l=0
n
aμi (Γ, g, s)εi .
i=1
We find: Proposition 13. Using the subtraction scheme defined in terms of projector T−q on A, the exponential method applied to the Feynman rules character ψ gives a regular character: ψq+ = Υ− ∞,q ∗ ψ, where we use a self-explaining notation for the counter-term Υ− ∞,q and the renormalized character ψq+ . Now we would like to prove that the exponential method using the projector T−q on A gives local counter-terms. That is, we want to prove that the − counter-factor Υ− n,q for all n, and hence Υ∞,q , are strongly local. In the following: − − − ψ− = Υ− ∞ = · · · ∗ Υn ∗ · · · ∗ Υ2 ∗ Υ1
stands for the multiplicative decomposition of Bogoliubov’s strongly local counter-term character following from the exponential method using the minimal subtraction scheme T− . Whereas: − − − Υ− ∞,q = · · · ∗ Υn,q ∗ · · · ∗ Υ2,q ∗ Υ1,q
stands for the counter-term character following from the exponential method according to the modified subtraction scheme T−q . The following Lemma is instrumental in this section.
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Lemma 14. For a substraction scheme such that the image of P+ is a subalgebra, let φ be a n-regular character and ξ be a regular character, then: P− ◦ (φ ∗ ξ)n+1 = P− ◦ φn+1 . In particular, the counter-factor Υ− n+1 associated to φ is equal to the counterfactor associated to φ ∗ ξ. It follows that, if the exponential decomposition of a character ψ is given + by: ψ = Υ− ∞ ∗ ψ , the exponential decomposition of the convolution product of + ψ with a regular character ξ is given by: ψ ∗ ξ = Υ− ∞ ∗ (ψ ∗ ξ). Proof. Indeed, for a n + 1-loop graph Γ, φ ∗ ξ(Γ) = φ(Γ) + ξ(Γ) + c, where c is a linear combination of products of the image by φ and ξ of graphs of loop-order strictly less than n + 1. The regularity hypothesis and the hypothesis that the image of P+ is a subalgebra imply P− (ξ(Γ)+c) = 0, hence the first assertion of the Lemma. The others follow from the definition of the exponential methods by recursion. Lemma 15. Let ψ be a regular character for the minimal substraction scheme (T− ◦ψ = 0). Using the subtraction scheme defined in terms of projector T−q on A, the exponential method applied to ψ gives ψq+ = Υ− ∞,q ∗ ψ, where, for each (Γ) is a polynomial with constant coefficients in the perturbation graph Γ, Υ− ∞,q is strongly local. parameter ε. In particular, Υ− ∞,q The Lemma follows from the definition of the substraction map T−q : by its very definition, since ψ(Γ) is a formal power series in the parameter ε (without singular part), T−q ◦ ψ(Γ) is a polynomial (of degree less or equal to n) with constant coefficients in the perturbation parameter ε. As usual, this behavior is preserved by convolution exponentials, and goes therefore recursively over − to the Υ− i,q and to Υ∞,q . Proposition 16. With the above hypothesis, i.e. a dimensionally regularized Feynman rules character ψ which is local with respect to the minimal subtraction scheme, the counter-factors and counter-term of the exponential method, Υ− i,q respectively, Υ∞,q , obtained using the subtraction scheme defined in terms of the projector T−q are strongly local. Proof. Indeed, we have, using the MS scheme, the BWH decomposition ψ = −1 −1 ψ− ∗ ψ+ , where ψ− is strongly local. Applying the exponential method with respect to the projector T−q to ψ + we get, according to Lemma 15, a decom− position ψ + = Υ− + ∗ ψ++ , where we write Υ+ (resp. ψ++ ) for the counterterm and renormalized character and where Υ− + is strongly local. We get: −1 ψ = ψ− ∗ Υ− ∗ ψ , where ψ is regular with respect to T−q . ++ ++ + From Lemma 14, we know that the counter-factors and counter-term for ψ in the exponential method for T−q are equal to the counter-factors and −1 counter-term for ψ− ∗ Υ− + , which is a product of strongly local characters, and therefore is strongly local. The Proposition follows then from Lemma 12 and its proof.
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4.2. A Toy-Model Calculation In the following example we apply the above introduced local non-Rota–Baxter type subtraction scheme within dimensional regularization. We exemplify it by means of a simple toy model calculation. We work with the bicommutative Hopf algebra H lad = k≥0 Hklad of rooted ladder trees. Let us recall the general co-product of the tree tn with n vertices: Δ(tn ) = tn ⊗ 1 + 1 ⊗ tn +
n−1
tn−k ⊗ tk .
k=1
The regularized toy model is defined by a character ψ ∈ G(A) mapping the tree tn to an n-fold iterated Riemann integral with values in A := C[[ε, ε−1 ]: ∞ p dx 1 ε ψ(p; ε, μ)(tn ) := μ ψ(x; ε, μ)(tn−1 ) 1+ε = exp −nε log , (22) x n!εn μ p
∞ with ψ(p; ε, μ)(t1 ) := με p xdx 1+ε , with μ, ε > 0, and where p denotes an external momenta. Recall that μ (’tHooft’s mass) has been introduced for dimensional reasons, so as to make the ratio μp a dimensionless scalar. In the following we will write a := log( μp ) and b := log( μq ), where q is fixed. For later use we write out the first three values: 1 1 1 1 ψ(p; ε, μ)(t1 ) = − a + εa2 − ε2 a3 + ε3 a4 − O(ε4 ) ε 2 3! 4! 1 1 2 1 2 ψ(p; ε, μ)(t2 ) = 2 − a + a2 − εa3 + ε2 a4 − ε3 a5 + O(ε4 ) 2ε ε 3 3 15 1 1 3 3 9 27 ψ(p; ε, μ)(t3 ) = − 2 a + a2 − a3 + εa4 − ε2 a5 + O(ε3 ). 3!ε3 2ε 4ε 4 16 80 ∞ Now, for a Laurent series α(p/μ) := n=−N αn (p/μ)εn , where the coefficients αn = αn (p/μ) are functions of p/μ, we define the following projector P− : ∞ −1 n P− αn (p/μ)ε αn (p/μ)εn + α1 (q/μ)ε, (23) := n=−N
n=−N
where q is fixed and chosen appropriately. We get: ∞ ∞ n P+ αn (p/μ)ε = α0 +(α1 (p/μ)−α1 (q/μ))ε + αn (p/μ)εn ∈ C[[ε]]. n=2
n=−N
One verifies that: P±2 = P±
and P± ◦ P∓ = P∓ ◦ P± = 0.
Let us emphasize that P− is not a Rota–Baxter map. This implies that we are not allowed to apply formulae (2) in Corollary 1 for the renormalization of ψ(p; ε, μ). However, we will show explicitly that the exponential method applies in this case, giving at each order a local counter-term(-factor) character as well as
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a finite renormalized character. At first order we apply the 1-regular character, ψ1+ , to the one vertex tree: ψ1+ (t1 ) = Υ(1) ∗ ψ(t1 ) = (exp∗ (−P− ◦ ψ ◦ π1 ) ∗ ψ) (t1 ) = −P− ◦ ψ ◦ π1 (t1 ) + ψ(t1 ) = −P− (ψ(t1 )) + ψ(t1 ) 1 1 2 1 1 1 + εb + − a + εa2 − ε2 a3 + O(ε3 ) =− ε 2 ε 2 3! 1 = −a + ε(a2 − b2 ) + O(ε2 ). 2 Observe that the counter-factor, and hence counter-term at order one is: 1 1 2 1 1 2 2 ∗ − εb ε Υ(1)(t1 ) = Υ− (t ) = exp (−P ◦ ψ ◦ π )(t ) = − = − b , 1 + − 1 1 1 1 ε 2 ε 2 which is local, i.e. does not contain any log(p/μ) terms. Let us define f = f (ε; q) := 1 + 12 ε2 b2 . Now, calculate the 2-regular character, ψ2+ , on the two vertex tree: ψ2+ (t2 ) = Υ(2) ∗ ψ(t2 ) = exp∗ (−P− ◦ ψ1+ ◦ π2 ) ∗ exp∗ (−P− ◦ ψ ◦ π1 ) ∗ ψ (t2 ) = ψ(t2 ) + Υ(1)(t1 )ψ(t1 ) + Υ(2)(t2 ) 1 = ψ(t2 ) − P− (ψ(t1 ))ψ(t1 ) − P− (ψ1+ (t2 )) + P− (ψ(t1 ))P− (ψ(t1 )) 2 = ψ(t2 ) − P− (ψ(t1 ))ψ(t1 ) − P− (ψ(t2 ) − P− (ψ(t1 ))ψ(t1 )) 1 1 − P− P− (ψ(t1 ))P− (ψ(t1 )) + P− (ψ(t1 ))P− (ψ(t1 )) 2 2 1 = P+ (ψ(t2 ) − P− (ψ(t1 ))ψ(t1 )) + P+ (P− (ψ(t1 ))P− (ψ(t1 ))) . 2 We first calculate the counter-term: Υ(2)(t2 ) = exp∗ (−P− ◦ ψ1+ ◦ π2 ) ∗ exp∗ (−P− ◦ ψ ◦ π1 )(t2 ) 1 = −P− ◦ ψ1+ (t2 ) + P− (ψ(t1 ))P− (ψ(t1 )) 2 f2 = −P− (Υ− ∗ ψ(t )) + 2 1 2ε2 Now observe that: 1 Υ− 1 ∗ ψ(t2 ) = ψ(t2 ) − P− (ψ(t1 ))ψ(t1 ) + P− (ψ(t1 ))P− (ψ(t1 )) 2 1 1 2 = 2 − a + a2 − εa3 + O(ε2 ) 2ε ε 3 1 1 2 1 1 2 1 2 3 3 + εb − a + εa − ε a + O(ε ) − ε 2 ε 2 3!
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1 1 2 1 1 1 2 + εb + εb + 2 ε 2 ε 2 1 1 = a2 − ε a3 − ab2 + O(ε2 ). 2 2 We get: − −P− (ψ(t2 ) − P− (ψ(t1 ))ψ(t1 ) + Υ− 1 (t2 )) = Υ2 (t2 ) = 0.
Hence, we find: − − Υ(2)(t2 ) = Υ− 2 ∗ Υ1 (t2 ) = Υ1 (t2 ) =
1 2 f , 2ε2
which is local, and: 1 1 ψ2+ (t2 ) = ψ1+ (t2 ) = a2 − ε a3 − ab2 + O(ε2 ) 2 2 2 1 1 = −a + ε(a2 − b2 ) + O(ε2 ) . 2 2 + − At third order, using Υ− 2 (t2 ) = P− (ψ1 (t2 )) = 0 and Υ2 (t1 ) = 0, a direct computation shows that similarly the order 3 counter-factor, Υ− 3 , evaluated on the order 3 tree, t3 , is zero: 2 ∗ 2 Υ− 3 (t3 ) = exp (−P− ◦ ψ+ ◦ π3 )(t3 ) = −P− ψ+ (t3 ) = 0
whereas ψ3+ (t3 ) = ψ2+ (t3 ) =
1 3!
3 1 −a + ε(a2 − b2 ) + O(ε2 ) , 2
and the counter-term at order 3 is: 1 3 f . 3!ε3 This pattern is general and encoded in the following proposition. Υ(3)(t3 ) = −
Proposition 17. The renormalization of the toy-model (22) via the exponential method, in the context of DR together with the general non-RB scheme (21) gives the nth-order counter-factor Υ− n (tn ) = 0 and counter-term: 1 Υ(n)(tn ) = (−f )n , n!εn with: 1 1 (−1)m+1 m+1 m+1 f = f (ε; q) := 1 + ε2 b2 − ε3 b3 + · · · + ε b 2 3! (m + 1)! m corresponding the Taylor jet operator (21), δε,q , say, of fixed order m ∈ N+ . The nth-regular, i.e. renormalized character is given by: 1 1 1 ψn+ (tn ) = −a + ε(a2 − b2 ) − ε2 (a3 − b3 ) n! 2 3! n m (−1) εm (am+1 − bm+1 ) − O(εm+1 ) +··· + (m + 1)!
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∞ Proof. Let us write T := 1 + n=1 tn for the formal sum of all rooted ladder trees. This sum is a group-like element (Δ(T ) = T ⊗ T ). It follows that n −ε 1 1 p −ε 1 p ψ(T ) = = exp n! ε μ ε μ n can be rewritten as the convolution exponential of the infinitesimal character η: η(t1 ) := 1ε ( μp )−ε , n = 1 η(tn ) := 0 else. Then: ψ(T ) = exp∗ (η)(T ). Let us write η − := P− (η) and η + := P+ (η), so that, in particular η − (t1 ) = 1 2 3 ε (a − b3 ) + · · · + −Υ(1)(t1 ) = fε and η + (t1 ) = −a + 12 ε(a2 − b2 ) − 3! (−1)m m m+1 m+1 m+1 −b ) − O(ε ). (m+1)! ε (a We get finally (recall that the convolution product of linear endomorphisms of a bicommutative Hopf algebra is commutative): ψ = exp∗ (η) = exp∗ (η− ) ∗ exp∗ (η+ ), where Υ(1)−1 = exp∗ (η− ) and where (by direct inspection) exp∗ (η+ ) is regular. It follows that ψ is renormalized already at the first order of the exponential algorithm, that is: Υ∞ = exp∗ (−η− ) = Υ(1) and ψ + = exp∗ (η+ ). The Proposition follows from the group-like structure of T which implies that: 1 Υ∞ (tn ) = exp∗ (−η− )(tn ) = (−η− (t1 ))n , n! and similarly for ψ + (tn ). Notice that in the classical MS scheme, one gets simply f = 1 in the above formulas. One recovers then by the same arguments the well-known result following from the BPHZ method in DR and MS.
Acknowledgements The first named author is supported by a de la Cierva grant from the Spanish government. We thank warmly J. Gracia-Bond´ıa. Long joint discussions on QFT in Nice and Zaragoza were seminal to the present work, which is part of a common long-term project.
References [1] Bellon, M., Schaposnik, F.: Renormalization group functions for the WessZumino model: up to 200 loops through Hopf algebras. Nuclear Phys. B 800, 517 (2008) [2] Brouder, Ch., Fauser, B., Frabetti, A., Krattenthaler, Ch.: Non-commutative Hopf algebra of formal diffeomorphisms. Adv. Math. 200, 479 (2006)
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[3] Brown, L. (ed.): Renormalization: From Lorentz to Landau (and Beyond). Springer, New York (1993) [4] Cartier, P.: Hyperalg`ebres et groupes de Lie formels. In: S´eminaire “Sophus Lie” de la Facult´e des Sciences de Paris, 1955–56. Secr´etariat math´ematique, 11 rue Pierre Curie, Paris, 61 pp (1957) [5] Caswell, W.E., Kennedy, A.D.: A simple approach to renormalization theory. Phys. Rev. D 25, 392 (1982) [6] Collins, J.: Renormalization. Cambridge monographs in mathematical physics, Cambridge (1984) [7] Connes, A., Kreimer, D.: Hopf algebras, renormalization and noncommutative geometry. Commun. Math. Phys. 199, 203 (1998) [8] Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem. Commun. Math. Phys. 210, 249 (2000) [9] Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann-Hilbert problem II: the β-function, diffeomorphisms and the renormalization group. Commun. Math. Phys. 216, 215 (2001) [10] Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives Colloquium Publications, vol. 55. American Mathematical Society, Providence (2008) [11] Delamotte, B.: A hint of renormalization. Am. J. Phys. 72, 170 (2004) [12] Dyson, F.: The S matrix in quantum electrodynamics. Phys. Rev. 75, 1736 (1949) [13] Ebrahimi-Fard, K., Gracia-Bond´ıa, J.M., Patras, F.: A Lie theoretic approach to renormalization. Commun. Math. Phys. 276, 519 (2007) [14] Ebrahimi-Fard, K., Gracia-Bond´ıa, J.M., Patras, F.: Rota–Baxter algebras and new combinatorial identities. Lett. Math. Phys. 81(1), 61 (2007) [15] Ebrahimi-Fard, K., Manchon, D., Patras, F.: A noncommutative Bohnenblust– Spitzer identity for Rota–Baxter algebras solves Bogoliubov’s recursion. J. Noncommutative Geom. 3(2), 181 (2009) [16] Ebrahimi-Fard, K., Patras, F.: A Zassenhaus-type algorithm solves the Bogoliubov recursion. In: Doebner, H.-D., Dobrev, V.K. (eds.) Proceedings of VII International Workshop“Lie Theory and Its Applications in Physics”, Varna, June 2007 [17] Figueroa, H., Gracia-Bond´ıa, J.M.: Combinatorial Hopf algebras in quantum field theory I. Rev. Math. Phys. 17, 881 (2005) [18] Itzykson, C., Zuber, J.-B.: Quantum Field Theory. McGraw-Hill, New York (1980) [19] Joni, S.A., Rota, G.-C.: Coalgebras and bialgebras in combinatorics. Stud. Appl. Math. 61, 93 (1979) [20] Kreimer, D.: Anatomy of a gauge theory. Ann. Phys. 321, 2757 (2006) [21] Kreimer, D.: Chen’s iterated integral represents the operator product expansion. Adv. Theor. Math. Phys. 3, 627–670 (1999) [22] Manchon, D.: Hopf algebras in renormalisation. In: Hazewinkel, M. (ed.) Handbook of Algebra, vol. 5, pp. 365–427. Elsevier, Oxford (2008) [23] Reutenauer, C.: Free Lie Algebras. Oxford University Press, Oxford (1993)
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[24] van Suijlekom, W.: Multiplicative renormalization and Hopf algebras. In: Ceyhan, O., Manin, Yu.-I., Marcolli, M. (eds.) Arithmetic and Geometry Around Quantization. Birkh¨ auser, Basel (2008) [25] van Suijlekom, W.: Renormalization of gauge fields: a Hopf algebra approach. Commun. Math. Phys. 276, 773 (2007) Kurusch Ebrahimi-Fard Departamento de F´ısica Te´ orica Universidad de Zaragoza 50009 Zaragoza Spain e-mail:
[email protected];
[email protected] Fr´ed´eric Patras Universit´e de Nice Laboratoire J.-A. Dieudonn´e UMR 6621, CNRS Parc Valrose 06108 Nice Cedex 02 France e-mail:
[email protected] Communicated by Vincent Rivasseau. Received: March 17, 2010. Accepted: May 12, 2010.
Ann. Henri Poincar´e 11 (2010), 973–990 c 2010 Springer Basel AG 1424-0637/10/050973-18 published online October 16, 2010 DOI 10.1007/s00023-010-0048-1
Annales Henri Poincar´ e
On the Lipschitz Continuity of Spectral Bands of Harper-Like and Magnetic Schr¨ odinger Operators Horia D. Cornean Abstract. We show for a large class of discrete Harper-like and continuous magnetic Schr¨ odinger operators that their band edges are Lipschitz continuous with respect to the intensity of the external constant magnetic field. We generalize a result obtained by Bellissard (Commun Math Phys 160:599–613, 1994), and give examples in favor of a recent conjecture of G. Nenciu.
1. Introduction and the Main Results Harper-like operators. Let Γ ⊂ R2 be a (possibly irregular) lattice which has the property that there exists an injective map F : Γ → Z2 such that |F (γ) − γ| < 1/2. The Hilbert space is l2 (Γ). The elements of the canonical basis in l2 (Γ) are denoted by {δx }x∈Γ , where δx (y) = 1 if y = x and zero otherwise. In the discrete case, to any bounded self-adjoint operator H ∈ B(l2 (Γ)) it corresponds a bounded and symmetric kernel H(x, x ) = Hδx , δx = H(x , x). We will extensively use the Schur–Holmgren upper bound for the norm of a self-adjoint operator: |H(x, x )|. (1.1) ||H|| ≤ sup x ∈Γ
x∈Γ α
Denote by x − x0 α = [1 + (x − x0 )2 ] 2 , α ≥ 0. We define C α to be the set of bounded and self-adjoint operators H ∈ B(l2 (Γ)) which have the property that their kernels obey a weighted Schur–Holmgren type estimate: x − x α |H(x, x )| < ∞. (1.2) ||H||C α := sup x ∈Γ
x∈Γ
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We also define the space Hα which contains bounded and self-adjoint operators H which obey: 12 x − x 2α |H(x, x )|2 < ∞. (1.3) ||H||Hα := sup x ∈Γ
x∈Γ
The flux of a unit magnetic field orthogonal to the plane through a triangle generated by x, x and the origin is given by: 1 (1.4) ϕ(x, x ) := − (x1 x2 − x2 x1 ) = −ϕ(x , x). 2 Note the important additive identity: ϕ(x, y) + ϕ(y, x ) = ϕ(x, x ) + ϕ(x − y, y − x ), (1.5) 1 |ϕ(x − y, y − x )| ≤ |x − y| |y − x |. 2 Let K ∈ C 0 . Let its kernel be K(x, x ). We are interested in a family of Harper-like operators {Kb }b∈R given by the kernels eibϕ(x,x ) K(x, x ). Clearly, {Kb }b∈R ⊂ C 0 . The usual Harper operator lives in l2 (Z2 ), and its generating kernel has the form K(x, x ) = k(x − x ) where k(x) equals 1 if |x| = 1, and 0 otherwise. In Lemma 2.1, we will show that Hα ⊂ C 0 if α > 1. Now here is the first main result of our paper: Theorem 1.1. Let α > 3 and K ∈ Hα . Construct the corresponding family of Harper-like operators {Kb }b∈R . Then we have: i. The resolvent set ρ(Kb ) is stable; more precisely, if dist(z, σ(Kb0 )) ≥ > 0 then there exist δ > 0 and η > 0 such that dist(z, σ(Kb )) ≥ η whenever |b − b0 | < δ. ii. Define E+ (b) := sup σ(Kb ) and E− (b) := inf σ(Kb ). Then E± are Lipschitz functions of b. iii. Let α > 4. Assume that Kb0 has a gap in the spectrum of the form (e− (b0 ), e+ (b0 )), where e± (b0 ) ∈ σ(Kb0 ) are the gap edges. Then as long as the gap is not closing by varying b in a closed interval I containing b0 , the operator Kb will have a gap (e− (b), e+ (b)) whose edges are Lipschitz functions of b on I. Remark. Denoting by δb = b − b0 , then according to our notations we have that Kb = (Kb0 )δb . It means that it is enough to prove spectral stability and Lipschitz properties near b0 = 0. We can complicate the setting by allowing the generating kernel to depend on b. Corollary 1.2. Assume that the generating kernel K(x, x ; b) obeys all the spatial localization conditions of Theorem 1.1, uniformly in b ∈ R. Moreover, assume that it also satisfies an extra condition: |K(x, x ; b) − K(x, x ; b0 )| ≤ C |b − b0 |, |b − b0 | ≤ 1. (1.6) sup x ∈Γ
x∈Γ
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Consider the family {Kb }b∈R generated by eibϕ(x,x ) K(x, x ; b). Then Theorem 1.1 holds true for Kb . Continuous Schr¨ odinger operators. Let us consider the operator in L2 (R2 ) H(b) := (p − ba)2 + V,
p = −i∇x ,
a(x) = (−x2 /2, x1 /2),
b ∈ R. (1.7)
where we assume that the scalar potential V is smooth and bounded together with all its derivatives on R2 . This very strong condition is definitely not necessary for the result given below, but it simplifies the presentation. For the same reason we formulate the result only near b0 = 0. Theorem 1.3. Assume that the spectrum of H(0) has a finite and isolated spectral band σ0 , where σ0 = [s− (0), s+ (0)]. Then if |b| is small enough, σ0 will evolve into a still isolated spectral island σb ⊂ σ(H(b)). Denote by s− (b) := inf σb and s+ (b) := sup σb . Then these edges are Lipschitz at b = 0, i.e. there exists a constant C such that |s± (b) − s± (0)| ≤ C |b|. Remark. We do not exclude the appearance of gaps inside σb . Moreover, the formulation of this result is slightly different from the one we gave in the discrete case. Here we look at the edges of a finite part of the spectrum, and not at the edges of a gap. In the discrete case both formulations are equivalent. However, our proof does not work in the continuous case if σ0 is infinite. 1.1. Previous Results and Open Problems Spectrum stability is a fundamental issue in perturbation theory. It is well known that if W is relatively bounded to H0 , then the spectrum of Hλ = H0 + λW is at a Hausdorff distance of order |λ| from the spectrum of H0 . But this is in general not true for perturbations which are not relatively bounded. And the magnetic perturbation coming from a constant field is not relatively bounded, neither in the discrete nor in the continuous case. With the notable exception of a recent paper by Nenciu [29], all previous results on the discrete case we are aware of deal with the situation in which Γ = Z2 and the generating kernel obeys K(x, x ) = k(x − x ), where k is sufficiently fast decaying at infinity. Maybe the first proof of spectral stability of Harper operators is due to Elliott [12]. The result is refined in [6] where it is shown that the gap boundaries are 13 -H¨older continuous in b. Later results by Avron et al. [2,3], Helffer and Sj¨ ostrand [17,18], and Haagerup and Rørdam [15] pushed the exponent up to 12 . In fact they prove more, they show that the 1 Hausdorff distance between spectra behaves like |b − b0 | 2 . These results are optimal in the sense that the H¨ older constant is independent of the length of the eventual gaps, and it is known that these gaps can close down precisely 1 like |b − b0 | 2 near rational values of b0 [16,18]. Note that Nenciu [29] proves a similar result for a much larger class of Harper-like operators. Many other spectral properties of Harper operators can be found in a paper by Herrmann and Janssen [19]. In the continuous case, the stability of gaps was first shown by Avron and Simon [1], and Nenciu [28]. Nenciu’s result implicitly gives a 12 -H¨older
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continuity in b for the Hausdorff distance between spectra. Then in [5] the H¨ older exponent of gap edges was pushed up to 23 . The first proof of Lipschitz continuity of gap edges for Harper-like operators was given by Bellissard [4] (later on Kotani [22] extended his method to more general regular lattices and dimensions larger than two). The configuration space is Γ = Z2 and the generating kernel is of the form K(x, x ) = k(x − x ; b), where k(x; b) decays polynomially in |x| and is allowed to depend smoothly on b. This extra-dependence is not central for our discussion, so we will consider that k is b independent. Bellissard’s innovative idea uses in an essential way that the Harper operators generated by translation invariant and fast decaying kernels k(x − x ) can be written as linear combinations of magnetic translations: Kb = k(γ)Wb (γ), γ∈Z2
[Wb (γ)ψ](x) = eibϕ(x,γ) ψ(x − γ),
Wb (γ)Wb (γ ) = eibϕ(γ,γ ) Wb (γ + γ ). Bellissard’s crucial observation was that the C ∗ algebra Ab0 +δ generated by {Wb0 +δ (γ)}γ∈Z2 is isomorphic with a sub-algebra of Ab0 ⊗ Aδ which is gen b +δ erated by {Wb0 (γ) ⊗ Wδ (γ)}γ∈Z2 . Thus one can construct an operator K 0 which is isospectral with Kb0 +δ . The new operator lives in the space l2 (Z2 ) ⊗ b = Kb ⊗ Id. It turns out that it is more convenient to study the L2 (R), and K 0 0 spectral edges of the new operator. The reason is that the singularity induced by the magnetic perturbation is hidden in the extra-dimension. But the proof breaks down in case of irregular lattices or if the generating kernel K(x, x ) is not just a function of x − x . Coming back to our proof, its crucial ingredient consists in expressing the magnetic phases with the help of the heat kernel of a continuous Schr¨ odinger operator, see (5.8)–(5.12). Moreover, the proof in the discrete case also works for continuous kernels living on R2 and not just on lattices. This is what we use in the last step of the proof of Theorem 1.3 dealing with continuous magnetic Schr¨ odinger operators. A limitation of our method consists in the fact that the phases ϕ(x, x ) are generated by a constant magnetic field. A more general discrete problem was formulated by Nenciu [29] where he proposed to replace the explicit formulas in (1.4) and (1.5) with more general real and antisymmetric phases obeying φ(x, x ) = φ(x, x ) = −φ(x , x) and |φ(x, y) + φ(y, x ) + φ(x , x)| ≤ area Δ(x, y, x ) where Δ(x, y, x ) is the triangle generated by the three points. These phases appear very naturally in the continuous case, see [8,9,20,23–27], where it is shown that if a(x) is the transverse gauge generated by a globally bounded magnetic field |b(x)| ≤ 1, then φ(x, x ) can be chosen to be the path integral of a(x) on the segment linking x with x. This is the same as the magnetic flux of b through the triangle generated by x, x and the origin.
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Using a completely different proof method, Nenciu shows among other things in [29] that the gap edges are Lipschitz up to a logarithmic factor, and he conjectures that they are actually Lipschitz. His method relies on the theory of almost convex functions, and the result provided by this technique is optimal in the sense that it cannot be improved in order to get rid of the logarithm. A new idea would be necessary in order to prove Nenciu’s Lipschitz conjecture. Our current paper supports this conjecture because it provides examples of phases not coming from a constant magnetic field which still generate Lipschitz gap edges. Let us show this here. Consider an irregular lattice Γ ⊂ R2 which is a local deformation of Z2 , that is there exists a bijective map F : Γ → Z2 such that |F (γ) − γ| < 12 . Define the phases ϕ(x, x ) := ϕ(F −1 (x), F −1 (x )) where ϕ is given by (1.4). Choose any self-adjoint operator K ∈ B(l2 (Z2 )) given by a kernel K(x, x ) sufficiently fast decaying outside the diagonal. The same operator can be seen in B(l2 (Γ)) given by K(γ, γ ) := K(F (γ), F (γ )). Thus the opera ibϕ(x,x ) tor Kb generated by Kb (x, x ) := e K(x, x ) is unitary equivalent with 2 an operator in B(l (Γ)) with a kernel b (γ, γ ) := eibϕ(γ,γ ) K(γ, K γ ). In this case, we know from Theorem 1.1 that the edges of the spectral gaps of b and thus Kb will have a Lipschitz behavior. But the general case remains K open.
2. Proof of Theorem 1.1 This section is dedicated to the proof of our first theorem. Parts of this proof will be later on adapted to the continuous case in Theorem 1.3. 2.1. Proof of (i) Let us start by showing the existence of natural embeddings of C α ’s in Hα ’s given by the following short lemma: Lemma 2.1. Let H ∈ Hα with α > 1. Then H ∈ C β with β < α − 1. In particular, if α > 3 then the kernel x − x 2 |H(x, x )| obeys a Schur–Holmgren estimate and thus defines a bounded operator. Proof. Choose some small enough > 0 such that α > β + 1 + . We write: x − x β |H(x, x )| ≤ x − x −1− x − x α |H(x, x )| and see that the Cauchy–Schwarz inequality gives ||H||C β ≤ Cα,β ||H||Hα .
(2.1)
Another technical estimate to be proved in “Appendix” claims that if H has a kernel which is localized near the diagonal, then the resolvent’s kernel will also have such a localization.
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Proposition 2.2. Let H ∈ C α , with α > 0. Let z ∈ ρ(H). Then for every 0 ≤ α < α we have (H − z)−1 ∈ Hα , and there exists a constant C independent of z such that ||(H − z)−1 ||Hα ≤ C (1 +
||H||α+1 Cα )
1 1 + α+2 {dist(z, σ(H))} dist(z, σ(H))
.
(2.2)
Remark. This proposition is related to what specialists in von Neumann algebras would call dual action, see [30–32]. Stronger localization results have been earlier obtained by Jaffard [21], later generalized by Gr¨ ochenig and Leinert [14]. We choose for completeness to give an elementary proof in “Appendix”; our proof also highlights the uniformity in z ∈ ρ(H). Now let us start the proof of (i). Constants only depending on will be named C even though they might have different values. Remember that it is enough to prove the stability result near b0 = 0. Let K ∈ Hα with α > 3. Lemma 2.1 gives us some β > 2 such that K ∈ C β . Prop osition 2.2 says that (K − z)−1 ∈ Hβ with some 2 < β < β, while Lemma 2.1 insures that there exists γ > 1 such that (K − z)−1 ∈ C γ . Denote by G(x, x ; z) the kernel of (K − z)−1 . From (2.1) and (2.2), we obtain a constant C such that: x − x |G(x, x ; z)| ≤ C if dist(z, σ(K)) ≥ . (2.3) sup x ∈Γ
x∈Γ
Define the operator Sb (z) to be the one corresponding to the kernel eibϕ(x,x ) G(x, x ; z). Using the Schur–Holmgren criterion we can write ||Sb (z)|| ≤ C ,
b ∈ R,
dist(z, σ(K)) ≥ .
Using (1.5) we can write: (Kb − z)Sb (z) =: 1 + Tb (z), where Tb (z) is given by the kernel (eibϕ(x−y,x −y) − 1)K(x, y) G(y, x ; z). eibϕ(x,x )
(2.4)
(2.5)
y∈Γ
Note that
|eibϕ(x−y,x −y) − 1| ≤ |b| |ϕ(x − y, x − y)| ≤
|b| |x − y| |y − x |. 2
Then for any f ∈ l2 (Γ) with compact support we can write: |Tb (z)f |(x) ≤ |b| |x − y| |K(x, y)| |y − x | |G(y, x ; z)| |f (x )| y∈Γ
and after applying the Schur–Holmgren criterion we get: ||Tb (z)|| ≤ |b| ||K||C 1 ||(K − z)−1 ||C 1 ≤ |b| C .
(2.6)
(2.7)
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Thus if |b| is small enough, ||Tb (z)|| ≤ 1/2 whenever dist(z, σ(K)) ≥ . Now if Imz = 0 we know that Kb − z is invertible, and from (2.4) we conclude that there exists a constant C such that (Kb − z)−1 = Sb (z) (1 + Tb (z))−1 , ||(Kb − z)−1 || ≤ C
whenever |b| ≤ b and dist(z, σ(K)) ≥ ,
uniformly in the imaginary part of z. This means that dist(z, σ(Kb )) ≥ whenever |b| ≤ b and dist(z, σ(K)) ≥ , and the proof of (i) is over.
(2.8) 1 C
>0
2.2. Proof of (ii) As before, we only need to consider b0 = 0. We give the proof just for the upper spectral limit E+ , since the argument for E− is similar. 2.2.1. Reduction to Localized Operators. We start with an abstract lemma. Lemma 2.3. Let M (b) and N (b) be two families of bounded and self-adjoint operators on some Hilbert space H, such that ||M (b)−N (b)|| ≤ C |b| if |b| ≤ 1. Then: | sup σ(M (b)) − sup σ(N (b))| ≤ ||M (b) − N (b)|| ≤ C |b|,
|b| ≤ 1, (2.9)
and a similar estimate holds for the infimum of their spectra. In particular, if sup σ(N (b)) is Lipschitz at b = 0 then the same is true for sup σ(M (b)). Remark. Note the important thing that we do not require from M (b) and N (b) to converge in norm to M (0) = N (0) when b tends to zero. Proof. For every ψ ∈ H with ||ψ|| = 1 we can write M (b)ψ, ψ ≤ N (b)ψ, ψ + ||M (b) − N (b)|| ≤ sup σ(N (b)) + ||M (b) − N (B)|| which means that sup σ(M (b)) − sup σ(N (b)) ≤ ||M (b) − N (b)||. By interchanging M (b) with N (b) we obtain the inequality: | sup σ(M (b)) − sup σ(N (b))| ≤ ||M (b) − N (b)||.
(2.10)
A similar argument shows the same estimate for the infimum of the spectra. Regarding the Lipschitz property, we use that sup σ(M (0)) = sup σ(N (0)) and then we apply the triangle inequality: | sup σ(M (b)) − sup σ(M (0))| ≤ | sup σ(N (b)) − sup σ(N (0))| + ||M (b) − N (b)|| ≤ C |b|.
(2.11)
Getting back to our theorem, we now want to reduce the problem to operators with kernels supported near the diagonal. Denote by χ the charac b the operator given by the teristic function of the √ interval [0, 1]. Denote by K b the operator given by kernel Kb (x, x √ ) := χ( b|x − x |)K(x, x ) and by K ibϕ(x,x ) Kb (x, x ) := χ( b|x − x |)e K(x, x ). Since K ∈ Hα with α > 3, according to Lemma 2.1 we have the bound: sup x − x 2 |K(x, x )| = ||K||C 2 < ∞. (2.12) x ∈Γ
x∈Γ
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Via the Schur–Holmgren criterion we obtain: b ||, ||Kb − K b ||} max{||K − K
√ 1 − χ( b|x − x |) |K(x, x )| ≤ |b| ||K||C 2 . ≤ sup x ∈Γ
(2.13)
x∈Γ
b we obtain |E+ (0) − sup(σ(K b ))| ≤ Using Lemma 2.3 for the pair K and K b ))|≤ |b| ||K||C 2 . The same lemma for the pair Kb and Kb gives |E+ (b)−sup(σ(K |b| ||K||C 2 . Then the triangle inequality leads to: b )) − sup(σ(K b ))|. |E+ (b) − E+ (0)| ≤ 2|b| ||K||C 2 + | sup(σ(K
(2.14)
b Thus we have reduced the problem to the study of the spectral edges of K and Kb . b (x, x ) = eibϕ(x,x ) 2.2.2. Study of the Operators with Cut-Off. Clearly, K b (x, x ). Without loss, assume that b > 0. Take ψ ∈ l2 (Γ) with compact K support and compute (use (5.10) in the second equality): b (x, x )ψ(x )ψ(x) b ψ, ψ = K eibϕ(x,x ) K x,x ∈Γ
dy
= R2
× exp
ψ(x )ψ(x)
x,x ∈Γ
4π sinh(2bt) Kb (x, x ) b
b|x − x |2 Gb (x, y; t)Gb (y, x ; t). 4 tanh(2bt)
(2.15)
Now denote by Ab (t) the operator with kernel
b|x − x |2 Ab (x, x ; t) := Kb (x, x ) exp 4 tanh(2bt)
√ b|x − x |2 = K(x, x ) × exp χ( b|x − x |). 4 tanh(2bt) The crucial observation is that Eq. (2.15) leads to: 4π sinh(2bt) Kb ψ, ψ = dyAb (t)Gb (y, ·; t)ψ, Gb (y, ·; t)ψ b R2 4π sinh(2bt) ≤ sup σ(Ab (t)) dy||Gb (y, ·; t)ψ||2 b R2 4π sinh(2bt) = sup σ(Ab (t)) dy |Gb (y, x; t)|2 |ψ(x)|2 b R2
2
= sup σ(Ab (t)) ||ψ|| ,
x∈Γ
(2.16)
b ) ≤ sup(σ(Ab (t))) where in the last line we used (5.12). It means that sup σ(K for all t. Now let us show that the operator Ab (t)− Kb has a norm proportional
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with b if t is large enough (say t = b−1 ). Indeed, we can write
√ b|x − x |2 −1 |Ab (x, x ; b ) − Kb (x, x )| ≤ |K(x, x )|χ( b|x − x |) exp −1 4 tanh(2)
√ b|x − x |2 b|x − x |2 exp ≤ |K(x, x )|χ( b|x − x |) (2.17) 4 tanh(2) 4 tanh(2) and on the support of χ we can bound the above difference with: b (x, x )| ≤ const b |x − x |2 |K(x, x )|. |Ab (x, x ; b−1 ) − K
(2.18)
The right-hand side defines an operator whose norm behaves like b. Thus (2.16) and (2.18) imply: b ) ≤ sup σ(Ab (b−1 )) sup σ(K
b || ≤ C b. and ||Ab (b−1 ) − K
(2.19)
b we arrive at: Using (2.10) for the pair Ab (b−1 ) and K b ) ≤ sup σ(K b ) + C b. sup σ(K
(2.20)
b and K b in the above inequalWe now want to change places between K ity, which would lead to sup σ(Kb ) ≤ sup σ(Kb ) + C b and thus: b ) − sup σ(K b )| ≤ C b, | sup σ(K which together with (2.14) would imply: |E+ (b) − E+ (0)| ≤ C b,
b ≥ 0.
b (x, x ) = The key step in the proof of (2.20) was (2.15). Since K b (x, x ) we can write (use (5.11) in the second line): e−ibϕ(x,x ) K b (x, x )ψ(x )ψ(x) b ψ, ψ = e−ibϕ(x,x ) K K x,x ∈Γ
dy
= R2
× exp
x,x ∈Γ
ψ(x )ψ(x)
4π sinh(2bt) Kb (x, x ) b
b|x − x |2 Gb (x , y; t)Gb (y, x; t). 4 tanh(2bt)
(2.21)
Now everything will work as before, because the phase eibϕ(x,x ) changes neither the localization nor the C 2 norm of the operators. The proof for the upper spectral edges is over. The proof for the lower spectral edges is based on an estimate which is very similar with (2.16), in which we reverse the inequality and show that b ) ≥ inf σ(Ab (t)) for all t. We give no further details. inf σ(K 2.3. Proof of (iii) The idea is to reduce the problem to the previous case. Again it is enough to consider b0 = 0 and b > 0 small enough. Assume that K has a gap in its spectrum of the form (e− , e+ ), with e± ∈ σ(K). Then due to (i) we know that if b is small enough the gap will survive: we can choose a positively oriented
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circle L in the complex plane containing Σ+ (b) := σ(Kb ) ∩ [e+ (b), ∞) such that dist(z, σ(Kb )) ≥ η > 0
whenever z ∈ L and 0 < b < bη .
The orthogonal projector Pb corresponding to Σ+ (b) can be written as a Riesz integral and we have: i i (Kb − z)−1 dz, Kb Pb = z(Kb − z)−1 dz, b ≥ 0. (2.22) Pb := 2π 2π L
L
If we consider Kb Pb as an operator living on the whole space l2 (Γ), then its spectrum is given by the union {0} ∪ Σ+ (b). If we choose λ := 1 + sup σ(K), then for b small enough the operator Db := Kb Pb − λPb will have inf σ(Db ) = e+ (b) − λ ≤ −1/2. Thus e+ (b) = λ + inf σ(Db ), hence e+ (b) is Lipschitz at b = 0 if inf σ(Db ) has the same property. This is what we prove next: Lemma 2.4. Let Db = Kb Pb − λPb with λ := 1 + sup σ(K). Then there exists b1 > 0 small enough and a constant C > 0 such that for every 0 < b < b1 we have | inf σ(Db ) − inf σ(D0 )| ≤ C b. Proof. Remember that we imposed α > 4. We have that ||Kb ||Hα = ||K||Hα < ∞ for all b. According to Lemma 2.1, there exists β > 3 such that ||Kb ||C β = ||K||C β < ∞. Then if b is smaller than some constant only depending on L, Proposition 2.2 tells us that (Kb − z)−1 ∈ Hβ for some 3 < β < β, for all z ∈ L and supz∈L ||(Kb − z)−1 ||Hβ ≤ C. Thus both Pb and Db belong to Hβ with β > 3 if b is small enough. More precisely, there exists b2 > 0 sufficiently small such that max{||Pb ||Hβ , ||Db ||Hβ } ≤ C,
0 ≤ b ≤ b2 .
(2.23)
−1
If G(x, x ; z) is the integral kernel of (K − z) , then we introduced at point (i) the operator Sb (z) given by the kernel eibϕ(x,x ) G(x, x ; z). Using (2.8) we can write: sup ||(Kb − z)−1 − Sb (z)|| ≤ C b, z∈L
(2.24)
provided b is small enough. Denoting by D0 the operator given by the integral kernel i (z − λ)G(x, x ; z) dz D0 (x, x ) := 2π L
and by (D0 )b the operator generated by eibϕ(x,x ) D0 (x, x ), then using (2.24) we arrive at the estimate: ||Db − (D0 )b || ≤ C b whenever 0 ≤ b < b2 .
(2.25)
It follows from Lemma 2.3 that inf σ(Db ) is Lipschitz at b = 0 if inf σ((D0 )b ) has the same property. But for the operator (D0 )b we can apply point (ii), and the proof is over.
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3. Proof of Corollary 1.2 In order to keep the notation simple, we will only consider b0 = 0. Here the generating kernel K(x, x ; b) depends on b and (1.6) at b0 = 0 reads as: sup |K(x, x ; b) − K(x, x ; 0)| ≤ C |b|, |b| ≤ 1. (3.1) x ∈Γ
x∈Γ
b where their kernels are given by eibϕ(x,x ) Let us introduce the family K b || ≤ C |b| K(x, x ; 0). Clearly, K0 = K0 . Moreover, (3.1) implies that ||Kb − K around b = 0. We know that Theorem 1.1 (ii) applies for Kb around b = 0, so the only thing we have left is to extend it to Kb . From Lemma 2.3 we immediately conclude that sup σ(Kb ) and inf σ(Kb ) are Lipschitz at b = 0. The spectral stability of Kb can be shown with the same strategy as the one used in (2.3)–(2.8). The operator Sb (z) must be constructed starting from the kernel of (K0 − z)−1 which gets multiplied with the phase eibϕ(x,x ) . When we act with Kb − z on Sb (z) as in (2.4), we obtain an extra term which enters b )Sb (z). This error is again proportional with |b| if in Tb (z), which is (Kb − K z is at some distance from the spectrum of K0 . Thus (2.8) holds again. For the case of gaps, the proof is identical with the case independent of b.
4. Proof of Theorem 1.3 There are important similarities between the proof strategies in the discrete and continuous cases. Although the stability of the resolvent set of H(b) is known, we will sketch a short proof which will also provide some ingredients for the proof of the Lipschitz behavior of the band edges. 4.1. Stability of Gaps Assume that M ⊂ ρ(H(0)) is a compact set and dist(M, σ(H(0))) > 0. The resolvent (H(0) − z)−1 is an integral operator given by an integral kernel Q0 (x, x ; z). The singularities of Q0 (x, x ; z) are the same as in the case of the free Laplacian and there exists some δ > 0 and CM < ∞ such that uniformly in x = x [13,11]:
sup |Q0 (x, x ; z)| ≤ CM (1 + | ln(|x − x |)|)e−δ|x−x | , z∈M 1 sup |∇x Q0 (x, x ; z)| ≤ CM 1 + e−δ|x−x | . |x − x | z∈M In particular we have the following Schur–Holmgren type property: sup sup |Q0 (x, x ; z)| dx ≤ C(M ) < ∞. z∈M x ∈R2
R2
(4.1)
(4.2)
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This allows us to define for every z ∈ M a bounded operator Sb (z) whose integral kernel is given by:
Sb (x, x ; z) := eibϕ(x,x ) Q0 (x, x ; z),
sup ||Sb (z)|| ≤ C(M ) < ∞. (4.3)
z∈M
Define Tb (z) to be the operator with the integral kernel:
Tb (x, x ; z) := beibϕ(x,x ) {2ia(x − x )∇x Q0 (x, x ; z) + b|a(x − x )|2 Q0 (x, x ; z)}.
(4.4)
The kernel Tb (x, x ; z) is bounded because the inequality |a(x − x )| ≤ |x − x | compensates the local singularities of ∇x Q0 (x, x ; z) and Q0 (x, x ; z) when |x − x | is small, while when |x − x | is large we have the exponential decay which comes into play. In fact, using (4.1) we see that the kernel Tb (x, x ; z) obeys a Schur-Holmgren estimate. We get: sup ||Tb (z)|| ≤ C(M ) |b|,
z∈M
|b| ≤ 1.
(4.5)
Note the important identity valid on Schwartz functions:
{−i∇x − ba(x)}eibϕ(x,x ) = eibφ(x,x ) {−i∇x − ba(x − x )}.
(4.6)
Let us note that Sb (z) leaves the Schwartz space invariant and for such two functions f and g we have (using (4.6)): {(p − ba)2 + V − z}Sb (z)f, g = eibϕ(x,x ) ({−i∇x − ba(x − x )}2 + V (x) − z) R2 R2
×Q0 (x, x ; z)f (x )g(x) dxdx = f, g + Tb (z)f, g.
(4.7)
The operator H(b) is essentially self-adjoint on the Schwartz space, and after a density argument we conclude that the range of Sb (z) is contained in the domain of H(b) and (H(b) − z)Sb (z) = 1 + Tb (z). Now there exists b1 > 0 small enough such that if |b| ≤ b1 we have supz∈M ||Tb (z)|| ≤ 1/2 [see (4.5)]. Then after a standard argument we conclude (H(b) − z)−1 = Sb (z)(1 + Tb (z))−1 , sup ||(H(b) − z)−1 || ≤ CM ,
z∈M
|b| ≤ b1 .
(4.8)
This means that the gaps in the spectrum of H(0) are preserved. In particular, for every > 0 there exists b2 () > 0 such that: s− (0) − ≤ s− (b) ≤ s+ (b) ≤ s+ (0) + whenever |b| ≤ b2 ().
(4.9)
Choose a positively oriented circle L isolated from σ(H(0)) such that L completely contains the finite band σ0 . Then if |b| is small enough L will completely contain σb and remain separated from σ(H(b)).
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4.2. The Reduction to Harper-like Operators As in the discrete case, we construct the Riesz integrals i i −1 (H(b) − z) dz, K(b) := H(b)Pb = z(H(b) − z)−1 dz. Pb := 2π 2π L
L
2
2
The operator H(b)Pb seen in the whole space L (R ) will have the spectrum σb ∪ {0}. Fix λ+ := 1 − s− (0). If |b| ≤ b2 ( 12 ) [see (4.9)], then we know that s+ (b) + λ+ ≥ s− (b) + λ+ ≥ 12 > 0. It means that s+ (b) + λ+ = sup σ{H(b)Pb + λ+ Pb }. Similarly, choosing λ− := −1 − s+ (0) we have s− (b) + λ− < − 12 < 0 hence s− (b) + λ− = inf σ{H(b)Pb + λ− Pb }. In other words, the band edges s± will be Lipschitz at b = 0 if the spectral edges of the operators i K± (b) := H(b)Pb + λ± Pb = (z + λ± )(H(b) − z)−1 dz 2π L
have the same property. Note that the operator K± (0) has an integral kernel given by: i K± (0)(x, x ) = (z + λ± )Q0 (x, x ; z) dz, |K± (0)(x, x )| ≤ Ce−δ|x−x | , 2π L
(4.10)
where the local singularity at x = x disappears due to the integral with respect to z. Now using (4.8), (4.3) and (4.5) we have: i K± (b) − (4.11) (z + λ± )Sb (z) dz ≤ C |b|. 2π L
According to Lemma 2.3, the spectral edges of K± (b) are Lipschitz at b = 0 if the same property is true for i (K± (0))b := (z + λ± )Sb (z) dz. 2π L
This notation wants to highlight the fact that (K± (0))b is given by the integral kernel
(K± (0))b (x, x ) := eibϕ(x,x ) K± (0)(x, x ). At this point we are in a situation which is completely similar to the discrete case, with the difference that the Hilbert space is L2 (R2 ) and the sums over Γ have to be replaced by integrals. The unperturbed kernel K± (0)(x, x ) has an exponential localization. We can mimic the proof of Theorem 1.1 (ii) and conclude that the spectral edges of (K± (0))b are Lipschitz at b = 0, and we are done.
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Acknowledgements H.C. acknowledges support from the Danish F.N.U. grant Mathematical Physics. The author is deeply indebted to Gheorghe Nenciu for his encouragements and for many fruitful discussions.
5. Appendix 5.1. Proof of Proposition 2.2 Denote by G(x, x ; z) the integral kernel of (H − z)−1 . If α = 0 we have 1 |G(x, x ; z)|2 = ||(H − z)−1 δx ||2 ≤ {dist(z, σ(H))}2 x∈Γ
uniformly in x , an estimate which is in fact much better than (2.2). So from now on we may assume that 0 < α < α. For k ∈ R2 define the unitary multiplication operator Uk by (Uk f )(x) = ik·x f (x). Define the family of isospectral operators Hk = Uk HUk∗ , with intee gral kernels given by Hk (x, x ) = eik·(x−x ) H(x, x ). We need the following technical result: Lemma 5.1. Let H be an element of C α . Let n be the integer part of α. Then the mapping R2 k → Hk ∈ B(l2 (Γ)) is n times continuously differentiable in the norm topology. Moreover, any n’th order mixed partial derivative of Hk is α − n H¨ older continuous at k = 0 in the norm topology.
Proof. Assume that k = (k1 , k2 ). The integral kernel of Hk is eik·(x−x ) H(x, x ). Let n be the integer part of α. Then Hk is n times differentiable in the norm topology with respect to kj , j ∈ {1, 2}, and its n’th mixed Hk is given by the integral kernel in (x1 −x1 )m (x2 − partial derivative ∂km1 ∂kn−m 2 x2 )n−m eik·(x−x ) H(x, x ). This integral kernel defines a bounded operator because |(x1 − x1 )m (x2 − x2 )n−m | ≤ x − x n and then we can use (1.2). For the H¨ older continuity statement, we use the estimate |eik·(x−x ) −1| ≤ 21−β |k|β |x − x |β which holds for every 0 ≤ β ≤ 1. Now let z ∈ ρ(H). Denote by Gk (x, x ; z) the integral kernel of (Hk −z)−1 . Due to the identity Uk (H − z)−1 Uk∗ = (Hk − z)−1 we have:
Gk (x, x ; z) = eik·(x−x ) G(x, x ; z).
(5.1)
Let us denote by n the integer part of α. We can suppose that n ≥ 1 since the case 0 < α < 1 is covered by the argument below. From the identity (Hk − z)−1 − (Hk − z)−1 = −(Hk − z)−1 [Hk − Hk ](Hk − z)−1
(5.2)
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and from Lemma 5.1 we conclude that the map R2 k → (Hk − z)−1 ∈ B(l2 (Γ)) is continuous in the norm topology, and also differentiable. We have: Dk (Hk − z)−1 = −(Hk − z)−1 [Dk Hk ](Hk − z)−1 .
(5.3)
Using this identity at k = 0 in (5.1) leads to: (x − x )G(x, x ; z) = −(H − z)−1 [Dk Hk ]k=0 (H − z)−1 δx , δx which gives: ||(H − z)−1 ||H1 ≤ C (1 + ||H||C 1 )
1 1 + dist(z, σ(H))2 dist(z, σ(H))
.
This is true because we have the pointwise bound x − x 2α ≤ (1 + |x1 − x1 | + |x2 − x2 |)2α ≤ (3 max{1, |x1 − x1 |, |x2 − x2 |)2α ≤ 32α +
2
32α |xj − xj |2α .
(5.4)
j=1
By induction we obtain the following rough estimate: 1 1 −1 n n ||(H − z) ||H ≤ Cn (1 + ||H||Cn ) + . dist(z, σ(H))n+1 dist(z, σ(H)) (5.5) Now let us assume that n < α < n + 1. The integral kernel of the n’th partial derivative of (Hk − z)−1 with respect to k1 is given by in eik·(x−x ) (x1 − x1 )n G(x, x ; z). Moreover, using (5.3) and Lemma 5.1 we conclude that the older continuous at k = 0. Let k = (k1 , 0). operator ∂kn1 (Hk − z)−1 is α − n H¨ We also have the identity:
in (eik1 (x1 −x1 ) − 1)(x1 − x1 )n G(x, x ; z) = [∂kn1 (Hk1 − z)−1 − ∂kn1 (Hk1 − z)−1 |k=0 ]δx , δx . If |k1 | ≤ 1 the following norm estimate holds true according to Lemma 5.1: ||[∂kn1 (Hk1 − z)−1 − ∂kn1 (Hk1 − z)−1 |k=0 ]|| 1 1 ≤ C|k1 |α−n (1 + ||H||n+1 ) + . α C dist(z, σ(H))n+2 dist(z, σ(H))
(5.6)
Choose n < α < α < n + 1. Then the following integral converges in norm and defines a bounded operator: ∞ 1 H := [∂ n (Hk1 − z)−1 − ∂kn1 (Hk1 − z)−1 |k=0 ]dk1 . 1+α −n k1 k1 0
Its integral kernel is given by x ; z) := in (x1 − x1 )n G(x, x ; z) G(x,
∞ 0
1 k11+α −n
(eik1 (x1 −x1 ) − 1) dk1 .
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Assuming without loss of generality that x1 − x1 = 0, and by a change of variable s = k1 |x1 − x1 | we obtain: x ; z) = |x1 − x |α −n (x1 − x )n G(x, x ; z) G(x, 1 1 ∞ 1 n is sign(x1 −x1 ) − 1) ds. × −n i (e 1+α s 0
Notice that the above integral only has two possible values C± both different x ; z) = Hδ x , δx = from zero, depending on the sign of x1 − x1 . Since G(x, α −n n C± (x1 , x1 ) |x1 − x1 | (x1 − x1 ) G(x, x ; z) with |C± (x1 , x1 )| ≥ C it follows that 2. sup |x1 − x1 |2α |G(x, x ; z)|2 ≤ C −2 ||H|| x ∈Γ
x∈Γ
This argument can be repeated for the other coordinate and bound the l2 norm of · − x α G(·, x ; z) using (5.4). The proof of Proposition 2.2 is over. 5.2. A Few Identities from the Continuous Case We list here a few well known facts about the continuous two dimensional magnetic Schr¨ odinger operator with constant magnetic field equal to b in L2 (R2 ): Hb = (p − ba(x))2 ,
p = −i∇x ,
a(x) = (−x2 /2, x1 /2).
(5.7)
The integral kernel of the semi-group e−tHb is denoted with Gb (x, x ; t) and is given by the following explicit formula:
b|x − x |2 b b (x, x ; t). exp − Gb (x, x ; t) = eibϕ(x,x ) =: eibϕ(x,x ) G 4π sinh(bt) 4 tanh(bt) (5.8) The semigroup property insures the following identity: Gb (x, x ; 2t) = Gb (x, y; t)Gb (y, x ; t) dy.
(5.9)
R2
Then we can write: ibϕ(x,x )
e
=
1 b (x, x ; 2t) G
Gb (x, y; t)Gb (y, x ; t) dy
R2
b|x − x |2 4π sinh(2bt) = exp Gb (x, y; t)Gb (y, x ; t) dy. b 4 tanh(2bt) R2
(5.10) Taking the complex conjugation in both sides gives:
b|x − x |2 4π sinh(2bt) −ibϕ(x,x ) exp = Gb (y, x; t)Gb (x , y; t) dy. e b 4 tanh(2bt) R2
(5.11)
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Again the semi-group property gives that: b = Gb (x, x; 2t) = Gb (x, y; t)Gb (y, x; t) dy 4π sinh(2bt) 2 R = |Gb (y, x; t)|2 dy
989
(5.12)
R2
which is clearly x independent.
References [1] Avron, J.E., Simon, B.: Stability of gaps for periodic potentials under variation of a magnetic field. J. Phys. A Math. Gen. 18, 2199–2205 (1985) [2] Avron, J., van Mouche, P.H.M., Simon, B.: On the measure of the spectrum for the almost Mathieu operator. Commun. Math. Phys. 132, 103–118 (1990) [3] Avron, J., van Mouche, P.H.M., Simon, B.: On the measure of the spectrum for the almost Mathieu operator. Erratum Commun. Math. Phys. 139, 215 (1991) [4] Bellissard, J.: Lipshitz Continuity of Gap Boundaries for Hofstadter-like Spectra. Commun. Math. Phys. 160, 599–613 (1994) [5] Briet, P., Cornean, H.D.: Locating the spectrum for magnetic Schr¨ odinger and Dirac operators. Comm. Partial Differ. Equ. 27(5–6), 1079–1101 (2002) [6] Choi, M.D., Elliott, G.A., Yui, N.: Gauss polynomials and the rotation algebra. Invent. Math. 99, 225–246 (1990) [7] Cornean, H.D.: On the magnetization of a charged Bose gas in the canonical ensemble. Commun. Math. Phys. 212(1), 1–27 (2000) [8] Cornean, H.D., Nenciu, G.: On eigenfunction decay for two dimensional magnetic Schr¨ odinger operators. Commun. Math. Phys. 192, 671–685 (1998) [9] Cornean, H.D., Nenciu, G.: Two-dimensional magnetic Schr¨ odinger operators: width of mini bands in the tight binding approximation. Ann. Henri Poincar´e 1(2), 203–222 (2000) [10] Cornean, H.D., Nenciu, G., Pedersen, T.G.: The Faraday effect revisited: general theory. J. Math. Phys. 47(1), 013511 (2006) [11] Cornean, H.D., Nenciu, G.: The Faraday effect revisited: Thermodynamic limit. J. Funct. Anal. 257(7), 2024–2066 (2009) [12] Elliott, G.: Gaps in the spectrum of an almost periodic Schrodinger operator. C. R. Math. Rep. Acad. Sci. Canada 4, 255–259 (1982) [13] Germinet, F., Klein, A.: Operator kernel estimates for functions of generalized Schr¨ odinger operators. Proc. Am. Math. Soc. 131, 911–920 (2003) [14] Gr¨ ochenig, K., Leinert, M.: Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices. Trans. AMS 358, 2695–2711 (2006) [15] Haagerup, U., Rørdam, M.: Perturbations of the rotation C ∗ -algebras and of the Heisenberg commutation relation. Duke Math. J. 77, 627–656 (1995) [16] Helffer, B., Kerdelhue, P., Sj¨ ostrand, J.: M´emoires de la SMF. S´erie 2 43, 1–87 (1990) [17] Helffer, B., Sj¨ ostrand, J.: Equation de Schr¨ odinger avec champ magn´etique et ´equation de Harper. Springer Lect. Notes Phys. 345, 118–197 (1989)
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[18] Helffer, B., Sj¨ ostrand, J.: Analyse semi-classique pour l’´equation de Harper. II. Bull. Soc. Math. France 117(4), 40 (1990) [19] Herrmann, D.J.L., Janssen, T.: On spectral properties of Harper-like models. J. Math. Phys. 40(3), 1197 (1999) [20] Iftimie, V., M˘ antoiu, M., Purice, R.: Magnetic Pseudodifferential Operators. Publ. Res. Inst. Math. Sci. 43(3), 585–623 (2007) [21] Jaffard, S.: Propri´et´es des matrices ‘bien localis`es’ pr`es de leur diagonale et quelques applications. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 7(5), 461–476 (1990) [22] Kotani, M.: Lipschitz continuity of the spectra of the magnetic transition operators on a crystal lattice. J. Geom. Phys. 47(2–3), 323–342 (2003) [23] Lein, M., M˘ antoiu, M., Richard, S.: Magnetic pseudodifferential operators with coefficients in C ∗ -algebras. http://arxiv.org/abs/0901.3704v1 (2009) [24] M˘ antoiu, M., Purice, R.: Strict deformation quantization for a particle in a magnetic field. J. Math. Phys. 46(5), 052105 (2005) [25] M˘ antoiu, M., Purice, R.: The magnetic Weyl calculus. J. Math. Phys. 45(4), 1394–1417 (2004) [26] M˘ antoiu, M., Purice, R., Richard, S.: Spectral and propagation results for magnetic Schrodinger operators; a C ∗ -algebraic framework. J. Funct. Anal. 250(1), 42–67 (2007) [27] Nenciu, G.: On asymptotic perturbation theory for quantum mechanics: almost invariant subspaces and gauge invariant magnetic perturbation theory. J. Math. Phys. 43(3), 1273–1298 (2002) [28] Nenciu, G.: Stability of energy gaps under variation of the magnetic field. Lett. Math. Phys. 11, 127–132 (1986) [29] Nenciu, G.: On the smoothness of gap boundaries for generalized Harper operators. In: Advances in Operator Algebras and Mathematical Physics. Theta Series in Advanced Mathematics, vol. 5, pp. 173–182. Theta, Bucharest (2005). arXiv:math-ph/0309009v2 [30] Takai, H.: Dualit´e dans les produits crois´es de C ∗ -alg`ebres. C. R. Acad. Sci. Paris S´er. A 278, 1041–1043 (1974) [31] Takai, H.: On a duality for crossed products of C ∗ -algebras. J. Funct. Anal. 19, 25–39 (1975) [32] Takesaki, M.: Duality for crossed products and the structure of von Neumann algebras of type III. Acta Math. 131, 249–310 (1973) Horia D. Cornean Department of Mathematical Sciences Aalborg University Fredrik Bajers Vej 7G 9220 Aalborg, Denmark e-mail:
[email protected] Communicated by Jean Bellissard. Received: December 3, 2009. Accepted: June 24, 2010.
Ann. Henri Poincar´e 11 (2010), 991–1005 c 2010 Springer Basel AG 1424-0637/10/050991-15 published online October 16, 2010 DOI 10.1007/s00023-010-0052-5
Annales Henri Poincar´ e
Wegner Estimate for Discrete Alloy-type Models Ivan Veseli´c Abstract. We study discrete alloy-type random Schr¨ odinger operators on 2 (Zd ). Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. If the single site potential is compactly supported and the distribution of the coupling constant is of bounded variation a Wegner estimate holds. The bound is polynomial in the volume of the box and thus applicable as an ingredient for a localisation proof via multiscale analysis.
1. Main Results A discrete alloy-type model is a family of operators Hω = H0 + Vω on 2 (Zd ). Here, H0 denotes an arbitrary symmetric operator. In most applications, H0 is the discrete Laplacian on Zd . The random part Vω is a multiplication operator ωk u(x − k) (1) Vω (x) = k∈Zd
defined in terms of an i.i.d. sequence ωk : Ω → R, k ∈ Zd of random vari1 d ables each having a density f , and a single site potential u ∈ (Z ; R). It follows that the mean value u ¯ := k∈Zd u(k) is well defined. We will assume throughout the paper that u does not vanish identically and that f ∈ BV . Here BV denotes the space of functions with bounded total variation and · BV denotes the corresponding norm. The mathematical expectation w.r.t. the product measure associated with the random variables ωk , k ∈ Zd will be denoted by E . The estimates we want to prove do not concern the operator Hω , ω ∈ Ω but rather its finite box restrictions. Thus for the purposes of the present paper, domain and selfadjointness properties of Hω are irrelevant. For L ∈ N, we denote the subset [0, L]d ∩ Zd by ΛL , its characteristic function by χΛL ,
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the canonical inclusion 2 (ΛL ) → 2 (Zd ) by ιL and the adjoint restriction 2 (Zd ) → 2 (ΛL ) by πL . The finite cube restriction of Hω is then defined as Hω,L := πL H0 ιL + Vω χΛL : 2 (ΛL ) → 2 (ΛL ). For any ω ∈ Ω and L ∈ N the restriction Hω,L is a selfadjoint finite rank operator. In particular, its spectrum consists entirely of real eigenvalues E(ω, L, 1) ≤ E(ω, L, n) ≤ · · · ≤ E(ω, L, ΛL ) counted including multiplicities. Note that if u has compact support, then there exists an n ∈ N and an x ∈ Zd such that supp u ⊂ Λ−n + x, where Λ−n := {−k | k ∈ Λn }. We may assume without loss of generality x = 0 without restricting the model (1). The number of points in the support of u is denoted by rank u. Now we are in the position to state our bounds on the expected number of eigenvalues of finite box Hamiltonians Hω,L in a compact energy interval [E − , E + ]. Theorem 1. Assume that the single site potential u has support in Λ−n . Then there exists a constant cu depending only on u such that for any L ∈ N, E ∈ R and > 0 we have E Tr χ[E−,E+] (Hω,L ) ≤ cu rank u f BV (L + n)d·(n+1) Remark 2. 1. By the assumption on the support of the single site potential rank u ≤ (n + 1)d . 2. The constant cu is given in terms of derivatives of a finite array of polynomials constructed in terms of values of the function u. 3. A bound of the type as it is given in Theorem 1 is called Wegner estimate. If such a bound holds one is interested in the dependence of the RHS on the length of the energy interval (in our case 2) and on the volume of the cube ΛL (in our case Ld ). More precisely, a general Wegner estimate is of the form ∀ L ∈ N, E ∈ R, > 0 : E Tr χ[E−,E+] (Hω,L ) ≤ constant(2)a (Ld )b with some a ≤ 1 and b ≥ 1. The best possible estimate is obtained in the case a = 1 and b = 1. Such a bound is, for instance, encountered in Corollary 4 below. 4. Our bound is linear in the energy-interval length and polynomial in the volume of the cube. This implies that the Wegner bound can be used for a localisation proof via multiscale analysis, see e.g. [5,8,13]. More precisely, if an appropriate initial scale estimate is available, the multiscale analysis—using as an ingredient the Wegner estimate as given in Theorem 1—yields Anderson localisation. As the Wegner bound is valid on the whole energy axis, one can prove Anderson localisation in any energy region where the initial scale estimate holds. 5. One might ask whether the exponent d · (n + 1) of the length scale is optimal for the model under consideration. To give an answer to this question one has to be more precise: it seems that this exponent is the best one can obtain using a conventional scheme of proof which at its heart only uses local averaging over one random variable. There are more elaborate techniques, used e.g in the proof of a Wegner estimate for an multidimensional model with Bernoulli disorder [4] where averaging over
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local families of random variables gives estimates which are impossible to obtain using just varying a single parameter. Such techniques could yield a better volume dependence than the one in Theorem 1. At the end of the paper, we discuss how to derive spectral and exponential localisation in the large disorder regime with the help of Theorem 1. If the single site potential u does not have compact support, one has to use an enhanced version of the multiscale analysis and so-called uniform Wegner estimates to prove localisation, see [14]. However, there exist criteria which allow one to turn a standard Wegner estimate into a uniform one, see, e.g., Lemma 4.10.2 in [28]. The main point of the theorem is that no assumption on u (apart from the compact support) is required. In particular, the sign of u can change arbitrarily. The single site potential may be even degenerate in the sense that u ¯ = 0. Also, note that the result holds on the whole energy axis. These two properties are in contrast to earlier results on Wegner estimates for sign-changing single site potentials. See the discussion of the previous literature at the end of this section. If u does satisfy the assumption u ¯ = 0 we obtain an even better bound. This is the content of Theorem 3 below.
The next Theorem applies to single site potentials u ∈ 1 (Zd ) with non u/2|. Here, vanishing mean u ¯ = 0. Let m ∈ N be such that k≥m |u(k)| ≤ |¯ k = k∞ denotes the sup-norm. Theorem 3. Assume u ¯ = 0 and that f has compact support. Then we have for any L ∈ N, E ∈ R and > 0 8 E Tr χ[E−,E+] (Hω,L ) ≤ min Ld , rank u f BV (L + m)d u ¯ In the case that the support of u is compact, we have an important Corollary 4. Assume u ¯ = 0 and supp u ⊂ Λ−n . Then we have for any L ∈ N, E ∈ R and > 0 4 E Tr χ[E−,E+] (Hω,L ) ≤ rank u f BV (L + n)d u ¯ In particular, the function R E → E Tr χ(−∞,E] (Hω,L ) is Lipschitz continuous. If the operator Hω has a well defined integrated density of states N : R → R, meaning that 1 lim E Tr χ(−∞,E] (Hω,L ) = N (E) L→∞ Ld at all continuity points of N , then Corollary 4 implies that the integrated density of states is Lipschitz continuous. Consequently its derivative, the density of states, exists for almost all E ∈ R. Remark 5. The situation that the two cases u ¯ = 0 and u ¯ = 0 have to be distinguished occurs also in other contexts, see for instance the paper [16] on weak disorder localisation.
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When looking at Theorems 1 and 3 one might wonder what kind of Wegner bound holds for non-compactly supported single site potentials with vanishing mean. To apply the methods of the present paper, in this case, it seems that one has to require that u tends to zero exponentially fast. In this situation one can hope to treat the decaying potential as a sufficiently small perturbation of a compactly supported potential. So far, only the case of one space dimension is settled. Theorem 6. Assume that f has compact support and that there exists s ∈ (0, 1) and C ∈ (0, ∞) such that |u(k)| ≤ Cs|k| for all k ∈ Z. Then there exist cu ∈ (0, ∞) and D ∈ N0 depending only on u such that for each β > D/| log s| there exists a constant Kβ ∈ (0, ∞) such that for all L ∈ N, E ∈ R and > 0 8 f BV L (L + β log L + Kβ )D+1 E Tr χ[E−,E+] (Hω,L ) ≤ cu Let us discuss the relation of the above theorems to previous results [10,15,20,26,27] on Wegner estimates with single site potentials which are allowed to change sign. The papers [10,15] concern alloy-type Schr¨ odinger operators on L2 (Rd ). The main result is a Wegner estimate for energies in a neighbourhood of the infimum of the spectrum. It applies to arbitrary nonvanishing single site potentials u ∈ Cc (Rd ) and coupling constants with a piecewise absolutely continuous density. The upper bound is linear in the volume of the box and H¨ older-continuous in the energy variable. This means in the notation of Remark 2 that a ∈ (0, 1) and b = 1. The papers [20,26,27] establish Wegner estimates for both alloy-type odinger operSchr¨ odinger operators on L2 (Rd ) and discrete alloy-type Schr¨ ators on 2 (Zd ). Since the present paper concerns the latter model we will discuss here first the results of [20,26,27] referring to operators on the lattice. For the discrete alloy-type model on 2 (Zd ), [26] establishes a Wegner estimate analogous to Corollary 4 above, under the additional assumption that the function u(k)e−ik·θ does not vanish on [0, 2π)d . (2) s : θ → s(θ) := k∈Zd
To be able to compare the two results, note that u ¯ := k∈Zd u(k) = s(0). Thus assumption (2) requires that the image of the set [0, 2π)d under s does not meet 0 ∈ C whereas the assumption in Corollary 4 requires this property for the image of the set {0} only. The later condition is generically satisfied. Let us now turn to the situation when u ¯ = 0. Special cases of this class of single site potentials are covered by Theorem 2 in [20] and Exp. 10 in [27]. They correspond to special cases of Theorem 1 and do not give an as explicit control over the volume dependence of the Wegner bound. Let us say a few words, which ideas are used in the proofs to overcome the restrictions imposed on the single site potentials in [20,26]. There a transformation of the random variables is used to construct a non-negative linear combination of translates of single site potentials. The price to pay is that the
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new transformed random variables are no longer independent. The argument of [20,26] uses the inverse transformation on the probability space to recover in a later step of the proof independence again. This leads to an uniform invertibility requirement for a sequence of a certain auxiliary Toeplitz or circulant matrices constructed from the values of the single site potential u. Condition (2) on the function s ensures that this invertibility property holds. The proof of the present paper uses a similar transformation of the coordinates of the product probability space, but the inverse transformation is no longer needed. This leads to less stringent conditions on the single site potential u. Contrary to the present paper [20,26,27] give Wegner estimates for continuum alloy-type Schr¨ odinger operators on L2 (Rd ) as well. The bounds are linear in the volume of the box and Lipschitz continuous in the energy variable. The bound is valid for all compact intervals along the energy axis. These d bounds are valid for single site potentials u ∈ L∞ c (R ) which have a generalised step function form and satisfy a condition analogous to (2). Let us stress that Wegner estimates for sign changing single site potentials are harder to prove for operators on L2 (Rd ) than for ones on 2 (Zd ). The reason is that for discrete models we have in the randomness a degree of freedom for each point in the configuration space Zd . For the continuum alloy-type model the configuration space is Rd while the degrees of freedom are indexed by a much smaller set, namely Zd . The role played by a Wegner estimate in the framework of a localisation proof using the multiscale analysis is analogous to role played by the finiteness of the expectation of fractional powers of the Green’s function for fractional moment method. Recently a fractional moment bound for the alloy-type model on 2 (Z) has been proven in [6]. (See also [24] for a related result.) It holds for arbitrary compactly supported single site potentials. The result can be extended to the one-dimensional strip, while the extension to Zd is unclear at the moment. Another important class of random Hamiltonians exhibiting nonmonotone dependence on the random variables are Schr¨ odinger operators with random magnetic fields. Wegner estimates for such models are established [10,19,25]. In particular, [19] gives a Wegner estimate for a random magnetic field Hamiltonian on the lattice 2 (Z2 ) and is thus comparable with results in the present paper. It is not clear whether our methods can be used to treat the model of [19] since is is necessary to find a set of transformed random variables which produces a perturbation of fixed sign. (For discrete alloy-type models studied here this is done in Sects. 3 and 4.) Since the structure of the randomness is different in disordered magnetic field models, it is not clear whether such an transformation exists. Let us also mention the random displacement model as an important example of random Schr¨ odinger operators exhibiting non-monotone parameter dependence. For such models in the continuum, the location of the minimum of the spectrum, Lifschitz tails and Wegner estimates have been studied in [1,2,7,17,18,23]. These models do not have a direct analog on the space 2 (Zd ) due to the lack of continuous deformations.
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Very recently Kr¨ uger [21] has obtained results on localisation for a class of discrete alloy type models which includes the ones considered here. The results rely on the multiscale analysis and the use of Cartan’s lemma in the spirit as is has been used earlier, e.g. in [3].
2. An Abstract Wegner Estimate and the Proof of Theorem 3 An important step in the proofs of the Theorems of the last section is an abstract Wegner estimate which we formulate now. We abbreviate in the sequel the characteristic function χΛL by χΛ . Lemma 7. Let L ∈ N, E ∈ R, > 0 and I := [E −, E +]. Denote by E(ω, L, n) the n-th eigenvalue of the operator Hω,L . Assume that there exist an δ > 0 and aL ∈ 1 (Zd ) such that for all n ∂ aL (k) E(ω, L, n) ≥ δ (3) ∂ω k d k∈Z
Then E (Tr χI (Hω,L )) ≤
4 |aL (k)| f BV rank(χΛ u(· − k)) δ d k∈Z
1
∂ Since aL ∈ and the derivatives ∂ω E(ω, L, n) are uniformly bounded, k the sum (3) is absolutely convergent. Note that one can always replace the d sum k∈Zd by k∈Λ+ . Here, Λ+ = {k ∈ Z | u(· − k) ∩ ΛL = ∅} denotes the L L set of lattice points such that the corresponding coupling constant influences the potential in the box ΛL . In particular, if the support of u is contained in [−n, . . . , 0]d , the sum reduces to k∈ΛL+n . Note that the sequence aL may be chosen differently for different cubes ΛL . In our applications, namely the proofs of Theorems 1, 3, and 6, we will find a fixed sequence a, not necessarily in 1 (Zd ), such that appropriate finite truncations give the desired coefficients aL (k) adapted for a cube ΛL of size L. Note that for u with compact support, the function k → rank(χΛ u(· − k)) already implements the truncation: the terms with k outside Λ+ L do not contribute to the sum. In this situation, the condition aL ∈ 1 is not needed. We give a simple sufficient condition which ensures the hypothesis of Lemma 7.
Corollary 8. Let L ∈ N, > 0 and I := [E − , E + ]. Assume that there exist an δ > 0 and aL ∈ 1 (Zd ) such that for all x ∈ ΛL aL (k)u(x − k) ≥ δ k∈Zd
Then E (Tr χI (Hω,L )) ≤
4 |aL (k)| f BV rank(χΛ u(· − k)) δ d k∈Z
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Proof. By first order perturbation theory, respectively the Hellmann–Feynman formula we have ∂ E(ω, L, n) = ψn , u(· − k)ψn ∂ωk where ψn is the normalised eigen solution to Hω,L ψn = E(ω, L, n)ψn . Thus ∂ aL (k) E(ω, L, n) = aL (k)ψn , u(· − k)ψn ≥ δ ∂ωk d d k∈Z
k∈Z
The proof of Lemma 7 relies on quite standard techniques, see e.g. [10,12, 22,29]. The main point of the Lemma is that it singles out a relation between properties of linear combinations of single site potentials and a Wegner estimate. In the course of the proof we will need the following estimate, which is related to the spectral shift function. Recall that n → E(ω, L, n) is an enumeration of the eigenvalues of Hω,L . Lemma 9. Let f : R → R be a function in BV ∩ L1 (R), ρ ∈ C ∞ (R), k ∈ Zd and s ∈ R. Then
∂ ρ(E(ω, L, n) + s) ≤ f BV rank(χΛ u(· − k)) |ρ (x)|dx dωk f (ωk ) ∂ωk n∈N
∂ Note that if k ∈ Λ+ L then ∂ωk E(ω, L, n) = 0. Also note that the sum over n is in fact finite since Hω,L is defined on a finite dimensional vector space.
Proof. We will use that if g ∈ C ∞ and f ∈ BV ∩ L1 the partial integration bound
f (x)g (x)dx ≤ g∞ f BV holds. Denote by E(ω, ωk = 0, L, n) the nth eigenvalue of the operator Hω,ωk =0,L := Hω,L − ωk u(· − k) on 2 (ΛL ). Partial integration yields
∂ ρ(E(ω, L, n) + s) dωk f (ωk ) ∂ωk n∈N
∂ (ρ(E(ω, L, n) + s) − ρ(E(ω, ωk = 0, L, n) + s)) = dωk f (ωk ) ∂ωk n∈N (ρ(E(ω, L, n) + s) − ρ(E(ω, ωk = 0, L, n) + s)) ≤ f BV sup ωk ∈supp f n∈N
Here, we used that ωk → E(ω, L, n) is an infinitely differentiable function, cf. [11]. Now ρ(E(ω, L, n) + s) − ρ(E(ω, ωk = 0, L, n) + s) n∈N
= Tr (ρ((Hω,L + s) − ρ((Hω,ωk =0,L + s))
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can be expressed in terms of the spectral shift function ξ(·, Hω,L , Hω,ωk =0,L ) of the operator pair Hω,L , Hω,ωk =0,L as
ρ (x)ξ(x, Hω,L , Hω,ωk =0,L )dx. Since ξ∞ is bounded by the rank of the perturbation χΛ u(· − k), we obtain
ρ(E(ω, L, n) + s) − ρ(E(ω, ωk = 0, L, n) + s) ≤ rank(χΛ u(· − k)) |ρ | n∈N
and the proof of the Lemma is completed. Now we turn to the proof of Lemma 7.
Proof of Lemma 7. Let ρ ∈ C ∞ (R) be a non-decreasing function such that on (−∞, −] it is identically equal to −1, on [, ∞) it is identically equal to zero and ρ ∞ ≤ 1/. By the chain rule we have ∂ aL (k) ρ(E(ω, L, n) − E + t) ∂ωk d k∈Z
= ρ (E(ω, L, n) − E + t)
k∈Zd
aL (k)
∂ E(ω, L, n) ∂ωk
The assumption (3) implies now ρ (E(ω, L, n) − E + t) ≤ Since χI ≤
2 −2
1 ∂ aL (k) ρ(E(ω, L, n) − E + t) δ ∂ω k d k∈Z
dt ρ (x − E + t) for I := [E − , E + ] we have
Tr χI (Hω,L ) ≤
1 δ
2 dt −2
aL (k)
n∈N k∈Zd
∂ ρ(E(ω, L, n) − E + t) ∂ωk
Note that for a random variable F : Ω → R we have E (F ) = E ( f (ωk ) dωk F (ω)) Thus using Lemma 9 and |ρ (x)|dx = 1 we obtain 4 E (Tr χI (Hω,L )) ≤ |aL (k)| f BV rank(u · χΛ ) δ d k∈Z
Now we are in the position to give a Proof of Theorem 3. Let ψn be a normalised eigenfunction associated to d d k + [−m, m] . W.l.o.g. we may ∩ Z E(ω, L, n) and Q(L, m) = k∈ΛL assume u ¯ > 0. Then k∈Q(L,m) u(k) ≥ u ¯/2. Choose now the coefficients in Corollary 8 in the following way: aL (k) = 1 for k ∈ Q(L, m) and aL (k) = 0 for k in the complement of Q(L, m). Then
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aL (k) ψn , u(· − k)ψn =
k∈Zd
ψn ,
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u(· − k)ψn
≥u ¯/2.
k∈Q(L,m)
Proof of Corollary 4. Set aL (k) = 1 for k ∈ ΛL+n and aL (k) = 0 for k in the complement of ΛL+n . Then aL (k)ψn , u(· − k)ψn = ψn , u(· − k)ψn = u ¯ k∈ΛL+n
k∈Zd
An application of Corollary 8 now completes the proof.
3. Proof of Theorem 1 In this section, we give a proof of Theorem 1. In view of Theorem 3, it is sufficient to consider the case that the single site potential u : Zd → R, u ∈ 1 (Zd ) is degenerate in the sense that x∈Zd u(x) = 0. We explain how to find in this situation an appropriate linear combination of single site potentials—or, equivalently, an appropriate linear transformation of the random variables—which can be efficiently used for averaging. The aim of the linear transformation is to extract a perturbation potential which is strictly positive on the box Λ. Let us first consider the case d = 1. Then we can assume without loss of generality that supp u ⊂ {−n, . . . 0}. For a given cube ΛL = {0, . . . , L}, we are looking for an array of numbers ak , k ∈ ΛL+n such that we have ak u(x − k) = constant > 0 for all x ∈ ΛL (4) k∈ΛL+n
In fact, we will find a sequence of numbers ak , k ∈ N such that we have ak u(x − k) = constant > 0 for all x ∈ N
(5)
k∈N
If we truncate this sequence, we obtain an array of numbers satisfying (4). For a function F : (1 − , 1 + ) → R with > 0 we say that it has a root of order m ∈ {0, . . . , n} at t = 1 iff it is in C m (1 − , 1 + ) and j d F (t) = 0 for j = 0, . . . , m − 1 (6) j dt t=1
c(F ) :=
dm F (t)
= 0 m dt t=1
(7)
In particular, m = 0 means that F (1) = 0. If F is a polynomial of degree not exceeding m, if (6) holds and in addition c(F ) = 0, then F ≡ 0. In this case, we say that F has a root of infinite order at t = 1. Given a function w : Z → R such that Fw (t) := ν∈Z tν w(−ν) converges for t ∈ (1 − , 1 + ) we call (1 − , 1 + ) t → F (t) := Fw (t) the accompanying (Laurent) series of w. If supp w ⊂ {−n, . . . , 0} we call t → p(t) := pw (t) := n ν ν=0 t w(−ν) the accompanying polynomial of w.
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Lemma 10. Let D ∈ N0 and ak = k D for all k ∈ N. Let m be the order of the root t = 1 of the Laurent series F accompanying the function w : Z → R with convergent series ν∈Z tν w(−ν) for t ∈ (1 − , 1 + ). (a) If m > D then k∈Z ak w(x − k) = 0 for all x ∈ N. (b) If m = D then k∈Z ak w(x − k) = c(F ) for all x ∈ N. An important and well known special case is Corollary 11. Let D ∈ N0 and ak = k D for all k ∈ N. Let m be the order of the root t = 1 of the polynomial p accompanying the function w : Z → R with supp w ⊂ {−n, . . . , 0}. x+n (a) If m > D then k=x ak w(x − k) = 0 for all x ∈ N. x+n (b) If m = D then k=x ak w(x − k) = c(p) for all x ∈ N. x+n Due to the support condition k∈N ak w(x − k) = k=x ak w(x − k) for all x ∈ N. Proof of Lemma 10. First, note that for arbitrary ν ∈ N and s ∈ R we have ν dν s F (e ) = cκ F (κ) (es ) eκs dsν κ=1 with some c1 , . . . , cν−1 ∈ N0 and cν = 1. For the value s = 0 it follows from dν dm s s (6) that ds ν F (e ) = 0 for ν = 0, . . . , m − 1 and from (7) that dsm F (e ) = F (m) (es ) ems = c(F ). dD ks for s = 0 and insert this into the LHS of (5) We note that ak = ds De to obtain dD dD ak w(x − k) = w(x − k) D eks = w(−ν) D e(ν+x)s ds ds ν∈Z k∈Z k∈Z D−r D r d D d dD s xs = D (exs F (es )) = F (e ) e . r ds dsr dsD−r r=0 (8) For s = 0, (8) vanishes if D < m and equals c(F ) if D = m.
Thus, we have found in the case d = 1 and w = u a linear combination with the desired property (5). In the multidimensional situation, we will reduce the dimension one by one and construct from a non-vanishing single site potential in dimension j a non-vanishing one in dimension j − 1. In each reduction step, we apply Corollary 11. Let w(j) : Zj → R be compactly supported and not identically vanishing. W.l.o.g. we assume supp w(j) ⊂ [−n, 0]j ∩ Zj . Next we define a ‘projected’ single site potential w(j−1) : Zj−1 → R as follows. Consider the family of polynomials p(x1 , . . . , xj−1 , ·) : R → R, indexed by (x1 , . . . , xj−1 ) ∈ {−n, . . . , 0}j−1 and defined by n p(x1 , . . . , xj−1 , t) := tν w(j) (x1 , . . . , xj−1 , −ν). (9) ν=0
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Let m(x1 , . . . , xj−1 ) ∈ {0, . . . , n, ∞} be the order of the root t = 1 of the polynomial p(x1 , . . . , xj−1 , ·) and M := Mj := min m(x1 , . . . , xj−1 ) | x1 , . . . , xj−1 ∈ {−n, . . . , 0} the minimal degree occurring in the family. Since w(j) does not vanish identically, Mj ≤ n. Set Ij−1 := {(x1 , . . . , xj−1 ) ∈ {−n, . . . , 0}j−1 | m(x1 , . . . , xj−1 ) = Mj } and Jj−1 := {(x1 , . . . , xj−1 ) ∈ {−n, . . . , 0}j−1 | m(x1 , . . . , xj−1 ) > Mj } Lemma 12. For all (x1 , . . . , xj−1 ) ∈ {−n, . . . , 0}j−1 we have the equality M d k M w(j) (x1 , . . . , xj−1 , xj − k) = p(x , . . . , x , t) . (10) 1 j−1 M dt t=1 k∈N
We denote the function in (10) by w(j−1) : Zj → R. Then w(j−1) is independent of the variable xj and therefore we call it the single site potential in reduced dimension and consider it sometimes as a function w(j−1) : Zj−1 → R. Its support is contained in {−n, . . . , 0}j−1 . Moreover, w(j−1) (x1 , . . . , xj−1 ) = 0 if (x1 , . . . , xj−1 ) ∈ Jj−1 and (j−1) (x1 , . . . , xj−1 ) = 0 if (x1 , . . . , xj−1 ) ∈ Ij−1 . w Remark 13. The lemma establishes in particular that • M is an element of {0, . . . , n}. If we had M ≥ n + 1, then all polynomials p(x1 , . . . , xj−1 , ·) would vanish identically and thus w(j) ≡ 0 contrary to our assumption. • w(j−1) does not vanish identically. In fact supp w(j−1) = Ij−1 = ∅ by definition. Proof. Consider first the case (x1 , . . . , xj−1 ) ∈ Jj−1 . Then for any xj ∈ N w(j−1) (x1 , . . . , xj−1 ) = k M w(j) (x1 , . . . , xj−1 , xj − k) = 0 k∈N
by Lemma 11, part (a), since t = 1 is a root of order M + 1 or higher of the accompanying polynomial p(x1 , . . . , xj−1 , ·). Now, if (x1 , . . . , xj−1 ) ∈ Ij−1 then the order of the root t = 1 of the polynomial p(x1 , . . . , xj−1 , ·) equals M . Thus by part (b) of Lemma 11 w(j−1) (x1 , . . . , xj−1 ) = k M w(j) (x1 , . . . , xj−1 , xj − k) k∈N
= for all xj ∈ N.
dM p(x , . . . , x , t) 1 j−1 M dt t=1
In the last step j = 1 → j − 1 = 0 of the induction we obtain a reduced single site potential M1 d (0) w = p(t) = c(p) M 1 dt t=1 which is simply a non-zero real.
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Now we describe the result which is obtained after the reduction is applied d times. Given a single site potential u : Zd → R with supp u ⊂ [−n, 0]d ∩ Zd , set w(d) = u and w(0) = k1M1 w(1) (x1 − k1 ) (11) k1 ∈N
=
k1M1 . . .
k1 ∈N
kd ∈N
kdMd w(d) (x1 − k1 , . . . , xd − kd )
(12)
Thus, we have produced a linear combination of single site potentials bk w(d) (x1 − k1 , . . . , xd − kd ) where bk := k1M1 . . . kdMd k∈ΛL+n
which is a constant, non-vanishing function on the cube ΛL . Moreover, the coefficients satisfy the bound |bk | ≤ k1n . . . kdn ≤ (L + n)d·n
for all k ∈ ΛL+n
Now an application of Corollary 8 with the choice aL (k) = bk for k ∈ ΛL+n and aL (k) = 0 for k in the complement of this set completes the Proof of Theorem 1.
4. Proof of Theorem 6 The assumption on the exponential decay of u implies that F (z) = ν ν∈Z z u(−ν) is an absolutely and uniformly convergent Laurent series on the annulus {z ∈ C | r1 ≤ |z| ≤ r2 } for some 0 < r1 < 1 < r2 < ∞ and represents there a holomorphic function. This implies that there exists a D ∈ N0 ∂D such that c(F ) := ∂z D F (z) |z=1 = 0. Otherwise F would be identically vanishing, implying that u vanishes identically. Thus the root z = 1 of F has a well defined, finite order D ∈ N0 and Lemma 10 can be applied. The problem is now that the series k∈Z k D is not absolutely convergent. For this reason we will replace it with an appropriate finite cut-off sum. Assume in the following w.l.o.g. that c(F ) > 0. A lengthy but easy calculation shows that for all β > D/| log s| there exists a constant Kβ ∈ (0, ∞) such that for all L ∈ N c(F ) |k|D |u(x − k)| ≤ ∀ x ∈ ΛL : 2 k∈{−K/2,...,m}
where m = L + β log L + Kβ /2. Consequently c(F ) k D u(x − k) ≥ ∀ x ∈ ΛL : 2 k∈{−K/2,...,m}
Thus we can apply Corollary 8 with the choice aL (k) = k D for k ∈ {−k, . . . , m} and aL (k) = 0 for k ∈ {−k1 , . . . , m + 1} and obtain 8 f BV L(L + β log L + K)D+1 E Tr χ[E−,E+] (Hω,L ) ≤ c(F )
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5. Discussion: Localisation for Large Disorder In the case that H0 = Δ is the finite difference Laplacian and the disorder is sufficiently strong, Theorem 1 can be used to prove exponential localisation on the whole energy axis R, i.e. to show that almost surely Hω , ω ∈ Ω has no continuous spectral component and that all eigenfunctions of Hω decay exponentially at infinity. We do not discuss localisation near spectral edges which is a more delicate issue. Note that Theorem 1 assumes in particular that the single site potential u is of compact support. The results described below are based on the multiscale analysis, cf. e.g. [5,8,9]. It is an induction procedure over increasing length scales Lk , k ∈ N. The induction step uses a Wegner estimate to deduce from probabilistic decay estimates on the Green’s function of the random operator restricted to a box of size Lk corresponding decay estimates on the larger scale Lk+1 . The induction anchor is provided by the initial scale estimate, a probabilistic statement on the decay of the Green’s function of the random operator restricted to a box on first scale L0 . A very strong form of the initial scale estimate is that for some p > d, m > 0 P{(Hω,L − E)−1 ≤ exp(−mL/2)} ≥ 1 − Lp0 .
(13)
Together with a Wegner estimate as in Theorem 1, the bound (13) yields exponential localisation for Hω , ω ∈ Ω in a small neighbourhood of E, provided that L0 is larger than a certain critical length scale L∗ , depending on the parameters of the model. Here, we present a simple idea how to derive the initial scale estimate from the Wegner estimate in the case of large disorder, which we learned from Klein and which has almost the same proof as Theorem 11.1 in [13] although the statements and the models under consideration are somewhat different. For an earlier related result see Proposition A.1.2 in [5]. Lemma 14. Let the assumptions of Theorem 1 hold. Let H0 = Δ and p ∈ N. −1/2 Choose L ∈ N such that eL ≥ (cu rank u)Lp (L + n)d(n+1) . If f BV ≥ eL , then ∀E ∈ R : P (Hω,L − E)−1 > e−L ≤ L−p Here, the quantity f −1 BV is a measure for the disorder: if it is large the values of the corresponding random variable are spread out over a large interval. Thus the assumption f BV ≤ e−2L describes a large disorder regime. Proof. Since (Hω,L − E)−1 = d(σ(Hω,L ), E)−1 , we have P{(Hω,L − E)−1 > e−L } = P{d(σ(Hω,L ), E) < eL } = P (E − eL , E + eL ) ∩ σ(Hω,L ) = ∅ ≤ E Tr χ[E−eL ,E+eL ] (Hω,L ) ≤ cu rank uf BV eL (L + n)d(n+1) ≤ e−L cu rank u (L + n)d(n+1) which is bounded by L−p by our assumption.
This lemma establishes an initial scale decay estimate (13) for a small neighbourhood of an arbitrary energy. However increasing the disorder means changing the model and in particular increasing the sup-norm of the single site
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potential. Thus one has to check on which parameters of the single site the critical scale L∗ depends. Indeed, L∗ does depend on the size of the support, but not on the supremum norm of the single site potential. This fact can be seen from the original proof of [5]. A detailed analysis how L∗ depend on various model parameters has been worked out for continuum random operators in [9] and applies to discrete operators analogously.
Acknowledgements The author would like to thank A. Klein for pointing out the reasoning behind Lemma 14 and anonymous referees for helpful comments. It was a pleasure to have stimulating discussions concerning discrete alloy type models with A. Elgart, H. Kr¨ uger, G. Stolz, and M. Tautenhahn.
References [1] Baker, J., Loss, M., Stolz, G.: Minimizing the ground state energy of an electron in a randomly deformed lattice. Commun. Math. Phys. 283(2), 397–415 (2008) [2] Baker, J., Loss, M., Stolz, G.: Low energy properties of the random displacement model. J. Funct. Anal. 256(8), 2725–2740 (2009) [3] Bourgain, J.: An approach to Wegner’s estimate using subharmonicity. J. Stat. Phys. 134(5–6), 969–978 (2009) [4] Bourgain, J., Kenig, C.E.: On localization in the continuous Anderson–Bernoulli model in higher dimension. Invent. Math. 161(2), 389–426 (2005) [5] Dreifus, H.V., Klein, A.: A new proof of localization in the Anderson tight binding model. Commun. Math. Phys. 124(2), 285–299 (1989) [6] Elgart, A., Tautenhahn, M., Veseli´c, I.: Localization via fractional moments for models on Z with single-site potentials of finite support. preprint. http://arxiv. org/abs/0903.0492 [7] Ghribi, F., Klopp, F.: Localization for the random displacement model at weak disorder. Ann. Henri Poincar´e 11(1–2) (2010) [8] Fr¨ ohlich, J., Spencer, T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88, 151–184 (1983) [9] Germinet, F., Klein, A.: Bootstrap multiscale analysis and localization in random media. Commun. Math. Phys. 222(2), 415–448 (2001) [10] Hislop, P.D., Klopp, F.: The integrated density of states for some random operators with nonsign definite potentials. J. Funct. Anal. 195(1), 12–47 (2002). doi:10.1007/s11040-010-9081-z [11] Kato, T.: Perturbation Theory of Linear Operators. Springer, Berlin (1966) [12] Kirsch, W.: Wegner estimates and Anderson localization for alloy-type potentials. Math. Z. 221, 507–512 (1996) [13] Kirsch W.: An invitation to random Schr¨ odinger operators. In: Random Schr¨ odinger Operators. Panor. Synth`eses, vol. 25, pp. 1–119. Society of Mathematics, France, Paris (2008) (With an appendix by Fr´ed´eric Klopp) [14] Kirsch, W., Stollmann, P., Stolz, G.: Anderson localization for random Schr¨ odinger operators with long range interactions. Commun. Math. Phys. 195(3), 495–507 (1998)
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[15] Klopp, F.: Localization for some continuous random Schr¨ odinger operators. Commun. Math. Phys. 167, 553–569 (1995) [16] Klopp, F.: Weak disorder localization and Lifshitz tails: continuous Hamiltonians. Ann. Henri Poincar´e 3(4), 711–737 (2002) [17] Klopp F., Nakamura S.: Lifshitz tails for some non monotonous random models. ´ ´ In: S´eminaire: Equations aux D´eriv´ees Partielles. 2007–2008, S´emin. Equ. D´eriv. ´ Partielles, pages Exp. No. XIV, 9. Ecole Polytech., Palaiseau (2009) [18] Klopp, F., Nakamura, S.: Spectral extrema and Lifshitz tails for non-monotonous alloy type models. Commun. Math. Phys. 287(3), 1133–1143 (2009) [19] Klopp, F., Nakamura, S., Nakano, F., Nomura, Y.: Anderson localization for 2D discrete Schr¨ odinger operators with random magnetic fields. Ann. Henri Poincar´e 4(4), 795–811 (2003) [20] Kostrykin, V., Veseli´c, I.: On the Lipschitz continuity of the integrated density of states for sign-indefinite potentials. Math. Z. 252(2), 367–392 (2006) [21] Kr¨ uger, H.: Localization for random operators with non-monotone potentials with exponentially decaying correlations. http://math.rice.edu/∼hk7/papers. html [22] Lenz, D., Peyerimhoff, N., Post, O., Veseli´c, I.: Continuity properties of the integrated density of states on manifolds. Jpn. J. Math. 3(1), 121–161 (2008) [23] Lott, J., Stolz, G.: The spectral minimum for random displacement models. J. Comput. Appl. Math. 148(1), 133–146 (2002) [24] Tautenhahn, M., Veseli´c, I.: Spectral properties of discrete alloy-type models. In: Proceedings of the XVth International Conference on Mathematical Physics, Prague, 2009. pp. 551–555, World Scientific, 2010 [25] Ueki, N.: Wegner estimate and localization for random magnetic fields. Osaka J. Math. 45(3), 565–608 (2008) [26] Veseli´c, I.: Wegner estimates for sign-changing single site potentials. Math. Phys. Anal. Geom. (to appear). doi:10.1007/s11040-010-9081-z, http://arxiv.org/abs/ 0806.0482 [27] Veseli´c, I.: Wegner estimate and the density of states of some indefinite alloy type Schr¨ odinger operators. Lett. Math. Phys. 59(3), 199–214 (2002) [28] Veseli´c, I.: Existence and regularity properties of the integrated density of states of random Schr¨ odinger Operators. Lecture Notes in Mathematics, vol. 1917. Springer, Berlin (2007) [29] Wegner, F.: Bounds on the DOS in disordered systems. Z. Phys. B 44, 9–15 (1981) Ivan Veseli´c Fakult¨ at f¨ ur Mathematik, 09107 TU Chemnitz, Germany URL: http://www.tu-chemnitz.de/mathematik/stochastik Communicated by Claude Alain Pillet. Received: January 26, 2010. Accepted: June 25, 2010.
Ann. Henri Poincar´e 11 (2010), 1007–1021 c 2010 Springer Basel AG 1424-0637/10/061007-15 published online August 10, 2010 DOI 10.1007/s00023-010-0046-3
Annales Henri Poincar´ e
On the Renormalization Group Approach to Perturbation Theory for PDEs Walid K. Abou Salem Abstract. We investigate the rigorous application of the renormalization group method to (singular) perturbation theory for nonlinear partial differential equations. As a paradigm, we consider the concrete example of the nonlinear Schr¨ odinger equation with quadratic nonlinearity in three spatial dimensions. We obtain an approximate solution using the RG method together with an estimate of the difference between the true and approximate solutions. Our analysis applies to cases where (space–time) resonances are present.
1. Introduction The Chen–Goldenfeld–Oono renormalization group (RG) approach to singular perturbation theory, introduced in [1,2], has proven to be very versatile in handling different classes of perturbations in a unified manner. Moreover, it is conceptually simple and elegant: a naive perturbative expansion together with the RG condition is generally sufficient to give all the relevant scales in the problem. The method has been put on a rigorous footing for a wide class of ODEs in [3,4], and for some PDEs defined on bounded intervals with periodic boundary conditions in [5–7]. Here, we show that the method is rigorously applicable to nonlinear PDEs in the continuum and in the presence of (space–time) resonances. For the sake of concreteness, we consider as an example the nonlinear Schr¨ odinger equation with quadratic nonlinearity in three spatial dimensions. We obtain a long-time approximation for the true solution using the RG method. Our analysis in handling resonances relies on the application of the Coifman–Meyer theorem [8], together with fractional integration. A similar approach to resonances has been used in studying the well-posedness of the quadratic nonlinear Schr¨ odinger equation in three dimensions with small data, see [9], and the Gross–Pitaevski equation [10]. We hope that the conceptual simplicity and elegance of the RG method will motivate
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further rigorous applications. One such application where a similar space–time resonance arises is the derivation of the effective equations of water waves over a variable bottom. The organization of this note is as follows. In Sect. 2, we recall the RG approach to singular perturbation theory in the setting of PDEs. We then discuss our model and state the main result (Theorem 1) in Sect. 3. After discussing some properties of the approximate solution in Sect. 4, we prove the main result in Sect. 5. In order to simplify the presentation, we defer to the Appendix the proof of a technical estimate that relies on the Coifman–Meyer theorem and fractional integration.
2. The RG Method We start with a general discussion about the RG approach to singular perturbation theory. Suppose u belongs to some Banach space X, and suppose that A is a generator of a semigroup on X. We consider the following equation ∂t u = Au + f (u),
(1)
with initial condition ut0 ∈ X, where f (u) is a polynomial nonlinearity, and 1. We are interested in finding approximate solutions of the above initial value problem in the limit → 0.1 The weak solution of the above equation is given by the Duhamel formula t ds e−As f (u(s)). u(t) = eA(t−t0 ) ut0 + eAt t0
Performing a naive perturbative expansion in , we have u = u0 + u1 + 2 u2 + · · · , where u0 (t) = eA(t−t0 ) ut0 t At u1 (t) = e ds e−As f (u0 (s)) t0 At
t
u2 (t) = e ···
ds e−As ∇u f (u0 (s)) · u1 (s)
t0
Depending on the nonlinearity, we have e−As f (eAs v) = fres (v) + fosc (s, v) 1
This is equivalent to studying the asymptotics of 1 ∂τ u = Au + f (u) where τ = t.
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where fres stands for the resonance part, and fosc stands for the oscillatory part. Formally, the solution is given by u(t) = eA(t−t0 ) ut0 + (t − t0 )eAt fres (ut0 ) + eAt Fosc (t, ut0 ) + O(2 ), t where Fosc (t, v) := t0 dsfosc (t, v). Note that the second term in the expansion, the so called secular term, grows with t, and the naive perturbative expansion will break down with time. The purpose of the CGO method is to renormalize this term so that the main contribution coming from the secular term is taken into account. Let W satisfy ∂t W = fres (W ),
(2)
−At0
with initial condition W (t0 ) = e ut0 . Eq. (2) is called the renormalization group equation. The approximate solution, to first order, is given by2 u(t) := eAt {W (t) + Fosc (t, W (t))}.
(3)
Differentiating (3) with respect to t gives ∂t u = Au + f (u) + R (u, W ), where the remainder term R is given by R (u, W ) = (f (eAt W (t)) − f (u)) + 2 eAt ∇W Fosc (t, W (t))fres (W (t)). (4) Note that the initial condition u(t0 ) = ut0 . In what follows, we apply the above abstract analysis to the concrete example of a nonlinear Schr¨ odinger equation with quadratic nonlinearity, for which the resonant term is nontrivial. We will show that indeed, u is an approx| imation of u in a suitable Banach space, up to time of order O( | log ).
3. The Model and Statement of the Main Result We consider the nonlinear Schr¨ odinger equation with quadratic nonlinearity in three spatial dimensions, i∂t u = −Δu − u2 ,
(5)
with initial condition ut0 . Here, A = iΔ, where Δ is the spatial Laplacian in R3 , and f (u) = iu2 . For τ > 0, we let Bτ be the Banach space with norm −iΔt u ∞ uBτ = uL∞ ([t0 ,t0 +τ ];L2 (R3 )) + e L ([t0 ,t0 +τ ];L∞ (R3 )) 2 x −iΔt x −iΔt √ e e + u + u log t ∞ t ∞ 2 3 L
([t0 ,t0 +τ ];L (R ))
L
([t0 ,t0 +τ ];L2 (R3 ))
+ t3/2 uL∞ ([t0 ,t0 +τ ];L∞ (R3 )) , 2 The RG method can be applied recursively to obtain an approximate solution to higher orders (see, for example [4]), but we restrict the discussion to first order in order to simplify the presentation.
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where the hat stands for the Fourier transform. Without loss of generality, we replace t appearing in the denominator in assume that t0 = 2 (alternatively, √ the B-norm with t = 1 + t2 ). The first term in the above norm corresponds odinger equation. The second term to the usual L2 control for the linear Schr¨ is effectively a strong condition of spatial localization of the initial data, while the third and fourth terms correspond to spatial localization with weak time growth. The last term corresponds to the L∞ decay of the solution of the linear Schr¨ odinger equation in three spatial dimensions. The well-posedness of the nonlinear Schr¨ odinger equation with quadratic nonlinearity in B∞ for small initial data has been proven in [9]. Small initial data corresponds to small in (5) after a simple rescaling of u. Therefore, if eiΔ(t−t0 ) ut0 B∞ < ∞, there exists 0 > 0 such that for all || < 0 , (5) is well-posed in B∞ . The following is the main result of this note. Theorem 1. Consider (5) with initial condition ut0 such that φ0 := eiΔ(t−t0 ) ut0 B∞ < ∞. Then there exists 0 > 0, that depends on φ0 , such that, for all || < 0 and δ ∈ (0, 1), u − uL∞ ([t0 ,t0 +δ | log | ];L2 (R3 )) < C1−δ eφ0
for some positive constant C that is independent of and δ.
4. Approximate Solution In this section, we investigate some properties of W and u, which are sufficient to prove Theorem 1. Since 2 2 2 −iΔs iΔs −1 dξeis(k −ξ −(k−ξ) ) u f (e u) = iF (ξ) u(k − ξ) e where F −1 stands for the inverse Fourier transform, we have u(0)u, fres (u) = i and fosc (u) = iF −1
dξ eis(k
2
−ξ 2 −(k−ξ)2 )
(6) u (ξ) u(k − ξ).
(7)
ξ=k
The RG equation corresponding to (5) is given by (t, 0)W (t, x) ∂t W (t, x) = iW
(8)
with initial condition W (t0 ) = e−iΔt0 ut0 . We have the following straightforward result, which follows from Gronwall lemma and a boot-strap argument. Lemma 1. Suppose that φ0 < ∞. Then sup
s∈[t0 ,t0 + eφ1 ] 0
eiΔt W (s)B∞ ≤ eφ0 .
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Proof. Let T be the maximal time such that sup eiΔt W (s)B∞ ≤ eφ0 .
s∈[t0 ,T ]
It follows from the RG equation (8) and Gronwall lemma that3
eiΔt W (s)B∞ ≤ e
s
t0
∞ ds W L ([t0 ,T ];L∞ )
≤ e(s−t0 ) sups ∈[t0 ,s] e (s−t0 )eφ0
≤e for all s ≤ t0 +
1 eφ0 .
iΔt
eiΔ(t−t0 ) ut0 B∞
W (s )B∞
φ0
φ0 ≤ eφ0
Hence, the maximal time T ≥ t0 +
1 eφ0 .
We now introduce the Banach space Cτ whose norm is given by uCτ := uL∞ ([t0 ,t0 +τ ];L2 (R3 )) + t3/2 uL∞ ([t0 ,t0 +τ ];L∞ (R3 )) .
(9)
The approximate solution u defined in (3) is explicitly given by u(t) = eiΔt (W (t) t +i
dsF −1 (
t0
dξei(k
2
−ξ 2 −(k−ξ)2 )s
ξ=k
(t, k − ξ))), W (t, ξ)W
Fosc (t,W (t))
(10) where F −1 stands for the inverse Fourier transform. It follows from Lemma 1 that eiΔt W (t)C
1 eφ0
≤
eiΔt W (s)B∞ < C
sup
s∈[t0 ,t0 + eφ1 ] 0
(11)
for some constant C that is independent of . We also have the following estimate, which follows from the Coifman–Meyer theorem and fractional integration, and which we prove in the Appendix, eiΔt Fosc (t, W (t))C
1 eφ0
≤ C,
(12)
for some constant C that is independent of , but that depends on φ0 . The following result follows from (10)–(12). Lemma 2. Suppose that φ0 < ∞. Then u ∈ C eφ1 . 0
3
(s,0) i tt ds W 0 e−iΔt0 u
The (weak) solution of (8) is given by W (t, x) = e
t0 (x).
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5. Proof of Theorem 1 We start by estimating the L2 -norm of the difference between the approximate solution u and the true solution u up to time of order O(−1 ), and then we iterate our estimates to longer times. We have the following proposition. Proposition 1. Suppose that φ0 < ∞. Then there exists 0 > 0 such that, for all || < 0 , u(t) − u(t)L∞ ([t0 ,t0 + eφ1 ];L2 (R3 )) < C, 0
for some constant C > 1 that is independent of . Proof. Let v(t) := u(t) − u(t). The Duhamel formula yields iΔ(t−t0 )
v(t) = e
I
iΔt
iΔt
v(t0 ) + ie
t
+ ie
t0
t
ds e−iΔs (f (u(s)) − f (u(s)))
t0
II
ds e−iΔs R (u(s), W (s)) .
III
(13)
It follows from the unitarity of the free evolution that IL2 = v(t0 )L2 .
(14)
Furthermore, it follows from an energy estimate that IIL2
t −iΔs = ds e (u(s) + u(s))(u(s) − u(s)) t0
t ≤
L2
ds(u(s) + u(s))(u(s) − u(s))L2 t0
t ≤ t0
ds (uCt + uCt )u − uL∞ ([t0 ,t];L2 (R3 )) s3/2
≤ C(uBt + uCt )u − uL∞ ([t0 ,t];L2 (R3 ))
(15)
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where C is a positive constant that is independent of and t. We also have t −iΔs ds e ∇ F (s, W (s))f (W (s)) IIIL2 = 2 W osc res t0
t ≤ 22
L2
s
ds t0
t0
(s, 0)e−iΔs (eiΔs W (s))2 L2 ds W
(s, 0)L∞ ([t ,t]) ≤ 2 W 0
t
2
s ds
t0 iΔs
× e
t0
ds 3/2 iΔs s e W (s)L∞ (R3 ) s3/2
W (s)L2 (R3 ) .
Now, applying the stationary phase estimate eiΔt gL∞ (R3 ) ≤
1 t3/2
g L∞ (R3 ) +
1 t7/4
x2 gL2 (R3 ) ,
we have
t IIIL2 ≤ C2
ds t0
1 1/2 t0
−
1
sup eiΔt W (s )2B∞
s1/2
s ∈[t0 ,t]
≤ C,
(16)
where C is a constant that is independent of and t. Estimates (14)–(16) and Lemmata 1 and 2 yield, for small enough and depending on φ0 , u(t) − u(t)L2 ≤ C(u(t0 ) − u(t0 )L2 + ) uniformly in t ∈ [t0 , t0 + of .
1 eφ0 ],
where C > 1 is a constant that is independent
We now iterate Proposition 1 in order to prove the main result of this note. Proof of Theorem 1. Let n ∈ N to be chosen below. It follows from applying the result of Proposition 1 n times that u(t) − u(t)L∞ ([t0 ,t0 + eφn ];L2 (R3 )) ≤ 0
n
j=1
≤C Now, for δ ∈ (0, 1), choose n such that C this choice, it follows from (17) that
n+1
C j ( + u2 − u(2)L2 )
=0
n+1
.
(17)
−δ
, i.e., n < δ| log | − 1. For
<
u(t) − u(t)L∞ ([t0 ,t0 + δ| log | ];L2 (R3 )) ≤ C1−δ eφ0
which is the claim of the theorem.
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We note that it is possible to iterate the estimate of Proposition 1 an asymptotically large number of times as 0, as long as the hypotheses on n successive φ0 continue to hold.
Acknowledgements WAS thanks Nigel Goldenfeld for pointing out references [3] and [4].
6. Appendix For the convenience of the reader, we recall in this Appendix certain standard results, such as the Coifman–Meyer theorem and fractional integration, see [8], before proving estimate (12). Theorem 2 (Coifman–Meyer Theorem). Consider the Fourier multiplier m(k, ξ) and the associated operator Tm defined by Tm (f, g) := F −1 dξ m(k, ξ)f(ξ) g (k − ξ) where F is the inverse Fourier transform. Suppose that the Fourier multiplier m satisfies C |∂kα ∂ξβ m(k, ξ)| ≤ (|k| + |ξ|)|α|+|β| for sufficiently many multi-indices (α, β). Then the operator Tm : Lp × Lq → Lr is bounded for 1 1 1 + = , p q r
1 < p, q ≤ ∞,
0 < r < ∞.
We also have the following standard result on fractional integration. α
α
1 2 Lemma 3 (Fractional integration). Let Λα := (−Δ) 2 and Λα t := ( t − Δ) , t > 0. Then the following holds. (i) If α ≥ 0 and 1 < p, q < ∞, 1q − p1 = α3 , then
Λ−α f Lp ≤ Cf Lq for some constant C > 0. (ii) If α ≥ 0 then Λ−α f L∞ ≤ Cf
3
L α ,1
for some constant C > 0, where Lp,q is the standard Lorenz space. (iii) If α ≥ 0 and 1 ≤ p, q ≤ ∞, 0 ≤ 1q − p1 < α3 , then α
3
1
1
2 + 2 ( p − q ) f q Λ−α L t f Lp ≤ Ct
for some constant C > 0.
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6.1. Proof of the Estimate (12) We start by estimating eiΔt Fosc (t, W (t))L∞ ([t0 ,t0 + eφ1 ];L2 ) . 0
Using an energy estimate together with H¨older’s inequality, we have eiΔt Fosc (t, W (t))L2 = Fosc (t, W (t))L2 t ≤ ds(eiΔs W (t))2 L2 t0
t
dseiΔs W (t)L∞ eiΔs W (t)L2
≤ t0
iΔs
≤ e
t W (t)L∞ ([t0 ,t];L2 ) t0
≤C
iΔt
sup
t ∈[t0 ,t0 + eφ1 ]
e
ds 3/2 iΔs s e W (t)L∞ s3/2
W (t )2B∞
0
≤ Cφ20 ,
(18)
uniformly in t ∈ [t0 , t0 + eφ10 ] and || < 0 . Here we used Lemma 1 in the last inequality. Estimating t3/2 eiΔt Fosc (t, W (t))L∞ ([t0 ,t0 + eφ1 ];L∞ ) is some more 0 work. It involves integration by parts and the application of the Coifman– Meyer theorem. In what follows, C denotes a positive constant that is independent of , and that might change from line to line. Let θ = k 2 − ξ 2 − (k 2 − ξ)2 1 P = −ξ + k 2 Z = θ + P · ∂ξ θ. Then 1 t
1 + iZ
1 + P · ∂ξ + ∂t eiθt = eiθt . t
(19)
Integrating by parts in time, we have Fosc (t, W (t)) =
t ds t0
dξ 1
ξ=k
= g(k) + h(k)
s
1 + iZ
1+P · ∂ξ (t, ξ)W (t, k−ξ) + ∂s eisθ W s
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where t
1 s
g(k) = t0 ξ=k
+ P ∂ξ θ isθ (t, k − ξ) e W (t, ξ)W + iZ
1 s
t
s−2
−i t0 ξ=k
1 (t, ξ)W (t, k − ξ) eisθ W ( 1s + iZ)2
and the boundary term h(k) =
1 t
ξ=k
1 (t, ξ)W (t, k − ξ) eitθ W + iZ
h1
− ξ=k
1 t0
1 (t, ξ)W (t, k − ξ). eit0 θ W + iZ
h2
We start by estimating the boundary terms. By the Coifman–Meyer theorem and fractional integration, we have 1 + ξ 2 iΔt −1 1 itΔ t iΔt e W (t)(ξ) e W (t)(k − ξ) e h1 L∞ = F 1 1 2 + ξ + iZ t t ∞ ξ=k L = Λ−2 T 1t +ξ2 (eiΔt W (t), eiΔt W (t)) t 1 +iZ L∞ t 3/4 iΔt iΔt ≤ Ct T 1t +ξ2 (e W (t), e W (t)) 1 +iZ L6
t
≤ Ct ≤
3/4
iΔt
e
iΔt
W (t)L6 e
W (t)L∞
Ct−7/4 φ20
(20)
where we have used Lemma 1 in the last inequality. Similarly, eitΔ h2 L∞ ≤ Ct−7/4 φ20 .
(21)
In order to estimate eiΔt gL∞ , we use the following three estimates uniformly in t ∈ [t0 , t0 + eφ10 ], which we prove below. hL∞ ≤ Cφ20 ,
(22)
Fosc (t, W (t))L∞ ≤ Cφ20 , x g(t)L2 ≤ Ct 2
γ
(23) (24)
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for arbitrary γ > 0. It follows from the stationary phase estimate that 1 1 g L∞ + 7/4 x2 gL2 t3/2 t 1 1 ≤ 3/2 (Fosc (t, W (t)))L∞ + hL∞ ) + 7/4 x2 gL2 t t 1 ≤ C 3/2 φ20 . t
eiΔt gL∞ ≤
(25)
Estimate (12) follows from (18), (20), (21) and (25). We now prove estimates (22)–(24). • Estimate (22).
1 (t, k − ξ)| |W (t, ξ)||W ξ2 ≤ CeiΔt W (t)2L2 ∩L∞
hL∞ ≤ 2
≤ Cφ20 . • Estimate (23). t 2 2 isk /2 −2isξ (t, k/2−ξ)W (t, k/2+ξ) |Fosc (t, W (t))(k)| = dse W dξe t0 ξ=k t ≤C t0
t ≤C t0
t ≤C t0
1
(t, k/2)|2 + 1 ∂ 2 (W (t, k/2 − ξ)W (t, k/2 + ξ))L2 |W s3/2 s7/4 ξ
(t)L2 W (t)2 4 ) (t, k/2)|2 + 1 (∂ 2 W (t)L∞ + ∂ξ W | W ξ L s3/2 s7/4 1
1
(t)L2 W (t, k/2)|2 + 1 ∂ 2 W (t)L∞ |W 3/2 s s7/4 ξ
⎛ ≤ C ⎝φ20 + φ20
t t0
⎞ s1/2 ⎠ ds 7/4 ≤ Cφ20 , s
where we have used the Gagliardo-Nirenberg inequality in the fourth line and Lemma 1 in the last line.
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• Estimate (24). In what follows, Pn denotes a homogeneous polynomial in (k, ξ) of degree n. Applying ∂k2 to g gives terms of the form t (a) :
P2k−2j−2 isθ (t, k − ξ) e W (t, ξ)W ( 1s + iZ)k
s−j
t0 ξ=k
k ≥ 0, k − 1 ≥ j ≥ 0. t (b) :
P2k−2j−1 isθ (t, k − ξ) e W (t, ξ)∂k W ( 1s + iZ)k
s−j
t0 ξ=k
k ≥ 0, k − t (c) :
1 ≥ j ≥ 0. 2 P2k−2j isθ (t, k − ξ) e W (t, ξ)∂k2 W + iZ)k
s−j
( 1s
t0 ξ=k
k ≥ 0, k − 1 ≥ j ≥ 0. t (d) :
s t0 ξ=k
( 1s
t (e) :
s t0 ξ=k
s2 t0 ξ=k
P2k+1 isθ (t, k − ξ) e W (t, ξ)∂k W + iZ)k
( 1s
t (f ) :
P2k (t, ξ)W (t, k − ξ) eisθ W + iZ)k
∂k2 θP · ∂ξ θ isθ (t, k − ξ). e W (t, ξ)W ( 1s + iZ)
We estimate (a) using the Coifman–Meyer theorem and fractional integration. We have t (a)L2 ≤ dss−j e−iΔs T P2k−2−2j (eisΔ W (t), eisΔ W (t))L2 ( 1 +iZ)k s
t0
t ≤C
dss−j Λ−2−2j eiΔs W (t)L2 eiΔs W (t)L∞ s
t0
t ≤C
1
dss−j sj+ 2 eiΔs W (t)L6/5 eisΔ W (t)L∞ .
t0
Now, Lemma 1 and the interpolation between different norms in B gives eiΔs W (t)L6/5 ≤ Csγ
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for arbitrary γ > 0, and hence t (a)L2 ≤ C
s1/2 sγ s−3/2 s3/2 eiΔs W (t)L∞
t0
≤ Ctγ
(26)
where we have used Lemma 1 in the last inequality. Similarly, using the Coifman–Meyer theorem and fractional integration, we have t (b)L2 ≤
s−j e−isΔ T P2k−1−2j (eisΔ W (t), eisΔ xW (t))L2 ( 1 +iZ)k s
t0
t ≤C
s−j {Λ−2j−1 eisΔ xW (t)L2 eisΔ W (t)L∞ s
t0
+ eisΔ xW (t)L2 Λ−2j−1 eisΔ W (t)L∞ } s t 1 ≤C dss−j sj+ 2 xW (t)L2 eiΔs W (t)L∞ t0
t ≤C
1
3
dss 2 − 2 log s ≤ Ctγ
(27)
t0
and t (c)L2 ≤
s−j e−isΔ T
t0
t ≤C
P2k−2j ( 1 +iZ)k s
(eisΔ W (t), eisΔ x2 W (t))L2
s−j {Λ−2j eisΔ x2 W (t)L2 eisΔ W (t)L∞ s
t0
+ eisΔ x2 W (t)L2 Λ−2j eisΔ W (t)L∞ } s t ≤C dss−j sj x2 W (t)L2 eiΔs W (t)L∞ t0
t ≤C
1
s2 t0
1 3
s2
≤ C log t.
Using the identity (19), we have t (d) =
s t0 ξ=k
P2k ( 1s + iZ)k+1
1 P (t, ξ)W (t, k − ξ), + ∂s + ∂ξ eisθ W s s
(28)
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which, after integrating by parts in s and ξ, gives terms of the form (a) − (c) plus boundary terms that are easily controlled. Hence, we also have (d)L2 ≤ Ctγ .
(29)
Similarly, t (e) = t0 ξ=k
P2k+1 s 1 ( s + iZ)k+1
1 P (t, ξ)∂k W (t, k − ξ), + ∂s + ∂ξ eisθ W s s
which, after integrating by parts in s and ξ, gives terms of the form (a) − (d), except for the term t t0 ξ=k
( 1s
P2k+1 (t, ξ)∂k W (t, k − ξ). P eisθ ∂ξ W + iZ)k+1
(30)
We control (30) using the Coifman–Meyer theorem, fractional integration, and the Gagliardo–Nirenberg inequality eisΔ (xW (t))L4 ≤ eiΔs W (t)L∞ eiΔs x2 W (t)L2 , see estimate of (c). Therefore, (e)L2 ≤ Ctγ .
(31)
To estimate (f ), we integrate by parts in ξ, which gives terms of the form (d) and (e), yielding (f )L2 ≤ Ctγ .
(32)
Estimate (24) follows from (26)–(32).
References [1] Chen, L.-Y., Goldenfeld, N., Oono, Y.: Renormalization group theory for global asymptotic analysis. Phys. Rev. Lett. 73(10), 1311–1315 (1994) [2] Chen, L.-Y., Goldenfeld, N., Oono, Y.: Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory. Phys. Rev. E 543(1), 376–394 (1996) [3] Ziane, M.: On a certain renormalization group method. J. Math. Phys. 41(5), 3290–3299 (2000) [4] De Ville, R., Harkin, A., Holzer, M., Josic, K., Kaper, T.: Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations. Physica D 237, 1029–1052 (2008) [5] Moise, I., Temam, R.: Renormalization group method. Applications to Navier– Stokes equation. Discret. Continuous Dyn. Syst. 6, 191–200 (2000) [6] Moise, I., Ziane, M.: Renormalization Group Method. Applications to Partial Differential Equations. J. Dyn. Differ. Equ. 13, 275–321 (2001) [7] Petcu, M., Temam, R., Wirosoetisno, D.: Renormalization group method applied to the primitive equations. J. Differ. Equ. 208, 215–257 (2005)
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[8] Coifman, R., Meyer, Y.: Au del` a des op´erateurs pseudo-diff´erentials. In: Ast´erisque, vol. 57. Soci´et´e Math´ematique de France, Paris (1978) [9] Germain, P., Masmoudi, N., Shatah, J.: Global solutions for 3D quadratic Schr¨ odinger equations (2008, preprint) [10] Gustafson, S., Nakanishi, K., Tsai, T.-P.: Global dispersive solutions for the Gross–Pitaevski equation in two and three dimensions. Ann. Henri Poincar´e 8, 1303–1331 (2007) Walid K. Abou Salem Department of Mathematics and Statistics University of Saskatchewan 106 Wiggins Road Saskatoon, SK S7N 5E6, Canada e-mail:
[email protected] Communicated by Rafael D. Benguria. Received: January 3, 2009. Accepted: June 29, 2010.
Ann. Henri Poincar´e 11 (2010), 1023–1052 c 2010 Springer Basel AG 1424-0637/10/061023-30 published online October 1, 2010 DOI 10.1007/s00023-010-0054-3
Annales Henri Poincar´ e
On Blowup for Time-Dependent Generalized Hartree–Fock Equations Christian Hainzl, Enno Lenzmann, Mathieu Lewin and Benjamin Schlein Abstract. We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree–Fock and Hartree–Fock–Bogoliubovtype equations, which describe the evolution of attractive fermionic systems (e.g. white dwarfs). Our main results are twofold: first, we extend the recent blowup result of Hainzl and Schlein (Comm. Math. Phys. 287:705–714, 2009) to Hartree–Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree–Fock–Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree–Fock–Bogoliubov theory.
1. Introduction In this paper, we study time-dependent generalized Hartree–Fock equations that arise as quantum fermionic models for the dynamics of relativistic stars (like white dwarfs and neutron stars). Here, our particular interest is devoted to the existence of finite-time blowup solutions, which physically indicate the onset of “gravitational collapse” of a very massive star due to its own gravity. Indeed, such a catastrophic scenario was already predicted by the physicist S. Chandrasekhar in 1930s, based on an intriguing and heuristic combination of special relativity and Newtonian gravity. However, it is fair to say that a rigorous understanding of the dynamics of the gravitational collapse of a star—in terms of relativistic quantum mechanics and Newtonian gravity to start with—is still in its beginning. In fact, the proposed mathematical models (such as the generalized Hartree–Fock equations considered below) are formulated as nonlinear dispersive evolution equations, which exhibit the essential feature of
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L2 -mass-criticality and thus reflect the physical fact of a so-called Chandrasekhar limit mass M∗ ≈ 1.4MSun , beyond which gravitational collapse of a cold star can occur. For a detailed discussion of Hartree–Fock type theory in astrophysics, we refer to [10,12] and references given there. See also [16,17] for rigorous work on Chandrasekhar’s theory in the time-independent setting. Let us now briefly outline the main content of the present paper. As a relativistic quantum model of the dynamics of star, we consider time-dependent generalized Hartree–Fock equations with relativistic kinetic energy and an attractive two-body interaction of Newtonian type. More specifically, we study equations of Hartree–Fock (HF) and Hartree–Fock–Bogoliubov (HFB) type, and our main results on finite-time blowup of spherically symmetric solutions for these models will be stated in Sects. 2 and 3, respectively. In fact, this paper aims at two directions: first, we consider the HF equations and we extend the recent blowup results of [12] to solutions with infinite-rank density matrices and a broader class of two-body interactions. Second, we address the finite-time blowup for HFB type evolution equations, which also incorporate the physically important phenomenon of Cooper pairing for attractive fermion systems. However, it turns out that the analysis of HFB equations brings in new key difficulties, which were completely absent for HF. Most importantly, the non-conservation of higher moments of angular momenta in HFB theory makes an adaptation of the strategies used in [10,12] an insurmountable task at the present. To circumvent this difficulty, we introduce an HFB model with an angular momentum cutoff and prove a finite-time blowup result for this simplified model; see Sect. 3 for more details. In the long run, we do hope that our arguments developed in this paper will be of use to understand the finite-time blowup for the full-fledged HFB model without any cutoffs. Outline of the Paper This paper is organized as follows. In Sects. 2 and 3, we first collect some preliminary facts and we then state our main results on finite-time blowup for spherically symmetric solutions of HF and HFB type models. Furthermore, we explain the key difficulties encountered for HFB theory. The proofs of the main theorem will be given in Sect. 4, and Appendix A contains the proof of a technical lemma. Conventions and Notations For the physically inclined reader, we remark that we employ units such that Planck’s constant and the speed of light c both satisfy = c = 1. As usual, the letter C denotes a positive constant depending only on fixed quantities (such as initial data etc.) and C is allowed to change from line to line.
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2. Blowup in Hartree–Fock Theory In this section, we consider the Hartree–Fock (HF) model describing a system of pseudo-relativistic fermions with average particle number λ > 0 and attractive interactions of Newtonian-type (like classical gravity). With regard to physical applications, we may think of such system representing a white dwarf star, where effects due to special relativity play a significant role but gravity can still be treated in the Newtonian framework. Of course such a model lacks Poincar´e covariance, and hence it is usually referred to as pseudo-relativistic. Nonetheless, many essential effects (such as the existence of a Chandrasekhar limit mass of white dwarfs) are already predicted—even quantitatively within good approximation—by such pseudo-relativistic models. Indeed, it should not go unmentioned here that S. Chandrasekhar’s original and acclaimed physical theory of gravitational collapse rests on a pseudo-relativistic model of a star in its semi-classical regime; see [7]. According to HF theory, the state of the system is described in terms of a density matrix γ which is a self-adjoint operator acting on L2 (R3 ; C)1 , satisfying 0 γ 1 (reflecting the Pauli principle for fermions) and the normalization condition Tr(γ) = λ fixing the (average) number of particles. Let V denote the interaction potential between the particles and, for notational convenience, let κ > 0 be a coupling constant controlling the strength of the interaction. Then the corresponding HF energy reads κ V (x − y) ργ (x)ργ (y) − |γ(x, y)|2 dx dy. (2.1) EHF (γ) = Tr(Kγ) + 2 Here γ(x, y) is the integral kernel of the trace-class operator γ and ργ (x) = γ(x, x) is the associated density function. The pseudo-differential operator K = −Δ + m2 describes the kinetic energy of a relativistic quantum particle with rest mass m 0. In the present section, we are interested in the corresponding timedependent HF equations, which can be formulated as the following initialvalue problem: d i γt = [Hγt , γt ], (2.2) dt γ|t=0 = γ0 ∈ KHF . Here [A, B] = AB − BA denotes the commutator and Hγ := −Δ + m2 + κ (V ∗ ργ ) − κV (x − y)γ(x, y)
(2.3)
is the so-called mean-field operator which depends on γ and acts on the onebody space L2 (R3 ; C). Here and henceforth, the symbol ∗ stands for convolution of functions on R3 . For expositional convenience, we use a slight abuse of 1 For simplicity, we discard the spin of the particles throughout. But all our arguments can be easily generalized to particles having q internal degrees of freedom, where L2 (R3 ; C) has to be replaced by L2 (R3 ; Cq ) etc.
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notation by writing V (x − y)γ(x, y) for the operator whose integral kernel is the function (x, y) → V (x − y)γ(x, y). We remark that the number of particles Tr(γt ) and the energy EHF (γt ) are both conserved along the flow given by (2.2), as we will detail later on. Also, the appropriate set KHF of initial data for the evolution equation (2.2) will be defined below. More specifically, we consider radial interaction potentials V = V (|x|) that are essentially Newtonian at small and large distances. By this, we mean that V is of the form V (|x|) = −
1 + w(|x|). |x|
(2.4)
Here w is a smooth and decaying function, satisfying the following conditions that we impose throughout this paper: w ∈ C 1 [0, ∞),
r |w(r)| C
and
(1 + r2 )1+ε |w (r)| C
(2.5)
for all r 0 and some ε > 0. Physically, one may think of w as describing screening effects due to internal degrees of freedom of the gravitating fermions. Indeed, as remarked above, our arguments carry over to the case where spin is taken into account, and w = w(|x|) would be a q × q matrix instead of a multiple of the identity (with an adequate redefinition of the HF-energy function EHF (γ) above). In the special case of γ being an orthogonal projection of finite rank N (i.e., we have γ = i=1 |ψi ψi |) and vanishing w ≡ 0, the existence of finitetime blowup for the time-dependent HF equation (2.2) was recently studied by Hainzl and Schlein [12], by extending the arguments developed by Fr¨ ohlich and Lenzmann [10]. The main result of [12] established that any (sufficiently regular and rapidly decaying) initial datum γ with spherical symmetry (see Definition 1) and negative energy EHF (γ) < 0 leads to finite-time blowup of the solution γt . Note that the assumption of finite rank was important for the method used in [12] to work. In this paper, we present a new technical approach which allows us to prove a blowup for γ having an infinite rank. Moreover, we show here that the existence of a blowup solution is stable with respect to a certain class of perturbations w of the Newton interaction, satisfying (2.5). After this preliminary remarks, we now state our results on the HF equation (2.2). To this end, we denote by Sp the Schatten space (see [18,20]) of p operators A acting on L2 (R3 , C) such that ||A||Sp = Tr|A|p < ∞. Further we introduce a set of fermionic density matrices KHF := {γ = γ ∗ ∈ XHF : 0 γ 1},
(2.6)
where the Sobolev-type space XHF is defined by XHF := {γ ∈ S1 : γ XHF < ∞} with the norm
γ XHF := (1 − Δ)1/4 γ(1 − Δ)1/4
(2.7)
S1
.
(2.8)
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In fact, it turns out that the initial-value problem (2.2) is locally well-posed in KHF . Theorem 1 (Well-posedness in Hartree–Fock theory). Suppose that w satisfies (2.5). For each initial datum γ0 ∈ KHF , there exists a unique solution γt ∈ ) with maximal time of existence 0 < T ∞. C 0 ([0, T ), KHF ) ∩ C 1 ([0, T ); XHF Moreover, we have conservation of energy and expected particle number: EHF (γt ) = EHF (γ0 )
and
Tr(γt ) = Tr(γ0 )
for
0 t < T.
Finally, if the maximal time satisfies T < ∞, then Tr(−Δ)1/2 γt → ∞ as t → T −. This theorem was proved in [10] in the special case w = 0 and for γ with finite rank. However, it is straightforward to extend the arguments of [10] to the case w = 0 and γ with infinite rank; see also [5,6]. Remark 1. (i) By using the conservation laws, we can deduce that sufficiently small solutions γt extend to all time. More precisely, there exists a universal constant C∗ > 0 such that Tr(γ0 )κ3/2 < C∗ implies global-in-time existence and a priori bounds in energy space; i.e., we have T = +∞ and that γt XHF C for all times t 0; see [14]. Physically, this global well-posedness result states that white dwarfs of sufficiently small mass have a well-defined global dynamics and, in particular, no gravitational collapse can occur. (ii) Particular global-in-time solutions of (2.2) are stationary states satisfying [Hγ , γ] = 0. Important examples for stationary states are given by the minimizers of EHF (γ), subject to the constraint Tr(γ) = λ with λκ3/2 not too large, which were proven to exist in [14]. (iii) Well-posedness can also be shown to hold true in Sobolev-type spaces Hs with the norm γ Hs = (1 − Δ)s/2 γ(1 − Δ)s/2 S for s 1/2, pro1 vided the initial datum γ0 belongs to Hs . Indeed, we will make tacitly use of this fact about this persistence of higher regularity, guaranteeing the well-definedness of our calculations below. We now define spherically symmetric states for which we will be able to prove blowup under some assumptions: Definition 1 (Spherically symmetric HF states). We say that γ ∈ KHF is spherically symmetric when γ(Rx, Ry) = γ(x, y) for all x, y ∈ R3 and all R ∈ SO(3). It is easy to verify (see Lemma 4 below) that spherically symmetry of γt is preserved under the flow (2.2). We also note that, if γ0 is sufficiently regular, then the condition of spherical symmetry can also be written as the commutator condition [γt , L] = 0.
(2.9)
Here L = −ix ∧ ∇x is the angular momentum operator, and ∧ denotes the cross product on R3 .
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Our main result of this section is the following blowup criterion. The proof will be given in Sect. 4.2 below. Theorem 2 (Blowup in Hartree–Fock theory). Suppose that w is a radial function satisfying (2.5) and that κ > 0. Let γ0 ∈ KHF be spherically symmetric and suppose that Tr |x|4 γ0 + Tr (−Δ)γ0 + Tr |L|2 γ0 < ∞ where L = −ix ∧ ∇x denotes the angular momentum operator. Our conclusion is the following: If γ0 has sufficiently negative energy, that is EHF (γ0 ) < −
κ 2 (Tr(γ0 )) sup |w(r) + r w (r)|− 2 r0
(2.10)
(where |f |− 0 denotes the negative part of f ), then the corresponding solution γt to (2.2) blows up in finite time; i.e., we have Tr (−Δ)1/2 γt → ∞ as t → T − for some T < ∞. Remark 2. By scaling, it is simple to show the existence of γ0 ∈ KHF satisfying the condition (2.10) for sufficiently large coupling constants κ, or equivalently for a sufficiently high particle number Tr(γ0 ). Note also that Tr(γ0 )κ3/2 must be sufficiently large, since otherwise T = +∞ holds, as mentioned in Remark 1 above. Regarding the proof of Theorem 2, we remark that we follow a virial-type argument developed in [9,10] for the Hartree equation, which was recently extended in [12] to Hartree–Fock equations, by using an additional conservation law for the square of the angular momentum Tr(|L|2 γt ) for radial solutions. More specifically, it was shown in [12] that spherically symmetric solutions γt of the Hartree–Fock equation (2.2) with V = −1/|x| satisfy the inequality Tr(M γt ) 2EHF (γ0 )t2 + CTr(γ0 )Tr((1 + |L|2 γ0 )t + C,
(2.11)
provided that γ0 is finite-rank. Since the relativistic “virial” operator M=
3
xi
−Δ + m2 xi
j=1
is non-negative, this yields a finite maximal time of existence T < ∞ and hence the desired blowup result. In the arguments developed below, we remove the finite-rank condition, by expanding the solution γt in terms of sectors of angular momentum. This approach is also essential when addressing the blowup for HFB theory, as done in the next section. However, the conservation law for Tr(|L|2 γt ), which plays an essential role in the argument sketched above, is no longer at our disposal in HFB theory, as will be detailed below.
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3. Blowup in Hartree–Fock–Bogoliubov Theory In this section, we now turn to the Hartree–Fock–Bogoliubov (HFB) model, which generalizes the well-known Hartree–Fock model studied in the previous section. It is fair to say that HFB theory is a widely used tool for (partially) attractive quantum systems, which are found–for example–in nuclear and stellar physics; see [3,8,19] for a detailed exposition. As a notable fact, we mention that HFB yields (in some limiting approximation) the well-known Bardeen–Cooper–Schrieffer (BCS) theory, which plays a fundamental role for the understanding of superconductivity and superfluidity. The outline of this section is briefly as follows. First, we introduce the HFB energy and its associated time-dependent equations. Second, we point out the key difficulties in the blowup analysis encountered within HFB theory. Finally, we state our main result on finite-time blowup for the HFB model with an angular momentum cutoff; see Theorem 4 below. 3.1. HFB Theory: Preliminaries and Difficulties 3.1.1. HFB Energy. The HFB model involves two variables: the same onebody density matrix γ as in HF theory, and an additional two-body fermionic wavefunction α(x, y) ∈ L2 (R3 ; C) ∧ L2 (R3 ; C)2 describing
the so-called Cooper pairs. The amount of Cooper pairing is given by |α(x, y)|2 dx dy; 2 and it must vanish when γ = γ is a pure Hartree–Fock state. For simplicity, we will always use the same notation for the two-body wavefunction α and the corresponding (non-self-adjoint but compact) operator acting on the one-body space L2 (R3 ; C), whose integral kernel is the function (x, y) → α(x, y). Note that the assumption that α is fermionic can then be expressed as αT = −α. We remind the reader that again we have neglected any spin degrees of freedom to simplify our presentation. Within the HFB framework, the operators γ and α are, by assumption, linked through the following operator constraint: 1 0 0 0 γ α (3.1) on L2 (R3 ; C) ⊕ L2 (R3 ; C). 0 1 0 0 α∗ 1 − γ In fact, this inequality guarantees that the pair (γ, α) is associated with a unique quasi-free state in Fock space; see [2]. The average number of particles is given (as in HF theory) by the trace Tr(γ). The HFB energy functional, for a system interacting through a radial two-body potential V , is given by κ EHFB (γ, α) := EHF (γ) + |α(x, y)|2 V (|x − y|) dx dy. (3.2) 2 R3 ×R3
It is then obvious that a ground state (if it exists) always has α ≡ 0 in the case of a purely repulsive potential V 0. Therefore, the use of HFB theory is only meaningful when V is not everywhere positive. In fact, in this case, it 2
Here ∧ denotes the antisymmetric tensor product.
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is often observed that there is a phase transition from an HF ground state to an HFB ground state when the coupling constant κ is increased. As in the previous section, we consider radial potentials V = V (|x|) of the form (2.4) above. Furthermore, we remark that the HFB energy EHFB (γ, α) can be appropriately defined (see [14]) on the set of generalized density matrices:
0 0 γ α 1 0 KHFB = (γ, α) ∈ XHFB : . (3.3) 0 0 α∗ 1 − γ 0 1 The space XHFB is the correct energy space for generalized density matrices, which is defined as (3.4) XHFB = (γ, α) ∈ S1 × S2 : γ ∗ = γ, αT = −α, (γ, α) XHFB < ∞ , and is equipped with the norm (γ, α) XHFB = (1 − Δ)1/4 γ(1 − Δ)1/4
S1
+ (1 − Δ)1/4 α
S2
.
(3.5)
We remind the reader that S1 and S2 denote the space of trace-class and Hilbert–Schmidt operators on L2 (R3 ; C), respectively. Also, note that (1 − Δ)1/4 α S2 is equivalent to the Sobolev norm α(·, ·) H 1/2 (R3 ×R3 ) where α(x, y) is regarded as a two-body wavefunction defined on R3 × R3 . The existence of minimizers for the HFB energy EHFB (γ, α), under the constraint that Trγ = λ for λκ3/2 not too large, was given in [14]. Moreover, some important properties were derived in [1,14]. We note it is not known whether a ground state always has α = 0 for V taking the form (2.4), but this is expected to hold true (at least for sufficiently large local potentials w). In view of (3.1) and following [2,14], we introduce a generalized one-body density matrix γ α , (3.6) Γ= α∗ 1 − γ as well as the corresponding HFB mean-field operator Hγ Πα FΓ := Π∗α −Hγ
(3.7)
which acts on L2 (R3 ; C) ⊕ L2 (R3 ; C), with Hγ denoting the conjugate of Hγ . Here Hγ := K + κ(V ∗ ργ )(x) − κV (x − y)γ(x, y)
(3.8)
denotes the Hartree–Fock mean-field operator introduced before in (2.3); and Πα is given by its kernel Πα (x, y) = κV (x − y)α(x, y). Moreover, it turns out to be convenient to define the number operator as 1 0 N = . (3.9) 0 −1
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Note that Γ commutes with N if and only if α = 0, i.e., if and only if the corresponding quasi-free state in Fock space also commutes with the number operator N ; see, e.g., [2]. Remark 3. It was shown in [2,14] that any minimizer of the HFB energy (3.2) solves the following self-consistent equation: Γ = χ(−∞,0) (FΓ − μN ) + D,
(3.10)
for some Lagrange multiplier μ < 0 chosen to ensure the constraint Tr(γ) = λ, and where D is a finite rank operator of the same matrix form as Γ, satisfying ran(D) ⊂ ker(FΓ − μN ). It was also shown in [1,14] that if α = 0 and w ≡ 0, then γ must be infinite rank. 3.1.2. The HFB Evolution Equation. Let us now turn to the time-dependent HFB theory. This evolution equation can be concisely written in terms of the generalized density matrix Γ = Γ(γ, α) introduced in (3.6). The corresponding initial-value problem reads as follows: d i Γt = [FΓt , Γt ] (3.11) dt (γ|t=0 , α|t=0 ) = (γ0 , α0 ) ∈ KHFB . This evolution equation can also be regarded as obtained from “forcing” the many-body Schr¨ odinger evolution in Fock space to stay on the manifold of HFB states, by means of the Dirac–Frenkel principle in quantum mechanics. With regard to the well-posedness of its initial-value problem, we note that a straightforward adaptation of [4–6,10,11,13] yields the existence and uniqueness of a maximal solution. We record the well-posedness fact as follows. Theorem 3 (Well-posedness in Hartree–Fock–Bogoliubov Theory). Assume that w is a radial function satisfying (2.5) and that 0 κ < 4/π holds. For each initial datum (γ0 , α0 ) ∈ KHFB , there exists a unique solution (γt , αt ) ∈ ) solving (3.11) with maximal time of exisC 0 ([0, T ); KHFB ) ∩ C 1 ([0, T ); XHFB tence 0 < T ∞. Moreover, we have conservation of energy and expected number of particles, i.e., E(γt , αt ) = E(γ0 , α0 )
and
Tr(γt ) = Tr(γ0 )
for
0 t < T,
as well as the a priori bound for the pairing wave function: Tr(αt∗ αt ) C
for
0 t < T.
Finally, if T < ∞ holds, then we have Tr(−Δ)1/2 γt → ∞ as t → T − . Remarks 4. (i) It was shown in [14] that when Tr(γ0 )κ3/2 is sufficiently small, then one has T = +∞ and (γt , αt ) is bounded in XHFB . (ii) Note the condition κ < 4/π as opposed to HF theory; see Sect. 3.1.3 for more details on this. (iii) It is straightforward to verify that if the generalized matrix Γ0 associated with the initial condition (γ0 , α0 ) is an orthogonal projector on L2 (R3 ; C) ⊕ L2 (R3 ; C), then so is Γt for all times t ∈ [0, T ). Similarly, if α0 = 0 (i.e., the initial datum is a Hartree–Fock state) then we have
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αt ≡ 0 for all time. Therefore the equation (3.11) is a generalization of the HF evolution equation (2.2). (iv) The a priori control on Tr(αt∗ αt ) simply follows from the operator inequality γ 2 + α∗ α γ (since Γ2 Γ) and the conservation of Tr(γ). See Sect. 3.1.3 for more details on the time evolution of α. (v) As for stationary solutions of (2.1), we remark that it can be verified that if Γ0 satisfies [FΓ0 − μN , Γ0 ] = 0 for some μ, then Γ(t) = e−iμN t Γ0 eiμN t is solution of equation (3.11). In particular, we see that HFB minimizers give rise to stationary solutions of (3.11), as one naturally expects. Remark 5 (Spin symmetry). There is a very natural spin symmetry in HFB theory, which was already used before in minimization problems for the Hubbard model in [2] and for general systems with an interaction potential of negative type in [1]. In the case of the Newtonian interaction, this was used in [14]. Let us assume only in this remark that the problem is now posed in L2 (R3 ; C2 ) and let us introduce the unitary operators for k = 1, 2, 3 iσk 0 0 σk Σk = =i 0 −σk iσk 0 where σk are the Pauli matrices. A simple calculation shows that [Σk , Γ] = 0 is equivalent to [σk , γ] = 0 and σk α + ασk = 0. It can then be checked3 that [Σk , Γ] = 0 for k = 1, 2 if and only if γ↑↓ = γ↓↑ = 0, γ↑↑ = γ↓↓ , α↑↑ = α↓↓ = 0 and α↓↑ = −α↑↓ . Let us now assume that our initial state Γ0 takes the special form γ0 1 0 0 1 α0 Γ0 = = τ ⊗ and α = a ⊗ with γ 0 0 0 0 0 1 −1 0 α0∗ 1 − γ 0 (3.12) where a0 and τ0 act on L2 (R3 ; C) with τ0∗ = τ0 and aT0 = a0 . If w is a multiple of the 2 × 2 identity matrix, then both the kinetic energy K and our interaction potential do not depend on the spin. It is then clear that we have Σk FΓ Σk = FΣk ΓΣ∗k for k = 1, 2, 3. Hence we see that Σk Γt Σ∗k ∈ KHFB solves the same time-dependent equation as Γt . By uniqueness we deduce that Γt has the same form as Γ0 for all times. Therefore if one seeks for solutions of the above special form, one can reduce the spin-1/2 case to a no-spin model with the condition αT = −α replaced by a singlet-type condition aT = a. All our results also hold true under this constraint. A similar reduction can be done for a system with q 1 degrees of freedom. The special form (3.12) is sometimes called time-reversal symmetry as the time reversal symmetry operators are usually defined through Tk = eiσk π/2 K = Σk K, where K is the conjugation (anti-linear) operator. Indeed, one verifies that Tk Γt Tk−1 solves the backward time-dependent equation. 3
The condition Σk ΓΣ∗k = Γ for k = 1, 2, 3 is indeed equivalent to γ↑↓ = γ↓↑ = 0, γ↑↑ = γ↓↓ and α ≡ 0, as may be seen by a simple computation. We do not want to impose such a condition.
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3.1.3. HFB: The Difficulties. In the previous section we have studied the blowup of solutions in the Hartree–Fock case. As we will explain now, the HFB generalization brings in new substantial difficulties as follows. • The pairing term coming from αt is difficult to control. • Powers of the angular momentum Tr(|L|s γt ) are generally not conserved along the HFB flow, even for spherical symmetric solutions (γt , αt ) defined below. This is a striking contrast to HF theory. It is enlightening to rewrite the time-dependent equation (3.11) as a coupled system with a von Neumann-type equation for the one-body density matrix γt and a two-body Schr¨ odinger equation for the pairing wavefunction αt . In fact, a simple calculation yields the following system: ⎧ ⎪ ⎪ ⎪ ⎨
d γt = [Hγt , γt ] + iGαt , dt d i αt (x, y) = (Hγt )x + (Hγt )y + κV (|x − y|) ⎪ dt ⎪ ⎪ ⎩ − κ ((γt )x + (γt )y ) V (|x − y|) αt (x, y). i
(3.13)
The time-dependent equation for γt contains the trace-class coupling term Gα arising from the pairing wavefunction α. By an elementary calculation, this term is found to be Gα = i(Πα α∗ − αΠ∗α ) or, in terms of its integral kernel, Gα (x, y) = i κ
α(x, z)α(y, z) (V (y − z) − V (x − z)) dz.
R3
Note that Tr(γt ) is constant along the trajectories, because of Tr(Gα ) = 0. It is fair to say that the two-body wavefunction αt evolves through a time-dependent two-body Schr¨ odinger equation with Hamiltonian (Hγt )x + (Hγt )y + κV (|x − y|) − κ ((γt )x + (γt )y ) V (|x − y|).
(3.14)
Here, the first two terms in this Hamiltonian form a one-body operator arising from the mean-field operator Hγ , which itself contains the potentials induced by γ as external sources. The third term κV (|x − y|) is a usual two-body Newtonian interaction term (perturbed by our local interaction w). The last term κ ((γt )x + (γt )y ) V (|x − y|) in (3.14) is a non-self-adjoint (but compact) operator which is responsible for the fact that αt does not have a constant L2 norm along a given trajectory (except of course if αt ≡ 0). We note that since 0 κ < 4/π, the two-body operator (3.14) is actually equal to a well-defined self-adjoint operator, plus a compact non-self-adjoint operator. Therefore, the two-body equation for αt (x, y) is well defined. Let us now detail the behavior of moments of angular momentum along the HFB flow. We recall for the reader’s orientation that, in HF theory, higher moments of angular momentum are conserved, provided the initial state is spherically symmetric. This fact is one of the main tools used in blowup proof
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for HF, as we have already seen. First, we define spherically symmetric HFB states in a similar fashion as in the HF case. Definition 2 (Spherically symmetric HFB states). We say that (γ, α) ∈ KHFB is spherically symmetric when γ(Rx, Ry) = γ(x, y) and α(Rx, Ry) = α(x, y) for all x, y ∈ R3 and all R ∈ SO(3). When the HFB initial datum (γ0 , α0 ) is spherically symmetric, it can easily be seen that (γt , αt ) stays spherically symmetric on its time interval of existence. In terms of the angular momentum operator (provided that γ and α are smooth enough) the condition of spherical symmetry may also be written as [L, γt ] = 0 and
(Lx + Ly )αt (x, y) = 0.
(3.15)
Moreover, note also that we find that [|L|2 , γt ] = 0 holds. However, we do not obtain any other information about the two-body wavefunction αt . In particular, we do not have that (|Lx |2 + |Ly |2 )αt (x, y) = 0, which would mean that αt (Rx, y) = αt (x, Ry) = αt (x, y) for all rotations R ∈ SO(3). Let us elaborate more explicitly the fact that Tr(|L|2 γt ) is not conserved for spherically symmetric solutions in HFB theory. To this end, we calculate the variation of the expectation value of |L|2 , assuming that [L, γt ] = 0 and (Lx + Ly )αt = 0 for all times t ∈ [0, T ). We obtain d Tr(|L|2 γt ) dt κi 1 = . αt , |Lx |2 + |Ly |2 , − w(|x − y|) αt 2 |x − y| L2 (R3 ;C)⊗L2 (R3 ;C)
(3.16)
As |x − y|−1 and w(|x − y|) do not commute with |Lx |2 + |Ly |2 , the above term is usually not zero, and hence the square of the angular momentum is not conserved. This is a major difficulty that prevents us to use the same proof as in the HF case! To see that the square of the angular momentum does actually vary in time, let us consider a sufficiently smooth initial datum (γ0 , α0 ) with α0 = 0 such that γ0 (x, y) = γ0 (|x|, |y|) and α0 (x, y) = α0 (|x|, |y|). The latter property means exactly that (|Lx |2 + |Ly |2 )α0 (x, y) = 0. It is clear from the above formula that the derivative of Tr(|L|2 γt ) vanishes at time t = 0. We have the following identities:
|Lx |2 + |Ly |2 ((γ0 )x + (γ0 )y ) α0 (|x|, |y|)V (x − y) = ((γ0 )x + (γ0 )y ) |Lx |2 + |Ly |2 α0 (|x|, |y|)V (x − y) = 2 ((γ0 )x + (γ0 )y ) Lx · Ly α0 (|x|, |y|)V (x − y) = 2 ((γ0 L)x · Ly + Lx · (γ0 L)y ) α0 (|x|, |y|)V (x − y) =0
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since γ0 L = Lγ0 = 0 and (Lx + Ly )V (x − y) = 0. Using this, we can compute the second derivative of the expectation value of |L|2 . We obtain d2 2 2 α0 (|x|, |y|) 2 2 α0 (|x|, |y|) , |L Tr(|L| γ ) = κ | + |L | t x y |t=0 dt2 |x − y| |x − y| L2 ⊗L2 2 2 2 + κ β0 (x, y), |Lx | + |Ly | β0 (x, y) L2 ⊗L2 (3.17) where β0 (x, y) = α0 (|x|, |y|)w(|x − y|). It is easy to see that (3.17) is always a positive quantity, except when α0 = 0. To sum up: Even when we start at time t = 0 with a state in the zero angular momentum sector, it immediately acquires a change of the square of angular momentum, in stark contrast to HF theory [10, p. 743]. 3.2. Blowup with Angular Momentum Cutoff As we have explained, the second moment of the angular momentum, Tr(|L|2 γt ), is not conserved for spherical states in general. In fact, it seems quite difficult to get any useful bound on this quantity. There is yet another difficulty that we have not mentioned so far: the pairing term is difficult to control and our proof reveals that we actually need a bound on Tr(|L|6+ε γt ) for some ε > 0; see Lemma 7 below. It seems a very interesting question to investigate whether the angular momentum can indeed substantially alter the collapsing behavior of a gravitational system in HFB theory, or actually prevent collapse at all (which, however, is very unlikely in our opinion). For all these reasons, we will now introduce a simpler model for which we are able to prove blowup of spherical solutions with sufficiently negative energy. The idea is to enforce an angular momentum cutoff in the model as follows. Let Λ 0 be a fixed integer, and define the (infinite-dimensional) orthogonal projector4 PΛ := χ[0,Λ(Λ+1)] |L|2 , which projects L2 (R3 ; C) onto sectors with angular momentum Λ(Λ + 1). We will now impose that our density matrices γ and α satisfy for all times PΛ γ = γPΛ = γ,
(PΛ ⊗ PΛ ) α(x, y) = α(x, y).
(3.18)
Note that the assumption on α is equivalent to ((PΛ )x + (PΛ )y ) α(x, y) = 2α(x, y), or in operator terms to PΛ α = αPΛ = α. Our assumption (3.18) is the same as saying that we restrict ourselves to HFB states in Fock space F that actually belong to the restricted Fock space FΛ = ⊕N 0 ∧N 1 HΛ ⊂ F, where HΛ = PΛ L2 (R3 ; C). Hence, introducing the angular momentum cutoff is the same as replacing the one-body space L2 (R3 ; C) by HΛ . This procedure does not change the HFB energy EHFB . But it changes the HFB evolution 4
As usual χA (x) is the characteristic function of the set A.
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equation, because the constraint (3.18) needs to be conserved for all times. The time-dependent equation in the cutoff model then becomes d i Γt = [PΛ FΓt PΛ , Γt ] (3.19) dt (γ|t=0 , α|t=0 ) = (γ0 , α0 ) ∈ KHFB satisfying (3.18), where
PΛ :=
PΛ 0
0 . 1 − PΛ
The coupled system (3.13) can be rewritten in a similar fashion. Let us recall that the HFB dynamics introduced in the last section was obtained by restricting the (formal) unitary evolution of the Hamiltonian in Fock space to the submanifold of HFB states, by means of the Dirac–Frenkel principle. By introducing a cutoff in angular moment, we simply restrict the time-evolution to the even smaller manifold of states belonging to HΛ . We remark that, in numerical simulations, an angular momentum cutoff (as we introduced) will be in fact always imposed. The following main result of this paper now establishes blowup in HFB theory with an angular momentum cutoff. Theorem 4 (Blowup for HFB with angular momentum cutoff). Suppose w satisfies assumption (2.5). Let (γ0 , α0 ) ∈ KHFB be a radially symmetric initial datum satisfying (3.18) and such that Tr |x|4 γ0 + Tr (−Δ)γ0 < ∞. Our conclusion is the following: If (γ0 , α0 ) has sufficiently negative energy, that is κ 2 EHFB (γ0 , α0 ) < − sup |w(r) + rw (r)|− (Trγ0 ) + Trγ0 , (3.20) 2 r0 then the unique maximal local-in-time solution (γt , αt ) of (3.19) blows up in finite time, i.e., we have T < ∞ and Tr(−Δ)1/2 γt → ∞ as t → T − . The proof of Theorem 4 is provided below in Sect. 4.3. Remark 6 (Removing the angular momentum cutoff). It is not obvious how to get any information on the initial model from the study of the model with an angular momentum cutoff. However, it is easy to grasp from our proof in Sect. 4.3 that if Tr(|L|6+ε γt ) stays bounded (or does not grow too fast), then blowup must occur under the same condition (3.20). Remark 7 (Case of the Laplacian). We mention, as a side remark, that in a non-relativistic HFB model, where K is replaced by the Laplacian −Δ, and with the scaling critical two-body potential V (x − y) = −1/|x − y|2 , higher powers of the angular momentum are also a priori not conserved. However, in 2 the non-relativistic setting, one can use √ in the proof the observable M = |x| , 3 2 instead of the operator M = i=1 xi m − Δxi , as done below. Hence, for non-relativistic systems, no angular momentum conservation is needed, and
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one easily sees that blowup always occurs under a negative energy condition, even for non-spherically symmetric initial data. Nevertheless, it should be mentioned that the Laplacian √ −Δ is qualitatively very different from the pseudo-relativistic operator −Δ + m2 , which models a relativistic system with a ‘finite speed of propagation’.
4. Proofs 4.1. Preliminaries In the proofs of our main results, we shall need several auxiliary lemmas, which we gather in this section for convenience. 4.1.1. Spherical Harmonics Expansion. An important tool will be the expansion of the time-dependent states in terms of spherical harmonics. As we restrict ourselves to spherically symmetric states, our states will be uniformly distributed in each angular momentum sector and we will only encounter Legendre polynomials. We recall that the th Legendre polynomial P (t) is a real polynomial of degree defined on [−1; 1] which satisfies −1 P (t) 1 and P (1) = 1, for all integers 0. It is linked to the usual spherical harmonics Ym by the formula P (ω · ω ) =
1 Ym (ω)Ym (ω ), 2 + 1
(4.1)
m=−
where ω, ω ∈ S2 = {x ∈ R3 : |x| = 1}. We define for all , the following function: 1
F, (r, r ) = 2π
P (t)P (t) V
r2 + r2 − 2rr t dt
−1
1
1 P (t)P (t) − √ 2 2 r + r − 2rr t −1 +w r2 + r2 − 2rr t dt.
= 2π
(4.2)
We recall that V (x) = −1/|x| + w(|x|). With the usual abuse of notation, we will henceforth write V (|x|) as well as V (x), whichever is more convenient for the case at hand. The function F, will often appear when integrating out the angle variable in the interaction potential between the projections of our states in, respectively, the th and the th angular momentum sector. We will need the following
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Lemma 1 (Estimates on F, ). We have the following bounds: 4π 1 + ||rw(r)||L∞ [0,∞) |F, (r, r )| max(r, r )
(4.3)
and |r2 ∂r F, (r, r )| C 1 + 2 + ( )2 + Cε (1 + r2 )1+ε w (r)L∞ [0,∞) (4.4) for some universal constant C and some constant Cε depending only on ε. Proof. Clearly we have 1
|F, (r, r )| 2π (1 + ||rw(r)||L∞ )
√
−1
r2
dt + r2 − 2rr t
4π (1 + ||rw(r)||L∞ ) = max(r, r ) which is nothing else but Newton’s Theorem. For the second bound (4.4), we treat w and the Newton interaction separately. Recall that √
1 r2 + r2 − 2rr t
=
m0
min(r, r )m Pm (t). max(r, r )m+1
Differentiating explicitly we find 1 1 2 (t) P (t)P r ∂r √ (1 + m) P P Pm dt 2 2 r + r − 2rr t m0 −1 −1 C 1 + 2 + ( )2
1 where we have used that the integral −1 P P Pm vanishes when m < | − |
1 and when m > + , and that −1 P P Pm 2. For the term involving w, we write 1 2 r ∂r P (t)P (t)w r2 + r2 − 2rr t dt −1
1 r r2 + r2 − 2rr t dt w 2
−1
(1 + r2 )1+ε w (r)L∞ r2
1
−1
dt (1 +
r2
+
r2
1+ε .
− 2rr t)
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Then we explicitly compute 1 r
dt
2 −1
(1 +
r2
+
1+ε
− 2rr t)
1 1 − (1 + (r − r )2 )ε (1 + (r + r )2 )ε ε ε 2rr 2rr 2 2 ε 1 + (1 + r + (r ) ) − 1 − 2 2 2 2 1+r +(r ) 1+r +(r ) r = Cε 2ε r (1 + (r − r )2 )ε (1 + (r + r )2 )ε
r = 2ε r
r2
which ends the proof of Lemma 1. The following is a simple consequence of Lemma 1:
Corollary 1. Let ρ ∈ L1 (R3 ) be spherically symmetric function. Then we have
1 + ||rw(r)||L∞ (0,∞) R3 |ρ| |V ∗ ρ(x)| (4.5) |x| and
∇x |x|2 (V ∗ ρ) (x) C 1+||rw(r)||L∞ (0,∞) +Cε (1 + r2 )1+ε w (r)L∞ (0,∞) |ρ|. R3
(4.6) Proof. Integrating out the angle, we find ∞ V ∗ ρ(r) = (r )2 dr F00 (r, r )ρ(r ) 0
hence the result is a simple application of Lemma 1, with = = 0 and using that ∇|x|2 (V ∗ ρ) 2|x| |V ∗ ρ(x)| + |x|2 |∇V ∗ ρ(x)|. 4.1.2. Pseudo-Relativistic Kinetic Energy. We now define the positive operator K acting on the space L2r := L2 ([0, ∞), r2 dr) by 1 ∂ 2 ∂ ( + 1) r + m2 + , (4.7) r2 ∂r ∂r r2 √ i.e. K is nothing but the restriction of m2 − Δ to the th angular momentum sector. We note that, for all k = −, . . . , , K2 := −
(m2 − Δ)Yk (ωx )u(|x|) = Yk (ωx )(K2 u)(|x|), and hence
m2 − ΔYk (ωx )u(|x|) = Yk (ωx )(K u)(|x|).
We will need two important results involving the operator K . The first one is the
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Lemma 2 (Commutator estimate). Suppose m > 0. We have, for all f ∈ W 1,∞ (R3 ), (4.8) m2 − Δ, f (x) 2 3 C ∇f ∞ . B(L (R ))
In particular, we have for all radial function f ∈ W 1,∞ (R3 ) and all nonnegative integer , ||[K , f (r)]||B(L2 ([0,∞),r2 dr)) C ||∂r f (r)||L∞ (0,∞) .
(4.9)
A proof of (4.8) can be found in [21]; see in particular the Corollary on page 309. The statement of this corollary does not give an effective bound on the norm of the commutator. However, the corollary is based on Theorem 3, on page 294 of [21], whose proof provides the effective control we need (a remark in this sense can be found in the paragraph 3.3.5, on page 305 of [21]). On the other hand, (4.9) can be proven by applying (4.8) in the subspace of L2 (R3 ) consisting√of functions of the form Yk (ωx )u(|x|), which is stabilized by the operator [ m2 − Δ, f ]. The second important result that will be very useful is the following: Lemma 3 (Estimating K − K ). There exists a constant C such that one has, for all nonnegative integers , , ||(K − K ) r||B(L2 ([0,∞),r2 dr)) C(1 + + )| − |.
(4.10)
The proof of Lemma 3 cannot be directly deduced from the literature, and we provide its proof in Section A below. 4.2. Hartree–Fock Theory: Proof of Theorem 2 Step 1. Conservation of Angular Momentum. First of all, the proof of Theorem 2 makes use of the fact that (2.2) preserves both the spherical symmetry and the total angular momentum. We recall that L = −ix ∧ ∇. Lemma 4 (Conservation of spherical symmetry and angular momentum). Let γ0 ∈ KHF be a spherically symmetric HF state. Then the unique maximal solution γt to (2.1) with γ|t=0 = γ0 is spherically symmetric for all times t ∈ [0; T ). If furthermore Tr(|L|2 γ0 ) < ∞, then γt satisfies for all times [L, γt ] = 0 and the total angular momentum is conserved: Tr |L|2 γt = Tr |L|2 γ0
for all t ∈ [0, T ).
Note in particular Lemma 4 says that Tr|L|2 γt < ∞ for all t ∈ [0, T ). The proof of this lemma can be found in [12] (it is trivial to see that the introduction of the regular and spherically symmetric potential w and the fact that γ may have infinite rank do not affect the proof of this lemma). The conservation of the squared angular momentum as expressed by Lemma 4 will be a key tool in our proof, as it was already in [12]. For the rest of the proof, we introduce the non-negative self-adjoint operator M :=
3 i=1
xi
−Δ + m2 xi .
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As in [10,12], the strategy of the proof consists in showing that the second derivative of the expectation TrM γt is negative, if the energy EHF (γt ) = EHF (γ0 ) satisfies a certain condition. Step 2. Estimate on Direct and Exchange Terms. We will prove in this step that there exists a constant C (depending on w) such that: d Tr M γt Tr Aγt + CTr(γ0 ) Tr(1 + |L|2 )γ0 (4.11) dt with A = x · p + p · x. Here and henceforth, we use the notation p = −i∇x for the momentum operator and we use the notation C to denote any constant which only depends on w. The explicit dependence in w is very easy to derive, as it only occurs from the estimates of Lemma 1. Remark 8. One can check that Tr((1 + |x|4 + |p|2 )γt ) < +∞ for t ∈ [0, T ), provided that initially Tr((1 + |x|4 + |p|2 )γ0 ) < +∞ holds. This fact may be used to show that all terms below are well defined. Furthermore, we note that: Tr(M γt ) CTr(1 + |x|4 + |p|2 )γt ) < ∞ and Tr(|A|γt ) CTr((1 + |x|4 + |p|2 )γt ) < ∞ for all times t ∈ [0, T ). Remark 9. In order to have well-defined terms, we impose the higher regularity condition γ0 H2 < ∞; see also Remark 1 iii) above. Note that a breakdown of the solution in · Hs for some s 1/2 implies breakdown of the solution in energy norm · H1/2 ; this claim follows easily by adapting the arguments in [13]. We start by computing: d TrM γt = TrAγt − iTr [M, V ∗ ργt ] γt + iTr [M, Rγt ] γt , dt where we denote: Rγ (x, y) = V (|x − y|)γ(x, y) = −
(4.12)
γ(x, y) + γ(x, y)w(|x − y|). |x − y|
To prove (4.11), we have to bound the last two terms on the r.h.s. of the last equation, arising, respectively, from the direct and the exchange term in the Hartree–Fock equation (2.2). We will estimate TrAγt later in Step 3. The appropriate estimate on the direct term is given in the following: Lemma 5 (Estimating the direct term). We have for all spherically symmetric γ ∈ KHF |Tr [M, (V ∗ ργ )] γ| C (Tr(γ)) where C only depends on w.
2
(4.13)
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For the exchange term, we have to face the problem that γt can a priori be infinite rank, hence the method of [12] does not apply directly. However, an estimate similar to what was proved in [12] holds true, as expressed in the following: Lemma 6 (Estimating the exchange term). We have for all spherically symmetric γ ∈ KHF such that Tr|L|2 γ < ∞ |Tr [M, Rγ ] γ| C Tr(γ) Tr(1 + |L|2 )γ
(4.14)
where C only depends on w. Inserting the estimates (4.13) and (4.14) in the expression (4.12), one gets the bound (4.11). We now provide the: Proof of Lemma 5. Following the strategy of [10], we find: iTr [M, V ∗ ργ ] γ = i Tr |p|2 + m2 , |x|2 (V ∗ ργ ) γ p · x(V ∗ ργ )γ − 2 Re Tr |p|2 + m2
(4.15)
which implies that: |Tr [M, (V ∗ ργ )] γt | |p|2 + m2 , |x|2 (V ∗ ργ ) + 2 |x|(V ∗ ργ ) ∞ Tr(γ). Applying Lemma 2 to the first term in the parenthesis, we obtain: |Tr [M, (V ∗ ργ )] γ| ∇x |x|2 V ∗ ργ ∞ + |x|(V ∗ ργ ) ∞ Tr(γ). Using the spherical symmetry of ργ , (4.13) is then a consequence of Corollary 1. We write the: Proof of Lemma 6. First we write as before: iTr [M, Rγ ] γ = iTr |p|2 + m2 |x|2 Rγ γ
− iTr Rγ |x|
2
|p|2 + m2 γ +2ReTr Rγ x ·
!
p |p|2 + m2
γ .
(4.16) We claim that: |p|2 + 1|x|2 Rγ γ − Tr Rγ |x|2 |p|2 + m2 γ Tr CTr(γ) Tr(1 + |L|2 )γ, and that:
! p γ C(Trγ)2 . Tr Rγ x · 2 2 |p| + m
(4.17)
(4.18)
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To prove (4.17) and (4.18), we observe that, because of the spherical symmetry of γ, we can expand the kernel of γ as: γ(x, y) = g (|x|, |y|)(2 + 1)P (ωx · ωy ) (4.19) 0
where ωx = x/|x|, ωy = y/|y| and P is the th Legendre polynomial whose formula was recalled in (4.1). We have |px |2 + 1 g (|x|, |y|) P (ωx · ωy ) = P (ωx · ωy ) (K )r g (|x|, |y|) where K was defined in (4.7). The subscript x in p2x and the subscript r = |x| in (K )r indicate that these operators act on the x, respectively, r = |x| variable. It follows that: |p|2 + m2 |x|2 Rγ γ Tr = γ(y, x) |px |2 + m2 |x|2 Rγ (x, y) dydx R6
=
g (|y|, |x|)P (ωx · ωy )
(2 + 1)(2 + 1)
,
R6
× |px |2 + m2 1 |x|2 g (|x|, |y|)P (ωx · ωy )V (x − y) dy dx = (2+1)(2 +1) g (r, r ), (K )r r2 F, (r, r )g (r, r ) L2 ⊗L2 , r
,
(4.20)
r
where F, was defined before in (4.2). Similarly to (4.20) we obtain: (2+1)(2 + 1) g , F, r2 (K )r g L2 ⊗L2 . Tr Rγ |x|2 |p|2 + m2 γ = r
,
r
(4.21) Therefore we can rewrite the left hand side of (4.17) in the form: Tr |p|2 + m2 |x|2 Rγ γ − Tr Rγ |x|2 |p|2 + m2 γ (2 + 1)(2 + 1) g , (K )r r2 F, − r2 F, (K )r g L2 ⊗L2 = ,
=
(2 + 1)(2 + 1)
r
g , (K − K )r r2 F, g
,
L2r ⊗L2r
# " + g , (K )r , r2 F, g L2 ⊗L2 . r
r
(4.22)
r
Now using both Lemmas 1 and 3, we get: [K − K ]r2 F, C(1 + + )| − |. This allows us to control the first term on the right side of (4.22). To control the second term, we remark that, from Lemma 2, we have: [K , |x|2 F, ] ∂r r2 F, ∞ 2 C. L
(0,∞)
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Inserting these estimates in (4.22) and noting that: ||g ||L2r ⊗L2r = ||g ||S2 (L2r ) ||g ||S1 (L2r ) = TrL2r g yields:
|p|2 + 1|x|2 Rγ γ − Tr Rγ |x|2 |p|2 + m2 γ Tr C (2 + 1)(2 + 1) (1 + (1 + + )| − |) TrL2r g TrL2r g ,
C
,
(2 + 1)(2 + 1) 1 + 2 + ( )2 TrL2r g TrL2r g
C Trγ Tr(1 + |L|2 )γ and thus implies (4.17). It remains to prove (4.18). Since x · p = r∂r , with |x| = r, we obtain: ! p γ Tr Rγ x · 2 |p| + m2 1 (2 + 1)(2 + 1) g , F, r∂r g K L2r ⊗L2r , (2+1)(2 +1) ||g ||S2 (L2r ) ||g ||S2 (L2r ) ||rF, ||L∞ ∂r (K )−1 B(L2 ) r
,
(Trγ)2 as was claimed. Note we have used that (∂r )∗ ∂r = −r−2 ∂r r2 ∂r K2 for that−1 all 0. Hence ∂r (K ) 1 as an operator acting on L2r . This ends the proof of Lemma 6. Step 3. Estimating Tr(Aγt ). We recall that A = x · p + p · x. We will show in this step that: d Tr Aγt 2 EHF (γ0 ) + κ (Trγ0 )2 sup |w(r) + rw (r)|− . dt r0
(4.23)
In order to prove (4.23), we compute as before: d Tr Aγt = −i TrA[ −Δ + m2 + κ(V ∗ ργt ) − κRγt , γt ] dt = −iTr [A, −Δ + m2 + κ(V ∗ ργt ) − κRγt ]γt p2 = 2 Tr γt p 2 + m2 − κ dx dy (x−y) · ∇V (x−y) γt (x, x)γt (y, y)−|γt (x, y)|2 . With V (x) = −|x|−1 + w(x), we obtain: x · ∇V (x) =
1 + x · ∇w(x) = −V (x) + x · ∇w(x) + w(x). |x|
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Since moreover p2 / p2 + m2 p2 + m2 , we find: d Tr Aγt 2 EHF (γt ) − κ dxdy ((x − y) · ∇w(x − y) + w(x − y)) dt × γt (x, x)γt (y, y) − |γt (x, y)|2 2 EHF (γt ) + κ (Trγt )2 sup |w(r) + rw (r)|− , r0
where |f |− 0 denotes the negative part of f . Here we used the fact that γt (x, x)γt (y, y) |γt (x, y)|2 since γ 0. Equation (4.23) thus follows because EHF (γt ) = EHF (γ0 ) and Tr(γt ) = Tr(γ0 ). Step 4. Conclusion of the Proof of Theorem 2. It follows from Step 2 and Step 3, that: κ 0 Tr M γt t2 EHF (γ0 ) + (Trγ0 )2 sup |w(r) + rw (r)|− 2 r0 + t Tr(Aγ0 ) + C Tr(γ0 ) Tr(1 + |L|2 )γ0 + TrM γ0 (4.24) for all t < T (recall that T is the time of existence of the maximal local solution) and where C is a constant which only depends on w. From the assumptions on the initial datum γ0 , we find: Tr(M γ0 ) CTr(1 + |x|4 + |p|2 )γ0 < ∞ and |Tr(Aγ0 )| = |Tr(x · p + p · x)γ0 | Tr(|x|2 + |p|2 )γ0 < ∞. Hence it is clear that, if EHF (γ0 ) + (κ/2)(Trγ0 )2 supr0 |w(r) + rw (r)|− < 0, our estimate (4.24) contradicts the positivity of the operator M , for t large enough. Therefore T < ∞, and, from the blowup alternative (see Theorem 1), it follows that γt XHF → ∞, as t → T − . 4.3. Hartree–Fock–Bogoliubov Theory: Proof of Theorem 4 The proof follows the same lines as the one of HF√case and not all the details the 3 will be provided. We take as before M = i=1 xi m2 − Δ xi and note that M commutes with L, hence it also commutes with our cutoff projector PΛ . The same calculation as in (4.12) yields: d Tr(M γt ) = −i Tr ([M, PΛ Hγt PΛ ]γt ) dt 1 i −w(|x−y|) PΛ ⊗ PΛ αt . + κ αt , Mx +My , PΛ ⊗ PΛ 2 |x−y| L2 (R6 ) (4.25) However, since [M, PΛ ] = 0, and γt and αt are easily seen to satisfy (3.18) for all times, we may simply rewrite this as: d Tr(M γt ) = Tr Aγt − i Tr [M, V ∗ ργt ] γt + i Tr [M, Rγt ] γt dt i − κ αt , [Mx + My , V (x − y)] αt L2 (R6 ) . (4.26) 2
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With the help of Lemmas 5 and 6, we deduce: 2
|Tr [M, V ∗ ργt ] γt | C (Tr γt ) and
|Tr [M, Rγt ] γt | C Tr γt Tr(1 + |L|2 )γt . Now, we need an estimate on the pairing term, i.e. the last term of (4.26). This is the goal of the: Lemma 7 (Estimating the pairing term). Let (γ, α) ∈ KHFB be a radially symmetric HFB state such that Tr(|L|6+ε γ) < ∞ for some ε > 0. Then one has: α, [Mx + My , V (x − y)] αL2 (R6 ) !1/2 1/2 1/2 1 3 6+ε C Tr(1+|L| )γ Tr(1+|L| )γ . (4.27) 1 + 1+ε
where C only depends on w. Assuming that Lemma 7 holds true, we can conclude the proof of Theorem 4. Combining estimates, we arrive at the bound: d Tr(M γt ) dt Tr Aγt + C Tr(γt ) Tr(1 + |L|2 )γt + CTr(1 + |L|3 )γt 1/2 × Tr(1 + |L|6+ε )γt 9+ε 2 (Tr γt ) Tr Aγt + C 1 + Λ 2
1/2
where we have used that PΛ γ = γ. The next step is to note that [PΛ , A] = 0, hence we have like in the HF case: d Tr Aγt 2 EHFB (γt , αt ) dt −κ
dx dy ((x − y) · ∇w(x − y) + w(x − y)) (γt (x, x)γt (y, y) − |γt (x, y)|2 + |αt (x, y)|2 . (4.28) 2
We may use as before that γt (x, x)γt (y, y) − |γt (x, y)|2 0 and that ||αt ||L2 (R6 ) Tr(γt ) to infer: d 2 Tr Aγt 2 EHFB (γt , αt ) + κ sup |w(r) + rw (r)|− (Trγt ) + Trγt . dt r0 The end of the proof is then the same as in the HF case, using that Trγt = Trγ0 for all times.
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It now rests to give the: Proof of Lemma 7. We write as usual g (|x|, |y|)(2 + 1)P (ωx · ωy ) γ(x, y) =
(4.29)
0
where g (r, r ) is the kernel of a self-adjoint operator 0 g 1 acting on radial functions in L2 (R3 ; C), and α(x, y) = a (|x|, |y|)(2 + 1)P (ωx · ωy ). (4.30) 0
This time the operator a is antisymmetric, a (r, r ) = −a (r , r), i.e. its kernel a (|x|, |y|) can be interpreted as a spherically symmetric, fermionic two-body wavefunction. We also recall [2] that the condition 0 Γ 1 gives αα∗ γ(1 − γ). The same inequality must hold on each angular momentum sector, hence we obtain the following inequality for operators acting on L2r : a a∗ g (1 − g ). Now, arguing as for the exchange term, we calculate: i α, [Mx + My , V (x − y)] αL2 (R6 ) $ % = 2i α, |px |2 + 1, |x|2 V (x − y) α 2 6 L (R ) ' & px · xV (x − y)α − 4 α, . |px |2 + 1 2 6 L (R ) For the second term on the right hand side of (4.31) we obtain: ∂r α, rV (x − y)α (K )r L2 (R6 ) ∂r = (2 + 1)(2 + 1) a , rF, a , (K )r L2 (R6 )
(4.31)
(4.32)
,
where F, was defined before in (4.2). This yields to: (2 + 1)(2 + 1) ||a ||S2 (L2 ) ||a ||S2 (L2 ) ||rF, ||L∞ (0,∞)2 . |(4.32)| r
,
r
By our assumptions, we have a a∗ g (1 − g ), hence, ||a ||S2 (L2 ) (TrL2r g )1/2 . r
Together with Lemma 1, we can further bound (4.32) using the Cauchy– Schwarz inequality as !2 1/2 |(4.32)| (2 + 1)(TrL2r ⊗C2 g ) CTr 1 + |L|4 γ.
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Note the previous bound can be improved to CTr 1 + |L|2+ε γ for ε > 0. Similarly as in the proof of Lemma 6, the first term in (4.31) can be expressed as: (2 + 1)(2 + 1) a , (K )r r2 F , − r2 F , (K )r a L2 ⊗L2 r
,
=
,
+
(2 + 1)(2 + 1) a , (K − K )r r2 F , a
,
r
L2r ⊗L2r
(2 + 1)(2 + 1) a , [(K )r , r2 F , ]a L2 ⊗L2 . r
(4.33)
r
Using Lemmas 1, 2 and 3 like for the exchange term, we obtain (2 + 1)(2 + 1) a , K |x|2 F , − |x|2 F , K a 2 2 Lr ⊗Lr , C (2 + 1)(2 + 1)(1 + ( + )| − |) ||a ||S2 ||a ||S2 ,
C
! 2
1/2
(2 + 1)(1 + )TrL2r (g )
1/2 1/2 C Tr(1 + |L|3 )γ Tr(1 + |L|6+ε )γ
!
1/2
(2 + 1)TrL2r (g )
1 1 + 1+ε
!1/2 .
This ends the proof of Lemma 7.
Acknowledgements Part of this work was done during the authors’ visit at the Erwin Schr¨ odinger Institute for Mathematical Physics in Vienna, Austria, and the hospitality and support during this visit is gratefully acknowledged. C. H. is partially supported by NSF grant DMS-0800906. E. L. acknowledges support from NSF grant DMS-070249 and a Steno fellowship from the Danish research council. M. L. acknowledges support from the ANR project ACCQuaRel of the French ministry of research.
Appendix A. Proof of Lemma 3 We have, using the well-known [15] integral formula for assuming that u, v are smooth enough,
√
−Δ + m2 − m and
(K − m) u, vL2 ([0,∞),r2 dr) $ % = ( m2 − Δ − m)Yk (ωx )u(|x|), Yk (ωx )v(|x|)
L2 (R3 )
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K2 (|x − y|) k 1 k (ω )u(|y|) = Y (ω )u(|x|) − Y x y 4π 2 |x − y|2 R6 × Yk (ωx )v(|x|) − Yk (ωy )v(|y|) dx dy where K2 is a Bessel function. Averaging over k and using that 1 Yk (ω)Yk (ω ) = P (ω · ω ) 2 + 1 k=−
where P is the th Legendre polynomial, we get (K − K ) r u, vL2 ([0,∞),r2 dr) K2 (|x − y|) 1 =− 2 |x|u(|x|)v(|y|) (P (ωx · ωy )−P (ωx · ωy )) dx dy. 2π |x−y|2 R6
(A.1) Note that we have used that for all , P (ωx · ωx ) = P (1) = 1. Using that |x|2 K2 (x) is bounded, this yields (K − K ) r u, vL2 ([0,∞),r2 dr) 1 |x| |u(|x|)| |v(|y|)| |P (ωx · ωy )−P (ωx · ωy )| 2 dx dy (A.2) 2π |x−y|4 R6
Passing first to spherical and then to polar coordinates gives (K − K ) r u, vL2 ([0,∞),r2 dr) 1 4
∞ dα 0
1
s2 ds
r dr
−1
=4
∞ 2
∞
r|u(r)| |v(s)| |P (α) − P (α)|
0
(r2 + s2 − 2rsα)
2
π/2 |u(t cos θ)| |v(t sin θ)| |P (α) − P (α)| t dt dθ cos3 θ sin2 θ . 2 (1 − 2 sin θ cos θα) 2
dα
−1
0
0
Using the Cauchy–Schwarz inequality for the t integration we get (K − K ) r u, vL2 ([0,∞),r2 dr) 1 4 −1
π/2 cos3/2 θ sin1/2 θ |P (α) − P (α)| dα dθ ||u||L2r ||v||L2r . 2 (1 − 2 sin θ cos θα) 0
Let us recall that P (1) = 1 for all and introduce C, :=
|P (α) − P (α)| 1−α α∈[−1,1] sup
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such that 1 ||(K − K ) r||B(L2 ([0,∞),r2 dr)) 4C, −1
which is finite as seen by computing 1 dα −1
1−α (1 − uα)
(1 + u) log 2
=
π/2 cos3/2 θ sin1/2 θ(1 − α) dα dθ 2 (1 − 2 sin θ cos θα) 0
1+u 1−u
− 2u
u2 (1 + u)
2 − log(1 − u).
Hence it remains to show that C, C(1 + + )| − |.
(A.3)
The Legendre polynomials satisfy the relation (1 − x2 )Pn (x) = −nxPn (x) + nPn−1 (x) which may be written Pn (x) − Pn−1 (x) P (x) = Pn (x) − (1 + x) n . 1−x n Assuming for convenience > and summing over n = + 1, . . . , , we obtain P (x) − P (x) Pn (x) = Pn (x) − (1 + x) . 1−x n n=+1
We have the formula Pn (x) which shows that (1 +
n(n + 1) = 1 − x2
x)|Pn (x)|
1 Pn (t) dt x
n(n + 1). Hence we obtain the bound
C,
(2 + n),
n=+1
which completes the proof of Lemma 2.
References [1] Bach, V., Fr¨ ohlich, J., Jonsson, L.: Bogolubov–Hartree–Fock mean field theory for neutron stars and other systems with attractive interactions. J. Math. Phys. 50, 102102, 22 (2009) [2] Bach, V., Lieb, E.H., Solovej, J.P.: Generalized Hartree–Fock theory and the Hubbard model. J. Statist. Phys. 76, 3–89 (1994) [3] Bender, M., Heenen, P.-H., Reinhard, P.-G.: Self-consistent mean-field models for nuclear structure. Rev. Mod. Phys. 75, 121–180 (2003) [4] Bove, A., Da Prato, G., Fano, G.: On the Hartree–Fock time-dependent problem. Commun. Math. Phys. 49, 25–33 (1976)
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[5] Chadam, J.M.: The time-dependent Hartree–Fock equations with Coulomb twobody interaction. Commun. Math. Phys. 46, 99–104 (1976) [6] Chadam, J.M., Glassey, R.T.: Global existence of solutions to the Cauchy problem for time-dependent Hartree equations. J. Math. Phys. 16, 1122–1130 (1975) [7] Chandrasekhar, S.: The maximum mass of ideal white dwarfs. Astrophys. J. 74, 81–82 (1931) [8] Dean, D.J., Hjorth-Jensen, M.: Pairing in nuclear systems: from neutron stars to finite nuclei. Rev. Mod. Phys. 75, 607–656 (2003) [9] Fr¨ ohlich, J., Lenzmann, E.: Blowup for nonlinear wave equations describing boson stars. Commun. Pure Appl. Math. 60, 1691–1705 (2007) [10] Fr¨ ohlich, J., Lenzmann, E.: Dynamical collapse of white dwarfs in Hartree- and Hartree–Fock theory. Commun. Math. Phys. 274, 737–750 (2007) [11] Hainzl, C., Lewin, M., Sparber, C.: Existence of global-in-time solutions to a generalized Dirac-Fock type evolution equation. Lett. Math. Phys. 72, 99–113 (2005) [12] Hainzl, C., Schlein, B.: Stellar collapse in the time dependent Hartree–Fock approximation. Commun. Math. Phys. 287, 705–717 (2009) [13] Lenzmann, E.: Well-posedness for semi-relativistic Hartree equations of critical type. Math. Phys. Anal. Geom. 10, 43–64 (2007) [14] Lenzmann, E., Lewin, M.: Minimizers for the Hartree–Fock–Bogoliubov theory of neutron stars and white dwarfs. Duke Math. J. 152, 257–315 (2010) [15] Lieb, E.H., Loss, M.: Analysis, Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence (2001) [16] Lieb, E.H., Thirring, W.E.: Gravitational collapse in quantum mechanics with relativistic kinetic energy. Ann. Phys. 155, 494–512 (1984) [17] Lieb, E.H., Yau, H.-T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112, 147–174 (1987) [18] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978) [19] Ring, P., Schuck, P.: The Nuclear Many-body Problem. Texts and Monographs in Physics. Springer, New York (1980) [20] Simon, B.: Trace ideals and their applications. London Mathematical Society Lecture Note Series, vol. 35. Cambridge University Press, Cambridge (1979) [21] Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton (1993) (With the assistance of Timothy S. Murphy) Christian Hainzl Department of Mathematics UAB Birmingham, AL 35294-1170, USA e-mail:
[email protected]
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Enno Lenzmann Department of Mathematical Sciences University of Copenhagen Universitetspark 5 2100 Copenhagen Ø, Denmark e-mail:
[email protected] Mathieu Lewin CNRS & Laboratoire de Math´ematiques (CNRS UMR 8088) Universit´e de Cergy-Pontoise 95300 Cergy-Pontoise, France e-mail:
[email protected] Benjamin Schlein Institute for Applied Mathematics University of Bonn Endenicher Allee 60 53115 Bonn, Germany e-mail:
[email protected] Communicated by Rafael D. Benguria. Received: October 6, 2009. Accepted: May 3, 2010.
Ann. Henri Poincar´e
Ann. Henri Poincar´e 11 (2010), 1053–1083 c 2010 Springer Basel AG 1424-0637/10/061053-31 published online August 21, 2010 DOI 10.1007/s00023-010-0051-6
Annales Henri Poincar´ e
Classical and Quantum Behavior of the Integrated Density of States for a Randomly Perturbed Lattice Ryoki Fukushima and Naomasa Ueki Abstract. The asymptotic behavior of the integrated density of states for a randomly perturbed lattice at the infimum of the spectrum is investigated. The leading term is determined when the decay of the single site potential is slow. The leading term depends only on the classical effect from the scalar potential. To the contrary, the quantum effect appears when the decay of the single site potential is fast. The corresponding leading term is estimated and the leading order is determined. In the multidimensional cases, the leading order varies in different ways from the known results in the Poisson case. The same problem is considered for the negative potential. These estimates are applied to investigate the long time asymptotics of Wiener integrals associated with the random potentials.
1. Introduction In this paper, we are concerned with the self-adjoint operator in the form of u( · − q − ξq ) (1.1) Hξ = −hΔ + 2
q∈Zd
defined on the L -space on R \ q∈Zd (q + ξq + K) with the Dirichlet boundary condition, where h is a positive constant and K is a compact set in Rd allowed to be empty. Our assumptions on the potential term are the following: (i) ξ = (ξq )q∈Zd is a collection of independent and identically distributed Rd -valued random variables with d
Pθ (ξq ∈ dx) = exp(−|x|θ )dx/Z(d, θ) R. Fukushima was partially supported by JSPS Fellowships for Young Scientists. N. Ueki was partially supported by KAKENHI (21540175).
(1.2)
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for some θ > 0 and the normalizing constant Z(d, θ); (ii) u is a nonnegative function belonging to the Kato class Kd (cf. [3, p-53] ) and satisfying u(x) = C0 |x|−α (1 + o(1))
(1.3)
as |x| → ∞ for some α > d and C0 > 0. Although we assume the equality in (1.2), it will be easily seen from the proofs that only the asymptotic relation Pθ (ξq ∈ x + [0, 1]d ) exp(−|x|θ ) is essential for our theory, where f (x) g(x) means 0 < lim|x|→∞ f (x)/ g(x) ≤ lim|x|→∞ f (x)/g(x) < ∞. In particular, we may replace |x|θ by (1+|x|)θ in (1.2). Then the point process {q + ξq }q∈Zd converges weakly to the complete lattice Zd as θ → ∞. Moreover, it is shown in Appendix A of [6] that this point process converges weakly to the Poisson point process with the intensity 1 as θ ↓ 0. Since the Poisson point process is usually regarded as a completely disordered configuration, our model gives an interpolation between complete lattice and completely disordered media. We will consider the integrated density of states N (λ) (λ ∈ R) of Hξ defined by the thermodynamic limit 1 Nξ,ΛR (λ) −→ N (λ) |ΛR |
as R → ∞.
(1.4)
In (1.4) we denote by ΛR a box (−R/2, R/2)d and by Nξ,ΛR (λ) the numD defined by ber of eigenvalues not exceeding λ of the self-adjoint operator Hξ,R restricting Hξ to ΛR \ q∈Zd (q +ξq +K) with the Dirichlet boundary condition. We here note that the potential term in (1.1) belongs to the local Kato class Kd,loc (cf. [3], p-53) as we will show in Sect. 7 below. It is then well known that the above limit exists for almost every ξ and defines a deterministic increasing function N (λ) (cf. [3,11]). The following are first two main results in this paper. Theorem 1.1. If d < α ≤ d + 2 and ess inf |x|≤R u(x) is positive for any R ≥ 1,
(1.5)
log N (λ) −λ−κ ,
(1.6)
then we have
where κ = (d + θ)/(α − d), and f (λ) g(λ) means 0 < limλ↓0 f (λ)/g(λ) ≤ limλ↓0 f (λ)/g(λ) < ∞. Moreover if α < d + 2, then we have ⎫ ⎧ ⎬κ+1 ⎨ κ C −κ 0 lim λκ log N (λ) = dq inf + |y|θ , (1.7) ⎭ λ↓0 (κ + 1)κ+1 ⎩ |q + y|α y∈Rd Rd
where the right hand side is finite by the assumption α > d.
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Theorem 1.2. If d = 1 and α > 3, then we have lim λ(1+θ)/2 log N (λ) = − λ↓0
π 1+θ h(1+θ)/2 . (1 + θ)2θ
(1.8)
If d = 2 and α > 4, then we have log N (λ) −λ
−1−θ/2
1 log λ
−θ/2 .
(1.9)
If d ≥ 3 and α > d + 2, then we have log N (λ) −λ−(d+μθ)/2 ,
(1.10)
where μ = 2(α − 2)/(d(α − d)). These results are generalizations of Corollary 3.1 in [6] to the case that supp(u) is not compact (cf. Theorem 3.11 below). The results in Theorem 1.1 are independent of the constant h. In fact these asymptotics coincide with those of the corresponding classical integrated density of states defined by √ Nc (λ) = Eθ [|{(x, p) ∈ ΛR × Rd : Hξ,c (x, p) ≤ λ}|](2π hR)−d for any R ∈ N, where | · | is the 2d-dimensional Lebesgue measure and Hξ,c (x, p) =
d
p2j + Vξ (x)
j=1
is the classical Hamiltonian (cf. [16]). Therefore we may say that only the classical effect from the scalar potential determines the leading term for α < d + 2 and the leading order for α ≤ d+2. To the contrary, the right hand side of (1.8) depends on h and the right hand sides of (1.9) and (1.10) are strictly less than that of (1.6). Therefore we may say that the quantum effect appears in Theorem 1.2. We here note that the right hand side of (1.6) gives an upper bound and the asymptotics of the classical counterpart not only for α ≤ d + 2 but also for α > d + 2 (see Proposition 2.1 below). For the critical case α = d + 2, the quantum effect appears at least in some cases. We shall elaborate on this in Sect. 4 below. In our model, the single site potentials are randomly displaced from the lattice. As is mentioned in [6], such a model describes the Frenkel disorder in solid state physics and is called the random displacement model in the theory of random Schr¨ odinger operator. Despite of the appropriateness of this model in physics, there are only a few mathematical studies and in particular the displacements have been assumed to be bounded in almost all works. For that case, Kirsch and Martinelli [12] discussed the existence of band gaps and Klopp [14] proved spectral localization in a semi-classical limit. More recently, Baker et al. [1,2] studied which configuration minimizes the spectrum of (1.1) and also showed that the corresponding integrated density of states increases rapidly at the minimum in a one-dimensional example. On the other hand, our displacements are unbounded. Then the infimum of the spectrum is easily shown to be 0 opposed to the bounded cases. This is an essential condition for
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our method, by which we investigate the behavior of N (λ) at λ = 0. All our results show that N (λ) increases slowly. In a slightly broader class of models where the potentials are randomly located, the most studied model is the Poisson model, where the random points (q + ξq )q∈Zd are replaced by the sample points of the Poisson random measure (cf. [3,20]). In the limit of θ ↓ 0, the above results coincide with the corresponding results for the Poisson model obtained by Pastur [21], Lifshitz ˆ [17], Donsker and Varadhan [4], Nakao [18], and Okura [19]. As in the Poisson model, the critical value is always α = d + 2 and, in the one-dimensional case, the leading order increases continuously as α increases to d + 2 and does not depend on α ≥ d + 2. However in contrast to the Poisson case, the leading order jumps at α = d + 2 for d = 2, and it depends on α ≥ d + 2 for d ≥ 3. These phenomena are due to the fact that the effect from states which have many tiny holes including {q + ξq }q in their supports appears in the leading term of the asymptotics, as observed in [6]. This is a characteristic difference with the Poisson case. On the other hand, the decay rates of N (λ) explode in the limit θ → ∞. This reflects the fact that the infimum of the spectrum is positive in the case of a finitely perturbed lattice including the case of the unperturbed lattice. On the subjects of this paper, we have more results for the alloy type model Hω = −hΔ +
ωq u(x − q)
q∈Zd
and the same critical value α = d + 2 is obtained, where ω = (ωq )q∈Zd is a collection of independent and identically distributed nonnegative real valued random variables. As for the results, further developments and the relation with other models, refer to a recent survey by Kirsch and Metzger [13]. Our proof of Theorem 1.1 is an extension of that of the corresponding result for the Poisson case (cf. [20,21]). For the proof of the multidimensional results in Theorem 1.2, we use a method based on a functional analytic approach (cf. [3,11]). This is different from the method in [6], where a coarse graining method following Sznitman [24] is applied. The method employed here can also be used to give a simpler proof of the results in the compact case in [6]. We will present it in Sect. 3 below. For the 1-dimensional result, we use a simple effective estimate of the first eigenvalue in [24]. As an application, we study the survival probability of the Brownian motion in a random environment. This was the main motivation in [6]. We recall the connection between this and the integrated density of states, and extend the theory to the present settings. For the results, see Theorem 6.3 below. In the proof, we take the hard obstacles K appropriately so that the local singularity of the potential u does not bring difficulty. This is our only motivation to introduce the hard obstacles, and the hard obstacles do not affect the results.
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We also consider the operator Hξ− = −hΔ −
u( · − q − ξq )
1057
(1.11)
q∈Zd
obtained by replacing the potential u in Hξ by −u. For this operator, we assume K = ∅ since we are interested only in the effect of the negative potential. The spectrum of this operator extends to −∞. For the asymptotic distribution, we show the following: Theorem 1.3. Suppose K = ∅, sup u = u(0) < ∞ and u(x) is lower semi-continuous at x = 0. Then the integrated density of states N − (λ) of Hξ− satisfies log N − (λ) −C1 = , 1+θ/d λ↓−∞ (−λ) u(0)1+θ/d lim
(1.12)
where C1 = d1+θ/d /{(d + θ)|S d−1 |θ/d } and |S d−1 | is the volume of the (d − 1)dimensional surface S d−1 . For the Poisson model, Pastur [21] showed that the corresponding inte− grated density of states NPoi (λ) satisfies − log NPoi (λ) −1 = . λ↓−∞ (−λ) log(−λ) u(0)
lim
The power of λ in (1.12) tends to that of the Poisson model as θ ↓ 0. However, the logarithmic term is not recovered. Therefore, we cannot interchange the limits λ ↓ −∞ and θ ↓ 0 in this case. Both for the Poisson and our cases, only the classical effect from the scalar potential determines the leading terms. The lower semi-continuity of u at 0 is a sufficient condition for the classical behavior: by this condition, the tunneling effect is suppressed. For this subject, refer to Klopp and Pastur [15]. Let us briefly explain the organization of this paper. We prove Theorems 1.1, 1.2, and 1.3 in Sects. 2, 3, and 5, respectively. In Sect. 3 we also give a simple proof of the corresponding results for the case that supp(u) is compact. In Sect. 4, we discuss the critical case α = d + 2. In Sect. 6 we study the asymptotic behaviors of certain Wiener integrals.
2. Proof of Theorem 1.1 2.1. Upper Estimate To derive the asymptotics of the integrated density of states, one of the standard ways is to estimate its Laplace transform and use the Tauberian theorem (cf. [5,18]). We here say the Tauberian theorem by the theorem deducing the
(t) be the asymptotics from that of the Laplace–Stieltjes transform. Let N Laplace–Stieltjes transform of the integrated density of states N (λ): ∞
N (t) = e−tλ dN (λ). 0
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Then, in view of the exponential Tauberian theorem due to Kasahara [10], the proof of the upper bound is reduced to the following: Proposition 2.1. If K = ∅ and (1.5) is satisfied, then we have
(t) C0 log N θ lim ≤− dq inf + |y| t↑∞ t(d+θ)/(α+θ) |q + y|α y∈Rd
(2.1)
Rd
for any α > d. Proof. We use the bound
(t) ≤ N
1 (t)(4πth)−d/2 , N where
1 (t) = N
⎡
⎛
dxEθ ⎣exp ⎝−t
Λ1
(2.2) ⎞⎤
u( x − q − ξq )⎠⎦ .
q∈Zd
This is a simple modification of the bound in Theorem (9.6) in [20] for Zd -stationary random fields. By replacing the summation by integration, we have
log N1 (t) ≤ dq log Eθ exp −t inf u(x − q − ξ0 ) . x∈Λ2
Rd
We pick an arbitrary L > 0 and restrict the integration to |q| ≤ Ltη . The assumption (1.3) tells us that for any ε1 > 0, there exists R1 such that u(x) ≥ C0 (1 − ε1 )|x|−α whenever |x|∞ ≥ R1 , where |x|∞ = max1≤i≤d |xi |. Thus the right hand side is dominated by ⎧ ⎪ ⎨ dy C0 (1 − ε1 ) θ exp −t inf dq log − |y| x∈Λ2 |x − q − y|α ⎪ Z(d, θ) ⎩ |q|≤Ltη |q+y|∞ ≥R1 +1 ⎫ ⎪ ⎬ . + exp −t inf u Λ2R1 +4 ⎪ ⎭ Thanks to the assumption (1.5), the second term makes only negligible contribution to the asymptotics. By changing the variables (q, y) to (t−η q, t−η y) with η = 1/(α + θ), we see that this equals dη
t dq log N2 (t, q) + exp −t inf u , |q|≤L
where
2 (t, q) N
= tdη |q+y|∞ ≥(R1 +1)t−η
Λ2R1 +4
dy C0 (1−ε1 ) θη θ exp −tθη inf −t |y| . x∈Λ2t−η |x−q−y|α Z(d, θ)
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We take L as an arbitrary constant independent of t. Then, taking ε2 , ε3 > 0
3 (q))ε−d/θ for large
2 (t, q) by exp(−tθη N sufficiently small, we can dominate N 2 enough t, where
3 (q) = inf C0 (1 − ε1 ) + (1 − ε2 )|y|θ : x ∈ Λε , y ∈ Rd . N 3 |x − q − y|α Therefore we obtain
(t) log N
3 (q) dq. N lim (d+θ)η ≤ − t↑∞ t |q|≤L
Since ε1 , ε2 , ε3 and L are arbitrary, this completes the proof. 2.2. Lower Estimate To prove the lower estimate, we have only to show the following: Proposition 2.2. If α < d + 2, then we have
(t) C0 log N θ + |y| . lim (d+θ)/(α+θ) ≥ − dq inf |q + y|α y∈Rd t↑∞ t
(2.3)
Rd
Moreover, this bound remains valid for α = d + 2 with a smaller constant in the right hand side. The case α = d + 2 will be discussed in more detail in Sect. 4 below. Proof of Proposition 2.2. We use the bound
(t) ≥ R−d exp −th ∇ψR 22 N
1 (t) N
(2.4)
C0∞ (ΛR )
such that ψR 2 = 1, where which holds for any R ∈ N and ψR ∈ · 2 is the L2 -norm, and ⎡ ⎛ ⎞
1 (t) = Eθ ⎣exp ⎝−t N dxψR (x)2 u( x − q − ξq )⎠ q∈Zd
:
⎤
(q + ξq + K) ∩ ΛR = ∅⎦ .
q∈Zd
This can be proven by the same method as for the corresponding bound in Theorem (9.6) in [20] for Rd -stationary random fields. By replacing the summation by integration, we have
2 (t, q) dq,
1 (t) ≥ N log N Rd
where
2 (t, q) = log Eθ exp −t N dxψR (x)2 sup u(x − q − z − ξ0 ) z∈Λ1 : (q + ξ0 + K) ∩ ΛR = ∅ .
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For any ε1 > 0, there exists R1 such that K ⊂ B(R1 ) and u(x) ≤ C0 (1 + ε1 )|x|−α for any |x| ≥ R1 by the assumption (1.3). To use this bound in the above right hand side, we need inf{|x − q − z − ξ0 | : x ∈ ΛR , z ∈ Λ1 } ≥ R1 . However we shall √ deal with a simpler sufficient condition |ξ0 | ≤ |q|/2 and |q| ≥ 2(R1 +√ dR) instead. Now fix β > 0 and take t large enough so that tβ > 2(R1 + dR). Then we obtain tC0 (1 + ε1 )2α
√ N2 (t, q) dq ≥ dq − + log Pθ (|ξ0 | ≤ |q|/2) . (|q| − 2 dR)α |q|≥tβ
|q|≥tβ
(2.5) By a simple estimate using log(1 − X) ≥ −2X for 0 ≤ X ≤ 1/2, we can bound the right hand side from below by −c1 t1−β(α−d) − c2 exp(−c3 tβθ ). The other part is estimated as dy
2 (t, q) dq ≥ N d log Z(d, θ) √ |q|≤tβ
|q|≤tβ
|q+y|≥R1 + dR
tC0 (1 + ε1 ) θ − |y| . × exp − inf{|x − q − z − y|α : x ∈ ΛR , z ∈ Λ1 }
(2.6)
By changing the variables, we find that the right hand side equals dytdη dη
3 (y, q)), exp(−tθη N dq log t Z(d, θ) √ |q|≤tβ−η
|q+y|≥(R1 + dR)t−η
where η = 1/(α + θ) and
3 (y, q) = N
C0 (1 + ε1 ) + |y|θ . inf{|x − q − z − y|α : x ∈ ΛRt−η , z ∈ Λt−η }
(2.7)
Let us take γ > 0 and restrict the integration with respect to y to the ball B(y0 , t−γ ) with center y0 and radius t−γ . Then we can bound the integrand with respect to q from below by log where
4 (q, t) = inf N
|B(0, 1)|td(η−γ)
4 (q, t), − tθη N Z(d, θ)
(2.8)
sup y∈B(y0 ,t−γ )
3 (y, q) N
: y0 ∈ Rd , d(B(y0 , t−γ ), −q) ≥ (R1 +
√
dR)t−η
.
(2.9)
We now specify R as the integer part of ε2 tη , where ε2 is an arbitrarily fixed positive number. We take ψR as the nonnegative and normalized ground state
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of the Dirichlet Laplacian on the cube ΛR and take β between η and η(1+θ/d). Then, for α < d + 2, we obtain
(t) log N
4 (q, t), dq N (2.10) lim (d+θ)η ≥ − lim t↑∞ t↑∞ t |q|≤tβ−η
since th ∇ψR 2 tR−2 and (2.5) is negligible compared with t(d+θ)η . When |q| ≤ tβ−η , we can dominate 1/t by a power of q. Thus, for large |q|, by taking
4 (q, t) by |q|−α + |q|−γθ/(β−η) . This is integrable y0 as 0, we can dominate N if we take γ large enough so that γθ/(β − η) > d. Thus, by the Lebesgue convergence theorem, we have
4 (q, t) dq N lim t↑∞ |q|≤tβ−η
=
dq inf Rd
√ C0 (1 + ε1 ) + |y|θ : y ∈ Rd , d(y, q) ≥ ε2 d . inf x∈Λε2 |x − q − y|α
Since ε1 and ε2 are arbitrary, this completes the proof of the former part of Proposition 2.2. For the case α = d + 2, we take ε2 = 1. Then we have th ∇ψR 2 t(d+θ)η and the latter part of Proposition 2.2 follows from the same argument as above.
3. Proof of Theorem 1.2 and the Compact Case In this section, we use some additional notations to simplify the presentation. For any self-adjoint operator A, let λ1 (A) be the infimum of its spectrum and, for any locally integrable function V and R > 0, let (−hΔ + V )D R and 2 (−hΔ+V )N be the self-adjoint operators −hΔ+V on the L -space on the cube R ΛR with the Dirichlet and the Neumann boundary conditions, respectively. 3.1. Proof of Theorem 1.2 (I): One-Dimensional Case To obtain the upper estimate, we have only to show the following: Proposition 3.1. If d = 1, K = ∅, supp(u) is compact, x
0 u(y) dy/x > 0,
lim inf x↓0
and
lim inf x↓0
0
then we have lim
(t) log N
t↑∞ t(1+θ)/(3+θ)
≤−
3+θ 1+θ
hπ 2 4
u(y) dy/x > 0,
(1+θ)/(3+θ) .
Proof. We assume h = 1 for simplicity. In the well known expression
(t) = Eθ [exp(−tHξ )(x, x)] dx, N Λ1
(3.1)
−x
(3.2)
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we apply the Feynman–Kac formula and an estimate for the exit time of the Brownian motion (cf. [9]) to obtain ! D
(t) ≤ Eθ exp −tHξ,t (x, x) dx + c1 e−c2 t , N Λ1 D where exp(−tHξ )(x, y) and exp(−tHξ,t )(x, y), t > 0, x, y ∈ R, are the integral D , respectively. By the kernels of the heat semigroups generated by Hξ and Hξ,t eigenfunction expansion of the integral kernel, we have
(t) ≤ c3 tN
1 (t) + c4 e−c5 t , N
1 (t) = Eθ [exp(−tλ1 (H D ))]. Thus we have only to prove (3.2) with where N ξ,t
(t) replaced by N
1 (t). Now we use Theorem 3.1 in the page 123 in [24], N which states 2 D ≥ π 2 / sup |Ik | + c6 λ1 Hξ,t k
for large enough t under"the assumption (3.1), where {Ik }k are the random open intervals such that k Ik = Λt \ {q + ξq : q ∈ Z} and |Ik | is the length of Ik . If supk |Ik | ≥ s for some 0 ≤ s ≤ t, then there exists p ∈ Z ∩ Λt such that {q + ξq : q ∈ Z} ∩ [p, p + s − 2] = ∅. The probability of this event is estimated as # Pθ (q + ξq ∈ [p, p + s − 2]) Pθ sup |Ik | ≥ s ≤ k
p∈Z∩Λt q∈Z∩[p,p+s−2]
≤t
#
exp(−(1 − ε)d(q, [p, p + s − 2]c )θ )/ε1/θ
q∈Z∩[p,p+s−2]
⎞ s−3 1 s ≤ t exp ⎝−(1 − ε) d(q, [0, s − 3]c )θ dq + log ⎠ θ ε 0 $ % θ+1 2(1 − ε) s − 3 1 s ≤ t exp − + log θ+1 2 θ ε ⎛
if s ≥ 3, where 0 < ε < 1 is arbitrary. Therefore we have 1 (1 − ε) R π2 2 θ+1
N1 (t) ≤ c7 t exp − inf t (R − 3) + θ − log R>3 (R + c6 )2 2 (θ + 1) θ ε + c8 e−c9 t for large t. Now it is easy to see that the infimum in the right hand side is attained by R ∼ 2(π 2 t/4)1/(3+θ) and we obtain (3.2). Remark 3.2. We put the additional assumption (3.1) only to use Theorem 3.1 in the page 123 in [24]. These assumptions are not restrictive at all since we can always find a z ∈ R such that u( · + z) satisfies them by the fundamental theorem of calculus and such a finite translation of u does not affect the above argument.
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Proposition 3.3. If d = 1 and α > 3, then we have (1+θ)/(3+θ)
(t) log N 3 + θ hπ 2 . lim (1+θ)/(3+θ) ≥ − 1+θ 4 t↑∞ t
(3.3)
Proof. This is proven by modifying our proof of Proposition 2.2. We take ψR as the nonnegative and normalized ground state of (−Δ)D R . In (2.6), we restrict the integral with respect to y to |q + y| ≥ R1 + (R + 1)/2. In (2.8), we take η = 1/(3 + θ) and R as the integer part of Rtη for a positive number R > 0. Then since t ∇ψR 22 ∼ t(1+θ)η (π/R)2 is not negligible, (2.10) is modified as & π '2
(t) log N
4 (q, t), lim (1+θ)η ≥ −h − lim dq N t↑∞ R t↑∞ t |q|≤tβ−η
4 (q, t) is defined by replacing N
3 (y, q) and R1 + where N t(α−3)η
√
d by
C0 (1 + ε1 ) + |y|θ inf{|x − q − z − y|α : x ∈ ΛRt−η , z ∈ Λt−η }
and R1 + (R + 1)/2, respectively, in (2.9). Since
4 (q, t) ≤ lim N
t↑∞
inf
y ∈ΛR (−q)
|y|θ = d(q, ΛcR )θ ,
we obtain & π '2
(t) log N Rθ+1 , ≥ −h − θ (1+θ)η R 2 (θ + 1) t↑∞ t lim
by the Lebesgue convergence theorem. By taking the supremum over R > 0, we obtain the result. 3.2. Proof of Theorem 1.2 (II): Upper Estimate for the Multidimensional Case In the two-dimensional case, we can simply use Corollary 3.1 in [6] to get the upper bound. Indeed, the integrated density of states increases if we truncate the tail of u and hence the bound for the compactly supported potentials yields N (λ) ≤ c1 exp(−c2 λ−1−θ/2 (log(1/λ))−θ/2 ),
(3.4)
for 0 ≤ λ ≤ c3 , where c1 , c2 and c3 are positive constants depending on h and C0 . We give another proof for Corollary 3.1 in [6] in Sect. 3.4 below. In the rest of this subsection we assume d ≥ 3. Then our goal is the following: Proposition 3.4. Let α ≥ d + 2 and K = ∅. There exist finite positive function k1 (h) and k2 (h) of h and a positive constant c such that N (λ) ≤ k1 (h) exp(−c((h ∧ h(α−d)/(α−2) )/λ)(d+μθ)/2 ) for 0 ≤ λ ≤ k2 (h).
(3.5)
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We first see that Proposition 3.4 follows from the following: Proposition 3.5. For sufficiently small ε1 , ε2 > 0, there exist a positive constant c independent of (h, R), and positive constants c and c independent of (c0 , h, R) such that #{q ∈ Zd ∩ ΛR : |ξq | ≥ ε1 Rμ } ≤ ε2 Rd , Rμd ≥ c h/c0 and Rμ(α−2−d) ≥ c c0 /h imply ⎛⎛ ⎞N ⎞ c0 1B(q+ξq ,R0 )c (x) ⎜ (α−d)/(α−2) ⎠ ⎟ λ1 ⎝⎝−hΔ + )/R2 , ⎠ ≥ c(h ∧ h α |x − q − ξ | q d q∈Z ∩ΛR
R
(3.6) where c0 and R0 are arbitrarily fixed positive constants and 1D is the characteristic function of D ⊂ Rd . Proof of Proposition 3.4. It is well known that c1 N √ N (λ) ≤ Pθ (λ1 (HR ) ≤ λ) (R ∧ h)d (cf. (10.10) in [20]). We can take c0 and R0 so that u(x) ≥ c0 1B(R0 )c (x)|x|−α . Thus by Proposition 3.5, there exists a constant c2 such that N (c2 (h ∧ h(α−d)/(α−2) )/R2 ) c1 √ Pθ (#{q ∈ Zd ∩ ΛR : |ξq | ≥ ε1 Rμ } ≥ ε2 Rd ). ≤ (R ∧ h)d We here should take c0 sufficiently small so that the conditions of Proposition 3.5 are satisfied if α = d + 2. When the event in the right hand side occurs, we have |ξq |θ ≥ εθ1 ε2 Rd+μθ . q∈Zd ∩ΛR
Thus it is easy to show N (c2 (h ∧ h(α−d)/(α−2) )/R2 ) ≤
c3 √ exp(−c4 Rd+μθ ), (R ∧ h)d
and (3.5) follows immediately.
We next proceed to the proof of Proposition 3.5. We start with the following: −d Lemma 3.6. inf{λ1 ((−Δ + 1B(b,1) )N . R ) : b ∈ ΛR } ≥ cR
This lemma follows immediately from the Proposition 2.3 of Taylor [25] using the scaling with the factor R−1 . That proposition is stated in terms of the scattering length. We here give an elementary proof following a lemma in the page 378 in Rauch [22] for the reader’s convenience.
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N Proof. We rewrite as λ1 ((−Δ + 1B(b,1) )N R ) = λ1 ((−Δ + 1B(1) )R,b ), where, for N any locally integrable function V and R > 0, (−Δ + V )R,b is the self-adjoint operator −Δ + V on the L2 space on the cube ΛR (b) = b + ΛR with the the Neumann boundary condition, and B(1) = B(0, 1). For any smooth function ϕ on the closure of ΛR (b), we have ⎛ R(b) ⎜ ϕ2 (x) dx = drrd−1 dS ⎝ϕ(g(r), θ) 1
ΛR (b)
θ∈S d−1 :(r,θ)∈ΛR (b)
r +
⎞2
⎟ ∂s ϕ(s, θ) ds⎠ +
ϕ2 (x) dx,
B(1)∩ΛR (b)
g(r)
where (r, θ) is the polar coordinate, R(b) = sup{|x| : x ∈ ΛR (b)}, dS is the volume element of the (d − 1)-dimensional surface S d−1 and g(r) = {(r − 1)/ (R(b) − 1) + 1}/2. By the Schwarz inequality and a simple estimate, we can show ⎛ ⎞2 R(b) r ⎜ ⎟ drrd−1 dS ⎝ ∂s ϕ(s, θ) ds⎠ 1
θ∈S d−1 :(r,θ)∈ΛR (b)
≤ cR(b)d
g(r)
|∇ϕ|2 (x) dx,
ΛR (b)
where c is a constant depending only on d. By changing the variable, we can also show R(b)
dSϕ(g(r), θ)2 ≤ c R(b)d
drrd−1 1
ϕ2 (x) d,
B(1)∩ΛR (b)
θ∈S d−1 :(r,θ)∈ΛR (b)
where c is also a constant depending only on d. Since supb∈ΛR R(b) ≤ we can complete the proof.
√
dR,
Lemma 3.7. There exist positive constants c, c , and c such that ⎧ ⎛⎛ ⎫ ⎞N ⎞ ⎪ ⎪ n ⎨ ⎬ c0 1B(bj ,R0 )c (x) ⎜⎝ ⎟ ⎠ inf λ1 ⎝ −hΔ + , . . . , b ∈ Λ : b ⎠ 1 n R ⎪ ⎪ |x − bj |α ⎩ ⎭ j=1 R
(d−2)/(α−2) (α−d)/(α−2)
≥ c(c0 n)
h
/R
d
for n ≥ c h/c0 and R ≥ c (c0 n/h)1/(α−2) . Proof. Since λ1 (A + B) ≥ λ1 (A) + λ1 (B) for any self-adjoint operators A and B, the left hand side is bounded from below by inf{λ1 ((−hΔ + c0 n1B(b,R0 )c (x)|x − b|−α )N R ) : b ∈ ΛR }.
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A change of the variable shows that this equals hk −2 inf{λ1 ((−Δ + c0 nk 2−α h−1 1B(b,R0 /k)c (x)|x − b|−α )N R/k ) : b ∈ ΛR/k } for any k > 0. We can bound this from below by hk −2 inf{λ1 ((−Δ + c0 nk 2−α h−1 3−α 1B(b ,1) (x))N R/k ) : b ∈ ΛR/k } √ for k ≥ R0 and R > 4 dk, and we can use Lemma 3.6 to complete the proof by taking k as (c0 n3−α h−1 )1/(α−2) . Indeed, for each b ∈ ΛR/k , we set b := b − (1 + R0 /k)b/|b| if b is not the zero vector. If b is the zero vector, we set b as an arbitrarily chosen vector with the norm 1 + R0 /k. Since R0 /k ≤ |x − b| ≤ 2 + R0 /k on B(b , 1), we have
1B(b,R0 /k)c (x)|x − b|−α ≥ (2 + R0 /k)−α 1B(b ,1) (x). We bound this from below by 3−α 1B(b ,1) (x) by assuming k ≥ R0 . Moreover we claim b ∈ ΛR/k for all b ∈ ΛR/k . A sufficient condition for this is √ R ≥ 2 d(R0 + k), since b for√b with |b| ≥ 1 + R0 /k is a contraction of b and sup{|b |∞ : |b| ≤ 1 + R0 /k} = d(1 + R0 /k). Lemma 3.8. Let V be any locally integrable nonnegative function on Rd . Then any eigenfunction φ of (−hΔ + V )N R satisfies * φ ∞ ≤ c(1/R + λ/h)d/2 φ 2 , where c is a finite constant depending only on d, λ is the corresponding eigenvalue, and · ∞ and · 2 are L∞ and L2 norms, respectively. The proof of this lemma is same as that of (3.1.55) in [24]. Now we prove Proposition 3.5: Proof of Proposition 3.5. We use the following classification: F = {a ∈ ΛR ∩ Rμ Zd : #(ΛRμ (a) ∩ {q + ξq : q ∈ Zd ∩ ΛR }) < Rμd /2} and N = {a ∈ ΛR ∩ Rμ Zd : #(ΛRμ (a) ∩ {q + ξq : q ∈ Zd ∩ ΛR }) ≥ Rμd /2}. By Lemma 3.7, $ % −α N λ1 (−hΔ + c0 1B(q+ξq ,R0 )c (x)|x − q − ξq | )Rμ ,a ≥ ch(α−d)/(α−2) /R2 q
for any a ∈ N . Let us write " ϕ for the nonnegative and normalized ground state of the operator (−hΔ + q c0 1B(q+ξq ,R0 )c (x)|x − q − ξq |−α )N R . Then, applying the Rayleigh–Ritz variational formula, we have ⎛$ %N ⎞ (α−d)/(α−2) c0 1B(q+ξq ,R0 )c (x) ⎠ ≥ ch ϕ2 dx. λ1 ⎝ −hΔ + α 2 |x − q − ξ | R q q "
R
a∈N
ΛRμ (a)
2 If we assume λ1 ((−hΔ + q c0 1B(q+ξq ,R0 )c (x)|x − q − ξq |−α )N Rμ ,a ) ≤ M h/R , then Lemma 3.8 implies that the right hand side is bounded from below by
cR−2 h(α−d)/(α−2) (1 − c M d/2 R(μ−1)d #F).
(3.7)
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Since #(ΛRμ (a) ∩ {q + ξq : q ∈ Zd ∩ ΛR }) ≥ #{q ∈ Λ(1−2ε1 )Rμ (a) ∩ Zd : |ξq | ≤ ε1 Rμ }, we have #{q ∈ Λ(1−2ε1 )Rμ (a) ∩ Zd : |ξq | ≤ ε1 Rμ } < Rμd /2 and #{q ∈ Λ(1−2ε1 )Rμ (a) ∩ Zd : |ξq | ≥ ε1 Rμ } > {(1 − 2ε1 )d − 1/2}Rμd for a ∈ F. Thus, by the assumption of this proposition, we have ε2 Rd ≥ (#F){(1 − 2ε1 )d − 1/2}Rμd and #F ≤ Rd(1−μ) ε2 /{(1 − 2ε1 )2 − 1/2}. By substituting this to (3.7), we complete the proof. 3.3. Proof of Theorem 1.2 (III): Lower Estimate for the Multidimensional Case We shall work with h = C0 = 1 for simplicity. Proposition 3.9. Suppose d = 2 and α > 4 or d ≥ 3 and α ≥ d + 2. Then there exist positive constants c1 , c2 , and c3 such that ' & −θ/2 c1 exp −c2 λ−1−θ/2 (log(1/λ)) (d = 2), N (λ) ≥ (3.8) −(d+μθ)/2 c1 exp(−c2 λ ) (d ≥ 3), for 0 ≤ λ ≤ c3 . Proof. We consider the event {For any p ∈ R1 Zd ∩ Λ3R and q ∈ Zd ∩ ΛR1 (p) ∩ Λ2R , q + ξq ∈ Λ1 (p)} ∩{For any q ∈ Zd \ Λ2R , |ξq | ≤ |q|/4} (3.9) √ μ where R1 = R for d ≥ 3 and R1 = R/ log R for d = 2. Then we have ⎛ ⎞ ⎛ u(x − q − ξq )ΦR ⎠ ≤ λ N (λ) ≥ R−d Pθ ⎝ ∇ΦR 22 + ⎝ΦR , q∈Zd
⎞
and the event (3.9) occurs⎠ ,
(3.10)
whereΦR is an element of the domain of the Dirichlet Laplacian on the cube ΛR \ p∈R1 Zd ∩Λ3R (p + K) such that ΦR 2 = 1 (cf. Theorem (5.25) in [20]). We take ΦR as φR ψR / φR ψR 2 , where ψR is the nonnegative and normalized ground state of the Dirichlet Laplacian on ΛR and ⎧& & " ' ' −ν ⎪ ν (p) 2d x, ∧ 1 (d ≥ 3), Λ R ⎪ μ d ∞ R p∈R Z ∩ΛR ⎨ (3.11) φR (x) = &log d∞ (x,ΛR ∩ √RZ2 )− 4 log R' α ⎪ log R + ⎪ ⎩ (d = 2). log √ R − 4 log R 2
log R
α
In (3.11), d∞ (·, ·) is the distance function with respect to the maximal norm, ν = 2/(α − d), and (·)+ is the positive part. Then it is not difficult to see ∇ΦR 22 ≤ c4 R−2 . On the event (3.9), we have in addition that q∈Zd
u(x − q − ξq ) ≤
d(x,
"
c5 R1d
p∈R1 Zd ∩Λ2R
−(α−d)
Λ1 (p))α
+ c6 R1
(3.12)
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in ΛR . Hence we have ⎛
⎝ΦR ,
Ann. Henri Poincar´e
⎞ u(x − q − ξq )ΦR ⎠ ≤ c7 R−2 .
q∈Zd
On the other hand, the probability of the event (3.9) can be estimated as log Pθ (the event (3.9) occurs) ≥ −#(R1 Zd ∩ Λ3R ) +
log Pθ (ξ0 ∈ Λ1 (q))
q∈Zd ∩ΛR1
log(1 − Pθ (|ξ0 | ≥ |q|/4))
q∈Zd \Λ2R
≥ −c8 Rd R1θ by using log(1 − X) ≥ −2X for 0 ≤ X ≤ 1/2 in the last line. Therefore, we have N (c9 R−2 ) ≥ R−d exp −c10 Rd R1θ
and the proof is finished.
Remark 3.10. For the manner of taking the function φR in (3.11) and the event in (3.9), we refer the reader to the notion of the “constant capacity regime” (cf. Sect. 3.2.B of [24]). The same technique is used in Appendix B of [6]. 3.4. Compact Case In this section, we adapt the methods in the preceding sections to give a simple proof of the following results in [6]: Theorem 3.11. Assume Λr1 ⊂ supp(u) ∪ K ⊂ Λr2 for some 0 < r1 ≤ r2 < ∞ instead of (1.3). Then we have ⎧ 2 (1+θ)/2 ⎪ (1 + θ)−1 2−θ (d = 1), ⎨∼ −(π h/λ) log N (λ) −λ−1−θ/2 (log(1/λ))−θ/2 (d = 2), ⎪ ⎩ (d ≥ 3) −λ−(d/2+θ/d) as λ ↓ 0, where f (λ) ∼ g(λ) means limλ↓0 f (λ)/g(λ) = 1 and f (λ) g(λ) means 0 < limλ↓0 f (λ)/g(λ) ≤ limλ↓0 f (λ)/g(λ) < ∞. Remark 3.12. The assumption on u in this theorem is only for giving a simple proof in the multidimensional case. If d = 1, then the assumption in Proposition 3.1 is sufficient. If d ≥ 3, then this theorem can be extended to the case that the scattering length of u is positive. The proof for d = 1 is given in Sect. 3.1. The lower estimate for d = 2 is given in Sect. 3.3. To prove the lower estimate for d ≥ 3, we replace Rν by 2r2 + 1 in the proof of Proposition 3.9. Then the rest of the proof is simpler than that of the proposition since ⎛ ⎞ ⎝ΦR , u(x − q − ξq )ΦR ⎠ = 0 q∈Zd
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under the event in (3.9) with R1 = R2/d . To prove the upper estimate for d ≥ 3, we have only to apply the following instead of Proposition 3.5 in the proof of Proposition 3.4: Proposition 3.13. For sufficiently small ε1 , ε2 > 0, there exists a finite constant c such that #{q ∈ Zd ∩ ΛR : |ξq | ≥ ε1 R2/d } ≤ ε2 Rd implies ⎛⎛ ⎞N ⎞ ⎜ ⎟ λ1 ⎝⎝−Δ + c0 1B(q+ξq ,r0 ) ⎠ ⎠ ≥ c/R2 , (3.13) q∈Zd ∩ΛR
R
where c0 and r0 are arbitrarily fixed positive constants. Proof. We use the classification F0 = {a ∈ ΛR ∩ R2/d Zd : ΛR2/d (a) ∩ {q + ξq : q ∈ Zd ∩ ΛR } = ∅} and N0 = {a ∈ ΛR ∩ R2/d Zd : ΛR2/d (a) ∩ {q + ξq : q ∈ Zd ∩ ΛR } = ∅}, instead of F and N in the proof of Proposition 3.5. Then we complete the proof by Lemmas 3.6 and 3.8 without using Lemma 3.7. To prove the upper estimate for d = 2, we have only to apply the following instead of Proposition 3.5 in the proof of Proposition 3.4: Proposition 3.14. For sufficiently small ε√ 1 , ε2 > 0, there exists a finite constant c such that #{q ∈ Z2 ∩ ΛR : |ξq | ≥ ε1 R/ log R} ≤ ε2 R2 implies ⎛⎛ ⎞N ⎞ ⎜ ⎟ λ1 ⎝⎝−Δ + c0 1B(q+ξq ,r0 ) ⎠ ⎠ ≥ c/R2 . (3.14) q∈Z2 ∩ΛR
R
√ To prove this, we replace R2/d by R/ log R in the proof of Proposition 3.13 and we further need to extend Lemma 3.6 to the 2-dimensional case. By a simple modification of the proof of Lemma 3.6, we have the following, which is sufficient for our purpose: Lemma 3.15. If d = 2, then we have inf{λ1 ((−Δ + c0 1B(b,r0 ) )N R ) : b ∈ ΛR } ≥ c/(R2 log R).
4. Critical Case In this section we discuss the case of α = d + 2. By modifying our proof of Proposition 2.2, we can prove the following: Proposition 4.1. If α = d + 2, then we have lim t↑∞
(t) log N t(d+θ)/(d+2+θ)
≥ −K0 (h, C0 ),
(4.1)
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where
⎧ ⎛ ⎞ ⎨ 2 dxC ψ(x) 0 ⎝ K0 (h, C0 ) = inf h ∇ψ 22 + dq inf +|y|θ ⎠ ⎩ |x − q − y|d+2 y ∈supp(ψ)−q d Rd ⎫ R ⎬ : ψ ∈ W21 (Rd ), ψ 2 = 1 (4.2) ⎭
and W21 (Rd ) = {ψ ∈ L2 (Rd ) : ∇ψ ∈ L2 (Rd )}. Proof. In (2.4), we replace ψR by an arbitrary function ϕ ∈ H01 (ΛR ) with ϕ 2 = 1, where H01 (ΛR ) is the completion of C0∞ (ΛR ) in W21 (Rd ). Then (2.6) is modified as
2 (t, q) dq N |q|≤tβ
≥
dq log |q|≤tβ
× exp −
√ y∈[supp(ϕ):R1 + d/2]c −q
dy Z(d, θ)
dxϕ(x)2 tC0 (1 + ε1 ) θ − |y| , inf{|x − q − z − y|d+2 : z ∈ Λ1 }
where [A : r] = {x ∈ Rd : d(x, A) < r} for any A ⊂ Rd and r > 0. We take η as 1/(d + 2 + θ). Then, by changing the variables, we see that the right hand side equals dytdη
3 (y, q; ϕη )), exp(−tθη N dq log tdη Z(d, θ) √ |q|≤tβ−η
y∈[supp(ϕη ):(R1 + d/2)/tη ]c −q
where
3 (y, q; ϕη ) = N
dxϕη (x)2 C0 (1 + ε1 ) + |y|θ inf{|x − q − z − y|d+2 : z ∈ Λt−η }
and ϕη (x) = tdη/2 ϕ(tη x). We take R as the integer part of Rtη for a positive number R and take ϕ so that ϕη = ψ is a t-independent element of H01 (ΛR ). Since t ∇ϕ 22 = t(d+θ)η ∇ψ 22 is not negligible, (2.10) is modified as
(t) log N
4 (q, t), dq N lim (d+θ)η ≥ −h ∇ψ 22 − lim t↑∞ t↑∞ t |q|≤tβ−η
where
4 (q, t) N = inf
+
,c √ + d/2 R 1 1
3 (y, q; ψ) : y0 ∈ supp(ψ) : N + γ −q . sup tη t y∈B(y0 ,t−γ )
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Since
4 (q, t) ≤ lim N
t↑∞
inf
y∈(supp(ψ))c −q
dxψ(x)2 C0 (1 + ε1 ) θ + |y| , |x − q − y|d+2
we obtain
(t) log N (d+θ)η t↑∞ t lim
≥ −h ∇ψ 22 −
dy Rd
inf
y∈(supp(ψ))c −q
dxψ(x)2 C0 (1 + ε1 ) + |y|θ |x − q − y|d+2
by the Lebesgue convergence theorem. By taking the supremum with respect to ε1 , ψ and R, we obtain the result. If we apply Donsker and Varadhan’s large deviation theory without caring about the topological problems, then the formal upper estimate lim
(t) log N
t↑∞ t(d+θ)/(d+2+θ)
≤ −K(h, C0 )
(4.3)
is expected, where K(h, C0 ) is the quantity obtained by removing the restriction y ∈ supp(ψ)−q in the definition (4.2) of K0 (h, C0 ). For the corresponding ˆ Poisson case, this is rigorously established in Okura [19]. In that case, the space d R can be replaced by a d-dimensional torus and the Feynman–Kac functional becomes a lower semi-continuous functional, so that Donsker and Varadhan’s theory applies. However, verifications of both the replacement of the space and the continuity of the functional seem to be difficult in our case. From the conjecture (4.3), we expect that the quantum effect appears in the leading term. By Proposition 3.4 in Sect. 3, we can justify this if d ≥ 3 and h is large: Proposition 4.2. If d ≥ 3 and α = d + 2, then we have lim lim λ(d+θ)/2 log N (λ) = −∞.
h→∞ λ→0
(4.4)
In the one-dimensional case we can show the same statement with a more explicit bound lim λ(1+θ)/2 log N (λ) ≤ −
λ→0
π 1+θ h(1+θ)/2 (1 + θ)2θ
by Theorem 1.2, since the leading order does not depend on α ≥ 3. In the two-dimensional case we have no such results.
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5. Proof of Theorem 1.3 5.1. Upper Estimate
− (t) be the Laplace–Stieltjes transform of the integrated density of states Let N − N (λ): ∞ −
N (t) = e−tλ dN − (λ). −∞
To prove the upper estimate, we have only to show the following: Proposition 5.1. Under the condition that u ≥ 0, sup u = u(0) < ∞ and sup |x|α u(x) < ∞ for some α > d, we have
− (t) log N 1+d/θ lim 1+d/θ ≤ u(0) dq(1 − |q|θ ). (5.1) t↑∞ t |q|≤1
Proof. We use the bound
− (t) ≤ N
− (t)(4πth)−d/2 N 1 as in (2.2), where
− (t) = N 1
⎡
⎛
dxEθ ⎣exp ⎝t
⎞⎤ u( x − q − ξq )⎠⎦ .
q∈Zd
Λ1
− (t) in Theorem VI.1.1 of Here we have used the path integral expression of N [3]. The assumption required in that theorem will be checked in Lemma 7.2 in Sect. 7. By replacing the summation by integration, we have −
− (t, q), log N1 (t) ≤ dq log N 2 Rd
where
− (t, q) = Eθ exp t sup u(x − q − ξ0 ) . N 2 x∈Λ2
Now we fix an arbitrary small number ε > 0 and let C = sup |x|α u(x). When |q| > (1 + ε)(u(0)t)1/θ , we estimate as
− (t, q) ≤ exp(t sup{u(x − y) : x ∈ Λ2 , |y| ≥ δ|q|}) N 2 + exp(tu(0))Pθ (|ξ0 | ≥ (1 − δ)|q|),
(5.2)
where δ > 0 is chosen to satisfy (1 − δ)θ+2 (1 + ε)θ = 1. For the first term in the right hand side, we use an obvious bound √ sup{u(x − y) : x ∈ Λ2 , |y| ≥ δ|q|} ≤ C(δ|q| − d)−α . For the second term, it is easy to see Pθ (|ξq | ≥ (1 − δ)|q|) ≤ M (δ, θ) exp(−(1 − δ)θ+1 |q|θ )
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for some large M (δ, θ) > 0. Moreover, we have (1 − δ)θ+1 |q|θ = (1 − δ)θ+2 |q|θ + δ(1 − δ)θ+1 |q|θ ≥ u(0)t + δ(1 − δ)θ+1 |q|θ thanks to |q| > (1 + ε)(u(0)t)1/θ and our choice of δ. Combining above three estimates, we get √
− (t, q) ≤ exp(tC(δ|q| − d)−α )(1 + M (δ, θ) exp(−δ(1 − δ)θ+1 |q|θ )) (5.3) N 2 and thus
− (t, q) ≤ tC(δ|q| − log N 2
√ −α d) + M (δ, θ) exp(−δ(1 − δ)θ+1 |q|θ ),
(5.4)
using log(1 + X) ≤ X. Since the integral of the right hand side over {|q| > (1 + ε)(u(0)t)1/θ } is easily seen to be o(t1+d/θ ), we can neglect this region. For q with |q| ≤ (1 + ε)(u(0)t)1/θ , we estimate as
− (t, q) ≤ exp(t sup{u(x − y) : x ∈ Λ2 , |y| ≥ L}) N 2 + exp(tu(0))Pθ (|q + ξ0 | ≤ L),
(5.5)
where L = 2ε(u(0)t)1/θ . We use obvious bounds sup{u(x − y) : x ∈ Λ2 , |y| ≥ L} ≤ C(L −
√ −α d)+
for the first term and Pθ (|q + ξ0 | ≤ L) ≤ exp(−(|q| − L)θ+ )|B(0, L)|/Z(d, θ) for the second term. Note also that we have √ θ tc(L − d)−α + ≤ tu(0) − (|q| − L)+ for large t, from |q| ≤ (1 + ε)(u(0)t)1/θ and our choice of L. Using these estimates, we obtain
− (t, q) dq log N 2 |q|≤(1+ε)(u(0)t)1/θ
≤
|B(0, L)| θ + 1 + tu(0) − (|q| − L)+ . dq log Z(d, θ)
|q|≤(1+ε)(u(0)t)1/θ
By changing the variable and taking the limit, we arrive at
(t) log N ≤ u(0)1+d/θ 1+d/θ t↑∞ t
dq{1 − (|q| − 2ε)θ+ }.
lim
|q|≤1+ε
This completes the proof of Proposition 5.1 since ε > 0 is arbitrary.
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5.2. Lower Estimate To prove the lower estimate, we have only to show the following: Proposition 5.2. Suppose u ≥ 0, sup u = u(0) < ∞, and u(x) is lower semicontinuous at x = 0. Then we have
− (t) log N dq(1 − |q|θ ). (5.6) lim 1+d/θ ≥ u(0)1+d/θ t t↑∞ |q|≤1
Proof. For any ε > 0, there exists Rε > 0 such that u(x) ≥ u(0) − ε
for |x| < Rε
(5.7)
by the lower semi-continuity of u. We use the bound
− (t) ≥ exp(−th ∇ψε 2 )N
− (t), N 1 for any ψε ∈ C0∞ (Λε ) such that the L2 -norm of ψε is 1, where ⎡ ⎛ ⎞⎤
− (t) = Eθ ⎣exp ⎝t N inf u( x − q − ξq )⎠⎦ . 1 q∈Zd
x∈Λε
(5.8)
This is proven by the same estimate as used in (2.4). We take ψε as the nonnegative and normalized ground state of the Dirichlet Laplacian on the cube √ Λε . Since a sufficient condition for supx∈Λε |x−q−ξq | ≤ Rε is |q+ξq | ≤ Rε −ε d/2, we restrict the expectation to this event and deduce from (5.7) that dy
− (t) ≥ exp(t(u(0) − ε) − |y|θ ). log log N 1 Z(d, θ) d √ q∈Z
|q+y|≤Rε −ε d/2
Since − ε)√− |y|θ ≤ Rε : |q + y| ≤ Rε − √ a sufficient condition for inf{u(0) 1/θ ε d/2} ≥ 0 is |q| ≤ {t(u(0) − ε)} − Rε + ε d/2, we restrict the range of q and deduce
− (t) log N 1 √ |B(0, Rε − ε d/2)| θ ≥ +t(u(0)−ε)−(|q| + Rε − c)) c log Z(d, θ) |q|≤h(t) d
= h(t)
√ |B(0, Rε −ε d/2)| θ + t(u(0) − ε) − (h(t)|q| + Rε + c)) c log Z(d, θ)
|q|≤1
for large t and small ε, where h(t) = {t(u(0) − ε)}1/θ − Rε − c and c and c are positive constants. Then we obtain
− (t) log N dq(1 − |q|θ ). lim 1+d/θ ≥ (u(0) − ε)1+d/θ t t↑∞ |q|≤1
Since ε is arbitrary, this completes the proof of Proposition 5.2.
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6. Asymptotics for Associated Wiener Integrals In the previous work [6], the asymptotic behaviors of the integrated density of states were derived from those of certain Wiener integrals. In this section, we recall the connection and derive the asymptotic behaviors of the associated Wiener integrals in our settings. Let h = 1/2 for simplicity and Ex denote the expectation with respect to the standard Brownian motion (Bs )0≤s≤∞ starting at x. Then the Laplace–Stieltjes transform of the integrated density of states can be expressed as follows (cf. Chapter VI of [3]): ⎧ ⎫ ⎡ ⎨ t ⎬
(t) = (2πt)−d/2 N dxEθ ⊗ Ex ⎣exp − u(Bs − q − ξq ) ds ⎩ ⎭ d 0 q∈Z Λ1 ⎤ : Bs ∈ (q + ξq + K) for 0 ≤ s ≤ t-- Bt = x⎦ . (6.1) q∈Zd
− (t) in the same form by changing the sign of u and We can also express N
(t) seems, and indeed setting K = ∅ in the right hand side. In view of (6.1), N will be proven below, to be asymptotically comparable to the Wiener integral ⎧ ⎫ ⎡ ⎨ t ⎬ u(Bs − q − ξq ) ds St, x = Eθ ⊗ Ex ⎣exp − ⎩ ⎭ d 0 q∈Z ⎤ : Bs ∈ (q + ξq + K) for 0 ≤ s ≤ t⎦ , (6.2) q∈Zd
which was the main object in [6]. This quantity is of interest itself since not only it gives the average of the solution of a heat equation with random sinks but also can be interpreted as the annealed survival probability of the Brownian
− (t) is asymptotically comparamotion among killing potentials. Similarly, N ble to the average of the solution ⎧ t ⎫⎤ ⎡ ⎨ ⎬ St,−x = Eθ ⊗ Ex ⎣exp u(Bs − q − ξq ) ds ⎦ , (6.3) ⎩ ⎭ d 0 q∈Z
of a heat equation with random sources which can also be interpreted as the average number of the branching Brownian motions in random media. We refer the readers to [7,8,24] about the interpretations of St, x and St,−x . The connec (t) and St, x can be found in the literature tion between the asymptotics of N for the case that {q + ξq }q is replaced by an Rd -stationary random field (see e.g. [18,23]). However our case is only Zd -stationary. We first prepare a lemma which gives upper bounds on log St, x and −
(t) and log N
− (t), respectively. We shall state the log St, x in terms of log N results only for x ∈ Λ1 since they automatically extend to the whole space by the Zd -stationarity.
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Lemma 6.1. For any x ∈ Λ1 and ε > 0, we have
(t − ε)(1 + o(1)) log St, x ≤ log N
(6.4)
− (t − t−2d/θ )(1 + o(1)) log St,−x ≤ log N
(6.5)
and
as t → ∞. Proof. We give the proof of (6.5) first. Let Vξ (x) denotes the potential " q∈Zd u(x − q − ξq ) for simplicity. We divide the expectation as ⎧ t ⎫ ⎡ ⎤ ⎨ ⎬ St,−x = Eθ ⊗ Ex ⎣exp Vξ (Bs ) ds : sup |Bs |∞ < [t1+d/θ ]⎦ ⎩ ⎭ 0≤s≤t 0 ⎧ t ⎫ ⎡ ⎤ ⎨ ⎬ + Eθ ⊗ Ex ⎣exp Vξ (Bs ) ds : n−1 ≤ sup |Bs |∞ < n⎦. ⎩ ⎭ 0≤s≤t 1+d/θ n>[t
]
0
(6.6) The summands in the second term can be bounded from above by Eθ exp t sup Vξ (y) Px n − 1 ≤ sup |Bs |∞ 0≤s≤t
y∈Λ2n
d ≤ c1 n Eθ exp t sup Vξ (y) exp{−c2 n2 /t} y∈Λ1
≤ c1 n exp{c3 t d
1+d/θ
− c2 n2 /t},
(6.7)
where we have used a standard Brownian estimate (cf. [9] Sect. 1.7) and the Zd -stationarity in the second line, and Lemma 7.2 below in the third line. Then, it is easy to see that the second term in (6.6) is bounded from above by
− (t). a constant and hence it is negligible compared with N Now let us turn to the estimate of the first term in (6.6). Note first that we can derive an upper large deviation bound % $ Vξ (y) ≥ v ≤ [t1+d/θ ]d Pθ sup Vξ (y) ≥ v ≤ exp(−c4 v 1+θ/d ) sup Pθ y∈Λ[t1+d/θ ]
y∈Λ1
(6.8) which is valid for all sufficiently large t and v ≥ t, from the exponential moment estimate in Lemma 7.2 below. Using this estimate, we get ⎧ t ⎫ ⎡ ⎨ ⎬ Eθ ⊗ Ex ⎣exp Vξ (Bs ) ds : sup |Bs |∞ < [t1+d/θ ], ⎩ ⎭ 0≤s≤t 0 , sup y∈Λ2[t1+d/θ ]
Vξ (y) ≥ t2d/θ
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⎡
≤ Eθ ⎣exp t ≤
n≥t2d/θ
≤
1077
⎤
sup
Vξ (y)
:
y∈Λ2[t1+d/θ ]
$
exp{tn}Pθ
n−1≤
sup y∈Λ2[t1+d/θ ]
sup
Vξ (y) ≥ t2d/θ ⎦ %
Vξ (y) < n
y∈Λ2[t1+d/θ ]
. / exp tn − c4 (n − 1)1+θ/d .
(6.9)
n≥t2d/θ
Since the last expression converges to 0 as t → ∞, we can restrict ourselves on the event {sup Vξ (x) ≤ t2d/θ }. Hereafter, we let T = [t1+d/θ ] since its exact form will be irrelevant in the sequel. Then, the Markov property at time ε = t−2d/θ yields ⎧ t ⎫ ⎡ ⎤ ⎨ ⎬ Vξ (Bs ) ds : sup |Bs |∞ < T, sup Vξ (y) < t2d/θ ⎦ Eθ ⊗ Ex ⎣exp ⎩ ⎭ 0≤s≤t y∈Λ2T 0 dy |x − y|2 ≤e exp − d/2 2ε (2πε) Λ2T ⎧ t−ε ⎫ ⎡ ⎤ ⎨ ⎬ Vξ (Bs ) ds : sup |Bs |∞ < T ⎦ × Eθ ⊗ Ey ⎣exp ⎩ ⎭ 0≤s≤t−ε 0 e −, D dy dzEθ [exp(−(t − ε)Hξ, (6.10) ≤ 2T )(y, z)], (2πε)d/2 Λ2T Λ2T
−, D where exp(−tHξ, 2T )(x, y), t > 0, x, y ∈ Λ2T , is the integral kernel of the heat semigroup generated by the self-adjoint operator Hξ− on the L2 -space on the cube Λ2T with the Dirichlet boundary condition. Finally, we use the estimate 11/2 0 D D D exp(−tHξ, 2T )(y, z) ≤ exp(−tHξ, 2T )(y, y) exp −tHξ, 2T (z, z)
for the kernel of self-adjoint semigroup and the Schwarz inequality to dominate
− (t − ε) multiplied by some constant. the right hand side in (6.10) by T 2d N Combining all the estimates above, we finish the proof of (6.5). We can also prove (6.4) in the same way as (6.10). However it is much simpler since we do not have to care about sup Vξ ( · ) and thus we omit the details.
(t), The next lemma gives the converse relation between log St, x and log N while the lower estimate of log St,−x will be derived directly. (See the proof of Theorem 6.3). Lemma 6.2. For any x ∈ Λ1 and ε > 0, we have
(t) ≤ log S v,K (1 + o(1)) log N t−ε, x
(6.11)
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as t → ∞, where St,v,K is the expectation defined by replacing K and u by x √ K = {x ∈ K : d(x, K c ) ≥ d} and v(y) = inf{u(y − x + z) : z ∈ Λ1 }, respectively in (6.2). Note that if u is a function satisfying the conditions in Theorem 1.1 or 1.2, then so is v. Proof. Let ε > 0 be an arbitrarily small number. By the ChapmanKolmogorov identity, we have ⎧ t−ε ⎫ ⎡ ⎨ ⎬
(t) ≤ (2πε)−d/2 N dzEθ ⊗ Ez ⎣exp − u(Bs − q − ξq ) ds ⎩ ⎭ d 0 q∈Z Λ1 ⎤ : Bs ∈ (q + ξq + K) for 0 ≤ s ≤ t − ε⎦ . q∈Zd
v,K The right hand side is dominated by (2πε)−d/2 St−ε, x and the proof of (6.11) is completed.
We now state our results on the asymptotics of St, x and St,−x : (i) Assume d = 1 and (1.5) if α ≤ 3. Then we have
Theorem 6.3.
& ' ⎧ 2 C0 θ ⎪ ∼ −t(1+θ)/(α+θ) R dq inf y∈R |q+y| α + |y| ⎪ ⎨ (1+θ)/(3+θ) log St, x −t & ' ⎪ ⎪ ⎩ ∼ −t(1+θ)/(3+θ) 3+θ π2 (1+θ)/(3+θ) 1+θ
(1 < α < 3), (α = 3),
(6.12)
(α > 3)
8
as t → ∞, where f (t) ∼ g(t) means limt→∞ f (t)/g(t) = 1 and f (t) g(t) means 0 < limt→∞ f (t)/g(t) ≤ limt→∞ f (t)/g(t) < ∞. (ii) Assume d = 2 and (1.5) if α ≤ 4. Then we have & ' ⎧ 2 C0 (2+θ)/(α+θ) θ ⎪ dq inf y∈R2 |q+y| ⎨ ∼ −t α + |y| R2 log St, x −t(2+θ)/(4+θ) ⎪ ⎩ −t(2+θ)/(4+θ) (log t)−θ/(4+θ)
(2 < α < 4), (α = 4), (α > 4) (6.13)
as t → ∞. (iii) Assume d ≥ 3 and (1.5) if α ≤ d + 2. Then we have
log St, x
∼ −t(d+θ)/(α+θ) −t
2
&
dq inf y∈Rd
Rd (d+θμ)/(d+2+θμ)
C0 |q+y|α
+ |y|θ
' (d < α < d + 2), (α ≥ d + 2) (6.14)
as t → ∞, where μ = 2(α − 2)/(d(α − d)) as in Theorem 1.2. (iv) Assume sup u = u(0) < ∞ and the existence of Rε > 0 for any ε > 0 such that ess inf B(Rε ) u ≥ u(0) − ε. Then we have log St,−x ∼ t1+d/θ u(0)1+d/θ dq(1 − |q|θ ) (6.15) |q|≤1
as t → ∞.
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(t) and N
− (t): the Proof. We first consider the corresponding results for N − −
estimates (6.12)–(6.15) with St, x and St, x replaced by N (t) and N (t), respectively. These are already proven in earlier sections except for the case of α > d + 2 and d ≥ 2. The results for the remaining case follow from Propositions 3.4 and 3.9 and Abelian theorems in [10]. Then by Lemma 6.1, we obtain the upper estimates of St, x and St,−x . For the lower estimates of St, x , we set u# (y) = sup{u(y + x + z) : z ∈ Λ1 }1B(R1 )c (y) + 1B(R1 ) (y) with R1 ≥ 0. If u satisfies the conditions in Theorems 1.1 and 1.2, and R1 is sufficiently large, then u# also satisfies the same conditions. Therefore we obtain the correspond (t) where K is replaced by B(R2 ) with any R2 ≥ R1 ing lower estimates of N and u is replaced by u# . Then by Lemma 6.2, we obtain the corresponding √ v # ,B(R2 + d) lower estimates of St, x , where v # (y) = inf{u# (y − x + z) : z ∈ Λ1 }. √ Since K ⊂ B(R2 + d) and v # ≥ u on B(R2 )c for some R2 ≥ R1 , we obtain the corresponding lower estimates of St, x . For the lower estimate of St,−x , we restrict the expectation to the event Bs ∈ Λε for any s ∈ [1, t] to obtain D
− (t − 1),
− (t − 1) ≥ c1 e−c2 t N St,−x ≥ dyeΔ/2 (x, y) dze(t−1)Δε /2 (y, z)N 1 1 Λε
Λε
− (t) is the function defined in (5.8), and exp(tΔ/2)(x, y), (t, x, y) ∈ where N 1 (0, ∞) × Rd × Rd and exp(tΔD ε /2)(x, y), (t, x, y) ∈ (0, ∞) × Λε × Λε are the integral kernels of the heat semigroups generated by the Laplacian and the Dirichlet Laplacian on Λε , respectively, multiplied by −1/2. Therefore the lower estimate of St,−x is given by our proof of Proposition 5.2.
7. Appendix We here state and prove two lemmas which we used before. The first one is to define the integrated density of states N (λ) and to represent it by the Feynman–Kac formula: Lemma 7.1. Let u be a nonnegative function belonging to the class Kd and satisfying (1.3). Let ξ = (ξq )q∈Zd be a collection of independently and identically distributed Rd -valued random variables satisfying (1.2). " Then almost all sample functions of the random field defined by Vξ (x) = q∈Zd u(x − q − ξq ) belong to the class Kd,loc . Proof. For any ε, δ > 0, by the Chebyshev inequality, we have Pθ (|ξq | ≥ |q|ε ) ≤ Eθ [(|ξq |/|q|ε )δ ] ≤ c1 /|q|εδ . For any ε, there exists δ such that Pθ (|ξq | ≥ |q|ε ) < ∞. q∈Zd
By the Borel–Cantelli lemma, for almost all ξ, we have Nξ ∈ N such that |ξq | < |q|ε < |q|/3 for any q ∈ Z − B(Nξ ). By the condition (1.3) we also have
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Rε such that u(x) ≤ (C0 + ε)/|x|α for any x ∈ B(Rε )c . We now take R > 0 arbitrarily. If x ∈ B(R) and q ∈ Zd − B(3(R ∨ Rε ) ∨ Nξ ), then |x − q − ξq | ≥ |q| − |ξq | − |x| ≥ |q|/3 ≥ Rε and
Vξ (x) ≤
u(x − q − ξq ) + c2 .
q∈Zd ∩B(3(R∨Rε )∨Nξ )
Since the right hand side is a finite sum, we have 1B(R) Vξ ∈ Kd . Since R is arbitrary, we complete the proof. The second is to define the integrated density of states N − (λ) and represent it by the Feynman–Kac formula. The following is enough to apply Theorem VI.1.1 in [3]. This lemma was also used in (6.8). Lemma 7.2. Let u be a bounded nonnegative function satisfying (1.3). Then there exist finite constants c1 and c2 such that Eθ exp r sup Vξ (x) ≤ c1 exp(c2 r1+d/θ ) x∈Λ1
for any r ≥ 0, where ξ and Vξ are same as in the last lemma. Proof. We first dominate as log Eθ exp r sup Vξ (x) ≤ log I(q) dq, x∈Λ1
Rd
where I(q) = Eθ exp r sup u(x − q − ξ0 ) . x∈Λ2
For sufficiently large R > 0, we have u(x) ≤ 2C0 |x|−α for |x| ≥ √ R0 . A suffi|x − q − ξ | ≥ R is |q + ξ | ≥ R + d. Then, for cient condition for inf x∈Λ2 0 0 √ q ∈ B(2(R + d))c , we dominate as 2rC0 |q| I(q) ≤ Eθ exp sup : |q + ξ0 | ≥ α 2 x∈Λ2 |x − q − ξ0 | |q| + er sup u Pθ |q + ξ0 | < 2 2rC0 √ ≤ exp (1 + c1 exp(r sup u − c2 |q|θ )) (|q|/2 − d)α
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Since log(1 + X) ≤ X for any X ≥ 0, we have log I(q) dq √ B(2(R+ d))c
≤ √ B(2(R+ d))c
2rC0 √ dq (|q|/2 − d)α
+
c1 exp(r sup u − c2 |q|θ )) dq
√ B(2(R+ d))c
c3 r + c4 exp(r sup u − c5 Rθ ). Rα−d By a simple uniform estimate, we have log I(q) dq ≤ c6 r sup uRd . ≤
√ B(2(R+ d))
Setting R = (r sup u/c5 )1/θ , we have log I(q) dq ≤ c7 r1+d/θ for sufficiently large r > 0.
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[10] Kasahara, Y.: Tauberian theorems of exponential type. J. Math. Kyoto Univ. 18(2), 209–219 (1978) [11] Kirsch, W., Martinelli, F.: On the density of states of Schr¨ odinger operators with a random potential. J. Phys. A 15(7), 2139–2156 (1982) [12] Kirsch, W., Martinelli, F.: On the spectrum of Schr¨ odinger operators with a random potential. Commun. Math. Phys. 85(3), 329–350 (1982) [13] Kirsch, W., Metzger, B.: The integrated density of states for random Schr¨ odinger operators. In: Spectral Theory and Mathematical Physics (a Festschrift in honor of Barry Simon’s 60th birthday). Proc. Sympos. Pure Math. vol. 76, pp. 649–696. American Mathematical of Society, Providence (2007) [14] Klopp, F.: Localization for semiclassical continuous random Schr¨ odinger operators. II. The random displacement model. Helv. Phys. Acta 66(7–8), 810–841 (1993) [15] Klopp, F., Pastur, L.: Lifshitz tails for random Schr¨ odinger operators with negative singular Poisson potential. Commun. Math. Phys. 206(1), 57–103 (1999) [16] Leschke, H., M¨ uller, P., Warzel, S.: A survey of rigorous results on random Schr¨ odinger operators for amorphous solids. Markov Process. Relat. Fields 9(4), 729–760 (2003) [17] Lifshitz, I.M.: Energy spectrum structure and quantum states of disordered condensed systems. Sov. Phys. Uspekhi 7, 549–573 (1965) [18] Nakao, S.: On the spectral distribution of the Schr¨ odinger operator with random potential. Jpn. J. Math. (N.S.) 3(1), 111–139 (1977) ˆ [19] Okura, H.: An asymptotic property of a certain Brownian motion expectation for large time. Proc. Jpn Acad. Ser. A Math. Sci. 57(3), 155–159 (1981) [20] Pastur, L., Figotin, A.: Spectra of random and almost-periodic operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 297. Springer, Berlin (1992) [21] Pastur, L.A.: The behavior of certain Wiener integrals as t → ∞ and the density of states of Schr¨ odinger equations with random potential. Teoret. Mat. Fiz. 32(1), 88–95 (1977) [22] Rauch, J.: The mathematical theory of crushed ice. In: Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974). Lecture Notes in Math., vol. 446, pp. 370–379. Springer, Berlin (1975) [23] Sznitman, A.S.: Lifschitz tail and Wiener sausage. I, II. J. Funct. Anal. 94(2), 223–246, 247–272 (1990) [24] Sznitman, A.S.: Brownian motion, obstacles and random media. Springer Monographs in Mathematics. Springer, Berlin (1998) [25] Taylor, M.E.: Scattering length and perturbations of −Δ by positive potentials. J. Math. Anal. Appl. 53(2), 291–312 (1976)
Vol. 11 (2010)
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Ryoki Fukushima Department of Mathematics Kyoto University Kyoto 606-8502 Japan Current address: Department of Mathematics Tokyo Institute of Technology Tokyo 152-8551 Japan e-mail:
[email protected] Naomasa Ueki Graduate School of Human and Environmental Studies Kyoto University Kyoto 606-8501 Japan e-mail:
[email protected] Communicated by Bernard Nienhuis. Received: December 21, 2009. Accepted: July 2, 2010.
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Ann. Henri Poincar´e 11 (2010), 1085–1116 c 2010 Springer Basel AG 1424-0637/10/061085-32 published online September 29, 2010 DOI 10.1007/s00023-010-0055-2
Annales Henri Poincar´ e
Entropy of Semiclassical Measures for Nonpositively Curved Surfaces Gabriel Rivi`ere Abstract. We study the asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of nonpositive sectional curvature. To do this, we look at sequences of distributions associated to them and we study the entropic properties of their accumulation points, the so-called semiclassical measures. Precisely, we show that the Kolmogorov–Sinai entropy of a semiclassical measure μ for the geodesic flow g t is bounded from below by half of the Ruelle upper bound, i.e. 1 hKS (μ, g) ≥ χ+ (ρ)dμ(ρ), 2 S∗ M
+
where χ (ρ) is the upper Lyapunov exponent at point ρ. The main strategy is the same as in Rivi`ere (Duke Math J, arXiv:0809.0230, 2008) except that we have to deal with weakly chaotic behavior.
1. Introduction Let M be a compact, connected, C ∞ riemannian manifold without boundary. For all x ∈ M , Tx∗ M is endowed with a norm .x given by the metric over M . The geodesic flow g t over T ∗ M is defined as the Hamiltonian flow correξ2 sponding to H(x, ξ) := 2 x . This quantity corresponds to the classical kinetic energy in the case of the absence of potential. As any observable, this quantity can be quantized via pseudodifferential calculus and the quantum operator 2 corresponding to H is − 2Δ where is proportional to the Planck constant and Δ is the Laplace Beltrami operator acting on L2 (M ). Our main concern in this article will be to study the asymptotic behavior, as tends to 0, of the following sequence of distributions: ∞ ∗ a(x, ξ)dμ (x, ξ) := ψ , Op (a)ψ L2 (M ) , ∀a ∈ Co (T M ), μ (a) = T ∗M
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where Op (a) is a -pseudodifferential operator of symbol a [8] and ψ satisfies −2 Δψ = ψ . An accumulation point (as → 0) of such a sequence of distribution μ is called a semiclassical measure. Moreover, one knows that a semiclassical measure is a probability measure on S ∗ M := {ξ2x = 1} which is invariant under the geodesic flow g t on S ∗ M . For manifolds of negative curvature, the geodesic flow on S ∗ M satisfies strong chaotic properties (Anosov property, ergodicity of the Liouville measure) and as a consequence, it can be shown that almost all the sequences (μ )→0 converge to the Liouville measure on S ∗ M [7,20,23]. This phenomenon is known as the quantum ergodicity property. A main challenge concerning this result would be to answer the Quantum Unique Ergodicity Conjecture [17], i.e. determine whether the Liouville measure is the only semiclassical measure or not (at least for manifolds of negative curvature). In [2], Anantharaman used the Kolmogorov–Sinai entropy to derive properties of semiclassical measures on manifolds of negative curvature.1 In particular, she showed that the Kolmogorov–Sinai entropy of any semiclassical measure is positive. This result implies that the support of a semiclassical measure cannot be restricted to a closed geodesic, i.e. eigenfunctions of the Laplacian cannot concentrate only on closed geodesics in the high energy limit. In subsequent works, with Nonnenmacher and Koch, more quantitative lower bounds on the entropy of semiclassical measures were given [3,4]. 1.1. Kolmogorov–Sinai Entropy Let us recall a few facts about the Kolmogorov–Sinai (also called metric) entropy (see [22] or Appendix B for more details and definitions). It is a nonnegative number associated to a flow (g t )t and to a (g t )t -invariant measure μ, that estimates the complexity of μ with respect to this flow. For example, a measure carried by a closed geodesic will have entropy zero while the Liouville measure has large entropy. Recall also that a standard theorem of dynamical systems due to Ruelle [18] asserts that, for any invariant measure μ under the geodesic flow: χ+ (1) hKS (μ, g) ≤ j (ρ)dμ(ρ) S∗ M
j
with equality if and only if μ is the Liouville measure in the case of an Anosov flow [15]. In the previous inequality, the χ+ j denoted the positive Lyapunov exponents of (S ∗ M, g t , μ) [6]. Regarding these properties, the main result of Anantharaman, Koch and Nonnenmacher was to show that, for a semiclassical measure μ on an Anosov manifold, one has d−1 (d − 1)λmax . hKS (μ, g) ≥ χ+ j (ρ)dμ(ρ) − 2 j=1 S∗ M
1
In fact, her result was about manifolds with Anosov geodesic flow, for instance manifolds of negative curvature.
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where λmax := limt→±∞ 1t log supρ∈S ∗ M |dρ g t | is the maximal expansion rate of the geodesic flow and the χ+ j ’s are the positive Lyapunov exponents [6]. Compared with the original result from [2], this inequality gives an explicit lower bound on the entropy of a semiclassical measure. For instance, for manifolds of constant negative curvature, this lower bound can be rewritten as d−1 2 . However, it can turn out that λmax is a very large quantity and in this case, the previous lower bound can be negative (which would imply that it is an empty result). Combining these two observations [4], they were lead to formulate the conjecture that, for any semiclassical measure μ, one has d−1 1 χ+ hKS (μ, g) ≥ j (ρ)dμ(ρ). 2 j=1 S∗M
They also asked about the extension of this conjecture to manifolds without conjugate points [4]. In a recent work [16], we were able to prove that their conjecture holds for any surface with an Anosov geodesic flow (for instance surfaces of negative curvature). Regarding our proof and the nice properties of surfaces of nonpositive curvature [11,19], it became clear that our result can be adapted in the following way: Theorem 1.1. Let M be a compact, connected, C ∞ riemannian surface without boundary and of nonpositive sectional curvature. Let μ be a semiclassical measure. Then, 1 hKS (μ, g) ≥ χ+ (ρ)dμ(ρ), (2) 2 S∗ M
where hKS (μ, g) is the Kolmogorov–Sinai entropy and χ+ (ρ) is the upper Lyapunov exponent at point ρ. In particular, this result shows that the support of any semiclassical measure cannot be reduced to closed unstable geodesics. We underline that our inequality is also coherent with the quasimodes constructed by Donnelly. In [9], he considered the question of Quantum Unique Ergodicity for packets of eigenfunctions and he proved that for this generalized question, you can construct exceptional sequences of quasimodes that concentrate on flat parts of the surface (even if the Liouville measure is ergodic) and that have in particular zero entropy. Our theorem on the entropy of semiclassical measures holds for sequences of eigenfunctions of the Laplacian. So the two situations are slightly different but our inequality on the entropy (if generalized to quasimodes) would be consistent with Donnelly’s construction. We can make a last observation on the assumptions on the manifold: it is not known whether the Liouville measure is ergodic or not for the geodesic flow on a surface of nonpositive curvature. In fact, if the genus of the surface is larger than 1, then the best known result in this direction is that there exists an open and dense invariant subset U of positive Liouville measure such that the restriction L|U of the Liouville measure is ergodic with respect to g|U [6]. The extension of this result on the entropy of semiclassical measures raises the
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question of knowing whether one could obtain an analogue of this result for weakly chaotic systems. For instance, regarding the construction from [14], it would be interesting to have a lower bound for ergodic billiards. Our purpose in this article is to prove Theorem 1.1. Our strategy will be the same as in [16] (and also [4]) so it is probably better (and easier) for the reader to have a good understanding of the methods from these two references where the geometric situation is “simpler”. We will focus on the main differences and refer the reader to [4,16] for the details of several lemmas. The crucial observation is that as in the Anosov case, surfaces of nonpositive curvature have continuous stable and unstable foliations and no conjugate points. These properties were at the heart of the proofs in [3,4,16] and we will verify that even if the properties of these stable/unstable directions are weaker for surfaces of nonpositive curvature, they are sufficient to answer the question of Anantharaman–Nonnenmacher in this weakly chaotic setting. In [3,4,16], there was a dynamical quantity which was crucially used: the unstable Jacobian of the geodesic flow. In the case of surfaces of nonpositive curvature, one can introduce an analogue of it. This quantity comes from the study of Jacobi fields and is called the unstable Riccati solution U u (ρ) [19]. In the case of surfaces without conjugate points, Freire and Ma˜ n´e have shown that this quantity is related to the upper Lyapunov exponent at point ρ [12]. In fact, for any (g t )t -invariant probability measure on S ∗ M , one has 1 μ a.e., χ (ρ) = lim T →+∞ T
T
+
U u (g s ρ)ds, 0
where χ+ (ρ) is the upper Lyapunov exponent at point ρ. Thanks to the Birkhoff ergodic theorem, the Ruelle inequality can be then rewritten as follows: U u (ρ)dμ(ρ). hKS (μ, g) ≤ S∗ M
And also, the lower bound of Theorem 1.1 can be rewritten as 1 U u (ρ)dμ(ρ). hKS (μ, g) ≥ 2
(3)
S∗ M
The main advantage of this new formulation is that the function in the integral of the lower bound is defined everywhere (and not almost everywhere). Remark. One could also ask whether it would be possible to extend this result to surfaces without conjugate points. In fact, these surfaces also have a stable and unstable foliations with nice properties [19] (and of course no conjugate points). The main difficulty is that the continuity of U u (ρ) is not true anymore [5] and at this point, we do not see any way of escaping this difficulty.
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1.2. Organization of the Article In Sect. 2, we will give a precise survey2 on surfaces of nonpositive curvature and highlight the properties we will need to make the proof works. As rewriting all the details of the proofs from [4,16] would be very long and very similar to what was already done in these earlier works, we will refer the reader to them for the proofs of some lemmas and we will explain precisely which points need to be modified at the different stages of the argument. In Sect. 3, we will draw a precise outline of the proof. Then, in Sect. 4, we will explain how the main result from [4] can be adapted in the setting of surfaces of nonpositive curvature. In Sect. 5, we follow the same strategy as in [16] to derive a crucial estimate on the quantum pressures. Finally, in the appendix, we recall some results on quantum pressure from [4] and some facts about the Kolmogorov– Sinai entropy.
2. Classical Setting of the Article 2.1. Background on Surfaces of Nonpositive Curvature In this first section, we recall some facts about nonpositively curved manifolds [19] (chapter 3), [11]. From this point of the article, we fix M to be a smooth, compact and connected riemannian surface of nonpositive sectional curvature which has no boundary. 2.1.1. Stable and Unstable Jacobi Fields. We define π : S ∗ M → M the canonical projection π(x, ξ) := x. The vertical subspace Vρ at ρ = (x, ξ) is the kernel of the application dρ π. We underline that it is in fact the tangent space at ρ of the 1-dimensional submanifold Sx∗ M . We can also define the horizontal subspace at ρ. Precisely, for Z ∈ Tρ S ∗ M , we consider a smooth curve c(t) = (a(t), b(t)), t ∈ (−, ), in S ∗ M such that c(0) = ρ and c (0) = Z. Then, we define the horizontal space Hρ as the kernel of the application Kρ (Z) = ∇a (0) b(0) = ∇dρ π(Z) b(0), where ∇ is the Levi-Civita connection. This subspace contains XH (ρ) the vector field tangent to the Hamiltonian flow and it is of dimension 2. We know that we can use these two subspaces to split the tangent space Tρ S ∗ M = Hρ ⊕ Vρ . This allows us to define the Sasaki metric on S ∗ M [19] (p. 18) that splits these two subspaces into orthogonal spaces, namely for every ρ = (x, ξ) ∈ S ∗ M and for every X, Y ∈ Tρ S ∗ M , X, Y ρ := gx (dρ π(X), dρ π(Y )) + gx (Kρ (X), Kρ (Y )), where gx is the metric at x on the riemannian manifold M . Using this decomposition, we would like to recall an important link between the linearization of the geodesic flow and the Jacobi fields on M . To do this, we underline that for each point ρ in S ∗ M , there exists a unique unit speed geodesic γρ . Then we define a Jacobi field in ρ (or along γρ ) as a solution of the differential equation: J”(t) + R(γρ (t), J(t))γρ (t) = 0, 2
We refer the reader to [11, 19] for more details.
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where R(X, Y )Z is the curvature tensor applied to the vector fields X, Y and Z and J (t) = ∇γρ (t) J(t). Recall that we can interpret Jacobi fields as geodesic variation vector fields [11]. Precisely, consider a C ∞ family of curves cs : [a, b] → M , s in (−, ). We say that it is a smooth variation of c = c0 . It defines a correspond∂ ing variation vector field Y (t) = ∂s (cs (t))|s=0 that gives the initial velocity of s → cs (t). If we suppose now that c is a geodesic of M , then a C 2 vector field Y (t) on c is a Jacobi vector field if and only if Y (t) is the variation vector field of a geodesic variation of c (i.e. ∀s ∈ (−, ), cs is a geodesic of M ). For instance, γρ (t) and tγρ (t) are Jacobi vector fields along γρ . Consider now a vector (V, W ) in Tρ S ∗ M given in coordinates Hρ ⊕ Vρ . Using the canonical identification given by dρ π and Kρ , there exists a unique Jacobi field JV,W (t) in ρ whose initial conditions are JV,W (0) = V and JV,W (0) = W , such that dρ g t (V, W ) = (JV,W (t), JV,W (t)) in coordinates Hgt ρ ⊕ Vgt ρ [19, Lemma 1.4]. Define Nρ the subspace of Tρ S ∗ M of vectors orthogonal to XH (ρ) and Hρ the intersection of this subspace with Hρ . Using the previous property on Jacobi fields, we know that the subbundle N perpendicular to the Hamiltonian vector field is invariant by g t and that we have the following splitting [19] (Lemma 1.5): Tρ S ∗ M = RXH (ρ) ⊕ Hρ ⊕ Vρ . These properties can be extended to any energy layer E(λ) := {ξ2x = λ} for any positive λ. Following [19, Lemma 3.1], we can construct two particular Jacobi fields along γρ . We denote (γρ (t), e(t)) an orthonormal basis defined along γρ (t). Given a positive T and because there are no conjugate points on the surface M (Hadamard–Cartan theorem [11,19]), there exists a unique Jacobi field JT (t) such that JT (0) = e(0) and JT (T ) = 0. Moreover, JT (t) is perpendicular to γρ (t) for all t in R [19, p. 50]. As a consequence, JT (t) can be identified with its coordinate along e(t) (as Tγρ (t) M is of dimension 2). A result due to Hopf (Lemma 3.1 in [19]) tells us that the limits lim JT (t)
T →+∞
and
lim JT (t)
T →−∞
exist. They are denoted Jsρ (t) and Juρ (t) (respectively the stable and the unstable Jacobi field). They satisfy the simplified one dimensional Jacobi equation: J”(t) + K(t)J(t) = 0, where K(t) = K(γρ (t)) is the sectional curvature at γρ (t). They are never vanishing Jacobi fields with J∗ρ (0) = e(0) and for all t in R, they are perpendicular √ to γρ (t). Moreover, we have J∗ρ (t) ≤ K0 J∗ρ (t) for every t in R (where −K0 is some negative lower bound on the curvature). Using the previous link between geodesic flow and Jacobi fields, we can lift these subspaces to invariant subspaces E s (ρ) and E u (ρ) called the Green stable and unstable subspaces. These subspaces have dimension 1 (as M is a surface) and are included in Nρ . A basis of E s (g t ρ) is given by (Jsρ (t), Jsρ (t)) in
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coordinates Hgt ρ ⊕ Vgt ρ . We can underline that both subspaces are uniformly transverse to Vρ and that it can happen that they are equal to each other (which was not the case in the Anosov setting). In the case of nonpositive curvature, these subspaces depend continuously in ρ and are integrable as in the Anosov case [11]. Remark. We underline that we could develop the same construction for manifolds without conjugate points and the same properties would be true except the continuity of the stable/unstable foliation [5]. In the case where the Green subspaces attached to ρ are linearly independent, a splitting of Nρ is given by E u (ρ) ⊕ E s (ρ) and the splitting holds for all the trajectory. For the opposite case, we know that the Green subspaces attached to ρ (and hence to a geodesic γρ ) are linearly dependent if and only if the sectional curvature is vanishing at every point of the geodesic γρ [19]. As a consequence, we cannot use the same kind of splitting. However, there exists a splitting of Nρ that we can use in both cases, precisely E u (ρ) ⊕ Vρ . 2.1.2. Riccati Equation. The one dimensional Jacobi equation defined earlier gives rise to the Riccati equation: U (t) + U 2 (t) + K(t) = 0, where U (t) = J (t)J(t)−1 for non vanishing J. Then, we define the corresponding unstable Riccati solution associated to the unstable Jacobi field as Uρu (t) := Juρ (t)(Juρ (t))−1 . It is a nonnegative quantity that controls the growth of the unstable Jacobi field (in dimension 2) as follows: t
Juρ (t) = Juρ (0)e
0
Uρu (s)ds
.
The same works for the stable Jacobi field. Both quantities are continuous3 with respect to ρ. We underline that, we can use the previous results to obtain √ t u t the bound dρ g|E 1 + K0 e 0 Uρ (s)ds [19, pp. 53–54]. So the unstable u (ρ) ≤ Riccati solution describe the infinitesimal growth of the geodesic flow along the unstable direction. As for the unstable Jacobian, Freire and Ma˜ n´e showed that the unstable Riccati solutions are related to the Lyapunov exponents [12]. In fact, they proved that in the case of nonpositive curvature (and more generally for surfaces without conjugate points), the upper Lyapunov exponent at point ρ of a (g t )t -invariant measure μ is given by 1 μ a.e., χ (ρ) = lim T →+∞ T
T
+
U u (g s ρ)ds. 0
3
The continuity in ρ is a crucial property that we will use in our proof. We underline that it is not true if we only suppose the surface to be without conjugate points [5].
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2.1.3. Divergence of Vanishing Jacobi Fields. A last point we would like to recall is a result due to Green [13] and to Eberlein in the general case [10]. It asserts that for any positive c there exists a positive T = T (c) such that for any ρ in S ∗ M and for any nontrivial Jacobi field J(t) along γρ such that J(0) = 0 and J (0) ≥ 1, for all t larger than T , we have J(t) ≥ c [19, Proposition 3.1]). In the case of manifolds without conjugate points, this property of uniform divergence only holds in dimension 2 and it will be crucially used in the following (for manifolds without conjugate points of higher dimension, the same result holds but without any uniformity in ρ). Finally, all these properties allow to prove the following lemma: Lemma 2.1. Let v = (0, V ) be a unit vertical vector at ρ. Then for any c > 0, there exists T = T (c) > 0 (independent of ρ and of v) such that for any t ≥ T , dρ g t v ≥ c. As g t preserves the riemannian volume on S ∗ M (given by the Sasaki metric), we know that the jacobian of dρ g t from Nρ = E u (ρ) ⊕ Vρ to Ngt ρ = E u (g t ρ) ⊕ Vgt ρ is uniformly bounded. Combining the fact dρ g t v u is nondecreasing for every v u in E u (ρ) (and every t ≥ 0) and Lemma 2.1, we find that, for κ > 0, there exists T = T (κ) such that the angle between E u (g t ρ) and Rdρ g t v is bounded by some κ for every t ≥ T , for every ρ in S ∗ M and for every unit vector v in Vρ . As, it will be useful in the article, we would like to show that our discussion allows to have a control of dρ g t (for t ≥ 0) in terms of the unstable Riccati solution. In order to obtain this control, we use the splitting of Tρ S ∗ M given by RXH (ρ) ⊕ E u (ρ) ⊕ Vρ . These three subspaces are uniformly transverse so we t only have to give an estimate of dρ gE→T ∗ M when E is one of them. In the t g ρ
case where E = RXH (ρ), it is bounded by 1 and in the case where E = E u (ρ), √ t u it is bounded by 1 + K0 e 0 Uρ (s)ds . In the last case, we fix a small step of time η > 0. Then, we consider e0 a unit vector in Vρ and for 0 ≤ pη ≤ t, we d g pη e define the epη as the unit vector dρρ gpη e00 . We can write, for k := [t/η], dρ g kη e0 gkη ρ = |dρ g kη e0 , ekη gkη ρ | = |dg(k−1)η ρ g η e(k−1)η , ekη gkη ρ · · · dρ g η e0 , eη gη ρ |. We also define the corresponding sequence eupη :=
dρ g pη eu 0 dρ g pη eu 0
of unit unsta-
(J u (0),J u (0)) ble vectors, where eu0 := (J uρ(0),J uρ (0)) . From Lemma 2.1, we know that ρ ρ ρ epη becomes uniformly close (in ρ) to eupη . So, log |dg(p−1)η ρ g η e(p−1)η , epη gpη ρ | becomes uniformly close to log |dg(p−1)η ρ g η eu(p−1)η , eupη gpη ρ |. In particular, for
every δ > 0, there exists a constant C > 0 such that
dρ g kη e0 gkη ρ ≤ Cekηδ |dg(k−1)η ρ g η eu(k−1)η , eukη gkη ρ · · · dρ g η eu0 , euη gη ρ |.
Again, this last quantity is equal to Cekηδ dρ g kη eu0 gkη ρ . From the properties
kη
of the unstable Riccati solution, this quantity is bounded by Cekηδ e
0
Uρu (s)ds
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(with C uniform in ρ). As the subspaces RXH (ρ), Vρ and E u (ρ) are uniformly transverse to each other, we finally deduce that for every δ > 0, there exists t ∗ t tδ 0 Uρu (s)ds C > 0 such that for every ρ ∈ S M , dρ g ≤ Ce e . 2.2. Discretization of the Unstable Riccati Solution For θ small positive number (θ will be fixed all along the paper), one defines E θ := (x, ξ) ∈ T ∗ M : 1 − 2θ ≤ ξ2x ≤ 1 + 2θ . From previous section, we know that there exists a constant b0 such that ∀ρ ∈ E θ ,
0 ≤ U u (ρ) ≤ b0 .
This function will replace the logarithm of the unstable Jacobian log J u in the proof from [16]. The situation is slightly different from the case of an Anosov flow as we do not have that U u is uniformly bounded from below by some positive constant, a property that was crucially to prove Theorem 1.2 in [16]. We solve this problem by introducing a small positive parameter 0 and defining an auxiliary function U0u (ρ) := sup{U u (ρ), 0 }. We also fix and η two small positive constants lower than the injectivity radius of the manifold (that we suppose to be larger than 2). We choose η small enough to have (2 + b 00 )b0 η ≤ 2 (as in [16], this property is only used in the proof of Lemma 3.1). We underline that there exists d0 > 0 such that if ∀(ρ, ρ ) ∈ E θ × E θ ,
d(ρ, ρ ) ≤ d0 ⇒ |U u (ρ) − U u (ρ )| ≤ 0 .
(4)
We also choose η small enough to have ∀ρ ∈ E θ ,
∀0 ≤ s ≤ η,
|U u (ρ) − U u (g s ρ)| ≤ 0 .
We make the extra assumption that the small parameter used for the continuity is smaller than 0 , i.e. b0 (5) b0 η ≤ 0 . 2+ 0 2 In particular, d0 can (and will) be chosen independently of 0 (by taking 2 /2 instead of 0 in the previous continuity relations). Discretization of the Manifold. As in the case of Anosov surfaces, our strategy to prove Theorem 1.1 will be to introduce a discrete reparametrization of the geodesic flow. Regarding this goal, we cut the manifold M and precisely, we K consider a partition M = i=1 Oi of diameter smaller than some positive δ. Let (Ωi )K i=1 be a finite open cover of M such that for all 1 ≤ i ≤ K, Oi Ωi . For γ ∈ {1, . . . , K}2 , define the following open subset of T ∗ M : Vγ := (T ∗ Ωγ0 ∩ g −η T ∗ Ωγ1 ) ∩ E θ . K We choose the partition (Oi )K i=1 and the open cover (Ωi )i=1 of M such that 4 (Vγ )γ∈{1,...,K}2 is a finite open cover of diameter smaller than d0 of E θ . For 4
In particular, the diameter of the partition depends on θ and (but not on 0 ).
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γ := (γ0 , γ1 ), we define f (γ) and f0 (γ) as in the case of an Anosov flow i.e. f0 (γ) := η inf{U0u (ρ) : ρ ∈ Vγ } and
f (γ) := η inf{U u (ρ) : ρ ∈ Vγ },
if Vγ is nonempty, ηb0 otherwise. Compared with the Anosov case, we will have slightly different properties for the function f (γ), i.e.
η
(6) ∀ρ ∈ Vγ ,
Uρu (s)ds − f (γ)
≤ 2η0 .
0
We also underline that the function f0 satisfies the following bounds, for γ ∈ {1, . . . , K}2 , 0 η ≤ f0 (γ) ≤ b0 η. Finally, let α = (α0 , α1 , . . .) be a (finite or infinite) sequence of elements in {1, . . . , K} whose length is larger than 1 and define: f+ (α) := f0 (α0 , α1 ) ≤ and f (α) := f (α0 , α1 ) ≤ . (7) 2 2 In the following, we will also have to consider negative times. To do this, we define the analogous functions, for β := (. . . , β−1 , β0 ) of finite (or infinite) length, f− (β) := f0 (β−1 , β0 ) and f (β) := f (β−1 , β0 ). Remark. We underline that the functions f+ and f− are defined from U0u while f is defined from U u . This distinction will be important in the following.
3. Proof of Theorem 1.1 Let (ψk ) be a sequence of orthonormal eigenfunctions of the Laplacian corresponding to the eigenvalues −−2 k such that the corresponding sequence of distributions μk on T ∗ M converges as k tends to infinity to the semiclassical measure μ. For simplicity of notations and to fit semiclassical analysis notations, we will denote tends to 0 the fact that k tends to infinity and ψ and −−2 the corresponding eigenvector and eigenvalue. To prove the inequality of Theorem 1.1, we will give a symbolic interpretation of a semiclassical measure and apply results on suspension flows to this measure [1]. Let > 4 be a positive number, where was defined in Sect. 2.2. As in the Anosov setting, the link between the two quantities and is only used to obtain a theorem on product of pseudodifferential operators from Sects. 6 and 7 in [16] (here Theorem 3.2). In the following of the article, the Ehrenfest time nE () will be the quantity nE () := [(1 − )| log |].
(8)
We underline that it is an integer time and that, compared with usual definitions of the Ehrenfest time, there is no dependence on the Lyapunov exponent. We also consider a smaller non integer time TE () := (1 − )nE ().
(9)
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Before entering the details of the proof, we would like to say a few words about the Ehrenfest time and about ideas that are behind our strategy from [16]. In order to prove Theorem 1.1, we need to have a precise understanding of the range of validity of the semiclassical approximation. For instance, if one considers a smooth symbol a compactly supported in a small neighborhood of S ∗ M , it can be shown [4] that ıtΔ ıtΔ TE () , e− 2 Op (a)e 2 − Op (a ◦ g t ) 2 = oa (1), ∀|t| ≤ 2λmax L (M )→L2 (M ) where λmax := limt→±∞ 1t log supρ∈S ∗ M |dρ g t | is the maximal expansion rate of the geodesic flow. This result tells us that the semiclassical approximation remains valid for times of order TE ()/(2λmax ). This was the version of the Egorov theorem that was used by Anantharaman, Koch and Nonnenmacher in [3,4] and the λmax term in their lower bound came from this Egorov property. In [16], we managed to overcome this problem by observing that the range of validity of the semiclassical approximation depends also on the symbol you consider. In order to compute entropy, the symbols we will be interested in will in fact be of the form Qα0 × · · · Qαk ◦ g kη where Qαj is compactly supported in T ∗ Ωαj ∩ E θ (see (41) for instance). An important aspect of the proof is that this symbol remains in a nice class of symbols amenable to pseudodifferential calculus as long as k−2
f0 (αj , αj+1 ) ≤
j=0
TE () . 2
(10)
An analogous property was used in [16, Sect. 7] in order to prove a subadditivity property (here Theorem 3.2). This property means that there exists a local time for which the range of validity of the semiclassical approximation is longer than the usual Ehrenfest time TE ()/λmax . Precisely, the largest integer k for which relation (10) is true will be the local Ehrenfest time for the symbol Qα0 × · · · Qαk ◦ g kη . In order to prove our main theorem, we will introduce a “suspension of the quantum dynamics” for which the sum in (10) will appear naturally. We draw now a precise outline of the proof of Theorem 1.1 which is similar to the one we used in the Anosov case [16]. We will refer the reader to this reference for the proof of several lemmas. The main differences with the Anosov case is that we have to introduce a thermodynamical formalism to treat the problem of flat parts of the surface. 3.1. Quantum Partitions of Identity In order to find a lower bound on the metric entropy of the semiclassical measure μ, we would like to apply the uncertainty principle for quantum pressure (see Appendix A) and see what informations it will give (when tends to 0) on the metric entropy of the semiclassical measure μ. To do this, we define quantum partitions of identity corresponding to a given partition of the manifold.
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3.1.1. Partitions of Identity. In Sect. 2.2, we considered a partition of small K diameter (Oi )K i=1 of M . We also defined (Ωi )i=1 a corresponding finite open cover of small diameter of M . By convolution of the characteristic functions 1Oi , we obtain P = (Pi )i=1,...,K a smooth partition of unity on M , i.e. for all x∈M K
Pi2 (x) = 1.
i=1
We assume that for all 1 ≤ i ≤ K, Pi is an element of Cc∞ (Ωi ). To this classical partition corresponds a quantum partition of identity of L2 (M ). In fact, if Pi denotes the multiplication operator by Pi (x) on L2 (M ), then one has: K
Pi∗ Pi = IdL2 (M ) .
(11)
i=1
3.1.2. Refinement of the Quantum Partition Under the Schr¨ odinger Flow. Like in the classical setting of entropy, we would like to make a refinement of the quantum partition. To do this refinement, we use the Schr¨ odinger propagaıtΔ tion operator U t = e 2 . We define A(t) := U −t AU t , where A is an operator on L2 (M ). To fit as much as possible with the metric entropy, we define the following operators: τα = Pαk (kη) · · · Pα1 (η)Pα0
(12)
πβ = Pβ−k (−kη) · · · Pβ−2 (−2η)Pβ0 Pβ−1 (−η),
(13)
and
where α = (α0 , . . . , αk ) and β = (β−k , . . . , β0 ) are finite sequences of symbols such that αj ∈ [1, K] and β−j ∈ [1, K]. We can remark that the definition of πβ is the analogue for negative times of the definition of τα . The only difference is that we switch the two first terms β0 and β−1 . The reason of this choice relies on the application of the quantum uncertainty principle (see Appendix A). One can see that for fixed k and using rules of pseudodifferential calculus, Pαk (kη) · · · Pα1 (η)Pα0 ψ 2 → μ(Pα2k ◦ g kη ×· · · Pα21 ◦ g η ×Pα20 ) as → 0. (14) This last quantity is the one used to compute hKS (μ, g η ) (with the notable difference that the Pj are here smooth functions instead of characteristic functions). As in [16], we will study for which range of times, the operator τα is a pseudodifferential operator of symbol Pαk ◦ g kη × · · · Pα1 ◦ g η × Pα0 . In [4] and [3], they only considered kη ≤ | log |/λmax where λmax := limt→±∞ 1t log supρ∈S ∗ M |dρ g t |. This choice was not optimal and in the following, we will define sequences α for which we can say that τα is a pseudodifferential operator.
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3.1.3. Index Family Adapted to the Variation of the Unstable Riccati Solution. Let α = (α0 , α1 , . . .) be a (finite or infinite) sequence of elements of {1, . . . , K} whose length is larger than 1. We define a natural shift on these sequences σ+ ((α0 , α1 , . . .)) := (α1 , . . .). For negative times and for β := (. . . , β−1 , β0 ), we define the backward shift σ− ((. . . , β−1 , β0 )) := (. . . , β−1 ). In the following, we will mostly use the symbol x for infinite sequences and reserve α and β for finite ones. Then, using notations of Sect. 2.1, index families depending on the value of the unstable Riccati solutions can be defined as follows: k−2 k−1 η i i f+ σ+ α ≤ TE () < f+ σ+ α , (15) I () := (α0 , . . . , αk ) : i=1
K η () := (β−k , . . . , β0 ) :
k−2
i=1
i f− σ− β ≤ TE () <
i=1
k−1
i f− σ− β
.
(16)
i=1
We underline that f+ , f− ≥ 0 η ensures that we consider finite sequences. These sets define the maximal sequences for which we can expect rules from symbolic calculus to hold for the corresponding τα . The sums used to define these sets were already used in [16]. We can think of the time |α|η as a stopping time for which τα remains a nice pseudodifferential operator in a nice class of symbols. A good way of thinking of these families of words is by introducing the sets Σ+ := {1, . . . , K}N
and
Σ− := {1, . . . , K}−N .
Once more, the sets I η () (resp. K η ()) lead to natural partitions of Σ+ (resp. Σ− ). Families of operators can be associated to these families of index: (τα )α∈I η () and (πβ )β∈K η () . One can show that these partitions form quantum partitions of identity (Lemma 5.1 in [16]): τα∗ τα = IdL2 (M ) and πβ∗ πβ = IdL2 (M ) . α∈I η ()
β∈K η ()
3.2. Symbolic Interpretation of Semiclassical Measures Now that we have defined these partitions of variable size, we want to show that they are adapted to compute the pressure of a certain measure with respect to some reparametrized flow associated to the geodesic flow. To do this, we provide a symbolic interpretation of the quantum partitions. We denote Σ+ := {1, . . . , K}N . We also denote Ci the subset of sequences (xn )n∈N such that x0 = i. Define also: −k [α0 , . . . , αk ] := Cα0 ∩ · · · ∩ σ+ Cαk ,
(17)
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where σ+ is the shift σ+ ((xn )n∈N ) = (xn+1 )n∈N (it fits the notations of the previous section). The set Σ+ is then endowed with the probability measure (not necessarily σ+ -invariant): Σ Σ −k Cαk = Pαk (kη) · · · Pα0 ψ 2 . μ + ([α0 , . . . , αk ]) = μ + Cα0 ∩ · · · ∩ σ+ Using the property of partition of identity, it is clear that this definition ensures the compatibility conditions to define a probability measure [22]: Σ Σ μ + ([α0 , . . . , αk+1 ]) = μ + ([α0 , . . . , αk ]) . αk+1
Then, we can define a suspension flow, in the sense of Abramov, associated to this probability measure. To do this, the suspension set is defined as Σ+ := {(x, s) ∈ Σ+ × R+ : 0 ≤ s < f+ (x)}.
(18)
Recall that the roof function f+ is defined as f+ (x) := f0 (x0 , x1 ). We define a Σ probability measure μ + on Σ+ : Σ
Σ
μ + =
μ + × dt Σ
Σ+
f+ dμ +
.
(19)
The suspension semi-flow associated to σ+ is for time s: ⎛ ⎞ n−2 j n−1 σ s+ (x, t) := ⎝σ+ (x), s + t − f+ σ+ x ⎠,
(20)
j=0
where n is the only integer such that
n−2 j=0
j f+ (σ+ x) ≤ s + t <
n−1 j=0
j f+ (σ+ x).
Remark. We underline that we used the fact that f+ > 0 to define the suspension flow. If we had considered f , we would not have been able to construct the suspension flow as f could be equal to 0. Remark. It can be underlined that the same procedure holds for the partition (πβ ). The only differences are that we have to consider Σ− := {1, . . . , K}−N , σ− ((xn )n≤0 ) = (xn−1 )n≤0 and that the corresponding measure is, for k ≥ 1: Σ Σ −k Cβ−k ∩ · · · ∩ Cβ0 μ − ([β−k , . . . , β0 ]) = μ − σ− = Pβ−k (−kη) · · · Pβ0 Pβ−1 (−η)ψ 2 . For k = 0, one should take the only possibility to assure the compatibility condition: Σ
μ − ([β0 ]) =
K
Σ
μ − ([β−1 , β0 ]).
j=1
The definition is quite different from the positive case but in the semiclassical limit, it will not change anything as Pβ0 and Pβ− 1 (−η) commute. Finally, the “past” suspension set can be defined as Σ− := {(x, s) ∈ Σ− × R+ : 0 ≤ s < f− (x)}.
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+ Consider the partition C˜ := ([α])α∈I η () of Σ+ . A partition C of Σ+ can be defined starting from the partition C˜ and [0, f+ (α)[. An atom of this + − suspension partition is an element of the form C α = [α] × [0, f+ (α)[. For Σ (the suspension set corresponding to Σ− ), we define an analogous partition − C = ([β] × [0, f− (β)[)β∈K η () . In the case of the Anosov geodesic flows [16], we used these partitions and show that they could be used to interpret some quantum entropy as the + entropy of a refined partition of Σ . Then, we used an entropic uncertainty principle taken from [4] to derive a lower bound on the quantum entropy. We will do the same thing here but we will have to be more careful and we will apply the entropic uncertainty principle for quantum pressures as in [4] (see Sect. A for a brief reminder). We introduce the weights ⎛ ⎞ ⎛ ⎞ k−1 k−1 1 1 j j f (σ+ α)⎠ and Wβ− := exp ⎝ f (σ− β)⎠ . Wα+ := exp ⎝ 2 j=1 2 j=1
We underline that the weights depends on f and not on f+ or f− . It came from the fact that f is the function that appears in Theorem 4.1. We introduce the associated quantum pressures5 + + + Σ Σ Σ p μ + , C := H μ + , C − 2 μ + C α log Wα+ (21) α∈I η ()
and
− − Σ Σ p μ − , C := H μ − , C − 2
− Σ μ − C β log Wβ− .
(22)
Thanks to Proposition 5.1, we know that + − Σ Σ p μ + , C + p μ − , C ≥ − log C − (1 + + 4)nE (),
(23)
β∈K η ()
where C is a constant that does not depend on . Remark. This last inequality is a crucial step to prove Theorem 1.1. We will recall how one can get such a lower bound in Sect. 5. This inequality corresponds to Proposition 5.3 in [16]. The strategy of the proof is exactly the same except that we have to deal with quantum pressures and not quantum entropies (see Sect. 5). However, we can follow the same lines as in Sect. 5.3.2 in [16] and obtain a lower bound that depends on the bound from Theorem 4.1. At this point, there is a difference because Theorem 4.1 was proved in [4] for Anosov flows. In Sect. 4, we will show that the proof of this result from [4] can be adapted in the setting of nonpositively curved surfaces. +
−
The partitions C and C are not exactly refined partitions of the suspension flow (as in definition (41) for instance). However, as in the Anosov setting, one can prove that they are refinements of “true refined partitions” of the suspension flow. A notable difference is that we will not consider time 5
We refer the reader to Appendix B for the definition of H.
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1 of the suspension flow. Instead of it, we fix a large integer N0 (such that6 1/N0 0 ) and consider time 1/N0 of the flow and its iterates. Precisely, as in [16], one can prove the following lemma: Lemma 3.1. Let N0 be a positive integer defined as previously. There exists an i n ()N0 −1 − N0 σ+
+
E explicit partition C N0 of Σ+ , independent of such that ∨i=0
is a refinement of the partition
+ C .
C+
Moreover, let n be a fixed positive integer. −
i
N0 C + is of the form [α] × B(α), Then, an atom of the refined partition ∨n−1 i=0 σ + where α = (α0 , . . . , αk ) is a k+1-uple such that (α0 , . . . , αk ) verifies Nn0 (1−) ≤ k−1 j n j=0 f+ (σ+ α) ≤ N0 (1 + ) and B(α) is a subinterval of [0, f+ (α)[.
Remark. This lemma is the exact analogue of Lemma 4.1 in [16] and its proof is the same: the only difference is that we consider times 1/N0 instead of time + 1. In particular, in the proof, the partition C N0 is constructed from7 I˜η (1/N0 ) and not from I˜η (1). We also underline that we have only stated the result in the case of σ+ . The same results holds for σ− : there exists an adapted partition − C N0 with the same properties. As in the Anosov case, we would like to use this lemma to rewrite the quantum pressure in terms of the pressure of a refined partition. To do this, we use basic properties of the classical entropy (see Appendix B) to find that 1 + + Σ Σ (24) H μ + , C ≤ HN0 nE () μ + , σ +N0 , C N0 , where Hn (.) is defined by (41). Consider now an atom A of the partition n ()N −1 −
j
+
E 0 σ + N0 C N0 . There exists an unique family (γ0 , . . . , γnE ()N0 −1 ) in ∨j=0 η N I˜ (1/N0 ) 0 nE () corresponding to this atom and we define the associated weight as
N0 nE ()−1
WA+ :=
Wγ+j .
j=0
From Lemma 3.1, we know that for every such A, there exists α in I η () such that A is a subset of [α] × [0, f+ (α)[. From the proof of this lemma (see Sect. 5.2.3 in [16]), we know that α is of the form (˜ γ0 , . . . , γ˜nE ()N0 −1 ) where every γ˜j is given by γj where we have erased at most the last b0 /0 + 1 letters. In particular, this implies that b0 + + 1 b0 η Wα+ . WA ≤ exp 2N0 nE () 0 6
To summarize the relations between the different parameters, we have
4
<
Moreover η depends on and 0 and tends to 0 when tends to 0 and 0 is fixed. k−2 k−1 7 We define I˜η (t) = {α = (α , . . . , α ) : i i 0 k i=1 f+ (σ+ α) ≤ t < i=1 f+ (σ+ α)}.
1 N0
0 .
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Recall that we have taken ( b 00 + 1)b0 η ≤ /2. One can then write the following inequality + Σ −2 μ + C α log Wα+ α∈I η ()
≤ −2 N n
0 E A∈∨j=0
()−1
Σ
μ + (A) log WA+ + 2N0 nE (). σ
−
j N0
(25)
+
C N0
We introduce the refined pressure at time n 1 + Σ+ N0 pn μ , σ + , C N0 1 + Σ+ N0 := Hn μ , σ + , C N0 − 2 A∈∨n−1 j=0 σ
−
Σ
μ + (A) log WA+ . j N0
+
CN
0
Finally, combining inequalities (24) and (25) with (23), we derive that − log C − (1 + + 4(1 + N0 ))nE () 1 1 + − Σ Σ N0 ≤ pnE ()N0 μ + , σ +N0 , C N0 + pnE ()N0 μ − , σ − , C N0 .
(26)
This estimate is crucial in our proof as we have derived from a quantum relation a lower bound on the classical pressure of a dynamical system associated to the geodesic flow. 3.3. Subadditivity of the Quantum Pressure As in [16], we would like to let tends to 0 in inequality (26). The main difficulty to do this is that everything depends on . In order to overcome this problem, we have to prove a subadditivity property for the quantum pressure: +
Theorem 3.2. Let C N0 be the partition of Lemma 3.1. There exists a function R(n0 , ) on N × (0, 1] and R(N0 ) independent of n0 such that ∀n0 ∈ N,
lim sup |R(n0 , )| = R(N0 ). →0
Moreover, for any ∈ (0, 1] and any n0 , m ∈ N such that n0 + m ≤ N0 nE (), one has 1 1 + + Σ Σ pn0 +m μ + , σ N0 , C N0 ≤ pn0 μ + , σ N0 , C N0 1 + Σ +pm μ + , σ N0 , C N0 + R(n0 , ). Proof. To prove this subadditivity property, we will prove subadditivity of the quantum entropy and subadditivity of the pressure term. As in Sect. 6 from [16], we write for the entropy part that 1 + Σ Hn0 +m μ + , σ N0 , C N0 1 1 −m + + Σ Σ ≤ Hn0 μ + ◦ σ + N0 , σ N0 , C N0 + Hm μ + , σ N0 , C N0 .
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As in [16], we have to show that the measure of the atoms of the partition is 1
almost invariant under σ +N0 for the range of times we have considered (Proposition 6.1 in [16]). Consider now the pressure term in the quantum pressure. Using the multiplicative structure of the WA+ , one has Σ μ + (A) log WA+ n +m−1
0 A∈∨j=0
=
σ
−
j N0
A∈∨m−1 j=0 σ
+
−
+
C N0 Σ
μ + (A) log WA+ j N0
n −1
0 A∈∨j=0 σ
−
+
C N0
j N0
−m Σ μ + σ + N0 A log WA+ . +
C N0
So, once more, the additivity property of the pressure term derives from the almost invariance of the measure for the range of times we consider.8 Precisely, according to the last two inequalities, we only need to verify that + Proposition 6.1 in [16] remains true for the partition C N0 in the setting of surfaces of nonpositive curvature. We will not reproduce the proof here which is similar except that we consider time 1/N0 instead of 1 and that we look at surfaces of nonpositive curvature. The first difference is not a problem and the proof from [16] can be adapted straightforward. The main difference comes from the fact that the geometric situation is slightly different. We will briefly explain here which points need to be modified in this new setting. We recall that Proposition 6.1 in [16] relied on a theorem for products of pseudodifferential operators (Theorem 7.1 in [16]) and we need to verify that the proof we gave still works in the case of surfaces of nonpositive curvature. The key point of the proof of this theorem is that in the allowed range of times, dρ g t is bounded by some −ν (with ν < 1/2) (see Sect. 7.2 in [16]). Precisely, following Sects. 6 and 7 in [16], we need to verify that this bound on the growth of dρ g t holds for ρ in T ∗ Ωα0 ∩ · · · g (k−1)η T ∗ Ωαk−1 ∩ E θ k−1 j (where α satisfies j=0 f+ (σ+ α) ≤ nE2() ) and for 0 ≤ t ≤ kη. In fact, if we take Op (χ) to be a “good” truncation operator in a neighborhood of S ∗ M , it allows to verify that Pαk−1 ((k − 1)η) · · · Pα0 Op (χ) satisfies the usual rules of pseudodifferential calculus (see Theorem 7.1 in [16]) and then to derive the Σ property of almost invariance of the measure μ + (Proposition 6.1 in [16]). From Sect. 2.1.3 and if we take δ = 0 , we find that dρ g kη is bounded kη u by Cekη 0 e 0 Uρ (s)ds (with C uniform in ρ). For the allowed words, 2ekη 0 is of order − (as kη0 ≤ 1/2nE ()). To conclude, we can estimate:
kη
(j+1)η
k−1
k−1
j u j
≤
Uρu (s)ds − f (σ α) U (s)ds − f (σ α)
.
ρ
j=0
j=0
0
8
jη
+ We underline that R(N0 ) will be equal to supA∈C + log WA which only depends on N0 . N0
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To bound this sum, we can use the continuity of U u (see inequality (6)) to show that this quantity is bounded by | log |. By definition of the allowed k−1 words α, we know that j=0 f (σ j α) ≤ 1/2nE (). This allows to conclude that |dρ g t | is bounded by some C−ν (with C independent of ρ and ν < 1/2). Remark. We underline that here we need to use the specific properties of surfaces of nonpositive curvature to prove this theorem. It is not really surprising that Theorem 7.1 from [16] can be extended in our setting as the situation can only be less “chaotic”. We also mention that we have to use the continuity of U u (ρ) which is for instance false for surfaces without conjugate points [5]. 3.4. The Conclusion 3.4.1. The Semiclassical Parameter Tends to 0. Thanks to the subadditivity property of the quantum pressure, we can proceed as in [16] and write, for a fixed n0 , the euclidean division N0 nE () = qn0 + r. We find, after applying the subadditivity property and letting tends to 0, −
1 R(N0 ) − (1 + + 4(1 + N0 )) n0 N0 1 1 1 + − N0 N0 Σ+ Σ− ≤ pn0 μ , σ + , C N0 + pn0 μ , σ − , C N0 . n0
As in [2,4,16], we can replace the smooth partitions by true partitions of the manifold in the previous inequality. We would like now to transform the previous inequality on the metric pressure into an inequality on the Kolmogorov–Sinai entropy. To do this, we write the multiplicative property of WA+ and + we use the fact that C N0 is a partition of Σ+ . It allows us to derive that μΣ+ (A) log WA+ = n0 μΣ+ (A) log WA+ n −1
0 A∈∨j=0 σ
−
j N0 + + CN 0
+
A∈C N0
The same property holds for the backward side. After letting n0 tends to infinity, we find that −
1 (1 + + 4(1 + N0 )) N0 ⎞ ⎛ ⎟ ⎜ +2 ⎝ μΣ+ (A) log WA+ + μΣ− (A) log WA− ⎠ +
− 0
A∈C N
A∈C N
0
≤
1 hKS μΣ+ , σ + + hKS μΣ− , σ − . N0
3.4.2. Lower Bound on
(27)
+
+
A∈C N0
+ µΣ (A) log WA . Before applying Abramov
theorem in inequality (27), we would like to give a lower bound on the pressure + term in this inequality. Precisely, we know that, by construction of C N0 and
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by invariance of the measure μΣ+ , one has 1 μΣ+ (A) log WA+ = f dμΣ+ ˜η + Σ+ 0
f0 (γ)μΣ+ ([γ]) log Wγ+ .
γ∈I (1/N0 )
A∈C N
0
To obtain a lower bound on this quantity, we use the notations of Sect. 2.1 and introduce, for ρ ∈ S ∗ M , the application F0 (ρ) := f0 (γ) log Wγ+ 1Oγ0 (ρ) · · · 1Oγk ◦ g kη (ρ). γ∈I˜η (1/N0 )
This allows us to rewrite f0 (γ)μΣ ([γ]) log Wγ+ = γ∈I˜η (1/N0 )
Define also (η) X0
F0 (ρ)dμ(ρ).
S∗ M
:= ρ ∈ S ∗ M : ∀0 ≤ t ≤
1 u t + η, U (g ρ) > 20 . N0 0
We can verify that F0 (ρ) ≥
1 f0 (γ)1X (η) (ρ)1Oγ0 (ρ)1Oγ1 ◦ g η (ρ), 0 2N0 γ ,γ 0
(28)
1
for all ρ in E θ . In order to prove this property, we can restrict ourselves to (η) the case: ρ ∈ X0 (otherwise the inequality is trivial). In this case, F0 (ρ) = f+ (γ) log Wγ+ , where γ is the unique element in I˜η (1/N0 ) such that ρ belongs to Oγ0 ∩ · · · g −kη Oγk . As γ belongs to I˜η (1/N0 ), it satisfies k−2
1 < f+ (σ j γ). N0 j=1 k−1
f+ (σ j γ) ≤
j=1
As f+ ≥ η0 , one has (k − 2)η ≤ 1/(N0 0 ). Using the fact that ρ belongs to (η) X0 ∩ Oγ0 ∩ · · · g −kη Oγk and using the relation of continuity (4), we find that for every 1 ≤ j ≤ k − 1, f+ (σ j γ) = f (σ j γ). In particular, one has 1 1 f+ (σ j γ) ≥ . 2 j=1 2N0 k−1
log Wγ+ = Then, we can derive +
A∈C N0
(η) −η f (γ)μ X ∩ O ∩ g O 0 γ γ 0 1 0 γ0 ,γ1 . 2N0 Σ+ f+ dμΣ+
μΣ+ (A) log WA+ ≥
We underline that the same lower bound holds for
− 0
A∈C N
μΣ− (A) log WA− .
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3.4.3. Applying Abramov Theorem. We use this last property in inequality (27) and combine it with the Abramov theorem [1] (see relation (44)). We find that the Kolmogorov Sinai entropy ηhKS (μ, g) is bounded from below by (η) f0 (γ) Γ(, , N0 )μΣ ([γ]) + μ X0 ∩ Oγ0 ∩ g −η Oγ1 , γ=(γ0 ,γ1 )
where
1 Γ(, , N0 ) := − + − − 2(1 + N0 ) . 2 2
3.4.4. The Different Small Parameters Tend to 0. We have obtained a lower bound on the Kolmogorov–Sinai entropy of the measure μ. This lower bound depends on several small parameters that are linked to each other in the following way: < 4
1 0 . N0
Moreover the small parameter η depends on and 0 . For a fixed 0 , it tends to 0 when tends to 0. We have now to be careful to transform our lower bound on the entropy of μ into the expected lower bound. First, we let the diameter of the partition tends to 0 (and then θ to 0) and we divide by η. This gives us (Γ(, , N0 ) + 1X (η) (ρ))U0u (ρ)dμ(ρ) ≤ hKS (μ, g). 0
S∗ M
Finally, we let and tend to 0 (in this order). We obtain the following bound on the entropy of μ: 1 u U0 (ρ)dμ(ρ) + U0u (ρ)1X (0) (ρ)dμ(ρ) ≤ hKS (μ, g). − 0 2 S∗ M
S∗ M
We let now N0 tend to infinity and then 0 tend to 0 (in this order). We find the expected lower bound, i.e. 1 U u (ρ)dμ(ρ) ≤ hKS (μ, g). 2 S∗M
4. Proof of the Main Estimate from [4] In the previous section, we have been able to apply the method we used for Anosov surfaces in order to prove Theorem 1.1. As in [16], the strategy relied on a careful adaptation of an uncertainty principle. In particular, to derive inequality (23) (Sect. 5), we have to use the following equivalent of Theorem 3.1 from [3]:
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Theorem 4.1. Let M be a surface of nonpositive sectional curvature and , 0 and η be small positive parameters as in Sect. 2.2. For every K > 0 (K ≤ Cδ0 ), there exists K and CK (, η, 0 ) such that uniformly for all ≤ K , for all k ≤ K| log |, for all α = (α0 , . . . , αk ), Pαk U η Pαk−1 · · · U η Pα0 Op (χ(k) )L2 (M ) ⎛ ⎞ k−1 1 1 j ≤ CK (, η, 0 )− 2 −cδ0 e2kη 0 exp ⎝− f (σ+ α)⎠ , 2 j=0
(29)
where c depends only on the riemannian manifold M . Remark. We underline two facts about this theorem. The first one is that Op (χ(k) ) is a cutoff operator that was already defined in [16, Paragraph 5.3] and in the appendix of [4] (we describe briefly its construction in Sect. 5.1). The second one is that it is function f and not f+ that appears in the upper bound. This theorem is the analogue for surfaces of nonpositive curvature of a theorem from [4]. As the geometric situation is slightly different from [4], we will recall the main lines of the proof where the geometric properties appear and focus on the differences. We refer the reader to [4] for the details. In [4], the proof of the analogue of Theorem 4.1 (Sect. 3 and more precisely Corollary 3.5) relies on a study of the action of Pαk U η Pαk−1 · · · U η Pα0 on a particular family of Lagrangian states. This reduction was possible because of the introduction of the cutoffs operators Op (χ(k) ) (see Sect. 3 in [4] for the details). 4.1. Evolution of a WKB State ı
Consider u (0, x) = a (0, x)e S(0,x) a Lagrangian state, where a (0, •) and S(0, •) are smooth functions on a subset Ω in M and a (0, •) ∼ k k ak (0, •). This represents a Lagrangian state which is supported on the Lagrangian manifold L(0) := {(x, dx S(0, x) : x ∈ Ω}. According to [4], if we are able to understand the action of Pαk U η Pαk−1 · · · U η Pα0 on Lagrangian states (with specific initial Lagrangian manifolds: see next paragraph), then we can derive our main theorem. A strategy to estimate this action is to use a WKB Ansatz. Recall ˜(t) can be that if we note u ˜(t) := U t u (0), then, for any integer N , the state u approximated to order N by a Lagrangian state u(t) of the form ı
ı
u(t, x) := e S(t,x) a (t, x) = e S(t,x)
N −1
k ak (t, x).
K=0
As u is supposed to solve ı Δ 2 u = ∂t u (up to an error term of order N ), we know that S(t, x) and the ak (t, x) satisfy several partial differential equations. In particular, S(t, x) must solve the Hamilton–Jacobi equation ∂S + H(x, dx S) = 0. ∂t
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Assume that, on a certain time interval (for instance s ∈ [0, η]), the above equations have a well defined smooth solution S(s, x), meaning that the transported Lagrangian manifold L(s) = g s L(0) is of the form {(x, dx S(s, x))}, where S(s) is a smooth function on the open set πL(s). As in [4], we shall say that a Lagrangian manifold L is “projectible” if the projection π : L → M is a diffeomorphism onto its image. If the projection of L on M is simply connected, L is the graph of dS for some function S: we say that L is generated by S. Suppose now that, for s ∈ [0, η], the Lagrangian L(s) is “projectible”. Then, this family of Lagrangian manifolds defines an induced flow on M , i.e. t : x ∈ πL(s) → πg t (x, dx S(s, x)) ∈ πL(s + t). gS(s) t+τ t τ This flow satisfies a property of semi-group: gS(s+τ ) ◦ gS(s) = gS(s) . Using this flow, we define an operator that sends functions on πL(s) into functions on πL(s + t): 12 −t −t t (a)(x) := a ◦ gS(s+t) (x) JS(s+t) (x) , (30) TS(s) t t where JS(s) (x) is the Jacobian of the map gS(s) at point x (w.r.t. the riemannian volume). This operator allows to give an explicit expression for all the ak (t) [4], i.e.
t ak (t) :=
t TS(0) a0 (0)
and ak (t) :=
t TS(0) ak (0)
t−s TS(s)
+
ıΔak−1 (s) 2
ds.
0
(31) Regarding the details of the proof in [4], we know that there are two main points where the geometric properties of the manifold are used: • the evolution of the Lagrangian manifold under the action of the operator Pαk U η Pαk−1 · · · U η Pα0 (Sect. 3.4.1 in [4]); t • the value of JS(0) for large t (Sect. 3.4.2 in [4]). We will discuss these two points in the two following paragraphs. We will recall what was proved for these two questions in Sect. 3.4 of [4] and see how it can be translated in the setting of surfaces of nonpositive curvature. 4.2. Evolution of the Lagrangian Manifolds The first thing we need to understand is how the Lagrangian manifolds evolve under the action of the operator Pαk U η Pαk−1 · · · U η Pα0 . According to [4], we know that the introduction of the cutoff operator Op (χ) implies that we can restrict ourselves to a particular family of Lagrangian states. Precisely, we fix some small parameter η1 and we know that they must be localized on a piece of ∗ M (where Lagrangian manifold L0 (0) which is included in the set ∪|τ |≤η g τ Sz,η 1 ∗ 2 Sz,η1 M := {(z, ξ) : ξz = 1 + 2η1 }). If we follow the method developed in [4], we are given a sequence of Lagrangian manifolds Lj (0) as follows:
∀t ∈ [0, η], ∀j, L0 (t) := g t L0 (0) and Lj (t) := g t Lj−1 (η) ∩ T ∗ Ωαj .
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The manifold Lj (0) is obtained after performing Pαj U η Pαk−1 · · · U η Pα0 on the initial Lagrangian state. To show that the procedure from [4] is consistent (i.e. performing several WKB Ansatz), we need to verify that the Lagrangian manifold Lj (t) does not develop caustics and remains “projectible”. The only geometric properties which were used to derive these two properties were: • M has no conjugate points (to derive that S j will not develop caustics); • the injectivity radius is larger than 2 (to ensure the “projectible” property). In our setting, these two properties remain true (in particular, a surface of nonpositive curvature has no conjugate points [11,19]). Finally, we underline that, thanks to the construction of the strong unstable foliation for surfaces ∗ M becomes uniformly close to of nonpositive curvature, any vector in Sz,η 1 the unstable subspace under the action of dρ g t (see Lemma 2.1). As a consequence, under the geodesic flow, a piece of sphere becomes uniformly close to the unstable foliation as j tends to infinity. This point is the main difference with [4]. In fact, if we consider an Anosov geodesic flow, we have the stronger property that a piece of sphere becomes exponentially close to the unstable foliation, as j tends to infinity. However, we will check that this property is sufficient for our needs. Remark. At this point of the proof, we can ask about an extension of these results to manifolds without conjugate points. According to [19], the “uniform divergence” property (given by Lemma 2.1) is true for surfaces without conjugate points and so a piece of sphere also becomes uniformly close to the unstable foliation in this more general setting. So this crucial aspect of the proof can be transposed in the setting of surfaces without conjugate points. In the next paragraph, we will use an additional argument which is specific to surfaces of nonpositive curvature. 4.3. Estimates on the Induced Jacobian As was already mentioned, the Jacobian JSt j of the map gst j appears in the WKB expansion of a Lagrangian state evolved under the action of the operator Pαj U η Pαj−1 · · · U η Pα0 . Precisely, by iterating the WKB Ansatz, we have to estimate the following quantity (see Eq. (3.22) in [4]): 12 −η −η −η (−k+2)η (x)) . Jk (x) := JS−η k−1 (x)JS k−2 (gS k (x)) · · · JS 1 (gS k
(32)
This Jacobian appears in each term of the WKB expansion of a Lagrangian state evolved under the operator Pαk U η Pαk−1 · · · U η Pα0 as every ap in the t expansion is defined using the operator TS(.) (see Definitions (30) and (31)). It is necessary to provide a way to bound this quantity as it will appear in the control of every derivatives of the WKB expansion. According to the proof in [4], if we are able to bound uniformly this quantity, the bound we will obtain is the one that will appear in Theorem 4.1. This point of the proof is the main difference with the proof in the Anosov case. Our goal in this paragraph is to provide an upper bound on (32). The quantity Jk (x) can be rewritten as
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1 −η −η −η −η (−k+2)η exp (x)) . log JS k−1 (x) + log JS k−2 (gS k (x)) · · · + log JS 1 (gS k 2 As the Lagrangian Lj become uniformly close to the unstable foliation when j tends to infinity, we know that, for every ε > 0, there exists some integer j(η, ε ) such that ∀j ≥ j(η, ε ),
−η ∀ρ = (x, ξ) ∈ Lj (0), | log JS−η j (x) − log JS u (ρ) (x)| ≤ ε ,
where S u (ρ) generates the local unstable manifold at point ρ (which is a Lagrangian submanifold). Therefore, we find that there exists a constant C(ε , η) (depending only on ε and η) such that, uniformly with respect to k and to ρ in Lk (0),
Jk (x) ≤ C(ε , η)ekε
k−1
JS−η u (g (−j+1)η ρ) (gS k
(−j+1)η
(x)) = C(ε , η)ekε JS u (ρ) (x). (1−k)η
j=0
In the following, we will take ε = η0 . The Jacobian JS−η u (ρ) measures the −η contraction of g along the unstable direction. From the construction of the unstable Riccati solution Uρu (s), we know that Uρu (s) also measures the contraction of g −η along E u (ρ). In fact, according to Sect. 2.1, one has −t u −t 1 + K0 e 0 Uρ (s)ds . dρ g|E u (ρ) ≤ As a consequence, there exists an uniform constant C (depending only on the manifold) such that: (1−k)η
(1−k)η
JS u (ρ) (x) ≤ Ce
0
Uρu (s)ds
.
Using then relation (6) between the discrete Riccati solution f and the continuous one, we find that there exists a constant C(, η, 0 ) such that, uniformly in k, ⎛ ⎞ k−1 1 f (σ j α)⎠ . sup Jk (x) ≤ C(, η, 0 )e2kη
0 exp ⎝− 2 j=0 x∈πLk (0) Finally, this last inequality gives us a bound on the quantity (32). This estimate is not as sharp as the one derived in [4] (Eq. 3.23 for instance) however it is sufficient as the correction term is not too large: it is of order − . Remark. We underline that we used the continuity of U u to go from the continuous representation of the upper bound of Jk to the one in terms of the discrete Riccati solution. This property fails for surfaces without conjugate points [5].
5. Applying the Uncertainty Principle for Quantum Pressures In this section, we would like to prove inequality (23) which was a crucial step of our proof. To do this, we follow the same lines as in [16, Sect. 5.3] and prove the following proposition:
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Proposition 5.1. With the notations of Sect. 3, one has: + − Σ Σ p μ + , C + p μ − , C ≥ − log C − (1 + + 4)nE (),
(33)
where p is defined by (21) and where C ∈ R∗+ does not depend on (but depends on the other parameters (, 0 , η)). To prove this result, we will proceed in three steps. First, we will introduce an energy cutoff in order to get the sharpest bound as possible in our application of the uncertainty principle. Then, we will apply the uncertainty + − Σ Σ principle and derive a lower bound on p(μ + , C ) + p(μ − , C ). Finally, we will use the estimate of Theorem 4.1 to conclude. 5.1. Energy Cutoff Before applying the uncertainty principle, we proceed to sharp energy cutoffs so as to get precise lower bounds on the quantum pressure (as it was done in [2–4]). These cutoffs are made in our microlocal analysis in order to get as good exponential decrease as possible of the norm of the refined quantum partition. This cutoff in energy is possible because even if the distributions μ are defined on T ∗ M , they concentrate on the energy layer S ∗ M . The following energy localization is made in a way to compactify the phase space and in order to preserve the semiclassical measure. Let δ0 be a positive number less than 1 and χδ0 (t) in C ∞ (R, [0, 1]). Moreover, χδ0 (t) = 1 for |t| ≤ e−δ0 /2 and χδ0 (t) = 0 for |t| ≥ 1. As in [4], the sharp -dependent cutoffs are then defined in the following way: ∀ ∈ (0, 1), ∀n ∈ N, ∀ρ ∈ T ∗ M, χ(n) (ρ, ) := χδ0 (e−nδ0 −1+δ0 (H(ρ) − 1/2)). For n fixed, the cutoff χ(n) is localized in an energy interval of size 2enδ0 1−δ0 centered around the energy layer E. In this paper, indices n will satisfy 2enδ0 1−δ0 1. It implies that the widest cutoff is supported in an energy interval of microscopic length and that n ≤ Kδ0 | log |, where Kδ0 ≤ δ0−1 . Using then a non standard pseudodifferential calculus (see [4] for a brief reminder of the procedure from [21]), one can quantize these cutoffs into pseudodifferential operators. We will denote Op(χ(n) ) the quantization of χ(n) . The main properties of this quantization are recalled in the appendix of [16]. In particular, the quantization of these cutoffs preserves the eigenfunctions of the Laplacian: Proposition 5.2. [4] For any fixed L > 0, there exists L such that for any ≤ L , any n ≤ Kδ | log | and any sequence β of length n, the Laplacian eigenstate verify 1 − Op χ(n) πβ ψ ≤ L ψ . 5.2. Applying Theorem A.1 Let ψ = 1 be a fixed element of the sequence of eigenfunctions of the Laplacian defined earlier, associated to the eigenvalue − 12 . To get bound on the pressure of the suspension measure, the uncertainty principle should not be applied to the eigenvectors ψ directly but it will be
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applied several times. Precisely, we will apply it to each Pγ ψ := Pγ1 Pγ0 (−η)ψ where γ = (γ0 , γ1 ) varies in {1, . . . , K}2 . In order to apply the uncertainty principle to Pγ ψ , we introduce new families of quantum partitions corresponding to each γ. Let γ = (γ0 , γ1 ) be an element of {1, . . . , K}2 . We define γ.α = (γ0 , γ1 , α ). Introduce the following families of indices: I (γ) := {(α ) : γ.α ∈ I η ()} , K (γ) := {(β ) : β .γ ∈ K η ()} . We underline that each sequence α of I η () can be written under the form γ.α where α ∈ I (γ). The same works for K η (). Operators can be associated to these new families, for α ∈ I (γ) and β ∈ K (γ), τ˜α = Pαn (nη) · · · Pα2 (2η), (−2η). (−nη) · · · Pβ−2 π ˜β = Pβ−n
πβ )β ∈I (γ) form quantum partitions of identity The families (˜ τα )α ∈I (γ) and (˜ [16]. Given these new quantum partitions of identity, the uncertainty principle should be applied for given initial conditions γ = (γ0 , γ1 ) at times 0 and 1. We underline that, for α ∈ I (γ) and β ∈ K (γ): τ˜α U −η Pγ = τγ.α U −η
and π ˜β Pγ = πβ .γ ,
(34)
where γ.α ∈ I () and β .γ ∈ K () by definition. Equality (34) justifies that the definitions of τ and π were slightly different (see (12) and (13)). It is due to the fact that we want to compose τ˜ and π ˜ with the same operator Pγ . Suppose now that Pγ ψ is not equal to 0. We apply the quantum uncertainty principle A.1 using that πβ )β ∈K (γ) are partitions of identity; • (˜ τα )α ∈I (γ) and (˜ • the cardinal of I (γ) and K (γ) is bounded by N −K0 where K0 is some fixed positive number (depending on the cardinality of the partition K, on a0 , on b0 and η); • Op(χ(k ) ) is a family of bounded operators Oβ (where k is the length of β ); η
η
b0
− 2 + − 0 ; • the constants Wγ.α and Wβ.γ are bounded by −1 L • the parameter δ can be taken equal to Pγ ψ where L is such that b0
L−K0 − 0 e2kη
0 −1/2−cδ0 for every k and the upper bound in Theorem 4.1); • U −η is an isometry; P ψ • ψ˜ := Pγγ ψ is a normalized vector.
1
η | log |
(see Proposition 5.2
Applying the uncertainty principle A.1 for quantum pressures, one gets: Corollary 5.3. Suppose that Pγ ψ is not equal to 0. Then, one has b0 pτ˜ (U −η ψ˜ ) + pπ˜ (ψ˜ ) ≥ −2 log cγχ (U −η ) + L−K0 − 0 Pγ ψ −1 ,
+ − where cγχ (U −η ) = maxα ∈I (γ),β ∈K (γ) (Wγ.α τα U −η π ˜β∗ Op(χ(k ) )). Wβ .γ ˜
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Under this form, the quantity Pγ ψ −1 appears several times and we would like to get rid of it. First, remark that the quantity cγχ (U −η ) can be bounded by + − cχ (U −η ) := max Wγ.α τα U −η π ˜β∗ Op(χ(k ) ) , (35) Wβ .γ ˜ (∗)
where (∗) means (γ ∈ {1, . . . , K}2 and α ∈ I (γ), β ∈ K (γ)). This last quantity is independent of γ. Then, one has the following lower bound: b0 −2 log cγχ (U −η ) + L−K0 − 0 Pγ ψ −1 b0 ≥ −2 log cχ (U −η ) + L−K0 − 0 + 2 log Pγ ψ 2 . (36) as Pγ ψ ≤ 1. Now that we have given an alternative lower bound, we rewrite the entropy term hτ˜ (U −η ψ˜ ) of the quantum pressure pτ˜ (U −η ψ˜ ) as follows: hτ˜ (U −η ψ˜ ) = − ˜ τα U −η ψ˜ 2 log ˜ τα U −η Pγ ψ 2 α ∈I (γ)
+
˜ τα U −η ψ˜ 2 log Pγ ψ 2 .
α ∈I (γ)
τα )α ∈I (γ) is a Using the fact that ψ is an eigenvector of U η and that (˜ partition of identity, one has: 1 hτ˜ (U −η ψ˜ ) = − τγ.α ψ 2 log τγ.α ψ 2 + log Pγ ψ 2 . 2 Pγ ψ α ∈I (γ)
The same holds for the entropy term hπ˜ (ψ˜ ) of the quantum pressure pπ˜ (ψ˜ ) (using here equality (34)): 1 πβ .γ ψ 2 log πβ .γ ψ 2 + log Pγ ψ 2 . hπ˜ (ψ˜ ) = − Pγ ψ 2 β ∈K (γ)
Combining these last two equalities with (36), we find that + − τγ.α ψ 2 log τγ.α ψ 2 − 2 τγ.α ψ 2 log Wγ.α α ∈I (γ)
−
β ∈K (γ)
α ∈I (γ)
πβ .γ ψ 2 log πβ .γ ψ 2 − 2
≥ −2Pγ ψ 2 log cχ (U −η ) +
b L−K0 − 0 0
πβ .γ ψ 2 log Wβ− .γ
β ∈K (γ)
.
(37)
This expression is very similar to the definition of the quantum pressure. We also underline that this lower bound is trivial in the case where Pγ ψ is equal to 0. Using the following numbers: f (γ) , 2 γ ∈{1,...,K}2 f (γ )Pγ ψ
cγ.α = cβ .γ = cγ =
one can derive, as in [16], the following property:
(38)
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Corollary 5.4. One has: b0 + − Σ Σ p μ + , C + p μ − , C ≥ −2 log cχ (U −η ) + L−K0 − 0 − C,
(39)
where C := log(maxγ cγ ). As expected, by a careful use of the entropic uncertainty principle, we Σ have been able to obtain a lower bound on the pressures of the measures μ + Σ
and μ − . 5.3. The Conclusion To conclude the proof of Proposition 5.1, we use Theorem 4.1 to give an upper b0
bound on cχ (U −η ). From our assumption on L, we know that L−K0 − 0 cχ (U −η ). As kη ≤ nE ()/0 , we also have that 1
cχ (U −η ) ≤ CK (, η, 0 )− 2 −cδ0 e4 nE () .
For δ0 small enough, we find the expected property.
Acknowledgements I would like to sincerely thank my advisor Nalini Anantharaman for introducing me to this question and for encouraging me to extend the result from [16] to nonpositively curved surfaces. I also thank her for many helpful discussions about this subject.
Appendix A. Uncertainty Principle for the Quantum Pressure In [4], generalizations of the entropic uncertainty principle were derived for quantum pressures. We saw that the use of this thermodynamic formalism was crucial in our proof and we recall in this section the main results from [4, Sect. 6] on quantum pressures. Consider two partitions of identity (πk )N k=1 2 and (τj )M j=1 on L (M ), i.e. N
πk∗ πk = IdL2 (M )
k=1
and
M
τj∗ τj = IdL2 (M ) .
j=1
M We also introduce two families of positive numbers: (Vk )N k=1 and (Wj )j=1 . We denote A := maxk Vk and B := maxj Wj . One can then introduce the quantum pressures associated to these families, for a normalized vector ψ in L2 (M ),
pπ (ψ) := −
N
πk ψ2L2 (M ) log πk ψ2L2 (M ) − 2
k=0
N
πk ψ2L2 (M ) log Vk
k=0
and pτ (ψ) := −
M j=0
τj ψ2L2 (M ) log τj ψ2L2 (M ) − 2
M
τj ψ2L2 (M ) log Wj .
j=0
The main result on these quantities that was derived in [4] was Theorem 6.5:
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Theorem A.1. Under the previous setting, suppose U is an isometry of L2 (M ) and suppose (Ok )N k=1 is a family of bounded operators. Let δ be a positive number and ψ be a vector in H of norm 1 such that (Id − Ok )πk ψL2 (M ) ≤ δ . Then, one has pτ (Uψ) + pπ (ψ) ≥ −2 log (cO (U) + max(N , M)ABδ ) , where cO (U) := supj,k {Vk Wj τj Uπk∗ Ok }.
Appendix B. Kolmogorov–Sinai Entropy Let us recall a few facts about Kolmogorov–Sinai (or metric) entropy that can be found for example in [22]. Let (X, B, μ) be a measurable probability space, I a finite set and P := (Pα )α∈I a finite measurable partition of X, i.e. a finite collection of measurable subsets that forms a partition. Each Pα is called an atom of the partition. Assuming 0 log 0 = 0, one defines the entropy of the partition as: μ(Pα ) log μ(Pα ) ≥ 0. (40) H(μ, P ) := − α∈I
Given two measurable partitions P := (Pα )α∈I and Q := (Qβ )β∈K , one says that P is a refinement of Q if every element of Q can be written as the union of elements of P and it can be shown that H(μ, Q) ≤ H(μ, P ). Otherwise, one denotes P ∨ Q := (Pα ∩ Qβ )α∈I,β∈K their join (which is still a partition) and one has H(μ, P ∨ Q) ≤ H(μ, P ) + H(μ, Q) (subadditivity property). Let T be a measure preserving transformation of X. The n-refined partition −i P of P with respect to T is then the partition made of the atoms ∨n−1 i=0 T (Pα0 ∩ · · · ∩ T −(n−1) Pαn−1 )α∈I n . We define the entropy with respect to this refined partition: S μ(Pα0 ∩ · · · ∩ T −(n−1) Pαn−1 ) , (41) Hn (μ, T, P ) := |α|=n
where S(x) := −x log x. Using the subadditivity property of entropy, we have for any integers n and m: Hn+m (μ, T, P ) ≤ Hn (μ, T, P ) + Hm (μ ◦ T −n , T, P ) = Hn (μ, T, P ) + Hm (μ, T, P ).
(42)
For the last equality, it is important to underline that we really use the T -invariance of the measure μ. A classical argument for subadditive sequences allows us to define the following quantity: Hn (μ, T, P ) . (43) n It is called the Kolmogorov Sinai entropy of (T, μ) with respect to the partition P . The Kolmogorov Sinai entropy hKS (μ, T ) of (μ, T ) is then defined as the supremum of hKS (μ, T, P ) over all partitions P of X. hKS (μ, T, P ) := lim
n→∞
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A property of entropy we used in the paper is the so-called Abramov property [1]. Using the notations of the article, one has ⎞ ⎛
⎜ +⎟ hKS μΣ+ , σ+ = ⎝ f+ dμΣ ⎠ hKS μΣ+ , σ + . (44) Σ+
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[20] Shnirelman, A.: Ergodic properties of eigenfunctions. Usp. Math. Nauk. 29, 181– 182 (1974) [21] Sj¨ ostrand, J., Zworski, M.: Asymptotic distribution of resonances for convex obstacles. Acta Math. 183, 191–253 (1999) [22] Walters, P.: An Introduction to Ergodic Theory. Springer, Berlin (1982) [23] Zelditch, S.: Uniform distribution of the eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55, 919–941 (1987) Gabriel Rivi`ere Centre de Math´ematiques Laurent Schwartz (UMR 7640) ´ Ecole Polytechnique 91128 Palaiseau Cedex, France e-mail:
[email protected] Communicated by Viviane Baladi. Received: December 4, 2009. Accepted: July 19, 2010.
Ann. Henri Poincar´e 11 (2010), 1117–1169 c 2010 Springer Basel AG 1424-0637/10/061117-53 published online August 19, 2010 DOI 10.1007/s00023-010-0047-2
Annales Henri Poincar´ e
Lorentz Gas with Thermostatted Walls Nikolai Chernov and Dmitry Dolgopyat Abstract. In a planar periodic Lorentz gas, a point particle (electron) moves freely and collides with fixed round obstacles (ions). If a constant force (induced by an electric field) acts on the particle, the latter will accelerate, and its speed will approach infinity (Chernov and Dolgopyat in J Am Math Soc 22:821–858, 2009; Phys Rev Lett 99, paper 030601, 2007). To keep the kinetic energy bounded one can apply a Gaussian thermostat, which forces the particle’s speed to be constant. Then an electric current sets in and one can prove Ohm’s law and the Einstein relation (Chernov and Dolgopyat in Russian Math Surv 64:73–124, 2009; Chernov et al. Comm Math Phys 154:569–601, 1993; Phys Rev Lett 70:2209–2212, 1993). However, the Gaussian thermostat has been criticized as unrealistic, because it acts all the time, even during the free flights between collisions. We propose a new model, where during the free flights the electron accelerates, but at the collisions with ions its total energy is reset to a fixed level; thus our thermostat is restricted to the surface of the scatterers (the ‘walls’). We rederive all physically interesting facts proven for the Gaussian thermostat in Chernov, Dolgopyat (Russian Math Surv 64:73–124, 2009) and Chernov et al. (Comm Math Phys 154:569–601, 1993; Phys Rev Lett 70:2209–2212, 1993), including Ohm’s law and the Einstein relation. In addition, we investigate the superconductivity phenomenon in the infinite horizon case.
1. Introduction and Historic Overview We study a two-dimensional periodic Lorentz gas. It consists of a particle that moves on a plane between a periodic array of fixed scatterers (the latter are disjoint convex domains with C 3 boundaries). The particle bounces off the scatterers according to the rule ‘the angle of incidence is equal to the angle of reflection’. The particle may be subject to an external force; see Fig. 1. Lorentz gas models the motion of electrons in metals [25]. The authors are grateful to the anonymous referees for many useful remarks and suggestions. N. Chernov acknowledges the hospitality of the University of Maryland. N. Chernov was partially supported by NSF grant DMS-0652896. D. Dolgopyat was partially supported by NSF grant DMS-0555743.
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X
Figure 1. Lorentz gas with a finite horizon and an external force ˜ the area available to the moving particle, i.e., the plane We denote by D R minus the union of all the scatterers. Due to the periodicity, the plane can be covered by replicas of a fundamental domain (unit cell) K ⊂ R2 so that ˜ is the union of disjoint copies of D = D ˜ ∩ K. We denote those copies by D ˜ = ∪i Di . Let π Di (we number them arbitrarily), i.e., D ˜ denote the natural ˜ onto D. We will use tildas for objects on the unbounded table. projection of D ˜ by q ˜ and those in D by q. In particular, we denote points in D 2
Lorentz gases without external forces. First we review some known facts. Suppose the particle moves freely between collisions, then its speed remains con˜ the trajectory of stant (and we set it to unity). Due to the periodicity of D, the particle can be projected onto D, and one gets a billiard in the compact table D with periodic boundary conditions. It is known as Sinai billiard and has been intensively studied since 1970; see [29]. Let q = q(t) denote the position and v = dq/dt the velocity of the moving particle in D. Since v = 1, the phase space is a 3D compact manifold Ω = D × S1 . The resulting flow Φt on Ω is Hamiltonian; it preserves the Liouville measure μ, which has a uniform density on Ω. It is common to study flows by using a cross-section and the respective return map. In billiards, those can be constructed naturally on the collision space M = {(q, v) : q ∈ ∂D, v = 1, v points inside D} consisting of all post-collisional vectors. The return map F : M → M takes the particle right after a collision to its state right after the next collision; F is called the collision map. We use the standard coordinates (r, ϕ) in M, where r is the arclength parameter on ∂D and ϕ ∈ [−π/2, π/2] the angle between the outgoing velocity vector v and the outward normal to ∂D at the collision point q; cf. [7,17] and Fig. 3 below. The map F preserves a smooth probability measure ν on M given by
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dν = cν cos ϕ dr dϕ,
1119 −1
cν = [2 · length(∂D)]
(1.1)
here cν is just the normalizing factor. For every X = (q, v) ∈ M let τ (X) = min{t > 0 : Φt (X) ∈ M}
(1.2)
denote the time of the first collision of the trajectory starting at X. Now Φt becomes a suspension flow constructed over the base map F : M → M under the ceiling function τ . Sinai proved [29] that the billiard flow Φt and the collision map F are hyperbolic (i.e., have non-zero Lyapunov exponents), ergodic, and K-mixing. Gallavotti and Ornstein [23] derived the Bernoulli property. Bunimovich and Sinai [6,7] studied the Central Limit Theorem and other statistical laws. Young [31] and Chernov [8] established an exponential decay of correlations for the map F; see [10] for bounds on correlations for the flow Φt . In other words, the Sinai billiards are highly chaotic in every mathematical sense. Diffusion. We are mostly interested in the dynamics of the particle on the ˜ We denote by q ˜ ˜ (t) the position of the moving particle in D, infinite table D. ˜ ˜ n the point of its nth collision with ∂ D. Typically, both ˜ and by q q(t) and ˜ qn grow to infinity as time goes on. ˜ n+1 − q ˜ n denote the displacement vector between collisions. Let Δn = q ˜ 0 + Δ0 + Δ√ ˜n = q Note that q 1 + · · · + Δn−1 , thus based on the central limit ˜ n / n to converge to a normal distribution, which theorem one may expect q would be just a classical diffusion law. This indeed happens when the horizon is finite, i.e., when the free path between collisions with scatterers is bounded. Figure 1 illustrates this situation—the scatterers are large enough to block the particle from all sides, so that it cannot move indefinitely without collisions. (We note, however, that Fig. 1 shows a trajectory affected by an external field, while in this subsection we describe a field-free model whose trajectories between collisions must be straight lines.) ˜ have finite horizon. Suppose that the initial position Theorem 1 ([6,7]). Let D ˜ (0) and velocity v(0) are chosen according to a smooth, compactly supported q probability measure. Then √ ˜ n / n converges, as n → ∞, to a normal distribution, i.e., (a) q √ ˜ n / n ⇒ N (0, D) q with a non-degenerate covariance matrix D called diffusion matrix given by the Green-Kubo formula: D=
∞
ν (Δ0 ⊗ Δn )
(1.3)
n=−∞
where u ⊗ v denotes the ‘tensor product’ of two vectors, i.e., the product of the√column-vector u and the row-vector v. ˜ (t)/ t converges, as t → ∞, to another normal distribution, (b) q √ ˜ (t)/ t ⇒ N (0, τ¯−1 D) q
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x
Figure 2. Lorentz gas with infinite horizon where τ¯ = ν(τ ) is the mean free path given by τ¯ = ν(τ ) =
π Area(D) . length(∂D)
(1.4)
The series (1.3) converges exponentially fast. See a recent exposition of these facts in [17, Chapter 7]; for the proof of (1.4), see [17, Sect. 2.13]. Super-diffusion. When the horizon is not finite, the particle can move freely without collisions along infinite corridors stretching between the scatterers, see Fig. 2 (though see the remark before Theorem 1, it applies here, too). Now Theorem 1 cannot hold because the central term in the Green-Kubo series (1.3) diverges: ν (Δ0 ⊗ Δ0 ) = ∞; thus the diffusion matrix (1.3) turns infinite. In this case the particle exhibits an abnormal diffusion (often called ‘superdiffusion’), namely the correct √ √ scaling factor is now t log t, rather than t: ˜ have infinite horizon. Suppose that the initial Theorem 2 ([15,30]). Let D ˜ (0) and velocity v(0) are chosen according to a smooth compactly position q supported probability measure. Then, (a) We have the following weak convergence, as n → ∞: ˜ q √ n ⇒ N (0, D∞ ), n log n where D∞ is called superdiffusion matrix given by the formula (1.5) below. D∞ is non-degenerate iff there are two non-parallel corridors. (b) We have the following weak convergence, as t → ∞: ˜ (t) q √ ⇒ N (0, τ¯−1 D∞ ), t log t where τ¯ is again given by (1.4).
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Part (a) is proved by Sz´ asz and Varj´ u [30], and part (b) by the present authors [15]. The matrix D∞ is given by a simple explicit formula in terms of the geometric characteristics of infinite corridors. Namely, suppose that each ˜ is bounded by two straight lines, each of which is tangent infinite corridor in D to an infinite row of scatterers that are copies of one scatterer in D, then such lines are trajectories of some fixed points X ∈ M : F(X) = X (one such point is shown in Fig. 8). Now we have cν w2 X Δ(X) ⊗ Δ(X). (1.5) D∞ = 2 Δ(X) X
where the summation is taken over all corridors and all respective fixed points (note that there are four points X for each corridor). Here, Δ(X) = Δ0 (X) is the displacement vector of X introduced earlier and wX is the width of the corridor bounded by the trajectory of X. Lorentz gases under external forces. Suppose a constant force E acts on the particle, i.e., its motion is governed by equations d˜ q/dt = v,
dv/dt = E.
(1.6)
It is common to interpret E as an electrical field that drives electrons and produces electrical current. Another popular physical model of this sort is Galton board [24]. It is commonly pictured as an upright wooden board with rows of pegs on which a ball rolls down under the gravitation force and bounces off the pegs. See recent studies in [13,14]. For convenience we choose the coordinate system so that the x axis is aligned with the field, i.e., E = (E, 0) for some constant E > 0. Illustrations in Figs. 1 and 2 show the trajectories of the particle affected by an external force. Equations (1.6) preserve the total energy 1 ˜ = E = const. v2 − E, q (1.7) 2 In our coordinates the conservation law takes form 1 v2 − Ex = E = const, (1.8) 2 where x denotes the x-coordinate of the particle. As the particle is driven forward by the field, its x-coordinate grows, and so does its speed v = v. Despite the periodicity of the scatterers, the dynamics is not periodic, i.e., it is not a factor of any dynamical system in the domain D with periodic ˜ of the system is an unbounded boundary conditions. In fact, the phase space Ω 3D manifold ˜ E = {(˜ ˜ v2 = 2(Ex + E)}. ˜ = (x, y) ∈ D, Ω q, v) : q ˜ depends One can say that the motion of the particle on each domain Di ⊂ D on where Di is located. More precisely, it depends on the x-coordinate of points ˜ ∈ Di ; the larger their x-coordinates, the faster the particle moves. q Suppose the particle is √ in Di at time t0 , its x-coordinate is x0 and its speed v0 (it is roughly v0 ∼ 2Ex0 ). If we change the time variable by tˆ = v0 t,
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then the particle will have unit speed vˆ0 = 1 at time tˆ0 , and it will move in a ˆ = E/v0 . Otherwise its trajectory will be the same. Therefore, rescaled field E we can approximate the particle’s trajectories in the domain Di with large x-coordinates by those of the particle moving in D with speed close to one, but in a weak external field. In other words, the particle moves as if it always had unit speed but the ˆ is space inhomogeneous—it gets weaker as the x-coordinate grows, so field E √ ˆ that E = O(1/ x). Hence, as the particle moves on, it will behave more and more like a billiard particle. Since the net displacement of the latter is zero, the overall drift of our particle will tend to slow down. As a result, the x-coordinate will not grow linearly in t. A detailed analysis shows that x will only grow as t2/3 , on average: ˜ have finite horizon. Suppose that the initial position Theorem 3 ([13]). Let D ˜ (0) and velocity v(0) are chosen according to a smooth compactly supported q measure on an energy surface {E = E0 } where E0 is sufficiently large. Then there is a constant c > 0 such that ct−2/3 x(t) converges, as t → ∞, to a random variable with density 3 exp −z 3/2 , z ≥ 0. 2Γ(2/3) An explicit formula for c is given in [13,15]. There exists a limiting distribution for t−2/3 y(t), too, but it is given by a more complicated expression ˜ with infinite horizon remains [14]. The extension of this theorem to tables D an open problem. Our paper is motivated by an attempt to solve this problem—we believe our results will be useful. Gaussian thermostat. Equations (1.6) and the resulting dynamics (Theorem 3) clearly do not give a realistic description of the electrical current—the real electrons do not accelerate indefinitely, and their average drift must be linear in t. To keep the energy of the moving particle fixed (and to make its drift proportional to t), Moran and Hoover [27] modified equations (1.6) as follows: d˜ q/dt = v,
dv/dt = E − ζv,
(1.9)
where ζ = E, v /v . The friction term ζv is called the Gaussian thermostat; it ensures that v = const at all times; again we will assume that v = 1. We also assume that the field is weak, i.e., E = (ε, 0) for a small ε > 0. The resulting dynamics is now truly periodic—it can be projected onto the domain D with periodic boundary conditions; then, we obtain a system with the same phase space Ω = D × S1 and the same collision space M as for the billiards discussed earlier. We denote by Φtε the resulting flow on Ω and by Fε the resulting collision map on M. Also let τε (X) denote the time of the first collision of the trajectory starting at X ∈ M. And we denote ˜1 − q ˜ 0 the displacement of the particle moving in the infinite by Δε (X) = q ˜ before its next collision at ∂ D. ˜ domain D 2
˜ have finite horizon and ε be sufficiently small. Theorem 4 ([18,19]). Let D Then Fε is a hyperbolic map; it preserves a Sinai–Ruelle–Bowen (SRB)
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measure (a steady state) νε , which is ergodic and mixing. It is singular but positive on open sets. The flow Φtε also preserves an SRB measure με on Ω, which is ergodic, mixing, and positive on open sets. The electrical current ˜ (t)/t = με (v) = νε (Δε )/¯ τε J = lim q t→∞
is well defined; here τ¯ε = νε (τε ) is the mean free path for which we have τ¯ε = τ¯ + O(ε). We have 1 DE + o(ε), (1.10) J= 2¯ τ where D is the diffusion matrix of Theorem 1. We also have the following weak convergence, as t → ∞: ˜ (t) − Jt q √ ⇒ N (0, D∗ε ), t where D∗ε is the corresponding diffusion matrix satisfying D∗ε = τ¯−1 D + o(1).
(1.11)
In physical terms, (1.10) can be regarded [18,19] as classical Ohm’s law: the electrical current J is proportional to the voltage E (to the leading order). 1 The fact that the electrical conductivity, i.e., 2¯ τ D in (1.10), is proportional to the diffusion matrix D is known as Einstein relation. If the horizon is not finite, then there are infinite corridors between scatterers where the electrons move without collisions. Due to this ballistic motion, the current and diffusion become abnormal in the following exact sense: ˜ have infinite horizon. Assume that the force E = (ε, 0) Theorem 5 ([15]). Let D is not parallel to any of the infinite corridors and ε is sufficiently small. Then, Fε is a hyperbolic map; it preserves an SRB measure (steady state) νε , which is ergodic, mixing, and positive on open sets. The flow Φtε also preserves an SRB measure με on Ω, which is ergodic, mixing, and positive on open sets. The electrical current ˜ (t)/t = με (v) = νε (Δε )/¯ τε J = lim q t→∞
(1.12)
is well defined; here τ¯ε = νε (τε ) is the mean free path for which we have τ¯ε = τ¯ + O(εa ) for some a > 0. We have 1 | log ε| D∞ E + O(ε) (1.13) J= 2¯ τ where D∞ is the superdiffusion matrix of Theorem 2. We also have the following weak convergence, as t → ∞: ˜ (t) − Jt q √ (1.14) ⇒ N (0, D∗ε ), t where D∗ε is the corresponding diffusion matrix satisfying D∗ε = τ¯−1 | log ε| D∞ + O(1).
(1.15)
Observe that now the current becomes proportional to ε| log ε|, rather than ε, i.e., Ohm’s law fails in this regime. This mathematical fact may be
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related to the physical phenomenon of ‘superconductivity’. At low temperatures (near absolute zero), ions tend to form an almost perfect crystal structure with long corridors in between resembling our infinite horizon model. Thus, the electron tends to travel fast and one observes superconductivity. On the other hand, at normal temperatures, ions are somewhat agitated; their configuration is more randomized, which creates an effect of finite horizon. It slows the electron down and one observes a normal current. To summarize, the Gaussian thermostatted dynamics (1.9) allows us to obtain an adequate description of the ordinary electrical current when the horizon is finite (Theorem 4). It also can be related to superconductivity in the infinite horizon case (Theorem 5). However, the Gaussian thermostat may be criticized as an unrealistic and artificial device—it acts on the particle all the time, even during the ‘free flights’ between collisions (thus violating Newton’s law (1.6)). It is perhaps more reasonable to allow the electron move freely (and accelerate naturally) between collisions, but remove the excess of its energy when it collides with heavy immovable ions. In other words, the thermostat should be placed on the ˜ boundaries of the ions (i.e., at the walls of D). Our objectives. In this paper we propose a dynamics of that kind. Our electrons move freely, according to Newton’s law (1.6), so that their kinetic energy may grow. But at every collision with ions the speed of the electron is reset so that it remains bounded. In fact if the electron was transported back to the original cell after every collision, then its total energy would remain fixed. ˜ are kept specular, i.e., the angle of incidence is still Reflections off the wall ∂ D equal to the angle of reflection. Our ultimate goal is to show that all the conclusions of Theorems 4 and 5 remain valid in this new context. Thus, we present a more realistic version of the Lorentz gas in a constant external field than the Gaussian thermostatted model studied in [13,18,19]. Our dynamics with thermostatted walls constitutes the physical novelty of the paper. Besides being physically more realistic, our dynamics also has mathematical advantages compared with the Gaussian thermostat. Namely, the Galton board can be regarded as a slow–fast system where the kinetic energy is a slow variable, while particle’s position in D and its velocity direction are fast variables. The velocity vector of the Gaussian thermostatted model (1.9) is simply obtained by projecting the velocity vector (1.6) onto the surface of constant kinetic energy; therefore, the Gaussian thermostatted model represents the so-called frozen system (its slow variable is rigidly fixed, i.e., frozen). Frozen systems may provide good approximations to slow–fast systems, and such an approximation played a fundamental role in our studies of the Galton board with finite horizon [13]. However, in the infinite horizon case this approximation turns out to be poor, because during long free flights in infinite corridors small errors accumulate and grow too much. Here we reset the energy only at the time of collisions, so our new model provides a much better approximation to the Galton board dynamics.
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Our mathematical constructions and arguments are also unusual in many ways. In all the previous studies [9,11,13,15,18] of billiard-like particles moving under external forces, the dynamics was time-reversible, or at least invertible. That is, the flow Φt was well defined for all −∞ < t < ∞, and the collision map F was invertible, i.e., F −1 existed. Thus, the Lyapunov exponents could be computed and the dynamics could be (and usually were) hyperbolic. Then, all the mathematical tools developed for hyperbolic systems (Markov partitions, Young tower construction, Coupling method, etc.) could be applied with a remarkable success resulting in Theorems 3–5. Our model is different. The energy of the electron is ‘reset’ at every collision to a fixed level; thus it ‘forgets’ its past. As a result, its past trajectory cannot be uniquely recovered from its present state. The dynamics ceases to be invertible—some phase points have multiple preimages, while others have none. Lyapunov exponents can only be computed under forward iterations (but not under backward iterations), and the system cannot be hyperbolic in the ordinary sense. More precisely, our collision map F : M → M will be piecewise smooth in the sense that M will be a finite union of subdomains, M = ∪i M+ i , on each of which F will be smooth and diffeomorphic, but the images F(M+ i ) + and F(M+ j ) will overlap for some i = j. Furthermore, their union ∪i F(Mi ) will not cover M entirely—there will be open gaps (or ‘cracks’) in between. In the gaps, F −1 will not be defined; thus there will be no unstable manifolds. At points in the overlapping regions, F −1 will be multiply defined; thus unstable manifolds will not be unique. Dynamically, our map F resembles a hyperbolic map with singularities, in particular it has stable and unstable cones (though defined only under forward iterations). Stable manifolds are well defined, but unstable manifolds may not exist or may not be unique. In a way, our map F can be compared to non-invertible expanding maps where only the future evolution is uniquely defined. In other words, our map F does not belong to any standard class of chaotic dynamical systems; it is somewhere ‘between’ invertible hyperbolic maps and non-invertible expanding maps (though much closer to the former than to the latter). Such maps are interesting from a purely dynamical point of view, but very little is known about them. Many examples of such maps can be easily constructed. For one, let F0 : T → T be a hyperbolic toral automorphism of a unit 2-torus; let T = M1 ∪ · · · Mk be a finite partition of T into domains with piecewise smooth boundaries, and let F1 : T → T be a map that is smooth on each Mi and its restriction to Mi is a C 2 -perturbation of the identity map on Mi . Then F = F0 ◦ F1 is a map of the sort described above – it has strong hyperbolic features, but the images of Mi may overlap and/or leave uncovered gaps. Despite the simplicity of such examples, they were rarely studied in the past. In recent papers [2,3], Baladi and Gou¨ezel considered non-invertible maps with stable and unstable cones and bounded derivatives; they used operator techniques to derive the existence and ‘finitude’ of physical invariant measures.
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A particular countable-to-one map with stable and unstable cones was investigated in [16], where natural extensions were used to cope with non-invertibility. In our current model, the derivatives are unbounded, so we have to use methods different from [2,3]. We develop a general approach to the construction of SRB-like (“physical”) invariant measures which works for countable-to-one maps and does not require strong a priori bounds on the probabilities of particular inverse branches which were needed in [16]. So, we develop alternative methods that hopefully will be useful for future similar studies. Our ideas are similar to those developed in [1] in the context of partially hyperbolic diffeomorphisms. We also note that [2,3,16] deal with statistical properties of individual systems, while we study a continuous family of systems. Its dependence on the parameter ε plays an important role. The question of smoothness of SRB measures for piecewise hyperbolic systems is far from being settled, see discussions in [4,28]. Our model provides a new example where transport theory works (i.e., Ohm’s law and the Einstein relation hold), so we hope it could be useful for the development of a more general theory. In the next section we precisely describe our dynamics with ‘thermostatted walls’ and state our main results.
2. The Model and Main Results We define the dynamics in D as follows: First we fix an external field E = (ε, 0); here ε > 0 is a small constant. In our dynamics the particle has a fixed total energy so that v2 − 2εx = 1, cf. (1.8), and moves under the field E between collisions. More precisely, let q = (x, y) ∈ ∂D be a point where the particle collides with a scatterer and v denotes the outgoing velocity vector. Since the total energy is fixed, we have √ (2.1) v = v = 1 + 2εx. Leaving the scatterer, the particle moves under the external field E according to the standard equations dq/dt = v,
dv/dt = E,
(2.2)
until it collides with another scatterer. The motion between collisions is smooth, so that it is better visualized if we allow the particle move on the ˜ In particular, if the particle crosses the border of the unbounded domain D. fundamental cell K, it leaves D and enters another domain Di , etc. In this way the x-coordinate of the moving particle changes continuously, and so does its speed, according to (2.1). But when the particle collides with a scatterer in ˜ it is instantaneously projected back onto D, under R2 , i.e., when its hits ∂ D, the natural projection π ˜ . As we apply the projection π ˜ , the x-coordinate of the particle can change; thus we need to adjust its speed, too, in order to keep the total energy fixed, according to (2.1); when adjusting the speed, we keep the direction of the velocity vector unchanged. Then the motion continues.
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In this way our obstacles act not only as scatterers but also as ‘heat baths’ or ‘thermostats’, which remove the excess of the particle’s kinetic energy and keep it bounded (but not constant). Remark. It is tempting to define a thermostat simply by resetting the kinetic energy (i.e., the speed) of the particle to a constant value after every collision. We will show that such dynamics would cause formidable complications, though. In fact the only way to keep the features of the model tractable is to reset the total energy in the above sense; see Appendix. Remark. Our dynamics clearly depends on the choice of the fundamental domain, which is far from unique, but this choice does not affect our main results. For example, if K = [0, 1] × [0, 1] is a unit square, then one can alternatively define K = [a, 1 + a] × [b, 1 + b] for any a, b > 0. In that case the trajectories of our particle would change by O(ε2 ), which is too small to affect our results. (Loosely speaking, our dynamics is a O(ε)-perturbation of the field-free billiard system, and further changes of order O(ε2 ) are negligible.) One can also combine several copies of K with a bigger fundamental domain. For example, if K = [0, 1] × [0, 1], then one can define K = [0, m] × [0, n] for any positive integers m, n ≥ 1. The resulting changes in the dynamics are not essential either, as we will explain at the end of Sect. 5. It is important to estimate how far the particle can travel between collisions and thus how much its speed can change. In the finite horizon case, the free path is bounded; hence the speed remains 1 + O(ε). In the infinite horizon case, we assume that the corridors between scatterers are not parallel to the field E. Lemma 2.1. The longest free path (i.e., the length of the trajectory segment between consecutive collisions) is O(ε−1/2 ). Thus, the speed of the particle between collisions remains 1 + O(ε1/2 ). At the same time, its speed right after each collision is 1 + O(ε), according to (2.1). Proof. Between collisions the particle moves along parabolas. The proof requires elementary calculations which we leave to the reader. We see that the magnitude of the velocity vector remains close to one (and is completely determined by the x-coordinate of the particle). Hence the velocity vector is specified uniquely by its direction, the fact we use below. We can define the collision space by √ M = (q, v) : q = (x, y) ∈ ∂D, v = 1 + 2εx, v points inside D . We will use standard coordinates r, ϕ on M, where r denotes the arclength parameter on the boundary ∂D and ϕ the angle between the outgoing velocity vector v and the normal vector to ∂D pointing inside D; thus ϕ ∈ [−π/2, π/2]. We orient the coordinates so that r increases when one traverses ∂D in such a way that the domain D remains on the left-hand side, and the value ϕ = π/2 corresponds to direction of growth of r; see Fig. 3.
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Figure 3. Orientation of r and ϕ Note that the domain of the variables r, ϕ does not depend on ε. In the coordinates r, ϕ the space M becomes a rectangle M = [0, L] × [−π/2, π/2],
where L = length(∂D)
(topologically, it is rather a union of cylinders, as r is a cyclic coordinate on each scatterer). Thus, we regard M as independent of ε, so the collision map Fε : M → M acts on the same space for all small ε. Note that F0 = F is the billiard collision map discussed earlier. In a sense, Fε is a small perturbation of F0 , see below. Next, for every point X ∈ M we denote by τε (X) the time it moves until ˜ Thus, we obtain a suspension flow Φt over the map the next collision with ∂ D. ε Fε under the function τε . We denote by Ω = {(X, t) : X ∈ M, 0 ≤ t ≤ τε (X)} the phase space of the suspension flow. Note that now our trajectories are parameterized by t. Last, we lift our dynamics from the compact domain D with periodic ˜ This can be done naturally boundary conditions to the unbounded table D. as follows: Recall that the particle starts at a point q = (x, y) on a scatterer √ in D with initial speed v0 = 1 + 2εx. It moves according to (2.2) until its ˜ 1 = (x1 , y1 ) on another trajectory lands at a point q √ scatterer in some domain − D(1) ⊂ R2 . Before the collision, its speed is v = 1 + 2εx1 . After the collision 1 (1)
(1)
its speed is reset to v1+ = 1 + 2ε(x1 − Zx ), where Zx denotes the displacement, in the x direction, between the domains D and D(1) (this means that D(1) is obtained from D by translation along vector Z(1) whose x-coordinate (1) is denoted by Zx ). ˜ 1 ∈ D(1) to q ˜ 1 −Z(1) ∈ Instead of translating the particle’s position from q (0) D , as we did before, we now let it continue its motion in the unbounded ˜ i.e., its trajectory will run from q ˜ 1 , with initial speed v1+ , until it domain D, (2) 2 hits a scatterer in another domain D ⊂ R , and we denote the collision point ˜ 2 = (x2 , y2 ). By that time its speed is v2− = 1 + 2ε(x2 − Zx ), but we by q (2) (2) reset it to v2+ = 1 + 2ε(x2 − Zx ), where Zx is the displacement, in the x direction, between the domains D and D(2) . Then the particle runs further, ˜ etc. ˜ 2 until its next collision with ∂ D, from the point q (1)
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˜ is a natural lift to the plane R2 of the The so defined motion on D motion on D constructed before. Thus, for each initial state (q, v) ∈ M we ˜ (t) of ˜ n ∈ R2 of collision points and a continuous trajectory q get a sequence q the particle, where the time variable t corresponds to our suspension flow Φtε . ˜1 − q We also have velocity v = d˜ q(t)/dt. We denote by Δε = (Δε,x , Δε,y ) = q the displacement vector between consecutive collisions; it is a function on M. ˜ (t) ∈ R2 represents the electrical current, Now the particle’s position q and we can state our main results. The first one is an analogue of Theorem 4: ˜ have finite horizon and ε be sufficiently small. Then Fε is Theorem 6. Let D a hyperbolic map; it preserves a unique Sinai–Ruelle–Bowen (SRB) measure (a steady state) νε , which is ergodic and mixing and enjoys exponential decay of correlations. The flow Φtε also preserves a unique SRB measure με on Ω, which is ergodic and mixing. The electrical current ˜ (t)/t = με (v) = νε (Δε )/¯ J = lim q τε t→∞
(2.3)
is well defined; here τ¯ε = νε (τε ) is the mean free path for which we have τ¯ε = τ¯ + O(εa ) for some a > 0. We have 1 DE + o(ε), (2.4) J= 2¯ τ where D is the diffusion matrix of Theorem 1. We also have the following weak convergence, as t → ∞: ˜ (t) − Jt q √ ⇒ N (0, D∗ε ), t where D∗ε is the corresponding diffusion matrix satisfying D∗ε = τ¯−1 D + o(1).
(2.5)
(2.6)
The second main result is an analog of Theorem 5: ˜ have infinite horizon. Assume that the force E = (ε, 0) is Theorem 7. Let D not parallel to any of the infinite corridors and ε is sufficiently small. Then, Fε is a hyperbolic map; it preserves a unique SRB measure (steady state) νε , which is ergodic and mixing and enjoys exponential decay of correlations. The flow Φtε also preserves a unique SRB measure με on Ω, which is ergodic and mixing. The electrical current ˜ (t)/t = με (v) = νε (Δε )/¯ J = lim q τε t→∞
(2.7)
is well defined; here τ¯ε = νε (τε ) is the mean free path for which we have τ¯ε = τ¯ + O(εa ) for some a > 0. We have 1 | log ε| D∞ E + O(ε) (2.8) J= 2¯ τ where D∞ is the superdiffusion matrix of Theorem 2. We also have the following weak convergence, as t → ∞: ˜ (t) − Jt q √ ⇒ N (0, D∗ε ), t
(2.9)
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where D∗ε is the corresponding diffusion matrix satisfying D∗ε = τ¯−1 | log ε| D∞ + O(1).
(2.10)
We note that the invariant measures in Theorems 6 and 7 are not positive on open sets (unlike SRB measures for the Gaussian thermostatted dynamics). We do not know if their support (i.e., the intersection of all closed sets of full measure) has a positive Lebesgue measure or not; we return to that issue in Sect. 8. We will prove Theorems 6 and 7 in parallel. Our arguments follow a general scheme developed in [15,18] for the Gaussian thermostatted Lorentz gases, but we have to modify many steps of that scheme to deal with the non-invertibility of our dynamics. In Sects. 3–5 we show that the map Fε is a small (in a certain sense) perturbation of the billiard map F0 ; it has stable and unstable cones and estimate the sizes of overlaps and gaps in M. Our technical analysis here is essentially different from that in all the previous studies [9,15,18], so we give a detailed presentation. In Sect. 6 we describe the singularities of Fε and in Sect. 7 build up our main tools—standard pairs and families—and prove the key growth lemma. Our arguments are similar to those of [15], so we only present them briefly. But the construction of the SRB measure νε in Sect. 8 is quite novel and is presented in full. All the formulas in Theorems 6 and 7 are then derived in Sects. 9 and 10, based on a substantial modification of the arguments of [15].
3. Perturbative Analysis Here we derive formulas for the differential of our collision map Fε and show that it is a small perturbation of the billiard map F0 . Let X = (r, ϕ) ∈ M be a phase point and X1 = Fε (X) = (r1 , ϕ1 ) denote its image. Denote by (x, y) ∈ ∂D and (x1 , y1 ) ∈ ∂D the coordinates of the boundary points corresponding to r and r1 . Denote by ω and ω1 the angles made by the outgoing velocity vector at the point X and the incoming velocity vector at the point X1 , respectively, with the x axis; see Fig. 4. Here and in what follows we use notation of [17, Sect. 2.11], where a similar analysis was done for the billiard dynamics. Our aim is to express dr1 and dϕ1 in terms of dr and dϕ, and we use differentials of other variables in the process. We denote by γ and γ1 the slopes of the tangents to ∂D at the points r and r1 , respectively. We also denote ψ = π/2 − ϕ
and
ψ1 = π/2 − ϕ1 .
ω =γ+ψ
and
ω1 = γ1 − ψ1 .
Observe that (3.1)
Our trajectory curves under the action of the field E, and so in general ω1 = ω. We put δ = ω1 − ω.
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Figure 4. Action of the map in coordinates Let K > 0 and K1 > 0 denote the curvature of ∂D at the points r and r1 , respectively. Note that, in terms of differentials, dγ = −K dr
dγ1 = −K1 dr1 .
and
(3.2)
Also note that dx = cos γ dr
and
dy = sin γ dr,
dx1 = cos γ1 dr1
and
dy1 = sin γ1 dr1 .
as well as
Let v denote the speed of the particle when it departs from the point (x, y), and v1 denote its speed right before it arrives at (x1 , y1 ). Of course, the conservation of the total energy implies v 2 − 2εx = v12 − 2εx1 = 1. In particular, v dv = ε cos γ dr.
(3.3)
Now solving the (2.2), we obtain v1 cos ω1 = v cos ω + ετ
and
v1 sin ω1 = v sin ω
(3.4)
and 1 2 ετ and y1 = y + vτ sin ω, (3.5) 2 where τ is the travel time between the points (x, y) and (x1 , y1 ). Equations (3.4) can be rewritten as x1 = x + vτ cos ω +
v1 = v cos δ + ετ cos ω1
and
v sin δ + ετ sin ω1 = 0.
(3.6)
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Differentiating (3.5) yields cos γ1 dr1 = cos γ dr + v cos ω dτ − vτ sin ω dω + τ cos ω dv + ετ dτ sin γ1 dr1 = sin γ dr + v sin ω dτ + vτ cos ω dω + τ sin ω dv.
(3.7)
Using (3.4) we can rewrite (3.7) as cos γ1 dr1 = cos γ dr + v1 cos ω1 dτ − vτ sin ω dω + τ cos ω dv sin γ1 dr1 = sin γ dr + v1 sin ω1 dτ + vτ cos ω dω + τ sin ω dv.
(3.8)
Next, we eliminate dτ from (3.8) using (3.1) and arrive at sin ψ1 dr1 + sin(ψ + δ) dr − vτ cos δ dω + τ sin δ dv = 0.
(3.9)
Similarly, eliminating dω from (3.7) gives cos(ψ1 + δ) dr1 = cos ψ dr + (v + ετ cos ω) dτ + τ dv.
(3.10)
Now the second equation in (3.6) implies v + ετ cos ω = −
ετ cos δ sin ω sin δ
(3.11)
thus, (3.10) can be written as sin δ cos(ψ1 + δ) dr1 = sin δ cos ψ dr − ετ cos δ sin ω dτ + τ sin δ dv.
(3.12)
Differentiating the second equation in (3.6) yields v cos δ(dω1 − dω) + sin δ dv + ε sin ω1 dτ + ετ cos ω1 dω1 = 0 and combining this with the first equation in (3.6) gives v1 dω1 − v cos δ dω + sin δ dv + ε sin ω1 dτ = 0.
(3.13)
Now eliminating dτ from (3.12) and (3.13) gives sin δ cos ψ sin ω1 dr − sin δ cos(ψ1 + δ) sin ω1 dr1 + v1 τ cos δ sin ω dω1 − vτ cos2 δ sin ω dω + τ sin δ sin ω1 dv + τ sin δ cos δ sin ω dv = 0.
(3.14)
Recall (3.3) and observe that dω = −K dr + dψ
and
dω1 = −K1 dr1 − dψ1 ;
thus, we can solve the two equations (3.9) and (3.14) for dr1 and dψ1 . We record their solutions by using the standard variables ϕ = π/2 − ψ and ϕ1 = π/2 − ψ1 : − cos ϕ1 dr1 = [Kvτ cos δ + cos(ϕ − δ)]dr + τ sin δ dv + vτ cos δ dϕ
(3.15)
and − v1 τ dϕ1 = [Kvτ cos δ + sin δ sin ϕ(1 + tan δ cot ω)] dr + τ sin δ [1 + sin ω1 /(cos δ sin ω)] dv + vτ cos δ dϕ − [K1 v1 τ + sin δ sin(ϕ1 − δ)(1 + tan δ cot ω)] dr1 , (3.16) where dv = (ε cos γ/v)dr
(3.17)
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according to (3.3). Equations (3.15)–(3.16) represent the derivative of the map Fε in the (r, ϕ) coordinates. When ε = 0, our system reduces to the usual billiard dynamics in which v = v1 = 1 and δ = 0, and we recover the derivative of the billiard collision map, in the rϕ coordinates (see (2.26) in [17]): −1 τ K + cos ϕ τ DF0 = . (3.18) τ K1 + cos ϕ1 cos ϕ1 τ KK1 + K cos ϕ1 + K1 cos ϕ Next, we estimate corrections due to the external field E. First, note that ετ = O(ε1/2 ) is small, by Lemma 2.1. Then, we note that v = 1 + O(ε)
and
v1 = 1 + O(ετ ),
and it follows from the second equation in (3.6) that ετ sin ω ; sin δ ∼ − 1 + O(ετ )
(3.19)
thus, all the cot ω factors in (3.16) will be neutralized by sin δ. Now a direct analysis shows that the derivative of Fε satisfies 1 DFε = DF0 + O2×2 (ετ ), (3.20) cos ϕ1 where O2×2 (ετ ) denotes a matrix 2 × 2 whose entries are O(ετ ) uniformly over M. We note that (3.20) does not really mean that Fε is C 1 -close to F0 . What it means is that the derivative of the map Fε at a point (r, ϕ), whose image is (r1 , ϕ1 ), is close to the derivative of a (hypothetical) billiard map that takes (r, ϕ) to the same image (r1 , ϕ1 ). But the real billiard map F0 on D may take (r, ϕ) to a quite different point, which may even be on a different scatterer. Still, the closeness in the sense of (3.20) has important implications.
4. ‘Cone’ Hyperbolicity Here we establish certain hyperbolic properties of the map Fε . Similar maps were studied recently by Baladi and Gou¨ezel [2,3] who described them by the term cone hyperbolicity. Recall that the billiard map F0 is hyperbolic [17]. Its unstable vectors dX = (dr, dϕ) can be defined by 0 < C1 ≤ dϕ/dr ≤ C2 < ∞,
(4.1)
where C1 < C2 are some constants. The expansion factor satisfies DF0 (dX)∗ /dX∗ ≥ 1 + a/ cos ϕ1
(4.2)
for some constant a = a(D) > 0, where ϕ1 again denotes the reflection angle at F(X) and dX∗ is an adapted metric. The latter can be defined in various ways, for example (see [17, Sect. 5.10]) dX∗ = |K dr + dϕ|, where K > 0 again denotes the curvature of ∂D at the given point.
(4.3)
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The estimate (3.20) shows that Fε expands the same unstable vectors (4.1), and the expansion factor satisfies (4.2), with a possibly smaller constant a > 0. More precisely, the expansion factor of unstable vectors satisfies τ dr1 (dr12 + dϕ21 )1/2 , 2 2 1/2 dr cos ϕ1 (dr + dϕ )
(4.4)
which for billiards is a standard formula [17, Eq. 4.20]; we use the notation A B in the sense that 0 < c1 ≤ A/B ≤ c2 < ∞ for some constants c1 , c2 that may depend on D, but not on ε. Unstable curves are described by functions ϕ = ϕ(r) whose slope is positive and bounded above and below, i.e., unstable curves are increasing in the rϕ coordinates. Let W ⊂ M be an unstable curve and Wn = Fεn (W ) is its image under Fεn (which is a union of unstable curves). For X ∈ W , we denote by JW Fε (X) the Jacobian of Fε restricted to W (i.e., the factor of expansion of W under the map Fε ) at the point X. We also denote by JW1 Fε−1 (X1 ) the Jacobian of the inverse map W1 → W at the point X1 = Fε (X). Next, we verify two standard regularity properties for unstable curves. Lemma 4.1 (Curvature bounds). Suppose that |d2 ϕ/dr2 | ≤ C0 at all points (ϕ, r) ∈ W , for some C0 > 0. Then, there is a constant C = C(C0 , D) > 0 such that for all n ≥ 1 we have |d2 ϕn /drn2 | ≤ C at all points (ϕn , rn ) ∈ Wn . Due to this lemma, we can (and will) assume that all our unstable curves have uniformly bounded curvature; this assumption is standard for billiards and their perturbations [9,17]. Proof. For the billiard maps and their perturbations by external forces (though different from ours) this lemma was proved in [9]. A more direct proof (for the billiard maps only) is given in [12, Equation (B.2)]. The last proof can be adapted to our case if we estimate the corrections to the second-order derivatives due to the external field E. In fact, all those corrections are of order O(ετ ). For example, in billiards dv/dr = 0 and in our model dv/dr = O(ετ ) due to (3.17). Differentiating the second equation in (3.6) gives dδ = O(ετ dr1 ). Solving (3.10) for dτ gives dτ = cos ψ1 dr1 − cos ψ dr + O(ετ dr1 ), while for the billiard map we have dτ = cos ψ1 dr1 − cos ψ dr, so the remainder O(ετ dr1 ) is due to the field; it is small. We should note that the quotient Q : = sin δ/ sin ω requires a special care, because its derivative is not small: dQ = O(ε dτ ), as it follows from (3.11). However, we note that in all the expressions (3.15)–(3.16) the fraction sin δ/ sin ω is multiplied either by sin δ = O(ετ ) or by dv = O(ετ dr); thus, dQ will be always suppressed by a small factor. Then the direct proof of curvature bounds (see the proof of equation (B.2) in [12]) can be carried over to our case; we leave the details to the reader. Thus, the corrections to the second-order derivatives due to the external field E are all relatively small. Next, the proof of (B.2) in [12] uses the value B, the curvature of the orthogonal cross-section of the wave front corresponding to the unstable curve right before the collision with ∂D. We cannot use it here, so it has to be
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eliminated from the above proof as follows. First, d2 ϕn /drn2 = cos ϕn dBn /drn + Dn where |Dn | ≤ D < ∞ are bounded (see page 169 in [12]). Second, it was proved on page 171 in [12] that |dBn /drn | = θn (wn /wn−1 )|dBn−1 /drn−1 | + Rn where |θn | ≤ θ < 1, |Rn | ≤ R < ∞, and 0 < wmin < wn < wmax < ∞ are bounded away from zero and infinity. Combining the above formulas we see that
2
2
d ϕn
≤ θ cos ϕn wn d ϕn−1 + Dn−1 + θ wn |Rn | + |Dn |.
dr2
cos ϕn−1 wn−1 dr2 wn−1 n
n−1
The lemma now follows easily.
Next, to ensure distortion control we need to construct homogeneity strips in M, in a standard way: π π ∀j ≥ j0 Hj = (r, ϕ) : − j −2 < ϕ < − (j + 1)−2 2 2 π π H0 = (r, ϕ) : − + j0−2 < ϕ < − j0−2 , (4.5) 2 2 π π ∀j ≥ j0 H−j = (r, ϕ) : − + (j + 1)−2 < ϕ < − + j −2 2 2 where j0 > 1 is a large constant, see [9, p. 216] and [17, Sect. 5.3]. We cut M along their boundaries, i.e., replace M with a countable union of the above strips. Accordingly, unstable curves must respect these new boundaries, i.e., each unstable curve must lie in one of the strips. If an unstable curve crosses several strips, it must be cut into pieces by the boundaries of the strips. Lemma 4.2 (Distortion bounds). There is a constant C = C(D) > 0 such that for any unstable curve W and any point X1 = (ϕ1 , r1 ) ∈ W1 on its image
d ln JW1 Fε−1 (X1 )
C
≤ (4.6)
|W |2/3 dr1 1 The first construction of homogeneity strips (4.5) and the first proof of distortion bounds were given in [7]. A more recent and direct proof is given in [12]; see equation (B.3) there. That proof can be adapted to our case because the corrections due to the field are small, as we explained earlier. We emphasize that all the constants in Lemmas 4.1 and 4.2 are uniform in ε. Next, we turn to stable vectors and stable curves. The estimate (3.20) is insufficient for the control over the contraction of stable vectors. To estimate contraction rates, we use the Jacobian of DFε : Lemma 4.3. In terms of differential forms, we have v1 cos ϕ1 dr1 ∧ dϕ1 = v cos ϕ dr ∧ dϕ.
(4.7)
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Proof. It follows from (3.15)–(3.16) that v1 τ cos ϕ1 dr1 ∧ dϕ1 = vτ [cos(ϕ − δ) cos δ − sin δ cos δ sin ϕ
− sin2 δ sin ϕ cot ω dr ∧ dϕ − vτ 2 sin δ sin ω1 / sin ω dv ∧ dϕ.
(4.8)
Observe that cos(ϕ − δ) cos δ = cos ϕ − sin2 δ cos ϕ + sin δ cos δ sin ϕ; therefore, (4.8) becomes v1 τ cos ϕ1 dr1 ∧ dϕ1 = vτ cos ϕ dr ∧ dϕ − vτ sin2 δ [cos ϕ + sin ϕ cot ω] dr ∧ dϕ − vτ 2 sin δ sin ω1 / sin ω dv ∧ dϕ.
(4.9)
Last, we note that cos ϕ + sin ϕ cot ω = sin(ω + ϕ)/ sin ω = cos γ/ sin ω,
and use (3.17) and the second equation in (3.6). When ε = 0, then v = v1 = 1 and δ = 0, and (4.7) becomes cos ϕ1 dr1 ∧ dϕ1 = cos ϕ dr ∧ dϕ, which shows that the billiard map F0 preserves the measure dν0 = cos ϕ dr dϕ
(4.10)
and is positively oriented in the rϕ coordinates (these are standard facts [17]). The map Fε , with respect to the measure (4.10) has (local) Jacobian ν0 (Fε (dX))/ν0 (dX) = v/v1 = 1 + O(ετ ).
(4.11)
One can also explain (4.11) as follows: the dynamics (2.2) is Hamiltonian and ˜ ×R2 . preserves Liouville measure (volume) in the 4-dimensional phase space D It obviously expands in the flow direction, between the two collisions, by a factor of v1 /v; thus, it contracts areas in the orthogonal directions by a factor of v/v1 , and the measure ν0 corresponds to the area in the directions orthogonal to the velocity vector. Now we can analyze the contraction of stable vectors. The billiard map F0 is known to have a family of stable cones defined by − ∞ < C1 ≤ dϕ/dr ≤ C2 < 0,
(4.12)
where C1 < C2 are some negative constants. Stable vectors dX = (dr, dϕ) are contracted by a factor DF0 (dX)∗ /dX∗ ≤ (1 + a/ cos ϕ)
−1
<1
(4.13)
for some constant a = a(D) > 0, where dX∗ is again the adapted metric (4.3). Our estimate (3.20) implies that Fε , being close to F0 , also contracts stable vectors defined by (4.12), and the contraction factor also satisfies (4.13), with a possibly smaller constant a > 0.
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For the billiard map F0 , the contraction factor of stable vectors satisfies a more accurate asymptotic formula (dr12 + dϕ21 )1/2 cos ϕ dr1 , 2 2 1/2 dr τ (dr + dϕ )
(4.14)
see [17, Sect. 4.4]. The same is true for our map. Indeed, due to (4.11) the inverse maps DFε−1 and DF0−1 are also close to each other; hence, they expand images of stable vectors at nearly the same rates. These facts also follow from the ‘local’ time reversibility of our dynamics described next.
5. Local Time Reversibility, Overlaps, and Gaps It is helpful to note that the map Fε is (locally) time reversible in the following sense. Let I : M → M denote the standard involution; it acts by I : (r, ϕ) → (r, −ϕ), i.e., it takes (q, v) ∈ M to (q, v ) ∈ M, so that the vector v is obtained by reflecting v across the normal line to ∂D at the point q. Note that I −1 = I. Now consider any point (r, ϕ) ∈ M that starts on a scatterer B ⊂ D and whose trajectory lands on another scatterer B1 ⊂ R2 , which is in another cell containing some domain Di (Fig. 5). Its initial velocity v and the final velocity v1 (right before landing) satisfy v2 − 2εx = v1 2 − 2εx1 = 1, where (x, y) ∈ ∂B denotes the starting point and (x1 , y1 ) ∈ ∂B1 the landing point. The domain Di is obtained from D by translation along some vector Zi = (Zi,x , Zi,y ). Our dynamics requires that we project the reflection point ˜ , i.e., the point (x1 , y1 ) will be moved to (x1 , y1 ) from ∂B1 back onto D under π so that x1 = x1 − Zi,x . Note also that B1 = π ˜ (B1 ) = B1 − Zi is a scatterer in the central cell K and (x1 , y1 ) ∈ ∂B1 . If we translate the precollisional vector v1 from ∂B1 to ∂B1 along the vector −Zi , its total energy becomes 1 1 v1 2 − ε(x1 − Zi,x ) = + εZi,x . 2 2 Let us now reverse the velocity vector v1 attached to ∂B1 and let it move under the field ε. Clearly, its trajectory will be the translate, under the vector −Zi , of the previous trajectory (from (x, y) to (x1 , y1 )), but now it is traversed backwards. It will land at point (x , y ) = (x − Zi,x , y − Zi,y ) on a scatterer B − Zi in a cell containing some domain Dj ; see Fig. 5. We emphasize that in the above analysis the new total energy E = Ei =
1 + εZi,x 2
(5.1)
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Figure 5. Weak time reversibility will be the same for all points (r, ϕ) starting on a given scatterer, B, and landing on another given scatterer, B1 . This implies that FE,ε ◦ I ◦ Fε = I,
(5.2)
where FE,ε denotes the collision map similar to Fε , but constructed for the dynamics with the total energy E, instead of 1/2. Clearly our choice of E = 1/2 for the total energy in (2.1) was quite arbitrary, and any other constant would give us a map with similar properties. Equation (5.2) can be rewritten as Fε−1 = I ◦ FE,ε ◦ I
(5.3)
which is only valid locally, on the set of points (r1 , ϕ1 ) that start from the scatterer B and land on the scatterer B1 ; the constant E is computed by (5.1) for the given pair of scatterers B and B1 . Still, the local time reversibility in the above sense is helpful. In particular, it shows that since FE,ε expands unstable vectors, Fε will contract stable vectors (i.e., those that are the images of unstable vectors under I). Given a pair of scatterers B and B1 , the set of points (r1 , ϕ1 ) that have come from B and land on B1 is bounded by the images of points that make grazing (tangential) collisions either with B or with some other scatterers on their way from B to B1 . Grazing collisions are described by the lines ϕ = ±π/2; hence, their images are unstable curves. So this set of points is bounded by unstable curves and possibly by some segments of the lines ϕ = ±π/2.
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Figure 6. How the images can overlap. Both the solid and dashed trajectories are tangent to the scatterer B1 (see the inset) Similarly, the set of points (r, ϕ) that start from B and land on B1 is bounded by stable curves (as well as some parts of the lines ϕ = ±π/2); this follows from our local time reversibility. Thus, the space M is naturally partitioned into a finite collection of domains, M = ∪M+ i , on each of which Fε is smooth (for a moment, let us forget that M was divided into homogeneity strips (4.5)). The domains M+ i are bounded by ∂M = {ϕ = ±π/2} and by some smooth stable curves. The set S + = ∪∂M+ i is the singularity set for the map Fε . + −1 The images M− is i = Fε (Mi ) are also domains in M on which Fε − (locally) defined and smooth. The domains Mi are bounded by ∂M = {ϕ = ±π/2} and by some smooth unstable curves. The billiard map F0 has all these properties, too, but its domains M− i make a partition of M, i.e., they are disjoint and cover the entire M. Their boundaries are singularities for the inverse map F0−1 . But for our map Fε , the domains M− i may overlap and they may not cover the entire M, i.e., some gaps are left in between. Figure 6 shows how overlaps occur. In that figure (unlike Fig. 5), we draw trajectories as if they started from different domains Di , Dj but landed on the same scatterer in the main domain D. Two trajectories land at a point q ∈ ∂B ⊂ K. One (the solid line) starts from a scatterer B1 (in Di ), and the other (the dashed line)—from another scatterer B2 (in Dj ), and it comes infinitesimally close to B1 . Since Zi,x < Zj,x , the first trajectory is less energetic (thus more curved) than the segment of the second trajectory running between B1 and B. Though
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Figure 7. Creation of a gap between the images. Both the solid and dashed trajectories are tangent to the scatterer B1 (see the inset) both trajectories land at the same point q, they arrive at different incidence angles ϕ1 and ϕ2 , and the difference |ϕ1 −ϕ2 | corresponds to the size of overlap − of the respective domains M− i1 and Mi2 (one consists of trajectories coming to ∂B from ∂B1 , and the other—from ∂B2 ). An elementary analysis shows that the size of the overlap (in the ϕ direction on M) is (5.4) Δϕ = O ε2 |Zij,x |(|Zi,y | + 1) , where Zij,x = Zi,x − Zj,x is the displacement in the x direction between the domains Di and Dj , and Zi,y is the displacement in the y direction between the domains Di and D. So, for finite horizon Lorentz gases we simply have Δϕ = O(ε2 ). In the infinite horizon case (5.4) leads to larger overlaps, to be described later. Similarly, Fig. 7 shows how gaps occur. The trajectory coming from B1 is less energetic (i.e., more curved) than the one coming from B2 . The difference between their angles of incidence at the landing point q corresponds to the size − of the gap between the respective domains M− i1 and Mi2 . Again an elementary analysis shows that the size of the gap is given by the same formula (5.4), so for finite horizon Lorentz gases the gaps are O(ε2 ). In the infinite horizon case, gaps may be larger, see below. Remark. As we mentioned in Sect. 2, the fundamental domain K can be doubled, tripled, etc. We describe how “multiplying” K would affect the gaps and overlaps. First, they occur only when trajectories cross from one copy of K to another. So if we make K larger, we would have fewer gaps and overlaps. But since Zij,x in (5.4) corresponds to the size of K (in the x direction), our gaps
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and overlaps would get wider. Roughly speaking, if we double the size of K, then the space M also doubles in size, and in the new, bigger M half of the gaps would close up and disappear, half of the overlaps would disappear, too, but the other half of the gaps and overlaps would get about twice as wide. It seems that such a change would not alter the basic properties of our dynamics, though.
6. Structure of Singularities Next we analyze the singularities of the map Fε . We begin by recalling the properties of the singularities of the billiard map F0 . When the horizon is finite, the singularity set S of F0 is a finite union of smooth compact stable curves. They have a continuation property ([17, Sect. 4.9]), i.e., each curve S ⊂ S either terminates on the boundary of M (i.e., on the lines ϕ = ±π/2), or it is a part of a longer monotonic continuous curve S ⊂ S that extends (continues) to the boundary of M (of course, then S is a union of several smooth components of S). In other words, S divides M into domains M+ i with piecewise smooth boundaries (‘curvilinear polygons’) such that at their corner points the interior angles are ≤ π (polygons are ‘convex’). The singularity set for our map Fε with finite horizon has all the same properties. This follows from the local time reversibility. + For the billiard map F0 , the images M− i = Fε (Mi ) are also ‘convex’ curvilinear polygons bounded by smooth compact unstable curves. The same remains true for the map Fε , except the domains M− i no longer make a partition of M: some of them overlap by O(ε2 ) near their boundaries, and between others there are gaps of size O(ε2 ). In the infinite horizon case, the singularity set S of the billiard map F0 is a countable union of smooth compact stable curves, which accumulate near finitely many points X ∈ M whose trajectories run along the borders of the infinite corridors ([17, Sect. 4.10]). These are exactly the points entering the formula (1.5); see one of them in Fig. 8. The singularity curves near the fixed points X form a cell structure [17] whose features essentially determine global properties of the map F0 . Unlike F0 , our map Fε has finitely many singularity lines even in the infinite horizon case, because the length of the free path is always bounded (Lemma 2.1). But the number of singularity lines is O(ε−1/2 ), i.e., it is not uniformly bounded. The structure of the singularity lines and the corresponding cells is similar to that of the collision map for the thermostatted dynamics [15]. We briefly describe it next. Trajectories leaving D into an infinite corridor can land on scatterers on either side of the corridor; see Fig. 8. The number of scatterers that our trajectories can reach is O(ε−1/2 ), due to Lemma 2.1. More precisely, let mU denote the number of reachable scatterers on the upper side of the corridor, and let mL denote those on the lower side; see Fig. 8. Since our trajectories
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Figure 8. A row of scatterers forming the border of an infinite corridor are parabolas, an elementary computation shows that √ √ mL ∼ C2 / ε, 0 < C1 < C2 mU ∼ C1 / ε, The singularity lines near the point marked by X in Fig. 8 are shown in Fig. 9. The long singularity curve S corresponds to grazing collisions with the very next scatterer in the corridor (marked by B2 in Fig. 8). The short singularity curves are made by grazing collisions with other scatterers in the corridor. The regions between singularity curves (cells) are made by trajectories landing on a particular scatterer. Thus, the cells can be naturally numbered by 1, . . . , mU (corresponding to the upper scatterers) and 1, . . . , mL (for the lower scatter(U ) (U ) (L) (L) ers). Accordingly, we denote the cells by D1 , . . . , DmU and D1 , . . . , DmL (here U and L stand for ‘Upper’ and ‘Lower’, respectively). In Fig. 9 the cells are depicted as follows. First (farther from S) come √ (U ) (U ) (U ) the cells D1 , . . . , DmU (in this order). The height of Dm is 1/ m and its width is 1/m2 , just like in classical billiards with infinite horizon, see, e.g., (U ) (U ) [17, Sect. 4.10]. Unstable curves inside Dm are expanded by a factor Λm ≥ 3/2 −1/2 cm for some c > 0, due to (4.4), in which τ m and cos ϕ1 = O(m ). (L) (L) Second (closer to S) come the cells DmL , . . . , D1 (in the reverse order; √ (L) (L) the cell D1 is adjacent to the point X). The height of Dm is m/mL (L) and its width is 1/m2L . Unstable curves inside Dm are expanded by a fac√ (L) tor Λm ≥ cm √ L m for some c > 0, again due to (4.4), in which τ m and cos ϕ1 = O( m/mL ). We record the following formulas for the measures of the cells: (U ) (L) 1/m3 , m/m4L mε2 ν0 Dm (6.1) ν0 Dm Indeed, the billiard measure ν0 is smooth and has density cos ϕ; thus, the measure of each cell is of the same order of magnitude as the product width×(height)2 .
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Figure 9. Singularity curves and cells. On the left, an ‘upper’ (U ) cell Dm is shown, with all its dimensions, in light gray; the (L) union of all the ‘lower’ cells Dm ’s is painted dark gray. On (L) the right, a ‘lower’ cell Dm is shown, with all its dimensions, (U ) in dark gray; the union of all the ‘upper’ cells Dm ’s is painted light gray
Figure 10. The map F0 transforms a cell Dm into a similar-looking domain. The corners of Dm and their respective images are numbered to indicate the action of F0 For the billiard map F0 , there are no ‘lower’ cells, but there are infinitely (U ) many ‘upper’ cells Dm = Dm , m = 1, 2, . . ., each having the shape and size as described earlier. The map F0 transforms cells into similar-looking domains, but the images are bounded by unstable curves; see Fig. 10 and more details in [17, Sect. 4.10]. The image F0 (Dm ) has the same dimensions as Dm , so the expansion factor of the map F0 acting on unstable curves in Dm is Λ∼
height of F0 (Dm ) height of Dm ∼ m3/2 width of Dm width of Dm
(6.2)
This rule can be used to verify our previous formulas for the expansion factor (U ) (L) of the map Fε in Dm and Dm .
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Figure 11. Images of cells under Fε . There is a narrow (white) gap between two (light grey) ‘upper’ cells. Two (dark gray) lower cells overlap along a black narrow strip between them. There is a wider white gap between the last (top) upper cell and the last (bottom) lower cell. All the cells overlap with the long curve S − and stretch slightly beyond (to the left of) it
(U )
(L)
For our map, the domains Fε (Dm ) and Fε (Dm ) also look like the cells (U ) (L) Dm and Dm , respectively, and the images are also bounded by unstable curves. But, unlike F0 , the images of cells under Fε can overlap and there may be gaps between them. Figure 11 shows the images of cells depicted in Fig. 9. There are gaps of (U ) (U ) size mε2 between the images of neighboring upper cells Dm and Dm+1 . (L) The image of each lower cell Dm overlaps by mε2 with the images of the two neighboring lower cells. (Since mε2 is much less than the width of the m-th cell, only neighboring cells can overlap.) The image of each cell Dm (upper or lower) overlaps by mε2 with the region beyond (i.e., to the left of) the long singularity curve S − . And there is a wide gap of size ε between the image of (U ) (L) the last upper cell DmU and that of the last lower cell DmL . All these formulas follow from (5.4), which describes the extent of overlaps and gaps. (U ) (L) In some other cases, the images Fε (Dm ) and Fε (Dm ) may form a different structure: images of the upper cells may overlap (instead of leaving gaps in between), and images of the lower cells may leave gaps (instead of overlapping); it is also possible that the lower cells (and their images) are missing altogether. But in all cases the sizes of gaps and overlaps are given by the same formula mε2 as above. The total ν0 -measure of all the gaps and overlaps is 2 √ 2 mU mL 1 mε2 m √ 2+ + mε2 ν0 (gaps + overlaps) ε √ mU mL ( m) m=1 m=1 ε3/2 + ε3/2 + ε3/2 ε3/2 ,
(6.3)
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where the first term accounts for the ‘big’ gap of width ε between the images of the last upper cell and the last lower cell (that gap has the largest measure of all our gaps and overlaps). We note that the height of every cell in (6.3) is squared to account for the density of ν0 , which is cos ϕ.
7. Growth Lemmas and Standard Families Next we derive a key fact about the growth of unstable curves, known as Growth Lemma. Given an unstable curve W , let us denote by Wi ⊂ W the connected components of W \S, i.e., the segments of W on which Fε is smooth, and by Λi the (minimal) factor of expansion of Wi under Fε in the adapted metric (4.3). Lemma 7.1 (One-step expansion). We have sup Λ−1 < 1, lim inf i δ0 →0
W : |W |<δ0
(7.1)
i
where the supremum is taken over unstable curves W of length < δ0 . The bound (7.1) is called the one-step expansion estimate [20, Sect. 5]; it shows how much unstable curves stretch under one iteration of the map. Proof. For dispersing billiards with finite horizon, the proof of (7.1) is standard (see [17, Lemma 5.56]), and it readily carries over to our dynamics with an external field. Without finite horizon, infinite corridors add new complications—it is possible that a short unstable curve W intersects many cells (U ) (L) described above. Suppose W intersects cells Dp , p0 ≤ p ≤ mU , and Dq , q0 ≤ q ≤ mL , then mU mL C C Λ−1 ≤ + i 3/2 p mL q 1/2 p=p q=q i 0
≤
C 1/2
p0
0
+
C 1/2
.
mL
If W is small, then p0 must be large, and the above bound is 1. This completes the proof of (7.1) in the infinite horizon case. For simplicity, we have ignored additional singularities that come from the artificially constructed boundaries of the homogeneity strips (4.5). Taking them into account would make the analysis somewhat more complicated, but the final result would remain the same as for billiards; see [8, Sect. 8] and [20, Sect. 7]. The one-step expansion estimate (7.1) implies several properties known collectively as Growth Lemmas, see [17, Chapter 5], [20, Sect. 4.7], [9, Proposition 5.3], [12, Lemma 4.10], and the proofs therein. We state their most common versions below. Given an unstable curve W , we denote by mW the Lebesgue measure on it. For every n ≥ 0, its image Fεn (W ) is a finite or countable union of
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unstable curves (components), and for every X ∈ W we denote by Wn (X) the component containing the point Fεn (X). Now let rn (X) = dist (Fεn (X), ∂Wn (X))
(7.2)
Fεn (X)
to the nearer endpoint of Wn (X). denote the distance from the point Clearly, rn is a function on W that characterizes the size of the components of Fεn (W ). We also denote by Λ > 1 the hyperbolicity constant, i.e., the minimal expansion factor of unstable curves in the adapted metric. Lemma 7.2 (“Growth lemma”). Unstable curves W ⊂ M have the following properties: (a) There are constants ϑ0 ∈ (0, 1) and c1 , c2 > 0, such that for all n ≥ 0 and ζ>0 mW (rn (X) < ζ) ≤ c1 (ϑ0 Λ)n mW (r0 < ζ/Λn ) + c2 ζ mW (W ) (b) There are constants c3 , c4 > 0, such that if n ≥ c3 |ln mW (W )|, then for any ζ > 0 we have mW (rn (X) < ζ) ≤ c4 ζ mW (W ). (c) There are constants ϑ1 ∈ (0, 1) c5 , c6 > 0, a small ζ0 > 0 such that for any n2 > n1 > c5 |ln mW (W )| we have mW max rn (X) < ζ0 ≤ c6 ϑn1 2 −n1 mW (W ). n1
For the proof and implications of this lemma we refer to [17,20]. We emphasize that the lim inf in (7.1) and all the constants in Growth Lemma are independent of ε, i.e., the respective properties hold uniformly in ε. In plain words, the Growth Lemma means that the images of any unstable curve W grow, on average, exponentially fast, until they reach a certain minimal average size, and then that minimal average size is maintained forever. More precisely, the proportion of small components (of size < ζ) among all the components of Fεn (W ) remains O(ζ). Such a fast expansion of unstable curves allows us to construct stable manifolds for the map Fε . A stable manifold W s is a stable curve such that Fεn (W s ) is also a stable curve for every n ≥ 1. In that case the size of Fεn (W s ) is ≤ Cλn , where λ < 1 is the weakest contraction factor of stable curves. Lemma 7.3. Let W be an unstable curve and mW the Lebesgue measure on it. For mW -almost every point X ∈ W there is a stable manifold W s (X) passing through X. Moreover, for any ζ > 0 mW (X ∈ W : rs (X) < ζ) ≤ Cζ, where rs (X) = dist (X, ∂W s (X)) denotes the distance from X to the nearer endpoint of the curve W s (X), and C > 0 a constant. Proof. It is standard (see, e.g., [17, Sect. 4.12]) that rs (X) ≥ c min Λn dist(Fεn (X), ∂M) n≥0
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for some constant c > 0, and we obviously have dist(Fεn (X), ∂M) ≥ c rn (X) for some constant c > 0. Thus if rs (X) < ζ, then rn (X) < c ζΛ−n for some n ≥ 0, where c > 0 is another constant. Due to Growth Lemma (a) such points make a set of mW -measure O(ε). As usual, stable manifolds W s ⊂ M cannot cross each other, and the foliation of M into stable manifolds is a measurable partition; see [17, Sect. 5.1]. That foliation is absolutely continuous in the following sense: Let W1 and W2 be two nearby unstable curves and W s a stable manifold crossing each Wi in a point Xi ; then, the Jacobian of the holonomy map h : W1 → W2 at X1 satisfies e−C(γ+[dist(X1 ,X2 )]
1/3
)
≤ J h(X1 ) ≤ e−C(γ+[dist(X1 ,X2 )]
1/3
)
(7.3)
where γ is the angle between the tangent vectors to W1 and W2 at the points X1 and X2 , respectively. The property (7.3) is proved for the billiard map F0 in [17, Theorem 5.42], and it extends to our case because the corrections to derivatives due to the field are O(τ ε), as we established in the proof of Lemma 4.1. Next, for any two points X, Y ∈ M we denote by s+ (X, Y ) ≥ 0 the future separation time: it is the first time when the images Fεn (X) and Fεn (Y ) land on different scatterers or in different homogeneity strips (i.e., the first time when Fεn−1 (X) and Fεn−1 (Y ) belong to different connected components of M\S). Observe that if X and Y lie on one unstable curve W ⊂ M, then dist(X, Y ) ≤ CΛ−s+ (X,Y ) , cf. [17, Eq. (5.32)]. Now (7.3) implies (just like in the case billiards; see Sect. 5.8, in particular Proposition 5.48 of [17]) that for any pair of nearby unstable curves W1 , W2 and any X, Y ∈ W1 | ln J h(X) − ln J h(Y )| ≤ Cϑs+ (X,Y ) ,
(7.4)
for some constant ϑ < 1 (in fact, ϑ = Λ−1/6 ). Following Young [31, p. 597], we call the property (7.4) the ‘dynamically defined H¨ older continuity’ of J h. Next, we define a class of probability measures supported on unstable curves. A standard pair = (W, ρ) is an unstable curve W ⊂ M with a probability measure P on it, whose density ρ (with respect to the Lebesgue measure on W ) satisfies | ln ρ(X) − ln ρ(Y )| ≤ Cr Λ−s+ (X,Y ) .
(7.5)
Here Cr > 0 is a sufficiently large constant (independent of ε). For any standard pair = (W, ρ) and n ≥ 1 the image Fεn (W ) is a finite or countable union of components on which the density of the measure Fεn (P ) satisfies (7.5), due to the distortion bounds (Lemma 4.2); see a proof in [17, Proposition 7.12]. Hence, the image of a standard pair under Fεn is a countable family of standard pairs (with a factor measure). More generally, a standard family is an arbitrary (countable or uncountable) collection G = {α } = {(Wα , ρα )}, α ∈ A, of standard pairs with a
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probability factor measure λG on the index set A. Such a family induces a probability measure PG on the union ∪α Wα (and thus on M) defined by ∀B ⊂ M. PG (B) = Pα (B ∩ Wα ) dλG (α) Any standard family G is mapped by Fεn into another standard family Gn = Fεn (G), and PGn = Fεn (PG ). For every α ∈ A, any point X ∈ Wα divides the curve Wα into two pieces, and we denote by rG (X) the length of the shorter one. Now the quantity ZG = sup ζ −1 PG (rG < ζ) ζ>0
reflects the ‘average’ size of curves Wα in G, and we have dλG (α) ZG ≤ C |Wα |
(7.6)
see [17, Exercise 7.15]. We only consider standard families with ZG < ∞. The growth lemma implies that for all n ≥ 0 and some constant θ ∈ (0, 1) ZGn ≤ C(θn ZG + 1),
(7.7)
see a proof in [17, Proposition 7.17]; this estimate effectively asserts that standard families grow under Fεn exponentially fast. We say that a standard pair (W, ρ) is proper if |W | ≥ δp , where δp > 0 is a small but fixed constant. We say that a standard family G is proper if ZG ≤ Cp , where Cp is a large but fixed constant (chosen so that a family consisting of a single proper standard pair is proper). We note that the image of a proper standard family under Fεn is proper for every n ≥ const = const(C, θ). We note that the measure ν0 can be represented by a proper standard family. Indeed, foliating M by unstable curves and conditioning the measure ν0 on them gives a standard family G0 such that PG0 = ν0 . (U ) In systems with infinite horizon, trajectories starting in cells Dm and (L) Dm have a long free flight–they travel the distance m before landing on another scatterer. It is important to estimate the integral effect of the long free flights. Let A be the ‘cell number’ function on M, i.e., A = m on every (U ) (L) cell Dm and Dm (and A = 0 on the rest of M). It easily follows from (6.1) that mU mL m + m2 ε2 = O(1), (7.8) ν0 (A) 3 m m=1 m=1 i.e., ν0 (A) is bounded uniformly in ε. Similarly, ν0 (A2 )
mU mL m2 + m3 ε2 = O(| ln ε|), 3 m m=1 m=1
(7.9)
and for any k ≥ 3 ν0 (Ak )
mU mL k−2 mk + mk+1 ε2 = O(ε− 2 ). 3 m m=1 m=1
(7.10)
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(U )
Let G be a proper standard family. Since Dm has width 1/m2 , in (U ) (L) the unstable direction, we have PG (Dm ) = O(1/m2 ). Similarly, PG (Dm ) = 2 O(1/mL ). Therefore, mU mL m m + PG (A) ≤ 2 = O(| ln ε|), 2 m m L m=1 m=1
(7.11)
and similarly PG (Ak ) = O(ε−
k−1 2
),
(7.12)
which is not much worse than (7.8)–(7.10). Next, we estimate possible effect of gaps and overlaps on our integral formulas. Let ν denote the measure ν0 restricted to all gaps and overlaps 3/2 ). Gaps and between domains M− i . Due to (6.3), its norm is ν = O(ε overlaps are strips stretching in the unstable direction. We can foliate them by unstable curves, condition ν on those curves, and get a ‘standard family’ G (though the norm of the corresponding measure PG will be ν , rather than one). We can see directly that ZG = sup ζ −1 PG (rG < ζ) ζ>0
ε1/4 ε +
mU
m−1/2 mε2 +
m=1
mL √
mε mε2
m=1
ε5/4 (in the middle line, the first term accounts for the gap of width ε between the last upper cell and the last lower cell, the first sum—for gaps/overlaps of width mε2 between the upper cells,√and the second—the same for the lower cells; the factors ε1/4 , m−1/2 , and mε simply account for the density cos ϕ of ν0 ). Thus, (7.11) now takes form PG (A) = O(ε5/4 | ln ε|)
(7.13)
and (7.12) takes form 5
PG (Ak ) = O(ε 4 −
k−1 2
) = O(ε
7−2k 4
).
(7.14)
Fεn (G )
Furthermore, the image is a ‘standard family’ (with the same norm of the total measure, though) whose Z-value can only decrease with n, due to (7.7); hence we have PG (A ◦ Fεn ) = O(ε5/4 | ln ε|).
(7.15)
for all n ≥ 1. In the same way we can estimate the measure of gaps and over(U ) (L) laps in each cell Dm or Dm . Since the width of the cell is 1/m2 or 1/m2L , respectively, we have for all n ≥ 0 (U ) (Fεn ν ) (Dm ) = O(ε5/4 /m2 )
(7.16)
(U ) (Fεn ν ) (Dm ) = O(ε5/4 /m2L )
(7.17)
and
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Last, let M− denote the union of the preimages of all the overlaps; this is the subset of M on which Fε fails to be injective. We have seen in (6.3) that (U ) (L) ν0 (Dm ∩ M− ) = O(ε2 ) and ν0 (Dm ∩ M− ) = O(ε2 m2 /m2L ); hence mU mL A dν0 mε2 + ε2 m3 /m2L = O(ε). (7.18) M−
m=1
m=1
We will use the estimates (7.8)–(7.18) later.
8. Physical Invariant Measure Here, we construct a natural (‘physical’) Fε -invariant measure on M that attracts Lebesgue a.e. point X ∈ M. Recall that the basin of attraction Bν of an ergodic Fε -invariant measure ν is the set of points X ∈ M such that
1 n−1 f (X) + f (Fε (X)) + · · · + f (Fε (X)) → f dν n M
for every bounded continuous function f . By the ergodic theorem, ν(Bν ) = 1. We say that ν is a physical measure if Leb(Bν ) > 0, i.e., there is a positive chance of ‘seeing’ this measure in a physical experiment where one observes an orbit of a randomly selected point X ∈ M. In hyperbolic systems like ours, the basin Bν of every measure ν is (mod 0) a union of stable manifolds. Thus, the condition Leb(Bν ) > 0 is equivalent to mW (W ∩ Bν ) > 0 for some unstable curve W . It is then natural to construct a physical measure by iterating the Lebesgue measure mW defined on an unstable curve, i.e., by taking a Cesaro limit point of the sequence of its images Fεn (mW ). Suppose we start with a standard pair 0 (or more generally, with a standard family G0 that has a finite ZG0 < ∞). Then, we consider the sequence of measures PGn = Fεn (PG0 ). For all n ≥ n0 ∼ C| ln ZG0 |, the measure PGn will be supported on a proper standard family Gn . As such, it will be mostly supported on long unstable curves. More precisely, for any ζ > 0 we have (8.1) PGn ∪Wα ∈Gn : |Wα |<ζ Wα ≤ Cζ for some constant C > 0. Next, for any ζ > 0, consider the class Cu (ζ) of unstable curves W ⊂ M of length ≥ ζ. Denote by Cu (ζ) its closure in the Hausdorff metric. Recall that our unstable curves are C 2 with uniformly bounded curvature, and their tangent vectors satisfy (4.1). Therefore, ([17, Lemma 4.60]) all the curves in the class Cu (ζ) are of length ≥ ζ, they are at least C 1 (but not necessarily C 2 , though their derivatives are Lipschitz continuous), and their tangent vectors also satisfy (4.1). We will call curves W ∈ ∪ζ>0 Cu (ζ) generalized unstable curves. Accordingly, we define generalized standard pairs and families as those supported on generalized unstable curves. (For brevity, we will omit the word ‘generalized’ most of the time).
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Proposition 8.1. For every generalized standard family G the sequence of Cesaro averages of the images Fεn (PG ) has a weakly convergent subsequence. The limit measure νε will be Fε -invariant, and it will be supported on a proper generalized standard family Gε . Proof. Our argument is similar to that used in [1] for construction of physical measures for partially hyperbolic diffeomorphisms. First we define a metric on the space of all generalized standard pairs. Given two pairs 1 = (W1 , ρ1 ) and 2 = (W2 , ρ2 ), we parameterize W1 and W2 by a normalized arclength parameter s, i.e., W1 = {r1 (s), ϕ1 (s)} and W2 = {r2 (s), ϕ2 (s)} for 0 ≤ s ≤ 1. We assume that the orientations of these parameterizations agree—for example, s = 0 corresponds to the bottom left endpoint and s = 1 to the top right endpoint of each curve (recall that the curves are monotonically increasing in the rϕ coordinates, i.e., they run from bottom left to top right). The metric is now defined as dist(1 , 2 ) = max {|r1 (s) − r2 (s)|, |ϕ1 (s) − ϕ2 (s)|, |ρ1 (s) − ρ2 (s)|}. 0≤s≤1
For any ζ > 0 we denote by Ψζ the metric space of standard pairs supported on unstable curves of length ≥ ζ. Observe that this space is compact. Now any standard family G is a measure on ∪ζ>0 Ψζ , we denote that measure by ˜ G (note that it is different from PG , as it is defined on the space of standard P pairs, while PG is defined on M). Now let {Gi } be any sequence of proper standard families; then ˜ G (Ψζ ) → 1 inf P i i
as ζ → 0,
˜ G has a weakly convergent subsequence. due to (8.1). Thus, the sequence P i Now let G be any standard family with ZG < ∞. Then, for all n ≥ nG the ˜ G ) will be a proper standard family. Due to the above, its Cesaro image Fεn (P averages will have a weak limit point, which will be supported on a proper standard family, we denote it by Gε . It will obviously be invariant under Fε , and its projection onto M will be an Fε -invariant probability measure νε . We will show later that the physical measure νε is unique, ergodic, and mixing. ˜ G . We say Next, we investigate the support of the so defined measure P ε ˜ that a standard pair = (W, ρ) belongs in the support of PGε if for any δ > 0 the measure of the δ-neighborhood of is positive. The δ-neighborhood consists of standard pairs on unstable curves W of length |W | ≥ |W | − 4δ that ˜ G is invariant, we see that (at least some of) the are δ-close to W . Since P ε curves W are images of other unstable curves (or components thereof). By continuity, W is an image of an unstable curve, too, we will call it W−1 . In other words, at least one branch of Fε−1 is defined and continuous on W , and it takes W to the unstable curve W−1 (of course, W−1 will be shorter than W ). In that case W−1 belongs to the support of νε as well.
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We say that an unstable curve W0 ⊂ M is an unstable manifold if there is a sequence of unstable curves W−i such that Fε (W−i−1 ) = W−i for all i ≥ 0. Our previous analysis implies Proposition 8.2. The proper generalized standard family PGε constructed above and corresponding to the invariant measure νε consists of unstable manifolds. In other words, νε is supported on a union of unstable manifolds. In this sense, our invariant measure νε is an analog of Sinai–Ruelle–Bowen (SRB) measures for hyperbolic systems—it is absolutely continuous on unstable manifolds and ergodic (see below). (We note, however, that the measure νε can be represented by many different generalized standard families, not all of them consisting of unstable manifolds). We also note that our unstable manifolds do not foliate M; in fact, they may cross each other (because our map Fε is not invertible). One can think of PGε as a family of unstable manifolds that are ‘scattered’ all over M with plenty of mutual intersections. Their distribution in M may be very inhomogeneous, i.e., they may ‘pile up’ in some places and completely avoid other places (in fact their union may possibly be nowhere dense in M). The above proposition is included here for the purpose of comparing our invariant measure νε with “usual” SRB states; our further analysis will not rely on it. Next, we turn to the Coupling Lemma. This is a useful tool in the studies of hyperbolic maps; and it is flexible enough to be applied to non-invertible maps like ours. We briefly describe the relevant constructions, see a detailed exposition in [17, Sect. 7.5] and [12, Appendix A]. First, every standard pair = (W, ρ) ˆ : = W × [0, 1] equipped with a probability is replaced with a ‘rectangle’ W ˆ defined by measure P ˆ (X, t) = dP (X) dt = ρ(X) dX dt, dP
(8.2)
ˆ is also ρ. The map F n can be naturally defined on i.e., the density of P ε ˆ ‘rectangles’ W by Fεn (X, t) = (Fεn X, t). Given a standard family G = (Wα , ρα ), α ∈ A, with a factor measure ˆ α , ρα ) the family of the corresponding rectangles λG , we denote by Gˆ = (W ˆG the induced equipped with the same factor measure λG , and denote by μ ˆ α. measure on the union ∪α W Lemma 8.3 (Coupling Lemma). Let G = (Wα , ρα ), α ∈ A, and E = (Wβ , ρβ ), β ∈ B, be two proper standard families. Then, there exist a bijection (a couˆ α → ∪β W ˆ β that preserves measure; i.e., Θ(ˆ pling map) Θ : ∪α W μG ) = μ ˆE , and ˆ α → N such that two properties hold: a (coupling time) function Υ : ∪α W ˆ α , α ∈ A, and Θ(X, t) = (Y, s) ∈ W ˆ β , β ∈ B. Then, the A. Let (X, t) ∈ W m m points Fε (X) and Fε (Y ) lie on the same stable manifold W s ⊂ M for m = Υ(X, t).
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B. There is a uniform exponential tail bound on the function Υ: μ ˆG (Υ > n) ≤ CΥ ϑnΥ ,
(8.3)
for some constants CΥ = CΥ (D) > 0 and ϑΥ = ϑΥ (D) < 1. We emphasize that the constants CΥ and ϑΥ are uniform in ε. Proof. The argument goes along the same lines as in [17, Sect. 7.5] and [12, Appendix A] (we have already prepared all the necessary technical tools). The only new issue is the uniformity in ε. To settle it, we recall that the proof uses a quadrilateral R ⊂ M bounded by two stable curves and two unstable curves (called a ‘rhombus’ in [17]) where the coupling is defined explicitly (R contains all the points Fεm (X) and Fεm (Y ) mentioned in part A). That rhombus must first be constructed for the billiard map F0 , and then, because Fε is a small perturbation of F0 , the rhombus R will have nearly the same properties for all small ε. In particular, the fact that there exists m0 such that for any proper unstable curve Fεm0 W R = ∅ is first proven for ε = 0 using mixing of F0 and then extended to ε = 0 using compactness of the set of proper unstable curves. This perturbative argument is described in detail in [9, pp. 229–233], and it works in our case as well. Remark. The above perturbative argument puts severe restrictions on ε, i.e., it works only for very small ε; see a remark on page 232 in [9] for explanations and related issues. One can expect that hyperbolicity, regularity conditions, and the existence of physical measures hold for reasonably small ε, but the uniqueness and strong statistical properties (see below) hold only for extremely small ε. We expect that if we increase ε continuously, one first observes a unique physical measure, then a finite collection of physical (SRB-like) measures, and then singular non-SRB stationary states. For models with Gaussian thermostats numerical experiments of this sort were done in [21]. In plain words, the Coupling Lemma means that the images of any two measures supported on proper standard families will get close together (exponentially fast), and their further images will become almost indistinguishable. In particular, one of the two standard families in Lemma 8.3 can be Gε supporting the invariant measure νε = PGε ; then the images of any other measure PG converge to νε exponentially fast (in the sense specified by the coupling map). Corollary 8.4. The measure νε = PGε described in Proposition 8.1 is unique and independent from the initial standard family G. It is ergodic and mixing. In addition, we have the following weak limit: lim Fεn (PG ) = νε
n→∞
(8.4)
for any standard family G with a finite ZG < ∞. Note that PG in (8.4) can be replaced with ν0 , as one can foliate M by unstable curves and condition the measure ν0 on them to get a PG .
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Proof. The uniqueness and ergodicity readily follow from the Coupling Lemma, and so does (8.4). To show that νε is mixing it is enough to verify that lim νε (f · (g ◦ Fεn )) = νε (f )νε (g)
(8.5)
n→∞
for any continuous functions f, g on M. Clearly, f and g can be approximated by smooth functions; then, f can be replaced with a linear combination of smooth functions with values in the interval [1 − δ0 , 1 + δ0 ] for a small δ0 > 0 and mean values =1, and then f PGε will be a proper standard family; thus (8.5) follows from (8.4). Now we address the statistical properties of the system (M, Fε , νε ). We say that a function f : M → R is dynamically H¨ older continuous if there are ϑf ∈ (0, 1) and Kf > 0 such that for any X, Y ∈ W lying on one unstable curve W s (X,Y )
|f (X) − f (Y )| ≤ Kf ϑf+
(8.6)
and for any X, Y ∈ W s lying on one stable manifold W s s (X,Y )
|f (X) − f (Y )| ≤ Kf ϑf−
,
(8.7)
where s− (X, Y ) is the largest m ≥ 0 such that W s = Fεm (W1s ) for another stable manifold W1s . The value s+ (X, Y ) is called the future separation time for the points X and Y , and we can naturally call s− (X, Y ) the past separation time. We denote the space of such functions by H. It contains every piecewise H¨older continuous function whose discontinuities coincide with those of Fεm for some m > 0. For example, the components Δε,x , Δε,y of the vector displacement function Δε belong in H (to be proven in Sect. 10). The following two propositions follow from the Coupling Lemma; see the proofs of Theorems 7.31 and 7.37 in [17]. Proposition 8.5 (Equidistribution). Let G be a proper standard family. For any dynamically H¨ older continuous function f ∈ H and n ≥ 0
n
f ◦ Fε dPG − f dνε
≤ Bf θfn (8.8)
M
M
where Bf = 2C (Kf + f ∞ ) and θf = [max{ϑΥ , ϑf }] a constant independent of G and ε.
1/2
< 1; here, C > 0 is
In other words, iterations of measures on standard pairs weakly converge to the measure νε , and the convergence is exponentially fast in the sense of (8.8). Proposition 8.6 (Exponential bound on correlations). For any pair of dynamically H¨ older continuous functions f, g ∈ H and n > 0 n |νε (f · (g ◦ Fεn )) − νε (f )νε (g)| ≤ Bf,g θf,g
(8.9)
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1/4 θf,g = max ϑΥ , ϑf , ϑg , e−1/c3 < 1,
where c3 > 0 is the constant from Lemma 7.2, and Bf,g = C (Kf g∞ + Kg f ∞ + f ∞ g∞ ). Remark. The invariant measure νε in (8.9) can be replaced with any measure PG supported on a proper standard family (in particular, by the F0 -invariant measure ν0 ). In that case (8.9) takes form: n |PG (f · (g ◦ Fεn )) − PG (f )νε (g)| ≤ Bf,g θf,g .
The proof of this is just a simple adaptation of the standard proof of the above theorem, see, e.g., the proof of Theorem 7.37 in [17]. This fact was used in [15]. The last theorem can be extended to multiple correlations, and it implies, via a standard argument, Central Limit Theorem for the map Fε , see [17, Chapter 7] and [11]. Last, we derive the Kawasaki-type formula for any dynamically H¨ older continuous function f ∈ H. Due to (8.4) νε (f ) = ν0 (f ) + lim
n→∞
n
ν0 (f ◦ Fεk ) − (f ◦ Fεk−1 ) .
(8.10)
k=1
Also, recall that ν0 = PG0 for a proper standard family G0 . Thus, due to Equidistribution property, ν0 (f ◦ Fεk ) converges to νε (f ) exponentially fast. Therefore, the series in (8.10) converges at an exponential rate, and we have the following tail bound:
∞
k k−1
ν0 (f ◦ Fε ) − (f ◦ Fε ) ≤ 2Bf θfn /(1 − θf ). (8.11)
k=n
older norm We note that if the H¨older exponent ϑf of the function f , its H¨ Kf , and its ∞-norm f ∞ are all independent of ε, then the above estimate is uniform in ε, too.
9. Electrical Current Now we turn to the physically interesting feature—electrical current. In this section we present our arguments in a relatively general way suppressing some model-specific details; the latter will be supplied in the next section. We first investigate the discrete-time current defined by ˆ = lim 1 q ˜ n = νε (Δε ), J n→∞ n ˜ 1 − q denotes the displacement vector, cf. (2.3). where Δε = (Δε,x , Δε,y ) = q We write Δε for both components Δε,x and Δε,y of Δε . The function Δε is
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bounded and dynamically H¨ older continuous (see Corollary 10.4); hence, the Kawasaki formula (8.10) applies: νε (Δε ) = ν0 (Δε ) +
∞
ν0 (Δε ◦ Fεn ) − (Δε ◦ Fεn−1 ) ,
(9.1)
n=1
in which the series converges exponentially. In fact, its H¨ older exponent θΔε ∈ (0, 1) is independent of ε and its norm is KΔε = O(ε−a ) with some constant a > 0 (see Corollary 10.4). Due to (8.11) we can choose N = L| ln ε| with a large constant L > 0 and rewrite (9.1) as νε (Δε ) = ν0 (Δε ) +
N
ν0 (Δε ◦ Fεn ) − (Δε ◦ Fεn−1 ) + χ1 ,
(9.2)
n=1
where χ1 = O(ε2 ). Here and in what follows we will denote by χ1 , χ2 , . . . various remainder terms; they will all satisfy χi = O(ε1+ai ) for some ai > 0. Now let M=
∞
Mk ,
k=0
where Mk consists of points with exactly k preimages under Fε−1 . In other words, Mk consists of points where exactly k domains M− i (described in Sect. 5) overlap. (In fact, all Mk = ∅ only for k ≤ k0 , with some k0 independent of ε). The set M1 is overwhelmingly large, and we denote its complement by M = M\M1 . In the finite horizon case, ν(M ) = O(ε2 ), and in the infinite horizon case ν0 (M ) = O(ε3/2 ) due to (6.3). The inverse map Fε−1 is uniquely defined on M1 , and we put M1− = Fε−1 (M1 ) and M− = M\M1− . We also split ν0 = ν01 + ν0 , where ν01 denotes ν0 restricted to M1 and − = Fε−1 (ν01 ) is supported on ν0 is ν0 restricted to M . Now the measure ν01 − M1 , and according to Lemma 4.3, for any X ∈ M1− we have − 1 + 2εx dν01 v (X) = = (9.3) dν0 v1 1 + 2εx + 2εΔε,x (using the notation of Sect. 3). Now Taylor expansion gives v/v1 = 1 − εΔε,x + εRε , where Rε is the remainder, whose contribution in the end will be negligible − denote the measure on M− = M\M1− (see discussion after (10.7)). Let ν0 − − − having the same density v/v1 as the measure ν01 on M1− . Note that ν01 + ν0 is not necessarily a probability measure; in fact, we have ∞ v dν0 = kν0 (Mk ) = 1 − χ3 v1 M
k=1
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− − where χ3 = O(ε3/2 ). Normalizing ν01 + ν0 gives a probability measure μ ˜0 on M with density
g(X) =
v d˜ μ0 = 1 − εΔε,x + εRε + χ4 , = dν0 v1 (1 − χ3 )
(9.4)
where χ4 = O(ε3/2 ). Summarizing our formulas, we obtain for every n ≥ 1 ˜0 (Δε ◦ Fεn ) + χ5,n , ν0 (Δε ◦ Fεn−1 ) = μ
(9.5)
where − χ5,n = ν0 (Δε ◦ Fεn−1 ) − ν0 (Δε ◦ Fεn ) − χ3 μ ˜0 (Δε ◦ Fεn )
= O(ε5/4 | ln ε|) + O(ε3/2 | ln ε|) (here, we used (7.11) and (7.13)). Now the Kawasaki formula (9.2) can be rewritten as νε (Δε ) = ν0 (Δε ) +
N
ν0 [(1 − g)(Δε ◦ Fεn )] + χ5 ,
(9.6)
n=1
where χ5 =
N
χ5,n = O ε5/4 | ln ε|2 .
n=1
Next, we observe that ν0 (gΔε ) =
1 1 − χ3
M1−
v 1 Δε dν0 + v1 1 − χ3
M−
v Δε dν0 v1
(9.7)
and note that the integral I := M− Δε dν0 is O(ε2 ) for systems with finite horizon, but I = O(ε) for systems with infinite horizon; recall (7.18). Now we define a new function Δε ◦ Fε−1 on M1 − Δε = (9.8) 0 on M and rewrite (9.7) as ν0 (gΔε ) =
1 1 − χ3
Δ− ε dν0 + I + χ6
M1
= ν0 (Δ− ε ) + I + χ7 ,
(9.9)
where χ6 and χ7 are O(ε3/2 ). Therefore, ν0 (Δε ) = ν0 (Δ− ε ) + ν0 [(1 − g)Δε ] + I + χ7 and so
1 1 ν0 (Δε ) + ν0 (Δ− ν0 [(1 − g)Δε ] + I/2 + χ7 /2. ε ) + 2 2 a We will show that ν0 (Δε ) + ν0 (Δ− ε ) = O(ε ), where a = 2 for systems with finite horizon and a = 1 for systems with infinite horizon (Lemma 10.1). ν0 (Δε ) =
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Combining all our formulas gives νε (Δε ) =
N 1 ν0 [(1 − g)Δε ] + ν0 [(1 − g)(Δε ◦ Fεn )] + O(εa ), 2 n=1
(9.10)
where a > 1 for systems with finite horizon and a = 1 for systems with infinite horizon. Using Taylor expansion (9.4) gives νε (Δε ) =
N 1 εν0 (Δε,x Δε ) + ε ν0 [(Δε ◦ Fεn )Δε,x ] 2 n=1
−
N 1 εν0 (Rε Δε ) − ε ν0 [(Δε ◦ Fεn )Rε ] + O(εa ). 2 n=1
(9.11)
The contribution of the remainder Rε is small and can be incorporated into the last term O(εa ). The correlations ν0 [Δε,x (Δε ◦ Fεn )] decay exponentially and uniformly in ε (see Proposition 10.6). The vector Δε = (Δε,x , Δε,y ) depends on ε continuously, so using a priori bounds of Lemma 10.2 we see that for every n ≥ 1 we have lim ν0 [Δε,x (Δε ◦ Fεn )] = ν0 [Δ0,x (Δ0 ◦ F0n )],
ε→0
(9.12)
where Δ0,x and Δ0 denote the components of the displacement vector Δ0 in the field-free (billiard) dynamics. In the finite horizon case, (9.12) holds for n = 0 as well. Thus, in the finite horizon case we arrive at ˆ = νε (Δε ) = 1 DE + o(ε), (9.13) J 2 where D is the diffusion matrix (1.3). In the infinite horizon case, the very first term in (9.11), i.e., 12 εν0 (Δε,x Δε ) is dominant; it is O(ε| ln ε|) due to (7.9). In fact, we will show in Sect. 10, see (10.12), that ν0 (Δε ⊗ Δε ) = | log ε| D∞ + O(1)
(9.14)
where D∞ is the super-diffusion matrix (1.5) for the respective field-free (billiard) system. Therefore, the current is given by ˆ = νε (Δε ) = 1 | log ε| D∞ E + O(ε). (9.15) J 2 Next, basic facts from the theory of suspension flows (see [22, pp. 292–295], [5, pp. 21–22], or [17, Sect. 2.9]) readily give (2.3) and (2.7). Thus, in order to complete the proofs of the main formulas (2.4) and (2.8) it suffices to show that, as stated in Theorems 6 and 7, τ¯ε = τ¯ + O(εa ),
(9.16)
for some a > 0. This will be done in Sect. 10 (see Lemma 10.8). It remains to prove the Central Limit Theorem and the corresponding formulas for the diffusion matrices, i.e., (2.5)–(2.6) and (2.9)–(2.10). According to general results, cf. [26] or [17, Theorem 7.68], the continuous-time CLT follows from its discrete-time counterpart whenever the corresponding ceiling
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function is bounded and dynamically H¨ older continuous. Our ceiling function τε has these properties, as we show in the next section. Now the discrete-time CLT (mentioned in Sect. 8) says that ˆ ˜ n − nJ q ˆ ∗ ), √ ⇒ N (0, D ε n
(9.17)
ˆ ∗ is given by the sum of correlations where the covariance matrix D ε ˆ ∗ = νε (Δε ⊗ Δε ) − νε (Δε ) ⊗ νε (Δε ) D ε ∞ +2 (νε [(Δε ◦ Fεn ) ⊗ Δε ] − νε (Δε ) ⊗ νε (Δε )).
(9.18)
n=1
The series in (9.18) converges exponentially fast, and we will show in the next section that the convergence is uniform in ε. We will also show (see Lemma 10.8) that, for each n ≥ 0, νε ((Δε ◦ Fεn ) ⊗ Δε ) = ν0 ((Δε ◦ Fεn ) ⊗ Δε ) + O(εa )
(9.19)
for some a > 0. (We also recall that νε (Δε ) is small, according to (9.13) and (9.15)). This, along with (9.12) and (9.14), implies that ˆ ∗ = D + o(1) D ε for systems with finite horizon and ˆ ∗ = | log ε| D∞ + O(1) D ε
(9.20)
ˆ ∗ by a standard in the infinite horizon case. We also recall that D∗ε = τ¯ε−1 D ε formula [17, Theorem 7.68]. This completes the proof of (2.5)–(2.6) and (2.9)– (2.10), and thus that of Theorems 6 and 7.
10. Finishing the Proofs Here, we prove various technical statements made in Sect. 9. Our arguments mostly follow the lines of Sects. 8–10 in [15] where Gaussian thermostatted Lorentz gases were treated. We sketch the steps that repeat those of [15] and give details whenever our arguments differ from those of [15], in particular when gaps and overlaps are involved. Recall that Δε denotes either component, Δε,x or Δε,y , of the displacement vector Δε . Our map Fε is not invertible, but the function Δ− ε defined by (9.8) plays the role of Δε ◦ Fε−1 . a Lemma 10.1. We have ν0 (Δε ) + ν0 (Δ− ε ) = O(ε ), where a = 2 for systems with finite horizon and a = 1 for systems with infinite horizon.
Proof. Let X = (r, ϕ) ∈ M1 be a point such that we also have I(X) = (r, −ϕ) ∈ M1 (recall that I denotes the involution on M). Since the density cos ϕ of the measure ν0 is equal at the points X and I(X), we can combine them and estimate the following sum: − Δε (r, ϕ) + Δ− ε (r, −ϕ) + Δε (r, −ϕ) + Δε (r, ϕ).
(10.1)
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The key observation is that Δε (r, ϕ) and Δ− ε (r, −ϕ) nearly cancel each other (and so do the other two terms). Indeed, they are projections (on the x or y axis) of two trajectories starting at the same point r, shooting at the same angle ϕ, but having slightly different initial velocities (according to the local time reversibility of our dynamics; see Sect. 5). In fact, the difference between their velocities is O(ε|Δε (X)|). Note that if the velocities were equal, then the trajectories would coincide, all the way up the next collision, and because they run in the opposite directions the first two terms in (10.1) would perfectly cancel out. But since the velocities differ a little, the trajectories run very closely, and the sum of the first two terms is (usually) small, see next. An elementary analysis shows that when the above two trajectories cover the distance |Δε (X)|, i.e., when they are about to come to the next collision, they are O(ε2 |Δε (X)|3 ) apart from each other. If they land on the same scatterer, at some angle ϕ1 , the landing points will be O(ε2 |Δε (X)|3 / cos ϕ1 ) apart. The overall contribution of such trajectories amounts to J := ν0 ε2 |Δε (X)|3 / cos ϕ1 . In the finite horizon case, J = O(ε2 ). For infinite horizon, m0.5 + ε2 m−3 m3.5 = O(ε5/4 ). J ε2 L m
m
If the above two trajectories land on different scatterers, then Δε (r, ϕ) + Δ− ε (r, −ϕ) is of order one, and we just need to estimate the measure of the set of such points X. Those points are characterized by the fact that their trajectories run O(ε2 m3 )-closely to the edge of a scatterer that is at distance m away from X. Thus, such points X are located in the (ε2 m2 )-neighborhood of (U ) (L) the borders of the cells Dm and Dm for all m. Thus, such points make a set of measure mU
mL √ 2 √ 2 ε2 m2 1/ m + ε 2 m2 m/mL = O(ε).
m=1
m=1
It remains to account for points X ∈ M and X ∈ I −1 (M ). In the finite horizon case, of measure O(ε2 ). For infinite horizon, we have they make a set 5/4 seen that M Δε dν0 = O(ε | ln ε|). A similar analysis for the set I −1 (M ) shows that Δε dν0 = O(ε), (10.2) I −1 (M )
which is an analog of (7.18). The lemma is proven. Lemma 10.2. For each n ≥ 0 we have (U ) 1/m3 , (Fεn ν0 ) Dm
(L) m/m4L (Fεn ν0 ) Dm
and the same estimates hold for the limit measure νε .
(10.3)
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Proof. The proof is almost identical to that of Lemma 8.1 in [15]: for small n (say, for n < ε−0.1 ), we use the fact that the (local) Jacobian (4.11) of the map Fε is very close to one; hence the (local) Jacobian of Fεn is still close to one (it is enough for us that the Jacobian of Fεn is between 0.9 and 1.1). For larger n we use the equidistribution (Theorem 8.5): it shows that the sequence of the above measures converges exponentially fast; hence it will already be close to its limit when n ≥ ε−0.1 . In addition, now we need to estimate the effect of gaps and overlaps on the above measures for small n. Due to (7.16), the measure of gaps and over(U ) laps that come to Dm during the first n iterations is O(nε5/4 /m2 ) which 3 is o(1/m ) for n < ε−0.1 . Similarly, due to (7.17), the measure of gaps and (L) overlaps that come to Dm during the first n iterations is O(nε5/4 /m2L ) which is o(m/m4L ) for n < ε−0.1 . Lemma 10.3. Let X, Y ∈ M be two nearby points such that their trajectories in D land, at the next collision, on the same scatterer in R2 . Then we have Δε (X) − Δε (Y ) ≤ C Δε (X) · dist(X, Y ), where C > 0 is a constant independent of ε. This is an analog of Lemma 8.2 in [15]. Its proof is an elementary calculation, because the trajectories between collisions are parabolas; we leave the details to the reader. older continuous function older Corollary 10.4. Δε is a dynamically H¨ √ with H¨ older constant KΔε Δε ∞ 1/ ε. exponent ϑΔε = 1/2 and H¨ This proves the Kawasaki formula (9.1) with the error estimate
n
Δε ◦ Fε dν0 − Δε dνε
≤ Cε−1/2 θn .
M
(10.4)
M
Furthermore, for every k ≥ 2 we have KΔkε + Δkε ∞ ε−k/2 , and so
k n k −k/2 n
Δ ◦ F dν − Δ dν θ . (10.5) 0 ε ≤ Cε ε ε ε
M
M
For k = 1, we can get a uniform bound, independent of ε: Proposition 10.5. For some constants C > 0 and θ < 1, independent of ε,
n
Δε ◦ Fε dν0 − Δε dνε
≤ Cθn . (10.6)
M
M
The proof repeats that of Proposition 8.3 in [15] verbatim. The uniform convergence here justifies our choice of N = L| ln ε| in (9.2). Next, we examine the remainder Rε : (9.3) implies v 1·3 2 1·3·5 3 η + η − ··· 1− =η− v1 2! 3!
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where η =
εΔε,x 1+2εx ;
Rε =
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thus,
1·3 εΔ2ε ε2 Δ3ε 2εxΔε 1·3·5 + · · − + ··· 1 + 2εx 2! (1 + 2εx)2 3! (1 + 2εx)3
(10.7)
This expansion is similar to (8.8) in [15]. One can easily see that Rε is dynamically H¨ older continuous with the same exponent and norm as the function Δε,x . Besides, Rε is bounded (uniformly in ε), and Rε → 0 pointwise, as ε → 0. For all these reasons its contribution to (9.11) will√ be much easier to handle than that of Δε,x . In particular, ν0 (Rε Δε ) = O( ε), as the main contribution comes from the second term of (10.7), and we can apply (7.10) with k = 3. Next we establish a uniform bound on correlations: Proposition 10.6. For some constants C > 0 and θ ∈ (0, 1) independent of ε and all n ≥ 1 |νε [(Δε ◦ Fεn )Δε,x ] − νε (Δε )νε (Δε,x )| ≤ Cθn
(10.8)
|ν0 [(Δε ◦ Fεn )Δε,x ] − νε (Δε )ν0 (Δε,x )| ≤ Cθn
(10.9)
and
This is our analog of Proposition 9.3 in [15], which in turn is an extension of the bounds on correlations in the infinite horizon Lorentz gas given in Proposition 9.1 in [15]. The proofs in [15] only use the estimates on the sizes, (U ) (L) shapes, and measures of the cells Dm and Dm , which are the same here, so they apply to the present situation without changes. Proposition 10.6 implies ν0 [(Δε ◦ Fεn )Δε,x ] = νε (Δε )ν0 (Δε,x ) + O(θn )
(10.10)
and similarly, replacing one Δε with a much milder function Rε we get ν0 [(Δε ◦ Fεn )Rε ] = νε (Δε )ν0 (Rε ) + O(θn ).
(10.11)
These uniform bounds and (9.11) imply our final results. Precisely, in the case of finite horizon they imply (9.13). For infinite horizon we obtain 1 J = νε (Δε ) = εν0 (Δε ⊗ Δε ) + O(ε), 2 so it remains to verify (9.14). This is done by direct integration, see below. (L) First, we note that the contribution from the lower cells Dm is O(1), see the second sum in (7.9), hence they can be ignored. Second, we can choose a large constant C 1 and ignore all phase points with Δε ≤ C. So, we only consider trajectories that shoot from the initial scatterer facing an infinite corridor (see B1 in Fig. 12) and landing on distant scatterers on the opposite side of that corridor (the upper row in Fig. 12). Next, let L1 and L2 denote two parallel lines bordering our corridor, i.e., the two common tangent lines to all the scatterers along the corridor, see Fig. 12. Our trajectories leave the scatterer B1 , cross L1 first, then after a long trip in the corridor they cross L2 and land very shortly on the next ˆ ε = (Δ ˆ ε,x , Δ ˆ ε,y ) denote the vector between the scatterer in the top row. Let Δ
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Figure 12. An infinite corridor bounded by two lines tangent to the scatterers
ˆ ε,x + O(1) two crossing points, on L1 and L2 respectively. Note that Δε,x = Δ ˆ and Δε,y = Δε,y + O(1); thus, ˆ 2 ) + O(1), ν0 (Δ2ε,x ) = ν0 (Δ ε,x
ˆ ε,y ) + O(1), ˆ ε,x Δ ν0 (Δε,x Δε,y ) = ν0 (Δ
ˆ ε,x , Δ ˆ ε,y . and so we can replace Δε,x , Δε,y with Δ Let γ denote the angle between the line L1 and the field direction (i.e., the x axis). The equation of the line L1 is ay − bx = 0, where a = cos γ and b = sin γ , and that of the other line L2 is ay − bx = w, where w denotes the width of the corridor. Let bf I ⊂ L1 be a segment of L1 between two consecutive scatterers, one of which is in D, and let r be the arclength parameter on bf I, so that 0 ≤ r ≤ rmax = |bf I|. We regard bf I as an ‘artificial’ part of the boundary of ˜ and then the measure ν0 on it would be a smooth measure with our table D, density cν cos ϕ dr dϕ, where ϕ is the angle made by the trajectory crossing bf I and the normal vector to L1 . Xr At every point Xr = (xr , yr ) ∈ bf I we consider trajectories leaving √ into the corridor, with velocity vector v such that v = v = 1 + 2εxr according to (2.1). Let ψ = π/2 − ϕ denote the angle that v makes with the line L1 . There is a minimum ψmin > 0 so that the trajectory crosses L2 before ˆ ε,x , yr + Δ ˆ ε,y ) denote the (first) intersection of coming back to L1 . Let (xr + Δ our trajectory with L2 . Proposition 10.7. We have π/2 rmax
ˆ ε,x )2 sin ψ dψ dr = (Δ 0
ψmin
1 2 2 a w rmax | ln ε| + O(1) 2
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and π/2 rmax
0
ψmin
ˆ ε,x Δ ˆ ε,y sin ψ dψ dr = 1 abw2 rmax | ln ε| + O(1). Δ 2
This proposition easily implies ν0 (Δε ⊗ Δε ) = | log ε| D∞ + O(1)
(10.12)
see details in the proof of Eq. (10.5) in [15]. Note that sin ψ = cos ϕ. Proof. This follows by an elementary calculation; we only outline the main steps. The initial velocity vector of the moving particle is (v cos(γ + ψ), v(sin γ + ψ)), and its position at time t is 1 2 xr + tv cos(γ + ψ) + εt , yr + tv sin(γ + ψ) 2 Substituting this into the equation ay − bx = w of the line L2 gives εbt2 − 2tv sin ψ + 2w = 0. Solving this quadratic equation for 1/t gives the time of intersection of the trajectory with the line L2 :
t= v sin ψ(1 +
2w 1 − 2εbw/(v 2 sin2 ψ))
Incidentally, we see that ψmin = sin−1 terms of order ε, we get t≈
2bwε/v 2
ˆ ε,x ≈ aw , Δ sin ψ
w , v sin ψ
√
ε. If we ignore small
ˆ ε,y ≈ bw . Δ sin ψ
(10.13)
We note that C √ c ε
1 dψ = | ln ε| + O(1) sin ψ 2
for any constants c > 0 and 0 < C < π/2; thus, the approximation (10.13) gives the right answer. To take care of terms of order ε we note that their contribution is of order C √ c ε
√ εdψ = O( ε); sin2 ψ
thus, they will not affect the final result.
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Lemma 10.8. Formulas (9.16) and (9.19) hold. We prove (9.16) in two steps: νε (τε ) = ν0 (τε ) + O(εa )
(10.14)
ν0 (τε ) = ν0 (τ0 ) + O(εa ).
(10.15)
and It is easy to verify directly that if a trajectory starts from a phase point (q, v) ∈ M and moves without collisions for a time τ , then it deviates from a billiard trajectory starting from the same phase point by O(ετ 2 ). Then, the argument used in [15, Sect. 10] gives (10.15). Now (10.14) and (9.19) are similar, except (10.14) applies to τε and (9.19) applies to (Δε ◦ Fεn ) ⊗ Δε for n ≥ 0. One can check directly that the function older continuous with τε has all the same properties as Δε —it is dynamically H¨ exponent ϑτε ∈ (0, 1) independent of ε and norm Kf = O(ε−a ) for some a > 0, and it has the same order of magnitude as Δε . Thus, all of these estimates can be treated in the same way. We work out the most difficult of them, (9.19) for n = 0: νε (Δε ⊗ Δε ) = ν0 (Δε ⊗ Δε ) + O(εa )
(10.16)
for some a > 0. To this end we apply the Kawasaki formula (9.1) to each component of the matrix Δ2ε,x Δε,x Δε,y Δε ⊗ Δε = . Δε,x Δε,y Δ2ε,y They are treated similarly, and we only show the formulas for Δ2ε,x : νε (Δ2ε,x ) = ν0 (Δ2ε,x ) +
∞
ν0 (Δ2ε,x ◦ Fεn ) − (Δ2ε,x ◦ Fεn−1 ) ,
(10.17)
n=1
in which the series converges exponentially. As we have shown, its H¨older exponent ϑΔ2ε,x ∈ (0, 1) is independent of ε and its norm is KΔ2ε,x + Δ2ε,x ∞ = O(ε−1 ). Hence, due to (8.11), the terms of the above series are O(ε−1 θn ) for some constant θ ∈ (0, 1). Thus, we can choose N = L| ln ε| with a large constant L > 0 and rewrite (10.17) as νε (Δ2ε,x ) = ν0 (Δ2ε,x ) +
N
ν0 (Δ2ε,x ◦ Fεn ) − (Δ2ε,x ◦ Fεn−1 ) + O(ε). (10.18)
n=1
Next, we apply the analysis developed in Sect. 9 to the function Δ2ε,x , instead of Δε . In particular, an analog of (9.5) will be ˜0 (Δ2ε,x ◦ Fεn ) + χ9,n , ν0 (Δ2ε,x ◦ Fεn−1 ) = μ where − χ9,n = ν0 (Δ2ε,x ◦ Fεn−1 ) − ν0 (Δ2ε,x ◦ Fεn ) − χ3 μ ˜0 (Δ2ε,x ◦ Fεn )
= O(ε3/4 ) + O(ε)
(10.19)
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(here we used (7.12) and (7.14) with k = 2). Now the Kawasaki formula (10.18) can be rewritten as N
2 2 νε (Δε,x ) = ν0 (Δε,x ) + ν0 (1 − g)(Δ2ε,x ◦ Fεn ) + χ9 , (10.20) n=1
where χ9 =
N
χ9,n = O ε3/4 | ln ε| .
n=1
Using Taylor expansion (9.4) gives νε (Δ2ε,x ) = ν0 (Δ2ε,x ) + ε
N
ν0 (Δ2ε,x ◦ Fεn )(Δε,x + Rε ) + χ10 ,
(10.21)
n=1
where the remainder is χ10 = O(ε3/2 | ln ε|2 ) because χ4 = O(ε3/2 ) and ν0 (Δ2ε,x ◦ Fεn ) = O(| ln ε|) for every n ≥ 1; see (7.9) and Lemma 10.2. Now using H¨ older inequality gives
2
ν0 (Δε,x ◦ Fεn )(Δε,x +Rε ) ≤ (ν0 ◦ Fεn ) |Δ3ε,x | 2/3 ν0 |Δε,x +Rε |3 1/3 , and each of these integrals is bounded by mU mL m3 m4 −1/2 C + C , 4 ≤ C(mU + mL ) ≤ Cε 3 m m L m=1 m=1 recall (7.10) for k = 3 and Lemma 10.2. Summing over n ≤ L| log ε| gives L| log ε|
ε
ν0 (Δ2ε,x ◦ Fεn )(Δε,x + Rε ) = O ε1/2 | log ε| ,
n=1
which completes the proof of (10.16).
Appendix: The Choice of a Thermostat In our model, we control the particle’s kinetic energy indirectly, via dealing with its total energy; see Sect. 2. It might be tempting to control the kinetic energy directly, by resetting the speed of the particle to a constant value (say, to one) after every collision. Here, we show that this would cause intractable complications. Let us consider trajectories arriving at a fixed point (x0 , y0 ) on the border of a scatterer B0 with a velocity vector making a fixed angle θ0 with the x-axis. The angle θ0 would completely determine the direction of the trajectory after the reflection at ∂B0 ; thus, all our trajectories would arrive at a single point X0 ∈ M determined by (x0 , y0 , θ0 ); we will see under what conditions there might be more than one preimage of X0 in M. Let the trajectory originate at a point (x, y) with initial velocity (u, v). Then, by the rules (2.2) we have 1 y0 = y + vt x0 = x + ut + εt2 , 2
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where t is the travel time. Note that (u + εt)/v = cot θ0 = : k, so that we can eliminate t and obtain x = x0 +
u2 − k 2 v 2 , 2ε
y = y0 +
uv − kv 2 . ε
(A.22)
Let us first suppose the particle starts with a unit speed, i.e., u = cos θ and v = sin θ. Then x, y in (A.22) become functions of θ, and we get a one-parameter family of phase points that will arrive at the same collision point X0 ∈ M. By elementary calculation, all the points (x, y) lie on an ellipse E, and the corresponding outgoing velocity vectors (u, v) make a vector field on E transversal to E. Now it is well possible that the boundary of another scatterer, B, crosses E more than once. Then, from every point (x, y) ∈ E ∩ ∂B a trajectory may originate that is mapped by Fε to X0 . Thus, the collision map Fε would take several points from the surface of the scatterer B to a single point X0 ∈ M. This means that the map Fε would fail to be one-to-one even on the set of points that travel from one given scatterer, B, to another given scatterer, B0 . As a result, the map Fε may create “wrinkles” and “folds” even within the domains where it is naturally continuous. The map Fε may have infinite contraction rates within the domains of continuity, and its Jacobian may be zero at some points. All this would severely complicate our analysis of the hyperbolicity of Fε . In order to avoid the above complications, let us abandon our assumption of the initial unit speed (i.e., u = cos θ and v = sin θ) and return to the more general situation where (A.22) hold. There will be still a family of phase points (x, y, u, v) that arrive at the given point (x0 , y0 ) with velocity directed at the given angle θ0 . Now a natural way to prevent the above complications (and guarantee that the map Fε diffeomorphic on the set of points traveling from one given scatterer, B, to another given scatterer, B0 ) is to make sure that this family of phase points is a single trajectory, i.e., y/ ˙ x˙ = v/u, where dots denote differentiation with respect to a family parameter. Differentiating (A.22) gives v uv ˙ + uv˙ − 2kv v˙ = uu˙ − k 2 v v˙ u from which v(u ˙ − kv)2 = 0. Note that u − kv = 0 implies x = x0 and y = y0 , which holds only at the last point (x0 , y0 ). Thus, in the rest of the family we must have v˙ = 0; hence, v = const (the constant may depend on x0 , y0 , θ0 and ε, of course). Combining with the first equation of (A.22) we obtain 1 2 (u + v 2 ) = const + εx. 2 This means that the total energy must be kept constant. This brings us back exactly to the dynamics defined in Sect. 2.
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References [1] Alves, J., Bonatti, C., Viana, M.: SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140, 351–398 (2000) [2] Baladi, V., Gou¨ezel, S.: Good Banach spaces for piecewise hyperbolic maps via interpolation. Ann. Inst. H. Poincar´e 26, 1453–1481 (2009) [3] Baladi, V., Gou¨ezel, S.: Banach spaces for piecewise cone hyperbolic maps. J. Modern Dyn. 4, 91–137 (2010) [4] Bonetto, F., Daems, D., Lebowitz, J.L.: Properties of stationary nonequilibrium states in the thermostatted periodic Lorentz gas. I. The one particle system. J. Stat. Phys. 101, 35–60 (2000) [5] Brin, M., Stuck, G.: Introduction to Dynamical Systems. Cambridge University Press, Cambridge (2002) [6] Bunimovich, L.A., Sinai, Ya.G.: Statistical properties of Lorentz gas with periodic configuration of scatterers. Comm. Math. Phys. 78, 479–497 (1980/81) [7] Bunimovich, L.A., Sinai, Ya.G., Chernov, N.I.: Statistical properties of twodimensional hyperbolic billiards. Russ. Math. Surv. 46, 47–106 (1991) [8] Chernov, N.: Decay of correlations and dispersing billiards. J. Stat. Phys. 94, 513–556 (1999) [9] Chernov, N.: Sinai billiards under small external forces. Ann. H. Poincar´e 2, 197–236 (2001) [10] Chernov, N.: A stretched exponential bound on time correlations for billiard flows. J. Stat. Phys. 127, 21–50 (2007) [11] Chernov, N.: Sinai billiards under small external forces II. Ann. H. Poincar´e 9, 91–107 (2008) [12] Chernov, N., Dolgopyat, D.: Brownian Brownian motion-I. Memoirs AMS 198(927), 193 (2009) [13] Chernov, N., Dolgopyat, D.: Galton board: limit theorems and recurrence. J. Am. Math. Soc. 22, 821–858 (2009) [14] Chernov, N., Dolgopyat, D.: Diffusive motion and recurrence on an idealized Galton Board. Phys. Rev. Lett. 99, paper 030601 (2007) [15] Chernov, N., Dolgopyat, D.: Anomalous current in periodic Lorentz gases with infinite horizon. Russ. Math. Surv. 64, 73–124 (2009) [16] Chernov, N., Dolgopyat, D.: Particle’s drift in self-similar billiards. Ergod. Theory Dyn. Syst. 28, 389–403 (2008) [17] Chernov, N., Markarian, R.: Chaotic billiards. Mathematical Surveys and Monographs, vol. 127, pp. xii+316. AMS Publishing Providence, RI (2006) [18] Chernov, N.I., Eyink, G.L., Lebowitz, J.L., Sinai, Ya.G.: Steady-state electrical conduction in the periodic Lorentz gas. Comm. Math. Phys. 154, 569–601 (1993) [19] Chernov, N.I., Eyink, G.L., Lebowitz, J.L., Sinai, Ya.G.: Derivation of Ohm’s law in a deterministic mechanical model. Phys. Rev. Lett. 70, 2209–2212 (1993) [20] Chernov, N., Zhang, H.-K.: Billiards with polynomial mixing rates. Nonlineartity 4, 1527–1553 (2005) [21] Dettmann, C.P., Morriss, G.P.: Crisis in the periodic Lorentz gas. Phys. Rev. E. 54, 4782–4790 (1996) [22] Kornfeld, I.P., Fomin, S.V., Sinai, Ya.G.: Ergodic Theory. Springer, New York (1982)
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[23] Gallavotti, G., Ornstein, D.: Billiards and Bernoulli schemes. Comm. Math. Phys. 38, 83–101 (1974) [24] Galton, F.: Natural Inheritance. MacMillan (1889) (facsimile available at www. galton.org) [25] Lorentz, H.A.: The motion of electrons in metallic bodies. Proc. Amst. Acad. 7, 438–453 (1905) [26] Melbourne, I., T¨ or¨ ok, A.: Statistical limit theorems for suspension flows. Israel J. Math. 144, 191–209 (2004) [27] Moran, B., Hoover, W.: Diffusion in a periodic Lorentz gas. J. Stat. Phys. 48, 709–726 (1987) [28] Ruelle, D.: A review of linear response theory for general differentiable dynamical systems. Nonlinearity 22, 855–870 (2009) [29] Sinai, Ya.G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russ. Math. Surv. 25, 137–189 (1970) [30] Sz´ asz, D., Varj´ u, T.: Limit Laws and Recurrence for the Planar Lorentz Process with Infinite Horizon. J. Stat. Phys. 129, 59–80 (2007) [31] Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147, 585–650 (1998) Nikolai Chernov Department of Mathematics University of Alabama at Birmingham Birmingham, AL 35294, USA e-mail:
[email protected] Dmitry Dolgopyat Department of Mathematics University of Maryland College Park, MD 20742, USA e-mail:
[email protected] Communicated by Domokos Szasz. Received: January 19, 2010. Accepted: April 6, 2010.
Ann. Henri Poincar´e 11 (2010), 1171–1200 c 2010 The Author(s). This article is published with open access at Springerlink.com 1424-0637/10/061171-30 published online October 21, 2010 DOI 10.1007/s00023-010-0053-4
Annales Henri Poincar´ e
Global Versus Local Casimir Effect Andrzej Herdegen and Mariusz Stopa Abstract. This paper continues the investigation of the Casimir effect with the use of the algebraic formulation of quantum field theory in the initial value setting. Basing on earlier papers by one of us (AH), we approximate the Dirichlet and Neumann boundary conditions by simple interaction models whose nonlocality in physical space is under strict control, but which at the same time are admissible from the point of view of algebraic restrictions imposed on models in the context of Casimir backreaction. The geometrical setting is that of the original parallel plates. By scaling our models and taking appropriate limit, we approach the sharp boundary conditions in the limit. The global force is analyzed in that limit. One finds in Neumann case that although the sharp boundary interaction is recovered in the norm resolvent sense for each model considered, the total force per area depends substantially on its choice and diverges in the sharp boundary conditions limit. On the other hand the local energy density outside the interaction region, which in the limit includes any compact set outside the strict position of the plates, has a universal limit corresponding to sharp conditions. This is what one should expect in general, and the lack of this discrepancy in Dirichlet case is rather accidental. Our discussion pins down its precise origin: the difference in the order in which scaling limit and integration over the whole space is carried out.
1. Introduction and the Main Idea The most natural setting for the consideration of the Casimir effect is the algebraic approach. This approach allows a mathematically rigorous analysis of the effect and gives a clear understanding of the sources of the difficulties one encounters in more traditional treatments. In application to quantum fields this analysis rests, in broad terms, on the following cornerstones. (i) A quantum relativistic theory is defined by an algebra of observables, in simple cases defined directly by ‘fields’ (scalar, electromagnetic). (ii) Each particular physical system obeying this theory is described by a Hilbert space representation of this algebra. Inequivalent representations
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refer to physically non-comparable systems or idealizations (such as a local isolated system and a thermodynamic limit system). (iii) The change of external conditions under which a quantum system is placed leads to a change of the state of the system. For the calculation of the global Casimir-type effects, as the backreaction force, one needs models which respect the above three constituents of a quantum theory. Thus the model of a quantum field should be based on one definite algebra, and the interaction with the external conditions should not lead to a change of its representation. If this condition is fulfilled then the Casimir force results from the change in the expectation value of one and the same energy observable, as defined by free field, as the state changes with changing external conditions (such as position of macroscopic bodies). This analysis has been conducted at length by one of us in [1,2], where also clear cut criterions for the admissibility of external interaction models for a class of systems were formulated. In application to the free quantum scalar field these amount to the following. Let φ(x) be a scalar field and denote by h2 the standard self-adjoint extension in L2 (R3 , d3 x) of −Δ. The free field dynamics is then (∂t2 + h2 )φ(t, x) = 0.
(1)
Suppose now that one introduces to the system external macroscopic bodies, such as conducting plates in the original Casimir system, which change the dynamics of the field. Let a denote free parameters of these bodies, such as separation of the plates, and let the modified dynamics (for fixed a) be given by (∂t2 + h2a )φ(t, x) = 0,
(2)
h2a
where is a positive self-adjoint operator. Now, the two settings can be described by one choice of observables algebra, and in common representation −1/2 of this algebra if, and only if, ha (ha −h)h−1/2 is a Hilbert–Schmidt operator in L2 (R3 , d3 x), that is −1/2 < ∞. (3) Tr h−1/2 (ha − h)h−1 (h − h)h a a Suppose this condition holds and let the algebra of the field be represented in some Hilbert space H. Let further H be the energy operator as defined by free field dynamics, and let Ωa be the minimal energy state vector as defined by the modified dynamics (2). The Casimir energy is then given by 1 Tr (ha − h)h−1 (4) a (ha − h) . 4 That this energy be finite is another condition on the model of ha , and only if both conditions are satisfied the Casimir problem has a finite solution and the Casimir force is then dEa . Fa = − da Ea = (Ωa , HΩa ) =
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These admissibility conditions say, roughly, that the modified dynamics h2a cannot differ much from the free dynamics h2 . Introducing sharp boundary conditions, such as Dirichlet/Neumann conditions on plates, violates these demands. One faces therefore the problem of an appropriate approximation for the description of such plates. We consider the simplest geometrical situation, the original Casimir problem of two infinite, parallel plane plates at a distance a from each other. We assume that z-axis is perpendicular to the planes and the modification of dynamics affects this direction only, thus h2 = h2z + h2⊥ ,
h2a = h2za + h2⊥ .
where h2⊥ is the free dynamics in the directions perpendicular to z-axis and h2za is a modification of h2z = −∂z2 in L2 (R, dz); we refer the reader for details to [2]. In this setting the conditions of finiteness of (3) and (4) cannot be expected to hold as they stand because of translation symmetry in the planes, and must be replaced by conditions ‘per unit area’ of the planes. It has been shown in [2] that this amounts to (5) Tr (hza − hz )2 < ∞, Tr [(hza − hz )hz (hza − hz )] < ∞, (6) where the trace refers to the Hilbert space L2 (R, dz). If these conditions are satisfied, the energy per unit area is finite and reads 1 Tr [(hza − hz )(2hz + hza )(hza − hz )] . εa = (7) 24π A class of models for h2za imitating the boundary conditions, but consistent with the above demands, was considered in [2]. The idea was to take B 2 2 h2za = (f unction of )(hz , hB za ), where (hza ) is −∂z with boundary conditions at z = ±a/2. The choice of functions assured that for small spectral val2 ues of h2z and (hB za ) the models reproduced the sharp boundaries, while for large spectral values tended to free dynamics. Moreover, one could introduce a scaling parameter μ such that for μ → ∞ the models approached the sharp boundaries in the whole spectrum. For the Casimir energy per area in the rescaled models ελa (we prefer to work with λ = 1/μ here) one then found ε∞ c π2 + − + (terms → 0 for λ → 0), (8) λ3 λa2 1440a3 where ε∞ and c are constants, c = 0 in Dirichlet case, but c = 0 in Neumann case. It was also shown that the direct sum of the two models describes the setting of the electromagnetic field between conducting plates. Thus the Casimir force per unit area is then ελa =
2c π2 d[ελa (D) + ελa (N )] = − + (terms → 0 for λ → 0). da λa3 240a4 The second term reproduces the well-known Casimir’s formula, but the first term is model-dependent and dominates for large a. Moreover, in typical situations there is c > 0 and the force becomes repulsive for large a. −
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In the present paper we want to find out whether these results will be confirmed in another class of models, constructed in a wholly different way. Rather than manipulate spectral properties directly, now we want to approximate interaction with the plates directly in the physical space. It is easy to see that strictly local potential interaction of the form [h2za ψ](z) = −∂z2 ψ(z)+V (z)ψ(z) violates our conditions. Therefore we replace V by a slightly nonlocal integral quasi-potential [V ψ](z) = V (z, z )ψ(z )dz with the kernel V (z, z ) concentrated around the position of the plates. We show that with an appropriately defined scaling of some simple kernels of this kind one can reproduce sharp boundary conditions on the plates in the limit. The Casimir energy can again be calculated and for a class of models the result (8) is confirmed. However, in general the Neumann case proves to be even more singular here than in the models considered in [2] and the universal term could be disturbed. Nonlocal quasi-potentials has been considered in the Casimir context by other authors before, but in different formalisms and with rather different motivations (see e.g. [3,4]). The present choice of models makes also possible a local analysis of the local energy density. We show that outside the interaction region the density tends in the scaling limit to a well defined universal form corresponding to sharp boundary conditions. The present mathematically rigorous setting allows the comparison and better understanding of the local–global relation. The model-dependent and divergent (in the limit) contributions to the global force are due to the interaction region. We discuss this point more fully in the Discussion section. For a more extensive discussion of the background of the present paper, as well as for more extensive literature we refer the reader to [1,2]. We define our models in Sect. 2. Appropriate scaling of these models is shown to reproduce the sharp boundary conditions in Sect. 3. Spectral properties of the models are discussed in Sect. 4 and the admissibility of the models in the sense mentioned above is proved in Sect. 5. It is shown in Sect. 6 that the Casimir energy of the scaled models is obtained by the expansion of the formula for energy in inverse powers of a, and this expansion (up to a significant order) is obtained in Sect. 7. For comparison, in Sect. 8 we obtain local results and their scaling limit. The discussion occupies Sect. 9. More technical points of our derivations are shifted to Appendices.
2. The Models We postulate for our analysis the following quasi-potentials V = σ (|Ub g Ub g| + |U−b g U−b g|) ,
b = a/2 > 0,
σ = ±1,
(9)
where U is the translation operator and σ = 1, −1 corresponds to Dirichlet (D) and Neumann (N) conditions respectively. These conditions will be achieved in the two cases by an appropriate scaling limit to be defined below. In all what follows one should keep in mind that unless stated otherwise we treat parallelly both cases, but the dependence of quantities on σ is suppressed.
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In position representation the quasi-potential is an integral operator (V ψ)(z) = V (z, z )ψ(z ) dz with the kernel V (z, z ) = σ g(z − b)g(z − b) + g(z + b)g(z + b) . For the functions g we assume that f (z) g(z) = −i d f (z) dz
if σ = 1,
(D)
if σ = −1,
(N)
(10)
where f is a complex, compactly supported smooth function, with the following properties f (−z) = f (z), suppf ⊆ −R, R, R < b, f = 1, (N)
f(0) = 0,
(11) (12)
where f is the Fourier-transformed function 1 f(p) = √ f (z)e−ipz dz, 2π and the last property is assumed only in the Neumann case. The first condition reflects the symmetry of each of the plates, the second says that the nonlocalities of the two interaction centers at z = ±b do not overlap, and the third and fourth are technical. We denote by hz and hza the self-adjoint, non-negative square roots of the operators h2z = −(d/dz)2 ,
h2za = h2z + V,
(13)
respectively. Operator h2z is the standard one-dimensional Laplace operator (with opposite sign), while h2za is its Kato–Rellich perturbation, with unchanged domain, as V is bounded. The (strict) positivity of h2za in the Dirichlet case is obvious, while in the Neumann case one has by (10) that for all ψ in the domain of h2z there is (ψ, h2za ψ) = ψ 2 − |(U+b f, ψ )|2 − |(U−b f, ψ )|2 ,
(N)
where ψ (z) = dψ(z)/dz. The functions U±b f do not overlap, and satisfy U±b f = f . Thus by (12) they form an orthonormal system, which implies (ψ, h2za ψ) ≥ 0 for each ψ. (Here a more strongly bounding condition f < 1 would produce a strictly positive operator; this, however, would not lead to the recovery of Neumann condition in a limit to be defined below.) For w2 ∈ C with Im w2 = 0 the resolvents denoted by G0 (w2 ) = (w2 − h2z )−1 ,
G(w2 ) = (w2 − h2za )−1 ,
(14)
are bounded operators. In all what follows for given w2 we fix w by Im w > 0. We also introduce the T -operator known from the stationary scattering theory T (w2 ) = V + V G0 (w2 )T (w2 ).
(15)
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This equation may be explicitly solved for T : making the Ansatz
U+b g|
2 2 , T (w ) = |U+b g |U−b g T (w ) U−b g| with T (w2 ) a numerical matrix, one easily finds −1 σ − (g, G0 (w2 )g) −(Ua g, G0 (w2 )g) 2 T (w ) = . −(Ua g, G0 (w2 )g) σ − (g, G0 (w2 )g)
(16)
(17)
The resolvent may be then expressed by G(w2 ) = G0 (w2 ) + G0 (w2 )T (w2 )G0 (w2 ). In momentum representation, taking into account (10) and denoting
−ibp +ibp 1−σ e e Fp = p 2 f(p) , Mp = |f(p)|2 ,
(18)
(19)
we have p| T (w2 ) |q = Fp T (w2 )Fq† , p| G(w2 ) − G0 (w2 ) |q =
Fq† Fp T (w2 ) 2 , 2 −p w − q2
w2
with elements in the matrix (17) given by 1−σ
Mp p 2 dp, σ − g, G0 (w )g = σ − w 2 − p2 iap 1−σ
Mp e p − Ua g, G0 (w2 )g = − dp = iπw−σ eiaw Mw . w 2 − p2
(20) (21)
(22) (23)
The integral in the last formula is calculated in the complex plane by residues, with the use of analyticity and asymptotic properties of Mw discussed at the beginning of Appendix A.
3. Reproduction of the Sharp Boundary Conditions We consider now a family of rescaled quasi-potentials Vλ , λ ∈ (0, 1, built as in (9), but with the use of rescaled functions gλ instead of g. We write the scaling in several equivalent forms: z z 3 σ σ , fλ (z) = λ−1− 2 f , fλ (p) = λ− 2 f(λp), gλ (z) = λ− 2 g λ λ and note also that Mp,λ = λ−σ Mp . The rescaled potentials give rise to the corresponding operators hza,λ . All quantities referring to these operators acquire the subscript λ. Let hB za be the self-adjoint, positive square root of the operator d2 B 2 [hza ] = − 2 (24) dz boundary conditions
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with standard domains in L2 (R), Dirichlet/Neumann (for σ = +1/−1) condi2 tions in z = ±b, and denote by GB (w2 ) the resolvent of [hB za ] . Our objective in this section is to show the limiting property lim Gλ (w2 ) − GB (w2 )HS = 0,
λ→0+
(25)
where the Hilbert–Schmidt norm is A2HS = Tr(A∗ A) ≥ A2 . From this relation it follows then the norm convergence lim F (hza,λ ) − F (hB za ) = 0
λ→0+
for each continuous and vanishing in infinity complex function F on R, and the strong convergence lim [F (hza,λ ) − F (hB za )]ψ = 0,
λ→0+
for each bounded continuous function F and vector ψ ∈ L2 . It is clear from the form of Eq. (21) that G(w2 ) − G0 (w2 ) and its scaled version Gλ (w2 ) − G0 (w2 ) are finite rank, hence Hilbert–Schmidt, operators. Thus it is sufficient to calculate the strong–L2 (R2 , dp dq) limit for λ → 0+ of the integral kernel p| Gλ (w2 ) − G0 (w2 ) |q. First we consider the numerical matrix Tλ , and for later use we also look at higher orders in λ. We observe that 1 Mλp − M0 dp = I0 + O(λ), (26) λ w 2 − p2 with M0 − Mp I0 = dp. (27) p2 This is shown by writing the difference of (26) and (27) as λw Mu − M0 du λw u2 (λw)2 − u2 and Fourier-transforming the integral as a scalar product of two L2 -functions. This integral is shown in this way to be bounded by a constant. Using (22), (23), (26) and the assumption (12) (in Neumann case), we find
2−σ σ − gλ , G0 (w2 )gλ = iπ(λw)−σ M0 + (λw)1−σ 1+σ ), 2 − I0 + O(λ (28)
2 −σ iaw 2−σ − Ua gλ , G0 (w )gλ = iπ(λw) e M0 + O(λ ). (29) From these we get Tλ (w2 ) =
−i(λw)σ 1 −eiaw πM0 (1 − e2iaw ) hboxeiaw 1
2+σ + (λw)1+σ 1+σ ). 2 − I0 × {matrix independent of λ} + O(λ
Next, we observe that (this is shown in Appendix A) σ 1−σ 3 λ 2 f (p)p 1−σ 2 const(w)λ 2 , (D) f(0)p 2 λ − 2 ≤ 1 w 2 − p2 w − p2 const(w) λ 2 , (N)
(30)
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and in addition, for Neumann case 1 λ− 2 f (p) 3 f(0) λ − 2 2 ≤ const(w)λ 2 w − p2 w − p2
Ann. Henri Poincar´e
(N)
(31)
(norms of functions of p as elements of L2 ). Now we easily obtain in the L2 (R2 , dp dq)–sense 1−σ
(pq) 2 iwσ s − lim p| Gλ (w2 ) − G0 (w2 ) |q = − 2iaw 2 + π(1 − e ) (w − p2 )(w2 − q 2 ) λ→0 −ibq ibp iaw ibq ibp ibq −ibp e −e e e +e e − eiaw e−ibq e−ibp . × e We transform this to position representation and get s − lim z| Gλ (w2 ) − G0 (w2 ) |z λ→0+ σ i(b+z)w −i(b+z)w θ(b + z)e + σθ(−b − z)e =− 2iw (1 − e2iaw ) × σθ(−b − z )e−i(b+z )w + θ(b + z )ei(b+z )w −σ θ(b − z )eiaw ei(b−z )w − θ(−b + z )eiaw e−i(b−z )w + θ(−b + z)e−i(b−z)w + σθ(b − z)ei(b−z)w × σθ(b − z )ei(b−z )w + θ(−b + z )e−i(b−z )w −σ θ(−b − z )eiaw e−i(b+z )w − θ(b + z )eiaw ei(b+z )w . We shall use also the explicit form of the unperturbed Green function in this representation i (32) θ(z − z )ei(z−z )w + θ(z − z)ei(z −z)w . z| G0 (w2 ) |z = − 2w In this way we find s − lim z| Gλ (w2 ) |z = z| GB (w2 ) |z , λ→0+
where z| GB (w2 ) |z i −i(z+z +a)w 2 = z| G0 (w ) |z + σ e χ(−∞,−b) (z)χ(−∞,−b) (z ) 2w i i(z+z −a)w 2 e χ(b,+∞) (z)χ(b,+∞) (z ) + z| G0 (w ) |z + σ 2w i cos(zw) cos(z w) sin(zw) sin(z w) 2 + + z| G0 (w ) |z + w 1 + σe−iaw 1 − σe−iaw ×χ(−b,b) (z)χ(−b,b) (z ), (33) and χΩ is the characteristic function of the set Ω.
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In the three regions z| GB (w2 ) |z differs from z| G0 (w2 ) |z only by solutions of homogeneous equation and satisfies the boundary conditions ±b| GB (w2 ) |z = z| GB (w2 ) |±b = 0, (D) d d z| GB (w2 ) |z |z=±b = z| GB (w2 ) |z |z =±b = 0, (N) dz dz so it is indeed the Green function of the Dirichlet/Neumann operator and therefore (25) is finally proven. We now want to acquire some information on the rate at which the limit (25) is achieved. Using (30) one finds for any ϕ, η ∈ L2
(1 − I0 )O(λ) + O(λ3/2 ), (D) 2 B 2 ϕ, Gλ (w )η = ϕ, G (w )η + (34) (N) O(λ1/2 ). The Neumann case turns out to be here, as in many other problems, more singular. However, we also note that if we assume that ϕ and η are in the domain of hz then the estimate (31) implies
ϕ, Gλ (w2 )η = ϕ, GB (w2 )η + I0 O(λ) + O(λ3/2 ). (N) (35) In the following two sections we treat unscaled models. The scaling is again considered in Sect. 6.
4. Spectral Analysis We add now some further assumptions on the choice of functions f . We denote for k ∈ R Mk − Mp Ik = dp, (36) p2 − k 2 and demand that 0 < Ik
for k = 0
(D,N),
Ik < 1 (D).
(37) (38)
Note that by continuity I0 ≥ 0 (this is the quantity introduced in (27)). We also denote 1−σ M (q) − k 1−σ M (k) q dq = 12 (1 + σ)|k|σ − |k|Ik . πNk = |k|σ σ + q2 − k2 (39) The operators h2za are non-negative, and outside a compact set in R they act as −∂z2 . Therefore their continuous spectrum covers the whole positive axis and thus the spectrum is 0, +∞). This does not resolve the question of point spectrum, and we treat it first. The eigenvector equation h2za ψk = k 2 ψk ,
ψk ∈ L2 (R),
k≥0
(40)
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is solved in momentum space. It is easily seen that the distributional solution which is square-integrable at infinity must have the form cb e−ibp + c−b eibp 1−σ p 2 f (p), ψk (p) = p2 − k 2
(41)
with constants c±b to be determined. Putting this form back into Eq. (40), one finds for k > 0 that the constants c±b have to satisfy the linear system Nk c+b − Mk sin(ak)c−b = 0,
Mk sin(ak)c+b − Nk c−b = 0,
(42)
where the integration leading to coefficients Mk sin(ak) is carried out with the use of analyticity and asymptotic behavior of Mk discussed in Appendix A. Now, for k > 0 the conditions (37) and (38) imply Nk = 0, thus non-trivial solutions to the system (42) exist only if Mk sin(ak) = ±Nk , and in that cases (41) takes, respectively, the form e ψk (p) = c−b
ibp
∓ e−ibp 1−σ p 2 f (p). p2 − k 2
The condition ψk ∈ L2 (R) requires that f(±k) = 0 or eibk ± e−ibk = 0. Each of these cases implies Mk sin(ak) = 0, which shows that there are no eigenvectors for k > 0. For k = 0 in Dirichlet case the solution (41) cannot be in L2 as f(0) = 0. For Neumann case one finds that (41) is a distributional solution for any constants c±b , so we are free to choose them so as to satisfy the square-integrability of (41). This happens only for c−b = −cb and then sin(bp) ψ0 (p) = N f (p), p
(N)
(43)
where N is a proportionality factor. The normalization condition ||ψ0 ||L2 = 1 gives 2 . (44) |N |2 = aπM0 − I0 Summarizing, there are no bound states for Dirichlet case, however, for Neumann case there is one bound state, which corresponds to the zero eigenvalue, described by (43) and (44). We now consider the continuous spectrum and for this purpose use the stationary scattering formalism. The improper eigenfunctions of scattering states in momentum representation are given in standard notation by p| T (k 2 + i0) |k , ψk (p) = p | k+ = δ(p − k) + k 2 − p2 + i0
(45)
where T (w2 ) is the operator discussed in Sect. 2. The variable k takes all values k = 0 and each spectrum point k 2 has two-fold degeneracy corresponding to ±k. Taking into account the results of Sect. 2, we can write −1 |k|σ Mk − iNk Mk eia|k| T (k 2 + i0) = . (46) Mk eia|k| Mk − iNk iπ
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For later use we write this in two alternative forms. We introduce matrix notation 2|k| M(p) † M(k) = Fk Fk , N (k) = dp , (47) σ1 + P π p2 − k 2 1 cos(ak) . (48) L(k) = M(k) + M(−k) = 2k 1−σ Mk cos(ak) 1 and then 2|k| −1 (N (k) + iL(k)) , T (k 2 + i0) = π p| T (k 2 + i0) |k2 = Tr M(p)T (k 2 + i0)M(k)T (k 2 + i0)† .
(49) (50)
Finally, we calculate the inverse in (46) and write the result in the form |k|σ sk 1 1 −qk eia|k| 2 T (k + i0) = , (51) 1 iπ 1 − (qk eia|k| )2 −qk eia|k| where 1 Mk , qk = = Mk sk . Mk − iNk Mk − iNk Some properties of sk function are shown in Appendix A. sk =
5. Hilbert–Schmidt Properties In this section we show that our model satisfies the admissibility conditions (5) and (6). If we write TRτ for the l.h.s. of these two conditions, with τ = 0 for (5) and τ = 1 for (6), then 2 TRτ = |p|τ | p| hza − hz |k+ |2 dk dp + 1−σ |p|2+τ ψ0 (p) dp, (52) 2 R
R2
where the first term results from the continuous spectrum space of hza and the second is the bound state contribution in Neumann case. The second term is evidently finite (by (43)), and we restrict attention to the first one, which we denote TRcont . In momentum representation we have τ p| T (k 2 + i0) |k , p| hza − hz |k+ = (|k| − |p|)ψk (p) = |p| + |k| thus by change of variables for negative arguments we get pτ ±p| T (k 2 + i0) |±k 2 dk dp, TRcont = τ 2 (p + k) ±±
(53)
R2+
where the signs in ‘bra’ and ‘ket’ are uncorrelated and the sum is over all four possibilities. From now on we assume that k, p ≥ 0. Using Eqs. (48), (50) we can write ±p| T (k 2 + i0) |±k 2 = Tr L(p)T (k 2 + i0)L(k)T (k 2 + i0)† , ±±
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and substituting here (from (49)) L(k) =
ik
T (k 2 + i0)†−1 − T (k 2 + i0)−1 π
we give Eq. (53) the form kpτ 2 TRcont = Re iTr L(p)T (k 2 + i0) dk dp. τ 2 π (p + k)
(54)
R2+
With the use of (48) and (51) we have 1 − cos(ap)qk eiak 4 (55) k Re iTr L(p)T (k 2 + i0) = p1−σ k 1+σ Mp Re sk π 1 − (qk eiak )2 Writing out the real part gives us the appropriate behavior of the nominator for k = 0 and the whole expression becomes proportional to Mk . Using are finite and the the estimates (92), (91) and (87), one finds that TRcont τ admissibility conditions (5) and (6) are satisfied.
6. Scaling We return to the scaling transformation to view it from a different point. If we make the dependence of the potential V on a explicit by writing it as Va and the rescaled potential as Va,λ then we have Va,λ (z, z ) = λ−3 Va/λ (z/λ, z /λ). It is then an easy exercise to show that this implies a simple scaling law of the eigenfunctions (43) and (45): λ ψk,b (p) = λψλk,b/λ (λp),
λ ψ0,b (p) = λ1/2 ψ0,b/λ (λp)
(different powers of λ reflect different normalizations: to the Dirac delta and to unity, respectively). Denoting the scaled versions of (52), with explicit dependence on a, by TRλτ,a we find TRλτ,a = λ−2−τ TRτ,a/λ . Thus the admissibility conditions are satisfied also for the rescaled potential (but not in the limit). In the same way one obtains the scaling law for the Casimir energy: ελa = λ−3 εa/λ .
(56)
Therefore to identify the scaling behavior of the energy in the limit it is sufficient to expand the unscaled energy in inverse powers of a up to the third order. This will be done in the next section.
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7. The Energy In this section we prove the following expansion of Casimir energy for large a: π2 + o(a−3 ), (D) 1440a3 ζ(3) 1 1 I0 π2 εa = ε ∞ + + + + o(a−3 ). − 2 48πM0 a2 8π 2 M0 a3 π2 3 1440a3 εa = ε ∞ −
(57) (N) (58)
We postpone the discussion of this result to the concluding section and here only note that there are functions in our class for which I0 = 0, and then the a−3 -term has the known universal form. The two conditions (5) and (6), considered in Sect. 5, imply already finiteness of the Casimir energy per unit area (7), as mentioned in the Introduction. This can be easily seen: we observe that conditions (5) and (6) mean that 1/2 Δ = hza − hz and hz Δ are Hilbert–Schmidt operators. Also, from Δhza Δ = Δhz Δ + Δ3 1/2
we infer that hza Δ is HS as well, which is sufficient for the claim. The expression (7) closely parallels that of the condition (6) and can be written in analogy to (52). We split the trace in (7) into two terms and calculate these traces in hza or hz (improper) basis in the first and second line below, respectively. Then we insert the spectral decomposition of the operators hz or hza , respectively. In this way we get 2 2 Tr [2Δhz Δ] = 2 |p| p| hza − hz |k+ dk dp + (1 − σ) |p|3 ψ0 (p) dp, R2
Tr [Δhza Δ] =
2 |k| p| hza − hz |k+ dk dp.
R
R2
Therefore the expression (7) may be written in the form 2 2 1 1−σ εa = (2|p| + |k|) p| hza − hz |k+ dk dp + |p|3 ψ0 (p) dp. 24π 24π R
R2
(59) —the second term—will be calculated in The bound state contribution εbound a the Neumann case subsection. Following the same steps as in Sect. 5 and using (55) we give the continuous spectrum contribution the form 1 − cos(ap)qk eiak 1 cont εa = χ(k, p)Mp Re sk dk dp, (60) 3π 3 1 − (qk eiak )2 R2+
where χ(k, p) =
k 1+σ p1−σ (2p + k) . (p + k)2
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We write this as the limit for → 0+ of the integral restricted to k ∈ , +∞) and expand the denominator into geometric power series (note that |qk | < 1 for k > 0) 1 2
1 − (qk eiak )
=
∞
(qk eiak )2n ,
n=0
getting εcont a
1 = lim 3π 3 →0+
∞
∞ Mp
χ(k, p) Re sk
0
−sk cos(ap)
qkn einak
n∈2N0
qkn einak dk dp.
n∈2N−1
We write the n = 0 term separately as 1 ε∞ = χ(k, p)Mp Mk |sk |2 dk dp. 3π 3
(61)
R2+
For other terms we observe that N n=1
|qk e
iak 2n
|
1 + k m−2σ |sk |2 Mk2 ≤ ≤ const Mk2 2 1 − |qk | km
m = 2 (D) m ≥ 2 (N)
(62)
[see (93) and (91)], so we can pull the infinite sum sign outside the integral by the Lebesgue dominated convergence theorem to obtain ⎡∞ ⎤ ∞ 1 = ε∞ + 3 lim Mp ⎣ χ(k, p)sk (qk eiak )n dk + c.c.⎦ dp εcont a 6π →0+ n∈2N 0 ⎡∞ ⎤ ∞ 1 iak n Mp cos(ap) ⎣ χ(k, p)sk (qk e ) dk + c.c.⎦ dp. − 3 lim 6π →0+ n∈2N−1 0
(63) We now split the analysis into separate cases. 7.1. Dirichlet Case (σ = 1) In this case when (62) is multiplied by χ(k, p) the k −2 singularity in k = 0 is cancelled by k 2 from the function χ. Therefore here the intermediate step with nonzero is not needed and (63) should be read with = 0. Moreover, there is no bound state here, so this formula represents the total Casimir energy εa . If we write (na)3 einak = (−i∂k )3 einak and integrate three times by parts we find ∞ ∞ 4Mp 3 iak n − i ∂k3 (χ(k, p)sk qkn ) einak dk, χ(k, p)sk (qk e ) dk = −i (na) M0 p 0
0
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and then obtain (the first term on the r.h.s. above is imaginary and falls out) ⎧ ⎪ i ⎨ 1 − εa − ε ∞ = Mp ∂k3 (χ(k, p)sk qkn ) einak dk dp 6π 3 a3 ⎪ n3 ⎩ n∈2N
+
n∈2N−1
1 n3
R2+
Mp cos(ap)∂k3 (χ(k, p)sk qkn ) einak dk dp
R2+
⎫ ⎪ ⎬ ⎪ ⎭
+ c.c.. (64)
R2+
Let now Ω be the intersection of with an arbitrary neighborhood of zero. We now use the results of Appendix B to infer that the following successive three operations on this formula lead only to the neglect of terms of order o(a−3 ): (i) replacement of ∂k3 (χ(k, p)sk qkn ) by sk qkn ∂k3 χ(k, p); (ii) restriction of the integration region to Ω; (iii) replacement (in the restricted region) of Mp sk qkn by M0 s0 q0n = 1. In this way we arrive at
2 p (k − 3p) 2 1 sin(nak)dk dp π 3 a3 n3 (k + p)5 n∈2N Ω 1 p2 (k − 3p) 1 2 cos(ap) sin(nak)dk dp + o − 3 3 . 3 5 π a n (k + p) a3
εa = ε∞ +
n∈2N−1
(65)
Ω
The integrals are bounded uniformly with respect to a. It is now sufficient to show that they have well defined limits for a → ∞; then those limits determine the a−3 -term. Consider the integrals (with k, p > 0) p2 4p3 C(n, , a) = − sin(nak + ap)dp dk, = 0, ±1. (k + p)4 (k + p)5 k+p≤1
It is easy to show that with the choice Ω = {k, p > 0, k + p ≤ 1} the integral in the first line of (65) is this integral with = 0, while the integral in the second line of (65) is one half of the sum of integrals with = ±1. By the change of variables r = a(k + p), t = p/(k + p) we bring this to the form 1
t − 4t
C(n, , a) =
2
3
0
and find lim C(n, , a) =
a→∞
a
1 sin [r (n + t( − n))] dr dt, r
0
π − 3 π 3
1+
n3 (n+1)3
−
3n4 (n+1)4
for = 0, 1, for = −1.
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Using these results we get finally 2 1 1 1 3n + − εa = ε∞ − 2 3 + o 3 3 4 3π a n (n + 1) (n + 1) a3 n∈2N n∈2N−1 1 1 ζ(4) π2 = ε∞ − + o − + o = ε , ∞ 2 3 3 3 16π a a 1440a a3 where ε∞ is defined in (61). 7.2. Neumann Case (σ = −1) We now use formula (63) with the replacement in notation qk eiak = q˜k ei˜ak , where α=
I0 , πM0
a ˜ = a − α,
q˜k = qk eiαk ,
which has the technical advantage that q˜0 = 0. After this modification the general scheme is very similar to the Dirichlet case. Integration by parts gives expansion in 1/˜ a but at the end we shall translate it to the 1/a expansion. Integrating by parts we obtain boundary terms, for which in the present case the addition of c.c. terms must be taken into account before the limit → 0+ is performed. For example, the first integration by parts in k in the first line in (63) gives a term proportional to (before p-integration) 1 χ(, p) Im sk q˜kn ()ein˜a n →0 n∈2N
πp = − lim+ χ(, p) Im s ln 1 − q˜2 e2i˜a = , M0 →0 lim+
where in the last equality we used the fact that in the neighborhood of zero ln |1 − q˜2 e2i˜a | ≤ ln const(a) [see (92)]. After integrating by parts three times in similar way we get the expression ⎡ ⎤ ∞ πp π2 3I0 ζ(3) lim+ ⎣ χ(k, p)sk q˜kn ein˜ak dk + c.c.⎦ = − + 2 + 3 a ˜M0 4˜ a M0 2˜ a πM02 →0 n∈2N ⎡ ⎤ ∞ 1 − lim+ ⎣ ∂k3 (χ(k, p)sk q˜kn ) ein˜ak dk + c.c.⎦ . (in˜ a)3 →0 n∈2N
For the sum over odd natural numbers the computations are similar. The integrals over p we treat in a similar way as in the Dirichlet case (steps (i) to (iii), for their permissibility here see Appendix B). The integrals of ∂k3 χ over the neighborhood of zero go similarly as in the Dirichlet case. In this way we find 1 1 1 CM 1 I0 ζ(3) π2 + εcont = ε − − − + , + o ∞ a 6π 2 M0 a ˜ 48πM0 a ˜2 a ˜3 6π 2 8π 4 M02 1440 a ˜3
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where ∞ CM =
pMp dp. 0
In the calculation we used the relations ∞ ∞ 1 1 M0 Mp dp = , pMp cos(ap)dp = − 2 + O . 2 a ˜ a ˜3 0
(66)
0
The first equality follows from normalization of f . We now return to the bound state contribution to the energy. From the second term of (59), (43) and (44), using (66) we get 1 1 CM bound + 2 3+ o = . εa 2 6π M0 a ˜ 6π a ˜ a ˜3 Adding the two contributions and changing the expansion parameter to a we obtain (58).
8. Local Properties In this section we consider the local algebras of fields supported outside the regions of support of the potential V . For the initial (unscaled) models this means that the z-support of fields is outside the set −b − R, −b + R ∪ b − R, b + R, but for λ → 0 eventually every support outside the planes z = ±b is admitted. Fields thus supported are also in the algebra of fields of the models with sharp boundary conditions at z = ±b, so one can also consider sharp boundary conditions for them. We recall from [1] that we use the initial value fields (smeared on a Cauchy surface of constant time) Φ(V ), where V is a pair of real test functions (v, u), and X(u) = Φ(0, u), P (v) = Φ(v, 0) have the interpretation of canonical variables. For the present choice of the algebras the test functions are assumed to be in the space of smooth functions of compact support outside z = ±b, which we denote by Db . The algebra of fields is formulated, more precisely, in Weyl form, which means that rather than Φ(V ) elements W (V ) = exp[iΦ(V )] are used, and the fields Φ are defined on the level of specific representations. The states on the algebra are given as normalized positive linear functionals on the algebra of Weyl elements. With the Hamiltonians of the perpendicular motion hza and hB za as defined in (13) and (24) we denote h2a = h2za + h2⊥ ,
2 B 2 2 [hB a ] = [hza ] + h⊥ ,
(67)
where −h2⊥ is the two-dimensional Laplacian in the directions parallel to the plates. Then the ground states of the fields corresponding to the models proposed in the present article and to the sharp boundary conditions are given, respectively, by (68) ωa (W (V )) = exp − 14 ja (V )2 , ωaB (W (V )) = exp − 14 jaB (V )2 ,
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where ja (V ) = ha1/2 v − ih−1/2 u, a
1/2 −1/2 jaB (V ) = hB v − i hB u. a a
(69)
(To be precise, to obtain formula (7) for the Casimir energy we start in [2] with the directions parallel to the plates restricted to a box, whose size then tends to infinity. This may be shown to reproduce the states given above, but we omit this step here.) The ground states ωa,λ of the scaled models are defined analogously with the use of ha,λ . We show in this section that the scaled states reproduce in the weak limit the sharp boundary state: lim ωa,λ (W (V )) = ωaB (W (V )) .
(70)
λ→0
Also, we consider the limit of the local energy density. 8.1. Local Limit of States u), thus to prove (70) it is sufficient to There is ja (V )2 = (v, ha v) + (u, h−1 Ba±1 show that (ϕ, h±1 ψ) tend to (ϕ, h ψ) for λ → 0 and ϕ, ψ ∈ Db . For such a a,λ ψ one has 2 ψ = h2 ψ ≡ −Δψ h2a,λ ψ = hB a for sufficiently small λ. Therefore (ϕ, ha,λ ψ) = −(ϕ, h−1 a,λ Δψ) and similarly for −1 B ha , and the problem is reduced to the ha,λ -case. Moreover, ϕ and ψ are in the −1/2
domain of h⊥
, so we can write −1/2 1/2 −1 1/2 −1/2 ψ) = (h ϕ), h h h (h ψ) (ϕ, h−1 ⊥ ⊥ ⊥ a,λ a,λ ⊥
and similarly for hB a . In this way the problem is reduced to the following: w − lim h⊥ h−1 a,λ h⊥ = h⊥ 1/2
1/2
1/2
λ→0
hB a
−1
1/2
h⊥ .
(71)
To show this, we first observe that h⊥ h−1 a,λ h⊥ ≤ 1, so it is sufficient to perform the limit between vectors from the total set of the form χ(x) = χ⊥ (x⊥ )χz (x3 ). Using the spectral representation of h2⊥ one has ⎞ ⎛ | | p ⊥ 1/2 −1 1/2 ρz ⎠ χp⊥ )ρ⊥ ( p⊥ ) d2 p⊥ . (72) (χ, h⊥ ha,λ h⊥ ρ) = ⎝χz , * ⊥ ( h2za,λ + p2⊥ 1/2
1/2
The scalar product under the integral is bounded by χz ρz and for each p⊥ , by the result of Sect. 3, tends to analogous expression with hza,λ replaced by hB za . This is sufficient to perform the limit, which ends the proof.
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8.2. Local Energy Density The state ωa (unlike the state ωaB ) is defined on the whole algebra. In the language used in [1] the energy density in this state (properly normally ordered with respect to the vacuum) is given in the whole space by point-splitting procedure by Ea (x) = Ta (x, x), where Ta (x, y ) is the distribution defined by −1 (h−1 Ta (ϕ, ψ) = 14 (ϕ, (ha − h)ψ) + 14 ∇ϕ, − h ) ∇ψ , a
(73)
(74)
with scalar product between the two gradients understood in the second term. The test functions are taken to be real. If one takes into account the translational symmetry in the directions parallel to the plates one realizes that the x⊥ , y⊥ -dependence of Ta may be only through the difference x⊥ − y⊥ . The removal of point-splitting in these directions means putting this variable equal to zero, or integrating the 2-dimensional Fourier transform of Ta over all space of p⊥ -variables of the spectral representation of h⊥ . In this way one finds (now x, y are variables in the direction orthogonal to plates and ϕ, ψ are one-dimensional) Ea (x) = Tza (x, x), where Tza (ϕ, ψ) =
(75)
1 ϕ, (h2za + p2⊥ )1/2 − (h2z + p2⊥ )1/2 ψ 2 16π + p2⊥ ϕ, (h2za + p2⊥ )−1/2 − (h2z + p2⊥ )−1/2 ψ + ϕ , (h2za + p2⊥ )−1/2 − (h2z + p2⊥ )−1/2 ψ d2 p⊥ ,
with prime in ϕ , ψ denoting the derivative. The integral is easily carried out explicitly and one finds 1 1 (ϕ, (h3za − h3z )ψ) − (ϕ , (hza − hz )ψ ). (76) 24π 8π Our objective is to find the limit of the energy density in the scaled models. Thus following the introductory remarks of the present section we assume now the supports of ϕ, ψ to be outside the set −b − R, −b + R ∪ b − R, b + R. For such functions there is h2za ψ = h2z ψ, so Tza (ϕ, ψ) =
1 1 (ϕ, (hza − hz )ψ ) − (ϕ , (hza − hz )ψ ). (77) 24π 8π Consider the general element (ϕ, (hza − hz )ψ) with test functions in the assumed class. For any non-negative real numbers a, b one has the identity ∞ 1 1 2 − a−b=− r2 dr. π (a2 + r2 ) (b2 + r2 ) Tza (ϕ, ψ) = −
0
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Using this and the spectral representations of hza and hz one finds 2 (ϕ, (hza − hz )ψ) = π
∞
ϕ, G(−r2 ) − G0 (−r2 ) ψ r2 dr.
0
To find the integral kernel [G(−r2 ) − G0 (−r2 )](x, y) of [G(−r2 ) − G0 (−r2 )], one needs only to transform the kernel (21) to the position space: [G(−r2 ) − G0 (−r2 )](x, y) Fq† Fp 1 2 −iqy dp T (−r dq. ) e = eipx 2 2π (p + r2 ) (q 2 + r2 ) For x, y in the assumed region the Fourier integrals may be closed to contour integrals in the complex plane (the half-circular contributions vanish either in the upper or the lower half-plane due to the bound (83)) and evaluated by residues. Because eventually we are interested in removal of point-splitting we assume that x and y are in the same connected part of the region considered. A B , where A and B are functions of r, and get We denote T (−r2 ) = BA
π G(−r2 ) − G0 (−r2 ) (x, y) = 1+σ |f(ir)|2 r ⎧ −|x+y|r [A cosh(ar) + B] for x, y > b + R or x, y < −b − R, ⎪ ⎨e × e−ar [A cosh ((x + y)r) + σB cosh ((x − y)r)] ⎪ ⎩ for x, y ∈ (−b + R, b − R).
We use this integral kernel in (77), evaluate derivatives by parts and remove the point-splitting. In this way we find in the assumed regions Ea (x) = Tza (x, x) . ∞ −2|x|r (A cosh(ar) + B) 2e 1 =− |f(ir)|2 r3−σ dr, 6π e−ar (2A cosh(2xr) − σB) 0
|x| > b + R, |x| < b − R.
We now want to consider the limit of this local energy in the scaled version of the model. Because of the appropriate convergence of the integral this limit may be performed inside the integral. The scaling of f(ir) and T (−r2 ) follows from Sect. 3. As for the scaled model the excluded position set shrinks to ||x| − b| ≤ λR, in the limit all x = ±b are admitted. A straightforward calculation yields lim Ea,λ (x)
λ→0+
=−
σ 6π 2
⎧ ∞ ⎪ ⎨ 0
e−(2|x|−a)r
⎫ ⎪ ⎬
3 e(2x−a)r + e−(2x+a)r + σe−2ar ⎪ r dr, ⎪ ⎩ ⎭ 1 − e−2ar
for |x| > b, for |x| < b.
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With the help of Eq. (94) in Appendix A we get EaB (x) ≡ lim Ea,λ (x) λ→0+ ⎧ σ ⎪ − ⎪ ⎨ 16π 2 (|x| − b)4 = π2 ⎪ − − ⎪ ⎩ 1440a4
for |x| > b,
n∈(2Z+1)
σ 16π 2 (nb − x)4
for |x| < b,
(78)
where 2Z + 1 denotes the set of odd whole numbers. Let us stress once more: the above result holds in the distributional sense only for test functions supported outside x = ±b, i.e., for functions in that class there is lim Ea,λ (x)ϕ(x)dx = EaB (x)ϕ(x)dx. λ→0+
For functions not in this class our assumptions leading to the above result cease to hold. Nevertheless, we shall attempt some comparison with our global results. For that purpose let us denote σ for x = 0. (79) E B (x) = − 16π 2 x4 Then (78) may be written as B (x), EaB (x) = E B (x + b) + E B (x − b) + Ea,int
where B Ea,int (x) =
⎧ σ ⎪ + ⎪ ⎨ 16π 2 (|x| + b)4 π2 ⎪ − ⎪ ⎩ 1440a4 −
(80)
for |x| > b,
n∈(2Z+1)\{+1,−1}
σ 16π 2 (nb − x)4
for |x| < b.
(81)
B (x) Ea,int
= a−4 F (x/a), where F (y) is an absolutely It is easy to see that bounded and integrable function. Thus for large separation of plates the energy outside the plates is concentrated in the first two terms in (80). Therefore (79) has the interpretation of the energy density produced around one single B (x) may be regarded as the energy density (locally outside plate, while Ea,int the boundaries) of the interaction. When integrated over x ∈ R it gives B Ea,int =−
π2 1440a3
(82)
for both Dirichlet and Neumann cases.
9. Discussion The models considered in this article do not pretend to describe the details of the interaction of quantum field with macroscopic bodies in realistic way. Their merit comes from the fact that (i) they are consistent with the restrictions imposed by the general algebraic analysis of any quantum backreaction setting [1] while (ii) being simple enough to allow explicit calculations, and
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(iii) approximating sharp boundary conditions in a controllably localized (in physical space) way. The last point distinguishes them from the class of models constructed in [2] and allows the comparison of the results of the global and local analyses. We summarize the results and lessons to be drawn from them. (i) The models discussed in the present article are defined on the Weyl algebra of the free theory, and the interaction introduced by the ‘nonlocal boundaries’ does not lead out of the vacuum representation of the theory. The energy of the field (as defined by the free theory) is finite in the ground state (as defined by the interaction with ‘boundaries’) without any arbitrary ‘renormalization’. The change of this ‘Casimir energy’ with the variation of the position of the boundaries determines the backreaction force. (ii) The scaled interaction with the ‘nonlocal boundaries’ approaches in the scaling limit the sharp Dirichlet/Neumann conditions in the Hilbert– Schmidt norm sense for the resolvents. This implies norm convergence (or strong convergence) for any continuous, vanishing in infinity (resp. any bounded) function of the first-quantized hamiltonian. The nonlocality of the boundaries is under control and tends to zero in the limit. (iii) The Casimir energy per area of the scaled models obeys the scaling law (56). Thus it is governed by the a-dependence of the energy given in Eqs. (57) and (58). The Casimir force per area is minus the derivative of those formulas. We have not discussed this point, but one can show that one can differentiate terms in these formulas one by one with o(a−3 ) going over to o(a−4 ). Thus one finds ⎧ π2 λ ⎪ ⎪ − + o , (D) ⎨ λ 4 dεa 480a a4 − = λ 3ζ(3) 1 I0 π2 da ⎪ ⎪ + 2 2 4 +1 − + o 4 . (N) ⎩+ 3 2 4 24πM0 λa 8π M0 a π 480a a One finds that in Dirichlet case the force has a well-defined limit, but in Neumann case depends on the model and diverges for sharp boundaries. This model-dependence occurs despite the fact that the models approximate in many respects the sharp boundaries very well (globally!). Neumann case models with I0 = 0 (which are among those admitted by our assumptions—see Appendix C) have faster convergence property [see Eq. (35)] and for them the additional a−4 term in the force vanishes. (iv) Algebras of fields localized outside the interaction region (test functions with supports not intersecting with that region) admit free vacuum state, ground states of our models, as well as ground state of the sharp boundaries. Restricted to these algebras our ground states tend weakly to the sharp boundaries states in the scaling limit. (v) The local energy density is unambiguously defined in our models in the whole physical space by point-splitting, with no ad hoc later renormalization. When restricted to regions not intersecting with the interaction
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area (which in scaling limit just means not intersecting with the boundaries) the local Casimir energy has a well defined limit given by a smooth function Eq. (78). This limit density has universal, model-independent form. Thus the ‘bulk’ contribution to the total Casimir energy has this universality and the model-dependent terms in Neumann case turn out to be squeezed in the limit inside the boundaries. The hope that these non-regular contributions may be removed in a model-independent way is therefore not justified. Further confirmation of our interpretation supply formulas and remarks ending the last section. Local energy density has been discussed by various authors before [5–8], but usually with the use of some regularizations, often of not quite clear status. The results to be found in literature are not consistent. The Dirichlet case for the region between the plates is discussed in [7] and modulo some infinite renormalization agrees with ours (formula (2.32) in that reference). On the other hand the authors of [8] obtain a different result (by a rather indirect way of ‘regularization’ and removal of cut-off). We are not aware of a complete rigorous discussion resulting in our formulas (78)–(82). It is also worth noting that the density EaB (x) can be also obtained directly by the use of sharp boundary conditions Hamiltonian hB za instead of hza in (77). This amounts to the use of the difference of (33) and (32) in the calculation. However, let us stress once more, this limit value of the density is correct only if smeared with a test function with support not touching the borders.
Acknowledgements We are grateful to the referee for careful reading and constructive editorial remarks. Open Access. This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Appendices If not stated otherwise we work here with the same assumptions as stated in (11) and (12).
Appendix A. Integrals and Estimates We gather here a few technical, separate points used in the main text. (i) We recall that f is a smooth function supported in −R, R if, and only if, its Fourier transform f is an entire analytic function satisfying the estimates
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const(N )eR| Im u| , (1 + |u|)N
(83)
|f(u)| ≤
for all u ∈ C and N ∈ N. Therefore Mp , which is the product of the Fourier transforms of f (z) and of f (−z) = f (z), extends to the analytic function Mu on C satisfying similar ˇ of Mp is smooth estimates with R replaced by 2R. The inverse transform M and supported in −2R, 2R. Using this we find for a > 2R / πˇ ˇ (−a) = 0. cos(ap)Mp dp = M (a) + M (84) 2 Mp − M k also extends to an analytic function and with the p2 − k 2 use of estimates on Mu one finds, by closing the contour of integration (as always for a > 2R), that cos(ap)(Mp − Mk ) dp = 0. (85) p2 − k 2
As Mu is even,
(ii) In order to prove the estimate (30), we start with Dirichlet case, we note that (remember that f (0) = 0, as f is even) λ λ f(λp) − f(0) = f (pξ)(λ − ξ)dξ ≤ λ f (pξ)dξ. p2 0
0
Moreover we have (norms of functions of p as elements of L2 ) λ 2 f (pξ) dξ ≤ 0
2 ≤ f
˜ dξdξ˜ f (pξ)f (pξ)
0,λ 2
˜ −1/2 dξdξ˜ ≤ constλ, (ξ ξ)
0,λ 2
where in the first step we have used the Schwartz inequality. Now because p2 is bounded we get the estimate for Dirichlet. The same considerw 2 − p2 ations show (31). The proof of Neumann case in (30) is almost the same but with the use of the formula f(λp) − f(0) = p
λ
f (pξ)dξ.
0
(iii) For the function Ik , (36), we first observe that for some 0 < 0 < 1 there is I0 < 2πRM0 (1 − 0 ).
(86)
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This follows from the identity ∞ 2 sin (Rp)Mp 4 dp = 2πRM0 − I0 , p2 0
(used already in the normalization of the bound state, see (43),(44)), as the l.h.s. is strictly greater than zero. Further, we need the following estimates: const(k∗ ) , k ≥ k∗ , k2 const , k ≥ 0, Ik ≤ (1 + k)2
Ik ≥
(87)
with arbitrary k∗ > 0. We write Ik as a principal value distribution calculated on test function M . In position space, using evenness of f , we have ∞ √ ˇ (x) sin(kx) dx. (88) Ik = 2π xM kx 0
Integrating once by parts we get ⎡ ⎤ √ ∞ 2π ˇ ˇ (x) cos(kx)dx⎦ . (0) + M Ik = 2 ⎣M k
(89)
0
The estimation from above is now trivial, whereas for the estimation from below we use the Riemann–Lebesgue lemma, the assumption that Ik > 0 for k = 0 [see (37)] and continuity of Ik . Expanding further in powers of 1/k the integral in (89) we find moreover const . (90) |∂kn Ik | ≤ (1 + k)n+2 (iv) The sk function, defined in the end of Sect. 4, is smooth for k ≥ 0 and satisfies |∂kn sk | ≤ const(n)(1 + k)−(n+σ) .
(91)
To show this, we note first that |Mk − iNk |−1 is bounded in a neighborhood of k = 0 as M0 > 0. Outside this neighborhood we have |Mk − iNk |−1 ≤ |Nk |−1 ≤ const(1 + k)−σ due to (38) and the first bound in (87). On the other hand, due to (90) there is |∂kn Nk | ≤ const(1 + k)−(n+1) for n ≥ 1 except for n = 1 in the Dirichlet case, when this is replaced by |∂k Nk | ≤ const. As Mk is Schwartz, this ends the proof. It now follows that qk = sk Mk and all its derivatives are bounded by Schwartz functions. (v) For both (D and N) cases and k ≥ 0 we note the bound 1+k 1 . ≤ const(a) 2 1 − (qk eiak ) k
(92)
At k = 0 there is 1 − (q0 )2 = 0 but the first derivative of the denominator at zero does not vanish (in Neumann case use (86)), which is sufficient for (92)
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in a neighborhood of zero. Outside that neighborhood, mainly due to (87), it follows that 1 1 Mk2 + Nk2 = ≤ const, ≤ 2 1 − (qk eiak ) 1 − |qk |2 Nk2 which ends the proof. We also note that for k ≥ 0 we have m = 2, (D) 1 1 + km ≤ const , 1 − |qk |2 km m ≥ 2, (N)
(93)
where the Neumann case depends on the behavior of Ik at zero (m = 2 for I0 = 0 and m = 2 + 4r when Ik k 2r , r ≥ 1 as Ik is even). k→0
(vi) Finally, we note the following identity using a known integral representation of the Hurwitz zeta function: for α > 0 there is 1 6
∞ 0
∞ α r3 e−αr 1 = dr = ζ 4, (α + 2an)−4 , 1 − e−2ar (2a)4 2a n=0
(94)
which is needed for the calculation of (78).
Appendix B. Operations (i)–(iii) from Section 7 In this appendix we prove the admissibility of the three operations (i)–(iii) performed in Sect. 7.1 for the Dirichlet case and mentioned in Sect. 7.2 for the Neumann case. The key tool for this is the following simple lemma. 0 Let cn for n ∈ N ⊆ N be complex measurable functions on D ⊆ R. If n∈N |cn (k)| is integrable on D then cn (k)einak dk = 0. (95) lim a→∞
n∈N D
The proof is straightforward and uses the Lebesgue dominated convergence theorem and the Riemann–Lebesgue lemma. Throughout this appendix Sk denotes some Schwartz function in k variable (each time it may be a different function). All we have to do is to check if the assumption of the mentioned lemma is fulfilled for the appropriate expressions. B.1. Dirichlet Case Let us first consider the following part of the expression (64) 1 i Mp ∂k3 (χ(k, p)sk qkn ) einak dk dp; a3 n3 n∈2N
R2+
we shall need its real part multiplied by −1/3π 3 in (64).
(96)
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(i) To justify the validity of operation (i) of Sect. 7.1 it is sufficient to check whether the sequence of functions ∞ −3 cn (k) = n Mp ∂k3 (sk qkn χ(k, p)) − sk qkn ∂k3 χ(k, p) dp 0
satisfies the assumption of lemma (95). For this we note that using properties of sk and qk described in Appendix A (iv) we have n−3 ∂k3 (sk qkn χ(k, p)) − sk qkn ∂k3 χ(k, p) 2
≤ Sk n−3 |qk |n−1 + n−2 |qk |n−1 |∂kj χ(k, p)| j=0
+Sk n−2 (n − 1)|qk |n−2
1 j=0
−2
+Sk n
|∂kj χ(k, p)|
(n − 1)(n − 2)|qk |n−3 |χ(k, p)|.
(97)
A simple calculation yields |∂kj χ(k, p)| ≤ const
k 2−j , k+p
j = 0, 1, 2,
and therefore ∞ Mp |∂kj χ(k, p)|dp ≤ constk 2−j log(1 + k −1 ),
j = 0, 1, 2.
(98)
(99)
0
Using this and remembering that |qk | ≤ 1 we obtain
|cn (k)| ≤ Sk log 1 + k −1 n−3 + n−2 + n−1 k|qk |n−2 + k 2 |qk |n−3 ,
(100)
where the third and the fourth terms inside the parentheses appear for n ≥ 2 and n ≥ 3 respectively. The summation gives
1 k2 |cn (k)| ≤ Sk log 1 + k −1 1 + k log + . (101) 1 − |qk |2 1 − |qk |2 n∈2N
The estimate (93) shows that the integral of the r.h.s. over R+ is finite, thus the assumption of lemma (95) is satisfied. (ii) For the operation (ii) the assumption of the lemma is fulfilled because 2
Mp sk qkn ∂k3 χ(k, p) ≤ 6p (k + 3p) Mp Sk ∈ L1 R2+ \Ω , 5 (k + p)
with Ω as defined in Sect. 7.1. (iii) The operation (iii) is admissible because |Mp sk qkn − M0 s0 q0n | ≤ const p + const(1 + n)k + const(1 + n)pk and k∂k3 χ(k, p), p∂k3 χ(k, p) are both integrable on Ω, so the lemma holds also in this case. This ends the proof for the terms (96).
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For the expression with the sum over odd natural numbers and with cos(ap) we use the following modification of the lemma. Let cn for n ∈ N ⊆ N 0 be complex measurable functions on D ⊆ R2 . If n∈N |cn (k, p)| is integrable on D then lim cn (k, p)einak e±iap dkdp = 0. (102) a→∞
n∈N D
Now, almost the same considerations as before show the admissibility of operations (i)–(iii) for this part of energy. B.2. Neumann Case First we note that for the terms which we consider here, the limit over can be easily performed. This follows from the estimations below. Therefore we use the same lemma as for Dirichlet case, i.e., (95) and (102), but we recall that we now replace qk eiak = q˜k ei˜ak , as mentioned at the beginning of Sect. 7.2. With this modification the estimation (97) for the terms in the sum (96) is still valid, but the bounds (98) change: k 2−j is replaced by p2−j . In consequence there is no k 2−j factor in front of log in (99) and no k, k 2 factors in (100) and (101). With this modification the sum of n−3 , n−2 and n−1 terms in (100) is still sufficiently well bounded, but the sum of n0 terms is to singular (no k 2 in the last term in (101)). Therefore these terms need a more detailed qk |n−3 |˜ qk |3 log(1 + k −1 ) treatment. In fact, they can be estimated by const|sk ||˜ (prime denotes here the derivative with respect to k) and their sum over n is bounded by const
qk |3 |sk ||˜ log(1 + k −1 ). 1 − |˜ qk |2
This function is indeed in L1 (R+ ), since outside the neighborhood of zero, qk |3 |sk ||˜ ≤ Sk , whereas in the neighborhood we have using (93), we have 1 − |˜ qk |2 q˜0 = 0 which is enough for I0 = 0 case [see (93) and the comment after it] and |˜ qk |3 k 2r−2 . The discussion of if Ik k 2r , r ≥ 1 (Ik is even), then k→0 1 − |˜ qk |2 k→0 admissibility of operations (ii) and (iii) goes in the same way as for Dirichlet case (estimates hold also for Neumann case, with q˜k instead of qk ). The analysis of the expression with the sum over odd natural numbers and with cos(ap) is also analogous to the Dirichlet case.
Appendix C. Comments on the Assumed Class of Models The models discussed in this paper are based on functions f subject to the conditions formulated in (11), (12), (37) and (38). In this appendix we exhibit a class of functions conforming to them. It is sufficient to satisfy (11) and (37) as the other two are then achieved by simple rescaling of the function by a constant factor.
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First, we note that each even function with the assumed support which in addition is real, non-negative and monotonically (weakly) decreasing for positive arguments satisfies the demands. Indeed, for each such function the ˇ (x) last condition in (11) is fulfilled. Moreover, it is easy to see that then M (being the convolution of the function with itself) is also even, positive, compactly supported and decreasing for x > 0. Thus from (89), since for k = 0 there is ∞ ∞ ˇ (x) cos(kx)dx < − M ˇ (x)dx = M ˇ (0), M 0
0
we have Ik > 0 for k = 0, which ends the proof of (37). For each function in the class defined in the previous paragraph there is ˇ (x) [see (88)]. We now extend the I0 > 0, which is due to the positivity of M class to include also functions for which I0 = 0 (see Discussion). Let f be a function in the class of the last paragraph and define a new function f r (z) = f (z) − μ (f (z − r) + f (z + r)) , where μ > 0 and r > R > 0. One finds Mpr = |fr (p)|2 = (1 − 2μ cos(rp)) Mp . 2
Using this and taking into account (85) one has
Mkr − Mpr sin(rk) Ikr = (1 − μ cos(rk)) Mk . dp = 1 + 2μ2 Ik − 4πμ 2 2 p −k k We impose the condition I0r = 0, which is a quadratic equation for μ: 2(I0 + 2πrM0 )μ2 − 4πrM0 μ + I0 = 0. For sufficiently large r the equation has two roots, and we take the smaller one, which is less then 1/2 and for large r tends to zero. Then Ikr = 4πrμ(1 − μ)M0
Ik [1 − η(rk)ξ(k)] , I0
where η(rk) =
sin(rk) 1 − μ cos(rk) , rk 1−μ
ξ(k) =
I0 Mk . M0 Ik
It is an exercise in function analysis to show that for sufficiently small μ there is η(u) < 1 for all u = 0. Below we show that f may be chosen such that also ξ(k) < 1 for k = 0, and then Ikr > 0 for k > 0, which is the condition (37). To reduce the size of the support of f r one can use the scaling defined in Sect. 3. Consider now the non-negative and even function ξ(k) for positive arguments. Suppose that ξ (0) < 0 (see below). Then there is k0 > 0 such that for k ∈ (0, k0 ) it is ξ(k) < 1. For k ≥ k0 , using (87), we have ξ(k) ≤ const ≡ ξmax . We now need to improve the estimation of η for k ≥ k0 . It is easy to see that 1 for k ≥ k0 . if r has been chosen large enough then η(rk) < ξmax
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Finally, we have to choose f so as to satisfy ξ (0) < 0. Let f be defined as f (z) = f0 = const on −R, R and zero outside. Straightforward calculation gives Mk =
2|f0 |2 2 sin2 (Rk) R , π (Rk)2
Ik = 8|f0 |2 R3
2Rk − sin(2Rk) , (2Rk)3
4 2 R < 0. 15 The function used here needs ‘rounding the corners’ to be in the class of the second paragraph. But this may be made by a small local variation, and both Mk and Ik and their derivatives depend continuously on such small variations of f [for Ik see (88)], which is sufficient to conclude the proof. ξ (0) = −
References [1] [2] [3] [4] [5] [6] [7] [8]
Herdegen, A.: Ann. Henri Poincar´e 6, 669–707 (2005) Herdegen, A.: Ann. Henri Poincar´e 7, 253–301 (2006) Jaffe, A., Williamson, L.R.: Ann. Phys. 282, 432 (2000) Fosco, C.D., Losada, E.: Phys. Lett. B 675, 252 (2009) L¨ utken, C.A., Ravndal, F.: Phys. Rev. A 31, 2082 (1985) Sopova, V., Ford, L.H.: Phys. Rev. D 66, 045026 (2002) Milton, K.A.: Phys. Rev. D 68, 065020 (2003) Kawakami, N.A., Nemes, M.C., Wreszinski, W.F.: J. Math. Phys. 48, 102302 (2007)
Andrzej Herdegen and Mariusz Stopa Institute of Physics Jagiellonian University Reymonta 4 30-059 Cracow Poland e-mail:
[email protected] e-mail:
[email protected] Communicated by Klaus Fredenhagen Received: April 12, 2010 Accepted: July 6, 2010
Ann. Henri Poincar´e 11 (2010), 1201–1224 c 2010 Springer Basel AG 1424-0637/10/071201-24 published online November 3, 2010 DOI 10.1007/s00023-010-0058-z
Annales Henri Poincar´ e
Chern–Simons Invariants of Torus Links S´ebastien Stevan Abstract. We compute the vacuum expectation values of torus knot operators in Chern–Simons theory, and we obtain explicit formulae for all classical gauge groups and for arbitrary representations. We reproduce a known formula for the HOMFLY invariants of torus knots and links, and we obtain an analogous formula for Kauffman invariants. We also derive a formula for cable knots. We use our results to test a recently proposed conjecture that relates HOMFLY and Kauffman invariants.
1. Introduction The idea of using Chern–Simons theory [5] to compute knot invariants goes back to Witten’s paper [32] in 1989, when he identified the skein relation satisfied by the Jones polynomial [12]. Though the theory is in principle exactly solvable, the computations are quite challenging in most cases. One convenient framework to address such problems is the formalism of knot operators [21]. For torus knots, an explicit operator formalism has been constructed by [15], that successfully reproduces the Jones polynomial for Wilson loops carrying the fundamental representation of SU (2). Several further works have generalized the computation to arbitrary representations of SU (2) [11], to the fundamental representation of U (N ) [16] and to arbitrary representations of U (N ) [17]. There have also been attempts to compute Kauffman invariants from Chern–Simons theory. With Wilson loops carrying the fundamental representation of SO(N ), Labastida and P´erez obtained a simple formula for the Kauffman polynomial [20]. For torus knots of the form (2, 2m + 1), there are formulae for arbitrary representations of SO(N ) [1,29], but they are not completely explicit due to the presence of a generally unknown group-theoretic sign. Recently, a simple formula for HOMFLY invariants of torus links has been obtained by using quantum groups methods [22]. For quantum Kauffman invariants, L. Chen and Q. Chen [4] had derived a similar formula but published it only after this paper was submitted. These results encouraged
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us to address the computation of torus link invariants from Chern–Simons point of view. In this paper, we carefully analyze the matrix elements of knot operators to produce simpler formulae. Our approach uses only group-theoretic data and is valid for any gauge group. As an application, we compute the polynomial invariants for all classical Lie groups and for arbitrary representations, and we reproduce the results of [22]. As explicit formulae are available, torus knots represent an useful ground to test the conjectured relationship between knot invariants and string theory. The equivalence of 1/N expansion of Chern–Simons theory to topological string theory [8] implies that the colored HOMFLY polynomial can be related to Gromov–Witten invariants and thus enjoys highly nontrivial properties [19,27]. This conjecture has been extensively checked [17,19,22] and is now proved [24]. The large-N duality of Chern–Simons theory with gauge group SO(N ) or Sp(N ) has also been studied [30]. In [3], partial conjectures on the structure of Kauffman invariants have been formulated. The complete conjecture, that also involves HOMFLY invariants for composite representations, has been stated by Mari˜ no [25]. The outline of the paper is as follows: in Sect. 2 we recall some important properties of Wilson loops. Section 3 is devoted to the matrix elements of torus knot operators. In Sects. 4, 5 and 6, we deduce explicit formulae for HOMFLY and Kauffman invariants of cable knots, torus knots and torus links. Finally, in Sect. 7 we provide some tests of Mari˜ no’s conjecture.
2. Chern–Simons Theory and Wilson Loop Operators Chern–Simons theory is a topological gauge theory on an orientable, boundariless 3-manifold M with a simple, simply connected, compact, nonabelian Lie group G and the action k 2 (2.1) S(A) = Tr A ∧ dA + A ∧ A ∧ A , 4π 3 M
where Tr is the trace in the fundamental representation and k is a real parameter. In this expression A is a g-valued 1-form on M , where g is the Lie algebra of the gauge group G. In the context of knot invariants, M is usually taken to be S3 and the relevant gauge-invariant observables are Wilson loop operators. Let K ⊂ S3 be a knot and Vλ an irreducible g-module of highest weight λ. The associated Wilson loop is ⎡ ⎤ (2.2) WλK (A) = TrVλ ⎣Pexp A⎦ , K λ (A) is obtained where Pexp is a path-ordered exponential. In other words, WK by taking the trace on Vλ of the holonomy along K.
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Figure 1. Products of Wilson loops with various orientations
As was realized first by Witten [32], the vacuum expectation value (VEV)
D[A] WλK11 (A) · · · WλKLL (A)eiS(A) K1 KL
Wλ1 · · · WλL = , (2.3) D[A] eiS(A) where the functional integration runs over the gauge orbits of the field, is a framing-dependent invariant of the link L = K1 ∪ · · · ∪ KL . Indeed, Wλ (K) = WλK reproduces the quantum invariant obtained from the category of Uq (g)-modules. In this paper we shall encounter colored HOMFLY invariants HλK (t, v) corresponding to the group U (N ) and colored Kauffman invariants KλK (t, v) corresponding to the groups SO(N ) and Sp(N ). The VEV (2.3) can be computed perturbatively or by nonperturbative methods based on surgery of 3-manifolds. In this paper we consider these later methods, in particular the formalism of knot operators. Before turning to knot operators, and restricting to torus knots, we review some properties of Wilson loops. 2.1. Product of Wilson Loops with the Same Orientation We provisorily take G to be U (N ) for definiteness. Representations that label Wilson loops are usually polynomial representations (those indexed by partitions). When we write WλK for a Wilson loop or Wλ (K) for an invariant, we implicitly assume that the representation with highest weight λ ∈ Λ+ W is polynomial, so that we can symbolize λ by a partition. The first relation to be mentioned is the well-known fusion rule for Wilson loops. For an oriented link made of two copies of the same knot, with the same orientation for both components (as in Fig. 1a for instance), one has ν Nλμ WνK , (2.4) WλK WμK = ν∈P ν are the coefficients in the where P is the set of nonempty partitions and Nλμ decomposition of the tensor product
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Vλ ⊗ Vμ =
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ν Nλμ Vν .
ν∈P
They are called Littlewood–Richardson coefficients for U (N ). Formula (2.4) is extremely useful, since it reduces any product of Wilson loops that share the same orientation to a sum of Wilson loops. It only applies to links composed by several copies of the same knot, but this is not a restriction for torus links. For other Lie groups the same formula holds with different coefficients. For SO(N ) and Sp(N ) they are given by [14,23] ν λ μ ν = Nαβ Nαγ Nβγ . (2.5) Mλμ α,β,γ
Here the sum runs over P ∪ {∅}. Remark 1. Formula (2.4) has to be understood as a regularization for the product of two operators evaluated at the same point. It extends the relation ν WλK (A)WμK (A) = Nλμ WνK (A) (2.6) ν∈P
WλK (A)
to the quantized Wilson loops. We derive between the functionals (2.6) by noting that the holonomy UK is an element of G; hence it is conjugate to an element of the maximal torus of G [13]. Furthermore, TrVλ is the character of Vλ as a function of the eigenvalues, and the product of characters is decomposed as the tensor product of representation. 2.2. Product of Wilson Loops with Different Orientations The need to consider all rational representations appears when one deals with both orientations for K (as in Fig. 1b for example). The product of two Wilson loops WλK and Wμ−K , where −K denotes K with the opposite orientation, cannot be decomposed as above. In the formalism of the HOMFLY skein of the annulus [9], one would have to use the basis of the full skein, indexed by two partitions. In Chern–Simons theory the same role is played by composite representations. Composite (or mixed tensor) representations μ λ (−1)|η| Nην Nηζ Vν ⊗ Vζ V[λ,μ] = η,ν,ζ
are the most general irreducible representations of U (N ), where the sum runs over partitions and η is the partition conjugate to η (the transpose Young diagram). More details on composite representations can be found in [10]. It is straightforward to derive a fusion rule for WλK Wμ−K by decomposing mixed tensor representations. Let UK be the holonomy along K; then WλK Wμ−K = TrVλ UK TrVμ U−1 K = TrVλ UK TrVμ UK = TrVλ ⊗Vμ UK .
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One has the following decomposition of Vλ ⊗ Vμ in terms of composite representations [14] μ λ Nην Nζν V[η,ζ] . Vλ ⊗ Vμ = η,ν,ζ K W[η,ζ]
the Wilson loop in the composite representation V[η,ζ] , If we denote by we get the fusion rule μ λ K WλK Wμ−K = Nην Nζν W[η,ζ] . (2.7) η,ν,ζ
Remark 2. Since V[λ,∅] = Vλ and V[∅,λ] = Vλ∗ , one has K = WλK W[λ,∅]
and
K W[∅,λ] = Wλ−K .
−K K More generally W[λ,μ] = W[μ,λ] .
We can as well consider product of Wilson loops carrying composite representations and write a fusion rule for them. It is given by [14]
μ ζ K K λ ν η ξ K W[λ,μ] W[η,ν] = Nκα Nκβ N δ N γ Nαγ Nβδ W[ξ,ζ] . α,β,γ,δ ξ,ζ
κ
2.3. Traces of Powers of the Holomony As will be illustrated later in this paper, traces of powers of the holonomy along a given knot play an important in the gauge theory approach to knot invariants. In fact, such composite observables can be decomposed by a grouptheoretic approach. Given a knot K, the holonomy UK is conjugate to an element in the maximal torus of G and we already mentioned that TrVλ UK = chλ (z1 , . . . , zr ),
(2.8)
where chλ is the character of g and z1 , . . . , zr are the variable eigenvalues of UK (r is the rank of G). The trace of the n-th power of the holonomy is then given by Trλ UnK = chλ (z1n , . . . , zrn ).
(2.9)
Let ΛW be the weight lattice and W the Weyl group of G. Equation (2.9) is obtained from (2.8) by applying the ring homomorphism Ψn : Z[ΛW ]W −→ Z[ΛW ]W eμ −→ enμ which is called the Adams operation. Since the characters form a Z-basis of Z[ΛW ]W , there exist integer coefficients cνλ,n univocally determined by the decomposition of Ψn chλ with respect to the basis (chν )ν∈Λ+ : W ν Ψn chλ = cλ,n chν . (2.10) ν∈Λ+ W
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Figure 2. Knot lying on a surface (torus knot) Hence we have obtained the following formula: cνλ,n Trν UK . Trλ UnK =
(2.11)
ν∈P
The coefficients cνλ,n depend on the gauge group, and for clarity we will denote those by aνλ,n for U (N ) and by bνλ,n for SO(N ). Remark 3. In the case of U (N ), the above formula is an easy generalization of Tr UnK = χλ (C(n) ) TrVλ UK , (2.12) λ∈Pn
where χλ is the character of the symmetric group SN in the representation indexed by the partition λ and C(n) is the conjugacy class of one n-cycle in SN . This formula is precisely (2.11) for the fundamental representation of U (N ). As we will see later, the coefficients aνλ,n can be expressed in terms of the characters of the symmetric group.
3. Knot Operators Formalism We move towards the study of Wilson loop operators associated with torus knots. The main result of this section is a formula for the matrix elements of torus knot operators that is much simpler than the one of Labastida et al. [15]. Eventually, we will provide a simple formula for the quantum invariants of torus knots. 3.1. Construction of the Operator Formalism If a knot K lies on a surface Σ, the Wilson loop associated with K can be represented by an operator WλK acting on a finite-dimensional Hilbert space H(Σ). For example, the trefoil knot pictured on Fig. 2 lies on the torus T2 , and hence can be represented by an operator on H(T2 ). In the case of torus knots, an important achievement of [15] is the construction of the operator formalism that was just alluded to. The original paper treats the case of U (N ) and arbitrary gauge groups are addressed in
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(1,0)
Figure 3. Wilson loop Wλ cycle of T2
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around the noncontractible
[20]. H(T2 ) is the physical Hilbert space of Chern–Simons theory on R × T2 , which is the finite-dimensional complex vector space with orthonormal basis |ρ + λ : λ ∈ Λ+ W
(3.1)
indexed by strongly dominant weights. Each of these states is obtained by inserting a Wilson loop in the representation λ along the noncontractible cycle of the torus (Fig. 3). The state |ρ associated with the Weyl vector ρ corresponds to the vacuum (no Wilson loop inserted). To be more rigorous, one should restrict (3.1) to integrable representations at level k. However, one can show that, provided k is large enough, all representations that arise from the action of knot operators are integrable. Hence, we formally work as if k were infinite. We denote by Tnm the (n, m)-torus link. Tnm is a knot if and only if n and (n,m) m are coprime. We denote by Wλ the corresponding torus knot operator. The following formula is due to [15] for the group U (N ) and to [20] for an arbitrary gauge group: (n,m)
Wλ
|p =
μ∈Mλ
exp iπ
nm m μ2 + 2πi p · μ |p + nμ. (3.2) 2yk + cˇ 2yk + cˇ
In this formula, Mλ denotes the set of weights of the irreducible G-module Vλ , y is the Dynkin index of the fundamental representation and cˇ is the dual Coxeter number of G. The quantization condition requires that 2yk is an integer. Expression (3.2) is actually more complicated than it seems, because not all weights p + nμ are of the form ρ + ν for some ν ∈ Λ+ W . Hence, it is very difficult to get tractable formulae for WλK from (3.2). To simplify the computation of the invariants, we shall provide simple expressions for the matrix elements. This result has been established in our master’s thesis [31] for the group SU (N ).
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3.2. Parallel Cabling of the Unknot To begin with, we consider an n-parallel cabling1 of the unknot represented (n,0) by the operator Wλ . It may look a bit awkward to consider such an operator, but if we manage to cope with the exponential factor we can reduce any (n,m) (n,0) to Wλ . From our considerations on powers of the holonomy, it is Wλ clear that (n,0) Wλ = cνλ,n Wν(1,0) ν∈Λ+ W (1,0)
As a result of this operator expansion, and since Wλ get the formula (n,0) cνλ,n |ρ + ν. Wλ |ρ =
|ρ = |ρ + λ, we (3.3)
ν∈Λ+ W (n,m)
This equality can also be proved from the explicit representation of Wλ on H(T2 ). More details are given in Appendix A. 3.3. Matrix Elements of Torus Knot Operators (n,m)
To deal with the generic torus knot operator Wλ operator
, we introduce a diagonal
m
D|ρ + λ = e2πi n hρ+λ |ρ + λ, where hp =
p2 − ρ2 2(2yk + cˇ) (n,m)
(n,0)
and Wλ is a conformal weight of the WZW model. The action of Wλ on |ρ + η differ only by an exponential factor, which is nm mπi 2m μ2 + p·μ = (p + nμ)2 − p2 . πi 2yk + cˇ 2yk + cˇ n(2yk + cˇ) It follows immediately that (n,m)
Wλ
(n,0)
= DWλ
D−1 .
(3.4)
(n,0)
Using this result and our discussion on Wλ , we obtain a simple formula for (n,m) the matrix elements of Wλ : m (n,m) |ρ = cνλ,n e2πi n hρ+ν |ρ + ν. (3.5) Wλ ν∈Λ+ W
Remark 4. This formula contains the same ingredients as Lin and Zheng’s formula [22] for the colored HOMFLY polynomial. One of our goals was to reproduce this formula in the framework of Chern–Simons theory. 1 Here parallel cabling is not to be understood in the classical sense. Usually the n-parallel cable of a knot is a n-component link, which should be represented by the product of operators (TrVλ U)n . In our case, the n-parallel cable is the quantum quantity TrVλ (Un ).
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3.4. Fractional Twists Formula (3.5) resembles a result of Morton and Manch´ on [26] on cable knots, to which we shall return in Sect. 4. Following their terminology, we shall refer to D as a fractional twist. In fact, there are intrinsic reasons in Chern–Simons theory to refer to D as a fractional twist. We recall that the mapping class group of the torus is SL(2, Z). It has two generators, T and S; the former represents a Dehn twist and the later exchanges the homology cycles. There is an unitary representation R : SL(2, Z) −→ GL H(T2 ) [6], and T acts by c
R(T)|p = e2πi(hp + 12 ) |p where c=
2yk dim g . 2yk + cˇ
If we redefine D to act as m
c
D|p = e2πi n (hp + 12 ) |p, formula (3.4) remains true and we can consider D as the Furthermore, SL(2, Z) acts by conjugation (n,m)
R(M)Wλ
R(M)−1 = Wλ
(n,m)M
m n -th
power of R(T).
,
(3.6)
where (n, m)M stands for the natural action by right multiplication. m If we define Tm/n = 10 1n and extend R to such elements, then D = R(Tm/n ) and formula (3.4) also extends to (n,0)
R(Tm/n )Wλ
(n,0)Tm/n
R(Tm/n )−1 = Wλ
(n,m)
= Wλ
.
(and its representative D) should With this identification it is clear why T be called a fractional twist. It is, however, less obvious that R extends to Tm/n . m/n
Remark 5. Any torus knot can be obtained from the unknot by a complicated sequence of Dehn twists along both homology cycles. With a fractional twist we obtain Tnm in one step from n-copies of the unknot. Our computations indicate that fractional twists have simple actions on Chern–Simons observables (at least on torus knot operators). Hopefully, fractional twists apply to more general knots.
4. Invariants of Cable Knots We extend our analysis to cable knots from the point of view of Chern–Simons theory. Consider a knot K ⊂ S3 and its tubular neighborhood TK . Let Q be a knot in the standard solid torus T and iK : T −→ TK the embedding of T into TK . The satellite K ∗ Q is the knot iK (Q) obtained by placing Q in the tubular neighborhood of K. In case the pattern Q is a torus knot, the satellite is called a cable. Figure 4 illustrates a cabling of the trefoil.
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Figure 4. Cabling of the trefoil knot by the (2, 1)-torus knot pattern We follow the procedure described in [32], translated in terms of knot operators. The path integral over the field configuration with support in M = S3 \TK gives a state φM | ∈ H(∂TK )∗ , since the boundary of M is ∂TK with the opposite orientation, and the path integral over T gives a state (n,m)
Wλ
|φT ∈ H(T2 )
when the pattern Tnm is inserted in the solid torus. The homeomorphism iK |T2 : T2 −→ ∂TK is represented by an operator FK : H(T2 ) −→ H(∂TK ). We deduce the formula (n,m)
Wλ (K ∗ Tnm ) =
φM |FK Wλ |φT . φM |FK |φT
In particular, when the trivial pattern T10 is placed in the neighborhood TK , the resulting satellite is K: (1,0)
Wλ (K) =
φM |FK Wλ |φT . φM |FK |φT (n,m)
(1,0)
Using our relation between Wλ and Wλ , we deduce the following formula for the invariant of cable knots: m Wλ (K ∗ Tnm ) = aνλ,n e−2πi n hρ+ν Wν (K) (4.1) ν∈Λ+ W
for U (N ) and the same formula with aνλ,n replaced by bνλ,n for SO(N ). This formula has been proved by Morton and Manch´ on [26] for HOMFLY invariants. The analogous for Kauffman invariants seems to be new.
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Figure 5. Heegaard splitting of S3 as two solid tori
5. Quantum Invariants of Torus Knots In the preceding we have not specified the 3-manifold M onto which the knots are embedded, but the construction of the operator formalism implicitly requires M to admit a genus-1 Heegaard splitting. The case of interest, which is M = S3 , admits the decomposition into two solid tori pictured on Fig. 5. The choice of a homeomorphism to glue both solid tori together determines Chern–Simons invariants through the following formula [16]: Wλ (Tnm ) =
ρ|FW(n,m) |ρ , ρ|F|ρ
(5.1)
where F is an operator on H(T2 ) that represents the homeomorphism. But this choice also determines a framing w(K) of the knot. We will correct Wλ (K) by the deframing factor e−2πiw(K)hρ+λ [32] to express the invariants in the standard framing. It is common to glue the solid tori along the homeomorphism represented by S in the mapping class group (the one that exchanges the two homology cycles of T2 ). The framing determined by this choice turns out to be mn for the (n, m)-torus knot. Its action on H(T2 ) is given by the Kac–Peterson formula [6] p|S|p =
Λ 2πi i|Δ+ | W (−1)w e− 2yk+ˇc p·w(p ) . 1/2 ΛR (2yk + cˇ) w∈W
(5.2)
Depending on the choice of the gauge group, several invariants can be computed. Our results apply to any semisimple Lie group, but we will restrict ourselves to classical Lie groups. As it turns out, the group U (N ) reproduces the colored HOMFLY invariants, whereas both groups SO(N ) and Sp(N ) reproduce the colored Kauffman invariants.
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5.1. Colored HOMFLY Polynomial The precise relation between colored HOMFLY invariants and Chern–Simons invariants with gauge group U (N ) is the following: −πi (5.3) HλK (t, v) = e−2πiw(K)hρ+λ Wλ (K) k+N N e
=t,t =v
−πi k+N
and v = tN are considered as independent variables. Since where t = e G = U (N ) has been fixed, we have replaced cˇ by N and y by 1/2. (n,m) We use the notation Hλ for the HOMFLY invariants of the (n, m)(λ) torus knot. It is easy to see that e2πihρ+λ = t−κλ v −|λ| , where κλ = i=1 (λi − 2i + 1)λi . By using the action of knot operators, (n,m) πi Hλ (t, v) = e−2πinmhρ+λ Wλ (Tnm ) − k+N e =t,tN =v m mnκλ mn|λ| ν −m κ =t v aλ,n t n ν v − n |ν| Wν (T10 ). ν∈Λ+ W
The invariant of the unknot Wν (T10 ) is called the quantum dimension of Vλ . Using the Kac–Peterson formula (5.2) and the Weyl character formula, one obtains ρ|S|ρ + λ 2πi = chλ − ρ . Wλ (T10 ) = ρ|S|ρ k+N This expression is a function of t and v given by the Schur polynomial sλ (x1 , . . . , xN ) evaluated at xi = tN −2i+1 . We denote this function by sλ (t, v). Finally, by showing that all ν ∈ P appearing in the sum satisfy |ν| = n|λ|, we obtain the following formula: m (n,m) (t, v) = tmnκλ v m(n−1)|λ| aνλ,n t− n κν sν (t, v). (5.4) Hλ |ν|=n|λ|
This formula has already been proved by Lin and Zheng [22] starting from the rigorous quantum group definition. This formula is much simpler than the one originally obtained by Labastida and Mari˜ no by using knot operators [17]. For actual calculations the following expression is useful: 1 aνλ,n = χλ (Cμ )χν (Cnμ ). zμ μ∈P|λ|
It is easily proved using Frob´enius formula for the characters of the symmetric group. Example 1. Apart from the examples found in [22], we obtained for (3, m)torus knots the following results: (3,m) H = t18m v 6m t−24m s(9) − t−18m s(8,1) + t12m s(7,12 ) + t−10m s(6,3) − t−8m s(6,2,1) − t−8m s(5,4) + t−4m s(5,22 ) + t−4m s(42 ,1) − t−2m s(4,3,2) + s(33 )
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H
(3,m)
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= v 6m t−10m s(6,3) − t−8m s(6,2,1) + t−6m s(6,13 ) − t−8m s(5,4) + t−4m s(5,22 ) − s(5,14 ) + t−4m s(42 ,1) − t−2m s(4,3,2) + t6m s(4,15 ) + 2s(33 ) − t2m s(32 ,2,1) + t4m s(32 ,13 )
H
(3,m)
+ t4m s(3,23 ) − t8m s(3,2,14 ) − t8m s(24 ,1) + t10m s(23 ,13 ) = t−18m v 6m s(33 ) − t2m s(32 ,2,1) + t4m s(32 ,13 ) + t4m s(3,23 ) − t8m s(3,2,14 ) + t12m s(3,16 ) −t8m s(24 ,1) + t10m s(23 ,13 ) − t18m s(2,17 ) + t24m s(19 )
Remark 6. For the sake of simplicity, we have restricted our analysis to polynomial representations of U (N ); analogous formulae, which will not be presented there, exist for composite representations. For example, Paul et al. [28] compute such invariants for (2, 2m + 1)-torus knots. 5.2. Colored Kauffman Polynomial Colored Kauffman invariant are obtained from Chern–Simons theory with gauge group SO(N ) by −πi (5.5) KλK (t, v) = e−2πiw(K)hρ+λ Wλ (K) 2k+N N −1 −2 e
=t,t
=v
For the Lie group SO(N ), one has cˇ = N − 2 and y = 1, regardless of parity. Using the fact that e2πihρ+λ = t−κλ v −|λ| , the procedure is very similar to the case of U (N ). The quantum dimension of Vλ , which is Wλ (T10 ), is a function of t and v that we denote dλ (t, v). Thanks to Weyl character formula, it is given by the character of SO(N ); there are explicit expressions in [2]. The final result is the exact analogous of (5.4), m m (n,m) (t, v) = tmnκλ v mn|λ| bνλ,n t− n κν v − n |ν| dν (t, v). (5.6) Kλ |ν|≤n|λ|
This formula had in fact been derived by L. Chen and Q. Chen [4]; the proof is similar to [22]. The main difference, as compared with (5.4), is that the coefficients bνλ,n are those of SO(N ), and they are nonzero also for |ν| = n|λ|. To express these coefficients in terms of the aνλ,n , we use relations between characters of SO(N ) and U (N ) obtained by Littlewood [23]. There are two formulae that give bνλ,n : |μ|−r(μ) λ τ bνλ,n = (−1) 2 Nμη aτη,n (−1)|ξ| Nξν |τ |=n|η|
η∈P μ=μ
=
η∈P γ∈C
|γ|/2
(−1)
λ Nγη
|τ |=n|η|
aτη,n
ξ∈P ν∈P
τ Nδν .
(5.7)
ν∈P δ∈D
More details, including notations, can be found in Appendix C. In principle the first formula applies to N odd and the second to N even, but they seem to give the same result. A similar situation occurs for tensor products where the decomposition does not depend on the parity of N .
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Example 2. For (2, m)-torus knots, the colored Kauffman invariants are given by (2,m) = v 2m t−m v −m d(2) − tm v −m d(12 ) + 1 K (2,m) K = t4m v 4m t−6m v −2m d(4) − t−2m v −2m d(3,1) +v −2m d(22 ) + t−m v −m d(2) − tm v −m d(12 ) + 1 (2,m) = t−4m v 4m v −2m d(22 ) − t2m v −2m d(2,12 ) K +t6m v −2m d(14 ) + t−m v −m d(2) − tm v −m d(12 ) + 1 (2,m) = t12m v 6m 1 + t−15m v −3m d(6) − t−9m v −3m d(5,1) K +t−5m v −3m d(4,2) − t−3m v −3m d(3,3) + t−6m v −2m d(4)
K
(2,m)
−t−2m v −2m d(3,1) + v −2m d(22 ) + t−m v −m d(2) − tm v −m d(12 ) = v 6m 1 + t−5m v −3m d(4,2) − t−3m v −3m d(4,12 ) − t−3m v −3m d(32 ) +t3m v −3m d(3,13 ) + t3m v −3m d(23 ) − t5m v −3m d(22 ,12 ) +t−6m v −2m d(4) − t−2m v −2m d(3,1) + 2v −2m d(22 )
K
(2,m)
−t2m v −2m d(2,12 ) + t6m v −2m d(14 ) + 2t−m v −m d(2) − 2tm v −m d(12 ) = t−12m v 6m 1 + t3m v −3m d(23 ) − t5m v −3m d(22 ,12 ) +t9m v −3m d(2,14 ) − t15m v −3m d(16 ) + t−2m d(22 ) −t2m v −2m d(2,12 ) + t6m v −2m d(14 ) + t−m v −m d(2) − tm v −m d(12 )
Example 3. For (3, m)-torus knots we further obtain K
(3,m)
K
(3,m)
K
(3,m)
= v 2m t−2m d(3) − d(2,1) + t2m d(13 ) = t6m v 6m t−10m v −2m d(6) − t−6m v −2m d(5,1) + t−2m v −2m d(4,12 ) +t−2m v −2m d(32 ) − v −2m d(3,2,1) + t2m v −2m d(23 ) + 1 = t−6m v 6m t−2m v −2m d(32 ) − v −2m d(3,2,1) + t2m v −2m d(3,13 ) +t2m v −2m d(23 ) − t6m v −2m d(2,14 ) + t10m v −2m d(16 ) + 1
Remark 7. These results are rather simple as compared with formula (5.7) for the Adams coefficients. We observed important cancellations of terms; thus it might be possible to simplify (5.7). In particular, Kauffman invariants present the following recursive structure: K appears in K , K appears in turn in K , and so on.
6. Quantum Invariants of Torus Links The formulae for HOMFLY and Kauffman invariants generalizes to links by using the fusion rule (2.4) and taking into account the framing correction. One
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obtains (Ln,Lm)
Hλ1 ,...,λL = tmn (Ln,Lm)
Kλ1 ,...,λL = tmn
L
α=1
κ λα
v
L
α=1
L
α=1
κ λα
Nλμ1 ,...,λL t−mnκμ Hμ(n,m)
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(6.1)
μ∈P mn|λα |
Mλμ1 ,...,λL t−mnκμ v −mn|μ| Kμ(n,m)
μ∈P
for the (Ln, Lm)-torus link. The first formula is equivalent to the formula of [22] for torus links. Example 4. For (4, 2m)-torus links, the colored Kauffman invariants are (4,2m) K , = v 4m 3 + t−6m v −2m d(4) − t−2m v −2m d(3,1) + 2v −2m d(22 ) K
(4,2m) ,
−t2m v −2m d(2,12 ) + t6m v −2m d(14 ) + 2t−m v −m d(2) − 2tm v −m d(12 ) = t4m v 6m t−15m v −3m d(6) − t−9m v −3m d(5,1) + 2t−5m v −3m d(4,2)
−t−3m v −3m d(4,12 ) − 2t−3m v −3m d(32 ) + t3m v −3m d(3,13 ) +t3m v −3m d(23 ) − t5m v −3m d(22 ,12 ) + 2t−6m v −2m d(4) −2t−2m v −2m d(3,1) + 3v −2m d(22 ) − t2m v −2m d(2,12 )
K
(4,2m) ,
+t6m v −2m d(14 ) + 4t−m v −m d(2) − 4tm v −m d(12 ) + 3 = t−4m v 6m t−5m v −3m d(4,2) − t−3m v −3m d(4,12 ) + t−3m v −3m d(32 ) +t3m v −3m d(3,13 ) + 2t3m v −3m d(23 ) − 2t5m v −3m d(22 ,12 ) +t9m v −3m d(2,14 ) − t15m v −3m d(16 ) + t−6m v −2m d(4) −t−2m v −2m d(3,1) + 3v −2m d(22 ) − 2t2m v −2m d(2,12 )
+t6m v −2m d(14 ) + 4t−m v −m d(2) − 4tm v −m d(12 ) + 3
7. Mari˜ no Conjecture for the Kauffman Invariants Many highly nontrivial properties of the Kauffman invariants as well as their relation to the HOMFLY invariants might be explained by a conjecture of Mari˜ no [25] that completes the prior partial conjecture of Bouchard et al. [3]. This new conjecture is similar to the Labastida–Mari˜ no–Ooguri–Vafa conjecture [19,27] for HOMFLY invariants, but it applies to Kauffman invariants and HOMFLY invariants with composite representations. 7.1. Statement of the Conjecture The conjecture contains two distinct statements, one for HOMFLY invariants including composite representations and one for both Kauffman and HOMFLY invariants. We first construct the generating functions L H[λ (t, v)sλ1 (x1 )sμ1 (x1 ) · · · sλL (xL )sμL (xL ) ZH (L) = 1 ,μ1 ],...,[λL ,μL ] λ1 , . . . , λL μ1 , . . . , μL
ZK (L) =
λ1 ,...,λL
KλL1 ,...,λL (t, v)sλ1 (x1 ) · · · sλL (xL ),
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where all sums run over partitions including the empty one. The reformulated invariants hλ1 ,...,λL (t, v) and gλ1 ,...,λL (t, v) are defined by log ZH =
∞
hλ1 ,...,λL (td , v d )sλ1 (xd1 ) · · · sλL (xdL )
d=1 λ1 ,...,λL
1 log ZK − log ZH = 2
(7.1)
d
gλ1 ,...,λL (t , v
d
)sλ1 (xd1 ) · · · sλL (xdL ).
d odd λ1 ,...,λL
All reformulated invariants can be expressed in terms of the original invariants through computing connected vacuum expectation values, following the procedure of [18]. We suggest an alternative procedure in Appendix B. For a knot, the lowest-order invariants are g (t, v) = K (t, v) − H (t, v) 1 1 g (t, v) = K (t, v) − K (t, v)2 − H (t, v) + H (t, v)2 − H[ , ] (t, v) 2 2 1 1 g (t, v) = K (t, v) − K (t, v)2 − H (t, v) + H (t, v)2 − H[ , ] (t, v). 2 2 More examples can be found in [25]. We now introduce the block-diagonal matrix Mλμ , which is
Mλμ (t) =
χλ (Cν )χμ (Cν )
n
(tν − t−ν ) i
i
i=1
ν∈Pn
for |λ| = |μ| = n and zero otherwise. We finally define hλ1 ,...,λL (t, v) = Mλ−1 (t) · · · Mλ−1 (t)hμ1 ,...,μL (t, v) 1 μ1 L μL μ1 ,...,μL
gλ1 ,...,λL (t, v) =
Mλ−1 (t) · · · Mλ−1 (t)gμ1 ,...,μL (t, v). 1 μ1 L μL
(7.2)
μ1 ,...,μL
The conjecture states that hλ1 ,...,λL ∈ z L−2 Z[z 2 , v ±1 ]
and
gλ1 ,...,λL ∈ z L−1 Z[z, v ±1 ],
with z = t − t−1 . In other words, there exist integer invariants Nλc1 ,...,λL ;g,Q (c = 0, 1, 2) such that Nλ01 ,...,λL ;g,Q z 2g−1 v Q (7.3) hλ1 ,...,λL (z, v) = z L−2 g≥0 Q∈Z
and gλ1 ,...,λL (z, v) = z L−1
Nλ11 ,...,λL ;g,Q z 2g v Q + Nλ21 ,...,λL ;g,Q z 2g+1 v Q . g≥0 Q∈Z
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Table 1. Integer invariants for the (3, 4)-torus knot
7.2. Direct Computations We now proceed to various tests of the conjecture for torus knots and links using formulae (5.4) and (5.6). Unfortunately, we cannot test the conjecture for all torus knots at once, and since the complexity increases rapidly, only the cases (2, m) and (3, m) are tractable. In principle the integer invariants can be computed as functions of m (though they are in infinite number if m is not fixed). In practice, however, we had to fix m to obtain results in a reasonable amount of time. We have obtained generic results in a few cases, to which we shall return later on. For (2, m)-torus knots, we have checked the conjecture for various values of m and for several low-dimensional representations. Most of these tests had already been made by [25], using the formulae of [1] for Kauffman invariants. Recently, analogous tests have also been made for this class of knots with nontrivial framing [28]. For (3, m)-torus knots, we were able to verify parts of the conjecture. As an illustration, we have compiled the integer invariants N 1 ,g,Q of the (3, 4)torus knot in Table 1. We further have proceeded to nontrivial checks of the conjecture for (2, 2m)- and (4, 2m)-torus links. For definiteness, we consider here the twocomponent trefoil link T46 . We have obtained g , = (36v 9 − 180v 7 + 288v 5 − 144v 3 )z + (57v 9 − 453v 7 + 912v 5 − 516v 3 )z 3 +(36v 9 −494v 7 +1286v 5 −828v 3 )z 5 +(10v 9 − 286v 7 + 1001v 5 − 725v 3 )z 7 +(v 9 − 91v 7 + 455v 5 − 365v 3 )z 9 − (15v 7 − 120v 5 + 105v 3 )z 11 −(v 7 − 17v 5 + 16v 3 )z 13 + (v 5 − v 3 )z 15 , from which the integer invariants can be read. We have also compiled the invariants N 2 , ;g,Q of the same link in Table 2.
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Table 2. Integer invariants for the (4, 6)-torus link
It is interesting to remark that in the above formula all N 2, ;g,Q vanish. For torus knots it is the case that N 2,g,Q = 0, because of Labastida–P´erez relation [20] 1 (n,m) (n,m) (n,m) K (z, v) − K (−z, v) = H (z, v) 2 between the HOMFLY and the Kauffman polynomials. But this relation does not hold in for torus links, and we suggest that an appropriate generalization is 1 (2n,2m) (2n,2m) (n,m) (n,m) (2n,2m) (2n,2m) −K K , +K , K = H[ ,∅],[ ,∅] + H[ ,∅],[∅, ] (7.4) 2 for two-components torus links, where the bar stands for the substitution z → − z. More generally, we are led to conjecture that N 2,..., ;g,Q = 0 for any torus link. We return to the computation of the integer invariants as functions of m. c is a polynomial in m with rational coefficients, enjoying the Formally Nλ,g,Q following properties: for each m such that gcd(n, m) = 1, (i) (ii)
c Nλ,g,Q is an integer; c Nλ,g,Q vanishes for large g and large |Q|.
For the (2, m)-torus knot we were able to perform the computation for the representation and for g = 0, 1, 2. The results are compiled in Table 3. The fact that these complicated expressions are indeed integers is not completely trivial: let us show for instance that N 1 ,2,3m =
m2 (m2 − 1)(2m + 1)(339m2 + 296m − 259) ∈ Z. 5760
Let p(m) = 339m2 + 296m − 259. We test the divisibility of the numerator by 5760 = 27 · 32 · 5 for m odd.
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Table 3. Integer invariants for the (2, m)-torus knot
(i) (ii) (iii)
Divisibility by 5: since p(m) ≡ 4m2 + m + 1 (mod 5), we see that {m, m − 1, 2m + 1, p(m), m + 1} always contains a multiple of 5. Divisibility by 32 : we observe that p(m) ≡ 2m + 2 (mod 3), hence both sets {m, 2m + 1, p(m)} and {m, m − 1, m + 1} contain a multiple of 3. Divisibility by 27 : one has to consider classes modulo 16, in particular p(m) ≡ 3m2 + 8m + 13 (mod 16). For m ≡ 1 (mod 8), we have two multiples of 8 (m − 1 and p(m)). Similarly for m ≡ 7 (mod 8). In both cases there is an additional even factor (m+1 resp. m−1). If now m ≡ 3
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(mod 8), then p(m) is a multiple of 16. Also m + 1 is a multiple of 4 and m − 1 is even. Similarly for m ≡ 5 (mod 8).
Acknowledgements We would like to thank Marcos Mari˜ no for suggesting the subject of our Master’s thesis, for helpful discussions on various topics, and for comments on the manuscript. We also thank Andrea Brini for helpful discussions on large-N duality and matrix models.
Appendix A. Action of the Knot Operators on H(T2 ) (n,0)
This appendix is devoted to the proof of formula (3.3) for the action of Wλ on |ρ. Though it can be deduced from generic considerations on Wilson loops, we provide an alternative derivation starting from the action of torus knot operators on H(T2 ). Our considerations are based on the following remark: the basis elements of H(T2 ) are anti-symmetrized sums over the Weyl group (−1)w f w(p) , (A.1) |p = w∈W
where f p is some complex function that admits a Fourier series expansion [15]. Hence we can work with the formal anti-symmetric elements (−1)w ew(p) Ap = w∈W
in Z[ΛW ] and translate the results to H(T2 ). We derive the required formula |ρ + nμ = cνλ,n |ν μ∈Mλ
(A.2)
ν∈ΛW
from simple properties of the Weyl group and of the weight lattice. Lemma 1. The following equality holds in Z[ΛW ]: Aρ+nμ = cνλ,n Aρ+ν , μ∈Mλ
where
cνλ,n
ν∈ΛW
are the coefficients of the Adams operation (2.10).
Proof. Using the fact that the set of weights is just permuted by the Weyl group, we immediately obtain Aρ+nμ = (−1)w ew(ρ+nμ) = enμ (−1)w ew(ρ) μ∈Mλ
μ∈Mλ w∈W
= (Ψn chλ )Aρ =
μ∈Mλ
cνλ,n
w∈W
chν Aρ
ν∈ΛW
and the conclusion follows from Weyl character formula.
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Some further properties of Wilson loops can be checked explicitly for torus knot operators using similar arguments [31].
Appendix B. Computation of the Reformulated Invariants In this appendix we give explicit formulae for the reformulated invariants hλ (t, v) and gλ (t, v). Since we shall be dealing with finite collections of all different partitions, it is convenient to introduce the set N[P] of finitely-supported functions P −→ N. If we use elementary functions eλ : P −→ N , μ −→ δλμ each Λ ∈ N[P] can be written as Λ=
nΛ (λ)eλ ,
λ∈P
where nΛ = (nΛ (λ))λ∈Λ is a sequence with finite support. Let also |n| = λ∈P nΛ (λ) and nΛ (λ)|λ|. Λ = λ∈P η We introduce the following combinatoric object: NΛ is defined as η n (λ) chλΛ = NΛ chη . λ∈P
η∈P
Clearly, the above sum is finite and only runs on elements such that |η| = Λ. Because of composite representations, we also need two-variables polynomials N[P, P]. Introducing the elementary functions eλ,μ : P × P −→ N , (α, β) −→ δλα δμβ we can write Λ ∈ N[P, P] as
Λ=
nΛ (λ, μ)eλ,μ .
λ,μ∈P
We define as before Λ =
(nΛ (λ, μ) + nΛ (μ, λ)) |λ|
λ,μ∈P η and NΛ by
λ,μ∈P
(chλ chμ )nΛ (λ,μ) =
η NΛ chη .
η∈P
We write d|λ if d divides |λ|, and we let μ(d) be the M¨ obius function.
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By expanding the logarithm in series, we obtained the following formulae: hλ =
µ(d) d d|λ
×
Nκη1 κ2
κ1 ,κ2 ∈P
η∈P|λ|/d
aλη,d
(−1)|nΛ |+|nΓ |+1 |nΛ | + |nΓ |
|nΛ | + |nΓ | κ1 NΓκ2 Hα (td , v d )nΛ (α) H[β,γ] (td , v d )nΓ (β,γ) NΛ nΛ nΓ
and µ(d) d
odd d|λ
odd d|λ
|nΛ |−1
×2
aλη,d
Λ=|η|
η∈P|λ|/d
µ(d) d
−
2|nΛ |
Λ∈N[P] Γ∈N[P,P] Λ=|κ1 | Γ=|κ2 |
α∈P
gλ =
(−1)|nΛ |−1 |nΛ | η Kα (td , v d )nΛ (α) NΛ |nΛ | nΛ α∈P
aλη,d
η∈P|λ|/d
β,γ∈P
Nκη1 κ2
κ1 ,κ2 ∈P
Λ∈N[P] Γ∈N[P,P] Λ=|κ1 | Γ=|κ2 |
(−1)|nΛ |+|nΓ |+1 |nΛ | + |nΓ |
|nΛ | + |nΓ | κ1 NΓκ2 Hα (td , v d )nΛ (α) H[β,γ] (td , v d )nΓ (β,γ) NΛ nΛ nΓ α∈P
β,γ∈P
Appendix C. Characters of SO(N ) The characters of SO(2r + 1) and SO(2r) can be represented by symmetric −1 polynomials in Z[x1 , . . . , xr , x−1 1 , . . . , xr ], whose explicit expression are given in [7]. They can be expressed as linear combination of Schur functions in 2r variables. The relations are [23] |μ|−r(μ) so(2r+1) λ = (−1) 2 Nμη sη chλ η∈P μ=μ so(2r) chλ
=
λ (−1)|γ|/2 Nγη sη
(C.1)
η∈P γ∈C
and the reciprocals sλ =
λ (−1)|ξ|/2 Nξη chso(2r+1) η
η∈P ξ∈P∪{∅}
sλ =
λ Nδη chso(2r) . η
(C.2)
η∈P δ∈D
In these formulae, μ is the partition conjugate to μ, r(μ) is the rank of μ, C is the set of partitions of the form (b1 + 1, b2 + 1, . . . |b1 , b2 , . . . ) in Frob´enius notation and D is the set of partitions into even parts only. Both sets include the empty partition, and so does the sum over self-conjugate partitions.
References [1] Borhade, P., Ramadevi, P.: SO(N ) reformulated link invariants from topological strings. Nucl. Phys. B 727, 471–498 (2005). arXiv:hep-th/0505008
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[2] Bouchard, V., Florea, B., Mari˜ no, M.: Counting higher genus curves with crosscaps in Calabi-Yau orientifolds. J. High Energy Phys. 12, 35 (2004). arXiv:hepth/0405083 [3] Bouchard, V., Florea, B., Mari˜ no, M.: Topological open string amplitudes on orientifolds. J. High Energy Phys. 2(2) (2005). arXiv:hep-th/0411227 [4] Chen, L., Chen, Q.: Orthogonal quantum group invariants of links (2010). arXiv:1007.1656 [math.QA] [5] Chern, S.-S., Simons, J.: Characteristic forms and geometric invariants. Ann. Math. 99(1), 48–69 (1974) [6] Fuchs, J., Schweigert, C.: Symmetries, Lie algebras and representations. In: Cambridge Monographs on Mathematical Physics. Cambridge University Press (1997) [7] Fulton, W., Harris, J.: Representation Theory: A First Course. Springer, Berlin (1991) [8] Gopakumar, R., Vafa, C.: On the gauge theory/geometry correspondence. Adv. Theor. Math. Phys. 3, 1415–1443 (1999). arXiv:hep-th/9811131 [9] Hadji, R.J., Morton, H.R.: A basis for the full Homfly skein of the annulus. Math. Proc. Camb. Philos. Soc. 141, 81–100 (2006). arXiv:math/0408078 [10] Halverson, T.: Characters of the centralizer algebras of mixed tensor representations of GL(r, C) and the quantum group Uq (gl(r, C)). Pac. J. Math. 174(2), 359–410 (1996). euclid.pjm/1102365176 [11] Isidro, J.M., Labastida, J.M.F., Ramallo, A.V.: Polynomials for torus links from Chern-Simons gauge theories. Nucl. Phys. B 398, 187–236 (1993). arXiv:hepth/9210124 [12] Jones, V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull. Am. Math. Soc. 12, 103–111 (1985). euclid.bams/1183552338 [13] Knapp, A.W.: Lie Groups Beyond an Introduction, 2nd edn. Birkh¨ auser, Basel (2005) [14] Koike, K.: On the decomposition of tensor products of the representations of the classical groups. Adv. Math. 74, 57–86 (1989) [15] Labastida, J.M.F., Llatas, P.M., Ramallo, A.V.: Knot operators in Chern-Simons gauge theory. Nucl. Phys. B 348, 651–692 (1991) [16] Labastida, J.M.F., Mari˜ no, M.: The HOMFLY polynomial for torus links from Chern-Simons gauge theory. Int. J. Mod. Phys. A 10(7), 1045–1089 (1995). arXiv:hep-th/9402093 [17] Labastida, J.M.F., Mari˜ no, M.: Polynomial invariants for torus knots and topological strings. Commun. Math. Phys. 217, 423–449 (2001). arXiv:hep-th/ 0004196 [18] Labastida, J.M.F., Mari˜ no, M.: A new point of view in the theory of knot and link invariants. J. Knot Theory Ramif. 11, 173–197 (2002). arXiv:math/0104180 [19] Labastida, J.M.F., Mari˜ no, M., Vafa, C.: Knots, links and branes at large N. J. High Energy Phys. 11(7), 42 (2000). arXiv:hep-th/0010102 [20] Labastida, J.M.F., P´erez, E.: A relation between the Kauffman and the HOMFLY polynomials for torus knots. J. Math. Phys. 37(4), 2013–2042 (1996). arXiv:q-alg/9507031
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[21] Labastida, J.M.F., Ramallo, A.V.: Operator formalism for Chern-Simons theories. Phys. Lett. B 227, 92 (1989) [22] Lin, X.-S., Zheng, H.: On the Hecke algebras and the colored HOMFLY polynomial (2006). arXiv:math/0601267 [23] Littlewood, D.E.: The Theory of Group Characters. Oxford University Press (1940) [24] Liu, K., Peng, P.: Proof of the Labastida-Mari˜ no-Ooguri-Vafa Conjecture (2007). arXiv:0704.1526 [math.QA] [25] Mari˜ no, M.: String theory and the Kauffman polynomial (2009). arXiv:0904.1088 [hep-th] [26] Morton, H.R., Manch´ on, P.M.G.: Geometrical relations and plethysms in the Homfly skein of the annulus. J. Lond. Math. Soc. 78, 305–328 (2008). arXiv:0707.2851 [math.GT] [27] Ooguri, H., Vafa, C.: Knot invariants and topological strings. Nucl. Phys. B 577, 419–438 (2000). arXiv:hep-th/9912123 [28] Paul, C., Borhade, P., Ramadevi, P.: Composite invariants and unoriented topological string amplitudes (2010). arXiv:1003.5282 [hep-th] [29] Ramadevi, P., Govindarajan, T., Kaul, R.: Three dimensional Chern-Simons theory as a theory of knots and links III: compact semi-simple group. Nucl. Phys. B 402, 548–566 (1993). arXiv:hep-th/9212110 [30] Sinha, S., Vafa, C.: SO and Sp Chern-Simons at large N (2000). arXiv:hep-th/ 0012136 [31] Stevan, S.: Knot operators in Chern-Simons gauge theory. Master’s thesis, University of Geneva (2009) [32] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–399 (1989). euclid.cmp/1104178138 S´ebastien Stevan Section de math´ematiques Universit´e de Gen`eve Case postale 64 1211 Geneva 4 Switzerland e-mail:
[email protected] Communicated by Marcos Marino. Received: June 4, 2010. Accepted: July 20, 2010.
Ann. Henri Poincar´e 11 (2010), 1225–1271 c 2010 Springer Basel AG 1424-0637/10/071225-47 published online December 7, 2010 DOI 10.1007/s00023-010-0063-2
Annales Henri Poincar´ e
On the Construction of a Geometric Invariant Measuring the Deviation from Kerr Data Thomas B¨ackdahl and Juan A. Valiente Kroon Abstract. This article contains a detailed and rigorous proof of the construction of a geometric invariant for initial data sets for the Einstein vacuum field equations. This geometric invariant vanishes if and only if the initial data set corresponds to data for the Kerr spacetime, and thus, it characterises this type of data. The construction presented is valid for boosted and non-boosted initial data sets which are, in a sense, asymptotically Schwarzschildean. As a preliminary step to the construction of the geometric invariant, an analysis of a characterisation of the Kerr spacetime in terms of Killing spinors is carried out. A space spinor split of the (spacetime) Killing spinor equation is performed to obtain a set of three conditions ensuring the existence of a Killing spinor of the development of the initial data set. In order to construct the geometric invariant, we introduce the notion of approximate Killing spinors. These spinors are symmetric valence 2 spinors intrinsic to the initial hypersurface and satisfy a certain second order elliptic equation—the approximate Killing spinor equation. This equation arises as the Euler-Lagrange equation of a non-negative integral functional. This functional constitutes part of our geometric invariant—however, the whole functional does not come from a variational principle. The asymptotic behaviour of solutions to the approximate Killing spinor equation is studied and an existence theorem is presented.
1. Introduction The Kerr spacetime is, undoubtedly, one of the most important exact solutions to the vacuum Einstein field equations [32]. It describes a rotating stationary asymptotically flat black hole parametrized by its mass m and its specific angular momentum a. One of the outstanding challenges of contemporary General Relativity is to obtain a full understanding of the properties and the structure of the Kerr spacetime and of its standing in the space of solutions to the Einstein field equations.
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There are a number of difficult conjectures and partial results concerning the Kerr spacetime. In particular, it is widely expected to be the only rotating stationary asymptotically flat black hole. This conjecture has been proved if the spacetime is assumed to be analytic (C ω )—see, e.g. [14] and references within. Recently, there has been progress in the case where the spacetime is assumed to be only smooth (C ∞ )—see [29]. Moreover, it has been shown that a regular, non-extremal stationary black hole solution of the Einstein vacuum equations which is suitably close to a Kerr solution must be that Kerr solution—i.e. perturbative stability among the class of stationary solutions [1]. Another of the conjectures concerning the Kerr spacetime is that it describes, in some sense, the late time behaviour of a spacetime with dynamical (that is, non-stationary) black holes—this is sometimes known as the establishment point of view of black holes, cfr. [40]. A step in this direction is to obtain a proof of the non-linear stability of the Kerr spacetime—this conjecture roughly states that the Cauchy problem for the vacuum Einstein field equations with initial data for a black hole which is suitably close to initial data for the Kerr spacetime gives rise to a spacetime with the same global structure as Kerr and with suitable pointwise decay. Numerical simulations support the conjectures described in this paragraph. A common feature in the problems mentioned in the previous paragraphs is the need of having a precise formulation of what it means that a certain spacetime is close to the Kerr solution. Due to the coordinate freedom in General Relativity, it is, in general, difficult to measure how much two spacetimes differ from each other. Statements made in a particular choice of coordinates can be deceiving. In the spirit of the geometrical nature of General Relativity, one would like to make statements which are coordinate and gauge independent. Invariant characterisations of spacetimes provide a way of bridging this difficulty. Most analytical and numerical studies of the Einstein field equations make use of a 3+1 decomposition of the equations and the unknowns. In this context, the question of whether a given initial data set for the Einstein field equations corresponds to data for the Kerr spacetime arises naturally—an initial data set will be said to be data for the Kerr spacetime if its development is isometric to a portion (or all) of the Kerr spacetime. A related issue arises when discussing the (either analytical or numerical) 3+1 evolution of a spacetime: do the leaves of the foliation approach, as a result of the evolution, hypersurfaces of the Kerr spacetime? In order to address these issues it is important to have a geometric characterisation of the Kerr solution which is amenable to a 3+1 splitting. A number of invariant characterisations are known in the literature, each with their own advantages and disadvantages. For completeness we discuss some which bear connection to the analysis presented in this article: The Simon and Mars-Simon tensors. A convenient way of studying stationary solutions to the Einstein field equations is through the quotient manifold of the orbits of the stationary Killing vector. The Schwarzschild spacetime is
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characterised among all stationary solutions by the vanishing of the Cotton tensor of the metric of this quotient manifold—see, e.g. [20]. In [43] a suitable generalisation of the Cotton tensor of the quotient manifold was introduced— the Simon tensor. The vanishing of the Simon tensor together with asymptotic flatness and non-vanishing of the mass characterises the Kerr solution in the class of stationary solutions. In [36,37] a spacetime version of the Simon tensor was introduced—the so-called Mars-Simon tensor. The construction of this tensor requires the a priori existence of a Killing vector in the spacetime. Accordingly, it is tailored for the problem of the uniqueness of stationary black holes. The vanishing of the Mars-Simon tensor together with some global conditions (asymptotic flatness, non-zero mass, stationarity of the Killing vector) characterises the Kerr spacetime. Characterisations using concomitants of the Weyl tensor. A concomitant of the Weyl tensor is an object constructed from tensorial operations on the Weyl tensor and its covariant derivatives. An invariant characterisation of the Kerr spacetime in terms of concomitants of the Weyl tensor has been obtained in [19]. This result generalises a similar result for the Schwarzschild spacetime given in [17]. These characterisations consist of a set of conditions on concomitants of the Weyl tensor, which if satisfied, characterise locally the Kerr/Schwarzschild spacetime. An interesting feature of the characterisation is that it provides expressions for the stationary and axial Killing vectors of the spacetime in terms of concomitants of the Weyl tensor. Unfortunately, the concomitants used in the characterisation are complicated, and thus, produce very involved expressions when performing a 3+1 split. Characterisations by means of generalised symmetries. Generalised symmetries (sometimes also known as hidden symmetries) are generalisations of the Killing vector equation—like the Killing tensors and conformal Killing-Yano tensors. These tensors arise naturally in the discussion of the so-called Carter constant of motion and in the separability of various types of linear equations on the Kerr spacetime—see, e.g. [11,31,42]. In particular, the existence of a conformal Killing-Yano tensor is equivalent to the existence of a valence-2 symmetric spinor satisfying the Killing spinor equation. An important property of the Schwarzschild and Kerr spacetimes is that they admit a Killing spinor. This Killing spinor generates, in a certain sense, the Killing vectors and Killing-Yano tensors of the exact solutions in question [27]. Moreover, as it will be discussed in the main part of this article, for a spacetime which is neither conformally flat nor of Petrov type N, the existence of a Killing spinor associated with a Killing-Yano tensor together with the requirement of asymptotic flatness renders a characterisation of the Kerr spacetime. To the best of our knowledge, this property has only been discussed in the literature—without proof—in [18]. Although at first sight independent, the characterisations of the Schwarzschild and Kerr spacetimes described in the previous paragraphs are interconnected—sometimes in very subtle manners. This is not too surprising as all these characterisations make use in a direct or indirect manner of the
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fact that the Kerr spacetime is a vacuum spacetime of Petrov type D—see, e.g. [45] for a discussion of the Petrov classification. The art in producing a useful characterisation of the Kerr spacetime lies in finding further conditions on type D spacetimes which are natural and simple to use. A Characterisation of Kerr Data Characterisations of initial data sets for the Schwarzschild and Kerr spacetimes have been discussed in [21,22,47]. These characterisations make use of a number of local and global ingredients. For example, in [22] it is necessary to assume the existence of a Killing vector on the development of the spacetime. In this article we present a rigorous and detailed discussion of a geometric invariant characterising initial data for the Kerr spacetime. A restricted version of this construction has been presented in [2]. The starting point of our construction is the observation that the existence of a Killing spinor in the Kerr spacetime is a key property. It allows to relate the Killing vectors of the spacetime with its curvature in a neat way. The reason for its importance can be explained in the following way: from a specific Killing spinor it is possible to obtain a Killing vector which in general will be complex. It turns out that for the Kerr spacetime this Killing vector is in fact real and coincides with the stationary Killing vector. It can be shown that the Kerr solution is the only asymptotically flat vacuum spacetime with these properties, if one assumes that there are no points where the Petrov type is either N or O. Given the aforementioned spacetime characterisation of the Kerr solution, the question now is how to make use of it to produce a characterisation in terms of initial data sets. For this, one has to encode the existence of a Killing spinor at the level of the data. The way of doing this was first discussed in [23] and follows the spirit of the well-known discussion of how to encode Killing vectors on initial data—see, e.g. [5]. The conditions on the initial data that ensure the existence of a Killing spinor in its development are called the Killing spinor initial data equations and are, like the Killing initial data equations (KID equations), overdetermined. In [15], a procedure was given on how to construct equations which generalise the KID equations for time symmetric data. These generalised equations have the property that for a particular behaviour at infinity they always admit a solution. If the spacetime admits Killing vectors, then the solutions to the generalised KID equations with the same asymptotic behaviour as the Killing vectors are, in fact, Killing vectors. Therefore, one calls the solutions to the generalised KID equations approximate symmetries. The total number of approximate symmetries is equal to the maximal number of possible Killing vectors on the spacetime. A peculiarity of this procedure is that if the spacetime is not stationary, the approximate Killing vector associated with a time translation does not have the same asymptotic behaviour as a time translation.1 1
Here, and in what follows, for a time translation it is understood a Killing vector which in some asymptotically Cartesian coordinate system has a leading term of the form ∂t .
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The Killing spinor initial data equations consist of three conditions: one of them differential (the spatial Killing spinor equation)2 and two algebraic conditions. Following the spirit of [15] we construct a generalisation of the spatial Killing spinor equation—the approximate Killing spinor equation. This equation is elliptic and of second order. This equation is the Euler-Lagrange equation of an integral functional—the L2 -norm of the exact spatial Killing spinor equation. For this equation it is possible to prove the following theorem: Theorem. For initial data sets to the Einstein field equations with suitable asymptotic behaviour, there exists a solution to the approximate Killing spinor equation with the same asymptotic behaviour as the Killing spinor of the Kerr spacetime. A precise formulation will be given in the main text. In particular, it will be seen that the conditions on the asymptotic behaviour of the initial data are rather mild and amount to requiring the data to be, in a sense, asymptotically Kerr data. Contrasted with the results in [15], this result is notable because, arguably, the most important approximate symmetry of [15] does not share the same asymptotic behaviour as the exact symmetry. The precise version of this theorem generalises the one discussed in [2] in that it allows for boosted data. This generalisation is only possible after a detailed analysis of the asymptotic solutions of the exact Killing spinor equation. The approximate Killing spinor discussed in the previous paragraphs can be used to construct a geometric invariant for the initial data. This invariant is global and involves the L2 norms of the Killing spinor initial data equations evaluated at the approximate Killing vector. It should be observed that only part of the invariant satisfies a variational principle—this is a further difference with respect to the construction of [15]. As the initial data set is assumed to be asymptotically Euclidean, one expects its development to be asymptotically flat. This renders the desired characterisation of Kerr data and our main result. Theorem. Consider an initial data set for the vacuum Einstein field equations whose development in a small slab is neither of Petrov type N nor O at any point, and such that the L2 norm of the Killing spinor initial data equations evaluated at the solution (with the same asymptotic behaviour as the Killing spinor of the Kerr spacetime) to the approximate Killing spinor equation vanishes. Then the initial data set is locally data for the Kerr spacetime. Furthermore, if the 3-manifold has the same topology as that of hypersurfaces of the Kerr spacetime, then the initial data set is data for the Kerr spacetime. There are several advantages of this characterisation over previous ones given in the literature. Most notably, it allows to condense the non-Kerrness of an initial data set in a single number. That this invariant constitutes a 2
The idea of using the spatial part of spinorial equations to characterise slices of particular spacetimes is not new. In [46] the spatial twistor equation has been used to characterise slices of conformally flat spacetimes. See also [7].
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good distance in the space of initial data sets will be discussed elsewhere. Furthermore, the way the invariant is constructed is fully amenable to a numerical implementation—the elliptic solvers that one would need to compute the solution to the approximate Killing spinor equation are, nowadays, standard technology. Detailed Outline of the Article The outline of the article is as follows: in Sect. 2 we study Killing spinors, and their influence on the algebraic type of the spacetime. We relate the Killing spinors to Killing vectors and Killing-Yano tensors. Using these results together with a characterisation of the Kerr spacetime by Mars [37], we conclude that the Kerr spacetime can be characterised in terms of existence of a Killing spinor related to a real Killing vector. This has previously been overlooked in the literature, but it is a key element in our analysis. Section 3 follows with an exposition of space spinors, which will be the main computational tool for the remainder of the paper. Following that, in Sect. 4 we study a 3+1 splitting of the Killing spinor equation. A similar analysis was carried out in [23], but here we manage to condense the result into three simple equations, the spatial Killing spinor equation and two algebraic equations. We also present general equations for the spatial derivatives of a general valence 2 spinor, which is not necessarily a Killing spinor. These equations are also used in later parts of the paper. In Sect. 5 we introduce the new concept of approximate Killing spinors. These are introduced as solutions to an elliptic equation formed by composing the spatial Killing spinor operator with its formal adjoint. That this composed operator is indeed elliptic and formally self adjoint is proved. We also see that the approximate Killing spinor equation can be derived from a variational principle. To get unique solutions to the approximate Killing spinor equation, we need to specify the asymptotic behaviour. For a rigorous treatment of this, we use weighted Sobolev spaces; these are described in Sect. 6. Here, we also study the asymptotics of a Killing spinor on a boosted slice of the Schwarzschild spacetime. In general, we study slices of an arbitrary spacetime with asymptotics similar to those of the Schwarzschild spacetime. Using these assumptions, we can then in Sect. 7 prove existence of spinors with the same asymptotics as the Killing spinor in the Schwarzschild spacetime. We later use these spinors as seeds for solutions to the approximate Killing spinor equation. In this way we get the desired asymptotic behaviour of our approximate Killing spinors. In Sect. 8 we study the approximate Killing spinor equation in our asymptotically Euclidean manifolds to gain existence and uniqueness of solutions with the desired asymptotics. This is done by means of the Fredholm alternative on weighted Sobolev spaces, transforming the existence problem into a study of the kernel of the Killing spinor operator. In this process we get the first part of the geometric invariant—the L2 norm of the approximate Killing spinor. This norm is proved to be finite. The geometric invariant is constructed in Sect. 9, by adding the L2 norms of the algebraic conditions. There follows our
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main theorem: the invariant vanishes if and only if the spacetime is the Kerr spacetime. The invariant is as a consequence of the construction proved to be finite and well defined. We also include two appendices. The first describes an alternative proof of finiteness of a particular boundary integral in Sect. 8. The other contains tensor versions of the invariant—this can be useful in applications. General Notation and Conventions All throughout, (M, gμν ) will be an orientable and time orientable globally hyperbolic vacuum spacetime. It follows that the spacetime admits a spin structure—see [24,25]. Here, and in what follows, μ, ν, · · · denote abstract 4-dimensional tensor indices. The metric gμν will be taken to have signature (+, −, −, −). Let ∇μ denote the Levi-Civita connection of gμν . The sign of the Riemann tensor will be given by the equation ∇μ ∇ν ξζ − ∇ν ∇μ ξζ = Rνμζ η ξη . The triple (S, hab , Kab ) will denote initial data on a hypersurface of the spacetime (M, gμν ). The symmetric tensors hab , Kab will denote, respectively, the 3-metric and the extrinsic curvature of the 3-manifold S. The metric hab will be taken to be negative definite—that is, of signature (−, −, −). The indices a, b, . . . will denote abstract 3-dimensional tensor indices, while i, j, . . . will denote 3-dimensional tensor coordinate indices. Let Da denote the Levi-Civita covariant derivative of hab . Spinors will be used systematically. We follow the conventions of [41]. In particular, A, B, . . . will denote abstract spinorial indices, while A, B, . . . will be indices with respect to a specific frame. Tensors and their spinorial counterparts are related by means of the solder form σμ AA satisfying gμν = σμAA σνBB AB A B , where AB is the antisymmetric spinor and A B its com plex conjugate copy. One has, for example, that ξμ = σμ AA ξAA . Let ∇AA denote the spinorial counterpart of the spacetime connection ∇μ . Besides the connection ∇AA , two other spinorial connections will be used: DAB , the spinorial counterpart of the Levi-Civita connection Da and ∇AB , the Sen connection of (M, gμν )—full details will be given in Sect. 3. The Kerr spacetime. For the Kerr spacetime the maximal analytic extension of the Kerr metric as described by Boyer and Lindquist [8] and Carter [10] will be understood. When regarding the Kerr spacetime as the development of Cauchy initial data, we will only consider its maximal globally hyperbolic development.
2. Killing Spinors: General Theory As mentioned in the introduction, our point of departure will be a characterisation of the Kerr spacetime based on the existence in the spacetime of a valence-2 symmetric spinor satisfying the Killing spinor equation. To the best of our knowledge, this characterisation of the Kerr spacetime has not explicitly
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been discussed in the literature, save for a side remark in [18]. In this section we provide a summary of this characterisation and fill in some technical details. 2.1. Killing Spinors and Petrov Type D Spacetimes A valence-2 Killing spinor is a symmetric spinor κAB = κ(AB) satisfying the equation ∇A (A κBC) = 0.
(1)
Killing spinors offer a way of relating properties of the curvature to properties of the symmetries of the spacetime. Taking a further derivative of equation (1), antisymmetrising and commuting the covariant derivatives one finds the integrability condition Ψ(ABC F κD)F = 0,
(2)
where ΨABCD denotes the self-dual Weyl spinor. The above integrability imposes strong restrictions on the algebraic type of the Weyl spinor. More precisely, it follows that if ΨABCD = 0 and κAB = 0, then ΨABCD = ψκ(AB κCD) ,
(3)
where ψ is a scalar. Thus, ΨABCD must be of Petrov type D or N—see, e.g. [23,30]. The converse is also true [28,42,48]. Summarising: Theorem 1 (Walker and Penrose [48]). A vacuum spacetime admits a valence-2 Killing spinor if and only if it is of Petrov type D, N or O. From (3) it can also be seen that ΨABCD is of Petrov type N if and only if κAB is algebraically special. That is, there exists a spinor αA such that κAB = αA αB . Thus, an algebraically general Killing spinor κAB = α(A βB) is always associated with a vacuum spacetime of Petrov type D. 2.2. The Killing Vector Associated with a Killing Spinor and the Generalised Kerr-NUT Metrics Given a Killing spinor κAB , the concomitant ξAA = ∇B A κAB ,
(4)
is a complex Killing vector of the spacetime: its real and imaginary parts are themselves Killing vectors of the spacetime [27]. In relation to this it should be pointed out that all vacuum Petrov type D spacetimes are known [33]. It follows from the analysis in the latter reference that all vacuum, Petrov type D spacetimes have a commuting pair of Killing vectors. A key property of the Kerr spacetime is the following (cfr. [27,42]): Proposition 2. Let (M, gμν ) be a vacuum Petrov type D spacetime. The Killing vector ξAA given by (4) is real in the case of the Kerr spacetime. Remark 1. In what follows, the class of Petrov type D spacetimes for which ξAA is real will be called the generalised Kerr-NUT class—cfr. [18]. This
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class can be alternatively characterised—see, e.g. [31]—by the existence of a Killing-Yano tensor Yμν = Y[μν] ,
∇(μ Yν)λ = 0.
The correspondence between the Killing spinor κAB and the spinorial counterpart YAA BB of the Killing-Yano tensor, Yμν , is given by ¯ A B ), YAA BB ≡ i (κAB A B − AB κ where the overbar denotes the complex conjugate. Remark 2. In terms of the Kinnersley list of type D metrics, the class of generalised Kerr-NUT metrics contains, in addition to the proper Kerr-NUT metrics (II.C), also the metrics II.E—see, [16]. An important property of the generalised Kerr-NUT metrics involves the Killing form, FAA BB = −FBB AA , of a real Killing vector ξAA defined by 1 FAA BB ≡ (∇AA ξBB − ∇BB ξAA ). (5) 2 Let 1 ∗ (6) FAA BB ≡ (FAA BB + iFAA BB ) 2 ∗ denote the corresponding self-dual Killing form, with FAA BB the Hodge dual of FAA BB . Due to the symmetries of the Killing form one can write FAA BB = FAB A B ,
(7)
with FAB ≡
1 FAQ B Q = FBA . 2
(8)
One has the following result: Lemma 3. For generalised Kerr-NUT spacetimes one has that FAB = κκAB , where κ is a non-vanishing scalar function, so that the principal spinors of FAB and ΨABCD are parallel. Equivalently, one has that ΨABP Q F P Q = ϕFAB , with ϕ a non-vanishing scalar. Proof. One proceeds by a direct computation. One notes that the expressions (5), (6) and (8) assume that the Killing vector ξAA is real. Using Eqs. (4) and (8) and the vacuum commutators for ∇AA one finds that 3 FAB = ΨABP Q κP Q . 4 As the spacetime is assumed to be of Petrov type D one has that κAB = α(A βB) with αA β A = ς, where ς is a non-vanishing scalar. From Eq. (3) one finds then that ΨABCD = ψα(A αB βC βD) , so that 1 ΨABP Q κP Q = − ψς 2 κAB , 3
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and finally that 1 FAB = − ψς 2 κAB , 4 from where the desired result follows.
The property that allows us to single out the Kerr spacetime out of the generalised Kerr-NUT class is given by the following result proved by Mars [36,37]: Theorem 4 (Mars [36,37]). Let (M, gμν ) be a smooth vacuum spacetime with the following properties: (i) (M, gμν ) admits a Killing vector ξAA such that, FAB , the spinorial counterpart of the Killing form of ξAA satisfies ΨABP Q F P Q = ϕFAB , with ϕ a scalar; (ii) (M, gμν ) contains a stationary asymptotically flat 4-end, and ξAA tends to a time translation at infinity, and the Komar mass of the asymptotic end is non-zero. Then (M, gμν ) is locally isometric to the Kerr spacetime. Remark. A stationary asymptotically flat 4-end is an open submanifold M∞ ⊂ M diffeomorphic to I × (R3 \ BR ), where I ⊂ R is an open interval and BR a closed ball of radius R such that in the local coordinates (t, xi ) defined by the diffeomorphism, the metric gμν satisfies |gμν − ημν | + |r∂i gμν | ≤ Cr−α , ∂t gμν = 0, with C, α constants, ημν is the Minkowski metric and r = 1 2 2 2 3 2 (x ) + (x ) + (x ) . In particular α ≥ 1. The definition of the Komar mass is given in [34]. In this context it coincides with the ADM mass of the spacetime. 2.3. Non-Degeneracy of the Petrov Type of the Kerr Spacetime Finally, we note the following result about the non-degeneracy of the Petrov type of the Kerr spacetime [37]. Proposition 5 (Mars [37]). The Petrov type of the Kerr spacetime is always D—there are no points where it degenerates to type N or O. 2.4. A Characterisation of the Kerr Spacetime Using Killing Spinors As a consequence of Theorem 1 and Propositions 2, 5 one obtains the following invariant characterisation of the Kerr spacetimes. From this characterisation we will extract, in the sequel, a characterisation of asymptotically Euclidean Kerr data.
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Theorem 6. Let (M, gμν ) be a smooth vacuum spacetime such that ΨABCD = 0,
ΨABCD ΨABCD = 0
on M. Then (M, gμν ) is locally isometric to the Kerr spacetime if and only if the following conditions are satisfied: (i) (ii)
there exists a Killing spinor, κAB , such that the associated Killing vector, ξAA , is real; the spacetime (M, gμν ) has a stationary asymptotically flat 4-end with non-vanishing mass in which ξAA tends to a time translation.
Proof. Clearly, the conditions (i) and (ii) are necessary to obtain the Kerr spacetime. For the sufficiency, assume that (i) holds, that is, the spacetime has a Killing spinor κAB such that the associated Killing vector ξAA is real. Accordingly, the spacetime must be of type D, N or O. As ΨABCD = 0 and ΨABCD ΨABCD = 0 by hypothesis, the spacetime cannot be of types N or O. By the reality of ξAA it must be a generalised Kerr-NUT spacetime and the conclusion of Lemma 3 follows. Now, if (ii) holds then by Theorem 4, the spacetime has to be locally the Kerr spacetime. Remark. It is of interest to see whether the conditions ΨABCD = 0 and ΨABCD ΨABCD = 0 can be removed. An analysis along what is done in the proof of Theorem 4 may allow to do this. This will be discussed elsewhere.
3. Space Spinors: General Theory As mentioned in the introduction, in this article we will make use of a space spinor formalism to project the longitudinal and transversal parts of the Killing spinor equation (1) with respect to the timelike vector field τ μ . The space spinor formalism was originally introduced in [44]. Here, we follow conventions and notations similar to those in [23]. For completeness, we introduce all the relevant notation here. 3.1. Basic Definitions Let τ μ be a timelike vector field on (M, gμν ) with normalisation τμ τ μ = 2. Define the projector 1 hμν ≡ gμν − τμ τν . 2 We also define the following tensors: Kμν = −hμ λ hν ρ ∇λ τρ , 1 K μ = − τ ν ∇ν τ μ . 2 Note that it is not being assumed that τ μ is hypersurface orthogonal. Thus, the tensor Kμν as defined above is not necessarily the second fundamental form of a foliation of the spacetime (M, gμν ).
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Let τ AA denote the spinorial counterpart of τ μ . One has that τ AA ≡ AA μ σμ τ so that
τAA τ AA = 2,
τ A A τ BA = AB .
The spinor τ AA allows to introduce the spatial solder forms
σμ AB ≡ σμ (A A τ B)A ,
σ μ AB ≡ τ(B A σ μ A)A ,
so that one has σ μ AB σν AB = hμ ν ,
μ ν ν gμν σ μ AB σCD = hμν σAB σCD =
1 (AC BD + AD BC ), 2
1 τAA τBB + hμν σ μ AE σ ν BF τ E A τ F B . 2 If τ μ is hypersurface orthogonal, then hab , Kab , K a , σa AB , σ a AB denote, respectively, the pull-backs to the hypersurfaces orthogonal to τ μ of hμν , Kμν , K μ , σμ AB , σ μ AB —note that these objects are spatial, in the sense that their contraction with τ μ vanishes, and thus, their pull-backs are well defined. The relevant properties of these tensors apply to their pull-backs. Often we will begin with a space-like hypersurface S, and define τ μ as the normal to this hypersurface; we then automatically get the desired properties. τμ σ μ AB = 0,
AB A B =
3.2. Space Spinor Splittings
The spinor τ AA can be used to construct a formalism consisting of unprimed indices. For example, given a spacetime spinor ζAA one can write 1 ζAA = τAA ζ − τA P ζP A , (9) 2 with
ζ ≡ τ P P ζP P ,
ζAB ≡ τ(A P ζB)P .
This decomposition can be extended in a direct manner to higher valence spinors. Any spatial tensor has a space-spinor counterpart. For example, if Tμ ν is a spatial tensor (i.e. τ μ Tμ ν = 0 and τν Tμ ν = 0), then its space spinor counterpart is given by TAB CD = σ μ AB σν CD Tμ ν . 3.3. Spinorial Covariant Derivatives Applying formally the space spinor split given by (9) to the spacetime spinorial covariant derivative ∇AA one obtains 1 ∇AA = τAA ∇ − τA B ∇AB , 2 where we have introduced the differential operators
∇ ≡ τ AA ∇AA , ∇AB ≡ τ A
(A ∇B)A
= σ μ AB ∇μ .
The latter is referred to as the Sen connection. Let KABCD denote the space spinor counterpart of the tensor Kμν . One has that
KABCD = τD C ∇AB τCC ,
KABCD = K(AB)(CD) .
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In the sequel, it will be convenient to write KABCD in terms of its irreducible components. For this, define ΩABCD ≡ K(ABCD) ,
ΩAB ≡ K(A C B)C ,
K ≡ K ABAB ,
so that one can write 1 1 1 KABCD = ΩABCD − A(C ΩD)B − B(C ΩD)A − A(C D)B K, 2 2 3
(10)
If τ μ is hypersurface orthogonal, then ΩAB = 0, and thus Kμν can be regarded as the extrinsic curvature of the leaves of a foliation of the spacetime (M, gμν ). Let KAB denote the spinorial counterpart of the acceleration Kμ . It has the symmetry KAB = K(AB) and satisfies
KAB = τB A ∇τAA . If τ μ is hypersurface orthogonal then the pull-back, Da , of Dμ ≡ hν μ ∇ν corresponds to the Levi-Civita connection of the intrinsic metric of the leaves of the foliation of hypersurfaces orthogonal to τ μ . Its spinorial counterpart is given by DAB = D(AB) = σ a AB Da . The Sen connection, ∇AB , and the LeviCivita connection, DAB , are related to each other through the spinor KABCD . For example, for a valence 1 spinor πC one has that 1 ∇AB πC = DAB πC + KABC D πD , 2 with the obvious generalisations for higher valence spinors. 3.4. Hermitian Conjugation Given a spinor πA , we define its Hermitian conjugate via
¯E . π ˆ A ≡ τA E π The Hermitian conjugate can be extended to higher valence symmetric spinors in the obvious way. The spinors νAB and ξABCD are said to be real if νˆAB = −νAB ,
ξˆABCD = ξABCD .
It can be verified that νAB νˆAB , ξABCD ξˆABCD ≥ 0. If the spinors are real, then there exist real spatial tensors νa , ξab such that νAB and ξABCD are their spinorial counterparts. Notice that the differential operator DAB is real in the sense that D ˆC . AB πC = −DAB π Crucially, however, one has that 1 ∇ ˆC + KABC D π ˆD . AB πC = −∇AB π 2
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3.5. Commutators The analysis in the sequel will require intensive use of the commutators of the covariant derivative operators ∇ and ∇AB . These can be derived from a space spinor splitting of the commutator of ∇AA . Define AB ≡ τA A τB B A B = τA A τB B ∇C(A ∇B ) C . AB ≡ ∇C (A ∇B) C , The action of these operators on a spinor πA is given by 1 AB πC = τA A τB B ΦF CA B π F , AB πC = ΨABCQ π Q + ΛC(A πB) , 2 where ΦABA B and Λ denote, respectively, the spinor counterparts of the tracefree part of the Ricci tensor Rμν and the Ricci scalar R of the spacetime metric gμν . Clearly, the above expressions simplify in the case of a vacuum spacetime, where we have ΦABA B = 0, Λ = 0. AB , the commutators of ∇ and ∇AB read In terms of AB and AB − AB − 1 KAB ∇ + K D (A ∇B)D − KABCD ∇CD , (11a) [∇, ∇AB ] = 2 1 1 D)B + B(C D)A A(C D)B + B(C D)A + A(C [∇AB , ∇CD ] = 2 2 1 + (KCDAB ∇ − KABCD ∇) + KCDQ(A ∇B) Q − KABQ(C ∇D) Q . 2 (11b) 3.6. Decomposition of the Weyl Spinor The Hermitian conjugation can be used to decompose the Weyl spinor ΨABCD in terms of its electric and magnetic parts via 1 ˆ ABCD − ΨABCD , ˆ ABCD , BABCD ≡ i Ψ ΨABCD + Ψ EABCD ≡ 2 2 so that ΨABCD = EABCD + iBABCD .
The spinorial Bianchi identity ∇AA ΨABCD = 0 can be split using the space spinor formalism to render ∇ΨABCD = 2∇E A ΨBCDE , ∇
AB
ΨABCD = 0.
(12a) (12b)
Crucial for our applications is that the spinors EABCD and BABCD can be expressed in terms of quantities intrinsic to a hypersurface S. More precisely, if ΩAB = 0, one has that 1 1 EABCD = −r(ABCD) + Ω(AB P Q ΩCD)P Q − ΩABCD K, (13a) 2 6 (13b) BABCD = −i DQ (A ΩBCD)Q , where rABCD is the space spinor counterpart of the Ricci tensor of the intrinsic metric of the hypersurface S.
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3.7. Space Spinor Expressions in Cartesian Coordinates In some occasions it will be necessary to give spinorial expressions in terms of Cartesian or asymptotically Cartesian frames and coordinates. For this we make use of the spatial Infeld-van der Waerden symbols σ i AB , σi AB . Given xi , ξi ∈ R3 we shall follow the convention that xAB ≡ σi AB xi , with AB
x
1 =√ 2
−x1 + ix2 x3
x3 x1 + ix2
ξAB ≡ σ i AB ξi ,
,
ξAB
1 =√ 2
−ξ1 − iξ2 ξ3
ξ3 ξ1 − iξ2
. (14)
4. Killing Spinor Data In this section we review some aspects of the space spinor decomposition of the Killing spinor equation (1). A first analysis along these lines was first carried out in [23]. The current presentation is geared towards the construction of geometric invariants. 4.1. General Observations Given a symmetric spinor κAB (not necessarily a Killing spinor), it will be convenient to define the following spinors: ξ ≡ ∇P Q κP Q , 3 ξBF ≡ ∇(F D κB)D , 2 ξABCD ≡ ∇(AB κCD) ,
(15b)
ξAA ≡ ∇ A κAB , HA ABC ≡ 3∇A (A κBC) ,
(15d) (15e)
SAA BB ≡ ∇AA ξBB + ∇BB ξAA .
(15f)
B
(15a)
(15c)
We will use this notation throughout the rest of the paper. Clearly, for a Killing spinor one has HA ABC = 0,
SAA BB = 0.
The spinors ξ, ξAB and ξABCD arise in the space spinor decomposition of the spinors HA ABC and ξAA . To see this, let τ AA denote, as in Sect. 3, the spi norial counterpart of a timelike vector with normalisation τAA τ AA = 2. Some manipulations show that 1 2 1 ξAA = τAA ξ − τ B A ξAB + τ B A ∇κAB , (16a) 2 3 2 3 HA ABC = τA (A ξBC) + τA (A ∇κBC) − 3τA D ξABCD . (16b) 2 Furthermore, the spinors ξ, ξAB and ξABCD correspond to the irreducible components of ∇AB κCD so that one can write:
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1 1 1 ∇AB κCD = ξABCD − A(C ξD)B − B(C ξD)A − A(C D)B ξ. (17) 3 3 3 Using the commutator (11b) for vacuum, one can obtain equations for the derivatives of ξ and ξAB —these will be used systematically in the sequel. The irreducible components of the derivative ∇AB ξCD are given by 3 1 1 3 ∇AB ξAB = − Kξ + ΩABCD ξABCD + ΩAB ξAB − ΩAB ∇κAB , (18a) 2 4 2 4 3 2 1 C CD ∇ (A ξB)C = ∇AB ξ + ΨABCD κ − KξAB − ΩABCD ξ CD 2 3 2 3 3 CD 1 1 CDF − ξ(A ΩB)CDF − ∇ ξABCD − ΩAB ξ + Ω(A C ξB)C 2 2 2 2 3 CD 3 C + Ω ξABCD − Ω(A ∇κB)C , (18b) 4 2 1 ∇(AB ξCD) = 3ΨF (ABC κD) F + KξABCD − ΩABCD ξ + Ω(ABC F ξD)F 2 3 PQ 1 Q − Ω (AB ξCD)P Q + 3∇ (A ξBCD)Q + Ω(AB ξCD) 2 2 3 F 3 − Ω (A ξBCD)F + Ω(AB ∇κCD) . (18c) 2 2 We note the appearance of the term ∇AB ξ in (18b). Thus, there is no independent equation for the derivative of ξ. Finally, we consider the equations for the second-order derivatives of ξ. For the sake of simplicity, we restrict our attention to the case when ΩAB = 0 so that KABCD = KCDAB . For notational purposes we define ΩABCDEF ≡ ∇(AB ΩCDEF ) . One finds ∇AB ∇AB ξ 1 1 = − K 2 ξ − ΩABCD ΩABCD ξ + 3ΨA CDF ΩBCDF κAB + ξAB ∇AB K 6 2 3 ˆ ABCD 9 + Ψ ξABCD − ΨABCD ξABCD + 2KΩABCD ξABCD 4 4 15 9 − ΩABF H ΩCD F H ξABCD + ΩABCD ∇F D ξABCF 4 2 3 + ∇AB ∇CD ξABCD , (19a) 2 1 1 ∇C (A ∇B)C ξ = ΩABCD ∇CD ξ − K∇AB ξ, (19b) 2 3 ∇(AB ∇CD) ξ 1ˆ 5 2ˆ E = −4KΨ(ABC E κD)E + Ψ ABCD ξ − ΨABCD ξ − Ψ(ABC ξD)E 2 2 3 10 4 − Ψ(ABC E ξD)E + ΩABCDEL ξ EL + K 2 ξABCD + 3ΩEF L(ABC ξD) ELF 3 3 3 EL ELF H +3Ψ(AB ξCD)EL − ξ(A ΩBCD) ΩELF H − 3ΨEL(A F κEL ΩBCD)F 2 2 1 −ξ EL ΩELF (A ΩBCD) F + Kξ(A E ΩBCD)E + ξ ELF H ΩEL(AB ΩCD)F H 3 2
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−3ΨE(B LF κA E ΩCD)LF − 3ΨE(AB F κEL ΩCD)LF − ΩELF (B ξA E ΩCD) LF 1 3 −4Kξ(AB EL ΩCD)EL − ξΩ(AB EL ΩCD)EL + ξ ELF H ΩE(ABC ΩD)LF H 2 2 1 ELF H 1 H ELF −2ΩE(BC ξA ΩD)LF H + ξ ΩABCD ΩELF H − KξΩABCD 4 3 1 2 12 EL FH + ξ(AB ΩCD) ΩELF H + ξ(CD ∇AB) K + ξE(BCD ∇A) E K 2 5 5 3 3 E ELF −3ΩE(BCD ∇A) ξ − Ω(A ∇CD ξB)ELF − ΩF (A EL ∇D F ξBC)EL 2 2 9 9 3 EL L F E − Ω(AB ∇D ξC)ELF − ∇L(D ∇C ξAB)E − ∇L(D ∇EL ξABC)E 2 2 2 −6K∇E(D ξABC) E + 3ΩL(AB E ∇LF ξCD)EF − 3Ω(ABC E ∇LF ξD)ELF −3κEL ∇L(D ΨABC)E + 3κ(A E ∇D L ΨBC)EL .
(19c)
The equations presented in this section have been deduced using the tensor algebra suite xAct for Mathematica—see [38]. 4.2. Propagation of the Killing Spinor Equation A straightforward consequence of the Killing spinor equation (1) in a vacuum spacetime is that κAB = −ΨABCD κCD ,
(20)
where ≡ ∇AA ∇AA . The latter equation is obtained by applying the dif ferential operator ∇AA to Eq. (1) and then using the vacuum commutator relation for the spacetime Levi-Civita connection. The wave equation (20) plays a role in the discussion of the propagation of the Killing spinor equation. More precisely, one has the following result—cfr. [23] for further details: Lemma 7. Let κAB be a solution to Eq. (20). Then the corresponding spinor fields HA ABC and SAA BB will satisfy the system of wave equations (21a) HA ABC = 4 Ψ(AB P Q HC)P QA + ∇(A Q SBC)Q A , P QR P QR SAA BB = −∇AA ΨB HB P QR − ∇BB ΨA HA P QR P Q PQ ¯ +2ΨAB SP A QB + 2ΨA B SAP BQ . (21b) The crucial observation is that the right-hand sides of Eqs. (21a) and (21b) are homogeneous expressions of the unknowns and their first-order derivatives. The hyperbolicity of Eqs. (21a) and (21b) imply the following result— again, cfr. [23] for further details: Proposition 8. The development (M, gμν ) of an initial data set for the vacuum Einstein field equations, (S, hab , Kab ), has a Killing spinor in the domain of dependence of U ⊂ S if and only if the following equations are satisfied on U:
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HA ABC = 0,
(22a)
∇HA ABC = 0, SAA BB = 0,
(22b) (22c)
∇SAA BB = 0.
(22d)
4.3. The Killing Spinor Data Equations The Killing spinor data conditions obtained in Proposition 8 can be re-expressed in terms of conditions on the spinor κAB which are intrinsic to the hypersurface S. For this one uses the split of ξAA and HA ABC given by Eqs. (16a)–(16b). Extensive computations using the xAct suite for Mathematica render the following result: Theorem 9. Let (S, hab , Kab ) be an initial data set for the Einstein vacuum field equations, where S is a Cauchy hypersurface. Let U ⊂ S be an open set. The development of the initial data set will then have a Killing spinor in the domain of dependence of U if and only if ξABCD = 0,
(23a)
Ψ(ABC F κD)F = 0,
(23b)
3κ(A E ∇B F ΨCD)EF + Ψ(ABC F ξD)F = 0,
(23c)
are satisfied on U. The Killing spinor is obtained by evolving (20) with initial data satisfying conditions (23a)–(23c) and 2 ∇κAB = − ξAB 3
(24)
on U. Remark 1. Conditions (23a)–(23c) are intrinsic to U ⊂ S and will be referred to as the Killing spinor initial data equations. In particular, Eq. (23a), which can be written as ∇(AB κCD) = 0,
(25)
will be called the spatial Killing spinor equation, whereas (23b) and (23c) will be known as the algebraic conditions. Remark 2. Theorem 9 is an improvement on Proposition 6 of [23] where the interdependence of the equations implied by (22a)–(22d) was not analysed. Proof. The proof of Theorem 9 consists of a space spinor decomposition of the conditions (22a)–(22d) and of an analysis of the dependencies of the resulting conditions. All calculations are made on U ⊂ S. •
Decomposition of equation (22a). Splitting τF A HA ABC into irreducible parts gives that (22a) is equivalent to ξABCD = 0, ∇κAB
2 = − ξAB . 3
(26a) (26b)
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Decomposition of equation (22b). It follows that
τD A ∇HA ABC = ∇(τD A HA ABC ) + HA ABC KDF τ F A .
Hence, under the condition (22a), the irreducible parts of τD A ∇HA ABC are given by ∇ξABCD = 0, 2 ∇2 κAB = − ∇ξAB . 3
(27a) (27b)
From the commutator (11a) together with (26a) and (26b) we get ∇ξABCD = ∇∇(AB κCD) 1 1 = 2Ψ(ABC F κD)F − Ω(AB ξCD) − ΩABCD ξ 3 3 2 2 + Ω(ABC F ξD)F − ∇(AB ξCD) . 3 3 Equation (18c) and again (26a) and (26b) then yield ∇ξABCD = 4Ψ(ABC F κD)F .
(28)
Using the commutator (11a) one obtains that 1 2 2 Kξ + K AB ξAB + ΩAB ξAB 3 3 3 1 AB ABCD −Ω ξABCD − K ∇κAB (29a) 2 3 1 1 1 = ΨABCD κCD − KAB ξ − KξAB + K C (A ξB)C 2 2 3 2 3 CD 1 + K ξABCD − ξΩAB 4 2 1 C 3 CD 3 1 − ξ (A ΩB)C + Ω ξABCD + ξ(A CDF ΩB)CDF + ξ CD ΩABCD 2 4 2 2 3 − K C (A ∇κB)C + ∇C(A ∇κB) C (29b) 4
∇ξ = ∇AB ∇κAB −
∇ξAB
In terms of the normal derivative and the Sen connection, Eq. (20) reads ∇2 κAB = −2ΨABCD κCD − K∇κAB 2 4 − ∇AB ξ − ∇C(A ξ C B) − 2∇CD ξABCD 3 3 1 2 2 + KAB ξ − K C (A ξB)C + K CD ξABCD + ΩAB ξ 3 3 3 4 C CD + ξ (A ΩB)C + 2ξABCD Ω . 3
(30)
It is worth stressing that Eqs. (29a), (29b) and (30) are valid not only on U, but on the spacetime. Hence, it makes sense taking normal derivatives of these equations. Using (29b), (26b) and (26a), the wave equation (20) is seen to imply
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2 4 1 ∇2 κAB + ∇ξAB = −ΨABCD κCD + KξAB + ΩAB ξ + ξ C (A ΩB)C 3 9 3 1 2 2 + ΩABCD ξ CD − ∇AB ξ − ∇C(A ξ C B) . 3 3 3
•
Using Eqs. (18b), (26b), (26a), the latter equation reduces to (27b). This far we have that for all solutions to (20), the system (22a), (22b) is equivalent to the system (23a), (23b), (24). Decomposition of equation (22c). Splitting τC A τD B SAA BB into irreducible parts yields 4 ∇(AB ∇κCD) − ΩABCD ξ + K(ABC F ξD)F 3 4 −K(ABC F ∇κD)F − ∇(AB ξCD) = 0, 3 4 AB 2∇ξ − K ξAB + K AB ∇κAB = 0, 3 4 4 ∇AB ξ AB + Kξ − ΩAB ξAB + ΩAB ∇κAB − ∇AB ∇κAB = 0, 3 3 2 A 1 2 1 KBD ξ − K (B ξD)A + K A (B ∇κD)A − KBDAC ξ AC 2 3 2 3 1 2 1 + KBDAC ∇κAC + ∇ξBD − ∇2 κBD + ∇BD ξ = 0. 2 3 2
(31a) (31b) (31c)
(31d)
Using Eqs. (18a), (18c), (26a), (26b) and (27b), one sees that Eqs. (31a)– (31c) simplify to ΨF (ABC κD)F = 0,
(32a)
∇ξ = K AB ξAB ,
(32b)
1 ∇ξBD = − KBD ξ + K A (B ξD)A + KBDAC ξ AC − ∇BD ξ, (32c) 2
•
while Eq. (31d) is seen to be satisfied identically. Furthermore, employing equations (18a), (18b), (29a), (26a), (26b) and (29b) one obtains Eq. (32b) and (32c). Hence, they are a consequence of the commutators, (26b) and (26a). One concludes that for all solutions to (20), the Eqs. (23a), (23b) together with (24) are equivalent to (22a), (22b), (22c). Decomposition of equation (22d). A straightforward computation shows that
τC A τD B ∇SAA BB = ∇(τC A τD B SAA BB )
+KCF SAA BB τD B τ F A +KDF SAA BB τC A τ F B .
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Hence, if condition (22c) holds, the irreducible parts of τC A τD B ∇SAA BB are ∇-derivatives of (31a)–(31d). Using Eq. (27b), these components become 2 ΩABCD ∇ξ + Ω(AB ∇ξCD) − 2ΩF (ABC ∇ξD)F + ξ(AB ∇ΩCD) 3 1 4 − ∇κ(AB ∇ΩCD) + ξ∇ΩABCD + ξ F (A ∇ΩBCD)F 2 3 4 −(∇κF (A )∇ΩBCD)F + ∇∇(AB ξCD) − ∇∇(AB ∇κCD) = 0, (33a) 3 4 2∇2 ξ − 2K AB ∇ξAB + (∇K AB )∇κAB − ξ AB ∇KAB = 0, (33b) 3 4 ξ∇K + K∇ξ − 2ΩAB ∇ξAB − ξ AB ∇ΩAB + (∇κAB )∇ΩAB 3 4 + ∇∇AB ξ AB − ∇∇AB ∇κAB = 0, (33c) 3 2 4 4 ∇3 κBD + ∇2 ξBD = ξ A (B ∇κD)A + ξ∇κBD − ξ AC ∇KBDAC 3 3 3 +(∇K A (B )∇κD)A + (∇KBDAC )∇κAC + KBD ∇ξ − 2K A (B ∇ξD)A −2KBDAC ∇AC + 2∇2 ξBD + 2∇∇BD ξ.
(33d)
Now, using the commutator (11a), and Eqs. (27b) and (26b) it is easy so see that 2 ∇∇AB ∇κCD = − ∇∇AB ξCD . (34) 3 Taking the normal derivative of the spacetime equations (29a)–(29b) and using the relations (34), (18a), (18b), (26a), (26b), (27a) and (27b) one gets ∇2 ξ = ξ AB ∇KAB + K AB ∇ξAB , 1 1 1 ∇2 ξAB = − ξ∇KAB − ξ C (A ∇KB)C + ξAB ∇K − KAB ∇ξ 2 3 2 1 + K∇ξAB + K C (A ∇ξB)C − ΩC (A ∇ξB)C + ΩABCD ∇ξ CD 3 + ξ C (A ∇ΩB)C + ξ CD ∇ΩABCD − ∇∇AB ξ. Using these last two equations together with Eqs. (18a), (18c), (26a), (26b), (27a), (27b) and (32b) one finds that the system (33a)–(33d) reduces to 4ΨF (ABC ξD)F + 6κF (A ∇ΨBCD)F = 0, (35a) 2 (35b) ∇3 κBD + ∇2 ξBD = 0. 3 Taking the normal derivative of equation (30) and using Eqs. (18b), (26a), (26b), (27a), (27b) and (32b) one gets Eq. (35b). Finally, using the Bianchi equation (12a), one has that Eq. (35a) reduces to 3κ(A E ∇B F ΨCD)EF + Ψ(ABC F ξD)F = 0 This completes the proof.
(36)
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Remark. Note that the result is independent of KAB and ΩAB . 4.3.1. The Killing Spinor Initial Data Conditions in Terms of the Levi-Civita Connection. It should be stressed that the Killing spinor equations (23a)– (23c) are truly intrinsic to the hypersurface S. This can be more easily seen by expressing the Sen connection, ∇AB , in terms of the intrinsic (Levi-Civita) connection of the hypersurface, DAB , and the second fundamental form KABCD . One obtains the following completely equivalent set of equations: D(AB κCD) + Ω(ABC E κD)E = 0, Ψ(ABC F κD)F = 0, 3 3 3κ(A E DB F ΨCD)EF − ΨL(ABC DHL κD)H − ΨL(ABC DD) F κL F 4 4 3 3 3 + Ψ(ABC L ΩD)F HL κF H + Ψ(AB HL κC F ΩD)F HL − ΨF H(A L ΩBCD)L κF H 4 2 2 3 3 + ΨF H(AB κCD) ΩF H + ΨF H(AB ΩCD) κF H = 0, 8 4 where the last expression was simplified using the first algebraic condition, and the value of the Weyl spinor is expressed in terms of initial data quantities via formulae (13a)–(13b). 4.4. The Integrability Conditions of the Spatial Killing Spinor Equation For the rest of the paper we assume that the tensor Kab is symmetric—accordingly, ΩAB = 0. The condition ξABCD ≡ ∇(AB κCD) = 0 does not immediately give information about the other irreducible components of ∇AB κCD , namely ξ and ξAB . However, using ξABCD = 0 and ΩAB = 0 in the relations (18a)– (18c) one finds that ∇AB ξCD can be written in terms of ∇AB ξ and lower order derivatives of κAB . Furthermore, using ξABCD = 0 in the relations (19a)–(19c), we see that the second-order derivatives of ξ can be expressed in terms of lower order derivatives of κAB . This yields the following result which will play a role in the sequel: Lemma 10. Assume that ∇(AB κCD) = 0, then ∇AB ∇CD ∇EF κGH = HABCDEF GH , where HABCDEF GH is a linear combination of κAB , ∇AB κCD and ∇AB ˆ ABCD and KABCD . ∇CD κEF with coefficients depending on ΨABCD , Ψ Remark. It is important to point out that the assertion of the lemma is false if ∇(AB κCD) = 0.
5. The Approximate Killing Spinor Equation In what follows we will regard the spatial Killing spinor equation (23a) as the key condition of the Killing spinor initial data equations. Equation (23a) is an overdetermined condition for the 3 (complex) components of the spinor κAB : not every initial data set (S, hab , Kab ) admits a solution. One would like to
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deduce a new equation which always has a solution and such that any solution to Eq. (23a) is also a solution to the new equation. 5.1. The Approximate Killing Spinor Operator Let S2 and S4 denote, respectively, the spaces of totally symmetric valence 2 and valence 4 spinors. Given ζABCD , χABCD ∈ S4 , we introduce an inner product in S4 via
ζABCD , χEF GH = ζABCD χ ˆABCD dμ, S
where dμ denotes the volume form of the 3-metric hab . We introduce the spatial Killing spinor operator Φ via Φ : S2 → S4 ,
Φ(κ)ABCD = ∇(AB κCD) .
Now, consider the pairing
∇(AB κCD) ζˆABCD dμ
∇(AB κCD) , ζEF GH = S
∇AB κCD ζˆABCD dμ.
= S ∗
The formal adjoint, Φ , of the spatial Killing operator can be obtained from the latter expression by integration by parts. To this end we note the identity
∇AB κCD ζˆABCD dμ − κAB ∇CD ζABCD dμ + 2κAB ΩCDF A ζˆBCDF dμ U
U
U
nAB κCD ζˆABCD dS,
=
(37)
∂U
with U ⊂ S, and where dS denotes the area element of ∂U, nAB is the spinorial counterpart of its outward pointing normal, and ζABCD is a symmetric spinor. From (37) it follows that Φ∗ : S4 → S2 ,
Φ∗ (ζ)CD = ∇AB ζABCD − 2ΩABF (C ζD)ABF .
(38)
Definition. The composition operator L ≡ Φ∗ ◦ Φ : S2 → S2 given by L(κCD ) ≡ ∇AB ∇(AB κCD) − ΩABF (A ∇|DF | κB)C − ΩABF (A ∇B)F κCD = 0, (39) will be called the approximate Killing spinor operator, and Eq. (39) the approximate Killing spinor equation. Remark. Note that every solution to the spatial Killing spinor equation (25) is also a solution to Eq. (39).
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5.2. Ellipticity of the Approximate Killing Spinor Operator As a prior step to the analysis of the solutions to the approximate Killing spinor equation (39), we look first at its ellipticity properties. Lemma 11. The operator L defined by Eq. (39) is a formally self-adjoint elliptic operator. Proof. The operator is by construction formally self-adjoint as it is given by the composition of an operator and its formal adjoint. In order to verify ellipticity, it suffices to look at the operator L (κ)CD ≡ ∂ AB ∂(AB κCD) , corresponding to the principal part of L in some Cartesian spin frame. In the corresponding Cartesian coordinates (x1 , x2 , x3 ) one has that 1 1 ∂3 ∂3 −∂1 − i∂2 −∂1 + i∂2 AB ∂AB = √ =√ , ∂ . ∂3 ∂1 − i∂2 ∂3 ∂1 + i∂2 2 2 In particular, ∂ AB ∂AB = Δ ≡ ∂12 + ∂22 + ∂32 , the flat Laplacian. One notes that ∂ PQ ∂(PQ κAB) =
1 PQ 2 1 ∂ ∂PQ κAB + ∂ PQ ∂P(A κB)Q + ∂ PQ ∂AB κPQ . 6 3 6
Now, writing κ0 ≡ κ00 ,
κ1 ≡ κ01 ,
κ2 ≡ κ11 ,
one has that L can be expressed in matricial form as Aij ∂i ∂j u, where 1 Aij ∂i ∂j ≡ 12 ⎛ 7Δ − ∂32 ⎜ −∂1 ∂3 ⎜ 2 ⎜ ∂2 − ∂12 ×⎜ ⎜ 0 ⎜ ⎝ −∂2 ∂3 −2∂1 ∂2
−2∂1 ∂3 6Δ + 2∂32 2∂1 ∂3 2∂2 ∂3 0 ∂2 ∂3
∂22 − ∂12 ∂1 ∂3 7Δ − ∂32 2∂1 ∂2 −∂2 ∂3 0
0 ∂2 ∂3 2∂1 ∂2 7Δ − ∂32 −∂1 ∂3 ∂22 − ∂12
−2∂2 ∂3 0 −2∂2 ∂3 −2∂1 ∂3 6Δ + 2∂32 2∂1 ∂3
⎞ −2∂1 ∂2 ∂2 ∂3 ⎟ ⎟ ⎟ 0 ⎟ 2 2 ⎟, ∂2 − ∂1 ⎟ ∂1 ∂3 ⎠ 7Δ − ∂32 (40)
and
⎛
⎞ Re(κ0 ) ⎜ Re(κ1 ) ⎟ ⎜ ⎟ ⎜ Re(κ2 ) ⎟ ⎟ u≡⎜ ⎜ Im(κ0 ) ⎟ . ⎜ ⎟ ⎝ Im(κ1 ) ⎠ Im(κ2 )
(41)
The symbol, l(ξi ), of the operator given by (40) is then given by replacing ∂i with ξi ∈ R3 . One finds that 6 1 (ξ1 )2 + (ξ2 )2 + (ξ3 )2 , det l(ξi ) = 36
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so that det l(ξi ) = 0 if and only if ξi = 0. Accordingly, the operator L = Φ∗ ◦ Φ is elliptic. 5.3. A Variational Formulation We note that the approximate Killing spinor equation arises naturally from a variational principle. Lemma 12. The approximate Killing spinor equation (39) is the Euler-Lagrange equation of the functional
J = ∇(AB κCD) ∇AB κCD dμ. (42) S
Proof. This is a direct consequence of the identity (37).
6. Asymptotically Euclidean Manifolds After having studied some formal properties of the Killing spinor initial data equations (23a)–(23c),(24), and the approximate Killing spinor equation (39), we proceed to analyse their solvability on asymptotically Euclidean manifolds. In order to do this we introduce some relevant terminology and ancillary results. 6.1. General Assumptions In what follows, we will be concerned with vacuum spacetimes arising as the development of asymptotically Euclidean data sets. Let (S, hab , Kab ), denote a smooth initial data set for the vacuum Einstein field equations. The pair (hab , Kab ) satisfies on the 3-dimensional manifold S the vacuum constraint equations − 2r − K a a K b b + Kab K ab = 0, Da Kab − Db K a a = 0,
(43a) (43b)
where r and D denote, respectively, the Ricci scalar and the Levi-Civita connection of the negative definite 3-metric hab , while Kab corresponds to the extrinsic curvature of S. The unusual coefficients in the formulae above come from our normalisation of τ μ . For an asymptotic end it will be understood an open set diffeomorphic to the complement of a closed ball in R3 . In what follows, the 3-manifold S will be assumed to be the union of a compact set and two asymptotically Euclidean ends, i1 , i2 . 6.2. Weighted Sobolev Norms In order to discuss the decays of the various fields on the 3-manifold S we make use of weighted Sobolev spaces. In what follows, we follow the ideas of [9] written in terms of the conventions of [3]. Choose an arbitrary point O ∈ S, and let 1/2 , σ(x) ≡ 1 + d(O, x)2
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where d denotes the Riemannian distance function on S. The function σ is used to define the following weighted L2 norm: ⎛ ⎞1/2
uδ ≡ ⎝ |u|2 σ −2δ−3 dx⎠ , (44) S
for δ ∈ R. In particular, if δ = −3/2 one recovers the usual L2 norm. Different choices of origin give rise to equivalent weighted norms—as mentioned earlier, the convention of indices used in the definition of the norm (44) follows the one of Bartnik [3]. The fall-off conditions of the various fields will be expressed in terms of weighted Sobolev spaces Hδs consisting of functions for which the norm Dα uδ−|α| < ∞, us,δ ≡ 0≤|α|≤s
with s a non-negative integer, and where α = (α1 , α2 , α3 ) is a multiindex, |α| = α1 + α2 + α3 . We say that u ∈ Hδ∞ if u ∈ Hδs for all s. We will say that a spinor or a tensor belongs to a function space if its norm does. For instance, the notation ζAB ∈ Hδs is a shorthand notation for (ζAB ζˆAB + ζA A ζˆB B )1/2 ∈ Hδs . We will make use of the following result: Lemma 13. Let u ∈ Hδ∞ . Then u is smooth (i.e. C ∞ ) over S and has a fall off at infinity such that Dl u = o(rδ−|l| ). The smoothness of u follows from the Sobolev embedding theorems. The proof of the behaviour at infinity of u can be found in [3]—cfr. Theorem 1.2 (iv)—while the decay for the derivatives follows from the definition of the weighted Sobolev norms. Remark. Here, r is a radial coordinate on the asymptotic end—see the next section for details. We also note the following multiplication lemma—cfr. e.g. Theorem 5.6 in [9]. Lemma 14. Let u ∈ Hδ∞ , v ∈ Hδ∞ . Then 1 2 uv ∈ Hδ∞ , 1 +δ2 +ε
ε > 0.
Notation. We will often write u = o∞ (rδ ) for u ∈ Hδ∞ at an asymptotic end. For the present applications we will require a somehow finer multiplication lemma concerning the behaviour at infinity. For this we exploit the fact that we are working with smooth functions. More precisely Lemma 15. Let u = o∞ (rδ1 ), v = o∞ (rδ2 ) and w = O(rγ ). Then uv = o(rδ1 +δ2 ),
uw = o(rδ1 +γ ).
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Proof. Let ∂Sr denote the surfaces of constant r. For sufficiently large r (so that one is in an asymptotic end), the surface ∂Sr has the topology of the 2-sphere. Now, the functions u, v are continuous and the surfaces ∂Sr are compact. Therefore, for sufficiently large r the functions f (r) ≡ max |ur−δ1 |, ∂Sr
g(r) ≡ max |vr−δ2 |, ∂Sr
are finite and well defined. Furthermore rδ1 |u| ≤ f (r), rδ2 |v| ≤ g(r). By construction, one has that f (r) = o(1) and g(r) = o(1)—that is, f, g → 0 for r → ∞. One also has that |wr−γ | is bounded by a constant C. Hence, |uv| ≤ f (r)g(r)rδ1 +δ2 = o(rδ1 +δ2 ), |uw| ≤ f (r)rδ1 |w| ≤ Cf (r)rδ1 +γ = o(rδ1 +γ ),
from where the desired result follows.
Remark. The lemmas extend to symmetric spatial spinors with even number of indices by the Cauchy–Schwartz inequality. 6.3. Decay Assumptions As mentioned earlier, our analysis will be restricted to initial data sets (S, hab , Kab ) with two asymptotic ends. Without loss of generality one of the ends will be denoted by the subscript/superscript + on the relevant objects, while those of the other end by −. Often, when no confusion arises the subscript/superscript will be dropped. Remark. We do not need to assume any topological restriction apart from paracompactness, orientability and the requirement of two asymptotically flat ends. Hence, we can have an arbitrary number of handles. For black holes, this means that we can handle Misner-type data with several black holes [39]. The standard assumption for asymptotic flatness is that on each end it is possible to introduce asymptotically Cartesian coordinates xi± with r = ((x1± )2 + (x2± )2 + (x3± )2 )1/2 , such that the intrinsic metric and extrinsic curvature of S satisfy hij = −δij + o∞ (r−1/2 ), Kij = o∞ (r
−3/2
).
(45a) (45b)
Note that the decay conditions (45a) and (45b) allow for data containing nonvanishing linear and angular momentum. For the purposes of our analysis, it will be necessary to have a bit more information about the behaviour of leading terms in hij and Kij . More precisely, we will require the initial data to be asymptotically Schwarzschildean in some suitable sense. For example, in [2] the assumptions (46a) hij = − 1 + 2m± r−1 δij + o∞ (r−3/2 ), Kij = o∞ (r−5/2 ),
(46b)
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have been used. This class of data can be described as asymptotically nonboosted Schwarzschildean. Here, we consider a more general class of data which include boosted Schwarzschild data. Following [6,26] we assume 2A± α± 2xi xj −3/2 − δ ), (47a) hij = − 1 + δij − ij + o∞ (r r r r2 β± 2xi xj −5/2 − δ ), (47b) Kij = 2 ij + o∞ (r r r2 where α± and β± are smooth functions on the 2-sphere and A± denotes a constant. The functions α and β are related to each other via the vacuum constraint equations (43a) and (43b). We will later need to be more specific about their particular form. The decay assumption for the metric, Eq. (45a) and hence also (47a), is included in the analysis of [9]. Important for our analysis is that boosted Schwarzschild data are of this form—see [6]. It is noticed that a second fundamental form of the type given by (47b) is, in general, not trace-free: β± + o∞ (r−5/2 ). r2
Ki i =
Henceforth, we drop the superscripts/subscripts ± for ease of presentation. If ± appears in any formula, + is assumed for the (+)-end, − for the (−)-end. For the ∓ sign we assume the opposite. 6.4. ADM Mass and Momentum The ADM energy, E, and momentum, pi , at each end are given by the integrals
xk 1 δ ij (∂i hjk − ∂k hij ) dS, E= 16π r ∂S∞
1 pi = 8π
(Kij − Khij )
xj dS, r
∂S∞
so that the ADM 4-momentum covector is given by pμ = (E, pi ). In what follows it will be assumed that pμ is timelike—that is, pμ pμ > 0. The need of this assumption will become clear in the sequel. From the ADM 4-momentum, we define the constants √ m ≡ pν pν , p2 ≡ E 2 − m2 . 6.5. Asymptotically Schwarzschildean Data Boosted Schwarzschild data sets are initial data for the Schwarzschild spacetime for which pi = 0. They satisfy the decay assumptions (47a)–(47b). This type of data satisfy
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m , 1 − v2 −1/2 2m (n · v)2 (n · v)2 −√ , α = 2m 1 + 2 1+ 1 − v2 1 − v2 1 − v2 −3/2 3 (n · v)2 n·v (n · v)2 β = 2m + , 1 + 1 − v2 2 1 − v2 1 − v2
A= √
where ni ≡ xi /r, n · v ≡ ni vi , v 2 ≡ δ ij vi vj , vi is a constant 3-covector—cfr. [6], and m± = m. Note that if vi = 0 then (47a)–(47b) reduce to (46a)–(46b). It can be checked that m mvi E=√ , pi = √ . 2 1−v 1 − v2 Rewriting this in terms of (E, pi ), we get A = E,
2m2 + 4(n · p)2 − 2E, α= m2 + (n · p)2
β=
(n · p)E(3m2 + 2(n · p)2 ) , (m2 + (n · p)2 )3/2
(48)
where n · p = ni pi = r−1 xi pi . Assumption. In the sequel, we will restrict our analysis to initial data sets which are asymptotically Schwarzschildean to the order given by (47a)–(47b). For any asymptotically flat data that admit ADM 4-momentum, one can compute (E, pi ), and then try to find coordinates that cast the metric and extrinsic curvature into the form (47a)–(47b) with (A, α, β) given by (48) with m = m± . If this is possible, we will say that the data are asymptotically Schwarzschildean. We expect this to be the case for a large class of data. The initial data sets excluded by this assumption will be deemed pathological. Examples of such pathological cases can be found in [26]. We stress that all data of the form (46a)–(46b) are included in our more general analysis. The need to restrict our analysis to asymptotically Schwarzschildean data as defined in the previous paragraph will become evident in the sequel, where we need to find an asymptotic solution to the spatial Killing spinor equation.
7. Asymptotic Behaviour of the Spatial Killing Spinors In this section we discuss in some detail the asymptotic behaviour of solutions to the spatial Killing spinor equation on an asymptotically Euclidean manifold. We begin by studying the asymptotic behaviour of the appropriate Killing spinor in the Kerr spacetime. Then, we will impose the same asymptotics on the approximate Killing spinor on a slice of a much more general spacetime. In what follows, we concentrate our discussion on a particular asymptotic end. 7.1. Asymptotic Form of the Stationary Killing Vector As seen in Sect. 2, the Killing spinor of the Kerr spacetime gives rise to its stationary Killing vector ξ μ . It will be assumed that the spacetime is such that pμ = (E, pi ) is timelike at each asymptotic end. If this is the case, then
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√ pμ / pν pν gives the asymptotic direction of the stationary Killing vector at each end—see, e.g. [4]. Let √ m ≡ pν pν , p2 ≡ E 2 − m2 . Recall now, that ξ and ξAB denote the lapse and shift of the spinorial counter part, ξ AA , of the Killing vector ξ μ . One finds that for non-boosted initial data sets of the form (46a)–(46b), one has in terms of the asymptotic Cartesian coordinates and spin frame, that √ ξ = ± 2 + o∞ (r−1/2 ), ξAB = o∞ (r−1/2 ). √ The factor of 2 arises due to the particular normalisations used in the space spinor formalism. This particular form of the asymptotic behaviour of the Killing vector has been discussed in [2]. Now consider the more general case given by (47a)–(47b). Again, adopting asymptotically Cartesian coordinates, we extend pi to a constant covector field on the asymptotic end. In terms of the associated asymptotically Cartesian spin frame, we then define pAB ≡ σ i AB pi . One finds that √ √ 2E 2pAB −1/2 + o∞ (r + o∞ (r−1/2 ). ), ξAB = ± (49) ξ=± m m We see that the conditions (49) are well defined even if we do not have a Killing vector in the spacetime. Hence, for the general case when the metric satisfies (47a)–(47b) and the ADM 4-momentum is well defined, we can still impose the asymptotics (49) for our approximate Killing spinor. We will, however, need to assume that the functions in the metric are given by (48). ∞ , as we will do in Otherwise, we will not be able to assume ξABCD ∈ H−3/2 the next section. We will later see that this condition is important for the solvability of the elliptic equation (39). 7.2. Asymptotic Form of the Spatial Killing Spinor In the sequel, given an initial data set (S, hab , Kab ) satisfying the decay conditions (47a)–(47b) with A, α and β given by (48) with m = m± , it will be necessary to show that it is always possible to solve the equation ∇(AB κCD) = o∞ (r−3/2 ),
(50)
order by order without making any further assumptions on the data. A direct calculation allows us to verify that Lemma 16. Let (S, hab , Kab ) denote an initial data set for the vacuum Einstein field equations satisfying at each asymptotic end the decay conditions (47a)– (47b) with A, α and β given by (48) and m the ADM mass of the respective end. Then
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√
2E 2E 1+ xAB 3m r √ m2 + 2(n · p)2 2 2 4E − pQ(A xB) Q + o∞ (r−1/2 ), ± 1+ 3m r m2 + (n · p)2 r
κAB = ∓
(51) with xAB as in (14), and n · p = r
−1 AB
x
pAB satisfies equation (50).
Remark. Formula (51) implies the following expansions for ξ and ξAB : √
√ 2E 2E(m2 + 2(n · p)2 ) −1 r + o∞ (r−3/2 ), (52a) ∓ m m m2 + (n · p)2 √ √ 2(E 2 + 4(n · p)2 ) 2 2E mE 2 √ =± − + (n · p)r−2 xAB + m 2(m2 + (n · p)2 )3/2 m m2 + (n · p)2 √ √ √ 2 2 2E 2 2(m2 + 2(n · p)2 ) pAB + o∞ (r−3/2 ). ± (52b) + − m mr m m2 + (n · p)2 r ξ=±
ξAB
In the case of non-boosted data the expansions (51), (52a) and (52b) reduce to √ 2 2m κAB = ∓ 1+ xAB + o∞ (r−1/2 ), 3 r √ √ ξ = ± 2 ∓ 2mr−1 + o∞ (r−3/2 ), ξAB = o∞ (r−3/2 ), as discussed in [2]. 7.3. Existence and Uniqueness of Spinors with Killing Spinor Asymptotics In this section we prove that given a spinor κAB satisfying equation (49) and (50), then the asymptotic expansion (51) is unique up to a translation. Theorem 17. Assume that on an asymptotic end of the slice S, one has an asymptotically Cartesian coordinate system such that (47a)–(47b) hold. Then there exists κAB = o∞ (r3/2 ), such that −3/2
(53)
√ 2pAB + o∞ (r−1/2 ), =± m
ξABCD = o∞ (r ), ξAB √ 2E + o∞ (r−1/2 ). ξ=± m
(54)
The spinor κAB is unique up to order o∞ (r−1/2 ), apart from a (complex) constant term. Remark 1. The complex constant term arising in Theorem 17 contains six real parameters. In the sequel, given a particular choice of asymptotically Cartesian coordinates and frame, we will set this constant term to zero. Note
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that a change of asymptotically Cartesian coordinates would introduce a similar term containing only three real parameters—which by construction could be removed by a suitable choice of gauge. In what follows, we will use coordinate independent expressions, and therefore, this translational ambiguity will not affect the result. Remark 2. Note that ξABCD = o∞ (r−3/2 ) implies ξABCD ∈ L2 . The conditions in Theorem 17 are coordinate independent. Proof. A direct calculation shows that the expansion (51) yields (52a), (52b) and ξABCD = o∞ (r−3/2 ). Hence, (51) gives a solution of the desired form. In order to prove uniqueness we make use of the linearity of the integrability conditions (18a)–(18c) and (19a)–(19c). Note that the translational freedom gives an ambiguity of a constant term in κAB . Let √ 2E 2E ˚ κAB ≡ ∓ 1+ xAB 3m r √ m2 + 2(n · p)2 4E 2 2 1+ − pQ(A xB) Q . ± (55) 3m r m2 + (n · p)2 r Let κ ˘ AB , be an arbitrary solution to the system (49), (50). Furthermore, let ˘ AB − ˚ κAB . We then have κAB ≡ κ ξABCD = o∞ (r−3/2 ),
ξAB = o∞ (r−1/2 ),
ξ = o∞ (r−1/2 ),
κAB = o∞ (r3/2 ).
To obtain the desired conclusion we only need to prove that κAB = CAB + o∞ (r−1/2 ), where CAB is a constant. This is equivalent to DAB κCD = o∞ (r−3/2 ). Note that we now have coordinate independent statements to prove. We note that from (47a)–(47b) it follows that KABCD = o∞ (r−2+ε ),
ΨABCD = o∞ (r−3+ε ),
with ε > 0. From (17) and Lemma 14 we have DAB κCD 1 1 1 = ξABCD − A(C ξD)B − B(C ξD)A − A(C D)B ξ − KAB(C E κD)E 3 3 3 = o∞ (r−1/2+ε ). Integrating the latter yields κAB = o∞ (r1/2+ε ). The constant of integration is incorporated in the remainder term. Repeating this procedure allows to gain an ε in the decay so that DAB κCD = o∞ (r−1/2 ),
κAB = o∞ (r1/2 ).
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Estimating all terms in (19a), (19b) and (19c) gives ∇AB ∇AB ξ = ξAB ∇AB K + o∞ (r−7/2 ) = o∞ (r−7/2+ε ), (56a) 1 1 ∇C (A ∇B)C ξ = ΩABCD ∇CD ξ − K∇AB ξ 2 3 −7/2+ε ), (56b) = o∞ (r 1ˆ 5 2ˆ 10 E E ∇(AB ∇CD) ξ = + ΨABCD ξ − ΨABCD ξ − Ψ(ABC ξD)E − Ψ(ABC ξD)E 2 2 3 3 2 EL + ΩABCDEL ξ + ξ(CD ∇AB) K − 3ΩE(BCD ∇A) E ξ 5 EL − 3κ ∇L(D ΨABC)E + 3κ(A E ∇D L ΨBC)EL + o∞ (r−7/2 ) = o∞ (r−7/2+ε ).
(56c)
Hence, ∇AB ∇CD ξ = o∞ (r−7/2+ε ), and therefore DAB DCD ξ = o∞ (r−7/2+ε ). Integrating this yields DAB ξ = o∞ (r−5/2+ε ). In this step the constants of integration are forced to vanish by the condition DAB ξ = o∞ (r−3/2 ), which is a consequence of ξ = o∞ (r−1/2 ). Integrating DAB ξ = o∞ (r−5/2+ε ) and using ξ = o∞ (r−1/2 ) to remove the constants of integration yields ξ = o∞ (r−3/2+ε ). Estimating all terms in (18a), (18b) and (18c) yields ∇AB ξAB = o∞ (r−7/2+ε ), (57a) 3 2 1 ∇C (A ξB)C = ΨABCD κCD − KξAB − ΩABCD ξ CD + ∇AB ξ + o∞ (r−5/2 ) 2 3 2 (57b) = o∞ (r−5/2+ε ), ∇(AB ξCD) = 3ΨE(ABC κD) E − ΩE(ABC ξD) E + o∞ (r−5/2 ) = o∞ (r−5/2+ε ).
(57c)
Hence, ∇AB ξCD = o∞ (r−5/2+ε ), and therefore DAB ξCD = o∞ (r−5/2+ε ). Integrating and using ξAB = o∞ (r−1/2 ) to remove the constants of integration yields ξAB = o∞ (r−3/2+ε ). Now, DAB κCD 1 1 1 = ξABCD − A(C ξD)B − B(C ξD)A − A(C D)B ξ − KAB(C E κD)E 3 3 3 = o∞ (r−3/2+ε ). Integrating the latter we get κAB = CAB + o∞ (r−1/2+ε ),
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where CAB is a constant in some frame. To get a frame independent statement one can still use the estimate κAB = o∞ (rε ). Re-evaluating the estimates (56a), (56b) and (56c) yields ∇AB ∇AB ξ = o∞ (r−7/2 ), ∇C (A ∇B)C ξ = o∞ (r−9/2+ε ), ∇(AB ∇CD) ξ = o∞ (r−7/2 ). Hence, one obtains ∇AB ∇CD ξ = o∞ (r−7/2 ). Integrating as before, we get ξ = o∞ (r−3/2 ). Finally, we can reevaluate the estimates (57b) and (57c), to get ∇C (A ξB)C = o∞ (r−5/2 ), ∇(AB ξCD) = o∞ (r−5/2 ). Combining this with (57a), we obtain ∇AB ξCD = o∞ (r−5/2 ). Integrating as before, we get ξAB = o∞ (r−3/2 ). Hence, 1 1 1 DAB κCD = ξABCD − A(C ξD)B − B(C ξD)A − A(C D)B ξ − KAB(C E κD)E 3 3 3 = o∞ (r−3/2 ), from where the result follows.
From the asymptotic solutions we can obtain a globally defined spinor ˚ κAB on S that will act as a seed for our approximate Killing spinor. Corollary 18. There are spinors ˚ κAB , defined everywhere on S, such that the asymptotics at each end is given by (51), where opposite signs are used at each ∞ . end. Different choices of ˚ κAB can only differ by a spinor in H−1/2 Remark. The opposite signs at each end are motivated by looking at the explicit example of standard Kerr data. Proof. Theorem 17 gives the existence at each end. Smoothly cut off these functions, and paste them together. This gives a smooth spinor ˚ κAB defined ∞ κCD) ∈ H−3/2 . everywhere on S. Furthermore ∇(AB˚
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8. The Approximate Killing Spinor Equation in Asymptotically Euclidean Manifolds In this section we study the invertibility properties of the approximate Killing spinor operator L : S2 → S2 given by Eq. (39) on a manifold S which is asymptotically Euclidean in the sense discussed in Sect. 6. In order to do so, we first present some adaptations to our context of results for elliptic equations that can be found in [9,13,35]. 8.1. Ancillary Results of the Theory of Elliptic Equations on Asymptotically Euclidean Manifolds 8.1.1. Asymptotic Homogeneity of L. Let u be the vector given by Eq. (41). The elliptic operator defined by (39) can be written matricially in the form i (Aij + aij 2 )Di Dj u + a1 Di u + a0 u = 0,
where Aij corresponds to the matrix associated with the elliptic operator with j constant coefficients L given by Eq. (40), and aij 2 , a1 , a0 are matrix-valued functions such that ∞ aij 2 ∈ H−1/2 ,
∞ aj1 ∈ H−3/2 ,
∞ a0 ∈ H−5/2 .
Using the terminology of [9,35] we say that L is an asymptotically homogeneous elliptic operator.3 This is the standard assumption on elliptic operators on asymptotically Euclidean manifolds. It follows from [9], Theorem 6.3 that Theorem 19. The elliptic operator 0 , L : Hδ2 → Hδ−2
with δ is not a negative integer is a linear bounded operator with finite dimensional Kernel and closed range. 8.1.2. The Kernel of L. We investigate some relevant properties of the Kernel of L. This, in turn, requires an analysis of the Kernel of the operator of the Killing spinor equation (25). The following is an adaptation to the smooth spinorial setting of an ancillary result from [13].4 Theorem 20. Let νA1 B1 ···Ap Bp be a C ∞ spinorial field over S such that ∇Em+1 Fm+1 · · · ∇E1 F1 νA1 B1 ···Ap Bp = HEm+1 Fm+1 ···E1 F1 A1 B1 ···Ap Bp with m, p non-negative integers, and where HEm+1 Fm+1 ···E1 F1 A1 B1 ···Ap Bp is a linear combination of νA1 B1 ···Ap Bp , ∇E1 F1 νA1 B1 ···Ap Bp , . . . , ∇Em Fm · · · 3 The sharp conditions for a second-order elliptic operator to be asymptotically homogeneous are that ∞ aij 2 ∈ Hδ ,
∞ ai1 ∈ Hδ−1 ,
∞ a0 ∈ Hδ−2 ,
for δ < 0. As one sees, our operator L satisfies these conditions with a margin. The hypotheses in [13] are much weaker than the ones presented here. The adaptation to the smooth setting has been chosen for simplicity. 4
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∇E1 F1 νA1 B1 ···Ap Bp with coefficients bk where k denotes the order of the derivwith ative the coefficient is associated with. If bk ∈ Hδ∞ k k − m − 1 > δk , and νA1 B1 ···Ap Bp ∈
Hβ∞ ,
0≤k≤m
β < 0, then νA1 B1 ···Ap Bp = 0
on S.
This last result, together with Lemma 10 allows to show that there are no non-trivial Killing spinor candidates that go to zero at infinity—in [13] an analogous result has been proved for Killing vectors. More precisely, ∞ Proposition 21. Let νAB ∈ H−1/2 such that ∇(AB νCD) = 0. Then νAB = 0 on S.
Proof. From Lemma 10 it follows that ∇AB ∇CD ∇EF νGH can be expressed as a linear combination of lower order derivatives with smooth coefficients with the proper decay. Thus, Theorem 20 applies with m = 2 and one obtains the desired result. We are now in the position to discuss the Kernel of the approximate Killing spinor operator in the case of spinor fields that go to zero at infinity. The following is the main result of this section: ∞ . If L(νAB ) = 0, then νAB = 0. Proposition 22. Let νAB ∈ H−1/2
Proof. Using the identity (37) with ζABCD = ∇(AB νCD) and assuming that L(νCD ) = 0, one obtains
∇AB ν CD ∇(AB νCD) dμ = nAB ν CD ∇(AB νCD) dS, S
∂S∞
∞ . It where ∂S∞ denotes the sphere at infinity. Assume now, that νAB ∈ H−1/2 ∞ follows that ∇(AB νCD) ∈ H−3/2 and furthermore, using Lemma 15 that
nAB ν CD ∇(AB νCD) = o(r−2 ). The integration of the latter over a finite sphere of sufficiently large radius is of type o(1). Thus one has that
nAB ν CD ∇(AB νCD) dS = 0, ∂S∞
from where
∇AB ν CD ∇(AB νCD) dμ = 0.
S
Therefore, one concludes that ∇(AB νCD) = 0. That is, νAB has to be a spatial Killing spinor. Using Proposition 21 it follows that νAB = 0 on S.
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8.1.3. The Fredholm Alternative and Elliptic Regularity. We will make use of the following adaptation of the Fredholm alternative for second-order asymptotically homogeneous elliptic operators on asymptotically Euclidean manifolds—cfr. [9]: Theorem 23. Let A be an asymptotically homogeneous elliptic operator of order 2 with smooth coefficients. Given δ not a negative integer, the equation A(ζAB ) = fAB , has a solution ζAB ∈ Hδ2 if
0 fAB ∈ Hδ−2 ,
fAB νˆAB dμ = 0 S
for all νAB satisfying 0 νAB ∈ H−1−δ ,
A∗ (νAB ) = 0,
where A∗ denotes the formal adjoint of A. In order to assert the regularity of solutions, we will need the following elliptic estimate—see expression (62) in the proof of Theorem 6.3 of [9]: Theorem 24. Let A be an asymptotically homogeneous elliptic operator of order 2 with smooth coefficients. Then for any δ ∈ R and any s ≥ 2 there exists a s constant C such that for every ζAB ∈ Hloc ∩ Hδ0 , the following inequality holds: ζAB Hδs ≤ C A(ζAB )H s−2 + ζAB H s−2 . δ−2
δ
s Notation. denotes the local Sobolev space. That is, u ∈ Hloc if for an s arbitrary smooth function v with compact support, uv ∈ H . s Hloc
Remark. If A has smooth coefficients, and A(ζAB ) = 0 then it follows that all the Hδs norms of ζAB are bounded by the Hδ0 norm. Thus, it follows that if a solution to A(ζAB ) = 0 exists, it must be smooth—elliptic regularity. 8.2. Existence of Approximate Killing Spinors We are now in the position of providing an existence proof to solutions to Eq. (39) with the asymptotic behaviour discussed in Sect. 7.2. Theorem 25. Given an asymptotically Euclidean initial data set (S, hab , Kab ) satisfying the asymptotic conditions (47a)–(47b) and (48), there exists a smooth unique solution to Eq. (39) with asymptotic behaviour at each end given by (51). Proof. We consider the Ansatz κAB = ˚ κAB + θAB ,
2 θAB ∈ H−1/2 ,
with ˚ κ given by Corollary 18. Substitution into Eq. (39) renders the following equation for the spinor θAB : L(θCD ) = −L(˚ κCD ).
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By construction it follows that ∞ κCD) ∈ H−3/2 , ∇(AB˚
so that ∞ κCD ) ∈ H−5/2 . FCD ≡ −L(˚
Using Theorem 23 with δ = −1/2, one concludes that Eq. (58) has a unique 0 solution if FAB is orthogonal to all νAB ∈ H−1/2 in the Kernel of L∗ = L. Proposition 21 states that this Kernel is trivial. Thus, there are no restrictions on FAB and Eq. (58) has a unique solution as desired. Due to elliptic regularity, 2 ∞ any H−1/2 solution to the previous equation is in fact a H−1/2 solution—cfr. Lemma 24. Thus, θAB is smooth. To see that κAB does not depend on the κAB , be another choice. Let κAB be the correparticular choice of ˚ κAB , let ˚ ∞ κAB ∈ H−1/2 . sponding solution to (58). Due to Corollary 18, we have ˚ κAB − ˚ ∞ Hence, we have κAB − κAB ∈ H−1/2 and L(κAB − κAB ) = 0. According to Proposition 22, κAB − κAB = 0, and the proof is complete. The following is a direct consequence of Theorem 25, and will be crucial for obtaining an invariant characterisation of Kerr data: Corollary 26. A solution, κAB , to Eq. (39) with asymptotic behaviour given by (51) satisfies J < ∞ where J is the functional given by Eq. (42). Proof. The functional J given by Eq. (42) is the L2 norm of ∇(AB κCD) . Now, 0 if κAB is the solution given by Theorem 25, one has that ∇(AB κCD) ∈ H−3/2 . In Bartnik’s conventions one has that 0 < ∞. ∇(AB κCD) L2 = ∇(AB κCD) H−3/2
The result follows.
Remark. Again, let κAB be the solution to Eq. (39) given by Theorem 25. Using the identity (37) with ζABCD = ∇(AB κCD) one obtains that
nAB κCD ∇(AB κCD) dS < ∞. J= ∂S∞
Thus, the invariant J evaluated at the solution κAB given by Theorem 25 can be expressed as a boundary integral at infinity. A crude estimation of the integrand of the boundary integral does not allow directly to establish its boundedness. This follows, however, from Corollary 26. Hence, the leading order terms of nAB κCD and ∇(AB κCD) are orthogonal. For an independent proof of this fact, see Appendix A.
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9. The Geometric Invariant In this section we show how to use the functional (42) and the algebraic conditions (23b) and (23c) to construct the desired geometric invariant measuring the deviation of (S, hab , Kab ) from Kerr initial data. To this end, let κAB be a solution to Eq. (39) as given by Theorem 25. Furthermore, let ξAB ≡ 3 P 2 ∇ (A κB)P . Define
ˆ ABCG κ I1 ≡ Ψ(ABC F κD)F Ψ ˆ D G dμ, (59a) S
I2 ≡
3κ(A E ∇B F ΨCD)EF + Ψ(ABC F ξD)F
S
ˆ ABCP ξˆD P dμ. × 3ˆ κAP ∇BQ ΨCD P Q + Ψ
(59b)
The geometric invariant is then defined by I ≡ J + I1 + I2 .
(60)
Remark. It should be stressed that by construction I is coordinate independent and that I ≥ 0. We also have the following lemma. Lemma 27. The geometric invariant given by (60) is finite for an initial data set (S, hab , Kab ) satisfying the decay conditions (47a)–(47b). Proof. From Corollary 26 we already have J < ∞. From the form of the decay ∞ , ε > 0. By Lemma 14 and assumptions (47a)–(47b) we have ΨABCD ∈ H−3+ε ∞ κAB ∈ H1+ε we have ∞ . Ψ(ABC F κD)F ∈ H−3/2
Thus, again one finds that I1 < ∞. A similar argument shows that ∞ 3κ(A E ∇B F ΨCD)EF + Ψ(ABC F ξD)F ∈ H−3/2 ,
from where it follows that I2 < ∞. Hence, the invariant (60) is finite and well defined. Finally, we are in the position of stating the main result of this article. It combines all the results in the sections 2 to 7. Theorem 28. Let (S, hab , Kab ) be an asymptotically Euclidean initial data set for the Einstein vacuum field equations satisfying on each of its two asymptotic ends the decay conditions (47a)–(47b) and (48) with a timelike ADM 4-momentum. Furthermore, assume that ΨABCD = 0 and ΨABCD ΨABCD = 0 everywhere on S. Let I be the invariant defined by Eqs. (42), (59a), (59b) and (60), where κAB is given as the only solution to Eq. (39) with asymptotic behaviour on each end given by (51). The invariant I vanishes if and only if (S, hab , Kab ) is locally an initial data set for the Kerr spacetime.
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Proof. Due to our smoothness assumptions, if I = 0 it follows that Eqs. (23a)– (23c) are satisfied on the whole of S. Thus, the development of (S, hab , Kab ) will have, at least in a slab, a Killing spinor. Accordingly, it must be of Petrov type D, N or O on the slab—see Theorem 1. The types N and O are excluded by the assumptions ΨABCD = 0 and ΨABCD ΨABCD = 0 on S—by continuity, these conditions will also hold in a suitably small slab. Thus, the development of the data can only be of Petrov type D—at least on a suitably small slab. Now, from the general theory on Killing spinors, we know that ξAA = Q ∇A κAQ will be, in general, a complex Killing vector. In particular, both the real and imaginary parts of ξAA will be real Killing vectors. The Killing initial data for ξAA on S consist of the fields ξ and ξAB on S calculated from κAB using the expressions (15a) and (15b). It can be verified that ξ − ξˆ = o∞ (r−1/2 ), ξAB + ξˆAB = o∞ (r−1/2 ). The latter corresponds to the Killing initial data for the imaginary part of ξAA . It follows that the imaginary part of ξAA goes to zero at infinity. However, there are no non-trivial Killing vectors of this type [4,13]. Thus, ξAA is a real Killing vector. This means that the spacetime belongs, at least in a suitably small slab of S, to the generalised Kerr-NUT class. By construction, it tends to a time translation at infinity so that, in fact, it is a stationary Killing vector. By virtue of the decay assumptions (47a)–(47b) the development of the initial data will be asymptotically flat, and it can be verified that the Komar mass of each end coincides with the corresponding ADM mass—these are non-zero by assumption. Hence, Theorem 6 applies and the slab of S is locally isometric to the Kerr spacetime. Corollary 29. If furthermore, the slice S is assumed, a priori, to have the same topology as a slice of the Kerr spacetime one has that the invariant I vanishes if and only if (S, hab , Kab ) is an initial data set for the Kerr spacetime. Proof. This follows from the uniqueness of the maximal globally hyperbolic development of Cauchy data—see [12]. Remark 1. A improvement of Theorem 6 in which no a priori restrictions on the Petrov type of the spacetime are made—see the remark after Theorem 6— would allow to remove the conditions ΨABCD = 0 and ΨABCD ΨABCD = 0, and thus obtain a stronger characterisation of Kerr data. Remark 2. It is of interest to analyse whether the same conclusion of the corollary can be obtained without making a priori assumptions on the topology of the 3-manifold.
10. Future Prospects We have seen that one can construct a geometric invariant for a slice with two asymptotically flat ends. A natural extension of this work would be to also allow asymptotically hyperboloidal and asymptotically cylindrical slices. Furthermore, one would like to analyse parts of manifolds in the same way.
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In this case we need to find appropriate conditions that can be imposed on κAB on the boundary of the region we would like to study. A typical scenario would be to study the domain of outer communication for a black hole, or the exterior of a star. Another natural question to be asked is how the geometric invariant behaves under time evolution. A great part of this problem is to obtain a time evolution of κAB such that it satisfies (39) on every leaf of the foliation. If the geometric invariant is small, one could instead use (20) as an approximate evolution equation for the approximate Killing spinor. In this case the system (21a), (21b) could be used to gain control over the evolution. If some type of constancy or monotonicity property could be established for the geometric invariant, this would be a useful tool for studying non-linear stability of the Kerr spacetime and also in the numerical evolutions of black hole spacetimes.
Acknowledgements We thank A. Garc´ıa-Parrado and J.M. Mart´ın-Garc´ıa for their help with computer algebra calculations in the suite xAct [38], and M. Mars and N. Kamran for valuable comments. TB is funded by a scholarship of the Wenner-Gren foundations. JAVK is funded by an EPSRC Advanced Research fellowship.
Appendix A. An Alternative Estimation of the Boundary Integral In this section we present an alternative argument to show that the boundary integral
nAB κCD ∇(AB κCD) dS, ∂Sr
is finite as r → ∞—cfr. the remark after Corollary 26. For simplicity, we only consider the non-boosted case, so we have √ 2 rnAB + O(1). κAB = ± 3 A similar, but much lengthier argument can be implemented in the boosted case. It is only necessary to consider the finiteness of the integral
r nAB nCD ∇(AB κCD) dS as r → ∞. (61) ∂Sr
We begin by investigating the multipole structure of ξABCD ≡ ∇(AB κCD) in an asymptotically flat end U ⊂ S. The equation satisfied by ξABCD is ∇AB ξABCD − 2ΩABF (C ξD)ABF = 0,
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—see Eq. (39). As U ≈ (r0 , ∞) × S2 , with r0 ∈ R, it will be convenient to work in spherical coordinates. For simplicity, we adopt the point of view that all the angular dependence of the various functions involved is expressed in terms of (spin-weighted) spherical harmonics. Accordingly, we use the differential oper¯ ∈ TS2 —see, e.g. [41]. Let ω+ , ω− ∈ T∗ S2 denote the 1-forms dual ators ð, ð ¯ to ð and ð: ¯ ω− = 1.
ð,
ð, ω+ = 1,
¯ are extended into TU by In addition, we consider ∂r ∈ TU. The operators ð, ð requiring that ¯ ∂r ] = 0. [ð, ∂r ] = [ð, Again, let dr ∈ T∗ U denote the form dual to ∂r . One has that δij dxi ⊗ dxj = dr ⊗ dr + r2 (ω+ ⊗ ω− + ω− ⊗ ω+ ). Now, recalling that
2m hij = − 1 + r
δij + o∞ (r−3/2 ),
we introduce the following frame and coframe: m m ∂r + o∞ (r−3/2 ), σ 01 = 1 + dr + o∞ (r−3/2 ) e01 = 1 − r r m 1 m ð + o∞ (r−5/2 ), σ 00 = 1 + rω+ + o∞ (r−1/2 ) e00 = 1 − r r r m 1¯ m rω− + o∞ (r−1/2 ). e11 = 1 − ð + o∞ (r−5/2 ), σ 11 = 1 + r r r The fields eAB and σ AB satisfy
eAB , σ CD = hAB CD ,
h = hABCD σ AB ⊗ σ CD .
where hABCD ≡ −A(C D)B . Let μAB denote a smooth spinorial field. Its covariant derivative DEF μAB can be computed using DEF μAB = eEF (μAB ) − ΓEF Q A μQB − ΓEF Q B μAQ , where ΓEF Q A denote the spin coefficients of the frame eAB . The components of the spinor field ξABCD with respect to the frame eAB can be written as ξABCD = ξ0 0ABCD + ξ1 1ABCD + ξ2 2ABCD + ξ3 3ABCD + ξ4 4ABCD , where kABCD ≡ (A (E B F C G D) H)k , where (EF GH)k means that after symmetrisation, k indices are set to 1. In terms of this formalism, Eq. (62) is given by AP BQ eP Q (ξABCD ) − 4ΓABQ (A ξBCD)Q + 2K ABQ (A ξBCD)Q −2ΩABQ (C ξD)ABQ = 0.
(63)
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Recalling that by assumption ξABCD = o∞ (r−3/2 ), a lengthy but straightforward calculation shows that (63) implies the equations 1¯ 11 3 m ðξ2 + (64a) ∂r ξ1 − ðξ 1+ ξ1 = o∞ (r−5 ), 0+ r 6r r r 3 1¯ 31 3 m ðξ3 + ∂r ξ2 + (64b) ðξ1 + 1+ ξ2 = o∞ (r−5 ), 2r 2r r r m 1 1 1¯ 3 1+ ξ3 = o∞ (r−5 ). ∂r ξ3 + ðξ4 − (64c) ðξ2 + r 6r r r A computation shows that n(AB nCD) = 2ABCD , so that the boundary integral (61) involves only the component ξ2 . Furthermore, only the harmonic Y0,0 (monopole) contributes to the integral as 2ABCD is a constant spinor in our frame. From the Eqs. (64a)–(64c), it follows that the coefficient ξ2;0 of ξ2 associated with the harmonic Y0,0 satisfies the ordinary differential equation m 3 1− ∂r ξ2;0 + ξ2;0 = f (r), f (r) = o∞ (r−5 ). r r Consequently, one has that
α 1 ξ2;0 = + r(r − m)2 f (r)dr, α ∈ C. (r − m)3 (r − m)3 It follows that α + o∞ (r−4 ). r3 Using this last expression in the integral (61) and recalling that dS = O(r2 ), it follows that
nAB nCD ∇(AB κCD) dS = 4πα < ∞. r ξ2;0 =
∂Sr
It is worth noting that the constant α contains information of global nature, and it is only known after one has solved the approximate Killing spinor equation.
Appendix B. Tensor Expressions For many applications, it is useful to have tensor expressions for the invariants. To this end, define the following tensors on S: κa ≡ σa AB κAB ,
ζ ≡ ξ,
ζa ≡ σa ξAB , ζab ≡ σa AB σb CD ξABCD , Cac ≡ Eac + iBac . AB
Here abc , Eac and Bac are the pull-backs of √12 τ μ μαβγ , 12 τ γ τ δ Cαβγδ and 1 β δ μν , respectively. Observe that we are using a negative definite 4 μνγδ τ τ Cαβ metric. In this section we assume Kab = Kba .
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The tensorial versions of the Eqs. (15a), (15b), (15c) then read ζ = D a κa , 3 3 3 ζa = √ iabc Dc κb − Kab κb + Kb b κa , 4 4 2 2 1 1 ζab = D(a κb) − hab Dc κc − √ icd(a Kb) d κc . 3 2 Note that the spatial Killing spinor equation ζab = 0 reduces to the conformal Killing vector equation in the time symmetric case (Kab = 0). Expressed in terms of these tensors the elliptic equation (39) takes the form 1 Db ζab − √ iacd K bc ζb d = 0. 2
(66)
∞ Let κa ∈ H3/2 be the solution to (66) with the asymptotics
√
2E 2E 1+ xi 3m r m2 + 2(n · p)2 4E 2i 1+ − i jk pj xk + o∞ (r−1/2 ), ± 3m r m2 + (n · p)2 r
κi = ∓
√ at each end, where pμ = (E, pi ) is the ADM-4 momentum, m ≡ pμ pμ , and n · p = r−1 xi pi . The metric and extrinsic curvature are assumed to have the asymptotics (47a) and (47b), respectively. The integrand in (42) is J ≡ ξABCD ξˆABCD = ζab ζ¯ab . From the equation 1 σa AB σb CD Ψ(ABC F κD)F = √ icd(a Cb) d κc . 2 we get the integrand for the I1 part of the invariant ˆ ABCP κ I1 ≡ Ψ(ABC F κD)F Ψ ˆD P 1 1 1 = − C bc C¯bc κa κ ¯ a + Cb c C¯ac κa κ ¯ b + Ca c C¯bc κa κ ¯b. 2 2 4 In order to discuss the integrand of I2 we introduce the spinor ΣABCD ≡ ∇(A F ΨBCD)F and its tensor equivalent Σab = σa AB σb CD ΣABCD . One finds that i 0 = σa AB ∇CD ΨABCD = Db Cab − √ acd C bc Kb d , 2 i 1 cd 3 f d Σab = √ df (a D C b) + C Kcd hab + Cab K f f − C c (a Kb)c . 2 2 2
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The integrand for I2 is given by ˆ BCD P + Ψ ˆ ABCP ξˆD P ) I2 = (3κ(A F ΣBCD)F + Ψ(ABC F ξD)F )(3ˆ κAP Σ 9 9 3¯ 9 bc a ¯ bc κa κ ¯ ac κa κ ¯ bc κa κ ¯ a + Σb c Σ ¯ b + Σa c Σ ¯b + Σ ¯ ζa = − Σbc Σ bc C κ 2 2 4 2 3¯ 3¯ 3 3 c a b c a b − Σ ¯ ζ − Σ ¯ ζ + Σbc C¯ bc κa ζ¯a − Σac C¯b c κa ζ¯b ac Cb κ bc Ca κ 4 2 2 4 3 1 1 1 − Σbc C¯a c κa ζ¯b + C bc C¯bc ζ a ζ¯a + Cb c C¯ac ζ a ζ¯b + Ca c C¯bc ζ a ζ¯b . 2 2 2 4 The complete invariant is given by
I = (J + I1 + I2 )dμ. S
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[15] Dain, S.: A new geometric invariant on initial data for the Eistein equations. Phys. Rev. Lett. 93, 231101 (2004) [16] Debever, R., Kamran, N., McLenahan, R.: Exhaustive integration and a single expression for the general solution of the type D vacuum and electrovac field equations with cosmological constant for a singular aligned Maxwell field. J. Math. Phys. 25, 1955 (1984) [17] Ferrando, J.J., S´ aez, J.A.: An intrinsic characterization of the Schwarzschild metric. Class. Quantum Grav. 15, 1323 (1998) [18] Ferrando, J.J., S´ aez, J.A.: On the invariant symmetries of the D-metrics. J. Math. Phys. 48, 102504 (2007) [19] Ferrando, J.J., S´ aez, J.A.: An intrinsic characterization of the Kerr metric. Class. Quantum Grav. 26, 075013 (2009) [20] Friedrich, H.: Smoothness at null infinity and the structure of initial data. In: Chru´sciel, P.T., Friedrich, H. (eds.) 50 Years of the Cauchy Problem in General Relativity. Birkh¨ ausser, Boston (2004) [21] Garc´ıa-Parrado, A., Valiente Kroon, J.A.: Initial data sets for the Schwarzschild spacetime. Phys. Rev. D 75, 024027 (2007) [22] Garc´ıa-Parrado, A., Valiente Kroon, J.A.: Kerr initial data. Class. Quantum Grav. 25, 205018 (2008) [23] Garc´ıa-Parrado, A., Valiente Kroon, J.A.: Killing spinor initial data sets. J. Geom. Phys. 58, 1186 (2008) [24] Geroch, R.: Spinor structure of spacetimes in general relativity I. J. Math. Phys. 9, 1739 (1968) [25] Geroch, R.: Spinor structure of spacetimes in general relativity II. J. Math. Phys. 11, 343 (1970) [26] Huang, L.-H.: Solutions of special asymptotics to the Einstein Constraint equations. Class. Quantum Grav. 27, 245002 (2010) [27] Hughston, L.P., Sommers, P.: The symmetries of Kerr black holes. Commun. Math. Phys. 33, 129 (1973) [28] Hughston, L., Penrose, R., Sommers, P., Walker, M.: On a quadratic first integral for the charged partice orbits in the charged Kerr solutions. Commun. Math. Phys. 27, 303 (1972) [29] Ionescu, A.D., Klainerman, S.: On the uniqueness of smooth, stationary black holes in vacuum. Invent. Math. 175, 35 (2009) [30] Jeffryes, B.P.: Space-times with two-index Killing spinors. Proc. R. Soc. Lond. A 392, 323 (1984) [31] Kamran, N., McLenaghan, R.G.: Separation of variables and constants of the motion for the Dirac equation on curved spacetime. Acad. R. Belg. Bull. Cl. Sci. 70, 596 (1984) [32] Kerr, R.P.: Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11, 237 (1963) [33] Kinnersley, W.: Type D vacuum metrics. J. Math. Phys. 10, 1195 (1969) [34] Komar, A.: Covariant conservation laws in general relativity. Phys. Rev. 113, 934 (1958) [35] Lockhart, R.B.: Fredholm properties of a class of elliptic operators on noncompact manifolds. Duke Math. J. 48, 289 (1981)
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[36] Mars, M.: A spacetime characterization of the Kerr metric. Class. Quantum Grav. 16, 2507 (1999) [37] Mars, M.: Uniqueness properties of the Kerr metric. Class. Quantum Grav. 17, 3353 (2000) [38] Mart´ın-Garc´ıa, J.M.: http://www.xAct.es [39] Misner, C.W.: The method of images in geometrodynamics. Ann. Phys. 24, 102 (1963) [40] Penrose, R.: Naked singularities. Ann. N. Y. Acad. Sci. 224, 125–134 (1973) [41] Penrose, R., Rindler, W.: Spinors and space-time. In: Two-Spinor Calculus and Relativistic Fields, vol. 1. Cambridge University Press, Cambridge (1984) [42] Penrose, R., Rindler, W.: Spinors and space-time. In: Spinor and Twistor Methods in Space-Time Geometry, vol. 2. Cambridge University Press, Cambridge (1986) [43] Simon, W.: Characterizations of the Kerr metric. Gen. Rel. Grav. 16, 465 (1984) [44] Sommers, P.: Space spinors. J. Math. Phys. 21, 2567 (1980) [45] Stephani, H., Kramer, D., MacCallum, M.A.H., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations, 2nd edn. Cambridge University Press, Cambridge (2003) [46] Tod, K.P.: Three-surface twistors and conformal embedding. Gen. Rel. Grav. 16, 435 (1984) [47] Valiente Kroon, J.A.: Characterization of Schwarzschildean initial data. Phys. Rev. D 72, 084003 (2005) [48] Walker, M., Penrose, R.: On quadratic first integrals of the geodesic equation for type {22} spacetimes. Commun. Math. Phys. 18, 265 (1970) Thomas B¨ ackdahl and Juan A. Valiente Kroon School of Mathematical Sciences Queen Mary University of London Mile End Road London E1 4NS UK e-mail:
[email protected];
[email protected] Communicated by Piotr T. Chrusciel. Received: June 7, 2010. Accepted: September 29, 2010.
Ann. Henri Poincar´e 11 (2010), 1273–1283 c 2010 The Author(s). This article is published with open access at Springerlink.com 1424-0637/10/071273-11 published online December 14, 2010 DOI 10.1007/s00023-010-0061-4
Annales Henri Poincar´ e
Schwinger Functions in Noncommutative Quantum Field Theory Dorothea Bahns Abstract. It is shown that the n-point functions of scalar massive free fields on the noncommutative Minkowski space are distributions which are boundary values of analytic functions. Contrary to what one might expect, this construction does not provide a connection to the popular traditional Euclidean approach to noncommutative field theory (unless the time variable is assumed to commute). Instead, one finds Schwinger functions with twistings involving only momenta that are on the mass-shell. This explains why renormalization in the traditional Euclidean noncommutative framework crudely differs from renormalization in the Minkowskian regime.
1. Introduction A quantum field theoretic model is to a large part determined by the choice of a partial differential operator. For physical reasons, this operator has to be hyperbolic, and one of its fundamental solutions, the so-called Feynman propagator, is the building block in any perturbative calculation of physically relevant quantities. Nonetheless, ever since proposed by Symanzik in 1966 [10] based on ideas of Schwinger, the so-called Euclidean framework has played a very important role. In this framework, the building block is the so-called Schwinger function, a fundamental solution of an elliptic partial differential operator. The Euclidean formalism not only simplifies calculations, but seems to be indispensable in constructive quantum field theory. The remarkable theorem of Osterwalder and Schrader gives sufficient conditions for the possibility to recover the original hyperbolic (physically meaningful) field theory from a Euclidean framework, and therefore justifies the Euclidean framework in ordinary quantum field theory. It is recalled below how the Schwinger function of the Euclidean framework of free scalar field theory is derived by analytic
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continuation from the hyperbolic theory and how it relates to the Feynman propagator. To incorporate gravitational aspects into quantum field theory, one possibility is to study quantum fields on noncommutative spaces, the most popular of which is the noncommutative “Moyal space” whose coordinates are subject to commutation relations of the Heisenberg type [4]. Already in that early paper, a possible setting for (unitary) hyperbolic perturbative quantum field theory was proposed, where the field algebra is endowed with a noncommutative product, the twisted (convolution) product. Notwithstanding, the vast majority of publications on field theory on noncommutative spaces (“noncommutative field theory”) has been and still is formulated within a Euclidean setting. This setting was not derived from a hyperbolic noncommutative theory but directly from the Euclidean framework of ordinary field theory by replacing all products with twisted ones. I shall refer to this approach as the traditional noncommutative Euclidean framework. Despite some attempts, it has not been possible to relate this traditional noncommutative Euclidean setting to some hyperbolic noncommutative theory—in fact, there is evidence that it might be impossible to do so, unless the time variable commutes with all space variables. It became clear after some years that within the traditional Euclidean noncommutative framework, already the models built from the most harmless of fields, namely the scalar massive fields, have very peculiar properties. Most notably, the so-called ultraviolet–infrared mixing problem noted in [8] severely limits the type of model that can be defined at all [6,7]. It is at present not clear whether the ultraviolet–infrared mixing problem is present in a hyperbolic setting, as the calculations and the combinatorial aspects of hyperbolic noncommutative field theory are quite involved. It is therefore desirable to find a Euclidean framework that can actually be derived from a hyperbolic noncommutative setting in the hope that—as in ordinary quantum field theory—such a Euclidean setting might simplify the combinatorial aspects of perturbation theory and that the full Euclidean machinery of renormalization might be employed. In such a setting, it might be feasible to investigate a theory’s renormalizability and the ultraviolet-infrared mixing problem in general. As a very first step towards this goal, I will show in this note that one can indeed derive a noncommutative Euclidean framework from a hyperbolic theory of free fields on the Moyal space, and that this framework is not the traditional one that is investigated in the literature. In contrast to this traditional framework, the new Euclidean framework can moreover be related to a setting involving Feynman propagators via an analytic continuation similar to the one of ordinary quantum field theory. The main idea is to take the Wightman functions (of free fields) as the starting point. This is justified by the fact that within axiomatic quantum field theory, the Wightman functions are the fundamental objects from which the field content can be recovered by the celebrated reconstruction theorem. On Minkowski space, a simple construction allows to pass from the Wightman functions (of free fields) to the Schwinger functions (of free fields), and
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from there to expressions involving (hyperbolic) Feynman propagators. This procedure will be mimicked in the present note. The note is organized as follows: In the next section it is recalled how the Schwinger function is derived in ordinary massive scalar quantum field theory and how it is related to the Feynman propagator. In the third section, a Euclidean 4-point function (Schwinger function) is derived from a noncommutative hyperbolic Wightman function of 4 free massive scalar fields and the prescription how Schwinger functions of arbitrarily high order are calculated is given. It is shown that the Euclidean framework thus derived differs from the traditional noncommutative Euclidean approach. Moreover, the relation to Feynman propagators is clarified. In an outlook I will briefly comment on further possible research that ensues from these new results.
2. Euclidean Methods in Quantum Field Theory The hyperbolic partial differential operator of massive scalar field theory is the ∂2 2 4 massive Klein–Gordon operator P := ∂x 2 − Δx + m on R where Δx denotes 0 3 3 the Laplace operator on R , x ∈ R , and m > 0 is a real parameter, called the field’s mass. As mentioned in the introduction, all the relevant quantities of a scalar field theoretic model can be calculated from a fundamental solution of this operator. Recall here that a distribution E ∈ D (Rn ) is a fundamental solution (or Green’s function) of a partial linear differential operator P (∂) on Rn provided that in the sense of distributions, P (∂)E = −δ with δ denoting the δ-distribution. Our starting point here, however, is the 2-point-function Δ+ ∈ S (R4 ), a tempered distribution which is a solution (not a fundamental solution) of the Klein–Gordon equation, P Δ+ = 0 in the sense of distributions. For x = (x0 , x) ∈ R4 , x0 ∈ R, x ∈ R3 , it is given explicitly by 1 1 −iωk x0 +ikx 3 e d k, where ωk = k2 + m2 , Δ+ (x) = 3 (2π) 2ωk an expression which in fact makes sense as an oscillatory integral, see [9, Sect. IX.10] for details. Here and in what follows, boldface letters denote elements of R3 and an expression such as kx is shorthand for the canonical scalar product of k and x. It is well-known that Δ+ is the boundary value (in the sense of distributions) of an analytic function. To see this, let us first fix some notation. Let a ∈ Rn with |a| = 1, let θ ∈ (0, π/2), and let ay denote the canonical scalar product in Rn . Then the cone about a with opening angle θ is the set Γa,θ = {y ∈ Rn | ya > |y| cos θ} ⊂ Rn . Let Γ∗a,θ denote the dual cone, Γ∗a,θ := Γa, π2 −θ . For tempered distributions whose support is contained in the closure of a cone, the following general assertion holds: Theorem 1 ([9, Thm. IX.16]). Let u be a tempered distribution with support in the closure of a cone Γa,θ , a ∈ Rn , 0 < θ < π/2. Then its Fourier transform u ˜
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is the boundary value (in the sense of tempered distributions) of a function f which is analytic in the tube {z ∈ Cn | − im z ∈ Γ∗a,θ } =: Rn − iΓ∗a,θ ⊂ Cn . Observe that for u ˜ to be the boundary value of f as above in the sense of tempered distributions means that for any η ∈ Γ∗a,θ and for any test function g ∈ S(R4 ), we have for t ∈ R approaching 0 from above, f (x − itη)g(x) dx → u ˜(g) as tempered distributions. ˜ + of the 2-point function, The Fourier transform Δ ˜ + (p0 , p) = 1 δ(p0 − ωp ), Δ 2ωp
(2.1)
is a tempered distribution whose support (the positive mass shell) is contained in the closure of the cone Γ+ := Γ(1,0),π/4 (the forward light cone). Applied to ˜ + , Theorem 1 thus guarantees that u u := Δ ˜ = Δ+ is the boundary value of a function f which is analytic in R4 − iΓ+ (observe that Γ∗+ = Γ+ ). Explicitly, for x = (x0 , x) ∈ R4 and η = (x4 , 0) ∈ Γ+ (hence x4 > 0), we have in this case 1 1 +ikx−ωk (x4 +ix0 ) 3 f (x − iη) = e d k. (2.2) (2π)3 2ωk We now define a function s via s(x, x4 + ix0 ) := f (x − iη) 4
for x = (x0 , x) ∈ R and η = (x4 , 0) ∈ Γ+ as above. Making use of the identity ∞ ik4 x4 e 1 −ωk x4 1 e = dk4 for x4 > 0 (2.3) 2 2ωk 2π k + m2 −∞
4
2
where k = (k, k4 ) ∈ R , k = k2 + k42 , and setting x0 = 0 in (2.2), we then find that 1 eikx s(x) = d4 k where x = (x, x4 ) ∈ R4 , x4 > 0. (2.4) (2π)4 k 2 + m2 One now extends the function s to a distribution S ∈ S (R4 ), the so-called Schwinger function, by dropping the restriction on x4 . Then the integral still makes sense as an oscillatory integral 1 eikx d4 k 4 2 (2π) k + m2 Observe that S is the unique fundamental solution of the elliptic partial differential operator Δ − m2 with Δ the Laplace operator on R4 . As mentioned in the introduction, the building block in hyperbolic perturbation theory is the Feynman propagator ΔF , a fundamental solution for ∂2 2 the Klein–Gordon operator P = ∂x 2 − Δx + m . Without going into details, 0 let me mention that, remarkably, the Fourier transform S˜ of the Schwinger ˜ F of the function is the analytic continuation of the Fourier transform Δ
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Feynman propagator ΔF (up to a sign). In fact, formally, for the kernel w given by ˜ F (p0 + ip4 , p), w(p, p4 − ip0 ) := Δ we have ˜ p4 ) = −w(p, p4 ) S(p, ˜ for the Schwinger function’s Fourier transform S.
3. Analytic Continuation in the Noncommutative Case It would be beyond the scope of this note to explain the possible unitary perturbative setups for massive scalar fields on the noncommutative Moyal space with hyperbolic signature (see [3] for a comparison). Only two features of such noncommutative (hyperbolic) field theories matter here. The first is the fact that our starting point still is the Klein–Gordon operator and the 2-point-function Δ+ discussed in the previous section. The second important feature—and this feature is shared by the traditional noncommutative Euclidean formalism—is the fact that one has to consider not only products but also twisted products of distributions. To fix the notation, we note here that for two Schwartz functions f, g ∈ S(R4 ) this twisted product (Moyal product) is i (3.1) f ∗ g(x) = f˜(k)˜ g (p)e−i(p+k)x e− 2 pθk d4 k d4 p where f˜ and g˜ denote the Fourier transforms of f and g, respectively, and where θ is a nondegenerate antisymmetric 4 × 4-matrix. Observe that in a Euclidean theory, a product such as kx stands for the canonical scalar product on R4 , whereas in a hyperbolic setting, it denotes a Lorentz product, kx = k0 x0 − kx for x = (x0 , x) and k = (k0 , k), with kx denoting the canonical scalar product i on R3 . The oscillating factor e− 2 pθk is also called the twisting. 3.1. The Tensor Product of 2-Point Functions Since the 2-point function remains unchanged in noncommutative field theory, we have to consider higher order correlation functions in order to see a difference between field theory on Moyal space and ordinary field theory. Again, it would be beyond the scope of this note to explain the whole setup. It will be sufficient to consider as an example a particular contribution to the so-called 4-point function of free massive scalar field theory. In ordinary field theory, the distribution of interest here is the 2-fold tensor product of 2-point functions, 1 1 1 −i(ωk x0 +ωp y0 )+i(kx+py) 3 3 (2) e d kd p (3.2) Δ+ (x, y) = (2π)6 2ωk 2ωp The 4-point function (the vacuum expectation value of four fields) is given as a linear combination of such tensor products, i.e. (2) Δ+ (xi1 − xj1 , xi2 − xj2 ) Ω, φ(x1 )φ(x2 )φ(x3 )φ(x4 )Ω =
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where the sum runs over all pairs (i1 , j1 ), (i2 , j2 ) of indices with {i1 , i2 , j1 , j2 } = {1, 2, 3, 4} and i1 < j1 , i2 < j2 . Observe that each of these index sets labels one of the three possibilities to contract the four fields, i.e. (i1 , j1 ) = (1, 3), (i2 , j2 ) = (2, 4) corresponds to the contribution to the 4-point function where the first field is contracted with the third, and the second with the fourth. By standard arguments from microlocal analysis involving the wavefront set of distributions, it can be shown that even the pullback of this tensor product with respect to the diagonal map, that is, the product in the sense of H¨ormander, is a well-defined distribution ∈ D (R4 ) (see for instance [9, Chap. IX.10]) which can then be shown to be still tempered. For the kernel given by (3.2), this would amount to setting x = y. In order to avoid issues regarding renormalization later, in this note, however, only tensor products of distributions will be considered. It is well-known and, by application of Theorem 1, in fact not difficult to (2) see that Δ+ is again the boundary value of an analytic function: (2)
Lemma 2. The tempered distribution Δ+ is the boundary value of a function f2 which is analytic in R4 × R4 − iΓ+ × Γ+ . Explicitly, for z = (x0 , x, y0 , y) ∈ R4 × R4
and
η = (x4 , 0, y4 , 0) ∈ Γ+ × Γ+
(hence x4 and y4 > 0), we have 1 1 1 −ωk (x4 +ix0 )−ωp (y4 +iy0 )+ikx+ipy 3 3 e d kd p. f2 (z − iη) = (2π)6 2ωk 2ωp For the function s2 defined for η and z as above, by s2 (x, x4 + ix0 , y, y4 + iy0 ) := f2 (z − iη), we find for (x, y) = (x, x4 , y, y4 ) ∈ R4 × R4 , x4 and y4 > 0, the explicit form 1 1 1 e+ikx+ipy d4 k d4 p (3.3) s2 (x, y) = 8 2 2 2 (2π) k + m p + m2 where p2 = p2 + p24 , and likewise, k 2 = k2 + k42 . Proof. The first claim is a direct consequence of Theorem 1 applied with respect to x and y separately, and the second claim follows again from the identity (2.3). As in the previous section, one again drops the restrictions on x4 and y4 and thereby extends s2 to a distribution S2 , whose Fourier transform is the smooth function 1 1 ˜ S(p) ˜ S(k) = 2 2 2 k + m p + m2 Again, upon restriction of S2 to R3 × R>0 × R3 × R>0 , it is equal to the function s2 . As an aside, it is mentioned that when one considers the pullback of (2) Δ+ with respect to the diagonal map (such that, formally, one finds x = y in (3.3)), then the kernel S2 (x, x) is the Fourier transform of the convolution 1 1 ˜ S˜ × S(k) = d4 p. (k − p)2 + m2 p2 + m2
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Moreover, let us consider again, how Feynman propagators enter the game. As is well-known, S˜2 is the analytic continuation of a product of Feynman propagators. Explicitly, we find that its Fourier transform S˜2 is given in terms of the kernel ˜ F (k0 + ik4 , k)Δ ˜ F (p0 + ip4 , p) w2 (k, k4 − ik0 , p, p4 − ip0 ) := Δ as follows S˜2 (k, k4 , p, p4 ) = −w2 (k, k4 , p, p4 ). It is well-known that the procedure applied to the twofold tensor product in lemma 2 can be applied more generally. Each contribution to the (hyperbolic) 2n-point function (or Wightman function) is an n-fold tensor product of 2-point functions (n-point functions for odd n vanish). In order to find the corresponding higher order Schwinger function, one considers the analytic continuation according to Theorem 1 in each of the n variables and proceeds in the same manner as explained for the 4-point function above. 3.2. The Twisted Product of 2-Point Functions In [1,2], it was shown how 2n-point functions are calculated in hyperbolic massive scalar field theory on the noncommutative Moyal space (n-point functions for n odd still vanish). Their general properties were investigated in [2]. As it turns out, the first deviation from ordinary field theory shows up in the 4-point function, where one of the contributions is a twisted tensor product of two 2-point functions, 1 1 −i(ωk x0 +ωp y0 )+i(kx+py) −ipθ ˜ (2) e e ˜ k d3 kd3 p (3.4) Δ+ (x, y) := 2ωk 2ωp where k˜ = (ωk , k), and p˜ = (ωp , p). In the terminology of physics, this means that the momenta k and p in the oscillating factor are on-shell. This will turn out to be very important later on. It is also important to note that, while our starting point is the twisted product (3.1) on R4 , the vectors in the twisting are on-shell as a consequence of the support properties of ˜ + (k0 , k) = 1 δ(k0 − ωk ). Δ 2ωk Observe also that compared to the ordinary twisting in (3.1), the factor 2 in the oscillating factor appears, since in the calculations, two oscillating factors as in (3.1) either cancel or (in the above case) add up, see [1,2]. In fact, if we consider the three different contributions to the 4-point function as explained after equation (3.2), we find that two of them are the same as in the commutative case (the twistings cancel), and only in the contribution where the first field is contracted with the third, and the second with the fourth, the twistings add up (hence the factor 2). Once more, we now apply Theorem 1.
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(2)
Lemma 3. The tempered distribution Δ+ is the boundary value of a function f2θ which is analytic in R4 × R4 − iΓ+ × Γ+ . Explicitly, for z = (x0 , x, y0 , y) ∈ R4 × R4 and η = (x4 , 0, y4 , 0) ∈ Γ+ × Γ+ (hence x4 and y4 > 0), we have 1 1 1 −ωk (x4 +ix0 )−ωp (y4 +iy0 )+ikx+ipy −ipθ ˜ e e ˜ k d3 kd3 p f2θ (z − iη) = (2π)6 2ωk 2ωp where k˜ = (ωk , k), p˜ = (ωp , p). For the function sθ2 defined for η and z as above, by sθ2 (x, x4 + ix0 , y, y4 + iy0 ) := f2θ (z − iη) we then find for (x, y) = (x, x4 , y, y4 ) ∈ R4 × R4 with x4 , y4 > 0, 1 1 1 ˜ 4 ˜ k e+ikx e+ipy e−ipθ d k d4 p (3.5) sθ2 (x, y) = (2π)8 k 2 + m2 p 2 + m2 where p2 = p2 + p24 and p2 = p2 + p24 , and with k˜ = (ωk , k), p˜ = (ωp , p) as above. (2)
Proof. To see that Δ+ is a tempered distribution, we first note that the twisting is in the multiplier algebra of S(R8 ). Hence, for any Schwartz function f ∈ S(R8 ), we have (∗2)
(2)
Δ+ (f ) = Δ+ (gθ ) where gθ ∈ S(R8 ) is the (inverse) Fourier transform of the product of f˜ and (2) the twisting. The claim follows, since the tensor product Δ+ is tempered. (2) Since the support of the Fourier transform of Δ+ is still contained in (2) the closure of Γ+ × Γ+ , it follows from Theorem 1 that Δ+ indeed is the boundary value of the analytic function fθ . The explicit form of sθ again follows from the identity (2.3)—which, as should be noted, does not affect the twisting factor. Observe that sθ2 and s2 from Lemma 2 differ only by the oscillating factor e . As before, we now extend sθ2 to a distribution S2θ by dropping the restriction on x4 and y4 , such that S2θ is given by the Fourier transform of the smooth function 1 1 ˜ ˜ k S˜2θ (k, p) = 2 e−ipθ . (3.6) 2 2 2 k +m p +m Again, in the case of coinciding points, instead of S˜2θ (k, p) one considers the Fourier transform of the (now twisted) convolution 1 1 ˜ k−p 4 e−ipθ d p (k − p)2 + m2 p2 + m2 ˜ −ipθ ˜ k
where k − p = (ωk−p , k − p). It is very important to note that the momenta which appear in the oscillating factors in all the expressions above are on-shell, i.e. that they are of the form p˜ = (ωp , p), likewise for k or p − k. The oscillating factor therefore distinguishes the components of (p, p4 ) and is, in particular, independent of
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the fourth component p4 . The reason for this lies in the fact that the Fourier transform of the 2-point function forces the momenta in the oscillating factor to be on-shell, and this is not changed by the analytic continuation. These considerations turn out to be crucial in the following assertion: Remark 4. Since the oscillating factor in (3.6) is independent of one of the components of k and p ∈ R4 , it is obvious that S˜2θ is the analytic continuation a product of Feynman propagators with an on-shell twisting. Explicitly, we find that the Schwinger function’s S2θ Fourier transform S˜2θ is given in terms of the kernel ˜ ˜ k ˜ F (k0 + ik4 , k)Δ ˜ F (p0 + ip4 , p)e−ipθ wθ (k, k4 − ik0 , p, p4 − ip0 ) := Δ 2
as follows S˜2θ (k, k4 , p, p4 ) = −w2θ (k, k4 , p, p4 ). Remark 5. All this remains true when one calculates the higher order Schwinger functions from the 2n-point functions. These latter distributions are of a similar form as (3.4), i.e. they are twisted tensor products of 2-point functions where a certain combinatorics determines which combinations of momenta appear in the twistings, see [2]. The important point is that again, all momenta in the twistings are on-shell. Therefore, the same construction that was employed for the 4-point function above, i.e. an analytic continuation in the n variables separately, can be applied and again leads to Schwinger functions with twistings that remain on-shell. Finally, the analytic continuation of the corresponding Fourier transform can be performed as in Remark 4 and leads to (twisted products of) Feynman propagators with twistings still only involving mass-shell momenta. The fact that one starts from hyperbolic two-point functions which in turn force the momenta in the twistings to be on-shell is the essential difference to the traditional noncommutative Euclidean framework employed in the literature. In this latter framework, (Euclidean) Schwinger functions are the starting point, and of course, when twisted products appear, by (3.1) the oscillating factors depend on all four components of a momentum vector k = (k, k4 ). For instance, instead of finding s˜θ2 as in (3.6), one starts from the following expression 1 1 e˜θ2 (k, p) = 2 e−ipθk (3.7) k + m2 p 2 + m2 where k = (k, k4 ) and p = (p, p4 ). So far, it was not possible to relate this framework to a hyperbolic one, the main difficulty being the dependence of the oscillating factor on k4 . Naively copying the procedure sketched on the previous pages and in Remark 4 to pass to Feynman propagators (via the kernels w, w2 and w2θ , respectively) leads to exponentially increasing terms which render the integrals ill-defined. Not even the apparently easy way out to make the oscillating factor independent of one of the components in an ad hoc way, by requiring θ to be a matrix of rank 2 (“spacelike noncommutativity”), provides a sensible theory in the long run, cf. [5] and [1, Sect. 5].
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Remark 4 shows that such measures are unnecessary when the new noncommutative Euclidean framework derived from the hyperbolic n-point functions is employed.
4. Outlook Based on the above considerations, the most important question now is whether it is possible to set up a consistent Euclidean noncommutative framework with on-shell momenta in the twisting. An obstruction might be that, as can be easily seen already in the example S2θ discussed above, the higher order Schwinger functions are not symmetric with respect to reflections in the origin. Also, the new Euclidean on-shell product is not associative. This may jeopardize the possibility to set up a complete consistent perturbative framework using a Schwinger functional. Still, it is to be hoped that the results presented here open many interesting possibilities for future research. For one thing, one should try to generalize the Osterwader Schrader Theorem in this setting. Also, it would be most interesting to study whether the ultraviolet-infrared mixing problem appears in this setting at all. Certainly,there is reason to hope so, since the most prominent graph (the nonplanar tadpole) that exhibits this problem in the traditional Euclidean noncommutative approach, does not do so, when one simply replaces its twisting by an on-shell twisting. Last but not least, a thorough understanding of the new Euclidean setup (if feasible) should enable us to learn more about hyperbolic noncommutative models—which in themselves have proved to be quite difficult to treat. If a consistent Euclidean perturbative setup can be developed from the ideas presented here, general proofs of renormalizability of hyperbolic noncommutative field theory should at last be possible.
Acknowledgments Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of G¨ ottingen Open Access. This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
References [1] Bahns, D., Doplicher, S., Fredenhagen, K., Piacitelli, G.: Field theory on noncommutative spacetimes: quasiplanar wick products. Phys. Rev. D 710, 025022 (2005)
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[2] Bahns, D.: Local counterterms on the noncommutative Minkowski space. In: Rigorous Quantum Field Theory. Progr. Math., vol. 251, pp. 11–26. Birkh¨ auser, Basel (2007) [3] Bahns, D.: Perturbative methods on the noncommutative Minkowski space. PhD thesis, Hamburg [DESY-THESIS-2004-004] (2003) [4] Doplicher, S., Fredenhagen, K., Roberts, J.E.: The quantum structure of spacetime at the Planck scale and quantum fields. Commun. Math. Phys. 172(1), 187–220 (1995) [5] Gayral, V., Gracia-Bondia, J.M., Ruiz, F.R.: Trouble with space-like noncommutative field theory. Phys. Lett. B 610, 141–146 (2005) [6] Grosse, H., Wulkenhaar, R.: Renormalisation of φ4 -theory on noncommutative R4 in the matrix base. Commun. Math. Phys. 256(2), 305–374 (2005) [7] Gurau, R., Rivasseau, V., Tanasa, A.: A translation-invariant renormalizable non-commutative scalar model. Commun. Math. Phys. 287, 275–290 (2009) [8] Minwalla, S., Van Raamsdonk, M., Seiberg, N.: Noncommutative perturbative dynamics. J. High Energy Phys. 2, 20–31 (2000) [9] Reed, M., Simon, B.: Methods of modern mathematical physics. II. In: Fourier Analysis, Self-Adjointness. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1975) [10] Symanzik, K.: Euclidean quantum field theory. I. Equations for a scalar model. J. Math. Phys. 7, 510–525 (1966) Dorothea Bahns Courant Research Centre “Higher Order Structures in Mathematics” Universit¨ at G¨ ottingen Bunsenstr. 3-5 37073 G¨ ottingen Germany e-mail:
[email protected] Communicated by Raimar Wulkenhaar. Received: January 19, 2010. Accepted: October 6, 2010.
Ann. Henri Poincar´e 11 (2010), 1285–1302 c 2010 Springer Basel AG 1424-0637/10/071285-18 published online October 30, 2010 DOI 10.1007/s00023-010-0057-0
Annales Henri Poincar´ e
Large Newforms of the Quantized Cat Map Revisited Rikard Olofsson Abstract. We study the eigenfunctions of the quantized cat map, desymmetrized by Hecke operators. In the papers (Olofsson in Ann Henri Poincar´e 10(6):1111–1139, 2009; Math Phys 286(3):1051–1072, 2009) it was observed that when the inverse of Planck’s constant is a prime exponent N = pn , with n > 2, half of these eigenfunctions become large at some points, and half remains small for all points. In this paper we study the large eigenfunctions more carefully. In particular, we answer the question of for which q the Lq norms remain bounded as N goes to infinity. The answer is q ≤ 4.
1. Introduction The subject of quantum chaos studies the quantum analogs of classical dynamical systems. In particular, much interest concerns chaotic systems and what properties the corresponding quantum system must have or can be expected to have. One of the general heuristic ideas is, that since particles in a chaotic system often move in a space filling manner, the wave function measuring the probability of finding the particle at a specific place must be evenly spread in position space. The wave function is a linear combination of eigenstates corresponding to different energies and the expected behavior is that most eigenstates are nicely spread. We will, throughout the paper, assume all eigenstates and eigenfunctions to be L2 normalized, and given this normalization, all eigenstates should be quite similar to each other and to a constant function. To make rigorous statements in the line of this general idea one needs to find some property that measures this. There are several such properties at hand, but maybe the most natural objects to study are the possible limits as the energy goes to infinity (this is often called the semiclassical limit) of subsequences of the induced measures of the eigenstates. For large families R. Olofsson is supported by a grant from the Swedish Research Council.
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of quantum mechanical systems it is known that a sequence of full density, i.e., such that the proportion of elements left out from the sequence tends to zero, of these induced measures converge to the uniform measure [1,2,20]. This is known as Schnirelman’s theorem and a model having this property is called quantum ergodic. If the sequence of all induced measures tends to the uniform measure the model is called quantum uniquely ergodic. There are also other, more qualitative properties of the sequences, which are studied intensively, such including the Lq norms and the Shannon entropy of the eigenstates. We will study one of the most common “toy models” in quantum chaos, namely the cat map. Given a hyperbolic matrix A ∈ SL(2, Z) we get a chaotic, time discrete, dynamical system by the mapping taking the point x ∈ T2 = R2 /Z2 to Ax ∈ T2 and this is what is known as the cat map. There are several different quantizations of this system [3,4,6,7,10,11,14,20], but we will follow [12]. In particular, this means that our quantized cat map is desymmetrized by Hecke operators, and also that we make the further assumption A ≡ I (mod 2) on the matrix A. This assumption is related to the physical relevance of the model, not to the mathematical truth of our theorems. The quantized cat map is well studied and much is known about the semiclassical limit, which in this model corresponds to letting the integer N, the dimension of the Hilbert space of states, go to infinity. The integer N is often called the inverse of Planck’s constant. The most important result about the semiclassical limit is due to Kurlberg and Rudnick and states that the model is uniquely quantum ergodic [12]. It should be noted that the desymmetrization is necessary in the sense that other induced limits exists, if this extra condition is dropped [5]. Also the supremum norms of the eigenstates and the Shannon entropies are asymptotically known as N goes to infinity for our model. The upper bound for the supremum norm of all eigenstates is O(N 1/4 ) for “almost all” N (see [15] Theorem 1.1 for the exact formulation) and the lower bound for the Shannon entropy is 1/2 log N [16]. Both these inequalities are sharp, but equality only occurs for special N and very special eigenstates. In short, one might say that the number theoretic properties of N starts to come into play. We will restrict to odd N throughout this introduction because it simplifies the expressions below, but the idea is the same also for even N. If M > 1 is such that M 2 |N then the state space L2 (ZN ) = L2 (Z/N Z) ∼ = CN can be decomposed into two parts, where eigenstates in the first part are called oldforms and eigenstates in the second part are called newforms. The space spanned by the oldforms have dimension N/M 2 and is nothing but blown up images of states corresponding to the Planck’s constant N/M 2 living as N/M -periodic functions supported on the ideal M ZN/M ⊂ ZN . If N = M 2 , the unique oldform for M is the function that gives equality in the estimates above. The newforms are more interesting eigenstates. Here the number theoretic properties of N comes into play again. It is well known that one can use the Chinese remainder theorem to write any eigenstate ψN ∈ L2 (ZN ) as d a tensor product of eigenstates ψpi ni ∈ L2 (Zpi ni ) with N = i=1 pi ni . We therefore formulate the results for N = pn and the behavior for general N is given directly from the tensor decomposition.
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Before we state the known results for the newforms it seems in order to point out that there is a close connection between the newforms and complete character sums over ZN . Given a specific cat map, the value of a newform can in fact be written as a character sum for “half of all primes” p. These primes are often called split and corresponds to ( D p ) = 1, where D is defined in Definition 2.1. Changing point of evaluation corresponds to changing character sum. This was observed by Kurlberg and Rudnick [13] for the case when N = p is a prime, but the construction can be made for N = pn without any extra effort. More precisely, we have that N qb (y) C χ(y)e ψ(b) = √ , pn N y=1 where |C| = (1 − 1/p)−1/2 , χ(y) is a multiplicative character on the units of ZN and zero for other y and qb (y) is a second degree polynomial where the linear and constant term is dependent on b. All results stated below applies to this special family of character sums. A general treatment of character sums may be found in Chapter 12 of [9]. For N = pn with n = 1 all eigenstates are uniformly bounded at all points and the same is true for all newforms for n = 2. If, however, n > 2, then the set of newforms (defined up to multiplication by scalars) is divided into two parts which are of the same size. In the first part all newforms are uniformly bounded at all points, but in the second part there is a small number of points where the functions are large. At the points where the function is as large as possible, it is of size p[n/3]/2 , but there are also points where it assumes intermediate values. Since the proportion of points where the function is large goes to zero as p goes to infinity, it was proven in [15] that the value distribution is the same for all newforms, when p goes to infinity through the primes. That is, for both large and small newforms, and for any cat map and with any fixed n ≥ 2. In this paper, we study the large newforms and try to get more detailed information about the points where these functions are large. One simple way to characterize this is to study their impact on the Lq norm, i.e., study the size of 1/q 1 q |ψ(x)| . ψq = N x∈ZN
More precisely, we ask the following question: for which q > 0 does the Lq norm of the newforms stay bounded as N goes to infinity? We will call the supremum of all such q the critical exponent of the sequence. Since the measure of the whole space is one, it is obvious that the norm is bounded for all q ≤ 2, but a priori there is no reason why it is bounded away from zero. However, the theorem about the value distribution (Theorem 1.2) in [15] rules out this possibility since it shows that |ψ(x)| > 1 for a positive proportion of x ∈ ZN . Another simple calculation shows that the Lq norms of oldforms always go to zero or infinity, depending on if q is less than 2, or larger than
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2, respectively (as long as p goes to infinity). With these observations in mind it seems that the only non-trivial question that can be raised concerning the boundedness of Lq norms is for the large newforms. Before we state the theorem we have to make a technical restriction on N. The reason is simply that for a finite set of primes p we have no real definition of what a large newform in L2 (Zpn ) is. Thus we need to bound the impact of these primes trivially. We call primes that divide tr(A)2 − 4 or that divide the element in the lower left corner of A “bad”. We also say that 3 is bad. Fix a positive integer m. We say that N is “good” with respect to m if there is no “bad” prime p such that pm |N. In this language it is fair to say that if N is “good”, all newforms are small at the “bad” primes. This is the same assumption that was made in Theorem 1.1 in [15], expect that we now include 3 in the “bad” primes. Theorem 1.1. The critical exponent for a sequence of large newforms is 4. The L4 norm is bounded if and only if we restrict to the sequence of N where if pn N and the newform corresponding to pn is large, then n is bounded. For q > 4 the Lq norm is bounded by Om (N (1−4/q)/6 ). Remark. For a general N, a newform is said to be large, if it is large for some newform in its tensor decomposition. The question of Lq bounds for eigenfunctions of general operators has been studied intensively. For a typical compact manifold the critical exponent for the eigenfunctions of the Laplace–Beltrami operator is 2, and there are no q > 2 for which the Lq norm is bounded. More interesting is the question of finding a general upper bound for ψλ q as a function of the eigenvalue λ. This was settled by Sogge [19], who proved that ψλ q = O(λδ(q) ), where δ(q) = 1/8 − 1/(4q), for 2 ≤ q ≤ 6 and δ(q) = 1/4 − 1/q, for 6 ≤ q ≤ ∞ for compact manifolds without boundary. This estimate is sharp for some manifolds: the most well-known such example is S 2 with the usual metric. In fact, the result of Sogge is more general than it is stated here and may also be generalized even more, see for instance [18]. Although it is not the general behavior, there are manifolds where a nontrivial critical exponent exists. One such example was provided by Zygmund [21], who proved that the eigenfunctions of the Laplacian for the 2-torus has bounded L4 norm. In an arithmetic setting the Lq norms has been discussed by Iwaniec and Sarnak [8]. They studied the eigenfunctions of the Laplace–Beltrami operator for the so-called arithmetic surfaces, compact and non-compact, where the eigenfunctions also were assumed to be eigenfunctions of the Hecke operators. They proved that the power 1/4 in the general upper bound for the L∞ norm stated above can be replaced by 5/24 + for these eigenfunctions and also that ψλ ∞ → ∞ as λ → ∞. The proved rate at which the norm blew up was very slow, but they conjectured that this was in fact not so far from the truth. More precisely, they conjectured that ψλ ∞ , and consequently all ψλ q with
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2 < q ≤ ∞, is bounded by O(λ ) for all > 0. For the most famous arithmetic surface, i.e., the modular surface, there has been some progress on this conjecture. Sarnak and Watson have proven, in a still unpublished paper and assuming the Ramanujan conjectures, that the L4 norm of the Hecke eigenfunctions is in fact bounded by O(λ ). The interested reader will find more details about the arithmetic surfaces in [17]. We will derive Theorem 1.1 from an investigation that in fact gives us a much more detailed description. Assume once more that N = pn and let k = [n/2]. Given a fixed large newform ψ of a fixed quantized cat map we will, for each point b ∈ ZN , define the level of the point (see Definition 2.5 for the exact definition). If the level is odd, but not maximal, the value of the newform at the point is zero (see Lemma 5.1), but if the level l is even, |ψ(b)| is of the size pl/4 (see Theorems 1.2, 1.4 and Proposition 1.3 for exact statements). A small newform will only have points of level 0. Let b have level l = 2s for some l < k. If b ≡ b (mod pn−k+s ), it is easy to see that b also has level l. The values ψ(b) and ψ(b ) are related through the following theorem: Theorem 1.2. Let b ∈ Zpn have level 2s for some s < k/2, where k = [n/2] and assume that ψ(b) = 0. Then there exists an integer x0 such that (ps x0 )2 ≡ −C + Db2 (mod pk ) and x0 t −x0 t ps/2 n−k+s ψ(b + tp )= αψ (b)e + βψ (b)e
pk−2s pk−2s 1 1− D p p where |αψ (b)| = |βψ (b)| = 1 and C and D are given by Definition 2.3 and Definition 2.1, respectively. Using the theorem it is straightforward to prove value distribution results for the newform at each level individually. We will not formulate this in a theorem, but only observe that all levels behave exactly the same, but on different scales. In other words, they all behave as level 0 and the value distribution of this level is given by Theorem 1.2 in [15]. In order to derive Theorem 1.1 we need to bound the values for all levels l, not only for the levels where l < k. In general, we prove the following proposition: Proposition 1.3. If b has level 2s then |ψ(b)| ≤ 1−
2 s/2 p . D p
1 p
It is easy to calculate the number of points that have a fixed level l > 0 (see Lemma 2.1) and Theorem 1.1 follows from the fact that the value of ψ at a point of even level l is of size pl/4 . In fact, one can calculate the asymptotic relations of each level individually. Fix a normalized newform ψ and let ψs be the restriction of ψ to the points of level 2s, for some s ≤ n/3. In other words, we let
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if x has level 2s . else
We now have the following theorem: Theorem 1.4. If ψs denotes the restriction of ψ to the points of level 2s, for some s ≤ n/3, then p−s(1−4/q)/2 ψs q is uniformly bounded away from both zero and infinity as N goes to infinity. That the sequence is bounded follows directly from Proposition 1.3. On the other hand, Theorem 1.4 shows that for a positive proportion of points of level 2s, Proposition 1.3 is not off by more than a constant. Note also that the value of the newform at a point of maximal level does not have to be 0 even if this level is odd. One can formulate an analogous statement for the restriction to maximal level, but we will not do this.
2. Basic Assumptions and Definitions The main purpose of this chapter is to introduce the necessary notation and ideas from [12,15,16]. We will let A ∈ SL(2, Z) be the hyperbolic matrix that determines the cat map and assume that N = pn , where p > 3 is a prime and n ≥ 2. We will also use the notation k = [n/2]. In order for our quantization to be consistent, we assume that A is congruent to the identity modulo 2. We also assume that the element in the lower left corner of A is invertible modulo N . Having made these assumptions it turns out (or will turn out) that the results are more or less independent of A. One of the reasons for this is that the Hecke operators we will study only depend on the following parameter: Definition 2.1. Let D ≡ (tr(A)2 − 4)/(4c2 ) (mod N ), where c is the element in the lower left corner of A. Remark. The parameter D says if A is possible to diagonalize modulo N. If D is a non-zero square modulo p, we can diagonalize A. If D is not a square, we cannot diagonalize A. Observe also that a matrix is diagonalized simultaneously with A if and only if it commutes with A modulo N. Definition 2.2. Given D ∈ ZN we let a bD ; a, b ∈ ZN , a2 − Db2 = 1 . HD = b a Our state space is L2 (ZN ) = L2 (Z/N Z) with the inner product 1 φ(x)ψ(x).
φ, ψ = N x∈ZN
A quantization, like the quantized cat map, assigns to every smooth real-valued function of the classical phase space (this is usually called an observable) an Hermitian operator acting on the state space. It also prescribes how the system evolves in time through a unitary operator called the quantum propagator. We will denote the quantum propagator for the quantized cat map by UN (A). The
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most important properties of UN (A) is that it is well defined for A ∈ SL(2, ZN ), that the function mapping A to UN (A) is a representation of SL(2, ZN ), and that it has the so-called exact Egorov property (see [12, page 48]). The representation is known as the Weil representation, and it can be defined by explicit formulas for the generators of the group, see [12, Section 4.3], although there are more natural ways to introduce this representation in more abstract settings. The element r occurring in the formulas is nothing but the inverse of 2 modulo N and we will also use this notation. The Hecke operators corresponding to A and N are the commutative group of operators UN (g), such that g = xI + yA, x, y ∈ Z and det(g) ≡ 1 (mod N ). However, one may, without loss of generality, assume that the Hecke operators are given by {UN (h); h ∈ HD } (see [16, page 1060]) and we will do this. In HD there is a neighborhood of the identity which forms a cyclic subgroup of order pk , namely {h ∈ HD ; h ≡ I (mod pn−k )}. The joint eigenfunctions of the Weil representation restricted to this subgroup form subspaces of L2 (ZN ) which were called VC with the following convention in [15,16]: Definition 2.3. For C ∈ Zpk we say that ψ ∈ VC if C 1 2pn−k D UN ψ. ψ=e 2pn−k 1 pk
(1)
Different Hecke eigenfunctions in VC are very similar, but we also intro which was more detailed than C and eigenfunctions duced the notation C
having the same C, where even more similar.
∈Z Definition 2.4. Let α = pk/2 . Given a Hecke eigenfunction ψ, we define C n−k/2 ] such that to be the integer in the interval [1, p
C 2α + Dα3 D 1 + 2Dα2 ψ. (2) ψ=e UN 2α + Dα3 1 + 2Dα2 pn−k/2 Let B denote the matrix appearing in the left-hand side of (2) and B denote the matrix appearing in the left-hand side of (1). An easy calculation
shows that B m ≡ B (mod N ) for m = pn−k /α and this implies that C ≡ C k (mod p ). One of the main observations in [15,16] was that the value of a Hecke eigenfunction ψ at the point b can be written as an exponential sum over the solutions to the equation x2 ≡ −C + Db2 (mod pk ). How many solutions this equation has is more or less determined by how many times p divides −C + Db2 . A natural guess, which turns out to be more or less correct, is that the number of terms in the exponential sum determines the size of the sum. However, for the b such that −C + Db2 ≡ 0 (mod pk ) there are different sizes of cancelations for different b, so in this case we need to be careful with the definition of the level. Definition 2.5. Fix a Hecke eigenfunction ψ ∈ VC and let b ∈ ZN . For even n we define the level l at b to be the largest integer l ≤ (2n + 2)/3 such that 2
pl |−C+Db −pk+(k−2[k/2]) 3−1 rCD, and for odd n we define the level l at b to be
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+Db2 −pk+(k−2[k/2]) 3−1 CD. the largest integer l ≤ (2n+2)/3 such that pl |− C If l = [(2n + 2)/3] we say that b has maximal level. Remark. The levels l < k can be defined by the simpler formula pl −C +Db2 . Also note that if C/D is a quadratic non-residue modulo p, then all b have
level 0. One can show that the level is independent of the representative of C n−k/2 . modulo p We will mostly discuss even levels and for those we will use the notation l = 2s. This is the same s as appears in [15,16]. The following notation (already used in Theorem 1.2) was also used in these papers: Definition 2.6. For a point b of level 2s < k we let x0 be some integer such that (ps x0 )2 ≡ −C + Db2 (mod pk ) if such an integer exists. Remark. Note that p x0 . Definition 2.7. If C and D are both invertible modulo N and C/D is a quadratic residue, then a function ψ ∈ VC which is a joint eigenfunction for all UN (h) with h ∈ HD is called a large newform. Remark. That C and D are invertible and C/D is a square modulo N is equivalent to the same statement modulo p. Since the aim of the paper is to study the large newforms we will from now on assume that p C, D and that C/D is a square modulo p. Lemma 2.1. Let (2n − 1)/3 ≥ l > 0. The number of b ∈ ZN of level l is 2N 1 1− . pl p The number of b of level 0 is N (1 − 2/p) and the number of b of maximal level lmax = [(2n + 2)/3] is 2N/plmax . Proof. Let us first assume that (2n − 1)/3 ≥ l > 0. The condition that b have level l can be written on the form b2 ≡ E (mod pl ), but b2 ≡ E (mod pl+1 ), where E is easily derived from Definition 2.5. We observe that E ≡ C/D (mod p) and therefore b2 ≡ E (mod pl ) has 2 solutions modulo pl . This shows the total number of points in ZN of level l to be 2N/pl −2N/pl+1 . The number of b of level 0 is N − 2N/p and the number of b of level lmax is 2N/plmax by the same arguments.
3. Exponential Sums As we have mentioned earlier, the value of a Hecke eigenfunction may be calculated as the value of an exponential sum. The calculations focus on finding the absolute value of the following object in the special case when q is of degree three:
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Definition 3.1. Let m be a nonnegative integer. For q ∈ Zpm [x] we define pm q(z) e S(q, m) = . pm z=1 In [16] the following three lemmata (Lemma 6.1, Lemma 6.2 and Lemma 6.3) were developed: Lemma 3.1. Let q(z) = a3 z 3 + a2 z 2 + a1 z + a0 and assume that p|a3 but p a2 . Then |S(q, m)| = pm/2 . Lemma 3.2. Let q(z) = a3 z 3 + a1 z + a0 and assume that p a3 and that p2 a1 . Then |S(q, m)| ≤ 2pm/2 . Lemma 3.3. Let q(z) = a3 z 3 + p2 a1 z + a0 and assume that p a3 . For m ≥ 3 we have that |S(q, m)| = p2 |S(q1 , m − 3)|, where q1 (z) = a3 z 3 + a1 z. We will also need a special case of Lemma 6.4 from the same paper. For the reader’s convenience we only formulate the part of the statement we need: Lemma 3.4. Let q(z) = a3 z 3 + a1 z + a0 and assume that p a3 and that m ≤ 2. Then |S(q, m)| ≤ 2pm/2 . As a final recollection of the work in [16], we observe that in the proof of Lemma 3.2, the special case d = 1 was shown for the following lemma: Lemma 3.5. Let q(z) = a3 z 3 + a1 z, assume that p a3 and that pd a1 . If d is odd and (3d + 1)/2 ≤ m we have that S(q, m) = 0. Proof. By the remark above we may assume that d ≥ 3. We use Lemma 3.3 (d − 1)/2 times (observe that d is small enough for this) and get |S(q, m)| = pd−1 |S(q1 , m − 3(d − 1)/2)|, where q1 is a polynomial of the same form as q and with d = 1. The lemma now follows from the fact that S(q1 , m−3(d−1)/2) fulfills the lemma. Observe in particular that m−3(d−1)/2 ≥ 2 = (3d +1)/2. Lemma 3.6. Let q(z) = a3 z 3 + a1 z, assume that p a3 and that pd a1 . If d is even |S(q, m)| ≤ 2p(2m+d)/4 . Proof. We use Lemma 3.3 c = min(d/2, [m/3]) times and get |S(q, m)| = p2c |S(q1 , m − 3c)|, where q1 is a polynomial of the same form as q and either d = 0 or m − 3c ≤ 2. According to Lemmas 3.2 and 3.4, we get |S(q, m)| ≤ p2c 2p(m−3c)/2 = 2p(m+c)/2 ≤ 2p(2m+d)/4 . Remark. Note that it is obvious from the proof that we always have the estimate |S(q, m)| ≤ 2p(m+[m/3])/2 . Lemma 3.7. Let Sm (t) = S(a3 z 3 + tz, m) where p a3 . There exists a constant C > 0 such that if 0 ≤ d ≤ 2m/3 is even, then a positive proportion of all t ∈ Zpm−d fulfills |Sm (pd t)| > Cp(2m+d)/4 . Proof. Define the inner product in L2 (Zpm ) to be 1
φ, ψ = m φ(z)ψ(z). p m z∈Zp
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2 m Observe that {e( −tz pm )} is an orthonormal basis for L (Zp ). This leads to 2 2 2 1 a3 z 3 a3 z 3 −tz e = = S (t) , e 1 = e m pm pm pm pm 2 m m t∈Zp
t∈Zp
Sm 22 = . pm In other words, the sum of |Sm (t)|2 is p2m . According to Lemma 3.3 we have that |Sm (p2 t)| = p2 |Sm−3 (t)| for m ≥ 3 and according to Lemma 3.5 Sm (pt) = 0 for p t and m ≥ 2. This shows that the sum over all |Sm (t)|2 with p t is p2m − p2m−1 ≥ 45 p2m . On the other hand, |Sm (t)|2 ≤ 4pm for these t by Lemma 3.2. Thus, for more than pm /7 of all t ∈ Zpm we must have the estimate |Sm (t)|2 ≥ pm /4. For m ≤ 2 Lemma 3.4 shows the estimate |Sm (t)|2 ≤ 4pm for all t, and we get our estimate by comparing this with the sum of all |Sm (t)|2 . This finishes the argument for d = 0. Since d ≤ 2m/3, we may use Lemma 3.3 d/2 times and get |Sm (pd t)| = d p |Sm−3d/2 (t)| > pd Cp(m−3d/2)/2 = Cp(2m+d)/4 for a positive proportion of t ∈ Zpm−d .
4. Evaluating Newforms In [15,16] methods for evaluating newforms were developed. The main ingredient in the proofs of the main theorems in this paper is the use of those methods. This chapter serves to recapitulate the main ideas of this evaluation procedure. We let δx denote the function defined on ZN which is 1 at x and 0 else. The following functions play an important role in the evaluation of the newforms: x1 Definition 4.1. Given x = ∈ Z2N we let ζx : ZN → C be defined by x2 x1 t ζx = e δx2 +pk t . pn−k t∈Zpn−k
Remark. In [15] these functions were called ζ0,x , but the extra index is not necessary in our presentation and we will therefore be omitted. The reason that we study these functions is that they transform in a simple manner when we apply UN (B) to them. For our purposes, the following statement, which combines Lemma 5.1 in [16] and parts of Lemma 3.1 in [15], will suffice: Lemma 4.1. Let B ∈ SL(2, ZN ) and if n is odd we also assume that B ≡ I (mod p). If x = Bx then r(x1 x2 − x1 x2 ) UN (B)ζx = e ζx . N
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From this lemma one may observe that ζx is an eigenfunction of UN (h) as long as h ≡ I (mod pn−k ), or in other words, each function ζx belongs to some space VC . In fact, ζx ∈ VC if and only if x21 − Dx22 ≡ −C (mod pk ). Moreover, we can calculate the coefficients for a given Hecke eigenfunction ψ in some basis of ζx −functions. Lemma 4.2. Let ψ ∈ VC be a normalized Hecke eigenfunction and assume that p does not divide C or D. If n is odd we also assume that p x2 . Then 1 if x21 − Dx22 ≡ −C (mod pk ) 1 N (1−( D p )p) | ψ, ζx | = . 0 if x21 − Dx22 ≡ −C (mod pk ) Remark. For odd n this is a part of the statement of Lemma 3.3 in [15], and for even n it follows from Lemma 4.1 since all elements x such that x21 −Dx22 ≡ −C (mod pk ) forms an orbit of the group HD . Fixing a point b ∈ ZN , we may now observe that for most of the functions ζx , the value at b is zero. In fact, Lemma 4.2 shows that ψ(b) is a sum over all solutions to the equation x2 ≡ −C + Db2 (mod pk ). Note that this is a very short sum in this context. In particular, if x2 ≡ −C + Db2 (mod pk ) has no solutions, then ψ(b) = 0. If x2 ≡ −C + Db2 (mod pk ) has some solutions (but not the maximal number) Theorem 5.4 in [16] and Theorem 3.5 in [15] tell us how to evaluate ψ(b). These theorems can be summarized by the following theorem: Theorem 4.3. Let ψ ∈ VC be a normalized Hecke eigenfunction and assume that p does not divide C or D. Let b ∈ ZN and assume that the equation x2 ≡ −C + Db2 (mod pk ) has the solutions x ≡ ±x0 ps + pk−s Zps (mod pk ) for some x0 and s such that p x0 and 0 ≤ s < k/2. If n is even, then ps ps q+ (z) q− (z) 1 ψ(b) = e e αψ (b) + βψ (b) , (3)
ps ps D 1 z=1 z=1 1− p p and if n is odd and p b, then ⎛ ⎞ s+1 s+1 p p q q 1 (z) (z) + − ⎝ ⎠. (4) ψ(b) = e e + βψ (b) αψ (b) s+1 s+1 p p D z=1 z=1 p− p Here, q± (z) = r(Θψ (b)z ± x0 Dbz 2 + pk−2s 3−1 D2 b2 z 3 ), |αψ (b)| = |βψ (b)| = 1 and the function Θψ (b) is given by
+ Db2 − p2(k−s) 3−1 rD2 b2 Θψ (b)pk ≡ −x20 p2s − C
(mod pn−k+s ).
The careful reader may have observed that the case when n is odd and p|b was left out from the theorem. However, that case will always be covered by Theorem 3.4 in [15], which states that:
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Theorem 4.4. Assume that n is odd and let ψ ∈ VC be a normalized Hecke eigenfunction. If p does not divide C or D and b fulfills that −C + Db2 ≡ x20 (mod pk ) for some p x0 , then 1 ψ(b) = (αψ (b) + βψ (b)). 1 1− D p p αψ and βψ are functions satisfying |αψ (b)| = |βψ (b)| = 1, αψ (b + pk+1 t) = 0t e( xp0kt )αψ (b) and βψ (b + pk+1 t) = e( −x )βψ (b). pk Finally, we turn to the case when the equation x2 ≡ −C + Db2 (mod pk ) has the maximal number of solutions, i.e., when −C + Db2 ≡ 0 (mod pk ). In this case, Theorem 5.5 in [16] and Theorem 3.6 in [15] can be summarized in the following way: Theorem 4.5. Let ψ ∈ VC be a normalized Hecke eigenfunction and assume that p does not divide C or D. Let b ∈ ZN and assume that −C + Db2 ≡ 0 (mod pk ). If n is even then [k/2] p q(z) αψ (b) ψ(b) = e , (5)
p[k/2] D 1 z=1 1− p p where q(z) = r(Θψ (b)z + pk−2[k/2] 3−1 CDz 3 ), |αψ (b)| = 1 and
+ Db2 − pk+(k−2[k/2]) 3−1 rCD Θψ (b)pk ≡ −C
(mod p[3k/2] ).
If n is odd, then p q(z) αψ (b) ψ(b) = e ,
p[k/2]+1 D z=1 p− p [k/2]+1
(6)
where q(z) = Θψ (b)z − pk−2[k/2] 3−1 2CDz 3 , |αψ (b)| = 1 and
+ Db2 − p2k/2 3−1 CD Θψ (b)pk ≡ −C
(mod p[3k/2]+1 ).
5. The Proofs of the Main Theorems Lemma 5.1. If the level l ≤ (2n − 1)/3 of b is odd, we have that ψ(b) = 0. Proof. If the level l is less than k, the equation x2 ≡ −C + Db2 (mod pk ) has no solutions and so we are done. If k ≤ l, ψ(b) is given by Theorem 4.5. We have to treat the cases n even and n odd separately. Assume first that n is even. In this case ψ(b) is given by (5), recall in particular that
+ Db2 − pk+(k−2[k/2]) 3−1 rCD Θψ (b)pk ≡ −C
(mod p[3k/2] ).
(7)
The right-hand side of (7) is divisible by p exactly l times; thus pl−k Θψ (b). The coefficient in front of z 3 is divisible by p exactly k − 2[k/2] times and this number is always less than or equal to l − k. Hence, the exponential sum is
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actually an exponential sum over Zp3[k/2]−k , where the coefficient in front of z 3 is invertible and the linear term is divisible by p exactly l − 2k + 2[k/2] times. This number is odd and the claim now follows from Lemma 3.5 since 3(l − 2k + 2[k/2]) + 1 ≤ 3[k/2] − k. 2 Let us now assume that n is odd. In this case ψ(b) is given by (6), where this time
+ Db2 − pk+(k−2[k/2]) 3−1 CD (mod p[3k/2]+1 ). Θψ (b)pk ≡ −C (8) The same reduction as above gives a sum over Zp3[k/2]−k+1 , where the coefficient in front of z 3 is invertible and the linear coefficient is divisible by p exactly l − 2k + 2[k/2] times. The claim now follows from Lemma 3.5, since 3(l − 2k + 2[k/2]) + 1 ≤ 3[k/2] − k + 1. 2 The idea behind the proof of Theorem 1.2 is that, in some sense, Theorem 1.2 is a special case of Theorem 4.3. Unfortunately, it is not so easy to read off our theorem from Theorem 4.3, and in reality a better description of the proof of Theorem 1.2 is that it is a close analysis of the proofs of Theorem 5.4 in [16] and Theorem 3.5 in [15]. The notation is these proofs are not well equipped for our situation and for that reason we will rewrite the proofs completely. Proof of Theorem 1.2. We fix b ∈ ZN and study the values of ψ(b ) for b = b + tpn−k+s . Theorem 4.4 proves the theorem for n odd and s = 0 and we may therefore assume p b for odd n. By Lemma 4.2 we know that we can write ψ = R ax ζx , where the sum is taken over all x ∈ Zpn−k × Zpk such that x21 − Dx22 ≡ −C (mod pk ). Actually, the functions ζx are dependent on the exact representative of x2 we choose, but we will postpone the exact choice of representative for now. If we let 1 R=
1 pn−2k 1 − D p p the coefficients ax (where p x2 if n is odd) will have absolute value 1. Note that ax ζx (b ), (9) ψ(b ) = R where the sum is over all x such that x2 ≡ b (mod pk ) and x1 solves the equation x21 ≡ −C + Db2 (mod pk ). Since b has level 2s and ψ(b) = 0 we know that there must exist an integer x0 , not divisible by p, such that (x0 ps )2 ≡ −C +Db2 (mod pk ). The solutions to x21 ≡ −C + Db2 ≡ (x0 ps )2 (mod pk ) are given by x1 ≡ ±x0 ps (mod pk−s ). Let us now define 1 + rDp2(k−s) pk−s D B(s) = . 1 + rDp2(k−s) pk−s
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By induction it is easy to show that k−s p z+3−1 rDp3(k−s) (z 3 −z) D 1+rDz 2 p2(k−s) z B(s) = . 1+rDz 2 p2(k−s) pk−s z+3−1 rDp3(k−s) (z 3 −z) Let m = n − 2k + s. We observe that the two orbits s z x0 p m B(s) ; z = 0, 1, . . . , p − 1 b and
s z −x0 p m B(s) ; z = 0, 1, . . . , p − 1 b
corresponds exactly to the x in (9). We may therefore choose these representatives for x in (9). In other words, we introduce ζ±,z = ζB(s)z (±x0 ps ) and b
write ψ(b ) = R
pm −1
a+,z ζ+,z (b ) +
m p −1
z=0
a−,z ζ−,z (b ) .
z=0
1t e( pxk−s )ζx (b)
this can be reduced to Since ζx (b ) = m p pm −1 −1 x0 t −x0 t a+,z ζ+,z (b) + e a−,z ζ−,z (b) . ψ(b ) = R e pk−2s z=0 pk−2s z=0 (10) The task is now to show that a±,z ζ±,z (b) = a±,0 e
q± (z) , pm
(11)
where q± (z) = r(Θψ (b)z ±x0 Dbz 2 +pk−2s 3−1 D2 b2 z 3 ) is the polynomial occurring in Theorem 4.3. This follows from Lemma 4.1 together with the definition of ζx , remembering that ψ is a Hecke eigenfunction. (11) was shown in the proofs of Theorem 5.4 in [16] and Theorem 3.5 in [15], and we therefore omit the (quite tedious) exact calculations proving this statement. At this point we have reproved Theorem 4.3 [combine (10) and (11)], also showing that the functions αψ and βψ appearing in the Eqs. (3) and (4) fulfill x0 t n−k+s ) = αψ (b)e αψ (b + tp and pk−2s −x0 t . βψ (b + tpn−k+s ) = βψ (b)e pk−2s To conclude our proof we must show that the exponential sums in (3) and (4) have absolute value ps/2 and p(s+1)/2 , respectively. If s = 0 this is trivial for even n; otherwise, we must have p b also in the even case. The claim now follows immediately from Lemma 3.1.
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Proof of Proposition 1.3. This follows immediately from Theorem 1.2 for s < k/2. For s ≥ k/2 we have to analyze Theorem 4.5. If we let q(z) denote the polynomial in this theorem we have to show that |S(q, [k/2])| ≤ 2ps/2 for even n and that |S(q, [k/2] + 1)| ≤ 2p(s+1)/2 for odd n. The level is defined in such a way that p2s−k divides the coefficient in the linear term of the polynomial. Assume that k is even. For such k the polynomial fulfills the assumptions of Lemma 3.6; thus for even n we get |S(q, [k/2])| = |S(q, k/2)| ≤ 2p(2k/2+2s−k)/4 = 2ps/2 and similarly for odd n we get |S(q, [k/2] + 1)| = |S(q, k/2 + 1)| ≤ 2p(2(k/2+1)+2s−k)/4 = 2p(s+1)/2 . If on the other hand we assume k to be odd, we know that both the linear coefficient (since this was divisible by p2s−k ) and the third degree coefficient is divisible by p. Thus we may cancel one p in the numerators and denominators to arrive at p shorter exponential sums of length p[k/2]−1 and p[k/2] , respectively. We now see that the new polynomial q1 fulfills the assumption of Lemma 3.6, and the linear coefficient is divisible by p at least 2s − k − 1 times. If n is even we get |S(q, [k/2])| = p|S(q1 , (k − 3)/2)| ≤ 2p1+(2(k−3)/2+2s−k−1)/4 = 2ps/2 and if n is odd we get |S(q, [k/2] + 1)| = p|S(q1 , (k − 1)/2)| ≤ 2p1+(2(k−1)/2+2s−k−1)/4 = 2p(s+1)/2 . Proof of Theorem 1.4. Let ψs be the restriction of ψ to the points of level 2s, for some s such that s ≤ n/3. The theorem is obvious for s = 0. Let us now assume that the level is 0 < l = 2s < k and take a point b of level l. We study the values of −C + Db2 , for b = b + tp2s as t runs though Zp . Since 2Db is invertible, we see that −C + D(b + tp2s )2 runs through all values modulo p2s+1 that are congruent to zero modulo p2s . Of these numbers, (p − 1)/2 are squares, (p − 1)/2 are non-squares and the last number is zero, saying that the corresponding point b has higher level than b . For the points b corresponding to non-squares ψ(b) = 0 and for b corresponding to squares we can use Theorem 1.2 (actually even if it were to happen that ψ(b) = 0). In other words, for exactly half the points of level 0 < l = 2s < k we may use Theorem 1.2 and for the other half ψ(b) = 0. Since x0 t −x0 t + βψ (b)e αψ (b)e pk−2s pk−2s is bounded and also bounded away from zero for most t, it is easy to see that p−s(q−4)/2 ψs qq is bounded and also bounded away from zero, by combining this observation with Lemma 2.1. This shows the theorem for levels l < k. Now assume l = 2s ≥ k. We can no longer use Theorem 1.2, but Proposition 1.3 works just as well for the upper bound. To prove the lower bound we once more have to study the expressions from Theorem 4.5. Let us therefore review the proof of Lemma 5.1: In our new case, l is even, which makes l − 2k + 2[k/2] even. We want to use Lemma 3.7 and in order to do this we note that 3/2(l − 2k + 2[k/2]) ≤ 3[k/2] − k
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for even n and 3/2(l − 2k + 2[k/2]) ≤ 3[k/2] − k + 1 for odd n. Fix a point b and study b = b + tpk for t ∈ Zpn−k . The polynomial q(z) is of the form q(z) = a3 z 3 + a1 z where a3 is fixed (but dependent on the parity of k and n) and a1 changes with b. Observe that Θ(b + tpk ) − Θ(b ) = 2Db t [see (7) and (8)] and since p 2Db , this shows that a1 takes all values of Zp[k/2] and Zp[k/2]+1 , respectively, the same number of times. The theorem now follows from Lemma 3.7. Proof of Theorem 1.1. We first note that since ψ1 ⊗ ψ2 q = ψ1 q ψ2 q it is enough to study the case where N = pn and p > 3 is such that D and the lower left element in A is invertible modulo p. If p|C we know that D is a square and by Proposition 4.4 in [15] we see that the newforms are uniformly bounded at all points. If p C and C/D is not a square, we get the same conclusion from Theorem 4.2 in [15]. The only thing left to study is the large newforms, i.e., where C/D is a non-zero square modulo p. Let ψs be the restriction of the large newform ψ to the points of level 2s for some s < n/3 and let ψmax be the restriction to the points of maximal level. Obviously, ψqq is the sum of all ψs qq . By Theorem 1.4 we know that ψs qq ≈ ps(q−4)/2 , where ψs qq ≈ ps(q−4)/2 should be understood in the sense that the p−s(q−4)/2 ψs qq is uniformly bounded away from both zero and infinity as N goes to infinity. According to Theorem 4.3 in [15] |ψ(b)| = O(p[n/3]/2 ) for all points, including those with maximal level. This, together with Lemma 2.1, gives us q[n/3]/2
p nq/6−2n/3 n(q−4)/6 = O p . = O p ψmax qq = O p[(2n+3)/3] These estimates show that ψqq is bounded for q < 4. Moreover, if n is bounded, so is the number of different levels, and therefore ψ44 is bounded. On the other hand, if q = 4 and n goes to infinity, the number of different levels, and therefore the sum over all ψs qq goes to infinity. Since n ≥ 3 the maximal level is at least 2. In other words, ψ1 is non-trivial (for n = 3 this was denoted ψmax above). Theorem 1.4 shows that ψ1 qq explodes for q > 4 and this causes ψq to explode. Finally, it is easy to see that for q > 4, ψqq , the sum of all ψs qq , is dominated by O(pn(q−4)/6 ) (corresponding to the term ψmax qq ), and this ends the proof.
Acknowledgements I would like to thank Nalini Anantharaman for bringing this problem to my attention. I also thank Christopher Sogge and Peter Sarnak for helpful comments.
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References [1] Bouzouina, A., De Bi`evre, S.: Equipartition of the eigenfunctions of quantized ergodic maps on the torus. Commun. Math. Phys. 178(1), 83–105 (1996) [2] Colin de Verdi`ere, Y.: Ergodicit´e et fonctions propres du laplacien. Commun. Math. Phys. 102(3), 497–502 (1985) [3] Degli Esposti, M.: Quantization of the orientation preserving automorphisms of the torus. Ann. Inst. H. Poincar´e Phys. Th´eor. 58(3), 323–341 (1993) [4] Degli Esposti, M., Graffi, S., Isola, S.: Classical limit of the quantized hyperbolic toral automorphisms. Commun. Math. Phys. 167(3), 471–507 (1995) [5] Faure, F., Nonnenmacher, S., De Bi`evre, S.: Scarred eigenstates for quantum cat maps of minimal periods. Commun. Math. Phys. 239(3), 449–492 (2003) [6] Gurevich S., Hadani, R.: The two dimensional Hannay–Berry model (preprint) [7] Hannay, J.H., Berry, M.V.: Quantization of linear maps on a torus—Fresnel diffraction by a periodic grating. Phys. D 1(3), 267–290 (1980) [8] Iwaniec, H., Sarnak, P.: L∞ norms of eigenfunctions of arithmetic surfaces. Ann. Math. (2) 141(2), 301–320 (1995) [9] Iwaniec, H., Kowalski, E.: Analytic number theory. American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Providence, RI, 2004 [10] Klimek, S., Le´sniewski, A., Maitra, N., Rubin, R.: Ergodic properties of quantized toral automorphisms. J. Math. Phys. 38(1), 67–83 (1997) [11] Knabe, S.: On the quantisation of Arnold’s cat. J. Phys. A 23(11), 2013–2025 (1990) [12] Kurlberg, P., Rudnick, Z.: Hecke theory and equidistribution for the quantization of linear maps of the torus. Duke Math. J. 103(1), 47–77 (2000) [13] Kurlberg, P., Rudnick, Z.: Value distribution for eigenfunctions of desymmetrized quantum maps. Int. Math. Res. Not. 18, 985–1002 (2001) [14] Mezzadri, F.: On the multiplicativity of quantum cat maps. Nonlinearity 15(3), 905–922 (2002) [15] Olofsson, R.: Hecke eigenfunctions of quantized cat maps modulo prime powers. Ann. Henri Poincar´e 10(6), 1111–1139 (2009) [16] Olofsson, R.: Large supremum norms and small Shannon entropy for Hecke eigenfunctions of quantized cat maps. Commun. Math. Phys. 286(3), 1051–1072 (2009) [17] Sarnak, P.: Spectra of hyperbolic surfaces. Bull. Am. Math. Soc. (N.S.) 40(4), 441–478 (2003, electronic) [18] Seeger, A., Sogge, C.D.: Bounds for eigenfunctions of differential operators. Indiana Univ. Math. J. 38(3), 669–682 (1989) [19] Sogge, C.D.: Concerning the Lp norm of spectral clusters for second-order elliptic operators on compact manifolds. J. Funct. Anal. 77(1), 123–138 (1988) [20] Zelditch, S.: Index and dynamics of quantized contact transformations. Ann. Inst. Fourier (Grenoble) 47(1), 305–363 (1997) [21] Zygmund, A.: On Fourier coefficients and transforms of functions of two variables. Studia Math. 50, 189–201 (1974)
1302 Rikard Olofsson Department of Mathematics Uppsala University P. O. Box 480 75106 Uppsala Sweden e-mail:
[email protected] Communicated by Jens Marklof. Received: February 4, 2010. Accepted: August 11, 2010.
R. Olofsson
Ann. Henri Poincar´e
Ann. Henri Poincar´e 11 (2010), 1303–1339 c 2010 Springer Basel AG 1424-0637/10/071303-37 published online November 3, 2010 DOI 10.1007/s00023-010-0059-y
Annales Henri Poincar´ e
Van Hove Limit for Infinite Systems David Taj Abstract. We study the van Hove limit for master equations on a Banach space, and propose a contraction semigroup as limit dynamics. The generator has a Lindblad form if specialized to C ∗ -algebras, is always well defined irrespectively of the subsystem spectrum, includes first-order contributions, and returns Davies averaged generator, when the latter is defined. The theory is applied to the case of a free particle in contact with a heat bath.
1. Introduction The Van Hove limit [1], also referred to as weak-coupling limit, amounts to study the time evolution of a perturbed hamiltonian system on the λ2 t time scale, as the coupling constant λ goes to zero. It has long been rigorously known [2,3] that the time evolution of a finite (or discrete) quantum system, interacting with an external steady environment, becomes markovian in such circumstances, as described by the celebrated Fermi Golden Rule [4,5]. However, nowadays technologies often require decay times and steadystate analysis [6] for infinitely extended open quantum systems [7], i.e. with continuous, and even mixed spectrum. Although phenomenological markovian laws for such systems have been studied (see e.g. [8–10]), a general recipe to construct a proper markovian generator in the van Hove limit, given the hamiltonian perturbation, is lacking. This appears to be because the limit dynamics that has been found [3] fails to provide a positive [11] (or even contractive [12]) evolution in general; moreover, the time averaging procedure [2], successfully employed to remedy for this in the discrete case, is not well defined anymore in the continuum. As a consequence, the resulting markovian law cannot be adopted as a consistent physical model per se, and large time behavior and steadiness cannot be addressed. Here, we propose a semigroup that (i) is contractive, (ii) is always well defined irrespectively of the subsystem spectrum, (iii) includes first-order
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dynamics, (iv) approaches the exact projected evolution in the van Hove limit, and upon specializing to operator algebras, (v) is (completely) positive. Also, (vi) Davies averaged generator is recovered (in a slightly generalized form including possible first-order dynamics) in case of discrete spectrum. Formalizing some ideas of a previous work of ours [13], as well as introducing new ones, on rigorous grounds, the proposed Quantum Fokker–Planck Equation in (16) is constructed through a “dynamical time averaging” that a posteriori mimics the procedure employed in [2], but differs from the latter, in that it scales with the coupling constant. Finally, we apply the theory to the case of free particle in contact with a heat bath through energy–energy couplings, and find sufficient conditions, different with respect to the discrete case, to prove that thermal distribution of diagonal observables (i.e. affiliated to the hamiltonian of the small system) is stationary. The analysis is far from being exhaustive and indicates that a plethora of new possibilities opens up.
2. General Framework We briefly report from [3] the general framework we will be involved with. We suppose that P0 is a linear projection on a Banach space B (that represents some global system), put P1 = 1 − P0 and Bi = Pi B, so that B = B0 ⊕ B1 , and we take B0 to be the subsystem of interest. We suppose that Z is the (densely defined) generator of a strongly continuous one-parameter group of isometries Ut on B with Ut P0 = P0 Ut for all t ∈ R, or equivalently [Z, P0 ] = 0, and put Zi = Pi Z. We suppose that A is a bounded perturbation of Z and put Aij = Pi APj . We let Utλ be the one-parameter group generated by (Z +λA00 +λA11 ) so that Utλ P0 = P0 Utλ for all t ∈ R, and let Vtλ be the one-parameter group generated by Z + λA. Then putting Xtλ = P0 Utλ , and defining the projected evolution as Wtλ = P0 Vtλ P0 , one obtains the all important closed and exact integral master equation: Wtλ = Xtλ + λ2
t
s ds
0
λ λ du Xt−s A01 Us−u A10 Wuλ ,
(1)
0
named after Nakajima, Prigogine, Resibois, and Zwanzig [14,15]. Now assume that Xtλ is a one-parameter group of isometries (see Lemma 1.1 in [3]). Then, changing variables in the integral in (1) to x = s−u, σ = λ2 u and introducing the time rescaled (and A00 -renormalized) interaction picture λ λ evolution Wτλ,i = X−λ −2 τ Wλ−2 τ , one is led to Wτλ,i
τ =1+
λ λ λ,i dσ X−λ −2 σ K(λ, τ − σ)Xλ−2 σ Wσ ,
(2)
0
where the slowly varying kernel K(λ, τ ) converges in the weak-coupling limit λ → 0 (under suitable hypotheses) to the celebrated Davies’ generator
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∞ KD =
U−x A01 Ux A10 dx.
(3)
0
3. A Family of New Generators In our first theorem, we shall find a new class of generators for the semigroup approximation in the van Hove limit. To this purpose, note that KD has been defined thanks to the change of variable σ 0 λ2 s = . (4) x 1 −1 u Here, instead we would like to allow for the most general linear change of variable that keeps a λ2 jacobian, proper of a second-order approximation, while fixing the relative variable to be s − u = x. Accordingly, we take 1 2 2 2 1 s λ q σ 2 −α λ 2 +α λ , (5) + = 0 u x 1 −1 for some real α and q. Some straightforward algebra shows that the integration domain s = 0 . . . λ−2 τ , u = 0 . . . s in (1) becomes the domain D(λ, τ, α, q) in the (σ, x) plane given by the triangle of vertices 1 2 2 2 −2 − α τ + λ q, λ τ D(λ, τ, α, q) = (λ q, 0), (τ + λ q, 0), (6) 2 in Fig. 1. Accordingly, (1), written for Wτλ,i = X−λ−2 τ Wλλ−2 τ , becomes λ λ λ dσdx X−λ Wτλ,i = 1 + −2 σ+q− α+ 1 x A01 Ux A10 Xλ−2 σ−q+ α− 1 x ( ) ( ) 2
2
D(λ,τ,α,q) λ,i ×Wσ−λ 2 (q+(1/2−α)x) .
(7)
We now consider the following facts, which will be made precise in Theorem 3.4: •
D(λ, τ, α, q) → [0, τ ] × [0, ∞) as λ → 0, for any real α and q. This justifies the approximation
τ dσ dx ≈
D(λ,τ,α,q)
•
∞ dσ
0
2
dx e
(x/2) − 2T (λ)2
λ ≈ 0,
0
for some real positive function T of the coupling constant, provided limλ→0 T (λ) = +∞. The choice of the gaussian is dictated by later purposes. In the weak-coupling limit λ → 0 one could approximate λ,i λ,i Wσ−λ 2 (q+(1/2−α)x) ≈ Wσ
(8)
in the integral kernel of (7), provided that T (λ) does not grow too fast.
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Figure 1. Integration domains D(λ, τ, α, q) and D(λ, τ, α, q) defined in (6) and (33), respectively (we have put q > 0, α < −1/2 for clarity). The arrow indicates the asymptotic behavior of the two domains as λ → 0 Then (7) becomes Wτλ,i
τ ≈1+
λ λ λ,i dσ X−λ −2 σ K(α,q,T (λ)) Xλ−2 σ Wσ
0 λ
or, which is the same, Wtλ ≈ W t , where we give the following. Definition 3.1. Let α, q ∈ R, and T : I˙ → R+ a real positive continuous function on I˙ = [−1, 1]\{0} ⊂ R. For λ ∈ I˙ define the linear operator K(α,q,T (λ)) on B0 as ∞ 2 − (x/2) K(α,q,T (λ)) = dx e 2T (λ)2 U−(α+ 1 )x+q A01 Ux A10 U(α− 1 )x−q . 2
2
0
Denote also with λ
W t = exp{(Z0 + λA00 + λ2 K(α,q,T (λ)) )t}
(9)
the associated semigroup on B0 . This is indeed the case, under the same kind of assumptions made in [3], namely Assumption 3.2. There exists some 0 < c < ∞ such that for every τ > 0 λ−2 τ
A01 Uxλ A10 dx ≤ c
0
is bounded uniformly on |λ| ≤ 1.
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Assumption 3.3. For every 0 < τ < ∞ λ−2 τ
lim
λ→0
A01 (Uxλ − Ux )A10 dx = 0.
0
Sufficient conditions for these assumptions to hold are given in Sect. 4.3. We denote here and in the sequel with I˙ the dotted interval [−1, 1]\{0} ⊂ R, and we state our first result: Theorem 3.4. Suppose that Xtλ is a one-parameter group of isometries. Let T : I˙ → R+ a real positive continuous function, assume T (λ) ∼ |λ|−ξ T ,
λ→0
for some T > 0 and 0 < ξ < 2 (strictly), and suppose that Assumptions 3.2 and 3.3 hold. Then for every τ > 0
lim
λ→0
sup 0≤t≤λ−2 τ
λ
Wtλ − W t
= 0.
We defer the proof to Sect. 8, by just commenting that the same philosophy of the proof in [3] is followed, the main difference being the delicate step sketched in (8), which requires a telescopic expansion and is controlled by our bound on T (λ). At this point a few comments are of order: •
•
The dynamical time T (λ) (named “dynamical” as it scales with the coupling constant) proves to be an essential, new and natural ingredient of the theory, whose necessity becomes evident as soon as one departs from Davies choices α = 1/2, q = 0. With respect to this, the theorem generalizes [3]. For each choice of α, q, T and nonzero λ, the corresponding generator Z0 + λA00 + K(α,q,T (λ)) is always well defined, irrespective of the spectrum of Z0 , and self-interactions A00 are taken into account.
However, the analysis in [12] shows that Davies generator KD = K(1/2,0,+∞) need not be dissipative, and it is not evident a priori that any of the K(α,q,T (λ)) we have found could prove to be dissipative. However, this new class of generators is sufficiently ample to allow for a new type of averaging procedure, close to the idea of temporal averaging introduced in [2], which solves the problem.
4. Dynamical Time Average The idea is to average among {K(α,q,T (λ)) }q∈R in the α-fibration, using a gaussian probability distribution, whose standard deviation is provided by the natural time scale T (λ). The latter goes to infinity in the van Hove limit, but furnishes a well-defined bounded generator for all nonzero λs, irrespective of spectral conditions on Z0 .
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We start from the simplest case α = 0, but we will see later on that the same averaging procedure can be applied for any real α, always leading to the same, α-independent, averaged generator. We give the following. Definition 4.1. For any real positive T > 0 put 1 KT := √ 2πT
∞ ∞ q2 q2 1 − 2T 2 dq e K(0,q,T ) = √ dq e− 2T 2 U−q K(0,0,T ) Uq 2πT
−∞
−∞
Then, we can prove a markovian approximation theorem for KT (λ) under the same conditions of Theorem 3.4, and this constitutes our first main result: Theorem 4.2. Suppose that Xtλ is a one-parameter group of isometries. Let T : I˙ → R+ be a real positive continuous function such that T (λ) ∼ |λ|−ξ T ,
λ∼0
for some T > 0 and 0 < ξ < 2 (strictly), and denote with tλ = exp{(Z0 + λA00 + λ2 KT (λ) )t} W
(10)
the associated semigroup on B0 . Make also Assumptions 3.2 and 3.3. Then for every τ > 0
λ λ t = 0. lim sup Wt − W λ→0
0≤t≤λ−2 τ
Again we defer the proof to Sect. 8, while just mentioning that the most delicate point is perhaps the control of uniform bounds on the q variable in the telescopic expansion. It turns out however that integration over q does not pose any problem, as the integration domain scales with T (λ) (hence with ξ), while the various Volterra integral operators involved in the proof of Theorem 3.4 approach each other either uniformly in q or with velocity 2 > ξ. It is worth here to emphasize once again that no specific spectral conditions on Z0 are assumed and self-interactions A00 are taken into account. 4.1. Case of Discrete Spectrum We make contact with the definition of the time average proposed in [2]. Proposition 4.3. Let B0 be finite dimensional, and A00 = 0. Then for b ∈ B0 and every τ > 0
lim
λ→0
sup 0≤t≤λ−2 τ
e(Z0 +λ
2
KT (λ) )t
b − e(Z0 +λ
2
KD )t
b
=0
Proof. Following Theorem 1.4 in [3], it is sufficient to show that = 0. To this end, let limλ→0 KT (λ) − KD
Z0 = iωα Πα α
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be the spectral decomposition of Z0 , with all ωα s distinct and real, and associated projectors Πα . Noting that K(0,q,T ) = U−q K(0,0,T ) Uq , we compute ∞ q2 1 lim KT = lim √ dq e− 2T 2 ei(ωα −ωβ )q Πα K(0,0,+∞) Πβ T →+∞ T →+∞ 2πT αβ −∞
= Πα KD Πα α
1 = lim T →+∞ 2T
T dq U−q KD Uq −T
which clearly shows that the time average in [2] coincides with our dynamical one, in the weak-coupling limit λ → 0, i.e., recalling that limλ→0 T (λ) = +∞, KD = lim KT (λ) λ→0
and the averaging map is well defined because B0 is finite dimensional.
The statement of the proposition is unchanged by only assuming that Z0 has discrete spectrum. More importantly, KD can be recovered as a particular (limit) case of our KT (λ) . Indeed, by choosing T (λ) = |λ|−ξ τ , for some 0 < ξ < 2 and τ > 0, one has lim KT (λ) = KD .
τ →+∞
where the limit exists for discrete Z0 spectrum due to the computation in the proof above. 4.2. Structure of KT for Nonzero α We denote A(t) = U−t AUt , and introduce1 t2 1 e− 2T 2 . δT (t) = √ 2πT
Then according to the change of variable t1 = q+x/2, t2 = q−x/2, and because of the properties of the gaussian, KT can be factorized in the following form: ∞ KT =
dt1 −∞
t1 δT (t1 ) A01 (t1 )
dt2
δT (t2 ) A10 (t2 ).
(11)
−∞
We wish to show here that there is nothing peculiar in the choice α = 0 that leads to the above factorized structure for KT , apart from simplicity in the definitions and in the proofs involved. In fact, one could equally well proceed along the following lines: in (1), change variable according to (5) as in Theorem 3.4. Proceed as in the proof of Theorem 3.4, but use exp{−(t1 (x, q)2 + t2 (x, q)2 − √ √This notation for the gaussian is motivated by the fact that the Fourier transform δω of δT , with ω = 1/(2T ), defines a nascent delta function δω in the van Hove limit λ → 0.
1
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2q 2 )/(4T (λ)2 )} instead of exp{−(x/2)2/(2T (λ)2)} as gaussian smoothing for the kernel in (52), where t1 (x, q) = q + (α + 1/2)x . t2 (x, q) = q + (α − 1/2)x Then, one can proceed as in the proof Theorem 3.4 to show that (under the (α,q,T (λ)) )t} λ = exp{(Z0 + λA00 + λ2 K same hypotheses) the semigroup W t satisfies the same markovian approximation theorem with ∞ t (x,q)2 +t2 (x,q)2 −2q 2 − 1 4T (λ)2 K(α,q,T (λ)) = dx e A01 (t1 (x, q))A10 (t2 (x, q)). 0
Averaging over q with a normalized gaussian distribution of standard deviation T (λ), as in Theorem 4.2, gives exactly the same result as in (11). 4.3. A Sufficient Condition for the Assumptions We provide a sufficient condition for the validity of the hypotheses in Theorems 3.4 and 4.2, which is a slight adaptation of Theorem 1.3 in [3], and is supported by perturbation arguments. First, for n ∈ N define the coefficients t n−1
t dt0 · · ·
an (t) = 0
dtn A01 Ut0 −t1 A11 Ut1 −t2 A11 . . . A11 Utn A10 , 0
which come from the expansion of Utλ in powers of λ. Theorem 4.4. Suppose that ∞ A01 Ut0 A10 dt0 < ∞.
(12)
0
Suppose that an (t) ≤ cn |t|n/2 ∞ for all t ∈ R and n ≥ 1, where the series n=1 cn z n has infinite radius of convergence. Suppose also that for some > 0, dn , and all t ≥ 0 an (t) ≤ dn |t|n/2−
Then, Assumptions 3.2 and 3.3 are satisfied. Proof. By expanding Uxλ in a λ power series, one obtains λ−2 τ
A01 Uxλ A10 dx ≤
∞
λn an (λ−2 τ ) ≤ a0 (λ−2 τ ) +
n=0
0
∞
n=1
which converges for any τ due to hypothesis (12). Similarly, λ−2 τ
0
A01 (Uxλ − Ux )A10 dx ≤
∞
n=1
λn an (λ−2 τ )
cn |τ |n/2
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∞ For |λ| < 1, the series is dominated by the convergent n=1 cn |τ |n/2 , and each term of the series is dominated by dn |λ|2 |τ |n/2− , which goes to zero when λ → 0, completing the proof.
5. A Contraction Semigroup
√ As a further useful manipulation of KT , we name ΦT (t) = δT (t)A(t) for short and ΦTij (t) = Pi ΦT (t)Pj . Then by usual algebraic manipulations KT can be cast in the form +∞ +∞ dt1 dt2 Φ01 (t1 ) √ Φ10 (t2 ) KT = π √ 2π 2π −∞
−∞
+∞ t1 dt1 dt √ 2 (Φ01 (t1 )Φ10 (t2 ) − Φ01 (t2 )Φ10 (t1 )) +π √ 2π 2π −∞
T
T
Now we put Φ = Φ −
−∞
ΦT00 , LT =
+∞ dt T √ Φ (t), 2π
−∞
T on B as and define the operator K +∞ t1 dt dt T 1 2 T = π L + π √ T (t2 )]. √ 2 [Φ K (t1 ), Φ T 2π 2π −∞
(13)
−∞
It follows that T P0 KT = P0 K
(14)
Now from expressions (14) and (13) one can already understand why KT gen T , so it generates contractions erates contractions:2 KT is the projection of K T is made of two parts, the first being the square of if the latter does. Now K a generator of isometries (and thus dissipative), and the second a generator of isometries (being a superposition of commutators of generators of isometries). It is remarkable that the explicit and symmetric forms above (13) and T P0 can be achieved only through factorization of integra(14) for KT = P0 K tion variables for the two time integrals, and this in turn has been possible due to the choice of the gaussians. If one starts from (13) and goes backward using other non-gaussian weights, integrals over q and x do not factor, forbidding a markovian approximation theorem like Theorem 4.2 for KT . Before stating our main result of this section, we report here without proof, for completeness, the part of the Hille–Yosida Theorem [16] we will be using in this section: most operators we are involved with are bounded, as we are assuming A to be bounded. 2
We recall that a semigroup Tt is a contractive if Tt ≤ 1 for all t ≥ 0.
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Theorem 5.1. Let S be a bounded operator on the Banach space B. Then S is the infinitesimal generator of a contraction semigroup if an only if there exist some α > 0 such that for every b ∈ B (1 − αS)b ≥ b .
(15)
We shall also need the following. Lemma 5.2. Let A and B be bounded generators of one-parameter groups of isometries on a Banach space B. Then C = [A, B] generates a one-parameter group of isometries on B. Proof. Define Ft = exp{At} exp{Bt} exp{−At} exp{−Bt} t ∈ R Clearly, Ft is an isometry for every t ∈ R. Now for t ∈ R\{0} define Cn (t) = n2 t−2 (Ft/n − 1). Then Cn (t) generates a one-parameter group of isometries. To show this we take α ≥ 0, b ∈ B, and prove inequality (15) by computing α 1 − F (1 − α(Ft/n − 1))b = (1 + α) t/n b 1+α α ≥ (1 + α) b − Ft/n b ≥ b , 1+α as Ft/n b = b . Since Cn (t) ≤ 2 n2 /t2 is bounded for all t > 0, it Cn (t) generates a one-parameter semigroup of contractions by the Hille–Yosida Theorem. Now a simple calculation [17] shows that limn→∞ Cn (t) = C uniformly, due to boundedness of A and B. Then C generates contractions, as for t ≥ 0 it follows that for any n > 0,
eCt ≤ eCn (t )t + eCt − eCn (t )t
≤ 1 + tet(2C+C−Cn (t )) C − Cn (t ) (see Chapter 3.1.1 in [18]). Inverting the role of A and B in all of the above shows that −C also generates contractions, thus proving the Lemma. We can now state the main result of this section (second main result): tλ is a contraction semigroup on B0 , for all Theorem 5.3. If P0 = 1, then W real λ = 0. Proof. Xtλ is a one-parameter group of isometries because of Lemma 1.1 in [3]. Now because of the Trotter product formula [19], one has n tλ = P0 e(Z0 +λA00 +λ2 KT (λ) )t = lim X λ eλ2 KT (λ) t/n , P0 W t/n n→∞
so that λ ≤ sup P0 eλ P0 W t
2
n
KT (λ) t/n n
,
and the theorem would follow if KT would generate a contraction semigroup on B0 , for all T > 0. To show this is indeed the case, we consider its form given by
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T P0 . Now again through the Trotter product formula, the (13) with KT = P0 K T generates a contraction semigroup theorem would follow from the fact that K on B. Indeed, T P0 t/n + O(n−2 )}n P0 eKT t = lim P0 {1 + P0 K n→∞
= lim {P0 eKT t/n P0 }n , n→∞
so that, since P0 = 1, we obtain
P0 eKT t ≤ eKT t T generates a contraction semigroup on for all t > 0. So we shall prove that K B by showing that each of the two terms in (13) does. T (t) generates isometries for all t, as In order to do that, we note that Φ for all real α and b ∈ B we have (1 − αΦT (t))b ≥ (1 − α δT (t)A)Ut b ≥ b because A generates isometries and U±t are isometries, and from Trotter formula, and P0 = 1 (see above), it follows that ΦT00 (t) generates isometries as well for all t. 1√ Then by linearity it follows that LT (bounded by ( π2 ) 4 T A ) generates isometries, as if ηb is any tangent functional at b ∈ B it follows that T (t)b) = dt ηb (Φ T (t)b) = 0 since Φ T (t) is conservative for every ηb ( dt Φ t (see e.g. Proposition 3.1.14 in [18]). It then follows that the bounded L2T generates a contraction semigroup, as √ √ (1 − α L2T )b = (1 − α LT )(1 + α LT )b ≥ b for any α > 0 and b ∈ B, by repeated use of the Hille–Yosida theorem. T (t) To treat the remaining term in (13), we have already noted that Φ T (t2 )] gen T (t1 ), Φ generates isometries for all t, so Lemma 5.2 implies that [Φ erates isometries as well. By linearity of the tangent functionals and the fact that +∞ t1 2T dt1 dt2 T T √ √ [ Φ A 2 (t ), Φ (t )] 1 2 ≤ √ 2π 2π 2π −∞
−∞
is bounded, it follows that the operator in the norm at the left hand side generates a one-parameter group of isometries. This completes the proof.
6. Quantum Fokker–Planck Equation In this section, we will address the problem of positivity of the contraction tλ , generated by Z0 + λA00 + KT (λ) . In order to give meaning to semigroup W that, we need some algebraic structure, so we will restrict our attention to the case B = A is a C ∗ -algebra with identity [20], and P0 is a (completely positive) conditional expectation projecting onto the C ∗ -subalgebra X → A.
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Definition 6.1. For strictly positive T and self-adjoint H ∈ A define the selfadjoint averaged perturbation LT ∈ A as ∞ LT = −∞
dt √ δT (t) Ut (H ) 2π
and put L T = P1 (LT ) ∈ A. Moreover, define GT ∈ A through i GT = 2
+∞
−∞
dt dt √ 1 √ 2 sign(t1 − t2 ) δT (t1 ) δT (t2 ) P1 (Ut1 (H ))P1 (Ut2 (H )) 2π 2π
We can now state our third main result: Theorem 6.2. Let P0 be a (completely) positive conditional expectation onto a C ∗ -subalgebra X → A. Let Ut = exp{Zt} be a one-parameter group of automorphisms on A. Assume that [Z, P0 ] = 0 and that Z0 generates a oneparameter group of automorphisms on X . Suppose also that A = i[H , ·] for some self-adjoint H ∈ A. Then (i) (ii) (iii)
Xtλ = e(Z0 +λA00 )t is a one-parameter group of automorphisms on X . tλ = exp{(Z0 + λA00 + λ2 KT (λ) )t} on X is The contraction semigroup W a dynamical semigroup [(completely) positive and identity preserving]. Its generator on X , that we name “Quantum Fokker–Planck Equation”, has the Lindblad form ∂t X = Z0 (X) + iλ[P0 (H ), X] + 2πλ2 i[P0 (GT (λ) ), X] 1 2 2 +2πλ − {(P0 (LT (λ) )) , X} + P0 (LT (λ) X LT (λ) ) 2
(16)
with P0 (GT (λ) ) ∈ X self-adjoint. Proof. To prove (i) note that P0 (H ) is self-adjoint, as H is self-adjoint and P0 is an adjoint map. Then A00 |X = i[P0 (H ), ·]|X generates a one-parameter group of automorphisms on X , and so does Z0 by hypothesis. The validity of (ii) follows from (iii), by just noting that equation (iii) is in the Lindblad form [21]. In fact, both X → P0 (X) and X → L λ X L λ are completely positive maps (the latter is completely positive since it has the Kraus form [22]), and so is their composition. Moreover G, and so P0 (G), is evidently self-adjoint. The remaining requirement ∂t 1X = 0 of identity preservation can be easily checked upon proving (iii). T (λ) P0 with K T (λ) defined in (13). To show (iii), we consider KT (λ) = P0 K In order to compute KT (λ) , let us also note that for every X, Y ∈ A, and real t, Ut (XY ) = Ut (X)Ut (Y ) since Ut is an automorphism. Then, it follows that Aij (t)(X) = U−t Aij Ut (X) = iPi ([U−t (H ), Pj (X)]).
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T P0 , for We will now separately consider each of the two terms of KT = P0 K KT in (13). Since for a generic X ∈ A we compute +∞ dt T √ Φ (t) (X) = i[LT , X] − iP0 ([LT , P0 (X)]) 2π
−∞
= i[P1 (LT ), X] + i[P0 (LT ), P1 (X)] because P0 is a conditional expectation, so P0 (Y LT Z) = Y P0 (LT )Z for Y, Z ∈ X . We take some X = P0 (X) ∈ X (in which case the second term at the right hand side in the equation above disappears), and using again the latter property we compute P0 L2T P0 (X) = −P0 ([P1 (LT ), [P1 (LT ), X]]) − P0 ([P0 (LT ), P1 ([P1 (LT ), X])]) =−{P0 ((P1 (LT ))2 ), X} + 2P0 (P1 (LT )XP1 (LT )),
since P0 P1 = 0. The second term at the right hand side of (13), applied to some X ∈ X , and projected, can be treated with in the same way: we denote = A(t) − P0 A(t)P0 where as before A(t) = U−t AUt , and Ht = Ut (H ), A(t) for X ∈ X we compute 1 )A(t 2 )(X) = −P0 (P1 (Ht )P1 (Ht ))X − XP0 (P1 (Ht )P1 (Ht )) P0 A(t 1 2 2 1
+P0 (P1 (Ht1 )XP1 (Ht2 )) + P0 (P1 (Ht2 )XP1 (Ht1 )). √ √ Multiplying by 12 δT (t1 ) δT (t2 )sign(t1 − t2 ) and integrating on all space (t1 , t2 ) ∈ R2 gives the projection P0 of the second term at the right hand side of (13), applied to any X ∈ X . Due to antisymmetry of the sign function the second line above disappears, so that dt dt T (t1 )Φ T (t2 )(X) √ 1 √ 2 sign(t1 − t2 ) Φ πP0 2π 2π R2 dt dt √ 1 √ 2 δT (t1 ) δT (t2 ) = −π 2π 2π R2
×P0 ([P1 (Ht1 ), P1 (Ht2 )])X + XP0 ([P1 (Ht2 ), P1 (Ht1 )]) Standard manipulation on the integration domain shows that this is equal to 2πi[P0 (GT ), X], and the proof is concluded by noting that P1 and P0 are adjoint maps, so that P0 (GT ) is (bounded and) self-adjoint. Corollary 6.3. If A is a W ∗ -algebra with identity, X is a W ∗ -subalgebra, and P0 is a normal completely positive conditional expectation, then under tλ is a Quantum Dynamical Semigroup the hypotheses of the last theorem W (QDS) in the sense of [21]. •
Comments: The dynamical system above is always well defined irrespective of spectral conditions on Z0 , and even when sp(Z0 ) is discrete, it generalizes literature in that first-order terms P0 (H ) need not vanish.
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If P0 is completely positive, then P0 = P0 (1) = 1 (see e.g. [21]), in agreement with the hypotheses of the validity of Theorem 5.3. P0 need not be the partial trace over a heat bath [13]. Put in the form of the above theorem, KT (λ) furnishes a dynamical measure of the obstruction for P0 to be an algebra homomorphism (as KT (λ) = 0 in that case), thus giving dynamical information on the physical subsystem X → A.
7. Example: A Free Particle Coupled to a Heat Bath We would like to discuss here the limit dynamics for the prototypical example in Quantum Open System, that of the partial trace on a bipartite system, in case Z0 has purely continuous spectrum. The situation can be easily generalized to more reservoirs and/or to different forms for the coupling hamiltonian. However, it turns out that the number of different possibilities is considerably larger with respect to the discrete case, and in order to address steadiness we shall eventually choose to focus on the concrete case of a free nonrelativistic point particle, whose interaction with a fermionic heat bath depends on its energy transitions only. Although the (physically realistic) example is chosen for its simplicity, it is clear from the analysis that the same conclusions would apply to more general situations (spin degrees of freedom could easily be dealt with, some potential profiles may be included, etc.): because of this reason, we shall now start our analysis at a fairly general level, and only at a second stage shall we specialize to our free particle and our heat bath. Let H = HA ⊗ HB be the tensor product of two Hilbert spaces and σ = |ΩΩ| a faithful normal state in T (HB ) (T stands for trace-class operators). Define the normal conditional expectation in the W ∗ -algebra A = B(H) (B stands for bounded operators) according to P0 (X ⊗ Y ) = Y σ X ⊗ 1, and extension by linearity, Y σ being the expectation of Y on the state σ. The range of P0 is identified with the W ∗ -subalgebra X = B(HA ). To discuss the dynamics, we suppose that τt = ei[HA ,·]t is a weakly continuous one-parameter group of automorphisms of the W ∗ -dynamical system (τ, X ), for an (unbounded) self-adjoint hamiltonian HA . We let the same hold for the W ∗ -dynamical system (ei[HB ,·]t , B(HB )), and suppose that σ is invariant under the latter, so that [Z, P0 ] = 0. Now consider the self-adjoint hamiltonian on H, of the form Hλ = HA ⊗ 1 + 1 ⊗ HB + λHI with interaction HI = Q ⊗ Φ
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where Q and Φ are (bounded) self-adjoint operators on the respective spaces. Name the connected bath continuous correlation function as h(t) = eiHB t (Φ − Φσ )e−iHB t (Φ − Φσ )σ , suppose h is integrable and let ∞ dt iωt 1ˆ √ e h(t) = h(ω) + is(ω). 2 2π 0
We have already shown that (16) in Theorem 6.2 defines a QDS, and it is evident that KT itself generates a norm-continuous (quantum) dynamical semigroup if P0 is the partial trace. However, to gain insight in the study of equilibrium states, an energy spectral resolution is imperative. To do that, we shall already at this point limit our analysis to subsystems for which the following assumption is valid: Assumption 7.1. There exist bounded Aω ∈ X , ω ∈ R such that ω → Aω is integrable and the following expansion (17) τt (Q) = dω e−iωt Aω holds true. Moreover τt (Aω ) = e−iωt Aω for every t ∈ R. It is reasonable to expect that the last condition should in some way be implied by the group property of τt , but this is not completely evident a priori, at least in the general case. Note that Assumption 7.1 can only hold if HA has purely continuous spectrum. Upon identifying X ∼ X ⊗ 1 for X ∈ B(HA ) we compute from (16) that Z0 (X) = i[HA , X] and A00 (X) = iΦσ [Q, X]. Then we give the following Proposition 7.2. For ω = (2T )−1 > 0 and under Assumption 7.1, dω s(ω) [A†ω,ω Aω,ω , X] KT (X) = −2πi √ 2π 1 † dω ˆ † Aω,ω Aω,ω , X + Aω,ω XAω,ω h(ω) − +2π √ (18) 2 2π is norm bounded, with bounded (19) Aω,ω = dω δω (ω − ω ) Aω . Proof. We shall only treat the term P0 (GT ) in (16), all the others being completely analogous. To this purpose, note that P1 (Ut (H )) = τt (Q) ⊗ eiHB t (Φ − Φσ )e−iHB t , so that i GT = 2
+∞
−∞
dt dt √ 1 √ 2 h(t1 − t2 )sign(t1 −t2 ) δT (t1 ) δT (t2 ) τt1 (Q)τt2 (Q) 2π 2π (20)
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Ann. Henri Poincar´e
Then we pass to t1,2 = q ± r/2 and use Assumption 7.1 above to compute dq −i(ωα +ωβ )q √ √ P0 (GT ) = πi e dωα dωβ δT ( 2q) 2π β √ dr −i ωα −ω r 2 × √ e δT (r/ 2) h(r) sign(r) Aωα Aωβ 2π √ √ √ √ √ √ because √ δT (t1 ) δT (t2 ) =√ δT ( 2q) δT (r/ 2). Now the √ over q is √ integral just 2−1/2 δω ((ωα + ωβ )/ 2) and Fourier transform for δT (r/ 2) gives ω −ω α β √ √ dω iωr √ ωα − ωβ −i r 2 e e δT (r/ 2) = 2 √ δω 2ω + √ 2π 2 Because of the properties of the gaussian we also have ω α + ω β √ ωα − ωβ √ δω (ω + ωα ) δω (ω − ωβ ) = δω δω 2ω + √ 2 2 (21) so that
dr iωr i dω √ √ e h(r) sign(r) A−ω,ω Aω,ω . 2π 2 2π The term inside the parentheses is just −s(ω), and observing that A−ω,ω = A†ω,ω gives the form of P0 (GT ) stated in the proposition. As said, the other terms all follow in the same way, and KT (X) written in terms of Aω,ω is norm bounded as (h(ω) + is(ω)) Aω,ω 2 is integrable.
P0 (GT ) = 2π
Note that the presence of first-order contributions manifests itself in that, with physical and diagrammatical terminology, only “connected” correlation functions h(t) appear to second order. Note also that the spectral theorem for ˆ ≥ 0 for all real ω. HB reveals h(ω) The assumption of boundedness for the Aω in (17), and even of their existence, can surely be relaxed, as no mention to Aω or restriction on the spectrum of Z0 is made in the following Proposition 7.3. The form of KT in Proposition 7.2 follows if 1. h(t) is integrable and h(0) = 0 ˆ 2. h(ω) and s(ω) are integrable by defining dt iωt √ e δT (t) τt (Q). Aω,ω = 2π Then Aω,ω is bounded uniformly on ω ∈ R and KT is norm bounded. Proof. We have
dω −iωt √ e s(ω) 2π because h(t)sign(t) admits Fourier transform (as h(t) is integrable by hypothesis), vanishes at infinity and is uniformly continuous (as h(t) is the inverse h(t)sign(t) =
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ˆ Fourier transform of h(ω) which is integrable by hypothesis and h(0) = 0). By plugging this into (20) we immediately obtain P0 (G) as in Proposition 7.2, with Aω,ω defined above, and the other terms√ follow analogously (with no 1 requirement on h(0)). Then Aω,ω ≤ (2/π) 4 T Q is bounded uniformly on ω and KT is norm bounded because of hypothesis (2). As a brief comment, the technical hypothesis h(0) = 0 avoids energy renormalization singularities in the time domain, corresponds to Φ2 σ = Φ2σ , and explicitly requires nonzero first-order terms (Φσ = 0 ⇔ A00 = 0). Despite of this, the hypothesis has nothing to do with the dissipative part of KT , as noted in the proof above and, as evident from Proposition 7.2, could surely be relaxed by providing suitable additional information on the subsystem X . In passing, we have already proven in the general case (Sect. 4.1) that the limit T → ∞ is well defined and corresponds to Davies averaged K , if sp(Z0 ) is discrete and A00 = 0 (which here corresponds to Φσ = 0). Here it is possible to see that even when Φσ = 0 one can write precisely Eq. (4.9) in [2] by just substituting the “connected” h and s we have defined.3 Indeed, this ω e−iωt in (20): the limit ω → 0 is found by expanding τt (Q) = iω∈spd (Z0 ) A of the right hand side of (21) gives δωβ ,−ωα δ(ω − ωα ), where the first is a Kronecker delta, and the second is a Dirac delta. Reasoning backwards in the frequency domain, it is evident how our new average procedure has smoothed the product of a Kronecker times a Dirac delta, by letting the singularity be shared symmetrically and without restrictions on the spectrum of Z0 ; the gaussian form of our weights then allows to factorize frequencies according to (21), and always because gaussians are involved, the factorization remains even in the time domain, both for t1,2 (proving that KT is dissipative) and for t1 ± t2 /2 (proving the markovian approximation theorem). The only side effect is that now the resonance of Aω,ω has been smoothed to roughly ω ± ω, so that KT does not preserve the set of diagonal elements D = {X ∈ B(HA ) | τt (X) = X, ∀t ∈ R}. Moreover, for purely continuous Z0 T →∞ spectrum one has KT −→ 0 strongly, as evident from the above discussion. However, there is still a means to compare with Eq. (4.9) in [2]: Definition 7.4. Under Assumption 7.1, h(t) integrable and ω → Aω square integrable, we define the operator L on X given by L(X) = −2πi dω s(ω) [A−ω Aω , X] 1 ˆ +2π dω h(ω) − {A−ω Aω , X} + A−ω XAω (22) 2 Proposition 7.5. Suppose h to be integrable. Under Assumption 7.1 suppose that R ω → Aω ∈ X is continuous and Aω = o(|ω|1/2 ), 3
The additional factor
√
|ω| → ∞
2π is due to different definitions of Fourier transform.
(23)
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Ann. Henri Poincar´e
Then L is bounded, generates a norm continuous QDS on X and L preserves D = {X ∈ B(HA ) | τt (X) = X, ∀t ∈ R} in the sense that eLt (X) ∈ D whenever X ∈ D, for all t ≥ 0. Moreover lim T KT − L = 0.
T →∞
ˆ and s(ω) are Proof. L is bounded as Aω is square integrable and both h(ω) integrable, and it is manifestly completely dissipative. From the stated form of the generator L, the fact that τt are automorphisms, and that τt (Aω ) = e−iωt Aω , one then sees that X ∈ D ⇒ (Lτt )(X) = (τt L)(X) = L(X), Lt proving the invariance of D under ∞: using √ e . We pass to the limit T → √ ω= † 1/2T we plug Aω,T = dωα δω (ω − ωα ) A−ωα and Aω,T = dωβ δω (ω − ω +ω ωβ ) Aωβ into (18), change variable with Ω = α 2 β and ν = ωα − ωβ , and use (21) to find 1 1 (X) = −2πi K 2ω dΩ sω (Ω) [(A−Ω AΩ )(ω) , X] 2ω 1 (ω) (ω) ˆ +2π dΩ hω (Ω) − {(A−Ω AΩ ) , X}+(A−Ω XAΩ ) 2
with
ˆ dω h(ω) δω (Ω − ω), i Ω−ω 2 ˆ δω (Ω − ω) i erf √ sω (Ω) = dω h(ω) π 2 ω (A−Ω XAΩ )(ω) = dν δω (ν) A−Ω−ν XAΩ−ν . ˆ ω (Ω) = h
Now
2 δω (ω) i erf π
i ω √ 2ω
1 , → P√ 2πω
ω → 0+
ˆω → h ˆ pointwise as ω → 0+ , in the sense of distributions, so that sω → s and h ˆ because of our hypotheses on s and h. The fact that dω Aω 2 < ∞ furnishes an integrable upper bound for the dominated convergence theorem, so that convergence 1 1 − Lω ω → 0+ , (X) 2ω K 2ω → 0, is proved uniformly on X = 1 for the obvious intermediate Lω (X) = −2πi dω s(ω) [(A−ω Aω )(ω) , X] 1 ˆ + 2π dω h(ω) − {(A−ω Aω )(ω) , X} + (A−ω XAω )(ω) 2
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ˆ as an ω (X) with Lω (X), we use h (where we renamed Ω as ω). To compare L integrable upper bound for the dominated convergence theorem applied to the pointwise convergence (A−ω XAω )(ω) − A−ω XAω → 0,
ω → 0+
In turn, this follows from the fact that the norm of the above difference is smaller than c X dν δω (ν) ( A−ω−ν − A−ω + Aω−ν − Aω ) , which goes to zero for ω → 0+ , because Aω − Aω+ν ≤ Aω + Aω+ν is uniformly bounded and continuous with respect to ν, due to continuity of Aω and the asymptotics (23). Proceeding similarly for the commutator part proves ω − L)(X) → 0 uniformly on X = 1 and thus the first claim of the that (L proposition. As a remark, we have shown, under the conditions of Proposition 7.5 above and for Φβ = 0, that for every τ > 0
lim
λ→0
sup 0≤t≤λ−2 τ
P0 e(Z+λA)t P0 − e(Z0 +
λ3 τ
L)t
= 0.
Indeed assumptions in Theorem 4.4 are satisfied, as proven in [2], and the markovian estimate is deduced from Theorem 4.2 and Proposition 7.5 with T (λ) = |λ|−1 τ for λ = 0. We are now in position to focus on the case of a free particle in the 3D-euclidean space weakly coupled to a heat bath, for which we shall prove the validity of the general framework above (namely Assumption 7.1 and the hypotheses on Aω of Proposition 7.5). As the bath is concerned, we shall assume it to be described, as in [2], by a quasi-free representation on HB of operators ϕ(f ), f in the Hilbert space V of test functions, satisfying the CAR algebra {ϕ(f ), ϕ(g)} = 2f, g1, and being compatible with single particle dynamics in the sense that Ut (1⊗ϕ(f )) = 1 ⊗ ϕ(eiSt f ) for a (densely defined) self-adjoint operator S on V. Moreover |Ω is cyclic, HB |Ω = 0, and the representation is fixed by Ω, ϕ(f )ϕ(g) Ω = g, e−βS (1+e−βS )−1 f +f, (1−e−βS (1+e−βS )−1 ) g
(24)
for inverse temperature β. The perturbation H = Q ⊗ Φ is chosen with Φ = iϕ(f1 )ϕ(f−1 ), for some f1 and f−1 in V. Even though for what follows it will be sufficient to assume that f±1 have disjoint energy spectra, implying Φσ = 0 (see [2]), possible first-order contributions could be included, as we report here in a slight generalization of Lemma 4.1 in [2]. Lemma 7.6. If h is integrable then its transform satisfies ˆ ˆ h(−ω) = e−βω h(ω) for all ω ∈ R.
(25)
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Ann. Henri Poincar´e
Proof. By quasi-freeness, Wick expansion of h(t) gives h(t) = h1 (t)h−1 (t) − |χ(t)|2 , with hi (t) = Ω, ϕ(eiSt fi )ϕ(fi )Ω and χ(t) = Ω, ϕ(eiSt f1 )ϕ(f−1 )Ω. The specˆ i (−ω) = e−βω h ˆ i (ω) and χ (−ω) = e−βω χ (ω)∗ , so tral theorem and (24) give h that the conclusion follows from ˆ 1 (ω )h ˆ −1 (ω − ω ) − χ ˆ (ω )∗ χ (ω − ω )). h(ω) = dω (h ∞ In [2], the conditions 0 |hi (t)|(1 + t ) dt < ∞ are shown to be sufficient for the hypotheses of Theorem 4.4 (and Lemma 7.6) to hold, in case χ(t) vanishes identically. We conjecture that there exist analogous conditions on χ(t) for the foretold hypotheses to hold, but leave the involved analysis to future work and, as already stated, we shall assume χ ≡ 0. We now focus on the particle, by specializing the coupling to depend on energy transitions only: Proposition 7.7. Let H = L2 (R3 ) be the Hilbert space of a free particle HA = P 2 /2 in three dimensions, and let the integral kernel q(p, p ) define (the bounded self-adjoint operator) Q in the representation where P is the multiplication operator. Let q(p, p ) = q (εp , εp ) for a complex function q in the Schwartz space S(R2 ), and εp = p2 /2. Then Assumptions 7.1 and the hypotheses on Aω of Proposition 7.5 are satisfied. Proof. Using the spectral theorem for τt (Q), Assumption 7.1 is satisfied if for test functions φ, ψ in the Schwartz space S (and denoting with θ the Heaviside step function) the equality φ, Aω ψ =
d3 p d3 p δ(ω − (εp − εp )) q(p, p ) φ(p)∗ ψ(p ),
(26)
(i) defines an operator Aω on S(R3 ); (ii) Aω extends to a bounded operator on H (that we again denote with Aω ); (iii) the integral at the right hand side of (17) is bounded. Now, point (i) follows, as the surfaces Sε = {p ∈ R3 | ε = ω + εp } are regular and compact (they are possibly degenerate spheres), the function q is in Schwartz space, the gradient ∇εp is zero only at the origin, and for every real ω the singularity is controlled by the jacobian passing to spherical coordinates. Point (ii) follows if Aω ψ is bounded uniformly on normalized ψ ∈ S(R3 ), because of density. We compute
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Aω ψ =
1323
d3 p d3 p δ(εp − εp )| q (εp , εp + ω)|2 η(ω + εp ) ψ(p )∗ ψ(p)
∞ = dξ ξ 3 η(ω + ξ 2 /2) | q (ξ 2 /2, ω + ξ 2 /2)|2 0
2 sin θ dθ dφ ψ(ξ, θ, φ) × with η(ε) = 4πθ(ε) 2|ε| being the density of states, and having passed to spherical coordinates in the second and third lines. The third line is overesti mated by 4π sin2 θ dθ dφ |ψ(ξ, θ, φ)|2 , so that Aω ψ 2 ≤ d3 p f (εp , ω + εp ) |ψ(p)|2 with continuous f (ε, ε ) = η(ε)η(ε ) | q (ε, ε )|2 . Now, for any ω the right hand side above is smaller than c ψ 2 for some constant c independent of ψ, because q is continuous and q (ε, ε ) = o(ε) for ε → +∞, and η(ε) = O(1) for ε → 0+. Point (iii) follows from the fact that the function ω → Aω is (a) continuous, and (b) Aω = o(ω + ) for some > 0 and for |ω| → ∞. To prove point (a), denote with ∂1 q (ε, ε ) the partial derivative with respect to the first argument, and compute (Aω+ν −Aω )ψ 2 = ν 2 d3 p d3 p δ(εp − εp ) |∂1 q (εp , εp + ω)|2 η(ω + εp ) ×ψ(p )∗ ψ(p) + o(ν 2 ) due to continuity of η, and the fact that p → q (εp , εp + ω) is in S(R3 ), and proceeding as above shows that the second-order term in ν is bounded at every real ω because q ∈ S(R3 ). Now, b) follows because f (ε, ε+ω) = o(ω 2 + + ) as |ω| → ∞ uniformly on all ε ∈ R+ , as can be seen directly for ε → 0+ , and by noting that q(ε, ω) := q (ε, ε+ω) is also in Schwartz space. Finally, (26) shows that τt (Aω ) = e−iωt Aω for all t, ω ∈ R. This proves the validity of Assumption 7.1 and at the same time all the other requirements on Aω , concluding the proof. We now consider the problem of stationary distribution for (at least a sufficiently ample subclass of) elements of D = {X ∈ B(HA )|τt (X) = X, ∀t ∈ R}. In T (HA ) consider the equivalence relation ρ1 ∼ ρ2 iff Tr(ρ1 A) = Tr(ρ2 A) for all A ∈ D, and for ρ ∈ T (HA ) denote with [ρ] its associated equivalence class. Then clearly Tr(ρA) = Tr([ρ]A) passes to the quotient for A ∈ D. Let ρβ be the density matrix (positive trace-class normalized operator) on HA , whose kernel in the momentum representation is given by μβ (εp1 ) μβ (εp2 ) e−βε for μβ (ε) = ∞ −βε (27) ρβ (p1 , p2 ) = η(εp1 ) η(εp2 ) e dε 0 √ and η(ε) = 4π 2ε being the density of states.
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Ann. Henri Poincar´e
Define the subset G ⊂ D of bounded operators G on HA whose kernel in momentum representation has the form G(p1 , p2 ) = g(p1 , p2 )δ(εp1 − εp2 ) for some g ∈ S(R6 ), and for G ∈ G define the L∞ (R+ ) function g by g (ε) := dp1 dp2 g(p1 , p2 ) δ(ε − εp1 )δ(ε − εp2 ). (28) ˆ be in the Proposition 7.8. Under the setting of Proposition 7.7, let moreover h Schwartz space S(R). Then for all G ∈ G, under Definitions 27 and 28, the expectation ∞ 3 (Z0 + λτ L)t Gβ (t) := Tr([ρβ ] e G) = g (ε) μβ (ε) dε (29) 0
is stationary (for all τ > 0). Proof. Clearly Z0 |D = 0, so Gβ (t) = Tr([ρβ ] eLt G) modulo rescaling time. Using bracket notation, the spectral decomposition δ(ω + ε − ε ) dEε QdEε Aω = for HA = ε dEε allows to compute LG according to † ˆ ˆ dω h(ω) Aω Aω G = dω h(ω) GA†ω Aω ˆ p − εp ) = dp dp1 |pp1 | δ(εp − εp1 )| q (εp , εp )|2 h(ε × dp2 g(p2 , p1 )δ(εp1 − εp2 ) and
ˆ dω h(ω) A†ω GAω =
×
ˆ p − εp ) dp dp1 |pp1 | δ(εp − εp1 )| q (εp , εp )|2 h(ε dp2 g(p2 , p )δ(εp − εp2 ).
ˆ we From these expressions, and from the regularity assumptions for q and h, Lt see that e preserves G at all (positive) times. Moreover, we compute ˆ − ε ) |q(ε, ε )|2 (η(ε) Tr([ρβ ]LG) = 2π dεdε μβ (ε) h(ε g (ε ) − η(ε ) g (ε)), or equivalently Tr([ρβ ]LG) = dε g (ε)(L∗ μβ )(ε) where L∗ acting on L1 (R+ ) gives the Boltzmann equation ∞ ∗ (L μ)(ε) = dε η(ε ) (r(ε , ε) μ(ε ) − r(ε, ε ) μ(ε)) (30) 0
with celebrated Fermi Golden Rule transition rates ˆ − ε) | q (ε, ε )|2 . r(ε , ε) = 2π h(ε
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The statement of the proposition then follows from the detailed balance condition
r(ε, ε ) = eβ(ε−ε ) r(ε , ε) due to Lemma 7.6 (and then by explicit computation of Tr([ρβ ]G)).
Remarks. • The above proposition only needs the function g to be in L∞ (R+ ), surely allowing to include a class of observables in D larger than G: for example, bounded and sufficiently regular functions of the energy should be included without problems, even if they don’t belong to G. • Unicity of the steady distribution for G ∈ G is not obvious from the above analysis, as the fact that Tr([ρβ ]LG) = dε g (ε)(L∗ μβ )(ε) depends on G only through g (ε) is conditional on [ρβ ], and it is not generally true for other density matrices. We argue that unicity could be restored for momentum conserving interactions H , but this would take us beyond the scope of the present example (whose aim is to motivate the general theory section through a simple analysis). • There are no KMS states for a free particle in Euclidean space (physically, one cannot measure position if the particle is “everywhere”), and therefore one is forced to consider equilibrium distribution for a suitable subclass of observables only.4
8. Proofs of Theorems 3.4 and 4.2 The following lemma is not new, and for example it is contained in Theorem 1.2 of [3], but we report it here as we shall make use of it repeatedly all throughout. Lemma 8.1. Let b ∈ B0 be given, together with some real τ > 0. Suppose Xtλ is λ a one-parameter group of isometries. Suppose Wtλ and W t are operators on λ λ B0 such that fλ (τ ) = X−λ −2 τ Wλ−2 τ b satisfies
fλ = Hλn b (32) n≥0
for a Volterra operator Hλ on the Banach space V = C 0 ([0, τ ], B0 ) of continuous B0 -valued functions on the interval [0, τ ] (assume the same holds also for λ W t , with associated f λ and Hλ ). Suppose there exists a real positive c such that Hλ V ≤ cτ and Hλ V ≤ cτ uniformly on |λ| ≤ 1. Put (i) (ii) (iii)
limλ→0 Hλ − Hλ V = 0; limλ→0 fλ − f λ ∞ = 0; λ limλ→0 sup0≤t≤λ−2 τ Wtλ b − W t b B0 = 0.
Then (i) ⇒ (ii) ⇒ (iii). 4
We are grateful to our referee for pointing that out.
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Ann. Henri Poincar´e
Proof. Of course (ii) ⇒ (iii) as Xtλ is group of isometries and one has sup 0≤t≤λ−2 τ
λ
Wtλ b − W t b B0 = sup fλ (τ ) − f λ (τ ) B0 = fλ − f λ ∞ . 0≤τ ≤τ
Then, subtracting the von Newmann expansions for fλ and f λ one obtains fλ − f λ ∞ ≤ =
∞
n=1 ∞
n=1
n
Hλn b − Hλ b ∞ n−1
Hλn−1 (Hλ − Hλ )b + · · · + (Hλ − Hλ )Hλ
≤ Hλ − Hλ V b B0
b ∞
∞
(τ c)n−1 (n − 1)! n=1
and the last series is (obviously) convergent and independent of λ. Note that we have used (and will use throughout) the important property that if H is Volterra and H ≤ C, then Hn ≤ C n /n! (see e.g. Eq. (1.28) in [3]). This shows that (i) ⇒ (ii) and thus finishes the proof. We still need a technical result that will allow us to perform approximation (8): its interpretation will become clear in the context of the proof of Theorem 3.4, but it deserves to be reported in the more autonomous environment of a Lemma, as it will find application also in the proof of Theorem 4.2. Lemma 8.2. Let B0 be a Banach space, τ > 0, and let V = C 0 ([0, τ ], B0 ) be the Banach space of continuous functions from [0, τ ] into B0 . For some real q and α, let D(λ, τ, α, q) be the triangle in the (σ, x)-plane of vertices 1 2 2 2 −2 − α τ + λ q, λ τ D(λ, τ, α, q) = (λ q, 0), (τ + λ q, 0), 2 and define the truncated domain5 (see Fig. 1) D(λ, τ, α, q) = D(λ, τ, α, q) ∩ [0, τ ] × [0, ∞).
(33)
Let T : I˙ → R+ a real positive continuous function, assume T (λ) ∼ |λ|−ξ T ,
λ→0
for some T > 0 and 0 < ξ < 2 (strictly). Let H(λαq) be a Volterra integral operator on V and assume it can be put in the form 2 − (x/2) dσdx e 2T (λ)2 Kαq (λ, σ, x) g σ−λ2 (q+(1/2−α) x) , (H(λαq) g)(τ ) = D(λ,τ,α,q)
(34) 5
D(λ, τ, α, q) is the empty set for (0 ≤)τ ≤ λ2 q, but this does not alter what follows.
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for a suitable kernel Kαq (λ, σ, x), and define the Volterra integral operator H(λαq) by
2
(H(λαq) g)(τ ) =
dσdx e
(x/2) − 2T (λ)2
Kαq (λ, σ, x) g(σ).
D(λ,τ,α,q)
Suppose that Kαq (λ, σ, x) = k(x) independently on σ, λ, α, q, and that ∞ dx k(x) = c
(35)
0
for some finite 0 < c < ∞, and assume H(λαq) < τ c uniformly on |λ| ≤ 1. Let b ∈ B0 and define the von Neumann series f(λαq) and f (λαq) through (32) in Lemma 8.1. Then H(λαq) ≤ τ c
(36)
lim f(λαq) − f (λαq) ∞ = 0.
(37)
uniformly on |λ| ≤ 1 and λ→0
Proof. Estimation (36) is a consequence of the definition of H(λαq) , as ∞ H(λαq) g(τ ) ≤ τ
k(x) g ∞ . 0
To show the validity of (37) we perform a telescopic expansion as in the proof of Lemma 8.1 to obtain the following estimate, f(λαq) −f (λαq) ∞ ≤
∞ n−1
(τ c)n−l−1 l (H(λαq) −H(λαq) ) H(λαq) b ∞ . (n − l)! n=1 l=1
Note that the case l = 0 has been dropped, since H(λαq) b = H(λαq) b trivially, as b is constant, as one can see from (34). Now if we could show that for every > 0 there exists some λ > 0 such that |λ| < λ implies that for every l ≥ 1 l
(H(λαq) − H(λαq) ) H(λαq) b ∞ ≤
(τ c)l−1 τ b B0 , (l − 1)!
we would be done, as, following estimation (38), we would have fλ − f λ ∞ ≤
∞ n−1
n=1 l=1
(τ c)n−2 τ b B0 , (n − l)!(l − 1)!
(38)
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Ann. Henri Poincar´e
Figure 2. Integration domains D(λ, σ, α, q) (rectangular triangle (1)), D(λ, σ−λ2 (q + (1/2 − α) x) , α, q) (rectangular triangle (2)) and S(λ, σ, x) (region (1) ∪ (2) = region (1) in this case). We have put α < −1/2, q > 0 for clarity. The graph shows that the projection on the σ1 -axis of integration domain (1)\(2) → 0 as λ → 0. Its complementary projection tends to the whole positive part of the x1 -axis, but it is controlled by the boundedness of the corresponding integral kernels and the series would obviously converge (because c > 0 by hypothesis). To show that property (38) holds, we take l > 0 and evaluate6 l (H(λαq) − H(λαq) ) H(λαq) b (τ ) 2 − (x/2) dσdx e 2T (λ)2 Kαq (λ, σ, x) = D(λ,τ,α,q)
l l × (H(λαq) b) σ − λ2 (q + (1/2 − α)x) − (H(λαq) b)(σ) 2 − (x/2) dσdx e 2T (λ)2 Kαq (λ, σ, x) = D(λ,τ,α,q)
×
dσ1 dx1 Δλ (σ1 , x1 ) e
2
(x/2) − 2T (λ)2
l−1
Kαq (λ, σ1 , x1 ) (H(λαq) b)(σ1 )
S(λ,α,q,σ,x) λ λ where we have put Δλ = χλ1 − χ2 , χ1 λbeing the characteristic func1 2 tion of D(λ, σ − λ q + 2 − α x , α, q), χ2 the characteristic function of D(λ, σ, α, q), and we have defined (see Fig. 2) S(λ, α, q, σ, x) = D λ, σ−λ2 (q+(1/2−α) x) , α, q ∪ D(λ, σ, α, q). 6
Note that σ − λ2 q + 12 − α x = u ≥ 0.
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Passing to the norms, we overestimate the norm of the left hand side of (38) with
sup
0≤τ ≤τ D(λ,τ,α,q)
≤
2
dσdx e
(x/2) − 2T (λ)2
(τ c)l−1 bB0 sup (l − 1)! 0≤τ ≤τ
l−1 dσ1 dx1 |Δλ (σ1 , x1 )| k(x1 ) H(λαq) b
k(x)
∞
S(λ,α,q,σ,x) 2
(x/2) − 2T (λ)2
dσdx e
k(x) Ξαq (λ, σ, x)
D(λ,τ,α,q)
l−1
(τ c) ≤ bB0 τ sup (l − 1)! 0≤σ≤τ
∞
2
dx e
(x/2) − 2T (λ)2
k(x) Ξαq (λ, σ, x)
(39)
0
where we named
Ξαq (λ, σ, x) =
dσ1 dx1 |Δλ (σ1 , x1 )| k(x1 )
(40)
S(λ,α,q,σ,x)
At this point let us consider the sector defined by q > 0 and α < −1/2. In this case, it is but a straightforward algebra to show that the two triangles D λ, σ − λ2 (q + (1/2 − α) x) , α, q and D(λ, σ, α, q) are similar (the same is true for the unbarred Ds for arbitrary (α, q)) and that their left edges lie on the same line of equation xl (λ, σ1 ) =
λ−2 (σ1 − λ2 q) −α
1 2
in the (σ1 , x1 )-plane (see Fig. 2). In particular D(λ, σ, α, q) contains D λ, σ−λ2 (q+(1/2−α) x) , α, q . For this reason we compute xl (λ,σ 1)
σ Ξαq (λ, σ, x) =
dx1 k(x1 )
dσ1 σ−λ2
0
(q+( )x) ∞ 1 2 − α x dx1 k(x1 ) ≤ λ q + 2 ≤ cλ
2
1 2 −α
0
1 |q| + − α x 2
uniformly on σ. According to the last line in estimation (39), we study ∞ 2 − (x/2) sup dx e 2T (λ)2 k(x) Ξαq (λ, σ, x) 0≤σ≤τ
0
∞ 2 1 − (x/2) ≤ c λ q + c − α dx k(x)e 2T (λ)2 λ2 x 2 2 2
(41)
0
The first term goes to zero with velocity ∼ λ2 q, whereas the dominated convergence theorem applies to the second term, showing convergence to zero
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uniformly on q, with velocity ∼ λ2−ξ (note that we have supposed ξ < 2 strictly). This, when put in (39), shows the validity of (38). A very similar analysis can be done for the remaining sectors q 0, |α| < 1/2 and α > 1/2. In each case, the net result is that the estimation (41) is always of the form ∞ sup
0≤σ≤τ
2
dx e
(x/2) − 2T (λ)2
k(x) Ξαq (λ, σ, x) ≤ C1 λ2 |q| + C2 λ2−ξ
(42)
0
for suitable real positive constants C1 and C2 . This shows the validity of (38), and thus concludes the proof. 8.1. Proof of Theorem 3.4 Let V be the Banach space of norm continuous B0 -valued functions on [0, τ ], and let b ∈ B0 . Define the “interaction picture” time rescaled solution of (1) λ λ fλ (τ ) = X−λ −2 τ Wλ−2 τ b.
Then fλ is a solution to the integral equation fλ = b + Hλ fλ , where the integral operator Hλ is defined (recall that Xtλ is a group of isometries) by (Hλ g)(τ ) = λ
2
λ−2 τ
s
ds 0
λ du Xsλ A01 Us−u A10 Xuλ g(λ2 u).
(43)
0
In [3], Davies shows that Hλ is a Volterra operator. Indeed, by changing coordinates according to (4), (43) can be given an explicit Volterra form, namely as τ λ λ (44) (Hλ g)(τ ) = dσX−λ −2 σ K(λ, τ − σ)Xλ−2 σ g(σ), 0
where we defined the “slowly varying” kernel λ−2 τ
K(λ, τ ) =
λ dx X−x A01 Uxλ A10 .
0
Because of this reason, and since K(λ, τ ) is manifestly bounded by c, uniformly on |λ| ≤ 1 thanks to Assumption 3.2, it follows that Hλn ≤ cn τ n /n!,
(45)
and also that the associated von Newmann series expansion fλ = b + Hλ b + Hλ2 b + · · · converges.
(46)
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We can proceed in similar fashion also for the semigroup (9): iteration gives indeed τ λ W λ−2 τ = Xλλ−2 τ + dσ Xλλ−2 (τ −σ) K(α,q,T (λ)) Wλλ−2 σ . 0
Accordingly, we define λ
λ f λ (τ ) = X−λ −2 τ W λ−2 τ b
so that it follows that f λ is a solution to the integral equation f λ = b + H(λαq) f λ , where we have defined τ (H(λαq) g)(τ ) =
λ λ dσ X−λ −2 σ K(α,q,T (λ)) Xλ−2 σ g(σ).
(47)
0
Now again, H(λαq) is a Volterra operator, and since K(α,q,T (λ)) ≤ c, (45) and (46) follow analogously for H(λαq) . Clearly (see Lemma 8.1), we must show that for any chosen b ∈ B0 , sup 0≤t≤λ−2 τ
λ
Wtλ b − W t b = fλ − f λ ∞ → 0,
λ → 0.
We shall do that by defining six suitable Volterra operators, denoted with (j) (0) (N ) H(λαq) , j = 0 . . . 6, such that H(λαq) = Hλ , H(λαq) = H(λαq) , for which either we show, according to Lemma 8.1, that (j)
(j−1)
H(λαq) − H(λαq) V → 0,
λ→0
or more directly that (j)
(j−1)
f(λαq) − f(λαq) ∞ → 0, where (j)
f(λαq) =
∞
n=0
(j)
H(λαq)
λ→0 n
b
is the associated von Neumann series.7 Then, our conclusion (see Lemma 8.1) would follow from N
(j) (j−1) fλ − f λ ≤ f(λαq) − f(λαq) → 0 λ → 0. (48) j=1
To follow our purpose, instead of (4), we perform the coordinate transformations (5) As we noted before, the integration domain s = 0 . . . λ−2 τ , u = 0 . . . s, becomes the domain D(λ, τ, α, q) in the (σ, x)-plane defined in (6), 7 Note that we have attached the subscript “(λαq)” all throughout: although some Volterra operator may not actually depend on α, nor q, this unifying notation will become useful in the sequel.
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and depicted in Fig. 1. Accordingly, the integral kernel in (43) is now written as λ λ λ (Hλ g)(τ ) = dσdx X−λ −2 σ+q− α+ 1 x A01 Ux A10 Xλ−2 σ−q− 1 −α x ( ) ( ) 2
2
D(λ,τ,α,q)
×g(σ − λ2 (q + (1/2 − α)x)).
(49)
We shall work throughout with the choice α < −1/2 and q > 0, without loss of generality (the structure of the proof is basically the same for all the remaining sectors, and actually the case |α| < 1/2 presents less difficulties). The first thing we shall be concerned with, is to find a way to substitute the free “polarization evolution” P1 Ux in place of the interacting P1 Uxλ in the middle of the kernel in (49). This step is accomplished by defining a related (1) integral operator Hλ by
(1)
(H(λαq) g)(τ ) =
λ λ dσdx X−λ −2 σ+q− α+ 1 x A01 Ux A10 Xλ−2 σ−q+ α− 1 x ( ) ( ) 2
2
D(1) (λ,τ,α,q)
×g(σ − λ2 (q + (1/2 − α)x)).
(50) (1)
Here, we have denoted D(1) = D for sake of notation. Note that H(λαq) does not depend on α, nor on q, although both the latter parameters appear in its definition. To compare the two integral operators, take a bounded g ∈ V and estimate (1) dσdx A01 (Uxλ − Ux )A10 g ∞ Hλ − H(λαq) g ≤ D(λ,τ,α,q) λ−2 τ
dx A01 (Uxλ − Ux )A10
≤ max{1, |1/2 − α|} τ g ∞ 0
(note for later purposes that the estimation does not depend on q). Our convergence hypothesis 3.3 on Uxλ allows then to conclude that this goes to zero when λ → 0 uniformly on every g = 1, so that we obtain (1) lim Hλ − H(λαq) = 0.
λ→0
(51)
We proceed along similar lines to smooth the kernel with T (λ): define (2)
(H(λαq) g)(τ ) =
dσdx e
−
(x/2)2 2T (λ)2
λ λ X−λ −2 σ+q−(α+ 1 )x A01 Ux A10 Xλ−2 σ−q+(α− 1 )x 2
2
D(2) (λ,τ,α,q)
×g(σ − λ2 (q + (1/2 − α)x)).
(52)
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(1)
Here we have denoted D(2) = D(1) . To compare with H(λαq) , take a bounded g ∈ V and estimate 2 − (x/2) (1) (2) dσdx |1 − e 2T (λ)2 | A01 Ux A10 g ∞ H(λαq) − H(λαq) g ≤ D(λ,τ,α,q)
∞ 2 1 − (x/2) ≤ max 1, − α τ g ∞ dx|1 − e 2T (λ)2 | A01 Ux A10 2 0
Hypothesis 3.2 furnishes an integrable upper bound to the last integrand, and so the integral goes to zero in the limit λ → 0 because of the dominated convergence theorem: in fact, one has pointwise convergence (x/2)2 1 − e− 2T (λ)2 → 0, λ → 0 due to our hypothesis 0 < ξ. Uniform convergence on all g = 1 in the last line of our estimation then shows that (1) (2) lim H(λαq) − H(λαq) = 0. (53) λ→0
(2) H(λαq) ,
(1)
We note that as H(λαq) and Hλ , is also a Volterra operator, as is easy to verify: it would suffice to apply the inverse transform (5) for the specific choice of α and q, and subsequently apply the transform (5) for q = 0 and (2) α = 1/2, to find H(λαq) into its explicit Volterra form (and also find that it does not depend on (α, q)). Now define the restricted domain (see Fig. 1) D(3) (λ, τ, α, q) = D(2) (λ, τ, α, q) ∩ [0, τ ] × [0, ∞) (3)
(2)
and define H(λαq) accordingly, as the restriction of H(λαq) to D(3) (λ, τ, α, q), that is, (3) (H(λαq) g)(τ )
=
dσdx e
−
(x/2)2 2T (λ)2
λ λ X−λ −2 σ+q−(α+ 1 )x A01 Ux A10 Xλ−2 σ−q+(α− 1 )x 2
2
D(3) (λ,τ,α,q)
×g(σ − λ2 (q + (1/2 − α)x)).
(54)
This is again a Volterra operator, as the image of D(3) (λ, τ, α, q) under the composition of (5) and its inverse for q = 0 and α = 1/2, is inside the rectangular triangle, image of D(2) (λ, τ, α, q) through the same transformations. So (2) the composition of (5) its inverse for q = 0 and α = 1/2, will put both H(λαq) (3)
and H(λαq) in their explicit Volterra form. We must prove that (2) (3) lim H(λαq) − H(λαq) → 0
(55)
λ→0
(2)
(3)
(note that for |α| < 1/2 this is trivial, as one has H(λαq) = H(λαq) , for the upper vertex of the triangular domain projects on the horizontal edge of the
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latter (the one lying on the σ-axis), and thus D(2) (λ, τ, α, q) = D(3) (λ, τ, α, q)). To this end, we take as usual any g ∈ V with g = 1 and estimate (2) (3) dσdx A01 Ux A10 g ∞ Hλ − H(λαq) g (τ ) ≤ D(λ,τ,α,q)∩{τ ≤σ≤|1/2−α|τ +λ2 q}
⎫ ⎧ ( 12 −α)τ +λ2 q ∞ ⎪ ⎪ ∞ ⎬ ⎨ 2 ≤ g ∞ λ q dx A01 Ux A10 + dσ dx A01 Ux A10 ⎪ ⎪ ⎭ ⎩ 2 0
τ +λ q
xr (λ,q,σ)
Here xr (λ, q, σ) =
q + λ−2 (τ − σ) α + 12
is the x-coordinate of the right edge of the triangle D(λ, τ, α, q) (see Fig. 1). The first term in the curly brackets clearly goes to zero uniformly on τ (with speed ∼ λ−2 q), because of our boundedness hypothesis 3.2. In the second term in the curly brackets, we change coordinates according to σ → σ − λ2 q, so that it becomes equal to ( 12−α)τ dσ τ
+∞ dx A01 Ux A10 .
xr (λ,0,σ)
But this converges to zero as λ → 0, as a consequence of xr (λ, 0, σ) → ∞, λ → 0, of Assumption 3.2, and of the boundedness of the σ-integration domain. Uniform convergence on all 0 ≤ τ ≤ τ (and real q) follows from the fact that the σ-integration domain is compact. This shows the validity of (53). We now define the following “time localized” Volterra operator: (4)
(H(λαq) g)(τ ) =
dσdx e
−
(x/2)2 2T (λ)2
λ λ X−λ −2 σ+q−(α+ 1 )x A01 Ux A10 Xλ−2 σ−q+(α− 1 )x 2
2
D(4) (λ,τ,α,q)
×g(σ)
(56)
where we have put D(4) (λ, τ, α, q)) = D(3) (λ, τ, α, q). It turns out that proving (3) an operator convergence to H(λαq) is impossible, due to the strong requirement of uniform convergence with respect to any normalized g ∈ V. However, the (3) (4) Volterra operators H(λαq) and H(λαq) do fulfill the hypotheses of Lemma 8.2, and so we conclude that (3)
(4)
lim f(λαq) − f(λαq) = 0.
λ→0
We now go back to consider the domain D(4) (λ, τ, α, q) as a function of λ, and note that it tends to fill the strip D(5) (τ ) = [0, τ ] × [0, ∞].
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Accordingly, we define the following Volterra integral operator on this strip: τ
(5)
H(λαq) g =
∞ dσ
0
2
(x/2) − 2T (λ)2
dx e
λ λ X−λ −2 σ+q− α+ 1 x A01 Ux A10 Xλ−2 σ−q+ α− 1 x g(σ). ( ) ( ) 2
0
and note that (5) (4) H(λαq) − H(λαq) g (τ ) ≤
2
dσdx A01 Ux A10 g ∞ D (5) (τ )\D (4) (λ,τ,α,q)
⎧ ⎪ ∞ τ ⎨ 2 ≤ g ∞ λ q dx A01 Ux A10 + dσ ⎪ ⎩ 2 0
λ q
∞ xl (λ,q,σ)
⎫ ⎪ ⎬ dx A01 Ux A10 ⎪ ⎭
where xl (λ, q, σ) =
q − λ−2 σ α − 12
is the x-coordinate of the left edge of the triangular domain D(λ, τ, α, q) (see Fig. 1). Now the first term in the curly brackets clearly goes to zero as λ → 0 (with velocity ∼λ2 q), due to hypothesis 3.2 for λ = 0. In the second one, as before, we change coordinate according to σ → σ − λ2 q, obtaining τ −λ2 q
∞
dσ 0
τ dx A01 Ux A10 ≤
∞
0
xl (λ,0,σ)
dx A01 Ux A10 .
dσ
xl (λ,0,σ)
This last term can be seen to go to zero uniformly on 0 ≤ τ ≤ τ (and real q) by arguing, as before, that xl (λ, 0, σ) → ∞, λ → 0 for a.e. σ ∈ [0, τ ], and using hypothesis 3.2. So it follows that (4) (5) (57) H(λαq) − H(λαq) → 0, λ → 0. (6)
Now we can finally compare with H(λαq) := H(λαq) . By adding and subtracting obvious terms, we estimate ∞ (5) λ H(λαq) − H(λαq) g ≤ τ dx Xq−(α+ 1 )x − P0 Uq−(α+ 1 )x A01 Ux A10 g∞ 2
2
0
∞
+τ
dx A01 Ux A10 X(λα− 1 )x−q 2
0
−P0 U(α− 1 )x−q g∞ . 2
Uniform convergence to zero follows by hypothesis 3.2 together with the dominated convergence theorem, using the fact that for every x ∈ R lim Xxλ − P0 Ux = 0.
λ→0
This proves the estimation in (48) and thus finishes the proof.
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8.2. Proof of Theorem 4.2 We shall borrow most part of the proof of Theorem 3.4. Accordingly, we should (j) (j) denote for example with H(λq) the operator defined in Theorem 3.4 as H(λ0q) , D( λ, τ, q) ≡ D(λ, τ, α, q) and so on. Define (j) Hλ
1 =√ 2πT (λ)
∞ 2 − q (j) dq e 2T (λ)2 H(λq) . −∞
This is obviously Volterra, being an integral of Volterra operators. A closer inspection soon reveals that also 1 (j) Hλ ≤ √ 2πT (λ)
∞ 2 − q (j) dq e 2T (λ)2 H(λq) ≤ cτ , −∞
(j)
as for each j, H(λq) is bounded by c uniformly on q and 1 √ 2πT (λ)
∞
2
dq e
− 2Tq(λ)2
= 1.
(58)
−∞
We proceed on the very same lines of Theorem 3.4: the proofs that (j)
(j−1)
lim Hλ − Hλ
λ→0
=0
for j = 1, 2, and j = 6 are in fact identical to that of the foretold Theorem, as, for those values for j, one has (j)
(j−1)
lim H(λ,q) − H(λ,q) = 0
(59)
λ→0
uniformly on q. Normalization (58) can be exploited to state the validity of the above relation for j = 3 and j = 5. In fact, for these values of j, we can estimate (j) Hλ
−
(j−1) Hλ
1 ≤√ 2πT (λ)
∞
2
dq e
− 2Tq(λ)2
(j)
(j−1)
H(λ,q) − H(λ,q) .
−∞ (j)
(j)
Now, the norm in the integrand goes to zero as ∼ c1 (λ) + c2 λ2 q, with (j) c1 (λ) → 0 uniformly on q as λ → 0, as already noted in the proof of Theorem 3.4, so the whole integral goes to zero as λ → 0 precisely because T (λ) scales with |λ|−ξ and ξ < 2 by hypothesis. (j) (j) It remains to show that if fλ = n (Hλ )n b, for some initial condition b ∈ B0 , then (4)
(3)
lim fλ − fλ = 0.
λ→0
(60)
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Proceeding according to Lemma 8.2, we perform a telescopic expansion and estimate (3) fλ
−
(4) fλ ∞
∞ ∞ n−1 2
(τ c)n−l−1 1 − q √ ≤ dq e 2T (λ)2 (n − l)! 2πT (λ) n=1 l=1 −∞ l (3) (4) (4) × b (H(λq) − H(λq) ) Hλ .
(61)
∞
Now we compute $
(3)
(4)
H(λq) − H(λq)
−
=
dσdx e
(4) l
Hλ
(x/2)2 2T (λ)2
% b (τ )
Kq (λ, σ, x)
$
(4) l
Hλ
% $ % (4) l b (σ − λ2 (q+x/2))− Hλ b (σ)
D(λ,τ,q)
= √
∞
1 2πT (λ)
$ × = √
(4)
Hλq
q2 2T (λ)2
(4) l−1
Hλ
∞
−
−∞
1 2πT (λ)
dq e
dq e
−
−
dσdx e
Kq (λ, σ, x)
D(λ,τ,q)
% $ % (4) (4) l−1 b σ − λ2 (q + x/2) − Hλq b (σ) Hλ
q2 2T (λ)2
−∞
−
dσdx e
(x/2)2 2T (λ)2
Kq (λ, σ, x)
D(λ,τ,q)
dσ1 dx1 Δλ (σ1 , x1 ) e
×
(x/2)2 2T (λ)2
−
(x/2)2 2T (λ)2
Kq (λ, σ1 , x1 )
$
(4) l−1
Hλ
% b (σ1 )
,σ,x) S(λ,q,q
where we have put Δλ = χλ1 − χλ2 , χλ1 being the characteristic function of D(λ, σ − λ2 (q + x/2) , q ), χλ2 the characteristic function of D(λ, σ, q ), and we have defined q, q , σ, x) = D(λ, σ − λ2 (q + x/2) , q ) ∪ D(λ, σ, q ). S(λ, Passing to the norms we obtain (3) (H(λq)
−
(4) H(λq) )
(4) l Hλ
b ∞
× sup
0<τ <τ
dσdx e
1 (τ c)l−1 b B0 √ ≤ (l − 1)! 2πT (λ)
(x/2)2 − 2T (λ)2
k(x) Ξqq (λ, σ, x)
∞
dq e
2
− 2Tq(λ)2
−∞
(62)
D(λ,τ,q)
where the slight modification of the related definition of Ξq (λ, σ, x) in (40) is given by dσ1 dx1 |Δλ (σ1 , x1 )| k(x1 ). (63) Ξqq (λ, σ, x) = ,σ,x) S(λ,q,q
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Proceeding on the same lines as in Lemma 8.2 we find the asymptotic behavior 2 − (x/2) sup dσdx e 2T (λ)2 k(x) Ξqq (λ, σ, x) 0<τ <τ D(λ,τ,q)
∞ ≤ τ sup
0≤σ≤τ
2
dxe
(x/2) − 2T (λ)2
k(x) Ξqq (λ, σ, x)
0
≤ τ {c1 (λ) + c2 λ2 |q| + c3 λ2 |q |}
(64)
where c1 (λ) → 0 uniformly on q and q . This, plus the fact that ξ < 2 by our hypothesis, allows us conclude that (60) holds, as can be seen by putting result (64) into (62), and then back into (61), and by using the dominated convergence theorem. Collecting the results as in Theorem 3.4 concludes the proof.
Acknowledgements We wish to thank Prof. Fausto Rossi (Phys. Dept., Politecnic of Turin), Prof. Hisao Fujita Yashima and Prof Enrico Priola (Math. Dept., University of Turin), Prof. Claude-Alain Pillet (Centre de Physique Th´eorique, Marseille), Prof. Vladimir Gritsev (Phys. Dept., University of Fribourg) and finally Dr. Taj Mohammad (Phys. Dept., University of Turin) for precious help and stimulating discussions.
References [1] Van Hove, L.: Energy corrections and persistent perturbation effects in continuous spectra. Physica 21(6–10), 901–923 (1955) [2] Davies, E.B.: Markovian master equations. Commun. Math. Phys. 39, 91–110 (1974) [3] Davies, E.B.: Markovian master equations. II. Math. Ann. 219, 147–158 (1976) [4] Alicki, R.: The Markov master equations and the Fermi golden rule. Int. J. Theor. Phys. 16(5), 351–355 (1977) [5] Fermi, E.: Nuclear Physics, chap. 2–3, Revised edn., University of Chicago Press, Chicago (1950) [6] Taj, D., Genovese, L., Rossi, F.: Quantum-transport simulations with the Wigner-function formalism: Failure of conventional boundary-condition schemes. Europhys. Lett. 74(6), 1060–1066 (2006) [7] Attal, S., Joye, A., Pillet, C.-A.: Open Quantum Systems I, II and III, vol. 1880–1882. LNM (2006) [8] Davies, E.B.: Quantum Theory of Open Systems. Academic Press, London (1976) [9] Umanit` a, V.: Classification and decomposition of Quantum Markov Semigroups. Probab. Theory Relat. Fields 134(4), 603–623 (2006)
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[10] Fagnola, F.: Quantum Markov Semigroups and Quantum Flows. Proyecciones 18(3), 1–144 (1999) [11] D¨ umcke, R., Spohn, H.: The Proper Form of the Generator in the Weak Coupling Limit. Z. Physik B 34, 419–422 (1979) [12] Taj, D., Lotti, R.C., Rossi, F.: Microscopic modeling of energy relaxation and decoherence in quantum optoelectronic devices at the nanoscale. Eur. Phys. J. B 72, 305322 (2009) (Colloquium) [13] Taj, D., Rossi, F.: Completely positive Markovian quantum dynamics in the weak-coupling limit. Phys. Rev. A 78, 052113 (2008) [14] Nakajima, S.: On Quantum Theory of Transport Phenomena. Prog. Theor. Phys. 20(6), 948–959 (1958) [15] Zwanzig, R.: Ensemble Method in the Theory of Irreversibility. J. Chem. Phys. 33, 1338 (1960) [16] Yosida, K.: Functional Analysis. Springer, Berlin (1968) [17] Goldstein, J.: A Lie product formula for one parameter groups of isometries on Banach spaces. Math. Ann. 186, 299–306 (1970) [18] Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I, 2nd edn. Springer, Berlin (2002) [19] Chernoff, P.R.: Note on product formulae for operator semigroups. J. Funct. Anal. 2, 238–242 (1968) [20] Sakai, S.: C ∗ -algebras and W ∗ -algebras. Springer, Berlin (1998) [21] Lindblad, G.: On the Generators of Quantum Dynamical Semigroups. Commun. Math. Phys. 48, 119–130 (1976) [22] Kraus, K.: General state changes in quantum theory. Ann. Phys. 64, 311–335 (1970) David Taj Physics Department University of Fribourg Ch. du Mus´ee 3 1700 Fribourg Switzerland e-mail:
[email protected];
[email protected] Communicated by Jens Marklof. Received: April 26, 2010. Accepted: September 13, 2010.
Ann. Henri Poincar´e 11 (2010), 1341–1373 c 2010 Springer Basel AG 1424-0637/10/071341-33 published online November 3, 2010 DOI 10.1007/s00023-010-0056-1
Annales Henri Poincar´ e
Localization Properties of the Chalker–Coddington Model Joachim Asch, Olivier Bourget and Alain Joye We dedicate this work to the memory of our friend and colleague Pierre Duclos
Abstract. The Chalker–Coddington quantum network percolation model is numerically pertinent to the understanding of the delocalization transition of the quantum Hall effect. We study the model restricted to a cylinder of perimeter 2M . We prove first that the Lyapunov exponents are simple and in particular that the localization length is finite; secondly, that this implies spectral localization. Thirdly, we prove a Thouless formula and compute the mean Lyapunov exponent, which is independent of M .
1. Introduction We start with a mathematical then a physical description of the model. Fix the parameters r, t ∈ [0, 1],
such that, r2 + t2 = 1,
denote by T the complex numbers of modulus 1 and for q = (q1 , q2 , q3 ) ∈ T3 by S(q) the general unitary U (2) matrix depending on these three phases q1 q 2 0 t −r q3 0 S(q) := . 0 q1 q2 0 q3 r t 2 := ⊗(2Z)2 d6 l where be the probability space: Ω F, P := T6 (2Z) , P Let Ω, dl is the normalized Lebesgue measure on T, and F the σ-algebra generated by the cylinder sets. With p ∈ Ω,
p(2j, 2k) =: (p1 , p2 , p3 , p4 , p5 , p6 ) pe (2j,2k)
po (2j+1,2k+1)
(j, k ∈ Z)
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and the basis vectors eμ (ρ) := δμ,ρ (μ, ρ ∈ Z2 ), the family of unitary operators (p) : l2 (Z2 ) → l2 (Z2 ) U μ;ν = eμ , U eν : is defined by its matrix elements U
(p)(2j+1,2k);(2j,2k) U (p)(2j,2k+1);(2j,2k) U
(2j+2,2k+2);(2j+2,2k+1) U (2j+1,2k+1);(2j+2,2k+1) U
μ;ν := 0 except for the blocks U
(p)(2j+1,2k);(2j+1,2k+1) U (p)(2j,2k+1);(2j+1,2k+1) U
(2j+2,2k+2);(2j+1,2k+2) U (2j+1,2k+1);(2j+1,2k+2) U
:= S(pe (2j, 2k))
(1)
:= S(po (2j + 1, 2k + 1)).
is an ergodic family of random unitary operators; indeed, Note that U ∗ ∗ U U = I = U U because of the unitarity of the blocks; further denote by Θ 2 2 the action of Z on functions f on Z : (l,m) f )(μ) := f (μ + (2l, 2m)) (Θ
(μ ∈ Z2 , (l, m) ∈ Z2 ),
Then Θ is measure and, by abuse of notation, the corresponding shift on Ω. preserving and ergodic on Ω and (Θp) =Θ U (p)Θ −1 . U This model was introduced in the physics literature by Chalker and Coddington [10], see [21] for a review, in order to study essential features describes the dynamof the quantum Hall transition in a quantitative way. U ics of a 2D electron in a strong perpendicular magnetic field and a smooth bounded random electric potential which is supposed to have some array of hyperbolic fixed points forming the nodes of a graph. In this picture, the electron moves on the directed edges of the graph whose nodes are “even”: {(1/2, 1/2)+(2j, 2k), j, k ∈ Z} or “odd”: {(1/2, 1/2)+ (2j + 1, 2k + 1), j, k ∈ Z} with edges connecting the even (odd) nodes to their describes the evolution at time one of the nearest odd (even) neighbors. U electron. The edges are labeled by their midpoints. They are directed in such models the tunneling near the hyperbolic fixed points of the a way that U potential, see Fig. 1. The tunneling is described by the scattering matrices S associated with the even, respectively odd, nodes. The i.i.d. random phases associated with each node take into account the deviation of the random electric potential from periodicity. Following the literature on tunneling near a hamiltonian saddle point 1 [11,14], the parameter t is √1+e ε where ε is the distance of the electron’s energy to the nearest Landau Level. An application of a finite size scaling
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(2j+1,2k+1)
x-axis even node
odd node
y-axis
Figure 1. The network model with its incoming (solid arrows) and outgoing links
method to their numerical observations led Chalker and Coddington [10], see also [21], to conjecture that the localization length diverges as t/r → 1 as α 1 , ln | rt | where the critical exponent α exceeds substantially the exponent expected when a classical percolation model is applied to the problem [25]; the values advocated for α are 2.5 ± 0.5 for the quantum and 4/3 for the classical case. Because of its importance for the understanding of the integer quantum Hall effect the one electron magnetic random model in two dimensions was and continues to be heavily studied in the mathematical literature. Mathematical results concerning the full Schr¨ odinger Hamiltonian can be traced from the following contributions and their references: [27] for percolation, [16] for the existence of the localization–delocalization transition [2,8,15] for the general theory of the quantum Hall effect. For results concerning a 2D electron in a magnetic field and periodic potential, which corresponds to the absence of phases here, see [18,26]. For recent work on Lyapunov exponents on Hamiltonian strip models see [6,7,23]. Our results concern the restriction of the model to a strip of width 2M and periodic boundary conditions; they are presented as follows. In Sect. 2, we analyze the extreme cases, r = 0 and r = 1. Then, for the case where all phases are chosen to be 1, we give a description of the spectrum. Questions related to transfer matrix formalism are handled in Sects. 3, 4, 5. In Sect. 6, we prove simplicity of the Lyapunov spectrum and finiteness of the localization length. In Sect. 7, we prove a Thouless formula and show that the density of states is flat, which implies our results on the mean Lyapunov exponent. In Sect. 8, we prove complete spectral localization.
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2. Some Properties of the Model 2.1. Extreme Cases Note that in case of complete “reflection” or “transmission” the system localizes completely: the spectrum of U (p) is pure Proposition 2.1. Let rt = 0. Then, for any p ∈ Ω, point. and define the family of subspaces (Hj,k )(j,k)∈Z2 Proof. Assume r = 0, p ∈ Ω as: Hj,k = Ran(e2j,2k , e2j+1,2k , e2j+1,2k−1 , e2j,2k−1 ). (p) and These subspaces are invariant under U ⊕(j,k)∈Z2 Hj,k = l2 (Z2 ),
(2)
(p) is pure point. The case t = 0 is treated simiwhich means the operator U larly. On the other hand, one has complete propagation if all the phases are p = (. . . , 1, 1, 1, . . .) by p(2j, 2k) := (1, 1, 1, 1, 1, 1) equal to one; define Ω then we have: (. . . , 1, 1, 1, . . .) is purely Proposition 2.2. Let rt = 0. Then, the spectrum of U absolutely continuous. Proof. We make use of a decomposition similar to (2) and define the unitary V from l2 (Z2 ) to l2 (Z2 ) ⊗ C4 by V e2j,2k := ej,k ⊗ e1 , V e2j+1,2k+1 := ej,k ⊗ e2 , V e2j,2k+1 := ej,k ⊗ e3 , V e2j+1,2k := ej,k ⊗ e4 . Let P be the projection in (1) one reads that P := I ⊗ (|e1 e1 | + |e2 e2 |). From the definition of U 2 V −1 P is equivalent to 2 V −1 commutes with P and that P V U VU
rt(T0,1 − T1,0 ) r2 T1,0 + t2 T0,1 t2 T0,−1 + r2 T−1,0 2
rt(T−1,0 − T0,−1 )
2
with the translations on l (Z ) defined by Tn,m ψ(j, k) := ψ(j + n, k + m)
(n, m ∈ Z).
The Fourier transform F : l2 (Z2 ) → L2 (T2 ) transforms the translations to multiplication operators: FTn,m F −1 = exp (−i(nx + my)), thus the restric 2 V −1 P F −1 is equivalent to a matrix-valued tion to the range of P of FP V U multiplication operator
rt(e−iy − e−ix ) r2 e−ix + t2 e−iy . (3) t2 eiy + r2 eix rt(eix − eiy ) The trace of this matrix is not constant, its determinant is −1 hence the spec 2 is purely tral bands are not flat, thus the spectrum of the restriction of U absolutely continuous. By an analogous argument this also holds for the restric tion to P ⊥ .
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Remark that a more general periodic distribution of phases leads to matrix-valued translation operators with periodic coefficients thus to nontrivial Hofstadter like problems.
3. Restriction to a Cylinder, Transfer Matrices Let M ∈ N. Use the notation Z2M := Z/(2M Z) for the discrete circle of perimeter 2M . Consider the restriction of the model to the cylinder Z × Z2M : U (p) : l2 (Z × Z2M ) → l2 (Z × Z2M ) defined by its matrix elements with respect to the canonical basis (μ ,μ U (p)μ,ν := U 1 2
mod 2M );(ν1 ,ν2 mod 2M ) .
(4)
Remark that U (p) has the same spectral properties for the extreme cases (p), the model on the full lattice: as U Proposition 3.1. Let rt = 0. Then, for any p ∈ Ω, the spectrum of U (p) is pure point. Proof. Similar to the proof of Proposition 2.1.
Proposition 3.2. Let rt = 0. Then the spectrum of U (. . . , 1, 1, 1, . . .) is purely absolutely continuous. Proof. In the proof of Proposition 2.2 note that V now acts from l2 (Z × Z2M ) to l2 (Z × ZM ) ⊗ C4 and replace F by the Fourier transform from l2 (Z × ZM ) to L2 (T × ZM ) defined by 2π Fψ(x, κ) = ψj,k eixj ei M κk j∈Z,k∈ZM
which diagonalizes the translations. Then, setting y = 2π M κ (κ ∈ ZM ), the matrix-valued multiplication operator obtained in (3) is understood as a family of matrix-valued operators indexed over ZM . The spectral bands are not flat by the same argument. From now on we restrict the discussion to the case rt = 0. In the following z denotes a complex number; also, unless otherwise stated, all indices in the second variable are to be understood mod 2M , e.g.: ψ2j,2k+1 = ψ2j,2k+1
mod 2M
= ψ2j,2k+1[2M ] .
A standard approach to the spectral problem of U is the transfer matrix method. Though this is well known, we wish to recall the construction explicitly for the model at hand.
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Proposition 3.3. For z = 0, q = (q1 , q2 , q3 ) ∈ T3 define q1 q2 0 1 z −1 −r q3 0 , Teo (z, q) := 0 q3 t −r z 0 q1 q2 1 z q3 0 q1 q2 0 −t . Toe (z, q) := q3 0 q1 q2 r t −z −1 0 Then 1.
For ψ : Z × Z2M it holds: Uμν ψν = zψμ ν∈Z×Z2M
and
2.
ψ2j+1,2k ψ2j+1,2k+1
ψ2j+2,2k+1 ψ2j+2,2k+2
∀μ ∈ Z × Z2M
⇐⇒
= Teo (z, pe (2j, 2k))
ψ2j,2k ψ2j,2k+1
= Toe (z, po (2j + 1, 2k + 1))
ψ2j+1,2k+1 ψ2j+1,2k+2
.
For z ∈ T, it holds that Toe , Teo ∈ U (1, 1), the Lorentz group defined as a subset of the complex 2 × 2 matrices by 1 0 U (1, 1) := B ∈ M2,2 (C); B ∗ JB = J, J := 0 −1
Proof. By definition of U , we have for the “even” nodes: (U ψ)2j+1,2k ψ2j,2k ψ2j+1,2k = S (pe (2j, 2k)) =z , ψ2j+1,2k+1 ψ2j,2k+1 (U ψ)2j,2k+1 and, for the “odd” nodes:
S (po (2j + 1, 2k + 1)) For a matrix
S=
it holds:
S11 S21
ψ2j+2,2k+1 ψ2j+1,2k+2
S12 S22
=z
ψ2j+2,2k+2 ψ2j+1,2k+1
.
with S22 S21 = 0
a x a x x b =S ⇐⇒ = S ⇐⇒ = S˘ b y y b a y
with 1 S = S22 Now
t r
−r t
det S −S21
=
1 t
S12 1
1 −r
−r 1
1 S˘ = S21
,
;
t r
−r t
1 S11 ˘ =
−S22 − det S 1 r
.
1 −t t −1
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a q2 0 t −r q3 0 x = q1 b 0 q2 0 q3 r t y −1 a q q 0 1 z −r q3 0 x ⇐⇒ = 1 2 y 0 q3 t −r z 0 q1 q2 b 1 z x q 0 q1 q2 0 b −t ⇐⇒ = 3 a q3 0 q1 q2 r t −z −1 0 y
from which the first claim follows. Denote by I the identity matrix in C2 . S is a unitary matrix ifand only I 4 if the pullback of the quadratic form in C associated with Q = −I is zero. (blanks stand for 0 entries) to the graph of S: (u, Su) ∈ C4 ,u ∈ C2 J The mapping from (x, y, a, b) to (x, b, a, y) transforms Q to . The −J pullback of the corresponding form to the graph of Teo being zero, it follows that Teo and, by the analogous argument, Toe , belong to the Lorentz group. For later use we fix the following notation Definition 3.4. Denote by J the 2M × 2M block diagonal matrix consisting of M non-zero diagonal blocks equal to J and by UM (1, 1) := {B ∈ M2M,2M (C); B ∗ JB = J}. the unitary group of the hermitian form defined by J. Note that UM (1, 1) is isomorphic to the classical unitary group U (M, M ) of the hermitian form |z1 |2 + · · · + |zM |2 − |zM +1 |2 · · · − |z2M |2 .
4. Relevant Phases Because of the uniform distribution, it is possible to reduce the number of relevant phases in the model to two phases per node. Before proceeding we do this reduction. We shall repeatedly make use of Lemma 4.1. Let ϕ1 , . . . , ϕn be independent and uniformly distributed random ϕ variables on R/Z and let A ∈ Mm,n (Z). Then, θ1 , . . . , θm defined by θ = A are independent and uniformly distributed if and only if Rank A is maximal. Proof. For k ∈ Zm it holds
E(eik,θ ) = E(eik,Aϕ ) = E(eiA
t
k, ϕ
) = δAtk,0 .
Thus, the θ are independent and uniformly distributed if and only if E(eik,θ ) = δk,0 if and only if KerAt = {0}, equivalently, if and only if Rank A is maximal.
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→ TZ2 such that for p ∈ Ω the evolution Proposition 4.2. There exists g : Ω U (p) defined by (1) is unitarily equivalent to D(g(p))S
on l2 (Z2 )
(. . . , 1, 1, 1, . . .). Morewhere D(q) is diagonal, D(q)(j,k);(j,k) = qj,k , and S = U 6 over, the image measure of ⊗(2Z)2 d l by g is ⊗Z2 dl. (p) is of the form U (p) = D(1) (p)SD(2) (p) where D(j) (p) are Proof. By (1), U diagonal, and defined by their diagonal elements: D(1) (p)2j+1,2k = p1 p2 (2j, 2k), D(1) (p)2j+2,2k+2 = p4 p5 (2j + 1, 2k + 1), D(2) (p)2j,2k = p3 (2j, 2k), D(2) (p)2j+2,2k+1 = p6 (2j + 1, 2k + 1), D(1) (p)2j,2k+1 = p1 p¯2 (2j, 2k), D(1) (p)2j+1,2k+1 = p4 p¯5 (2j + 1, 2k + 1), D(2) (p)2j+1,2k+1 = p¯3 (2j, 2k), D(2) (p)2j+1,2k+2 = p¯6 (2j + 1, 2k + 1). (p) is unitarily equivalent to D(2) (p)D(1) (p)S which has the Hence, U asserted shape. Define q = g(p) by q(2j + 1, 2k) := p6 (2j + 1, 2k − 1)p1 p2 (2j, 2k), q(2j, 2k + 1) := p6 (2j − 1, 2k + 1)p1 p¯2 (2j, 2k), q(2j + 2, 2k + 2) := p3 (2j + 2, 2k + 2)p4 p5 (2j + 1, 2k + 1), q(2j + 1, 2k + 1) := p3 (2j, 2k)p4 p¯5 (2j + 1, 2k + 1). Now, an application of Lemma 4.1 shows the q’s are i.i.d. and uniformly distributed. Remark 4.3. Note that the unitary transformation just constructed is diagonal and thus does not affect the localization properties of the model. 2
In the following, we abuse notations and call for q ∈ TZ the matrix (q); same abuse for the restriction to the cylinder. operator D(q)S again U
5. Characteristic Exponents We now define and analyze the transfer matrices and in particular the localization length. Consider U (p) = D(p)S
on l2 (Z × Z2M )
(5)
with identically distributed uniformly distributed phases in TZ×Z2M , ⊗Z×Z2M dl and the cylinder set algebra.
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We use the unitary equivalence l (Z × Z2M ) ∼ = l2 (Z × {0, . . . , 2M − 1}) → l2 (Z; l2 (Z2M )) ∼ = l2 (Z, C2M ) ψ → Ψ (6) 2
Ψj := (ψj,0 , . . . , ψj,2M −1 ). Note that with the reduced phases the building blocks of the transfer matrices read with phases p,q 1 0 p 0 1 z −1 −r , z 0 q 0 1 t −r q 0 1 0 1 z −t . 0 1 0 p r t −z −1 As we shall explain below, the previous analysis leads us to deal with the following random dynamical system: 2Z Consider the probability space defined by Ω = (T4M ) , P = ⊗Z d4M l, and F, the cylinder set algebra. The shift Θ : Ω → Ω,
Θp(2m) := p(2(m + 1))
(m ∈ Z)
is measure preserving and ergodic. For p ∈ Ω define the following elements of 2M 2Z T pr = (1, p1 , 1, p3 , . . . , 1, p2M −1 ) pl = (p0 , 1, p2 , 1, . . . , p2M −2 , 1) pm = (p2M , p2M +1 , . . . , p4M −1 ). Denote for q ∈ T2M the unitary diagonal matrix ⎞ ⎛ q1 ⎟ ⎜ .. D(q) := ⎝ ⎠, . q2M (where 0 valued matrix entries are represented by blanks) and for z = 0 the 2M × 2M matrices ⎛ −1 ⎞ z −r ⎜ −r ⎟ z ⎟ 1⎜ ⎜ ⎟ . .. M1 (z) := ⎜ ⎟, ⎟ t⎜ −1 ⎝ −r ⎠ z −r z ⎛ −1 ⎞ −z t ⎜ ⎟ z −t ⎜ ⎟ −1 ⎜ ⎟ t −z ⎜ ⎟ 1⎜ ⎟ . .. M2 (z) := ⎜ ⎟. ⎟ r⎜ ⎜ ⎟ z −t ⎜ ⎟ −1 ⎝ ⎠ t −z −t z
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Define for a fixed z = 0 Az : Ω → UM (1, 1)
(7)
Az (p) := D (pl ) M2 (z)D (pm ) M1 (z)D (pr ) . Then A generates the cocycle Φ over the ergodic dynamical system (Ω, F, P, (Θn )n∈Z ) defined by Φz : Z × Ω → UM (1, 1) ⎧ n>0 ⎨ A(Θn−1 p) . . . A(p) n=0 . Φz (n, p) := I ⎩ −1 n A (Θ p) . . . A−1 (Θ−1 p) n < 0
Oseledets theorem holds for Φ, see [1], Theorem 3.4.11 and Remark 3.4.10 (ii): Definition 5.1. Let z = 0. There exists an invariant subset of full measure of p ∈ Ω such that the limits lim (Φ∗z (n, p)Φz (n, p))
n→∞
1/2n
= lim (Φ∗z (n, p)Φz (n, p)) n→−∞
1/2|n|
=: Ψz (p)
exist. Denote by γk (p, z), k ∈ {1, . . . , 2M }, the eigenvalues of Ψz (p) arranged in decreasing order. Due to ergodicity there exists γk (z) ≥ 0 such that γk (p, z) = γk (z) on an invariant subset of full measure. The characteristic exponents are defined by λk (z) := log γk (z). Due to the Lorentz symmetry of the transfer matrices for z ∈ T we have Proposition 5.2. 1. it holds
Let B ∈ UM (1, 1). Then for the singular values SV (B) γ ∈ SV (B) ⇐⇒
2.
1 ∈ SV (B). γ
For λj := log γj , γj ∈ SV (B) arranged in decreasing order it holds: λj+M = −λM −j+1
∀j ∈ {0, . . . , M }.
Proof. We have B ∗ JB=J. In particular, det B = 0, so γ = 0 and J−1 B ∗ = B −1 J−1 as well as BJ = JB ∗ −1 . Now, det B ∗ B − z 2 = 0 ⇐⇒ det J−1 B ∗ BJ − z 2 = 0 ⇐⇒ ∗ −1 1 2 4M ∗ −1 ∗ det (B B) − z = 0 ⇐⇒ z det(B B) det − B B = 0. z2 From which the two claims follow.
Thus, we restrict our discussions to the first M non-negative Lyapunov exponents λ1 ≥ λ2 ≥ · · · ≥ λM ≥ 0 which we shall call for simplicity “the” Lyapunov exponents in the sequel. We show that due to the translation invariance of the uniform distribution, the exponents are independent of z:
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Lemma 5.3. For any w ∈ T, Awz (p) = Az (w p), where w p is defined by w p2j := w−1 p2j , and w p2j+1 := wp2j+1 . Proof. Write D(pm ) in (7) as D(p l )D(p r ), where p r := (1, p2M +1 , 1, p2M +3 , . . . , 1, p4M −1 ) p l := (p2M , 1, p2M +2 , 1, . . . , p4M −2 , 1). Thus, Az (p) is the product of the block diagonal matrices D(pl )M2 (z)D(p l ) and D(p r )M1 (z)D(pr ) whose blocks are 1 z 1 0 1 z 1 0 −t −qt l az (p, q) := = 0 p r t −z −1 0 q r pt −pqz −1 −1 1 z −1 −qr 1 0 1 z −r 1 0 r az (p, q) := = . 0 p t −r z 0 q r −pr pqz For any w ∈ T, these matrices satisfy alwz (p, q) = walz (w−1 p, w−1 q),
arwz (p, q) = w−1 alz (wp, wq),
from which the result follows.
Therefore, for any fixed w ∈ T, the matrices Az (w p) have the same distribution as Awz (p). As a consequence, Corollary 5.4. All characteristic exponents λk (z) = λk are independent of z ∈ T. Proof. λk = E (log(γk (z, p))) = E log(γk (1, z1 p)) = E (log(γk (1, p))). Definition 5.5. The localization length ξM ∈ [0, ∞] is defined as 1 ξM := . λM Remark 5.6. In the physics literature, see [21], ξM in assumed to be finite for all parameters; a change of the asymptotic behavior as M → ∞ is conjectured when the parameters of the model approach the critical point t = r. This conjecture is supported by a numerical finite size scaling method and is supposed to reflect the divergence of the localization length of the full system at the critical point. Thus a first step to support these heuristics is to prove finiteness of ξM and to establish precise information of its behavior as a function of M . The announced equivalence to the propagation problem is the content of the following Proposition 5.7. Let U (p) be the ergodic family of unitary operators defined in (5) over the probability space Γ := TZ×Z2M , ⊗Z×Z2M dl and the cylinder set algebra. Let f : Γ → Ω be defined for j ∈ Z by f (p)(2j) := (p2j,0 , p2j+2,1 , p2j,2 , p2j+2,3 . . . p2j,2M −2 , p2j+2,2M −1 p2j+1,0 , p2j+1,1 . . . p2j+1,2M −1 ) .
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The image measure by f is the measure on Ω and it holds U (p)ψ = zψ ⇐⇒ Ψ2N = Φz (N, f (p))Ψ0 for Ψ defined in (6). Proof. The construction of f follows from Proposition (3.3). The image measure follows from Lemma (4.1).
6. Finiteness of the Localization Length Using the methods exposed in [5], see also [17], we prove that all Lyapunov exponents are distinct and in particular that the localization length for the cylinder in finite. Theorem 6.1. For rt = 0, z ∈ T it holds λ1 > λ2 > · · · > λM > 0. Proof. We follow the strategy exposed in [5] and prove the theorem in several steps making use of lemmata to be proven below. Denote by G := the smallest subgroup of UM (1, 1)
generated by {A(p), p ∈ Ω}.
By Lemma 6.2 G = UM (1, 1). In particular it is then known: G is connected. Furthermore, see also [23], G is isomorphic to the complex symplectic group. Indeed : denote by גthe 2M × 2M block diagonal matrix consisting of 0 −1 M non-zero blocks σ = ; we write: ⊕ = גM 1 σ for short; denote by 1 0 Sp(M, C) := {B ∈ M2M,2M (C); B ∗ גB = }ג the complex symplectic group. From ∗ 1 1 1 −i 1 −i √ J√ = iσ 2 1 i 2 1 i M 1 −i that it follows defining C := 1 √12 1 i G = UM (1, 1) = CSp(M, C)C ∗ . In order to freely use results in [5] we shall do our argument for real matrices. To this end we separate real and imaginary parts and consider τ : M2M,2M (C) → M4M,4M (R) a −b x = a + ib → . b a
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It holds: τ (x + y) = τ (x) + τ (y); τ (xy) = τ (x)τ (y); τ (x∗ ) = τ (x)t ; ker{τ } = {0}; det τ (x) = | det x|2 , thus τ (C ∗ GC) ⊂ Sp(2M, R) with the real symplectic group Sp(2M, R) := {B ∈ M4M,4M (R); B t גB = }ג for ⊕ = ג2M 1 σ. det τ (x) = | det x|2 implies that τ (x) shares its eigenvalues with x with the degeneracies doubled. So the Lyapunov exponents γ defined by the τ − C transformed products of transfer matrices are γ1 = γ2 = λ1 ≥ · · · ≥ γ2p−1 = γ2p = λp ≥ · · · ≥ γ2M −1 = γ2M = λM . As τ (C ∗ GC) is connected one can infer from [5] Theorem 3.4 and Exercice 2.9 for p ∈ {1, . . . , M }: ⎫ τ (C ∗ GC) L2p irreducible ⎬ and =⇒ γ2p = λp > λp+1 = γ2p+1 ⎭ τ (C ∗ GC) 2p contracting in particular for p = M : λM > 0. Now by Lemmas 6.3 and 6.4, the group τ (C ∗ GC) is 2p irreducible and 2p contracting for all p ∈ {1, . . . , M } so all Lyapunov exponents are distinct and λM > 0. The following lemmata complete the proof of Theorem 6.1, we use the notations introduced in the above proof. Lemma 6.2. G = UM (1, 1). Proof. By definition G ⊂ UM (1, 1) is a closed subgroup of Gl(2M, C) thus G is a Lie group. By connectedness of UM (1, 1) it is sufficient to show that the Lie algebras g and uM (1, 1) coincide. Now uM (1, 1) = {A ∈ M2M,2M (C); Ajk = −Akj (−1)k+j } whose dimension as a real vector space equals 4M 2 . Denote by Dj (t) = diag(1, 1, . . . , 1, eit , 1, . . . , 1) the unitary matrix where the phase sits at the j’th slot, for j = 1, 2, . . . , 2M and use the Mj as defined in Sect. 5. For z ∈ T the matrices i|jj|,
iM2 (z)|jj|M2 (z)−1 ,
iM1 (z)−1 |jj|M1 (z)
(8)
belong to g, for j = 1, 2, . . . , 2M as they are the generators of the curves Dj (t), M2 (z)Dj (t)M2−1 (z), M1−1 (z)Dj (t)M1 (z), which lie in G as Dj (t) = Dj (t)M2 (z)M1 (z)(M2 (z)M1 (z))−1 , M2 (z)Dj (t)M2−1 (z) = M2 (z)Dj (t)M1 (z)(M2 (z)M1 (z))−1 , M1−1 (z)Dj (t)M1 (z) = (M2 (z)M1 (z))−1 M2 (z)Dj (t)M1 (z).
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The generators in (8) have the same block structure as the Mj . We compute the relevant blocks. For iM2 (z)|jj|M2 (z)−1 we get i z −t i 1 −tz 1 0 z¯ −t = 2 z 0 0 t −z z −t2 r2 t −¯ r t¯ 2 i z −t i −t tz 0 0 z¯ −t . = 2 z 1 z 0 1 t −z r2 t −¯ r −t¯ Similarly, for iM1 (z)−1 |jj|M1 (z), the blocks take the form i z r i 1 1 0 z¯ −r −rz = 2 0 0 −r z z −r2 t2 r z¯ t r¯ 2 i z r i −r rz 0 0 z¯ −r . = z 1 0 1 −r z t2 r z¯ t2 −r¯ Now using these matrices for z ∈ / R and the diagonal matrix i|jj|, j = 1, 2 one gets by taking suitable real linear combinations of the matrices above that, in both cases, the relevant blocks are generated by 1 0 0 0 0 1 0 1 i ,i ,i , . 0 0 0 1 −1 0 1 0 −1 −1 −1 −1 For real z, use the curves Dj (t), De M 2 Dj (t)M 2 De , Do M1 Dj (t) · 1 0 w 0 and Do = ⊕ for w ∈ T, which amounts M1 Do , with De = ⊕ 0 w 0 1 to perform the change z → w−1 z. Taking into account the shift in the blocks and the period 2M of the indices in the matrices, we get that the restrictions of g and uM (1, 1) to their tridiagonal elements, mod 2M coincide. To go off the diagonals we use commutators, i.e. we exploit that X, Y ∈ g implies [X, Y ] ∈ g. Let Ak = |k + 1k| + |kk + 1| ∈ g, for k ∈ Z2M . Considering [Aj , Ak ] for all values of j, k, we generate a basis of all anti self-adjoint matrices that have non-zero real matrix elements at distance two away from the diagonal (and in the corners, by periodicity). By commuting Ak with A˜j = i(|j + 1j| − |jj + 1|) ∈ g, we get a basis of self-adjoint matrices with non-zero purely imaginary elements on the same upper and lower diagonals (plus corners) only. These matrices correspond to the restriction of all matrices in uM (1, 1) to these diagonals. We generalize the argument as follows: Assume we already generated a basis of all matrices A ∈ uM (1, 1) such that Ajk = 0 if |j − k| > m, m fixed. Again, periodicity is implicit here. Let uM (1, 1) Bj± (m) = |j + mj| ± |jj + m|. We compute
[Aj+m , Bj± (m)] = Bj∓ (m + 1). This way we generate all matrices A ∈ uM (1, 1) such Ajk = 0 if |j − k| > m + 1. Hence by induction, we see that g = uM (1, 1), so that G = UM (1, 1).
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Lemma 6.3. τ (C ∗ GC) = τ (Sp(M, C)) is L2p irreducible for p ∈ {1, . . . , M }. Proof. Denote ei , i ∈ {1, . . . , 4M } the canonical basis vectors of R4M . By definition (see [5] with adaptation to our symplectic form) Lq := span v ∈ Λq R4M ; v = M e1 ∧ M e3 ∧ · · · ∧ M e2q−1 , M ∈ Sp(2M, R) for q ≤ 2M . Remark that the set of directions in Lq corresponds to the set of isotropic subspaces of R4M . τ (C ∗ GC) is Lq -irreducible if there is no proper linear subspace V ⊂ Lq invariant under Λq τ (C ∗ GC). Consider M real numbers a1 > a2 > · · · aM > 1. The 4M × 4M diagonal matrix 1 1 1 1 1 1 A = diag a1 , , a2 , , . . . , aM , , a1 , , a2 , , . . . , aM , a1 a2 aM a1 a2 aM belongs to τ (C ∗ GC) and e1 ∧e3 ∧· · · ∧e2q−1 is an eigenvector of (Λq A) for all n with simple dominant eigenvalue > 1. Thus for an invariant subspace V of Lq either e1 ∧e3 ∧· · ·∧e2q−1 ∈ V which implies Λq M (e1 ∧e3 ∧· · ·∧e2q−1 ) ∈ V, ∀M , thus V = Lq , or e1 ∧ e3 ∧ · · · ∧ e2q−1 ∈ V ⊥ , which implies for all w ∈ V " ! 0 = Λq M t w, e1 ∧ e3 ∧ · · · ∧ e2q−1 = w, Λq M e1 ∧ e3 ∧ · · · ∧ e2q−1 n
thus V ⊥ = Lq ⇐⇒ V = {0}. Thus we conclude the claimed irreducibility for q = 2p. Lemma 6.4. τ (C ∗ GC) = τ (Sp(M, C)) is 2p contracting for p ∈ {1, . . . , 2M −1}. 2 2 Proof. For any a ∈ R\0 there exist x, y ∈ R with x − y = 1 such that x y , which belongs to U (1, 1), has eigenvalues a, 1/a. Taking such matriy x ces as blocks one sees that there exists an element of UM (1, 1) whose singular values are distinct: a1 > a2 > · · · > aM > 1 > 1/aM · · · and thus an element of τ (C ∗ GC) with 2M distinct singular values
b1 = b2 = a1 > · · · > b2p−1 = b2p = ap > · · · > b2M −1 = b2M = aM > 0. Thus bn2p+1 /bn2p →n→∞ 0 and it follows from Proposition 2.1, p. 81 of [5] that τ (C ∗ GC) is 2p contracting. Remark 6.5. To summarize, we have proved that if the transfer matrices generate the complex symplectic group Sp(M, C) then the results of [5] apply, i.e.: the Lyapunov spectrum is simple. The results in [5] are stated for real groups only. While it is remarked in their introduction that these results should hold in the complex case, this seems not to be obvious to specialists in the field.
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7. Thouless Formula and the Mean Lyapunov Exponent In this section we shall prove the announced identity in a series of lemmata. Theorem 7.1. Let M ∈ N. For the first M Lyapunov exponents associated with U defined in Definition (5.1) with z ∈ T it holds: M 1 1 1 1 ≥ log 2 λi (z) = log M i=1 2 rt 2
Proof. Let z ∈ T. Denoting by P2L (z) the propagator P2L (z) : Ω → UM (1, 1) P2L (z)(p) := Φz (L, p) (Φz (−L, p))
−1
we have for m ∈ {1, . . . , M } m i
λi = lim
L→∞
1 log ∧m P2L (z)(p) 4L
p a.e.
(9)
where ∧m denotes the mth exterior product (cf. [1], ch. 3). We analyze the above limit in Proposition 7.2 below and show: # M 1 1 1 λi = 2 log |z − x|dl(x) + log . M i 2 rt T
The assertion follows by an explicit calculation proving that # log |z − x|dl(x) = 0. T
Proposition 7.2. (Thouless formula) Let M ∈ N, z ∈ C\0 then # M 1 1 1 λi (z) = 2 log |z − x|dl(x) + log − log |z| M i 2 rt T
Proof. Will be done in Appendix 1.
Remark 7.3. We prove in particular that the density of states is the Lebesgue measure, see Lemma 9.3 below. 7.1. Bounds on the Localization Length We now use the Thouless formula and an M independent bound on the largest Lyapunov exponent to derive a bound on the localization length. We remark that this bound is very crude and that more involved techniques should be established to get more detailed information; cf. [23] and references therein. First observe that a lower bound on the mean Lyapunov exponent together with a tight upper bound on the largest, implies a lower bound on all.
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Lemma 7.4. Let κ > 0, δ > 0 such that ∀M ∈ N, z ∈ T M 1 λj ≥ κ, M j=1
and
λ1 ≤ κ + δ,
then, for all j = 0, 1, . . . , M − 1, λj+1 ≥ κ −
jδ . M −j
(10)
$M 1 Proof. First note that λ1 ≥ M j=1 λj . Thus λ1 ≥ κ, which corresponds to (10) for j = 0. Similarly, using also the upper bound on λ1 , we have for any 1 ≤ j ≤ M − 1, ⎛ ⎞ j M M ⎠ λk ≤ j(κ + δ) + Mκ ≤ ⎝ + λk k=1
k=j+1
k=j+1
so that λj+1 ≥
M 1 jδ . λk ≥ κ − M −j M −j k=j+1
Remark. In view of localization properties, the estimate is useful only if κ > (M − 1)δ.
(11)
We now estimate the cocycle to derive an upper bound on the largest Lyapunov exponent, which is uniform in the quasienergy and width of the strip M . Proposition 7.5. Let M ∈ N 1. For the generator of the cocycle defined in (7), it holds 1 A(p) ≤ (1 + r)(1 + t); rt 1 + log ((1 + r)(1 + t)). 2. It follows: 2λ1 ≤ log rt 3. There exists a c > 0 such that for M ∈ N it holds: dist(r, {0, 1}) < e−cM =⇒ 1 2 1 . ≤ ξM = λM − (M − 1) log ((1 + r)(1 + t)) log rt Proof. The estimate on A follows from its definition. The estimate on λ1 is obtained using the equality (9). Finally, from the estimate (10) it follows 1 2 1 . ≤ ξM = λM − (M − 1) log ((1 + r)(1 + t)) log rt The bound is symmetric around t = r = √12 and finite for r sufficiently away from the critical point √12 because of the singularity of log 1/rt.
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8. Spectral Localization We follow the strategy which was successfully employed for the case of one dimensional Schr¨ odinger operators: polynomial boundedness of generalized eigenfunctions, positivity of the Lyapunov exponent and spectral averaging. We lean on the work of [4,19]. Our result is: Theorem 8.1. Let M ∈ N, rt = 0. Then, the Chalker Coddington model on the cylinder exhibits spectral localization throughout the spectrum, almost surely. More precisely, 1. the almost sure : spectrum Σ, continuous spectrum Σc and pure point spectrum Σpp of U (p) satisfy Σ = Σpp = T 2.
and
Σc = ∅;
the eigenfunctions decay exponentially, almost surely.
Proof. We prove the theorem in Appendix 2.
Acknowledgements We should like to thank the referee for his constructive criticism and H. Schulz Baldes and H. Boumaza for enlightening discussions. We acknowledge gratefully support from the grants Fondecyt Grant 1080675; Anillo PBCT-ACT13; MATH-AmSud, 09MATH05; Scientific Nucleus Milenio ICM P07-027-F.
9. Appendix 1 We follow the strategy of [13] and first prove the lower bound # M 1 1 1 λi (z) ≥ 2 log |z − x|dl(x) + log − log |z| M i 2 rt
(12)
T
for 0 = z ∈ C\T which follows from Lemma 9.3 Eq. 15 below in the limit L → ∞. Lemma 9.1. Denote by U Dthe unitary operator defined by restriction of U to 2 l {−2L, . . . , 2L}, l2 (Z2M ) with reflecting boundary conditions: the scattering picture for the links which are incoming to walls at −(2L + 1) and 2L + 1 reads U D e−2L,2k+1 = e−2L,2k+2 ,
U D e2L,2k = e2L,2k+1 .
For z ∈ C let
Fz := ψ ∈ l2 (Z2M ); ψ2k+1 = zψ2k+2 , k ∈ ZM , Gz := ψ ∈ l2 (Z2M ); zψ2k+1 = ψ2k , k ∈ ZM
and denote by QF the orthogonal projection to a subspace F . It holds: z is an eigenvalue of U D ⇐⇒ Ψ2L = P2L (z)Ψ−2L and Ψ−2L ∈ Fz and Ψ2L ∈ Gz ⇐⇒ Ker QG⊥ P2L (z)QFz = {0} z
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Proof. It holds U D ψ = zψ =⇒ ψ−2L,2k+1 = zψ−2L,2k+2
and
zψ2L,2k+1 = ψ2L,2k
so Ψ−2L ∈ Fz
and
Ψ2L ∈ Gz .
The identity Ψ2L = P2L (z)Ψ−2L holds by construction of the transfer matrices so P2L (z)QFz Ψ = 0. U D ψ = zψ ⇐⇒ QG⊥ z Lemma 9.2. Denote the “even” subspace of l2 (Z2M ) by E := span{e2k ; k ∈ ZM }. For z = 0 there exist invertible operators Vz , Wz on l2 (Z2M ) such that Wz (E) = Fz and Vz (E) = G⊥ z such that 1. z is an eigenvalue of U D ⇐⇒ det QE Vz−1 P2L (z)Wz QE = 0 2.
where we understand the determinant to apply to the restriction to E. For z = 0; {z1 , . . . , z(4L+1)2M } the eigenvalues of U D it holds: |z|(4L+1)M | det QE Vz−1 P2L Wz QE | =
1 (4L+1)2M Π |z − zi |. (rt)2LM i=1
(13)
Proof. Fix 0 = z ∈ C. In the following Nj , Dj denote generic, z independent matrices whose precise values may change from line to line. The Dj are diagonal. The transfer matrix Az defined in (7) is of the form 1 2 z D1 QO + z −2 D2 QE + zN1 + N2 + z −1 N3 Az = rt where O denotes the “odd” subspace defined by O + E = l2 (Z2M ). Thus (rt)2L P2L = z 4L D1 QO + z −4L D2 QE +
4L−1
z j Nj .
j=−4L+1
Note that
1 √ (ze2k+1 + e2k+2 ) ; k ∈ ZM 2 1 Gz = span √ e2k + z −1 e2k+1 ; k ∈ ZM . 2 Fz = span
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On l2 (Z2M ) define the operators 1 Wz := √ |ze2k+1 + e2k+2 e2k+2 | + | − e2k+1 + z −1 e2k+2 e2k+1 | 2 k∈Z M 1 | − e2k+1 + ze2k e2k | + |e2k + z −1 e2k+1 e2k+1 |. Vz := √ 2 k∈Z M
Then Wz QE = QFz Wz and Vz QE = QG⊥ Vz . Moreover, one checks that z Wz2 = I,
$
Vz−1 = KVz K
(14)
|e2k+1 e2k | + |e2k e2k+1 |. It follows: z is eigenvalue of U D ⇐⇒ ker QE Vz−1 P2L Wz QE = {0}.
with K :=
k∈ZM
Now QE Vz−1 P2L Wz QE % & 1 1 = |e2k −e2k+1 + e2k , P2L (ze2m+1 + e2m+2 ) e2m+2 | 2 z k,m ⎛ ⎞ 1 ⎝ 4L+1 = z D1 QO + z −4L−1 D2 QE + z j Nj ⎠ . (rt)2L |j|<4L+1
Multiplication by z z
(4L+1)M
4L+1
implies that for some aj ∈ C
det QE Vz−1 P2L Wz QE =
1 (rt)2LM
(8L+2)M
z j aj .
0
(4L+1)M
D1 is unitary thus, in particular, |a(8L+2)M | = 1. z det . . . being a polynomial of degree (8L + 2)M whose leading coefficient has modulus (rt)−2LM and which is zero on the (4L + 1)2M eigenvalues of U D the formula for the determinant follows. We now prove convergence of the finite volume (L < ∞) density of states μM L as L → ∞ to a non-random measure: the density of state. Then we show that this measure equals the Lebesgue measure. (M )
Lemma 9.3. Denote μL 1 (4L + 1)2M
the measure defined by # D trf (U ) =: f (x)dμM L (x) T
Then 1. vaguely μM L →L→∞ dl
the Lebesgue measure on T.
(f ∈ C(T)).
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For M ∈ N there exists cM > 0 such that for all L ∈ N, 0 = z ∈ C\T 1 1 log ∧M P2L M 4L # 1 1 1 log |z − x|dμM ≥ log + 2 + L (x) 2 rt 2L T 1 cM − 1+ . log |z| − 4L L
(15)
Proof. 1. We first prove the existence of a non-random limit measure. The first step consists in showing that p a.e, #
1 {E (e0,0 , f (U )e0,0 ) + E (e1,1 , f (U )e1,1 ) L→∞ 4 T # + E (e1,0 , f (U )e1,0 )+E (e0,1 , f (U )e0,1 )} =: f dμM ∞, lim
f (U D (p))dμM L =
for all f ∈ C(T). This follows from a classical argument based on ergodicity, separability of C(T) and from the fact that U − U D has norm and rank uniformly bounded in L, see e.g. [20] for the details for the unitary case. In order to identify μM ∞ recall that the normalized Lebesgue measure dl on T is uniquely characterized by : # pn dl = δn,0
n∈Z
T
where pn (x) := xn . Consider the space of loops of euclidean length n starting at (0, 0) : Γ(0,0) = {γ : {0, . . . , n} → {−2L, . . . , 2L} × Z2M , γ(0) = γ(n) = (0, 0)}. Then because of the structure of U e0,0 , pn (U )e0,0 =
e0,0 , U eγ(1) . . . eγ(n−1) , U e0,0 .
γ∈Γ(0,0)
Now eγ(j) , U (p)eγ(j+1) = l(p)tα rβ for some α, β ≥ 0 and l a uniformly distributed random variable. Thus, E (e0,0 , pn (U )e0,0 ) = δn,0 . Applying the same argument to e1,1 , f (U )e1,1 , e0,1 . . ., we conclude: dμM ∞ = dl
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2. By formula (13): ' ' 1 log 'det QE Vz−1 P2L Wz QE ' 4LM # 1 1 1 1 = log + 2 + log |z − x|dμM (x) − 1 + log |z| L 2 rt 2L 4L T
1 log ∧M QE Vz−1 P2L Wz QE ≤ 4LM 1 1 1 1 log ∧M P2L + log ∧M QE Vz−1 + log ∧M Wz QE ≤ M 4L L 4M
=:cM
where we used the identity " ! det QE AQE = e0 ∧ · · · ∧ e2M −2 , ∧M Ae0 ∧ · · · ∧ e2M −2 .
From this, the claim follows. We turn now to the proof of the opposite inequality: # M 1 1 1 λi (z) ≤ 2 log |z − x|dl(x) + log − log |z| M i 2 rt
(16)
T
for 0 = z ∈ C\T: Proposition 9.4. Suppose for any choice of sets of vectors + that − − + + d0 , d2 , . . . , d− 2M −2 and d0 , d2 , . . . , d2M −2 in the “odd” subspace O 1 L→∞ M (4L + 1) ! − × log | (e0 + d− 0 ) ∧ · · · ∧ (e2M −2 + d2M −2 ),
lim sup
" + ∧M (Vz−1 P2L (z)Wz )(e0 + d+ 0 ) ∧ · · · ∧ (e2M −2 + d2M −2 ) | # 1 1 ≤ ln + 2 log |x − z|dl(x) − log |z| 2 rt
(17)
T
then, for all 0 = z ∈ C\T, # M 1 1 1 λi (z) ≤ log + 2 log |x − z|dl(x) − log |z| M i 2 rt T
Proof. The vectors of the form {(e0 +d0 )∧· · ·∧(e2M −2 +d2M −2 ); d0 , . . . , d2M −2 ∈ O} span ∧M C2M . On the other hand, given any spanning sets S1 and S2 in ∧M C2M , the mapping · S defined by AS ≡
sup φ∈S1 ,ψ∈S2
|φ, Aψ|
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defines a norm over the algebra of operators in ∧M C2M . It follows that there exists c > 0, which depends on S1 and S2 , such that for any matrix A, AS ≥ cA, hence that # M 1 1 1 λi (z) ≤ log + 2 log |ζ − z|dl(ζ) − log |z|. M i 2 rt T
We now prove that the inequality (17) is satisfied. This will be achieved in two steps. In order to keep track of the L dependence denote by ULD the former U D . Now reinterpret the left hand side of Eq. 17 as the characteristic polynomial of a deformation of ULD denoted VLD ; more precisely: we aim at Eq. 18 below. The problem is then reduced to the proof of the weak convergence of M ) towards μM . the associated sequence of counting measures (νL,z 9.1. Deformation of ULD
D Let L in N and define the matrix VL+1 on l2 ({−2L − 2, . . . , 2L + 2}, l2 (Z2M )) 2 2 by: ∀ψ ∈ l ({−2L − 2, . . . , 2L + 2}, l (Z2M )), D ψ)2L+2,2k = (VL+1
M −1
+ B2k,2l ψ2L,2l
l=0 D (VL+1 ψ)2L+1,k
= ψ2L+1,k
D (VL+1 ψ)2L,2k+1 =
M −1
+ C2k+1,2l+1 ψ2L+2,2l+1 ,
l=0 D ψ)−2L,2k = (VL+1
M −1
− B2k,2l ψ−2L−2,2l
l=0 D (VL+1 ψ)−2L−1,k
= ψ−2L−1,k
D (VL+1 ψ)−2L−2,2k+1 =
M −1
− C2k+1,2l+1 ψ−2L,2l+1 ,
l=0 D D with the same reflecting boundary conditions as UL+1 and (VL+1 ψ)μ,ν = D (UL+1 ψ)μ,ν for any values of (μ, ν) which were not described previously. The D D is a deformation of the matrix UL+1 , but its structure remains matrix VL+1 D . Note that span{e , e−2L−1,j ; j, k ∈ Z2M } close to the structure of U 2L+1,k L+1 D D − I . For ψ an eigenvector of VL+1 associated with the belongs to ker VL+1 eigenvalue z, D VL+1 ψ = zψ.
This implies that either ψ ∈ span{e2L+1,k , e−2L−1,j ; j, k ∈ Z2M } and z = 1, or ψ−2L−2 ∈ Fz and ψ2L+2 ∈ Gz and
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ψ2L = P2L (z)ψ−2L ψ2L+2 = A+ (z)ψ2L = z −1 QE B + QE ψ2L + zQO C + −
ψ−2L = A (z)ψ−2L−2 = z
−1
−1
−
QO ψ2L
QE B QE ψ−2L−2 + zQO C −
−1
QO ψ−2L−2 .
The transfer matrices A+ (z) and A− (z) are deformations of the matrices Az . This construction is useful to establish the following lemma. In the following, z will be fixed as a parameter. − Lemma 9.5. Let (d+ 2k )k∈{0,...,M −1} and (d2k )k∈{0,...,M −1} two families of vectors belonging to the “odd” subspace O. These families are the columns of two cor+ − responding matrices denoted D and D respectively. Assume that for z = 0, 1 − + max(D , D ) ≤ max |z| , |z| and consider the matrix VLD parametrized by z
Bz+ = z + T D+ Cz+ = z(1 − z −1 D+ T )−1 Bz− = z + z 2 D−∗ Cz− = (z −1 − D−∗ )−1 with
⎛
0 ⎜1 ⎜ ⎜ T =⎜ ⎜ ⎝
1 0 1
..
. 1
⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠
0
Then, − (e0 + d− 0 ) ∧ · · · ∧ (e2M −2 + d2M −2 ), + ∧M (Vz−1 P2L (z)Wz )(e0 + d+ 0 ) ∧ · · · ∧ (e2M −2 + d2M −2 )
= e0 ∧ · · · ∧ e2M −2 , ∧M (Vz−1 A− P2L (z)A+ Wz )e0 ∧ · · · ∧ e2M −2 .
(18)
Proof. By (14), Wz2 = I, Vz−1 = KVz K , z = 0. It follows for all k ∈ {0, . . . , M − 1} : A+ Wz e2k = Wz (e2k + d+ 2k ) A−∗ Vz−1∗ e2k = Vz−1∗ (e2k + d− 2k ). D Given z, D+ and D− and the associated matrix VL+1 , we consider the corresponding eigenvalue problem: D VL+1 ψ = z ψ
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D The complex number z is an eigenvalue of VL+1 iff −
+
(z − 1)4M e0 ∧ · · · ∧ e2M −2 , ∧M (Vz−1 Az (z )P2L (z )Az (z )Wz )
e0 ∧ · · · ∧ e2M −2 = 0 −1
−1 QE Bz± QE + z QO Cz± QO . where A± z (z ) = z (4L+5)M , the left-hand side is a polynomial of degree Once multiplied by z
2M (4L + 5) in z . Following the Thouless argument, we get for the logarithm of the modulus divided by 4M L: # 1 1 1 1
M log − log |z | + 2 + log |x − z |dνL,z (x) + O , 2 rt 2L L B(0,Rz )
M are supported on some closed ball B(0, Rz ), where the family of measures νL,z D + due to the fact that supL UL − VLD < ∞. Note that if z = z, A+ z (z) = A − − and Az (z) = A .
9.2. End of Proof of Inequality (17) We split the proof in the two following lemmas, whose proof is an adaptation of the argument given in [13]. Lemma 9.6. If (νL )L∈N and μ are measures supported on B(0, R) for some R > 0, and if (νL ) converge weakly to μ, then for any z ∈ C # # log |ζ − z|dνL (ζ) ≤ log |ζ − z|dμ(ζ). T
T
Proof. Given z ∈ C, let f be defined by: f (ζ) = log |ζ − z| if |ζ − z| ≥ f (ζ) = log || if |ζ − z| ≤ Since the support is compact, # # lim f (ζ)dνL (ζ) = f (ζ)dμ(ζ). L→∞
T
T
On the other hand, for any ζ in T, log |ζ − z| ≤ f (ζ) so that:
# L→∞
# log |ζ − z|dνL (ζ) ≤ lim sup
lim sup
L→∞
T
# =
f (ζ)dνL (ζ) T
f (ζ)dμ(ζ). T
The result follows by monotone convergence theorem when goes to zero.
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Remark. Let us note that the ∗-algebra of trigonometric polynomials F defined by: F = {f ∈ C(B(0, R)); f (r, θ) ak1 ,k2 rk1 eik2 θ , =
N ∈ N0 , k1 ∈ N0 , k2 ∈ Z}
k1 +|k2 |≤N
separates points and contains the constants. Its closure under the supremum norm is C(B(0, R)). The weak convergence of the measures is equivalent to have for all f in F, # # f (ζ)dνL (ζ) = f (ζ)dμ(ζ). lim L→∞ B(0,R)
B(0,R)
M Lemma 9.7. As a Borel measure on C, the sequence of measures (νL,z ) parametrized by z, M converges almost surely weakly to dl as L tends to infinity. 2M (4L+1)
2M (4L+1)
Proof. Let (rj,z eiξj,z )j=1 and (eiλj )j=1 be the eigenvalues of the problems with reflecting boundary conditions for V and UD , which correspond respectively to the modified and unmodified “potentials”. We have that: M νL,z =
μM L =
1 δ iξj,z 2M (4L + 1) j rj,z e 1 δ iλj . 2M (4L + 1) j e
These measures are supported on some B(0, Rz ). We will drop the z subscript in the sequel. Since we already know that (μM L ) converges almost surely weakly to dl, we only need to show that for any non-negative integer k1 and any integer k2 , 1 lim L→∞ 2M (4L + 1)
2M (4L+1)
eik2 λj − rjk1 eik2 ξj = 0.
j=1
Actually, it is enough to prove it for non-negative integers k1 , k2 . Let us fix such a couple (k1 , k2 ) and decompose the term on the left-hand side as follows: 1 2M (4L + 1) where
2M (4L+1)
eik2 λj − rjk1 eik2 ξj = T1 (L) + T2 (L)
j=1
T1 (L) = T2 (L) =
1 2M (4L + 1)
2M (4L+1)
(rjk2 − rjk1 )eik2 ξj
j=1
1 k2 Tr(UD − V k2 ). 2M (4L + 1)
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If k = min(k1 , k2 ) and l = max(k1 , k2 ), we have 2M (4L+1)
1 |T1 (L)| ≤ 2M (4L + 1) ≤
rk |rjl−k − 1|
j=1 2M (4L+1)
Rzk 2M (4L + 1)
|rjl−k − 1|.
j=1
Following [13], we first prove that: 1 L→∞ 2M (4L + 1)
2M (4L+1)
lim
|rj − 1| = 0.
j=1 2M (4L+1)
We know that there exists two orthonormal bases (φj )j=1 2M (4L+1) (φ j )j=1
and
such that: 2M (4L+1)
VLD =
μj (VLD )|φ j φj |,
j=1
(μj (VLD ))
are the singular values of the operator VLD . Actually, μj (VLD ) = where rj and we assume them to be ordered: μj+1 (VLD ) ≥ μj (VLD ) ≥ 0. Note that: ∗
{μ2j (VLD ); j ∈ {1, . . . , 2M (4L + 1)}} = σ(VLD VLD )\{0}. Since for each j ∈ {1, . . . , 2M (4L + 1)}, μj (ULD ) = 1 we deduce from the remark following Theorem 1.20 in [24] that: 2M (4L+1)
2M (4L+1)
|rj − 1| =
j=1
Since
VLD
2M (4L+1)
|μj (VLD )
−
μj (ULD )|
j=1
−
ULD
≤
μj (VLD − ULD ).
j=1
has rank and norm uniformly bounded in L, we obtain that: lim T1 (L) = 0.
L→∞
The term T2 (L) will be treated in a similar way. The operator ULD − VLD has rank and norm uniformly bounded in L. This implies that for all integer k2 , k k ULD 2 − VLD 2 has also rank and norm uniformly bounded in L. So, lim T2 (L) = 0,
L→∞
which concludes the proof.
The above lemmata together with Eq. 18 establish the inequality (17) which implies (16). We finish with the proof of the Thouless formula on T: Lemma 9.8. For all z ∈ T,
# M 1 1 1 λi (z) = log + 2 log |x − z|dl(x) M i 2 rt T
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1 Proof. We note( with [12] that limL→∞ 4L log ∧M (P2L )(z) is subharmonic in C\{0} and T log |z − x|dl(x) subharmonic on C\T. The two exceptional sets are of measure zero in C, these quantities must agree everywhere.
Remark 9.9. We note that the above proof does not depend on the specific form of the density of states.
10. Appendix 2 Now we prove Theorem 8.1 in several steps. By Theorem 6.1 the localization length is finite for all values of the parameters. Note that the spectrum is characterized by the existence of generalized eigenfunctions: Suppose that the support of Ep (·), the spectral resolution of U (p), is the whole circle T. Proposition 10.1. For M ∈ N, p ∈ Ω the spectrum of U (p) is the closure of the set Sp = {z ∈ T; U (p)φ = zφ has a non-trivial polynomially bounded solution} and Ep (T\Sp ) = 0. Proof. The stated behaviour at infinity of the generalized eigenvectors and the spectrum of U (p) are related by Sh’nol’s Theorem. This well known deterministic fact for self-adjoint operators was proven in [4] to hold in the unitary setup for band matrices on l2 (Z). It is straightforward to check that the result holds for band matrices on l2 (Z, C2M ), with M finite. Secondly we prove the existence of a finite cyclic subspace: Lemma 10.2. Let M ∈ N, rt = 0. Denote I0 := {0} × Z2M . The vectors {eμ ; μ ∈ I0 } span a cyclic subspace of l2 (Z × Z2M ). Proof. The only non-vanishing elements in U are the blocks given in Eq. 1. Denoting generically the elements of S by α β S =: γ δ −1 = Uν,μ we have and observing that Uμ,ν U(2j+1,2k);(2j,2k) U(2j+1,2k);(2j+1,2k+1) α β = U(2j,2k+1);(2j,2k) U(2j,2k+1);(2j+1,2k+1) γ δ U(2j+2,2k+2);(2j+2,2k+1) U(2j+2,2k+2);(2j+1,2k+2) = U(2j+1,2k+1);(2j+2,2k+1) U(2j+1,2k+1);(2j+1,2k+2)
and
−1 U(2j,2k);(2j+1,2k)
−1 U(2j,2k);(2j,2k+1)
= −1 −1 U(2j+1,2k+1);(2j+1,2k) U(2j+1,2k+1);(2j,2k+1) −1 −1 U2j+2,2k+1;2j+2,2k+2 U2j+2,2k+1;2j+1,2k+1 = . −1 −1 U2j+1,2k+2;2j+2,2k+2 U2j+1,2k+2;2j+1,2k+1
α β
γ δ
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Computing U e(0,2k) = αe(1,2k) + γe(0,2k+1) and the corresponding expressions for U −1 e(1,2k) , U e(0,2k+1) , U −1 e(−1,2k+1) we infer: 1 U e(0,2k) − γe(0,2k+1) α β γ = e(0,2k) − U −1 e(0,2k+1) α β 1 U e(0,2k+1) − αe(0,2k+2) = γ δ α = e(0,2k+1) − U −1 e(0,2k+2) . γ δ
e(1,2k) = e(1,2k+1) e(−1,2k+1) e(−1,2k+2)
Thus vectors with indices in {±1} × Z2M belong to the subspace generated by U ±1 (I0 ). The lemma follows by induction. 4M
Let I = {0, 1} × Z2M , Ω = TZ \I , P = ⊗k∈Z4M \I dl, p = {pj }j∈Z4M \I ∈ Ω, and ΘI = {θj }j∈I . We shall use the notation Ω p = (p, ΘI ). Denote 1 1 log( ∧M P2L (z)(p)) − log( ∧M −1 P2L (z)(p)) , λM (p, z) := lim L→∞ 4L 4L if the limit exists. By construction, 5.1, it holds for almost every p λM = λM (p) By definition λM (p, z) is independent of the finitely many ΘI , if p = (p, ΘI ). By Theorem 6.1 there exists Ω(z) ⊂ Ω with P(Ω(z)) = 1 such that for any z ∈ T\R λM ((p, ΘI ), z) = λM > 0, for all θj ∈ ΘI and all p ∈ Ω(z). We can apply Fubini to the measure P × dl to get the existence of Ω0 ∈ Ω with P(Ω0 ) = 1 such that for every p ∈ Ω0 there is Bp ∈ T with l(Bp ) = 0 and λM ((p, ΘI ), z) > 0 for all θj ∈ ΘI , and all z ∈ Bp C .
(19)
Then we show that for p ∈ Ω0 , Bp is a support of the spectral resolution of U ((p, ΘI )) for almost every θj ∈ ΘI w.r.t. d|I| l on T|I| . For any(fixed j ∈ I, we introduce the spectral measures μjp associated with U (p) = T xdEp (x) defined for all Borel sets Δ ∈ T by C
μjp (Δ) = ej |Ep (Δ)|ej . Since U (p) = D(p)S, where D(p) is diagonal, the variation of a random phase at one site is described by a rank one perturbation. More precisely, dropping ) by taking θj = 1 in the definition of the variable p temporarily, we define D D: ) = D + |ej ej |(1 − θj ) = elog(θj )|ej ej | D, D
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so that, with the obvious notations, ) = DS ) = elog(θj )|ej ej | U. U The unitary version of the spectral averaging formula, see [9] and [3], reads in our case: for any f ∈ L1 (T), # # # j dl(θj ) f (x)dμ(p,ΘI ) (x) = f (x)dl(x). T
T
T
Applied to f = χBp , the characteristic function of Bp , this yields # 0 = l(Bp ) = μj(p,ΘI ) (Bp ), dl(θj ).
(20)
T
Consequently, μj(p,ΘI ) (Bp ) = 0,
for every θk ∈ ΘI , k = j and Lebesgue-a.e. θj .
Therefore, for all p ∈ Ω0 , there exists Jp ⊂ T|I| s.t. l(Jp C ) = 0 and ΘI ⊂ Jp ⇒ μj(p,ΘI ) (Bp ) = 0,
∀j ∈ I.
(21)
Now fix p ∈ Ω0 and ΘI ⊂ Jp and consider p = (p, ΘI ). By Lemma 10.2 and (21) we deduce that Ep (Bp ) = 0. If Sp is the set from Sh’nol’s Theorem 10.1, then the set Sp ∩ Bp C is a support for Ep (·). Now take z ∈ Sp ∩ Bp C . By Theorem 10.1, U (p)ψ = zψ has a non-trivial polynomially bounded solution ψ. On the other hand, by (19), λM (p, z) > 0. Thus, by Osceledec’s Theorem, every solution which is polynomially bounded necessarily has to decay exponentially both at +∞ and −∞, and therefore it is an eigenfunction of U (p). In other words, every z ∈ Sp ∩Bp C is an eigenvalue of U (p), hence Sp ∩ Bp C is countable. Therefore Ep (·) has countable support thus U (p) has pure point spectrum. With Ω0 := {(p, {θj }j∈I ) s.t. p ∈ Ω0 , {θj }j∈I ⊂ ΘI ⊂ Jp }, we have p ∈ Ω0 ⇒ σc (U (p)) = ∅.
(22)
C
Also, from l(Jp ) = 0 we have (⊗j∈I dl)(Jp ) = (⊗j∈I dl)(T|I| ) = 1.
(23)
As P(Ω0 ) = 1, we conclude from (22) and (23) that # P(σc (U (p)) = ∅) ≥ P(Ω0 ) = dP(p)(⊗j∈I dl)(Jp ) = 1, Ω0
which proves that U (p) has almost surely pure point spectrum. The fact that the support of the density of state coincides with the almost sure spectrum, see [20], shows that Σpp = T. We finally show that almost surely all eigenfunctions decay exponentially. Note that we actually have shown above that the event “all eigenvectors of
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U (p) decay at the rate of the smallest Lyapunov exponent” has probability one, since this is true for all p ∈ Ω0 . Measurability of this event was proven for the case of ergodic one-dimensional Schr¨ odinger operators by Kotani and Simon in Theorem A.1 of [22]. The proof of this fact provided in [22] carries over to the CC model $ as well. It is enough to note that, due to Lemma 10.2, we may use ρp = j∈I μjp as spectral measures in their argument.
References [1] Arnold, L.: Random dynamical systems. Springer Monographs in Mathematics. Springer, Berlin (1998) [2] Avron, J.E., Seiler, R., Simon, B.: Charge deficiency, charge transport and comparison of dimensions. Comm. Math. Phys. 159, 399–422 (1994) [3] Bourget, O.: Singular continuous Floquet operator for periodic quantum systems. J. Math. Anal. Appl. 301, 65–83 (2005) [4] Bourget, O., Howland, J.S., Joye, A.: Spectral analysis of unitary band matrices. Commun. Math. Phys. 234, 191–227 (2003) [5] Bougerol, P., Lacroix, J.: Products of Random Matrices with Applications to Schr¨ odinger Operators. Progress in Probability and Statistics, vol. 8. Birkh¨ auser, Boston (1985) [6] Boumaza, H.: H¨ older continuity of the IDS for matrix-valued Anderson models. Rev. Math. Phys. 20, 873–900 (2008) [7] Boumaza, H., Stolz, G.: Positivity of Lyapunov exponents for Anderson-type models on two coupled strings. Elecron. J. Differ. Equ. 47, 11–18 (2007) [8] Bellissard, J., van Elst, A., Schulz-Baldes, H.: The noncommutative geometry of the quantum Hall effect. J. Math. Phys. 35, 5373–5451 (1994) [9] Combescure, M.: Spectral properties of a periodically kicked quantum Hamiltonian. J. Stat. Phys. 59, 679–690 (1990) [10] Chalker, J.T., Coddington, P.D.: Percolation, quantum tunneling and the integer Hall effect. J. Phys. C 21, 2665–2679 (1988) ´ [11] Colin de Verdi`ere, Y., Parisse, B.: Equilibre instable en r´egime semi-classique. I. Concentration microlocale. Commun. Partial Differ. Equ. 19, 1535–1563 (1994) [12] Craig, W., Simon, B.: Subharmonicity of the Lyaponov index. Duke Math. J. 50, 551–560 (1983) [13] Craig, W., Simon, B.: Log H¨ older continuity of the integrated density of states for stochastic Jacobi matrices. Commun. Math. Phys. 90(2), 207–218 (1983) [14] Fertig, H.A., Halperin, B.I.: Transmission coefficient of an electron through a saddle-point potential in a magnetic field. Phys. Rev. B 36, 7969–7976 (1987) [15] Graf, G.M.: Aspects of the integer quantum Hall effect. Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday. In: Proceedings of symposium of Pure Mathematics vol. 76, Part 1, pp. 429–442. American Mathematical Society, Providence, RI (2007) [16] Germinet, F., Klein, A., Schenker, J.: Dynamical delocalization in random Landau Hamiltonians. Ann. Math. 166, 215–244 (2007) [17] Goldshe˘ıd, I.Ya., Margulis, G.A.: Lyapunov exponents of a product of random matrices. Uspekhi Mat. Nauk 44, 13–60 (1989)
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[18] Helffer, B., Sj¨ ostrand, J.: Analyse semi-classique pour l’´equation de Harper. M´emoires de la S.M.F. 34, 1–113 (1988) [19] Hamza, E., Joye, A., Stolz, G.: Localization for random unitary operators. Lett. Math. Phys. 75, 255–272 (2006) [20] Joye, A.: Density of states and Thouless formula for random unitary band matrices. Ann. Henri Poincar´e 5, 347–379 (2004) [21] Kramer, B., Ohtsuki, T., Kettemann, S.: Random network models and quantum phase transitions in two dimensions. Phys. Rep. 417, 211–342 (2005) [22] Kotani, S., Simon, B.: Localization in general one-dimensional random systems. Commun. Math. Phys. 112, 103–119 (1987) [23] Roemer, R., Schulz-Baldes, H.: Random phase property and the Lyapunov spectrum for disordered multi-channel systems (preprint, 2009). http://de.arxiv.org/ abs/0910.5808 [24] Simon B.: Trace ideals and their applications, 2nd edn. Mathematical Surveys and Monographs, 120. American Mathematical Society (2005) [25] Trugman, S.A.: Localization, percolation, and the quantum Hall effect. Phys. Rev. B 27, 7539–7546 (1983) [26] Thouless, D.J., Kohmoto, M., Nightingale, M.P., den Nijs, N.: Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982) [27] Wang, W.M.: Microlocalization, percolation, and Anderson localization for the magnetic Schr¨ odinger operator with a random potential. J. Funct. Anal. 146, 1–26 (1997) Joachim Asch CPT-CNRS UMR 6207 Universit´e du Sud, ToulonVar BP 20132 83957 La Garde Cedex, France e-mail:
[email protected] Olivier Bourget Departamento de Matem´ aticas Pontificia Universidad Cat´ olica de Chile Av. Vicu˜ na Mackenna 4860 C.P. 690 44 11 Macul Santiago, Chile e-mail:
[email protected]
Vol. 11 (2010)
Localization Properties of the Chalker–Coddington Model
Alain Joye Institut Fourier Universit´e Grenoble 1 BP 74 38402 Saint-Martin d’H`eres, France e-mail:
[email protected] Communicated by Claude Alain Pillet. Received: January 22, 2010. Accepted: July 27, 2010.
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Annales Henri Poincar´ e
Ionization of Atoms by Intense Laser Pulses J¨ urg Fr¨ohlich, Alessandro Pizzo and Benjamin Schlein Dedicated to our friend and colleague Robert Schrader on the occasion of his 70th birthday
Abstract. The process of ionization of a hydrogen atom by a short infrared laser pulse is studied in the regime of very large pulse intensity, in the dipole approximation. Let A denote the integral of the electric field of the pulse over time at the location of the atomic nucleus. It is shown that, in the limit where |A| → ∞, the ionization probability approaches unity and the electron is ejected into a cone opening in the direction of −A and of arbitrarily small opening angle. Asymptotics of various physical quantities in |A|−1 is studied carefully. Our results are in qualitative agreement with experimental data reported in Eckle et al. (Science 322, 1525–1529; 2008, Nature (physics) 4, 565–570 2008).
1. Experimental Findings and Preliminary Theoretical Considerations In recent experimental work [1,2], Eckle et al. have investigated the ionization of Helium atoms by highly intense elliptically polarized infrared laser pulses of short duration. One of the purposes of their work has been to perform an (indirect) measurement of the tunneling delay time in strong-field ionization of Helium atoms. The experimental parameters in their work have been chosen as follows: the pulse duration, T , is around 5.5 femtoseconds; the peak intensity, I0 , is between 2.3 × 1014 and 3.5 × 1014 W/cm2 , and the center wave length is around 725 nm. The ionization potential, Ip , of a Helium atom in its groundstate is known to be Ip ≈ 24.6 eV. These parameter values yield a Keldysh parameter, γ, for circularly polarized light ranging from 1.17 to 1.45. The Keldysh parameter for circular polarization is given by Ip (eV) γ ≈ 0.33 . (1.1) 14 I0 (10 W/cm2 )[L(μm)]2 A. Pizzo was partially supported by NSF Grant DMS-0905988.
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If γ 1, i.e., for short wave lengths, L, and low intensity, I0 , the ionization process can be described in terms of multi-photon absorption, and one may attempt to treat the ionization problem perturbatively (for a theoretical analysis of a related problem, see, e.g., [3]). If γ 1, i.e., for high intensities and long wave lenghts, a regime is approached where the electromagnetic field can be treated classically. However, due to the high intensity of the pulse, the theoretical analysis of the ionization process is intrinsically non-perturbative in the coupling of the electrons to the electromagnetic field. This is the regime we study in this paper. For the values of γ between 1.17 and 1.45 realized in the experiments described in [1,2], reliable analytical calculations of the ionization process appear to be very difficult to come by, and it is advisable to perform numerical studies; see [4]. We find, however, that our analytical results are in good qualitative agreement with the experimental findings in [1,2]. One key point of these findings is that the ionization process of a Helium atom by a short, intense near-infrared laser pulse is essentially instantaneous, in contrast to theoretical predictions based on an approximate theoretical picture taken from [5,6]: experimentally, an upper bound on the time it takes to ionize a Helium atom (with experimental parameters chosen as discussed above) appears to lie between 12 and 34 attoseconds, while a theoretical prediction relying on [5,6] yields an ionization (or “barrier traversal”) time of 450–560 attoseconds. Obviously, there is a problem with either the interpretation of the experimental findings in terms of an “ionization time” or with the approximate theory of the ionization process based on [5,6]; but most likely with both. The purpose of our paper is to provide a qualitative theoretical interpretation of the data gathered in the experiments described in [1,2]. We start with a brief sketch of the picture on which the theoretical interpretation of the experimental results is based that the authors of [1] have advocated implicitly. We then describe our own approach and state our main results. Without harm, we may simplify our discussion by considering the ionization of Hydrogen atoms or Helium+ ions by elliptically polarized laser pulses. The direction of propagation of the pulses through a very dilute, cold gas of atoms or ions is chosen to be our z axis. The electric and magnetic field of the pulse are then parallel to the x−y plane. If E0 denotes the peak electric field of the pulse at the location of an atom or ion and T denotes the duration of the pulse then the field of the pulse is assumed to be homogeneous over a region of the x − y plane of large diameter, d, as compared to E0 T 2 , centered at the location of the atom or ion. Note that E0 T 2 has the dimension of length. This assumption partially justifies to use the dipole approximation. The Hamiltonian generating the time evolution of the electron in the atom or ion then only depends on the electric field, E(t), at the location of the atomic (or ionic) nucleus; (t denotes time). The vector E(t) can be chosen to have the form T T E(t) = E0 (t) cos ω t − , sin ω t − ,0 (1.2) 2 2
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where E0 (t) is a smooth envelope function with support in the interval [0, T ], ω = 2πc/L is the angular frequency of the pulse (with L cT ), and is a parameter describing the elliptical polarization of the pulse. To be concrete, we choose E0 (t) to be non-negative, symmetric-decreasing about t = T /2, with a maximum, E0 (T /2) =: E0 , at t = T /2. Apparently, the pulse arrives at the location of the nucleus at time t = 0 and lasts until time t = T . An important quantity is the vector potential t A(t) =
dτ E(τ ).
(1.3)
0
Clearly, A(t) = 0, for t ≤ 0, and A(t) ≡ A(T ), for t ≥ T . For our choice of the envelope function E0 (t), A(T ) = const · E0 (1, 0, 0),
(1.4)
where the constant depends on ω and on E0 (t); it tends to 0 rapidly, as ω → ∞, i.e., in the ultraviolet. In this regime, the Keldysh parameter γ becomes very large, and the analysis presented in our paper is not applicable. It does, however, apply to the situation where const · E0 , in Eq. (1.4), becomes large, meaning that γ becomes small. To anticipate our main result, we will show that, for a laser pulse of the form in Eq. (1.2), (i) the ionization probability approaches unity, as E0 → ∞ (with a rate that will be estimated explicitly), and (ii) the electron is ejected by the pulse into a cone with axis parallel to A(T ) and a small opening angle Θ = Θ(E0 ); its average velocity v = v(E0 ) is approximately parallel to A(T ). Moreover, Θ(E0 ) → 0, as E0 → ∞,
(1.5)
(with a rate that will be estimated), and v(E0 ) A(T ), as E0 → ∞,
(1.6)
with |v(E0 )| ∝ E0 . These theoretical results are in good qualitative agreement with the experimental findings described in [1,2]. In the experiments, the motion of the ions after ionization is measured. However, by momentum conservation, such measurements also determine the motion of the electron. In [1], data compatible with Eqs. (1.5) and (1.6) are interpreted as saying that the ionization process is nearly instantaneous. This interpretation is based, implicitly, on arguments that rely on the “Ritz Hamiltonian” for the motion of the electron: Z HRitz (t) = −Δ − − E(t) · x. (1.7) |x| Δ is the Laplacian, Z is the charge of the nucleus, and E(t) is the electric field of the laser pulse at the location of the nucleus, see Eq. (1.2). Here we work in units such that = 1, mel = 1/2 and e = 1, where mel is the mass of an
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electron and e is the elementary electric charge. Therefore, in our units, the numerical value of the speed of light, c, is around 137. Hereafter, we follow the convention that the dimension of a physical quantity is a function of the length only, namely: [length] = length; [mass] = length−1 ; [time] = length; the electric charge is dimensionless. At a fixed moment, t = t0 , of time, the potential Ut0 (x) := −
Z − E(t0 ) · x |x|
(1.8)
has a shape indicated in Fig. 1. Initially, the electron is localized near the nucleus placed at the origin, O, of our coordinate system and treated as static for the duration of the tunneling process. If E(t) depends slowly on time t, i.e., for rather large pulse duration T and long wave lengths, one may expect that an adiabatic approximation for the description of the tunneling process of the electron through the barrier of the potential Ut0 (x) to the point xT (see Fig. 1) is appropriate. If ΔtT denotes the barrier traversal time, the electric field acting on the (nearly free) electron, after it has traversed the barrier, is given by E(t), with t ≥ t0 + ΔtT . If we interpret t0 = 0 as the time of onset of barrier traversal then the electron, after barrier traversal will be ejected in a direction roughly parallel to the vector T dτ E(τ ). (1.9) X := t0 +ΔtT
For a pulse described by Eq. (1.2) and a strictly positive barrier traversal time, ΔtT , the direction of X in which the electron is ejected is not parallel to the direction of A(T ) (parallel to the x-axis, for our concrete choice of an envelope function E0 (t)). By tuning the direction of A(T ) and measuring the direction in which the electrons are ejected, one can determine the angle, φ, between X and A(T ). This angle then provides information on the barrier traversal time ΔtT . Experimentally, φ is very small, so that ΔtT is argued to be very short.
Figure 1. The potential Ut0 (x)
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The analysis presented in this paper shows that, for large E0 , φ is small. We have found the Ritz Hamiltonians in Eq. (1.7) to be rather inconvenient for an analysis of ionization processes. It is advantageous to, instead, consider the “Kramers Hamiltonians” Z , (1.10) H(t) = (p − A(t))2 − |x| where p = −i∇ is the usual electron momentum operator and A(t) is the vector potential at the location of the nucleus given in Eq. (1.3). The evolutions generated by HRitz (t) see (1.7) and H(t), as in (1.10), are related to each other by a time-dependent gauge transformation given by Λ(x, t) := A(t) · x.
(1.11)
If (E(T ) · x, 0) denotes the 4-vector potential before the gauge transformation (1.11) is made then, after this gauge transformation, it is given by (0, A(t)). Quantum mechanically, the gauge equivalence of the time evolutions generated by the Ritz Hamiltonians, Eq. (1.7), and the Kramers Hamiltonians, Eq. (1.10), can easily be verified using the Trotter product formula (see, e.g., [7]) for the propagators and the identity Z , (1.12) e−iΛ(x,t) H(t)eiΛ(x,t) = p2 − |x| with H(t) as in (1.10). Next, we sketch some key ideas in our analysis of the time evolution generated by the Kramers Hamiltonians. As an initial condition, ψ0 , for the electron we choose a bound state wave function, typically the atomic groundstate. In our units, it has a spatial spread of order O(Z −1 ). The quantum-mechanical propagator generated by the Kramers Hamiltonians H(t), defined in Eq. (1.10), is denoted by U(t, t0 ) = U(t, t0 ; Z). It evolves an electronic wave function from time t0 to time t and solves the equation i∂t U(t, t0 ; Z) = H(t)U(t, t0 ; Z),
(1.13)
with U(t0 , t0 ; Z) = 1, for an arbitrary t0 ; see [8]. We note that the propagator U0 (t, t0 ) ≡ U(t, t0 ; Z = 0) can be calculated explicitly: ⎡ ⎤ t U0 (t, t0 ) = exp ⎣−i (p − A(τ ))2 dτ ⎦ (1.14) t0
⎡
= eiφ(t,t0 ) e−i(t−t0 )p exp ⎣2ip · 2
t
⎤ A(τ )dτ ⎦.
(1.15)
t0
The first factor on the R. S. of (1.15) is a pure phase factor (with φ(t, t0 ) = t − t0 A(τ )2 dτ ), the second factor is the free time evolution, and the third factor t is a space translation by the vector 2 t0 A(τ )dτ . As our initial time, we choose t0 = 0, and the initial condition at t = 0 is chosen to be ψ0 , as described above. The laser pulse hits the atom at time t = 0 and lasts up to time T . Because of the space translation,
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⎡ TE0 := exp ⎣2ip ·
T
Ann. Henri Poincar´e
⎤ A(τ )dτ ⎦,
(1.16)
0
in the free propagator (1.15), which moves the initial wave function, ψ0 , far out of the potential well (described by −Z/|x|), provided E0 (the peak electric field) is large, one expects that U(T, 0; Z)ψ0 ≈ U0 (T, 0)ψ0 ,
(1.17)
with an error term that tends to 0, as E0 → ∞. Results of this type have first been proven by Fring, Kostrykin and Schrader in [9]. We will reproduce their results in Sect. 2, below. As noted in (1.3), A(t) = A(T ), for t ≥ T,
(1.18)
i.e., the vector potential is constant when the pulse has passed. We may therefore use a gauge transformation to remove it: e−iΛ(.,T ) U(t, T ; Z)eiΛ(.,T ) = UC (t, T ), for all t ≥ T,
(1.19)
where UC (t, T ) = exp [−i(t − T )HC ], and HC := p2 −
Z |x|
(1.20)
is the Coulomb Hamiltonian. Next we note that, by Eq. (1.11), e−iΛ(x,T ) = e−iA(T )·x ,
(1.21)
i.e., e−iΛ(x,T ) is a translation in momentum space: it translates ψ T (p) to ψ A(T ) (p) := ψ T (p + A(T )),
(1.22)
ψT (x) = (U(T, 0; Z)ψ0 )(x),
(1.23)
where and ψ T is the Fourier transform of ψT . An electron in the state given by ψA(T ) , T see Eq. (1.22), has a mean distance from the nucleus of order O(| 0 A(τ )dτ |) and a mean velocity in the direction of A(T ) of magnitude |A(T )|. Thus, the mean distance of ψA(t) from the nucleus and the mean velocity of the electron, parallel to A(T ), diverge, as the peak electric field, E0 , of the pulse tends to ∞. However, by Eqs. (1.17) and (1.15), the spread of the wave function ψA(t) in x−space around its mean position is of order O(T Z), which is independent of E0 . It is then almost obvious that, for t ≥ T , U(t, 0; Z)ψ0 = U(t, T ; Z)ψT iΛ(.,T )
=e
UC (t, T )ψA(T )
iΛ(.,T ) −i(t−T )p2
≈e
(1.24)
e
ψA(T ) ,
(1.25) (1.26)
with an error term that tends to 0, as E0 → ∞, uniformly in t ≥ T . This will be proven mathematically in Sect. 2.2., below. The phase factor, eiΛ(.,T ) , on
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the R.S. of (1.26) is unimportant. Moreover, exp [−i(t − T )p2 ]ψA(T ) is the free time evolution of an electron wave function initially located at a distance of T order O(| 0 A(τ )dτ |) from the nucleus and with a mean velocity parallel to A(T ) and of magnitude |A(T )|. Its spread in the direction perpendicular to A(T ) is of order O(t Z), which is independent of E0 . Thus, the state U(t, 0; Z)ψ0 propagates into a cone with axis parallel to A(T ) and with an opening angle of order O(Z/|A(T )|), which tends to 0, as E0 → ∞. In the technical sections of this paper, these claims are verified mathematically, and the asymptotics in 1/E0 is estimated quite carefully. This is crucial, because the Kramers Hamiltonians H(t) of Eq. (1.10) do not capture the physics of the ionization process correctly for very large values of E0 , for the following reasons: (1) Non-relativistic kinematics for the electron is justified in our study of the ionization process only if the (mean) electron speed after ionization, |A(T )|, is small compared to the speed of light, c, (with c ≈ 137, in our units). If this condition is violated relativistic kinematics would have to be employed, and electron-positron pair creation by the laser pulse in the Coulomb field of the nucleus would have to be incorporated in our analysis, i.e., the whole process would have to be studied by using methods of relativistic QED. (2) The dipole approximation used in the Hamiltonians defined in Eqs. (1.7) and (1.10) can only be justified under the following conditions: (i) The wave length L and the spatial extension, T c, of the laser pulse in the propagation direction (here the z−axis) must be large, as compared to the spatial spread in the z−direction of the electron wave function at time t = T , which is of order O(T Z). It follows right away that Z 137, i.e., our analysis only applies to light atoms, such as Hydrogen or Helium, which, of course, was to be expected. Thus, we must impose that T Z L T c.
(1.27)
(ii) In order to justify neglecting the spatial dependence of the vector potential, A(x, t), of the laser pulse in the Pauli-Fierz Hamiltonian HP F (t) := (p − A((x, t))2 −
Z |x|
(1.28)
that should be used in our analysis, instead of the Kramers Hamiltonian, Eq. (1.10), the laser pulse must be spatially homogenous in the x− and the y−directions up to a distance d from the nucleus large compared to the mean distance of the electron from T the nucleus at time T , which is given by 2| 0 A(τ )dτ |. (iii) Finally, terms like |A(x, t)2 − A(0, t)2 | should be small in the tales of the electron wave function, ψt , for all times. These conditions are satisfied if Z 137 and if E0 is fairly small compared to Z; e.g., Z and E0 of order 1.
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Since our analytical methods only yield asymptotics in 1/E0 , we would be lucky if our results gave reliable information about the ionization process for E0 of order 1, (i.e., γ ≈ 1), corresponding to the experimental situation and needed to justify the dipole approximation. More precise quantitative information can presumably only be obtained from extensive numerical simulations. Yet, it is gratifying to note that our results are in good qualitative agreement with the experimental findings. Moreover, our analysis, which is based on the Kramers Hamiltonian in Eq. (1.10), suggests that naive calculations of “barrier traversal times” based on an adiabatic approximation to the Ritz Hamiltonians, Eq. (1.7), may not yield reliable results.
2. Description of the Theoretical Setup We consider an electron bound to a nucleus by a static potential V (x) and under the influence of a laser pulse described, in the Coulomb gauge, by the time dependent vector potential A(t), which we assume to be independent of x. The Hamiltonian is given by H(t) = (p − A(t))2 + V (x) and acts on the Hilbert space L2 (R3 ). Here p = −i∇ is the momentum operator. We denote by U(t, s) the propagator generated by the time-dependent Hamiltonian H(t), that is i∂t U(t, s) = H(t)U(t, s),
with
U(s, s) = 1
for all s ∈ R.
(2.1)
2.1. The Pulse We consider a pulse with amplitude λ lasting for a time T > 0. We will be interested here in fixed T and large λ. The electric component of the pulse is given by E(t) =
λ f (t/T ) T
for a vector valued function f : R → R3 , with supp f ⊂ [0, 1]. (In Sec. 1, we have used the notation E0 λ/T ). The vector potential A(t) is then given by t A(t) =
ds E(s) = λF (t/T ) −∞
with s F (s) =
dτ f (τ ). −∞
By definition F (s) = 0, for all s < 0, and F (s) = F (1), for all s ≥ 1.
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The time integral of the vector potential will also play an important role in our analysis. We set s G(s) = dτ F (τ ). −∞
Then t A(s)ds = λT G(t/T ). −∞
By definition G(s) = 0, for all s < 0, and G(s) = G(1) + (s − 1)F (1), for all s > 1. Assumptions on pulse. We assume that |G(s)|−1 ∈ L1 ((s0 , 1)),
for all 0 < s0 < 1.
(2.2)
Moreover, we assume that F (1) = 0
(2.3)
and that |G(s)| ≥ Cs,
for all s ≥ 1.
(2.4)
Assuming that F (1) = 0, this last condition is satisfied if F (1) · G(1) ≥ 0; in other words, if the angle between F (1) and G(1) is less or equal to π. In fact, for arbitrary s ≥ 1, |G(s)|2 = |G(1) + (s − 1)F (1)|2 = |G(1)|2 + (s − 1)2 |F (1)|2 + 2(s − 1)G(1) · F (1) ≥ |G(1)|2 + (s − 1)2 |F (1)|2 ≥
min(|G(1)|2 , |F (1)|2 ) 2 s . 2
Examples. A simple example of a pulse satisfying the assumptions (2.3), (2.4) is obtained by setting f (s) = ε 1(0 ≤ s ≤ 1) for a fixed polarization vector ε ∈ R3 (pulse with linear polarization). Then F (s) = 0, for s ≤ 0, F (s) = ε s, for s ∈ [0, 1], and F (s) = ε for s ≥ 1. This gives G(s) = 0 for s ≤ 0, G(s) = (s2 /2)ε for s ∈ [0, 1], G(s) = (s − 1/2)ε for s ≥ 1. Another example is a pulse with modulated circular polarization. If the polarization is perpendicular to the z-axis, such a pulse is described by f (s) = h(s)(cos(ω(s − 1/2)), sin(ω(s − 1/2)), 0) where h(s) ≥ 0 is symmetric decreasing about s = 1/2, with supp h ⊂ [0, 1]. If the effect of the pulse does not average out to zero, it is simple to check that, in this case, too, the conditions (2.3) and (2.4) are satisfied; see Sect. 1.
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2.2. The Potential To describe the coupling of the electron to the nucleus, we consider a static potential V (x). We distinguish two sets of assumptions on the potential V . Short range potential. We assume that there is a constant V0 (with [V0 ] = length−1 ), a length scale D > 0, and an α > 0 such that |V (x)| ≤
1 V0 D . |x| (1 + (x/D)2 )α/2
(2.5)
From the physical point of view, it is important to also cover an attractive Coulomb potential. Coulomb potential. V (x) = −
Z , |x|
Z > 0.
(2.6)
2.3. The Initial Wave Function We require exponential decay of the wave function ψ and of its first and second derivatives. In other words, we assume that |ψ(x)| ≤ CR−3/2 e−|x|/R ,
|∇ψ(x)| ≤ CR−5/2 e−|x|/R ,
|Δψ(x)| ≤ CR−5/2 e−|x|/R (R−1 + |x|−1 )
(2.7)
for some R > 0 and some dimensionless constants C. Moreover, we will also need decay in momentum space. We assume that
|ψ(p)| ≤
CR3/2 (1 + (Rp)2 )γ/2
(2.8)
for a dimensionless constant C, and for some γ > 3/2. 2.4. The Observable For fixed δ, θ > 0, we are interested in the probability that the electron is ejected in the direction G(t/T ) of the pulse (with G(t/T ) → F (1), as t/T → ∞). To this end, we propose to estimate the norm N (t) = χδ,θ (t)U(t, 0)ψ , where the propagator U(t, 0) is defined in (2.1), and χδ,θ (t) = 1(|x| ≥ δt) 1(x · G(t/T ) ≥ |x||G(t/T )| cos θ) for some fixed positive δ, θ, with θ > 0 arbitrarily small. We will prove that if the dimensionless quantity Rλ is sufficiently large the norm N (t) can be made arbitrarily close to one. Note that our results are uniform in time t. In particular, they hold in the limit of large t/T . We observe that, for large t/T , the direction of G(t/T ) approaches the direction of F (1); in other words, the vector F (1) determines the direction in which the electron propagates asymptotically, after ionization, in the limit of large Rλ.
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3. Results and Proofs 3.1. Short Range Potentials We begin our analysis by considering an interaction potential decaying faster than Coulomb. That is, we assume, in this subsection, that V satisfies condition (2.5), for some α > 0. Notation. Throughout the paper, C will denote a universal constant, independent of the parameters λ, T, R, D, V0 characterizing the pulse, the initial wave function, and the interaction potential. Remark. Note that, with our conventions, [T ] = [D] = [R] = length, [V0 ] = [λ] = length−1 , and Z = Ze2 is dimensionless. We have chosen the numerical value mel = 1/2 for the electron mass. Therefore, in the formulae below, t stands for t/2mel , T stands for T /2mel , δ stands for 2δmel , V0 stands for 2V0 mel , and (in Section 3.2) Z stands for 2Zmel . Theorem 3.1. Assume that conditions (2.2), (2.3), (2.4), (2.5) for some α > 0, (2.7), (2.8) for some γ > 5/2, are satisfied. Then we have that, uniformly in t ≥ T, 1 1 R2 +
χδ,θ (t)U(t, 0)ψ ≥ 1 − C 1+ R(Cλ − δ) Rλ tan θ t R4 R4 C V0 T 1 + 2 − CV0 DR 1 + 2 κλ − α(λT /D)1+α T T where the dimensionless quantity κλ is given by ⎫ ⎧ 1 ⎨T dτ 1 1 ⎬ s0 + . κλ = inf 1+ 2 0<s0 <1 ⎩ R2 Rλ |G(τ )| τ ⎭
(3.1)
s0
Remark. It follows from assumption (2.2) that κλ → 0, as Rλ → ∞. In fact, it 1 follows from (2.2) that the function K(s0 ) = s0 dτ |G(τ )|−1 (1 + τ −2 ) is finite, for all s0 > 0. Clearly, K(s0 ) is monotonically decreasing in s0 and can therefore be inverted. Typically K(s0 ) → ∞, as s0 → 0. Thus, for Rλ large enough, we can choose s0 = K −1 ((Rλ)1/2 ). Then s0 → 0 and K(s0 )(λR)−1 → 0, as Rλ → ∞. To have more precise information about how fast κλ tends to zero, as Rλ → ∞, one needs more information about the pulse. Example. If f (s) = ε, for a fixed ε ∈ R3 , for all s ∈ [0, 1], and f (s) = 0 for s ∈ [0, 1], it follows that F (s) = sε and G(s) = (s2 /2)ε, for all s ∈ [0, 1]. Then we find that ⎫ ⎧ ⎬ 1 ⎨T dτ 2 1 κλ = inf s0 + 1+ 2 Rλ τ2 τ ⎭ s0 ∈(0,1) ⎩ R2 s0 T 2 2 ≤ inf s + + . 0 Rλs0 3Rλs30 s0 ∈(0,1) R2
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It is easy to check that the infimum is attained at s20 = (R/T λ)(1 + 1 + 2T λ/R). For Rλ 1 (and R2 /T 1), the infimum is attained at t0 (2T R5 /λ)1/4 and is given by 1/4 4 2T 3 . κλ 3 R7 λ Remark. It follows from Theorem 3.1, that, as t → ∞, the electron will propagate, with probability approaching one, as Rλ → ∞, into the cone with an opening angle smaller than an arbitrary θ > 0 around the direction of F (1). In other words, with χ δ,θ (t) = 1(|x| ≥ δt) 1(x · F (1) ≥ |x||F (1)| cos θ), we find lim inf χδ,θ (t)U(t, 0)ψ
t→∞ 1 1 ≥1−C + R(Cλ − δ) Rλ tan θ R4 R4 CV0 T 1 + 2 − CV0 DR 1 + 2 κλ . − (λT /D)1+α T T
(3.2)
To prove (3.2), observe that 1(x · G(t/T ) ≥ |x||G(t/T )| cos θ)ψ ≥ 1(x · F (1) ≥ |x||F (1)| cos(θ/2))ψ , if the angle between G(t/T ) and F (1) is smaller than θ/2. Since G(t/T ) = G(1) + (t/T − 1)F (1), the angle between G(t/T ) and F (1) is certainly smaller than θ/2, for sufficiently large t/T 1. To prove Theorem 3.1, we first show how the evolution up to time T can be approximated by the evolution generated by the time dependent Kramers Hamiltonian without potential. The next lemma is due to Fring, Kostrykin and Schrader; see [9]. t
Lemma 3.2. Let U0 (t, s) = e−i s dτ (p−A(τ )) . Assume that conditions (2.3), (2.4), (2.7), (2.8), and (2.5), for some α ≥ 0, are satisfied; (for α = 0 we recover the Coulomb potential (2.6) by taking V0 D = Z). Then there exists a constant C such that R4
(U(T, 0) − U0 (T, 0)) ψ ≤ CV0 DR 1 + 2 κλ T 2
with κλ defined in (3.1). Proof. We define the new propagator 0) = e−2ip· U(s,
s 0
dτ A(τ ) i
e
s 0
dτ A2 (τ )
U(s, 0).
Then i
d U(s, 0) U(s, 0) = H(s) ds
with H(s) = p2 + V (x − 2λT G(s/T )).
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Since U(T, 0) − U0 (T, 0) = e2iλT p·G(1) e−i
T 0
dτ A2 (τ )
2 U(T, 0) − e−iT p
we find that 2 0) − e−iT p ψ
(U(T, 0) − U0 (T, 0)) ψ = U(T, T ≤
2 ds V (x − 2λT G(s/T ))e−isp ψ . (3.3)
0
Now, we observe that, on one hand, by (2.5), V0 D −isp2
V (x − 2λT G(s/T ))e . ψ ≤ 2V0 D dx|∇ψ(x)|2 ≤ C R
(3.4)
On the other hand, by Lemma 3.3 (see (3.12) below), we have that −isp2
V (x − 2λT G(s/T ))e
CV0 D R4 ψ ≤ 1+ 2 . λT |G(s/T )| s
(We are neglecting here the factor (1 + λ2 T 2 |G(s/T )|2 /D2 )−α/2 on the r.h.s. of (3.12). This factor will play an important role for large times; here it would just give a faster decay in λ.) Thus t0 (U(T, 0) − U0 (T, 0)) ψ ≤ 0
CV0 D ds + R
T t0
CV0 D R4 ds 1+ 2 , λT |G(s/T )| s
(3.5)
for arbitrary t0 ∈ [0, T ], and hence R4
(U(T, 0) − U0 (T, 0)) ψ ≤ CV0 D 1 + 2 T ⎧ ⎫ ⎪ ⎪ 1 ⎨t 1 dτ 1 ⎬ 0 + × inf . 1+ 2 0
Proof of Theorem 3.1. We begin by writing χδ,θ (t)U(t, 0)ψ = χδ,θ (t)U(t, T )U(T, 0)ψ T 2 = χδ,θ (t)U(t, T ) U(T, 0) − e−i 0 dτ (p−A(τ )) ψ + χδ,θ (t)U(t, T )e−i
T 0
dτ (p−A(τ ))2
ψ.
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Therefore, by Lemma 3.2,
χδ,θ (t)U(t, 0)ψ ≥ χδ,θ (t)U(t, T )e−i − (U(T, 0) − e−i
T 0
T 0
dτ (p−A(τ ))2 dτ (p−A(τ ))
T
2
ψ
)ψ
≥ χδ,θ (t)U(t, T )e−i 0 dτ (p−A(τ )) ψ
R4 − CV0 DR 1 + 2 κλ , T 2
with κλ defined in (3.1). Since A(t) = A(T ), for all t > T , we obtain that
χδ,θ (t)U(t, 0)ψ ≥ χδ,θ (t)e−i(t−T )[(p−A(T )) +V (x)] e−i R4 −CV0 DR 1 + 2 κλ . T 2
T 0
dτ (p−A(τ ))2
ψ
(3.6)
Next, we notice that χδ,θ (t)e−i(t−T )[(p−A(T ))
2
+V (x)] −i 0T dτ (p−A(τ ))2
−i 0t dτ (p−A(τ ))2
e
ψ
ψ ≥ χδ,θ (t)e T 2 2 −i(t−T )[(p−A(T ))2 +V (x)] − e−i(t−T )(p−A(T )) e−i 0 dτ (p−A(τ )) ψ − e ≥ χδ,θ (t)e−i
t 0
dτ (p−A(τ ))2
t−T
ψ −
T +s 2 ds V (x)e−i 0 dτ (p−A(τ )) ψ .
(3.7)
0
We then use that t−T
T +s 2 ds V (x)e−i 0 dτ (p−A(τ )) ψ
0 t−T
=
2 ds V (x)e−i(T +s)p e2iλT p·G(1+s/T ) ψ
0 t−T
=
2 ds V (x − 2λT G(1 + s/T ))e−i(T +s)p ψ .
(3.8)
0
To bound the integrand, we observe that, by Lemma 3.3 (see (3.12) below), 2 V (x − 2λT G(1 + s/T ))e−i(T +s)p ψ ≤
CV0 R4 1 + (λT |G(1 + s/T )|/D)1+α T2
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for all s ≥ 0. Hence, by (2.4), t−T
2 ds V (x − 2λT G(1 + s/T ))e−i(T +s)p ψ
0
t−T CV0 1 R4 ≤ ds 1 + 1+α 2 (λT /D) T |G(1 + s/T )|1+α 0
.
∞ 4
CV0 T 1 R dτ 1+ 2 1+α (λT /D) T (1 + τ )1+α 0 CV0 T R4 ≤ 1+ 2 . α(λT /D)1+α T ≤
Therefore, from (3.7),
χδ,θ (t)e−i(t−T )[(p−A(T )) ≥ χδ,θ (t)e−i
t 0
2
T
+V (x)] −i
dτ (p−A(τ ))2
e
ψ −
0
dτ (p−A(τ ))2
CV0 T α(λT /D)1+α
ψ
R4 1+ 2 . T
(3.9)
The first term on the right hand side of (3.9) can be bounded by
χδ,θ (t)e−i
t 0
dτ (p−A(τ ))2
ψ
≥ 1 − 1(|x| ≤ δt)e2iλT p·G(t/T ) e−itp ψ
2
− 1(x · G(t/T ) ≤ |x||G(t/T )| cos θ)e2iλT p·G(t/T ) e−itp ψ
2
≥ 1 − 1(|x − 2λT G(t/T )| ≤ δt)e−itp ψ
− 1((x − 2λT G(t/T )) · G(t/T ) ≤ |x − 2λT G(t/T )||G(t/T )| cos θ) 2
×e−itp ψ . 2
(3.10)
Since, by (2.4), |G(s)| ≥ Cs for all s ≥ 1, we find that
χδ,θ (t)e−i
t 0
dτ (p−A(τ ))2
ψ ≥ 1 − 1(|x| ≥ (Cλ − δ)t)e−itp ψ
2
− 1(|x| ≥ Cλt tan θ)e−itp ψ . (3.11) 2
To conclude, we observe that 2 2 1 e−itp ψ, x2 e−itp ψ 2 (Kt) 1 ≤ ψ, (x + 2tp)2 ψ (Kt)2 2 ≤ ψ, (x2 + 4t2 p2 )ψ (Kt)2 C C R4 2 2 −2 (R + t R ) ≤ ≤ 1 + (Kt)2 (KR)2 t2
1(|x| ≥ Kt)e−itp ψ 2 ≤ 2
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using (2.7), and (2.8) for some γ > 5/2. Hence, (3.11) yields 1 1 R2 −i 0t dτ (p−A(τ ))2 +
χδ,θ (t)e ψ ≥ 1 − C 1+ . R(Cλ − δ) Rλ tan θ t
Together with (3.6) and (3.9), this concludes the proof of the theorem.
Lemma 3.3. Assume (2.5), for some α > 0, and (2.7), (2.8), for some γ > 3/2. Then
V (x − 2λT G(t/T ))e−itp ψ
1 CV0 D R4 ≤ 1+ 2 . 2T 2 2 α/2 λT |G(t/T )| (1 + λD t 2 |G(t/T )| ) 2
Proof. We notice that eix /4t (4πit)3/2 2
(e−itp ψ)(x) = 2
dye−iy·x/2t eiy
2
/4t
(3.12)
ψ(y)
eix /4t ψ(x/2t) (4πit)3/2 2 2 eix /4t dye−iy·x/2t eiy /4t − 1 ψ(y). + 3/2 (4πit) 2
=
In the second term, we perform integration by parts to obtain decay in the x-variable. 2 dye−iy·x/2t eiy /4t − 1 ψ(y) Δy e−iy·x/2t iy2 /4t e − 1 ψ(y) = − dy (x/2t)2 1 dye−iy·x/2t =− (x/2t)2 2 2 y × (Δψ)(y) eiy /4t − 1 + i(∇ψ)(y) · eiy /4t 2t y2 3i iy2 /4t +ψ(y) − 2 + . e 4t 2t Therefore, we obtain that dye−iy·x/2t eiy2 /4t − 1 ψ(y) 3 1 |y| |y|2 |y|2 ≤ + |∇ψ(y)| + |ψ(y)| + 2 dy |Δψ(y)| (x/2t)2 4t t 2t 4t 3/2 2 1 R R 1+ ≤C . 2 t (x/t) t Since, on the other hand, 2 dye−iy·x/2t eiy2 /4t − 1 ψ(y) ≤ CR3/2 R , t
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it follows that 2 1 R2 dye−iy·x/2t eiy2 /4t − 1 ψ(y) ≤ CR3/2 R 1 + . t 1 + (Rx/t)2 t Hence, using (2.5) and (2.8), 2 2 V (x − 2λT G(t/T ))e−itp ψ 2
|ψ(x/2t)| ≤ C dxV 2 (x − 2λT G(t/T )) t3 4 2 2 R 1 dx 2 R +CR3 2 1 + V (x − 2λT G(t/T )) 3 t t t (1 + (Rx/t)2 )2 2
|ψ(x)| CV02 D2 dx ≤ t2 |x − λ(T /t)G(t/T )|2 (1 + 4t2 D−2 |x − λ(T /t)G(t/T )|2 )α 2 2 2 R4 CV D R2 + 02 R3 2 1 + t t t (1 + (Rx)2 )−2 dx × 2 2 −2 |x − λ(T /t)G(T /t)| (1 + 4t D |x − λ(T /t)G(T /t)|2 )α 2 CV02 D2 R2 R4 R2 ≤ 1+ 2 1+ t2 t t dx (1 + x2 )−β × |x − 2Rλ(T /t)G(t/T )|2 (1 + 4t2 R−2 D−2 |x − 2Rλ(T /t)G(t/T )|2 )α where β = min(γ, 2) > 3/2. It follows that 2 2 V (x − 2λT G(t/T ))e−itp ψ ≤
CV02 D2 2 λ T 2 |G(t/T )|2 (1 +
2 1 R4 . 1 + λ2 T 2 2 α t2 D 2 |G(t/T )| )
3.2. Coulomb Potentials In this section we consider the physically more interesting case of a Coulomb interaction. The long range of the Coulomb potential requires some modification of the argument used in the previous section; in particular, to obtain results uniform in time, we need to approximate the long time evolution by a “Dollard-modified” free dynamics (see [10]). As initial data we consider here the ground state of the Schr¨ odinger operator with an attractive Coulomb interaction, which satisfies the assumptions (2.7), and (2.8), with γ = 4. (In the following theorem we will therefore assume (2.8) with γ = 4; but, of course, other values of γ can also be considered.) Theorem 3.4. Assume that conditions (2.3), (2.4), (2.6), and (2.8), for γ = 4, are satisfied. Suppose that there exists a constant C such that C −1 ≤ R2 /T ≤
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C, that Z ≤ λ, and that λR ≥ 1 is large enough. Then we have that, uniformly in t ≥ T ,
1 1 R2 +
χδ,θ (t)U(t, 0)ψ ≥ 1 − C 1+ Rλ tan θ R(Cλ − δ) t 4/7 ZT 3/2 C R4 − ZR 1 + 2 κλ − T R2 (Rλ)1/7 where the dimensionless quantity κλ was defined in (3.1). Since, by assumption (2.2), κλ → 0, as (λR) → ∞, it follows in particular that lim χδ,θ (t)U(t, 0)ψ = 1
λR→∞
uniformly in t ≥ T . Remark. Just like Theorem 3.1, Theorem 3.4 implies that
1 1 lim inf + χδ,θ (t)U(t, 0)ψ ≥ 1 − C t→∞ Rλ tan θ R(Cλ − δ) 4/7 ZT 3/2 C R4 − ZR 1 + 2 κλ − T R2 (Rλ)1/7 where χ δ,θ (t) = 1(|x| ≥ tδ)1(x·F (1) ≥ |x||F (1)| cos θ). In other words, it is the vector F (1) that determines, with probability approaching one, as Rλ → ∞, the direction in which the electron propagates asymptotically. Proof. By Lemma 3.2, we have that
χδ,θ (t)U(t, 0)ψ ≥ χδ,θ (t)U(t, T )e−i
T 0
dτ (p−A(τ ))2
R4 ψ − CZR 1 + 2 κλ . T (3.13)
To replace the unitary evolution U(t, T ) by a free evolution, we introduce, first of all, a cutoff in momentum space. We choose a smooth function χ ∈ C0∞ (R3 ), with χ(x) = 0 for all |x| ≥ 1 and χ(x) = 1 for all |x| ≤ 1/2. We define χ ¯ = 1 − χ. Then we have
χδ,θ (t)U(t, T )e−i
T 0
dτ (p−A(τ ))2 −i
≥ χδ,θ (t) U(t, T )e
≥ χδ,θ (t) U(t, T )e−i
T 0
T 0
ψ
dτ (p−A(τ ))2 dτ (p−A(τ ))
2
χ(p/K0 )ψ − χ(p/K ¯ 0 )ψ
C χ(p/K0 )ψ − (3.14) (RK0 )γ−3/2
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for arbitrary K0 > 0; we will later optimize the choice of K0 . Next, we let 2 ψT = e−iT p χ(p/K0 )ψ, and we observe that χδ,θ (t)U(t, T )e−i
T 0
ds(p−A(s))2
χ(p/K0 )ψ
−i(t−T )[(p−A(T ))2 −Z/|x|] 2iλT p·G(1) −i
= χδ,θ (t)e
= eix·A(T ) χδ,θ −i
=e
T 0
T
2
e e 0 dsA (s) ψT T 2 2 (t)e−i(t−T )[p −Z/|x|] e−ix·A(T ) e2iλT p·G(1) e−i 0 ds A (s) ψ
dsA2 (s) 2iλT G(1)·A(T ) ix·A(T )
e
e
χδ,θ (t)e
−i(t−T )[p2 −Z/|x−2λT G(1)|] −ix·A(T )
×e
T
2iλT p·G(1)
e
ψT
(3.15)
and we write e−i(t−T )[p
2
−Z/|x−2λT G(1)|] −ix·A(T ) iZ
t−T
e
ψT
dτ |2τ p−2λT G(1)|
= e−i(t−T )p e 0 e−ix·A(T ) ψT t−T 2 2 + e−i(t−T )[p −Z/|x−2λT G(1)|] − e−i(t−T )p eiZ 0 2
dτ |2τ p−2λT G(1)|
×e−ix·A(T ) ψT .
(3.16)
Observe here that ψT = χ(p/K0 )e−iT p ψ is supported, in momentum space, in the ball of radius K0 around the origin. This implies that |p+λF (1)| ≤ K0 for all p in the support of the Fourier transform of e−ix·A(T ) ψT . Therefore |2τ p − 2λT G(1)| ≥ 2λT |G(1 + τ /T )| − 2τ K0 ≥ CλT + (Cλ − K0 )τ for all τ ∈ [0, t − T ]. In particular, if we require that K0 ≤ Cλ/2, the inte t−T dτ |2τ p − 2λT G(1)|−1 is well defined (at the end, we will choose gral 0 K0 R (λR)2/35 , and therefore the condition K0 ≤ Cλ/2 is certainly satisfied for sufficiently large values of (λR)). It follows that 2
χδ,θ (t)U(t, T )ei
T 0
ds(p−A(s))2
χ(p/K0 )ψ
t−T
≥ χδ,θ (t)e2iλT p·G(1) e−i(t−T )p eiZ 0 |2τ p−2λT G(1)| e−ix·A(T ) ψT
t−T dτ 2 2 − e−i(t−T )[p −Z/|x−2λT G(1)|] − e−i(t−T )p eiZ 0 |2τ p−2λT G(1)| (3.17) ×e−ix·A(T ) ψT . 2
dτ
To bound the first term, we observe that
χδ,θ (t)e2iλT p·G(1) e−i(t−T )p eiZ 2
t−T 0
dτ |2τ p−2λT G(1)|
e−ix·A(T ) ψT
= χδ,θ (t)e2iλT (p−A(T ))·G(1) e−i(t−T )(p−A(T )) eiZ 2
iZ
= χδ,θ (t)e2iλT p·G(t/T ) e−itp e 2
t−T 0
t−T 0
dτ |2τ p−2λT G(1+τ /T )|
dτ |2τ (p−A(T ))−2λT G(1)|
χ(p/K0 )ψ .
ψT
(3.18)
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From (3.18), we obtain
χδ,θ (t)e2iλT p·G(1) e−i(t−T )p eiZ 2
t−T
dτ |2τ p−2λT G(1)|
0
e−ix·A(T ) ψT
= 1((x − 2λT G(t/T )) · G(t/T ) ≥ |x − 2λT G(t/T )||G(t/T )| cos θ) × 1(|x − 2λT G(t/T )| ≥ δt)e−itp eiZ 2
t−T
≥ χ(p/K0 )ψ
− 1(|x − 2λT G(t/T )| ≤ δt)e−itp eiZ 2
0
dτ |2τ p−2λT G(1+τ /T )|
t−T 0
χ(p/K0 )ψ
dτ |2τ p−2λT G(1+τ /T )|
χ(p/K0 )ψ
− 1((x − 2λT G(t/T )) · G(t/T ) ≤ |x − 2λG(t/T )||G(t/T )| cos θ) × e−itp eiZ 2
t−T 0
dτ |2τ p−2λT G(1+τ /T )|
χ(p/K0 )ψ
−5/2
≥ 1 − (RK0 )
− 1(|x| ≥ (Cλ − δ)t)e−itp eiZ 2
iZ
− 1(|x| ≥ Cλt tan θ)e−itp e 2
t−T 0
t−T 0
dτ |2τ p−2λT G(1+τ /T )| dτ |2τ p−2λT G(1+τ /T )|
χ(p/K0 )ψ
χ(p/K0 )ψ .
From Lemma 3.5, below, we find that t−T
χδ,θ (t)e2iλT p·G(1) e−i(t−T )p eiZ 0 |2τ p−2λT G(1)| e−ix·A(T ) ψT
1 1 R2 −5/2 ≥ 1 − (RK0 ) + −C 1+ . (3.19) Rλ tan θ R(Cλ − δ) t 2
dτ
As for the second term on the r.h.s. of (3.17), we use the bound t−T 2 −i(t−T )[p2 −Z/|x−2λT G(1)|] − e−i(t−T )p eiZ 0 e
dτ |2τ p−2λT G(1)|
1 1 − ds |x − 2λT G(1)| |2sp − 2λT G(1)| 0 s dτ 2 ×e−isp ei 0 |2τ p−2λT G(1)| e−ix·A(T ) ψT .
e−ix·A(T ) ψT
t−T
≤Z
(3.20)
We first handle small values of s ∈ [0, t − T ]. To this end, we observe that 2 s dτ 1 −isp2 iZ 0 |2τ p−2λT −ix·A(T ) G(1)| e e ψT |x − 2λT G(1)| e 2 s 1 −ix·A(T ) −is(p−A(T ))2 iZ 0 |2τ p−2λTdτ G(1+τ /T )| e = e e ψT |x − 2λT G(1)| 2 dx −is(p−A(T ))2 iZ 0s |2τ p−2λTdτG(1+τ /T )| ≤ e ψT (x) e 2 |x − 2λT G(1)| 2
≤4 dp|p|2 |ψ(p)| ≤ CR−2 |p|≤K0
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using (2.8), with γ = 4. On the other hand, we have that s dτ 1 −isp2 iZ 0 |2τ p−2λT −ix·A(T ) G(1)| e e ψT |2sp − 2λT G(1)| e 1 = |2sp − 2λT G(1 + s/T )| χ(p/K0 )ψ 1 1 ≤ ≤ 2λT |G(1 + s/T )| − sK0 CλT for all s ∈ [0, t], if K0 < Cλ/2; here we used the assumption (2.4). Therefore s dτ 1 1 −isp2 iZ 0 |2τ p−2λT −ix·A(T ) G(1)| e e ψT |x − 2λG(T )| − |2sp − 2λG(T )| e 1 1 ≤C + ≤ CR−1 R λT assuming λT ≥ R. In conclusion t−T
0
1 1 − ds |x − 2λT G(1)| |2sp − 2λT G(1)| s dτ 2 ×e−isp eiZ 0 |2τ p−2λT G(1)| e−ix·A(T ) ψT t−T
Ct0 ≤ + R
ds
t0
−isp2 iZ
×e
e
s 0
1 1 − |x − 2λT G(1)| |2sp − 2λT G(1)| dτ |2τ p−2λT G(1)| e−ix·A(T ) ψ T
(3.21)
for an arbitrary t0 > 0. To estimate the second term, we use the kernel representation 2 1 −isp2 ψ)(x) = (e dyei(x−y) /4s ψ(y) (4πis)3/2 implying that s 2 1 e−isp eiZ 0 |x − 2λT G(1)|
dτ |τ p−2λT G(1)|
eix /4s iZ 0s e (4πis)3/2 |x − 2λT G(1)|
e−ix·A(T ) ψT (x)
2
=
dτ |τ x/s−2λT G(1)|
ψ T (x/2s + λF (1))
(1)
+ Rλ (s, x), with eix /4s = (4πis)3/2 |x − 2λT G(1)| 2 s × dye−iy·x/2s eiy /4s −1 eiZ 0 2
(1) Rλ (s, x)
dτ |2τ p−2λT G(1)|
e−ix·A(T ) ψT (y).
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Similarly, we notice that s 2 1 e−isp eiZ 0 |2sp − 2λT G(1)|
dτ |2τ p−2λG(T )|
1 eix /4s iZ 0s e = (4πis)3/2 |x − 2λT G(1)|
e−ix·A(T ) ψT (x)
2
dτ |τ x/s−2λT G(1)|
ψ T (x/2s + λF (1))
(2)
+ Rλ (s, x), with (2)
Rλ (s, x) =
2 2 eix /4s −iy·x/2s iy /4s e − 1 dy e (4πis)3/2 dτ 1 iZ 0s |2τ p−2λT −ix·A(T ) G(1)| e e ψT (y). × |2sp − 2λT G(1)|
From (3.20), we find that t−T
t0
1 1 − ds |x − 2λT G(1)| |sp − 2λT G(1)| s dτ 2 ×e−isp eiZ 0 |2τ p−2T λG(1)| e−ix·A(T ) ψT t−T
≤
(1) (2) ds Rλ (s, x) + Rλ (s, x) .
t0
To control the first remainder term, we compute s dτ eiZ 0 |2τ p−2λT G(1)| e−ix·A(T ) ψT (y) s dτ 1 !T (k + λF (1)) = dkeik·y eiZ 0 |2τ k−2λT G(1)| ψ 3/2 (2π) s e−iλF (1)·y ik·y iZ 0 |2τ k−2λTdτ !T (k). G(1+τ /T )| ψ = e dke (2π)3/2 Hence eix /4s 2 3/2 (8π is) |x − 2λT G(1)| 2 × dye−iy·(x/2s+λF (1)) (eiy /4s − 1)hλ (s, y) 2
(1)
Rλ (s, x) =
with
hλ (s, y) =
dkeik·y eiZ
s
dτ 0 |2τ k−2λT G(1+τ /T )|
!T (k). ψ
In Lemma 3.6, below, we show that, for every multi-index β ∈ N3 , 2n −3/2 |β| β K0 T K0 Z 2n Dx hλ (s, x) ≤ CR log (1 + s/T ) . 1 + 1 + (x/R)2n R λ
(3.22)
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Therefore, on the one hand, (1) |Rλ (s, x)|
C ≤ 5/2 s |x − 2λG(T )| ≤
CR7/2 5/2 s |x − 2λG(T )|
dy|y|2 |hλ (s, y)|
T K0 R
2n Z 1 + log2n (1 + s/T ) λ
(3.23)
for all n > 5/2. On the other hand, from eix /4s = (8π 2 is)3/2 |x − 2λT G(1)| −iy·(x/2s+λF (1)) Δm 2 y e × dy (eiy /4s − 1)hλ (s, y) m 2m (−1) |x/2s + λF (1)| 2
(1) Rλ (s, x)
we find by integrating by parts that (1)
|Rλ (s, x)| ≤
C s3/2 |x − 2λT G(1)||x/2s + λF (1)|2m " 2 × dy Dα (eiy /4s − 1) |Dβ hλ (s, y)|. |α|+|β|=2m
Using that |Dα (eiy
2
/4s
− 1)| ≤
C sr
1+
|y|2r sr
(3.24)
if |α| = 2r, r ≥ 1, and that |Dα (eiy
2
/4s
− 1)| ≤
C|y| sr
1+
|y|2(r−1) sr−1
(3.25)
if |α| = 2r − 1, r ≥ 1, we arrive at (1)
|Rλ (s, x)| ≤
2n K0 T CR−3/2 K02m Z 2n log (1 + s/T ) 1 + R λ s3/2 |x − 2λT G(1)||x/2s + λF (1)|2m 2 |y| 1 × dy s 1 + (|y|/R)2n m " 1 1 y 2r + dy 1 + r 2 r (K0 s) s 1 + (|y|/R)2n r=1 # m " 1 1 y 2(r−1) dy(K0 |y|) 1 + r−1 + , (K02 s)r s 1 + (|y|/R)2n r=1
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where the first term in the parenthesis corresponds to |α| = 0, the second to |α| = 2r and the third to |α| = 2r − 1. It follows that (1)
|Rλ (s, x)| ≤
2n K0 T CR3/2 K02m R s3/2 |x − 2λT G(1)||x/2s + λF (1)|2m Z × 1 + log2n (1 + s/T ) λ % $ 2 2r &# m 2 R R2 " 1 R + 1+ × (3.26) 2r−1 s s r=1 (RK0 ) s 2n K0 T CR7/2 K02m ≤ 5/2 R s |x − 2λT G(1)||x/2s + λF (1)|2m % 2 2m & R Z × 1 + log2n (1 + s/T ) 1 + λ s
for all n > m + 3/2, and all m ≥ 1. Combining this bound with (3.23), we find that 2n K0 T CR7/2 (1) |Rλ (s, x)| ≤ 5/2 R s |x − 2λT G(1)|(1 + (|x/2s + λF (1)|/K0 )2m ) % 2 2m & R Z × 1 + log2n (1 + s/T ) 1 + λ s for all n > m + 3/2, and all m ≥ 1. We thus conclude that % 2 2m & 2n R Z 2n 1 + log (1 + s/T ) 1 + λ s 1/2 dx × . (3.27) 2 |x − 2λT G(1)| (1 + (|x/2s + λF (1)|/K0 )2m )2
(1)
Rλ (s, x)
Since
CR7/2 ≤ 5/2 s
K0 T R
dx 1 2 |x − 2λT G(1)| (1 + (|x/2s + λF (1)|/K0 )2m )2 C(sK0 )3 dx = 2sK0 ≤ λ2 T 2 |G(1 + s/T )|2 |x − λT G(1+s/T ) |2 (1 + x2m )2 sK0
we find that (1)
Rλ (s, x)
2n 3/2 K0 T CR7/2 K0 ≤ λsT |G(1 + s/T )| R % 2 2m & R Z 2n × 1 + log (1 + s/T ) 1 + λ s
(3.28)
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for any m ≥ 1. Since |G(1 + s/2T )| ≥ C(1 + s/2T ), we find t−T
(1)
ds Rλ (s, x)
t0
2n 3/2 CR7/2 K0 K0 T ≤ λT R & ∞ % 2 2m R ds 1 Z 2n × 1 + log (1 + s) 1 + s(1 + s) λ T s2m t0 /T
C(K0 R)3/2 ≤ λ
K0 T R
2n
R2 t0
% 1+
R2 t0
2m &
Z 1+ λ
t0 T
(3.29)
ε
for any m ≥ 1 and n > m + 3/2, and any ε > 0. To control the second remainder term on the r.h.s. of (3.22), we write dτ 1 iZ 0s |2τ p−2λT −ix·A(T ) G(1)| e e ψT (y) |2sp − 2λT G(1)| s dτ dk 1 ik·y iZ 0 |2τ k−2λT !T (k + λF (1)) G(1)| ψ e e = |2sk − 2λT G(1)| (2π)3/2 s e−iλF (1)·y dk ik·y iZ 0 |2τ k−2λTdτ !T (k). G(1+τ /T )| ψ e = e |2sk − 2λT G(1 + s/T )| (2π)3/2 Hence eix /4s (8π 2 is)3/2 2
(2)
Rλ (s, x) = with
gλ (s, y) =
dy e−iy·(x/2s+λF (1)) (eiy
s dk eik·y eiZ 0 |2sk − 2λT G(1)|
2
/4s
− 1)gλ (s, y)
dτ |2τ k−2λT G(1+τ /T )|
!T (k). ψ
In Lemma 3.7, we show that for every multi-index β ∈ N3 , 2n |β| β C K0 T R−3/2 K0 Z 2n Dx gλ (s, x) ≤ 1 + log (1 + s/T ) . λ(T + s) 1 + (x/R)2n R λ Therefore, on the one hand C (2) |Rλ (s, x)| ≤ 5/2 dy|y|2 |gλ (s, y)| s 2n T K0 CR7/2 Z 2n ≤ 5/2 1 + log (1 + s/T ) (3.30) R λ λs (s + T ) for all n > 3/2. On the other hand, from 2 −iy·(x/2s+λF (1)) Δm 2 eix /4s y e (2) Rλ (s, x) = (eiy /4s − 1)gλ (s, y) dy m 2m 3/2 (−1) |x/2s + λF (1)| (8π2is)
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we find by integrating by parts that (2)
|Rλ (s, x)| ≤
C s3/2 |x/2s + λF (1)|2m " 2 × dy Dα (eiy /4s − 1) |Dβ gλ (s, y)|. |α|+|β|=2m
Using the bounds (3.24), (3.25), we obtain, similarly to (3.26), the bound (2)
|Rλ (s, x)| ≤
2n K0 T CR7/2 K02m R λs5/2 (s + T )|x/2s + λF (1)|2m % 2 2m & R Z × 1 + log2n (1 + s/T ) 1 + λ s
for all n > m + 3/2, and all m ≥ 1. Combining the last bound with (3.30), we conclude that 2n K0 T CR7/2 (2) |Rλ (s, x)| ≤ R λ s5/2 (s + T )(1 + (|x/2s + λF (1)|/K0 )2m ) % 2 2m & R Z × 1 + log2n (1 + s/T ) 1 + . λ s Hence we have (2)
Rλ (s, x) ≤
2n K0 T CR7/2 Z 2n log (1 + s/T ) 1 + R λ λs5/2 (s + T ) % 2 2m & 1/2 R dx × 1+ s (1 + (|x/2s + λF (1)|/K0 )2m )2 2n 3/2 K0 T CR7/2 K0 Z ≤ 1 + log2n (1 + s/T ) λs(T + s) R λ % 2 2m & R (3.31) × 1+ s
for all m ≥ 1. Similarly to (3.29), this implies that t−T
(2)
ds Rλ (s, x) ≤ t0
2n 2 R C(K0 R)3/2 K0 T λ R t0 % 2 2m & ε R Z t0 × 1+ 1+ , t0 λ T
for any m ≥ 1, n > m + 3/2, and ε > 0.
(3.32)
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From (3.20), (3.21), (3.29), (3.32), we find that t−T dτ 2 −i(t−T )[p2 −Z/|x−2λT G(1)|] − e−i(t−T )p eiZ 0 |τ p−2λT G(1)| e−ix·A(T ) ψT e 2 2m & 2n 2 % ε R K0 T CZt0 Z R Z t0 3/2 ≤ + C (K0 R) 1+ 1+ R λ R t0 t0 λ T for all m ≥ 1, n > m + 3/2, and ε > 0. We now choose m = 1 and n = 3, and we set % ' (6 &1/4 (K0 R)3/2 KR0 T t0 = . R2 Rλ We will choose K0 so that K0 R and K0 T /R are large in the limit of large (Rλ) so that we may assume that t0 /R2 ≤ 1, (t0 /T )ε ≤ λ/(Z). Then t−T dτ 2 −i(t−T )[p2 −Z/|x−2λT G(1)|] − e−i(t−T )p eiZ 0 |τ p−2λT G(1)| e−ix·A(T ) ψT e % ' (6 &1/4 (K0 R)3/2 KR0 T ≤ CZR . Rλ This last bound, together with (3.19), implies that T 2 C R2
χδ,θ (t)U(t, T )e−i 0 dτ (p−A(τ )) ψ ≥ 1 − 1+ − C(RK0 )−5/2 Rλ t % ' (6 &1/4 (K0 R)3/2 KR0 T − CZR . Rλ We finally optimize our choice of K0 by setting (K0 T /R) = C0 (Rλ)2/35 , for an appropriate constant C0 . This yields C R2 −i 0T dτ (p−A(τ ))2 ψ ≥ 1 −
χδ,θ (t)U(t, T )e 1+ Rλ t 4/7 ZT 3/2 C , − R2 (Rλ)1/7 which, combined with (3.13), concludes the proof of the theorem.
Lemma 3.5. Suppose that χ ∈ C0∞ (R3 ), with χ(x) = 0 for all |x| ≥ 1 and χ(x) = 1 for all |x| ≤ 1/2. Assume that C −1 ≤ R2 /T ≤ C, Z ≤ λ, K0 ≤ Cλ for an appropriate constant C, and that λR ≥ 1 is large enough (at the end (K0 R) (λR)2/35 , and therefore the condition K0 ≤ Cλ is satisfied for sufficiently large (Rλ)). Then, for every t ≥ T , and for every constant D > 0, we have that t−T dτ C R2 −itp2 iZ 0 |2τ p−2λT G(1+τ /T )|
1(|x| ≥ Dt)e e χ(p/K0 )ψ ≤ 1+ . DR t (3.33)
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J. Fr¨ ohlich, A. Pizzo and B. Schlein
Proof. We notice that 2 iZ t−T 1(|x| ≥ Dt) e−itp e 0
dτ |2τ p−2λT G(1+τ /T )|
where W 2 (t) :=e−itp eiZ 2
t−T 0
x2 e−itp eiZ 2
0
2 χ(p/K0 )ψ ≤
dτ |2τ p−2λT G(1+τ /T )|
t−T
Ann. Henri Poincar´e
1 W 2 (t) (Dt)2 (3.34)
χ(p/K0 )ψ,
dτ |2τ p−2λT G(1+τ /T )|
χ(p/K0 )ψ.
Next we compute
) t−T 2 d 2 W (t) = e−itp eiZ 0 dt
dτ |2τ p−2λT G(1+τ /T )|
χ(p/K0 )ψ, Z 2 2 ,x i p + |2(t − T )p − 2λT G(t/T )| * t−T dτ 2 ×e−itp eiZ 0 |2τ p−2λT G(1+τ /T )| χ(p/K0 )ψ ) t−T dτ 2 = 2Im e−itp eiZ 0 |2τ p−2λT G(1+τ /T )| χ(p/K0 )ψ, 2(t − T )p − 2λT G(t/T ) x · 2p + 2Z(t − T ) |2(t − T )p − 2λT G(t/T )|3 * dτ 2 iZ t−T × e−itp e 0 |τ p−2λT G(1+τ /2T )| χ(p/K0 )ψ
which, using |2(t − T )p − 2λT G(t/T )| ≥ (Cλ − K0 )t, implies that d 2 W (t) ≤ CW (t) Z + |p|ψ ≤ CR−1 W (t) dt tλ2
(3.35)
for all t > T (because λR ≥ 1, and Z/λ ≤ 1, λT /R ≥ 1). By Gronwall’s Lemma we find that W (t) ≤ C(t − T )R−1 + W (T )
(3.36)
where W 2 (T ) = e−iT p χ(p/K0 )ψ, x2 e−iT p χ(p/K0 )ψ. 2
2
Similarly to (3.35), we find that d 2 ≤ 2W (T ) |p|ψ ≤ CR−1 W (T ) W (T ) dT which implies that
' ( W (T ) ≤ CT R−1 + W (0) ≤ C T R−1 + R
and thus, combining the last equation with (3.34) and (3.36), we obtain (3.33). Lemma 3.6. Let hλ (s, x) =
dkeik·y eiZ
s
dτ 0 |2τ k−2λT G(1+τ /T )|
!T (k) ψ
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with ψ T (k) = e−iT k χ(k/K0 )ψ, with χ ∈ C0∞ (R3 ) such that χ(y) = 1 for |y| ≤ 1/2 and χ(y) = 0 for |y| ≥ 1. Assume that R−1 + RT −1 ≤ K0 ≤ Cλ for an appropriate constant C (at the end, we will choose K0 R (Rλ)2/35 , and therefore these conditions are satisfied for large enough λR). Assume also Z ≤ λ. Then, for every β ∈ N3 , we have 2n −3/2 |β| β K0 T K0 Z 2n Dx hλ (s, x) ≤ CR log (1 + s/T ) . 1 + 1 + (x/R)2n R λ 2
Proof. We have Dxβ hλ (s, x)
dk(ik)β eik·x eiZ
=
Hence
s
dτ 0 |2τ k−2λT G(1+τ /T )|
ψ T (k).
(3.37)
dk |k|β χ(k/K0 )|ψ(k)| 1 ≤ R−3/2−|β| dk|k|β χ(k/RK0 ) (1 + k 2 )2 1 ≤ CR−3/2−|β| (1 + |k|)4−|β|
|Dxβ hλ (s, x)| ≤
|k|≤2RK0
≤ CR
−3/2
|β| K0 .
(3.38)
Integrating by parts in (3.37), we arrive at Δnk eik·x iZ 0s |2τ k−2λTdτG(1+τ /T )| β e ψT (k) Dx hλ (s, x) = dk(ik)β (−1)n |x|2n s eik·x n β iZ 0 |2τ k−2λTdτ
T (k) G(1+τ /T )| ψ (ik) = dk Δ e (−1)n |x|2n and therefore |Dxβ hλ (s, x)| C ≤ |x|2n
"
dk|k||β|−|α1 |
|α1 |+|α2 |+|α3 |=2n s dτ
α2 iZ
×|D e
0 |2τ k−2λT G(1+τ /T )|
||Dα3 ψ T (k)|.
(3.39)
Observe that, for all |α2 | ≥ 1, |Dα2 eiZ
s
dτ 0 |2τ k−2λT G(1+τ /T )|
|≤C
|α2 | "
"
m=1 j1 ,..,jm ≥1:j1 +..+jm =|α2 | m + i=1
s Z 0
dτ τ ji . |2τ k − 2λT G(1 + τ /T )|ji +1
Using the fact that |k| ≤ K0 on the support of ψ T , we find |2τ k − 2λT G(1 + τ /T )| ≥ 2λT |G(1 + τ /T )| − 2K0 τ ≥ CλT + (Cλ − K0 )τ . Therefore, assuming
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Ann. Henri Poincar´e
that K0 < Cλ/2, and that Z < λ, |Dα2 eiZ
s
dτ 0 |2τ k−2λT G(1+τ /T )|
|α2 |
"
≤C
| m +
"
m=1 j1 ,..,jm ≥1:j1 +..+jm =|α2 |
Z
λj1 +1 i=1
m C " Z log(1 + s/t) ≤ |α | λ λ 2 m=1
s
dτ T +τ
0
|α2 |
≤C
Z (1 + log|α2 | (1 + s/T )). λ|α2 |+1
(3.40)
On the other hand, by a simple computation, we have |Dα3 ψ T (k)| ≤ Cχ(k/K0 ) % ×
R3/2+|α3 | (1 + (Rk)2 )2
T 1/2 1 1 (1 + |k|T 1/2 ) + + R RK0 1 + (Rk)2
≤ Cχ(k/K0 )
&|α3 |
R3/2 (K0 T )|α3 | (1 + (Rk)2 )2
(3.41)
assuming that K0 R ≥ 1 and K0 T /R ≥ 1. From (3.39), it follows that |x|2n |Dxβ hλ (s, x)| " ≤C
|α1 |+|α3 |=2n|k|≤K
"
+C
dk|k||β|−|α1 | (K0 T )|α3 | 0
|α1 |+|α2 |+|α3 |=2n,|α2 |≥1
Z (1 + log|α2 | (1 + s/T )) λ|α2 |+1
dk|k||β|−|α1 | (K0 T )|α3 |
× |k|≤K0
"
≤C
R3/2 (1 + (Rk)2 )2
R−3/2−|β|+|α1 | (K0 T )|α3 |
|α1 |+|α3 |=2n
+C ×
|k|≤RK0
Z (1 + log2n (1 + s/T )) λ " R−3/2−|β|+|α1 |
|α1 |+|α2 |+|α3 |=2n |α2 |≥1
R3/2 (1 + (Rk)2 )2
λ|α2 |
(K0 T )|α3 |
dk (1 + |k|)4−|β|+|α1 |
|k|≤RK0
dk (1 + |k|)4−|β|+|α1 |
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and therefore |x|2n |Dxβ hλ (s, x)| ≤ CR
"
−3/2+2n−|β|
|α1 |+|α3 |=2n
+C
ZR
|α3 |
(1 + (K0 R)−1+|β|−|α1 |+ε )
−3/2+2n−|β|
λ "
×
K0 T R
|α1 |+|α2 |+|α3 |=2n |α2 |≥1 |β|
≤ CR−3/2+2n K0
(1 + log2n (1 + s/T )) |α3 | K0 T 1 (1 + (K0 R)−1+|β|−|α1 |+ε ) R (Rλ)|α2 |
K0 T R
2n Z 1 + log2n (1 + s/T ) . λ
Combining this bound with (3.38), we find, for arbitrary n ≥ 0, 2n |β| K0 T CR−3/2 K0 Z 2n β |Dx hλ (s, x)| ≤ 1 + log (1 + s/T ) . 1 + (x/R)2n R λ Lemma 3.7. Let
gλ (s, x) =
s
ik·x
dke
eiZ 0 |2τ k−2λT G(1+τ /T )| ψT (k) |2sk − 2λT G(1 + s/T )| dτ
with ψ T (k) = e−iT k χ(k/K0 )ψ, with χ ∈ C0∞ (R3 ) such that χ(y) = 1 for |y| ≤ 1/2 and χ(y) = 0 for |y| ≥ 1. Assume that R−1 + RT −1 ≤ K0 ≤ Cλ for an appropriate constant C (at the end, we will choose K0 R (Rλ)2/35 , and therefore these conditions are satisfied for large enough λR). Assume also Z ≤ λ. Then, for every β ∈ N3 , we have 2n |β| β C K0 T R−3/2 K0 Z 2n Dx gλ (s, x) ≤ log (1 + s/T ) . 1 + λ(T + s) 1 + (x/R)2n R λ 2
Proof. We have Dxβ gλ (s, x)
=
s
β ik·x
dk(ik) e
eiZ 0 |2τ k−2λT G(1+τ /T )| ψT (k). |2sk − 2λT G(1 + s/T )| dτ
(3.42)
Since, for |k| ≤ K0 , we have |sk − 2λT G(1 + s/2T )| ≥ Cλ(T + s) − sK0 ≥ CλT + s(Cλ − K0 ) ≤ Cλ(T + s), we find C
|Dxβ gλ (s, x)| ≤ dk|k|β χ(k/K0 )|ψ(k)| λ(T + s) CR−3/2−|β| 1 ≤ dk|k|β χ(k/RK0 ) λ(T + s) (1 + k 2 )2 |β|
≤
CR−3/2 K0 . λ(T + s)
(3.43)
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Integrating by parts in (3.42), we arrive at s dτ n ik·x eiZ 0 |2τ k−2λT G(1+τ /T )| β β Δk e Dx gλ (s, x) = dk(ik) ψT (k) (−1)n |x|2n |2sk − 2λT G(1 + s/T )| iZ 0s |2τ k−2λTdτ G(1+τ /T )| eik·x n β e
ψT (k) = dk Δ (ik) (−1)n |x|2n |2sk − 2λT G(1 + s/T )| and therefore |Dxβ hλ (s, x)| ≤
C |x|2n ×
"
dk|k||β|−|α1 | |Dα2 eiZ
s
dτ 0 |2τ k−2λT G(1+τ /T )|
|
|α1 |+···+|α4 |=2n
×
s|α3 | |Dα3 ψ T (k)|. |2sk − 2λT G(1 + s/T )||α3 |+1
(3.44)
From (3.40), (3.41), and since |sk − λT G(1 + s/T )| ≥ Cλ(T + s), we find |Dxβ hλ (s, x)| ≤
CR3/2 |x|2n ×
"
|α1 |+···+|α4 |=2n |α2 |=0
+ ×
(K0 T )|α4 | λ|α3 |+1 (T + s)
dk |k|≤K0
CR3/2 Z (1 + log2n (1 + s/T )) |x|2n λ " (K0 T )|α4 | |α1 |+···+|α4 |=2n |α2 |≥1
λ|α2 |+|α3 |+1 (T + s)
|k||β|−|α1 | (1 + (Rk)2 )2
dk |k|≤K0
|k||β|−|α1 | (1 + (Rk)2 )2
2n |β| K0 T CR−3/2 K0 Z 2n ≤ 1 + log (1 + s/T ) λ(T + s)(|x|/R)2n λ R where we used K0 T /R ≥ 1, Rλ > 1, Z < λ. Combining the last equation with (3.43), we conclude the proof of the lemma.
Acknowledgments We thank Patrissa Eckle and Ursula Keller for explaining their experiments to us and encouraging us to carry out the analysis presented in this paper.
References [1] Eckle, P., Pfeiffer, A.N., Cirelli, C., Staudte, A., D¨ orner, R., Muller, H.G., B¨ uttiker, M., Keller, U.: Attosecond ionization and tunneling delay time measurements in helium. Science 322, 1525–1529 (2008)
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[2] Eckle, P., Smolarski, M., Schlup, Ph., Biegert, J., Staudte, A., Sch¨ offer, M., Muller, H.G., D¨ orner, R., Keller, U.: Attosecond angular streaking. Nature (Physics) 4, 565–570 (2008) [3] Bach, V., Klopp, F., Zenk, H.: Mathematical analysis of the photoelectric effect. Adv. Theor. Math. Phys. 5(6), 969–999 (2001) [4] Supporting material for [1] can be found at http://www.sciencemay.org/cgi/ content/full/322/5907/1525 [5] Keldysh, L.V.: Ionization in the field of a strong electromagnetic wave. Sov. Phys. JETP 20, 1307 (1965) [6] B¨ uttiker, M., Landauer, R.: Traversal time for tunneling. Phys. Rev. Lett. 49, 1739 (1982) [7] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. 1, p. 297. Academic Press, New York (1972) [8] Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. 3, Theorem X.70. Academic Press, New York (1972) [9] Fring, A., Kostrykin, V., Schrader, R.: Ionization probabilities through ultra-intense fields in the extreme limit. J. Phys. A 30(24), 8599–8610 (1997) [10] Dollard, J.: Asymptotic convergence and the Coulomb interaction. J. Math. Phys. 5, 729 (1964) J¨ urg Fr¨ ohlich Institute of Theoretical Physics ETH Z¨ urich 8093 Z¨ urich Switzerland e-mail:
[email protected] Alessandro Pizzo Department of Mathematics University of California Davis One Shields Avenue Davis CA 95616 USA e-mail:
[email protected] Benjamin Schlein Institute for Applied Mathematics University of Bonn Endenicher Allee 60 53115 Bonn Germany e-mail:
[email protected] Communicated by Claude Alain Pillet. Received: May 21, 2010. Accepted: September 30, 2010.
Ann. Henri Poincar´e 11 (2010), 1409–1452 c 2010 Springer Basel AG 1424-0637/10/081409-44 published online December 16, 2010 DOI 10.1007/s00023-010-0068-x
Annales Henri Poincar´ e
Anomalous Behavior in an Effective Model of Graphene with Coulomb Interactions Alessandro Giuliani, Vieri Mastropietro and Marcello Porta Abstract. We analyze by exact Renormalization Group (RG) methods the infrared properties of an effective model of graphene, in which twodimensional (2D) massless Dirac fermions propagating with a velocity smaller than the speed of light interact with a 3D quantum electromagnetic field. The fermionic correlation functions are written as series in the running coupling constants, with finite coefficients that admit explicit bounds at all orders. The implementation of Ward Identities in the RG scheme implies that the effective charges tend to a line of fixed points. At small momenta, the quasi-particle weight tends to zero and the effective Fermi velocity tends to a finite value. These limits are approached with a power law behavior characterized by non-universal critical exponents.
1. Introduction and Main Result The charge carriers in graphene at half filling are effectively described by massless Dirac fermions constrained to move on a two-dimensional (2D) manifold embedded in three-dimensional (3D) space [10], with a Fermi velocity v that is approximately 300 times smaller than the speed of light. As a consequence, already without taking into account the interactions, the system displays highly unusual features as compared to standard 2D electron gases, such as an anomalous integer quantum Hall effect and the insensitivity to disordered-induced localization; most of these effects have already been experimentally observed [30,31]. The study of many-body interactions among the charge carriers in graphene is of course very important, particularly in view of recent experiments that suggest their relevant role in several physical properties of graphene [9,25,27,37]. The effect of a weak short range interaction in graphene is quite well understood: it turns out that the behavior of the ground state is qualitatively similar to the free one, except that the Fermi velocity and the wave function
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renormalization are renormalized by a finite amount. This was expected on the basis of a power counting analysis [22–24]; recently, it has been rigorously proven in [18,19], where the convergence of the perturbative series was established, using the methods of constructive Quantum Field Theory (QFT) and by taking into full account the lattice effects (i.e., by considering the Hubbard model on the honeycomb lattice). The situation in the presence of long range interactions is much more subtle and still not completely understood. Their effect in graphene is usually studied in terms of a model of Dirac fermions interacting via a static Coulomb potential; retardation effects of the electromagnetic (e.m.) field are neglected because the free Fermi velocity v is 300 times smaller than the speed of light c. At weak coupling, a logarithmic divergence of the effective Fermi velocity v(k) at the Fermi points p± F and a finite quasi-particles weight have been predicted, on the basis of one-loop [21] and two-loop [29] computations. An unbounded growth of the effective Fermi velocity was also confirmed by an analysis based on a large-N expansion [26,36], which predicted a power law divergence of v(k) at p± F . It is not clear how to reconcile the logarithmic divergence expected from two-loop perturbative computations with the power law behavior found by large-N expansions; moreover, the description in terms of Dirac fermions introduces spurious ultraviolet divergences that can produce ambiguities in the physical predictions [24,29]. These difficulties may be related to a basic inadequacy of the effective model of Dirac fermions with static Coulomb interactions: the fact that the effective Fermi velocity diverges at the Fermi points (as predicted by all the analyses of the model) signals that its physical validity breaks down at the infrared scale where v(k) becomes comparable with the speed of light; at lower scales, retardation effects must be taken into account, as first proposed in [20], where a model of massless Dirac fermions propagating with speed v c and interacting with an e.m. field was considered. In [20] it was found that, at small momenta, the wave function renormalization diverges as a power law; this implies that the ground state correlations have an anomalous decay at large distances. Moreover, it was found that the interacting Fermi velocity increases up to the speed of light, again with an anomalous power law behavior. Despite its interest, the model proposed in [20] has not been considered further. The results in [20] were found on the basis of one-loop computations and in the presence of an ultraviolet dimensional regularization scheme. It is interesting to investigate whether the predictions of [20] remain valid even if higher orders corrections are taken into account and in the presence of different regularization schemes closer to the lattice cut-off that is truly present in actual graphene. The model we consider describes massless Dirac fermions in 2 + 1 dimensions propagating with velocity v < c, and interacting with a 3 + 1 dimensional photon field in the Feynman gauge. We will not be concerned with the instantaneous case (c → ∞); therefore, from now on, for notational simplicity, we shall fix units such that = c = 1. The model is very similar to the one in [20], the main difference being the choice of the ultraviolet cut-off: rather than considering dimensional regularization, in order to mimic the presence of an
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underlying lattice, we explicitly introduce a (fixed) ultraviolet momentum cutoff both in the electronic and photonic propagators. The correlations can be computed in terms of derivatives of the following Euclidean functional integral: (1.1) eW(J,φ) = P (dψ)P (dA)eV (A,ψ)+B(J,φ) with, setting x = (x0 , x) and x = (x1 , x2 ) (repeated indexes are summed; Greek and Latin labels run, respectively, from 0 to 2, 1 to 2), V (A, ψ) := dx [e jμ,x Aμ,x − νμ Aμ,x Aμ,x ] Λ
B(J, φ) :=
dx jμ,x Jμ,x + φx ψ x + φx ψx ,
(1.2)
Λ
where Λ is a 3D box of volume |Λ| = L3 with periodic boundary conditions (playing the role of an infrared cutoff, to be eventually removed), the couplings e, νμ are real and ν1 = ν2 ; the couplings νμ are counterterms to be fixed so that the photon mass is vanishing in the deep infrared. Moreover, ψ x , ψx are four-component Grassmann spinors, and the μ-th component jμ,x of the current is defined as: j0,x = iψ x γ0 ψx ,
jx = iv ψ xγ ψx ,
(1.3)
where γμ are euclidean gamma matrices, satisfying the anticommutation relations {γμ , γν } = −2δμ,ν . The symbol P (dψ) denotes a Grassmann integration with propagator dk ikx ik0 γ0 + ivk · γ e χ0 (k). (1.4) g (≤0) (x) := (2π)3 k02 + v 2 |k|2 where (2π)−3 dk is a shorthand for |Λ|−1 k=2πn/L with n ∈ Z3 , and χ0 (k) = χ(|k|) plays the role of a prefixed ultraviolet cutoff (here χ(t) is a non-increasing C ∞ function from R+ to [0, 1] such that χ(t) = 1 if t ≤ 1 and χ(t) = 0 if t ≥ M > 1). Finally, Aμ,x are gaussian variables and P (dA) is a gaussian integration with propagator dp dp3 ipx χ0 (p) dp ipx χ0 (p) = w(≤0) (x) := e e . (1.5) 3 (2π) 2|p| (2π)4 p2 + p23 We perform an analysis based on the methods of constructive Renormalization Group (RG) for non-relativistic fermions, introduced in [3,14] (see [4,28,33,34] for updated introductions), which have already been proved effective in the study of several low-dimensional critical systems, such as one-dimensional (1D) interacting fermions [3,5,6], 2D critical Ising and vertex models [2,17], the 2D Hubbard model on the square lattice at positive temperatures [7,11,12], interacting fermions with asymmetric Fermi surfaces [13] and the 2D Hubbard model on the honeycomb lattice [18,19], just to mention a few. Compared to other RG approaches, such as those in [32,35], the advantage of the constructive methods we adopt is that they allow us to get a rigorous and complete
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treatment of the effects of the cut-offs and a full control on the perturbative expansion via explicit bounds at all orders; quite remarkably, in certain cases, such as the ones treated in [6,11,12,7,18,19], these methods even provide a way to prove the convergence of the resummed perturbation theory. Using these methods, we construct a renormalized expansion, allowing us to express the Schwinger functions, from which the physical observables can be computed, as series in the effective couplings (the effective charges and the effective photon masses, also called in the following the running coupling constants), with finite coefficients at all orders, admitting explicit N ! bounds (see Theorem 2.1 in Sect. 2 below). If the effective couplings remain small in the infrared, informations obtained from our expansion by lowest order truncations are reliable at weak coupling. The importance of having an expansion with finite coefficients should not be underestimated; the naive perturbative expansion in the fine structure constant is plagued by logarithmic infrared divergences and higher orders are more and more divergent. Of course the renormalized expansion is useful only as long as the running coupling constants are small. In fact, we do prove that they remain small for all infrared scales, by implementing Ward Identities (WIs) in the RG flow, using a technique developed in [6] for the rigorous analysis of Luttinger liquids in situations where bosonization cannot be applied (e.g., in the presence of an underlying lattice and/or of non-linear bands). The WIs that we use are based on an approximate local gauge invariance, the exact gauge symmetry being broken by the ultraviolet cut-off; its presence produces corrections to the “naive” (formal) WIs, which can be resummed and, again, explicitly bounded at all orders. The resulting modified WIs imply that the effective charges tend to a line of fixed points, exactly as in 1D Luttinger liquids. We note that this is one of the very few examples in which Luttinger liquid behavior is found in dimensions higher than 1. Let us denote by . . . = lim|Λ|→∞ . . . Λ the expectation value with respect to the interaction (1.2) in the infinite volume limit; our main result can be informally stated as follows (more rigorous statements will be found below). Main result. There exists a choice of νμ such that, for k small, ψk ψ k =
1 ik0 γ0 + iv(k)k · γ (1 + B(k)), Z(k) k02 + v(k)2 |k|2
(1.6)
where Z(k) ∼ |k|−η ,
veff − v(k) ∼ (veff − v)|k|η˜,
(1.7)
B(k), νμ , η, η˜, veff are expressed by series in the effective couplings with finite coefficients that admit N !-bounds at all orders. Moreover: (i) the first nontrivial contribution to B(k) is of second order in e; (ii) the first non-trivial contribution to νμ is of second order in e and positive; (iii) the first non-trivial contributions to η, η˜, veff are, respectively:
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η (2) = with 5 F (v) = 8
e2 , 12π 2
η˜(2) =
2e2 , 5π 2
(2)
veff = 1 − F (v)
ξ0 − arctan ξ0 1 1 arctan ξ0 −2 + 2 , 2v 2 ξ03 2v ξ0
e2 , 6π 2 √
ξ0 :=
1413
(1.8)
1 − v2 . (1.9) v
Note that the theory is not Lorentz invariant, because v = c; moreover, gauge symmetry is broken by the presence of the ultraviolet momentum cutoff. These two facts produce unusual features as compared to standard QFT models. In particular, the momentum cut-off produces correction terms in the WIs, which can be rigorously bounded at all orders; despite these corrections, one can still use the WIs to prove that the beta function for the effective charges is asymptotically vanishing at all orders, so that the model admits a line of (non-trivial) fixed points. The lack of gauge invariance due to the ultraviolet momentum cut-off makes it necessary (as in [8]) to introduce positive counterterms to keep the photon mass equal to zero. Similarly, it implies that the effective couplings with the temporal and spatial components of the gauge field are different and that the effective Fermi velocity veff is not equal to the speed of light. However, it is possible to introduce in the bare interaction two different charges, e0 and e1 , describing the couplings of the photon field with the temporal and spatial components of the current, which can be tuned so that the dressed charges are equal and veff = 1. A more realistic model for single layer graphene could be obtained by considering tight binding electrons hopping on the honeycomb lattice, whose lattice currents are coupled to a 3D photon field. A Renormalization Group analysis similar to the one in the present paper could be repeated for the lattice model, by extending the formalism in [18,19]. If the lattice model is chosen in such a way that lattice gauge invariance is preserved, we expect that its photon mass counterterms are exactly zero and that its effective Fermi velocity is equal to the speed of light. In any case (i.e., both in the presence or in the absence of lattice gauge invariance), we expect the lattice model to have the same infrared asymptotic behavior of the continuum model considered here, provided that the bare parameters eμ , νμ of the continuum model are properly tuned. Finally, let us comment about the possibility of providing a full non-perturbative construction of the ground state of the present model or, possibly, of a more realistic model of tight binding electrons hopping on the honeycomb lattice and interacting with e.m. forces. In this paper, we express the physical observables in terms of series in the running coupling constants (with bounded coefficients at all orders) and we show that the running coupling constants remain close to their initial value, thanks to Ward Identities and cancellations in the beta function. Thus, the usual problem that so far prevented the non-perturbative construction of the ground state of systems of interacting fermions in d > 1 with convex symmetric Fermi surface (namely,
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the presence of a beta function driving the infrared flow for the effective couplings out of the weak coupling regime) is absent in the present case. Therefore, a full non-perturbative construction of the ground state of the present model appears to be feasible, using determinant bounds for the fermionic sector and cluster expansion techniques for the bosonic sector. Of course, the construction is expected to be much more difficult than the one in [6] or [18,19], due to the simultaneous presence of bosons and fermions; if one succeeded in providing it, it would represent the first rigorous example of anomalous Luttinger-liquid behavior in more than one dimension. The paper is organized as follows: in Sect. 2 we describe how to evaluate the functional integrals defining the partition function and the correlations of our model in terms of an exact RG scheme (details are discussed in Appendices A and B); in Sect. 3 we describe the infrared flow of the effective couplings and prove the emergence of an effective Fermi velocity different from the speed of light (the explicit lowest order computations of the beta function are presented in Appendix C); in Sect. 4 we derive the Ward Identity allowing us to control the flow of the effective charges (details are discussed in Appendix D) and proving that the beta function for the charges is asymptotically vanishing; finally, in Sect. 5 we draw the conclusions.
2. Renormalization Group Analysis 2.1. The Effective Potential In this section we show how to evaluate the functional integral (1.1); the integration will be performed in an iterative way, starting from the momenta “close” to the ultraviolet cutoff moving towards smaller momentum scales. At the n-th step of the iteration the functional integral (1.1) is rewritten as an integral involving only the momenta smaller than a certain value, proportional to M −n , where M > 1 is the same constant (to be chosen sufficiently close to 1) appearing in the definition of the cut-off function (see lines after (1.4)), and both the propagators and the interaction will be replaced by “effective” ones; they differ from their “bare” counterparts because the physical parameters appearing in their definitions (the Fermi velocity v, the charge e, and the “photon mass” νμ ) are renormalized by the integration of the momenta on higher scales. In the following, it will be convenient to introduce the scale label h ≤ 0 as h := −n. Setting χh (k) := χ(M −h |k|), we start from the following identity: χ0 (k) =
0 h=−∞
fh (k),
fh (k) := χh (k) − χh−1 (k);
(2.1)
0 0 let ψ = h=−∞ ψ (h) and A = h=−∞ A(h) , where {ψ (h) }h≤0 , {A(h) }h≤0 are independent free fields with the same support of the functions fh introduced above. We evaluate the functional integral (1.1) by integrating the fields in an iterative way starting from ψ (0) , A(0) ; for simplicity, we start by treating the
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case J = φ = 0. We define V (0) (A, ψ) := V (A, ψ) and we want to inductively prove that after the integration of ψ (0) , A(0) , . . . , ψ (h+1) , A(h+1) we can rewrite: (h) (≤h) √ , Zh ψ (≤h) ) , (2.2) eW(0,0) = e|Λ|Eh P (dψ (≤h) )P (dA(≤h) )eV (A where P (dψ (≤h) ) and P (dA(≤h) ) have propagators g (≤h) (k) =
vh (k)k · γ χh (k) iγ0 k0 + i˜ , Z˜h (k) k02 + v˜h (k)2 |k|2
V (h) has the form V (h) (A, ψ) =
n,m≥0 ρ,μ n+m≥1
×
m
w(≤h) (p) =
χh (p) , 2|p|
(2.3)
⎤ ⎡ m 2n n
dki
dpj ⎦ ⎣ ψ ψk ,ρ , (2π)3 (2π)3 i=1 k2i−1 ,ρ2i−1 2i 2i i=1 j=1 ⎛
(h) Aμi ,pi Wm,n,ρ,μ ({ki }, {pj })δ ⎝
i=1
m
pj +
j=1
2n
⎞ (−1)i ki ⎠,
i=1
(2.4) (h) and Eh , Z˜h (k), v˜h (k) and the kernels Wm,n,ρ,μ will be defined recursively. In order to inductively prove (2.2), we split V (h) as LV (h) + RV (h) , where R = 1 − L and L, the localization operator, is a linear operator on functions of (h) the form (2.4), defined by its action on the kernels Wm,n,ρ,μ in the following way: (h)
(h)
(h)
LW0,1,ρ (k) := W0,1,ρ (0) + k∂k W0,1,ρ (0), (h)
(h)
LW1,1,ρ,μ (p, k) := W1,1,ρ,μ (0, 0), (h)
(h)
(h)
LW2,0,μ (p) := W2,0,μ (0) + p∂p W2,0,μ (0), (h)
(2.5)
(h)
LW3,0,μ (p1 , p2 ) := W3,0,μ (0, 0), (h)
and LWP := 0 otherwise. As it will be clear from the dimensional analysis performed in Sect. 2.4 below, these are the only terms that need renormaliza(h) tion; in particular, LW0,1,ρ (k) will contribute to the wave function renormal(h)
ization and to the effective Fermi velocity, LW2,0,μ (p) to the effective photon (h)
mass, and LW1,1,ρ,μ (p, k) to the effective charge. As a consequence of the symmetries of our model, see Appendix A, it turns out that (h)
W1,0,μ (0) = 0,
(h)
W3,0,μ (0, 0) = 0,
ˆ (h) (0) = −δμ ,μ M h νμ ,h , W 1 2 1 2,0,μ
(h)
W0,1,ρ (0) = 0, (h)
ˆ ∂p W 2,0,μ (0) = 0
(2.6)
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and, moreover, that (h)
ψ k k∂k W0,1 (0)ψk = −izμ,h kμ ψ k γμ ψk
(2.7)
(h)
ψ k+p W1,1,μ (0, 0)ψk Aμ,p = iλμ,h ψ k+p γμ ψk Aμ,p ,
with zμ,h , λμ,h real, and z1,h = z2,h , λ1,h = λ2,h . We can renormalize P (dψ (≤h) ) by adding to the exponent of its gaussian weight the local part of the quadratic terms in the fermionic fields; we get that √ (h) P (dψ (≤h) )P (dA(≤h) )eV (A, Zh ψ) √ (h) |Λ|th P(dψ (≤h) )P (dA(≤h) )eV (A, Zh ψ) , =e (2.8) where th takes into account the different normalization of the two functional (h) is given by integrals, V dk (h) (h) V (A, ψ) = V (A, ψ) + izμ,h kμ ψ k γμ ψk (2π)3 =: V (h) (A, ψ) − Lψ V (h) (A, ψ),
(2.9)
and P(dψ (≤h) ) has propagator equal to g˜(≤h) (k) =
vh−1 (k)k · γ χh (k) iγ0 k0 + i˜ , Z˜h−1 (k) k 2 + v˜h−1 (k)2 |k|2
(2.10)
0
with Z˜h−1 (k) = Z˜h (k) + Zh z0,h χh (k), Z˜h−1 (k)˜ vh−1 (k) = Z˜h (k)˜ vh (k) + Zh z1,h χh (k).
(2.11)
After this, defining Zh−1 := Z˜h−1 (0), we rescale the fermionic field so that (h) (A, Zh ψ) = Vˆ (h) (A, Zh−1 ψ); V (2.12) therefore, setting vh−1 := v˜h−1 (0),
e0,h :=
Zh λ0,h , Zh−1
e1,h vh−1 = e2,h vh−1 :=
Zh λ1,h , Zh−1 (2.13)
we have that:
LVˆ (h) (A(≤h) , Zh−1 ψ (≤h) ) (≤h) (≤h) (≤h) , = dx Zh−1 eμ,h jμ,x Aμ,x − M h νμ,h A(≤h) μ,x Aμ,x
(2.14)
Λ
where (≤h)
(≤h)
j0,x := iψ x
γ0 ψx(≤h) ,
jx(≤h) := ivh−1 ψ (≤h) γ ψx(≤h) . x
(2.15)
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After this rescaling, we can rewrite (2.8) as √ (h) P (dψ (≤h) )P (dA(≤h) )eV (A, Zh ψ) = e|Λ|th P (dψ (≤h−1) )P (dA(≤h−1) ) √ ˆ (h) (≤h−1) +A(h) , Zh−1 (ψ (≤h−1) +ψ (h) )) × P (dψ (h) )P (dA(h) )eV (A , (2.16) where ψ (≤h−1) , A(≤h−1) have propagators given by (2.3) (with h replaced by h − 1) and ψ (h) , A(h) have propagators given by vh−1 (k)k · γ g (h) (k) f˜h (k) iγ0 k0 + i˜ = , 2 Zh−1 Zh−1 k0 + v˜h−1 (k)2 |k|2 fh (k) = χh (k) − χh−1 (k),
f˜h (k) =
w(h) (p) =
fh (p) , 2|p|
Zh−1 fh (k). Z˜h−1 (k)
At this point, we can integrate the scale h and, defining √ (h−1) ˜ eV (A, Zh−1 ψ)+|Λ|Eh √ (h) (h) ˆ (h) := P (dψ (h) )P (dA(h) )eV (A+A , Zh−1 (ψ+ψ )) ,
(2.17)
(2.18)
our inductive assumption (2.2) is reproduced at the scale h − 1 with Eh−1 := ˜h . Notice that (2.18) can be seen as a recursion relation for the Eh + th + E effective potential, since from (2.9), (2.12) it follows that (h) (A, Zh ψ) Vˆ (h) (A, Zh−1 ψ) = V (2.19) = V (h) (A, Zh ψ) − Lψ V (h) (A, Zh ψ). The integration in (2.18) is performed by expanding in series the exponential in the r.h.s. (which involves interactions of any order in ψ and A, as apparent from (2.4)), and integrating term by term with respect to the gaussian integration P (dψ (h) )P (dA(h) ). This procedure gives rise to an expansion for the effective potentials V (h) (and to an analogous expansion for the correlations) in terms of the renormalized parameters {eμ,k , νμ,k , Zk−1 , vk−1 }h
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2.2. Tree Expansion The iterative integration procedure described above leads to a representation of the effective potentials in terms of a sum over connected Feynman diagrams, as explained in the following. The key formula, which we start from, is (2.18), which can be rewritten as ˜h + V (h−1) (A(≤h−1) , Zh−1 ψ (≤h−1) ) |Λ|E 1 EhT Vˆ (h) (A(≤h) , Zh−1 ψ (≤h) ); n , (2.20) = n! n≥1
with
EhT
the truncated expectation on scale h, defined as ∂n T (h) (h) (h) (h) λX(A(h) ,ψ (h) ) Eh (X(A , ψ ); n) := log P (dψ )P (dA )e ∂λn λ=0 (2.21)
If X is graphically represented as a vertex with external lines A(h) and ψ (h) , the truncated expectation (2.21) can be represented as the sum over the Feynman diagrams obtained by contracting in all possible connected ways the lines exiting from n vertices of type X. Every contraction corresponds to a propagator on scale h, as defined in (2.17). Since Vˆ (h) is related to V (h) by a rescaling and a subtraction, see (2.9) and (2.12), Eq. (2.20) can be iterated until scale 0, and V (h−1) can be written as a sum over connected Feynman diagrams with lines on all possible scales between h and 0. The iteration of (2.20) induces a natural hierarchical organization of the scale labels of every Feynman diagram, which will be conveniently represented in termsof tree diagrams. In fact, let us rewrite Vˆ (h) in the r.h.s. of (2.20) as Vˆ (h) (A, Zh−1 ψ) = √ √ LV (h) (A, Zh ψ) + RV (h) (A, Zh ψ), where L := L − Lψ , see (2.9). Let us graphically represent V (h) , LV (h) and RV (h) as in the first line of Fig. 1, and let us represent Eq. (2.20) as in the second line of Fig. 1; in the second line, the node on scale h represents the action of EhT . Iterating the graphical equation in Fig. 1 up to scale 0, we end up with a representation of V (h) in terms of a sum over Gallavotti–Nicol` o trees τ [3,15,16]: (h) (≤h) , Zh ψ (≤h) ) = V (h) (τ ), (2.22) V (A N ≥1 τ ∈Th,N
_ V(h) =
V(h−1) =
,
h
h−1
h
V(h) =
+
,
h
h−1
h
V(h) =
+
h−1
h
h
+
h−1
h
Figure 1. Graphical interpretation of Eq. (2.20). The graphical equations for LV (h−1) , RV (h−1) are obtained from the equation in the second line by putting an L, R label, respectively, over the vertices on scale h
+ ...
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v V (h − 1) =
v0 trees
h −1
h
hv
1
Figure 2. The effective potential V (h−1) can be represented as a sum over Gallavotti–Nicol` o trees. The black dots will be called vertices of the tree. All the vertices except the first (i.e. the one on scale h) have an R label attached, which means that they correspond to the action of REhTv , while the first represents EhT . The endpoints correspond to the graph elements in Fig. 3 associated to the two terms in (2.14) e)
Figure 3. The two possible graph elements associated to the endpoints of a tree, corresponding to the two terms in the r.h.s. of (2.14) where Th,N is the set of rooted trees with root r on scale hr = h and N endpoints, see Fig. 2. The tree value V (h) (τ ) can be evaluated in terms of a sum over connected Feynman diagrams, defined by the following rules. With each endpoint v of τ we associate a graph element of type e or ν, corresponding to the two terms in the r.h.s. of (2.14), see Fig. 3. We introduce a field label f to distinguish the fields associated to the graph elements e and ν (any field label can be either of type A or of type ψ); the set of field labels associated with the endpoint v will be called Iv . Analogously, if v is not an endpoint, we call Iv the set of field labels associated with the endpoints following the vertex v on τ . We start by looking at the graph elements corresponding to endpoints of scale 1: we group them in clusters, each cluster Gv being the set of endpoints attached to the same vertex v of scale 0, to be graphically represented by a box enclosing its elements. For any Gv of scale 0 (associated to a vertex v of scale 0 that is not an endpoint), we contract in pairs some of the fields in ∪w∈Gv Iw , in such a way that after the contraction the elements of Gv are connected; each
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v0 0 −2
−1
0
−1
1
Figure 4. A possible Feynman diagram contributing to V (−2) and its cluster structure
contraction produces a propagator g (0) or w(0) , depending on whether the two fields are of type ψ or of type A. We denote by Iv the set of contracted fields inside the box Gv and by Pv = Iv \Iv the set of external fields of Gv ; if v is not the vertex immediately following the root we attach a label R over the box Gv , which means that the R operator, defined after (2.4), acts on the value of the graph contained in Gv . Next, we group together the scale 0 clusters into scale-(−1) clusters, each scale-(−1) cluster Gv being a set of scale 0 clusters attached to the same vertex v of scale −1, to be graphically represented by a box enclosing its elements, see Fig. 4. Again, for each v of scale −1 that is not an endpoint, if we denote by v1 , . . . , vsv the vertices immediately following v on τ , we contract some of the v Pvi in pairs, in such a way that after the contraction the boxes fields of ∪si=1 associated to the scale 0 clusters contained in Gv are connected; each contraction produces a propagator g (−1) or w(−1) . We denote by Iv the set of fields v v Pvi contracted at this second step and by Pv = ∪si=1 Pvi \Iv the set of in ∪si=1 fields external to Gv ; if v is not the vertex immediately following the root we attach a label R over the box Gv (Table 1). Now, we iterate the construction, producing a sequence of boxes into boxes, hierarchically arranged with the same partial ordering as the tree τ . Each box Gv is associated to many different Feynman (sub-)diagrams, constructed by contracting in pairs some of the lines external to Gvi , with vi , i = 1, . . . , sv , the vertices immediately following v on τ ; the contractions are made in such a way that the clusters Gv1 , . . . , Gvsv are connected through propagators of scale hv . We denote by PvA and by Pvψ the set of fields of type A and ψ, respectively, external to Gv . The set of connected Feynman diagrams compatible with this hierarchical cluster structure will be denoted by Γ(τ ). Given these definitions, we can write: dkf dpf V (h) (τ ) = Val(G), (2π)3 (2π)3 A ψ G∈Γ(τ ) f ∈P f ∈Pv0 v0 ⎡ ⎤⎡ ⎤
(≤h)
(≤h) Val(G) = ⎣ Aμ(f ), pf ⎦ ⎣ Zh−1 ψkf ,ρ(f ) ⎦ δ(v0 ) Val(G), f ∈PvA
0
f ∈Pvψ0
(2.23)
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Table 1. List of the symbols introduced in Sects. 2.2, 2.4 Symbol
Description
τ r v0 hv Th,N Iv Gv Iv Pv vi sv Pv# Γ(τ )
Gallavotti–Nicol` o (GN) tree Root label of the tree First vertex of the tree, immediately following the root Scale label of the tree vertex v Set of GN trees with root on scale hr = h and with N endpoints Set of field labels associated with the endpoint of the tree v Cluster associated with the tree vertex v Set of contracted fields inside the box corresponding to the cluster Gv Set of external fields of Gv i-th vertex immediately following v on the tree Number of vertices immediately following the vertex v on the tree Set of fields of type # = A, ψ external to Gv Set of connected Feynman diagrams compatible with the hierarchical cluster structure of the tree τ Number of propagators contained in Gv but not in any smaller cluster Number of end-points of type ν immediately following v on the tree Vertex immediately preceding v on the tree Number of vertices of type # = e, ν following v on the tree Improvement on the scaling dimension due to the renormalization
n0v mνv v n# v zv
Val(G) = (−1)
π
v not e.p.
⎛ ⎡ ⎜ ⎢ ⎢ (hv ) ⎜ ⎜ ⎢ g ×⎢ ⎜ ⎝ ⎣ ∈v
Zhv −1 Zhv −2
|P2vψ |
v ∗ e.p. v ∗ >v, hv∗ =hv +1
Rαv sv ! ⎞⎤ ⎟⎥
(h ) ⎟⎥ ⎥ Kv ∗ v ⎟ ⎟⎥
⎠⎦
where (−1)π is the sign of the permutation necessary to bring the contracted ψ fermionic fields next to each other; in the product over f ∈ P v , ψ can be either ψ or ψ, depending on the specific field label f ; δ(v0 ) = δ( f ∈PvA pf − 0 ε(f ) is equal to ψ or k ), where ε(f ) = ± depending on whether ψ ψ (−1) f f ∈Pv0 ψ; the integral in the third line runs over the independent loop momenta; sv is the number of vertices immediately following v on τ ; R = 1 − L is the operator defined in (2.5) and preceding lines); αv = 0 if v = v0 , and otherwise αv = 1; (k) g is equal to g (k) or to w(k) depending on the fermionic or bosonic nature of the line , and ∈ v means that is contained in the box Gv but not in any (k) other smaller box; finally, Kv∗ is the matrix associated to the endpoints v ∗ on scale k + 1 (given by ie0,k γ0 if v ∗ is of type (a) with label ρ = 0, by iej,k vk γj if v ∗ is of type e with label ρ = j ∈ {1, 2}, or by −M k νμ,k if v ∗ is of type ν. In (2.23) it is understood that the operators R act in the order induced by the tree ordering (i.e., starting from the endpoints and moving toward the root); (k) moreover, the matrix structure of g is neglected, for simplicity of notations.
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e v1 e
v0
e k
h−1
e h+1
h
k
h+1 h
h+2
Figure 5. A possible Feynman diagram contributing to V (h−1) and its cluster structure 2.3. An Example of Feynman Graph To be concrete, let us apply the rules described above in the evaluation of a simple Feynman graph G arising in the tree expansion of V (h−1) . Let G be the diagram in Fig. 5, associated to the tree τ drawn in the left part of the figure; let us assume that the sets Pv of the external lines associated to the vertices of τ are all assigned. Setting e0,h := e0,h ,
e¯j,h := vh−1 ej,h ,
(2.24)
we can write: 1 Zh Zh−1 2 e¯ e¯2 M h νμ1 ,h ψ k 4!2! Zh−1 Zh−2 μ1 ,h μ2 ,h+1 # dp × |w(h) (p)|2 γμ1 g (h) (k + p) (2π)3
dq (h+1) (h+1) γμ g (k + p + q)γμ2 w (q) ×R (2π)3 2 $ × g (h) (k + p)γμ1 ψk ,
Val(G) = −
(2.25)
where R [F (k + p)] = F (k + p) − F (0) − (k + p) · ∇F (0) ≡ 12 (kμ + pμ )(kν + pν )∂μ ∂ν F (k∗ ). Notice that the same Feynman graph appears in the evaluation of other trees, which are topologically equivalent to the one represented in the left part of Fig. 5 and that can be obtained from it by: (i) relabeling the fields in Pv1 , Pv0 , (ii) relabeling the endpoints of the tree, (iii) exchanging the relative positions of the topologically different subtrees with root v0 . If one sums over all these trees, the resulting value one obtains is the one in Eq. (2.25) times a combinatorial factor 22 · 3 · 4 (22 is the number of ways for choosing the fields in Pv1 and in Pv0 ; 3 is the number of ways in which one can associate the label ν to one of the endpoints of scale h + 1; 4 is the number of distinct unlabeled trees that can be obtained by exchanging the positions of the subtrees with root v0 ). 2.4. Dimensional Bounds We are now ready to derive a general bound for the Feynman graphs produced N ;(h) by the multiscale integration. Let Wm,n,ρ,μ be the contribution from trees with
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(h)
N end-points to the kernel Wm,n,ρ,μ in 2.4, i.e. (h) Wm,n,ρ,μ ({ki }, {pj }) ∞
=
∗
Val(G) ≡
N =1 τ ∈Th,N G∈Γ(τ ) |PvA0 |=m,
∞
N ;(h) Wm,n,ρ,μ ({ki }, {pj }),
(2.26)
N =1
|Pvψ0 |=2n
where the * on the sum indicates the constraints that: ∪f ∈PvA {pf } = 0 2n ∪m j=1 {pj }; ∪f ∈Pvψ0 {kf } = ∪i=1 {ki }; ∪f ∈PvA0 {μ(f )} = μ; ∪f ∈Pvψ0 {ρ(f )} = ρ. The N -th order contribution to the kernel of the effective potential admits the following bound. Theorem 2.1. (N ! bound) Let ε¯h = maxh
0, then N N ;(h) N N (2.27) ||Wm,n,ρ,μ || ≤ (const.) ε¯h M h(3−m−2n) , 2 N ;(h)
N ;(h)
where ||Wm,n,ρ,μ || := sup{ki },{pj } |Wm,n,ρ,μ ({ki }, {pj })|. The factor 3 − 2n − m in (2.27) is referred to as the scaling dimension of the kernel with 2n external fermionic fields and m external bosonic fields; according to the usual RG terminology, kernels with positive, vanishing or negative scaling dimensions are called relevant, marginal or irrelevant operators, respectively. Notice that, if we tried to expand the effective potential in terms of the bare couplings e, νμ , the N -th order contributions in this “naive” perturbation series could not be bounded uniformly in the scale h as in (2.27), but rather by the r.h.s. of (2.27) times |h|N , an estimate which blows up order by order as h → −∞ (Table 1). Proof. Using the bounds % % % (h) % %g (k)% ≤ const · M −h , (h) w (k) ≤ const · M −h ,
% % % % dk %g (h) (k)% ≤ const · M 2h , dk w(h) (k) ≤ const · M 2h ,
(2.28)
and the assumptions on vk−1 and Zk /Zk−1 into (2.23), we find that, if τ ∈ Th,N and G ∈ Γ(τ ), | Val(G)| ≤ (const.)N ε¯N h ×
v not
2
e 2 ε¯h |Pv | −3hv (sv −1) 2hv n0v hv mνv M M M sv ! e.p. C
M −zv (hv −hv ) ,
ψ
(2.29)
v not e.p. v>v0
where n0v is the number of propagators ∈ v, i.e., of propagators contained in the box Gv but not in any smaller cluster; sv is the number of vertices
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immediately following v on τ ; mνv is the number of end-points of type ν immediately following v on τ (i.e., contained in Gv but not in any smaller cluster); v is the vertex immediately preceding v on τ and zv = 2 if |Pvψ | = |Pv | = 2, zv = 1 is |Pvψ | = 2|PvA | = 2 and zv = 0 otherwise. The last product in (2.29) is due to the action of R on the vertices v > v0 that are not end(h ) points. In fact, the operator R, when acting on a kernel W1,1v (p, k) associ(h )
ated to a vertex v with |Pvψ | = 2|PvA | = 2, extracts from W1,1v the rest of first order in its Taylor expansion around p = k = 0: if (h )
(h )
(h ) |W1,1v (p, k)| ≤ C(v), −hv +hv
C(v), then |RW1,1v (p, k)| = 12 |(p∂p +k∂k )W1,1v (p∗ , k∗ )| ≤ (const.)M where M −hv is a bound for the derivative with respect to momenta on scale hv and M hv is a bound for the external momenta p, k; i.e., R is dimensionally (h ) equivalent to M −(hv −hv ) . The same is true if R acts on a kernel W3,0v (p1 , p2 ). Similarly, if R acts on a terms with |Pv | = 2, it extracts the rest of second order in the Taylor expansion around k = 0, and it is dimensionally equivalent to k2 ∂k2 ∼ M −2(hv −hv ) . As a result, we get (2.29). Now, let nev (nνv ) be the number of vertices of type e (of type ν) following v on τ . If we plug in (2.29) the identities
(hv − h)(sv − 1) =
v not e.p.
v not e.p.
(hv −
v not e.p.
h)n0v
=
(hv − h)mνv =
v not e.p.
(hv − hv )(nev + nνv − 1) (hv − hv )
v not e.p.
v not e.p.
3 e |Pv | nv + nνv − 2 2
(2.30)
(hv − hv )nνv
we get the bound | Val(G)| ≤ (const.)N ε¯N h
1 sv0 !
M h(3−|Pv0 |)
2
e 2 ε¯h |Pv | (hv −hv )(3−|Pv |−zv ) M . sv ! e.p. C
ψ
v not v>v0
(2.31) In the latter equation, 3 − |Pv | is the scaling dimension of the cluster Gv , and 3−|Pv |−zv is its renormalized scaling dimension. Notice that the renormalization operator R has been introduced precisely to guarantee that 3−|Pv |−zv < 0 for all v, by construction. This fact allows us to sum over the scale labels h ≤ hv ≤ 1, and to conclude that the perturbative expansion is well defined at any order N of the renormalized expansion. More precisely, the fact that the renormalized scaling dimensions are all negative implies, via a standard argument (see, e.g., [3,16]), the following bound, valid for a suitable constant C (see (2.27) for a definition of the norm · ):
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1 M h(3−m−2n) sv0 !
e C2 ε¯2h |Pvψ | M (hv −hv )(3−|Pv |−zv ) , (2.32) s ! v not e.p.
N ;(h) ||Wm,n,ρ,μ || ≤ (const.)N ε¯N h
×
τ ∈Th,N G∈Γ(τ ) v v>v0 |PvA0 |=m, |Pvψ0 |=2n
from which, after counting the number of Feynman graphs contributing to the sum in (2.32), (2.27) follows. An immediate corollary of the proof leading to (2.27) is that contributions from trees τ ∈ Th,N with a vertex v on scale hv = k > h admit an improved h(3−|Pv0 |) bound with respect to (2.27), of the form ≤ (const.)N ε¯N h (N/2)! M M θ(h−k) , for any 0 < θ < 1; the factor M θ(h−k) can be thought of as a dimensional gain with respect to the “basic” dimensional bound in (2.27). This improved bound is usually referred to as the short memory property (i.e., long trees are exponentially suppressed); it is due to the fact that the renormalized scaling dimensions dv = 3 − |Pv | − zv in (2.31) are all negative, and can be obtained by taking a fraction of the factors M (hv −hv )dv associated to the branches of the tree τ on the path connecting the vertex on scale k to the one on scale h. Remark. All the analysis above is based on the fact that the scaling dimension 3 − |Pv | in (2.31) is independent of the number of endpoints of the tree τ ; i.e., the model is renormalizable. A rather different situation is found in the case of instantaneous Coulomb interactions, in which case the bosonic propagator is given by (2| p|)−1 rather than by (2|p|)−1 . In this case, choosing the bosonic −1 single scale propagator as w(h) (p) = p)(2| that the h ( p|) , one finds χ0(p)f (h) last bound in (2.28) is replaced by dp w (p) ≤ (const.) M h (dimensionally, this bound has a factor M h missing). Repeating the steps leading to (2.31), one finds a general bound valid at all orders, in which the new scaling dimension is 3 − |Pv | + nev + nνv ; this (pessimistic) general bound assumes that at each scale the loop lines of the graph are all bosonic. Perhaps, this bound can be improved, by taking into account the explicit structure of the expansion; however, it shows that the renormalizability of the instantaneous case, if true, does not follow from purely dimensional considerations and its proof will require the implementation of suitable cancellations. 2.5. The Schwinger Functions A similar analysis can be performed for the two-point function, see Appendix B. It turns out that, similarly to what we found above for the effective potentials, the two-point function can be written in terms of a renormalized perturbative expansion in the effective couplings {eμ,k , νμ,k }k≤0 and in the renormalization constants {Zk , vk }k≤0 , with coefficients represented as sums of Feynman graphs, uniformly bounded as |Λ| → ∞; in contrast, the graphs forming the naive expansion in e, νμ are plagued by logarithmic infrared divergences.
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More explicitly, if M h ≤ |k| ≤ M h+1 , we get (see Eqs. B.10–B.13): ψk ψ k =
h+1 j=h
g (j) (k) ˜ 1 + B(k) , Zj−1
(2.33)
˜ where B(k) is given by a formal power series in {eμ,k , νμ,k }k≤0 with coefficients depending on {Zk , vk }k≤0 , and starting from second order; under the same ˜ hypothesis of Theorem 2.1, the N -th order contribution to B(k) is bounded N N ε−∞ ) (N/2)! uniformly in k. Eq. (2.33) is equivalent to Eq. (1.7) by (const.) (¯ of the main result (see the end of Section 3.3 below for the explicit relation between Zh , vh and Z(k), v(k)). To prove our main result we need to control the flow of the effective charges at all orders in perturbation theory, and to do this we shall use Ward Identities, see Sect. 4. These are non-trivial relations for the three-point functions, which can be related to the renormalized charges in the following way. Consider a theory with a bosonic infrared cutoff M h∗ , that is assume that the bare bosonic propagator is given by (1.5) with χ0 (p) replaced by χ[h∗ ,0] (p) := ∗ χ0 (p) − χ0 (M −h p), which is vanishing for |p| ≤ M h∗ and it is equal to χ0 (p) ∗ for |p| ≥ M h +1 ; denote by . . . h∗ the expectation value in the presence of the bosonic infrared cutoff. As shown in Appendix B, setting e¯0,h := e0,h , ∗ ∗ ∗ e¯1,h = e¯2,h := vh−1 e1,h , and taking |q| = M h , |q + p| ≤ M h , |p| M h (we will be interested in the limit p → 0), the following result holds (see Eqs. B.14–B.16): jμ,−p ; ψq+p ψ q h∗ & ' e¯μ,h∗ ¯μ,h∗ (p, q) ψq ψ q ∗ , (2.34) ψq+p ψ q+p h∗ γμ + B = iZh∗ −1 h e ¯ ∗ ∗ where Bμ,h is given by a formal power series in {eμ,k , νμ,k }h
3. The Flow of the Effective Couplings 3.1. The Beta Function A crucial point for the consistency of our approach is that the running coupling constants eμ,h , νμ,h are small for all h ≤ 0, that the ratios Zh /Zh−1 are close to 1, and the effective Fermi velocity vh does not approach zero. Even if we do not prove the convergence of the series but only N ! bounds, we expect
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that our series gives meaningful information only as long as the running coupling constants satisfy these conditions. In this section we describe how to control their flow. We shall proceed by induction: we will first assume that 2 ε¯ = maxk≤0 {|eμ,k |} is small, that Zh /Zh−1 ≤ eC ε¯ and C −1 ≤ vh ≤ 1 for all h ≤ 0 and a suitable constant C > 0, and we will show that, by properly choosing the values of the counterterms νμ in (1.2), the constants νμ,h remain small: maxh≤0 {|νμ,h |} ≤ (const.) ε¯2 . Next, once that the flow of νμ,h is controlled, we will study the flow of Zh and vh under the assumption that the constants eμ,h remain bounded and small for all h ≤ 0; we will show that, asymptotically as h → −∞, Zh ∼ M −ηh , with η = O(e2 ) a positive exponent, while vh grows, approaching a limiting value veff close to the speed of light. Finally, we shall start to discuss the remarkable cancellations following from a Ward Identity that guarantee that the constants eμ,h remain bounded and small for all h ≤ 0; the full proof of this fact will be postponed to Sect. 4 and Appendix D. The renormalized parameters obey to recursive equations induced by the previous construction; i.e., (2.6), (2.7), (2.11), (2.13) imply the flow equations: Zh−1 = 1 + z0,h := 1 + βhz , Zh
vh−1 =
Zh (vh + z1,h ) := vh + βhv Zh−1
(3.1)
(h)
ν νμ,h = −M −h W2,0,μ,μ (0) := M νμ,h+1 + βμ,h+1 , Zh e λ0,h := e0,h+1 + β0,h+1 , e0,h = Zh−1 Zh λ1,h e := e1,h+1 + β1,h+1 , e1,h = Zh−1 vh−1
(3.2) (3.3) (3.4)
and e2,h = e1,h . The beta functions appearing in the r.h.s. of flow equations are N ;(h) related, see (2.7), to the kernels Wm,n,ρ,μ , so that they are expressed by series in the running coupling constants admitting the bound (2.27). For the explicit expressions of the one-loop contributions to the beta function, see below. 3.2. The Flow of νμ,h 2
Let us assume that ε¯ = maxk≤0 {|eμ,k |} is small, that Zh /Zh−1 ≤ eC ε¯ and C −1 ≤ vh ≤ 1 for a suitable constant C, for all h ≤ 0. Under these assumptions, the flow of νμ,h can be controlled by suitably choosing the counterterms νμ ; in fact, if νμ is chosen as νμ = −
0
ν M k−1 βμ,k ,
(3.5)
k=−∞
then the effective coupling νμ,h is νμ,h = −
h
ν M −h−1+k βμ,k ,
(3.6)
k=−∞
from which one finds that νμ,h can be expressed by a series in {eμ,k }k≤0 , starting at second order and with coefficients bounded uniformly in h. At lowest
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√ order, if h < 0 and setting ξh := ν,(2) β0,h
2 1−vh vh
Ann. Henri Poincar´e
(see Appendix C):
∞ & ' e20,h vh−2 ξh − arctan ξh = −(M − 1) dt 2χ(t) − χ2 (t) (3.7) 3 2 π ξh 0
ν,(2)
β1,h
e21,h arctan ξh ξh − arctan ξh = −(M − 1) 2 − 2π ξh ξh3 ∞ & ' × dt 2χ(t) − χ2 (t) .
(3.8)
0
By the above equations we see that lowest order contributions to νμ are positive, that is νμ can be interpreted as bare photon masses. Using the short memory property and symmetry considerations, one can also show that ν ν − β1,h is a sum of graphs whose contributions are of the order O(1 − vh ) β0,h or O(e0,h − e1,h ). 3.3. The Flow of Zh and vh In this section we show that, under proper assumptions on the flow of the effective charges, the effective Fermi velocity vh tend to a limit value veff = v−∞ and that both veff − vh and Zh−1 vanish as h → −∞ with an anomalous power law. Let us assume that the effective charges tend to a line of fixed points: eμ,h = eμ,−∞ + O(e3 (v−∞ − vh )) + O(e3 M θh ),
(3.9)
with 0 < θ < 1 and eμ,−∞ = e + O(e3 ); this is a remarkable property that will be proven order by order in perturbation theory using WIs, see the following section. Moreover, let νμ be fixed as in the previous subsection (under the proper inductive assumptions on Zk and vk ). We start by studying the flow of the Fermi velocity. At lowest order (see Appendix C), its beta function reads: −1 e20,h vh−1 ξh −arctan ξh log M e20,h vh arctan ξh v,(2) 2 − 2e1,h vh − . βh = 4π 2 2 ξh 2 ξh3 (3.10) Note that if e0,k := e1,k , then the r.h.s. of (3.10) is strictly positive for all ξh > 0 and it vanishes quadratically in ξh at ξh = 0. The higher order contributions to βhv have similar properties. This can be proved as follows: we observe that the beta function βhv is a function of the renormalized couplings and of the Fermi velocities on scales ≥ h, i.e.: (3.11) βhv = βhv {(e0,k , e1,k , e2,k ) , (ν0,k , ν1,k , ν2,k ), vk }k≥h .
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We can rewrite βhv as βhv,rel + βhv,1 + βhv,2 + βhv,3 , with: βhv,rel = βhv {(e0,k , e0,k , e0,k ), (ν0,k , ν0,k , ν0,k ), 1}k≥h , βhv,1 = βhv {(e0,k , e0,k , e0,k ), (ν0,k , ν0,k , ν0,k ), vk }k≥h − βhv {(e0,k , e0,k , e0,k ), (ν0,k , ν0,k , ν0,k ), 1}k≥h , βhv,2 = βhv {(e0,k , e0,k , e0,k ), (ν0,k , ν1,k , ν2,k ), vk }k≥h − βhv {(e0,k , e0,k , e0,k ), (ν0,k , ν0,k , ν0,k ), vk }k≥h , βhv,3 = βhv {(e0,k , e1,k , e2,k ), (ν0,k , ν1,k , ν2,k ), vk }k≥h − βhv {(e0,k , e0,k , e0,k ), (ν0,k , ν1,k , ν2,k ), vk }k≥h .
1429
(3.12)
By relativistic invariance it follows that βhv,rel = 0 and by the short memory property (see discussion after Eq. (2.32)) we get: & ' βhv,1 = O e20,h (1 − vh ) , & ' βhv,2 = O e20,h (ν0,h − ν1,h ) , (3.13) βhv,3 = O (e0,h (e0,h − e1,h )) . Using (3.6) and an argument similar to the one leading to Eq. (3.13), we also find that ν0,h − ν1,h can be written as a sum of contributions of order e0,h (e0,h − e1,h ) and of order e20,h (1 − vh ). Therefore, we can write:
vh−1 log M 8 2 4 e (1 − vh )(1 + Ah ) + e(1 + Bh )(e0,h − e1,h ) , (3.14) =1+ vh 4π 2 5 3 where the numerical coefficients are obtained from the explicit lowest order computation (3.10); A h is a sum of contributions that are finite at all orders in the effective couplings, which are either of order two or more in the effective charges, or vanishing at vk = 1; similarly, Bh is a sum of contributions that are finite at all orders in the effective couplings, which are of order two or more in the effective charges. From (3.14) it is apparent that vh tends as h → −∞ to a limit value 5 ) (3.15) veff = 1 + (e0,−∞ − e1,−∞ )(1 + C−∞ 6e with C−∞ a sum of contributions that are finite at all orders in the effective couplings, which are of order two or more in the effective charges. The fixed point (3.15) is found simply by requiring that in the limit h → −∞ the argument of the square brackets in (3.14) vanishes. Using Eq. (3.9), we find that the expression in square brackets in the r.h.s. of (3.14) can be rewritten as (8e2 /5)(veff − vh + Rh )(1 + A h ), where (i) A h is a sum of contributions that are finite at all orders in the effective couplings, which are either of order two or more in the effective charges, or vanishing at vk = veff ; (ii) Rh is a sum of contributions that are finite at all orders in the effective couplings, which are of order two or more in the effective charges and
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are bounded at all orders by M θh , for some 0 < θ < 1. Therefore, (3.14) can be rewritten as veff − vh + Rh 2e2 log M 2 (1 + Ah ) , (3.16) veff − vh−1 = (veff − vh ) 1 − vh veff − vh 5π from which, using the fact that Rh = O(e2 M θh ), we get that there exist two positive constants C1 , C2 such that 1 : veff − vh C1 M h˜η ≤ ≤ C2 M h˜η , (3.17) veff − v with
' 2e2 & η˜ = − logM 1 − veff log M 2 1 + A−∞ ; (3.18) 5π at lowest order, Eq. (3.18) gives η˜(2) = 2e2 /(5π 2 ). Similarly C1 M ηh ≤ Zh ≤ C2 M ηh for two suitable positive constants C1 , C2 , with η = limh→−∞ logM (1 + z0,h ); at lowest order we find (see Appendix C): z,(2)
βh
=
log M ξh − arctan ξh (2e21,h − e20,h vh−2 ) , 4π 2 ξh3
(3.19)
2
e so that η (2) = 12π 2. Before we conclude this section, let us briefly comment about the relation between Zh , vh and the functions Z(k) and v(k) appearing in the main result, see (1.6). If |k| = M h , we define Z(k) = Zh and v(k) = vh ; for general |k| ≤ 1, we let Z(k) and v(k) be smooth interpolations of these sequences. Of course, we can choose these interpolations in such a way that, if M h ≤ |k| ≤ M h+1 , Z(k) Zh+1 ≤ = O(η log M ), − 1 − 1 Zh Zh (3.20) v(k) − vh vh+1 − vh ≤ = O(˜ η log M ). veff − vh veff − vh
Therefore, we can replace in the leading part of the two-point Schwinger function (2.33) the wave function renormalization Zj and the effective Fermi veloc˜ in (2.33) is ity vj by Z(k) and v(k), provided that the correction term B(k) replaced by a quantity B(k) defined so to take into account higher order corrections satisfying the bounds (3.20). This leads to the main result Eq. (1.6). 3.4. The Flow of the Effective Charges at Lowest Order The physical behavior of the system is driven by the flow of eμ,h ; in the following section, using a WI relating the three- and two-point functions, we will show that eμ,h remain close to their initial values for all scales h ≤ 1 and limh→−∞ eμ,h = eμ,−∞ = e + Fμ , where Fμ can be expressed as series in the renormalized couplings starting at third order in the effective charges. In 1 Equation (3.17) must be understood as an order by order inequality: if we truncate the theory at order N in the bare coupling e, both sides of the inequality in Eq. (3.17) are verified asymptotically as e → 0, for all N ≥ 1.
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perturbation theory, this fact follows from non-trivial cancellations that are present at all orders. For illustrative purposes, here we perform the lowest order computation in non-renormalized perturbation theory, in the presence of an infrared cutoff on the bosonic propagator χ[h,0] (p) = χ0 (p) − χ0 (M −h p); at this lowest order, such a “naive” computation gives the same result as the renormalized one; for the full computation, see next section and Appendix D. If (¯ γ0 , γ¯1 , γ¯2 ) := (γ0 , vγ1 , vγ2 ),
(k¯0 , k¯1 , k¯2 ) := (k0 , vk1 , vk2 ),
(3.21)
the effective charges on scale h at third order are given by: (3) eμ,h − e γμ ) dk χ[h,0] (k) ( (≤0) (≤0) (≤0) γ ¯ = ie3 g (k)iγ g (k)¯ γ +¯ γ ∂ g (k)¯ γ ¯ ν μ ν ν kμ ν (2π)3 2|k| (3.22) where the first term in square brackets is the vertex renormalization, while the second term is due to the wave function and velocity renormalizations. Note that both integrals are well defined in the ultraviolet (thanks to the presence of an ultraviolet cutoff in the propagators), while for h → −∞ they are logarithmically divergent in the infrared. However, a remarkable cancellation takes place between the two integrals; in fact: g (≤0) (k)iγμ g (≤0) (k) + ∂k¯μ g (≤0) (k) =
∂k¯μ χ0 (k) i k
+ χ0 (k) (χ0 (k) − 1)
1 1 iγμ , i k i k
(3.23)
with k := k0 γ0 + vk · γ , so that & ' 1 k0 dk (3) 3 γ¯μ γ¯μ χ (k)χ[h,0] (k) + O e3 (M − 1) e0,h = e + ie (2π)3 i k 2|k|2 0 3
(3)
e1,h = e +
ie v
(3.24) ' 1 k1 dk γ¯μ γ¯μ χ (k)χ[h,0] (k) + O e3 (M − 1) . (2π)3 i k 2|k|2 0 &
Notice that the cancellation does not depend on the presence of the bosonic IR cutoff; this fact will play an important role in the analysis at all orders of the flow of the effective charges, see next section. An explicit computation of (3.24) says that, at third order in e, (3)
eμ,−∞ = e + eαμ(2) , where (2)
' ξ0 − arctan ξ0 e2 & −2 2 − v + O(e2 (M − 1)), 8π 2 ξ03 e2 1 arctan ξ0 ξ0 − arctan ξ0 = − + O(e2 (M − 1)); 16π 2 v 2 ξ0 ξ03
α0 = (2)
α1
(3.25)
(3.26) (3.27)
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the correction terms O(e2 (M −1)) can be made as small as desired, by choosing 0 < M − 1 1. Note that the two effective charges are different: (3)
(3)
e0,−∞ − e1,−∞ = −
& ' e3 F (v) + O e3 (M − 1) , 2 5π
(3.28)
where F (v) the function defined in (1.9). Combining (3.28) with (3.15) gives the last equation of (1.8) Of course, the one-loop computation that we just described does not say much: if we could not guarantee that a similar cancellation takes place at all orders there would always be the possibility that higher orders produce a completely different behavior, e.g. a vanishing or diverging flow for eμ,h , corresponding to completely different physical properties of the system. In order to obtain a control at all orders on eμ,h one needs to combine the multiscale evaluation of the effective potentials with Ward Identities. This is not a trivial task: Wilsonian RG methods are based on a multiscale momentum decomposition which breaks the local gauge invariance, which Ward Identities are based on. In Sect. 4 below, following a strategy recently proposed and developed in [6], we will prove (3.25). Remark. Note the unusual feature that e0,h = e1,h , an effect due to the presence of the momentum cut-off and the fact that v = 1. The discussion of this and previous sections can be repeated in the case that the bare interaction involves two different charges, e0 and e1 , describing the couplings of the photon field with the temporal and spatial components of the current. If e = (e0 + e1 )/2 and e0 − e1 = O(e3 ), the conclusion is that veff = 1 − (e2 /6π 2 )F (v) + (5/6)(e0 − e1 )/e + O(e4 ) and it is of course possible to fine tune the bare parameters e0 and e1 in such a way that e0,−∞ = e1,−∞ and veff = 1. Note that, in a more realistic model for graphene, describing tight binding electrons on the honeycomb lattice coupled with a 3D photon field via a lattice gauge invariant coupling, one expects that e0,−∞ = e1,−∞ and veff = 1.
4. Ward Identities In this section we prove that order by order in perturbation theory the effective charges eμ,h remain close to their original values eμ,0 = e; moreover, we prove that asymptotically as h → −∞, e0,h = e1,h , see (3.25)–(3.28). The proof is based on a suitable combination of the RG methods described in the previous sections together with Ward Identities; even though the momentum regularization breaks the local gauge invariance needed to formally derive the WIs, we will be able, following the strategy of [6], to rigorously take into account the effects of cutoffs, and to control the corrections generated by their presence. As anticipated at the end of Sect. 2, we consider a sequence of models, to be called reference models in what follows, with different infrared bosonic
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cutoffs on scale h, i.e. with bosonic propagator given by: w[h,0] (p) ≡
χ[h,0] (p) , 2|p|
χ[h,0] (p) ≡ χ0 (p) − χ0 (M −h p)
(4.1)
(the idea of introducing an infrared cutoff only in the bosonic sector is borrowed from Adler and Bardeen [1], who used a similar regularization scheme in order to understand anomalies in quantum field theory). The generating functional W[h,0] (J, φ) of the correlations of the reference model can be evaluated following an iterative procedure similar to the one described in Sect. 2 (see Appendix B for details), with the important difference that after the integration of the scale h we are left with a purely fermionic theory, which is superrenormalizable: in fact, setting m = 0 in the formula for the scaling dimension (see lines following (2.27) and recall that for scales smaller than h the reference model has no bosonic lines) we recognize that the scaling dimension of this fermionic theory is 3 − 2n, which is always negative once that the two-legged subdiagrams have been renormalized, see [18,19]. Let us denote by [h] {eμ,k }k≥h the effective couplings of the reference model; of course, if k ≥ h [h]
eμ,k = eμ,k ,
(4.2)
where {eμ,k }k≤0 are the running coupling constants of the original model. On the other hand, as proven in Appendix B, the vertex functions jμ,−p ; ψk+p ψ k h of the reference model with bosonic cutoff on scale h computed at external momenta k, k + p such that |k + p|, |k| M h and |p| M h are proportional [h] to the charges eμ,h = eμ,h , see (2.34); therefore, if we get informations on the vertex functions of the reference models, we automatically infer informations on the effective couplings of the original model. Such informations are provided by Ward Identities; by performing the change of variables ψx → eiαx ψx , ψ x → e−iαx ψ x in the generating functional W[h,0] (J, φ) of the reference model and using that the Jacobian of this transformation is equal to 1, see [6], we get: (J,φ) eW[h,0]
=
P (dψ)P[h,0] (dA)e−
dx ψ x (e−iαx Deiαx −D)ψx +V (A,ψ)+B(J,φe−iα )
,
(4.3)
where P[h,0] (dA) is the gaussian integration with propagator (4.1) and, if k = γ0 k0 + vγ · k, the pseudo-differential operator D is defined by: dk eikx (Dψ)x = i k ψk . (2π)3 χ0 (k) χ(k)>0
If we derive (4.3) with respect to α, φ and φ and then set α = φ = J = 0, we get the following identity: pμ jμ,−p ; ψk+p ψ k h = ψk ψ k h − ψk+p ψ k+p h + Δh (k, p)
(4.4)
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where
Δh (k, p) =
and
Ann. Henri Poincar´e
dk ψ C(k , p)ψk ; ψk+p ψ k h (2π)3 k +p
& ' & ' C(k, p) = i k χ0 (k)−1 − 1 − i( k + p) χ0 (k + p)−1 − 1 .
(4.5)
(4.6)
The correction term Δh (k, p) in (4.4) is due to the presence of the ultraviolet momentum cut-off, and it can be computed by following a strategy analogous to the one used to prove the vanishing of the beta function in one-dimensional Fermi systems [6]. We can write pμ Δh (k, p) = αμ pμ jμ,−p ; ψk+p ψ k h + Rμ,h (k, p), (4.7) Zh where the correction Rμ,h (k, p) is dimensionally negligible with respect to the first term, see Appendix D. More precisely, in Appendix D it is shown that: (i) Rμ,h (k, p) can be written as a sum over trees with N endpoints of contribu(N ) tions Rμ,h (k, p); (ii) it is possible to choose αμ in such a way that, under the same conditions of Theorem 2.1 and if |k| = M h , |k + p| ≤ M h and |p| M h , h N (N ) N |Rμ,h (k, p)| ≤ (const.) (4.8) !M −2h M 2 ε¯N h . 2 (2)
An explicit computation, see Appendix D, shows that at lowest order αμ is given by (3.26)–(3.27). Let us now show how to use the previous relations in order to derive bounds on the effective charges. Let us pick |k| = M h and |p| M h ; using Eqs. (2.33)–(2.34) and the fact that g (h) (k) − g (h) (k + p) = g (h) (k + p)(ip0 γ0 + ivh−1 p · γ )g (h) (k) + pμ rˆμ (k, p), (4.9) with rˆμ (k, p) = O(|p|M
−3h
), we find that 1
g (h) (k + p)(ip0 γ0 + ivh−1 p · γ )g (h) (k) Zh−1 pμ (˜ rμ (k, p) + rˆμ (k, p)) , (4.10) + Zh−1 1 g (h) (k + p)(ie0,h p0 γ0 + ivh−1 e1,h p · γ ) pμ jμ,−p ; ψk+p ψ k h = eZh−1 pμ rμ (k, p), (4.11) ×g (h) (k) + Zh−1
ψk ψ k h − ψk+p ψ k+p h =
with |rμ (k, p)|, |˜ rμ (k, p)| expressed by sums over trees of order N ≥ 2 of con(N ) (N ) tributions rμ (k, p), r˜μ (k, p) bounded by (see Appendix B, formulas (B.13) and (B.16)) N |rμ(N ) (k, p)| + |˜ rμ(N ) (k, p)| ≤ (const.)N ε¯N (4.12) ! M −2h . h 2
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Now, if we plug (4.7) into the Ward identity (4.4), and we use the relations (4.10)–(4.11), we get an identity that, computed at k = k0 := (M h , 0) and p = p0 := (p, 0), after taking the limit p → 0, reduces to: e0,h (1 − α0 ) = 1 + iM 2h [˜ r0 (k0 , 0) + R0,h (k0 , 0) − (1 − α0 )r0 (k0 , 0)] γ0 e ≡ 1 + A0,h , (4.13) with A0,h a sum of contributions associated to trees of order N ≥ 2 bounded εh )N (N/2)!, as it follows from the estimates at the N -th order by (const.)N (¯ on R0,h , r0 , r˜0 (note the crucial point that such estimate is proportional to ε−∞ )N ; this is the main reason why we chose to intro(¯ εh )N rather than to (¯ duce the infrared cutoff on the bosonic propagator, see the end of Sect. 2.5 and the beginning of this section). Eq. (4.13) combined with (3.26) implies, as desired, that the effective charge e0,h remains close to e0,0 = e at all orders in renormalized perturbation theory. Moreover, proceeding as in the derivation of Eq. (3.13), we find that |A0,h −A0,−∞ | = O(e2 (veff −vh ))+O(e2 M θh ), for some 0 < θ < 1, from which we get Eq. (3.9) for μ = 0. Similarly, if k1 := (0, M h , 0), we get: e1,h vh−1 (1 − α1 ) = + iM 2h vh−1 e vh × [˜ r1 (k1 , 0) + R1,h (k1 , 0) − (1 − α1 )r1 (k1 , 0)] γ1 ≡ 1 + A1,h , (4.14) with A1,h a sum of contributions associated to trees of order N ≥ 2 bounded εh )N (N/2)!, which implies that the effective at the N -th order by (const.)N (¯ charge e1,h remains close to e1,0 = e at all orders in renormalized perturbation theory. Moreover, as in the μ = 0 case, |A1,h − A1,−∞ | = O(e2 (veff − vh )) + O(e2 M θh ), for some 0 < θ < 1, from which we get Eq. (3.9) for μ = 1. Equations (4.13) and (4.14) not only imply the boundedness of the effective charges eμ,h but they also allow us to compute the difference e0,h − e1,h , asymptotically as h → −∞, at all orders in renormalized perturbation theory. (3) (3) (2) (2) At lowest order, e0,h − e1,h = e(α0 − α1 ), as anticipated in previous section.
5. Conclusions We considered an effective continuum model for the low energy physics of single-layer graphene, first introduced by Gonzalez et al. [20]. We analyzed it by constructive Renormalization Group methods, which have already been proved effective in the non-perturbative study of several low-dimensional fermionic models, such as one-dimensional interacting fermions [6], or the Hubbard model on the honeycomb lattice [18,19]. While in the present case we are not able yet to prove the convergence of the renormalized expansion, we can prove that it is order by order finite, see Theorem 2.1 above. Note that, on the contrary, the power series expansion in the bare couplings is plagued by logarithmic divergences and, therefore, informations obtained from it by lowest order truncation are quite unreliable. In perspective, the proof
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of convergence of the renormalized expansion appears to be much more difficult than the one in [6] or [18,19], due to the simultaneous presence of bosons and fermions, but it should be feasible (using determinant bounds for the fermionic sector and cluster expansion techniques for the boson sector). A key point of our analysis is the control at all orders of the flow of the effective couplings: this is obtained via Ward Identities relating three- and twopoint functions, using a technique developed in [6] for the analysis of Luttinger liquids, in cases where bosonization cannot be applied (like in the presence of an underlying lattice or of non-linear bands). The Ward Identities have corrections with respect to the formal ones, due to the presence of a fermionic ultraviolet cut-off. Remarkably, these corrections can be rigorously bounded at all orders in renormalized perturbation theory (see Sect. 4). Several questions remain to be understood. First of all, the effective model we considered is clearly not fundamental: a more realistic model for graphene should be obtained by considering electrons on the honeycomb lattice coupled to an electromagnetic field living in the 3D continuum. We believe that a Renormalization Group analysis, similar to the one we performed here, is possible also for the lattice model, by combining the techniques and results of [18,19] with those of the present paper; we expect that the lattice model is asymptotic to the continuum one considered here, provided that the bare parameters of the continuum model are properly tuned. Another important open problem is to understand the behavior of the system in the case of static Coulomb interactions; this case can be obtained by taking the limit c → ∞ together with a proper rescaling of the electronic charge in the model with retarded interactions. However, as discussed in the Remark at the end of Sect. 2.4, the static case seems to be much more subtle than the one considered in this paper, since it apparently requires cancellations even to prove renormalizability of the theory at all orders. We plan to come back to this case in a future publication.
Acknowledgements A.G. and V.M. gratefully acknowledge financial support from the ERC Starting Grant CoMBoS-239694. We thank D. Haldane and M. Vozmediano for many valuable discussions.
Appendix A. Symmetries In this Appendix we prove formulas (2.6) and (2.7); to do this, we exploit suitable symmetry transformations. We use the following explicit representation of the euclidean gamma matrices: 0 I 0 iσ2 0 iσ1 , γ1 = , γ2 = , (A.1) γ0 = −I 0 iσ2 0 iσ1 0
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where σ1 = 0 −iσ3 −iσ3 0 matrix
0 1
1437
1 0 −i and σ2 = . It is also useful to define: γ3 = 0 i 0 1 0 , and the corresponding fifth gamma , with σ3 = 0 −1 γ5 = γ0 γ1 γ2 γ3 =
I 0
0 , −I
(A.2)
which anticommutes with all the other gamma matrices: {γμ , γ5 } = 0, ∀μ = 0, . . . , 3. Finally, given ω ∈ {+, −}, we define the chiral projector Pω := (1 + ωγ5 )/2. It is straightforward to check that both the gaussian integrations P (dψ), P (dA) and the interaction V (A, ψ) are invariant under the following symmetry transformations, which are preserved by the multiscale integration: (1) (2)
Chirality: Pω ψk → e−iαω Pω ψk , ψ k P−ω → ψ k P−ω e+iαω , with αω ∈ R independent of k. θ θ Spatial rotations: ψk → e 4 [γ1 ,γ2 ] ψR[1,2] k , ψ k → ψ R[1,2] k e− 4 [γ1 ,γ2 ] and −θ −θ ) ( [1,2] Aμ,p → Rθ A·,R[1,2] p , with −θ
μ
⎛
[1,2]
Rθ
1 = ⎝0 0
0 cos θ sin θ
⎞ 0 − sin θ⎠ . cos θ
(A.3)
The invariance of the model under (2) is a simple consequence of the fact that e− 4 [γ1 ,γ2 ] (γ0 , γ1 , γ2 ) e 4 [γ1 ,γ2 ] = (γ0 , γ1 cos θ − γ2 sin θ, γ2 cos θ + γ1 sin θ). θ
(3)
(4.a) (4.b) (5) (6)
θ
(A.4)
Complex conjugation: ψk → (−iγ2 )ψ−k , ψ k → ψ −k (iγ2 ), Aμ,k → −Aμ,−k and κ → κ∗ , where κ is a generic constant appearing in P (dψ), P (dA) and/or in V (A, ψ). Horizontal reflections: ψk → (iγ3 γ1 )ψk˜ , ψ k → ψ k˜ (−iγ1 γ3 ) and Aμ,p → ˜ = (k0 , −k1 , k2 ). (−1)μ Aμ,˜p , where k Vertical reflections: ψk → (−iγ2 )ψk˜ , ψ k → ψ k˜ (iγ2 ) and Aμ,p → ˜ = (k0 , k1 , −k2 ). (−1)δμ,2 Aμ,˜p , where k T
Particle-hole: ψk → (−γ0 γ2 )ψ k˜ , ψ k → ψkT˜ γ2 γ0 , and Aμ,p → ˜ = (k0 , −k). (−1)1−δμ,0 Aμ,˜p , where k
Inversion: ψk → γ0 γ3 ψk˜ , ψ k → ψ k˜ γ3 γ0 and Aμ,k → (−1)δμ,0 Aμ,˜p , ˜ = (−k0 , k). where k In addition to the previous symmetries, if v = c = 1 the theory has an additional space–time invariance, namely:
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Relativistic invariance: ψk → e 4 [γ0 ,γ1 ] ψR[0,1] k , ψ k → ψ R[0,1] k e− 4 [γ0 ,γ1 ] −θ −θ ( ) −1 and Aμ,p → Rθ A·,R p , with θ
θ
μ
⎛
[0,1]
Rθ
θ
cos θ = ⎝ sin θ 0
⎞ 0 0⎠. 1
− sin θ cos θ 0
(A.5)
The invariance of the model under (7) is a simple consequence of the remark that e− 4 [γ0 ,γ1 ] (γ0 , γ1 , γ2 ) e 4 [γ0 ,γ1 ] = (γ0 cos θ−γ1 sin θ, γ1 cos θ+γ0 sin θ, γ2 ). (A.6) θ
θ
It is now straightforward to check that these symmetries imply (2.6), (2.7). In fact, the first two identities in the first line of (2.6) and the second identity in the second line of (2.6) easily follow from (4.a) + (4.b) + (6). Using (4.a) + (4.b) + (6) we also find that (h)
(h)
W2,0,μ,ν (0) = δμν W2,0,μ,μ (0),
(A.7)
while, from (2) + (3), we get (h)
(h)
W2,0,1,1 (0) = W2,0,2,2 (0),
(h)
W2,0,μ,μ (0) ∈ R,
(A.8)
which imply the first identity in the second line of (2.6) (notice that, if v = (h) (h) c = 1, from (7) we also get that W2,0,0,0 (0) = W2,0,1,1 (0)). (h)
Let us now consider the combination ψ k W0,1 (k)ψk . Using the fact that {I, γ5 , {γj }0≤j≤3 , {γj γ5 }0≤j≤3 , {γj1 γj2 }0≤j1 <j2 ≤3 } is a complete basis for the space of complex 4 × 4 matrices, we can rewrite it as: ⎧ 3 ( ⎨ ) (h) cj1 (k)γj + cj15 (k)γj γ5 ψ k W0,1 (k)ψk = ψ k c0 (k)I + ⎩ j=0 ⎫ ⎬ + cj21 j2 (k)γj1 γj2 + c5 (k)γ5 ψk . ⎭
(A.9)
0≤j1 <j2 ≤5
Now, using the invariance under (1), we find that, e.g., ψ k c0 (k)Iψk =
ω=±
ψ k Pω c0 (k)IPω ψk = e−2iαω
ψ k Pω c0 (k)IPω ψk ,
(A.10)
ω=±
for all αω ∈ R, which implies that c0 (k) = 0; similarly, using the invariance under (1) and the fact that [γ5 , Pω ] = [γj1 γj2 , Pω ] = 0, ∀0 ≤ j1 < j2 ≤ 3, we
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find that c5 (k) = 0 and cj21 j2 (k) = 0, ∀0 ≤ j1 < j2 ≤ 3. Therefore, 3 ( ) dk dk (h) ψ W (0)ψ = ψ k cj1 (0)γj + cj15 (0)γj γ5 ψk , k k 0,1 3 3 (2π) (2π) j=0
(A.11) dk (h) ψ k∂k W0,1 (0)ψk (2π)3 k 3 ( ) dk j j k∂ ψk . = ψ c (0)γ + k∂ c (0)γ γ k j k j 5 k 1 15 (2π)3 j=0
(A.12)
Let us first look at (A.11). Using the invariance under (4.a), we find that [cj1 (0)γj + cj15 (0)γj γ5 ] = γ3 γ1 [cj1 (0)γj + cj15 (0)γj γ5 ]γ1 γ3 , which implies that c11 (0) = c31 (0) = c115 (0) = c315 (0) = 0. Using (2), we find that also c21 (0) = c215 (0) = 0; finally, using (6), we find that c01 (0) = c015 (0) = 0. This concludes the proof of the third identity in the first line of (2.6). Let us now look at (A.12). The terms proportional to k0 in the r.h.s. of (A.12) are invariant under (2) + (4.a) + (4.b), which implies that ∂k0 c11 (0) = ∂k0 c21 (0) = ∂k0 c31 (0) = ∂k0 cj15 (0) = 0. The terms proportional to k1 are invariant under (4.b) + (6), while the terms proportional to k2 are invariant under (4.a) + (6); combining these transformations with (2), we find that ∂k1 c01 (0) = ∂k1 c21 (0) = ∂k1 c31 (0) = ∂k1 cj15 (0) = 0, that ∂k2 c01 (0) = ∂k2 c11 (0) = ∂k2 c31 (0) = ∂k2 cj15 (0) = 0, and that ∂k1 c11 (0) = ∂k2 c21 (0). Therefore, ( ) dk dk (h) k · γ ψk , (A.13) a ψ k∂ W (0)ψ = ψ k γ + a k k 0 0 0 1 k k 0,1 (2π)3 (2π)3 for two suitable constants a0 , a1 . Using the invariance under (3), we find that a0 = iz0,h and a1 = iz1,h , with zμ,h ∈ R, which concludes the proof of the first line of (2.7) (of course, if v = c = 1, then from (7) we also get that z0,h = z1,h , that is the speed of light is not renormalized). A completely analogous discussion can be repeated for the second line of (2.7), but we will not belabor the details here.
Appendix B. Multiscale Integration for the Correlation Functions The multiscale integration used to compute the partition function W(0, 0), described in Sect. 2, can be suitably modified in order to compute the two and three-point correlation functions in the reference model with bosonic infrared cutoff on scale h, see (4.1). We start by rewriting the two and three point Schwinger functions in the following way: ∂2 ψk ψ k h∗ = W[h∗ ,0] (J, φ)J=φ=0 , ∂φk ∂φk (B.1) ∂3 W[h∗ ,0] (J, φ) J=φ=0 jμ,−p ; ψk+p ψ k h∗ = ∂Jμ,p ∂φk+p ∂φk
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where W[h∗ ,0] (J, φ) is the generating function of the reference model. The two point Schwinger function ψk ψ k appearing in our main result is obtained as limh∗ →−∞ ψk ψ k h∗ . In order to compute W[h∗ ,0] (J, φ), we proceed in a way analogous to the one described in Sect. 2. We iteratively integrate the fields ψ (0) , A(0) , . . ., ψ (h+1) , A(h+1) , . . ., and after the integration of the first |h| scales we are left with a functional integral similar to (2.2), but now involving new terms depending on J, φ. Let us first consider the case h ≥ h∗ ; the regime h < h∗ will be discussed later. Case h ≥ h∗ . We want to inductively prove that eW[h∗ ,0] (J,φ) = e|Λ|Eh +S (h)
×eBφ
(≥h)
(J,φ)
P (dψ (≤h) )P[h∗ ,0] (dA(≤h) )eV
(h)
√ (A(≤h) +GA J, Zh ψ (≤h) )
√ √ (h) (A(≤h) +GA J, Zh ψ (≤h) ,φ)+WR (A(≤h) +GA J, Zh ψ (≤h) ,φ)
,
(B.2)
(h)
where S (≥h) (J, φ) is independent of (A, ψ), WR contains terms explicitly (h) depending on (A, ψ) and of order ≥ 2 in φ, while Bφ is given by: ) dk ( (h) (h+1) † (h+1) Bφ (A, Zh ψ, φ) = φ [Q (k)] ψ + ψ Q (k)φ k k k k (2π)3 dk (h+1) † ∂ (h) φ [G (k)] V (A, Zh ψ) + k ψ (2π)3 ∂ψ k
∂ (h) (h+1) V (A, Zh ψ)Gψ (k)φk . (B.3) + ∂ψk (h)
Moreover, the functions GA , Q(h) , Gψ are defined by the following relations: ∗
eGA,μ (p) := 1 + νμ w[h
,0]
(p),
(h)
Gψ (k) :=
0 g (i) (k) i=h
Q(h) (k) := Q(h+1) (k) −
Zi−1
(h+1) iZh zμ,h kμ γμ Gψ (k),
Q(i) (k), (B.4)
with Q(1) (k) ≡ 1, G(1) (k) ≡ 0. Note that, if k is in the support of g (h) (k), Q(h) (k) = 1 − izμ,h kμ γμ g (h+1) (k), (h) Gψ (k)
g (h) (k) (h) g (h+1) (k) = Q (k) + , Zh−1 Zh (h)
(B.5)
that is ||Q(h) (k) − 1|| ≤ (const.) ε¯2h and ||Gψ (k)|| ≤ (const.) Zh−1 M −h . More∗ over, by the compact support properties of w[h ,0] (p), GA,μ (p) ≡ e−1 for all ∗ |p| ≤ M h . In order to prove (B.2)–(B.4) by induction, let us first check them at the first step. The generating functional of the correlations is defined as
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(see (1.1)–(1.2)) eW[h∗ ,0] (J,φ) =
P (dψ (≤0) )P[h∗ ,0] (dA(≤0) )e
dp (2π)3
(≤0)
(≤0) (eA(≤0) , A(≤0) )+B(0,φ) μ,p +Jμ,p )jμ,−p −νμ (Aμ μ
(B.6) which, under the change of variables ∗
(≤0) −1 νμ w[h A(≤0) μ,p → Aμ,p + e
,0]
(p)Jμ,p ;
(B.7)
can be rewritten in the form (B.2), with Eh = 0, (0)
WR = 0,
∗
e2 S (≥0) = νμ (Jμ , Jμ ) + νμ2 (Jμ , w[h V (0) = V,
,0]
Jμ ),
(0)
Bφ (A + GA J, ψ, φ) = B(0, φ).
(B.8)
Let us now assume that (B.2)–(B.4) are valid at scales ≥ h, and let us prove that the inductive assumption is reproduced at scale h − 1. We proceed as in Sect. 2; first, we renormalize the free measure by reabsorbing into P(dψ (≤h) ) the term exp{Lψ V (h) }, see (2.8)–(2.11), and then we rescale the (h) fields as in (2.12). Similarly, in the definition of Bφ , Eq. (B.3), we rewrite V (h) = Lψ V (h) + Vˆ (h) , combine the terms proportional to Lψ V (h) with those proportional to Q(h+1) , and rewrite B (h) (A, Zh ψ, φ) = Bˆ(h) (A, Zh−1 ψ, φ) ) dk ( (h) † (h) φ [Q (k)] ψ + ψ Q (k)φ := k k k k (2π)3 ∂ ˆ (h) dk (h+1) V (A, Zh−1 ψ) φk [Gψ (k)]† + (2π)3 ∂ψ k
∂ ˆ (h) (h+1) V (A, Zh−1 ψ)Gψ (k)φk , + ∂ψk (h)
with Q(h) defined by (B.4). Finally, we rescale WR , by defining ˆ (h) (A + GA J, Zh−1 ψ) := W (h) (A + GA J, Zh ψ), W R R and perform the integration on scale h: √ ˆ (h) (≤h) +GA J, Zh−1 ψ (≤h) ) P (dψ (h) )P (dA(h) )eV (A √ ˆ (h) ˆ(h) (≤h) +GA J, Zh−1 ψ (≤h) )+W R ×eBφ (A √ ˜h +S (h−1) (J,φ)+V (h−1) (A(≤h−1) +GA J, Zh−1 ψ (≤h−1) ) |Λ|E ≡e √ (h−1)
×eBφ
(A(≤h−1) +GA J,
(h−1)
Zh−1 ψ (≤h−1) )+WR
,
where S (h−1) (J, φ) contains terms depending on (J, φ) but independent of A(≤h−1) , ψ (≤h−1) . Defining S (≥h−1) := S (h−1) + S (≥h) , we immediately see that the inductive assumption is reproduced on scale h − 1. Case h < h∗ . For scales smaller than h∗ , there are no more bosonic fields to be integrated out, and we are left with a purely fermionic theory, with scaling
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dimensions 3 − 2n, 2n being the number of external fermionic legs, see Theorem 2.1 and following lines. Therefore, once that the two-legged subdiagrams have been renormalized and step by step reabsorbed into the free fermionic measure, we are left with a superrenormalizable theory, as in [18,19]. In particular, the four fermions interaction is irrelevant, while the wave function renormalization and the Fermi velocity are modified by a finite amount with respect to their values at h∗ ; that is, if ε¯h∗ = maxk≥h∗ {|eμ,k |, |νμ,k |}: Zh = Zh∗ (1 + O(¯ ε2h∗ )),
vh = vh∗ (1 + O(¯ ε2h∗ )).
(B.9)
Tree expansion for the two-point function. As for the partition function (see Sect. 2), the kernels of the effective potentials produced by the multiscale integration of W[h∗ ,0] (J, φ) can be represented as sums over trees, which in turn can be evaluated as sums over Feynman graphs. Let us consider first the expansion for the two-point Schwinger function. After having taken functional derivatives with respect to φk , φk and after having set J = φ = 0, we get an (h∗ ) expansion in terms of a new class of trees τ ∈ Tk, , with k¯ ∈ (−∞, −1] the ¯ h,N ¯ ¯ > k; ¯ these trees are similar to the ones described in scale of the root and h section 2, up to the following differences. (1) There are N + 2 end–points and two of them, called v1 , v2 , are spe † (≤h −1) cial and, respectively, correspond to Q(hv1 −1) (k) ψk v1 or to (≤hv −1)
(2)
(3) (4) tion
ψ k 2 Q(hv2 −1) (k). The first vertex whose cluster contains both v1 , v2 , denoted by v¯, is on ¯ No R operation is associated to the vertices on the line joining v¯ scale h. to the root. There are no lines external to the cluster corresponding to the root. There are no bosonic lines external to clusters on scale h < h∗ . In terms of the new trees, we can expand the two-point Schwinger funcas: ψk ψ k h∗ =
h k +1
(j)
[Qψ (k)]†
j=hk
+
∞ 0
g (j) (k) (j) Q (k) Zj−1
¯ h−1
S2 (τ ; k),
(B.10)
(h∗ ) ¯ ¯ N =2 h=−∞ k=−∞ τ ∈T
¯ h,N ¯ k,
where hk < 0 is the integer such that M hk ≤ |k| < M hk +1 , and S2 (τ ; k) is defined in a way similar to V (h) (τ ) in (2.23), modulo the modifications described in items (1)–(4) above. Using the bounds described immediately after (B.5), which are valid for k belonging to the support of g (h) (k), and proceeding as in Sect. 2.4, we get bounds on S2 (τ ; k), which are the analogues of Theorem 2.1: ¯ −hk 0 h−1 N M N N ||S2 (τ ; k)|| ≤ (const.) ε¯h∗ . (B.11) ! 2 Zhk (h∗ ) ¯ ¯ h=−∞ k=−∞ τ ∈T ¯
¯ k,h,N
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Notice that the result (B.10) and the bound (B.11) are true for any k such ∗ that |k| ≥ M h ; being the bound (B.11) uniform in h∗ , our result on the two point function (1.6) and (2.33) is obtained by fixing k and taking the limit h∗ → −∞ in (B.10). In order to understand (B.11), it is enough to notice that, as far as dimensional bounds are concerned, the vertices v1 and v2 play the role of two ν ver−1/2 tices with an external line (the φ line) and an extra Zhk M −hk factor each. Moreover, since the vertices on the path Pr,¯v connecting the root with v¯ are not associated with any R operation, we need to multiply the value of the tree ¯ ¯ ¯ ¯ ¯ ¯ (h∗ ) τ ∈ Tk, by M (1/2)(h−k) M (1/2)(k−h) , and to exploit the factor M (1/2)(k−h) ¯ h,N ¯ in order to renormalize all the clusters in Pr,¯v . Therefore, 0
¯ h−1
||S2 (τ ; k)|| ≤ (const.)N
(h∗ ) ¯ ¯ h=−∞ k=−∞ τ ∈T ¯ ¯
N ! 2
k,h,N
ε¯N∗ k¯ h−h ¯ ¯ ¯ k M M M (1/2)(h−k) M −2hk × h Zhk ¯ ¯ ¯
(B.12)
h≤hk k≤h
¯
where the factor M k is due to the fact that graphs associated to the trees ¯ (h∗ ) τ ∈ Tk, have two external lines; the factor M h−hk is given by the product ¯ h,N ¯ of the two short memory factors associated to the two paths connecting v¯ with ¯ ¯ v1 and v2 , respectively; the “bad” factor M (1/2)(h−k) is the price to pay to and the last M −2hk are due to the renormalize the vertices in Pr,¯v ; the Zh−1 k −1/2
fact that v1 , v2 behave dimensionally as ν vertices times an extra Zhk M −hk ¯ in (B.12), we get (B.11). Note factor. Performing the summation over k¯ and h ∗ also that, if k is on scale hk h , then the derivatives of ||S2 (τ ; k)|| can be dimensionally bounded as 0
¯ h−1
(h∗ ) ¯ ¯ h=−∞ k=−∞ τ ∈T ¯ ¯
||∂kn S2 (τ ; k)|| ≤ (const.)N ε¯N h∗
−(1+n)hk N M , ! 2 Zhk
(B.13)
k,h,N
(N )
from which the bound on r˜μ (k, p) stated in (4.12) immediately follows. ∗ Tree expansion for the three-point function. Let us pick |k| = M h , ∗ ∗ |k + p| ≤ M h and |p| M h , which is the condition that we need in order to apply Ward Identities in the form described in Sect. 4. In this case, the expansion of three-point function jμ,−p ; ψk+p ψ k h∗ is very similar to the one just described for the two-point function. The result can be written in the form ∗ ∗ e¯μ,h∗ (h∗ −1) [Gψ (k + p)]† γμ g (h ) (k)Q(h ) (k) e S3 (τ ; k, p) (B.14) +
jμ,−p ; ψk+p ψ k h∗ = i
¯ h, ¯ τ ∈T (h∗ ) N ≥1, k< ¯ h,h ¯ v ,N ∗ k, ¯ h≤h 3 hv3 >h∗
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is a new class of trees, with k¯ < 0 the scale of the root,
where Tk, ¯ h,h ¯
v3 ,N
(h∗ )
, up to the fact that they have N + 3 endpoints similar to the trees in Tk, ¯ h,N ¯ rather than N + 2 (see item (1) in the list preceding (B.10)); three of them are special: v1 and v2 are associated to the same contributions described in item (≤h ) (1) above, while v3 is associated to a contribution Zhv¯3 −1 (eμ,hv¯3 /e)jμ,−pv¯3 − M hv¯3 (νμ,hv¯3 /e)Aμ,−p , with v¯3 the vertex immediately preceding v3 on τ (which the endpoint v3 is attached to) and hv3 > h∗ . The value of the tree, S3 (τ ; k, p), is defined in a way similar to S2 (τ ; k), modulo the modifications described above. S3 (τ ; k, p) admits bounds analogous to (B.11)–(B.12); recalling that ∗ ∗ ∗ |k| = M h , |k + p| ≤ M h and |p| M h , we find: ¯ h−1
∗
h
1
||S3 (τ ; k, p)||
¯ ¯ hv3 =h∗ +1 τ ∈T (h∗ ) h=−∞ k=−∞ ¯ ¯
k,h,hv ,N 3
N 1 ≤ (const.)N ! ε¯N h∗ 2 Zh∗ −1 ∗ ∗ ∗ ¯ h) ¯ ¯ (1/2)(k− × M M h−h M (1/2)(h −hv3 ) M −2h ,
(B.15)
∗ ¯ h≤h ¯ h ¯ k< hv3 >h∗
¯ ¯
where M (1/2)(k−h) is the short memory factor associated to the path between ∗ ¯ the root and v¯; M h−h is the product of the two short memory factors associ∗ ated to the paths connecting v¯ with v1 and v2 , respectively; M (1/2)(h −hv3 ) is ∗ the short memory factor associated to a path between h∗ and v3 ; M −2h /Zh∗ −1 −1/2 is the product of two factors M −hk Zhk −1 associated to the vertices v1 and v2 (see the discussion following (B.11) and recall that in this case hk = h∗ ). We remark that in this case, contrary to the case of the two-point function, the fact that there is no R operator acting on the vertices on the path between the root and v¯ does not create any problem, since those vertices are automatically irrelevant (they behave as vertices with at least five external lines, i.e., J, φ, φ¯ and at least two fermionic lines) and, therefore, R = 1 on them. Note also that the vertices of type Jφψ, which have an R operator acting on, can only be on scale h∗ − 1 or h∗ (by conservation of momentum) and, therefore, the action of the R operator on such vertices automatically gives the usual dimensional ¯ h, ¯ hv in gain of the form const. M hv −hv . Performing the summations over k, 3 (B.15), we find the analogue of (B.11): ∗
h
¯ h−1
1
||S3 (τ ; k, p)||
¯ ¯ hv3 =h∗ +1 τ ∈T (h∗ ) h=−∞ k=−∞ ¯ ¯
≤ (const.)N (N )
k,h,hv ,N 3 ∗
N M −2h , ! ε¯N h∗ 2 Zh∗ −1
from which the bound on rμ (k, p) stated in (4.12).
(B.16)
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Appendix C. Lowest Order Computations In this Appendix we reproduce the details of the second order computations leading to (3.7)–(3.10). z,(2)
C.1. Computation of βh
(h)
By definition, see (2.7) and (3.1), βhz = z0,h = −iγ0 ∂k0 W0,1 (0). At one-loop, defining e¯0,h = e0,h and e¯1,h = vh−1 e1,h , we find: fh+1 (p) dp z,(2) (2) 2 = z0,h = −iγ0 e¯μ,h+1 ∂p βh γμ g (h+1) (p)γμ (2π)3 0 2|p| 2 Zh+1 fh+2 (p) dp − iγ0 e¯2μ,h+2 ∂ γμ g (h+1) (p)γμ p Zh (2π)3 0 2|p| fh+1 (p) Zh+1 dp − iγ0 e¯2μ,h+2 ∂ γμ g (h+2) (p)γμ . (C.1) p Zh (2π)3 0 2|p| Using inductively the beta function equations for Zh+1 , vh+1 , eμ,h+2 , and neglecting higher order terms, we can rewrite (C.1) as 1 dp p20 iγμ γ0 γμ (2) z0,h = iγ0 e¯2μ,h+1 2 (2π)3 |p|3 p20 + vh2 | p|2 × (fh+1 (p) − |p|fh+1 (p))(fh+1 (p) + fh+2 (p)) (C.2) + (fh+2 (p) − |p|fh+2 (p))fh+1 (p) . Passing to radial coordinates, p = p(cos θ, sin θ cos ϕ, sin θ sin ϕ), and using the fact that dp(fh+1 fh+1 + fh+1 fh+2 + fh+2 fh+1 ) = 0, we find: ⎡∞ ⎤ dp 1 (2) z0,h = (2vh2 e21,h − e20,h ) 2 ⎣ (f 2 + 2fh+1 fh+2 )⎦ 8π p h+1 0 ⎤ ⎡ 1 2 θ cos ⎦. (C.3) × ⎣ d cos θ cos2 θ + vh2 sin2 θ −1
The integral over the radial coordinate p can be computed using the definition (2.1): ∞ 0
dp 2 (f + 2fh+1 fh+2 ) p h+1 ∞
= 0
dp [2(χ(p) − χ(M p)) − (χ2 (p) − χ2 (M p))] = lim ε→0 p
M ε
dp = log M. p
ε
(C.4) Finally, an explicit evaluation of the integral over d cos θ leads to (3.19).
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(2)
C.2 Computation of z1,h v,(2)
By definition, see formulas (2.7) and (3.1), βh (h)
(2)
(2)
= z1,h − vh z0,h , with z1,h =
−iγ1 ∂k1 W0,1 (0). At second order, proceeding as in the derivation of (C.2), we find: 1 (2) z1,h = iγ1 e¯2μ,h+1 2 dp p21 iγμ vh γ1 γμ × (fh+1 (p) − |p|fh+1 (p))(fh+1 (p)+fh+2 (p)) (2π)3 |p|3 p20 + vh2 | p|2 + (fh+2 (p) − |p|fh+2 (p))fh+1 (p) ⎤ ⎡∞ 1 ⎣ dp 2 2 = e0,h vh (fh+1 + 2fh+1 fh+2 )⎦ 16π 2 p 0 ⎤ ⎡ 1 2 sin θ ⎦. (C.5) × ⎣ d cos θ cos2 θ + vh2 sin2 θ −1
An explicit evaluation of the integral leads to (2) z1,h
=
log M e20,h vh−1 2
8π
v,(2)
which, combined with βh
(2)
arctan ξh ξh − arctan ξh − ξh ξh3
,
(C.6)
(2)
= z1,h − vh z0,h , leads to (3.19).
ν,(2)
C.3 Computation of βμ,h
(h−1)
ν By definition, see (2.6) and (3.1), βμ,h = −M −h+1 W2,0,μ,μ (0) − M νμ,h . At second order, we find:
ν,(2) βμ,h
dp (h) (h) Tr γ g (p)γ g (p) μ μ 2 (2π)3 Zh dp (h+1) (h) Tr γ g (p)γ g (p) . −M −h+1 e¯2μ,h+1 μ μ Zh−1 (2π)3
= −M
e¯2 −h+1 μ,h
(C.7)
Using inductively the beta function equations for eμ,h , Zh−1 , vh−1 , and neglecting higher orders, we can rewrite (C.7) as ν,(2) β0,h ν,(2)
β1,h
dp fh (p)2 + 2fh (p)fh+1 (p) (−p20 + vh2 | p|2 ), (2π)3 (p20 + vh2 | p|2 )2 (C.8) dp fh (p)2 + 2fh (p)fh+1 (p) 2 −h+1 2 = −2M e¯1,h p0 , (2π)3 (p20 + vh2 | p|2 )2 =
−2M −h+1 e20,h
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where we used that Tr (γμ γα γμ γα ) = −4 if μ = α and 4 otherwise; passing to radial coordinates we find ⎤ ⎡∞ ' & −2 ν,(2) β0,h = M −h+1 e20,h ⎣ dp fh2 + 2fh fh+1 ⎦ (2π)2 0
1 ×
d cos θ
−1
ν,(2)
β1,h
− cos2 θ + vh2 sin2 θ , (cos2 θ + vh2 sin2 θ)2
⎤ ⎡∞ ' & −2 = M −h+1 e¯21,h ⎣ dp fh2 + 2fh fh+1 ⎦ (2π)2
(C.9)
0
1 ×
d cos θ
−1
cos2 θ . (cos2 θ + vh2 sin2 θ)2
The integral over the radial coordinate p can be rewritten as, using the definition (2.1): ∞ dp
&
fh2
'
+ 2fh fh+1 = M
0
∞ h−1
(M − 1)
& ' dp 2χ(p) − χ2 (p) . (C.10)
0
Finally, an explicit evaluation of the integral over d cos θ leads to (3.7).
Appendix D. Multiscale Integration of the Correction Term to the WI In this Appendix we prove (4.7) and the bound (4.8). We assume that |k| = M h and |p| M h . We start by rewriting pμ ∂3 0[h,0] (J, ˜ φ) ˜ W Rμ,h (k, p) = , J=φ=0 ˜ Zh ∂ Jp ∂φk+p ∂φk ˜ φ) defined as: 0[h,0] (J, with W 0 ˜ ˜ eW[h,0] (J,φ) := P (dψ)P[h,0] (dA) eV (A,ψ)+B(J,φ) , and J, ˜ φ) = B(
dp ˜ dk J ψ C(k, p)ψ − α p j p k μ μ μ,−p (2π)3 (2π)3 k+p dk + φk ψ k + φk ψk . 3 (2π)
(D.1)
(D.2)
(D.3)
The main difference with respect to the generating functional of the correlation functions is the presence of the correction term proportional to C(k, p), see (4.6) for a definition. Equation (D.2) can again be studied by RG methods, see
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[6] for further details. A crucial role is played by the properties of the function C(k, p); it is easy to verify that g (i) (k + p)C(k, p)g (j) (k)
(D.4)
is non-vanishing only if at least one of the indices i, j is equal to 0; moreover, when it is non-vanishing, it is dimensionally bounded from above by (const.)|p|M −i−j . We start by integrating the scale 0, and we find: 0
˜
(≥−1) (J,φ) ˜
(J,φ) = e|Λ|E−1 +S eW[h,0]
×
P (dψ (≤−1) )P[h,−1] (dA(≤−1) ) eV
(−1)
(A(≤−1) ,
√
Z−1 ψ (≤−1) )+B(−1)
,
(D.5)
˜ φ but independent of A, ψ, where S(≥−1) collects the terms depending on J, and (−1) (−1) 0 (−1) , B(−1) (A, ψ) = BJ (A, ψ) + Bφ (A, ψ) + W (D.6) R (−1) (−1) with: BJ (A, ψ) linear in J˜ and independent of φ; Bφ (A, ψ) given by (B.3); 0 (−1) the rest, which is at least quadratic in (J, ˜ φ). With respect to the compuW R ¯ which ˜ tation of W[h,0] (J, φ), we now have new marginal terms of the form J˜ ψψ,
are contained in BJ (A, ψ) and need to be renormalized. Let us symbolically (−1) 0m,n ({ki }, {qi }, p) the generic non-trivial kernel appearing in represent by W (−1) BJ (A, ψ); m is the number of bosonic external lines while 2n is the number of ψ fields; {ki }, {qi } are, respectively, the fermionic/bosonic momenta and p ˜ As usual, these new kernels can be repreis the momentum flowing through J. sented as sums over Feynman graphs. The J˜ external line can be attached to a (≤0) simple vertex, corresponding to the monomial −αμ pμ J˜p jμ,−p , or to a “thick” vertex, representing J˜p ψ k+p C(k, p)ψk (the “small circle” associated to the ver(−1)
(−1),C
tex represents the matrix kernel C(k, p), see Fig. 6). Let us denote by Wm,n (−1) 0m,n the contribution to W coming from graphs with the J˜ line attached to a thick vertex, see Fig. 6. By the properties of the C(k, p) function, see [6] for (−1) (−1) ¯ m,n,μ 0m,n details, it follows that W ({ki }, {qi }, p) =: pμ W ({ki }, {qi }, p), with (−1) (−1) ¯ Wm,n,μ dimensionally bounded as a Wm+1,n kernel, uniformly in p. We define (−1) ¯ m,n,μ in a way similar to (2.5)–(2.7). the action of the R ≡ 1 − L operator on W ¯ (−1) (k, p) := W ¯ (−1) (0, 0) and, by symmetry, In particular, LW 0,1,μ
0,1,μ
¯ (−1) ψk Z−1 J˜p ψ k+p LW 0,1,μ
(≤−1)
= −Z−2 J˜p αμ,−1 jμ−p ,
(D.7)
for a real constant αμ,−1 , which is by definition the effective α-coupling on scale −1. Note that the last two graphs in Fig. 6 do not contribute to αμ,−1 simply because they are one-particle reducible and, therefore, they are vanishing at zero external momenta. We now iterate the same procedure, and step by step the local parts of ¯ are collected together to form a new running couthe kernels of type J˜ψψ pling constant, αμ,k ; in order to show that Rμ,h is dimensionally negligible as
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q+p (−1),C W0,1 =
p +
+
+
k q
Figure 6. Schematic representation of the expansion for (−1),C W0,1 ; the small circle represents C(k, p)
h → −∞, we need to show that it is possible to fix the initial data αμ = αμ,0 in such a way that αμ,h goes exponentially to zero as h → −∞, which is proved in the following. The flow of αμ,k . The new marginal running coupling constants αμ,h α evolve according to the flow equation: αμ,k−1 = αμ,k + βμ,k , where αμ,0 = αμ are the counterterms appearing in the bare interaction (D.3). The beta funcα can be split as tion βμ,h α,1 α,2 α βμ,k = βμ,k + βμ,k ,
(D.8)
α,1 where βμ,k collects the contributions independent of αμ,k (which, therefore, are associated to graphs with the J˜ external line emerging from the α,2 thick vertex representing C(k, p)), and βμ,k collects the terms from graphs with one vertex of type αμ,k for some k > k. It is crucial to recall α,1 that by the properties of C(k, p), the graphs contributing to βμ,k have at least one propagator on scale 0 or −1; by the short memory property, this means that they can be dimensionally bounded by (const.)¯ ε2k M θk , for any α,2 0 < θ < 1. Similarly, the contributions to βμ,k associated to graphs with at least one vertex of type αμ,k for some k > k can be bounded by (const.)¯ ε2k |αμ,k |M θ(k−k ) . The counterterms αμ are fixed in such a way that 0 α,1 α,2 αμ,−∞ = 0, i.e., αμ = − k=−∞ (βμ,k + βμ,k ). Finally, using the fact that α,1 α,2 |βμ,k | ≤ (const.)¯ ε2k M θk and |βμ,k | ≤ (const.) k >k ε¯2k |αμ,k |M θ(k−k ) , we find that |αμ,h | ≤ (const.)¯ ε2h∗ M (θ/2)h . This dimensional estimate on αμ,h easily implies the desired estimate on Rμ,h (k, p) stated in (4.8) and we will not belabor the details here. (2) α,1,(2) Lowest order computation of αμ . At lowest order, αμ = − k≤0 βμ,k , α,1,(2)
where βμ,k
α,1,(2)
α,1 is the one-loop contribution to βμ,k . Moreover, βμ,k
all k ≤ −1. Therefore, neglecting higher order terms, we find
(2) αμ
= 0 for α,1,(2)
= −βμ,0
,
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J
0
Figure 7. Lowest order contribution to α0 , α1 that is (see Fig. 7): (2) α0
(2)
α1
=
−iγ0 e¯2ν,0
( ) dk (0) (0) g γ ∂ (k + p)C(k, p)g (k) ν p 0 (2π)3 p=0
×γν w(0) (k), ( ) iγ1 2 dk e¯ν,0 =− γν ∂p1 g (0) (k + p)C(k, p)g (0) (k) 3 v (2π) p=0 ×γν w(0) (k).
(D.9)
(D.10)
After a straightforward computation, using the fact that ( ) ∂pμ g (0) (k + p)C(k, p)g (0) (k) p=0
1 1 [−i¯ γμ χ0 (k) (1 − χ0 (k)) + i k∂μ χ0 (k)] , = i k i k where k = k0 γ0 + vk · γ and (¯ γ0 , γ¯1 , γ¯2 ) = (γ0 , vγ1 , vγ2 ), we finally get (3.26)– (3.27).
References [1] Adler, S., Bardeen, W.: Absence of higher-order corrections in the anomalous axial-vector divergence equation. Phys. Rev. 182, 1517–1536 (1969) [2] Benfatto, G., Falco, P., Mastropietro, V.: Universal relations for nonsolvable statistical models. Phys. Rev. Lett. 104, 075701 (2010) [3] Benfatto, G., Gallavotti, G.: Perturbation theory of the Fermi surface in a quantum liquid. A general quasiparticle formalism and one-dimensional systems. J. Stat. Phys. 59, 541–664 (1990) [4] Benfatto, G., Gallavotti, G.: Renormalization Group. Princeton University Press, NJ (1995) [5] Benfatto, G., Gallavotti, G., Procacci, A., Scoppola, B.: Beta function and Schwinger functions for a many fermions system in one dimension. Anomaly of the fermi surface. Commun. Math. Phys. 160, 93–171 (1994) [6] Benfatto, G., Mastropietro, V.: Ward identities and chiral anomaly in the Luttinger liquid. Commun. Math. Phys. 258, 609–655 (2005)
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[7] Benfatto, G., Giuliani, A., Mastropietro, V.: Fermi liquid behavior in the 2D Hubbard model at low temperatures. Ann. Henri Poincar´e 7, 809–898 (2006) [8] Bonini, M., D’Attanasio, M., Marchesini, G.: Ward identities and Wilson renormalization group for QED. Nucl. Phys. B 418, 81–112 (1994) [9] Bostwick, A., Ohta, T., Seyller, T., Horn, K., Rotenberg, E.: Quasiparticle dynamics in graphene. Nature Phys. 3, 36–40 (2007) [10] Castro Neto, A.H., Guinea, F., Peres, N., Novoselov, K., Geim, K.: The electronic properties of graphene. Rev. Mod. Phys. 81, 109 (2009) [11] Disertori, M., Rivasseau, V.: Interacting Fermi liquid in two dimensions at finite temperature. Part I: convergent attributions Commun. Math. Phys. 215, 251– 290 (2000) [12] Disertori, M., Rivasseau, V.: Interacting Fermi liquid in two dimensions at finite temperature. Part II: renormalization 215, 291–341 (2000) [13] Feldman, J., Knoerrer, H., Trubowitz, E.: Commun. Math. Phys. 247, 1–320 (2004) [14] Feldman, J., Trubowitz, E.: Perturbation theory for many fermion systems. Helvetica Phys. Acta 63, 156–260 (1990) [15] Gallavotti, G.: Renormalization theory and ultraviolet stability for scalar fields via renormalization group methods. Rev. Mod. Phys. 57, 471–562 (1985) [16] Gentile, G., Mastropietro, V.: Renormalization group for one-dimensional fermions. A review on mathematical results. Phys. Rep. 352, 273–437 (2001) [17] Giuliani, A., Mastropietro, V.: Anomalous critical exponents in the anisotropic Ashkin–Teller model. Phys. Rev. Lett. 93, 190603 (2004) [18] Giuliani, A., Mastropietro, V.: The two-dimensional Hubbard model on the honeycomb lattice. Commun. Math. Phys. 293, 301–346 (2010) [19] Giuliani, A., Mastropietro, V.: Rigorous construction of ground state correlations in graphene: renormalization of the velocities and Ward identities. Phys. Rev. B 79, 201403(R) (2009) [20] Gonz´ alez, J., Guinea, F., Vozmediano, M.A.H.: Non-Fermi liquid behavior of electrons in the half-filled honeycomb lattice (a renormalization group approach). Nucl. Phys. B 424, 595–618 (1994) [21] Gonz´ alez, J., Guinea, F., Vozmediano, M.A.H.: Marginal-Fermi-liquid behavior from two-dimensional Coulomb interaction. Phys. Rev. B 59, R2474 (1999) [22] Gonz´ alez, J., Guinea, F., Vozmediano, M.A.H.: Electron–electron interactions in graphene sheets. Phys. Rev. B 63, 134421 (2001) [23] Herbut, I.F.: Interactions and phase transitions on graphene’s honeycomb lattice. Phys. Rev. Lett. 97, 146401 (2006) [24] Herbut, I.F., Juricic, V., Roy, B.: Theory of interacting electrons on the honeycomb lattice. Phys. Rev. B 79, 085116 (2009) [25] Jiang, Z., Henriksen, E.A., Tung, L.C., Wang, Y.-J., Schwartz, M.E., Han, M., Kim, P., Stormer, H.L.: Infrared spectroscopy of Landau levels of graphene. Phys. Rev. Lett. 98, 197403 (2007) [26] Kotov, V.N., Uchoa, B., Castro Neto, A.H.: Electron–electron interactions in the vacuum polarization of graphene. Phys. Rev. B 78, 035119 (2008) [27] Li, G., Luican, A., Andrei, E.: Scanning tunneling spectroscopy of graphene on graphite. Phys. Rev. Lett. 102, 176804 (2009)
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[28] Mastropietro, V.: Non-Perturbative Renormalization. World Scientific, Singapore (2008) [29] Mishchenko, E.G.: Effect of electron–electron interactions on the conductivity of clean graphene. Phys. Rev. Lett 98, 216801 (2007) [30] Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Katsnelson, M.I., Grigorieva, I.V., Dubonos, S.V., Firsov, A.A.: Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197 (2005) [31] Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Zhang, Y., Dubonos, S.V., Gregorieva, I.V., Firsov, A.A.: Electric field effect in atomically thin carbon films. Science 306, 666 (2004) [32] Polchinski, J.: Renormalization and effective lagrangians. Nucl. Phys. B 231, 269 (1984) [33] Rivasseau, V.: From Perturbative to Constructive Renormalization. Princeton University Press, NJ (1991) [34] Salmhofer, M.: Renormalization: An Introduction, Texts and Monographs in Physics. Springer, Berlin (1999) [35] Shankar, R.: Renormalization-group approach to interacting fermions. Rev. Mod. Phys. 66, 129 (1994) [36] Son, D.T.: Quantum critical point in graphene approached in the limit of infinitely strong Coulomb interaction. Phys. Rev. B 75, 235423 (2007) [37] Zhou, S., Siegel, D., Fedorov, A., Lanzara, A.: Kohn anomaly and interplay of electron– electron and electron–phonon interactions in epitaxial graphene. Phys. Rev. B 78, 193404 (2008) Alessandro Giuliani Universit` a di Roma Tre L.go S. L. Murialdo 1 00146 Rome, Italy e-mail: [email protected] Vieri Mastropietro Universit` a di Roma Tor Vergata V.le della Ricerca Scientifica 00133 Rome, Italy e-mail: [email protected] Marcello Porta Universit` a di Roma La Sapienza P.le Aldo Moro 2 00185 Rome, Italy e-mail: [email protected] Communicated by Jean Bellissard. Received: June 9, 2010. Accepted: October 5, 2010.
Ann. Henri Poincar´e 11 (2010), 1453–1485 c 2010 Springer Basel AG 1424-0637/10/081453-33 published online December 9, 2010 DOI 10.1007/s00023-010-0067-y
Annales Henri Poincar´ e
Symmetry Breaking in Quasi-1D Coulomb Systems Michael Aizenman, Sabine Jansen and Paul Jung Abstract. Quasi 1D systems are systems of particles in domains which are of infinite extent in one direction and of uniformly bounded size in all other directions, e.g., a cylinder of infinite length. The main result proven here is that for such particle systems with Coulomb interactions and neutralizing background, the so-called “jellium”, at any temperature and at any finite-strip width, there is translation symmetry breaking. This extends the previous result on Laughlin states in thin, 2D strips by Jansen et al. (Commun Math Phys 285:503–535, 2009). The structural argument which is used here bypasses the question of whether the translation symmetry breaking is manifest already at the level of the one particle density function. It is akin to that employed by Aizenman and Martin (Commun Math Phys 78:99–116, 1980) for a similar statement concerning symmetry breaking at all temperatures in strictly 1D Coulomb systems. The extension is enabled through bounds which establish tightness of finite-volume charge fluctuations.
1. Introduction In this work, we investigate symmetry breaking in classical quasi 1D “jellium”, i.e., particle systems with Coulomb repulsion and attractive neutralizing background (also known as “one-component plasma”) and in quantum systems whose states may be described by such ensembles. The particles are of equal charge −q and move in domains which are of infinite extent in one direction and of uniformly bounded size in all other directions, e.g., cylinder or tube of M. Aizenman supported in part by NSF grant DMS-0602360, and BSF grant 710021 on a visit to the Weizmann Institute. S. Jansen supported in part by DFG Forschergruppe 718 “Analysis and Stochastics in Complex Physical Systems”, NSF grant PHY-0652854 and a Feodor Lynen research fellowship of the Alexander von Humboldt-Stiftung. P. Jung supported in part by Sogang University research grant 200910039.
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infinite length and a finite, uniform, cross-section. The Coulomb potential is the solution to a Poisson equation with Neumann or periodic boundary conditions in the confined directions. It corresponds to the situation where not only the particles but also the electric field is confined to the tube. Our main result is that such systems display translational symmetry breaking in the long direction, e.g., the cylinder axis, which is denoted here by x. This generalizes previously known results in one and two dimensions. The proof is by a structural argument which is not limited to low temperatures or small tube cross-sections. For the 1D jellium, symmetry breaking was shown in [2,4,7]. In that case (but not for d > 1, see the discussion in Sect. 9), the phenomenon discussed here expresses the formation of a “Wigner lattice”. Roughly, the Coulomb interaction leads to a strong suppression of large scale deviations from neutrality. One finds that each particle, as ranked by the x coordinate, fluctuates with only bounded mean value for its distance from the lattice site corresponding to its rank. In two dimensions, with periodic boundary conditions in the confined direction, symmetry breaking is known for some special cases corresponding to even-integer values of the so-called “plasma parameter” βq 2 (with β the inverse temperature). When βq 2 = 2, the model is explicitly solvable, and periodicity was shown in [5]. A proof of symmetry breaking for other even-integer values βq 2 = 2p and sufficiently small values of the strip width (compared to (density)−1/2 ) was given in [6], where the focus was on the Laughlin states in cylindrical geometry. Numerical results may be found in [14]. The case of even-integer values of the plasma parameter is of an additional interest, and further tools are available for it as it relates to Laughlin’s wave function [8] which is frequently used as an approximate ground state for electrons in the context of the fractional quantum Hall effect. The function models the state of an electron gas whose filling factor (a quantity related to the electron density) is a simple fraction 1/p with p an integer. The function’s modulus squared is proportional to the Boltzmann weight of a classical onecomponent plasma: |Ψ|2 ∝ exp(−βU ). The filling factor of an electron gas and the plasma parameter of a classical Coulomb system are related by βq 2 = 2p. The solvable case βq 2 = 2 corresponds to p = 1 (filled lowest Landau level), which models non-interacting fermions. The integrality of p enters the proof in [6] in a crucial way, allowing to expand pth powers of polynomial into monomials. The proof given here follows an altogether different approach, along the lines of [1,2]. A key point is that due to the strong tendency of Coulomb systems to maintain bulk neutrality an interesting phenomenon occurs in 1D and quasi 1D Coulomb systems: the total charge in cylinders of arbitrary length is of bounded variance. Even in the infinite-volume limit the question “what is the total charge at points with x ≤ x0 ?” has a well-defined answer (denoted here by qK(x, ω)). As was pointed out in [1] that fact in itself implies symmetry breaking, through the long-range correlations in the values of the phase ei2πK(x,ω) , or equivalently through the fractional part of
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the total charge below x0 . This approach to symmetry breaking was developed in the earlier work on strictly 1D Coulomb systems [2], in which case qK(x, ω) yields the electric field at x. In [1], this was extended into a general criterion that in one dimension, “tightness” of charge fluctuations implies symmetry breaking. To make this argument applicable in our setting, one needs to first establish the tightness estimates. These are somewhat more involved than in the strictly 1D case. Curiously, in both [6] and here, symmetry breaking is explained through a combination of analytical and topological arguments. However, these seem to be of somewhat different nature in these two works. The paper’s outline is as follows. The model is introduced in Sect. 2. The main results are stated in Sect. 3 that includes the statement of Theorem 3.1 and its Corollary 3.2 which addresses the symmetry breaking in the model’s infinite-volume Gibbs states. The results are enabled by a pair of estimates which play an essential role: Theorems 3.3 and 3.4. We then turn to the mathematical framing of the particle-excess function K(x, ω), which it is convenient to view alternatively as a random element of a Skorokhod space and as a cocycle in the sense of dynamical systems. These terms are discussed in Sect. 4 and used in Sect. 5 for a conditional proof of symmetry breaking, assuming the tightness bounds. The proof of the latter involves a different set of considerations. These are outlined in Sect. 6, which reviews the corresponding question in one dimension. Finally, the proof of the enabling bounds is spelled out in Sect. 7, where we establish tightness of the distribution of K(x, ω) at fixed x, and in Sect. 8 where it is shown that the volume-averages of K(x, ω) tend to null. We end with a few additional remarks in Sect. 9.
2. Coulomb Interaction in Quasi 1D Systems 2.1. The Potential Function The quasi 1D systems considered here are systems of particles which, along with the electric field they generate, are confined to a tubular region of the form T = R × D where D is a compact subset of Rk , possibly with some periodic boundary conditions. The precise technical assumption is spelled out below. In the simplest example, T is a strip whose cross-section is D = [0, W ] with periodic boundary conditions. Points on T will be denoted z = (x, y) with x ∈ R and y ∈ D. For simplicity, we denote the volume-form on D by dy and its total measure by W = D dy. The Coulomb potential between two points is given by a symmetric function, V (z, z ) = V (z , z), which satisfies: − ΔV (z, z ) = δ(z − z ) 2
(1)
∂ for −Δ = − ∂x 2 − ΔD , with ΔD the Laplacian on D which is taken here to be defined with either periodic or Neumann boundary conditions.
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Since T has locally the structure of Rd with d = 1 + k, the short distance behavior of the potential at interior points of D is [(d − 2)Cd ]−1 dist(z, z )−(d−2) for d = 2 V (z, z ) ≈ . (2) for d = 2 −(2π)−1 ln dist(z, z ) Yet, at long distances V (z, z ) behaves as a 1D Coulomb potential: V (z, z ) ≈ −|x − x |/(2W ),
(3)
in a sense which we shall now make more explicit. In the example of the 2D periodic strip, it is convenient to use the complex notation: z = x + iy, in terms of which V (z, z ) = −(2π)−1 log |2 sinh(π(z − z )/W )|.
(4)
This function is clearly periodic in y = Imz and harmonic throughout the (periodic) strip except at z = 0, and it can be easily seen to satisfy (1). At short distances, V (z, z ) behaves as the 2D Coulomb potential, with logarithmic divergence, but its long distance behavior is close to that in one dimension and better described by the decomposition: V (z, z ) = −|x − x |/(2W ) + V2 (y, y ; |x − x |)
(5)
with the correction to the linear term given by
V2 (y, y ; |x − x |) = −(2π)−1 log |1 − e−2π(|x−x |+i|y−y |)/W |,
(6)
which decays exponentially in |x − x |. In the more general case, the potential admits the eigenfunction expansion: √ V (z, z ) = −|x − x |/(2W ) + (2 En )−1 e−|x−x | En ϕn (y)ϕn (y ) n≥1
=: −|x − x |/(2W ) + V2 (y, y ; |x − x |).
(7)
in terms of the eigenfunctions of Δ: −ΔD ϕn (y) = En ϕn (y). We may now state our assumptions on D, which are: 1.
2.
The correspondingly periodic/Neumann Laplacian ΔD is a self adjoint operator with a non-degenerate ground state (ϕ(y) = W −1/2 ) and compact resolvent. The Coulomb potential in T is of the form (7) whose second term averages to zero over D: V2 (y, y ; |x − x |) dy = 0, (8) D
3.
for any x, x ∈ R and y ∈ D. The term V2 admits bounds of the form V2 (y, y ; |x|) ≥ −g(|x|)
(9)
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and for each δ > 0 there exists c(δ) > 0 such that V2 (y, y ; |x|) ≤ cg(|x|) for all |x| ≥ δ
(10)
for g(x) a positive decreasing function on [0, ∞), which satisfies the ‘finite energy condition’: ∞ x g(x) dx < ∞.
(11)
0
The boundary conditions ensure that the electric flux lines (i.e., lines tangent to ∇V ) do not leave the tube, see Fig. 1. The averaging condition (8) is equivalent to saying that the interaction of a uniform slab with a point charge does not depend on the charge’s position within the tube’s cross-section. In terms of the eigenfunction expansion, this is implied by the constancy of the ground state ϕ0 , to which all other eigenstates ϕn are orthogonal. The decay in |x| of the individual terms is due to the spectral gap. However, one has to control the sum in (7), whose terms need not be uniformly bounded (in L∞ ). We owe to Rupert Frank the comment that conditions (9) and (10) hold for compact Lipschitz domains. In R2 , this class includes compact domains with piecewise differentiable boundary, which may exhibit discontinuities in the tangent’s direction, but no “horn singularities” of vanishing angle. For bounded domains D with smooth boundary, these conditions hold with g(x) of exponential decay. 2.2. Quasi 1D Jellium Jellium in a finite segment, corresponding to T[L1 ,L2 ] = [L1 , L2 ] × D
with L1 , L2 ∈ R,
consists of a collection of N ≈ ρ(L2 − L1 )W particles of charge (−q) each, with ρ the mean number of particles per volume, moving in a neutralizing background of homogeneous charge density qρ. We shall denote a particle configuration by ω = (z1 , . . . , zN ), with zj = (xj , yj ) ∈ T[L1 ,L2 ] and the labeling chosen in the increasing order of the (N )
x-coordinates. Thus, the configuration space is Ω[L1 ,L2 ] = (N, [L1 , L2 ]) × DN with (N, [L1 , L2 ]) the simplex {x ∈ RN : L1 < x1 < · · · < xN < L2 }.
Figure 1. Flux lines of the potential with Neumann boundary conditions
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Using (7) and (8), one gets for the system’s energy, discarding the finite self-interaction of the fixed background: U (z1 , . . . , zN ) =
1≤j
=
1≤j
2
q V (zj , zk ) −
N j=1
2
q ρ
V (zj , z) dz
T[L1 ,L2 ]
N 1 L1 + L2 2 xj − q 2 V (zj , zk ) + q 2 ρ + Const(N, L), (12) 2 2 j=1
with dz = dx dy the Lebesgue measure on T[L1 ,L2 ] and L = L2 − L1 . The jellium’s Gibbs equilibrium state, at the inverse temperature β ≡ (N ) is the probability measure on Ω[L1 ,L2 ] : e−βU (ω) dω[L1 ,L2 ] /Z(β, N, L)
1 kT
,
(13) (N ) Ω[L1 ,L2 ] .
The norwhere dω[L1 ,L2 ] denotes the product Lebesgue measure on malizing factor Z(β, N, L) is the finite volume partition function. One may note that for D = [0, W ] with the periodic boundary conditions and βq 2 = 2p with p an integer, this measure coincides with the Laughlin states, |Ψp (z1 , . . . , zN )|2 , which were considered in [16]. Dimensionless parameters and choice of units. One could, for convenience, rescale the length units (uniformly in all directions) so that the particle density is ρ = 1, and, since q affects the state only in the combination βq 2 , absorb q 2 in the units of β thereby setting q = 1. With this choice, one is still left with dependency on two parameters, β and W . We shall follow this choice when referring to constants which appear in bounds throughout the paper as C(β, W ). However, in other places, we shall leave the ρ and q dependence explicit. An explicitly relevant length scale for us is λ := (ρW )−1 .
(14)
This is the length of a cylindrical cell in which the mean number of particles is one. It is also the basic scale for the translation symmetry breaking proven here. It may be added that for dimensionless parameters of the model one may chose W 1/(d−1) /λ = ρW d/(d−1) and the “plasma parameter”, sometimes called the coupling constant, Γd = βq 2 ρ(d−2)/d .
3. Statement of the Main Results 3.1. Symmetry Breaking Our main result is the following statement concerning the infinite-volume limits of the finite volume Gibbs equilibrium states of the quasi 1D systems in the regions T[L1 ,L2 ] .
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As is explained by the result, it is natural to take the boundaries of the finite regions as: L1 = (−n1 − θ) λ,
L2 = (n2 − θ) λ
(15)
with n1 , n2 ∈ N, and θ ∈ [0, 1). The corresponding Gibbs equilibrium measure for the neutral systems of N = n1 +n2 particles in such domains will be denoted (β,W,θ) here μn1 ,n2 . For the purpose of convergence statements, these can be viewed as point processes on the common space T = R × D. Theorem 3.1. At all β > 0 and W > 0: 1. For any θ ∈ [0, 1) and any sequence of integer pairs {(n1 (j), n2 (j))}, with (β,W,θ) n1 (j), n2 (j) → ∞, there is a subsequence for which μn1 ,n2 converges to a limit, in the sense of convergence of probability measures on the space of configurations in T (i.e., of point processes). 2. Any two limiting measures which correspond to different values of θ are mutually singular. 3. More explicitly: there exists a measurable function on the space of configurations, such that for each θ ∈ [0, 1): Φ(ω) = ei2πθ
(16)
almost surely with respect to each measure which is a limit of a sequence of finite-volume ‘θ Gibbs states’. Furthermore, under the shifts (induced by Tu : x → x + u) and reflections (induced by R : x → −x): Φ(Tu ω) = Φ(ω) ei2πu/λ
and
¯ Φ(Rω) = Φ(ω).
(17)
This has the following elementary consequence: Corollary 3.2 (Symmetry breaking). 1. None of the limiting measures discussed above is invariant under the shifts / λZ (at λ ≡ (W ρ)−1 ). Tu , with u ∈ 2. Except for the two cases θ = 0, 1/2 (for which θ ≡ 1 − θ(mod 1)), the limiting measures are also not invariant under the reflection R. The derivation of the above statements combines three elements: “statistical mechanical” bounds on the distribution of the ‘particle-excess function’, which are presented below, dynamical systems’ concepts of tight cocycles, and simple topological considerations. A fundamental role in the analysis is played by the ‘particle-excess’ function K(u), which expresses the difference (below u) of the total background charge times q −1 and the number of particles: ⎧ 1 ⎨ u−L − |{j ≤ N : xj ≤ u}| if u ∈ [L1 , L2 ] λ K(u, ω) := (18) ⎩ 0 if u ∈ R\[L1 , L2 ] with | · | the cardinality of the set (see Fig. 2). For a point process on the line, it is rather exceptional that such a quantity has a good infinite-volume limit
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π
−π L1
L2
K(x) L1
L2
Figure 2. A particle configuration on [L1 , L2 ] × D (here D = [−π, π], ρ = 1) along with its associated particle-excess function K(x) (L1 → −∞, L2 → ∞). When it does, translation symmetry breaking follows by the general argument of [1]. 3.2. Bounds on the Distribution of the Particle-Excess Function The results stated in Theorem 3.1 are enabled by the following auxiliary bounds. The first bound expresses the tightness of the distribution of the particle-excess function. Theorem 3.3 (Tightness bound). For each β > 0 and W > 0, the following bound holds for all n1 , n2 ∈ Z, θ ∈ [0, 1), and x ∈ [L1 , L2 ] [with Lj defined in (15)], (β,W,θ)
μ[n1 ,n2 ] ({|K(x; ω)| ≥ γ}) ≤ C(β, W ) e−A(β,W )γ
2
(19)
at some C(β, W ) < ∞ and A(β, W ) > 0. This bound can be made intuitive by noting that the Coulomb systems’ 2 energy can be presented as q 2ρ K(x)2 dx plus the short-range interaction corresponding to V2 . In fact, a stronger statement is valid, with the exponent in (19) replaced by −A(β, W )|γ|3 . The bound asserted in (19) applies also to the two component Coulomb system. It is stated here in this weaker form in order to make the discussion of its implications applicable to also such systems. The second bound expresses the asymptotic vanishing of the translation averages of K(x, ω):
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Theorem 3.4 (The vanishing of K’s volume-averages). For each β, W, δ > 0, = C(β, W, δ) < ∞ and α = α(β, W ) > 0 with which there exist C ⎫⎞ ⎛⎧ r ⎬ ⎨ 1 (β,W,θ) e−α δ2 r K(x + u; ω) du ≥ δ ⎠ ≤ C μ[n1 ,n2 ] ⎝ (20) ⎭ ⎩r 0
for any n1 , n2 ∈ Z, x ∈ R, and any r > 0. Since K(x, ω) = 0 for x ∈ / [L1 , L2 ], it suffices to verify (20) for all x ∈ [L1 , L2 ] and r ∈ [0, L2 − x]. To simplify the notation, we shall sometimes write (β,W,θ) P instead of μ[n1 ,n2 ] . Before presenting the derivation of these estimates (Sects. 7, 8), we shall show how they imply symmetry breaking, i.e., give a conditional proof of Theorem 3.1. First however, let us frame the discussion of the function K(x; ω) within a convenient setup.
4. Basic Properties of the Particle-Excess Function 4.1. A Convenient Topological Setup It is convenient to view the particle-excess function K(x; ω) as a random variable with values ranging over a subset of the Skorokhod space [3], which we denote by Sλ , of functions on R which 1. 2. 3.
have the c` adl` ag property (continuity from the right, and existence of limits from the left), have only integer valued discontinuities, are piecewise differentiable with a constant derivative, given by ρW ≡ 1/λ.
ag continuity modulus wK (δ), It may be noted that for elements of Sλ the c`adl` which expresses the maximal variation of K(x; ω) between pairs of points at distance δ omitting a finite number of discontinuities [3], takes the nonfluctuating values: wK (δ) = δ/λ ,
for all δ > 0
and K ∈ Sλ .
(21)
We shall make use of the following two observations, which are simple consequences of standard arguments. Lemma 4.1. A sufficient condition for tightness of a family of probability measures supported on Sλ is the tightness with respect to this family of supx∈R |K(x; ω)/F (x)|, for some continuous function F (x) which diverges at ∞. The proof is by a standard argument (which uses the Arzel` a-Ascoli theorem), and will be omitted here. Let us however note that the standard criterion for tightness of probability measures on the Skorokhod space requires as a second condition also the vanishing in probability of the continuity modulus wK (δ) for δ → 0. That, however, is directly implied for functions in Sλ by (21).
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Lemma 4.2. For x ∈ R and K ∈ Sλ , the following ‘phase functional’ Φ(x, K) := ei2πK(x)
(22)
is continuous in K with respect to the Skorokhod topology (restricted to Sλ ). The point here is that while the evaluation function K → K(x) itself is not continuous, since the location of the jump discontinuities may change, the above phase is not affected by the location of the jumps when they are by integer amounts. 4.2. The Charge Cocycle The total charge in an interval (0, u], i.e., Q(u; ω) :=
qu − q|{j : xj ∈ (0, u]}|, λ
(23)
can be expressed as the difference: Q(u; ω) = q K(0; Tu ω) − qK(0; ω).
(24)
In the dynamical systems terminology, Q(u; ω) is a cocycle under the action on Sλ by the group of shifts, i.e., it transforms as: Q(u + s; ω) = Q(u; Ts ω) + Q(s; ω).
(25)
Equation (24) states that this cocycle is the coboundary of K as long as the latter is well defined (a point which is not to be taken for granted in the infinite-volume limit). For background, it may be of relevance to recall the following general principle (To avoid excessive notation, we do not change here the notation from the specific to the general.). Proposition 4.3 (K. Schmidt [15]). A cocycle Q(x; ω), which transforms as (25) under the group of measure preserving transformations {Tu }u∈λZ , is a coboundary, i.e., representable as (24) with some measurable function K0 (ω), if and only if the collection of variables {Q(u; ω)}u∈λZ is tight. Tightness means in this context that the following bound holds uniformly for u ∈ λZ P (|Q(u; ω)| ≥ t) ≤ p(t)
(26)
with some p(t) which vanishes for t → ∞. The less elementary part of the proposition concerns the ‘only if ’ direction, for which a constructive argument can be provided [15] (as discussed also in [1]). While the above proposition sheds light on our discussion, we shall bypass here the requirement of shift invariance by making use of the additional information given by Theorem 3.4.
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5. Conditional Proof of Theorem 3.1 Since the derivation of the enabling Theorems 3.3 and 3.4, which is presented (independently) in Sects. 7 and 8, would take the discussion into a different arena than the one introduced above, let us first present their implication for our main results. Proof of Theorem 3.1—assuming Theorems 3.3 and 3.4. Existence of Limits for Subsequences Our strategy is to first discuss the question of convergence of the probability distributions at the level of Sλ , that is of the distribution of the random particle-excess function K(x, ω), which takes values in that Skorokhod space. Since sup
|K(x; ω)| ≤ max{|K(n; ω)|, |K(n + 1; ω)|} + ρW
(27)
x∈[n,n+1]
the bound of Theorem 3.3 yields:
|K(x; ω)| ≥t sup ln(2 + |x|) x∈R √ 2 ≤ 2 C(β, W ) e−A(β,W )[t ln n−ρW ]
(β,W,θ) μ[L1 ,L2 ]
(28)
n∈N
where the upper bound is finite when t2 A(β, W ) > 1 and vanishes, uniformly in {θ, L1 , L2 }, for t → ∞. By Lemma 4.1, this implies tightness of the probability measures on Sλ which are induced by the Gibbs measures μ(β,W,θ) . Hence, for every sequence of such measures, there exists a subsequence for which the induced probability measures on Sλ converge. Since the charge configuration in any finite interval I ⊂ R is a continuous function of K (in the Skorokhod topology), this convergence implies also convergence of the corresponding point process. To keep the discussion simple, we ignored so far (in this section) the existence of the internal degree of freedom yj . To incorporate that, one may consider a variant of the above argument, with a ‘decorated’ version of the function K(x; ω), for which values of the variable yj are associated with the discontinuities of the function K (Although this is not true for the full range of c`adl` ag functions, for K ∈ Sλ the collection of discontinuities varies continuously with K in the Skorokhod topology which is employed here.). The above argument applies then mutatis mutandis. This may be a place to note that the strategy employed here for the proof of convergence of the point process has its roots in the proof by A. Lenard of convergence of the Gibbs states of 1D Coulomb systems, based on the analysis of the corresponding electric field ensemble [10]. For the proof of symmetry breaking, we employ additional arguments which were introduced in [1,2].
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Mutual Singularity of Limiting Measures at Different Values of θ: Reconstruction of the Phase Φ(ω) The above construction of the limiting measures for the point process in T proceeds through the construction of a limiting measure for the variable K ∈ Sλ . In terms of this variable, the quantity we have after this is Φ(ω) = ei2πK(0;ω) .
(29)
This expression, however, does not suffice: for our purpose, it is essential to establish that the phase Φ can be evaluated as a measurable (and thus quasi local) function of the point configuration. That may not be obvious at first sight since a shift of the function by a constant: K(x) → K(x) + C changes the value of Φ, without affecting the point configuration (that is the set of discontinuities of K). However, the information provided by Theorem 3.4 is of help here. Using the coboundary relation (24), for any R > 0 1 K(0; ω) = − R 1 =− R
=
R R 1 [K(u; ω) − K(0; ω)] du + K(u; ω) du R 0
0
R q
−1
0
j: 0<xj
1 Q(u; ω) du + R
R K(u; ω) du 0
R R xj 1 + K(u; ω) du. 1 − − R 2λ R
(30)
0
Summing the probability bound (20), we find that for any of the finite-volume Gibbs measures: ⎫⎞ ⎛⎧ ⎬ ⎨ 1 (β,W,θ) μ[n1 ,n2 ] ⎝ sup K(u; ω) du − 1/ ≥ δ ⎠ ⎭ ⎩ ≥R 0
C ≤ e−α 1 − e−αδ2
δ2 R
,
(31)
where the insignificant 1/ correction allows to relate the maximum of |K(x; ω)| within intervals of length λ to the end-point values. It now easily follows (by considering the implications of (20) and then applying the Borel Cantelli lemma) that the following limit converges almost surely with respect to any probability measure which is an accumulation point of the finite-volume θ Gibbs states (the statement denoted here by ‘θ−a.s.’) ⎧ ⎤⎫ ⎡ ⎬ ⎨ R x θ−a.s. j ⎦ (32) lim exp i2π ⎣ Φ(ω) := 1 − − R→∞ ⎩ R 2λ ⎭ j: 0<xj
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The construction also guarantees that Φ(ω) satisfies Φ(ω)
θ−a.s.
=
ei2πθ
and by implication also the relations which were claimed in (17).
(33)
Remarks: 1. Extending the above considerations, one may obtain from Eqs. (30) and (31) an algorithm for the almost sure reconstruction, with respect to the limiting measure, of the function K from the location of its discontinuities. A fully deterministic quasi-local reconstruction is not possible, as the observation made above shows. 2. The argument used above bypasses the question of translation invariance. Assuming it, or a slightly weaker regularity statement [11], the reconstruction of K(x; ω) could be done using Proposition 4.3 through the combination of tightness of the charge cocycle, which follows from the bounds of Theorem 3.3, and the normalization of the reference level which is implied by Theorem 3.4.
6. The 1D Case It is now left to derive the bounds which enable the above analysis. To introduce some of the ideas which are used in the proof in a somewhat simpler context, let us first consider the analogous question for strictly 1D Coulomb systems. Thus, we consider particles of Coulomb charge −1 (q = 1) on a line in the presence of a uniform positive background charge of density given by the Lebesgue measure (ρ = 1). We start with a bound on the probability of a uniformly large charge imbalance in such 1D systems with appended fixed charges at the endpoints. We denote here by {x} = x − x the fractional part of x ∈ R. Lemma 6.1. For a 1D jellium system of R = r particles in an interval [0, r] with an ‘external’ charge γ ∈ N affixed at x = 0 and charge −(γ + {r}) affixed at x = r, the Gibbs probability P satisfies at any γ ∈ (0, r): 2 1 e− 2 β(γ −1)(R−γ) . (34) P min K(x) ≥ γ ≤ 0≤x≤r e−(2R+γ ln γ) Proof. We recall that in one dimension the energy of a neutral configuration is given simply by r 1 K(x; ω)2 dx. (35) U (ω) = 2 0
With the affixed boundary charges, we have that K(x; ω) = γ+x−|{i : xi < x}|. Let be the R-simplex {0 < x1 < · · · < xR < r} of all possible particle configurations. The edges of in the axes directions have length r. Consider the two subsets + consisting of all configurations for which K(x; ω) ≥ γ uniformly in x, and − consisting of all configurations for which:
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Figure 3. The function K(x) for configurations in + and − • There are γ + 1 particles in unit interval [0, 1]; hence, K(1) = K(0) + 1 − (γ + 1) = 0). • There is exactly one particle per unit interval {(k − 1, k) : 2 ≤ k ≤ R − γ}; hence, K(k) = 0 for those k. • There are no particles in [R − γ, r], and the charge imbalance increases linearly from 0 to γ + {r} in this interval. For the Gibbs measure with the boundary charges as specified above, we have: exp − 12 β K(x; ω)2 dx μ(dω) + . P min K(x) = γ ≤ 0≤x≤R exp − 21 β K(x; ω)2 dx μ(dω) − μ( + ) ≤ exp − inf βU (ω) + sup βU (ω) , (36) μ( − ) ω∈ + ω∈ − with μ(dω) the Lebesgue measure on .We shall now estimate the energy and volume factors of the RHS of (36) separately. Energy estimate: For ω ∈ + , we have (see Fig. 3) γ + {x}, x ∈ [0, 1] ∪ [R, r), K(x; ω) ≥ γ, x ∈ [1, R],
(37)
thus 2U (ω) ≥ A + γ 2 (R − 1) + B, r 1 with A = 0 (γ + {x})2 dx and B = R (γ + {x})2 dx. For ω ∈ − , ⎧ ⎪ x ∈ [0, 1] ∪ [R, r), ⎨γ + {x}, |K(x)| ≤ 1, x ∈ [1, R − γ + 1], ⎪ ⎩ γ, x ∈ [R − γ + 1, R],
(38)
(39)
whence 2U (ω) ≤ A + (1)2 (R − γ) + γ 2 (γ − 1) + B.
(40)
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Putting this together, we get the following bound on the “improvement in energy”, i.e., its lowering of configurations in − in comparison to those in
+ : inf U − sup U ≥ (R − γ)(γ 2 − 1).
+
(41)
−
Volume factors: Applying the two-sided Stirling approximation √ √ 2πn(n/e)n ≤ n! ≤ 2πn(n/e)n e1/12 we find that (R + 1)R rR e2R . ≤√ ≤√ R! 2πR(R/e)R 2πR
(42)
eγ+1−1/12 1 ≥ . (γ + 1)! 2π(γ + 1)(γ + 1)γ+1
(43)
μ( + ) ≤ and μ( − ) =
Combining Eqs. (41), (42), and (43) we obtain ! $ # β" 2 μ( + ) exp − (γ − 1)(R − γ) P min K(x) = γ ≤ 0≤x≤R μ( − ) 2 ! $ 2R+(γ+1) ln(γ+1) # β" 2 e (γ × exp − − 1)(R − γ) ≤ 2 eγ+1−1/12 ! $ " # β ≤ e2R+γ ln γ × exp − (γ 2 − 1)(R − γ) (44) 2 We now consider the 1D jellium on [L1 , L2 ], with L1 < 0 < L2 (∈ Z). First, let us define two ‘crossing events’ (analogous to ± of the previous lemma) which play a key role in the sequel: Definition 6.1. For γ, ∈ N and r > 0, let G + ( , r) be the set of configurations satisfying (+a) K(0) > 3γ and K(x) > γ for − < x < r (b) K(− ) = γ (c) K(r−) > γ ≥ K(r+). Equivalently, K(r−) = γ + {r} and K(r+) = γ + {r} − 1. Definition 6.2. Let G = G( , r) be the set defined by the two conditions (b) and (c) above, but with (+a) replaced with (a) K(x) does not equal or cross the value γ for x ∈ [− + 1, R). Condition (a) of G ensures that the sets G( , r) ( ∈ N, r > 0) are disjoint. In particular, suppose H is the event that K(x) ≥ γ for at least one x-value both to the left and right of x = 0. Then [− , − + 1) and [R, R + 1) are the first unit intervals to the right and left of the origin for which K(x) “crosses” γ. The events G( , r) are the decomposition of H into such events, thus % G( , r) (45) H= ,r
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Obviously, G + ⊂ G. We will write G (+) when a statement applies to both G and G + . Theorem 6.2. The probability distribution of the charge imbalance, K(x), in an overall neutral 1D Jellium in [L1 , L2 ], with L1 < 0 < L2 (∈ Z), satisfies: 3
P (|K(0)| > 3γ) ≤ c2 e−c1 γ ,
(46)
for some c1 (β), c2 (β) > 0. Proof. The strategy of proof is as follows: by symmetry, it is enough to consider the case K(0) ≥ 3γ for some fixed γ. Because of the boundary conditions K(L1 ) = K(L2 ) = 0 (overall neutrality with no boundary charges), the charge imbalance must cross the line K(x) = γ to the left and right of the origin. We group together configurations that have the same crossing points. We may suppose without loss of generality that there are no particles at integer coor/ Z, and xi < xj dinates and no two particles have the same coordinate, xk ∈ for i < j. Denote the simplex of configurations of N = L2 − L1 particles by ' & (47)
(N, [L1 , L2 ]) := (x1 , . . . , xN ) ∈ [L1 , L2 ]N : x1 < x2 < · · · < xN and by dˆ ω the Lebesgue integration with respect to all x-variables except the kth one pinned at xk = r, where k = |L1 | + R + 1 − γ (recall that R = r). Let also: exp(−βU (ω))dˆ ω p(G( , r)) := G . (48) exp(−βU (ω))dω Note that by the disjointness of the events G( , r) and (45), |L1 |
L2
dr p(G( , r)) = P (H) ;
=1 0
thus, p(G( , r))/P (H) is a probability density. It follows that P (K(0) > 3γ) ≤
|L1 | L2 =2γ+1 0
dr p(G + |G)p(G) ≤ sup p(G + |G).
(49)
,r
since {K(0) > 3γ} = ∪ >2γ,r>0 G + ( , r). Here p(G + |G) is using the RadonNikodym derivative, p(G + |G) := p(G + )/p(G) where p(G + ) is defined in the fashion of (48). The density p(G + | G) is equal to the probability of (+a) occurring on [− , r] with boundary charges as prescribed in (b) and (c); this is the so-called Markov property of 1D Coulomb systems. Thus, p(G + | G) is in the form required by Lemma 6.1.
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additional improved energy
K(x)
γ
1469
improved energy R
0
Figure 4. Excesses in the energy of ω ∈ G + , as estimated in (51) Note that K(0) ≥ 3γ implies ≥ 2γ + 1. Suppose now that γ ≥ which in turn implies 12 β(γ 2 − 1) ≥ 3. A little algebra gives us
( 1+
6 β
2
1
e− 2 β(γ −1)(R+ −γ) P (K(0) > 3γ) ≤ sup −(2(R+ )+γ ln γ) R, e ≤ sup e−c1 γ
2
(R+ −γ)
R,
≤ e−c1 γ
3
(50)
for some c1 (β) > 0. We can then choose c2 (β) > 0 large enough so that for all 3 γ, P (K(0) > 3γ) ≤ c2 e−c1 γ . Before moving on to the quasi-1D systems, let us give a refinement of the bound (38) which will be useful for proving an analogous result to Theorem 6.2 for the strip. Lemma 6.3 (Improved energy bound). Let ω ∈ G + . If nk (ω) is the number of particles in [k, k + 1] for k = − , . . . , R − 1 and nR (ω) is the number of particles in [R, r), then n− (ω) = 0 and r −
K(x; ω)2 dx ≥ γ 2 ( + r) +
R "
# γnk (ω)2 + nk (ω)3 /3 .
(51)
k=−
and the integrality Proof. n− = 0 follows from K(− + 1) = γ + 1 − n− > γ ) m of the involved quantities. More generally, nk ≤ k + and k=−l nk ≤ m + for all m ≤ R. The energy estimate follows from the observation that any
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non-zero nk adds at least a triangle (see Fig. 4) with side length nk to the field K(x), x ≤ k, on top of γ: K(k) = γ + (k + ) − nj ≥ γ + nk . j
This gives additional factors in the energy nk nk 2γxdx + x2 dx = γnk (ω)2 + nk (ω)3 /3. 0
(52)
0
7. Tightness Bounds We now turn to the proof of Theorem 3.3 for jellium in the tube T . The strategy is as demonstrated in the 1D case. However, the energy estimates are complicated by the V2 -interaction. We denote by V2 (ω1 , ω2 ) the sum of the V2 -interaction terms of pairs of distinct particles, one in ω1 and the other in ω2 , and by V2 (ω) := V2 (ω, ω) such a sum for a single configuration, omitting self interactions. Following the decomposition (7), the energy of a configuration ω = (z1 , . . . , zN ) is a sum of a 1D part and a short-range interaction energy. q2 U (ω) = 2W
L2
|K(x; ω)|2 dx + q 2
V2 (yj , yk ; |xj − xk |)
1≤j
L1 1
= U (ω) + V2 (ω).
(53)
Given > 0, r > 0, the tube T splits into subsystems Y = Y ,r := [− , r] × D,
LR := T \Y,
(54)
and the total energy naturally decomposes into U (ω) = ULR (ωLR ) + UY (ωY ) + V2 (ωY , ωLR ).
(55)
Note that in one dimension V2 ≡ 0 so that the last term on the RHS of (55) vanishes. This yields the Markov property for the function K(x; ω) in 1D Coulomb systems (where it plays the role of the electric field). In the current setting, we must control V2 (ωY ) + V2 (ωY , ωLR ).
(56)
Following are two useful perspectives on the arguments which are presented below. An alternative viewpoint: replacing particles with rods. To analyze V2 interactions, we can employ the procedure of replacing a particle at zk with a rod, or in dimension d ≥ 3 a D-shaped charge slab, of vanishing thickness x ¯k = {z : x = xk }, in which the charge (−q) is uniformly distributed with charge density −qW −1 dy. By (8) the ‘rod-particle’ or ‘rod-rod’ interaction is completely free of the term V2 , and thus when the particles in Y are replaced
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by rods, the rod’s distribution conditioned on the rest of the system is identical to that of a strictly 1D system. In this setting, one will broaden the notion of a configuration of particles to a generalized configuration of particles and rods. ¯k corresponds The x-values will still be ordered, and the replacement zk → x to the map ω = (z1 , . . . , zN ) → ω {¯xk } = (z1 , . . . , zk−1 , x ¯k , zk+1 , . . . , zN ).
(57)
Each replacement changes the energy by an amount equal to (56): Thus, denoting by ω → ω Y (generalized) configuration in which all particles in Y were replaced by rods, we have U (ω) − U (ω Y ) = V2 (ωY ) + V2 (ωY , ωLR ).
(58)
Therefore there are two equivalent points of view for the estimates of the next section: we can either think of V2 -energy estimates for a standard system of point particles, or we can think of the estimate as bounding the replacement effect on the energy incurred when all particles in Y are replaced by rods. Jellium as a system of unbounded spins. One may also regard the system discussed here as one of unbounded spins, with additional internal degrees of freedom. There is, however, a significant difference, which does not make the existing bounds for such systems with ‘superstable interactions’ [9,12,13] directly applicable. The “spins” are the charge imbalances at integer x coordinates, K(k; ω), k ∈ Z. The internal degrees of freedom are the exact locations, including y coordinates, of particles in [k, k + 1) × D. The V2 -interaction between two disjoint regions X and Y is bounded below by −q 2 g(dist(X, Y ))n(X)n(Y ), with n(X) the number of particles in X. This is reminiscent of lower regularity as defined in [12]. Upper regularity, i.e., a (pointwise) converse inequality, does not hold since V2 = ∞ is possible. However, a suitable substitute can be obtained through the zero mean of V2 over y (applying the Jensen inequality). ) The representation (53) together with V2 (ω) ≥ −q 2 j
for suitable A, B > 0 might hold, i.e., the system of spins might be superstable. This is really close to situations considered in [12,13]. In fact, the proof of Theorem 3.3 is close in spirit to the proofs in those papers. One complication arising for jellium is that we cannot simply decrease one spin K(k; ω) without affecting other spins: point charges cannot be removed without affecting an infinite change in the energy, but they can be moved from one place to another. In other words, the overall neutrality fixes the total number of particles, in contrast to the grand canonical setting of [12]. The procedure employed here circumvents this difficulty: as in the 1D case, we look not at an individual large spin but at a selected subsystem [− , r] × D = Y, inside of which charges are rearranged leaving the charge distribution outside unchanged. A technical
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difficulty here, shared with the systems of [12,13], is that making some spin smaller may actually increase its interaction energy with other spins. (β,W,θ) In the rest of the section, we shall write P instead of μ[n1 ,n2 ] , and without loss of generality, we will assume θ = 0. 7.1. V2 -Energy Estimates We would like bounds on what a replacement does to the total energy and to the Boltzmann weight of a configuration ω. The first step towards this end is the zero average property (8). Together with Jensen’s inequality applied to the exponential function, it yields xk } 1 −βU (ω {¯ ) ≤ e−βU (ω) dyk . (60) e W D
Our goal for the rest of this subsection will be to prove a sort of converse to (60). Let ∈ N, r > 0 and Y ⊂ T as in (54). Let R = r/λ. We divide Y into cells Yk = [kλ, (k + 1)λ) × D for k = − , . . . , R − 1 and
YR = [Rλ, r] × D, (61)
and denote by nk (ω) the number of particles in Yk . We start by noting that V2 is “lower regular” in the sense that it can be bounded below in terms of the (ω-dependent) particle numbers nk (ω). In this statement, a role is played by: Lemma 7.1 (Lower bound on V2 (ωY )). For all ω, the V2 -energy satisfies V2 (ωY ) ≥ −q 2 C
R
nk (ωY )2
(62)
k=−
for some constant C = C(W ) > 0. Proof. By the assumption which was made in (9), the interaction V2 between particles, in cells Yk1 , Yk2 (not necessarily distinct) which are distance Dλ ≥ 0 from each other, is bounded below by − q 2 g(Dλ) nk1 (ω) nk2 (ω),
(63)
with g(u) the non-negative monotone decreasing function which appears there. The integrability ) assumption on g implies that the sum over integer multiples of λ is finite, D∈N g(Dλ) < ∞. Let k0 be the value of k such that nk (ω) ≤ nk0 (ω) for all k = − , . . . , R. The interaction between particles in Yk (or YR = [Rλ, r] × D if k = R) with
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themselves and with others is lower bounded by −q 2 g(0)nk0 (ω)2 /2 − q 2
k 0 −1
g([k0 − k − 1]λ)nk0 (ω)nk (ω) − q 2
k=−
×
R
g([k − k0 − 1]λ)nk0 (ω)nk (ω)
k=k0 +1
≥ −q 2
∞ 5 g(0) + 2 g(Dλ) nk0 (ω)2 =: −q 2 Cnk0 (ω)2 . 2
(64)
D=1
Next, we can find k1 maximizing nk (ω) among k = k0 , and estimate interactions between the remaining cells, k = − , . . . , R, k = k0 , in a strictly analogous way. Iterating the procedure until all cells have been taken care of, yields the lemma. Next, we need to control the interaction between Y and the rest of the tube (LR). Obviously, as in (63), the interaction between particles in, e.g., Y− and the left subsystem may be bounded below by −q
2
− −1 L1
g(Dλ) n− (ω) n− −D−1 (ω).
(65)
D=0
However, for long tubes, with L1 >> 1, this bound can in principle become very negative. That happens in case there is an accumulation of particles of LR close to Y. We will need to control the number of occurrences of such “irregular” configurations. Let γ ∈ N. Let G ⊂ Ω be defined as in Sect. 6 with suitable y degrees of freedom added. Recall that for ω ∈ G, K(− ; ω) = γ,
K(r−; ω) > γ ≥ K(r+; ω).
(66)
We will call a configuration ω ∈ G regular if its charge imbalance is wellbehaved outside Y in the sense that |K(x; ω)| ≤ γ + dist(x, [− , R])
for x ∈ / [− , r].
(67)
The set of regular configurations will be denoted here Greg ⊂ G. Remark. On G, we automatically have that K(x; ω) ≥ −γ − |x + | for x < − and that K(x; ω) ≤ γ+|x−{r}| for x > R (see the regions corresponding to the dotted triangular regions in Fig. 5); thus, the condition (67) really amounts to restricting K(x; ω) from entering the shaded triangular regions of Fig. 5 below. For configurations in Greg , we can lower bound the V2 -interaction and the total energy change due to a replacement: Lemma 7.2 (Lower bound for V2 (ωY , ωLR )). For all ω ∈ Greg , the V2 -energy satisfies
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K(x)
γ 0 −γ
x
R
−l
Figure 5. Regular and irregular configurations ω ∈ G, in the sense of (67). For regular configurations, the graph of K does not enter the shaded region. Within [− , R + 1], the function K can cross γ only in the first and last unit intervals, marked by black strips V2 (ωY , ωLR ) ≥ −q 2 Cγ max nk (ω) − ≤k≤R
(68)
for some constant C = C(W ) > 0. Proof. For k = − , . . . , R, let nk (ω) be the particle numbers introduced above. We extend the definition to k ∈ Z, L1 ≤ kλ ≤ L2 − λ in the obvious way. Call nr (ω) the number of particles in (r, (R + 1)λ) × D. Using the monotonicity of g(u), the interaction between a particle z ∈ Y with the particles from ω in x < − is lower bounded by − q2
− −1 L1
g([k + x + ]λ) n− −k−1 (ω) ≥ −2q 2
k=0
∞
g([k + x + ]λ).
(69)
k=0
The inequality is obtained as follows: sum by parts in order to rewrite the sum with differences of g and sums of particle numbers, use that g is decreasing and ω regular, and sum by parts again. Thus, the total interaction of all particles in Y with the left substrip is bounded by R
− 2q 2
np (ω)
p=−
≥q
2
∞
g([p + + k]λ)
k=0
max
k=− ,...,R
∞ nk (ω) 2(k + 1)g(kλ). k=0
The interaction with the right tube x > r can be bounded in a similar way. An additional summand γ will appear because for regular configurations, there may be as many as 2γ particles accumulated near x = r (see Fig. 4 above).
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Lemma 7.3 (Coupling lemma). For all ωY (a configuration of particles inside Y), 3/2 3/2 e−βU (ω) dωLR ≤ eC(γ +nmax (ω) +γnmax (ω)) e−βU (ω) dωLR (70) G
Greg
for some constant C = C(β, W ) > 0 and nmax (ω) = max− ≤k≤R nk (ω). The integrals are over configurations ωLR where ω := ωLR ∪ ωY ∈ G(reg) . In somewhat loose notation, Eq. (70) may be thought of as a lower bound for the probability of regular configurations, given that they are in G and that the configuration inside Y is ωY P (Greg | G, ωY ) ≥ exp −C(γ 3/2 + nmax (ωY )3/2 + γnmax (ωY )) . (71) In this sense, Lemma 7.3 says that regular configurations have high enough probability. We defer the proof of this lemma to Sect. 7.3. Using Lemmas 7.1,7.2, and 7.3, we can now formulate the “converse” to Eq. (60). Note that U (ω Y ) is the energy of the system with all particles in Y replaced with rods. Lemma 7.4 (Replacement lemma). For all ωY , )R 3/2 2 Y e−βU (ω) dωLR ≤ eC(γ +γnmax (ωY )+ k=− nk (ωY ) ) e−βU (ω ) dωLR , G
(72)
G
for some constant C = C(β, W ). Thus, on average, the Boltzmann weight before replacement is smaller than the Boltzmann weight after replacement, up to some (controllable) function of ωY . 7.2. Proof of Theorem 3.3 The strategy of proof is exactly the same as in the 1D case. Let γ, −L1 , L2 ∈ λN. We want to estimate the probability of the event K(0; ω) ≥ 3γ. Let G (+) ( , r) be events defined as in Sect. 6. Note that since the events were defined in terms of charge imbalances which depend on x-coordinates solely, we can extend the definitions to the quasi-1D case by adding the y degrees of freedom. The definition of conditional densities p(G (+) ), p(G (+) |G) is extended in the natural way as well. By the same argument as in Sect. 6, P (K(0; ω) ≥ 3γ) ≤ sup p(G + ( , r)|G( , r)).
(73)
,r
On Greg , Lemmas 7.1 and 7.2 give us control over the V2 -interactions, enough so that the “1D-portion” of the energy implies the desired exponential decay of the RHS of (73). However, to deduce the exponential decay for all of G, we need Lemma 7.4 (which combines Lemmas 7.1, 7.2, and 7.3).
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Recall that U (ω Y ) = UY1 (ωY ) + ULR (ωLR ) is the energy of the system with particles in Y replaced with rods. By Lemma 7.4, )R 3/2 2 Y e−βU (ω) dˆ ω≤ e−βU (ω ) eC(γ +γnmax (ωY )+ k=− nk (ωY ) ) dˆ ω (74) G + ( ,r)
G + ( ,r)
where we recall that dˆ ω means Lebesgue measure of all variables except the x-variable pinned at x = r, and by Jensen’s inequality and Eq. (8) exp(−βU (ω)) dˆ ω≥ exp(−βU (ω Y )) dˆ ω. G( ,r)
It follows that +
p(G |G) ≤
G+
G( ,r)
exp(−βU (ω Y )) exp[C(γ 3/2 + γnmax (ω) + exp(−βU (ω Y ))dˆ ω G
)R k=−
nk (ωY )2 ]dˆ ω
.
Note that this upper bound is independent of L1 and L2 . Since in U (ω Y ) there are no interactions between Y and LR, we have again a Markov property, and the estimates reduce to estimates for the finite tube Y. We can proceed as in Lemma 6.1, using Lemma 6.3 in order to take )R care of the extra factor exp[C(γ 3/2 + γnmax (ωY ) + k=− nk (ωY )2 ]. 7.3. Regular Configurations Have Positive Probability In this section, we prove Lemma 7.3. The strategy of proof is to define a map G → Greg ,
ω → ω
(75)
by shifting some of the particles in L and R (the left and right substrips in the complement of Y) away from Y, and to substitute G dω by Greg dω . Complications arise because the map is not one-to-one and has a non-trivial Jacobian, resulting in a compression of phase space, i.e., entropy loss. However regular configurations have lower (1D) energy, and this energetic improvement is enough to compensate for the entropy loss. Remark. The astute reader will notice the parallel between the above map and the comparison (in the 1D case) of + and − . Configurations in − are energetically more favorable but have smaller Lebesgue volume μ( − ). Fix γ, ∈ N and r > 0. Let R = r/λ. For ω ∈ G, denote ωY , ωL and ωR the projections onto Y, L and R. To simplify matters, let us focus on R and pretend first that = L1 (which really cannot happen since K(− ; ω) = γ > 0 = K(L1 ; ω)). There are NR = L2 − R − 1 + γ particles in R. Write ωR = {z1 , . . . , zL2 −R−1+γ } with particles labeled from left to right, xj < xj+1 . We observe that if for all k xk − r ≥ (k − γ − 1)λ/2,
(76)
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then ω ∈ Greg . We will call particles regular if they satisfy Eq. (76), irregular otherwise. Thus, regularity of all particles implies regularity of the configuration. The first γ + 1 particles are always regular. For ω ∈ G, let ω ∈ G be such that ωY = ωY (no changes in Y) and ωR the collection of points {zk } with regular particle, xk , (77) yk = yk , xk = xk −r , irregular. (R + k − γ)λ + (k−γ−1)/2 The idea behind the map is the following: for fixed charge imbalance K(r) ≈ γ, the 1D part of the energy in R is minimal if γ particles accumulate near x = r and the remaining L2 − R − 1 particles occupy each one of the “equilibrium” cells YR+1 , . . . , YL2 −1 . The map ω → ω simply shifts an irregular particle closer to its equilibrium position, thereby decreasing the (1D) energy. Lemma 7.5 (Energy estimates). Let Iirr (ω) ⊂ {γ + 2, . . . , NR } be the set of irregular particle labels. The map G ω → ω ∈ Greg decreases the 1D-energy by an amount which is at least q 2 λ (k − γ)2 − 1 . (78) U 1 (ω) − U 1 (ω ) ≥ W 8 k∈Iirr (ω)
Furthermore,
" V2 (ωY , ωR ) ≥ −q 2 Cnmax (ω) |Iirr (ω)| + γ + 1) V2 (ωR ) ≥ −q 2 C (k + 1)
(79) (80)
k∈Iirr (ω)
for some constant C = C(W ) > 0. Note that we do not evaluate V2 (ω)−V2 (ω ). Instead we estimate directly V2 (ω). Jensen’s inequality will take care of V2 (ω ). Proof of Lemma 7.5. The 1D-energy term: We shift irregular particles successively by the order of their subscript, starting with the highest one. In this way, one obtains a sequence ω (n) starting at ω (0) = ω. The shift xk → xk increases the particle imbalance in [xk , xk ) by 1 and leaves it unchanged elsewhere. This decreases the energy by an amount
xk*
q2 U 1 (ω (n) ) − U 1 (ω (n−1) ) = 2W
2 + K(x; ω (n) )2 − K(x; ω (n) ) + 1 dx
xk
q2 =− W
xk* K(xk ; ω)+
q2 λ 1+ dx ≥ [(k − γ)2 −1] 2 8W
(81)
xk
where we have used that the integrand is bounded above by an affine function with slope 1/λ, end value ≤ 1/2, integrated over an interval of length xk − xk ≥ (k − γ)λ/2. The sum of these bounds yields (78).
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γ
−γ
a
r
−l
b
Figure 6. A possible energy improvement for an irregular ω ∈ G, which is obtained through the displacement of a particle from x = a to x = b. The modified K(x) is described by the dotted line
V2 -energy inside R: The individual terms V2 are bounded below by )|Iirr |−1 −q 2 g(0). Since there are k=1 k pairs of irregular particles, we get
V2 (yj , yk ; |xj − xk |) ≥ −q 2 g(0)
j,k∈Iirr j
k.
(82)
k∈Iirr
The interaction between a given irregular particle zk and the regular particles in R is lower bounded by (Fig. 6) − q2
g(|xj − xk |λ) ≥ −(k − 1)q 2 g(0) − q 2
j reg
g((j − k)λ/2). (83)
j>k+1
It follows that
V2 (yj , yk ; |xj − xk |) ≥ −q 2
j reg k irr
g(0)k +
k irr.
∞
g(nλ/2) .
(84)
n=1
V2 -interaction between Y and R: Eq. (79) holds provided C is large enough so that ∞ ∞
g(mλ + [n/2]λ) < C.
(85)
m=0 n=0
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Lemma 7.6. For all ω, γ and , r, L1 , L2 ,
exp(−βU (ω))dy1 · · · dyNR DNR
⎛ ≤ C exp C (γ 3/2 + nmax (ω)3/2 + γnmax (ω)) exp ⎝−C
⎞ (k − γ)2 )⎠
k∈Iirr (ω)
×
exp(−βU (ω ))dy1 · · · dyNR ,
(86)
DNR
for some constants C = C(β, W ), C = C (β, W ), C = C (β, W ). Proof.
e−βU (ω) dy ≤ e−β[U
1
(ω)−U 1 (ω )+inf y (V2 (ωR )+V2 (ωY ,ωR ))]
DNR
e−βU
1
(ω)
dy
DNR
≤ e−β[U
1
(ω)−U 1 (ω )+inf y (V2 (ωR )+V2 (ωY ,ωR ))]
e−βU (ω ) dy.
(87)
DNR
We might think of this inequality as a result of a three step procedure: (a) Replace particles with rods. The replacement error is quantified by Eqs. (80) and (79). (b) Shift irregular rods away from Y. The energy decrease is given by Eq. (78). (c) Replace rods with particle averages using Jensen’s inequality. Next, by Lemma 7.5, letting η = k − γ U 1 (ω) − U 1 (ω ) + inf (V2 (ωR ) + V2 (ωY , ωR )) ≥
η: (η+γ) ∈Iirr
y
* q2 λ 8W
+ η 2 − C(γ + η + nmax + 1) − Cnmax (ω)(γ + 1).
(88)
The Cη is dominated by η 2 except for small η. The γ term inside the sum is dominated by η 2 except for η 2 const γ, whence a negative contribution of the order of γ 3/2 , and similarly for nmax (ω). Thus, the RHS of (88) is greater than (k − γ)2 − C2 (γ 3/2 + nmax (ω)3/2 + γnmax (ω)) C1 k∈Iirr (ω)
for suitable constants C1 = C1 (β, W ), C2 = C2 (β, W ) (at this point the constants depend on β only through their dependence on q; see the remarks concerning dimensionless parameters in Sect. 2.2). We are now equipped for the proof of Lemma 7.3: Proof of Lemma 7.3. Let γ, , r be fixed. As in previous proofs, we will deal only with the region R (L is dealt with analogously). Define the map ω → ω from G to Greg as above. Let I ⊂ {γ + 2, . . . , NR }. Conditioned on the event that Iirr (ω) = I, the regularizing map is injective with the Jacobian , dω 2 = exp − ln[(k − γ − 1)/2] . (89) dω = k−γ−1 k∈I
k∈I
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Thus, for suitable constants, e−βU (ω) dωR ≤ C e−A(I) e−βU (ω ) dωR ,
(90)
Greg
Iirr (ω)=I
with A(I) := βC
3/2 (k − γ)2 − βC (γ 3/2 + nmax + γnmax )
k∈I
− C
ln[(k − γ − 1)/2],
(91)
k∈I
where the last sum comes from the Jacobian. We obtain a lower bound on A(I) by dropping the last sum and adjusting the constants C and C . It follows that ∞ , 3/2 3/2 2 exp(−A(I)) ≤ eC (γ +nmax +γnmax ) (1 + e−Ck ). (92) I⊂{γ+2,...,NR }
k=1
Summing Eq. (90) over I yields Lemma 7.3, up to the simplification = L1 . In the general case < L1 , the regularizing map is extended by defining its action on particles in L in a similar way: particles too close to Y get shifted away. The energy estimates of Lemma 7.5 extend in the natural way. From there one can proceed as in the fictitious case = L1 .
8. Convergence of the Volume-Averages of K(x; ω) In this section, we prove Theorem 3.4. In order to acclimate ourselves, let us first prove the 1D case via a shift coupling for fixed δ > 0, −L1 , L2 ∈ N, and r as in Theorem 3.4. As before, without loss of generality, we assume θ = 0. Recall that xi denotes the position of the ith particle with i < j implying ˜ is defined by moving the that xi < xj almost surely. The coupling ω → ω particles in the following manner: ˜i = xi /2. • If xi < 2δ then x ˜i = xi − δ. • If 2δ < xi < r then x By symmetry, we may think of Theorem 3.4 as a statement concerning the exponential decay (in r) of the probability of r Aδ := {ω : K(x; ω) dx ≥ δr}. 0
To prove this exponential decay, we use the observation that using the above coupling, for 2δ < x < r, K(x; ω) − δ = K(x − δ; ω ˜ ).
(93)
Proposition 8.1. Consider a 1D Coulomb system. For all δ > 0 and c < 1, there is a constant r0 (δ, c) such that r ≥ r0 implies 2
P (Aδ ) ≤ e−cβδ r .
(94)
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Proof. If we define the set F = {ω : |K(k; ω)| ≤ r2/5 − 1 for each integer k ∈ [0, r]},
(95)
then by Theorem 6.2 we have P (F c ) ≤ (r + 1)c2 (β)e−c1 (β)r
6/5
.
˜ )| ≤ r2/5 + δ for all Note that if ω ∈ F then |K(x; ω)| ≤ r2/5 and |K(x; ω x ∈ [0, r]. Thus for ω ∈ Aδ ∩ F: r
K(x; ω) dx ≥ δr − 2δr2/5 ,
(96)
2δ
and using (93), r
K(x; ω)2 − K(x; ω ˜ )2 dx
0
r ≥
2
K(x; ω ˜ )2 dx.
(2δK(x; ω) − δ ) dx − 2δ
[0,δ]∪[r−δ,r]
≥ 2δ(δr − 2δr
2/5
2
K(x; ω ˜ )2 dx
) − δ (r − 2δ) − [0,δ]∪[r−δ,r]
2
≥ δ (r + 2δ − 4r
2/5
) − 2δ(r
4/5
+ 2δr2/5 + δ 2 )
= δ 2 (r − 8r2/5 − 2δ −1 r4/5 ).
(97)
To get a bound on the probability, we now only need a bound on the change in the Jacobian or “volume-factor” caused by going from ω to ω ˜ . The only place where volume is lost is in the first step of the coupling when [0, 2δ] is mapped to [0, δ]. For ω ∈ F, there are at most 2r2/5 + 2δ particles in [0, 2δ]; 2/5 thus, the volume factor is bounded above by e(2r +2δ) log 2 . Altogether, we have P (Aδ ∩ F c ) + P (Aδ ∩ F) ≤ (r + 1)c2 e−c1 r + e−βδ
2
6/5
[r−(8+2 log 2/(βδ 2 ))r 2/5 −2δ −1 r 4/5 −(2 log 2)(βδ)−1 ]
from which (94) readily follows.
(98)
For the general quasi-1D setting, the coupling ω → ω ˜ maps xi → x ˜i just yi = yi ). We need to adjust the proof as above and leaves the yi ’s unchanged (˜ of Proposition 8.1 in two places. The first trivial adjustment is to bound the probability of F c with Theorem 3.3 (replacing the 1D bound of Theorem 6.2). The non-trivial adjustment is to show that on F, any increase in the shortrange V2 -interaction energy caused by the coupling is insignificant compared to the decrease in 1D-energy given in (97).
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Proof of Theorem 3.4. Let F be as in (95) and define the ‘regular set’ Freg similar to the definition of Greg in Sect. 7.1, with r2/5 playing the role of γ. By Lemma 7.3 we have that P (Freg ) > e−cr
4/5
P (F)
(99)
for some c > 0. We now condition on Freg and bound exp(−βV2 (ω)) dy/ exp(−βV2 (˜ ω )) dy on this event. Recall (from Sect. 7) that Ya,b = [a, b] × D and that ω Ya,b is ω with all particles in Ya,b replaced with rods. If there are k particles in the region Y0,δ (for ω ˜ ) and the region Y0,2δ (for ω), and y denotes the vector of y-coordinates of those particles, then for every ω ∈ F exp(−βV2 (ω))dy Dk
≤ exp(−β inf [V2 (ω) − V2 (˜ ω Y0,δ )]) y∈Dk
exp(−βV2 (˜ ω Y0,δ ))dy
Dk
ω Y0,δ )] ≤ exp −β inf [V2 (ω) − V2 (ω Y0,2δ )] − β[V2 (ω Y0,2δ ) − V2 (˜ y∈Dk ω ))dy, (100) × exp(−βV2 (˜ Dk
where we have compared the integrals in the last two lines using Jensen’s inequality applied to rod replacements. We have slightly abused notation in that the infimum over y ∈ Dk is at different x-values for the coupled configurations ω ˜ and ω. Note that the coupling does not affect V2 -interactions between two particles that are both in Y2δ,r (pre-coupling). Therefore, on the event Freg , the affected V2 -interaction can be lower bounded by V2 (ω) − V2 (ωLR ) − V2 (ωY\Y0,2δ ) = V2 (ωY , ωLR ) + V2 (ωY0,δ ) + V2 (ωY0,δ , ωY\Y0,δ ) ≥ −cr4/5 .
(101)
If we can now show that conditioned on Freg , V2 (ω Y0,2δ ) − V2 (ω) ≤ Cr4/5
(102)
ω Y0,δ ) − V2 (ω Y0,2δ ) ≤ Cr4/5 , V2 (˜
(103)
and
then the 1D decrease in energy caused by the coupling [see (97)] overwhelms the coefficients of the right-hand sides of both (100) and (99) which would prove the theorem. Let us now show (102) and (103). To bound V2 (ω Y0,2δ )−V2 (ω) on Freg , note that there are at most 2r2/5 +2δ particles in Y0,2δ . Replacing the role of max nk with 2r2/5 + δ and the role of γ with r2/5 in Lemma 7.2, we obtain
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Symmetry Breaking in Quasi-1D Coulomb Systems
V2 (ω Y0,2δ ) − V2 (ω) < C(r4/5 + δr2/5 )
1483
(104)
for some C > 0. To bound V2 (˜ ω Y0,δ ) − V2 (ω Y0,2δ ), first recall that ωY0,r and ωYc0,r denote the sets of particles of ω that are in Y0,r and its complement, respectively. Using (58), since the V2 -interactions within Y0,r are the same for both ω ˜ Y0,δ Y0,2δ and ω , we have Y
Y
Y
Y
0,δ 0,δ 0,2δ 0,2δ V2 (˜ ω Y0,δ ) − V2 (ω Y0,2δ ) = V2 (˜ ωY0,r ,ω ˜ Yc0,r ) − V2 (ωY0,r , ωYc0,r ).
By (10) we have that for each δ > 0 there exists C(δ) > 0 such that V2 (z1 , z2 ) ≤ Cg(|x1 − x2 |)
(105) Y
0,δ whenever |x1 − x2 | ≥ δ. The condition |x1 − x2 | ≥ δ is satisfied by z1 ∈ ω ˜ Y0,r
Y
0,δ and z2 ∈ ω ˜ Yc0,r , so again following the proof of Lemma 7.2, we get that
V2 (ω Y0,2δ ) − V2 (ω) < Cr4/5 for some C > 0.
9. Discussion The results presented here confirm a conjecture which was stated in [1]. Let us point out some related questions which were not addressed in this work. It may be worth stressing that the length λ ≡ (ρW )−1 , with which translation symmetry breaking is proven here for shifts by u ∈ / λZ, does not correspond to the interparticle spacing. The difference between the two length scales shows up only for values greater than 1 of the dimensionless parameter W/λ(d−1) , but it becomes very pronounced when W λ(d−1) . In particular, this means that no statement is made here about translation symmetry breaking in the d dimensional system (i.e., W = ∞), and about the conditions for the formation of a Wigner lattice in dimensions d > 1. 2. A question which was not addressed is whether the symmetry breaking stops at the level which is proven here, e.g., whether for each θ the Gibbs (β,W,θ) measures μn1 ,n2 have a unique limit, for n1 , n2 → ∞. The answer would be negative if, for instance, for at least certain values of W the system is in a lattice like state wrapped on the cylinder. 3. Let us express here the conjecture that the answer to the above question is positive. More explicitly: we expect that for each θ there is a unique limiting state. Furthermore, we expect this state to be not only invariant under the shift Tλ (which would be implied by the uniqueness) but also ergodic with respect to it. In particular, this would mean that the states do not admit any further cyclic decomposition, i.e., the symmetry breaking stops at the level which is proven here. Can that be shown in a brief argument? 4. Adding a comment to the above question: in case of the cylinder, we do not expect the symmetries of rotation in the compactified dimensions to be broken. 1.
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5.
As is the case for point processes which are not too singular, the θ-states discussed can be characterized through their correlation functions, {ρn (z1 , . . . , zn )}. Thus, the state’s non-invariance under shifts implies that at least some of these correlation functions (if not all) are not shift invariant. In [6], it was shown that in narrow enough strips translation symmetry breaking occurs already at the level of the one-point density function. Does such a statement extend to the entire regime covered by the non-perturbative argument presented here? 6. Other examples of particles with Coulomb potential include two component systems of particles of charges ±q (and not necessarily equal masses). For such systems on a line, there is no translation symmetry breaking, but there is a related phenomenon of phase non-uniqueness [2]. In essence, in one dimension, the fractional part of a charge placed at the boundary cannot be screened by any readjustment of the integer charges. Does this phenomenon also persist to the quasi 1D Coulomb systems? We conjecture that the answer is affirmative even though the rigidity of the 1D Coulomb interaction is somewhat ‘softened’ in the quasi 1D extension.
Acknowledgements S. Jansen wishes to thank E. H. Lieb and Princeton University for making possible the stay during which this work was initiated. Some of the work was done when M. Aizenman was visiting IHES at Bures-sur-Yvette, and the Center for Complex Systems at the Weizmann Institute of Science, Israel. He wishes to thank both institutions for their hospitality. Finally, P. Jung thanks UCLA and IPAM for their hospitality during visits at which some of this work was done.
References [1] Aizenman, M., Goldstein, S., Lebowitz, J.L.: Bounded fluctuations and translation symmetry breaking in one-dimensional particle systems. J. Stat. Phys. 103, 601–618 (2001) [2] Aizenman, M., Martin, P.: Structure of Gibbs states of one-dimensional coulomb systems. Commun. Math. Phys. 78, 99–116 (1980) [3] Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1999) [4] Brascamp, H.J., Lieb, E.H.: Some inequalities for Gaussian measures and the long-range order of the one-dimensional plasma. In: Arthurs, A.M. Functional Integration and Its Applications, Clarendon Press, Oxford (1975) [5] Choquard, P., Forrester, P.J., Smith, E.R.: The two-dimensional one-component plasma at Γ = 2: the semiperiodic strip. J. Stat. Phys. 33, 13–22 (1983) [6] Jansen, S., Lieb, E.H., Seiler, R.: Symmetry breaking in Laughlin’s state on a cylinder. Commun. Math. Phys. 285, 503–535 (2009) [7] Kunz, H.: The one-dimensional classical electron gas. Ann. Phys. 85, 303–335 (1974)
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[8] Laughlin, R.B.: Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395–1398 (1983) [9] Lebowitz, J.L., Presutti, E.: Statistical mechanics of systems of unbounded spins. Commun. Math. Phys. 50, 195–218 (1976) [10] Lenard, A.: Exact statistical mechanics of a one-dimensional system with coulomb forces. III. Statistics of the electric field. J. Math. Phys. 4, 533 (1963) [11] Moore, C., Schmidt, K.: Coboundaries and homomorphisms for non-singular actions and a problem of H. Helson. Proc. Lond. Math. Soc. 40, 443–475 (1980) [12] Ruelle, D.: Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127–159 (1970) [13] Ruelle, D.: Probability estimates for continuous spin systems. Commun. Math. Phys. 50, 189–194 (1976) ˇ [14] Samaj, L., Wagner, J., Kalinay, P.: Translation symmetry breaking in the one-component plasma on the cylinder. J. Stat. Phys 117, 159–178 (2004) [15] Schmidt, K.: Cocycles on Ergodic Transformation Groups. Macmillan, Delhi (1977) [16] Thouless, D.J.: Theory of the quantized Hall effect. Surf. Sci. 142, 147–154 (1984) Michael Aizenman Departments of Physics and Mathematics Princeton University Princeton, NJ 08544, USA e-mail: [email protected] Sabine Jansen Weierstrass Institute for Applied Analysis and Stochastics Mohrenstr. 39 10117 Berlin, Germany e-mail: [email protected] Paul Jung Department of Mathematics Sogang University Seoul 121-742, Korea e-mail: [email protected] Communicated by Jean Bellissard. Received: August 21, 2010. Accepted: September 26, 2010.
Ann. Henri Poincar´e 11 (2010), 1487–1544 c 2010 Springer Basel AG 1424-0637/10/081487-58 published online December 14, 2010 DOI 10.1007/s00023-010-0064-1
Annales Henri Poincar´ e
Quantitative Estimates on the Binding Energy for Hydrogen in Non-Relativistic QED Jean-Marie Barbaroux, Thomas Chen, Vitali Vougalter and Semjon Vugalter Abstract. In this paper, we determine the exact expression for the hydrogen binding energy in the Pauli–Fierz model up to the order α5 log α−1 , where α denotes the fine structure constant, and prove rigorous bounds on the remainder term of the order o(α5 log α−1 ). As a consequence, we prove that the binding energy is not a real analytic function of α, and verify the existence of logarithmic corrections to the expansion of the ground state energy in powers of α, as conjectured in the recent literature.
1. Introduction For a hydrogen-like atom consisting of an electron interacting with a static nucleus of charge eZ, described by the Schr¨ odinger Hamiltonian −Δ − αZ |x| , αZ (Zα)2 . inf σ(−Δ) − inf σ −Δ − = |x| 4 corresponds to the binding energy necessary to remove the electron to spatial infinity. The interaction of the electron with the quantized electromagnetic field is accounted for by adding to −Δ − αZ |x| the photon field energy operator Hf , and an operator I(α) which describes the coupling of the electron to the quantized electromagnetic field; the small parameter α is the fine-structure constant. Thereby, one obtains the Pauli–Fierz Hamiltonian described in detail in Sect. 2. The systematic study of this operator, in a more general case involving more than one electron, was initiated by Bach et al. [3–5]. In this case, determining the binding energy αZ + Hf + I(α) inf σ (−Δ + Hf + I(α)) − inf σ −Δ − (1) |x|
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is a very hard problem. A main obstacle emerges from the fact that the ground-state energy is not an isolated eigenvalue of the Hamiltonian, and cannot be determined with ordinary perturbation theory. Furthermore, the photon form factor in the quantized electromagnetic vector potential occurring in the interaction term I(α) contains a critical frequency space singularity that is responsible for the infamous infrared problem in quantum electrodynamics. As a consequence, quantities such as the ground-state energy do not a priori exist as a convergent power series in the fine-structure constant α with coefficients independent of α. In recent years, several rigorous results addressing the computation of the binding energy have been obtained [10,15,16]. In particular, the coupling to the photon field has been shown to increase the binding energy of the elec2 tron to the nucleus, and that up to normal ordering, the leading term is (αZ) 4 [14,16,10]. Moreover, for a model with scalar bosons, the binding energy is determined in [15], in the first subleading order in powers of α, up to α3 , with error term α3 log α−1 . This result has inspired the question of a possible emergence of logarithmic terms in the expansion of the binding energy; however, this question has so far remained open. In [2], a sophisticated rigorous renormalization group analysis is developed in order to determine the ground-state energy (and the renormalized electron mass) up to any arbitrary precision in powers of α, with an expansion of the form ε0 +
2N
εk (α)αk/2 + o(αN ),
(2)
k=1
(for any given N ) where the coefficients εk (α) diverge as α → 0, but are smaller in magnitude than any positive power of α−1 . The recursive algorithms developed in [2] are highly complex, and explicitly computing the ground-state energy to any subleading order in powers of α is an extensive task. While it is expected that the rate of divergence of these coefficient functions is proportional to a power of log α−1 , this is not explicitly exhibited in the current literature; for instance, it can a priori not be ruled out that terms involving logarithmic corrections cancel mutually. The choice of atomic units in [2], inherited from [3], and motivated by the necessity to keep the α-dependence only in the interaction term, but not in the Coulomb term, introduces a dependence between the ultraviolet cutoff and the fine-structure constant. This makes it challenging to compare the result derived for (2) to our estimate of the binding energy stated in Theorem 2.1, since we employ different units with an ultraviolet cutoff independent of α. However, such comparison would require to apply the procedure used in this paper to the model described in [2]. The goal of the current paper is to develop an alternative method (as a continuation of [7]) that determines the binding energy up to several
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subleading orders in powers of α, with rigorous error bounds, and proving the presence of terms logarithmic in α. The main result established in the present paper (for Z = 1) states that the binding energy can be estimated as α2 + e(1) α3 + e(2) α4 + e(3) α5 log α−1 + o(α5 log α−1 ), 4
(3)
where e(i) (i = 1, 2, 3) are independent of α, e(1) > 0, and e(3) = 0. Their explicit values are given in Theorem 2.1. As a consequence, we conclude that the binding energy is not analytic in α. In addition, our proof clarifies how the logarithmic factor in α5 log α−1 is linked to the infrared singularity of the photon form factor in the interaction term I(α). We note that for some models with a mild infrared behavior, [12], the ground-state energy is proven to be analytic in α. Notice that on the basis of the present computations, it is impossible to determine whether the α5 log α−1 term comes from inf σ(−Δ + Hf + I(α)), or inf σ(−Δ − αZ |x| + Hf + I(α)) or both. However, this question has been recently answered in [6]. Organization of the Proof The strategy consists mainly in an iteration procedure based on variational estimates. The derivation of the main result (3) is accomplished in two main steps, with first an estimate up to the order α3 with an error of the order O(α4 ), and then up to the order α5 log α. Each of these steps requires both an upper and a lower bound on the binding energy. From the knowledge of an approximate ground state for −Δ + Hf + I(α) (see [7]), the remaining hard task consists mainly in establishing a lower bound for the ground-state energy for H yielding an upper bound for the binding energy. A lower bound for the binding energy can be obtained by choosing a bona fide trial state. It is easy to see that the first term in the expansion for the binding 2 energy is not smaller than the Coulomb energy α4 by taking a trial state, which is a product state of the electronic ground state uα (15) of the α and the photon ground state of the Schr¨ odinger–Coulomb operator −Δ − |x| self-energy operator given by (7) with zero total momentum. Indeed, as shown in [10] there is an increase of the binding energy. However, the first term is exactly the Coulomb energy [15]. The proof of the upper bound, up to the order α3 and to the order 5 α log α−1 reduces to a minimization process. Because of the soft photon problem, the set of minimizing sequences does not have a clear structure which helps to find the minimizer, or approximate minimizers. Therefore, prior to the above two steps, we have to appropriately restrict the set of states which we are looking at. For that purpose, we first establish, in Lemma 3.1, an a priori bound on the expected photon number in α +Hf +I(α)), namely the ground state ΨGS of the Pauli–Fierz operator (−Δ− |x| GS GS 2 −1 Ψ , Nf Ψ = O(α log α ), where Nf is the photon number operator.
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Moreover, in Theorem 4.1, we estimate the contribution to the binding energy odinger stemming from states orthogonal to the ground state uα of the Schr¨ operator. It enables us to show that the projection of the ground state ΨGS onto the subspace of functions orthogonal to uα is small in several norms. Estimate up to the order α3 . To find a lower bound for the binding energy (Theorem 5.1), we pick a trial function Φtrial which is the sum of two states. The first odinger–Coulomb operstate is the product of the ground state uα of the Schr¨ ator with the approximate ground state Ψ0 of the self-energy operator (given by (7) below) with zero total momentum which was derived in [7]. The second state Φu α (42) is a term orthogonal to uα and has been chosen to minimize, up α + Hf + I(α))uα Ψ0 , Φu α to the order α3 , the sum of the cross term (−Δ − |x| 2
α and the term (Hf +Pf2 −Δ− |x| + α4 )Φu α , Φu α stemming from the quadratic form for H on the trial state Φtrial = uα Ψ0 + Φu α . At that stage, we emphasize that the contribution of these two states yields not only an α3 term, but also an α5 log α−1 term, which is in fact the only contribution to the binding energy for this order as the detailed proof of the next order estimate shall show. To recover the upper bound, we take an arbitrary state Ψ satisfying the condition on the expected photon number. Then we consider uα Ψuα , the projection of this state onto uα , and Ψ⊥ the projection onto the subspace orthogonal to uα . We estimate in Lemma 7.1 the quadratic form of H on uα Ψuα , in Lemma 7.4 the quadratic form for H on Ψ⊥ , and in Lemmata 7.2–7.3 the cross term. Collecting these estimates in (82) below yields that the function Ψ can be an approximate minimizer up to the order α3 , with error term O(α4 log α−1 ) of the functional HΨ, Ψ only if the difference R between this function Ψ and 1
the previous trial state Φtrial satisfies the condition Hf2 R2 = O(α4 log α−1 ) (see (87)) According to Lemma 7.5, the last estimate itself allows to improve the expected photon number estimate for the ground state of H given by Lemma 3.1, replacing there α2 log α−1 by α2 . Then, repeating the above steps for the upper bound with this improved expected photon number yields the upper bound of the binding energy with an error O(α4 ), instead of O(α4 log α−1 ). Estimate up to the order α5 log α−1 . In order to derive the lower bound for the binding energy, we improve the previous trial function by adding several terms that where irrelevant for the estimate up to the order α3 . Due to the improved estimates in Lemma 7.5, the expected photon number for the difference of the 33 true ground state and this new trial function cannot exceed O(α 16 ). Furthermore, we infer from Theorem 5.2 that this difference satisfies several smallness conditions. Using these conditions, we estimate once more the quadratic form and minimize it with respect to this difference in a similar way to what is done in Sect. 5 for the previous step. The paper is organized as follows: In Sect. 2, we give a detailed definition of the model and state the main result. In Sect. 3 we establish an a priori
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bound on the expected photon number in the ground state of the Pauli–Fierz operator. In Sect. 4 we estimate the contribution to the binding energy stemming from states orthogonal to the ground state of the Schr¨ odinger operator. The Sect. 5 is devoted to the statements of the lower and the upper bounds for the binding energy up to the order α3 , as well as to the proof of the lower bound. The difficult part of the proof, namely the upper bound on the binding energy, is presented in four parts and postponed to Sect. 7. In Sects. 7.1–7.3, we estimate separately the terms according to the splitting of the variational state. We collect these results in Sect. 7.4 and establish then the proof of the upper bound up to the order α3 . In Sect. 6, we establish the main steps of the proof of the upper bound for the binding energy up to the order α5 log α−1 . We start with some useful definitions, the statement for the upper bound, and two propositions (proved in Appendices A and B), that gives estimates of the contributions to the binding energy according to a refined splitting of the variational state. Details of the remaining straightforward computations are given in Sect. 8. Finally, Sect. 9 provides the proof for a lower bound on the binding energy. In Appendix C, we provide some technical lemmata whose proof are straightforward but rather long.
2. The Model We study a scalar electron interacting with the quantized electromagnetic field in the Coulomb gauge, and with the electrostatic potential generated by a fixed nucleus. The Hilbert space accounting for the Schr¨ odinger electron is given by Hel = L2 (R3 ). The Fock space of photon states is given by Fn , F= n∈N
n
2 3 2 where the n-photon space Fn = is the symmetric tensor s L (R ) ⊗ C product of n copies of one-photon Hilbert spaces L2 (R3 ) ⊗ C2 . The factor C2 accounts for the two independent transversal polarizations of the photon. On F, we introduce creation- and annihilation operators a∗λ (k), aλ (k) satisfying the distributional commutation relations [aλ (k), a∗λ (k )] = δλ,λ δ(k − k ),
[aλ (k), aλ (k )] = 0,
where aλ denotes either aλ or a∗λ . There exists a unique unit ray Ωf ∈ F, the Fock vacuum, which satisfies aλ (k)Ωf = 0 for all k ∈ R3 and λ ∈ {1, 2}. The Hilbert space of states of the system consisting of both the electron and the radiation field is given by H := Hel ⊗ F. We use units such that = c = 1, and where the√mass of the electron equals m = 1/2. The electron charge is then given by e = α, where the fine-structure constant α will here be considered as a small parameter. Let x ∈ R3 be the position vector of the electron.
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We consider the normal ordered Pauli–Fierz Hamiltonian on H for Hydrogen, 2 √ (4) : i∇x ⊗ If − αA(x) : +V (x) ⊗ If + Iel ⊗ Hf . where : · · · : denotes normal ordering, corresponding to the subtraction of a normal ordering constant cn.o. α, with cn.o. If := [A− (x), A+ (x)] is independent of x. The electrostatic potential √ V (x) is the Coulomb potential for a static point nucleus of charge e = α (i.e., Z = 1) α V (x) = − . |x| We will describe the quantized electromagnetic field by use of the Coulomb gauge condition. The operator that couples an electron to the quantized vector potential is given by χΛ (|k|)
A(x) = ελ (k) eikx ⊗ aλ (k) + e−ikx ⊗ a∗λ (k) dk 1/2 2π|k| λ=1,2 R3
where by the Coulomb gauge condition, divA = 0. The vectors ελ (k) ∈ R3 are the two orthonormal polarization vectors perpendicular to k, (k2 , −k1 , 0) ε1 (k) = 2 k1 + k22
and
ε2 (k) =
k ∧ ε1 (k). |k|
The function χΛ implements an ultraviolet cutoff on the momentum k. We assume χΛ to be of class C 1 , with compact support in {|k| ≤ Λ}, χΛ ≤ 1 and χΛ = 1 for |k| ≤ Λ − 1. For convenience, we shall write A(x) = A− (x) + A+ (x), where A− (x) =
χΛ (|k|) ε (k)eikx ⊗ aλ (k) dk 1/2 λ 2π|k| λ=1,2 R3
is the part of A(x) containing the annihilation operators, and A+ (x) = (A− (x))∗ . The photon field energy operator Hf is given by Hf = |k|a∗λ (k)aλ (k)dk. λ=1,2 R3
We will, with exception of our discussion in Sect. 3, study the unitarily equivalent Hamiltonian
2 √ (5) H = U : i∇x ⊗ If − αA(x) : +V (x) ⊗ If + Iel ⊗ Hf U ∗ ,
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where the unitary transform U is defined by U = eiPf .x , and Pf =
k a∗λ (k)aλ (k)dk
λ=1,2
is the photon momentum operator. We have U i∇x U ∗ = i∇x + Pf
and
U A(x)U ∗ = A(0).
In addition, the Coulomb operator V , the photon field energy Hf , and the photon momentum Pf remain unchanged under the action of U . Therefore, in this new system of variables, the Hamiltonian reads as follows: 2 √ α H = : (i∇x ⊗ If − Iel ⊗ Pf ) − αA(0) : +Hf − , (6) |x| where : ... : denotes again the normal ordering. Notice that the operator H can be rewritten, taking into account the normal ordering and omitting, by abuse of notations, the operators Iel and If , α H = −Δx − + (Hf + Pf2 ) − 2Re (i∇x .Pf ) |x| √ − 2 α(i∇x − Pf ).A(0) + 2αA+ (0).A− (0) + 2αRe (A− (0))2 . For a free spinless electron coupled to the quantized electromagnetic field, the self-energy operator T is given by 2 √ T = : i∇x ⊗ If − αA(x) : +Iel ⊗ Hf . We note that this system is translationally invariant; that is, T commutes with the operator of total momentum Ptot = pel ⊗ If + Iel ⊗ Pf , where pel and Pf denote, respectively, the electron and the photon momentum operators. Let C ⊗ F be the fiber Hilbert space corresponding to conserved total momentum p ∈ R3 . For fixed value p of the total momentum, the restriction of T to the fibre space is given by (see, e.g. [8]) √ T (p) = : (p − Pf − αA(0))2 : +Hf (7) where by abuse of notation, we again dropped all tensor products involving the identity operators If and Iel . Henceforth, we will write A± ≡ A± (0). Moreover, we denote Σ0 = inf σ(T ) and
Σ = inf σ(H) = inf σ(T + V ).
It is proven in [1,8] that Σ0 = inf σ(T (0)) is an eigenvalue of the operator T (0).
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Our main result is the following theorem: Theorem 2.1. The binding energy fulfills 1 Σ0 − Σ = α2 + e(1) α3 + e(2) α4 + e(3) α5 log α−1 + o(α5 log α−1 ), 4 where ∞ 2 χΛ (t) 2 (1) dt, e = π 1+t
(8)
0
e
(2)
3 2 = Re (A− )i (Hf + Pf2 )−1 A+ .A+ Ωf , (Hf + Pf2 )−1 (A+ )i Ωf 3 i=1 3 1 1 (Hf + Pf2 )− 2 2A+ .Pf (Hf + Pf2 )−1 (A+ )i 3 i=1 −Pfi (Hf + Pf2 )−1 A+ .A+ Ωf 2 2 − 12 3 2 − 1 1 2 −1 + i 2 2 ⊥ − A (Hf +Pf ) (A ) Ωf +4a0 Q1 −Δ− + Δu1 , 3 |x| 4
+
i=1
a0 = e(3)
2 k12 + k22 χΛ (|k|) dk1 dk2 dk3 , 4π 2 |k|3 |k|2 + |k| 2 1 1 1 2 1 + =− ∇u1 , −Δ − 3π |x| 4
and Q⊥ 1 is the projection onto the orthogonal complement to the ground state 1 odinger operator −Δ − |x| (for α = 1). u1 of the Schr¨
3. Bounds on the Expected Photon Number Lemma 3.1. Let
√
α α = v 2 + Hf − , |x| |x| be the Pauli–Fierz operator defined without normal ordering, where v = i∇ − √ αA(x). Let Ψ ∈ H denote the ground state of K, K = (i∇ −
αA(x))2 + Hf −
KΨ = EΨ, normalized by Ψ = 1. Let Nf :=
a∗λ (k)aλ (k) dk
λ=1,2
denote the photon number operator. Then, there exists a constant c independent of α, such that for any sufficiently small α > 0, the estimate
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Ψ, Nf Ψ ≤ cα2 log α−1 is satisfied. Proof. Using [aλ (k), Hf ] = |k|aλ (k),
and
[aλ (k), v] =
(k) ik.x (α) 1 χλ (k)e 2π|k| 2
and the pull-through formula, aλ (k)EΨ = aλ (k)KΨ 1 = (Hf + |k|)aλ (k) − aλ (k) + [v, aλ (k)]v + v[v, aλ (k)] Ψ, |x| we get aλ (k)Ψ = −
√
√ αχΛ (|k|) 2 i∇ − αA(x) · λ (k)eik.x Ψ. 2π |k| K + |k| − E
(9)
From (9), we obtain 2 2 2 √ 1 αχΛ (|k|) i∇ − αA(x) Ψ aλ (k)Ψ ≤ 2 π |k| K + |k| − E α αχΛ (|k|) Ψ, KΨ + Ψ, Ψ , ≤ π 2 |k|3 |x| √ α since K − E ≥ 0, and (i∇ − αA(x))2 ≤ (K + |x| ). 2 2 Since Ψ, : K : Ψ = −α /4 + o(α ) ([10,14,16]) and : K := K − cn.o. α, we have Ψ, KΨ ≤ cn.o. α. (10) Moreover, for the normal ordered hamiltonian defined in (4), we have √ α : K : = −Δx − 4 αRe i∇x .A− (x) + α : A(x)2 : +Hf − |x| √ √ α ≥ (1 − c α)(−Δx ) + α : A(x)2 : +(1 − c α)Hf − |x| 1 α 1 α α ≥ −cα2 − Δx − = −cα2 + 2 − Δx − . + 2 |x| 4 |x| |x|
(11)
where in the last inequality we used (see, e.g. [13]) √ α : A(x)2 : +(1 − c α)Hf ≥ −cα2 . Since 1 α ≥ −4α2 , − Δx − 4 |x| inequality (11) implies α 1 Ψ ≤ −Δx Ψ, Ψ + α2 Ψ2 ≤ cα2 Ψ2 . Ψ, |x| 4
(12)
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Collecting (10), (12) and (9) we find cαχ (|k|) Λ . aλ (k)Ψ ≤ 3 |k| 2
(13)
This a priori bound exhibits the L2 -critical singularity in frequency space. It does not take into consideration the exponential localization of the ground state due to the confining Coulomb potential and appears in a similar form for the free electron. To account for the latter, we use the following two results from the work of Griesemer et al. [13]. Equation (58) in [13] provides the bound: c√αχ (|k|) Λ |x|Ψ . aλ (k)Ψ < 1 |k| 2 Moreover, Lemma 6.2 in [13] states that 2 1 Ψ2 , exp[β|x|]Ψ ≤ c 1 + Σ0 − E − β 2 for any β 2 < Σ0 − E = O(α2 ). For the 1-electron case, Σ0 is the infimum of the self-energy operator, and E is the ground-state energy of : K :. Choosing β = O(α), 1 14 3 (4!) 4 |x|Ψ ≤ |x| Ψ Ψ ≤ exp[β|x|]Ψ Ψ 4 β 18 c 1 ≤ Ψ 1+ β Σ0 − E − β 2 1 4
4
3 4
5
≤ c1 α− 4 . Notably, this bound only depends on the binding energy of the potential. Thus, cα− 34 χ (|k|) Λ . aλ (k)Ψ < 1 |k| 2
(14)
We see that binding to the Coulomb potential weakens the infrared singularity by a factor |k|, but at the expense of a large constant factor α−2 . For the free electron, this estimate does not exist. Using (13) and (14), we find 2 Ψ, Nf Ψ = dk aλ (k)Ψ 3 cα− 2 cα2 + ≤ dk dk 3 |k| |k| |k|<δ
δ≤|k|≤Λ
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1 δ 9 2 −1 4 ≤ cα + c α log α . 3
≤ cα− 2 δ 2 + c α2 log
15
for δ = α 8 . This proves the lemma.
4. Estimates on the Quadratic Form for States Orthogonal to the Ground State of the Schr¨ odinger Operator Throughout this paper, we will denote by π n the projection onto the nth photon sector (without distinction for the n-photon sector of F and the n-photon n−1 sector of H). We also define π ≥n = 1 − j=0 π j . Starting with this section, we study the Hamiltonian H defined in (6), written in relative coordinates. In particular, i∇x now stands for the operator unitarily equivalent to the operator of total momentum, which, by abuse of notation, will be denoted by P . Let 1 (15) uα (x) = √ α3/2 e−α|x|/2 8π be the normalized ground state of the Schr¨ odinger operator α . hα := −Δx − |x| 2
2
We will also denote by −e0 = − α4 and −e1 = − α16 the two lowest eigenvalues of this operator. Theorem 4.1. Assume that Φ ∈ H fulfils π k Φ, uα L2 (R3 ,dx) = 0, for all k ≥ 0. Then there exists 1 > ν > 0 and α0 > 0 such that for all 0 < α < α0 1
HΦ, Φ ≥ (Σ0 − e0 )Φ2 + δΦ2 + νHf2 Φ2 , where δ = (e0 − e1 )/2 =
(16)
3 2 32 α .
Remark 4.1. All photons with momenta larger than the ultraviolet cutoff do not contribute to lower the energy. More precisely, due to the cutoff function χΛ (|k|) in the definition of A(x), and since we have √ α , H = (i∇x − Pf )2 − 2 αRe (i∇x − Pf ).A(0) + α : A(0)2 : +Hf − |x| it follows that for any given normalized state Φ ∈ H, there exists a normalized state Φ≤Λ such that ∈ R3 , for all n ∈ {1, 2,. . .}, for all ((k1 , λ1 ), ∀x n 3 (k2 , λ2 ), . . . , (kn , λn )) ∈ (R \{k, |k| ≤ Λ + 1}) × {1, 2} , we have π n Φ≤Λ (x, (k1 , λ1 ), (k2 , λ2 ), . . . , (kn , λn )) = 0 and Φ≤Λ , HΦ≤Λ ≤ Φ, HΦ
and
Φ≤Λ , T Φ≤Λ ≤ Φ, T Φ.
(17)
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A key consequence of this remark is that throughout the paper, all states will be implicitly assumed to fulfill condition (17). This is crucial for the proof of Corollary 4.2. To prove Theorem 4.1, we first need the following Lemma: Lemma 4.1. There exists c0 > 0 such that for all α small enough we have 1 1 H − (P − Pf )2 − Hf ≥ −c0 α2 . 2 2 A straightforward consequence of this lemma is the following result: Corollary 4.1. Let ΨGS be the normalized ground state of H. Then Hf ΨGS , ΨGS ≤ 2c0 α2 ΨGS 2
(18)
Proof. This follows from HΨGS , ΨGS ≤ (Σ0 − e0 )ΨGS 2 < 0. The last inequality holds since e0 = −α2 /4 and Σ0 is the infimum for the normal ordered Hamiltonian for the free electron, and thus Σ0 ≤ 0. Moreover, from Theorem 4.1 and Lemma 4.1, we obtain Corollary 4.2. Assume that Φ ∈ H is such that π n Φ, uα L2 (R3 ,dx) = 0 holds for all n ≥ 0. Then, for ν and δ defined in Theorem 4.1, there exists ζ > 0, and α0 > 0 such that for all 0 < α < α0 we have HΦ, Φ ≥ (Σ0 − e0 )Φ2 + M [Φ],
(19)
where M [Φ] :=
1 δ ν Φ2 + Hf2 Φ2 + ζ(P − Pf )Φ2 + ζπ n≤4 P Φ2 . 2 2
(20)
Proof. According to Remark 4.1, there exists c > 1 such that the operator cHf π n≤4 holds on the set of states for which (17) is inequality Pf2 π n≤4 ≤ satisfied. The value of c only depends on the ultraviolet cutoff Λ. Thus, π n≤4 P Φ2 ≤ 2(P − Pf )Φ2 + 2π n≤4 Pf Φ2 1
≤ 2(P − Pf )g2 + 2 cHf2 π n≤4 Φ2 , which yields (P − Pf )Φ2 ≥
1 1 n≤4 π P Φ2 − cHf2 π n≤4 Φ2 . 2
Therefore, it suffices to prove Corollary 4.2 with M [Φ] replaced by 1 [Φ] := δ Φ2 + 3 νH 2 Φ2 + 2ζ(P − Pf )Φ2 , M (21) f 2 4 and ζ small enough so that cζ < ν4 . Now we consider two cases. Let c1 := max{8c0 , 8δ/α2 }. If (P − Pf )Φ2 ≤ c1 α2 Φ2 , Theorem 4.1 and the above remark imply (19).
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If (P − Pf )Φ2 > c1 α2 Φ2 , Lemma 4.1 implies that 1 (P − Pf )2 Φ, Φ + 2 1 ≥ (P − Pf )2 Φ, Φ + 4
HΦ, Φ ≥
1 Hf Φ, Φ − c0 α2 Φ2 2 1 1 Hf Φ, Φ + c1 α2 Φ2 , 2 8
which concludes the proof since Σ0 − e0 < 0.
Proof of Lemma 4.1. Recall the notation A± ≡ A± (0). The following holds: 1 1 H − (P − Pf )2 − Hf 2 2 √ 1 α 2 − 2 αRe ((P − Pf ).A(0)) + 2αRe (A− )2 = (P − Pf ) − 2 |x| 1 + 2αA+ .A− + Hf , 2 1 α (P − Pf )2 − ≥ −4α2 . 4 |x|
(22)
and √ √ √ 2 α|(P − Pf ).A(0)ψ, ψ| ≤ 2 α(P − Pf )ψ2 + 2 αA− ψ2 .
(23)
By the Schwarz inequality, there exists c1 independent of α such that 1
A− ψ2 ≤ c1 Hf2 ψ2 .
(24)
Inequalities (23)–(24) imply that for small α, √ 1 1 2 α|(P − Pf ).A(0)ψ, ψ| ≤ (P − Pf )ψ2 + Hf ψ, ψ. 4 4
(25)
Moreover, using (24) and 1
A+ ψ2 ≤ c2 ψ2 + c3 Hf2 ψ2 ,
(26)
we arrive at α(A− )2 ψ, ψ = αA− ψ, A+ ψ ≤ A− ψ2 + −1 α2 A+ ψ2 1
1
≤ c1 Hf2 ψ2 + −1 α2 (c2 ψ2 + c3 Hf2 ψ2 ).
(27)
Collecting the inequalities (22), (25) and (27) with < 1/(8c1 ) and α small enough, completes the proof. Proof of Theorem 4.1. Let Φ := Φ1 + Φ2 := χ(|P | < p2c )Φ + χ(|P | ≥ p2c )Φ, where P =i∇x is the total momentum operator (due to the transformation (5)) and pc = 13 is a lower bound on the norm of the total momentum for which [8, Theorem 3.2] holds.
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α Since P commutes with the translation invariant operator H + |x| , we have for all ∈ (0, 1), α HΦ, Φ = HΦ1 , Φ1 + HΦ2 , Φ2 − 2Re Φ1 , Φ2 |x| α α −1 ≥ HΦ1 , Φ1 + HΦ2 , Φ2 − Φ1 , Φ1 − Φ2 , Φ2 . (28) |x| |x| • First, we have the following estimate: √ : (P − Pf − αA(0))2 : +Hf √ = (P − Pf )2 − 2Re (P − Pf ). αA(0) + α : A(0)2 : +Hf √ √ √ ≥ (1 − α)(P − Pf )2 + (α − α) : A(0)2 : +Hf − cn.o. α √ √ √ ≥ (1 − α)(P − Pf )2 + (1 − O( α))Hf − O( α)
where in the last inequality we used (24) and (26). Therefore, √ 1− α α α (P − Pf )2 − (1 + −1 ) H − −1 Φ2 , Φ2 ≥ Φ2 , Φ2 |x| 2 |x| 1 − √α √ √ (29) + (P − Pf )2 + (1 − O( α))Hf − O( α) Φ2 , Φ2 2 √ (1+−1 )α The lowest eigenvalue of the Schr¨ odinger operator −(1 − O( α)) Δ 2 − |x| is larger than −c α2 . Thus, using (29) and denoting √ √ √ 1− α (P − Pf )2 + (1 − O( α))Hf − O( α) − c α2 , L := 2 we get α −1 HΦ2 , Φ2 ) − Φ2 , Φ2 ≥ (LΦ2 , Φ2 . (30) |x| Now we have the following alternative: Either |Pf | < p3c , in which case we √ p2 have LΦ2 , Φ2 ≥ ( 24c − O( α))Φ2 2 , or |Pf | ≥ p3c , in which case, using √ pc Φ2 = χ(|P | > 2 )Φ2 and Hf ≥ |Pf |, we have L ≥ ( p6c − O( α))Φ2 2 . In both cases, for α small enough, this yields the bound p2c 7 2 Φ2 ≥ Σ0 − e0 + (e0 − e1 ) Φ2 2 (31) LΦ2 , Φ2 ≥ 48 8 since, for α small enough, the right-hand side tends to zero, whereas pc is a constant independent of α. Inequalities (30) and (31) yield α 7 −1 Φ2 , Φ2 ≥ Σ0 − e0 + (e0 − e1 ) Φ2 2 . (32) HΦ2 , Φ2 − |x| 8 • For T (p) being the self-energy operator with fixed total momentum p ∈ R3 defined in (7), we have from [8, Theorem 3.1 (B)] inf σ(T (p)) − p2 − inf σ(T (0)) ≤ c0 αp2 . Therefore, T (p) ≥ (1 − oα (1))p2 + Σ0 .
(33)
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Case 1. If Φ2 2 ≥ 8Φ1 2 , we first do the following estimate, using (33): α Φ1 , Φ1 HΦ1 , Φ1 − |x| α 2 ≥ (1 − oα (1))(P Φ1 , Φ1 − (1 + ) Φ1 , Φ1 + Σ0 Φ1 2 |x| ≥ (Σ0 − (1 + O(α) + O( ))e0 ) Φ1 2 .
(34)
Therefore, together with Φ2 2 ≥ 8Φ1 2 and (32), for α and small enough this implies 3 HΦ, Φ ≥ (Σ0 − e0 )Φ2 + (e0 − e1 )Φ2 . 4
(35)
Case 2. If Φ2 2 < 8Φ1 2 , we write the estimate α Φ1 , Φ1 HΦ1 , Φ1 − |x| α 2 ≥ (1 − oα (1))(P Φ1 , Φ1 − (1 + ) Φ1 , Φ1 + Σ0 Φ1 2 |x| ∞ k 2 ≥ (1+oα (1)+O( )) −e0 π Φ1 , uα L2 (R3 ,dx) − e1 Φ1 2 k=0
−
∞
k
π Φ1 , uα L2 (R3 ,dx)
2
+ Σ0 Φ1 2 .
(36)
k=0
Now, by orthogonality of Φ and uα in the sense that for all k, π k Φ, uα L2 (R3 ,dx) = 0, we get ∞
uα , π k Φ1 L2 (R3 ,dx) 2 =
k=0
∞
uα , π k Φ2 L2 (R3 ,dx) 2
k=0
pc pc uα 2 ≤ 8Φ1 2 χ |P | ≥ uα 2 →α→0 0 ≤ Φ2 χ |P | ≥ 2 2 2
(37)
Thus, for α and small enough, (36), (37) and (32) imply also (35) in that case. • Let c = max{δ, |c0 |α2 }. If Hf Φ, Φ < 8 cΦ2 , (16) follows from (35) with ν = δ/(16 c). cΦ2 , using Lemma 4.1, we obtain If Hf Φ, Φ ≥ 8 1 1 Hf Φ, Φ − c0 α2 Φ2 ≥ Hf Φ, Φ + cΦ2 2 4 1 ≥ Hf Φ, Φ + δΦ2 + (Σ0 − e0 )Φ2 , 4
HΦ, Φ ≥
since Σ0 − e0 ≤ 0, which yields (16) with ν = 14 . This concludes the proof of (16).
(38)
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5. Estimate of the Binding Energy up to α3 Term Definition 5.1. Let uα be the normalized ground state of the Schr¨ odinger operator hα , as defined in (15). We define the projection Ψuα ∈ F of the normalized ground state ΨGS of H, onto uα as follows: ΨGS = uα Ψuα + Ψ⊥ , where for all k ≥ 0, uα , π k Ψ⊥ L2 (R3 ,dx) = 0.
(39)
Remark 5.1. The definition implies that for all m (π m Ψuα )(k1 , λ1 ; k2 , λ2 ; . . . ; km , λm ) = (π m ΨGS )(y; k1 , λ1 ; . . . ; km , λm )uα (y)dy. R3
Definition 5.2. Let Φ2∗ := −(Hf + Pf2 )−1 A+ · A+ Ωf Φ3∗ := −(Hf + Pf2 )−1 Pf · A+ Φ2∗ Φ1∗ := −(Hf + Pf2 )−1 Pf · A− Φ2∗ where evidently, the state Φi∗ contains i photons. Definition 5.3. On F, we define the positive bilinear form v, w∗ := v, (Hf + Pf2 )w,
(40)
1/2 v, v∗ .
and its associated semi-norm v∗ = We will also use the same notation for this bilinear forms on Fn , H and Hn . Similarly, we define the bilinear form . , . on H as u, v := u, (Hf + Pf2 + hα + e0 ) v 1/2
and its associated semi-norm v = v, v . Definition 5.4. Let 1
Φu∗ α := 2α 2 ∇uα .(Hf + Pf2 )−1 A+ Ωf ,
(41)
and 1
Φu α := 2α 2 (Hf + Pf2 + hα + e0 )−1 A+ .∇uα Ωf .
(42)
Remark 5.2. The function Φu∗ α is not a vector in the Hilbert space H because of the infrared singularity of the photon form factor. However, in the rest of the paper, we only used the vectors Hfγ Φu∗ α or Pfγ Φu∗ α , with some γ > 0, which are always well defined. In particular, all expressions involving (Hf + Pf2 )−1 are always well-defined in the sequel. The next theorem gives an upper bound on the binding energy up to the term α3 with an error term O(α4 ).
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Theorem 5.1 (Lower bound on the binding energy). We have Σ ≤ Σ0 − e0 − Φu α 2 + O(α4 )
(43)
Proof. Using the trial function in H 3
3
Φtrial := uα (Ωf + 2α 2 Φ1∗ + αΦ2∗ + 2α 2 Φ3∗ ) + Φu α , and from [7, Theorem 3.1] which states Σ0 = −α2 Φ2∗ 2∗ + α3 (2A− Φ2∗ 2 − 4Φ1∗ 2∗ − 4Φ3∗ 2∗ ) + O(α4 ),
the result follows straightforwardly. We decompose the function Ψuα defined in Definition 5.1 as follows: Definition 5.5. Let η1 , η2 , η3 and Δu∗ α be defined by 3
3
Ψuα =: π 0 Ψuα + 2η1 α 2 Φ1∗ + η2 αΦ2∗ + 2η3 α 2 Φ3∗ + Δu∗ α , with the conditions π 0 Δu∗ α =0 and Φi∗ , π i Δu∗ α ∗ =0 (i = 1,2,3), where Φ1∗ , Φ2∗ , Φ3∗ are given in Definition 5.2. We further decompose Ψ⊥ into two parts. Definition 5.6. Let Ψ⊥ =: κ1 Φu α + Δ⊥ , be defined by Φu α , π 1 Δ⊥ = 0. The following theorem provides an upper bound of the binding energy with an error term of the order O(α4 ). Together with Theorem 5.1, it establishes an estimate up to the order α3 with an error term O(α4 ). Theorem 5.2 (Upper bound on the binding energy). (1) Let Σ = inf σ(H). Then Σ ≥ Σ0 − e0 − Φu∗ α 2∗ + O(α4 ), Σ0 = −α2 Φ2∗ 2∗ + α3 (2A− Φ2∗ 2 − 4Φ1∗ 2∗ − 4Φ3∗ 2∗ ) + O(α4 ), (2)
(44)
and Φu∗ α defined by (41). uα uα of the ground state ΨGS , and the For the components Δ⊥ , Ψ , Δ∗ coefficients η1 , η2 , η3 and κ1 defined in Definitions 5.1–5.5, holds 33
2 16 Δ⊥ = O(α ),
1
2 4 Hf2 Δ⊥ = O(α ),
2 4 (P − Pf )Δ⊥ = O(α ),
(45) uα 2
| ≥ 1 − cα ,
|π Ψ 0
Δu∗ α 2∗ 2
= O(α ),
|η1,3 − 1| ≤ cα,
4
2
Δu∗ α 2 2 2
|η2 − 1| ≤ cα ,
(46) 33 16
= O(α ), |κ1 − 1| ≤ cα. 2
(47) (48)
To prove this theorem, we will compute HΨGS , ΨGS according to the decomposition of ΨGS introduced in Definitions 5.1 to 5.6. Using
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H=
α P − |x|
2
+ : (Pf +
√
Ann. Henri Poincar´e
αA(0))2 : +Hf − 2Re P · (Pf +
√ αA(0)),
and due to the orthogonality of uα and Ψ⊥ , we obtain HΨGS , ΨGS = Huα Ψuα , uα Ψuα + HΨ⊥ , Ψ⊥ √ −4Re P.(Pf + αA(0))uα Ψuα , Ψ⊥ .
(49)
We will estimate separately each term in (49) in Sects. 7.1–7.3. These estimates will be used to establish in Sect. 7.4 the proof of Theorem 5.2.
6. Estimate of the Binding Energy up to o(α5 log α−1 ) Term We develop here the proof of the difficult part in Theorem 2.1 which is the upper bound in (8), and which is stated in Theorem 6.1 below for convenience. Some technical aspects of this proof are detailed in Sect. 8 and Appendices A and B. Theorem 6.1 (Upper bound up to the order α5 log α−1 for the binding energy). For α small enough, we have 1 Σ0 − Σ ≥ α2 + e(1) α3 + e(2) α4 + e(3) α5 log α−1 + o(α5 log α−1 ), (50) 4 (1) (2) where e , e and e(3) are defined in Theorem 2.1. In order to prove this result, we need to refine the splitting of the function Ψ⊥ orthogonal to uα Ψuα defined in Definition 5.1. Therefore, we consider the following decomposition: Definition 6.1. (1)
κ2 =
Pick
Ψ⊥ ,Φ2 π 0 Ψ⊥
3
α−1 Φ2 π0 Ψ⊥∗,Φ2 π0 Ψ⊥
if π 0 Ψ⊥ > α 2 ,
0
if π 0 Ψ⊥ ≤ α 2 .
∗
∗
3
⊥ n ⊥ n ⊥ n ⊥ We split Ψ⊥ into Ψ⊥ 1 and Ψ2 as follows: ∀n ≥ 0, π Ψ = π Ψ1 + π Ψ2 and for n = 0,
(2)
0 ⊥ π 0 Ψ⊥ 1 =π Ψ
and π 0 Ψ⊥ 2 = 0,
for n = 1, uα π 1 Ψ⊥ 1 = κ1 Φ
uα and π 1 Ψ⊥ 2 , Φ = 0,
for n = 2, π
2
Ψ⊥ 1
=
ακ2 Φ2∗ π 0 Ψ⊥ 1
with Wi =
+
3
ακ2,i (Hf + Pf2 )−1 Wi
i=1 i 2 Pf Φ∗ − 2A+ .Pf (Hf
2 ⊥ 2 ⊥ π 2 Ψ⊥ 2 = π Ψ − π Ψ1 ,
∂uα , ∂xi
+ Pf2 )−1 (A+ )i Ωf ,
2 −1 with π 2 Ψ⊥ Wi 2 , (Hf +Pf )
2 0 ⊥ and π 2 Ψ⊥ 2 , Φ∗ π Ψ1 = 0,
∂uα = 0 (i = 1, 2, 3), ∂xi
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for n = 3, 2 −1 + π 3 Ψ⊥ A .A+ Φu α , 1 = ακ3 (Hf + Pf ) 3 ⊥ 3 ⊥ 3 ⊥ 2 −1 + A .A+ Φu α = 0, π 3 Ψ⊥ 2 = π Ψ − π Ψ1 , π Ψ2 , (Hf + Pf )
and for n ≥ 4, π n Ψ⊥ 1 =0
n ⊥ and π n Ψ⊥ 2 =π Ψ .
The next step consists in establishing, in the next lemma, some a priori estimates concerning the function Ψ⊥ that give additional information to those obtained in (45) of Theorem 5.2. Lemma 6.1. The following estimates hold: 1
κ1 = 1 + O(α 2 ), |κ2 | π 0 Ψ⊥ 1 = O(α), κ2,i = O(1), π Ψ⊥ 1 0 ⊥ P π Ψ1 0
(i = 1, 2, 3),
= O(α), = O(α2 ).
Proof. To derive these estimates, we use Theorem 5.2. The first equality is a consequence of (48). To derive the next two estimates, we first notice that (45) yields 2 ⊥ 2 2 ⊥ 2 P π 2 Δ⊥ ≤ 2(P − Pf )Δ + 2Pf π Δ 1
2 2 ⊥ 2 4 2 ≤ 2(P − Pf )Δ⊥ + 2cHf π Δ = O(α ),
therefore, using again (45), we obtain 1
2 2 ⊥ 2 2 ⊥ 2 4 (hα + e0 ) 2 π 2 Δ⊥ ≤ P π Δ + e0 π Δ = O(α ), 1
2 4 and thus, using from (45) that Hf2 π 2 Δ⊥ = O(α ), we get 2 2 ⊥ 2 4 π 2 Δ⊥ = π Ψ = O(α ).
We then write, using (51) and the ·, · -orthogonality of π π
2
2 Ψ⊥ 1
≤ π
2
Ψ⊥ 2
= O(α ). 4
(51) 2
Ψ⊥ 1
and π
2
Ψ⊥ 2, (52)
2 ⊥ Since π 2 Ψ⊥ 1 ≤ π Ψ1 ∗ , and using (206) of Lemma C.2, we obtain 2 ∂uα 4 2 ⊥ 2 2 0 ⊥ 2 2 −1 O(α ) = π Ψ1 ∗ = ακ2 Φ∗ π Ψ1 ∗ + α κ2,i (Hf + Pf ) Wi ∂xi i
=α
2
2 Φ2∗ 2∗ |κ2 |2 π 0 Ψ⊥ 1
∗
α ∇uα 2 + |κ2,i |2 (Hf + Pf2 )−1 Wi 2∗ . 3 i 2
(53) which gives κ2,i = O(1)
and |κ2 | π 0 Ψ⊥ 1 = O(α).
The last two estimates are consequences of (45).
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Eventually, to derive the lower bound on the quadratic form of H(uα Ψuα +Ψ⊥ ), uα Ψuα +Ψ⊥ , yielding the upper bound (50) of Theorem 6.1, we will follow the same strategy as in Sect. 5, the only difference being that now we have better a priori estimates on Ψuα and Ψ⊥ . The two main results to achieve this are stated below, with the computation of the contribution to the ground-state energy of the cross term (Proposition 6.1) and of the direct term HΨ⊥ , Ψ⊥ (Proposition 6.2). Equipped with this two propositions, and using Theorem 5.2 and Lemma 6.1, the proof of Theorem 6.1 is only a straightforward computation which is detailed in Sect. 8. Proposition 6.1. We have 3 1 κ2,i (Hf + Pf2 )−1 Pfi Φ2∗ , Wi 2Re HΨ⊥ , uα Ψuα ≥ − α4 Re 3 i=1 uα 2 0 uα − 4αRe ∇uα .Pf Φ2∗ , π 2 Ψ⊥ 2 − 2Re κ1 π Ψ Φ 3 1 1 2 − α4 Re (Hf + Pf2 )− 2 (A− )i Φ2∗ , (Hf + Pf2 )− 2 (A+ )i Ωf 3 i=1
5 a 2 ⊥ 5 −1 − α2 (Ψ⊥ − α5 |κ3 |2 −|κ1 −1|cα4 +O(α5 ). 1 ) − M [Ψ2 ] − α log α 8 (54) The proof of this Proposition is detailed in Appendix A Proposition 6.2. 1
− uα 2 HΨ⊥ , Ψ⊥ ≥ (Σ0 − e0 )Ψ⊥ 2 − 4α(hα + e0 )− 2 Q⊥ α P A Φ∗
+ |κ1 |2 Φu α 2 + 2αA− Φu∗ α 2 +
3 α4 |κ2,i |2 (Hf + Pf2 )−1 Wi 2∗ 12 i=1
3 2 + α4 Re κ2,i Pf .A− (Hf + Pf2 )−1 Wi , (Hf + Pf2 )−1 (A+ )i Ωf 3 i=1 1
+ uα ⊥ + 4α 2 Re π 2 Ψ⊥ 2 , A .Pf Φ∗ + M1 [Ψ ],
(55)
where 1
a 2 M1 [Ψ⊥ ] = (1−c0 α)(hα + e0 ) 2 π 0 (Ψ⊥ 1) +
|κ3 + 1|2 2 2 2 uα 2 α Φ∗ ∗ Φ 2
3 5 −1 −|κ1 − 1|cα4 + M [Ψ⊥ ), 2 ] + o(α log α 4
(56)
and Q⊥ α is the projection onto the orthogonal complement to the ground state α odinger operator hα = −Δ − |x| uα of the Schr¨ The proof of this Proposition is detailed in Appendix B.
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7. Proof of Theorem 5.2 We prove Theorem 5.2 by bounding the individual terms in the expression for the binding energy. 7.1. Estimate of the Term Huα Ψuα , uα Ψuα Lemma 7.1. Huα Ψuα , uα Ψuα ≥ −e0 Ψuα 2 − α2 Φ2∗ 2∗ π 0 Ψuα 2 + α2 |η2 − π 0 Ψuα |2 Φ2∗ 2∗ + α3 |η2 |2 2A− Φ2∗ 2 − 4Φ1∗ 2∗ − 4Φ3∗ 2∗ + 4α3 |η1 − η2 |2 Φ1∗ 2∗ + |η3 − η2 |2 Φ3∗ 2∗ 1 + cα4 log α−1 |η1 |2 + |η2 |2 + |η3 |2 + π 0 Ψuα 2 + Δu∗ α 2∗ . 2 Proof. The proof is a trivial modification of the one given for the lower bound in [7, Theorem 3.1]. The only modification is that we have a slightly weaker estimate in Lemma 3.1 on the photon number for the ground state. This is accounted for by replacing the term of order α4 in [7, Theorem 3.1] by a term of order α4 log α−1 . In addition, we need to use the equality P.(Pf + √ αA(0))uα Ψuα , uα Ψuα = 0, due to the symmetry of uα . 7.2. Estimates for the Cross Term −4Re P.(Pf +
√
αA(0))uα Ψuα , Ψ⊥
Lemma 7.2 (−4Re P.Pf uα Ψuα , Ψ⊥ term). −4Re P.Pf uα Ψuα , Ψ⊥ 1
1
2 uα 2 2 ≥ −cα4 (|η1 |2 + |η2 |2 + |η3 |2 ) − Hf2 Δ⊥ − cαHf Δ∗ .
Proof.
• For n = 1 photon, 3
P.Pf π 1 uα Ψuα , Ψ⊥ = P.Pf (η1 α 2 Φ1∗ + π 1 Δu∗ α )uα , κ1 Φu α + π 1 Δ⊥ . (57) Obviously 3
|P.Pf (η1 α 2 Φ1∗ + π 1 Δu∗ α )uα , π 1 Δ⊥ | 1
1
2 5 2 2 1 uα 2 2 ≤ Hf2 π 1 Δ⊥ + cα |η1 | + cα Hf π Δ∗ .
(58)
1
Due to Lemma C.5 holds Hf2 (Φu α − Φu∗ α )2 = O(α5 ), which implies 3
|P.Pf (η1 α 2 Φ1∗ + π 1 Δu∗ α )uα , κ1 Φu α | 1
≤ |κ1 |2 cα5 + c|η1 |2 α5 + cα2 Hf2 π 1 Δu∗ α 2 3
+|P.Pf (η1 α 2 Φ1∗ + π 1 Δu∗ α )uα , κ1 Φu∗ α |.
(59)
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For the last term on the right hand side of (59), due to the orthogonality ∂uα ∂uα ∂uα α of ∂u ∂xi and ∂xj , i = j, and the equality ∂xi = ∂xj , holds 3
|P.Pf (η1 α 2 Φ1∗ + π 1 Δu∗ α )uα , κ1 Φu∗ α | 3 ∂uα 2 3 1 1 uα i + i 2 = ∂xi |η1 α Φ∗ + π Δ∗ , κ1 Pf (A Ωf ) | i=1 3
= c|η1 α 2 Φ1∗ + π 1 Δu∗ α , κ1 A+ .Pf Ωf | = 0. •
(60)
For n = 2 photons, |P.Pf π 2 uα Ψuα , Ψ⊥ | = |P.Pf (η2 αΦ2∗ + π 2 Δu∗ α )uα , π 2 Δ⊥ | 1
1
2 2 uα 2 2 ≤ cα4 |η2 |2 + Hf2 π 2 Δ⊥ + αHf π Δ∗ .
•
For n = 3 photons, a similar estimate yields 3
|P.Pf π 3 uα Ψuα , Ψ⊥ | = |P.Pf (η3 α 2 Φ3∗ + π 3 Δu∗ α )uα , π 3 Δ⊥ | 1
1
2 3 uα 2 2 ≤ cα4 |η3 |2 + Hf2 π 3 Δ⊥ + αHf π Δ∗ .
•
For n ≥ 4 photons, 1
1
2 n≥4 uα 2 2 |P.Pf π n≥4 uα Ψuα , Ψ⊥ | ≤ Hf2 π n≥4 Δ⊥ Δ∗ . + αHf π
√ Lemma 7.3 (−4Re αP.A(0)uα Ψuα , Ψ⊥ term). √ −4Re αP.A(0))uα Ψuα , Ψ⊥
1
2 2 ⊥ 2 ≥ −2Re κ1 π 0 Ψuα Φu α 2 − Hf2 Δ⊥ − α Δ 1
−cαHf2 Δu∗ α 2 − cα4 (|η1 |2 + |η2 |2 + |η3 |2 + |κ1 |2 ) + O(α5 log α−1 ). Proof. We first estimate the term 1
Re α 2 P.A+ uα Ψuα , Ψ⊥ = αRe
∞
P.A+ uα π n Ψuα , π n+1 Ψ⊥ .
n=0
•
For n = 0 photon, using the orthogonality Φu α , π 1 Δ⊥ = 0, yields 1
•
−Re α 2 P.A+ π 0 uα Ψuα , κ1 Φu α + π 1 Δ⊥
1 = − Re (π 0 Ψuα )Φu α , κ1 Φu α + π 1 Δ⊥ 2
1 = − Re κ1 π 0 Ψuα Φu α , Φu α . (61) 2 For n ≥ 1 photons, 1 1 + n uα n+1 ⊥ 2 2 Re α P.A π uα Δ∗ , π Δ ≤ cα3 Δu∗ α 2 + Hf2 Δ⊥ n≥2 1
2 5 2 2 2 5 −1 ≤ Hf2 Δ⊥ ), + cα (|η1 | + |η2 | + |η3 | ) + O(α log α
(62)
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where we used from Lemma 3.1 that Δu∗ α 2 ≤ O(α2 log α−1 )+cα3 (|η1 |2 + |η3 |2 ) + cα2 |η2 |2 . We also have 1
3
3
|Re α 2 P (2η1 α 2 Φ1∗ + η2 αΦ2∗ + 2η3 α 2 Φ3∗ )uα , A− Δ⊥ | 1
2 6 2 5 2 6 2 ≤ Hf2 Δ⊥ + c(α |η1 | + α |η2 | + α |η3 | ).
(63)
1
We next estimate the term 2Re α 2 P.A− uα Ψuα , Ψ⊥ . We first get 1
1
2 ⊥ 2 uα 2 2 |α 2 Re P.A− Δu∗ α uα , Δ⊥ | ≤ α Δ + cαHf Δ∗ .M
(64)
Then we write 1
3
3
|α 2 Re (2η1 α 2 A− Φ1∗ + 2η3 α 2 A− Φ3∗ )∇uα , Δ⊥ | 2 4 2 2 ≤ α2 Δ⊥ + cα (|η1 | + |η3 | ).
(65)
We also have 1
|α 2 Re η2 αA− Φ2∗ .∇uα , κ1 Φu α | −1
1
1
= |α 2 Re η2 αHf 2 A− Φ2∗ .∇uα , Hf2 κ1 Φu α | ≤ cα4 (|η2 |2 + |κ1 |2 ), (66) −1
1
since Hf 2 A− Φ2∗ ∈ L2 (R3 ) and Hf2 Φu α = O(α 2 ). Finally, we get 3
1
|α 2 Re η2 αA− Φ2∗ .∇uα , π 1 Δ⊥ | −1
1
1
1
1 ⊥ 2 5 2 2 = |α 2 Re η2 αHf 2 A− Φ2∗ .∇uα , Hf2 π 1 Δ⊥ | ≤ Hf π Δ + cα |η2 | .
(67) Collecting (61) to (67) concludes the proof.
7.3. Estimate for the Term HΨ⊥ , Ψ⊥ Lemma 7.4. 1 uα 2 2 4 2 ⊥ 2 HΨ⊥ , Ψ⊥ ≥ (Σ0 − e0 )Δ⊥ − cα |κ1 | + M [Δ ] + |κ1 | Φ , 2 where M [ . ] is defined in Corollary 4.2. Proof. Recall that
H=
α P2 − + (Hf + Pf2 ) − 2Re (P.Pf ) |x| √ −2 α(P − Pf ).A(0) + 2αA+ .A− + 2αRe (A− )2 .
(68)
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Due to the orthogonality Φu α , Δ⊥ = 0, we get HΨ⊥ , Ψ⊥ uα = Hr, r + |κ1 |2 Φu α 2 − e0 |κ1 |2 Φu α 2 − e0 κ1 Δ⊥ , Φ
+2α|κ1 |2 A− Φu α 2 uα −2Re P.Pf Φu α , Φu α + 2αRe A− .A− Δ⊥ , κ1 Φ √ uα −4 αRe P.A− Δ⊥ , κ1 Φ √ √ uα uα ⊥ −4 αRe P.A+ Δ⊥ , κ1 Φ + 4Re αPf .A(0)Δ , κ1 Φ uα uα ⊥ +4αRe A+ .A− Δ⊥ , κ1 Φ − 4Re P.Pf Φ , Δ .
(69)
For the first term on the right-hand side of (69), we have, from Corollary 4.2 ⊥ ⊥ 2 ⊥ HΔ⊥ , Δ ≥ (Σ0 − e0 )Δ + M [Δ ].
(70)
According to Lemma C.4, we obtain uα uα 2 2 2 ⊥ 2 2 5 −1 − e0 κ1 Δ⊥ . (71) , Φ − e0 |κ1 | Φ ≥ − α Δ − c|κ1 | α log α
The next term, namely 2α|κ1 |2 A− Φu α 2 , is positive. Due to the symmetry in x-variable, P.Pf Φu α , Φu α = 0.
(72)
uα The term 2αRe A− .A− Δ⊥ , κ1 Φ is estimated as uα 2αRe A− .A− Δ⊥ , κ1 Φ 1 1 1 1 2 ⊥ 2 2 5 −1 2 . ≥ −cα2 κ1 Φu α 2 − νHf2 Δ⊥ = − νHf Δ − c|κ1 | α log α 4 4 (73)
Due to Lemma C.4 and [13, Lemma A4], 1 1 ν uα 2 2 6 −1 |α 2 A− Δ⊥ Hf2 Δ⊥ . , P κ1 Φ | ≤ + c|κ1 | α log α 8 The next term we have to estimate fulfils
(74)
1
uα |α 2 P.A+ Δ⊥ , κ1 Φ | 2 2 − uα 2 ⊥ 2 2 4 ≤ P π 0 Δ⊥ + cα|κ1 | A Φ ≤ (P − Pf )Δ − c|κ1 | α . (75)
We have
√ uα αPf .A(0)Δ⊥ , κ1 Φ √ √ uα uα + 0 ⊥ = Re αPf .A− π 2 Δ⊥ , κ1 Φ + Re αPf .A π Δ , κ1 Φ √ √ uα uα + 0 ⊥ = Re αPf .A− π 2 Δ⊥ , κ1 Φ + Re αA .Pf π Δ , κ1 Φ (76)
Re
uα Obviously, A+ .Pf π 0 Δ⊥ , κ1 Φ = 0, and the first term is bounded by 1 √ uα uα 2 2 ⊥ 2 2 2 |Re αPf .A− π 2 Δ⊥ , κ1 Φ | ≤ Hf π Δ + cα|κ1 | Pf Φ 1
2 2 4 ≤ Hf2 π 2 Δ⊥ + c|κ1 | α .
(77)
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uα For the term αRe A+ .A− Δ⊥ , κ1 Φ we obtain uα αRe A+ .A− Δ⊥ , κ1 Φ 1
1
1
2 5 2 ⊥ 2 2 ≥ −cα2 Hf2 κ1 Φu α 2 |κ1 |2 − Hf2 Δ⊥ = −cα |κ1 | − Hf Δ .
(78)
According to (213) of Lemma C.4, the last term we have to estimate fulfils Re P.Pf κ1 Φu α , Δ⊥ 1
1
1
uα 2 2 1 ⊥ 2 5 2 2 2 ≤ Hf2 π 1 Δ⊥ + cP |Pf | κ1 Φ ≤ Hf π Δ + cα |κ1 | .
(79)
Collecting the estimates (69) to (79) yields (68).
7.4. Upper Bound on the Binding Energy with Error Term O(α4 ) We first establish a lemma that we shall need in the proof of Theorem 5.2 in order to improve the error term from O(α4 log α−1 ) to O(α4 ). 1
2 4 −1 Lemma 7.5. If Δu∗ α 2∗ = O(α4 log α−1 ) and Hf2 Δ⊥ ) hold, = O(α log α then we have 1
33
Nf2 Δu∗ α 2 = O(α 16 ) 1 2
(80)
33 16
2 Nf Δ⊥ = O(α )
(81)
Proof. We note that from Definition 5.1, we have 3
aλ (k)Ψuα 2 ≤ aλ (k)ΨGS 2 ≤
cα− 2 χΛ (|k|) , |k|
where in the last inequality, we used (14). Taking into account that 3
3
Δu∗ α = Ψuα − 2η1 α 2 Φ1∗ − η2 αΦ2∗ − 2η3 α 2 Φ3∗ , where aλ (k)Φ1∗ 2 ≤
cχΛ (|k|)(1 + |log |k||) |k| (|k|) cχ Λ , aλ (k)Φ3∗ 2 ≤ |k|
cχΛ (|k|) , |k|
aλ (k)Φ2∗ 2 ≤
and using (85), we arrive at 3
aλ (k)Δu∗ α 2
cα− 2 χΛ (|k|)(1 + | log |k| |) . ≤ |k|
For the expected photon number of Δu∗ α thus holds 1
Nf2 Δu∗ α 2 = aλ (k)Δu∗ α 2 dk λ
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≤
λ
|k|≤α
3
15 8
17
cα− 2 (1 + |log |k||) dk + |k|
|k|>α 15
1 2
Ann. Henri Poincar´e
|k|−1 |k|aλ (k)Δu∗ α 2 dk 15 8
33
≤ cα 8 + cα− 8 Hf Δu∗ α 2 ≤ cα 16 , using (87) in the last inequality. The relation (81) can be proved similarly, using aλ (k)Φu α 2 ≤ c
α−1 . |k|
This concludes the proof of the lemma.
7.5. Concluding the Proof of Theorem 5.2 The proof of Theorem 5.2 is obtained in the following two steps: We first show that the estimate holds with an error term O(α4 log α−1 ). In a second step, using Lemma 7.5, we improve to an error term O(α4 ). Then, we derive the estimates (45)–(48) that shall be used in the next section for the computation of the binding energy to higher order. • Step 1: We first show that (44) holds with an error estimate of the order α4 log α−1 . Collecting Lemmata 7.1, 7.2, 7.3 and Lemma 7.4 yields HΨGS , ΨGS ≥−e0 Ψuα 2 − α2 Φ2∗ 2∗ |π 0 Ψuα |2 + α3 |η2 |2 (2A− Φ2∗ 2 − 4Φ1∗ 2∗ − 4Φ3∗ 2∗ ) +α2 |η2 − π 0 Ψuα |2 Φ2∗ 2∗ + 4α3 (|η1 − η2 |2 Φ1∗ 2∗ + |η3 − η2 |2 Φ3∗ 2∗ ) 1 −cα4 log α−1 (|η1 |2 + |η2 |2 + |η3 |2 + |π 0 Ψuα |2 ) + Δu∗ α 2∗ 4 1 2 ⊥ 4 2 +|κ1 |2 Φu α 2 + (Σ0 − e0 )Δ⊥ + M [Δ ] − cα |κ1 | 4 −2Re (κ1 π 0 Ψuα )Φu α 2 + O(α5 log α−1 ).
(82)
We first estimate |κ1 |2 Φu α 2 − 2Re (κ1 π 0 Ψuα )Φu α 2 − cα4 |κ1 |2 |κ1 − π 0 Ψuα |2 uα 2 Φ + O(α4 ). (83) 2 Moreover, since |π 0 Ψuα | ≤ 1, we replace in (82) −α2 Φ2∗ 2∗ |π 0 Ψuα |2 by −α2 Φ2∗ 2∗ and in (83) −Φu α 2 |π 0 Ψuα |2 by −Φu α 2 . In addition, using the inequalities 1 |η2 −ηj |2 ≥ |ηj −π 0 Ψuα |2 −|η2 − π 0 Ψuα |2 and |ηj |2 ≤ 2|ηj − π 0 Ψuα |2 + 2 2 for j = 1, 2, 3 yields that for some c > 0, ≥ −Φu α 2 |π 0 Ψuα |2 +
HΨGS , ΨGS ≥ −e0 Ψuα 2 − α2 Φ2∗ 2∗ + α3 |η2 |2 (2A− Φ2∗ 2 −4Φ1∗ 2∗ − 4Φ3∗ 2∗ ) − Φu α 2
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+cα2 |η2 − π 0 Ψuα |2 Φ2∗ 2∗ + cα3 (|η1 − π 0 Ψuα |2 + |η3 − π 0 Ψuα |2 ) 1 2 +cα3 |κ1 − π 0 Ψuα |2 + Δu∗ α 2∗ + (Σ0 − e0 )Δ⊥ 4 1 4 −1 + M [Δ⊥ ). (84) ] + O(α log α 4 Comparing this expression with (43) of Theorem 5.1 gives max{|η1 |, |η2 |, |η3 } ≤ 2.
(85)
Σ = Σ0 − e0 − Φu α 2 + O(α4 log α−1 ).
(86)
and Φu α 4
by Φu∗ α ∗ −1
Finally, by Lemma C.5, we can replace which proves (44) with an error term O(α log α
in the above equality,
).
• Step 2: We now show that the error term does not contain any log α−1 term. From (84) we obtain 1
2 4 −1 Δu∗ α 2∗ = O(α4 log α−1 ) and Hf2 Δ⊥ ). = O(α log α
(87)
According to Lemma 7.5, this implies 1
33
Nf2 Δu∗ α 2 = O(α 16 )
1
33
2 16 and Nf2 Δ⊥ = O(α ).
Thus, we have 1 2
Nf Ψuα 2 1
1
1
1
≤ 4|η1 |2 α3 N 2 Φ1∗ 2 + |η2 |2 α2 N 2 Φ2∗ 2 + 4|η3 |2 α3 N 2 Φ3∗ 2 + N 2 Δu∗ α 2 = O(α2 ), which implies that in Lemma 7.1 we can replace the term cα4 log α−1 (|η1 |2 + |η2 |2 + |η3 |2 + |π 0 Ψuα |2 ) with cα4 (|η1 |2 + |η2 |2 + |η3 |2 + |π 0 Ψuα |2 ) and consequently in (84) and (86), the term O(α4 log α−1 ) can be replaced by O(α4 ). This proves (44). The estimates (45)–(48) follow from (44) and (84) with O(α4 log α−1 ) replaced with O(α4 ). We thus arrive at the proof of Theorem 5.2.
8. Proof of Theorem 6.1 In this section, we prove the upper bound on the binding energy up to the order α5 log α−1 provided in Theorem 6.1. We have HΨGS , ΨGS ≥ (Σ0 − e0 )Ψuα 2 + 2Re HΨ⊥ , uα Ψuα + HΨ⊥ , Ψ⊥ .
(88)
The estimates for the last two terms in (88) are given in Propositions 6.1 and 6.2. We will bound below this expression by considering separately the terms involving the parameters κ1 , κ2,i , and κ3 .
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• We first estimate the second term on the right-hand side of (54) together with the seventh term on the right-hand side of (55). We have 1
2 ⊥ + uα 2 −4αRe ∇uα .Pf Φ2∗ , π 2 Ψ⊥ 2 + 4α Re π Ψ2 , A .Pf Φ∗ 3 ∂uα 2 ⊥ i 2 + 2 −1 + i = −4αRe π Ψ2 , Pf Φ∗ − 2A .Pf (Hf + Pf ) (A ) Ωf ∂xi i=1 3 ∂uα 2 −1 = −4α Re π 2 Ψ⊥ Wi . (89) 2 , (Hf + Pf ) ∂xi ∗ i=1 2 −1 α Wi ∂u Using the ·, · -orthogonality of π 2 Ψ⊥ 2 and (Hf +Pf ) ∂xi , the last expression can be estimated as 3 ∂uα 2 ⊥ 2 −1 − 4α Re π Ψ2 , (hα + e0 )(Hf + Pf ) Wi . (90) ∂xi i=1
By the Schwarz inequality, this term is bounded below by 2 3 ∂uα 2 2 ⊥ 2 2 2 ⊥ 2 6 − α π Ψ2 − c (hα + e0 ) ∂xi = − α π Ψ2 − O(α ). (91) i=1 • Next, we collect all the terms involving κ1 in (54) and (55). This yields −2Re κ1 π 0 Ψuα Φu α 2 + |κ1 |2 Φu α 2 − |κ1 − 1|cα4 ≥ −|π 0 Ψuα |2 Φu α 2 + |κ1 − π 0 Ψuα |2 Φu α 2 − |κ1 − 1|cα4
(92)
Notice that from Theorem 5.2 we have |π 0 Ψuα |2 = 1 + O(α2 ); moreover, we have |π 0 Ψuα | = 1 + O(α2 ). This yields −2Re κ1 π 0 Ψuα Φu α 2 + |κ1 |2 Φu α 2 − |κ1 − 1|cα4 ≥ −Φu α 2 + |κ1 − 1|2 c α3 − |κ1 − 1|cα4 + O(α5 ) ≥ −Φu α 2 + O(α5 ). (93) • We now collect and estimate the terms in (54) and (55) involving κ2,i . We get 3 3 1 α4 − α4 Re κ2,i (Hf + Pf2 )−1 Wi , Pfi Φ2∗ + |κ2,i |2 (Hf + Pf2 )−1 Wi 2∗ 3 12 i=1 i=1 3 2 + α4 Re κ2,i (Hf + Pf2 )−1 Wi , Pf .A+ (Hf + Pf2 )−1 (A+ )i Ωf 3 i=1 3 3 1 α4 = − α4 Re κ2,i (Hf + Pf2 )−1 Wi 2∗ + |κ2,i |2 (Hf + Pf2 )−1 Wi 2∗ 3 12 i=1 i=1
≥−
3 α4 (Hf + Pf2 )−1 Wi 2∗ 3 i=1
(94)
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• Collecting in (54) and (55) the terms containing κ3 yields |κ3 + 1|2 2 2 2 uα 2 α Φ∗ ∗ Φ − α5 |κ3 |2 2 ≥ c1 α5 log α−1 |κ3 + 1|2 − α5 |κ3 |2 ≥ −c2 α5 ,
(95)
where c1 and c2 are positive constants. • The fifth term on the right-hand side of (54) and the first term on the right-hand side of (56) are estimated, for α small enough, as 1 δ 0 ⊥ a 2 2 ⊥ a 2 2 a 2 2 − α (Ψ⊥ (1 − c0 α)(hα + e0 ) π (Ψ1 ) − α (Ψ1 ) ≥ 1 ) ≥ 0, 2 (96) 3 2 a with δ = 32 α , and where we used that (Ψ⊥ 1 ) is orthogonal to uα . • Substituting the above estimates in (88) yields
HΨGS , ΨGS ≥ (Σ0 − e0 )Ψuα 2 + (Σ0 − e0 )Ψ⊥ 2 − Φu α 2 −α4
3
(Hf + Pf2 )−1 Wi 2∗
i=1 3 1 1 2 4 − α Re (Hf + Pf2 )− 2 (A− )i Φ2∗ , (Hf + Pf2 )− 2 (A+ )i Ωf 3 i=1 1
− uα 2 − uα 2 5 −1 −4α(hα + e0 )− 2 Q⊥ ), α P A Φ∗ + 2αA Φ∗ + o(α log α
(97)
where Q⊥ α is the projection onto the orthogonal complement to the ground α odinger operator hα = −Δ − |x| . state uα of the Schr¨ To complete the proof of Theorem 6.1 we first note that Ψuα 2 + Ψ⊥ 2 = ΨGS 2 .
(98)
Moreover, according to Lemma C.5 − Φu α 2 = −Φu∗ α 2∗ +
1 1 1 (h1 + ) 2 ∇u1 2 α5 log α−1 +o(α5 log α−1 ), 3π 4
(99)
and Φu∗ α 2∗ =
α3 2π
∞
χΛ (t) dt = e(1) α3 . 1+t
(100)
0
In addition, we have the following identities (i = 1, 2, 3): 1
(Hf + Pf2 )−1 Wi 2∗ = (Hf + Pf2 )− 2 × 2A+ .Pf (Hf + Pf2 )−1 (A+ )i − Pfi (Hf + Pf2 )−1 A+ .A+ Ωf 2 ,
(101)
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and 3 1 1 2 − α4 Re (Hf + Pf2 )− 2 A− Φ2∗ , (Hf + Pf2 )− 2 (A+ )i Ωf 3 i=1 3 2 = − α4 Re A− (Hf + Pf2 )−1 A+ .A+ Ωf , (Hf + Pf2 )−1 (A+ )i Ωf . 3 i=1
(102) We also have − 12
− 4α(hα + e0 )
− uα 2 4 2 Q⊥ α P A Φ∗ = −4α a0
2 − 12 1 1 ⊥ Q1 Δu1 , −Δ− + |x| 4 (103)
with
a0 =
2 k12 + k22 χΛ (|k|) dk1 dk2 dk3 , 2 3 2 4π |k| |k| + |k|
and 2αA− Φu∗ α 2 =
3 2 4 − α A (Hf + Pf2 )−1 (A+ )i Ωf 2 . 3 i=1
(104)
Substituting (98)–(104) into (97) finishes the proof of Theorem 6.1 and thus the proof of the upper bound in Theorem 2.1.
9. Proof of Theorem 2.1: Lower Bound up to o(α5 log α−1 ) for the Binding Energy In this section, we prove a lower bound for Σ0 − Σ in Theorem 2.1 which coincides with the upper bound given in (97). To this end, it suffices to compute trial , Φ trial (H − Σ0 + e0 )Φ , trial 2 Φ with the following trial function: trial = uα Ψ0 + Ψ ⊥ Φ 0, where Ψ0 is ground state of the operator T (0) (defined in (7)), with the norα , and malization π 0 Ψ0 =Ωf , uα is the normalized ground state of hα = − Δ − |x| ⊥ Ψ is defined by 0
⊥ = 2α 2 (hα + e0 )−1 Q⊥ P.A− Φuα , π 1 Ψ ⊥ = Φuα , π0 Ψ ∗ 0 α 0 1
⊥ = αΦ2 π 0 Ψ ⊥ + π2 Ψ 0 ∗ 0
3
2α(Hf + Pf2 )−1 Wi
i=1
⊥ = −α(Hf + P 2 )−1 A+ .A+ Φuα . π3 Ψ 0 f
∂uα , ∂xi
(105)
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Where Φu∗ α , Φu α , Φ2∗ and Wi are defined as in Sects. 5 and 6. Technical Lemmata used in this proof are given in Appendix C. We compute trial , Φ trial = Huα Ψ0 , uα Ψ0 + 2Re Huα Ψ0 , Ψ ⊥ + H Ψ ⊥, Ψ ⊥ , H Φ 0 0 0
(106)
and we recall 1
1
H = hα + (Hf + Pf2 ) − 2Re P.Pf − 2α 2 P.A(0) + 2α 2 Pf .A(0) +2αA+ .A− + 2α(A− )2 .
(107)
• For the first term in (106), a straightforward computation shows Huα Ψ0 , uα Ψ0 = (Σ0 − e0 )uα 2 Ψ0 2 .
(108)
• We estimate the second term on the right-hand side of (106) by computing each term that occurs in the decomposition (107). Using the symmetry of uα , the only non-zero terms in 2Re ⊥ are given by Huα Ψ0 , Ψ 0 ⊥ 2Re Huα Ψ0 , Ψ 0
⊥ − 4Re P.A+ uα Ψ0 , Ψ ⊥ − 4Re P.A− uα Ψ0 , Ψ ⊥ . = −4Re P.Pf uα Ψ0 , Ψ 0 0 0 (109)
The first term on the right-hand side of (109) is estimated with similar arguments as in Lemma 7.2, and using Δu∗ α 2 = O(α3 ) (Lemma C.1) and Δu∗ α 2∗ = O(α4 ) ([7, Theorem 3.2]). We obtain 3 ⊥ = − 2 α4 − 4Re P.Pf uα Ψ0 , Ψ Pfi Φ2∗ , (Hf + Pf2 )Wi +O(α5 log α−1 ). 0 3 i=1
(110) The second and third terms on the right-hand side of (109) are estimated as in Lemma 7.3, and using again Δu∗ α 2 = O(α2 ) and Δu∗ α 2∗ = O(α4 ). This yields uα 2 5 −1 ⊥ − 4Re P.A+ uα Ψ0 , Ψ ), 0 = −2Φ + o(α log α
(111)
and ⊥ −4Re P.A− uα Ψ0 , Ψ 0 3 2 (A− )i Φ2∗ , (Hf + Pf2 )−1 (A+ )i Ωf + O(α5 log α−1 ). = − α4 3 i=1
(112)
• Next, we estimate the third term on the right-hand side of (106). For that sake, we also use the decomposition (107) for H.
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⊥ = O(α3 ) (since For the term involving hα , using (hα + e0 )π 0 Ψ 0 5 ∂u uα 3 α P.A Φ∗ = O(α 2 )), and (hα + e0 ) ∂xi = O(α ), we directly obtain −
⊥, Ψ ⊥ = (hα + e0 )Ψ ⊥, Ψ ⊥ − e0 Ψ ⊥ 2 hα Ψ 0 0 0 0 0 1
− uα 2 = 4α(hα + e0 )− 2 Q⊥ α P.A Φ∗ 2 5 ⊥ +(hα + e0 )Φuα , Φuα − e0 Ψ 0 + O(α ).
(113)
For the term with Hf + Pf2 , we use the estimate (222) of Lemma C.6, and the ·, ·∗ -orthogonality (see (206) of Lemma C.2) of the two vectors 3 2 ⊥ α αΦ2∗ π 0 g and i=1 2α(Hf + Pf2 )−1 Wi ∂u ∂xi that occur in π Ψ0 . We therefore obtain ⊥ 2 ⊥, Ψ ⊥ = (Hf + P 2 )Φuα , Φuα + α2 Φ2 π 0 Ψ (Hf + Pf2 )Ψ 0 0 f ∗ 0 ∗ 2 3 ∂uα + 2α(Hf + Pf2 )−1 Wi ∂xi i=1
∗
+Φ2∗ 2∗ Φu α 2 + o(α5 log α−1 ).
(114)
⊥, Ψ ⊥ are zero, Using the symmetry of uα , all terms in Re P.Pf Ψ 0 0 2 ⊥ 2 ⊥ except the expression Re P.Pf π Ψ0 , π Ψ0 , which is estimated as follows: 2 ⊥ ⊥ Re P.Pf π 2 Ψ 0 , π Ψ0
⊥ P.Pf αΦ2∗ π 0 Ψ 0,
= 2Re
3
2α(Hf + Pf2 )−1 Wi
i=1
∂uα ∂xi
= O(α5 ),
where we used Lemma C.2 in the first equality to prove that only the crossed term remains. Therefore, we obtain 5 ⊥ ⊥ − 2Re P.Pf Ψ 0 , Ψ0 = O(α ).
(115)
⊥, Ψ ⊥ is estimated as in the The terms involving −2α 2 Re P.A(0)Ψ 0 0 proof of Lemma B.6. This yields 1
⊥, Ψ ⊥, Ψ ⊥ = −4Re α 2 P.A+ Ψ ⊥ −2α 2 P.A(0)Ψ 0 0 0 0 1 ⊥ 5 ⊥ log α−1 ) = −4Re P.A+ π 0 Ψ 0 , π Ψ0 + O(α − 12 ⊥ − uα 2 = −8α(hα + e0 ) Qα P.A Φ∗ + O(α5 log α−1 ). 1
1
(116)
⊥, Ψ ⊥ , we proceed as in the proof of Lemma B.7, For 2α Re Pf .A(0)Ψ 0 0 and obtain 1 2
⊥, Ψ ⊥ 2α 2 Pf .A(0)Ψ 0 0 1 ⊥ ⊥ = 4Re α 2 Pf .A− Ψ 0 , Ψ0 3 1 ∂uα uα − 2 2(Hf + Pf2 )−1 Wi ,Φ Pf .A α = 4α ∂xi ∗ i=1 1
+ O(α5
log α−1 ). (117)
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⊥ , the term 2αRe A− .A− Ψ ⊥, Ψ ⊥ Using the symmetry of uα and π 0 Ψ 0 0 0 is estimated as follows: ⊥, Ψ ⊥ 2αRe A− .A− Ψ 0 0 = 2αRe
−
−
−
−
A .A
⊥ αΦ2∗ π 0 Ψ 0
+2αRe A .A (−α(Hf +
+
3
2α(Hf +
∂uα Pf2 )−1 Wi ∂xi
⊥ , π0 Ψ 0
i=1 2 −1 + Pf ) A .A+ Φu α ), Φu α
2 2 2 2 2 −1 + ⊥ A .A+ Φu α 2 = −2α2 π 0 Ψ 0 Φ∗ ∗ − 2α (Hf + Pf )
⊥ 2 Φ2 2 − 2α2 Φ2 2 Φuα 2 + o(α5 log α−1 ), = −2α2 π 0 Ψ 0 ∗ ∗ ∗ ∗
(118)
where in the last inequality we used (222) of Lemma C.6. Finally, a straightforward computation yields − uα 2 5 − uα 2 5 ⊥ ⊥ 2αRe A− .A+ Ψ 0 , Ψ0 = 2αA Φ + O(α ) = 2αA Φ∗ + O(α ),
(119) where in the last equality, we used Lemma C.5. • Before collecting (109)–(119), we show that gathering some terms yield simpler expressions. Namely, we have 3 2 i 2 − α4 P Φ , (Hf + Pf2 )Wi 3 i=1 f ∗ 3 1 ∂uα uα +4α 2 Pf .A− α 2(Hf + Pf2 )−1 Wi ,Φ ∂xi ∗ i=1 2 3 3 ∂u 1 α 2α(Hf + Pf2 )−1 Wi (Hf + Pf2 )−1 Wi 2∗ . + = − α4 ∂x 3 i i=1 i=1 ∗
(120) We also have, using −α2 Φ2∗ 2∗ = Σ0 + O(α3 ) (see, e.g., [7]) ⊥ 2 − α2 Φ2 2 Φuα 2 ⊥ 2 − α2 Φ2 2 π 0 Ψ (Σ0 − e0 )Ψ0 2 − e0 Ψ 0 ∗ ∗ 0 ∗ ∗ 2 5 ⊥ = (Σ0 − e0 )(Ψ0 2 + Ψ 0 ) + O(α ).
(121)
Therefore, collecting (109)–(119), and using the two equalities (120)–(121), we obtain ⊥ ⊥ H(uα Ψ0 + Ψ 0 ), uα Ψ0 + Ψ0 3 1 4 2 ⊥ α ) − (Hf + Pf2 )−1 Wi 2∗ = (Σ0 − e0 )(Ψ0 2 + Ψ 0 3 i=1 3 2 − α4 (A− )i Φ2∗ , (Hf + Pf2 )−1 (A+ )i Ωf − Φu α 2 3 i=1 1
− uα 2 − uα 2 5 −1 − 4α(hα + e0 )− 2 Q⊥ ). α P.A Φ∗ + 2αA Φ∗ + o(α log α
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With the definition e(1) , e(2) , and e(3) , of Theorem 2.1 this expression can be rewritten as trial , Φ trial (H − Σ0 + e0 )Φ = e(1) α3 + e(2) α4 + e(3) α5 log α−1 + o(α5 log α−1 ).
(122)
Using Lemma C.1 yields Ψ0 = 1 + O(α ), which implies, due to the orthog ⊥ and uα in L2 (R3 , dx), onality of Ψ 0 2
2
2 2 ⊥ trial 2 = uα 2 Ψ0 2 + Ψ Φ 0 = 1 + O(α ).
Therefore, together with (122), this gives trial , Φ trial (H − Σ0 + e0 )Φ trial 2 Φ = e(1) α3 + e(2) α4 + e(3) α5 log α−1 + o(α5 log α−1 ). which concludes the proof of the lower bound in Theorem 2.1.
Acknowledgements J.-M. B. thanks V. Bach for fruitful discussions. The authors gratefully acknowledge financial support from the following institutions: The European Union through the IHP network Analysis and Quantum HPRN-CT-200200277 (J.-M. B., T. C., and S. V.), the French Ministry of Research through the project ANR HAM-MARK No. ANR-09-BLAN-0098-01 (J.-M. B.), and the DFG grant WE 1964/2 (S. V.). The work of T.C. was supported by NSF grant DMS-0704031/DMS-0940145.
Appendix A. Proof of Proposition 6.1 In this Appendix, we provide proofs of results that have a high level of technicality. To begin with, we establish Proposition 6.1. Lemma A.1. The following holds: uα
−4Re P.Pf uα Ψ
3 4 2 , Ψ ≥ − α Re κ2,i ∇uα 2 (Hf + Pf2 )−1 Pfi Φ2∗ , Wi 3 i=1 ⊥
1
⊥ 2 5 2 5 2 −4αRe ∇uα .Pf Φ2∗ , π 2 Ψ⊥ 2 − Hf Ψ2 − α |κ3 | + O(α )
(123)
Proof. For n = 2, 3, with the estimates from the proof of Lemma 7.2 and using that due to Theorem 5.2 we have 1
Hf2 Δu∗ α 2 = O(α4 ), |η1 | = O(1),
and
|κ1 | = O(1),
1 ⊥ n≥4 ⊥ and since π 1 Δ⊥ Δ = π n≥4 Ψ⊥ 2 , we obtain = π Ψ2 and π 1 2 5 −4Re P.Pf uα π n Ψuα , π n Ψ⊥ ≥ − Hf2 Ψ⊥ 2 + O(α ). n =2,3
(124)
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For n = 2, −4Re ∇uα .(αη2 Pf Φ2∗ + Pf π 2 Δu∗ α ), π 2 Ψ⊥ ≥ −4Re ∇uα .αη2 Pf Φ2∗ , π 2 Ψ⊥ 1 1
1
2 uα 2 2 ⊥ 2 2 2 −4Re ∇uα .αη2 Pf Φ2∗ , π 2 Ψ⊥ 2 − cαHf π Δ∗ − cαHf π Ψ . (125)
Using Theorem 5.2, the last two terms on the right-hand side of (125) can be estimated by O(α5 ). For the first term on the right-hand side of (125), using from Lemma C.2 that Pfi Φ2∗ , Φ2∗ = 0, from Theorem 5.2 that η2 = 1 + O(α), and from Lemma 6.1 that κ2,i = O(1), holds −4αRe ∇uα .η2 Pf Φ2∗ , π 2 Ψ⊥ 1 = −4Re α∇uα .η2 Pf Φ2∗ , ακ2 Φ2∗ π 0 Ψ⊥ ∂uα −4Re α ∇uα .η2 Pf Φ2∗ , ακ2,i (Hf + Pf2 )−1 Wi ∂xi i 4 = − Re α2 ∇uα 2 κ2,i (Hf + Pf2 )−1 Pfi Φ2∗ , Wi 3 i
(126)
3 1 = − α4 Re κ2,i (Hf + Pf2 )−1 Pfi Φ2∗ , Wi . 3 i=1 ∂uα ∂uα α ∂uα We also used that ∂u ∂xi , ∂xj = 0 for i = j and ∂xi = ∂xj for all i and j. Finally, the second term on the right-hand side of (125) gives the second term on the right-hand side of (123) plus O(α5 ), using from Theorem 5.2 that 1
2 |η2 − 1|2 = O(α2 ) and Hf2 π 2 Ψ⊥ 2 = O(α ). To complete the proof, we shall estimate now the term for n = 3, 3
4Re P.Pf uα π 3 Ψuα , π 3 Ψ⊥ = 4Re α 2 2η3 P.Pf uα Φ3∗ , π 3 Ψ⊥ 1 3
3 uα 3 ⊥ + 4Re α 2 2η3 P.Pf uα Φ3∗ , π 3 Ψ⊥ 2 + 4Re P.Pf uα π Δ∗ , π Ψ . 1 2
(127)
1 2
The inequalities Hf Δu∗ α ≤ cα2 and Hf π 3 Ψ⊥ ≤ cα2 (see Theorem 5.2) imply that the last term on the right-hand side of (127) is O(α5 ). For the second term on the right-hand side of (127) holds 1
3
3 ⊥ 2 5 2 Re α 2 η3 P.Pf uα Φ3∗ , π 3 Ψ⊥ 2 ≥ − Hf π Ψ2 + O(α ),
(128)
since from Theorem 5.2 we have η3 = O(1). Finally to estimate the first term on the right-hand side of (127), we note that 1
1
1
|k1 |− 6 |k2 |− 6 |k3 |− 6 Φ3∗ (k1 , k2 , k3 ) ∈ L2 (R9 , C6 ), and from Lemma C.6, 1
1
1
|k1 | 6 |k2 | 6 |k3 | 6 (Hf + Pf2 )−1 A+ .A+ Φu α 2 = O(α3 ). This implies, using again |η3 | = O(1), and the explicit expression of π 3 Ψ⊥ 1 3
5
3
5 2 5 2 2 |α 2 2η3 P.Pf uα Φ3∗ , π 3 Ψ⊥ 1 | ≤ cα |κ3 |α |η3 |P uα ≤ α |κ3 | + O(α ). (129)
Collecting (124)–(129) concludes the proof.
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Lemma A.2. The following estimate holds: √ −4 αRe P.A+ uα Ψuα , Ψ⊥ 1 5 2 5 = −2Re κ1 π 0 Ψuα Φu α 2 − M [Ψ⊥ 2 ] − α |κ3 | + O(α ), 4 Proof. Obviously
(130)
1
−4Re α 2 P.A+ uα Ψuα , Ψ⊥ 1
3
3
= −4Re α 2 P.A+ uα (π 0 Ψuα + 2η1 α 2 Φ1∗ + η2 αΦ2∗ + 2η3 α 2 Φ3∗ + Δu∗ α ), Ψ⊥ . (131) Step 1. From (131), let us first estimate the term 1
1
− 4Re α 2 P.A+ uα Δu∗ α , Ψ⊥ = −4α 2 Re
∞
P.A+ π n uα Δu∗ α , π n+1 Ψ⊥ . (132)
n=0
For n = 0, the corresponding term vanishes since π 0 Δu∗ α = 0. For n > 2, we can use (62) where the term O(α5 log α−1 ) can be replaced 33 with O(α5 ) because we know from Theorem 5.2 that Δu∗ α 2 = O(α 16 ). For n = 1, we have 1
1
2 ⊥ 3 1 uα 2 2 ⊥ 2 2 |4α 2 Re P.A+ π 1 uα Δu∗ α , π 2 Ψ⊥ 1 + π Ψ2 | ≤ cα π Δ∗ + Hf π Ψ2 3 1 ∂u 2 α + 4α κ2,i (Hf +Pf2 )−1 Wi ∇uα π 1 Δu∗ α , A− ακ2 Φ2∗ π 0 Ψ⊥ 1 +α ∂xi i=1
.
(133) To estimate the last term on the right-hand side we note that and
−1 Hf 2 A− (Hf
−1 Hf 2 A− Φ2∗
∈ L2
+ Pf2 )−1 Wi ∈ L2 which thus gives for this term the bound 1 2 6 cαHf2 Δu∗ α 2 + α4 |κ2 |2 π 0 Ψ⊥ |κ2,i |2 = O(α5 ), (134) 1 + α i
using Theorem 5.2 and Lemma 6.1. The inequalities (133) and (134) imply 1
1
2 ⊥ 2 ⊥ 2 5 2 |Re α 2 P.A+ π 1 uα Δu∗ α , π 2 Ψ⊥ 1 + π Ψ2 | ≤ Hf π Ψ2 + O(α ). (135) 1
To complete the estimate of the term 4α 2 Re P.A+ uα Δu∗ α , Ψ⊥ we have to 1 estimate the term for n = 2 in (132), namely 4Re α 2 P.A+ uα π 2 Δu∗ α , π 3 Ψ⊥ 1 + π 3 Ψ⊥ 2 . Obviously, 1
1
⊥ 2 5 2 |Re α 2 P.A+ uα π 2 Δu∗ α , π 3 Ψ⊥ 2 | ≤ Hf Ψ2 + O(α ).
For the term involving π
3
Ψ⊥ 1
(136)
we have
1 2
|Re α P uα π 2 Δu∗ α , ακ3 A− (Hf + Pf2 )−1 A+ .A+ Φu α | 1
1
≤ cα3− 16 π 2 Δu∗ α 2 + |κ3 |2 α2+ 16 Φu α 2 = |κ3 |2 α5 + O(α5 ), (137) using Theorem 5.2 and Lemma C.6. Collecting (132)–(137) yields 1
1
2 5 2 5 |Re α 2 P.A+ uα Δu∗ α , g| ≤ Hf2 Ψ⊥ 2 + α |κ3 | + O(α ).
(138)
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Step 2. We next estimate in (131) the term −4Re α 2 P.A+ uα (π 0 Ψuα + 3 3 α 2 2η1 Φ1∗ + αη2 Φ2∗ + α 2 2η3 Φ3∗ ), Ψ⊥ . First using (62) yields 1
− 4Re α 2 P.A+ uα π 0 Ψuα , Ψ⊥ = −2Re (κ1 π 0 Ψuα )Φu α 2 .
(139)
We also have, using Theorem 5.2 1
3
3
|α 2 P uα (α 2 2η1 Φ1∗ + α 2 2η3 Φ3∗ ), A− Ψ⊥ | 1
1
≤ αHf2 π 2 Ψ⊥ 2 + αHf2 π 4 Ψ⊥ 2 + O(α5 ) = O(α5 ),
(140)
and 1
3 ⊥ |α 2 P uα αη2 Φ2∗ , A− (π 3 Ψ⊥ 1 + π Ψ2 )| 1
3
1
1
1
1
2 −4 5 2 ≤ Hf2 π 3 Ψ⊥ |k2 |− 4 Φ2∗ , |k1 | 4 |k2 | 4 A− π 3 Ψ⊥ 2 +|α P uα η2 |k1 | 1 |+O(α ) 1
2 2 5 5 ≤ Hf2 π 3 Ψ⊥ 2 + |κ3 | α + O(α ). − 14
(141)
− 14
1 4
1 4
Here we used |k1 | |k2 | Φ2∗ ∈ L2 and |k1 | |k2 | A− (Hf + Pf2 )−1 A+ .A+ Φu α 2 = O(α3 ) (see Lemma C.6). Collecting (138)–(141) yields 1
−4Re α 2 P.A+ uα Ψuα , Ψ⊥ 1
2 5 2 5 ≥ −2Re κ1 π 0 Ψuα Φu α 2 − Hf2 Ψ⊥ 2 − α |κ3 | + O(α ). (142)
Lemma A.3. √ −4 αRe P.A− uα Ψuα , Ψ⊥ 3 1 1 2 (Hf + Pf2 )− 2 (A− )i Φ2∗ , (Hf + Pf2 )− 2 (A+ )i Ωf ≥ − α4 Re 3 i=1
1 2 ⊥ a 2 5 −1 − M [Ψ⊥ (α|κ3 |2 +1) − |κ1 −1|cα4 +O(α5 ), 2 ]− α (Ψ1 ) − α log α 4 (143) a 0 ⊥ 0 ⊥ 0 ⊥ where (Ψ⊥ 1 ) (x) := (π Ψ1 (x) − π Ψ1 (−x))/2 is the odd part of π Ψ1 .
Proof. Since from Lemma C.2 we have ∇uα .A− Φ1∗ = 0, we have 1
3
4Re α 2 P.A− uα Ψuα , Ψ⊥ = 4α 2 Re η2 A− Φ2∗ .P uα , π 1 Ψ⊥ 1
+ 4α2 Re 2η3 A− Φ3∗ .P uα , π 2 Ψ⊥ + 4α 2 A− Δu∗ α .P uα , Ψ⊥ .
(144)
For the first term on the right-hand side of (144) we have 3
uα 4α 2 Re η2 A− Φ2∗ .P uα , π 1 Ψ⊥ 2 + κ1 Φ 3
−1
1
3
uα − 2 2 = 4α 2 Re η2 Hf 2 A− Φ2∗ .P uα , Hf2 π 1 Ψ⊥ 2 + 4α Re η2 A Φ∗ .P uα , κ1 Φ .
(145) The first term on the right-hand side of (145) is bounded from below by 1
2 5 − Hf2 π 1 Ψ⊥ 2 + O(α ).
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Applying Lemma C.5, we can replace Φu α in the second term of the right-hand side of (145) by Φu∗ α , at the expense of O(α5 ). More precisely, 3
|α 2 η2 A− Φ2∗ .P uα , κ1 (Φu α − Φu∗ α )| 1
≤ cα5 |η2 |2 (Hf + Pf2 )− 2 A− Φ2∗ 2 + |κ1 |2 Φu α − Φu∗ α 2∗ = O(α5 ). Moreover, 3
4α 2 Re η2 κ1 A− Φ2∗ .P uα , Φu∗ α =
3 8 2 α ∇uα 2 Re η2 κ1 (A− )i Φ2∗ , (Hf + Pf2 )−1 (A+ )i Ωf 3 i=1
=
3 2 4 α Re η2 κ1 (A− )i Φ2∗ , (Hf + Pf2 )−1 (A+ )i Ωf 3 i=1
≥
3 2 4 α Re (A− )i Φ2∗ , (Hf + Pf2 )−1 (A+ )i Ωf − |κ1 − 1|cα4 , 3 i=1
(146)
where we used κ1 = O(1) (Lemma 6.1) and η2 = 1 + O(α) (Theorem 5.2). 1 Note that the right-hand side of (146) is well defined since (Hf + Pf2 )− 2 A+ Ωf 1 ∈ F and (Hf + Pf2 )− 2 A− Φ2∗ ∈ F. Collecting the estimates for the first and the second terms in the righthand side of (145), we arrive at 3
−4α 2 Re η2 A− Φ2∗ .P uα , π 1 Ψ⊥ 8 ≥ − α2 ∇uα 2 Re κ1 (A− )i Φ2∗ , (Hf + Pf2 )−1 A+ Ωf 3 1
2 5 − Hf2 π 1 Ψ⊥ 2 + O(α ).
(147)
Here we used also η2 = 1 + O(α). As the next step, we return to (144) and estimate the second term on the right-hand side as 4α2 Re 2η3 A− Φ3∗ .P uα , π 2 Ψ⊥ −1
1
= 8α2 Re η3 Hf 2 A− Φ3∗ .P uα , Hf2 π 2 Ψ⊥ = O(α5 ), −1
1
(148)
1
2 2 ⊥ 2 4 2 where we used Hf 2 A− Φ3∗ ∈ L2 and π 2 Hf2 Δ⊥ = π Hf Ψ = O(α ) from Theorem 5.2. For the last term on the right-hand side of (144), we have 1
4α 2 Re A− Δu∗ α · ∇uα , Ψ⊥ 1
1
uα − uα 2 = 4α 2 Re A− Δu∗ α · ∇uα , π 0 Ψ⊥ 1 + 4α Re κ1 A Δ∗ · ∇uα , Φ 1
1
1
1
− uα 2 ⊥ 2 + 4α 2 Re A− Δu∗ α · ∇uα , π 1 Ψ⊥ 2 + 4α Re A Δ∗ · ∇uα , π Ψ
+ 4α 2 Re A− Δu∗ α · ∇uα , π 3 Ψ⊥ + 4α 2 Re A− Δu∗ α .∇uα , π n≥4 Ψ⊥ . (149)
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⊥ s ⊥ a ⊥ s We write the function π 0 Ψ⊥ 1 = (Ψ1 ) + (Ψ1 ) where (Ψ1 ) (respectively ⊥ a 0 ⊥ (Ψ1 ) ) denotes the even (respectively odd) part of π Ψ1 . Obviously, we have 1
|α 2 Re A− Δu∗ α · ∇uα , π 0 Ψ⊥ 1 | 1
a 2 2 ⊥ a 2 5 ≤ cαHf2 R2 + α2 (Ψ⊥ 1 ) = α (Ψ1 ) + O(α ).
(150)
The constant can be chosen small for large c. For the second term on the right-hand side of (149), we have 1
|α 2 κ1 A− Δu∗ α · ∇uα , Φu α | 1
≤ α5 log α−1 |κ1 |2 + cαHf2 Δu∗ α 2 = α5 log α−1 + O(α5 ). For the third term on the right-hand side of (149), we have, since δ =
(151) 3 2 32 α
1
|4α 2 A− Δu∗ α · ∇uα , π 1 Ψ⊥ 2 | 1 δ δ 1 ⊥ 2 2 5 ≤ π Ψ2 + cαHf2 Δu∗ α 2 = π 1 Ψ⊥ 2 + O(α ). 8 8
(152)
Similarly, δ n≥4 ⊥ 2 π Ψ2 + O(α5 ). (153) 8 To complete the estimate of the last term in (149), we have to estimate 1 1 two terms: −4Re α 2 A− Δu∗ α .P uα , π 2 Ψ⊥ and −4Re α 2 A− Δu∗ α .P uα , π 3 Ψ⊥ . For the first one we have 1
|4α 2 A− Δu∗ α · ∇uα , π n≥4 Ψ⊥ | ≤
1
1
2 |Re α 2 A− Δu∗ α .P uα , π 2 Ψ⊥ | ≤ cαHf2 Δu∗ α 2 + α2 π 2 Ψ⊥ 2
+ α4 |κ2 |2 π 0 Ψ⊥ 2 + α6
3
2 5 |κ2,i |2 = α2 π 2 Ψ⊥ 2 + O(α ).
i=1
Similarly, 1
|Re α 2 A− Δu∗ α .P uα , π 3 Ψ⊥ | 1
2 6 −1 ≤ cαHf2 Δu∗ α 2 + α2 π 3 Ψ⊥ |κ3 |2 + O(α5 ). (154) 2 + α log α
Collecting the estimates (149)–(154) yields 1
|4Re α 2 A− Δu∗ α .P uα , Ψ⊥ | 1 δ 2 ⊥ 2 5 −1 2 (1 + α|κ3 |2 ) + O(α5 ). (155) ≤ Ψ⊥ 2 + Hf Ψ2 + α log α 8 Collecting (138), (142), (147), (148) and (155) concludes the proof. A.1. Concluding the Proof of Proposition 6.1 We can now prove the estimate on Re Hg, uα Ψuα asserted in Proposition 6.1. Using the orthogonality (39) of uα and Ψ⊥ , yields √ 2Re HΨ⊥ , Ψuα = −4Re P.Pf uα Ψuα , Ψ⊥ − 4 αRe P.A(0)uα Ψuα , Ψ⊥ . Together with Lemmata A.1–A.3, this concludes the proof of Proposition 6.1.
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Appendix B. Proof of Proposition 6.2 In this section, we present the proof of Proposition 6.2. To begin with, we establish the estimate Proposition B.1. We have HΨ⊥ , Ψ⊥
− ⊥ 2 ⊥ ⊥ ⊥ − ⊥ 2 − ⊥ 2 ≥ HΨ⊥ 1 , Ψ1 + HΨ2 , Ψ2 + 2α A Ψ − A Ψ1 − A Ψ2 1
⊥ ⊥ ⊥ ⊥ ⊥ 2 − 4Re P.Pf Ψ⊥ 2 , Ψ1 − 4α Re P.A(0)Ψ2 , Ψ1 + 4Re Pf .A(0)Ψ2 , Ψ1 1
6 −1 2 5 − M [Ψ⊥ |κ3 |2 − c0 α(hα + e0 ) 2 π 0 Ψ⊥ 2 ] − cα log α 1 + O(α ). (156)
Proof. Recall that H = P2 −
1 α + T (0) − 2Re P. Pf + α 2 A(0) , |x|
(157)
and 1
T (0) =: (Pf + α 2 A(0))2 : +Hf .
(158)
Due to the orthogonality n ⊥ π n Ψ⊥ 1 , π Ψ2 = 0,
n = 0, 1, . . . ,
and (157), (158), we obtain (H + e0 )Ψ⊥ , Ψ⊥ ⊥ ⊥ ⊥ = (H + e0 )Ψ⊥ 2 , Ψ2 + (H + e0 )Ψ1 , Ψ1
+
3
n ⊥ 2αRe A− .A− π n+2 Ψ⊥ 2 , π Ψ1
n=0 1 ⊥ − ⊥ 2 − ⊥ 2 − ⊥ 2 + 2αRe A− .A− π 3 Ψ⊥ 1 , π Ψ2 + 2α(A Ψ − A Ψ1 − A Ψ2 ) 1
1
⊥ ⊥ ⊥ ⊥ ⊥ 2 2 − 4Re P.Pf Ψ⊥ 2 , Ψ1 −4α Re P.A(0)Ψ2 , Ψ1 +4α Re Pf .A(0)Ψ2 , Ψ1 .
(159) We have 1
3 ⊥ 5 ⊥ 2 2 3 ⊥ 2 2 2αRe A− .A− π 5 Ψ⊥ 2 , π Ψ1 ≥ − Hf π Ψ2 − cα π Ψ1 1
2 7 −1 ≥ − Hf2 π 5 Ψ⊥ |κ3 |2 . (160) 2 − cα log α
Similarly, 2 ⊥ 2αRe A− .A− π 4 Ψ⊥ 2 , π Ψ1 1 2
4
2 Ψ⊥ 2
− cα |κ2 | π
1 2
4
2 Ψ⊥ 2
+ O(α ).
≥ − Hf π ≥ − Hf π
4
2
0
2 Ψ⊥ 1
−
3
cα6 |κ2,i |2
i=1 5
1 ⊥ To estimate the term 2αRe A− .A− π 3 Ψ⊥ 2 , π Ψ1 we rewrite it as 2αRe uα 3 ⊥ + + 3 ⊥ π Ψ2 , A .A κ1 Φ and use that π Ψ2 , (Hf + Pf2 )−1 A+ .A+ Φu α = 0.
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This yields, using Lemma C.6 1 ⊥ αRe A− .A− π 3 Ψ⊥ 2 , π Ψ1
2 −1 + A .A+ κ1 Φu α = −αRe π 3 Ψ⊥ 2 , (hα + e0 )(Hf + Pf ) 2 7 −1 . ≥ − α2 π 3 Ψ⊥ 2 + cα log α
(161)
3 2
Similarly, if π 0 Ψ⊥ 1>α , 0 ⊥ 2αRe A− .A− π 2 Ψ⊥ 2 , π Ψ1
2 −1 + A .A+ π 0 Ψ⊥ = −2αRe π 2 Ψ⊥ 2 , (hα + e0 )(Hf + Pf ) 1 1
1
2 2 −1 + 2 2 A .A+ π 0 Ψ⊥ ≥ −cα(hα + e0 ) 2 π 2 Ψ⊥ 2 − cα(hα + e0 ) (Hf + Pf ) 1 1
2 −2 2 ⊥ 2 ≥ −cαP π 2 Ψ⊥ π Ψ2 2 + cα|x| 1
2 0 ⊥ 2 2 −cαe0 π 2 Ψ⊥ 2 − c0 α(hα + e0 ) π Ψ1 1
2 2 2 ⊥ 2 0 ⊥ 2 2 ≥ −cαP π 2 Ψ⊥ 2 − α π Ψ2 − c0 α(hα + e0 ) π Ψ1 .
(162)
3 2
If π 0 Ψ⊥ 1 ≤ α , we have instead 1
0 ⊥ 2 ⊥ 2 2 0 ⊥ 2 2 2αRe A− .A− π 2 Ψ⊥ 2 , π Ψ1 ≥ − Hf π Ψ2 − cα π Ψ1 1
2 5 ≥ − Hf2 π 2 Ψ⊥ 2 + O(α ).
(163)
Finally, using Lemma C.7 yields 1
1 ⊥ ⊥ 2 5 2 2αRe A− .A− π 3 Ψ⊥ 1 , π Ψ2 ≥ − Hf Ψ2 + O(α ).
Collecting (159)–(164) concludes the proof of the proposition.
(164)
In the rest of this section, we estimate further terms in (156). ⊥ B.1. Estimate of Crossed Terms Involving Ψ⊥ 1 and Ψ2 Lemma B.1. 2 − ⊥ 2 ⊥ 5 2α(A− Ψ⊥ 2 − A− Ψ⊥ 1 − A Ψ2 ) ≥ − M [Ψ2 ] + O(α ).
(165)
Proof. Obviously, the left-hand side of (165) is equal to 1 1 − ⊥ n ⊥ 2 4αRe A− Ψ⊥ Hf2 π n Ψ⊥ 1 , A Ψ2 ≥ −cα 1 Hf π Ψ2 n
1 2
1
− cα |κ1 | Hf2 Φu α 2 ≥ − Hf π
1 1 2 n ⊥ 2 2 − cα 3Hf2 π n Ψ⊥ 2 + 2Hf π Ψ 1
2 Ψ⊥ 2
2
2
n =1
1
2 5 ≥ − Hf2 Ψ⊥ 2 + O(α ),
(166)
where in the last inequality we used (210) of Lemma C.4, and (45) of Theorem 5.2. Lemma B.2. ⊥ ⊥ 7 −1 |P.Pf Ψ⊥ |κ3 |2 + O(α5 ) 2 , Ψ1 | ≤ M [Ψ2 ] + cα log α
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Proof. We have ⊥ P.Pf Ψ⊥ 2 , Ψ1 1 ⊥ 2 ⊥ 2 ⊥ 3 ⊥ 3 ⊥ = Pf π 1 Ψ⊥ 2 , P π Ψ1 + Pf π Ψ2 , P π Ψ1 + Pf π Ψ2 , P π Ψ1 . (167)
Obviously, using Lemma C.4 and the equality κ1 = O(1) from Lemma 6.1, yields 1 ⊥ Pf π 1 Ψ⊥ 2 , P π Ψ1 | 1
1
1
uα 2 2 2 ⊥ 2 5 2 2 ≤ Hf2 Ψ⊥ 2 + |κ1 | P |Pf | Φ ≤ Hf Ψ2 + O(α ). (168)
We also have, by definition of π 2 Ψ⊥ 1 and using the estimates κ2,i = O(1) from Lemma 6.1, 1
2 ⊥ ⊥ 2 2 2 0 ⊥ 2 6 2 Pf π 2 Ψ⊥ 2 , P π Ψ1 | ≤ Hf Ψ2 + c|κ2 | α P π Ψ1 + O(α ). (169)
We next bound the second term on the right-hand side of (169). Notice that 2 3 by definition of κ2 , this term is nonzero only if π 0 Ψ⊥ 1 > α , which implies, 2 0 ⊥ 2 ≤ cαπ Ψ . The inequality (169) can with Lemma 6.1, that P π 0 Ψ⊥ 1 1 thus be rewritten as 2 ⊥ |P.Pf π 2 Ψ⊥ 2 , π Ψ1 | 1
1
2 2 3 0 ⊥ 2 6 ⊥ 2 5 2 ≤ Hf2 Ψ⊥ 2 + c|κ2 | α π Ψ1 + O(α ) ≤ Hf Ψ2 + O(α ),
(170) using in the last inequality that |κ2 |π 0 Ψ⊥ 1 = O(α) (see Lemma 6.1). For the second term on the right-hand side of (167), using (212) from Lemma C.4 yields 1
3 ⊥ 3 ⊥ 2 7 −1 2 π 3 Pf Ψ⊥ |κ3 |2 . 2 , π P Ψ1 ≤ π Hf Ψ2 + cα log α
(171)
The inequalities (167), (170) and (171) prove the lemma.
Lemma B.3. 1
⊥ ⊥ 5 −4α 2 Re P.A+ Ψ⊥ 2 , Ψ1 ≥ − M [Ψ2 ] + O(α ). 0 ⊥ Proof. Since π n>3 Ψ⊥ 1 = 0, π Ψ2 = 0 and 1
1
1
2 n =1 ⊥ 2 2 ⊥ 4 2 Ψ + 2Hf2 π n =1 Ψ⊥ Hf2 π n =1 Ψ⊥ 1 ≤ 2Hf π 2 ≤ cM [Ψ2 ] + O(α ),
(see (45) of Theorem 5.2) we obtain 1
⊥ 4α 2 Re P.A+ Ψ⊥ 2 , Ψ1 1
2 n≥2 2 ⊥ 5 ≤ π n≤2 P Ψ⊥ Hf2 Ψ⊥ 2 + cαπ 1 ≤ M [Ψ2 ] + O(α ).
Lemma B.4. 1
⊥ ⊥ 5 −4α 2 Re P.A− Ψ⊥ 2 , Ψ1 ≥ − M [Ψ2 ] + O(α ).
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Proof. 1
1
⊥ ⊥ 2 ⊥ 2 2 −4α 2 Re P.A− Ψ⊥ 2 , Ψ1 ≥ − Hf Ψ2 − cαP Ψ1 1 n=0,2,3,4 2 2 ≥ − Hf2 Ψ⊥ P Ψ⊥ 2 + π n=0,2,3,4 P Ψ⊥ 2 − cα π 2
−cα|κ1 |2 P Φu α 2 1
2 n≤4 2 6 −1 ≥ − Hf2 Ψ⊥ P Ψ⊥ ) 2 − π 2 + O(α log α 6 −1 ≥ − M [Ψ⊥ ), 2 ] + O(α log α
using (212) of Lemma C.4. Lemma B.5. 1
1
⊥ 2 ⊥ + uα ⊥ 5 2 4α 2 Re Pf .A(0)Ψ⊥ 2 , Ψ1 ≥ 4α Re π Ψ2 , Pf .A Φ∗ − M [Ψ2 ] + O(α ).
Proof. We have 1
⊥ 4α 2 Re Pf .A− Ψ⊥ 2 , Ψ1 1
1
uα 2 n≥2 2 − 2 ⊥ 2 Pf Ψ⊥ ≥ − Hf2 Ψ⊥ 2 − cαπ 1 + 4α Re Pf .A π Ψ2 , κ1 Φ 1
1
1
uα 2 n=2,3 ≥ − Hf2 Ψ⊥ Hf2 Ψ⊥ 2 + 4α 2 Re Pf .A− π 2 Ψ⊥ 2 − cαπ 2 , κ1 Φ 1
uα − 2 ⊥ 5 2 ≥ − M [Ψ⊥ 2 ] + 4α Re Pf .A π Ψ2 , κ1 Φ + O(α ).
We estimate the second term on the right-hand side as follows: 1
uα 4α 2 Re Pf .A− π 2 Ψ⊥ 2 , κ1 Φ 1
1
1
1
uα uα 2 2 ⊥ 2 2 2 ≥ 4α 2 Re Pf .A− π 2 Ψ⊥ 2 , Φ − Hf π Ψ2 − cα|κ − 1| Pf Φ uα 2 ⊥ 2 5 2 ≥ 4α 2 Re Pf .A− π 2 Ψ⊥ 2 , Φ∗ − Hf π Ψ2 + O(α ),
where we used |κ1 − 1|2 = O(α) from Lemma 6.1, Φu α 2∗ = O(α3 ) from Lemma C.4, and Pf (Φu α − Φu∗ α )2 = O(α4 ) from Lemma C.5. We also have, using Pf .A+ = A+ .Pf , 1
1
⊥ n≤2 − n≥2 ⊥ 2 4α 2 Re Pf .A+ Ψ⊥ Pf Ψ⊥ Ψ1 2 , Ψ1 = 4α Re π 2 ,A π 1
1
2 n≥2 ⊥ 2 5 2 Ψ1 ≥ − M [Ψ⊥ ≥ − Hf2 Ψ⊥ 2 − cαHf π 2 ] + O(α ), 1
⊥ where Hf2 π n≥2 Ψ⊥ 1 has been estimated as P Ψ1 in the proof of Lemma B.4. ⊥ B.2. Estimates of the Term (H + e0 )Ψ⊥ 1 , Ψ1 Due to (157) and (158), one finds 1
⊥ ⊥ ⊥ ⊥ ⊥ 2 (H + e0 )Ψ⊥ 1 , Ψ1 = Ψ1 , Ψ1 − 2Re P.(Pf + α A(0))Ψ1 , Ψ1 1
⊥ − ⊥ 2 − − ⊥ ⊥ +2α 2 Re Pf .A(0)Ψ⊥ 1 , Ψ1 + 2αA Ψ1 + 2αRe A .A Ψ1 , Ψ1
(172) We estimate the terms in (172) below.
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Lemma B.6. We have 1
⊥ −2Re P.(Pf + α 2 A(0))Ψ⊥ 1 , Ψ1 1
s ⊥ 4 5 ≥ −4α 2 Re A− Φu α , P π 0 (Ψ⊥ 1 ) − M [Ψ2 ] − |κ1 − 1|cα + O(α ), s 0 ⊥ 0 ⊥ a 0 ⊥ where π 0 (Ψ⊥ 1 ) = π Ψ1 − π (Ψ1 ) is the even part of π Ψ1 .
Proof. Using Φ2∗ , Pfi Φ2∗ = 0 (see Lemma C.2), the symmetry of uα , P.Pf Φu α , Φu α = 0, we obtain 3 ∂uα ⊥ ⊥ |2P.Pf Ψ1 , Ψ1 | = 2 Pf ακ2 Φ2∗ π 0 Ψ⊥ , P α κ2,i (Hf + Pf2 )−1 Wi 1 ∂xi
and
i=1
2 5 ≤ cα3 |κ2 |2 π 0 Ψ⊥ 1 + cα
3
|κ2,i |2 = O(α5 ),
(173)
i=1
where in the last inequality we used Lemma 6.1. We also have, using again the symmetry of uα , 1
1
⊥ − ⊥ ⊥ 2 −2α 2 Re P.A(0)Ψ⊥ 1 , Ψ1 = −4α Re P.A Ψ1 , Ψ1 1
1
uα s − 2 0 ⊥ 2 = −4α 2 Re A− κ1 Φu α , P π 0 (Ψ⊥ 1 ) − 4α Re A κ2 αΦ∗ π Ψ1 , P κ1 Φ 1
2 0 ⊥ −4α 2 Re A− π 3 Ψ⊥ 1 , P ακ2 Φ∗ π Ψ1 1
1
s 3 ⊥ 2 2 0 ⊥ 2 2 2 ≥ −4α 2 Re A− κ1 Φu α , P π 0 (Ψ⊥ 1 ) − cαHf π Ψ1 − cα P π Ψ1 |κ2 | 1
1
1
uα 6 −4α 2 Re |k|− 6 A− κ2 αΦ2∗ π 0 Ψ⊥ 1 , |k| κ1 P Φ 1
1
1
s 3 ⊥ 2 3 ⊥ 2 2 2 ≥ −4α 2 Re A− κ1 Φu α , P π 0 (Ψ⊥ 1 ) − cαHf π Ψ − cαHf π Ψ2 3
1
uα 6 −cα5 − cα 2 |κ2 |π 0 Ψ⊥ 1 |k| P Φ 1
s ⊥ 5 ≥ −4α 2 Re A− κ1 Φu α , P π 0 (Ψ⊥ 1 ) − M [Ψ2 ] − cα ,
(174)
where we used Theorem 5.2 and Lemma C.4. 3 2 Moreover, because A− Φu∗ α = O(α 2 ) and P π 0 Ψ⊥ 1 = O(α ), we obtain 1
s −4Re α 2 Re A− κ1 Φu α , P π 0 (Ψ⊥ 1) 1
s 4 ≥ −4α 2 Re A− Φu α , P π 0 (Ψ⊥ 1 ) − |κ1 − 1|cα .
(175)
This estimate, together with (173) and (174), proves the lemma. Lemma B.7. We have 1
⊥ 2α 2 Re Pf .A(0)Ψ⊥ 1 , Ψ1 2 κ2,i Pf .A− (Hf + Pf2 )−1 Wi , (Hf + Pf2 )−1 (A+ )i Ωf ≥ α4 Re 3 i
−|κ1 − 1|cα4 + O(α5 ).
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Proof. The following holds: 1
⊥ 2α 2 Re Pf .A(0)Ψ⊥ 1 , Ψ1 1
uα 2 ⊥ − 2 ⊥ = 4α 2 Re Pf .A− π 3 Ψ⊥ 1 , π Ψ1 + 4αRe Pf .A π Ψ1 , κ1 Φ
1 1 ∂uα − 3 ⊥ − 16 − 16 2 −1 6 6 κ2,i |k1 | |k2 | A π Ψ1 , Pf |k1 | |k2 | (Hf +Pf ) Wi = 4α Re ∂xi i=1 3
3 2
3
1
1
1
1
−6 + 4α 2 Re κ2 |k1 | 6 |k2 | 6 A− π 3 Ψ⊥ |k2 |− 6 Pf Φ2∗ π 0 Ψ⊥ 1 , |k1 | 1 1
uα + 4α 2 Re Pf .A− π 2 Ψ⊥ 1 , κ1 Φ .
(176) 1
1
Applying the Schwarz inequality and the estimates |k1 | 6 |k2 | 6 A− π 3 ⊥ 2 Ψ1 = O(α5 ) (Lemma C.6), κ2,i = O(1) (Lemma 6.1), and ∇uα 2 = O(α2 ), we see that the first term on the right-hand side of (176) is O(α5 ). Applying also the estimate |κ2 | π 0 Ψ⊥ 1 = O(α) (Lemma 6.1), we obtain that the second term on the right-hand side of (176) is also O(α5 ). 1 uα Finally, we estimate 4α 2 Re Pf .A− π 2 Ψ⊥ 1 , κ1 Φ . The following inequality holds: 1
1
uα uα − 2 ⊥ 4 2 4α 2 Re Pf .A− π 2 Ψ⊥ 1 , κ1 Φ | ≥ 4α Re Pf .A π Ψ1 , Φ − |κ1 − 1|cα ,
(177) whose proof is similar to the one of (175). Next we get 1
1
uα − 2 ⊥ uα 2 |4α 2 Re Pf .A− π 2 Ψ⊥ 1 , Φ − 4α Re Pf .A π Ψ1 , Φ∗ | 1
1
1
uα uα 6 2 ≤ α 2 Hf2 π 2 Ψ⊥ 1 Pf (Φ − Φ∗ ) = O(α | log α| ),
(178) 1
using Pf (Φu α − Φu∗ α ) = O(α 2 | log α| 2 ) (see Lemma C.5) and Hf2 π 2 Ψ⊥ 1= O(α2 ). Moreover, 7
1
1
uα 4α 2 Re Pf .A− π 2 Ψ⊥ 1 , Φ∗ 3 2
= 4α Re
−
Pf .A
3
Pf .A−
i
=
2 4 α Re 3
+
κ2,i (Hf +
i
= 4α 2 Re
κ2 Φ2∗ π 0 Ψ⊥ 1
κ2,i (Hf + Pf2 )−1 Wi
∂uα Pf2 )−1 Wi ∂xi
, Φu∗ α
∂uα uα ,Φ ∂xi ∗
κ2,i Pf .A− (Hf + Pf2 )−1 Wi , (Hf + Pf2 )−1 (A+ )i Ωf
(179)
i
where we used (207) of Lemma C.2 in the second equality. Collecting (176)–(199) concludes the proof.
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Lemma B.8. We have 0 ⊥ 2 ⊥ − − ⊥ ⊥ 1 ⊥ 2 Ψ⊥ 1 , Ψ1 + 2αRe A .A Ψ1 , Ψ1 ≥ Σ0 π Ψ1 + π Ψ1 3 α4 |κ2,i |2 (Hf + Pf2 )−1 Wi 2∗ + α2 |κ3 + 1|2 Φ2∗ 2∗ Φu α 2 + 12 i=1 2 0 ⊥ 2 5 −1 +π 1 Ψ⊥ ). 1 + π Ψ1 + o(α log α
Proof. Obviously, we have ⊥ Ψ⊥ 1 , Ψ1 =
3
i ⊥ π i Ψ⊥ 1 , π Ψ1 ,
(180)
i=0
and using Lemma C.2 π
2
2 ⊥ 2 2 2 0 ⊥ 2 2 Ψ⊥ 1 , π Ψ1 = α |κ2 | Φ∗ π Ψ1 +α
3
|κ2,i |2 (Hf +Pf2 )−1 Wi
i=1
∂uα 2 , ∂xi (181)
2 2 2 0 ⊥ 2 Moreover, from the inequality Φ2∗ π 0 Ψ⊥ 1 > Φ∗ ∗ π Ψ1 we obtain 2 − − 2 ⊥ 0 ⊥ α2 |κ2 |2 Φ2∗ π 0 Ψ⊥ 1 + 2αRe A .A π Ψ1 , π Ψ1 2 − − 2 ⊥ 0 ⊥ ≥ α2 |κ2 |2 Φ2∗ 2∗ π 0 Ψ⊥ 1 + 2αRe A .A π Ψ1 , π Ψ1 2 2 2 2 0 ⊥ 2 = α2 |κ2 |2 Φ2∗ 2∗ π 0 Ψ⊥ 1 − 2α Re κ2 Φ∗ ∗ π Ψ1 3 ∂uα 0 ⊥ +2αRe α A− .A− κ2,i (Hf + Pf2 )−1 Wi , π Ψ1 ∂xi i=1 2 2 2 0 ⊥ 2 5 0 ⊥ 2 5 ≥ Σ0 π 0 Ψ⊥ 1 + |κ2 − 1| cα π Ψ1 + O(α ) ≥ Σ0 π Ψ1 + O(α ). (182)
where we used Σ0 = −α2 Φ2∗ 2∗ +O(α3 ) and A− .A− (Hf +Pf2 )−1 Wi , π 0 Ψ⊥ 1= 2 −π 0 Ψ⊥ 1 Wi , Φ∗ = 0. 3 ⊥ Similarly, π 3 Ψ⊥ 1 > π Ψ1 ∗ yields 2 − − 3 ⊥ 1 ⊥ π 3 Ψ⊥ 1 + 2αRe A .A π Ψ1 , π Ψ1
≥ α2 |κ3 |2 (Hf + Pf2 )−1 A+ .A+ Φu α 2∗ +2Re α2 κ3 A− .A− (Hf + Pf2 )−1 A+ .A+ Φu α , κ1 Φu α ≥ −α2 (Hf + Pf2 )−1 A+ .A+ κ1 Φu α 2∗ +|κ3 + 1|2 α2 (Hf + Pf2 )−1 A+ .A+ Φu α 2∗ uα 2 2 2 2 2 2 5 −1 ≥ Σ0 π 1 Ψ⊥ ), 1 + α |κ3 + 1| Φ∗ ∗ Φ + o(α log α
(183) 1
where in the last inequality, we used (222) from Lemma C.6, κ1 = 1 + O(α 2 ) from Lemma 6.1, Φu α 2 = O(α3 log α−1 ) from Lemma C.4 and −α2 Φ2∗ 2∗ = Σ0 + O(α3 ).
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The second term on the right-hand side of (181) is estimated as 2 2 3 3 ∂uα ∂uα 2 2 2 −1 2 2 2 −1 |κ2,i | (Hf + Pf ) Wi |κ2,i | (Hf + Pf ) Wi α ≥α ∂x ∂x i i ∗ i=1 i=1 =α
=
21
3
∇uα
2
3
|κ2,i |2 (Hf + Pf2 )−1 Wi 2∗
i=1
3 α4 |κ2,i |2 (Hf + Pf2 )−1 Wi 2∗ . 12 i=1
(184)
Collecting (180)–(184) concludes the proof. Proposition B.2. We have ⊥ (H + e0 )Ψ⊥ 1 , Ψ1 1
1
− uα 2 0 ⊥ a 2 2 ≥ −4α(hα + e0 )− 2 Q⊥ α P.A Φ∗ + (hα + e0 ) π (Ψ1 ) 1
s 2 0 ⊥ 2 1 ⊥ 2 + c0 α(hα + e0 ) 2 π 0 (Ψ⊥ 1 ) + Σ0 (π Ψ1 + π Ψ1 )
+
3 1 α4 uα + |κ2,i |2 (Hf + Pf2 )−1 Wi 2∗ + 4α 2 Re π 2 Ψ⊥ 1 , A .Pf Φ 12 i=1
+ κ1 Φu α 2 + 2αA− Φu∗ α 2 + α2 |κ3 + 1|2 Φ2∗ 2∗ Φu α 2 4 5 −1 − M [Ψ⊥ ), 2 ] − |κ1 − 1|cα + o(α log α
(185)
where c0 is the same positive constant as in Proposition B.1 and • •
⊥ Q⊥ α is the orthogonal projection onto Span(uα ) , ⊥ a 0 ⊥ (Ψ1 ) is the odd part of π Ψ1 .
Proof. Collecting Lemmata B.6, B.7, and B.8 yields 1
⊥ − uα 0 ⊥ s 2 (H + e0 )Ψ⊥ 1 , Ψ1 ≥ −4α Re A Φ , P π (Ψ1 ) uα + + 4αRe π 2 Ψ⊥ 1 , A .Pf Φ uα 2 2 1 ⊥ 2 2 2 2 2 + Σ0 (π 0 Ψ⊥ 1 + π Ψ1 ) + α |κ3 + 1| Φ∗ ∗ Φ 1
uα 2 2 1 ⊥ 2 − 4 + (hα + e0 ) 2 π 0 Ψ⊥ 1 + π Ψ1 + 2αA κ1 Φ − |κ1 − 1|cα
+
3 α4 5 −1 |κ2,i |2 (Hf + Pf2 )−1 Wi 2∗ − M [Ψ⊥ ). (186) 2 ] + o(α log α 12 i=1
Obviously, 1
1
1
2 0 ⊥ s 2 0 ⊥ a 2 2 2 (hα + e0 ) 2 π 0 Ψ⊥ 1 = (hα + e0 ) π (Ψ1 ) + (hα + e0 ) π (Ψ1 ) . (187) ⊥ a As before, we write π 0 Ψ⊥ 1 as the sum of its odd part (Ψ1 ) and its even part ⊥ s 0 ⊥ a (Ψ1 ) . Since π Ψ1 is orthogonal to uα by definition of Ψ⊥ , and (Ψ⊥ 1 ) is ⊥ s orthogonal to uα by symmetry of uα , we also have (Ψ1 ) orthogonal to uα . s ⊥ 0 ⊥ s Therefore, one can replace π 0 (Ψ⊥ 1 ) by Qα π (Ψ1 ) in (186) and (187). Thus,
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as the next step, given a constant c0 > 0, we minimize 1
1
0 ⊥ s 2 ⊥ − uα ⊥ 0 ⊥ s 2 (1 − c0 α)(hα + e0 ) 2 Q⊥ α π (Ψ1 ) − 4α Re Qα P.A Φ , Qα π (Ψ1 ) 1 4α − uα 2 (hα + e0 )− 2 Q⊥ (188) ≥− α P.A Φ . 1 − c0 α
Obviously, −
1 4α − uα 2 (hα + e0 )− 2 Q⊥ α P.A Φ 1 − c0 α 1 4(1 + α)α − uα 2 (hα + e0 )− 2 Q⊥ ≥− α P.A Φ∗ 1 − c0 α 1 4(1 + α−1 )α uα − uα 2 (hα + e0 )− 2 Q⊥ − α P.A (Φ − Φ∗ ) . 1 − c0 α
(189)
There exist γ1 and γ2 positive, independent of α, such that −1 ⊥ Qα ≤ (γ1 P 2 + γ2 α2 )−1 , Q⊥ α (hα + e0 ) −1 ⊥ Qα P is a bounded operator. In addition, since and thus P Q⊥ α (hα + e0 ) uα − uα 2 A (Φ − Φ∗ ) = O(α7 log α−1 ) (Lemma C.4), this shows that 1 4(1 + α−1 )α uα − uα 2 6 −1 (hα + e0 )− 2 Q⊥ ). α P.A (Φ − Φ∗ ) = O(α log α 1 − c0 α
(190)
In addition, using A− Φu∗ α ≤ A− (Φu∗ α − Φu α ) + A− Φu α ≤ cα 2 −1 ⊥ (Lemma C.4 and Lemma C.5), and the fact that P Q⊥ Qα P is α (hα + e0 ) bounded, yields 3
−
1 (1 + α)α − uα 2 (hα + e0 )− 2 Q⊥ α P.A Φ∗ 1 − c0 α 1
− uα 2 5 = −4(hα + e0 )− 2 Q⊥ α P.A Φ∗ + O(α ).
(191)
Collecting (189)–(191), one gets −
1 4α − uα 2 (hα + e0 )− 2 Q⊥ α P.A Φ 1 − c0 α 1
− uα 2 5 ≥ −4(hα + e0 )− 2 Q⊥ α P.A Φ∗ + O(α ).
(192)
Finally, using A− Φu∗ α 2 = O(α3 ), we obtain 2α|κ1 |2 A− Φu∗ α 2 ≥ 2αA− Φu∗ α 2 − c|κ1 − 1|α4 .
(193)
Substituting (187), (188), (192) and (193) into (186) concludes the proof. We can now collect the above results to prove Proposition 6.2.
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B.3. Concluding the Proof of Proposition 6.2 Collecting the results of Proposition B.1, Lemmata B.1–B.5 and Proposition B.2 yields directly the following bound: HΨ⊥ , Ψ⊥ ⊥ ⊥ 2 4 0 ⊥ 2 6 −1 2 |κ3 |2 − e0 Ψ⊥ ≥ HΨ⊥ 2 , Ψ2 − M [Ψ2 ] − c|κ2 | α π Ψ1 − cα log α 1 1
1
− uα 2 0 ⊥ a 2 2 − 4α(hα + e0 )− 2 Q⊥ α P.A Φ∗ + (1 − c0 α)(hα + e0 ) π (Ψ1 )
+ Σ0 (π
0
2 Ψ⊥ 1
+ π
1
2 Ψ⊥ 1 )
3 α4 + |κ2,i |2 (Hf + Pf2 )−1 Wi 2∗ 12 i=1
1
1
uα + uα 2 ⊥ + 2 + 4α 2 Re π 2 Ψ⊥ 2 , A .Pf Φ∗ + 4α Re π Ψ1 , A .Pf Φ
+ |κ1 |2 Φu α 2 + 2αA− Φu∗ α 2 + α2 |κ3 + 1|2 Φ2∗ 2∗ Φu α 2 − |κ1 − 1|cα4 + o(α5 log α−1 ).
(194)
Comparing this expression with the statement of the Proposition, we see that it suffices to show that 2 4 0 ⊥ 2 6 −1 2 ⊥ ⊥ − M [Ψ⊥ |κ3 |2 − e0 Ψ⊥ 2 ] − c|κ2 | α π Ψ1 − cα log α 1 + HΨ2 , Ψ2 uα 2 2 1 ⊥ 2 2 2 2 2 + Σ0 (π 0 Ψ⊥ 1 + π Ψ1 ) + α |κ3 + 1| Φ∗ ∗ Φ
≥ (Σ0 − e0 )Ψ⊥ 2 + (1 − )M [Ψ⊥ 2]+
|κ3 + 1|2 2 2 2 uα 2 α Φ∗ ∗ Φ + O(α5 ). 2 (195)
⊥ ⊥ 2 ⊥ Using from Corollary 4.2 that HΨ⊥ 2 , Ψ2 ≥ (Σ0 − e0 )Ψ2 + M [Ψ2 ], we first estimate the following terms in (195):
2 1 ⊥ 2 2 ⊥ ⊥ ⊥ − e0 Ψ⊥ Σ0 π 0 Ψ⊥ 1 + π Ψ1 1 + HΨ2 , Ψ2 − M [Ψ2 ] ⊥ 2 ⊥ 2 n≥2 ⊥ 2 Ψ1 ≥ (1 − )M [Ψ⊥ 2 ] + (Σ0 − e0 )(Ψ1 + Ψ2 ) − Σ0 π
⊥ 2 ≥ (1 − )M [Ψ⊥ 2 ] + (Σ0 − e0 )Ψ
2 ⊥ 2 n≥2 ⊥ 2 −(Σ0 − e0 )(Ψ⊥ 2 − Ψ⊥ Ψ1 . 1 − Ψ2 ) − Σ0 π
(196)
We have obviously 2 ⊥ 2 Ψ⊥ 2 − Ψ⊥ 1 − Ψ2
1 ⊥ 2 ⊥ 2 ⊥ 3 ⊥ 3 ⊥ = 2Re (π 1 Ψ⊥ 1 , π Ψ2 + π Ψ1 , π Ψ2 + π Ψ1 , π Ψ2 ).
(197)
Since |Σ0 − e0 | ≤ cα2 , by definition of π 3 Ψ⊥ 1 and Lemma C.6, we obtain 3 ⊥ |(Σ0 − e0 )2Re π 3 Ψ⊥ 1 , π Ψ2 |
2 4 2 2 −1 + A .A+ Φu α 2 ≤ α2 π 3 Ψ⊥ 2 + cα |κ3 | (Hf + Pf )
2 2 7 −1 ≤ α2 π 3 Ψ⊥ . 2 + c|κ3 | α log α
(198)
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Similarly, for the two-photon sector, we find 2 ⊥ |(Σ0 − e0 )π 2 Ψ⊥ 1 , π Ψ2 | 2 4 2 0 ⊥ 2 ≤ α2 π 2 Ψ⊥ 2 + cα |κ2 | π Ψ1 + c
3
|κ2,i |2 α6
i=1 2 6 = α2 π 2 Ψ⊥ 2 + O(α ),
(199)
1 ⊥ where we used Lemma 6.1. For the term π 1 Ψ⊥ 1 , π Ψ2 , one gets 1
1
uα 1 ⊥ −2 2 |(Σ0 − e0 )π 1 Ψ⊥ κ1 Φu α | 2 , κ1 Φ | = |(Σ0 − e0 )(|k| π Ψ2 , |k| 1
1
1
2 4 2 − 2 uα 2 2 5 ≤ Hf2 π 1 Ψ⊥ Φ ≤ Hf2 π 1 Ψ⊥ 2 + cα |κ1 | |k| 2 + O(α ),
(200)
since |k|− 2 Φu α 2 = O(α) (Lemma C.4) and κ1 = O(1) (Lemma 6.1). Collecting (197), (198), (199) and (200) yields 2 ⊥ 2 2 7 −1 ≤ M [Ψ⊥ |Σ0 − e0 | Ψ⊥ 2 − Ψ⊥ + O(α5 ). 1 − Ψ2 2 ] + c|κ3 | α log α (201) 1
Therefore, together with (196), one finds 2 1 ⊥ 2 2 ⊥ ⊥ ⊥ Σ0 π 0 Ψ⊥ − e0 Ψ⊥ 1 + π Ψ1 1 + HΨ2 , Ψ2 − M [Ψ2 ] ⊥ 2 ≥ (1 − 2 )M [Ψ⊥ 2 ] + (Σ0 − e0 )Ψ
2 5 − c|κ3 |2 α7 log α−1 − Σ0 π n≥2 Ψ⊥ 1 + O(α ),
(202)
2 0 ⊥ By definition of Ψ⊥ 1 and using Σ0 = O(α ), |κ2 |π Ψ1 = O(α) (Lemma 6.1) and Inequality (223) of Lemma C.6, we straightforwardly obtain 2 6 2 7 −1 Σ0 π n≥2 Ψ⊥ . 1 ≤ cα + c|κ3 | α log α
Substituting this in (202) yields 2 1 ⊥ 2 2 ⊥ ⊥ ⊥ Σ0 π 0 Ψ⊥ − e0 Ψ⊥ 1 + π Ψ1 1 + HΨ2 , Ψ2 − M [Ψ2 ] ⊥ 2 2 7 −1 + O(α5 ). ≥ (1 − 2 )M [Ψ⊥ 2 ] + (Σ0 − e0 )Ψ − 2c|κ3 | α log α
(203)
To conclude the proof of (195), and thus of the Proposition, we first note that according to Lemma 6.1, 2 6 − c|κ2 |2 α4 π 0 Ψ⊥ 1 = O(α ).
(204)
Similarly, taking into account that Φu α 2 = cα3 log α−1 (see (209) in Lemma C.4), we get for some c2 > 0, |κ3 + 1|2 2 2 uα 2 Φ∗ ∗ Φ − cα6 log α−1 |κ3 |2 − 2cα7 log α−1 |κ3 |2 2 ≥ −c2 α6 log α−1 . (205)
α2
Collecting (203), (204) and (205) yields the bound (195), and thus concludes the proof of the Proposition 6.2.
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Appendix C. Proof of Theorem 2.1: Auxiliary Lemmata The proof of Theorem 2.1 uses Lemmata C.2–C.7 below. Lemma C.1. Let Ψ0 be the ground state of T (0), with |π 0 Ψ0 | = 1. Then we have 3
3
Ψ0 = Ωf + 2η1 α 2 Φ1∗ + η2 αΦ2∗ + 2η3 α 2 Φ3∗ + Δ0∗ , Δ0∗ , Φi∗ ∗ = 0 (i = 1, 2, 3), π 0 Δ0∗ = 0
with and
Δ0∗ 2 = O(α3 ). Proof. It follows from a similar argument as in [9, Proposition 5.1] that cα aλ (k)Δ0∗ ≤ . |k| This yields
Δ0∗ 2 ≤
aλ (k)Δ0∗ 2 dk c2 α2 dk + ≤ |k|2 |k|≤α
|k|>α
−1
≤ c α +cα 2 3
|k|aλ (k)Δ0∗ 2 χΛ (|k|) dk |k|
1
Hf2 Δ0∗ 2 = O(α3 ), 1
where in the last equality we used the estimate Δ0∗ 2∗ = (Hf + Pf2 ) 2 Δ0∗ 2 = O(α4 ) proved in [7, Theorem 3.2]. Lemma C.2. We have P.A− uα Φ1∗ = 0, and Φ2∗ , ζ(Hf , Pf2 )Pfi Φ2∗ = 0,
and
Φ2∗ , ζ(Hf , Pf2 )Wi = 0,
(206)
for any function ζ for which the scalar products are defined. Similarly, we have + uα Φ2∗ π 0 Ψ⊥ 1 , A .Pf Φ∗ = 0.
(207)
Proof. Straightforward computations using the symmetries of A− Φ1∗ and Φ2∗ . Lemma C.3. We have P.A− Φu∗ α = where
a0 =
√
αa0 Δuα ,
2 k12 + k22 χΛ (|k|) dk1 dk2 dk3 . 4π 2 |k|3 |k|2 + |k|
Proof. Straightforward computations using the symmetries of A− Φu∗ α .
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Lemma C.4. P.Pf π 1 ϕuα , Φu α = 0,
∀ϕ ∈ F,
Φu α 2 = O(α3 log α−1 ), Φu α 2∗ − 12
|k|
= O(α ),
Φu α 2
|k|
1 6
(210)
= O(α).
(211)
= O(α ),
P Φu α 2
(209)
3
P Φu α 2 = O(α5 log α−1 ), P Φu α 2∗
(208)
5
= O(α ). 5
(212) (213) (214)
Proof. The proof of (208) is as follows: P.Pf π 1 ϕuα , Φu α 3 j 3 λ (k)χΛ (|k|) ∂uα 1 i ∂uα k ϕ(k) 2 dxdk = 1 ∂x k + |k| + h + e ∂xj 2π|k| 2 i α 0 j=1 i=1 λ=1,2 3 1 ∂uα ∂uα = dk dx ∂xi k 2 + |k| + hα + e0 ∂xi i=1 ×
k i i (k)χΛ (|k|) λ 1
λ=1,2
2π|k| 2
ϕ(k)dk = 0,
(215)
using that the integral over x is independent of the value of i, and since k. λ (k) = 0. To prove (211), we note that − 12
|k|
Φu α 2
2 χΛ (|k|) dkdx ∇u ≤ cα α |k|(|k| + k 2 + hα + e0 ) χΛ (|k|)2 ≤ cα3 3 2 2 dk = O(α). |k|2 (|k| + 16 α )
The proofs of (209) and (210) are similar, but simpler. We next prove (212). P Φu α 2 = cα
2 3 χΛ (|k|) ∂uα P 1 dk. |k| 2 (|k| + k 2 + h + e ) ∂xi α 0 L2 (R3 ) i=1
(216)
The function ∂uα /∂xi is odd. On the subspace of antisymmetric functions on α > −e0 for some γ0 > 0, which implies L2 (R3 ), one has that −(1 − γ0 )Δ − |x| 2 on this subspace that hα + e0 > γ0 P , and thus, P 2 < γ0−1 (hα + e0 ).
(217)
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The relation (217) yields, for all k 2 ∂uα 1 |k| 2 (|k| + k 2 + hα + e0 ) ∂xi 2 1 ∂uα χΛ (|k|) 2 ≤ γ0−1 + e ) (h 0 |k| 12 (|k| + k 2 + h + e ) α ∂xi α 0 2 (|k|) χ Λ ≤ γ0−1 cα4 . 1 3 2 |k| 2 (|k| + k 2 + 16 α )
P
χΛ (|k|)
(218)
Substituting (218) into (216) and integrating over k proves (212). The proof of (213) is similar.
Lemma C.5. We have 1 1 1 (h1 + ) 2 ∇u1 2 α5 log α−1 + o(α5 log α−1 ), 3π 4 − Φu α 2∗ = O(α5 ),
Φu∗ α 2∗ − Φu α 2 = Φu∗ α
A− (Φu α − Φu∗ α )2 = O(α7 log α−1 ). Proof. We have Φu∗ α 2∗ − Φu α 2 2 1 χ (|k|) α5 4 1 Λ 2 ) ∇u1 = 1 (h1 + , 2 1 1 (2π) 3 |k| 2 (|k| + k 2 ) 2 |k| + k 2 + α2 h + 1 2 4 1 4 (219) where the norm here is obviously taken on L2 (R3 , dx) ⊗ L2 (R3 , dk). Since (h1 + 14 )∇u1 ∈ L2 , it implies that for sufficiently large c > 0 independent of α, 1 1 χ(h1 > c)(h1 + ) 2 ∇u1 2 < , 4 and 2 χΛ (|k|) 1 1 2 ∇u ) χ(h > c)(h + 1 1 1 1 1 1 |k| 2 (|k| + k 2 ) 2 (|k| + k 2 + α2 (h1 + 1 )) 2 4 4 ∞ 2 χΛ (t) ≤ dt = log α−1 + O(1). (220) t + cα2 0
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1
For the contribution of χ(h1 ≤ c)(h1 + 14 ) 2 ∇u1 in (219), the following inequalities hold: 1 1 1 1 −1 log α + O(1) χ(h1 < c)(h1 + ) 2 ∇u1 2 π 3 4 1 1 = (2π)2 3 2 χΛ (|k|) 1 1 2 ∇u ) χ(h < c)(h + 1 1 1 1 1 1 |k| 2 (|k| + k 2 ) 2 (|k| + k 2 + (c + 1 )α2 )) 2 4 4 1 1 ≤ (2π)2 3 2 χΛ (|k|) 1 1 2 ∇u ) χ(h < c)(h + 1 1 1 1 |k| 2 (|k| + k 2 ) 12 (|k| + k 2 + (h1 + 1 )α2 )) 12 4 4 2 χΛ (|k|) 1 1 1 1 2 ∇u ) ≤ χ(h < c)(h + 1 1 1 3 2 12 (2π)2 3 |k| 12 (|k| + k 2 ) 12 (|k| + k 2 + 16 4 α )) 1 1 1 1 log α−1 + O(1) χ(h1 < c)(h1 + ) 2 ∇u1 2 . ≤ (221) π 3 4 The inequalities (220) and (221) prove the first equality of the Lemma. The proofs of the last two equalities are similar but simpler.
Lemma C.6. (Hf + Pf2 )−1 A+ .A+ Φu α 2∗ − Φ2∗ 2∗ Φu α 2 = o(α3 log α−1 ), (Hf + 1 6
1 6
Pf2 )−1 A+ .A+ Φu α 2
= O(α log α 3
−1
)
1 6
|k1 | |k2 | |k3 | (Hf + Pf2 )−1 A+ .A+ Φu α 2 = O(α3 ) (hα + e0 )(Hf +
Pf2 )−1 A+ .A+ Φu α 2
= O(α log α 7
(222) (223) (224)
−1
).
(225)
Proof. Denoting by σn the set of all permutations of {1, 2, . . . , n}, we have (Hf + Pf2 )−1 A+ .A+ Φu α 2∗ 4α 1 λ=1,2 ελ (ki ). ν=1,2 εν (kj ) = √
12 3 (2π) 6 (i,j,n)∈σ 3 3 2 n |k | + ( k ) p=1 p p=1 p 2 κ=1,2 εκ (kn )χΛ (|k1 |)χΛ (|k2 |)χΛ (|k3 |) × ∇uα 1 1 1 |ki | 2 |kj | 2 |kn | 2 (|kn | + kn2 + (hα + e0 )) If we pick two triples (i, j, n) and (i , j , n ), such that n = n , then we get a product which is integrable at k1 = k2 = k3 = 0, even without the term (hα + e0 ). The contribution of such terms is cα∇uα 2 = O(α3 ). Moreover,
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using the symmetry in k1 , k2 , k3 , the 12 remaining terms give the same contribution. This yields (Hf + Pf2 )−1 A+ .A+ Φu α 2∗ =
8α (2π)3
2 λ ελ (k3 ). ν εν (k2 ) κ εκ (k1 )χΛ (|k1 |)χΛ (|k2 |)χΛ (|k3 |) .∇u × α 1 2 3 3 1 1 1 2 2 2 |k | 2 |k | 2 (|k |+k +(h |k |+( k ) |k | + e )) 3 2 1 1 α 0 1 i=1 i i=1 i +O(α3 ).
(226)
On the other hand, we have 8α (2π)3 λ ελ (k3 ). ν εν (k2 ) κ εκ (k1 )χΛ (|k1 |)χΛ (|k2 |)χΛ (|k3 |)
Φ2∗ 2∗ Φu α 2 =
2 .∇uα . × 1 1 1 1 (|k2 |+|k3 |+(k2 +k3 )2 ) 2 |k3 | 2 |k2 | 2 |k1 | 2 (|k1 |+k 2 + (hα + e0 )) 1 Therefore, we obtain 8α (Hf + Pf2 )−1 A+ .A+ Φu α 2∗ − Φ2∗ 2∗ Φu α 2 = − (2π)3 (|k1 |2 + 2|k1 | |k3 | + 2|k1 | |k2 |) × |k2 | |k3 | (|k1 | + |k2 | + |k3 | + |k1 + k2 + k3 |2 )(|k2 | + |k3 | + |k2 + k3 |2 ) 2 2 2 × ελ (k3 ). εν (k2 )χΛ (|k1 |)χΛ (|k2 |) χΛ (|k3 |) |u(k1 , x)| dk1 dk2 dk3 dx , λ
ν
(227) where
u(k1 , x) =
1
2 κ εκ (k1 )χΛ (|k1 |) .∇uα . 1 2 |k1 | 2 (|k1 | + k1 + (hα + e0 ))
For fixed δ, we first compute the integral in (227) over the regions |k1 | > δ. This yields a term cδ α3 , where cδ is independent on α. Next, integrating (227) over the regions |k1 | ≤ δ yields a bound O(δ)α3 log α, with O(δ) independent of α. This concludes the proof of (222). The proof of (223) is a straightforward computation showing 8α (Hf + Pf2 )−1 A+ .A+ Φu α 2 = (2π)3 2 ε (k ). ε (k ) ε (k )χ (|k |)χ (|k |)χ (|k |) λ 3 Λ 1 Λ 2 Λ 3 ν ν 2 κ κ 1 .∇u × λ α 3 1 1 1 3 2 2 2 2 2 |k3 | |k2 | |k1 | (|k1 |+k1 +(hα + e0 )) i=1 |ki |+( i=1 ki ) = O(α3 log α−1 ). The proofs of (224) and (225) are similar to those above.
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⊥ Lemma C.7. For Ψ⊥ 1 and Ψ2 given in Definition 6.1, we have 1
1 ⊥ 5 ⊥ 2 2 |2αRe A− .A− π 3 Ψ⊥ 1 , π Ψ2 | ≤ cα + Hf Ψ2 .
(228)
1 ⊥ 3 ⊥ + + 1 ⊥ Proof. Using 2αRe A− .A− π 3 Ψ⊥ 1 , π Ψ2 = 2αRe π Ψ1 , A .A π Ψ2 , we − − 3 ⊥ 1 ⊥ obtain that 2αRe A .A π Ψ1 , π Ψ2 can be rewritten as a linear combination of the following two integrals I1 and I2 : κ (k ). ν (k ) 5 dk dk dk dx I1 = α 2 2 |k|+|k |+|k | + |k+k +k | κ,ν |k | 12 |k | 12 κ (k ). η (k ) λ (k) 1 × ∇u (π 1 Ψ⊥ α 1 2 )(k, x), | 12 |k | 12 |k|+|k|2 + (hα +e0 ) |k| 2 |k κ,η λ
(229) and I2 is defined as I1 , except that in the last sum, we reverse the role of k and k , namely 5 dk dk dk dxχΛ (|k|)χΛ (|k |)χΛ (|k |) κ (k ). ν (k ) I2 = α 2 12 12 |k|+|k |+|k | + |k+k +k |2 κ,ν |k | |k | κ (k ). η (k) λ (k) 1 × ∇u (π 1 Ψ⊥ α 1 2 )(k , x). | 12 |k| 12 |k|+|k|2 + (hα + e0 ) |k| 2 |k κ,η λ
(230) 2 2b .
To bound I2 , we use the Schwarz inequality, |ab| ≤ + This yields 5 2 dk dk dk dxχ2Λ (|k|)χ2Λ (|k |)χ2Λ (|k |) |I2 | ≤ α 2 (|k|+|k |+|k | + |k+k +k |2 )2 |k | |k | 12 2 1 1 2− 2− 1 ∇u |k| |k | × α |k|+|k|2 + (hα +e0 ) |k| 12 |k | 12 2 2 2 dk dk dk dx 1 ⊥ 2 χΛ (|k|)χΛ (|k |)χΛ (|k |) |(π Ψ2 )(k , x)| |k | × 2 |k ||k| |k|2− |k |2− 1 2 2 a
1
2 ≤ cα7 + Hf2 Ψ⊥ 2 ,
Similarly, we bound I1 as follows: 5
|I1 | ≤ α 2
2 dk dk dk dx + |k+k +k |2 )2 |k | |k |
(|k|+|k |+|k |
1 2 2 2 2 2 1 1 2− 2− χΛ (|k|)χΛ (|k |)χΛ (|k |) ∇u |k | |k | α |k|+|k|2 + (hα +e0 ) |k| 12 |k| 1 dk dk dk dx 1 ⊥ χ2Λ (|k|)χ2Λ (|k |)χ2Λ (|k |) 2 2 × 2 Ψ )(k, x)| |k| |(π 2 |k ||k | |k |2− |k |2− χ2Λ (|k|) 1 2 ≤ cα5 dk dk |k |− |k |− χ2Λ (|k |)χ2Λ (|k |) dk 2 1 2 2 ∇uα |k| (|k| + 16 α ) ×
1
1
2 5 ⊥ 2 2 +Hf2 π 1 Ψ⊥ 2 ≤ cα + Hf Ψ2 ,
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α where we took into account that ∂u ∂xi is orthogonal to uα , and on the subspace 1 2 α . of such functions we have (h − α + e0 ) ≥ 16
References [1] Bach, V., Chen, T., Fr¨ ohlich, J., Sigal, I.M.: The renormalized electron mass in non-relativistic QED. J. Funct. Anal. 243(2), 426–535 (2007) [2] Bach, V., Fr¨ ohlich, J., Pizzo, A.: Infrared-finite algorithms in QED: the groundstate of an atom interacting with the quantized radiation field. Commun. Math. Phys. 264(1), 145–165 (2006) [3] Bach, V., Fr¨ ohlich, J., Sigal, I.M.: Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field. Commun. Math. Phys. 207(2), 249–290 (1999) [4] Bach, V., Fr¨ ohlich, J., Sigal, I.M.: Quantum electrodynamics of confined relativistic particles. Adv. Math. 137(2), 299–395 (1998) [5] Bach, V., Fr¨ ohlich, J., Sigal, I.M.: Mathematical theory of radiation. Found. Phys. 27(2), 227–237 (1997) [6] Barbaroux, J.-M., Vugalter, S.: Non-analyticity of the ground state energy of the Hydrogen atom in nonrelativistic QED. J. Phys. A: Math. Theor. 43, 474004 (2010) [7] Barbaroux, J.-M., Chen, T., Vougalter, V., Vugalter, S.A.: On the ground state energy of the translation invariant Pauli-Fierz model. Proc. Am. Math. Soc. 136, 1057–1064 (2008) [8] Chen, T.: Infrared renormalization in non-relativistic QED and scaling criticality. J. Funct. Anal. 354(10), 2555–2647 (2008) [9] Chen, T., Fr¨ ohlich, J.: Coherent infrared representations in non-relativistic QED. In: Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday. Proc. Symp. Pure Math., vol. I. AMS (2007) [10] Chen, T., Vougalter, V., Vugalter, S.A.: The increase of binding energy and enhanced binding in non-relativistic QED. J. Math. Phys. 44(5), 1961–1970 (2003) [11] Fr¨ ohlich, J.: On the infrared problem in a model of scalar electrons and massless, scalar bosons. Ann. Inst. H. Poincar´e Sect. A (N.S.) 19, 1–103 (1973) [12] Griesemer, M., Hasler, D.: Analytic perturbation theory and renormalization analysis of matter coupled to quantized radiation. Ann. Henri Poincar´e 10(3), 577–621 (2009) [13] Griesemer, M., Lieb, E.H., Loss, M.: Ground states in non-relativistic quantum electrodnamics. Inv. Math 145, 557–595 (2001) [14] Hainzl, C.: Enhanced binding through coupling to a photon field. Mathematical results in quantum mechanics (Taxco, 2001). Contemp. Math., vol. 307, pp. 149–154. Amer. Math. Soc., Providence (2002) [15] Hainzl, C., Hirokawa, M., Spohn, H.: Binding energy for hydrogen-like atoms in the Nelson model without cutoffs. J. Funct. Anal. 220(2), 424–459 (2005) [16] Hainzl, C., Vougalter, V., Vugalter, S.A.: Enhanced binding in non-relativistic QED. Commun. Math. Phys. 233, 13–26 (2003)
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Jean-Marie Barbaroux Centre de Physique Th´eorique Luminy Case 907 13288 Marseille Cedex 9 France and D´epartement de Math´ematiques Universit´e du Sud Toulon-Var 83957 La Garde Cedex France e-mail: [email protected] Thomas Chen Department of Mathematics University of Texas at Austin 1 University Station C1200 Austin TX 78712 USA e-mail: [email protected] Vitali Vougalter Department of Mathematics and Applied Mathematics University of Cape Town Private Bag Rondebosch 7701 South Africa e-mail: [email protected] Semjon Vugalter Mathematisches Institut Ludwig-Maximilians-Universit¨ at M¨ unchen Theresienstrasse 39 80333 Munich Germany e-mail: [email protected] Communicated by Volker Bach. Received: March 26, 2009. Accepted: October 1, 2010.
Ann. Henri Poincar´e
Ann. Henri Poincar´e 11 (2010), 1545–1589 c 2010 Springer Basel AG 1424-0637/10/081545-45 published online December 14, 2010 DOI 10.1007/s00023-010-0066-z
Annales Henri Poincar´ e
Absence of Embedded Mass Shells: Cerenkov Radiation and Quantum Friction Wojciech De Roeck, J¨ urg Fr¨ohlich and Alessandro Pizzo Abstract. We show that, in a model where a non-relativistic particle is coupled to a quantized relativistic scalar Bose field, the embedded mass shell of the particle dissolves in the continuum when the interaction is turned on, provided the coupling constant is sufficiently small. More precisely, under the assumption that the fiber eigenvectors corresponding to the putative mass shell are differentiable as functions of the total momentum of the system, we show that a mass shell could exist only at a strictly positive distance from the unperturbed embedded mass shell near the boundary of the energy–momentum spectrum.
1. Introduction The model studied in this paper describes a system consisting of a nonrelativistic quantum particle coupled to a quantized relativistic field of scalar massless bosons through an interaction term linear in creation and annihilation operators. The system is invariant under space translations. Therefore, its total momentum is conserved. In states where the initial particle momentum is larger than mc, where m is the mass of the non-relativistic particle and c the propagation speed of the bosonic modes, we expect that the particle will emit Cerenkov radiation, because its group velocity is larger than the speed of the bosons. We are thus interested in the spectral region (E, P ) with |P | > 1, using units such that m = c = 1. Here, E, P are the spectral variables of the Hamiltonian and of the total momentum operator, respectively. In this region, we expect that a mass shell of the non-relativistic particle does not exist. Presented differently, we expect that the mass shell, which in the unperturbed system described by the equation E = P 2 /2, disappears as soon as the interaction is switched on. This would show that one-particle states of the non-relativistic particle are unstable for values of |P | larger than 1. Our main result is as follows. We assume that, for |P | > 1, a mass shell exists with the property that the corresponding fiber eigenvectors are differentiable as functions of the total momentum of the system. Then we show that,
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for sufficiently small values of the coupling constant, such a mass shell may exist only at a strictly positive distance (> O(1)) from the unperturbed mass shell in the energy–momentum spectrum. More precisely, one-particle states might only exist in a region around the three-dimensional surface E = |P | − 12 , the width of which tends to zero, as the coupling constant approaches 0. Our results are proven for models with a fixed ultraviolet cutoff that turns off interactions with high-energy bosons, and under the assumption of appropriate infrared regularity of the form factor that models the interaction. In the literature, many results are concerned with the existence of a mass shell for |P | < 1, depending on the behavior of the coupling between the nonrelativistic particle and the relativistic boson field in the infrared region. These results clarify and extend the notion of stable particle by providing a scattering picture for infraparticles, for which a mass shell does not exist (i.e., the singleparticle states are not normalizable in the Hilbert space of pure states of the system); see [3,4,6,7,10,11,13,16,18,19]. To our knowledge, for the spectral region studied in this paper, no rigorous results have yet appeared in the literature concerning the existence or non-existence of an embedded mass shell. However, in [8], for the model studied in this paper, it is proven that the electron motion in the kinetic limit is described by a Boltzmann equation that exhibits the slowdown of the particle by emitting Cerenkov radiation, as long as its velocity is greater than 1. This supports the thesis that there is no mass shell for |P | > 1. We also stress that the conclusions of our paper leave open an interesting question: our analysis does not exclude the existence of single-particle states near the boundary of the energy–momentum spectrum (which, for |P | > 1, is approximately linear in |P |). In this respect, we recall that the existence of the ground-state eigenvalue for the fiber Hamiltonians, in the region |P | > 1, has been studied in [20] and [17] (see also [1,2] for some related spectral problems), but under some assumptions on the boson dispersion relation that change the physical phenomenon we are interested in. In fact, in these papers, the bosons are massive and their energy dispersion relation is strictly subadditive (see [17]). In particular, in [17], it is proven that, for spatial dimension d = 3, the fiber Hamiltonian has no ground state whenever the infimum of its spectrum equals the infimum of its essential spectrum. However, because of the assumptions above, this result does not apply to the model studied in this paper. In the following, the spin of the electron is neglected, and the bosons are scalar.
2. Description of the Model and Result 2.1. Hilbert Space The Hilbert space of pure states of the system is given by H = L2 (R3 ) ⊗ F,
(2.1)
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where F is the Fock space of scalar bosons, F :=
∞
F (N ) ,
F (0) = CΩ,
(2.2)
N =0
with Ω the vacuum vector, i.e., the state without any bosons, and the state space, F (N ) , of N bosons is given by F (N ) := SN h⊗ N ,
N ≥ 1.
(2.3)
Here, the Hilbert space, h, of state vectors of a single boson is given by h := L2 [R3 ],
(2.4)
and SN denotes symmetrization. We introduce the usual creation- and annihilation operators, a∗k and ak , obeying the canonical commutation relations [a∗k , a∗k ] = [ak , ak ] = 0, [ak , a∗k ]
(2.5)
= δ(k − k ),
ak Ω = 0,
(2.6) (2.7)
for all k, k ∈ R3 . 2.2. Fiber Decomposition We may write H as a direct integral ⊕ H=
HP d3 P.
(2.8)
Given any P ∈ R3 , there is an isomorphism, IP , IP : HP −→ F b ,
(2.9)
from the fiber space HP to the Fock space F b , acted upon by the annihilation and creation operators bk , b∗k , where bk corresponds to eik·x ak , and b∗k to
e−ik·x a∗k , and with vacuum Ωf := IP (eiP ·x ). To define IP more precisely, we consider a vector ψ(f (n) ;P ) ∈ HP with a definite total momentum describing an electron and n bosons. Its wave function in the variables (x; k1 , . . . , kn ) is given by
ei(P −k1 −···−kn )·x f (n) (k1 , . . . , kn ),
(2.10)
where f (n) is totally symmetric in its n arguments. The isomorphism IP acts by way of IP ei(P −k1 −···−kn )·x f (n) (k1 , . . . , kn ) 1 d3 k1 . . . d3 kn f (n) (k1 , . . . , kn )b∗k · · · b∗k Ωf . (2.11) =√ 1 n n!
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2.3. Hamiltonians We consider a non-relativistic particle moving in a medium of relativistic bosons. The Hamiltonian of the system is given by H :=
1 2 p + gφ(ρx ) + H f , 2
(2.12)
where: • The operators x, p describe the electron position and momentum, respectively; • H f := dΓ(ω(|k|)) (see Sect. 2.5), where ω(|k|) := |k| is the free field Hamiltonian. In physicist’s notation H f = d3 k|k|a∗k ak . • The real number g, |g| > 0 is a coupling constant. • The interaction Hamiltonian is φ(ρx ) := d3 kρ(k)(a∗k e−ik·x + ak eik·x ),
(2.13)
where the form factor ρ(k) ∈ R satisfies the following conditions 1. There is an ultraviolet cutoff Λ, i.e., ρ(k) = 0 whenever |k| > Λ. 2. The function ρ is rotationally invariant, i.e., ρ(k) = ρ(|k|), continuously differentiable, ρ ∈ C 1 . For expository convenience, when we describe the decay mechanism in Theorem 5.1, we will also assume that ρ(k) = 0 for 0 < |k| < Λ. Actually, this assumption is not necessary to state the main result of the theorem, but simplifies the construction of the trial state in Eq. (5.2) of Theorem 5.1. 3. The following infrared regularity condition holds: |ρ(k)| ≤ O(|k|β ),
ρ(k)| ≤ O(|k|β−1 ), and |∇ k
as k → 0
(2.14)
for an exponent β > 11/2. We believe that the critical value, β = 11/2, is not optimal. From physical considerations, the result concerning the instability of the mass shell should hold for any exponent β ≥ −1/2. For β = −1/2, the Hamiltonian describes the interaction of the electron with the quantized relativistic field with no infrared regularization. The operator H is self-adjoint, because φ(ρx ) is an infinitesimal perturbation 2 of H 0 := H f + p2 , and Dom(H) = Dom(H 0 ), i.e., the domains of self-adjointness coincide. Since the Hamiltonian H commutes with the total momentum, it preserves the fiber spaces HP , for all P ∈ R3 . Thus, we can write ⊕ H=
HP d3 P,
(2.15)
where HP : HP −→ HP .
(2.16)
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In terms of the operators bk , b∗k , and of the variable P , the fiber Hamiltonian HP is given by HP := HP0 + gφb (ρ), with
HP0 :=
P − P f
2
+ Hf , 2 where, as operators on the fiber space HP , P f = d3 kkb∗k bk , f b H = dΓ (ω(|k|)) = d3 kω(|k|)b∗k bk , and b
φ (ρ) :=
(2.17)
d3 kρ(k)(b∗k + bk ).
(2.18)
(2.19) (2.20)
(2.21)
2.4. Result The absence of a mass shell for |P | > 1 is expressed by the following statement. The equation HP ΨP = EP ΨP
(2.22)
has no normalizable solution for any value of EP and for almost every P ∈ R3 , |P | > 1. What we actually prove in this paper is the absence of regular mass shells as formulated in the theorem below (see also Fig. 1). More concretely, we address the question whether, for a given region I × ΔI in the momentum–energy space (see (ii) below), there is an open interval Ig , Ig ⊂ I, of size at least O(|g|γ ), γ > 0, where the mass shell exists, with EP ∈ ΔI and with the regularity property specified in the theorem. Recall that β determines the infrared behavior of the form factor ρ, see (2.14). Theorem 2.1. Assume that the form factor ρ satisfies (2.14), with β > 11/2, and fix an interval I of the form I := (1 + δ, σ), δ > 0, σ < ∞ and a bounded interval ΔI . Fix constants 0 < CI , cI < ∞ and exponents 0 < γ < 1/4 and 0 < < γ/4. Then, there is a g∗ > 0 such that, for all g satisfying 0 < |g| < g∗ , the following is ruled out: There exist normalizable solutions to equation (2.22), for all |P | ∈ Ig , such that: (i) Ig is an interval of length larger than |g|γ/2 (|Ig | ≥ |g|γ/2 ). (ii) Ig ⊂ I and EP ⊂ ΔI , for all |P | ∈ Ig . (iii) For all |P | ∈ Ig , ∇P ΨP ,E < CI . P
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Figure 1. The joint energy–momentum spectrum. By the rotation symmetry, it suffices to plot the (E, |P |)-plane. In the leftmost figure, we have drawn the spectrum of the uncoupled system. The parabola 12 |P |2 (in boldface) is the mass shell and the spectrum lies above the three-dimensional surface consisting of 12 |P |2 , for |P | < 1 and |P | − 1/2, for |P | > 1. Hence, for |P | > 1, the mass shell is embedded in the continuum. In the middle figure, we represent the situation when the coupling is switched on, according to formal perturbation theory. The mass shell has disappeared (drawn as a dashed line) for |P | > 1. For |P | < 1, the mass shell persists but gets deformed (mass renormalization). In the rightmost figure, we represent what is known rigorously: a regular mass shell is excluded in the colored area (result of the present paper) and there is a renormalized mass shell for small |P | (earlier works, see Sect. 1) (iv) For all |P | ∈ Ig ,
E − |P | − 1 > cI |g|γ/4− . P 2
We note that it is an interesting open problem to understand whether single-particle states could emerge at the boundary of the energy–momentum spectrum, i.e., near Ep = |P | − 12 . Our results only rule out the existence of single-particle states whose energies are embedded in the energy–momentum spectrum and with suitable regularity properties as far as their dependence on P is concerned. Remark. In the following theorems, lemmas and corollaries, we always assume that the Main Hypothesis in Sect. 3.1.1 holds. Furthermore, |g| “sufficiently small” means 0 < |g| < g∗ , where g∗ depends only on I, on ΔI and on γ, but with the form factor ρ and the ultraviolet cutoff Λ kept fixed.
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2.4.1. Main Ingredients of the Proof. |−1| > 3 |g|γ/3 , (a) If ΨP ,E existed with properties (i)–(iv) above, and ||∇E P 2 P then Ψ0P − ΨP ,E ≤ O(|g|(1−2γ)/6 ), P
where Ψ0P is the bare one-particle state (i.e., Ψ0P = Ωf ), and P 2 − EP ≤ O(|g|(1−2γ)/6 ). 2
(2.23)
(2.24)
(b) If ΨP ,E , as in (a), existed then it could decay into a state consisting of P an unperturbed single-particle state and a boson with momentum k in a region of momentum space away from the ray {λP |0 < λ ≤ ∞}. | − 1| ≤ 3 |g|γ/3 then |E − (|P | − 1 )| < const |g|γ/4 . In other (c) If ||∇E P P 2 2 words, a mass shell with group velocity close to one necessarily lies near the boundary of the energy momentum spectrum. 2.5. Notation Here is a list of notations used in subsequent sections. ˆ := |uu| . 1. Given any vector, u ∈ R3 , u 2. Ffin is the dense subspace of F obtained as the span of vectors containing finitely many bosons. 3. 1(a,b) (k) is the characteristic function of the set {k ∈ R3 : |k| ∈ (a, b)}. 4. For any function w ∈ h, w 2 is the corresponding L2 -norm. 5. dΓ(A) is the second quantization of an operator A acting on h; dΓ(A) is an operator on F. Analogously, dΓb (A) is defined on F b . 6. We define the (boson) number operators by N := dΓ(1(k)) and N b := dΓb (1(k)), where 1(k) is the identity operator on L2 (R3 ; d3 k). 7. We use the notation a∗ (fx ) := d3 kfx (k)a∗k , a(fx ) := d3 kfx (k)ak for smeared creation/annihilation operators, depending also on the (electron) position x. 8. Expressions like (P , EP ) ∈ Ig × ΔI are interpreted as follows: P ∈ R3 with |P | ∈ Ig and EP ∈ ΔI . 2.6. Structure of the Paper In Sect. 3, we state a Main Hypothesis (Sect. 3.1). The upshot of our analysis is Theorem 5.4 in Sect. 5. This theorem describes the possible location of
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a mass shell, under the assumption that the Main Hypothesis holds true. In other words, the implication Main Hypothesis
=⇒
Assumptions in Sect. 2.3
Theorem 5.4
(2.25)
is our main result, and this implication gives rise to Theorem 2.1. In the remainder of Sect. 3.1, we state some immediate consequences of the Main Hypothesis, and in Sect. 3.2, we put the technical tools in place. Section 3.3 contains a rather detailed description of the strategy of our proofs. The proofs themselves are presented in Sects. 4 and 5. An appendix contains the proofs of some preliminary results used in Sect. 4.
3. Strategy of the Proof 3.1. Main Hypothesis and Key Properties The proof of our result, Theorem 2.1, is by contradiction. We will assume that a regular mass shell exists and, subsequently, we derive that it cannot be located anywhere else than near the boundary of the energy–momentum spectrum. Our assumption is stated in Sect. 3.1.1 and it will be referred to as the Main Hypothesis. Throughout the rest of the paper, we assume that the Main Hypothesis holds. In Sect. 3.1.2, we derive some consequences of the Main Hypothesis, namely Properties P1, P2 and P3. 3.1.1. Main Hypothesis. Let R be a rotation matrix in R3 , and U (R) the unitary operator implementing the transformation bk
→
bR−1k =: U ∗ (R)bk U (R).
(3.1)
The identity U ∗ (R)HRP U (R) = HP ,
(3.2)
implies that if ΨP ,E is a normalized eigenvector of HP with eigenvalue EP , P then U (R)ΨP ,E is an eigenvector of HRP with the same eigenvalue, i.e., P
HRP U (R)ΨP ,E = EP U (R)ΨP ,E . P
(3.3)
P
In particular, the existence of an eigenvector, ΨP ,E , of HP for all P in a given P direction, Pˆ , yields a mass shell with energy function E ≡ E . P
|P |
Main hypothesis: We temporarily assume that single-particle states, ΨP ,E , P exist, i.e., HP ΨP ,E = EP ΨP ,E , P
P
ΨP ,E = 1, P
such that the vector ΨP ,E is differentiable in P with P ∇P ΨP ,E < CI , P
(3.4)
(3.5)
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where the constant CI < ∞, for all P such that |P | ∈ Ig ⊂ I and for EP ∈ ΔI , where ΔI is a bounded interval. Here, Ig is an open interval, |Ig | > |g|γ/2 , and I := (1 + δ, σ), σ − 1 > δ > 0. From the assumption in Eq. (3.5), the following properties follow for |P | ∈ Ig , EP ∈ ΔI . 3.1.2. Properties (P1), (P2) and (P3). (P1) EP = E|P | is differentiable and the Feynman–Hellman formula holds = (Ψ f ∇E P P ,E , (P − P )ΨP ,E ). P
(3.6)
P
The expression on the RHS (right-hand side) of (3.6) is continuous in is a continuous function of P . Moreover, |∇E | < C P . Thus ∇E I P P and P are for some CI < ∞, and, because of rotation invariance, ∇E P colinear. (P2) For some 0 < CI < ∞, 2 ∂ EP |P | (3.7) ≤ CI . ∂|P |2 Starting from the derivative of the RHS of (3.6), this bound can be easily obtained using (3.5) and that HP0 is HP -bounded. (P3) From H f = dΓb (|k|) and Eq. (3.5), it follows that
b (Ψ |∇ , dΓ (1 1 1 (k))Ψ )| ≤ O(n CI sup |E | +1 ; (3.8) P
P ,EP
( n+1 , n )
P ,EP
∈I P
P
here we use the inequality f 1 1 ( dΓb (1( n+1 ,n ) k))ψ ≤ (n + 1) H ψ ,
∀ψ ∈ Dom(H f ),
and that H f is HP -bounded. 3.2. Technical Tools We will use two different virial arguments to expand ΨP ,E in the coupling P constant g, |g| 1. For this purpose, we must introduce single-particle “wave packets”, Ψf g , defined below. Q
(I) Single-particle “wave packets”, Ψf g , and the interval Ig . Q
For |g| sufficiently small, we define the open interval Ig such that (|Ig | >)|Ig | > 4|g|γ ,
(3.9)
with the property that + |z| ∈ Ig , |Q| ∈ Ig and for all z such that |z| < |g|γ . for all |Q|
(3.10)
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We consider single-particle “wave packets”, Ψf g , with wave function, Q g |Q| ∈ Ig . The vector Ψf g is defined by , centered around vectors Q, fQ Q g 3 Ψf g := fQ (3.11) ,E d P (P )ΨP Q
P
|−|Q| θQP ˆ R( |P|g| )A( |g| γ γ
g where fQ ), θQP ˆ is the angle between Q and P , (P ) := and R(z), A(θ) are defined as follows. (1) R(z), z ∈ R, is non-negative, smooth and compactly supported in the interval (−1, 1), R(z) = 1 for z ∈ (− 12 , 12 ), (2) A(θ), θ ∈ R, is non-negative, smooth and compactly supported in the interval (−1, 1), A(θ) = 1 for θ ∈ (− 12 , 12 ). Therefore, the angular restriction Pˆ · Pˆ ≥ cos(|g|γ ) g holds for any P , P ∈ supp fQ . > 1, it follows from the definitions of R(z) and A(θ) that: (3) Since |Q|
f g (P )| ≤ O(|g|−γ ), |∇ P Q g for any P ∈ supp fQ . (II) Multiscale virial argument on the Hilbert space H for the Hamiltonian H. We define dilatation operators on the one-particle space h, constrained to a suitable range of frequencies and to a suitable angular sector around a direction u ˆ. We introduce the conjugate operator ˆ
ˆ
u ˆ ,Q u ˆ ,Q Dn,⊥ := dΓ(dn,⊥ ),
(3.12)
with ˆ ˆ ,Q ˆ 1 (k⊥ · i∇ ˆ n (|k|), + i∇ · k⊥ )ξ g (k)χ dun,⊥ := χn (|k|)ξugˆ (k) (3.13) u ˆ k⊥ k⊥ 2 where: i.e., k⊥ := (a) k⊥ is the component of the vector k orthogonal to Q, k − k·Q 2 Q; |Q| χn (|k|), n ∈ N, are non-negative, C ∞ (R+ ) functions with the properties: 1 3 and for |k| ≥ 2n ; (i) χn (|k|) = 0 for |k| ≤ 2(n+1) 1 1 ≤ |k| ≤ ; (ii) χn (|k|) = 1 for n+1
n
(iii) |χn (|k|)| ≤ Cχ n, for all n ∈ N, where the constant Cχ is independent of n. ˆ (see Fig. 2), 0 ≤ ξ g (k) ˆ ≤ 1, is a smooth function with support (b) ξugˆ (k) u ˆ in the g-dependent cone Cuˆ := {kˆ : kˆ · u ˆ ≥ cos(|g|γ )}, (3.14) such that:
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Figure 2. The cone Cuˆ containing the support of the smooth ˆ characteristic function ξugˆ (k) (i)
1 γ |g| kˆ : kˆ · u ˆ ≥ cos ; 2
ˆ =1 ξugˆ (k)
for
ˆ =0 ξugˆ (k)
for {kˆ : kˆ · u ˆ < cos(|g|γ )};
(3.15)
(ii) (3.16)
(iii) ˆ ≤ Cξ |g|−γ , ξugˆ (k)| |∂θku ˆ
(3.17)
ˆ where θku ˆ, and the constant Cξ ˆ is the angle between k and u is independent of g. We also define that 1 ˆ k · i∇ ˆ + i∇ · k]χn (|k|)ξ g (k), dunˆ := χn (|k|)ξugˆ (k)[ (3.18) u ˆ k k 2 and we introduce the second quantized operator Dnuˆ := dΓ(dunˆ ).
(3.19)
Later on in the paper, when we implement the virial argument, we will make use of the creation/annihilation operators ∗ u ˆ a (idn ρx ) = d3 k i(dunˆ ρx )(k)a∗k a(idunˆ ρx ) = d3 k i(dunˆ ρx )(k)ak , ˆ
ˆ
ˆ ,Q u ˆ ,Q ρx ), a∗ (idn,⊥ ρx ). In Lemma 4.1, we show that and, analogously, a(idun,⊥ the vector Ψf g belongs to the form domain of these operators. Q
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(III) Virial argument in each fiber space HP . Here, we consider Db1 ,κ := dΓb (d κ1 ,κ ) κ
as the conjugate operator, where 1 + i∇ · k)χ[ 1 ,κ] (|k|). d κ1 ,κ := χ[ κ1 ,κ] (|k|) (k · i∇ (3.20) k k κ 2 χ[ κ1 ,κ] (|k|), ∞ > κ > max{Λ, 1}, are non-negative, C ∞ (R+ ) functions with the properties: 1 ; (i) χ[ κ1 ,κ] (|k|) = 0 for |k| ≥ 2κ, |k| ≤ 2κ 1 (ii) χ 1 (|k|) = 1 for ≤ |k| ≤ κ; [ κ ,κ]
κ
(iii) for some C > 0, |χ[ 1 ,κ] (|k|)| < Cκ. κ
Analogously to a∗ (idunˆ ρx ), a(idunˆ ρx ), we will use ∗ b (id κ1 ,κ ρ) = d3 k(id κ1 ,κ ρ)(k)b∗k b(id κ1 ,κ ρ) = d3 k(id κ1 ,κ ρ)(k)bk . We will also consider the g-dependent cones (see Fig. 3), CPaˆ := {k : |kˆ · Pˆ | ≤ cos(a|g|γ/8 )},
(3.21)
ˆ a = 1 , 2, defined below. and use the smooth functions ξCg a (k), 2 ˆ P ˆ 0 ≤ ξ g a (k) ˆ ≤ 1, are chosen such that The functions ξCg a (k), C ˆ P
ˆ P
(i) ˆ =1 ξCg a (k) ˆ P
for {kˆ : |kˆ · Pˆ | ≤ cos(2a|g|γ/8 )};
(3.22)
(ii) ˆ = 0 for ξCg a (k)
{kˆ : |kˆ · Pˆ | > cos(a|g|γ/8 )};
ˆ P
(3.23)
(iii) ˆ ≤ Cξ |g|−γ/8 , ξCg a (k)| |∂θkP ˆ ˆ P
(3.24)
ˆ for a constant Cξ independent of g, where θkP ˆ is the angle between k ˆ and P . 3.3. Description of Strategy To exclude the existence of eigenvalues EP , |P | > 1, we elaborate on an argument introduced in [12]. The idea of the proof is as follows. One assumes that an eigenvector ΨP ,E ∈ HP of HP exists, for some energy EP in a compact P set. Then, using a multiscale virial argument, one intends to prove that (ΨP ,E , N b ΨP ,E ) ≤ O(g 2 ), P
P
(3.25)
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Figure 3. The double cone CPaˆ is the complement in R3 of the inner double cone around Pˆ of angular width a|g|γ/8 where N b is the boson number operator in the fiber spaces. The multiscale virial argument involves the dilatation operators 1 + i∇ · k]χn (|k|), dn := χn (|k|)[k · i∇ (3.26) k k 2 on the one-particle space h, where χn is a suitable smooth approximation to 1 , n1 ] contained in the positive the characteristic function of the interval [ n+1 frequency half axis, n = 0, 1, 2, . . . (cfr. Eq. (3.13)). After introducing the second quantized dilatation operators Dnb := dΓb (dn ), one starts from the formal virial identity 0 = (ΨP ,E , i[HP , Dnb ]ΨP ,E ) P
P
(3.27)
to establish the scale-by-scale inequality below, in a rigorous way: 2 1 3 ( (ΨP ,E , Nnb ΨP ,E ) ≤ O(g 2 n2 |k|β 1( 2(n+1) (3.28) , 2n ) k) 2 ), P P where Nnb := d3 k a∗ (k)χ2n (|k|)a(k), n = 1, 2, 3, . . .. If (3.28) holds true, for sufficiently large values of the exponent β in the form factor ρ, one can sum over n and conclude that (ΨP ,E , N b ΨP ,E ) ≤ O(g 2 ). Next, the eigenvalue P P equation (2.22) and the inequality in Eq. (3.25) can be combined to conclude that the vector ΨP ,E and the eigenvalue EP must fulfill the following estiP mates: ΨP ,E − Ψ0P 2 ≤ O(g 2 ), P
(3.29)
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where Ψ0P := Ωf is the unperturbed eigenstate, and P 2 | ≤ O(g 2 ). (3.30) 2 This result would imply that putative eigenvalues of HP lie in an O(g 2 )neighborhood of the eigenvalue of the Hamiltonian HPg=0 . Then the argument proceeds with the construction of suitable trial states of the type (P − k)2 /2 + |k| − EP 3 1 ηP := b∗k Ψ0P , d k 1h (3.31) 2 |EP −
where > 0 and h(z) ∈ C0∞ (R), h(z) ≥ 0. One then exploits the identity (ηP , (HP − EP )ΨP ,E ) = 0 P
(3.32)
that must hold true if ΨP ,E is an eigenvector of HP . Starting from P Eqs. (3.29)–(3.30), and using that the equation (P − k)2 /2 + |k| − EP = 0
(3.33)
has solutions for |P | > 1, provided |g| is sufficiently small, one arrives at a contradiction, for and |g| sufficiently small. However, the procedure just outlined (mimicking the treatment of atomic resonances in [12]) will not work without some important modifications. We will therefore implement an analogous, but more elaborate, strategy. The first problem encountered is that we cannot control the expectation value (ΨP ,E , N b ΨP ,E ) P
P
by a multiscale virial argument in the fiber space HP , because of the term (P f )2 in HP . The commutator of (P f )2 with dΓb (dn ), formally given by P f · dΓb (χ2n (k)k) + dΓb (χ2n (k)k) · P f ,
(3.34)
cannot be controlled in terms of the commutator of H f with dΓb (dn ). Consequently, the estimate in Eq. (3.28) cannot be justified starting from the virial identity in Eq. (3.27). At the price of limiting our analysis to regular mass shells (see Main Theorem in Sect. 2.4), this problem can be circumvented by implementing a multiscale virial argument in the full Hilbert space, by using single-particle “wave packets” rather than fiber eigenvectors, i.e., vectors in H of the type (3.35) Ψf := f (P )ΨP ,E d3 P, P
where f (P ) is a smooth function with support in Ig (the region of momenta for g which an eigenstate was assumed to exist). In practice, we choose f = fQ to see definition below (3.11). be sharply peaked around a given momentum Q;
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In the full Hilbert space, we can essentially mimic the treatment of atomic resonances to derive the following result (see Sect. 4). Theorem (4.3). For |g| sufficiently small, (Ψf g , Nn,C 1/2 Ψf g ) Q
Q
ˆ Q
(Ψ
f g Q
,Ψ
f g Q
)
2 1 3 ( ≤ O(g 2(1−2γ) n2 |k|β 1( 2(n+1) , 2n ) k) 2 ),
(3.36)
2 ˆ ∈ Ig . where Nn,C 1/2 := dΓ(χ2n (|k|)ξ g 1/2 (k)) and |Q| CQ ˆ
ˆ Q
g Furthermore, if for all P ∈ suppfQ the inequality
| − 1| > |g|γ/3 ||∇E P holds true, then (Ψf g , Nn Ψf g ) Q
Q
(Ψf g , Ψf g ) Q
2 1 3 ( ≤ O(g 2(1−2γ) n2 |k|β 1( 2(n+1) , 2n ) k) 2 ),
(3.37)
Q
where Nn := dΓ(χ2n (|k|). ∈ I ⊂ |Q| The constants in (3.36), (3.37) can be chosen uniformly in Q, g I (Ig is defined in Sect. 3.2, Eqs. (3.9), (3.10)). They only depend on I and on ΔI . g By exploiting the g-dependence of the wave functions fQ and the assumption on the regularity in P of ΨP ,E , one can convert a bound for the number P operator N on single-particle wave packets to a bound that holds pointwise in P on the number operator N b acting on the fiber eigenvectors ΨP ,E . In P essence, this follows from the fundamental theorem of calculus. These arguments are implemented in Sect. 4 and give the following results.
Theorem (4.5). For |g| sufficiently small and (P , EP ) ∈ Ig × ΔI , b (ΨP ,E , Nn,C 2 Ψ P ,E ) ≤ O(|g| P
ˆ P
(1−2γ) 3
P
1
4 3 1 3 ( |g|−γ/8 n 3 |k|β 1( 2(n+1) , 2n ) k) 2 ), (3.38)
where g2 ˆ b b 2 Nn,C 2 := dΓ (χn (|k|)ξ 2 (k)). C ˆ P
(3.39)
ˆ P
| − 1| > 3 |g|γ/3 then Furthermore, if in addition ||∇E P 2 (ΨP ,E , Nnb ΨP ,E ) ≤ O(|g| P
P
(1−2γ) 3
1
4 3 1 3 ( n 3 |k|β 1( 2(n+1) , 2n ) k) 2 ))
(3.40)
where Nnb := dΓb (χ2n (|k|)).
(3.41)
The constants in (3.38), (3.40) can be chosen uniformly in P , |P | ∈ Ig ⊂ I (Ig is defined in Sect. 3.2, Eqs. (3.9), (3.10)). They only depend on I and on ΔI .
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We now comment on the contents of Theorem 4.5. Inequality (3.38) means that we can bound the boson number operator if we exclude a double cone (see Fig. 3) and the definition of CP2ˆ in Eqs. (3.22)–(3.24), Sect. 3.2) around the direction of the particle velocity, provided the form factor ρ(k) scales like |k|β with β > 11/2, i.e., (2.14). The second result (see (3.40)) says that, for putative mass shells (P , EP ) | − 1| is not too small (i.e., ||∇E | − 1| > 3 |g|γ/3 ), we can such that ||∇E P P 2 bound the boson number without any angular restrictions, again using that ρ(k) scales like |k|β with β > 11/2. The constraint means that the forward emission of bosons by the (massive) particle cannot be controlled if its speed is too close to the boson propagation speed. The estimates on the number operator obtained in Sect. 4 are used in Sect. 5, where we will establish the following two results regarding the region Ig × ΔI , where Ig is any open interval contained in I such that |Ig | > |g|γ/2 . (i) The first result is that we can exclude all the regular mass shells except those with slope close to 1, i.e., all the regular mass shells such that (P , EP ) ∈ Ig × ΔI
| − 1| > and ||∇E P
3 γ/3 |g| . 2
(3.42)
(ii) The second result shows that a regular mass shell might exist only for (P , EP ) such that 1 EP = |P | − + O(|g|γ/4 ). 2
(3.43)
More precisely, we use that: (1) The expectation value in ΨP ,E , |P | ∈ Ig , of the operator N b restricted P to the angular sector CP2ˆ vanishes as g → 0 (see Theorem 4.5). P | − 1| > O(|g|γ/3 ), the expectation (2) For (P , E ) ∈ I × ΔI such that ||∇E P
g
value of the number operators N b on ΨP ,E vanishes as g → 0; see TheP g ) ∈ Ig × ΔI | ||∇E P | − 1| > orem 4.5. Analogously, if suppfQ ⊂ {(P , EP
O(|g|γ/3 )} then the expectation value of the number operator N on Ψgf Q vanishes as g → 0; see Theorem 4.3. (3) The results in (2) imply that the putative fiber eigenvectors ΨP ,E , |P | ∈ P Ig , and the corresponding energies EP are perturbative in g (see Corollary 4.6), provided that β > 11/2. We derive (i) in Theorem 5.1 by mimicking the argument with the trial states employed for the treatment of the atomic resonances [12], which was anticipated in Sect. 3.3, Eqs (3.31)–(3.33). To this end, we make us of (2) and (3). The result in (ii) follows thanks to a stronger version of (1) (for details, see Lemma 5.2, Lemma 5.3) where only the forward cone around the direction of the particle velocity is excluded in the definition of the restricted number operator, and by combining the eigenvalue equation with a standard (i.e., not a
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multiscale analysis) virial argument in the fiber space HP , where the conjugate operator is Db1 ,κ := dΓb (d κ1 ,κ ); see (3.20) and Theorem 5.4. κ The virial identity exploited in Theorem 5.4 is actually sufficient to exclude that, fiber by fiber, the eigenvalue lies at a distance larger than O(g) above the unperturbed eigenvalue. This observation is explained in the Remark after Theorem 5.4 in Sect. 5. However, the instability of the unperturbed mass shell proven in this paper requires a detailed analysis of the configuration of bosons in the putative eigenvector, the momenta of which are contained in different cones of momentum space. The decay mechanism exploited in Theorem 5.1 combined with the assumed continuity of the mass shell is responsible for the absence of single-particle states except for the region EP = |P |− 12 +O(|g|γ/4 ). P| = This is because if the particle propagated at the critical velocity, i.e., |∇E 1, then there would be no kinematical constraint preventing the emission of an arbitrarily large number of soft bosons in the forward direction (the direction of P ).
4. Boson Number Estimates The main results in this section are Theorem 4.3, Theorem 4.5 and Corollary 4.6. Two preparatory results, contained in Sect. 4.1, are needed. In particular, in Lemma 4.2, we provide a rigorous justification of a virial identity employed in Lemma 4.4 and in Theorem 4.3. Since the proof of Theorem 4.3 is lengthy, we present it in two different smaller sections: (a) In Sect. 4.2.1, we outline the proof of the theorem and, in Lemma 4.4, we introduce an important ingredient used later on. (b) In Sect. 4.2.2, we complete the steps of the proof by assuming the result obtained in Lemma 4.4. In Sect. 4.3, by using the regularity properties that follow from the Main Hypothesis, we derive some estimates for the number operator N b evaluated on the fiber eigenvectors ΨP ,E analogous to those obtained in Theorem 4.3 P for the number operator N evaluated on the single-particle states Ψf g . In Q Corollary 4.6, we then finally show that EP and ΨP ,E are perturbative in g, P | − 1| > 3 |g|γ/3 . provided |P | ∈ Ig , E ∈ ΔI , β > 11/2, and ||∇E P
P
2
4.1. Preparatory Results on Virial Identities The following two lemmas are repeated and proven in Sects. 6.1 and 6.2 of the appendix, respectively. Lemma 4.1. The vector Ψf g belongs to the domain of the position operator x Q and xj Ψf g ≤ O(|g|−γ Ψf g ), Q
Proof. See the Appendix.
Q
j = 1, 2, 3.
(4.1)
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Lemma 4.2 states a virial theorem for our model. We observe that by formal steps one can derive the identity · dΓ(i[k, dunˆ ]) i[H − EP , Dnuˆ ] = dΓ(i[|k|, dunˆ ]) − ∇E P − g[a∗ (idunˆ ρx ) + a(idunˆ ρx )],
(4.2)
are operator-valued functions of the total momentum where EP and ∇E P operator P . Another formal step would imply that 0 = (Ψf g , i[H − EP , Dnuˆ ]Ψf g ), Q
(4.3)
Q
and, hence, · dΓ(i[k, dunˆ ])Ψf g ) 0 = (Ψf g , dΓ(i[|k|, dunˆ ])Ψf g ) − (Ψf g , ∇E P Q
Q
∗
− g(Ψf g , [a Q
(idunˆ ρx )
+
Q
Q
a(idunˆ ρx )]Ψf g ). Q
(4.4)
The next Lemma shows that all terms on the RHS of Eq. (4.4) can be given a well-defined meaning such that the equality is true. Lemma 4.2. The identity ˆ k|)Ψf g ) 0 = (Ψf g , dΓ(χ2n (|k|)ξugˆ 2 (k)| Q
Q
− (Ψ
f g Q
− g(Ψ
ˆ k)Ψf g ) · dΓ(χ2 (|k|)ξ g 2 (k) , ∇E n u ˆ P Q
f g Q
∗
, [a
(idunˆ ρx )
+
a(idunˆ ρx )]Ψf g ) Q
(4.5)
holds true. As the one-particle state Ψf g belongs to the form domain of all Q
operators on the RHS of (4.5), this RHS is well defined.
Proof. See the Appendix
Note that the formal equality of the RHS of (4.4) and (4.5) is straightforward. The virial identity of the Lemma above is first used in item (i) of 4.2.1. A similar virial identity in item (ii) is proven analogously. 4.2. Number Operator Estimates in Putative Single-Particle States We now proceed to proving the following theorem, where the expectation of the boson number operator in the state Ψf g is bounded scale by scale. Q
Theorem 4.3. For |g| sufficiently small, (Ψf g , Nn,C 1/2 Ψf g ) Q
Q
ˆ Q
(Ψf g , Ψf g ) Q
2 1 3 ( ≤ O(|g|2(1−2γ) n2 |k|β 1( 2(n+1) , 2n ) k) 2 ),
Q
2 ˆ ∈ Ig . where Nn,C 1/2 := dΓ(χ2n (|k|)ξ g 1/2 (k)) and Q ˆ Q
CQ ˆ
g Furthermore, if for all P ∈ suppfQ the inequality
| − 1| > |g|γ/3 ||∇E P
(4.6)
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holds true then (Ψf g , Nn Ψf g ) Q
2 1 3 ( ≤ O(|g|2(1−2γ) n2 |k|β 1( 2(n+1) , 2n ) k) 2 ),
Q
(Ψf g , Ψf g ) Q
(4.7)
Q
where Nn := dΓ(χ2n (|k|). The (implicit) constants in (4.6)–(4.7) can be chosen to be uniform in Q, ∈ I ⊂ I; for the definition of I see Eqs. (3.9), (3.10)). They only depend |Q| g g on I and ΔI . 4.2.1. Outline of the Proof of Theorem 4.3. To prove inequalities (4.6), (4.7) we exploit two different virial arguments and properties (P1), (P2), and (P3) of Sect. 3.1.2. More precisely, we employ both conjugate operators Dnuˆ := dΓ(dunˆ ) ˆ ˆ ˆ u ˆ ,Q ˆ ,Q ˆ ,Q := dΓ(dun,⊥ ), with dunˆ and dun,⊥ defined in Eqs. (3.18) and (3.13), and Dn,⊥ respectively. The virial identities (see Lemma 4.2 for a rigorous treatment of ˆ u ˆ ,Q the identities below) corresponding to Dnuˆ and Dn,⊥ are: (i) 0 = (Ψf g , dΓ(i[|k|, dunˆ ])Ψf g ) Q
(4.8)
Q
· dΓ(i[k, dunˆ ])Ψf g ) − (Ψf g , ∇E P Q
(4.9)
Q
− g(Ψf g , [a∗ (idunˆ ρx ) + a(idunˆ ρx )]Ψf g ) Q
(4.10)
Q
ˆ k|)Ψf g ) = (Ψf g , dΓ(χ2n (|k|)ξugˆ 2 (k)| Q
(4.11)
Q
ˆ k)Ψf g ) · dΓ(χ2 (|k|)ξ g 2 (k) − (Ψf g , ∇E n u ˆ P
(4.12)
− g(Ψf g , [a∗ (idunˆ ρx ) + a(idunˆ ρx )]Ψf g );
(4.13)
Q
Q
Q
Q
(ii) ˆ ˆ ,Q 0 = (Ψf g , dΓ(i[|k|, dun,⊥ ])Ψf g ) Q
(4.14)
Q
· dΓ(i[k, duˆ,Qˆ ])Ψf g ) − (Ψf g , ∇E n,⊥ P Q
(4.15)
Q
ˆ
ˆ
ˆ ,Q ˆ ,Q − g(Ψf g , [a∗ (idun,⊥ ρx ) + a(idun,⊥ ρx )]Ψf g ) Q Q 2 ˆ |k⊥ | = Ψf g , dΓ χ2n (|k|)ξugˆ 2 (k) Ψf g Q Q |k| ˆ k⊥ )Ψf g ) · dΓ(χ2 (|k|)ξ g 2 (k) − (Ψf g , ∇E Q
n
P
ˆ
u ˆ
Q
ˆ
ˆ ,Q ˆ ,Q ρx ) + a(idun,⊥ ρx )]Ψf g ). − g(Ψf g , [a∗ (idun,⊥ Q
Q
(4.16) (4.17) (4.18) (4.19)
ˆ see (a) and (b), in Sect. 3.2). (For the definition of the functions χn (|k|), ξugˆ (k) Next, we explain in detail the key role of the virial identities. To arrive at inequalities (4.6), (4.7), we study (see Lemma 4.4) the number operator restricted to the sector associated with the unit vector u ˆ and derive the
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estimate 2 1 3 ( (Ψf g , Nn,ˆu Ψf g ) ≤ (Ψf g , Ψf g )O(|g|2(1−γ−˜γ ) n2 |k|β 1( 2(n+1) , 2n ) k) 2 ), (4.20) Q
Q
Q
Q
ˆ for some γ˜ , 0 < γ˜ < γ; we will eventually where Nn,ˆu := dΓ(χ2n (|k|)ξugˆ 2 (k)), choose γ˜ = γ/2. In doing this, we start from the bound ˆ k|)Ψf g ) − (Ψf g , ∇E ˆ k)Ψf g )| · dΓ(χ2 (|k|)ξ g 2 (k) |(Ψf g , dΓ(χ2n (|k|)ξugˆ 2 (k)| n u ˆ P Q
Q
Q
Q
ˆ k|)Ψf g ) ≥ O(|g|γ˜ )(Ψf g , dΓ(χ2n (|k|)ξugˆ 2 (k)| Q
(4.21)
Q
g ˆ that holds if, for all P ∈ suppfQ and for all k in the sector,
| > |g|γ˜ > 0. |1 − kˆ · ∇E P
(4.22)
Given (4.21), it is straightforward to control the term [see (4.13)] associated with the interaction part of the Hamiltonian and to derive the inequality in (4.20). Therefore, the bound in (4.22) is crucial, and we must identify the sec is collinear to P , and we may tors where it is violated. We recall that ∇E P assume that they are parallel; the other case can be treated in the same way. g First, note that the angle between P ∈ suppfQ and Q, as well as the angle between u ˆ and a vector k that belongs to the sector associated with u ˆ, is g and the cones Cuˆ , O(|g|γ ). This follows from the definitions of the function fQ ˆ given in Sect. 3.2. It implies that, roughly speaking, we can identify Pˆ = Q γ γ ˜ ˆ and k = u ˆ, since, for |g| sufficiently small, |g| is much smaller than |g| in (4.22). The vectors k for which (4.22) fails satisfy ˆ ˆ |−1 ≤ O(|g|γ˜ (=γ/2) ), k · P − |∇E P
| > 0. |∇E P
(4.23)
| is bounded away from 1, either—for |∇E | < 1; see also (B) Hence, if |∇E P P | > 1; see in Sect. 4.2.2—the condition (4.22) is always satisfied, or—for |∇E P k⊥ , (of order also (C) in Sect. 4.2.2—such k have a nonvanishing component, |−2 ) in the orthogonal complement of Q(= 1 − |∇E P ). In particular, they P
satisfy |k |2 |k⊥ | ˆ ⊥ − k⊥ · ∇EP > O(|g|γ/3 ) > 0. |k|2 |k|
(4.24)
Note that the second term on the LHS of (4.24) actually vanishes if our approxˆ were to hold exactly. In Sect. 4.2.2, we establish (4.24) rigorimation Pˆ = Q ously. The bound (4.24) immediately implies that, for the sectors u ˆ for which
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(4.22) fails, the following bound holds true 2 g 2 ˆ |k⊥ | 2 Ψf g Ψf g , dΓ χn (|k|)ξuˆ (k) Q Q |k|
ˆ k⊥ )Ψf g ) · dΓ(χ2n (|k|)ξ g 2 (k) −(Ψf g , ∇E u ˆ P Q
Q
γ/3
≥ O(|g|
ˆ k|)Ψf g ), )(Ψf g , dΓ(χ2n (|k|)ξugˆ 2 (k)| Q Q
(4.25)
Starting from this bound, we can use the second virial identity (4.17, 4.18, 4.19) to derive the inequality (4.20) for the sectors u ˆ for which (4.22) fails. |, P ∈ suppf g , The conclusion is that, under the condition that |∇E P Q
differs from 1 by a quantity > O(|g|γ/3 ), we can cover all the sectors by the |, these arguments two virial identities above. Without the restriction on |∇E P only show that Eq. (4.20) holds for all u ˆ-dependent sectors contained in the 1/2 cone CQˆ . In implementing this strategy, we make use of the following lemma. g Lemma 4.4. Fix a unit vector u ˆ and assume that, for all P ∈ suppfQ and for g all kˆ ∈ supp ξ , u ˆ
| > |g|γ˜ > 0, |1 − kˆ · ∇E P
0 < γ˜ < γ,
(4.26)
where γ˜ is g- and Q-independent. Then, for |g| sufficiently small, the following bound holds true 2 1 3 ( (Ψf g , Nn,ˆu Ψf g ) ≤ (Ψf g , Ψf g )O(|g|2(1−γ−˜γ ) n2 |k|β 1( 2(n+1) (4.27) , 2n ) k) 2 ), Q
Q
Q
Q
ˆ where Nn,ˆu := dΓ(χ2n (|k|)ξugˆ 2 (k)). Proof. We assume that (4.26) holds with < 0; 1 − kˆ · ∇E P > 0, can be treated similarly. We get the other case, 1 − kˆ · ∇E P ˆ k|)Ψf g ) 0 = (Ψf g , dΓ(χ2n (|k|)ξugˆ 2 (k)| Q
(4.28)
Q
ˆ k)Ψf g ) · dΓ(χ2 (|k|)ξ g 2 (k) − (Ψf g , ∇E n u ˆ P Q
Q
− g(Ψf g , [a∗ (idunˆ ρx ) + a(idunˆ ρx )]Ψf g ) Q
Q
ˆ k|)Ψf g ) ≤ − |g|γ˜ (Ψf g , dΓ(χ2n (|k|)ξugˆ 2 (k)| Q
Q
ˆ 1/2 Ψf g + c|g|1−γ Ψf g dΓ(χ2n (|k|)ξugˆ 2 (k)) Q Q 1/2 3 1 3 ( × |k|2β 1( 2(n+1) , 2n ) k) d k
(4.29) (4.30)
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|Q| ∈ Ig . To do the step from (4.28) for some constant c, c > 0, uniform in Q, to (4.29), we split ˆ k · ∇ ˆ χn (|k|)ξ g (k) i(dunˆ ρx )(k) = −χn (|k|)ξugˆ (k) ρ(k)e−ik·x u ˆ k ˆ k · ∇ ρ(k) e−ik·x −χ2n (|k|)ξugˆ 2 (k) k 1 2 g 2 ˆ −ik·x − χn (|k|)ξuˆ (k) ∇k · k ρ(k)e 2 ˆ k) k · ∇ e−ik·x (4.31) −χ2n (|k|)ξugˆ 2 (k)ρ( k and we may justify this step for each of the four terms separately, using the Schwarz inequality and > |g|γ˜ for P ∈ suppf g ; (i) the assumption 1 − kˆ · ∇E P Q (ii) the infrared behavior of ρ(k) as assumed in (2.14), i.e., |ρ(k)| ≤ O(|k|β ) ρ(k)| ≤ O(|k|β−1 ); and |∇ k (iii) Lemma 4.1. As an example, for the term proportional to ˆ k) k · ∇ e−ik·x , −χ2n (|k|)ξugˆ 2 (k)ρ( k we proceed as follows: g2 ˆ k) k · ∇ e−ik·x Ψf g ) (Ψf g , a∗ −χ2n (|k|)ξuˆ (k)ρ( k Q Q g2 ˆ 2 −i k· x 3 = χn (|k|)ξuˆ (k)ρ(k)k · (ak Ψf g , ∇k e Ψf g )d k Q Q ˆ k)||k||(a Ψf g , ∇ e−ik·x Ψf g )|d3 k ≤ χ2n (|k|)ξugˆ 2 (k)|ρ( k k Q Q ˆ k)| |k| a Ψf g ∇ e−ik·x Ψf g d3 k ≤ χ2n (|k|)ξugˆ 2 (k)|ρ( k k Q
ˆ Ψf g 2 d3 k χ2n (|k|)ξugˆ 2 (k) a k
≤
Q
1/2
Q
×
e−ik·x Ψf g 2 |k|2β 1( 1 , 3 ) (k)|k|2 d3 k ∇ k 2(n+1) 2n
1/2
Q
.
(4.32)
We notice that 1/2 g2 ˆ 2 2 3 ˆ 1/2 Ψf g (4.33) g = dΓ(χ2n (|k|)ξugˆ 2 (k)) χn (|k|)ξuˆ (k) ak Ψf d k Q
Q
e−ik·x Ψf g ≤ O(|g|−γ Ψf g ) (Lemma 4.1), and, since ∇ k
Q
Q
e−ik·x Ψf g 2 |k|2β 1( 1 , 3 ) (k)|k|2 d3 k ∇ k 2(n+1) 2n
1/2 (4.34)
Q
≤ C|g|−γ
3 1 3 ( |k|2β 1( 2(n+1) , 2n ) k) d k
1/2 Ψf g . Q
(4.35)
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Then, starting from (4.29), the bound from above takes the form 2 1 3 ( (Ψf g , Nn,ˆu Ψf g ) ≤ (Ψf g , Ψf g )O(|g|2(1−γ−˜γ ) n2 |k|β 1( 2(n+1) , 2n ) k) 2 ), (4.36) Q
Q
Q
Q
ˆ because where Nn,ˆu := dΓ(χ2n (|k|)ξugˆ 2 (k)), ˆ k| ≥ χ2n (|k|)ξugˆ 2 (k)|
1 ˆ χ2 (|k|)ξugˆ 2 (k). 2(n + 1) n
(4.37)
4.2.2. Proof of Theorem 4.3. Notice that, starting from Lemma 4.4, we can fill the region g ∀P ∈ suppfQ }
| > |g γ˜ | {kˆ : |1 − kˆ · ∇E P
(4.38)
with sectors corresponding to functions ξugˆj where 1 ≤ j ≤ ¯j ≤ O(|g|−γ ), so that we obtain ¯
j j=1
(Ψf g , Nn,ˆuj Ψf g ) Q
Q
≤ (Ψf g , Ψf g )O(|g|2(1− Q
3γ 2
−˜ γ) 2
Q
2 1 3 ( n |k|β 1( 2(n+1) , 2n ) k) 2 ).
(4.39)
g We observe that if, for some P ∈ suppfQ ,
| − 1| ≤ |g|γ/3 ||∇E P then, for |g| sufficiently small, | − 1| ≤ 2|g|γ/3 ||∇E P g for all P ∈ suppfQ . This holds because g • of the constraints on the support of fQ (see Sect. 3.2); ∂2E
• | ∂|P |P2 | ≤ CI ; (see Property (P2) in Sect. 3.1). After the result in Eq. (4.39), which holds for sectors such that (4.26) [Lemma (4.4)] is fulfilled, we may distinguish three possible situations, (A), (B) and , P ∈ suppf g . (C), depending on the length of the vector ∇E P Q g | − 1| ≤ |g|γ/3 . (A) For some P ∈ suppfQ , ||∇EP
1/2 In this case, ∀kˆ ∈ CQˆ , the inequality in (4.26) holds true for γ˜ = γ/2 and |g| sufficiently small, because | − 1| ≤ 2|g|γ/3 , ∀ P ∈ suppf g . (i) ||∇E Q
P
(ii) by definition 1/2
CQˆ :=
ˆ ≤ cos kˆ : |kˆ · Q|
1 γ/8 |g| 2
.
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Thus, we can use the estimate in Eq. (4.39) with γ˜ = γ/2, (Ψf g , Nn,C 1/2 Ψf g ) Q
Q
ˆ Q
(Ψf g , Ψf g ) Q
2 1 3 ( ≤ O(|g|2(1−2γ) n2 |k|β 1( 2(n+1) , 2n ) k) 2 ),
(4.40)
Q
2 ˆ ˆ are defined in (k)) (χn (|k|) and ξ g 1/2 (k) where Nn,C 1/2 := dΓ(χ2n (|k|)ξ g 1/2 ˆ Q
CQ ˆ
CQ ˆ
(a) and (b) of Sect. 3.2). g | < 1 − |g|γ/3 . (B) For some P ∈ suppfQ , |∇EP The constraint (4.26) with γ˜ = γ/2 is fulfilled for all angular sectors. g | > 1 + |g|γ/3 . (C) For all P ∈ suppfQ , |∇EP First we notice that we can restrict our analysis to an angular sector g labeled by a direction u ˆ such that, for some P ∈ suppfQ , the inequality | ≤ |g|γ˜ (=γ/2) |1 − kˆ · ∇E P
(4.41)
holds true for some kˆ belonging to the sector under consideration. This is because, if | > |g|γ˜ (=γ/2) , (4.42) |1 − kˆ · ∇E P for all kˆ belonging to the given sector, then the result in (4.27) holds, as we have proven in Lemma 4.4. | > 1 + |g|γ/3 and (4.41) We now show that the combination of |∇E P yields the useful inequality (4.46) below. We notice that, assuming the bound in Eq. (4.41) for some kˆ belonging to the sector, for |g| sufficiently small, | ≤ 2|g|γ˜ (=γ/2) , |1 − kˆ · ∇E P
(4.43)
ˆ ≥ cos(|g|γ ), for all for all kˆ in the sector labeled by u ˆ. Furthermore, Pˆ · Q g is parallel to P ∈ suppfQ , by construction, and we may assume that ∇EP = −|P ||∇E |, can be treated in an analogous P ; the other case, P · ∇E P P ˆ ˆ Then, (4.43) means that way. Let η be the angle between k and Q. | ≤ 2|g|γ/2 , − 2|g|γ/2 ≤ 1 − cos(η + )|∇E P
(4.44)
where = O(|g|γ ) and, for |g| sufficiently small, 1 − c |g|γ/3 ≥
[1 + 2|g|γ/2 ] [1 − 2|g|γ/2 ] ≥ cos(η + ) ≥ | | |∇E |∇E P P
(4.45)
for some constant c > 0. Hence, we have η ≥ c |g|γ/6 > 0 where c > 0, and we find that |k |2 |k⊥ | ˆ ⊥ − (4.46) k⊥ · ∇EP > O(sin2 (η)) ≥ O(|g|γ/3 ) > 0, |k|2 |k| for all k in the sector, where k⊥ := k − ˆ > cos(|g|γ ); (i) by assumption, Pˆ · Q
k·Q 2 Q, |Q|
because
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is parallel (or antiparallel) to P and |∇E | < C ; (ii) ∇E I P P (iii)
| k⊥ | | k|
= sin(η); ˆ | ≤ |∇E | × O(|g|γ ) using that Pˆ · Q ˆ > cos(|g|γ ). (iv) |k⊥ · ∇E P P Assuming, for example, that (4.46) holds, because |k⊥ | ˆ |k⊥ |2 − k⊥ · ∇EP < −c|g|γ/3 , |k|2 |k|
(4.47)
where c > 0, we use the second virial identity [see Eq. (4.14)] to obtain 2 g 2 ˆ |k⊥ | 2 0 = Ψf g , dΓ χn (|k|)ξuˆ (k) (4.48) Ψf g Q Q |k| ˆ k⊥ )Ψf g ) E · dΓ(χ2 (|k|)ξ g 2 (k) −(Ψf g , ∇ Q
P
n
P
u ˆ,Pˆ
∗
u ˆ
−g(Ψf g , [a (idn,⊥ ρx ) + Q
Q
ˆ,Pˆ a(idun,⊥ ρx )]Ψf g ), Q
ˆ k|)Ψf g ) ≤ −c|g|γ/3 (Ψf g , dΓ(χ2n (|k|)ξugˆ 2 (k)| Q
Q
ˆ 1/2 Ψf g + c |g|1−γ Ψf g dΓ(χ2n (|k|)ξugˆ 2 (k)) Q Q 1/2 3 1 3 ( × |k|2β 1( 2(n+1) , 2n ) k) d k
(4.49) (4.50)
for some c > 0. Now, we estimate (4.49) similarly to (4.29) in Lemma 4.4. Conclusions For |g| sufficiently small, we have proven that: (i) by combining cases (A), (B) and (C), (Ψf g , Nn,C 1/2 Ψf g ) Q
Q
ˆ Q
(Ψ
f g Q
,Ψ
f g Q
)
2 1 3 ( ≤ O(|g|2(1−2γ) n2 |k|β 1( 2(n+1) , 2n ) k) 2 ).
(4.51)
Note that, in cases (B) and (C), the angular restriction was not used. (ii) Under the assumption that | − 1| > |g|γ/3 , ||∇E P we have (Ψf g Nn Ψf g ) Q
Q
(Ψf g , Ψf g ) Q
g for all P ∈ fQ ,
2 1 3 ( ≤ O(|g|2(1−2γ) n2 |k|β 1( 2(n+1) , 2n ) k) 2 ),
(4.52)
Q
where Nn := dΓ(χ2n (|k|). This follows from cases (B) and (C).
4.3. Number Operator Estimates in Putative Fiber Eigenvectors Using the results in Theorem 4.3 and Property (P3), we are now in a position to state some bounds on the expectation value of the boson number operator restricted to the fiber spaces. These bounds hold pointwise in P , for |P | in the open interval Ig ⊂ Ig introduced in Sect. 3.2; see Eqs. (3.9, 3.10).
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Theorem 4.5. For |g| sufficiently small, and (P , EP ) ∈ Ig × ΔI : b (ΨP ,E , Nn,C 2 Ψ P ,E ) ≤ O(|g| ˆ P
P
(1−2γ) 3
P
1
4 3 1 3 ( |g|−γ/8 n 3 |k|β 1( 2(n+1) , 2n ) k) 2 ),
(4.53)
where g2 ˆ b b 2 Nn,C 2 := dΓ (χn (|k|)ξ 2 (k)). C ˆ P
(4.54)
ˆ P
| − 1| > 3 |g|γ/3 , then Furthermore, if in addition ||∇E P 2 (ΨP ,E , Nnb ΨP ,E ) ≤ O(|g| P
(1−2γ) 3
P
1
4 3 1 3 ( n 3 |k|β 1( 2(n+1) , 2n ) k) 2 )
(4.55)
where Nnb := dΓb (χ2n (|k|)).
(4.56) The constants in (4.53), (4.55) can be chosen uniformly in P , |P | ∈ Ig ⊂ I [Ig is defined in Sect. 3.2, Eqs. (3.9), (3.10)]. They only depend on I and ΔI . g Proof. First of all, we observe that, for P such that fQ (P ) = 1, the inequality b (1−2γ) 1 3 ( n |k|β 1( 2(n+1) (ΨP ,E , Nn,C 1/2 ΨP ,E ) ≤ O(|g| , 2n ) k) 2 ) P
P
ˆ Q
(4.57)
can fail to hold true only for P in a set If∗g of measure bounded above by Q
(Ψf g , Ψf g )O(g Q
i.e.,
(1−2γ)
Q
1 3 ( n |k|β 1( 2(n+1) , 2n ) k) 2 ),
1 3 ( d3 P ≤ (Ψf g , Ψf g )O(|g|(1−2γ) n |k|β 1( 2(n+1) , 2n ) k) 2 ). Q
Q
(4.58)
(4.59)
I ∗g f Q
This follows from inequality (4.6), which we can write as g g b d3 P f¯Q ,E , Nn,C 1/2 ΨP ,E ) (P )fQ (P )(ΨP P
ˆ Q
P
2 1 3 ( ≤ (Ψf g , Ψf g )O(|g|2(1−2γ) n2 |k|β 1( 2(n+1) , 2n ) k) 2 ). Q
Q
(4.60) (4.61)
Next, we make use of the following inequality, which holds in the sense of quadratic forms, b b Nn,C 2 ≤ N 1/2 n,C ˆ P
(4.62)
ˆ Q
g for P in the support of fQ . This inequality can be easily derived from the
definitions of the smooth functions ξ g 1/2 , ξCg 2 (see Sect. 3.2) with support in CQ ˆ
the sets
ˆ P
ˆ ≤ cos 1 |g|γ/8 kˆ : |kˆ · Q| , 2 CP2ˆ := kˆ : |kˆ · Pˆ | ≤ cos 2|g|γ/8 ,
1/2
CQˆ :=
ˆ ≥ cos(|g|γ ). respectively, and from the constraint Pˆ · Q
(4.63) (4.64)
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g g ∗ Hence, for P ∈ suppfQ \ If g such that fQ (P ) = 1, we have that Q
b b (ΨP ,E , Nn,C 2 Ψ ,E , Nn,C 1/2 ΨP ,E ) P ,E ) ≤ (ΨP ˆ P
P
P
P
(4.65)
P
ˆ Q
1 3 ( ≤ O(|g|(1−2γ) n |k|β 1( 2(n+1) , 2n ) k) 2 ),
(4.66)
by definition of If∗ . Q
Because of Eq. (4.59), any point P belonging to the set If∗g , and such Q
g that fQ (P ) = 1, is at a distance at the most of
(Ψf g , Ψf g )1/3 O(|g| Q
(1−2γ) 3
Q
≤ O(|g|
(1−2γ) 3
1
1
3 1 3 ( n 3 |k|β 1( 2(n+1) , 2n ) k) 2 )
(4.67)
1
1 3 1 3 ( n 3 |k|β 1( 2(n+1) , 2n ) k) 2 )
(4.68)
g ∗ from an arbitrary point in suppfQ \ If g . Thus, we consider a slightly modified Q
b version of property (P3) for the operator Nn,C 2 , namely ˆ P
b −γ/8 (Ψ |∇ ) ,E )| ≤ O(nCI [(sup |EP |) + 1]|g| P P ,E , Nn,C 2 ΨP ˆ P
P
P
(4.69)
∈I P
where, following the derivation of property (P3), the term |g|−γ/8 comes from the derivative of the smooth function ξCg 2 . Using the fundamental theorem of ˆ P
calculus, we can finally state that b (ΨP ,E , Nn,C 2 Ψ P ,E ) P
≤ O(|g|
ˆ P
(1−2γ) 3
P
−γ/8
|g|
1
4 ∗ 3 1 3 ( n 3 |k|β 1( 2(n+1) , 2n ) k) 2 |), P ∈ If g .
(4.70)
Q
|Q| ∈ Ig . We remark that the bounds in Eqs. (4.65), (4.70) hold uniformly in Q, because f (Q) = 1 by defThe bounds in Eqs. (4.65), (4.70) hold for P ≡ Q, Q inition. Thus, we arrive at the estimate in Eq. (4.53) for any P ∈ Ig . |−1| > 3 |g|γ/3 . Now, assume that for (P∗ , E ) ∈ I ×ΔI , we have ||∇E g
P∗
P∗
2
g Then we can consider a wave function fQ with Q ≡ P∗ . Thanks to Property P2, and for |g| sufficiently small, i.e., less than some value |¯ g | uniform in P∗ , γ/3 | − 1| > |g| , for all P ∈ f g . Thus, we can |P∗ | ∈ I, we have that ||∇E ∗ P
P
apply Theorem 4.3. Finally, following the same steps used before, one arrives at the inequality in Eq. (4.55) for P ≡ P∗ . Notice that, in this case, since there is no angular restriction, no term proportional to |g|−γ/8 appears on the RHS of Eq. (4.55). The bound in Eq. (4.55) trivially implies the corollary below. | − 1| > Corollary 4.6. For β > 11/2, and for (P , EP ) ∈ Ig × ΔI with ||∇E P 3 γ/3 , the putative eigenvector ΨP ,E (up to a suitable phase) is asymptotic 2 |g| P
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to the vacuum vector Ψ0P in HP , as g tends to 0. Likewise, the energy EP is asymptotic to P 2 /2. More precisely, Ψ0P − ΨP ,E ≤ O(|g|(1−2γ)/6 )
(4.71)
P
and
P 2 − EP ≤ O(|g|(1−2γ)/6 ). 2
(4.72)
Proof. The norm estimate in (4.71) follows from Theorem 4.5. Without loss of generality, we can start from the identity below, for some real and positive coefficient c(g), (≥1)
ΨP ,E = c(g)Ψ0P + ΨP ,E P
(4.73)
P
(≥1)
where ΨP ,E and Ψ0P are normalized, and ΨP ,E contains at least one boson. P
P
Then we can write:
(≥1)
ΨP ,E − Ψ0P 2 = (c(g) − 1)2 + ΨP ,E 2 P
(4.74)
P
(≥1)
= c(g)2 + 1 − 2c(g) + ΨP ,E 2 . P
(4.75)
Using the normalization condition, (≥1)
ΨP ,E 2 = 1 = c(g)2 + ΨP ,E 2 , P
P
(4.76)
we have (≥1)
c(g) = |1 − ΨP ,E 2 |1/2
(4.77)
ΨP ,E − Ψ0P 2 = 2 − 2c(g).
(4.78)
P
and P
From Theorem 4.5, it follows that (≥1)
(≥1)
ΨP ,E 2 ≤ (N b )1/2 ΨP ,E 2 ≤ O(|g|(1−2γ)/3 ), P
P
since the sum over n in (4.55) can be estimated as 4 4 1 2β+3 3 1 3 ( n 3 |k|β 1( 2(n+1) n 3 n− 6 ≤ const., , 2n ) k) 2 ≤ n≥1
(4.79)
if β > 11/2
n≥1
(4.80) − 2β+3 2
1 3 ( where we use that |k|β 1( 2(n+1) ), as follows by the size , 2n ) k) 2 = O(n O(1/n) of the support of the function χn , and the spatial dimension d = 3. We remind the reader that the expectation value in ΨP ,E of the number operator P associated with boson momenta above |k| = 1 can be bounded above by using the form inequality H f < aHP + b, for some a, b > 0. Consequently, the estimate in Eq. (4.71) is easily obtained.
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For the inequality in Eq. (4.72), consider EP −
P 2 = (ΨP ,E , HP (ΨP ,E − Ψ0P )) P P 2 +(ΨP ,E , (HP − HP0 )Ψ0P ) P
+(ΨP ,E − Ψ0P , HP0 Ψ0P ). P
Then, use (4.72) and the fact that HP − HP0 = gφb (ρ) is HP0 -bounded.
(4.81) (4.82) (4.83)
5. Absence of Regular Mass Shells In this section, we first make use of the results obtained in Sect. 4 to arrive at an argument that shows a contradiction to the existence of a mass shell | − 1| > 3 |g|γ/3 for P ∈ I . In imple(P , EP ) ∈ Ig × ΔI assuming that ||∇E g P 2 menting the argument, we employ suitable trial states; see Theorem 5.1. Then | − 1| > 3 |g|γ/3 , we proceed to show that, if we remove the assumption ||∇E P 2 a mass shell might exist for (P , EP ) ∈ Ig × ΔI such that 1 (5.1) EP = |P | − + O(|g|γ/4 ). 2 This result is completed in Theorem 5.4. We recall that so far we have assumed the existence of a mass shell for P in the open interval Ig , and we have defined another open interval Ig ⊂ Ig with the properties specified in Sect. 3.2. The results of Corollary 4.6, which will be used in the following theorem, hold for P ∈ Ig . Theorem 5.1. For β > 11/2, and for |g| sufficiently small, no regular (i.e., fulfilling the Main Hypothesis in Sect. 3.1.1) mass shell (P , EP ) can exist with the properties: (i) |P | ∈ Ig , |Ig | > |g|γ/2 ; (ii) |EP | ∈ ΔI ; | − 1| > 3 |g|γ/3 for P ∈ I . (iii) ||∇E g P 2 Proof. The proof is by contradiction. For |g| sufficiently small (depending on the exponent γ), we pick an open interval Ig ⊂ Ig fulfilling the following properties: (a) |Ig | > |g|γ ; ∈ Ig then |P | ∈ Ig for any P ∈ suppf g . (b) If |Q| Q Notice that the definition of Ig is meaningful for |g| sufficiently small. For ∈ Ig , we introduce the trial vector |Q| k)2 /2 + |k| − E ( P − 1 g P ηQ d3 P d3 kfQ b∗k Ψ0P , (5.2) := (P ) 1 h 2
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where: • > 0; • h(z) ∈ C0∞ (R), h(z) ≥ 0. Since Ψf g is a single-particle state, we have that Q
0 (ηQ , (H − EP )Ψf g ) = −(ηQ , gφ(ρ x )Ψf g ), Q
Q
(5.3)
2
where H 0 := p2 + H f and EP is a (operator-valued) function of the total momentum operator P . This equation implies that g(ηQ , φ(ρ x )PΩ Ψf g )
(5.4)
Q
0 ⊥ = −(ηQ , (H − EP )PΩ Ψf g ) Q
⊥ −g(ηQ , φ(ρ x )PΩ Ψf g ),
(5.5) (5.6)
Q
where, as usual, the expressions PΩ , PΩ⊥ acting on H stand for 1Hel ⊗ PΩ , 1Hel ⊗ PΩ⊥ , respectively. We observe that (ηQ , φ(ρ x ) PΩ Ψf g ) Q (P − k)2 /2 + |k| − EP g 2 1 3 3 = c(g) d P d k|fQ ρ(|k|), 1 h (P )| 2 (5.7) where c(g) → 1, as g → 0, because of Corollary 4.6. Notice that, for |P | > 1+δ, where δ > 0 is g-independent, the equation (P − k)2 /2 + |k| − P 2 /2 = 0,
|k| > 0
(5.8)
has the one-parameter family of solutions |k| = 2(|P | cos θ − 1) > 0
> 0, where cos θ = |PP||·kk| . Notice that, for P ∈ I, ρ(2(|P | cos θ − 1)) = 0 for some 0 < θ < π; see the conditions on ρ in Sect. 2.3. Hence, using (4.72), for and |g| sufficiently small, we arrive at the following bound for cos(θ) −
1 | |P
1
g 2 |(ηQ , φ(ρ x )PΩ Ψf g )| > D1 2 fQ 2 , Q
(5.9)
where D1 is an - and g- independent (positive) constant (hint: for each θ in Eq. (5.8), implement the change of variable |k| → zθ with zθ := [(P − k)2 /2 + |k| − EP ]/ ). Using the Schwarz inequality, we find that 1
0 ⊥ |(ηQ , N 2 (H − EP )PΩ Ψf g )|
(5.10)
Q
1 2
⊥ ≤ (H 0 − EP )ηQ N PΩ Ψf g . Q
(5.11)
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We then observe that g (H 0 − EP )ηQ ≤ O( fQ 2 ).
(5.12)
Using Eq. (4.7), one may easily derive the inequalities 1
N 2 PΩ⊥ Ψf g ≤ O(|g|
2(1−2γ) 2
Q
g fQ 2 )
(5.13)
and ⊥ |(ηQ , φ(ρ x ) PΩ Ψf g )|
(5.14)
Q
⊥ = |(ηQ , N φ(ρ x )PΩ Ψf g )|
(5.15)
Q
≤ O(|g|
2(1−2γ) 2
g fQ 2 ).
(5.16)
For the step from (5.15) to (5.16), one may use that ⊥ (−) (ηQ (ρx ) PΩ⊥ Ψf g ) , N φ(ρ x )PΩ Ψf g ) = (ηQ , Nφ Q
Q
(5.17)
where φ(−) (ρx ), φ(+) (ρx ) stand for the part proportional to the annihilationand to the creation operator, respectively; i.e., φ(ρx ) = φ(−) (ρx ) + φ(+) (ρx ). Then, we observe that (−) (ηQ (ρx )PΩ⊥ Ψf g ) , Nφ
(5.18)
Q
(−) (ρx )N PΩ⊥ Ψf g ) = (ηQ ,φ
(5.19)
Q
(−) (ρx ), N ]PΩ⊥ Ψf g ), −(ηQ , [φ
(5.20)
Q
and we finally use Theorem 4.3 together with the estimates 1
g N 2 φ(+) (ρx )ηQ ≤ O( fQ 2 ),
(5.21)
1 2
[φ(−) (ρx ), N ]PΩ⊥ Ψf g ≤ O( N PΩ⊥ Ψf g ). Q
Q
(5.22)
Finally, we arrive at 1
g 2 g 2 g 2 (1−2γ) 2−2γ fQ fQ D1 |g| 2 fQ 2 ≤ O( |g| 2 ) + O(|g| 2 ).
(5.23)
This inequality is violated whenever 1
c1 |g|1−2γ < 2 < c2 |g|2γ
(5.24)
for some c1 , c2 > 0. We note that the inequality in Eq. (5.24) can be fulfilled if 0 < γ < 1/4 and |g| is sufficiently small. From the argument above, we conclude that, for sufficiently small |g|, a mass shell cannot exist in Ig × ΔI with the assumed regularity properties, because Ig ⊂ Ig ⊂ Ig . We need two preparatory lemmas to state our final result: Theorem 5.4, concerning the absence of a mass shell anywhere but near the boundary of the energy–momentum spectrum.
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is collinear to P . In the From property (P1), we know that the vector ∇E P |−1| ≤ 3 |g|γ/3 first of the two lemmas below, Lemma 5.2, assuming that ||∇E P 2 and P are in fact parallel. and β > 11/2, we show that ∇E P The second lemma, Lemma 5.3, states that the boson number operator, restricted to the cone {kˆ : −kˆ · Pˆ < cos(2|g|γ/8 )} and evaluated on the puta(1−2γ) tive fiber eigenvector ΨP ,E , |P | ∈ Ig , is also bounded above by O(|g| 3 P
|g|−1/8 ), for β > 11/2.
Lemma 5.2. For β > 11/2, and for g in an interval {g : 0 < |g| ≤ g∗ } with g∗ > 0 sufficiently small, if (P , EP ) ∈ Ig × ΔI fulfills the constraint | − 1| ≤ ||∇E P then the bound
∂EP | ∂|P
3 γ/3 |g| , 2
(5.25)
≥ 1 − 32 |g|γ/3 holds true.
Proof. The proof is indirect. We assume that there exists g∗ > 0 such that, for some |g| < g∗ and for some P∗ ∈ Ig , ∂EP 3 |P =P∗ < −1 + |g|γ/3 < 0. 2 ∂|P |
(5.26)
We also assume that g∗ is sufficiently small to apply Lemma 4.4 and Theorem 4.5 later on. We shall show that the assumption in Eq. (5.26) yields a g contradiction. Consider the function fQ≡ P . By Property P2, ∗
∂EP < c|g|γ , ∂|P |
with c > 0,
(5.27)
g for all P ∈ suppfQ≡ ˆ-dependent sectors such that P . Now, for all u ∗
u ˆ · Pˆ∗ > 0, we consider the first virial identity of Sect. 4.2.1 [see Eqs. (4.8)–(4.13)] and observe that ˆ k)Ψf g ) · dΓ(χ2n (|k|)ξ g 2 (k) −(Ψf g , ∇E u ˆ P Q
(5.28)
Q
ˆ k|)Ψf g ), ≥ −c|g|γ (Ψf g , dΓ(χ2n (|k|)ξugˆ 2 (k)| Q
Q
(5.29)
g for all P ∈ suppfQ≡ P ∗ . Then, for |g| < g∗ and g∗ sufficiently small, one can proceed as in Lemma 4.4 and finally apply the argument used in Theorem 4.5 to obtain that b (ΨP∗ ,E , Nn,ˆ ∗ ,E ) u ΨP P∗
P∗
(5.30)
can be summed over n, yielding a quantity bounded by O(|g|γ/2 ). This result readily implies that (ΨP∗ ,E , P f ΨP∗ ,E ) · Pˆ∗ ≤ C|g|γ/2 P∗
P∗
(5.31)
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for some positive constant C; hence, − (ΨP∗ ,E , P f ΨP∗ ,E ) · Pˆ∗ ≥ −C|g|γ/2 . P∗
(5.32)
P∗
Using the Feynman–Hellman formula, = ∂EP Pˆ = P − (Ψ f ∇E ,EP ), P P ,EP , P ΨP ∂|P |
(5.33)
∂EP |P =P∗ ≥ |P∗ | − C|g|γ/2 > 1 − C|g|γ/2 . ∂|P |
(5.34)
we deduce that
This yields a a contradiction for g∗ sufficiently small; therefore, we conclude that the bound ∂EP 3 (5.35) |P =P∗ ≥ 1 − |g|γ/3 2 ∂|P | holds for {g | 0 < |g| ≤ g ∗ }, for some g ∗ > 0, because of (5.25).
We are now in a position to extend the result in Eq. (4.53). | − 1| ≤ Lemma 5.3. For (P , EP ) ∈ Ig × ΔI , with ||∇E P β > 11/2 and |g| sufficiently small, b (ΨP ,E , Nn,C 2 ∪C 2,− ΨP ,E ) ≤ O(|g| ˆ P
P
ˆ P
P
(1−2γ) 3
4 |g|−1/8 n 3 |k|β 1(
3 γ/3 , 2 |g|
and for 1
1 , 3 ) 2(n+1) 2n
(k)23 ), (5.36)
where g2 b b 2 ˆ (k)) Nn,C 2 ∪C 2,− := dΓ (χn (|k|)ξ g 2 C ∪C 2,− ˆ P
ˆ P
ˆ P
(5.37)
ˆ P
ˆ 0 ≤ ξg ˆ ≤ 1 is a smooth function with support in (k) and ξCg 2 ∪C 2,− (k), C 2 ∪ C 2,− ˆ P
ˆ P
ˆ P
ˆ P
CP2ˆ ∪ CP2,− ˆ ,
(5.38)
ˆ is defined as follows: where CP2,− := {kˆ : −kˆ · Pˆ ≥ cos(2|g|γ/8 )}. ξCg 2 ∪C 2,− (k) ˆ ˆ P
ˆ P
(i) g ˆ ξC 2 ∪C 2,− (k) = 1
for
{kˆ : kˆ · Pˆ ≤ cos(4|g|γ/8 )};
(5.39)
g ˆ ξC 2 ∪C 2,− (k) = 0
for
{kˆ : kˆ · Pˆ > cos(2|g|γ/8 )};
(5.40)
ˆ P
ˆ P
(ii) ˆ P
ˆ P
(iii) g −γ/8 ˆ ξC , |∂θkP 2 ∪C 2,− (k)| ≤ Cξ |g| ˆ ˆ P
(5.41)
ˆ P
ˆ ˆ where θkP ˆ is the angle between k and P , and the constant Cξ is independent of g.
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2,− g Proof. Because of Lemma 5.2, for kˆ in the sector CQ≡ ˆ P and P ∈ suppfQ≡ P , with Q ≡ P ∈ Ig , the condition in (4.26) of Lemma 4.4 is fulfilled. Then one can repeat the arguments of Theorem 4.5 for the number operator restricted to 2,− the sector CQ≡ ˆ Pˆ and derive the inequality in Eq. (5.36) for all Q ≡ P ∈ Ig .
Theorem 5.4. For β > 11/2 and |g| sufficiently small, if a regular mass shell (i.e., fulfilling the Main Hypothesis in Sect. 3.1.1) exists in an interval Ig , and if for some (P , EP ) ∈ Ig × ΔI | − 1| ≤ ||∇E P
3 γ/3 |g| , 2
(5.42)
then, for all P ∈ Ig , EP = |P | −
1 + O(|g|γ/4 ). 2
(5.43)
Proof. We consider (P , EP ) ∈ Ig × ΔI , such that 3 γ/3 |g| . 2 From the Feynman–Hellman formula [see Eq.(3.6)], | − 1| ≤ ||∇E P
(5.44)
= |P |2 − P · (Ψ f P · ∇E P P ,E , P ΨP ,E ). P
(5.45)
P
From the result in Lemma 5.2, we can derive the following identity = |P |(1 + O(|g|γ/3 )). P · ∇E P
(5.46)
From Lemma 5.3, for the expectation values in the equation below, we can restrict P f and H f to the sector CP2,+ := {kˆ : kˆ · Pˆ ≥ cos(2|g|γ/8 )} up to an ˆ o((|g|γ/4 ) remainder, and we deduce that Pˆ · (ΨP ,E , P f ΨP ,E ) = (ΨP ,E , H f ΨP ,E ) + O(|g|γ/4 ). P
P
P
P
(5.47)
Hence, by combining (5.45)–(5.47), one arrives at (ΨP ,E , H f ΨP ,E ) − P · (ΨP ,E , P f ΨP ,E ) P
P
P
P
= |P | − 1 + |P | − |P |2 + O(|g|γ/4 ).
(5.48)
Next, starting from the formal virial identity (ΨP ,E , i[HP , Db1 ,κ ]ΨP ,E ) = 0, κ
P
(5.49)
P
where Db1 ,κ = dΓb (d κ1 ,κ ) is defined in Sect. 3.2 (III), we derive κ
0 = (ΨP ,E , dΓb (i[|k|, d κ1 ,κ ])ΨP ,E ) P
P
+(ΨP ,E , dΓb (i[k, d κ1 ,κ ]) · dΓb (k)ΨP ,E ) P
P
−P · (ΨP ,E , dΓb (i[k, d κ1 ,κ ])ΨP ,E ) P
P
−g(ΨP ,E , [b∗ (id κ1 ,κ ρ) + b(id κ1 ,κ ρ)]ΨP ,E ). P
P
(5.50)
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The virial identity in Eq. (5.50) needs to be justified and this is done in Sect. 6.3 in the Appendix. By taking the limit κ ↑ +∞ on the RHS of (5.50), it follows that 0 = (ΨP ,E , H f ΨP ,E ) P
P
+(ΨP ,E , P f · P f ΨP ,E ) P
P
−P · (ΨP ,E , P f ΨP ,E ) P
P
−g(ΨP ,E , [b∗ (id∞ ρ) + b(id∞ ρ)]ΨP ,E ), P
P
(5.51)
where d∞ :=
1 · k). (k · i∇k + i∇ k 2
(5.52)
Equation (5.51) follows from (5.50) thanks to 1. the infrared behavior of the form factor ρ(k), namely for any β > −1; 2. the ultraviolet cutoff Λ; see Eq. (2.14); 3. the fact that dΓb (i[|k|, d κ1 ,κ ]) and dΓb (i[k, d κ1 ,κ ]) bounded by H f and ΨP ,E P
belong to the domain of H f .
Therefore, we can express the expectation value of (P f )2 in the state ΨP ,E P as a function of |P | up to g-dependent corrections (ΨP ,E , (P f )2 ΨP ,E ) = (|P | − 1)2 + O(|g|γ/4 ). P
P
(5.53)
Using the eigenvalue equation (2.22), we obtain 1 (|P | − 1)2 + 2|P | − |P |2 + O(|g|γ ) + |P | − 1 + O(|g|γ/4 ) EP = 2 1 = |P | − + O(|g|γ/4 ). (5.54) 2 (see Property P1, Sect. 3.1), if Finally, because of the constraint on ∇E P Eq. (5.54) holds for |P | ∈ Ig , either it is also true for |P | ∈ Ig or the mass shell cannot be defined on Ig with the assumed regularity properties. This can be explained considering the following two cases: | < C and conclude that Eq. (5.54) holds on (a) if |Ig | < 2|g|γ/4 , use |∇E I P Ig ; (b) if |Ig | ≥ 2|g|γ/4 , write Ig as Ig = ∪j Igj , with {Igj } disjoints, and 2|g|γ/4 > |Igj | > |g|γ/2 . For each Igj , either one can repeat the argument developed in Eqs. (5.44)–(5.54), and proceed as in a), or conclude that the mass shell does not exists for P ∈ Igj . In the latter case, since Igj ⊂ Ig , the mass shell does not exist in Ig with the assumed regularity properties. Remark. It is easy to see that EP ≤
P 2 + O(|g|). 2
(5.55)
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The proof follows from Eq. (5.51) by adding and subtracting P 2 on the righthand side. In fact, one gets 0 = (ΨP ,E , HP ΨP ,E ) − P
P
P 2 2
1 + (ΨP ,E , P f · P f ΨP ,E ) P P 2 −g(ΨP ,E , [b∗ (id∞ ρ) + b(id∞ ρ)]ΨP ,E ). P
P
(5.56)
Furthermore, assuming the validity of the Feynman–Helman formula, we see that = P · (Ψ f P · ∇E ,E ) P P ,E , (P − P )ΨP P
2
P
f
≥ P − |P | P ΨP ,E .
(5.57) (5.58)
P
From Eq. (5.56), P f ΨP ,E 2 ≤ P 2 − 2EP + C|g|, P
and then
C > 0,
≥ P 2 − |P | P 2 − 2E + C|g|. P · ∇E P P
For |P | ≥ 1 + δ, because of the constraint EP ≥ |P | − conclude that ≥ 1 − C |g| Pˆ · ∇E P
(5.59)
(5.60) 1 2
+ O(|g|), we can (5.61)
for some positive constant C . This yields an alternative proof of Lemma 5.2.
Acknowledgements We thank an anonymous referee who pointed out the Remark at the end of Sect. 5. At the time when this work was finished, W.D.R. was supported by the European Research Council and the Academy of Finland. A.P. was supported by NSF grant DMS-0905988.
6. Appendix In Sects. 6.1 and 6.2, we provide the proofs of Lemmas 4.1 and 4.2 in Sect. 4. For the convenience of the reader, these lemmas are repeated below. In Sect. 6.3, we prove the equality (5.50) in Sect. 5. Lemma 4.2 and the equality (5.50) are virial identities, the justification of which is, in general, a difficult task. We refer the reader to [9,14] and [15] for more background.
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6.1. Proof of Lemma 4.1 Lemma (4.1). The vector Ψf g belongs to the domain of the position operator Q x and xi Ψf g ≤ O(|g|−γ Ψf g ), Q
Q
i = 1, 2, 3
(6.1)
Proof. It suffices to estimate, in the limit Δi → 0, e−iΔi xi Ψf g − Ψf g
Q Q (6.2) Δi 1 g g 3 3 = fQ e−iΔi xi fQ ,E d P − ,E d P (P ) ΨP (P ) ΨP P P Δi 1 g g −iΔi xi 3 3 = ΨP ,E d P − fQ d P fQ −Δiˆi,E (P ) e (P ) ΨP P P −Δiˆ i Δi (6.3) 1 g g ˆ + d3 P (6.4) (fQ −Δiˆi,E (P ) − fQ (P − Δi i))ΨP P −Δiˆ i Δi 1 g g 3 3 ˆi)Ψ ( P − Δ d P − f ( P )Ψ d P . fQ + i P −Δiˆi,EP −Δ ˆi P ,EP Q Δi i (6.5)
We notice that e−iΔi xi ΨP ,E ∈ HP −Δiˆi [in (6.3)], and P
IP −Δiˆi (e−iΔi xi ΨP ,E ) = IP (ΨP ,E ) P
(6.6)
P
as vectors in F b . The term in (6.5) is identically zero, by a change of variables. We now derive bounds for (6.3), (6.4), as Δi → 0. By item (iii) in the Main Hypothesis (which, strictly speaking, means that I (Ψ ∇ P P P ,EP ) ≤ CI ) and the Cauchy–Schwartz inequality, we conclude that (6.3) is bounded by CI f g (P ) . 2
Q
For (6.4), we use again Cauchy–Schwartz and the bound (for some constant C) f g (P )| f g (P ) = O(|g|−γ f g (P ) ), f g (P ) ≤ C| sup ∇ ∇ 2 2 2 P Q P Q Q Q
(6.7)
g which can be checked from the construction of the functions fQ (see Eq. (3.11)). Collecting the bounds on (6.3, 6.4, 6.5), we have proven the lemma.
6.2. Proof of Lemma 4.2 We now proceed with the proof of Lemma 4.2 in Sect. 4. Lemma (4.2). The identity ˆ k|)Ψf g ) 0 = (Ψf g , dΓ(χ2n (|k|)ξugˆ 2 (k)| Q
(6.8)
Q
ˆ k)Ψf g ) · dΓ(χ2n (|k|)ξ g 2 (k) − (Ψf g , ∇E u ˆ P Q
Q
− g(Ψf g , [a∗ (idunˆ ρx ) + a(idunˆ ρx )]Ψf g ) Q
Q
(6.9) (6.10)
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holds true. As the one-particle state Ψf g belongs to the form domain of all Q
operators in (6.8, 6.9, 6.10), this RHS is well defined. Since the dilation operator is unbounded, we must check that a regularized expression for the commutator i[H − EP , Dnuˆ ] in Eq. (4.2) is well defined and that, upon the removal of the regularization, the expectation value of that commutator in the state Ψf g corresponds to the right-hand side above, i.e. Q
(6.8, 6.9, 6.10). We show that, provided β is sufficiently large, the same strategy as implemented in [12] justifies this identity. Most of the arguments below, with the exception of the one in Sect. 6.2.5, are standard in literature. However, compared to the literature, our virial theorem has a little twist. This is due to the fact that we do not attempt to rule out any eigenvector, but merely an eigenvector with a certain regularity property. This is exploited in Lemma 4.1 and it is a crucial ingredient of the justification of the virial identity in Lemma 4.2. In Sect. 6.2.1, we prove that the expressions in (6.8, 6.9, 6.10) are well defined. In Sect. 6.2.2, we start the proof of the equality in Lemma 4.2. 6.2.1. Well-Definedness of the Terms (6.8, 6.9, 6.10). The operators ˆ k|) dΓ(χ2n (|k|)ξugˆ 2 (k)|
ˆ k) E · dΓ(χ2 (|k|)ξ g 2 (k) and ∇ n u ˆ P P
(6.11)
is surely are bounded by a (multiple of) H f . In fact, the operator ∇E P bounded if we restrict the total Hilbert space to the fibers P ∈ I. This restricg tion can be done since the function fQ has support in I. Since Ψf g ∈ Dom(H) ⇒ Ψf g ∈ Dom(H f ), Q
Q
(6.12)
the expressions (6.8) and (6.9) are well defined. Next, from the expression in (4.31) and the fact that ρ ∈ C1 , we have 2 1 1 (dunˆ ρx )(k) < ∞ (6.13) d3 k sup |k| x |x| + 1 and, hence, by a standard argument for bounding creation/annihilation operators, 1 1 u ˆ (6.14) |x| + 1 a(idn ρx ) (H f + 1) < ∞. Since Ψf g ∈ Dom(x) ∩ Dom(H f ) by Lemma 4.1 and (6.12), it follows that also Q
the expression (6.10) makes sense. 6.2.2. Virial Identity with a Regularized Dilation Operator. We introduce the regularized gradient := ∇k , ∇ (6.15) k 1 − Δk where the parameter > 0 will be eventually removed. Consequently, we also replaced by ∇ . define Dnuˆ, := dΓ(dunˆ, ) where dunˆ, corresponds to dunˆ with ∇ k k
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Since, thanks to the regularization, Dnuˆ, is bounded w.r.t. to H f , we deduce that Ψf g ∈ Dom(Dnuˆ, ) [cfr. (6.12)]. Q We claim that i((H − EP )Ψf g , Dnuˆ, Ψf g ) − i(Dnuˆ, Ψf g , (H − EP )Ψf g ) Q
Q
Q
Q
(6.16)
= (Ψf g , dΓ(i[|k|, dunˆ, ])Ψf g )
(6.17)
− i(Ψf g [EP , Dnuˆ, ]Ψf g )
(6.18)
Q
Q
Q
Q
− g(Ψf g , [a∗ (idunˆ, ρx ) + a(idunˆ, ρx )]Ψf g ) Q
(6.19)
Q
where the LHS makes sense since Ψf g ∈ Dom(Dnuˆ, ) and the RHS is obtained Q
by formal evaluation of the commutator [H − EP , Dnuˆ, ]. All terms on the RHS are well defined by similar (but easier) arguments as those in Sect. 6.2.1 (for example, note that [|k|, dunˆ, ] is a bounded operator). Nevertheless, the equality above requires a justification. In the case at hand, a pedestrian way to provide such a justification is to introduce cutoffs in x, k and N (the number operator), such that all operators involved are bounded, calculate the commutator and finally remove the cutoffs. Since (H − EP )Ψf g = 0 by assumption, the expression (6.16) vanishes. Q
Thus, it is sufficient to prove that the expressions (6.17, 6.18, 6.19) converge to (6.8, 6.9, 6.10), respectively, as tends to 0. These three convergence statements will be established in Sects. 6.2.4, 6.2.5 and 6.2.6, respectively. 6.2.3. Some Properties of the Regularized Dilation Operator. In this preparatory section, we state some estimates on
eiz·P Dnuˆ, e−iz·P − Dnuˆ,
(6.20)
that will be useful in taking the limit → 0. First, we remark that
ˆ, eiz·P Dnuˆ, e−iz·P = dΓ(dun, z ),
ˆ, i z ·k u dun, dnˆ, e−iz·k z := e
(6.21)
on the appropriate domain. Explicitly, ˆ, g (k) ˆ 1 (k · F (i∇ ˆ n (|k|) (6.22) + z) + F (i∇ + z) · k)ξ g (k)χ dun, ˆ u ˆ z = χn (|k|)ξu k k 2 and F is the family of R3 → R3 functions given by [cfr. (6.15)] F (y ) =
y . 1 + |y |2
(6.23)
ˆ, We define the vector operator dun, z such that it satisfies ˆ, u ˆ, u ˆ, z · dun, z = dn, z − dn .
Namely, ˆ , (dun, z )j
:=
ˆ 1 χn (|k|)ξugˆ (k) 2
l
⎛ ⎝kl
1
(6.24) ⎞
ˆ n (|k|) + h.c. ,l )j (i∇ + tz)⎠ ξ g (k)χ dt(∇F u ˆ k
0
(6.25)
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where the subscripts l and j label vector components. To check that (6.24) holds, we substitute the line integral 1 ,l (y + tz), dt ∇F
F,l (y + z) − F,l (y ) = z ·
l = 1, 2, 3,
(6.26)
0
into the explicit expression for (6.22), using functional calculus. We derive immediately the following properties: 1. The operator norms, ˆ, dun, z ,
and hence also dΓ(duˆ, ) n, z
1 (H f + 1)
ˆ kχ2n (|k|)ξugˆ 2 (k) ,
(6.27)
dΓ(kχ2n (|k|)ξ g 2 (k)) u ˆ
1 , (H f + 1)
(6.28)
are bounded uniformly in and z ∈ R3 . For the operators on the left (involvˆ, ing dun, y ) is y , ∇F,j ( z ), this follows from the fact that sup bounded. For the operators on the right, this is a trivial consequence of the momentum cutoff functions. 2. For each z, ˆ 2 g 2 (k). duˆ, strongly ˆ n, z −→ kχn (|k|)ξu
(6.29)
→0
ˆ, dΓ(dun, z)
1 (H f + 1)
strongly
−→ dΓ(kχ2n (|k|)ξugˆ 2 (k)) →0
1 . (H f + 1)
(6.30)
) and Dom(dΓ(∇ )) ∩ Ffin follows by This convergence on Dom(∇ k k ,j (y ) → yˆj , as → 0, pointwise in y . Convergence on all vectors then ∇F follows by using the uniform boundedness (6.27, 6.28) above. u ˆ 6.2.4. The Term [H f , Dn ]. In this section, we show that (6.17) converges to (6.8), as → 0. We derive 1 1 strongly ˆ (6.31) −→ dΓ |k|χ2n (|k|)ξugˆ 2 (k) dΓ(i[|k|, dunˆ, ]) f H + 1 →0 Hf + 1
in exactly the same way as we did to arrive at (6.30). That is, we first establish (using properties of F ) that sup i[|k|, dunˆ, ] < ∞,
), the operator i[|k|, duˆ, ] converges to and that, on the dense domain Dom(∇ n k ˆ Since Ψf g ∈ Dom(H f ), we conclude that |k|χ2n (|k|)ξugˆ 2 (k). Q ˆ Ψf g dΓ i[|k|, dunˆ, ] Ψf g −→ dΓ |k|χ2n (|k|)ξugˆ 2 (k) (6.32) Q
→0
Q
We have proven that the difference between (6.17) and (6.8) vanishes as → 0.
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u ˆ 6.2.5. The Term [EP , Dn ]. In this section, we show that (6.18) converges to (6.9), as → 0. We consider an extension of the function EP that is twice differentiable (see Sect. 3.1.2) and of compact support K (i.e., {P | |P | ∈ I} ⊂ K). We use the same symbol, EP , for the function extended to K, and we write ˆ z )eiz·P , EP = d3 z E( (6.33)
ˆ z ) is the Fourier transform of E (up to the prefactor (2π)−3/2 ). where E( P ˆ z ) belongs to Since EP is twice differentiable and of compact support, |z|2 E( 2 3 3 1 3 3 ˆ L (R ; d z) and, by Cauchy–Schwartz, E(z) is in L (R ; d z). Therefore, using functional calculus, we can write, ˆ z )(Ψf g , [eiz·P , Duˆ, ]Ψf g ). (Ψf g , [EP , Dnuˆ, ] Ψf g ) = d3 z E( (6.34) n Q
Q
Q
Q
Then we observe that, on e.g., the domain Dom(H f ), ˆ, i z ·P eiz·P Dnuˆ, − Dnuˆ, eiz·P = (eiz·P Dnuˆ, e−iz·P − Dnuˆ, )eiz·P = z · dΓ(dun, z )e
(6.35) ˆ, with the bounded operator dun, z , as defined in Sect. 6.2.3. We are now ready to compare (6.18) with (6.9):
ˆ k)Ψf g ) · dΓ(χ2 (|k|)ξ g 2 (k) (6.36) i(Ψf g , [EP , Dnuˆ, ]Ψf g ) − (Ψf g , ∇E n u ˆ P Q Q Q Q ˆ z )(Ψf g , dΓ(duˆ, ) − dΓ(kχ2n (|k|)ξ g2 (k) · zeiz·P Ψf g ) = i d3 z E( u ˆ n, z Q
= −i
(6.37) ˆ z )(xΨf g , [dΓ(duˆ, ) − dΓ(kχ2 (|k|)ξ g2 (k))]eiz·P Ψf g ) d3 z E( n u ˆ n, z Q
+i
Q
Q
(6.38) ˆ z )(Ψf g , [dΓ(duˆ, ) − dΓ(kχ2n (|k|)ξ g2 (k))]eiz·P xΨf g ). d3 z E( u ˆ n, z Q
Q
(6.39) The first equality follows by (6.34, 6.35, 6.24) and the fact that the Fourier transform sends differentiation into multiplication. To obtain the second equality, we used the canonical commutation relation
zeiz·P = [eiz·P , x],
(6.40)
which holds e.g., on Dom(x) ∩ Dom(H f ). ˆ z ) ∈ L1 (R3 ; d3 z), we can estimate (6.38) Since E( |(6.38)| ˆ z )| ≤ d3 z |E(
u ˆ, g2 (xΨf g , [dΓ(dn,z ) − dΓ(kχ2n (|k|)ξuˆ (k))]eiz·P Ψf g ) Q
Q
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For each z, the second factor vanishes as → 0 by (6.30) and the fact that Ψf g ∈ Dom(x) ∩ Dom(H f ). Hence, we conclude that (6.38) tends to zero as Q
tends to zero, by dominated convergence. Obviously, (6.39) can be treated in exactly the same way and hence we have proven that (6.37) vanishes as → 0. Hence, we have shown that the difference between (6.18) and (6.9) vanishes, as → 0. u ˆ ]. In this section, we prove that (6.19) converges 6.2.6. The Term [gφ(ρx ), Dn to (6.10) as ↓ 0. First, we note that 2 1 3 1 u ˆ, (dn ρx )(k) < ∞. (6.41) sup d k sup |k| x |x| + 1
This follows in the same way as (6.13), established in Sect. 6.2.1. Together, (6.41) and (6.13) imply that the operator norms of 1 1 a∗ (i(dunˆ, − dunˆ )ρx ) Hf + 1 |x| + 1 1 1 u ˆ ∗ (Rn, a(i(dunˆ, − dunˆ )ρx ) f ) := |x| + 1 H +1 u ˆ := Rn,
(6.42) (6.43)
are uniformly bounded in . We can now take advantage of the fact that Ψf g ∈ Dom(x) ∩ Dom(H f ) to write Q
(Ψf g , [a∗ (i(dunˆ, − dunˆ )ρx ) + a(i(dunˆ, − dunˆ )ρx )]Ψf g ) Q
Q
u ˆ = ((H f + 1)Ψf g , Rn, χKδ (|x| + 1)Ψf g ) Q
f
+ ((H +
(6.45)
Q
u ˆ 1)Ψf g , Rn, (|x| Q
+ 1)(1 − χKδ )Ψf g )
(6.46)
Q
u ˆ ∗ + ((|x| + 1)Ψf g , χKδ (Rn, ) (H f + 1)Ψf g ) Q
(6.47)
Q
u ˆ ∗ + ((1 − χKδ )(|x| + 1)Ψf g , (Rn, ) (H f + 1)Ψf g ) Q
(6.44)
Q
(6.48)
where χKδ = χKδ (x) is the characteristic function of a compact set Kδ ⊂ R3 , chosen such that the |(6.46)|, |(6.48)| are smaller than δ. This can be done by u ˆ and the fact that (χKδ − 1)(|x| + 1)Ψf g can the uniform bound on Rn, Q be made arbitrarily small by choosing Kδ to be sufficiently big. Moreover, for any compact K, 1 (6.49) lim d3 k [ sup |((idunˆ, − idunˆ )ρx )(k)|]2 = 0. →0 |k| x∈K u ˆ u ˆ ∗ This implies that χK Rn, , χK (Rn, ) and hence (6.45), (6.47) vanish, as → 0. Together, the bounds on (6.45), (6.47) and on (6.46), (6.48) prove that (6.44) vanishes in the limit → 0. Hence, the difference of (6.19) and (6.10) vanishes as ↓ 0.
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6.3. Proof of the Fiber Virial Identity in (5.50) The justification of the virial identity in (5.50) is largely analogous to that of the virial identity in Lemma 4.2. To avoid repetitive arguments, we only sketch the main strategy of the proof. First, one introduces a regularized dilation operator d1 ,κ and the correκ
sponding second quantized operator Db, := dΓb (d1 ,κ ). The operator d1 ,κ is 1 κ κ κ ,κ , with obtained from d 1 ,κ [see Eq. (3.20)] by replacing the gradient, ∇ k
κ
:= ∇k , ∇ k 1 − Δk
> 0.
(6.50)
Then one exploits the following properties: ) ∈ h, (i) On the dense subspace Dom(∇ k
i[|k|, d 1 ,κ ] → |k|χ2[1 ,κ] (|k|),
(6.51)
i[k, d1 ,κ ] → kχ2[1 ,κ] (|k|)
(6.52)
κ
κ
κ
κ
as → 0 (strong convergence on the whole of h follows than from ii) below). (ii) The operator norms [|k|, d1 ,κ ] , κ b 1 , dΓ (i[|k|, d1 ]) f κ ,κ (1 + H )
[k, d1 ,κ ] κ b dΓ (i[k, d1 ]) ,κ κ
1 f (1 + H )
(6.53) (6.54)
are bounded uniformly in . (iii)
lim
→0
(iv) the operator norm
d3 k
1 |(id1 ,κ − id κ1 ,κ )ρ(k)|2 = 0. κ |k|
b(id1 ρ) ,κ κ
1 f 1/2 (H + 1)
(6.55)
(6.56)
is uniformly bounded in . (v)
2 1 b(id 1 − id 1 ,κ )ρ(k)) ( κ ,κ f 1/2 κ (1 + H ) 1 ≤ d3 k |(id1 ,κ −id κ1 ,κ )ρ(k))|2 κ |k|
(6.57)
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[2] Angelescu, N., Minlos, R.A., Zagrebnov, V.A.: Lower spectral branches of a spin-boson model. J. Math. Phys. 49, 102105 (2008) [3] Bach, V., Chen, T., Fr¨ ohlich, J., Sigal, I.M.: The renormalized electron mass in non-relativistic quantum electrodynamics. J. Funct. Anal. 243(2), 426–535 (2007) [4] Chen, T.: Infrared renormalization in non-relativistic QED and scaling criticality. J. Funct. Anal. 254(10), 2555–2647 (2007) [5] Chen, T., Fr¨ ohlich, J.: Coherent infrared representations in nonrelativistic QED. In: Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday. Proc. Symp. Pure Math. AMS (2007) [6] Chen, T., Fr¨ ohlich, J., Pizzo, A.: Infraparticle scattering states in QED: II. Mass shell properties. J. Math. Phys. 50, 012103 (2009) [7] Chen, T., Fr¨ ohlich, J., Pizzo, A.: Infraparticle scattering states in QED: I. The Bloch-Nordsieck paradigm. Commun. Math. Phys. doi:10.1007/ s00220-009-0960-x [8] Erd¨ os, L.: Linear Boltzmann equation as the long time dynamics of an electron weakly coupled to a phonon field. J. Stat. Phys 107(5–6), 1043–1127 (2002) [9] Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schr¨ odinger Operators, with Applications to Quantum Mechanics and Global Geometry. Springer-Verlag, Berlin (1987) [10] Fr¨ ohlich, J.: On the infrared problem in a model of scalar electrons and massless, scalar bosons. Inst. Henri Poincar´e, Section Physique Th´eorique 19(1), 1–103 (1973) [11] Fr¨ ohlich, J.: Existence of dressed one electron states in a class of persistent models. Fortschritte der Physik 22, 159–198 (1974) [12] Fr¨ ohlich, J., Pizzo, A.: On the absence of excited eigenstates in QED. Commun. Math. Phys. 286(3), 803–836 (2009) [13] Fr¨ ohlich, J., Pizzo, A.: The renormalized electron mass in non-relativistic QED. Commun. Math. Phys. 294(3), 761–825 (2010) [14] Fr¨ ohlich, J., Griesemer, M., Sigal, I.M.: Spectral theory for the standard model of non-relativistic QED. Commun. Math. Phys. 283, 613–646 (2008) [15] Georgescu, V., G´erard, C.: On the virial theorem in quantum mechanics. Commun. Math. Phys. 208(2), 275–281 (1999) [16] Hasler, D., Herbst, I.: Absence of ground states for a class of translation invariant models of non-relativistic QED. Commun. Math. Phys. 279(3), 769–787 (2008) [17] Schach-Møller, J.: The translation invariant Nelson model: I. The bottom of the spectrum. Ann. H. Poincar´e 6(6), 1091–1135 (2005) [18] Pizzo, A.: One-particle (improper) states in Nelson’s massless model. Ann. H. Poincar´e 4(3), 439–486 (2003) [19] Pizzo, A.: Scattering of an infraparticle: the one particle sector in Nelson’s massless model. Ann. H. Poincar´e 6, 553–606 (2005) [20] Spohn, H.: The polaron at large total momentum. J. Phys. A 21, 1199–1212 (1988)
Vol. 11 (2010)
Absence of Embedded Mass Shells
Wojciech De Roeck Institut f¨ ur Theoretische Physik Universit¨ at Heidelberg Philosophenweg 19 69120 Heidelberg Germany e-mail: [email protected] J¨ urg Fr¨ ohlich Institute of Theoretical Physics ETH Z¨ urich 8093 Zurich Switzerland e-mail: [email protected] Alessandro Pizzo Department of Mathematics University of California Davis One Shields Avenue Davis CA 95616 USA e-mail: [email protected] Communicated by Christian G´erard. Received: March 3, 2010. Accepted: October 21, 2010.
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Ann. Henri Poincar´e 11 (2010), 1591–1627 c 2010 Springer Basel AG 1424-0637/10/081591-37 published online December 14, 2010 DOI 10.1007/s00023-010-0065-0
Annales Henri Poincar´ e
On a Waveguide with Frequently Alternating Boundary Conditions: Homogenized Neumann Condition Denis Borisov, Renata Bunoiu and Giuseppe Cardone Abstract. We consider a waveguide modeled by the Laplacian in a straight planar strip. The Dirichlet boundary condition is taken on the upper boundary, while on the lower boundary we impose periodically alternating Dirichlet and Neumann condition assuming the period of alternation to be small. We study the case when the homogenization gives the Neumann condition instead of the alternating ones. We establish the uniform resolvent convergence and the estimates for the rate of convergence. It is shown that the rate of the convergence can be improved by employing a special boundary corrector. Other results are the uniform resolvent convergence for the operator on the cell of periodicity obtained by the Floquet–Bloch decomposition, the two terms asymptotics for the band functions, and the complete asymptotic expansion for the bottom of the spectrum with an exponentially small error term.
1. Introduction During last decades, models of quantum wave guides attracted much attention by both physicists and mathematicians. It was motivated by many interesting mathematical phenomena of these models and also by the progress in the semiconductor physics, where they have important applications. Much efforts were exerted to study the influence of various perturbations on the spectral This work was partially done during the visit of D.B. to the University of Sannio (Italy) and of G.C. to LMAM of University Paul Verlaine of Metz (France). They are grateful for the warm hospitality extended to them. D.B. was partially supported by RFBR (09-0100530), by the Grants of the President of Russia for young scientists–doctors of sciences (MD-453.2010.1) and for Leading Scientific School (NSh-6249.2010.1), by Federal Task Program “Research and educational professional community of innovation Russia” (contract 02.740.11.0612), FCT (ptdc/mat/101007/2008), and by the project “Progetto ISA: Attivit` a di Internazionalizzazione dell’Universit` a degli Studi del Sannio”.
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properties of the wave guides. One of such perturbations is a finite number of openings coupling two lateral wave guides (see, for instance [7–9,12,15,18,19]) and such openings are usually called “windows”. If the coupled wave guides are symmetric, one can replace them by a single waveguide with the opening(s) modeled by the change in boundary condition (see [9,12,15]). The main phenomenon studied in [7–9,12,15,18,19]) is the appearance of new eigenvalues below the essential spectrum, which is stable with respect to windows. A close model was suggested in [3], where the number of openings was infinite. The waveguide was modeled by a straight planar strip, where the Dirichlet Laplacian was considered. On the upper boundary, the Dirichlet condition was imposed. On the lower boundary, the Neumann condition was settled on a periodic set, while on the remaining part of the boundary the Dirichlet condition is involved. In other words, on the lower boundary one had the alternating boundary conditions. The main assumption was the smallness of the sizes of Dirichlet and Neumann parts on the lower boundary. They were described by two parameters: the first one, ε, was supposed to be small, while the other, η = η(ε), could be either bounded or small. The main difference between the models studied in [3] and in [7–9,12, 15,18,19] is the influence of the perturbation on the spectral properties: while in the latter papers the essential spectrum remained unchanged and discrete eigenvalues appeared below its bottom, in [3] the spectrum was purely essential and had band structure. Moreover, it depended on the perturbation and, for example, the bottom of the spectrum moved as ε → +0. Assuming that ε ln η(ε) → −0
as ε → +0,
(1.1)
it was shown in [3] that the homogenized operator is the Laplacian with the previous boundary condition on the upper boundary, while the alternation on the lower boundary should be replaced by the Dirichlet one. More precisely, it was shown that the uniform resolvent convergence for the perturbed operator holds true and the rate of convergence was estimated. Other main results were the two terms asymptotics for first band functions of the perturbed operator and the complete two-parametric asymptotic expansion for the bottom of the spectrum. In the present paper, we consider a different case: we assume that the homogenized operator has the Neumann condition on the lower boundary, which is guaranteed by the condition ε ln η(ε) → −∞
as ε → +0.
(1.2)
We observe that this condition is not new, and it was known before that it implied the homogenized Neumann boundary condition for the similar problems in bounded domains, see ([13,14,16,17,20,24]). We obtain the uniform resolvent convergence for the perturbed operator and we estimate the rate of convergence. We also obtain similar convergence for the operator appearing on the cell of periodicity after Floquet decomposition and provide two-term asymptotics for the first band function. The last main result is the complete asymptotic expansion for the bottom of the spectrum.
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Similar results were obtained [3] under the assumption (1.1), and now we want to underline the main differences. We first observe that in [3] the estimate of the rate of convergence for the perturbed resolvent was obtained for the difference of the resolvents of the perturbed and homogenized operator and this difference was considered as an operator from L2 into W21 . In our case, to have a similar good estimate, we have to consider the difference not with the resolvent of the homogenized operator, but with that of an additional operator depending on boundary condition on an additional parameter μ = μ(ε) := −
1 → +0 ε ln η(ε)
as ε → +0.
(1.3)
Moreover, we also have to use a special boundary corrector, see Theorem 2.1. Omitting the corrector and estimating the difference of the same resolvents as an operator in L2 , we can still preserve the mentioned good estimate. Omitting the corrector or replacing the additional operator mentioned above by the homogenized one, one worsens the rate of convergence. At the same time, this rate can be improved partially by considering the difference of the resolvents as an operator in L2 and such situation was known to happen in the case of the operators with the fast oscillating coefficients (see [1,2,6,30,31,34–36,38,39]) and the references therein for further results). From this point of view, the results of the present paper are closer to the cited paper in contrast to the results of [3] and [29, Ch. III, Sect. 4.1]. One more difference to [3] is the asymptotics for the band functions and the bottom of the essential spectrum. The second term in the asymptotics for the band functions is not a constant, but a holomorphic in μ function. In fact, it is a series in μ and this is why the mentioned two-term asymptotics can be regarded as the asymptotics with more terms, see (2.8). Even more interesting situation occurs in the asymptotics for the bottom of the spectrum. Here the asymptotics contains just one first term, but the error estimate is exponential. The leading term depends on ε and μ holomorphically and can be represented as the series in ε with the holomorphic in μ coefficients. For the bounded domains, the complete asymptotic expansions for the eigenvalues in the case of the homogenized Neumann problem were constructed in [4,25]. These asymptotics were power in ε [25] with the holomorphic in μ coefficients [4]. At the same time, the error terms were powers in ε and the convergence of these asymptotic series was not proved. In our case, the first term in the asymptotics for the bottom of the essential spectrum is the sum of the asymptotic series analogous to those in [4,25]. In other words, we succeeded to prove that in our case, this series converges is holomorphic in ε and μ and gives the exponentially small error term that for singularly perturbed problems in homogenization is regarded as a strong result. Eventually, we point out that the technique we use is different: in addition to the boundary layer method [37] used also in [3], here we also have to employ the method of matching of the asymptotic expansions [27] and such combination was borrowed from [4,23–25]. We use this combination to construct the
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aforementioned corrector to obtain the uniform resolvent convergence. Similar correctors were also constructed in [13,20,24], but to obtain either weak or strong resolvent convergence. We also employ the same corrector in the combination of the technique developed in [21] for the analysis of the uniform resolvent convergence for thin domains. In conclusion, we describe briefly the structure of the paper. In the next section, we formulate precisely the problem and give the main results. The third section is devoted to the study of the uniform resolvent convergence. In the fourth section, we make the similar study for the operator appearing after the Floquet decomposition, and we also establish two-term asymptotics for the first band functions. In the last, fifth section, we construct the complete asymptotic expansion for the bottom of the spectrum.
2. Formulation of the Problem and the Main Results Let x = (x1 , x2 ) be Cartesian coordinates in R2 , and Ω := {x : 0 < x2 < π} be a straight strip of width π. By ε, we denote a small positive parameter, and η = η(ε) is a function satisfying the estimate 0 < η(ε) <
π . 2
We indicate by Γ+ and Γ− the upper and lower boundary of Ω, and we partition Γ− into two subsets (cf. Fig. 1), γε := {x : |x1 − επj| < εη, x2 = 0, j ∈ Z},
Γε := Γ− \γε .
The main object of our study is the Laplacian in L2 (Ω) subject to the Dirichlet boundary condition on Γ+ ∪ γε and to the Neumann one on Γε . We introduce this operator as the non-negative self-adjoint one in L2 (Ω) associated with the sesquilinear form hε [u, v] := (∇u, ∇v)L2 (Ω)
˚ 1 (Ω, Γ+ ∪ γε ), on W 2
˚ 1 (Q, S) indicates the subset of the functions in W 1 (Q) having zero where W 2 2 trace on the curve S. We denote the described operator as Hε . The aim of this paper is to study the asymptotic behavior of the resolvent and the spectrum of Hε as ε → +0.
Figure 1. Waveguide with frequently alternating boundary conditions
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Let H(μ) be the non-negative self-adjoint operator in L2 (Ω) associated with the sesquilinear form h(μ) [u, v] := (∇u, ∇v)L2 (Ω) + μ(u, v)L2 (∂Ω)
˚ 1 (Ω, Γ+ ), on W 2
where μ 0 is a constant. Reproducing the arguments of [5, Sect. 3], one can show that the domain of H(μ) consists of the functions in W22 (Ω) satisfying the boundary condition ∂u − μu = 0 on ∂x2
Γ− ,
u=0
on
Γ+ ,
(2.1)
and H(μ) u = −Δu.
(2.2)
By · L2 (Ω)→L2 (Ω) and · L2 (Ω)→W21 (Ω) we denote the norm of an operator acting from L2 (Ω) into L2 (Ω) and into W21 (Ω), respectively. Our first main result describes the uniform resolvent convergence for Hε . Theorem 2.1. Suppose (1.2). Then (Hε − i)−1 − (H(μ) − i)−1 L2 (Ω)→L2 (Ω) Cεμ| ln εμ|,
(2.3)
(Hε − i)−1 − (H(0) − i)−1 L2 (Ω)→W21 (Ω) Cμ1/2 ,
(2.4)
−1
(Hε − i)
− (H
(0)
−1
− i)
L2 (Ω)→L2 (Ω) Cμ,
(2.5)
where the constants C are independent of ε and μ, and μ = μ(ε) was defined in (1.3). There exists a corrector W = W (x, ε, μ) defined explicitly by (3.17) such that (Hε − i)−1 − (1 + W )(H(μ) − i)−1 L2 (Ω)→W21 (Ω) Cεμ| ln εμ|,
(2.6)
where the constant C is independent of ε and μ. (0) The spectrum of the operator H is purely essential and coincides with 4 , +∞ . By [RS1, Ch. VIII, Sect. 7, Ths. VIII.23, VIII.24] and Theorem 2.1 we have
1
(0) if λ ∈ Theorem 2.2. The spectrum of Hε converges to that of H 1 . Namely, 1 , +∞ , then λ ∈ σ(H ) for ε small enough. If λ ∈ , +∞ , then there ε 4 4 exists λε ∈ σ(Hε ) so that λε → λ as ε → +0. The convergence of the spectral projectors associated with Hε and H(0)
P(a,b) (Hε ) − P(a,b) (H(0) ) → 0,
ε → 0,
is valid for a < b. The operator Hε is periodic since the sets γε and Γε are periodic, and we employ the Floquet decomposition to study its spectrum. We denote επ , 0 < x2 < π , Ωε := x : |x1 | < 2 ˚ Γ± := ∂Ωε ∩ Γ± . ˚ γε := ∂Ωε ∩ γε , Γε := ∂Ωε ∩ Γε , ˚
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˚ε (τ ) we indicate the self-adjoint non-negative operator in L2 (Ωε ) associBy H ated with the sesquilinear form ∂u ∂v ∂ τ τ ∂ ˚ i − − + , hε (τ )[u, v] := u, i v ∂x1 ε ∂x1 ε ∂x2 ∂x2 L2 (Ωε ) L2 (Ωε ) ˚ 1 (Ωε , ˚ ˚ 1 (Ωε , ˚ on W Γ+ ∪ ˚ γε ), where τ ∈ [−1, 1). Here W Γ+ ∪ ˚ γε ) is the set 2,per 2,per 1 ˚ ˚ γε ) satisfying periodic boundary conditions of the functions in W2 (Ωε , Γ+ ∪ ˚ ˚ε (τ ) has a compact resolvent, on the lateral boundaries of Ωε . The operator H 1 since it is bounded as that from L2 (Ωε ) into W2 (Ωε ), and the space W21 (Ωε ) is ˚ε (τ ) consists of its compactly embedded into L2 (Ωε ). Hence, the spectrum of H ˚ discrete part only. We denote the eigenvalues of Hε (τ ) by λn (τ, ε) and arrange them in the ascending order with the multiplicities taking into consideration λ1 (τ, ε) λ2 (τ, ε) · · · λn (τ, ε) · · · By [3, Lemma 4.1] we know that σ(Hε ) = σe (Hε ) =
∞
{λn (τ, ε) : τ ∈ [−1, 1)},
n=1
where σ(·) and σe (·) indicate the spectrum and the essential spectrum of an operator. By Lε we denote the subspace of L2 (Ωε ) consisting of the functions independent of x1 , and we shall make use the decomposition L2 (Ωε ) = Lε ⊕ L⊥ ε , where L⊥ ε is the orthogonal complement to Lε in L2 (Ωε ). Let Qμ be the selfadjoint non-negative operator in Lε associated with the sesquilinear form du dv ˚ 1 ((0, π), {π}), , + μu(0)v(0) on W q[u, v] := 2 dx2 dx2 L2 (0,π) 2
d i.e., Qμ is the operator − dx 2 in L2 (0, π) with the domain consisting of the 2 2 functions in W2 (0, π) satisfying the boundary conditions
u(π) = 0,
u (0) − μu(0) = 0.
˚ε (τ ) and Our next results are on the uniform resolvent convergence for H two-term asymptotics for the first band functions. Theorem 2.3. Let |τ | < 1 − κ, where 0 < κ < 1 is a fixed constant and suppose (1.2). Then for sufficiently small ε the estimate −1 τ2 ˚ − Q−1 Cκ −1/2 (ε1/2 μ + ε) (2.7) Hε (τ ) − 2 μ ⊕ 0 ε L2 (Ωε )→L2 (Ωε )
holds true, where the constant C is independent of ε, μ, and κ.
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Theorem 2.4. Let the hypothesis of Theorem 2.3 holds true. Then given any N , for ε < 2κ 1/2 N −1 the eigenvalues λn (τ, ε), n = 1, . . . , N , satisfy the relations τ2 + Λn (μ) + Rn (τ, ε, μ), ε2 |Rn (τ, ε, μ)| Cκ −1/2 n4 ε1/2 μ, λn (τ, ε) =
(2.8)
where Λn (μ), n = 1, . . . , N , are the first N eigenvalues of Qμ , and the constant C is the same as in (2.7). The eigenvalues Λn (μ) solve the equation √ √ √ Λ cos Λπ + μ sin Λπ = 0, (2.9) are holomorphic with respect to μ, and 2 1 μ + O(μ2 ). Λn (μ) = n − +
2 π n − 12
(2.10)
Let θ(β) := −
+∞ j=1
1 . 2 n 4j − β(2j + 4j 2 − β)
(2.11)
It will be shown in Lemma 5.2 that the function θ(β) is holomorphic in β and its Taylor series is θ(β) = −
+∞ (2j − 1)!!ζ(2j + 1)
8j j!
j=1
β j−1 ,
(2.12)
where ζ is the Riemann zeta-function. Our last main result provides the asymptotic expansion for the bottom of the essential spectrum of Hε . Theorem 2.5. For ε small enough, the first eigenvalue λ1 (τ, ε) attains its minimum at τ = 0, inf
τ ∈[−1,1)
λ1 (τ, ε) = λ1 (0, ε).
(2.13)
The asymptotics −1
λ1 (0, ε) = Λ(ε, μ) + O(με−1/2 e−2ε
+ ε1/2 η 1/2 )
holds true, where Λ(ε, μ) is the real solution to the equation √ √ √ √ Λ cos Λπ + μ sin Λπ − ε3 μΛ3/2 θ(ε2 Λ) cos Λπ = 0
(2.14)
(2.15)
satisfying the restriction Λ(ε, μ) = Λ1 (μ) + o(1),
ε → 0.
(2.16)
The function Λ(ε, μ) is jointly holomorphic with respect to ε and μ and can be represented as the series Λ(ε, μ) = Λ1 (μ) + μ2
+∞ j=1
ε2j+1 K2j+1 (μ) + μ3
+∞ j=2
ε2j K2j (μ),
(2.17)
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where the functions Kj (μ) are holomorphic with respect to μ, and, in particular, K3 (μ) = −
Λ21 (μ) ζ(3) , 4 πΛ1 (μ) + μ + πμ2
K4 (μ) = 0, Λ31 (μ) 3ζ(5) , 64 πΛ1 (μ) + μ + πμ2 ζ(3)2 Λ31 (μ)(2π 2 Λ21 (μ) + 7πμΛ1 (μ) + 2π 2 μ2 Λ1 (μ) + 7μ2 + 7πμ3 ) K6 (μ) = 64 (πΛ1 (μ) + μ + πμ2 )3 4 Λ1 (μ) 5ζ(7) K7 (μ) = − , 512 πΛ1 (μ) + μ + πμ2 3ζ(3)ζ(5) K8 (μ) = 512 Λ41 (μ)(2π 2 Λ21 (μ) + 9πμΛ1 (μ) + 2π 2 μ2 Λ1 (μ) + 9μ2 + 9μ3 π) . × (πΛ1 (μ) + μ + πμ2 )3 (2.18) K5 (μ) = −
˚ε (0) reads as The asymptotic expansion for the associated eigenfunction of H follows, −1
˚ ε) − Ψ ˚ε W 1 (Ω ) = O(μe−2ε ψ(·, ε 2
+ εη 1/2 ),
(2.19)
˚ε is defined in (5.27). where the function Ψ Remark 2.6. All other coefficients of (2.17) can be determined recursively by substituting this series and (2.12) into (2.15), expanding then (2.15) in powers of ε, and solving the obtained equations with respect to Ki .
3. Uniform Resolvent Convergence for Hε In this section, we prove Theorem 2.1. Given a function f ∈ L2 (Ω), we denote uε := (Hε − i)−1 f,
u(μ) := (H(μ) − i)−1 f.
The main idea of the proof is to construct a special corrector W = W (x, ε, μ) with certain properties and to estimate the norms of vε := uε − (1 + W )u(μ) and u(μ) W . In fact, the function W reflects the geometry of the alternation of the boundary conditions for Hε , and this is why it is much simpler to estimate independently vε and u(μ) W than trying to get directly the estimate for uε − u(μ) and uε − u(0) . Next lemma is the first main ingredient in the proof of Theorem 2.1 and it shows how W is employed. Lemma 3.1. Let W = W (x, ε, μ) be an επ-periodic in x1 function belonging to C(Ω) ∩ C ∞ (Ω\{x : x2 = 0, x1 = ±εη + επn, n ∈ Z}) satisfying boundary conditions ∂W = −μ on Γε , (3.1) W = −1 on γε , ∂x2
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and having differentiable asymptotics θ± + O(ρ± ), r± → +0. (3.2) 2 Here (r± , θ± ) are polar coordinates centered at (±εη, 0) such that the values θ± = 0 correspond to the points of γε . Assume also that ΔW ∈ C(Ω). Then ˚ 1 (Ω, Γ+ ∪ γε ), and (1 + W )u(μ) belongs to W 2 1/2
W (x, ε, μ) = c± (ε, μ)r± sin
∇vε 2L2 (Ω) + ivε 2L2 (Ω) = (f, vε W )L2 (Ω) + (u(μ) ΔW, vε )L2 (Ω) −2i(u(μ) W, vε )L2 (Ω) − 2(W ∇u(μ) , ∇vε )L2 (Ω) −μ(u(μ) , W vε )L2 (Γε ) .
(3.3)
Proof. We write the integral identities for uε and u(μ) , (∇uε , ∇φ)L2 (Ω) + i(uε , φ)L2 (Ω) = (f, φ)L2 (Ω) for all φ ∈
˚ 1 (Ω, Γ+ W 2
(3.4)
∪ γε ), and
(∇u(μ) , ∇φ)L2 (Ω) + μ(u(μ) , φ)L2 (Γ− ) + i(u(μ) , φ)L2 (Ω) = (f, φ)L2 (Ω)
(3.5)
˚ 1 (Ω, Γ+ ). Employing the smoothness of W , (3.1), (3.2), and for all φ ∈ W 2 proceeding as in the proof of Lemma 3.2 in [3], we check that (1 + W )φ ∈ ˚ 1 (Ω, Γ+ ∪γε ), if φ belongs to the domain of Hε or H(μ) . Hence, (1+W )u(μ) ∈ W 2 ˚ W21 (Ω, Γ+ ∪ γε ). Thus, ˚21 (Ω, Γ+ ∪ γε ). (1 + W )vε ∈ W
(3.6)
We take φ = (1 + W )vε in (3.5), (∇u(μ) , ∇(1 + W )vε )L2 (Ω) + μ(u(μ) , (1 + W )vε )L2 (Γ− ) + i(u(μ) , (1 + W )vε )L2 (Ω) = (f, (1 + W )vε )L2 (Ω) , (∇u(μ) , (1 + W )∇vε )L2 (Ω) + i(u(μ) , (1 + W )vε )L2 (Ω) = (f, (1 + W )vε )L2 (Ω) − (∇u(μ) , vε ∇W )L2 (Ω) − μ(u(μ) , (1 + W )vε )L2 (Γ− ) , (∇(1 + W )u(μ) , ∇vε )L2 (Ω) + i((1 + W )u(μ) , vε )L2 (Ω) = (f, (1 + W )vε )L2 (Ω) − (∇u(μ) , vε ∇W )L2 (Ω) + (u(μ) ∇W, ∇vε )L2 (Ω) − μ(u(μ) , (1 + W )vε )L2 (Γ− ) . We deduct (3.4) with φ = vε from the last identity, ∇vε 2L2 (Ω) + ivε 2L2 (Ω) = −(f, W vε )L2 (Ω) + (∇u(μ) , vε ∇W )L2 (Ω) − (u(μ) ∇W, ∇vε )L2 (Ω) + μ(u(μ) , (1 + W )vε )L2 (Γ− ) .
(3.7)
We integrate by parts taking into consideration (3.1), (3.5), and (3.6), (∇u(μ) , vε ∇W )L2 (Ω) − (u(μ) ∇W, ∇vε )L2 (Ω)
∂W = (∇u(μ) , vε ∇W )L2 (Ω) + u(μ) v ε dx1 + (div u(μ) ∇W, vε )L2 (Ω) ∂x2 Γε
= 2(∇u
(μ)
, vε ∇W )L2 (Ω) − μ(u(μ) , vε )L2 (Γε ) + (u(μ) ΔW, vε )L2 (Ω) ,
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and (∇u(μ) , vε ∇W )L2 (Ω) = (∇u(μ) , ∇W vε )L2 (Ω) − (∇u(μ) , W ∇vε )L2 (Ω) = (f, W vε )L2 (Ω) − i(u(μ) , W vε )L2 (Ω) (μ) − μ(u(μ) , W vε )L2 (˚ , W ∇vε )L2 (Ω) . Γ− ) − (∇u
We substitute the obtained identities into (3.7) and this completes the proof. As it follows from (3.3), to prove the smallness of vε in W21 (Ω)-norm, it is sufficient to construct a function W satisfying the hypothesis of Lemma 3.1 so that the quantities W and ΔW are small in certain sense. This is why we introduce W as a formal asymptotic solution to the equation ΔW = 0
in Ω,
(3.8)
satisfying (3.1), (3.2) and other assumptions of Lemma 3.1. To construct such solution, we shall employ the asymptotic constructions from [4,25], based on the method of matching of asymptotic expansions [27] and the boundary layer method [37]. We also mention that similar approach was used in [24, Lemma 1] for constructing a different corrector. First we construct W formally, and after that we shall prove rigorously all the required properties of the constructed corrector. Denote ξ = (ξ1 , ξ2 ) = (j) (j) (j) (j) xε−1 , ς (j) = (ς1 , ς2 ), ς1 = (ξ1 − πj)η −1 , ς2 = ξ2 η −1 . Outside a small neighborhood of γε we construct W as a boundary layer W (x, ε, μ) = εμX(ξ). We pass to ξ in (3.8) and let η = 0 in the boundary conditions. It yields a boundary value problem for X, Δξ X = 0, ξ2 > 0,
∂X = −1, ∂ξ2
ξ ∈ Γ0 := {ξ : ξ2 = 0}
+∞
{(πj, 0)},
j=−∞
(3.9) where the function X should be π-periodic in ξ1 and decay exponentially as ξ2 → +∞. It was shown in [23] that the required solution to (3.9) is X(ξ) := Re ln sin(ξ1 + iξ2 ) + ln 2 − ξ2 . It was also shown that X ∈ C ∞ ({ξ : ξ2 0, ξ = (πj, 0), j ∈ Z}), and this function satisfies the differentiable asymptotics X(ξ) = ln |ξ − (πj, 0)|+ln 2 − ξ2 + O(|ξ − (πj, 0)|2 ), ξ → (πj, 0),
j ∈ Z. (3.10)
In view of the last identity we rewrite the asymptotics for X as ξ → (πj, 0) in terms of ς (j) ,
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εμX(ξ) = εμ (ln |ξ − (πj, 0)| + ln 2 − ξ2 ) + O(εμ|ξ − (πj, 0)|2 ) (j) = −1 + εμ ln |ς (j) | + ln 2 − εμης2 + O(εμη 2 |ς (j) |2 ).
1601
(3.11)
In accordance with the method of matching of asymptotic expansions it follows from the obtained identities that in a small neighborhood of each interval of γε we should construct W as an internal layer, (j)
W (x, ε, μ) = −1 + εμWin (ς (j) ),
(3.12)
where (j)
Win (ς (j) ) = ln |ς (j) | + ln 2 + o(1),
ς (j) → +∞.
(3.13)
We substitute (3.12) into (3.8), (3.1), which leads us to the boundary value (j) problem for Win , (j)
Δς (j) Win = 0, (j)
Win = 0,
ς (j) ∈ γ 1 ,
(j)
ς2
(j) ∂Win (j) ∂ς2
γ 1 := {ς : |ς1 | < 1, ς2 = 0},
> 0, ς (j) ∈ Γ1 ,
= 0,
(3.14)
Γ1 := Oς1 \γ 1 .
It was shown in [23] that the problem (3.13), (3.14) is solvable and (j) Win (ς (j) ) = Y (ς (j) ), Y (ς) := Re ln(z + z 2 − 1), z = ς1 + iς2 , (3.15) √ where the branch of the root is fixed by the requirement 1 = 1. It was also shown that Y (ς) = ln |ς| + ln 2 + O(|ς|−2 ),
ς → ∞.
(3.16)
(j)
As it follows from the last asymptotics, the term −εμς2 in (3.11) is not matched with any term in the boundary layer. At the same time, it was found in [4,24,25] that such terms should be either matched or cancelled out to obtain a reasonable estimate for the error terms. This is also the case in our problem. In contrast to [4,24,25], to solve this issue we shall not construct additional terms in W , but employ a different trick. Namely, we add the function εμξ2 to the boundary layer and add also −μx2 as the external expansion. It changes neither equations nor boundary conditions for W but allows us to cancel out the mentioned term in (3.11). The final form of W is as follows, W (x, ε, μ) = −μx2 + εμ(X(ξ) + ξ2 )
+∞
1 − χ1 |ς (j) |η α
j=−∞
+
+∞
χ1 |ς (j) |η α
−1 + εμY (ς (j) ) ,
(3.17)
j=−∞
where α ∈ (0, 1) is a constant, which will be chosen later, and χ1 = χ1 (t) is an infinitely differentiable cut-off function taking values in [0, 1], being one as t < 1, and vanishing as t > 3/2. It can be easily seen that the sum and the product in the definition of (3.17) are always finite.
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Let us check that the function W satisfies the hypothesis of Lemma 3.1. By direct calculations we check that the function W is επ-periodic with respect to x1 , belongs to C(Ω) ∩ C ∞ (Ω\{x : x2 = 0, x1 = ±εη + επn, n ∈ Z}), and satisfies (3.2). The boundary condition on γε in (3.1) is obviously satisfied. Taking into consideration the boundary conditions (3.9), (3.13), we check +∞ ∂X ∂W = −μ + εμ 1 − χ1 (|ς (j) |η α ) 0 +1 ∂x2 x∈Γε ∂ξ2 ξ∈Γ j=−∞ +∞ ∂Y + εμ χ1 |ς (j) |η α = −μ, (j) ∂ς2 ς (j) ∈Γ1 j=−∞ i.e., the boundary condition on Γε in (3.1) is satisfied, too. Let us calculate ΔW . To do it, we employ Eqs. (3.9), (3.13), +∞
ΔW (x) = 2
(j) ∇x χ1 |ς (j) |η α · ∇x Wmat (x, ε, μ)
j=−∞ +∞
+
(j) Wmat (x, ε, μ)Δx χ1 |ς (j) |η α ,
(3.18)
j=−∞
(j) Wmat (x, ε, μ) = −1 + εμ Y (ς (j) ) − X(ξ) − ξ2 . It follows from the definition of ξ, ς (j) , χ1 , X, Y , and the last formula that ΔW ∈ C ∞ (Ω). Thus, we can apply Lemma 3.1. To estimate the right hand side of (3.3), we need two auxiliary lemmas. Given any δ ∈ (0, π/2), denote Ωδ :=
+∞
Ωδj ,
Ωδj := {x : |x − (πj, 0)| < εδ} ∩ Ω.
j=−∞
Lemma 3.2. For any u ∈ W21 (Ω) and any δ ∈ (0, π/4) the inequality uL2 (Ωδ ) Cδ | ln δ|1/2 + 1 uW21 (Ω)
(3.19)
holds true, where the constant C is independent of δ and u. Proof. We begin with the formulas u2L2 (Ωδ ) =
+∞ j=−∞
u2L2 (Ωδ ) , j
u2L2 (Ωδ ) j
|u(x)| dx = ε 2
= Ωδj
|u(εξ)|2 dξ
2
= ε2 |ξ−(πj,0)|<δ, ξ2 >0
|ξ−(πj,0)|<δ, ξ2 >0
|χ2 (ξ − (πj, 0))u(εξ)|2 dξ,
(3.20)
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where χ2 = χ2 (ξ) is an infinitely differentiable function being one as |ξ| < δ and vanishing as |ξ| > π/3. We also suppose that the functions χ2 , χ2 are bounded uniformly in ξ and δ. Hence, ˚ 1 (Π1 , ∂Π1 ), Π1 := ξ : |ξ1 − πj| < π , 0 < ξ2 < 1 . χ2 (· − (πj, 0))u ∈ W 2 j j j 2 By [28, Lemma 3.2], we obtain
|∇ξ χ2 u|2 + |χ2 u|2 dξ |χ2 u|2 dξ Cε2 δ 2 (| ln δ| + 1) ε2 Π1j
|ξ−(πj,0)|<δ, ξ2 >0
Cε2 δ 2 (| ln δ| + 1) ∇ξ uL2 (Π1j ) + uL2 (Π1j ) Cδ 2 (| ln δ| + 1)u2W 1 ({x:|x1 −επj|<επ/2,0<x2 <π}) , 2
where the constants C are independent of j, ε, δ, μ, and u. We substitute these inequalities into (3.20) and arrive at (3.19). Lemma 3.3. For any u ∈ W22 (Ω) and any δ ∈ (0, π/2) the inequality γεδ := {x : |x1 − επj| < εδ, x2 = 0},
uL2 (γεδ ) Cδ 1/2 uW22 (Ω) ,
holds true, where the constant C is independent of ε, δ, and u. Proof. It is clear that uL2 (γεδ ) =
+∞ j=−∞
uL2 (γε,j δ ),
δ γε,j := {x : |x1 − επj| < εδ, x2 = 0}. (3.21)
It follows from the definition of χ2 (see the proof of Lemma 3.2) that
2 x1 2 − πj u(x1 , 0) dx1 . (3.22) uL2 (γ δ ) = χ 2 ε,j ε δ γε,j
Since χ2
x
1
ε
x1
− πj u(x1 , 0) = επj− επ 2
∂ x1 χ2 − πj u(x1 , 0) dx1 , ∂x1 ε
by the Cauchy–Schwartz inequality we get ∂ x1 − πj u(x1 , 0) χ2 ∂x1 ε x x ∂u 1 1 = χ2 − πj − πj u(x1 ), (x1 , 0) + ε−1 χ2 ε ∂x1 ε 2 x 1 − πj, 0 u(x1 , 0) χ2 ε⎛ ⎞ 2
⎟ ⎜ ∂u (x1 , 0) dx1 + ε−1 C ⎝ε |u(x1 , 0)|2 dx1 ⎠, ∂x1 γε,j
γε,j
γε,j
επ , x2 = 0 , := x : |x1 − επj| < 2
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where the constants C are independent of j, ε, δ, and u. The last estimate and (3.22) imply ∂u 2 + u2L2 (γε,j ) , u2L2 (γ δ ) Cδ ε,j ∂x1 L2 (γε,j ) where the constant C is independent of j, ε, δ, and u. We substitute the obtained inequality into (3.21) and employ the standard embedding of W22 (Ω) into W21 (Γ− ) that completes the proof. Lemma 3.4. The estimates |ΔW | Cε−1 μ(1 + η 4α−2 ), |W | Cεμ(| ln δ| + 1),
x ∈ Ω,
x ∈ Ω\Ωδ ,
(3.23) 3 α π η <δ< , 2 2
(3.24)
3 α π η <δ< , (3.25) 2 2 are valid, where the constants C are independent of ε, μ, η, δ, and x. |W | C, x ∈ Ωδ ,
Proof. Since W is επ-periodic with respect to x1 , it is sufficient to prove the estimates only for |x1 | < επ/2, 0 < x2 < π. It follows directly from the definition of X, Y , and (3.13), (3.16) that for any δ ∈ (0, π/2) π |X(ξ)| C (| ln δ| + 1), |ξ1 | < , ξ2 > 0, |ξ| δ,
2
|Y (ς)| C | ln δη −1 | + 1 C | ln δ| + ε−1 μ−1 , |ς| δη −1 , where the constants C are independent of ε, μ, η, δ, and x. These estimates and (3.17) imply (3.24), (3.25). It follows from the definition of χ1 that ΔW is non-zero only as 3 η −α < |ς (1) | < η −α . 2 For the corresponding values of x due to (3.13), (3.15) the differentiable asymptotics (1) Wmat (x, ε, μ) = O εμ(|ς (1) |−2 + |ξ|2 ) , η −α < |ς (1) | <
3 −α η , 2
3 1−α η , 2 holds true. Hence, for the same values of ξ and ς (1)
(1) Wmat = O εμ(η 2α + η 2−2α ) ,
(1) ∇x Wmat = O μ(η −1 |ς (1) |−3 + |ξ|) = O μ(η 1−α + η 3α−1 ) . η 1−α < |ξ| <
Substituting the identities obtained into (3.18) and taking into consideration the relations ∇x χ1 |ς (j) |η α = O(ε−1 η α−1 ), Δx χ1 |ς (j) |η α = O(ε−2 η 2α−2 ), we arrive at (3.23).
Vol. 11 (2010)
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Let us estimate the right hand side of (3.3). We have |(f, W vε )L2 (Ω) | f L2 (Ω) W vε L2 (Ω) , Let δ ∈
3
2η
α π ,2
W vε 2L2 (Ω) = W vε 2L2 (Ω\Ωδ ) + W vε 2L2 (Ωδ ) .
(3.26)
. Applying Lemma 3.2 and using (3.24), (3.25), we have
vε W 2L2 (Ω\Ωδ ) Cε2 μ2 (| ln δ|2 + 1)vε 2L2 (Ω\Ωδ ) , vε W 2L2 (Ωδ ) Cδ 2 (| ln δ| + 1)vε 2W 1 (Ω) .
(3.27)
2
Here and till the end of this section we indicate by C various non-essential constants independent of ε, μ, η, δ, x, vε , u(μ) , and f . The inequalities (3.27) yield |(f, vε W )L2 (Ω) | C εμ| ln δ| + δ| ln δ|1/2 + δ vε W21 (Ω) f L2 (Ω) . (3.28) It follows from the definition of u(μ) that u(μ) W22 (Ω) Cf L2 (Ω) .
(3.29)
Taking into consideration this inequality, we proceed in the same way as in (3.26), (3.27), (3.28), u(μ) W L2 (Ω) C(εμ| ln δ| + δ| ln δ|1/2 + δ)u(μ) W21 (Ω) C(εμ| ln δ| + δ| ln δ|1/2 + δ)f L2 (Ω) , W ∇u
(μ)
L2 (Ω) C(εμ| ln δ| + δ| ln δ|
1/2
+ δ)u
(μ)
W22 (Ω)
C(εμ| ln δ| + δ| ln δ|
+ δ)f L2 (Ω) , , W ∇vε )L2 (Ω) 1/2
(μ) (u , W vε )L
2 (Ω)
u
(μ)
+ (∇u
(μ)
(3.30)
(3.31)
W L2 (Ω) vε L2 (Ω) + W ∇u(μ) L2 (Ω) ∇vε L2 (Ω)
C(εμ| ln δ| + δ| ln δ|1/2 + δ)f L2 (Ω) vε W21 (Ω) .
(3.32)
Employing (3.23) instead of (3.24), (3.25), and applying then Lemma 3.2 with δ = η α , we get u(μ) ΔW L2 (Ω) = u(μ) ΔW L2 (Ω2ηα ) Cη α ε−3/2 μ1/2 (1 + η 4α−2 )u(μ) W21 (Ω) Cη α ε−3/2 μ1/2 (1 + η 4α−2 )f L2 (Ω) .
(3.33)
Using (3.24), (3.25), (3.28), Lemma 3.3 with δ = δ ∈ (η α , π/2), the embedding of W22 (Ω) in W21 (Γ− ), and proceeding as in (3.26), (3.27), (3.28), we obtain (μ) (u , W vε )L (Γ ) u(μ) W L (Γ ) vε L (Γ ) Cu(μ) W L (Γ ) vε W 1 (Ω) , 2 ε 2 ε 2 − 2 ε 2 u(μ) W 2L2 (Γε ) = u(μ) W 2L
δ 2 (Γε \γε )
δ 2 (γε )
(μ) 2 2 Cε μ (| ln δ| + + C δu W2 (Ω) (3.34) 2 + 1) f 2 C δ + ε2 μ2 (| ln δ| L2 (Ω) , (μ) + 1) f L (Ω) . (u , W vε )L (Γ ) C δ1/2 + εμ(| ln δ| 2 ε 2 2 2
2
+ u(μ) W 2L
1)u(μ) 2L2 (Γε )
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Ann. Henri Poincar´e
Let α ∈ (1/2, 1). The last obtained estimate, (3.28), (3.32), (3.33), and (3.3) yield + μδ1/2 )f L (Ω) vε W 1 (Ω) , vε 2W 1 (Ω) C(δ| ln δ|1/2 + εμ| ln δ| + εμ2 | ln δ| 2 2 2
and it is assumed here that → +0 as ε → +0. η α < δ < π/2, η α < δ < π/2, δ = δ(ε) → +0, δ = δ(ε) Thus, taking δ = εμ, δ = ε2 μ2 , we get vε W21 (Ω) Cεμ| ln εμ|f L2 (Ω) , and it proves (2.6). We take δ = εμ in (3.30) and employ (2.6), (Hε − i)−1 f − (H(μ) − i)−1 f L2 (Ω) = uε − u(μ) L2 (Ω) uε − (1 + W )u(μ) L2 (Ω) + u(μ) W L2 (Ω) Cεμ| ln εμ|f L2 (Ω) , which proves (2.3). Lemma 3.5. The estimate ∇(u(μ) W )L2 (Ω) Cμ1/2 f L2 (Ω)
(3.35)
holds true. Proof. We integrate by parts employing (3.1), (3.2), (2.1), (2.2), ∇(u(μ) W )2L2 (Ω) ∂ (μ) =− u W, u(μ) W − Δ(u(μ) W ), u(μ) W ∂x2 L2 (Ω) L (Γ )
2 − ∂W = −μu(μ) W 2L2 (Γ− ) + |u(μ) |2 dx1 + μ(u(μ) , u(μ) W )L2 (Γε ) ∂x2 γε −(W Δu(μ) , W u(μ) )L2 (Ω) − 2 W ∇u(μ) , u(μ) ∇W L2 (Ω) − u(μ) ΔW, u(μ) W . L2 (Ω)
We take the real part of this identity, ∇(u(μ) W )2L2 (Ω)
|u(μ) |2
= μ(u(μ) , u(μ) W )L2 (Γε ) + γε
∂W dx1 ∂x2
−μu(μ) W 2L2 (Γ− ) − Re(W Δu(μ) , W u(μ) )L2 (Ω) −2 Re W ∇u(μ) , u(μ) ∇W − u(μ) ΔW, u(μ) W L2 (Ω)
L2 (Ω)
.
(3.36)
Vol. 11 (2010)
On a Waveguide with Frequently Alternating Boundary
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Let us calculate the fifth term in the right hand side of the last equation. We integrate by parts employing (2.1), 2 Re W ∇u(μ) , u(μ) ∇W L2 (Ω)
1 = ∇W 2 · ∇|u(μ) |2 dx 2 Ω
1 1 2 ∂ (μ) 2 =− W |u | dx1 − W 2 Δ|u(μ) |2 dx 2 ∂x2 2 Γ−
= −μu
Ω
(μ)
W 2L2 (Γ− )
− Re(W u
(μ)
, W Δu(μ) )L2 (Ω)
−W ∇u(μ) 2L2 (Ω) . We substitute the last identity into (3.36),
∂W dx1 ∂x2 γε (μ) 2 +W ∇u L2 (Ω) − u(μ) ΔW, u(μ) W
∇(u(μ) W )2L2 (Ω) = μ(u(μ) , u(μ) W )L2 (Γε ) +
|u(μ) |2
L2 (Ω)
.
(3.37)
Taking δ = εμ in (3.31), we get W ∇u(μ) L2 (Ω) Cεμ| ln εμ|f L2 (Ω) . It follows from (3.30) with δ = εμ and (3.33) that (μ) (u ΔW, u(μ) W )L (Ω) Cη α ε−1/2 μ3/2 | ln εμ|f 2 L2 (Ω) , 2
(3.38) α ∈ (1/2, 1). (3.39)
Employing (3.17), (3.15), by direct calculations we check that
+∞ ∂W ∂W |u(μ) |2 dx1 = |u(μ) |2 dx1 ∂x2 ∂x2 j=−∞ γε
γε,j
+∞
= εμ
|u(μ) |2
j=−∞γ ε,j
and
|u(μ) |2 γε,j
∂ x1 − επj dx1 , arcsin ∂x1 εη
∂ x1 − επj dx1 arcsin ∂x1 εη
επj |u
=
∂ | ∂x1
επj−εη επj+εη
|u(μ) |2
+ επj
(μ) 2
∂ ∂x1
π x1 − επj + arcsin εη 2 arcsin
π x1 − επj − εη 2
dx1 dx1 ,
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= π|u
(μ)
2
(επj, 0)| + γε,j
×
Ann. Henri Poincar´e
π x1 − επj − sgn(x1 − επj) arcsin εη 2
∂ |u(μ) |2 dx1 , ∂x1
where π|u
(μ)
1 (επj, 0)| = ε
επj
2
επ(j−1)
∂ (x1 − επ(j − 1))|u(μ) |2 dx1 . ∂x1
Thus, in view of the embedding of W22 (Ω) into W21 (Γ− ) and (3.29)
επj +∞ ∂ ∂W (μ) 2 |u(μ) |2 dx1 μ (x1 − επ(j − 1))|u | dx1 ∂x ∂x 2 1 j=−∞ γε
επ(j−1)
+∞ ∂ (μ) 2 2 + εμπ ∂x1 |u | dx1 Cμf L2 (Ω) . j=−∞
γεj
We substitute the obtained estimate, (3.34) with δ = ε2 μ2 , (3.38), (3.39) into (3.37) and arrive at (3.35). The proven lemma and (2.6), (3.30) with δ = εμ imply (Hε − i)−1 − (H(μ) − i)−1 L2 (Ω)→W21 (Ω) C1 μ1/2 .
(3.40)
The resolvent (H(μ) − i)−1 is obviously analytic in μ and thus (H(μ) − i)−1 − (H(0) − i)−1 L2 (Ω)→W21 (Ω) Cμ. This inequality, (3.40), and (2.3) yield (2.4), (2.5).
˚ε (τ ) 4. Uniform Resolvent Convergence for H This section is devoted to the proof of Theorems 2.3, 2.4. The proof of the first theorem is close in spirit to that of Theorem 2.3 in [3]. The difference is that here we employ the corrector W as we did in the previous section. This is why an essential modification of the proof of Theorem 2.3 in [3] is needed. We begin with several auxiliary lemmas. The first one was proved in [3], see Lemma 4.2 in this paper. Lemma 4.1. Let |τ | < 1 − κ, where 0 < κ < 1, and 2 −1 ˚ε (τ ) − τ Uε = H f, f ∈ L2 (Ωε ). ε2
Vol. 11 (2010)
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Then Uε L2 (Ωε ) 4f L2 (Ωε ) , ∂U ε 2f L2 (Ωε ) , ∂x2 L2 (Ωε ) ∂U 2 ε 1/2 f L2 (Ωε ) . ∂x1 L2 (Ωε ) κ
(4.1)
If, in addition, f ∈ L⊥ ε , then ε ε f L2 (Ωε ) . (4.2) Uε L2 (Ωε ) 1/2 f L2 (Ωε ) , ∇Uε L2 (Ωε ) 2κ κ It was also shown in [3] in the proof of the last lemma that for any 1 ˚ u ∈ W2,per (Ωε , ˚ Γ+ ) and |τ | 1 − κ ∂ ∂u 2 τ τ2 2 − − 2 u2L2 (Ωε ) κ , u i ∂x1 ε ε ∂x1 L2 (Ωε ) L2 (Ωε ) (4.3) ∂u 1 uL2 (Ω) . ∂x2 L2 (Ωε ) 2 Lemma 4.2. Let F ∈ L2 (0, π). Then |(Q−1 μ F )(0)| 5F L2 (0,π) . Proof. We can find Q−1 μ F explicitly
π 1 x2 − π −1 (1 + μ(t − π)) F (t) dt. (Qμ F )(x2 ) = − |x2 − t| − π + 2 1 + πμ 0
Hence, by the Cauchy–Schwartz inequality
π 1 −1 |(Qμ F )(0)| (2π − t)|F (t)| dt 5F L2 (0,π) , 2(1 + πμ) 0
that completes the proof. Proof of Theorem 2.3. Let f ∈ L2 (Ωε ), f = 1 Fε (x2 ) = επ
Fε +fε⊥ ,
where Fε ∈
Lε , fε⊥
επ 2
fε (x) dx1 , − επ 2
(4.4)
επFε 2L2 (0,π) + fε⊥ 2L2 (Ωε ) = f 2L2 (Ωε ) . Then
2 ˚ε (τ ) − τ H ε2
−1 f=
∈
L⊥ ε ,
2 ˚ε (τ ) − τ H ε2
−1
2 ˚ε (τ ) − τ Fε + H ε2
−1
fε⊥ .
By (4.2), (4.4), we obtain 2 −1 ε ε ⊥ ˚ε (τ ) − τ H fε 1/2 fε⊥ L2 (Ωε ) 1/2 f L2 (Ωε ) . ε2 κ κ L2 (Ωε )
(4.5)
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We denote
Uε :=
2 ˚ε (τ ) − τ H 2 ε
−1 Fε ,
Ann. Henri Poincar´e
Uε(μ) := Q−1 μ Fε ,
Vε (x) := Uε (x) − Uε(μ) (x) − Uε(μ) (0)W (x, ε, μ)χ1 (x2 ), where, we remind, the function χ1 was introduced in the third section. In view ˚ 1 (Ωε , ˚ of (3.1) and the definition of Uε the function Vε belongs to W Γ+ ∪˚ γε ). 2,per (μ)
We write the integral identities for Uε and Uε , τ2 ˚ hε (τ )[Uε , φ] − 2 (Uε , φ)L2 (Ωε ) = (Fε , φ)L2 (Ωε ) ε 1 ˚ ˚ for all φ ∈ W2,per (Ωε , Γ+ ∪ ˚ γε ), and (μ) dUε dφ , + μUε(μ) (0)φ(0) = (Fε , φ)L2 (0,π) dx2 dx2
(4.6)
(4.7)
L2 (0,π)
˚ 1 (Ωε , ˚ ˚ 1 ((0, π), {π}). Given any φ ∈ W Γ+ ), for a.e. x1 ∈ for all φ ∈ W 2 2,per 1 ˚ (−επ/2, επ/2) we have φ(x1 , ·) ∈ W2 ((0, π), {π}). We take such φ in (4.7) and integrate it over x1 ∈ (−επ/2, επ/2), (μ) dUε ∂φ , + μ(Uε(μ) , φ)L2 (˚ Γ− ) = (Fε , φ)L2 (Ωε ) . dx2 ∂x2 L2 (Ωε )
(μ)
The function Uε
is independent of x1 , and hence τ ∂ τ ∂ (μ) − − i Uε , i φ ∂x1 ε ∂x1 ε L2 (Ωε ) τ ∂ τ =− − Uε(μ) , i φ ε ∂x1 ε L2 (Ωε )
τ 2 (μ) (U , φ)L2 (Ωε ) . ε2 ε The sum of two last equations is as follows, (μ) τ ∂ τ ∂Uε ∂φ ∂ (μ) − − + , i Uε , i φ ∂x1 ε ∂x1 ε ∂x2 ∂x2 L2 (Ωε ) =
2
L2 (Ωε )
τ (U (μ) , φ)L2 (Ωε ) + μ(Uε(μ) , φ)L2 (˚ (4.8) Γ− ) = (Fε , φ)L2 (Ωε ) ε2 ε We let φ = Vε in (4.6), (4.8) and take the difference of these two equations, τ ∂ τ ∂ − − i (Uε − Uε(μ) ), i Vε ∂x1 ε ∂x1 ε L2 (Ωε ) 2 ∂ ∂V τ ε + (Uε − Uε(μ) ), − (Uε − Uε(μ) , Vε )L2 (Ωε ) ∂x2 ∂x2 L2 (Ωε ) ε2 −
= μ(Uε(μ) , Vε )L2 (˚ Γ− ) .
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(μ)
We represent Uε − Uε as Vε + Uε (0)W χ1 and substitute it into the last equation, 2 ∂Vε 2 τ2 i ∂ − τ Vε + − 2 Vε 2L2 (Ωε ) ∂x1 ε ∂x2 L2 (Ωε ) ε L2 (Ωε ) ∂ τ ∂ τ (μ) (μ) = μ(Uε , Vε )L2 (˚ i − − W χ1 , i Vε Γε ) − Uε (0) ∂x1 ε ∂x1 ε L2 (Ωε ) 2 ∂W χ1 ∂Vε τ − Uε(μ) (0) , − U (μ) (0)(W χ1 , Vε )L2 (Ωε ) ∂x2 ∂x2 L2 (Ωε ) ε2 ε ∂W χ 2iτ 1 = Uε(μ) (0) μ(W, Vε )L2 (˚ , Vε . Γε ) −(∇W χ1 , ∇Vε )L2 (Ωε ) − ε ∂x1 L2 (Ωε ) (4.9) We integrate by parts employing (3.1), ∂Vε 2iτ 2iτ ∂W χ1 , Vε = , − W, χ1 ε ∂x1 ε ∂x1 L2 (Ωε ) L2 (Ωε ) and μ(W, Vε )L2 (˚ Γε ) − (∇(W χ1 ), ∇Vε )L2 (Ωε ) ∂W = μ(W, Vε )L2 (˚ , Vε + (Δ(W χ1 ), Vε )L2 (Ωε ) Γε ) + ∂x2 L2 (˚ Γε ) = (ΔW χ1 , Vε )L2 (Ωε ) . Together with (4.9) it yields 2 ∂Vε 2 τ2 i ∂ − τ Vε + − Vε 2L2 (Ωε ) ∂x1 ∂x2 ε ε2 L2 (Ωε ) L2 (Ωε ) 2iτ ∂Vε (μ) = Uε (0) (Δ(W χ1 ), Vε )L2 (Ωε ) + . W χ1 , ε ∂x1 L2 (Ωε ) It follows from Lemmas (4.2) and (4.4) that |Uε(μ) (0)| 5πε−1/2 f L2 (Ωε ) . Hence, we can estimate the right hand side of (4.10) as follows, 2iτ ∂Vε (μ) W χ1 , Uε (0) (Δ(W χ1 ), Vε )L2 (Ωε ) + ε ∂x1 L2 (Ωε ) 5πε−1/2 f L2 (Ωε ) Δ(W χ1 )L2 (Ωε ) Vε L2 (Ωε ) +2ε
−1
∂Vε W χ1 L2 (Ωε ) ∂x1
L2 (Ωε )
(4.10)
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1 50π 2 ε−1 Δ(W χ1 )2L2 (Ω) f 2L2 (Ωε ) + Vε 2L2 (Ωε ) 8 ∂Vε 2 +25π 2 κ −1 ε−3 W 2L2 (Ωε ) f 2L2 (Ωε ) + κ . ∂x1 L2 (Ωε ) We substitute this inequality and (4.3) into (4.10), ∂Vε 2 1 κ + Vε 2L2 (Ωε ) 50π 2 ε−1 f 2L2 (Ωε ) Δ(W χ1 )2L2 (Ωε ) ∂x1 4 L2 (Ωε ) ∂Vε 2 1 +25π 2 κ −1 ε−3 W 2L2 (Ωε ) f 2L2 (Ωε ) + Vε 2L2 (Ωε ) + κ , ∂x1 8 L2 (Ωε ) Vε 2L2 (Ωε ) C ε−1 f 2L2 (Ωε ) Δ(W χ1 )2L2 (Ωε ) +κ −1 ε−3 f 2L2 (Ωε ) W 2L2 (Ωε ) , Vε L2 (Ωε ) C ε−1/2 Δ(W χ1 )L2 (Ωε ) +κ −1/2 ε−3/2 W L2 (Ωε ) f L2 (Ωε ) , where the constants C are independent of ε, μ, κ, and f . Combining the last inequality, Eq. (4.4) and Lemma 4.2, we arrive at Uε − Uε(μ) L2 (Ωε ) Vε L2 (Ωε ) + |U (μ) (0)|W L2 (Ωε ) Vε L2 (Ωε ) + Cε−1/2 f L2 (Ωε ) W L2 (Ωε ) C ε−1/2 ΔW χ1 L2 (Ωε ) + κ −1/2 ε−3/2 W L2 (Ωε ) f L2 (Ωε ) ,
(4.11)
where the constants C are independent of ε, μ, κ, and f . Let us estimate W L2 (Ωε ) and Δ(W χ1 )L2 (Ωε ) . We have W 2L2 (Ωε ) = W 2L2 (Ωε \Ωδ ) + W 2L2 (Ωε ∩Ωδ ) . We take δ = 32 η α and in view of the definition (3.17) of W we obtain
|X(ξ)|2 dξ Cε4 μ2 , W 2L2 (Ωε \Ωδ ) = ε2 μ2 X2L2 (Ωε \Ωδ ) ε4 μ2 |ξ1 |< π 2 , ξ2 >0
where the constant C is independent of ε, μ, κ, and f . It follows from (3.25) that W 2
3
α
L2 (Ωε ∩Ω 2 η )
Cε2 η 2α ,
α ∈ (0, 1),
where the constant C is independent of ε and η. Hence, W L2 (Ωε ) Cε2 μ,
(4.12)
where the constant C is independent of ε and μ. The definition (3.17) of W , Eqs. (3.9), (3.14), the estimate (3.23), and the exponential decay of X, X(ξ) = O(e−2ξ1 ),
ξ2 → +∞
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On a Waveguide with Frequently Alternating Boundary
yield that Δ(W χ1 )2L2 (Ωε )
2ΔW 2L2 (Ωε )
2 ∂W + 2 2 χ + W χ1 ∂x1 1
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L2 (Ωε )
,
ΔW 2L2 (Ωε ) Cμ2 η 2−2α , α ∈ (1/2, 1), 2 ∂W −1 2 Cμ2 e−2ε , ∂x1 χ1 + W χ1 L2 (Ωε ) where C are positive constants independent of ε, η, and μ. We substitute the last estimates and (4.12) into (4.11), Uε − Uε(μ) L2 (Ωε ) Cκ −1/2 με1/2 f L2 (Ωε ) , where the constant C is independent of ε, μ, and κ. Together with (4.5) it completes the proof. Proof of Theorem 2.4. First we obtain the upper bound for the eigenvalues λn . To do this, we employ standard bracketing arguments (see, for instance [33, ˚ε (τ ) by those of Ch. XIII, Sect. 15, Prop. 4]), and estimate the eigenvalues of H the same operator but with η = π/2, i.e., with Dirichlet boundary condition on ˚ Γ− . The lowest eigenvalues of the latter operator are τ2 (2 + τ )2 − τ 2 (2 − τ )2 − τ 2 2 2 + n , + n , + n2 , n = 1, 2, . . . ε2 ε2 ε2 Hence, for n2 < 4κε−2 the lowest eigenvalues among mentioned are τ 2 ε−2 +n2 , and thus 1 τ2 λn (τ, ε) − 2 n2 , n < 2κ 1/2 ε−1 . (4.13) 4 ε The lower estimate was obtained by replacing the boundary conditions on ˚ Γ− by the Neumann one. In the same way we can estimate the eigenvalues of Qμ replacing the boundary condition at x2 = 0 by the Dirichlet and Neumann one, 0 Λn (μ) n2
(4.14)
uniformly in μ for all n ∈ Z. By [29, Ch. III, Sect. 1, Theorem 1.4], Theorems (2.3), and (4.13), (4.14) we get 1 1 − Cκ −1/2 ε1/2 μ, λn (τ, ε) − τ 22 Λ (μ) n ε 2 2 λn (τ, ε) − τ − Λn (μ) Cκ −1/2 (με1/2 + ε)|Λn (μ)| λn (τ, ε) − τ ε2 ε2 Cn4 κ −1/2 (με1/2 + ε), which proves (2.8). The eigenvalues √ Λn (μ) are solutions to Eq. (2.9), and the associated eigenfunctions are sin Λn (x2 − π). Hence, these eigenvalues are holomorphic with
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respect to μ by the inverse function theorem. The formula (2.10) can be checked by expanding Eq.(2.15) and Λn (μ) with respect to μ.
5. Bottom of the Spectrum In this section, we prove Theorem 2.5. The proof of (2.13) reproduces word by word the proof of similar equation (2.5) in [3] with one minor change, namely, one should use here identity λ1 (0, ε) =
1 + o(1), 4
ε → +0,
(5.1)
instead of similar identity in [3]. The identity (5.1) follows from (2.8), (2.10). In order to construct the asymptotic expansion for λ1 (0, ε), we employ the approach suggested in [4,23–25] for studying similar problems in bounded domains. ˚ ε) of H ˚ε (0) The eigenvalue λ1 (0, ε) and the associated eigenfunction ψ(x, satisfy the problem ˚ ε) = λ1 (0, ε)ψ(x, ˚ ε) − Δψ(x,
in
˚ ε) = 0 on ˚ γε , ψ(x, Γ+ ∪ ˚
Ωε , ˚ ∂ψ (x, ε) = 0 on ˚ Γε . ∂x2
(5.2)
and periodic boundary conditions on the lateral boundaries of Ωε . We construct the asymptotics for λ1 (0, ε) as λ1 (0, ε) = Λ(ε, μ), where Λ = Λ(ε, μ) is a function to be determined. It view of (2.8) with τ = 0 the function Λ should satisfy (2.16). ˚ε is constructed as the The asymptotics of the associated eigenfunction ψ sum of three expansion, namely, the external expansion, the boundary layer, and the internal expansion. The external expansion has a closed form, √ (5.3) ψεex (x, Λ) = sin Λ(x2 − π). It is clear that for any choice of Λ(ε, μ) this function solves Eq. (5.2), and satisfies the periodic boundary conditions on the lateral boundaries of Ωε . The boundary layer is constructed in terms of the variables ξ, i.e., ψεbl = bl ψε (ξ, μ). The main aim of introducing the boundary layer is to satisfy the boundary condition on ˚ Γε . We construct ψεbl by the boundary layer method. In accordance with this method, the series ψεbl should satisfy Eq. (5.2), the periodic boundary condition on the lateral boundaries of Ωε , the boundary condition ∂ψεex ∂ψεbl + = 0 on ˚ Γε , ∂x2 ∂x2 and it should decay exponentially as ξ2 → +∞.
(5.4)
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It follows from (5.3) and the definition of ξ that ψεbl should satisfy the boundary condition √ √ ∂ψεbl = − Λ cos Λπ on ˚ Γ0 , ∂ξ2 (5.5) π ˚ Γ0 := ξ : 0 < |ξ1 | < , ξ2 > 0 . 2 Here we passed to the limit η → +0 in the definition of ˚ Γε . We substitute ψεbl into Eq. (5.2) and rewrite it in the variables ξ, π − Δξ ψεbl = ε2 Λψεbl , ξ ∈ Π, Π := ξ : |ξ1 | < , ξ2 > 0 . (5.6) 2 To construct ψεbl , in [4,23–25], the authors used the standard way. Namely, they sought ψεbl and Λ(ε, μ) as asymptotic series power in ε. Then these series were substituted into (5.5), (5.6), and equating the coefficients at like powers of ε implied the boundary value problems for the coefficients of the mentioned series. In our case we do not employ this way. Instead of this we study the existence of the required solution to the problem (5.5), (5.6) and describe some of its properties needed in what follows. By V we denote the space of π-periodic even in ξ1 functions belonging to C ∞ (Π\{0}) and exponentially decaying as ξ2 → +∞ together with all their derivatives uniformly in ξ1 . We observe that X ∈ V. Lemma 5.1. The function X can be represented as the series X(ξ) = −
+∞ 1 −2nξ2 e cos 2nξ1 , n n=1
(5.7)
which converges in L2 (Π) and in C k (Π ∩ {ξ : ξ R}) for each k 0, R > 0. Proof. Since X ∈ V, for each ξ2 > 0 and each k 0 we can expand it in C k [−π/2, π/2], X(ξ) =
+∞
Xn (ξ2 ) cos 2nξ1 ,
n=1
+∞ π 2 X(·, ξ2 )2L (− π , π ) = X (ξ2 ), 2 2 2 2 n=1 n
π
Xn (ξ2 ) =
2 π
2
(5.8)
X(ξ) cos 2nξ1 dξ1 . −π 2
Integrating the second equation in (5.8) with respect to ξ2 , we obtain the Parseval identity X2L2 (Π) =
+∞ π Xn 2L2 (0,+∞) . 2 n=1
It yields that the first series in (5.8) converges also in L2 (Π), since N N 2 π 2 X − X cos 2nξ = X − Xn 2L2 (0,+∞) . n 1 L2 (Π) 2 L2 (Π) n=1 n=1
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The harmonicity of X and the exponential decay as ξ2 → +∞ yield π
Xn (ξ2 ) = −
2
−π 2
Xn (ξ2 ) = kn e
∂2X cos 2nξ1 dξ1 = −n2 Xn (ξ2 ), ∂ξ12
−2nξ2
,
2 kn = π
Xn cos 2nξ1 dξ1 . ˚ Γ0
Denote Πδ := Π\{ξ : |ξ| < δ}. Employing (3.9) and the harmonicity of X, we integrate by parts,
0 = − lim e−2nξ2 cos 2nξ1 Δξ X dξ δ→+0
= ˚ Γ0
Π
∂X + 2nX cos 2nξ1 cos 2nξ1 ∂ξ2
+ lim
δ→+0 |ξ|<δ, ξ2 >0
e
−2nξ2
dξ1
∂ −2nξ2 ∂X −X e cos 2nξ1 cos 2nξ1 ∂|ξ| ∂|ξ|
ds
=−
cos 2nξ1 dξ1 + πnkn + π.
(5.9)
˚ Γ0
Thus, kn = −1/n, which implies (5.7). The convergence of this series in C k (Π∩ {ξ : ξ2 R}) follows from the exponential decay of its terms in (5.6) as n → +∞. Lemma 5.2. For small real β the problem − Δξ Z − β 2 Z = β 2 X,
ξ ∈ Π,
∂Z = 0, ∂ξ2
ξ ∈˚ Γ0 ,
(5.10)
has a solution in W22 (Π) ∩ V. This solution and all its derivatives with respect to ξ decay exponentially as ξ2 → +∞ uniformly in ξ1 and β. The differentiable asymptotics Z(ξ, β) = Z(0, β) + O(|ξ|2 ln |ξ|),
ξ → 0,
(5.11)
holds true uniformly in β. The function (X +Z) is bounded in L2 (Π) uniformly in β. The identity Z(0, β) = β 2 θ(β 2 )
(5.12)
is valid, where the function θ is defined in (2.11). The function θ is holomorphic and its Taylor series is (2.12). Proof. Let W be the subspace of W22 (Π) consisting of the functions satisfying periodic boundary conditions on the lateral boundaries of Π, the Neumann boundary condition on ˚ Γ0 , and being orthogonal in L2 (Π) to all functions φ = φ(ξ2 ) belonging to L2 (Π). The space W is the Hilbert one.
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By B we denote the operator in L2 (Π) acting as −Δξ on W. This operator is symmetric and closed. It follows from the definition of W that each v ∈ W satisfies the equation π
2
v(ξ) dξ1 = 0 for a.e. ξ2 ∈ (0, +∞). −π 2
Using this fact, one can check easily that B 4, and therefore the bounded inverse operator exists, and B −1 1/4. Hence, (B − β 2 )−1 = B −1 (I − β 2 B −1 )−1 , i.e., the inverse operator (B − β 2 )−1 exists and is bounded uniformly in β. We let Z := β 2 (B−β 2 )−1 X. It is clear that the function Z ∈ W22 (Π) solves (5.10) and satisfies the periodic boundary conditions on the lateral boundaries of Π. By the standard smoothness improving theorems and the smoothness of X we conclude that Z ∈ C ∞ (Π\{0}). Using Lemma 5.1, for ξ2 > 0 we can also construct Z by the separation of variables, +∞ √ 1 2n −2nξ2 − 4n2 −β 2 ξ2 − e cos 2nξ1 . (5.13) Z(ξ, β) = e n 4n2 − β 2 n=1 In the same way as in the proof of Lemma 5.1 one can check that this series converges in L2 (Π) and C k (Π ∩ {ξ : ξ2 R}) for each k 0, R > 0. Thus, this function and all its derivatives with respect to ξ decay exponentially as ξ2 → +∞ uniformly in ξ1 and β, and Z ∈ V. By (5.7), (5.13) we have X +Z = −
+∞ n=1
X +
Z2L2 (Π)
2 4n2
+∞
−
π = 2 4n − β 2 n=1
β2
e−
√
4n2 −β 2 ξ2
cos 2nξ1 ,
+∞
+∞ √ 2 2 e−2 4n −β ξ2 dξ2 = n=1
n=1
2(4n2
π . − β 2 )3/2
Hence, the function (X + Z) is bounded in L2 (Π) uniformly in β. Reproducing the proof of Lemma 3.2 in [22], one can show easily that the function Z satisfies differentiable asymptotics (5.11) uniformly in β. Let us calculate Z(0, β). The function β) := X(ξ) + Z(ξ, β) + β −1 sin βξ2 Z(ξ,
(5.14)
solves the boundary value problem ∂Z = 0, ξ ∈ ˚ Γ0 , ∂ξ2 is bounded, satisfies periodic boundary condition on the lateral boundaries of Π, and has the asymptotics β) = ln |ξ| + O(1), ξ → 0. Z(ξ, = 0, (Δξ + β 2 )Z
ξ ∈ Π,
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Using these properties and (5.10), we integrate by parts in the same way as in (5.9),
ξ + β 2 )Z dξ dξ = − lim Z(Δ β2 X Z δ→+0 Πδ
Π
= lim
δ→+0 |ξ|=δ, ξ2 >0
∂Z − Z ∂ Z Z ∂|ξ| ∂|ξ|
and hence β2 Z(0, β) = − π
ds = −πZ(0, β),
dξ. XZ
Π
We substitute (5.7), (5.13), (5.14) into the last identity, Z(0, β) = −β
2
+∞ n=1
= −β 2
+∞
√ 2 2 e−(2n+ 4n −β )ξ2 dξ2
1
n 4n2 − β 2
0
+∞
1 2 2 4n2 − β 2 ) n=1 n 4n − β (2n +
that proves (5.12). The series in the definition of θ converges uniformly in β, and by the first Weierstrass theorem this function is holomorphic in small β. It is easy to see that 1 2n − 4n2 − β = n 4n2 − β(2n + 4n2 − β) βn 4n2 − β ⎛ ⎞ +∞ 2 1 1 1 (2j − 1)!!β j−1 1 1 . = ⎝ − = − ⎠= β n β n 1− β n 8j n2j+1 j! 4n2 − β j=1 4n2
We substitute this identity into the definition of θ(β), θ(β) = −
+∞ +∞ (2j − 1)!!β j−1 n=1 j=1
8j n2j+1 j!
=−
+∞ (2j − 1)!!ζ(2j + 1)β j−1 j=1
8j j!
which yields (2.12). The proof is complete. We choose the boundary layer as √ √ √ ψεbl (ξ, Λ) = ε Λ cos Λπ X(ξ) + Z(ξ, ε Λ) .
,
(5.15)
It is clear that this function satisfies all the aforementioned requirements for the boundary layer. In accordance with Lemma 5.2, the boundary layer has a logarithmic singularity at ξ = 0, and the sum of the external expansion and the boundary layer does not satisfy the boundary condition on ˚ γε in (5.2). This is the reason of introducing the internal expansion. We construct it as depending on
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1619
ς := ς (1) and employ the method of matching of the asymptotic expansions. It follows from (5.3), (2.9) that ∂ψεex (0, μ)x2 + O(|x|2 ), ψεex (x, μ) = ψεex (0, μ) + ∂x2 ψεex (0, μ) = − sin Λ(ε, μ)π,
x → 0,
(5.16) (5.17)
where the asymptotics is uniform in Λ(ε, μ). Using the definition of ς = ξη −1 and (1.3), by (5.15), (5.11), (3.10) we obtain √ √ 1 ψεbl (ξ, Λ) = Λ cos Λπ − + ε(ln |ς| + ln 2) − x2 μ √ 3 3/2 2 +ε Λ θ(ε Λ) cos Λπ + O(ε|ξ|2 ln |ξ|), ξ → 0, uniformly in ε and Λ. In view of (5.5), (5.16), (5.17) we have ψεex (x, Λ) + ψεbl (ξ, Λ) √ √ √ √ Λ cos Λπ − sin Λπ + ε3 Λ3/2 θ(ε2 Λ) cos Λπ =− μ √ √
+ε Λ cos Λπ(ln |ζ| + ln 2) + O εη 2 |ζ|2 (| ln |ζ|| + | ln η|) , as x → 0. Hence, in accordance with the method of matching of asymptotic expansions we conclude that the internal expansion should be as follows, ψεin (ς, Λ) = ψ0in (ζ, Λ, ε) + εψ1in (ζ, Λ, ε), where the coefficients should satisfy the asymptotics √ √ √ Λ in ψ0 (ς, Λ, ε) = − cos Λπ − sin Λπ μ √ 3 3/2 +ε Λ1 θ(ε2 Λ) cos Λπ + o(1), ς → ∞, √ √ ψ1in (ς, Λ) = ε Λ cos Λπ(ln |ζ| + ln 2) + o(1), ς → ∞.
(5.18)
(5.19)
We substitute (5.18) into (5.2) and pass to the variables ς. It yields the boundary value problems for ψiin , ∂ψiin = 0, ς ∈ ˚ Γ1 . (5.20) ∂ς2 For i = 0 this problem has the only bounded solution which is trivial, Δς ψiin = 0,
ς2 > 0,
ψiin = 0,
ς ∈˚ γ1,
ψ0in = 0.
(5.21)
Thus, by (5.19) we obtain Eq. (2.15) for Λ(ε, μ). In view of the properties of the function Y described in the third section the function ψ1in should be chosen as √ √ (5.22) ψ1in (ζ, Λ, ε) = ε Λ cos ΛπY (ζ). ˚ε is complete. The formal constructing of λ1 (0, ε) and ψ We proceed to the studying of Eq. (2.15). Since the function θ is holomorphic by Lemma 5.2, the function √ √ √ √ T (ε, μ, Λ) := Λ cos Λπ + μ sin Λπ − ε3 μΛ3/2 θ(ε2 Λ) cos Λπ
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is jointly holomorphic with respect to small ε, μ, and Λ close to 1/4. Employing the formula (2.12), we continue T analytically to complex values of ε, μ, and Λ. As ε = μ = 0, Eq. (2.15) becomes √ √ Λ cos Λπ = 0, and it has the root Λ = 1/4. It is clear that ∂T 1 0, 0, = 0. ∂Λ 4 Hence, by the inverse function theorem there exists the unique root of Eq. (2.15). This root is jointly holomorphic in ε and μ and satisfies (2.16). We represent this root as Λ(ε, μ) = Λ0 (μ) +
+∞
j (μ), εj K
(5.23)
j=1
j (μ) are holomorphic in μ functions. We choose the leading term in where K this series as Λ1 (μ), since as ε = 0 Eq. (2.15) coincides with (2.9). We substitute (5.23) and (2.12) into (2.15) and equate the coefficients i , i = 1, . . . , 8. Solving these at εi , i = 1, . . . , 8. It implies the equations for K 1 = K 2 = 0 and (2.18). equations, we obtain K 2j+1 (μ) = μ2 K2j+1 (μ), K 2j (μ) = μ3 K2j (μ), where Let us prove that K Kj (μ) are holomorphic in μ functions. It is sufficient to prove that j (0) = K (0) = 0, K j
(0) = 0. K 2j
We take μ = 0 in (2.15) and (5.23), Λ(0, ε) cos Λ(0, ε)π = 0, (5.24) 1 (5.25) Λ(0, ε) = . 4 j (0) = 0. We differentiate Eq. (2.15) with respect By (2.10), (5.23) it implies K to μ and then we let μ = 0. It implies the equation 1 π Λ(ε, 0) sin Λ(ε, 0)π − cos Λ(ε, 0)π ∂Λ (ε, 0) − 2 ∂μ Λ(ε, 0) −ε3 Λ3/2 (ε, 0)θ(ε2 Λ(ε, 0)) cos Λ(ε, 0)π + sin Λ(ε, 0)π = 0. We substitute here the identity (5.25) and arrive at the equation −
π ∂Λ (ε, 0) + 1 = 0, 2 ∂μ
which by (2.10) implies ∂Λ 2 ∂Λ1 (ε, 0) = = (0). ∂μ π ∂μ (0) = 0. These identities and (5.23) yield K j
(5.26)
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1621
We differentiate Eq. (2.15) twice with respect to μ and then we let μ = 0 taking into consideration the identities (5.25), (5.26), and (2.12), 2 ε ε3 4 π ∂2Λ − + θ (ε, 0) = 0, − π 2 4 2 ∂μ2 2 ε ∂2Λ 1 3 (ε, 0) = πθ −8 + ε ∂μ2 π2 4 ⎛ ⎞ +∞ π (2j − 1)!!ζ(2j + 1) 2j+1 ⎠ 1 ⎝ ε . = − 2 8+ π 8 j=1 32j−1 j! (0) = 0, j 1. Hence, K 2j To calculate all other coefficients of (2.17) we substitute this series and (2.12) into Eq. (2.15) and then equate the coefficients of like powers of ε. It implies certain equations, which can be solved with respect to Ki . Since all the coefficients in the expansion in ε of θ and other terms in Eq. (2.15) are real, the functions Ki are real, too. Hence, by (2.17) the function Λ is real-valued for real ε and μ. We proceed to the justification of the asymptotics. Denote
˚ε (x) := ψεex (x, Λ(ε, μ)) + χ1 (x2 )ψεbl (ξ, Λ(ε, μ)) 1 − χ1 (|ς|η 1/2 ) Ψ (5.27) +χ1 |ς|η 1/2 ψεin (ς, Λ(ε, μ)). where, we remind, χ1 is the cut-off function introduced in the third section. ˚ε ∈ C ∞ (Ωε \{x : x1 = ±εη, x2 = 0}) belongs to Lemma 5.3. The function Ψ ˚ε (0), satisfies the convergence the domain of H ˚ε − sin x2 − π Ψ = O(ε1/2 μ), ε → +0, (5.28) 2 L2 (Π)
and solves the equation
˚ε (0) − Λ(ε, μ) Ψ ˚ ε = hε , H
(5.29)
where for the function hε ∈ L2 (Ωε ) an uniform in ε, μ, and η estimate −1
hε L2 (Ωε ) C(μe−2ε
+ εη 1/2 )
(5.30)
holds true. ˚ε that Proof. It follows from the definition of Ψ ∞ 1 ˚ε ∈ C (Ωε \{x : x1 = ±εη, x2 = 0}) ∩ W ˚2,per Ψ (Ωε , ˚ Γ+ ).
(5.31)
˚ε , The boundary condition (5.4), (5.17), and (3.14) for Y yield those for Ψ ˚ε = 0 on ˚ Ψ Γ+ ∪ ˚ γε ,
˚ε ∂Ψ =0 ∂x2
on ˚ Γε .
(5.32)
Let us show that ˚ ε = hε , − (Δξ + Λ(ε, μ))Ψ
x ∈ Ωε ,
(5.33)
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where hε ∈ L2 (Ωε ) satisfies (5.30). Employing Eqs. (5.6), (5.20), we obtain ˚ε = hε , hε = −(h(1) + h(2) + h(3) ), −(Δξ + Λ)Ψ ε ε ε ∂ bl (1) bl hε (x) = 2χ1 (x2 ) ψ (ξ, Λ(ε, μ)) + χ1 (x2 )ψε (ξ, Λ(ε, μ)), ∂x2 ε 1/2 h(2) )ψεin (ς, Λ(ε, μ)), ε (x) = Λ(ε, μ)χ1 (|ς|η 1/2 ˚mat ˚(mat) ) · ∇x Ψ (x) + Ψ (x)Δx χ1 (|ς|η 1/2 ), h(3) ε (x) = 2∇x χ1 (|ς|η ε ε
(5.34)
(5.35)
˚(mat) Ψ (x) := ψεin (ς, Λ(ε, μ)) − ψεex (x, Λ(ε, μ)) − ψεbl (ξ, Λ(ε, μ)). ε (i)
It is clear that hε ∈ L2 (Ωε ) that implies the same for hε . Due to (2.15) the function ψεbl can be rewritten as follows, ψεbl (ξ, Λ(ε, μ)) = μ ε3 Λ3/2 (ε, μ)θ(ε2 Λ(ε, μ)) cos Λ(ε, μ)π − sin Λ(ε, μ)π X(ξ) + Z(ξ, ε Λ(ε, μ)) . Thus,
3 3/2 2 h(1) (x) = μ ε Λ (ε, μ)θ(ε Λ(ε, μ)) cos Λ(ε, μ)π − sin Λ(ε, μ)π ε ∂ + χ1 (x2 ) X(ξ) + Z(ξ, ε Λ(ε, μ)) . × 2χ1 (x2 ) ∂x2
The functions χ1 (x2 ), χ1 (x2 ) are non-zero only for 1 < x2 < 32 that corresponds to ε−1 < ξ2 < 32 ε−1 . For such values we can use the series (5.7),
of ξ −1 k (5.13) for X and Z which converge in C ξ : ε ξ2 32 ε−1 , |ξ1 | π2 . (1) It yields the exponential estimate for hε , −1
−2ε h(1) , ε L2 (Ωε ) Cμe
(5.36)
where the constant C is independent of ε and μ. Taking into consideration (5.21), and replacing in (5.22) the factor √ √ Λ cos Λπ by μ ε3 Λ3/2 (ε, μ)θ(ε2 Λ(ε, μ)) cos Λ(ε, μ)π − sin Λ(ε, μ)π as (2)
we did it in (5.34), we estimate hε ,
2 4 2 2 h(2) ε L2 (Ωε ) Cε μ η
|Y (ς)|2 dς
|ς|<η −1/2 , ς2 >0
Cε μ η| ln2 η| Cε2 η, 4 2
(5.37)
where the constants C are independent of ε, μ, and η. The asymptotics (3.10), (5.11), (3.16), Eq. (2.15), and the identities (5.3), ˚mat , (5.15), (5.18), (5.21), (5.22) imply the differentiable asymptotics for Ψ ε √ √ √
mat −2 ˚ Ψε (x) = ε Λ cos Λπ ln |ς| + ln 2 + O(|ς| ) − sin Λ(x2 − π) √ √
−ε Λ cos Λπ ln |ξ| + ln 2 + ε2 Λθ(ε2 Λ) − ξ2 + O(|ξ|2 ) √ √ √ √
= − sin Λ(x2 − π) − sin Λπ + Λx2 cos Λπ + O εμ(|ξ|2 + |ς|−2 )
= O |x|2 + εμ(|ξ|2 + |ς|−2 )
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uniformly in ε, μ, and η as εη 1/2 < |x| <
3 1/2 εη , 2
x ∈ Ωε .
(5.38)
Thus, for such x ˚mat (x)| C(ε(ε + μ)η), |Ψ ε ˚mat (x)| C((ε + μ)η 1/2 ), |∇x Ψ ε where the constants C are independent of x, ε, μ, and η. Since the functions ∇x χ1 (|ς|η 1/2 ), Δx χ1 (|ς|η 1/2 ) are non-zero only for x satisfying (5.38), the last (3) ˚mat ˚mat inequalities for Ψ and ∇x Ψ enable us to estimate hε , ε ε 1/2 h(3) ), ε L2 (Ωε ) C((ε + μ)η
where the constant C is independent of ε, μ, and η. We sum the last estimate and (5.36), (5.37), −1
hε L2 (Ωε ) C(μe−2ε
+ εη 1/2 ),
where the constant C is independent of ε, μ, and η. This estimate imply (5.30). ˚ε , the boundary value conditions (5.32), Due to the smoothness (5.31) of Ψ ˚ε is a generalized solution to the boundary value and Eq. (5.33), the function Ψ ˚ε belongs to the domain of H ˚ε (0). problem (5.33), (5.32). Hence, Ψ Let us prove the estimate (5.28). Completely as in the estimating hε , we check that χ1 (x2 )ψεbl 1 − χ1 (|ς|η 1/2 ) + χ1 |ς|η 1/2 ψεin − ψεex χ1 (|ς|η 1/2 )L2 (Ωε ) = O(ε2 μ). In view of (2.10) and the definition (5.3) of ψεex the estimate ex ψε − sin x2 − π = O(ε1/2 μ) 2 L2 (Π)
˚ε imply (5.28). holds true. Two last estimates and the definition (5.27) of Ψ We proceed to the estimating of the error terms. The core of these estimates are Lemmas 12, 13 in [37]. We employ these results in the form they were formulated in [29, Ch. III, Sect. 1.1, Lemma 1.1]. For the reader’s convenience we provide this lemma below. Lemma 5.4. Let A : H → H be a continuous linear compact self-adjoint operator in a Hilbert space H. Suppose that there exist a real M > 0 and a vector u ∈ H, such that uH = 1 and Au − M uH κ,
α = const > 0.
Then there exists an eigenvalue Mi of operator A such that |Mi − μ| κ. Moreover, for any d > κ there exists a vector u such that u − uH 2κd−1 ,
uH = 1,
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and u is a linear combination of the eigenvectors of the operator A corresponding to the eigenvalues of A from the segment [M − d, M + d]. ˚ε (0) is non-negative and self-adjoint in L2 (Ωε ) and Since the operator H ˚ε−1 (0) exists, is bounded and self-adjoint, and satisfies (4.1), the inverse A := H satisfies the estimate A 4.
(5.39)
and in view The operator A is also bounded as that from L2 (Ωε ) into of the compact embedding of W21 (Ωε ) in L2 (Ωε ) the operator A is compact in L2 (Ωε ). We rewrite Eq. (5.29) as follows, W21 (Ωε )
˚ ε = AΨ ˚ε + hε , Λ−1 (ε, μ)Ψ
hε := Λ−1 (ε, μ)Ahε .
By (2.16), (2.10), (5.39), (5.30) the function hε satisfies the estimate −1 hε L2 (Ωε ) = O(μe−2ε + εη 1/2 ).
Hence, by (5.28) −1/2 −2ε−1 ˚ε −1 e + ε1/2 η 1/2 ). hε L2 (Ωε ) Ψ L2 (Ωε ) = O(με
Taking this estimate into consideration, we apply Lemma 5.4 with ˚ε Ψ H = L2 (Ωε ), u= , ˚ Ψε L2 (Ωε ) ˚ε −1 M = Λ−1 (ε, μ), κ = hε L2 (Ωε ) Ψ L2 (Ωε ) ,
(5.40)
(ε, μ) of A satisfying the estiand conclude that there exists an eigenvalue M mate (ε, μ) − Λ−1 (ε, μ)| = O(με−1/2 e−2ε−1 + ε1/2 η 1/2 ). |M Thus, by (2.16), (2.10) (ε, μ)| |Λ−1 (ε, μ)| − O(με−1/2 e−2ε−1 + ε1/2 η 1/2 ) 3, |M −1 (ε, μ)| 1 , |M 3 −1 −1/2 −2ε−1 1/2 1/2 −1 e + ε η )|Λ(ε, μ)||M (ε, μ)| |M (ε, μ) − Λ(ε, μ)| = O (με −1
= O(με−1/2 e−2ε
+ ε1/2 η 1/2 ).
(5.41)
−1 (ε, μ) is an eigenvalue of H ˚ε (0). Due to (2.8), (2.10) there The number M exists exactly one eigenvalue of this operator satisfying (5.41), and this eigenvalue is λ1 (0, ε). Thus, −1
|λ1 (0, ε) − Λ(ε, μ)| = O(με−1/2 e−2ε
+ ε1/2 η 1/2 )
(5.42)
that proves (2.14). The asymptotics (2.8), (2.10), (2.16), (2.14) imply that for ε small enough ˚ε , the segment [Λ(ε, μ) − 1, Λ(ε, μ) + 1] contains exactly one eigenvalue of H which is λ1 (0, ε). Bearing in mind this fact and (5.30), we apply Lemma 5.4 with d = 1 and other quantities given by (5.40) and conclude that the
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˚ ε) associated with λ1 (0, ε) satisfies normalized in L2 (Ωε ) eigenfunction φ(x, the estimate −2ε−1 1/2 C μe + εη ˚ 2hε L2 (Ωε ) Ψε ˚ ε) − φ(·, , ˚ε L (Ω ) ˚ε L (Ω ) ˚ε L (Ω ) Ψ Ψ Ψ 2
ε
L2 (Ωε )
2
ε
2
ε
where the constant C is independent of ε, μ, and η. Hence, for the eigenfunction ˚ ε) := Ψ ˚ ε) associated with λ1 (0, ε) we have ˚ε L (Ω ) φ(x, ψ(x, 2 ε ˚ ε) − Ψ ˚ε L (Ω ) = O μe−2ε−1 + εη 1/2 . (5.43) ψ(·, 2 ε ˚ ε). Equations (5.29) and the eigenvalue ˚ε (x) − ψ(x, Denote ˚ Φε (x) := Ψ ˚ equation for ψ(x, ε) imply the equation for ˚ Φε , ˚ε (0)˚ ˚ε . H Φε = λ1 (0, ε)˚ Φε + (λ1 (0, ε) − Λ(ε, μ)) Ψ Hence, we can write the integral identity ˚ε , ˚ ∇˚ Φε 2L2 (Ωε ) = λ1 (0, ε)˚ Φε 2L2 (Ωε ) + (λ1 (0, ε) − Λ(ε, μ)) (Ψ Φε )L2 (Ωε ) . Thus, by (5.43), (5.42), (5.28), (2.14), (2.16), (2.10) ˚ε , ˚ ∇˚ Φε 2L2 (Ωε ) λ1 (0, ε)˚ Φε 2L2 (Ωε ) + (λ1 (0, ε) − Λ(ε, μ)) (Ψ Φε )L2 (Ωε ) ˚ε L (Ω ) ˚ ˚ Φε 2L2 (Ωε ) + |λ1 (0, ε) − Λ(ε, μ)|Ψ Φε L2 (Ωε ) 2 ε −1 C μ2 e−4ε + ε2 η . The last estimate and (5.43) prove the asymptotics (2.19). Theorem 2.5 is proved.
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[29] Olejnik, O.A., Shamaev, A.S., Yosifyan, G.A.: Mathematical problems in elasticity and homogenization. In: Studies in Mathematics and its Applications, vol. 26. North-Holland, Amsterdam (1992) [30] Pastukhova, S.E. Tikhomirov R.N.: Operator estimates in reiterated and locally periodic homogenization. Dokl. Math. 76, 548–553 (2007) [31] Pastukhova, S.E.: Some estimates from homogenized elasticity problems. Dokl. Math. 73, 102–106 (2006) [32] Reed, M., Simon, B.: Methods of modern mathematical physics I: functional analysis. Academic Press, New York (1980) [33] Reed, M., Simon, B.: Methods of modern mathematical physics IV: analysis of operators. Academic Press, New York (1978) [34] Suslina, T.A.: Homogenization with corrector for a stationary periodic Maxwell system. St. Petersburg Math. J. 19, 455–494 (2008) [35] Suslina, T.A.: Homogenization in Sobolev class H 1 (Rd ) for periodic elliptic second order differential operators including first order terms. Alg. Anal. 22, 108–222 (2010) (in Russian) [36] Suslina, T.A., Kharin, A.A.: Homogenization with corrector for a periodic elliptic operator near an edge of inner gap. J. Math. Sci. 159, 264–280 (2009) [37] Vishik, M.I., Lyusternik, L.A.: Regular degeneration and boundary layer for linear differential equations with small parameter. Transl. Ser. 2 Am. Math. Soc. 20, 239–364 (1962) [38] Zhikov, V.V.: On operator estimates in homogenization theory. Dokl. Math. 72, 534–538 (2005) [39] Zhikov, V.V.: Some estimates from homogenization theory. Dokl. Math. 73, 96–99 (2006) Denis Borisov Bashkir State Pedagogical University October Revolution St. 3a 450000 Ufa, Russia e-mail: [email protected] Renata Bunoiu LMAM UMR 7122 Universit´e de Metz et CNRS Ile du Saulcy 57045 Metz Cedex 1, France e-mail: [email protected] Giuseppe Cardone Department of Engineering University of Sannio Corso Garibaldi, 107 82100 Benevento, Italy e-mail: [email protected] Communicated by Jan Derezinski. Received: 28 June, 2010. Accepted: 4 October, 2010.
