Ann. Henri Poincar´e 5 (2004) 1 – 73 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/010001-73 DOI 10.1007/s00023-004-0160-1
Annales Henri Poincar´ e
Non-Selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I Michael Hitrik and Johannes Sj¨ ostrand Abstract. This is the first in a series of works devoted to small non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2. In the present work we treat the case when the classical flow of the unperturbed part is periodic and the strength of the perturbation is h (or sometimes only h2 ) and bounded from above by hδ for some δ > 0. We get a complete asymptotic description of all eigenvalues in certain rectangles [−1/C, 1/C] + i[F0 − 1/C, F0 + 1/C].
1 Introduction In [20], A. Melin and the second author observed that for a wide and stable class of non-selfadjoint operators in dimension 2 and in the semi-classical limit (h → 0), it is possible to describe all eigenvalues individually in an h-independent domain in C, by means of a Bohr-Sommerfeld quantization condition. This result is quite remarkable since the corresponding conclusion in the selfadjoint case seems to be possible only in dimension 1 or under strong (and unstable) assumptions of complete integrability. The underlying reason for this result is the absence of small denominators which allows us to avoid the usual trouble with exceptional sets in the KAM theorem. As a next step, the second author noticed ([22]) that for non-selfadjoint operators of the form P (x, hDx ) + iQ(x, hDx ) it is possible to find a similar result, when P is selfadjoint, > 0 small and fixed and the classical bicharacteristic flow is periodic on each real energy surface. (Again, it is important that we are in dimension 2.) The method is similar to the one in [20] and uses non-linear CauchyRiemann equations, now in an “-degenerate” form. (See also [24] for a different extension.) It soon became quite clear that we run into a fairly vast program, and that logically one should start with even smaller perturbations, say = O(hδ ), for some δ > 0. The present work is planned to be the first in a series, devoted to small perturbations of selfadjoint operators in dimension 2. In addition to the challenge of doing plenty of things in dimension 2, that can usually only be done in dimension 1, we have been motivated by recent progress around the damped wave equation ([19], [2], [25], [14]), as well as the problem of barrier top resonances for the semi-classical Schr¨ odinger operator ([17]) where more complete results than the corresponding ones for eigenvalues of potential wells ([26], [3], [21]) seem possible. One long term goal of this series is to get improved results on the distribution of
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resonances for strictly convex obstacles in R3 . See [30] (and references given there) for a first result on Weyl asymptotics for the real parts inside certain bands. In the case of analytic obstacles, much more can probably be said, especially in dimension 3 (and 2). Let M denote R2 or a compact real-analytic manifold of dimension 2. When M = R2 , let (1.1) P = P (x, hDx , ; h) be the Weyl quantization on R2 of a symbol P (x, ξ, ; h) depending smoothly on ∈ neigh (0, R) with values in the space of holomorphic functions of (x, ξ) in a tubular neighborhood of R4 in C4 , with |P (x, ξ, ; h)| ≤ Cm(Re (x, ξ))
(1.2)
there. Here m is assumed to be an order function on R4 , in the sense that m > 0 and (1.3) m(X) ≤ C0 X − Y N0 m(Y ), X, Y ∈ R4 . We also assume that m ≥ 1.
(1.4)
We further assume that P (x, ξ, ; h) ∼
∞
pj, (x, ξ)hj , h → 0,
(1.5)
j=0
in the space of such functions. We make the ellipticity assumption |p0, (x, ξ)| ≥
1 m(Re (x, ξ)), |(x, ξ)| ≥ C, C
for some C > 0. When M is a compact manifold, we let P = aα, (x; h)(hDx )α ,
(1.6)
(1.7)
|α|≤m
be a differential operator on M , such that for every choice of local coordinates, centered at some point of M , aα, (x; h) is a smooth function of with values in the space of bounded holomorphic functions in a complex neighborhood of x = 0. We further assume that aα, (x; h) ∼
∞
aα,,j (x)hj , h → 0,
(1.8)
j=0
in the space of such functions. The semi-classical principal symbol in this case is given by p0, (x, ξ) = aα,,0 (x)ξ α , (1.9)
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and we make the ellipticity assumption |p0, (x, ξ)| ≥
1 ξm , (x, ξ) ∈ T ∗ M, |ξ| ≥ C, C
(1.10)
for some large C > 0. (Here we assume that M has been equipped with some Riemannian metric, so that |ξ| and ξ = (1 + |ξ|2 )1/2 are well defined.) Sometimes, we write p for p0, and simply p for p0,0 . Assume P=0 is formally selfadjoint.
(1.11)
In the case when M is compact, we let the underlying Hilbert space be L2 (M, µ(dx)) for some positive real-analytic density µ(dx) on M . Under these assumptions, P will have discrete spectrum in some fixed neighborhood of 0 ∈ C, when h > 0, ≥ 0 are sufficiently small, and the spectrum in this region will be contained in a band |Im z| ≤ O(). The purpose of this work and later ones in this series, is to give detailed asymptotic results about the distribution of individual eigenvalues inside such a band. Assume for simplicity that (with p = p=0 ) p−1 (0) ∩ T ∗ M is connected.
(1.12)
∂ ∂ − px · ∂ξ be the Hamilton field of p. In this work, we will always Let Hp = pξ · ∂x assume that for E ∈ neigh (0, R):
The Hp -flow is periodic on p−1 (E) ∩ T ∗ M with period T (E) > 0 depending analytically on E.
(1.13)
∂ Let q = 1i ( ∂ )=0 p , so that
p = p + iq + O(2 m),
(1.14)
in the case M = R2 and p = p + iq + O(2 ξm ) in the manifold case. Let T (E)/2 1 q = q ◦ exp tHp dt on p−1 (E) ∩ T ∗ M. (1.15) T (E) −T (E)/2 Notice that p, q are in involution; 0 = Hp q =: {p, q}. In Section 3, we shall see how to reduce ourselves to the case when p = p + iq + O(2 ), −1
(1.16)
∗
near p (0) ∩ T M . An easy consequence of this is that the spectrum of P in {z ∈ C; |Re z| < δ} is confined to ] − δ, δ[+i]Re qmin,0 − o(1), Re qmax,0 + o(1)[, when δ, , h → 0, where Re qmin,0 = minp−1 (0)∩T ∗ M Re q and similarly for qmax,0 . We will mainly think about the case when q is real-valued but we will work under the more general assumption that Im q is an analytic function of p and Re q, in the region of T ∗ M , where |p| ≤ 1/|O(1)|.
(1.17)
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Let F0 ∈ [Re qmin,0 , Re qmax,0 ]. The purpose of the present work is to determine all eigenvalues in a rectangle ]−
1 1 1 1 , [ + i ]F0 − , F0 + [, |O(1)| |O(1)| |O(1)| |O(1)|
(1.18)
for h ≤ O(hδ ),
(1.19)
where δ > 0 is any fixed number. (When the subprincipal symbol of P is zero, we can treat even smaller values of : h2 ≤ O(hδ ).) We will achieve this under the general assumption that T (0) is the minimal period of every Hp -trajectory in Λ0,F0 , where
Λ0,F0 := {ρ ∈ T ∗ M ; p(ρ) = 0, Re q(ρ) = F0 },
(1.20)
(1.21)
in the following three cases: I) The first case is when dp, dRe q are linearly independent at every point of Λ0,F0 .
(1.22)
This implies that every connected component of Λ0,F0 is a two-dimensional Lagrangian torus. For simplicity, we shall assume that there is only one such component. Notice that in view of (1.20), the space of closed orbits in p−1 (0) ∩ T ∗ M ; Σ := (p−1 (0) ∩ T ∗ M )/ ∼, where ρ ∼ µ if ρ = exp tHp µ for some t ∈ R, becomes a 2-dimensional symplectic manifold near the image of Λ0,F0 , and (1.22) simply means that Re q, viewed as a function on Σ, has non-vanishing differential along the image of Λ0,F0 . The image of Λ0,F0 is just a closed curve. The main results in this case are Theorems 6.2, 6.4 and they show that the eigenvalues form a distorted lattice. II) The second case is when F0 ∈ {Re qmin,0 , Re qmax,0 }. In this case, we again view Re q as a smooth function on Σ near the image of Λ0,F0 and assume that The Hessian of Re q is non-degenerate (positive or negative) at every point ρ ∈ Σ, with Re q(ρ) = F0 .
(1.23)
The main results in this case are given by Theorems 6.6, 6.7 which tell us that the eigenvalues form a distorted half-lattice. III) The third natural case would be when F0 is a critical value of Re q corresponding to a saddle point. We hope to study this case in the near future. The analyticity assumptions are introduced, because the optimal spaces are deformations of the usual L2 -space obtained by adding exponential weights with
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exponents that are O(), and there are closely related Fourier integral operators with complex phase some of which have associated complex canonical transformations that are -perturbations of the identity. When ∼ hδ , 0 < δ < 1, appropriate Gevrey type assumptions would probably suffice, but in the case ∼ h we seem to need analyticity assumptions at one point, even though standard C ∞ -microlocal analysis would suffice for most of the steps. At the opposite extreme, small but independent of h, the analyticity assumptions seem necessary, and in order to avoid technicalities, we have chosen to assume analyticity independently of the size of . In the selfadjoint case there have been many works about operators whose associated classical flow is periodic ([31], [8], [5], [11], [9], [16]), and we follow one of the main ideas in those works, namely to use some sort of averaging procedure in order to reduce the dimension by one unit, so that in our case, we come down to a one-dimensional problem. The implementation of this is more complicated in our case because of the need to work in modified exponentially weighted spaces (after suitable FBI-transforms). It should also be pointed out that in the case when is small but independent of h ([22]), this does not seem to work and the problem remains two-dimensional. The same seems to be the case (for the whole scale of ) in other situations, when the Hp -flow is completely integrable without being periodic, or more generally when the energy surface p−1 (0) ∩ T ∗ M contains certain invariant Lagrangian tori. We intend to treat such situations later in this series. The plan of the paper is the following: In Section 2, we reexamine the Egorov theorem in a form suitable for us, and complete some observations of [13] about the two term version of this result. In Section 3 we perform dimension reduction by averaging. In Section 4 we make a complete reduction in the torus case (I) and determine the corresponding quasi-eigenvalues. In Section 5 we do the analogous work in the extreme case (II). In Section 6 we justify the earlier computations by treating an auxiliary global (Grushin) problem, and we obtain the two main results. In Section 7, we give a first application to barrier top resonances. In the appendix, we review some standard facts about FBI-transforms on manifolds. The next work(s) in this series (in addition to [22]) will remain in the case when the classical flow of the unperturbed part is periodic. We intend to study the saddle point case (III), and the case when q vanishes.
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2 Quantization of canonical transformations between non-simply connected domains in phase space We first give an affirmative answer to a question asked in Appendix A of [13]. Let κ : neigh ((y0 , η0 ), T ∗ Rn ) → neigh ((x0 , ξ0 ), T ∗ Rn ) be an analytic canonical transformation and consider a corresponding Fourier integral operator n+N U u(x) = h− 2 (2.1) eiφ(x,y,θ)/ha(x, y, θ; h)u(y)dydθ, with a = a0 + O(h), a classical symbol in S 0,0 (see the appendix), and φ nondegenerate phase function in the sense of H¨ ormander [15] (without the homogeneity requirement in θ) which generates the graph of κ. (Since we work microlocally, φ, a are assumed to be defined near a fixed point (x0 , y0 , θ0 ) with φθ (x0 , y0 , θ0 ) = 0, (x0 , ξ0 ) = (x0 , φx (x0 , y0 , θ0 )), (y0 , η0 ) = (y0 , −φy (x0 , y0 , θ0 )).) We require U to be unitary: (2.2) U ∗ U = 1, microlocally near (y0 , η0 ), and we are interested in the improved Egorov property: If P U = U Q, where P = P w , Q = Qw are h-pseudodifferential operators of order 0, then P ◦ κ = Q + O(h2 ).
(2.3)
Here and in what follows we use the same letter to denote an operator and a corresponding Weyl symbol. In Appendix A of [13], it was shown that such U ’s exist and we shall answer the question raised there, by establishing the following proposition. (We learned from C. Fefferman that Jorge Silva has obtained essentially the same result in the framework of classical Fourier integral operators.) Proposition 2.1 Within the class of operators satisfying (2.1) and (2.2), the property (2.3) is equivalent to: a0| C has constant argument. φ
(2.4)
Here φ is defined in some open set D(φ) ⊂ R2n+N and Cφ = {(x, y, θ) ∈ D(φ); φθ (x, y, θ) = 0}. Proof. We first consider the special case of pseudodifferential operators, i.e., the case when κ is the identity. Then a0 is the principal symbol and (2.2) implies that |a0 | = 1 (after inserting an additional factor (2π)−n in front of the integral and taking the standard phase φ = (x − y) · θ). Write U −1 P U = P + U −1 [P, U ]. We see that (2.3) holds iff {p, a0 } = 0 for all p, i.e., iff a0 = Const. The proposition follows in the case of pseudodifferential operators since we also know in general
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that the property (2.4) is invariant under changes of (φ, a) in the representation of the given operator. When φ is quadratic and a is constant, we have a metaplectic operator and κ is linear. In that case, we know that (2.3) holds, and using the special case of h-pseudodifferential operators, we see that we have equivalence between (2.3) and (2.4) in the case when κ is linear. Consider a smooth deformation of canonical transformations [0, 1] t → κt , with a deformation field Ha(t) , so that ∂t κt (ρ) = Ha(t) (κt (ρ)) where a(t) = a(t, x, ξ) is smooth and independent of h. Let A(t) = aw (x, hDx ) and consider a corresponding family of Fourier integral operators U (t) associated to κt : hDt U (t) + A(t) ◦ U (t) = 0.
(2.5)
Since A(t) are selfadjoint, unitarity of U (t) is conserved under the flow of (2.5). Let U (t) be such a unitary family. Proposition 2.2 We have (2.3) for one value of t iff we have it for all values of t. Proof. Suppose we have (2.3) for U (0). From (2.5) we get hDt (U (t)−1 ) = U (t)−1 A(t). Consider a family P (t) = U (t)P U (t)−1 . Then hDt P (t) + [A(t), P (t)] = 0, and on the level of Weyl symbols, we get ∂t P (t) + {a(t), P (t)} = O(h2 ), or in other words, (∂t + Ha(t) )P (t) = O(h2 ). This means that P (t) ◦ (κt (ρ)) = P (0) ◦ κ0 + O(h2 ) = P (ρ) + O(h2 ), where we used (2.3) for U (0) in the last step. Then P (t) fulfills (2.3) for all t. On the other hand, if U (t) fulfills (2.5), we know, using that the subprincipal symbol of A(t) is 0, that if we represent n+N i U (t) = h− 2 e h φt (x,y,θ)at (x, y, θ; h)u(y)dydθ, with φt , at depending smoothly on t, then the argument of at,0 |
C φt
is constant
along every curve in {(t, x, θ); (x, θ) ∈ Cφt } corresponding to a Ha(t) -trajectory:
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t → (κt (ρ), κ0 (ρ)). This can be seen either by a direct computation leading to a real transport equation for the leading symbol, (using that e−iφ(x)/h ◦ aw (x, hDx ) ◦ eiφ(x)/h = (a(x, φ (x) + hDx ))w + O(h2 ), see Appendix A in [13]), or by using H¨ ormander’s definition ([15]) of the principal symbol of a Fourier integral operator, as well as a result of Duistermaat-H¨ ormander giving a real transport equation for the principal symbol for the evolution problem (2.5). In particular, if at | C has constant argument for one value of t, the same φt holds for all other values. For a given U associated to κ, choose κt and U (t) as in (2.5), so that κ0 is linear and U (1) = U . (We may assume for simplicity that (y0 , η0 ) = (x0 , ξ0 ) = (0, 0) and take κt (y, η) = 1t κ(t(y, η)).) Then using Proposition 2.2 and the above remark, we get the equivalences: [U satisfies (2.3).] ⇔ [U (0) satisfies (2.3).] ⇔ [The principal symbol of U (0) has constant argument.] ⇔ [The principal symbol of U has constant argument.] This gives Proposition 2.1. Let X, Y be analytic manifolds of dimension n equipped with analytic integration densities L(dx) = LX (dx), L(dy) = LY (dy). Let κ : ΩY → ΩX be a canonical transformation (and diffeomorphism), analytic for simplicity, where ΩY ⊂⊂ T ∗ Y, ΩX ⊂⊂ T ∗ X, are connected, open with smooth boundary. We do not assume ΩX , ΩY to be simply connected, so we may have finitely many closed cycles γ1 , . . . , γN ⊂ ΩY which generate the homotopy group of ΩY . T : L2 (Y ) → HΨ (Y ) be corresponding FBILet S : L2 (X) → HΦ (X), Y denote tubular complex neighborhoods transforms as in the appendix, where X, of X, Y and with associated canonical transformations: κS : T ∗ X ∩ {|ξ| < C} → ΛΦ , κT : T ∗ Y ∩ {|η| < C} → ΛΨ , where we equip HΦ , HΨ with the scalar products that make S, T unitary, and we can have C > 0 as large as we like. Choose C large enough, so that κS , κT are well defined on ΩX , ΩY respectively, and let X = πx κS ΩX ⊂ X, Ω Y = πy κT ΩY ⊂ Y . Ω Let κ : ΛΨ → ΛΦ be the lift of κ, so that κ = κS ◦ κ ◦ κ−1 T . Here ΛΦ,Ψ are restricted 2 2 X }. to ΩX,Y : ΛΨ = {(y, i ∂y Ψ); y ∈ ΩY }, ΛΦ = {(x, i ∂x Φ); x ∈ Ω
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We shall define a multi-valued “Floquet periodic” Fourier integral operator U : L2 (Y ) → L2 (X) which is only microlocally defined from ΩY to ΩX and associated to κ. Requiring that U be microlocally unitary with the improved Egorov property, we will see that we can have the Floquet periodicity: γ∗ U = eiθ(γ) U,
(2.6)
where γ is a closed loop in ΩY joining some point ρ to itself, U denotes the operator U as it is defined near ρ and the left-hand side of (2.6) denotes the operator obtained from U by following the loop γ. We will then achieve (2.6) with θ(γ) = h−1 S(γ) + k(γ)π/2, where S(γ) = κ◦γ ξdx − γ ηdy is the difference of the actions of κ ◦ γ and γ, and k(γ) ∈ Z is a “Maslov index”, both quantities depending only on the homotopy class of γ. (Requiring only the unitarity of U , we could take θ(γ) = S(γ)/h.) When discussing the improved property (2.3), recall from [13] and [29], that on a manifold with a preferred positive density, we can define the Weyl symbol of a 0-th order h-pseudodifferential operator modulo O(h2 ) by taking the ordinary Weyl symbol for some system of local coordinates x1 , . . . , xn for which the preferred density reduces to the Lebesgue measure. Clearly Proposition 2.1 extends to this situation. We first notice that if − n+N 2 eiφ(x,y,θ)/h a(x, y, θ; h)u(y)dydθ V u(x) = h is an elliptic Fourier integral operator with leading symbol a0 (x, y, θ) = 0 on Cφ , then we can obtain V ∗ V = 1 + O(h) by multiplying a0 by a positive real-analytic function. The same remark applies to loc X ), V : HΨ (ΩY ) → HΦloc (Ω
where we put V = S ◦ V ◦ T −1 and represent it as in [20] by V u(x) = h−n eiψ(x,y)/h b(x, y; h)u(y)e−2Ψ(y)/h L(dy).
(2.7)
Here ψ(x, y) is the multi-valued grad-periodic function near πx,y Γ, with ∂x,y ψ = 0, ∂x,y b = 0 near πx,y (Γ), ∂x ψ(x, y) =
2 2 ∂x Φ(x), ∂y ψ(x, y) = ∂y Ψ(y) on πx,y (Γ), i i
Φ(x) + Ψ(y) + Im ψ(x, y) ∼ dist ((x, y), πx,y (Γ))2 , where Γ denotes the graph of κ . (In [20] the first equation holds only to infinite order on πx,y (Γ) and the present improvement follows from the analyticity of κ .)
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Recall that Im ψ is single-valued, and that ξdx − ηdy, var(κ◦γ,γ) ψ = κ◦γ
(2.8)
γ
is the action difference, when γ is a closed curve in ΛΨ and ( κ ◦ γ, γ) denotes the Y , Ω X whenever so curve t → ( κ(γ(t)), γ(t)). Here we also identify ΛΨ , ΛΦ with Ω is convenient. Thus after multiplying b| π (Γ) by a positive real-analytic function, we may x,y assume that V ∗ V = 1 + O(h). (2.9) In order to have the improved Egorov property, we further need that locally on πx,y (Γ): arg b0 (x, y) = K(y) + Const., (notice that x = x(y) on Γ),
(2.10)
where K(y) is a grad-periodic function on πx,y (Γ), that we do not try to compute here, but whose existence we infer from Proposition 2.1 and the computation of V as S ◦ V ◦ T −1 , with V written microlocally with a real phase as in (2.1). We can find b0 satisfying (2.10) everywhere if we accept that b0 | π (Γ) is x,y multi-valued. More precisely, K is not globally well defined on πx,y (Γ) ΩY , but ω = dK is a well defined closed real 1-form on ΩY and we can find b0 | π (Γ) , x,y unique up to a constant factor of modulus 1, such that (2.9), (2.10) hold, though b0 will be multi-valued: γ∗ b0 = exp (i ω)b0 , (2.11) γ
where γ∗ b0 denotes the new locally defined symbol obtained by following b0 around the closed loop γ in πx,y (Γ) ΩY . Proposition 2.3 We have γ ω = k(γ) π2 for some integer k(γ) ∈ Z, for every closed loop γ ⊂ πx,y (Γ). Proof. Let γ be a closed loop and cover γ by small open topologically trivial sets 0, Ω 1, . . . , Ω N −1 with increasing index corresponding to the orientation of γ in Ω N = Ω 0 . Let Ωj be the corresponding regions in ΩY . In Ωj , the natural way. Let Ω we represent V by n+Nj Vj u(x) = h− 2 eiφj (x,y,θ)/haj (x, y, θ; h)u(y)dydθ. (2.12) θ∈RNj
For a given point in Ωj ∩ Ωj+1 , we have φj = φj+1 ,
aj+1 = rj+1,j eiαj+1,j π/2 aj + O(h),
rj+1,j > 0, αj+1,j ∈ Z,
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at the corresponding points in Cφj , Cφj+1 , provided that we require all the fibervariable dimensions Nj to have the same parity. (Cf. [15].) This last property is easy to achieve since we can always add one fiber-variable. We conclude that π ω = (α1,0 + α2,1 + · · · + αN,N −1 ), 2 γ
and the proposition follows. Take V as above with b = b0 in (2.7), so that (2.9), (2.10) hold. Put 1
U = V (V ∗ V )− 2 .
(2.13)
= SU T −1 is of the form (2.7) with b = b0 + O(h). We have U ∗ U = 1 and Then U U satisfies (2.3). Since the unitarization is a local operation which commutes with multiplication by a constant factor of modulus 1, (2.11) becomes valid also for b: π
γ∗ b = eik(γ) 2 b.
(2.14)
Here we also used Proposition 2.3. Summing up, we get Theorem 2.4 Under the assumptions above on κ, we can find a microlocally defined multi-valued Fourier integral operator U associated to κ, and a corresponding lift = SU T −1 of the form (2.7), such that U is unitary: U ∗ U = 1 + O(e−1/(Ch) ), U satisfies the improved Egorov property (2.3), and γ∗ U = ei(S(γ)/h+k(γ)π/2) U, for every closed loop in ΩY , where k(γ) ∈ Z and S(γ) = ξdx − ηdy. κ◦γ
γ
3 Reduction by averaging along trajectories Let P , M be as in the introduction. We work in a neighborhood of p−1 (0) ∩ T ∗ M , and recall that P = P has the semi-classical principal symbol p = p + iq + O(2 ),
(3.1)
in a complex neighborhood of p−1 (0)∩T ∗ M . Let G0 be an analytic function defined near p−1 (0) ∩ T ∗ M such that Hp G0 = q − q,
(3.2)
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where q is the trajectory average, defined in (1.15). We may take 1 G0 = T (E)
T (E)/2
−T (E)/2
(1R− (t)(t +
T (E) T (E) ) + 1R+ (t)(t − ))q ◦ exp tHp dt, (3.3) 2 2
on p−1 (E). We replace R4 by the new IR-manifold ΛG0 = exp (iHG0 )(R4 ),
(3.4)
which is defined in a complex neighborhood of p−1 (0) ∩ T ∗ M . Writing (x, ξ) = exp (iHG0 )(y, η), and using ρ = (y, η) as real symplectic coordinates on ΛG0 , we get p | Λ
G0
= p (exp (iHG0 )(ρ)) =
(3.5)
∞ (iHG0 )k (p ) = p + iq + O(2 ). k! k=0
Iterating this procedure, or looking more directly for G(x, ξ, ) as an asymptotic sum ∞ k Gk (x, ξ) (3.6) G∼ 0 −1
in some complex neighborhood of p (0) ∩ T ∗ M , we see that we can find G1 , G2 . . . such that if (3.7) ΛG = exp (iHG )(R4 ), and we again write ΛG (x, ξ) = exp (iHG )(y, η) and parametrize by the real variables (y, η), then p | Λ
G
= p + iq + 2 q2 + 3 q3 + · · · ,
(3.8)
where qj = qj , j ≥ 2. This means that we can transform p to p ◦ exp (iHG ) in such a way that we get a new leading symbol which Poisson commutes with the unperturbed leading symbol. As is well known in the selfadjoint case, this construction can be extended to the level of operators, and we may develop this globally in another paper. In the present work we will do it only after a reduction to a torus-like situation. After replacing p by p ◦ exp (iHG0 ) and correspondingly P , by U−1 ◦ P ◦ i U , where U is the Fourier integral operator U = e− h iG0 (x,hDx ) = e h G0 (x,hDx ) (defined microlocally near p−1 (0) ∩ T ∗M ), we may assume that our operator P is microlocally defined near p−1 (0) ∩ T ∗ M and has the h-principal symbol p = p + iq + O(2 ).
(3.9)
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Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
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This can be done in such a way that P=0 remains the original unperturbed operator. We refer to the beginning of Section 6 for the construction of U by means of an FBI-transform. Let γ0 ⊂ p−1 (0) ∩ T ∗ M be a closed Hp -trajectory and assume that T (0) is the minimal period of γ0 . Let g : neigh (0, R) → R be the analytic function defined by T (E) , g(0) = 0. (3.10) g (E) = 2π Then Hg◦p = g (p)Hp has a 2π-periodic flow and the same closed trajectories as Hp . Clearly 2π is the minimal period of γ0 when viewed as a Hg◦p -trajectory. Proposition 3.1 There exists an analytic canonical transformation κ : neigh ({τ = x = ξ = 0}, T ∗(St1 × Rx )) → neigh (γ0 , T ∗ M ), mapping {τ = x = ξ = 0} onto γ0 , such that g ◦ p ◦ κ = τ . Proof. Fix a point ρ0 ∈ γ0 and choose local symplectic coordinates (t, τ ; x, ξ) centered at ρ0 , with g ◦ p = τ . This means that {ξ, x} = 1, {t, x} = {t, ξ} = 0
(3.11)
Hτ t = 1, Hτ x = Hτ ξ = 0.
(3.12)
Now extend the definition of t, τ, x, ξ to a full neighborhood of γ0 , by putting τ = g ◦ p and requiring t, x, ξ to solve (3.12). Since the Hτ -flow is 2π-periodic (with 2π as the minimal period) near γ0 , we see that x, ξ are well defined singlevalued functions, while t becomes multi-valued in such a way that it increases by 2π each time we make a loop in the increasing time direction. (3.11) extends to a full neighborhood of γ0 . This is equivalent to the proposition. Notice that p ◦ κ = f (τ ),
(3.13)
where f := g −1 . From (3.9) we infer that p ◦ κ = f (τ ) + iq(τ, x, ξ) + O(2 ),
(3.14)
for a new function q which is independent of t (and obtained from the earlier one by composition with κ). If we let the Fourier integral operator U quantize κ as in Section 2, we get a new operator U −1 P U with leading semi-classical symbol p ◦ κ as in (3.14). (Here P is the new version of P ; P,new = U −1 P,old U .) Now write simply p, p , P for the transformed objects. Then P = P (t, x, hDt,x , ; h) is the formal Weyl quantization of a symbol P (t, x, τ, ξ, ; h) which has an asymptotic expansion (1.5) in the space of holomorphic functions in a fixed complex
14
M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
neighborhood of {Im t = τ = x = ξ = 0} in T ∗ (S1 × C), with S1 = S 1 + iR, and we will use the same notation as in Section 1. (An exact value of the new symbol P (t, x, τ, ξ, ; h) cannot be easily defined, but we know how to define it mod O(e−1/(Ch) ). We shall however avoid using the full power of analytic pseudodifferential operators, and content ourselves with the knowledge of P mod O(h∞ ).) Now look for G(1) = G1 (t, τ, x, ξ) + 2 G2 (t, τ, x, ξ) + · · · such that p ◦ exp iHG(1) = f (τ ) + iq(τ, x, ξ) + O(2 ) is independent of t. Here the left-hand side can be written ∞ 1 (iHG(1) )k p , k!
k=0
and we get p + i2 HG1 (f (τ )) + O(3 ) = f (τ ) + iq(τ, x, ξ) − i2 f (τ )
∂ G1 + O(2 ) + O(3 ), ∂t
where the O(2 ) term is the same as in (3.14). It is clear that we can find G1 so that the 2 -term in this expression is independent of t. Looking at the O(3 )-term we then determine G2 and so on. (In this construction, we could have applied κ at the very beginning before replacing q by q by averaging, and then incorporated G0 into the expression G = G0 + G1 + · · · , and as already indicated, this could also have been done entirely (and in a full neighborhood of p−1 (0) ∩ T ∗ M ), before applying κ.) After replacing p by p ◦ exp iHG(1) , we are now reduced to the case when p = f (τ ) + iq(τ, x, ξ) + O(2 )
(3.15)
is independent of t, up to O(∞ ). Finally we remove the t-dependence from the lower order terms. After conjugating P by a Fourier integral operator V , which quantizes exp iHG(1) , we may assume that p in (3.15) is the principal symbol of P (and that it is independent of t). Look for an h-pseudodifferential operator A(t, x, hDt,x , ; h) with symbol A(t, x, τ, ξ, ; h) ∼
∞
ak (t, x, τ, ξ, )hk ,
(3.16)
k=1
such that the full (Weyl) symbol of e
i hA
P e
− hi A
=e
i h adA
∞ 1 i ( adA )k P P = k! h k=0
(3.17)
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Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
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is independent of t. Since A = O(h) we know that hi adA lowers the order in h by one (with the convention that a symbol = O(h−j ) is of order j), so (3.17) makes sense asymptotically. The subprincipal symbol of (3.17) is h(p1, (x, ξ) + {p , a1 }) = h(p1, (x, ξ) + f (τ )
∂ a1 (t, τ, x, ξ, ) + O()), ∂t
and we make this independent of t by successively determining the coefficients in the asymptotic series a1 (t, τ, x, ξ, ) =
∞
a1,j (t, τ, x, ξ)j .
j=0
After that we return to (3.17) and see that the construction of a2 , a3 , . . . is essentially the same. Actually, we do not have to do this construction in 2 steps, and we can view G(1) above as (a constant factor times) the leading symbol a0 = O(2 ) in A∼
∞
ak (t, x, τ, ξ, )hk ,
(3.18)
k=0
such that if P denotes the very first operator we get on S 1 × R, then the left-hand side of (3.17) has a symbol which is well defined as an asymptotic series in (, h) and is independent of t, up to O(h∞ ). This can be seen by first determining a0 from (3.17) (leading to a repetition of what we already did) and then the other terms. (When is small but fixed, the problem becomes more subtle and the break-up into two steps is more natural, with the first step being the one containing the new difficulties.) Summing up the discussion of this section, we have Proposition 3.2 Let P , M be as in Section 1. Let γ0 ⊂ p−1 (0) ∩ T ∗ M be a closed Hp -trajectory where T (0) is the minimal period and let κ be the canonical transformation of Proposition 3.1. Let U be a corresponding elliptic Fourier integral operator as in Section 2. Then there exist G(x, ξ, ) (independent of γ0 , κ, U ) with the asymptotic expansion (3.6) in the space of holomorphic functions in some fixed complex neighborhood of p−1 (0) ∩ T ∗ M and a symbol A(t, x, τ, ξ, ; h) as in (3.16), where ∞ ak,j (t, x, τ, ξ)j (3.19) ak ∼ j=0
in the space of holomorphic functions in a fixed complex neighborhood of Im t = τ = x = ξ = 0 in T ∗ (S1 × C)), such that if G, A also denote the corresponding Weyl quantizations, the operator i i P = e h A U −1 e− h G P e h G U e− h A = Ad
i
e h A U −1 e− h G
P
(3.20)
16
M. Hitrik and J. Sj¨ ostrand
has a symbol P (x, τ, ξ, ; h) ∼
∞
Ann. Henri Poincar´e
pk (x, τ, ξ, )hk
(3.21)
0
∞ independent of t (up to O(h∞ )). Here each pk = pk (x, τ, ξ, ) ∼ j=0 pk,j (x, τ, ξ)j in the space of holomorphic functions in a fixed complex neighborhood of τ, x, ξ = 0. Moreover (3.22) p0, = f (τ ) + iq(τ, x, ξ) + O(2 ). If q has a non-degenerate extreme value along γ0 , then the proposition is directly applicable (see Section 5), while in other situations (such as in Section 4), it is not global enough.
4 Normal forms and quasi-eigenvalues in the torus case Let P, M, p, q, q, Λ0,F0 be as in Section 1. After replacing q by q − F0 , we may assume that F0 = 0, so we consider Λ0,0 : p = 0, Re q = 0.
(4.1)
Notice that Λ0,0 is invariant under the Hp -flow. We assume that T (0) is the minimal period for all the closed trajectories in Λ0,0 and that dp, dRe q are independent at the points of Λ0,0 ,
(4.2)
so that Λ0,0 is a Lagrangian manifold and also a union of tori. Assume for simplicity that Λ0,0 is connected, so that it is equal to one single Lagrangian torus. In this section we work microlocally near Λ0,0 and proceed somewhat formally. In Section 6 we follow up with suitable function spaces and see how to justify the computation of the spectrum via a global Grushin problem. We have seen that we can reduce ourselves to the case when p = p + iq + O(2 ).
(4.3)
Assume from now on that q is real-valued or more generally that q is a function of p and Re q. We can make a real canonical transformation κ : neigh (ξ = 0, T ∗T2 ) → neigh (Λ0,0 , T ∗ M ), T2 = (R/2πZ)2 ,
(4.4)
such that p ◦ κ = p(ξ1 ), q ◦ κ = q(ξ) (with a slight abuse of notation). Recall that this can be done in the following way: Let ΛE,F be the Lagrangian torus given by p = E, Re q = F , for (E, F ) ∈ neigh (0, R2 ). Let γ1 (E, F ) be the cycle in ΛE,F corresponding to a closed Hp -trajectory with minimal period, and let γ2 (E, F ) be a second cycle so that γ1 , γ2 form a fundamental system of cycles on the torus ΛE,F . Necessarily γ2 maps to the simple loop given by Re q = F in the abstract quotient manifold p−1 (E)/RHp . Now it is classical (see [1]) that
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Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
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we can find a real analytic canonical transformation κ : neigh (η = 0, T ∗ T2 ) (y, η) → (x, ξ) ∈ neigh (Λ0,0 , T ∗ M ), such that 1 ( ξdx − ξdx), ηj = 2π γj (E,F ) γj (0,0) where E, F depend on (x, ξ) and are determined by (x, ξ) ∈ ΛE,F , i.e., by E = p(x, ξ), F = Re q(x, ξ). We also know that here η1 = η1 (E) is a function of E only. Let us also recall that κ can be constructed as follows: We start by taking a first canonical transformation κ0 : neigh (ξ = 0, T ∗ T2 ) → neigh (Λ0,0 , T ∗ M ) such that the zero section is mapped to Λ0,0 and the lines {x2 = Const, ξ = 0} are mapped onto the closed Hp -trajectories in Λ0,0 . Then using κ0 , we can consider p, q as living on T ∗ T2 . ΛE,F is then given by ξ = φx , φ = φper (x, E, F ) + η1 x1 + η2 x2 , with det φx,(E,F ) = 0, with ηj = ηj (E, F ) as above (now being the actions/2π with respect to ξdx), and φper being (2πZ)2 -periodic. Moreover, φx (x, η) = 0, η = 0 for E = F = 0. It is easy to check, using that our functions are real-valued, that (E, F ) → (η1 (E, F ), η2 (E, F )) is a local diffeomorphism, so we can use η1 , η2 as new parameters replacing E, F , and write φ = φ(x, η). Consider κ1 : (
∂φ ∂φ , η) → (x, ) ∂η ∂x
which maps the zero section to itself. Then κ := κ0 ◦κ1 has the required properties. Let U be a corresponding Fourier integral operator, implementing κ, so that if we denote by P also the conjugated operator U −1 P U , we have a new operator with leading symbol (4.5) p = p(ξ1 ) + iq(ξ) + O(2 ). For the conjugated operator, we still have the property that P=0 is selfadjoint. From the assumption (4.2) about linear independence, we get ∂ξ1 p(0) = 0, ∂ξ2 Re q(0) = 0.
(4.6)
As in the section, we can find an h-pseudodifferential operator A with preceding ν symbol ∞ h a (x, ξ, ), a0 = O(2 ), such that formally ν ν=0 i
i
i
e h A P e− h A = e h adA (P ) =
∞ 1 i ( adA )k (P ) =: P , k! h
k=0
with P (x, ξ, ; h) independent of x1 , and leading symbol p = p(ξ1 ) + iq(ξ) + O(2 )
(4.7)
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M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
also independent of x1 . We recall that the symbol A(x, ξ, ; h) is a formal power series both in and h with coefficients all holomorphic in the same complex neighborhood of ξ = 0. This construction can be done in such a way that P=0 is selfadjoint. We next look for a further conjugation that eliminates the x2 -dependence in the symbol. a) We start by considering the general case, when the subprincipal symbol of P=0 is not necessarily 0, so that the complete symbol of P takes the form P (x2 , ξ; h) =
∞
hν pν (x2 , ξ, ),
(4.8)
ν=0
with p0 (x2 , ξ, ) = p = p(ξ1 ) + iq(ξ) + O(2 ),
(4.9)
and p1 (x2 , ξ, 0) not necessarily identically equal to 0. The easiest case is when h/ ≤ O(hδ1 ) for some δ1 > 0, so that we can consider h/ as an asymptotically small parameter. Look for ∞ h h hν bν (x2 , ξ, , ), B(x2 , ξ, , , h) = ν=0
(4.10)
with bν = O( + h/), such that on the operator level (with hDx instead of ξ), i i h e h B P e− h B =: P (hDx , , , h)
(4.11)
has a symbol independent of x. Notice that B(x2 , hDx , ; h) and p(hDx1 ) commute. On the symbol level we write h h P = p(ξ1 ) + (iq(ξ) + O() + p1 (x2 , ξ, ) + h p2 (x2 , ξ, ) + · · · ) (4.12) h h = p(ξ1 ) + (r0 (x2 , ξ, , ) + hr1 (x2 , ξ, , ) + · · · ), with
h h h r0 (x2 , ξ, , ) = iq(ξ) + O() + p1 = iq(ξ) + O() + O( ), r1 =
h p2 (x2 , ξ, ), . . .
Notice that rj = O(h/) for j ≥ 1. We shall treat h/ as an independent parameter. We use this and develop (4.11) to get, with adb c denoting the symbol of adb(x,hDx ) c(x, hDx ) = [b(x, hD), c(x, hD)],
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p(ξ1 ) +
Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
∞ ∞
···
k=0 j1 =0
∞ ∞
h+j1 +···+jk
jk =0 =0
19
∞ i 1 i ( adbj1 )...( adbjk )r = p(ξ1 ) + hn rn , k! h h n=0
with rn being equal to the sum of all coefficients for hn resulting from all the expressions 1 i i h+j1 +···+jk ( adbj1 ) · · · ( adbjk )r , (4.13) k! h h with + j1 + · · · + jk ≤ n. The first term is r0 =
1 H k r0 = r0 ◦ exp (Hb0 ), k! b0
where we want r0 to be independent of x2 (in addition to x1 ). We get with b0 = O( + h/): h r0 = iq(ξ) + O( + h/) − i∂ξ2 q∂x2 b0 + O((, )2 ),
(4.14)
and using that ∂ξ2 q = 0, it is clear how to construct b0 = O( + h/) as a formal Taylor series in , h/, so that r0 = iq(ξ)+O(+h/) is independent of x (modulo a term O(h∞ )). i Assume for simplicity that the conjugation by e h b0 (x2 ,hDx ,,h/) has already been carried out, so that we are reduced to the case when r0 = iq(ξ)+O(+h/) is independent of x2 , and rj = O( + h/) for jν ≥ 1. Then hlook for a new conjugation exp hi adB , with B(x2 , ξ, , h/; h) = ∞ ν=1 h bν (x2 , ξ, , ). The new expression for the left-hand side of (4.11) becomes p(ξ1 ) +
∞ ∞ k=0 j1 =1
···
∞ ∞ jk =1 =0
h+j1 +···+jk
∞ 1 i i ( adbj1 )...( adbjk )r = p(ξ1 ) + hn rn , k! h h n=0
(4.15) with rn equal to the sum of all coefficients for hn resulting from the expressions (4.13) with + j1 + · · · + jk ≤ n and jν ≥ 1. Then r0 = r0 , r1 = r1 + Hb1 r0 = r1 − Hr0 b1 , . . . , rn = rn − Hr0 bn + sn , where sn only depends on b1 , . . . , bn−1 and is the sum of all coefficients of hn arising in the expressions (4.13) with +j1 +· · ·+jk ≤ n, j1 , . . . , jk , < n, jν ≥ 1. It is therefore clear how to find b1 , b2 , . . . successively with bj = O( + h/), such that all the rj are independent of x and = O( + h/). This completes the proof of (4.11). Summing up the discussion so far, if we do not make any assumption on the subprincipal symbol of P=0 and restrict the attention to h/ ≤ O(hδ1 ) for some δ1 > 0, then we can find B0 = b0 (x2 , hDx , , h/), b0 = O( + h/),
20
M. Hitrik and J. Sj¨ ostrand
and B1 =
∞
Ann. Henri Poincar´e
bν (x2 , hDx , , h/)hν , bν = O( + h/),
ν=1
such that
i i P := e h adB1 e h adB0 P
(4.16)
has a symbol independent of x: h h P = p(ξ1 ) + (r0 (ξ, , ) + hr1 (ξ, , ) + · · · ),
(4.17)
with r0 = iq(ξ) + O( + h/), and rν = O( + h/) for ν ≥ 1. Remaining in the general case, without any assumption on the lower order terms, we now assume merely that h/ ≤ δ0 for some sufficiently small δ0 > 0. This means that we can no longer construct b0 by a formal Taylor series in h/, i and we shall replace e h b0 (x2 ,hDx ,,h/) by a Fourier integral operator, constructed directly. Look for φ = φ(x2 , ξ, , h/) solving h h r0 (x2 , ξ1 , ξ2 + ∂x2 φ, , ) = r0 (·, ξ, , ),
(4.18)
where · denotes the average with respect to x2 . By the implicit function theorem, (4.18) has a solution with ∂x2 φ single-valued and O( + h/). If we Taylor expand (4.18), we get h h h h (∂ξ2 r0 )(x2 , ξ, , )∂x2 φ + (r0 (x2 , ξ, , ) − r0 (·, ξ, , )) = O(( , )2 ), and using also that h h ∂ξ2 r0 (x2 , ξ, , ) = i∂ξ2 q(ξ) + O( + ), we get, φ = φper + x2 ζ2 , with ζ2 = ζ2 (ξ, , h ) = O((, h/)2 ), and φper = O((, h/)) periodic in x2 . Put η = η(ξ, , h/) = (ξ1 , ξ2 + ζ2 ), and h ψ(x, η, , ) = x · η + φper , where φper is viewed as a function of η rather than ξ. Consider the canonical transformation κ : (ψη , η) → (x, ψx ),
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Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
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which is ( + h/)-close to the identity and can be viewed as a family of transforms depending analytically on the parameter ξ1 . With ξ = ξ(η, , h ), we have by construction: h h h h (r0 ◦ κ)(y, η, , ) = r0 (·, ξ, , ) = r0 (·, η, , ) + O(2 + ( )2 ), (4.19) and this is a function of (y, η) which is independent of y. Notice that p(ξ1 ) is unchanged under composition with κ. We can quantize κ as a Fourier integral operator U and after conjugation by this operator, we may assume that we have a new operator P as in (4.12) with r0 = iq(ξ) + O( + h/) independent of x and with rj = O( + h/) . i As before, we can then make a further conjugation e h adB1 in order to remove the x-dependence completely and the conclusion is that if we make no assumption on the subprincipal symbol and restrict the attention to h/ ≤ δ0 , for δ0 > 0 small enough, then we can find a Fourier integral operator, i 1 −1 U u(x; h) = (4.20) e h (ψ(x,η)−y·η) a(x, η; h)u(y)dydη, (2πh)2 with ψ(x, η) = x · η + φper (x2 , η, , h/), φper = O( + h/), and B1 = such that
∞
h h bν (x2 , hDx , , )hν , bν = O( + ), ν=1 i P := e h adB1 AdU P
has a symbol independent of x as in (4.17), with the same estimates as there. b) We now assume that in the original problem, P=0 has subprincipal symbol 0. Then after a first time averaging, transportation to the torus, and the elimination of the x1 -dependence, we may assume that P (x2 , ξ, ; h) =
∞
hν pν (x2 , ξ, ),
(4.21)
ν=0
with p0 independent of x mod O(2 ): p0 (x2 , ξ, ) = p(ξ1 ) + iq(ξ) + O(2 ),
(4.22)
p1 (x2 , ξ, 0) = 0.
(4.23)
(Recall from Section 2 and the references given there, that the canonical transformations can be quantized in such a way that Egorov’s theorem holds modulo O(h2 ).) In analogy with (4.12), we have with p1 (x2 , ξ, ) = q1 (x2 , ξ, ), P
=
p(ξ1 ) + (iq(ξ) + O() + hq1 (x2 , ξ, ) +
=
p(ξ1 ) + (r0 (x2 , ξ, ,
h2 h2 p2 + h p3 + · · · )
h2 h2 ) + hr1 (x2 , ξ, , ) + h2 r2 + · · · ),
(4.24)
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M. Hitrik and J. Sj¨ ostrand
Ann. Henri Poincar´e
with h2 ) h2 r1 (x2 , ξ, , ) h2 r2 (x2 , ξ, , )
r0 (x2 , ξ, ,
= iq(ξ) + O() + = q1 (x2 , ξ, ) + =
h2 p2 ,
h2 p3 ,
h2 p4 , . . .
We first consider the case when h2 ≤ hδ1 ,
(4.25) i
h2
for some fixed δ1 > 0. A first conjugation by e h b0 (x2 ,hDx ,, ) , with b0 = O( + h2 /), allows us to make r0 independent of x2 , and we still have (4.24) with rj = O(1) for j ≥ 1. Then we look for a new conjugation exp hi adB1 with B1 (x2 , ξ, ,
∞ h2 h2 ; h) = hν bν (x2 , ξ, , ). ν=1
(4.26)
The conjugated operator (4.11) can be expanded as in (4.15) and as after that equation it is clear how to get bν = O(1) for ν ≥ 1, such that the resulting rn are independent of x2 , with r0 (ξ, , h2 /) = iq(ξ) + O( + h2 /). Summing up the discussion so far, if we assume that the subprincipal symbol of P=0 vanishes, and restrict the attention to the range (4.25) for some fixed 2 2 δ1 > 0, then we can find B0 = b0 (x2 , hDx , , h ) with b0 = O( + h ) and 2 B1 (x2 , hDx , , h ; h) with symbol (4.26), and bν = O(1), such that i i e h adB1 e h adB0 P = P
has the symbol p(ξ1 ) + (r0 (ξ, ,
h2 h2 ) + hr1 (ξ, , ) + · · · )
(4.27)
independent of x and with r0 = iq(ξ) + O( +
h2 ), rν = O(1), ν ≥ 1.
(4.28)
If we replace (4.25) by the weaker assumption, h2 ≤ δ0 , δ0 1,
(4.29)
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Non-selfadjoint Perturbations of Selfadjoint Operators in 2 Dimensions I
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i
then again we have to replace the conjugation by e h B0 by that by a Fourier integral operator constructed as earlier: We solve (4.18) (with h/ replaced by h2 /) and get ∂x2 φ single-valued and O( + h2 /). Taylor expanding (4.18) and using that ∂ξ2 r0 (x2 , ξ, ,
h2 h2 ) = i∂ξ2 q(ξ) + O( + ),
we get φ = φper + x2 ζ2 , 2
2
with ζ2 = ζ2 (ξ, , h ) = O((, h )2 ) and φper = O( + h2 /) periodic in x2 . Again we put η = η(ξ, , h2 /) = (ξ1 , ξ2 + ζ2 ) and ψ(x, η, ,
h2 ) = x · η + φper .
The canonical transformation κ : (ψη , η) →
(x, ψx ) is ( + h2 /)-close to the 2 identity and with ξ = ξ(η, , h /), we have by construction (r0 ◦ κ)(y, η, ,
h2 h2 h2 h2 ) = r0 (·, ξ, , ) = r0 (·, η, , + O((, )2 ),
(4.30)
which is a function independent of y. Let U −1 be the corresponding Fourier integral operator as before. Then after replacing P by AdU P , we still have (4.24), where now r0 = iq(ξ) + O( + h2 /) is independent of x and rj = O(1) for j ≥ 1. i We can then make a further conjugation by e h B1 as before, and we get the following conclusion: Assume that the subprincipal symbol of P=0 vanishes and restrict the attention to the range (4.29). Then we can find an elliptic Fourier integral operator U −1 of the form (4.20) with ψ as above and B1 (x2 , hDx , , h2 /; h) with symbol (4.26), and bν = O(1), such that i e h adB1 AdU P = P (hDx , , h2 /; h)
(4.31)
has a symbol P (ξ, , h2 /; h) of the form (4.27), such that (4.28) holds. We finish this section by discussing what spectral results can be expected from the reductions above. The first reduction (as in Section 3) was to conjugate the original operator P by a Fourier integral operator eiG(x,hD,)/h , with G(x, ξ, ) ∼ (G0 (x, ξ) + G1 (x, ξ) + · · · ), defined in some complex neighborhood of p−1 (0) ∩ T ∗ M , to achieve that the leading symbol of the conjugated operator is of the form p + iq + O(2 ) and Poisson commutes with p. At least formally, the new operator also acts on L2 (M ) and we have no Floquet type conditions to worry about. Geometrically, this corresponds to the fact that a canonical transformation κ = exp HG with asingle-valued generator G = O() preserves actions along closed loops: κ◦γ ξdx = γ ηdy, for every closed loop γ. The second reduction was to take κ in (4.4) and to conjugate by the inverse of the corresponding Fourier integral operator U . Let α1 (=γ0 ) and α2 be the
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fundamental cycles in Λ0,0 given by αj = κ ◦ βj , where β1 , β2 are the fundamental cycles in T2 {(x, 0) ∈ T ∗ T2 }, given by x2 = 0 and x1 = 0 respectively. Put Sj = ξdx, (4.32)
αj
so that Sj is the difference of actions, κ◦βj ξdx − βj ηdy, j = 1, 2. Since κ is a canonical transformation we know that if β is a closed loop homotopic to βj , then ξdx − ηdy = S . j κ◦β β As in [20] or as in Theorem 2.4, we see (at least formally) that if we want U u to be single-valued on M (possibly defined only microlocally near Λ0,0 ), then u should not necessarily be periodic on R2 (i.e., a function on T2 ) but a Floquet periodic function with iν·S
u(x − ν) = e 2πh +
iν·k0 4
u(x), ν ∈ (2πZ)2 , S = (S1 , S2 ), k0 ∈ Z2 .
(4.33)
The conjugated operator Ad −1 hi G P should therefore act on Floquet periodic U e functions as in (4.33). The further conjugations are by operators on the torus that conserve the property (4.33). This is clear from the definitions, and corresponds to the fact that a canonical transformation: (y, η) → (x, ξ), generated by ψ(x, η) = x·η + φper(x, η) and close to the identity, conserves actions. Indeed, on the graph of the transform, we have ξdx + ydη = dψ, so ξdx − ηdy = d(ψ − y · η) = d((x − y) · η + φper (x, η)), and (x − y) · η + φper (x, η) is single-valued on the graph. On the other hand the space of Floquet periodic functions as in (4.33), equipped with the L2 -norm over a fundamental domain of T2 , has the ON basis: i
ek (x) = e h x·(h(k−
k0 4
S )− 2π )
, k ∈ Z2 ,
(4.34)
and applying our reductions down to the operator P in the cases (a) and (b) above, we get formally (in the sense that we do not define the notion of quasi-eigenvalue): Proposition 4.1 Recall that we took F0 = 0 and that S, k0 are the actions and the Maslov indices in (4.32), (4.33). 1 1 a) In the general case, P has the quasi-eigenvalues in ] − |O(1)| , |O(1)| [+i] − 1 1 δ |O(1)| , |O(1)| [ for = O(h ), h/ 1: S h k0 , , ; h , k ∈ Z2 , P h(k − ) − 4 2π where P (ξ, , h ; h) is holomorphic in ξ ∈ neigh (0, C2 ), smooth in (0, R) and has the asymptotic expansion (4.17), when h → 0.
(4.35) h ,
∈ neigh
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b) If we assume that P=0 has subprincipal symbol 0, then P has the quasi-eigen1 1 1 1 values in ] − |O(1)| , |O(1)| [+i] − |O(1)| , |O(1)| [ for = O(hδ ), h2 / 1: S h2 k0 , , ; h , k ∈ Z2 , P h(k − ) − 4 2π
(4.36)
where P(ξ, , h2 /; h) is holomorphic in ξ ∈ neigh (0, C2 ), smooth in and h2 / ∈ neigh (0, R) and has the asymptotic expansion (4.27), (4.28), when h → 0.
5 Quasi-eigenvalues in the extreme cases We make the assumptions of the case II in the introduction and assume, in order to fix the ideas, that 0 = F0 = Re qmin,0 . (5.1) Apply Proposition 3.2 and reduce P near γ0 to P = P(x, hDt,x , ; h) with symbol described in that proposition. Recall that P has the leading symbol p = f (τ ) + iq(τ, x, ξ) + O(2 ),
(5.2)
where q(τ, x, ξ) is equal to the original averaged function q, composed with the canonical transformation κ of Proposition 3.1. The assumptions (1.23) and (5.1) imply that Re q(0, x, ξ) ∼ |(x, ξ)|2 (5.3) on the real domain. Also recall that we have the assumption (1.17) which with (5.3) implies that q(τ, x, ξ) = g(τ, Re q(τ, x, ξ)) (5.4) on the real domain, for some analytic function g(τ,q) with g(0,0) = 0, Re g(τ,q) = q. We conclude that (x, ξ) → iq(τ, x, ξ) + O(), appearing in (5.2), has a nondegenerate critical point (x(τ, ), ξ(τ, )) = O(|τ |+) depending analytically on τ, and real when τ ∈ R, = 0. After composition with the (τ, )-dependent (symplectic) translation (x, ξ) → (x − x(τ, ), ξ − ξ(τ, )) and subtracting the corresponding critical value, we may assume that the critical point is (0, 0) and hence that p (τ, x, ξ) = f (τ ) + iq(τ, x, ξ, ),
(5.5)
Re q(τ, x, ξ, ) ∼ |(x, ξ)|2
(5.6)
q(τ, x, ξ, 0) = g(τ, Re q(τ, x, ξ, 0)),
(5.7)
with on the real domain, and
on the real domain, where g(τ, 0) = 0, Re g(τ, q) = q.
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We shall next construct a (τ, )-dependent canonical transformation in the x, ξ-variables, which reduces p (τ, x, ξ) to a function of τ, , 12 (x2 + ξ 2 ). In doing so, we essentially follow Appendix B of [13], where the model was xξ rather than p0 := 12 (x2 + ξ 2 ). These two quadratic forms are equivalent up to a constant factor and composition by a linear complex canonical transformation, so the only difference is that the real domains are not the same. Let p(x, ξ) ∼ (x, ξ)2 be real and analytic in a neighborhood of (0,0). Lemma 5.1 There exists a real and analytic function f (E) defined near E = 0, with f (0) = 0, f (0) > 0, such that the Hamilton flow of f ◦ p is 2π-periodic, with 2π as its minimal period except at (0, 0). Proof. Consider, first for 0 < E 1, the action ξdx = E I(E) = p−1 (E)
−1 qE (1)
ηdy,
√ where qE (y, η) = E1 p( E(y, η)), so that q0 is a positive quadratic form (in the √ limit E → 0). Then qE is an analytic function of E in a neighborhood of 0 and consequently we have the same fact for I(E). If we let E describe a simple closed loop around 0 in neigh (0, C) \ {0}, then qE (y, η) transforms into qE (y, η) = qE (−y, −η) and it follows that I(E) transforms into itself. It follows that I(E) is analytic as a function of E. The period T (E) of the Hp -flow is given by T (E) = I (E) and the period of the Hf ◦p -flow is T (E)/f (E). It suffices to choose f with f (E) = T (E)/2π and f (0) = 0. In the following discussion, we replace p by f ◦ p, so that we get a reduction to the case when the Hp -flow is 2π-periodic. After composition with a real linear canonical transformation, we may assume that p(x, ξ) = p0 (x, ξ) + O((x, ξ)3 ), even though that is not really needed for the argument to follow. Consider the involution ι = exp (πHp ) with ι2 = id. Correspondingly, we have ι0 = exp (πHp0 ), so that ι0 (ρ) = −ρ. Let h(x, ξ) be a real-valued analytic function defined near (0, 0) with dh(0, 0) = 0, and put g = 12 (h − h ◦ ι). Then dg(0) = dh(0, 0) = 0, and g ◦ ι = −g.
(5.8)
Γ := g −1 (0) is a real curve passing through the origin, invariant under the action of ι. Let Γ also denote a corresponding complexification. If g0 , Γ0 are the corresponding objects for p0 , we may assume (though this is not essential), that dg(0, 0) = dg0 (0, 0) so that Γ, Γ0 are tangent at (0, 0). Since Γ is a curve, we have p| Γ = q 2 for some analytic function q, and similarly p0 | Γ = q02 . (We may assume that dq0 = dq = 0 at 0.) Let α : Γ0 → Γ be the 0 analytic diffeomorphism given by q ◦ α = q0 , so that p ◦ α = p0 on Γ0 . For neigh ((0, 0), C2 ) ρ = exp tHp0 (ν), ν ∈ Γ0 , t ∈ C,
(5.9)
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we put κ(ρ) = exp tHp (α(ν)).
(5.10)
With the precautions taken above, it is easy to see that the definition of κ(ρ) does not depend on how we choose ν ∈ Γ0 (unique up to the action of ι0 ) and t (unique mod (2π), once ν has been chosen.) As in [13], we see that some exceptional points ρ ∈ neigh ((0, 0), C2 ) cannot be represented as in (5.9), namely the ones = (0, 0) in the stable outgoing and incoming complex (Lagrangian) curves for the iHp0 flow, and if ρ converges to one of these lines, then in general |t| → ∞ for the t in (5.9), so a priori it is not clear then that the right-hand side of (5.10) is defined. These difficulties were analyzed and settled in [13], and at this point there is no difference with our situation, so we conclude that κ is a well-defined analytic map in a neighborhood of (0, 0): Lemma 5.2 With f, p as in Lemma 5.1, there exists an analytic canonical transformation κ : neigh ((0, 0), R2 ) → neigh ((0, 0), R2 ), with f ◦ p ◦ κ = p0 . If p depends smoothly (analytically) on some real parameters, and fulfills the assumptions above, then f, κ can be chosen to depend smoothly (analytically) on the same parameters. If p = p = O((x, ξ)2 ) is analytic in (x, ξ), depends smoothly on ∈ neigh (0, R) and satisfies the assumptions above for = 0, then we get f (E), κ (x, ξ), holomorphic in E and x, ξ, depending smoothly on with f ◦ p ◦ κ = p0 , but f , κ are no more necessarily real when = 0. Clearly Im f (E) = O(), Im κ (x, ξ) = O() when E, x, ξ are real. In our case the parameters are τ, and the above discussion gives: Proposition 5.3 For p (τ, x, ξ) in (5.5), we can find a canonical transformation (x, ξ) → κτ, (x, ξ) depending analytically on τ and smoothly on with values in the holomorphic canonical transformations: neigh ((0, 0), C2 ) → neigh ((0, 0), C2 ), and an analytic function g (τ, q) depending smoothly on such that κτ, (0, 0) = (x(τ, ), ξ(τ, )),
Moreover, κτ,0
1 p (τ, κτ, (x, ξ)) = f (τ ) + ig (τ, (x2 + ξ 2 )). 2 is real when τ is real and ∂ Re g (0, 0) > 0. ∂q
(5.11) (5.12)
(5.13)
As a matter of fact, as in Section 4, we will apply this result to a modification of p , containing also the leading lower order symbol. Before doing so, we recall how to treat lower order symbols in general for operators with leading symbol modelled on the 1-dimensional harmonic oscillator (similarly to what we did in Section 3 and as in [26]).
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Consider a formal h-pseudodifferential operator Q(x, hDx ; h) with symbol Q(x, ξ; h) ∼ q0 (x, ξ) + hq1 (x, ξ) + · · · ,
(5.14)
defined in a neighborhood of (0, 0) ∈ R2 . As usual, q0 , q1 , . . . are supposed to be smooth and we assume q0 (x, ξ) = g0 (p0 (x, ξ)), (5.15) where g0 ∈ C ∞ (neigh (0, R)) satisfies g0 (0) = 0, g0 (0) = 0. (We do not assume g0 to be real-valued.) As in Section 3 we find a smooth function a0 (x, ξ), defined in a neighborhood of (0, 0), such that Hq0 a0 = q1 − q1 , (5.16) 2π 1 q1 ◦ exp (tHp0 )dt. Adding lower order where q1 is the trajectory average 2π 0 corrections, we see that there exists A(x, ξ; h) ∼ a0 (x, ξ) + ha1 (x, ξ) + · · ·
(5.17)
with all aj smooth in some common neighborhood of (0, 0), such that hDx ; h) eiA(x,hDx ;h) Q(x, hDx ; h)e−iA(x,hDx ;h) =: Q(x,
(5.18)
∼ q0 + h has a symbol Q q1 + · · · , with q0 = q0 and Hq0 qj = 0, ∀j.
(5.19)
This means that qj is a smooth function of p0 (x, ξ) and as is well known (and exploited for instance in [26]), the facts (5.18), (5.19) can be reformulated by saying that we have found A as in (5.17) such that eiA(x,hD;h) Q(x, hD; h)e−iA(x,hD;h) = g(p0 (x, hD); h), ∞ where g(E; h) ∼ 0 gj (E)hj in C ∞ (neigh (0, R)), with g0 as before. When g0 , qj are holomorphic in fixed neighborhoods of E = 0 and (x, ξ) = (0, 0), we get the corresponding holomorphy for gk , q . Now return to the operator P of the beginning of this section. Write the full symbol as P (τ, x, ξ, ; h) ∼ p (τ, x, ξ) + h p1 (τ, x, ξ, ) + h2 p2 (τ, x, ξ, ) + · · ·
(5.20)
a) Consider first the general case without any assumptions on the subprincipal symbol, and assume that (5.21) h < hδ , for some fixed δ > 0. Following the strategy of Section 4, we rewrite (5.20) as h h h P (τ, x, ξ; h) = f (τ ) + [(iq(τ, x, ξ) + O() + p1 (τ, x, ξ)) + h p2 + h2 p3 + · · · ]. (5.22)
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As before, we now treat h/ as an additional small parameter. Proposition 5.3 extends to the case when p is replaced by p + h p1 , so we have a canonical transformation (x, ξ) → κτ,,h/ (x, ξ) depending analytically on τ and smoothly on , h , equal to κτ, when h = 0, such that h 1 ( p + p1 )(τ, κτ,, h (x, ξ)) = f (τ ) + ig, h (τ, (x2 + ξ 2 )), 2 with g,0 = g appearing in Proposition 5.3. As in Section 4, we therefore obtain an elliptic Fourier integral operator U,h/ , which is a convolution in t, and such that the Fourier transform with ,h/ (τ ), is a 1-dimensional Fourier integral operator in x quantizing respect to t, U κτ,,h/. After conjugation of P by U,h/ , we get a new operator P of the same type, with symbol 1 P (τ, x, ξ, , h/; h) = f (τ ) + [ig, h (τ, (x2 + ξ 2 )) + h p2 + h2 p3 + · · · ], 2
(5.23)
where p2 , p3 , . . . also depend on h/. h After a further conjugation by eiA(hDt ,x,hDx ,, ;h) , where each term Aj in the h-asymptotic expansion: h h h A(τ, x, ξ, , ; h) ∼ A0 (τ, x, ξ, , ) + hA1 (τ, x, ξ, , ) + · · · is holomorphic in τ, x, ξ in a fixed neighborhood of (0, 0, 0) ∈ C3 and smooth in , h/, we get a new operator of the form 1 h P = f (hDt ) + iG(hDt , (x2 + (hDx )2 ), , ; h), 2 where
(5.24)
∞
h h G(τ, q, , ; h) ∼ Gj (τ, q, , )hj , 0
(5.25)
with Gj holomorphic in τ, q in a j-independent neighborhood of (0, 0) and smooth in , h/. Moreover G0 is equal to the term g,h/ (τ, q) in (5.23). Recalling that 1 1 2 2 2 (x + (hDx ) ) has the eigenvalues h( 2 + k2 ), k2 ∈ N, we get the conclusion: Proposition 5.4 Make the assumptions of case II in the introduction, and assume that F0 = Re qmin,0 (the case when F0 is a maximum being analogous). Then in a 1 1 1 1 rectangle ] − |O(1)| , |O(1)| [+i]F0 − |O(1)| , F0 + |O(1)| [, P has the quasi-eigenvalues: f
h(k1 −
S1 k0 )− 4 2π
S1 1 k0 h , h( + k2 ), , ; h , + iG h(k1 − ) − 4 2π 2 (k1 , k2 ) ∈ Z × N. (5.26)
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Here f (τ ) is real-valued with f (0) = 0, f (0) > 0. The function G has the properties ∂ described in and after (5.25) and Re G0 (0, 0, 0, 0) = F0 , ∂q Re G0 (0, 0, 0, 0) > 0. Finally, k0 is a fixed integer. b) We next consider the case when the subprincipal symbol of P=0 vanishes, and assume that h2 < hδ , (5.27) for some fixed δ > 0. According to the improved Egorov theorem of Section 2, we know that p1 in (5.20) vanishes for = 0, so we can write h p1 (τ, x, ξ, ) = h p1 (τ, x, ξ, ) in (5.22) and treat this term as a lower order term, while we now 2 allow h p2 to be a correction to the leading terms. As in the corresponding case in Section 4, we get h2 / as an additional small parameter instead of h/, and the same procedure as in case a) now leads to (5.24), (5.25) with h/ replaced by h2 /. Proposition 5.5 Make the assumptions of Proposition 5.4 and assume in addition that the subprincipal symbol of P=0 vanishes. Then for in the range (5.27), P has the quasi-eigenvalues as described in the preceding proposition, with the only difference that “h/” in (5.26) should be replaced by “h2 /”.
6 Global Grushin problem Let P be as in Section 1. In Sections 4 and 5 we have constructed microlocal normal forms for P near a Lagrangian torus and near a closed Hp -trajectory, respectively. The purpose of this section is to justify the preceding microlocal constructions and computations, and to show that the quasi-eigenvalues of Proposition 4.1 and Propositions 5.4 and 5.5 give, modulo O(h∞ ), all of the true eigenvalues of P , in suitable regions of the complex plane. This will be achieved by studying an auxiliary global Grushin problem, well posed in a certain h-dependent Hilbert space, and the first and the main step for us will be to define this space globally. The actual setup of the Grushin problem and some of the details of the computations will be closely related to the corresponding analysis in [20]. When constructing the Hilbert space, we shall inspect all the steps of the microlocal reductions of Sections 3–5, and implement each step of the construction. In doing so, for simplicity, we shall concentrate on the case when M = R2 . In view of the results of the appendix, it will be clear how to extend the following discussion to the case of compact real-analytic manifolds. Also, in order to simplify the presentation, we shall assume throughout the section that the order function m, introduced in (1.2), is equal to 1. Again, it will be clear that the discussion below will extend to the case of a general order function. Throughout this section we shall assume that = O(hδ ), for some fixed δ > 0. Let G = G(x, ξ, ) be as in (3.6). We shall introduce an IR-manifold ΛG ⊂ C4 , which in a complex neighborhood of p−1 (0) ∩ R4 is equal to exp (iHG )(R4 ), and further away from p−1 (0) ∩ R4 agrees with the real phase space R4 . The
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manifold ΛG will be -close to R4 , and when defining it, it will be convenient to work on the FBI transform side. We shall use the FBI-Bargmann transform −3/2 (6.1) T u(x) = Ch eiϕ(x,y)/h u(y) dy, x ∈ C2 , C > 0, where ϕ(x, y) = i/2(x − y)2 . Associated to T there is a complex linear canonical transformation κT , given by C4 (y, −ϕy (x, y)) −→ (x, ϕx (x, y)) ∈ C4 . It is well known, see [28], that κT maps R4 onto
2 ∂Φ0 (Im x)2 . x, ΛΦ0 := , x ∈ C2 , Φ0 (x) = i ∂x 2 The IR-manifold ΛG has already been defined near p−1 (0) ∩ R4 , and when constructing it globally, we require that the IR-manifold κT (ΛG ) should agree with ΛΦ0 outside a bounded set and that it is -close to that manifold everywhere. We define therefore ΛG so that the representation
2 ∂Φ 2 κT (ΛG ) = x, (6.2) , x ∈ C =: ΛΦ i ∂x holds true. Here the function Φ ∈ C ∞ (C2 ; R) is uniformly strictly plurisubharmonic, and is such that Φ(x) = Φ0 (x) + g(x, ), with g(x, ) ∈ C ∞ in both arguments and with a uniformly compact support with respect to x. Associated to ΛG we then introduce the corresponding Hilbert space H(ΛG ) which agrees with L2 (R2 ) as a space, and which we equip with the norm || u || := || T u ||L2Φ . Here L2Φ = L2 (C2 ; e−2Φ/h L(dx)), with L(dx) being the Lebesgue measure on C2 . Performing a contour deformation in the integral representation of P on the FBI-Bargmann transform side, as in [20], [28], we see that P = O(1) : H(ΛG ) → H(ΛG ),
(6.3)
and the leading symbol on the FBI transform side is then p ◦ κ−1 T
. Continuing
ΛΦ
to work on the FBI-Bargmann transform side, as in Section 2 of [20], we introduce a microlocally unitary semiclassical Fourier integral operator eG(x,hDx,)/h : L2 (R2 ) → H(ΛG ),
(6.4)
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microlocally defined near p−1 (0) ∩ R4 , and associated to the complex canonical transformation exp (iHG ) : R4 → ΛG . The operator in (6.3) is then microlocally near p−1 (0), unitarily equivalent to the operator e−G(x,hDx,)/h P eG(x,hDx ,)/h : L2 → L2 , with the principal symbol p ◦ exp (iHG ) = p + iq + O(2 ).
(6.5)
This averaging procedure allows us therefore to reduce the further analysis to an operator P , microlocally defined near p−1 (0) ∩ R4 , which has the principal symbol (6.5), where q, as well as the O(2 )-term, are in involution with p. As explained in Section 4, at this stage the operator P acts on single-valued functions in L2 (R2 ). In the first part of this section we shall concentrate on the torus case of Section 4. We assume therefore that dp and dRe q are linearly independent on the set (6.6) Λ0,0 : p = 0, Re q = 0. We recall also the assumption that T (0) is the minimal period of every closed Hp trajectory in the Lagrangian torus Λ0,0 , and notice that in a neighborhood of Λ0,0 , p and Re q form a completely integrable system. Introduce a new Lagrangian 0,0 ⊂ ΛG defined by torus Λ 0,0 : p ◦ exp (−iHG ) = 0, Re q ◦ exp (−iHG ) = 0. Λ
(6.7)
0,0 by means of In what follows we shall often identify the tori Λ0,0 and Λ 0,0 when there is no risk of exp (iHG ), and we shall continue to write Λ0,0 for Λ confusion. Combining exp (iHG ) with the canonical transformation κ, introduced in (4.4), and given by the action-angle coordinates associated with p, Re q, we get a smooth canonical diffeomorphism
κ : neigh ξ = 0, T ∗ T2 → neigh (Λ0,0 , ΛG ) , (6.8) so that κ = exp (iHG ) ◦ κ. As in (4.32), we set ξ dx, j = 1, 2, Sj = αj
where α1 and α2 are the fundamental cycles in Λ0,0 , with α1 corresponding to a closed Hp -trajectory of the minimal period T (0). Introduce also the “Maslov indices” k0 (αj ) ∈ Z, j = 1, 2, of the cycles αj , defined as in Proposition 2.3. Let L2θ (T2 ) be the subspace of L2loc (R2 ) consisting of Floquet periodic functions u(x), satisfying u(x − ν) = eiθ·ν u(x),
2
ν ∈ (2πZ) ,
where θ =
k0 S + . 2πh 4
(6.9)
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Here S = (S1 , S2 ) and k0 = (k0 (α1 ), k0 (α2 )) ∈ Z2 . An application of Theorem 2.4 allows us to conclude that there exists a microlocally unitary multi-valued Fourier integral operator (6.10) U : L2θ (T2 ) → L2 (R2 ), microlocally defined from a neighborhood of ξ = 0 in T ∗ T2 to a neighborhood of Λ0,0 in R4 , and associated to κ in (4.4). Moreover, U satisfies the improved Egorov property (2.3). The composition eG(x,hDx,)/h ◦ U is then associated with κ in (6.8), and we have a Egorov’s theorem, still with the improved property (2.3). The operator P , acting in H(ΛG ) is therefore unitarily equivalent to an h-pseudodifferential operator microlocally defined near ξ = 0, acting in L2θ (T2 ), and which has the leading symbol p(ξ1 ) + iq(ξ) + O(2 ), independent of x1 . We shall continue to write P for the conjugated operator on T2 . From Section 4 we next recall that there exists an elliptic pseudodifferential operator of the form eiA/h , acting on L2θ (T2 ), such that after a conjugation by it, the full symbol of P becomes independent of x1 . Recall also that A is constructed as a formal power series in and h, with coefficients holomorphic in a fixed complex neighborhood of the zero section of T ∗ T2 . These formal power series are then realized as C ∞ -symbols, in view of our basic assumption = O(hδ ), δ > 0. Summing up the discussion so far, we have now achieved that, microlocally near Λ0,0 , the operator P : H(ΛG ) → H(ΛG ) is equivalent to an operator of the form P (x2 , ξ, ; h) ∼
∞
hν pν (x2 , ξ, )
(6.11)
ν=0
acting on L2θ (T2 ). Here pν (x2 , ξ, ) are holomorphic in a ν-independent complex neighborhood of ξ = 0, and p0 = p(ξ1 ) + iq(ξ) + O(2 ). Furthermore, P=0 is selfadjoint. Remark. It follows from the construction together with Theorem 2.4 that if the subprincipal symbol of P=0 vanishes, then p1 (x2 , ξ, 0) = 0. We must now implement the final conjugation of P , which removes the x2 dependence in the full symbol. In doing so, we shall first assume that we are in the general case, so that the subprincipal symbol of P=0 does not necessarily vanish. We shall work under the assumption h ≤ δ0 1.
(6.12)
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As in Section 4, we write h h P = p(ξ1 ) + r0 x2 , ξ, , + hr1 x2 , ξ, , + ··· , h h r0 x2 , ξ, , = iq(ξ) + O() + p1 (x2 , ξ, ),
where
and rj = Oj (h/), j ≥ 1. Let us introduce a complexification of the standard 2 = T2 + iR2 . From the constructions of Section 4 we know that there 2-torus, T exists a holomorphic canonical transformation 2 × C2 (6.13) κ : neigh Im y = η = 0, T 2 × C2 (y, η) → (x, ξ) ∈ neigh Im x = ξ = 0, T with the generating function of the form h h ψ x, η, , = x · η + φper x2 , η, , ,
φper
h =O + ,
and such that
(r0 ◦ κ ) (y, η, , h/) = r0 (·, η, , h/) + O
h ,
(6.14)
2 (6.15)
is independent of y – see (4.19). It follows from (6.14) that κ is ( + h/)-close to the identity, and has the expression (y1 , η1 ; y2 , η2 ) −→ (x1 (y2 , η), η1 ; x2 (y2 , η), ξ2 (y2 , η)). In particular it is true that h Im x = O + ,
h Im ξ2 = O + ,
Im ξ1 = 0,
2 2 2 on the image of T ∗ T
. We introduce now an IR-manifold Λ ⊂ T × C , which ∗ 2 is equal to κ T T in a complex neighborhood of the zero section of T ∗ T2 , and outside another complex fixed neighborhood of ξ = 0, coincides with T ∗ T2 . in such a way that it remains In the intermediate region, we shall construct Λ ∗ 2 we have the an ( + h/)-perturbation of T T , and such that everywhere on Λ property =⇒ Im ξ1 = 0. (6.16) (x1 , ξ1 ; x2 , ξ2 ) ∈ Λ
and describing the conjugation of P by a Fourier integral When constructing Λ operator associated to κ , it is convenient to work on the FBI transform side. As
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in Section 3 of [20], we notice that the FBI-Bargmann transformation introduced in (6.1) generates an operator from L2θ (T2 ) to the space of Floquet periodic holomorphic functions on C2 . We continue to denote this operator by T . Then after the application of the canonical transformation κT , associated to T , the cotangent 2 × C2 given by space T ∗ T2 becomes an IR-manifold ΛΦ1 ⊂ T 2 ∂Φ1 (Im x)2 = −Im x, Φ1 (x) = . i ∂x 2 Since T is a convolution operator acting separately in y1 and y2 , we see that ΛΦ1 : ξ =
= ΛΦ , κT (Λ)
ΛΦ : ξ =
2 ∂Φ , i ∂x
∂Φ where Φ is an ( + h/)-perturbation of Φ1 with the property that ξ1 = (2/i) ∂x 1 is real. It follows that Φ = Φ(Im x1 , x2 ) is independent of Re x1 . Using a standard cutoff function around Im x = 0, we modify Φ away from Im x = 0 to obtain a strictly plurisubharmonic function Φ which coincides with Φ1 further away from Im x = 0, in such a way that Φ remains an ( + h/)-perturbation of Φ1 and is still a function independent of Re x1 . We then define the global IR-manifold = κ−1 (ΛΦ ). Λ T Associated to κ , there is a Fourier integral operator U −1 introduced in (4.20),
U −1 = O(1) : L2 (T2 ) → H(Λ), is microlocally near ξ = 0 unitarily equivalent such that the action of P on H(Λ) to the operator U P U −1 : L2 (T2 ) → L2 (T2 ), whose Weyl symbol has the form h h p(ξ1 ) + r0 ξ, , + hr1 x2 , ξ, , + ··· .
(6.17)
h r0 = iq(ξ) + O +
Here
is independent of x, and
h rj = O + ,
j ≥ 1.
The corresponding statement is also true when considering the action on L2θ (T2 ), since U −1 preserves the Floquet property (6.9). on the torus side, there is an IR-manifold Associated to the IR-deformation Λ 4 Λ ⊂ C which is an (+ h/)-perturbation of ΛG near Λ0,0 , obtained by replacing exp (iHG ) ◦ κ(T ∗ T2 ) there by (T ∗ T2 ) = exp (iHG ) ◦ κ(Λ). exp (iHG ) ◦ κ ◦ κ
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, which is ( + h/)-close to In such a way we get a globally defined IR-manifold Λ 4 ΛG and agrees with R outside a neighborhood of p−1 (0) ∩ R4 . Associated with ), defined similarly to H(ΛG ), and obtained we then have a Hilbert space H(Λ Λ by modifying the standard weight Φ0 (x) on the FBI-Bargmann transform side. 0,0 ⊂ Λ , with the property We also get a corresponding new Lagrangian torus Λ that microlocally near Λ0,0 , the original operator ) → H(Λ ) P : H(Λ is equivalent to an operator on L2θ (T2 ), whose complete symbol has the form (6.17). Taking into account the conjugation by an elliptic operator eiB1 /h on the torus side, which was constructed in Section 4 and which eliminates the x2 -dependence also in the terms rj with j ≥ 1, we get the following result. Proposition 6.1 We make all the assumptions of case I in the introduction, and recall that we also take F0 = 0. Assume that = O(hδ ), δ > 0 is such that h/ ≤ δ0 , ⊂ C4 , and a smooth Lagrangian torus 0 < δ0 1. There exists an IR-manifold Λ 0,0 ⊂ Λ , such that when ρ ∈ Λ is away from a small neighborhood of Λ 0,0 in Λ Λ , we have |Re P (ρ, h)| ≥
1 |O(1)|
or
|Im P (ρ, h)| ≥
. |O(1)|
(6.18)
is an + h -perturbation of R4 in the natural sense, and it is The manifold Λ equal to R4 outside a neighborhood of p−1 (0) ∩ R4 . We have ) → H(Λ ). P = O(1) : H(Λ There exists a smooth canonical transformation 0,0 , Λ ) → neigh (ξ = 0, T ∗ T2 ), κ : neigh (Λ 0,0 ) = T2 ×{0}. Associated to κ , there is a Fourier integral operator such that κ (Λ ) → L2 (T2 ), U = O(1) : H(Λ θ which has the following properties: ), χ2 ∈ 1) U is concentrated to the graph of κ in the sense that if χ1 ∈ C0∞ (Λ ∞ ∗ 2 C0 (T T ), are such that 0,0 , Λ )} = ∅, (suppχ2 × suppχ1 ) ∩ {(κ (y, η), y, η); (y, η) ∈ neigh(Λ then
) → L2θ (T2 ). χ2 (x, hDx ) ◦ U ◦ χ1 (x, hDx ) = O(h∞ ) : H(Λ
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2) The operator U is microlocally invertible: there exists an operator V = O(1) : ) such that for every χ1 ∈ C0∞ (neigh(Λ 0,0 , Λ )), we have L2θ (T2 ) → H(Λ ) → H(Λ ). (V U − 1) χ1 (x, hDx ) = O(h∞ ) : H(Λ For every χ2 ∈ C0∞ (neigh(ξ = 0, T ∗ T2 )), we have (U V − 1) χ2 (x, hDx ) = O(h∞ ) : L2θ (T2 ) → L2θ (T2 ).
3) We have Egorov’s theorem: Acting on L2θ (T2 ), there exists P hDx , , h ; h with the symbol ∞ h h j P ξ, , ; h ∼ p(ξ1 ) + h rj ξ, , , j=0 with
|ξ| ≤
1 , |O(1)|
h r0 = iq(ξ) + O + ,
and rj = Oj
h +
,
j ≥ 1,
such that PU = U P microlocally, i.e., P U − U P χ1 (x, hDx ) = O(h∞ ), χ2 (x, hDx ) P U − U P = O(h∞ ), for every χ1 , χ2 as in 2). Remark. The estimate (6.18) holds true thanks to the property (6.16) of the final deformation, since then the term p(ξ1 ) does not contribute to the imaginary part of the symbol on the torus side. The bound (6.18) will allow us to reduce the 0,0 . spectral analysis of P to a small neighborhood of the Lagrangian torus Λ Using Proposition 6.1, we shall now proceed to describe the spectrum of P in a rectangle of the form
1 RC, = z ∈ C; |Re z| < , |Im z| < , (6.19) C C for a sufficiently large constant C > 0. We shall show that the eigenvalues in (6.19) are given by the quasi-eigenvalues of Proposition 4.1, modulo O(h∞ ). In doing so, let us consider the set of the quasi-eigenvalues, introduced in (4.35),
h k0 S + . Σ(, h) = P h(k − θ), , ; h ; k ∈ Z2 RC, , θ = 2πh 4
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Then the distance between 2 elements of Σ(, h) corresponding to k, l ∈ Z2 , k = l, is ≥ h |k − l| /|O(1)|. Introduce h h δ := 1/4 inf dist (P (h(k − θ), , ; h), P(h(l − θ), , ; h)) > 0, k=l and consider the family of open discs
h Ωk (h) := z ∈ RC, ; z − P (h(k − θ), , ; h) < δ ,
k ∈ Z2 .
The sets Ωk (h) are then disjoint, and dist (Ωk (h), Ωl (h)) ≥ h |k − l| /|O(1)|. As a warm-up exercise, we shall first show that Spec (P ) in the set (6.19) is contained in the union of the Ωk (h). When z ∈ C is in the rectangle (6.19), let us consider the equation (P − z) u = v,
). u ∈ H(Λ
(6.20)
We notice here that the symbol of Im P =
P − P∗ , 2i
), is O(), and from Proposition 6.1 we know taken in the operator sense in H(Λ it is true that |Im P (ρ, h)| > 0,0 in Λ that away from any fixed neighborhood of Λ /C, provided that |Re P (ρ, h)| ≤ 1/C, where C > 0 is sufficiently large. Here we are using the same letters for the operators and the corresponding (Weyl) symbols, and P + P∗ ) → H(Λ ). : H(Λ Re P = 2 ). We shall also write p to denote the leading symbol of P=0 , acting on H(Λ Let us introduce a smooth partition of unity on the manifold Λ , 1 = χ + ψ1,+ + ψ1,− + ψ2,+ + ψ2,− . ) is such that χ = 1 near Λ 0,0 , and supp χ is contained in a small Here χ ∈ C0∞ (Λ ) are sup neighborhood of Λ0,0 where U P = P U . The functions ψ1,± ∈ C0∞ (Λ ported in regions, invariant under the Hp -flow, where ±Im P > /C, respectively. ) are such that Finally ψ2,± ∈ Cb∞ (Λ
suppψ2,± ⊂ ρ; ±Re P (ρ, h) > 1/C . . Moreover, we arrange so that the functions ψ1,± Poisson commute with p on Λ We shall prove that 1 (6.21) || v || + O(h∞ )|| u ||, || (1 − χ)u || ≤ O ). In doing so, we shall first derive where we let || · || stand for the norm in H(Λ a priori estimates for ψ1,+ u.
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When N ∈ N, let ψ0 ≺ ψ1 ≺ · · · ≺ ψN ,
ψ0 := ψ1,+ ,
; [0, 1]), supported in an Hp -flow invariant region be cutoff functions in C0∞ (Λ where Im P ∼ , and which are in involution with p. Here standard notation f ≺ g means that supp f is contained in the interior of the set where g = 1. It is then true that in the operator norm, [P , ψj ] = [P=0 , ψj ] + O(h) = O(h2 ) + O(h) = O(h),
0 ≤ j ≤ N,
(6.22)
since ≥ h. For future reference we notice that in the case when the subprincipal symbol of P=0 vanishes, the Weyl calculus shows that [P=0 , ψj ] = O(h3 ), and since ≥ h2 , we still get (6.22). Here we have also used that the subprincipal symbol of ψj is 0, 0 ≤ j ≤ N . Near the support of ψj it is true that Im P ∼ , and an application of the semiclassical G˚ arding inequality allows us therefore to conclude that (Im (P − z)ψj u|ψj u) ≥
|| ψj u ||2 − O(h∞ )|| u ||2 . O(1)
). On the other hand, we have Here the inner product is taken in H(Λ (Im (P − z)ψj u|ψj u) = Im (ψj (P − z)u|ψj u) + ([P , ψj ]u|ψj u) , and since in the operator sense ψj (1 − ψj+1 ) = O(h∞ ), we see that the absolute value of this expression does not exceed O(1)|| (P − z)u || || ψj u || + O(h)|| ψj+1 u ||2 + O(h∞ )|| u ||2 . We get
≤
|| ψj u ||2 ≤ O(1)|| (P − z)u || || ψj u || + O(h)|| ψj+1 u ||2 + O(h∞ )|| u ||2 C O(1) || ψj u ||2 + || (P − z)u ||2 + O(h)|| ψj+1 u ||2 + O(h∞ )|| u ||2 , 2C
and hence, || ψj u ||2 ≤
O(1) || (P − z)u ||2 + O(h)|| ψj+1 u ||2 + O(h∞ )|| u ||2 . 2
Combining these estimates for j = 0, 1, . . . , N , we get || ψ0 u ||2 ≤
O(1) || (P − z)u ||2 + ON (1)hN || ψN u ||2 + O(h∞ )|| u ||2 , 2
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and therefore
O(1) || v || + O(h∞ )|| u ||. The same estimate can be obtained for ψ1,− u, microlocally concentrated in a flow invariant region where Im P ∼ −, and a fortiori such estimates also hold in regions where Re P ∼ 1 and Re P ∼ −1. The bound (6.21) follows. || ψ1,+ u || ≤
Write next (P − z) χu = χv + w, w = [P , χ]u, where w satisfies
(6.23)
h || w || ≤ O || v || + O(h∞ )|| u ||.
0,0 . Applying the operator U of Here we have used (6.21) with a cutoff closer to Λ Proposition 6.1 to (6.23), we get P − z U χu = U χv + U w + T∞ u, where
) → L2 (T2 ). T∞ = O(h∞ ) : H(Λ θ
Using an expansion in Fourier series (6.25) below, we see that the operator P − z : L2θ (T2 ) → L2θ (T2 ) is invertible, microlocally in |ξ| ≤ 1/|O(1)|, with a microlocal inverse of the norm O(1/h), provided that z ∈ RC, avoids the discs Ωk (h). Using also the uniform boundedness of the microlocal inverse V of U , we get || χu || ≤
O(1) || v || + O(h∞ )|| u ||. h
(6.24)
Combining (6.21) and (6.24), we see that when z ∈ RC, is in the complement of the union of the Ωk (h), the operator ) → H(Λ ) P − z : H(Λ is injective. Since the ellipticity assumption (1.6) implies that it is a Fredholm ) → H(Λ ) is bijective. operator of index zero, we know that P − z : H(Λ We shall now let z vary in the disc Ωk (h) ⊂ RC, , for some k ∈ Z2 . We shall show that z ∈ Ωk (h) is an eigenvalue of P if and only if z = P (h(k −θ), , h ; h)+r, where r = O(h∞ ). In doing so, we shall study a globally well-posed Grushin ). problem for the operator P − z in the space H(Λ As a preparation for that, we shall introduce an auxiliary Grushin problem for the operator P − z, defined microlocally near ξ = 0 in T ∗ T2 . From (4.34), let us recall the functions el (x) =
S 1 i(l−θ)x 1 hi (h(l− k40 )− 2π )x , e e = 2π 2π
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which form an ON basis for the space L2θ (T2 ), so that when u ∈ L2θ (T2 ), we have a Fourier series expansion, u(x) = u (l − θ)el (x). (6.25) l∈Z2
We also remark that el (x) are microlocally concentrated to the region of the phase space where ξ ∼ h l − k40 − S/2π. + : L2 (T2 ) → C and R − : C → L2 (T2 ), given Introduce rank one operators R θ θ − u− = u− ek . Here the inner product in the definition + u = (u|ek ) and R by R + is taken in the space L2 (T2 ). Using (6.25), it is then easy to see that the of R θ operator −z R − P := P (6.26) : L2θ (T2 ) × C → L2θ (T2 ) × C, + R 0 defined microlocally near ξ = 0, has a microlocal inverse there, which has the form + E(z) E E = (6.27) − E −+ (z) . E The following localization properties can be inferred from the construction of E: + = if ψ ∈ Cb∞ (T ∗ T2 ) has its support disjoint from ξ = 0, then it is true that ψ E − ψ = O(h∞ ) : L2 → C. We also find that O(h∞ ) : C → L2θ , and E θ −+ (z) = z − P h(k − θ), , h ; h . E (6.28) Using (6.25), we furthermore see that the following estimates hold true, = O(1) : L2 (T2 ) → L2 (T2 ), E θ θ h + = O(1) : C → L2θ (T2 ), E
− = O(1) : L2θ (T2 ) → C, E
−+ = O(h) : C → C, E so that h|| u || + || u− || ≤ O(1) (|| v || + h|| v+ ||) , when P
u u−
=
v v+
(6.29)
.
In (6.29), the norms of u and v are taken in L2θ (T2 ) and those of u− and v+ in C. ) → C and R− : C → H(Λ ) by Passing to the case of P , we define R+ : H(Λ + U χu = (U χu|ek ), R+ u = R
− u − = u − V ek . R− u− = V R
(6.30)
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It is then true that ), χR− = R− + O(h∞ ) : C → H(Λ
(6.31)
decreasing the support of χ if necessary. We now claim that for z ∈ Ωk (h), the Grushin problem (P − z) u + R− u− = v, (6.32) R+ u = v+ ) × C for every (v, v+ ) ∈ H(Λ ) × C, with an has a unique solution (u, u− ) ∈ H(Λ a priori estimate, h|| u || + || u− || ≤ O(1) (|| v || + h|| v+ ||) .
(6.33)
), and those of u− and v+ in C. To Here the norms of u and v are taken in H(Λ verify the claim, we first see that as in (6.21), we have 1 (6.34) || (1 − χ)u || ≤ O || v || + O(h∞ ) (|| u || + || u− ||) . Here we have also used (6.31). Applying χ to the first equation in (6.32) we get (P − z) χu + R− u− = χv + w + R−∞ u− , R+ u = v+ , where w = [P , χ]u satisfies || w || ≤ O
(6.35)
h || v || + O(h∞ ) (|| u || + || u− ||) ,
and R−∞ = O(h∞ ) in the operator norm. Applying U to the first equation in (6.35) and using (6.30), we get − u− = U χv + U w + w− (P − z)U χu + R (6.36) + U χu = v+ . R where the L2θ (T2 )-norm of w− is O(h∞ ) (|| u || + || u− ||). We therefore get a mi in (6.26), and in view of (6.29) we crolocally well-posed Grushin problem for P obtain, h|| χu || + || u− || ≤ O(1) (|| v || + h|| v+ ||) + O(h∞ ) (|| u || + || u− ||) .
(6.37)
Combining (6.34) and (6.37), we get (6.33). We have thus also proved that the operator P − z R− ) × C ) × C → H(Λ (6.38) : H(Λ P= R+ 0
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is injective, for z ∈ Ωk (h). Now P is a finite rank perturbation of P − z 0 , 0 0 which is a Fredholm operator of index zero. It follows that P is also Fredholm of index 0 and hence bijective, since we already know that it is injective. The inverse of P has the form E(z) E+ , (6.39) E= E− E−+ (z) and we recall that the spectrum of P in Ωk (h) will be the set of values z for which E−+ (z) = 0. We finally claim that the components E+ and E−+ (z) in (6.39) are given by
+ , and E−+ (z) = E −+ (z) = z − P h(k − θ), , h ; h , modulo terms that E+ = V E are O(h∞ ). Indeed, we need only to check that + ≡ 1, R+ V E
−+ ≡ 0, + + R− E (P − z) V E
(6.40)
∞
modulo O(h ), and at this stage the verification of (6.40) is identical to the corresponding computation from Section 6 of [20]. In particular, we get h E−+ (z) = z − P h(k − θ), , ; h + O(h∞ ), (6.41) and we have now proved the first of our two main results. Theorem 6.2 Let F0 be a regular value of Re q viewed as a function on p−1 (0) ∩ R4 . Assume that the Lagrangian manifold Λ0,F0 : p = 0, Re q = F0 is connected, and that T (0) is the minimal period of every closed Hp -trajectory in Λ0,F0 . When α1 and α2 are the fundamental cycles in Λ0,F0 with α1 corresponding to a closed Hp -trajectory of minimal period, we write S = (S1 , S2 ) and k0 = (k0 (α1 ), k0 (α2 )) for the actions and Maslov indices of the cycles, respectively. Assume furthermore that = O(hδ ), δ > 0, is such that h/ 1. Let C > 0 be sufficiently large. Then the eigenvalues of P in the rectangle |Re z| < are given by
1 , C
|Im z − F0 | <
C
(6.42)
h k0 S zk = P h k − , , ; h , k ∈ Z2 , − 4 2π
modulo O(h∞ ). Here P ξ, , h ; h is holomorphic in ξ ∈ neigh(0, C2 ), smooth in , h ∈ neigh(0, R) and has an asymptotic expansion in the space of such functions, h h h P ξ, , ; h ∼ p(ξ1 ) + r0 ξ, , + hr1 ξ, , + · · · , h → 0,
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with r0 = iq + O( + h/), and rν = O( + h/), ν ≥ 1. We have exactly one eigenvalue for each k ∈ Z2 such that the corresponding zk falls into the region (6.42). Keeping all the general assumptions of the torus case and still taking F0 = 0, we shall next consider the case when the subprincipal symbol of the unperturbed operator P=0 vanishes. It follows then from the previous arguments, now making use of the full strength of Theorem 2.4, that in this case, microlocally near Λ0,0 , P : H(ΛG ) → H(ΛG )
(6.43)
is equivalent to an operator of the form P (x2 , ξ, ; h) ∼
∞
hν pν (x2 , ξ, ),
(6.44)
ν=0
acting on L2θ (T2 ), with p0 (x2 , ξ, ) = p(ξ1 ) + iq(ξ) + O(2 ),
p1 (x2 , ξ, ) = q1 (x2 , ξ, ).
In what follows we shall discuss the range M h2 < = O(hδ ) M 1, δ > 0.
(6.45)
Recalling the operators eG(x,hDx,)/h and U from (6.4) and (6.10), respectively, we see, as in the general case, that the symbol of Im P on H(ΛG ) is O(), and away from any fixed neighborhood of Λ0,0 in ΛG , we have |Im P (ρ, h)| ∼ , if |Re P (ρ, h)| < 1/|O(1)|. We write, as in Section 4, h2 h2 P (x2 , ξ, , h) = p(ξ1 ) + r0 x2 , ξ, , + hr1 x2 , ξ, , + ··· , where
h2 h2 = iq + O() + p2 (x2 , ξ, ), r0 x2 , ξ, , 2 h h2 r1 (x2 , ξ, ) = q1 (x2 , ξ, ) + p3 (x2 , ξ, ), rj (x2 , ξ, ) = O , j ≥ 2.
Using the canonical transformation κ, generated by the function h2 h2 ψ x, η, , = x · η + φper x2 , η, , ,
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with φper = O( +
45
h2 ),
constructed in Section 4, we then argue similarly to the 2 × C2 which is an ⊂T general torus case. We thus introduce an IR-manifold Λ 2 ∗ 2 ∗ 2 ( + h /)-perturbation of T T , which agrees with κ(T T ) near ξ = 0, and we first further away from this set coincides with T ∗ T2 . When constructing Λ, ∗ 2 notice that κ(T T ) has the form Im x = Gξ (Re (x, ξ)),
Im ξ = −Gx (Re (x, ξ)),
2
where G = G(x2 , ξ, , h ) is such that h2 ∂ξ G, ∂x2 G = O + . As was observed in Section 4, the transformation κ conserves actions, and therefore the smooth function G is single-valued. We may assume that h2 G=O + . If we let χ(ξ) ∈ C0∞ (R2 ; [0, 1]) be a cutoff function with a small support and such by that χ = 1 in a small neighborhood of 0, we define Λ (Re (x, ξ)), G(Re (Re (x, ξ)), Im ξ = −G (x, ξ)) = χ(Re ξ)G(Re (x, ξ)). Im x = G ξ x such that Im ξ1 = 0 on We then obtain the desired globally defined IR-manifold Λ Λ. When acting on H(Λ), P is microlocally near ξ = 0 unitarily equivalent to an operator on L2 (T2 ), which has the form h2 h2 p(ξ1 ) + r0 ξ, , + hr1 x2 , ξ, , + ··· , where
h2 h2 r0 ξ, , = iq + O +
is independent of x. It follows, as in the general torus case, that on the Bargmann transform can be described by an FBI-weight Φ = Φ(Im x1 , x2 ) which does not side, Λ depend on Re x1 . Repeating the previous arguments, we obtain therefore a new associated to an IR-manifold Λ ⊂ C4 , and a globally defined Hilbert space H(Λ), → H(Λ) is Lagrangian torus Λ0,0 ⊂ Λ such that microlocally near Λ0,0 , P : H(Λ) 2 2 equivalent to an operator on Lθ (T ), described in (4.27), (4.28). Proposition 6.3 Assume that the subprincipal symbol of P=0 vanishes, and consider the range M h2 < = O(hδ ) for M 1, δ > 0. There exists an IR-manifold ⊂ C4 and a smooth Lagrangian torus Λ 0,0 ⊂ Λ such that when ρ ∈ Λ is away Λ
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0,0 in Λ and |Re P (ρ, h)| < 1/C, for a sufficiently from a small neighborhood of Λ large C > 0, it is true that |Im P (ρ, h)| ∼ . is ( + h2 /)-close to R4 and it coincides with R4 outside a neighThe manifold Λ −1 borhood of p (0) ∩ R4 . There exists a canonical transformation 0,0 , Λ) → neigh(ξ = 0, T ∗ T2 ), κ : neigh(Λ 0,0 onto T2 , and an elliptic Fourier integral operator U : H(Λ) → mapping Λ 2 2 Lθ (T ) associated to κ , such that, microlocally near Λ0,0 , U P = P U . Here 2
h P = P(hDx , , ; h) has the Weyl symbol, depending smoothly on , h2 / ∈ neigh(0, R), ∞ h2 h2 P ξ, , ; h ∼ p(ξ1 ) + hj rj ξ, , . j=0 We have r0 = iq(ξ) + O(1)( + h2 /),
rj = O(1), j ≥ 1.
Repeating the arguments, leading to Theorem 6.2, and using Proposition 6.3 instead of Proposition 6.1, we then find first that the spectrum of P in a region of the form (6.19) is contained in the union of disjoint discs of radii h/|O(1)| around
the quasi-eigenvalues P h(k − θ), , h2 /; h . Furthermore, when z varies in such a disc corresponding to k ∈ Z2 , such that the corresponding quasi-eigenvalue falls into the region (6.19), an inspection of the previous arguments shows that the Grushin operator P − z R− × C → H(Λ) ×C : H(Λ) R+ 0 is bijective with the inverse of the norm O((h)−1 ) – see (6.33) for the precise a and R+ : H(Λ) → C are defined as in (6.30). priori estimate. Here R− : C → H(Λ) This leads to the following result. Theorem 6.4 Keep all the assumptions and notation of Theorem 6.2, and in addition assume that the subprincipal symbol of P=0 vanishes. Let = O(1)hδ for some fixed δ > 0 be such that h2 . Then the eigenvalues of P in the rectangle 1 1 1 1 − , + i F0 − , F0 + C C C C
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are given by h2 k0 S zk = P h k − , , ; h + O(h∞ ), k ∈ Z2 . − 4 2π Here C > 0 is large enough, P (ξ, , h2 /; h) is holomorphic in ξ ∈ neigh(0, C2 ), smooth in and h2 / ∈ neigh(0, R), and as h → 0, there is an asymptotic expansion h2 h2 h2 P ξ, , ; h ∼ p(ξ1 ) + r0 ξ, , + hr1 ξ, , + ··· . We have h2 h2 r0 ξ, , = iq(ξ) + O + ,
h2 rj ξ, , = O(1), j ≥ 1.
We shall now turn to the case II from the introduction. Let us recall from Section 1, that if z ∈ Spec P is such that |Re z| ≤ δ → 0, then Im z ∈ inf Re q − o(1), sup Re q + o(1) , h → 0. (6.46) Σ
Σ
Here, as in Section 1, we write Σ = p−1 (0) ∩ R4 /exp (RHp ). Our purpose here is to show that the quasi-eigenvalues of Propositions 5.4 and 5.5 give, up to O(h∞ ), the actual eigenvalues in a set of the form |Re z| ≤
1 , |O(1)|
|Im z − F0 | ≤
, |O(1)|
when F0 ∈ {inf Σ Re q, supΣ Re q}. As we shall see, the analysis here will be parallel to the torus case just treated, so that in what follows we shall concentrate on the new features of the problem, and some of the computations that are essentially identical to the ones already performed, will not be repeated. In order to fix the ideas, we shall discuss the case when F0 = inf Re q, Σ
and we shall take F0 = 0. Recall from the beginning of this section that the original operator P acting on H(ΛG ), is microlocally unitarily equivalent to the operator P ∼
∞
hj pj (x, ξ, ),
j=0
acting on L2 and defined microlocally near p−1 (0) ∩ R4 , with p0 = p + iq + O(2 ),
(6.47)
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and the functions q and O(2 )-term are in involution with p. Let γ1 , . . . , γN ⊂ p−1 (0) ∩ R4 be the finitely many trajectories such that Re q = 0 along γj , 1 ≤ j ≤ N . We know that T (0) is the minimal period of each γj , and if ρj ∈ Σ is the corresponding point, then the Hessian of Re q at ρj is positive definite, 1 ≤ j ≤ N . Associated to γj , we have the quantities S = S(γj ) and k0 = k0 (γj ), the action along γj and the Maslov index, respectively, defined as in Section 2, and we recall from [11] that these quantities do not depend on j. In what follows we shall work microlocally near a fixed critical trajectory, say γ1 . We let L2S (S 1 × R) be the space of locally square integrable functions u(t, x) on R × R such that 2π
|u(t, x)|2 dx dt < ∞.
0
and u(t − 2π, x) = eiS/h+ik0 π/2 u(t, x). Applying Theorem 2.4 to the canonical transformation κ of Proposition 3.1, we see that there exists an analytic microlocally unitary Fourier integral operator U0 : L2S (S 1 × R) → L2 (R2 ), associated to κ,
and defined microlocally from a neighborhood of {τ = x = ξ = 0} in T ∗ S 1 × R to a neighborhood of γ1 in R4 , so that we have the two-term Egorov property (2.3). Combining exp (iHG ) with κ, we get a smooth canonical transformation
(6.48) κ : neigh τ = x = ξ = 0, T ∗ S 1 × R → neigh(γ1 , ΛG ), where abusing the notation slightly, we write here γ1 ⊂ ΛG also for the image of γ1 under the complex canonical transformation exp (iHG ). The operator eG(x,hDx ,)/h ◦ U0 is then associated with κ in (6.48), and an application of Egorov’s theorem shows that, microlocally near γ1 , we get a unitary equivalence between the operator P acting on H(ΛG ) and operator
an h-pseudodifferential microlocally defined near τ = x = ξ = 0 in T ∗ S 1 × R , with the leading symbol p0 (τ, x, ξ, ) = f (τ ) + iq(τ, x, ξ) + O(2 ), independent of t. Taking into account an additional conjugation by the elliptic operator eiA/h , acting on L2S (S 1 × R), with A∼
∞
ak (t, τ, x, ξ, )hk ,
k=1
constructed as a formal power series in , h in Proposition 3.2, we see that microlocally near γ1 , the operator P : H(ΛG ) → H(ΛG ) is equivalent to an operator of the form ∞ P (τ, x, ξ, ) ∼ hk pk (τ, x, ξ, ), (6.49) k=0
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acting on L2S (S 1 × R), whose full symbol is independent of t. We have p0 = f (τ ) + iq(τ, x, ξ) + O(2 ),
(6.50)
and Re q(0, x, ξ) ∼ x2 + ξ 2 on the real domain. We shall first consider the general case when the subprincipal symbol of the unperturbed operator P=0 does not necessarily vanish, and in doing so, it will be assumed that (6.51) h = O(1)hδ , δ > 0. As in Section 5, we write h h P = f (τ ) + iq(τ, x, ξ) + O() + p1 + h p2 + · · · . According to Proposition 5.3, there exists a holomorphic canonical transformation κσ,, h : neigh(0, C2 ) → neigh(0, C2 ),
depending analytically on σ ∈ neigh(0, C) and smoothly on , h ∈ neigh(0, R), such that h Im κσ,, h (y, η) = O + , when σ, y, η are real, and such that h y2 + η2 σ, κσ,, h (y, η) = f (σ) + ig, h σ, . p0 + p1 2 Here g, h (σ, q) is an analytic function, depending smoothly on , h/, for which
∂ Re g,0 (0, 0) > 0. ∂q We now lift the family of locally defined canonical transformations κσ,, h to a canonical transformation 1 × C (s, σ; y, η) Ξ, h : neigh Im s = 0, σ = y = η = 0, T ∗ S 1 × C
→ (t, τ ; x, ξ) ∈ neigh Im t = 0, τ = x = ξ = 0, T ∗ S given by Ξ, h (s, σ; y, η) = (t, τ ; x, ξ) = (s + h(y, σ, η), σ; κσ,, h (y, η)).
(6.52)
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Here h(y, σ, η) is uniquely determined up to a function g = g(σ), and if ϕσ,, h (x, y, θ) is an analytic family of non-degenerate phase functions (in the sense of H¨ ormander) locally generating the family κσ,, h , then
Φ, h (t, x, s, y, θ, σ) := ϕσ,, h (x, y, θ) + (t − s)σ
is a non-degenerate phase function with θ, σ as fiber variables, such that Φ, h generates the graph of Ξ, h . 1 × C , which in ⊂ T∗ S Associated to Ξ, h , we introduce an IR-manifold Λ
a complex neighborhood of τ = x = ξ = 0, is equal to Ξ, h T ∗ S 1 × R , and
further away from this set agrees with T ∗ S 1 × R . In the intermediate region, we
in such a way that it remains an ( + h )-perturbation of T ∗ S 1 × R , construct Λ it is true that and so that everywhere on Λ, =⇒ τ ∈ R. (t, τ ; x, ξ) ∈ Λ
(6.53)
If we now use the standard FBI-Bargmann transformation, viewed as a mapping on L2S (S 1 × R), so that under the associated canonical transformation, T ∗ (S 1 × R) 1 × C); (τ, ξ) = −Im (t, x)}, then as before we see is mapped to {(t, τ ; x, ξ) ∈ T ∗ (S is described by that after an application of such a transformation, the manifold Λ a weight function Φ = Φ(Im t, x) which does not depend on Re t. At this stage, the situation is similar to the previously analyzed torus case, and, in particular, we see again that the form of the weight Φ(Im t, x) implies that the term f (τ ) in (6.50) gives no contribution to the imaginary part of the operator. Summing up the discussion so far, we arrive to the following result. Proposition 6.5 Make the assumptions of case II in the introduction, and assume that F0 = inf Re q = 0. Σ
Assume that = O(h ), for some δ > 0, is such that h . There exists a closed IR-manifold Λ ⊂ C4 and finitely many simple closed disjoint curves γ1 , . . . , γN ⊂ Λ, which are ( + h/)-close to the closed Hp -trajectories ⊂ p−1 (0) ∩ R4 , along which Re q = 0, such that when ρ is outside a small neighborhood of ∪N j=1 γj in Λ, then 1 or |Im P (ρ, h)| ≥ . (6.54) |Re P (ρ, h)| ≥ |O(1)| |O(1)| δ
This estimate is true away from an arbitrarily small neighborhood of ∪N j=1 γj , provided that the implicit constant in (6.54) is chosen sufficiently large. The manifold Λ coincides with R4 away from a neighborhood of p−1 (0) ∩ R4 and is ( + h/)close to R4 everywhere. For each j with 1 ≤ j ≤ N , there exists a canonical transformation
κ,j : neigh (γj , Λ) → neigh τ = x = ξ = 0, T ∗(S 1 × R) ,
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whose domain of definition does not intersect the closure of the union of the domains of the κ,k for k = j, and an elliptic Fourier integral operator Uj = O(1) : H(Λ) → L2S (S 1 × R), associated to κ,j , such that, microlocally near γj , Uj P = Pj Uj . Here Pj = Pj (hDt , (1/2)(x2 + (hDx )2 ), , h ; h) has the Weyl symbol h h x2 + ξ 2 Pj τ, x, ξ, , ; h = f (τ ) + iGj τ, , , ; h , 2 with
∞ h h Gj τ, q, , ; h ∼ hl Gj,l τ, q, , ,
h → 0,
l=0
and Gj,l holomorphic in (τ, q) ∈ neigh(0, C2 ), smooth in , h/ ∈ neigh(0, R). Furthermore, Re Gj,0 (0, 0, 0, 0) = 0 and ∂ Re Gj,0 (0, 0, 0, 0) > 0. ∂q Take now small open sets Ωj ⊂ Λ, 1 ≤ j ≤ N , such that γj ⊂ Ωj and Ωj ∩ Ωk = ∅,
j = k.
Let χj ∈ C0∞ (Ωj ), 0 ≤ χj ≤ 1, be such that χj = 1 near γj , 1 ≤ j ≤ N . When z ∈ C satisfies 1 |Re z| ≤ , |Im z| ≤ , (6.55) C C and (P − z)u = v, it follows from (6.54) by repeating the arguments of the torus case, that N 1 χj u || ≤ O (6.56) || 1 − || v || + O(h∞ )|| u ||. j=1 We shall now discuss the setup of the global Grushin problem. Associated with each j , 1 ≤ j ≤ N , we have the quasi-eigenvalues given in Proposition normal form P 5.4, S S k0 k0 1 h , h k2 + z(j, k) := f h(k1 − ) − + iGj h(k1 − ) − , , ; h , 4 2π 4 2π 2 when 1 ≤ j ≤ N and k = (k1 , k2 ) ∈ Z × N. We also introduce an ON system of eigenfunctions of the (formally) commuting operators Pj , k0 S 1 i ek (t, x) = √ e h (h(k1 − 4 )− 2π )t ek2 (x), 2π
k = (k1 , k2 ) ∈ Z × N,
which forms an ON basis in L2S (S 1 × R). Here ek2 (x), k2 ∈ N, are the normalized eigenfunctions of 1/2(x2 + (hDx )2 ) with eigenvalues (k2 + 1/2)h.
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When 1 ≤ j ≤ N , let Mj = # z(j, k), |Re z(j, k)| <
Ann. Henri Poincar´e
1 , |Im z(j, k)| < |O(1)| |O(1)|
.
Then Mj = O(h−2 ) and we let k(j, 1), . . . , k(j, Mj ) ∈ Z × N be the corresponding half-lattice points. We introduce the auxiliary operator R+ : H(Λ) → CM1 × · · · × CMN , given by R+ u(j)(l) = (Uj χj u|ek(j,l) ),
1 ≤ j ≤ N, 1 ≤ l ≤ Mj .
Here the inner product in the right-hand side is taken in L2S (S 1 × R). Define also R− : CM1 × · · · × CMN → H(Λ), by R− u− =
Mj N
u− (j)(l)Vj ek(j,l) .
j=1 l=1
Here Vj is a microlocal inverse of Uj . We then claim that for z ∈ C satisfying (6.55), with a sufficiently large C > 0, the Grushin operator P=
P − z R+
R− 0
: H(Λ) × CM1 × · · · × CMN → H(Λ) × CM1 × · · · × CMN
(6.57)
is bijective. Indeed, when v ∈ H(Λ) and v+ ∈ CM1 × · · · × CMN , let us consider (P − z)u + R− u− = v, (6.58) R+ u = v+ . As in (6.56), we get N 1 χj u || ≤ O || 1 − || v || + O(h∞ ) (|| u || + || u− ||) . j=1 Applying χj and then Uj , 1 ≤ j ≤ N , to the first equation in (6.58), we get Mj (Pj − z)Uj χj u + l=1 u− (j)(l)ek(j,l) = (6.59) Uj (χj v + [P , χj ]u) + R∞ u + R−,∞ (j)u− , (Uj χj u|ek(j,l) ) = v+ (j)(l), 1 ≤ l ≤ Mj ,
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and here R∞ = R∞ (j) = O(h∞ ) and R−,∞ (j) = O(h∞ ) in the corresponding operator norms. For each j, 1 ≤ j ≤ N , we get a microlocally well-posed Grushin problem for Pj − z in L2S (S 1 × R), with inverse of the norm O(1/), and the global well-posedness of (6.58) follows. The inverse E of P in (6.57) has the form E(z) E+ E= , (6.60) E− E−+ (z) and a straightforward computation shows that E+ : CM1 × · · · × CMN → H(Λ) modulo O(h∞ ), is given by E+ v+ ≡
Mj N
v+ (j)(l)Vj ek(j,l) = R− v+ ,
j=1 l=1
and E−+ (z) ∈ L CM1 × · · · × CMN , CM1 × · · · × CMN is a block diagonal matrix with the blocks E−+ (z)(j) ∈ L(CMj , CMj ), 1 ≤ j ≤ N , of the form E−+ (z)(j)(m, n) ≡ (z − z(j, k(j, m))) δmn ,
1 ≤ m ≤ n ≤ Mj ,
modulo O(h∞ ). The computation of eigenvalues near the boundary of the band has therefore been justified, and we get the second of our two main results. Theorem 6.6 Assume that F0 = inf Re q Σ
is achieved along finitely many closed Hp -trajectories γ1 , . . . , γN ⊂ p−1 (0) ∩ R4 of minimal period T (0), and that the Hessian of Re q at the corresponding points ρj ∈ Σ, j = 1, . . . , N , is positive definite. Let us write S and k0 to denote the common values of the action and the Maslov index of γj , j = 1, . . . , N , respectively. Assume that = O(hδ ) for a fixed δ > 0, is such that h . Let C > 0 be sufficiently large. Then the eigenvalues of P in the set 1 1 1 1 − , + i F0 − , F0 + (6.61) C C C C are given by z(j, k) = f
k0 S h k1 − − 4 2π
1 k0 h S ,h + k2 , , ; h , + iGj h k1 − − 4 2π 2
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modulo O(h∞ ), when 1 ≤ j ≤ N and (k1 , k2 ) ∈ Z×N. Here f (τ ) is real-valued with f (0) = 0 and f (0) > 0. The function Gj (τ, q, , h/; h), 1 ≤ j ≤ N , is analytic in τ and q in a neighborhood of (0, 0) ∈ C2 , and smooth in , h/ ∈ neigh(0, R), and has an asymptotic expansion in the space of such functions, as h → 0, ∞ h h Gj τ, q, , ; h ∼ Gj,l τ, q, , , hl . l=0
We have Re Gj,0 (0, 0, 0, 0) = F0 and ∂ Re Gj,0 (0, 0, 0, 0) > 0, ∂q
1 ≤ j ≤ N.
Remark. With obvious modifications, Theorem 6.6 describes the eigenvalues in the region (6.61), when F0 = supΣ Re q, if we assume that F0 is attained along finitely many trajectories of minimal period T (0), such that the transversal Hessian of Re q along the trajectories is negative definite. The treatment of the remaining case of the eigenvalues near the boundary of the band (6.61), when the subprincipal symbol of P=0 vanishes proceeds in full analogy with the previously analyzed torus case. Thus, restricting attention to the region M h2 < = O(hδ ), M 1, we find that the symbol of Im P , acting on H(ΛG ) is O(), and away from an arbitrarily small but fixed neighborhood of ∪N j=1 γj we have that |Im P (ρ)| ≥ /C when we restrict the attention to the region |Re P (ρ)| ≤ 1/C. When working microlocally near τ = x = ξ = 0 in T ∗ (S 1 ×R) and simplifying the operator (6.49) further, we use Proposition 5.3 to find a holomorphic canonical transformation κσ,, h2 : neigh(0, C2 ) → neigh(0, C2 )
depending analytically on σ ∈ neigh(0, C) and smoothly on , h2 / ∈ neigh(0, R), such that h2 y2 + η2 σ, κσ,, h2 (y, η) = f (σ) + ig, h2 σ, p0 + p2 . 2 1 × C) As before, associated to κσ,, h2 , we construct an IR-submanifold of T ∗ (S
which is ( + h2 /)-close to T ∗ (S 1 × R), and which has the property that τ is real along this submanifold. This leads to a new IR-manifold Λ ⊂ C4 such that on Λ, Im P has a symbol of modulus ∼ in the region |Re P | < 1/C, when away from the union of small neighborhoods Ωj of γj ⊂ Λ, 1 ≤ j ≤ N . In Ωj , P is equivalent to an operator constructed in Section 5, which has the form h2 x2 + (hDx )2 , , ; h , f (hDt ) + iGj hDt , 2
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∞ h2 h2 Gj τ, q, , ; h ∼ Gj,l τ, q, , hl . l=1
Again we see that we have a globally well-posed Grushin problem for P − z in the h-dependent Hilbert space H(Λ). The following result complements Theorem 6.6. Theorem 6.7 Make the assumptions of Theorem 6.6, and assume in addition that the subprincipal symbol of P=0 vanishes. Then for in the range h2 < hδ ,
δ > 0,
the eigenvalues of P in the set of the form 1 1 1 1 − , + i F0 − , F0 + , C C C C
C 1,
are given by 1 k0 k0 h2 S S ,h + k2 , , ; h , f h k1 − − + iGj h k1 − − 4 2π 4 2π 2 modulo O(h∞ ), when 1 ≤ j ≤ N and (k1 , k2 ) ∈ Z×N. Here f (τ ) is real-valued with f (0) = 0 and f (0) > 0. The function Gj (τ, q, , h2 /; h) for 1 ≤ j ≤ N , is analytic in τ and q in a neighborhood of (0, 0) ∈ C2 , and smooth in , h2 / ∈ neigh(0, R), and has an asymptotic expansion in the space of such functions, as h → 0, ∞ h2 h2 Gj,l τ, q, , Gj τ, q, , ; h ∼ hl , l=0
where Re Gj,0 (0, 0, 0, 0) = F0 and ∂ Re Gj,0 (0, 0, 0, 0) > 0. ∂q
7 Barrier top resonances in the resonant case Consider P = −h2 ∆ + V (x),
p(x, ξ) = ξ 2 + V (x), x, ξ ∈ R2 ,
(7.1)
and let us assume that V (x) is real-valued, and that it extends holomorphically to a set {x ∈ C2 ; |Im x| ≤ Re x/C}, for some C > 0, and tends to 0 when x → ∞ in that set. The resonances of P can be defined in an angle {z ∈ C; −2θ0 < arg z < 0} for some fixed small θ0 > 0, as the eigenvalues of P in the same region. iθ 2 e
0R
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We shall assume that V (0) = E0 > 0, ∇V (0) = 0, and V (0) is a negative definite quadratic form. Assume also that the union of trapped Hp -trajectories in p−1 (E0 ) ∩ R4 is reduced to (0, 0) ∈ R4 . (We recall that a trapped trajectory is a maximal integral curve of the Hamilton vector field Hp , contained in a bounded set.) We are then interested in resonances of P near E0 , created by the critical point of V . After a linear symplectic change of coordinates, and a conjugation of P by means of the corresponding metaplectic operator, we may assume that as (x, ξ) → 0, p(x, ξ) − E0 =
2
λj 2 ξj − x2j + p3 (x) + p4 (x) + · · · , 2 j=1
λj > 0.
(7.2)
Here pj (x) is a homogeneous polynomial of degree j ≥ 3. For future reference we recall that according to the theory of resonances developed in [12], the resonances of P in a fixed h-independent neighborhood of E0 can also be viewed as the eigenvalues of P : H(ΛtG , 1) → H(ΛtG , 1), equipped with the domain H(ΛtG , ξ2 ). Here G ∈ C ∞ (R2 ; R) is an escape function in the sense of [12], t > 0 is sufficiently small and fixed, and ΛtG is a suitable IR-deformation of R4 , associated with the function G. The Hilbert space H(ΛtG , 1) consists of all tempered distributions u such that a suitable FBI transform T u belongs to a certain exponentially weighted L2 -space. We refer to [12] for the original presentation of the microlocal theory of resonances, and to [18] for a simplified version of the theory, which is applicable in the present setting of operators with globally analytic coefficients, converging to the Laplacian at infinity. Here we shall only remark that as in [17], the escape function G can be chosen such that G = x · ξ in a neighborhood of (0, 0), and such that Hp G > 0 on p−1 (E0 ) \ {(0, 0)}. Under the assumptions above, but without any restriction on the dimension and without any assumption on the signature of V (0), all resonances in a disc around E0 of radius Ch were determined in [23]. Here C > 0 is arbitrarily large and fixed. (See also [7].) Specializing the result of [23] to the present barrier top case, we may recall that in this disc, the resonances are of the form 1 1 λ1 h − i k2 + λ2 h + O(h3/2 ), h → 0, k = (k1 , k2 ) ∈ N2 . E0 − i k1 + 2 2 (7.3) Furthermore, in the non-resonant case, i.e., when λ · k = 0,
0 = k ∈ Z2 ,
(7.4)
a result of Kaidi and Kerdelhu´e [17] extended [23] to obtain all resonances in a disc around E0 of radius hδ , for each fixed δ > 0 and h > 0 small enough depending on δ. In this case, the resonances are given by asymptotic expansions in integer powers of h, with the leading term as in (7.3).
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Throughout this section we shall work under the following resonant assumption, λ · k = 0, for some 0 = k ∈ Z2 .
(7.5)
In this case we shall show how to obtain a description of all the resonances in an energy shell of the form h4/5 |E − E0 | < O(1)hδ ,
δ > 0,
provided that we avoid an arbitrarily small half-cubic neighborhood of E0 −i[0, ∞). The starting point is a reduction to an eigenvalue problem for a scaled operator, as in [17], [20], [24]. In these works it was shown how to adapt the theory of [12] so that P can be realized as an operator acting on a suitable H(Λ)-space, where
Λ ⊂ C4 is an IR-manifold which coincides with T ∗ eiπ/4 R2 near (0, 0), and further away from a neighborhood of this point, it agrees with ΛtG . Furthermore, Λ has the property that on this manifold, p − E0 is elliptic away from a small neighborhood of (0, 0), and this neighborhood can be chosen arbitrarily small, provided that the constant in the elliptic estimate is taken sufficiently large. Using a Grushin reduction exactly as in [20], we may and will therefore reduce the study of resonances of P near E0 to an eigenvalue problem for P after the complex x scaling, which near (0, 0) is given by x = eiπ/4 x , ξ = e−iπ/4 ξ, , ξ ∈ R. Using (7.2) and dropping the tildes from the notation, we see that the principal symbol of the scaled operator has the form E0 − i p2 (x, ξ) + ie3πi/4 p3 (x) + ie4iπ/4 p4 (x) + · · · , (x, ξ) → 0, (7.6) where p2 (x, ξ) =
2
λj 2 ξj + x2j 2 j=1
(7.7)
is the harmonic oscillator. In what follows we shall therefore consider an h-pseudodifferential operator P on R2 , microlocally defined near (0, 0), with the leading symbol p(x, ξ) = p2 (x, ξ) + ie3πi/4 p3 (x) + · · · , (x, ξ) → 0, (7.8) and with the vanishing subprincipal symbol. We extend P to be globally defined as a symbol of class S 0 (R4 ) = Cb∞ (R4 ), with the asymptotic expansion P (x, ξ; h) ∼ p(x, ξ) + h2 p(2) (x, ξ) + · · · , in this space, and so that |p(x, ξ)| ≥ outside a small neighborhood of (0, 0).
1 , C
C > 0,
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We shall be interested in eigenvalues E of P with |E| ∼ 2 , 0 < 1. It follows from [26] that the corresponding eigenfunctions are concentrated in a region where |(x, ξ)| ∼ , and so we introduce the change of variables x = y, hδ ≤ ≤ 1, 0 < δ < 1/2. Then 1 1 P (x, hDx ; h) = 2 P ((y, hDy ); h), 2
h h = 2 1.
The corresponding new symbol is 1 1 P ((y, η); h) ∼ 2 p((y, η)) + 2 h2 p(2) ((y, η)) + · · · , 2 to be considered in the region where |(y, η)| ∼ 1. The leading symbol becomes 1 p((y, η)) = p2 (y, η) + ie3πi/4 p3 (y) + O(2 ), 2 for (y, η) in a fixed neighborhood of (0, 0). Now the resonant assumption (7.5) implies that the Hp2 -flow is periodic on (E), for E ∈ neigh(1, R), with period T > 0 which does not depend on E. For p−1 2 z ∈ neigh(1, C), we shall then discuss the invertibility of 1/2 P (x, hDx ; h) − z in the range of , dictated by Theorem 6.4, and using h as the new semiclassical parameter. Indeed, all the assumptions of that theorem are satisfied in a fixed neighborhood of (0, 0), and outside such a neighborhood, we have ellipticity which guarantees the invertibility there. Proposition 7.1 Assume that (7.5) holds. When p3 is a homogeneous polynomial of degree 3 on R2 , we let p3 denote the average of p3 along the trajectories of the Hamilton vector field of p2 in (7.7), and assume that p3 is not identically zero. Let F0 ∈ R be a regular value of cos(3π/4)p3 restricted to p−1 2 (1), and assume that T is the minimal period of the Hp2 -trajectories in the manifold Λ1,F0 given by 3π Λ1,F0 : p2 = 1, cos p3 = F0 . 4 Assume that Λ1,F0 is connected. Let satisfy h2/5 = O(1)hδ ,
δ > 0.
(7.9)
Then for z in the set 1−
! ! 1 1 1 1 ,1 + , F0 + + i F0 − , |O(1)| |O(1)| |O(1)| |O(1)|
(7.10)
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the operator −2 P (x, hDx ; h) − z : L2 → L2 is non-invertible precisely when z = zk for some k ∈ Z2 , where the numbers zk satisfy 2 S h α h zk = P , , ; h + O(h∞ ), h(k − ) − h = 2. 4 2π 2 Here P ξ, , h ; h has an expansion as h → 0, ∞ h2 h2 j P ξ, , ; h ∼ p2 (ξ1 ) + , h rj ξ, , j=0
where r0 = ie
3πi/4
h2 p3 (ξ) + O +
.
The coordinates ξ1 = ξ1 (E) and ξ2 = ξ2 (E, F ) are the normalized actions of 3π ΛE,F : p2 = E, cos p3 = F, 4 for E ∈ neigh(1, R), F ∈ neigh(F0 , R), given by 1 η dy − η dy , ξj = 2π γj (E,F ) γj (1,F0 )
j = 1, 2,
(7.11)
with γj (E, F ) being fundamental cycles in ΛE,F , such that γ1 (E, F ) corresponds to a closed Hp2 -trajectory of minimal period T . Furthermore, Sj = η dy, j = 1, 2, S = (S1 , S2 ), (7.12) γj (1,F0 )
and α ∈ Z2 is fixed. Remark. In the case when the compact manifold Λ1,F0 has finitely many connected components Λj , 1 ≤ j ≤ M , with each Λj being diffeomorphic to a torus, the set of z in (7.10) for which the operator −2 P (x, hDx ; h) − z is non-invertible agrees with the union of the quasi-eigenvalues constructed for each component, up to an error which is O(h∞ ). In the following discussion, for simplicity it will be tacitly assumed that Λ1,F0 is connected. The reduction by complex scaling together with the scaling argument above and Proposition 7.1 allows us to describe the resonances E of the operator (7.1) in the set (7.13) h4/5 |E − E0 | = O(1)hδ , δ > 0,
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2 α S h h k− − , , ; h + O(h∞ ), E = E0 − i P 4 2π 2
(7.14)
where we choose > 0 with |E − E0 | /2 ∼ 1. The description (7.14) is valid provided that we exclude sets of the form 3/2 E ∈ C, Re E − E0 − F0 |Im E| <
1 3/2 |Im E| , |O(1)|
(7.15)
from the domain (7.13). Here F0 varies over the set of critical values of cos(3π/4) 2 p3 restricted to p−1 2 (1). Indeed, writing E = E0 − i z, we see that the set (7.15) in the E-plane corresponds to the set |Im z − F0 | < /|O(1)| in the z-plane. It is also clear that when F0 ∈ {inf p−1 (1) cos(3π/4)p3 , supp−1 (1) cos(3π/4)p3 }, an 2 2 application of Theorem 6.7 will allow us to extend a description of the resonances to a set of the form (7.15), provided that the assumptions of that theorem are satisfied. In what follows, we shall content ourselves by discussing an explicit example. Our starting point will be deriving an expression for p3 . Consider p2 (x, ξ) =
2 λj j=1
2
(xj + ξj2 ),
λj > 0,
where the λj satisfy (7.5). In order to describe the Hp2 -flow, it is convenient to introduce the action-angle variables Ij ≥ 0, τj ∈ R/2πZ for p2 , given by xj =
" " 2Ij cos τj , ξj = − 2Ij sin τj .
(7.16)
λj Ij and the Hamilton flow is given by R t → (I(t), τ (t)), with Then p2 = I(t) = I(0), τ (t) = τ (0) + tλ, λ = (λ1 , λ2 ). In the original coordinates, this gives " xj (t) = 2Ij (0) cos(τj (0) + λj t) " (7.17) ξj (t) = − 2Ij (0) sin(τj (0) + λj t), and we get a combination of two rotations in (xj , ξj ), j = 1, 2, with minimal periods 2π/λj (except in the degenerate cases when one of the (xj , ξj ) vanishes). Avoiding the totally degenerate case when I = 0, we get trajectories with • minimal period 2π/λ2 when I1 (0) = 0, • minimal period 2π/λ1 when I2 (0) = 0, • minimal period T = −k20 2π/λ1 = k10 2π/λ2 , when both I1 (0) and I2 (0) are = 0.
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Here we let k 0 = (k10 , k20 ) be the point satisfying (7.5), which has minimal norm and positive first component. The integers k in (7.5) are equally spaced on the straight line λ⊥ , and it will be convenient to represent them in the form nk 0 , n ∈ Z \ {0}. We shall now compute the averages xα along the Hp2 -trajectories of a 1 α2 monomial xα = xα 1 x2 . Using (7.17), we get α
xα = I(0) 2 2 α
I(0) 2 1 = |α| 2 2 T
T
|α| 2
1 T
T
(cos(τ1 (0) + λ1 t))α1 (cos(τ2 (0) + λ2 t))α2 dt
(7.18)
0
(ei(τ1 (0)+λ1 t) + e−i(τ1 (0)+λ1 t) )α1 (ei(τ2 (0)+λ2 t) + e−i(τ2 (0)+λ2 t) )α2 dt.
0
Here the integrand can be developed with the binomial theorem, α2 α1 α1 α2 i((2k1 −α1 )τ1 (0)+(2k2 −α2 )τ2 (0)) i((2k1 −α1 )λ1 +(2k2 −α2 )λ2 )t e , e k1 k2
k1 =0 k2 =0
and only the terms with (2k1 − α1 )λ1 + (2k2 − α2 )λ2 = 0 can give a non-vanishing contribution to the integral. This means that 2k − α = nk 0 for some n ∈ Z, i.e., α + nk 0 = 2k with 0 ≤ k ≤ α componentwise. We get I(0)α/2 x = |α|/2 2 α
α+nk0 =2k 0≤k≤α
α1 k1
α2 cos((2k1 − α1 )τ1 (0) + (2k2 − α2 )τ2 (0)), k2
(7.19) where it is understood that n ∈ Z, k ∈ N2 , and where we notice that if α + nk0 = 2k, 0 ≤ k ≤ α, then k := α − k also participates in the sum, since 0 ≤ k ≤ α and α − nk0 = 2k. Also notice that the cosine in (7.19) can be written in the form cos(nk0 · τ (0)). In order to find the non-vanishing terms in (7.19), we consider the “line” Z n → α + nk0 ∈ Z2 . The points on this line in the rectangle ([0, 2α1 ] × [0, 2α2 ]) ∩ N2 with even coordinates correspond to the terms in (7.19). Example 1. Let k 0 = (1, −1), corresponding for instance to λ = (1, 1). In this case the two components of α must have the same parity. For α = (2, 0) we have only one term with n = 0, k = (1, 0), and x21 = I1 (0). For α = (0, 2) we get similarly x22 = I2 (0). For α = (1, 1) we get two " terms with n = 1, k = (1, 0) and n = −1, k = (0, 1) respectively, and x1 x2 = I1 (0)I2 (0) cos(τ1 (0) − τ2 (0)). For |α| = 3 we get no non-vanishing terms. For α = (4, 0) we have one term with n = 0, k = (2, 0) and we get x41 = 32 I1 (0)2 . For α = (0, 4) we get similarly, x42 = 32 I2 (0)2 . For α = (2, 2) we get one term with n = 2, k = (2, 0) and one with n = −2, k = (0, 2), We also have a term with n = 0, k = (1, 1), and this leads to x21 x22 = I1 (0)I2 (0)(1 + 12 cos 2(τ1 (0) − τ2 (0))).
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It follows from Example 1 that Proposition 7.1 does not apply when λ = Const. (1, 1), since in this case p3 ≡ 0. We shall therefore consider a different choice of the resonant frequencies. Example 2. Let us take k 0 = (2, −1), corresponding for instance to λ = (1, 2), and let |α| = 3. For α = (3, 0), (0, 3), (1, 2) it follows from (7.19) that xα = 0. For α = (2, 1) we get two terms, one with n = 1, k = (2, 0) and one with n = −1, k = (0, 1). It follows that x21 x2 = 2−1/2 I1 (0)I2 (0)1/2 cos(2τ1 (0) − τ2 (0)).
(7.20)
For future reference, we shall also describe how the averages xα can be computed after a suitable complex linear change of symplectic coordinates. Introduce x = √12 (y + iη) y = √12 (x − iξ) , . η = i√1 2 (x + iξ) ξ = √i2 (y − iη) In these coordinates p =
2 j=1
iλj yj ηj , and
exp (tHp )(y, η) = (eitλ1 y1 , eitλ2 y2 , e−itλ1 η1 , e−itλ2 η2 ), so that 1 y α η β = T
T
eiλ·(α−β)t dty α η β = 0
We apply this to xα =
1 2|α|/2
and get x = 2 α
y α η β if λ · (α − β) = 0, 0 otherwise.
α y k (iη)α−k , k
0≤k≤α
α (x − iξ)k (x + iξ)α−k . k =2k
−|α|
(7.21)
α+nk0 0≤k≤α
As before we check that for each term present there is also the complex conjugate. The computations of Examples 1 and 2 can be written like (7.21). We shall only do it for the last example with k 0 = (2, −1), α = (2, 1): x21 x2 =
1 1 Re ((x1 + iξ1 )2 (x2 − iξ2 )) = (x21 x2 + 2x1 ξ1 ξ2 − x2 ξ12 ). 4 4
(7.22)
We may assume that λ = (1, 2), so that p2 =
1 2 (x + ξ12 ) + (x22 + ξ22 ), 2 1
and we may then check directly that Hp2 x21 x2 = 0.
(7.23)
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From (7.22) and (7.23) it is clear that dp2 and dx21 x2 are linearly independent except on some set of measure 0. When computing the critical points of x21 x2 on p−1 2 (1), we shall first make use of the (I, τ )-coordinates. From (7.20) we recall that √ 1 (7.24) p2 = I1 + 2I2 , 2x21 x2 = I1 I22 cos(2τ1 − τ2 ). It follows from the Hamilton equations that θ := 2τ1 − τ2 is invariant under the Hp2 -flow, and we can therefore work in the coordinates I1 , I2 , θ. We have √ 1 1 1 −1 dp2 = dI1 + 2dI2 , 2dx21 x2 = (I22 cos θ)dI1 + I1 I2 2 (cos θ)dI2 − I1 I22 (sin θ)dθ. 2 (7.25) If θ ∈ πZ, I1 , I2 = 0, we have ∂θ x21 x2 = 0, and hence the differentials are linearly independent. Still with I1 , I2 = 0, let θ ∈ πZ, so that cos θ = ±1. Then the differentials are linearly dependent iff 1 2 , i.e., iff I1 = 4I2 . 0 = det 1/2 1 − 12 I2 2 I1 I2 This gives two closed trajectories inside the energy surface p2 = 1 and the corresponding values for x21 x2 : I1 =
2 1 1 , I2 = , 2τ1 − τ2 = 0; x21 x2 = √ , 3 6 3 3
(7.26)
and
2 1 −1 , I2 = , 2τ1 − τ2 = π; x21 x2 = √ . (7.27) 3 6 3 3 When I1 = 0 or I2 = 0, the question of linear independence of the differentials should be analyzed directly in the (x, ξ)-coordinates (or (y, η)-coordinates), and here we shall use (7.22). On the plane I1 = 0, corresponding to x1 = ξ1 = 0, we have dx21 x2 = 0, so here we have linear dependence, with the corresponding critical value x21 x2 = 0. On the plane I2 = 0, corresponding to x2 = ξ2 = 0, we have dx21 x2 = 14 (x21 − ξ12 )dx2 + 12 x1 ξ1 dξ2 , dp2 = x1 dx1 + ξ1 dξ1 , I1 =
and these differentials are independent, since we avoid the point x = ξ = 0. We shall now look at the nature of the critical points of x21 x2 , when viewed as a function on Σ := p−1 2 (1)/exp (RHp ). For the trajectories found in (7.26) and (7.27), we use θ and I2 as local coordinates on Σ, and using√(7.24) together with I1 = 1 − 2I2 , we get for θ = kπ, k = 0, 1, I2 = 1/6 and f = 2x21 x2 , 1 3 − 12 1 − 32 2 k 2 k 2 I ∂θ ∂I2 f = 0, ∂θ f = −(1 − 2I2 )I2 (−1) , ∂I2 f = −(−1) + I2 . 4 2 2 For k = 0 we therefore have a non-degenerate maximum and for k = 1 we get a non-degenerate minimum.
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For the third trajectory, given by x1 = ξ1 = 0,
x22 + ξ22 = 1,
use that x21 x1 vanishes to the second order Hessian in p−1 2 (1) can be identified with the
we sal which is given by the matrix
1 2
x2 ξ2
(7.28)
there, and hence that the transverfree Hessian with respect to x1 , ξ1 ,
ξ2 . −x2
The eigenvalues are 12 and − 21 . Thus we have a non-degenerate saddle point. We summarize the discussion above in the following proposition. Proposition 7.2 Let p2 (x, ξ) =
1 2 x1 + ξ12 + (x22 + ξ22 ). 2
Then the Hp2 -flow is periodic in p−1 2 (E), for E ∈ neigh(1, R), with period T = 2π. If p3 (x) = a3,0 x31 + a1,2 x1 x22 + x21 x2 + a0,3 x32 , then we have
1 2 x1 x2 + 2x1 ξ1 ξ2 − x2 ξ12 . 4 The differential of p3 , restricted to p−1 2 (1), vanishes along three closed Hp2 trajectories, given by (7.26), (7.27), and (7.28). These critical trajectories are nondegenerate in the sense that the transversal √ Hessian of p3 is non-degenerate. The set of the critical values of p3 is {±(3 3)−1 , 0}, and the maximum and the minimum of p3 are attained along the trajectories (7.26) and (7.27), respectively. The transversal Hessian of p3 along (7.28) has the signature (1, −1). The minimal period of the trajectories in (7.26) and (7.27) is equal to T = 2π, and the minimal period in (7.28) is π. Let finally F0 be a regular value of p3 restricted to p−1 2 (1). Then the minimal period of every closed Hp2 -trajectory in the Lagrangian manifold p3 (x, ξ) =
Λ1,F0 : p2 = 1, p3 = F0 is equal to T = 2π. We now return to the operator P with principal symbol p in (7.1). Under the general assumptions from the beginning of this section, we shall assume that as (x, ξ) → 0, we have p(x, ξ) − E0 =
1 2 (ξ − x21 ) + (ξ22 − x22 ) + p3 (x) + O(x4 ), 2 1
where p3 (x) = a3,0 x31 + a1,2 x1 x22 + x21 x2 + a0,3 x32 . √ √ Let us write A1 = −(3 6)−1 , A2 = (3 6)−1 , and A3 = 0.
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Proposition 7.3 The resonances of P in the domain 3 # $ % 4/5 δ z ∈ C; h
|z − E0 | = O(1)h \ {z; Re z − E0 − Aj |Im z|3/2 j=1
< η |Im z|3/2 }, (7.29) where δ, η > 0 are arbitrary but fixed, are given by ∞ 2 S h α h − , , 5 , (7.30) hj −2j rj k− ∼ E0 − i h(k1 − α1 /4) + 3 2 4 2π j=0 with
h2 h2 r0 ξ, , 5 = ie3πi/4 p3 (ξ) + O + 5 , 2 2 h h rj ξ, , 5 = O + 5 , j ≥ 1
analytic in ξ ∈ neigh(0, C2 ), and smooth in , h2 / ∈ neigh(0, R). We have k = (k1 , k2 ) ∈ Z2 , S = (S1 , S2 ) with S1 = 2π, and α = (α1 , α2 ) ∈ Z2 is fixed, and we choose > 0 with |E − E0 | ∼ 2 . The resonances in the set # 3/2 3/2 $ and h4/5 |z − E0 | = O(1)hδ , z ∈ C, Re z − E0 − A1 |Im z| < η |Im z| (7.31) are given by E0 plus ∞ 2 h h 1 α1 α 1 h 1 k1 − h k1 − + i3 − 1, 2 k2 + hj −2j Gj , , 5 , i 4 2 4 2 j=0 (7.32) with (k1 , k2 ) ∈ Z × N, α1 ∈ Z, and |E − E0 | ∼ 2 . The function G0 (τ, q, , h2 /5 ) is ∂ such that Re G(0, 0, 0, 0) = A1 and ∂q Re G0 (0, 0, 0, 0) > 0. An analogous description of resonances is valid in the domain (7.31) with A1 replaced by A2 . Here in (7.30) we have also used that when expressed in terms of the action coordinates from (7.11), it is true that p2 (ξ1 ) = ξ1 + 1. Remark. If we replace rj (ξ, , h2 /5 ) in (7.30) by rj (ξ + S/2π, , h2 /5 ), then we get ∞ 2 h α h ∼ E0 − i h(k1 − α1 /4) + 3 k− , , 5 . hj −2j rj 2 4 j=0 Now let us notice that the choice of is not unique, and replacing by λ, with λ ∼ 1, does not affect the resonances. It follows therefore that ξ τ , λ, rj (ξ, , τ ) = λ3−2j rj . (7.33) λ2 λ5
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Using this, we define rj (ξ, 1, τ ) = 3−2j rj
ξ τ , , 5 2
,
when |ξ| ∼ 2 and |τ | ≤ O(5 ). Then (7.30) becomes
α ∼ E0 − i h(k1 − α1 /4) + , 1, h2 . hj rj h k − 4 j=0
A
∞
Function spaces and FBI-transforms on manifolds
Let X be a compact analytic manifold of dimension n. In this section we first review some parts of Section 1 in [27] about how to define global FBI-transforms on X, and function spaces associated to certain IR-deformations of the real cotangent space. After that we shall perform Bargmann type transforms which allow us to view the above-mentioned function spaces, microlocally in a bounded frequency region, as weighted spaces of holomorphic functions. The theory in [27] is an adaptation to the case of compact manifolds of the one in [12] and this as well as the Bargmann transform below are closely related to similar ideas and techniques, developed in [6], [4], [28], [32], [10]. We equip X with some analytic Riemannian metric so that we have a distance d and a volume density dy. Let φ(α, y) be an analytic function on {(α, y) ∈ T ∗ X × X; d(αx , y) < 1/C} (using the notation α = (αx , αξ ), αx ∈ X, αξ ∈ Tα∗x X) with the following two properties (A) and (B): (A) φ has a holomorphic extension to a domain of the form × X; |Im αx |, |Im y| < {(α, y) ∈ T ∗ X
1 1 1 , |Re αx − Re y| < , |Im αξ | < |αξ |} C C C (A.1)
and satisfies |φ| ≤ O(1)|αξ | there. denotes the cotangent space is some complexification of X and T ∗ X Here X in the sense of complex manifolds with pointwise fiber spanned by the pointwise & 2 2 (1,0)-forms. We write αξ = 1 + αξ with αξ defined by means of the dual metric, and as below, we shall often give statements in local coordinates whenever convenient and leave to the reader to check that the statements make sense globally. Notice that by the Cauchy inequalities, ∂αk x ∂α ξ ∂ym φ = Ok,,m (1)|αξ |1−|| , in a set of the form (A.1), with a slightly increased constant C.
(A.2)
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The second assumption is (B) φ(α, αx ) = 0, (∂y φ)(α, αx ) = −αξ , Im (∂y2 φ)(α, αx ) ∼ |Re αξ |I. By Taylor’s formula, we have φ(α, y) = αξ · (αx − y) + O(1)αξ |αx − y|2 ,
(A.3)
and on the real domain, for d(αx , y) ≤ 1/C, with C sufficiently large, we have: Im φ(α, y) ∼ αξ (αx − y)2 .
(A.4)
The following example was found in a joint discussion with M. Zworski: Let exp x : Tx X → X be the geodesic exponential map. Then we can take i 2 φ(α, y) = −αξ · exp −1 αx (y) + αξ d(αx , y) . 2
(A.5)
be a closed I-Lagrangian manifold which is close to T ∗ X in Let Λ ⊂ T ∗ X ∞ the C -sense and which coincides with this set outside a compact set. Recall that “I-Lagrangian” means Lagrangian for the real symplectic form −Im σ, where σ= dαξj ∧dαxj is the standard complex symplectic form. This means that if we choose (analytic) coordinates y in X and let (y, η) be the corresponding canonical then Λ is of the form {(y, η) + iHG (y, η); (y, η) ∈ coordinates on T ∗ X and T ∗ X, T ∗ X} for some real-valued smooth function G(y, η) which is close to 0 in the C ∞ sense and has compact support in η. Here HG denotes the Hamilton field of G. Since Λ is close to T ∗ X, it is also R-symplectic in the sense that the restriction to Λ of Re σ is non-degenerate. (We say that Λ is an IR-manifold.) It follows that dα| Λ = dαx1 ∧ · · · ∧ dαxn ∧ dαξ1 ∧ · · · ∧ dαξn | = Λ
1 n σ |Λ n!
is a real non-vanishing 2n-form on Λ, that we view as a positive density. We also need some symbol classes. A smooth function a(x, ξ; h), defined on is said to be of class S m,k , if Λ or on a suitable neighborhood of T ∗ X in T ∗ X ∂xp ∂ξq a = O(1)h−m ξk−q .
(A.6)
m,k A formal classical symbol a ∈ Scl is of the form a ∼ h−m (a0 + ha1 + · · · ) 0,k−j where aj ∈ S is independent of h. Here and in the following, we let 0 < h ≤ h0 for some sufficiently small h0 > 0. When the domain of definition is real or equal to Λ, we can find a realization of a in S m,k (denoted by the same letter a) so that
a − h−m
N 0
hj aj ∈ S −(N +1)+m,k−(N +1) .
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m,k When the domain of definition is a complex domain, we say that a ∈ Scl is a m,k formal classical analytic symbol (a ∈ Scla ) if aj are holomorphic and satisfy
|aj | ≤ C0 C j (j!)|ξ|k−j .
(A.7)
It is then standard, that we can find a realization a ∈ S m,k (denoted by the same letter a) such that ∂xk ∂ξ ∂ x,ξ a = Ok, (1)e−| ξ |/Ch, |a − h−m hj aj | ≤ O(1)e−| ξ| /C1 h ,
(A.8)
0≤j≤| ξ |/C0 h
where in the last estimate C0 > 0 is sufficiently large and C, C1 > 0 depend on m,k m,k and Scla also the classes of realizations of classical C0 . We will denote by Scl symbols. We say that a classical (analytic) symbol a ∼ h−m (a0 + ha1 + · · · ) is 3n n ,4
0,−k 4 elliptic, if a0 is elliptic, so that a−1 . Take such an elliptic a(α, y; h) ∈ Scla 0 ∈S and put i
e h φ(α,y) a(α, y; h)χ(αx , y)u(y)dy,
T u(α; h) =
(A.9)
where χ is smooth with support close to the diagonal and equal to 1 in a neighborhood of the same set. 3n n 4 ,4 , such that if According to [27] there exists b(α, x; h) ∈ Scla i ∗ Sv(x) = e− h φ (x,α) b(α, x; h)χ(αx , x)v(α)dα, (A.10) T ∗X
then ST u = u + Ru,
(A.11)
where R has a distribution kernel R(x, y; h) satisfying |∂xα ∂y R| ≤ Ck, e
− C1 h 0
.
(A.12)
Here we denote in general by f ∗ , the holomorphic extension of the complex conjugate of f . With Λ as above, we put TΛ u = T u| Λ ,
(A.13)
and define SΛ v by (A.10), but with T ∗ X replaced by Λ. Then, SΛ TΛ u = u + RΛ u,
(A.14)
where RΛ satisfies (A.12) (with a slightly larger C0 and under the assumption that Λ is sufficiently close to T ∗ X). In fact, using Stokes’ formula and the exponential decrease of ∂ of the symbols involved, we see that SΛ TΛ coincides up to an exponentially small error with ST .
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Since Λ is I-Lagrangian, we can find locally a real-valued smooth function H(α) on Λ, such that dH = −Im (αξ · dαx )| Λ . (A.15) Indeed, −Im (αξ · dαx ) is a primitive of −Im σ and the latter vanishes on Λ, so the right-hand side of (A.15) is closed. We assume: The equation (A.15) has a global solution H ∈ C ∞ (Λ; R). Notice that this property is equivalent to Im (αξ · dαx ) = 0, for all closed curves γ ⊂ Λ.
(A.16)
(A.17)
γ
When (A.16) is fulfilled, H is well defined up to a constant, and we shall always choose H to be zero for large αξ . As in [27] we notice that (A.16) is fulfilled in the case of IR-manifolds gen R) in the following way: Let HG = H Im σ erated by a weight G ∈ C ∞ (T ∗ X; G be the Hamilton field of G with respect to Im σ, and assume that G = 0 in the region where |αξ | is large. Then for t real with |t| small enough, we can consider the IR-manifold Λt = exp (tHG )(Λ0 ), where Λ0 = T ∗ X. Then we get (A.16) with H = Ht given by t (exp (s − t)HG )∗ (G + HG , ω)ds, (A.18) Ht = 0
where ω = −Im (αξ · dαx ) The function H appears naturally in connection with TΛ . We have dα φ = αξ · dαx + O(|αx − y|), so (dα φ)(α, αx ) = αξ · dαx and −Im (dα φ)(α, αx )| Λ = dα H.
(A.19)
Definition. For m ∈ R, put H(Λ; αξ m ) = {u ∈ D (X); TΛ u ∈ L2 (Λ; e−2H/h |αξ |2m dα)}.
(A.20)
When Λ = T ∗ X we get the usual h-Sobolev spaces, and in particular the case m = 0 just gives L2 (X). For general Λ we get the same spaces, but the equivalence of the norm (A.21) u H(Λ, αξ m ) = TΛ u L2 (Λ;e−2H/h | αξ |2m dα) with the h-m-Sobolev norm u H(T ∗ X, αξ m ) is no longer uniform with respect to h, in general. Recall from [27] that if we choose another FBI-transform T of the same type as T but with different phase φ and amplitude a, then for Λ close enough to T ∗ X,
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the definition (A.20) does not change if we replace T by T, and we get a new norm which is equivalent to the previous one, uniformly with respect to h. This follows from a fairly explicit description of TΛ TΛ−1 . We also know that T u = TT ∗ X u and TΛ u satisfy compatibility conditions similar to the Cauchy-Riemann equations for holomorphic functions. For the analysis in the most interesting region where ξ is bounded, it will be convenient to work with transforms which are holomorphic up to exponentially small errors, and for that we make a different choice of T , and take an FBI-transform as in [28], now with a global choice of phase (cf [4], [10], [32]). The function d(x, y)2 is analytic in a neighborhood of the diagonal in X × X, so we can consider it as a holomorphic function in a region × X; dist (x, y) < 1 , |Im x|, |Im y| < 1 }. {(x, y) ∈ X C C Put φ(x, y) = iλd(x, y)2 ,
(A.22)
where λ > 0 is a constant that we choose large enough, depending on the size of the neighborhood of the zero section in T ∗ X, that we wish to cover. |Im x| < 1/C, put For x ∈ X, i − 3n 4 (A.23) e h φ(x,y) χ(x, y)u(y)dy, u ∈ D (X), T u(x; h) = h × X; |Im x| < where χ is a smooth cut-off function with support in {(x, y) ∈ X 1/C, d(y, y(x)) < 1/C}. Here y(x) ∈ X is the point close to x, where X y → −Im φ(x, y) attains its non-degenerate maximum. We have the following facts ([28]): |Im x| < 1/C, is strictly The function Φ0 (x) = −Im φ(x, y(x)), x ∈ X, plurisubharmonic and is of the order of magnitude ∼ |Im x|2 . is an IR-manifold given by ΛΦ0 = κT (T ∗ X), ΛΦ0 := {(x, 2i ∂Φ0 ) ∈ T ∗ X} where κT is the complex canonical transform associated to T , given by with its (y, −φy (x, y)) → (x, φx (x, y)). Here and in the following, we identify X intersection with a tubular neighborhood of X which is independent of the choice of λ in (A.22). e−2Φ0 /h L(dx)), for L(dx) denoting a choice of Lebesgue meaIf L2Φ0 = L2 (X; sure (up to a non-vanishing continuous factor), then T = O(1) : L2 (X) → L2Φ0 , ∂ x T = O(e−1/Ch ) : L2 (X) → L2Φ0 . This means that up to an exponentially small error T u is holomorphic for u ∈ L2 (X) (and even for u ∈ D (X)). A natural is choice of Lebesgue measure might be (n!)−1 |π∗ (σ| ΛΦ )n |, where π : ΛΦ0 → X 0 the natural projection. ⊂ L2 (X) be the subspace of holomorphic functions. Assuming, Let HΦ0 (X) Φ0 is a Stein (“pseudoconvex”) domain, we can apply the wellas we may, that X 2 known L results of H¨ ormander for the ∂-operator and replace T by T = T + K,
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so that T : L2 (X) → HΦ0 (X). In the where K = O(e−1/(Ch) ) : L2 (X) → L2Φ0 (X), main text we do not distinguish between T and T . Unitarity: Modulo exponentially small errors and microlocally, T is unitary a0 e−2Φ0 /h L(dx)), where L(dx) is chosen as indicated above, and L2 (X) → L2 (X; a0 (x; h) is a positive elliptic analytic symbol of order 0. be an IR-manifold as before, satisfying (A.16) (or the equivalent Let Λ ⊂ T ∗ X condition (A.17)). Then κT (Λ) = ΛΦ , where Φ = ΦΛ , can be normalized by the (Here is where we have to requirement that Φ = Φ0 near the boundary of X. choose λ large enough, depending on Λ. In the applications, for a given elliptic operator, Λ and T ∗ X will coincide outside a fixed compact neighborhood of the zero section, and the whole study will be carried out with a fixed λ.) Let Ω ⊂ T ∗ X be the open neighborhood of the 0-section, given by πx κT Ω = X and view also Ω as a subset of Λ in the natural sense, assuming that T ∗ X and Λ coincide in a neighborhood of the closure of the complement of Ω. If χ ∈ C0∞ (Ω), then the norm u H(Λ, αξ m ) is equivalent to the norm T u L2Φ + (1 − χ)TΛ u L2 (Λ;e−2H/h | αξ |2m dα) uniformly with respect to h.
Acknowledgments We would like to thank Anders Melin and Maciej Zworski for useful discussions. The first author gratefully acknowledges the support of the Swedish Foundation for International Cooperation in Research and Higher Education (STINT) as well as of the MSRI postdoctoral fellowship.
References [1] V. Arnold, Mathematical methods of classical mechanics, Springer-Verlag, New York, 1989. [2] M. Asch and G. Lebeau, The spectrum of the damped wave equation, preprint, 1999. See http://www.math.u-psud.fr/∼biblio/rt/1999/. [3] D. Bambusi, S. Graffi and T. Paul, Normal forms and quantization formulae, Comm. Math. Phys. 207, 173–195 (1999). [4] L. Boutet de Monvel, Convergence dans le domaine complexe des s´eries de fonctions propres, C. R. Acad. Sci. Paris, S´erie A–B, 287, 855–856 (1978). [5] L. Boutet de Monvel and V. Guillemin, The spectral theory of Toeplitz operators, Annals of Math. Studies 99, Princeton University Press, 1981. [6] L. Boutet de Monvel and J. Sj¨ ostrand, Sur la singularit´e des noyaux de Bergman et de Szeg¨o, Ast´erisque 34–35, 123–164 (1976).
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[7] P. Briet, J.M. Combes, P. Duclos, On the location of resonances for Schr¨ odinger operators in the semiclassical limit. II. Barrier top resonances, Comm. P.D.E. 12, 201–222 (1987). [8] Y. Colin de Verdi`ere, Sur le spectre des op´erateurs elliptiques a bicaract´eristiques toutes p´eriodiques, Comment Math. Helv. 54, 508–522 (1979). [9] S. Dozias, Clustering for the spectrum of h-pseudodifferential operators with periodic flow on an energy surface, Journ. Funct. Anal. 145, 296–311 (1997). [10] F. Golse, E. Leichtnam, and M. Stenzel, Intrinsic microlocal analysis and inversion formulae for the heat equation on compact real-analytic Riemannian manifolds, Ann. Sci. Ecole Norm. Sup. 29, 669–736 (1996). [11] B. Helffer and D. Robert, Puits de potentiel g´en´eralis´es et asymptotique semiclassique, Ann. Inst. H. Poincar´e 41, 291–331 (1984). [12] B. Helffer and J. Sj¨ ostrand, R´esonances en limite semiclassique, M´em. Soc. Math. France (N.S.) 24–25 (1986). [13] B. Helffer and J. Sj¨ ostrand, Semiclassical analysis for Harper’s equation. III. Cantor structure of the spectrum, Mem. Soc. Math. France (N.S.) 39, 1–124 (1989). [14] M. Hitrik, Eigenfrequencies for damped wave equations on Zoll manifolds, Asymptot. Analysis 31, 265–277 (2002). [15] L. H¨ormander, Fourier Integral Operators I, Acta Math. 127, 79–183 (1971). [16] V. Ivrii, Microlocal analysis and precise spectral asymptotics, Springer-Verlag, Berlin, 1998. [17] N. Kaidi and P. Kerdelhue, Forme normale de Birkhoff et r´esonances, Asymptot. Analysis 23, 1–21 (2000). [18] A. Lahmar-Benbernou and A. Martinez, On Helffer-Sj¨ ostrand’s theory of resonances, IMRN 13, 697–717 (2002). [19] G. Lebeau, Equation des ondes amorties, in Algebraic and Geometric Methods of Mathematical Physics (Kaciveli 1993), 73–109, Math. Phys. Stud., 19 Kluwer Acad. Publ., Dordrecht, 1996. [20] A. Melin, J. Sj¨ostrand, Bohr-Sommerfeld quantization condition for non-selfadjoint operators in dimension 2, Ast´erisque 284, 181–244 (2003) . [21] G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms. II. Quantum Birkhoff normal forms, Ann. Henri Poincar´e 1, 249–279 (2000). [22] J. Sj¨ ostrand, Perturbations of selfadjoint operators with periodic classical flow, RIMS Kokyuroku 1315 (April 2003), “Wave Phenomena and asymptotic analysis”, 1–23. Also: http://xxx.lanl.gov/abs/math.SP/0303023.
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[23] J. Sj¨ ostrand, Semiclassical resonances generated by a non-degenerate critical point, Springer LNM, 1256, 402–429. [24] J. Sj¨ ostrand, Resonances associated to a closed hyperbolic trajectory in dimension 2, preprint, September 2002, http://xxx.lanl.gov/abs/math.SP/0209147, Asymptotic Analysis, a` paraˆitre. [25] J. Sj¨ ostrand, Asymptotic distribution of of eigenfrequencies for damped wave equations, Publ. Res. Inst. Math. Sci. 36, 573–611 (2000). [26] J. Sj¨ ostrand, Semi-excited states in non-degenerate potential wells, Asymptot. Analysis 6, 29–43 (1992). [27] J. Sj¨ ostrand, Density of resonances for strictly convex analytic obstacles, Can. J. Math. 48, 397–447 (1996). [28] J. Sj¨ ostrand, Singularit´es analytiques microlocales, Ast´erisque 85 (1982). [29] J. Sj¨ ostrand and M. Zworski, Quantum monodromy and semiclassical trace formulae, J. Math. Pure Appl. 81, 1–33 (2002). [30] J. Sj¨ ostrand and M. Zworski, Asymptotic distribution of resonances for convex obstacles, Acta Math. 183, 191–253 (1999). [31] A. Weinstein, Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J. 44, 883–892 (1977). [32] M. Zworski, Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces, Inv. Math. 136, 353–409 (1999). Michael Hitrik Department of Mathematics University of California Los Angeles, CA 90095–1555 USA email:
[email protected] Johannes Sj¨ostrand Centre de Math´ematiques UMR 7640 CNRS Ecole Polytechnique F-91128 Palaiseau France email:
[email protected] Communicated by Bernard Helffer submitted 13/03/03, accepted 06/10/03
Ann. Henri Poincar´e 5 (2004) 75 – 118 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/010075-44 DOI 10.1007/s00023-004-0161-0
Annales Henri Poincar´ e
Scattering of Dirac Particles by Electromagnetic Fields with Small Support in Two Dimensions and Effect from Scalar Potentials Hideo Tamura Abstract. We study the asymptotic behavior of scattering amplitudes for the scattering of Dirac particles in two dimensions when electromagnetic fields with small support shrink to point-like fields. The result is strongly affected by perturbations of scalar potentials and the asymptotic form changes discontinuously at half-integer fluxes of magnetic fields even for small perturbations. The analysis relies on the behavior at low energy of resolvents of magnetic Schr¨ odinger operators with resonance at zero energy. The magnetic scattering of relativistic particles appears in the interaction of cosmic string with matter. We discuss this closely related subject as an application of the obtained results.
1 Introduction We consider the relativistic massless particle moving in the two-dimensional space. We denote by x = (x1 , x2 ) a generic point in R2 and write D(A, V ) =
2
σj (−i∂j − Aj ) + V,
∂j = ∂/∂xj ,
j=1
for the Dirac operator, where A = (A1 , A2 ) : R2 → R2 and V : R2 → R are magnetic and scalar potentials respectively, and 0 1 0 −i 1 0 σ1 = , σ2 = , σ3 = 1 0 i 0 0 −1 are the Pauli spin matrices. The magnetic field b : R2 → R is defined by b = ∇ × A = ∂1 A2 − ∂2 A1 . The operator D(A, V ) acts on [L2 ]2 = [L2 (R2 )]2 . If A and V are bounded, then it is selfadjoint with domain [H 1 (R2 )]2 , where H s (R2 ) is the Sobolev space of order s. We also write L(A, V ) = (−i∇ − A)2 + V for the Schr¨ odinger operator. If A has further bounded derivatives, then L(A, V ) is selfadjoint with domain H 2 (R2 ) in L2 . If L(A, V )u = 0 has a bounded but not square integrable solution, then L(A, V ) is said to have a resonance at zero energy.
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Let b and V be given magnetic field and scalar potential. We assume that b, V ∈ C0∞ (R2 → R) are smooth functions with compact support. We define A(x) by (1.1) A(x) = (−∂2 ϕ(x), ∂1 ϕ(x)) , where ϕ(x) = (2π)−1
log |x − y| b(y) dy
(1.2)
and the integration without the domain attached is taken over the whole space. By definition, A satisfies ∇ × A = ∆ϕ = b, and hence it becomes the potential associated with field b. The function ϕ obeys ϕ(x) = α log |x| + O(|x|−1 ) as |x| → ∞, where α = (2π)−1
b(x) dx
is called the flux of b. The magnetic effect strongly appears when α ∈ Z is not an integer. We restrict ourselves to the case 0 < α < 1.
(1.3)
We make a brief comment on the other cases that α < 0 and α > 1 (Remark 8.1 at the end of Section 8). The potential A(x) is not necessarily expected to fall off rapidly and it has the long-range property at infinity even if b is of compact support. In fact, it behaves like A(x) = A0α (x) + O(|x|−2 ),
(1.4)
A0α (x) = α(−x2 /|x|2 , x1 /|x|2 ) = α(−∂2 log |x|, ∂1 log |x|)
(1.5)
where A0α is defined by
and it is often called the Aharonov-Bohm potential in physical articles. Let T = D(A, V ) = T0 + V , where T0 = D(A, 0) = σ1 ν1 + σ2 ν2 ,
(ν1 , ν2 ) = −i∇ − A,
is the Dirac operator without scalar potential V . We sometimes identify the coordinates ω = (ω1 , ω2 ) over the unit circle S with the azimuth angle from the positive x1 axis. According to this notation, we set τ (ω) = t (1, eiω ),
eiω = cos ω + i sin ω = ω1 + iω2 .
(1.6)
We denote by f (ω → ω ˜ ; E) the scattering amplitude of T for scattering from initial direction ω ∈ S to final one ω ˜ at energy E > 0. Roughly speaking, it is defined through the behavior at infinity of solution ψ = ψ(x; E, ω) to equation T ψ = Eψ, and the solution takes the asymptotic form ˜ ; E)τ (˜ ω )eiEr r−1/2 , ψ(rω ˜ ) ∼ ψin + f (ω → ω
r = |x| → ∞,
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along direction ω ˜ = ω, where the first term ψin = τ (ω)eiEx·ω is the wave incident from ω and the second term denotes the scattering wave. The precise representation of it is given in Section 4. We study the scattering by electromagnetic fields with small support. We set Aε (x) = ε−1 A(x/ε),
bε (x) = ε−2 b(x/ε),
Vε (x) = ε−1 V (x/ε)
(1.7)
for 0 < ε 1 small enough. Then Aε satisfies ∇ × Aε = bε . Our aim here is to analyze the asymptotic behavior as ε → 0 of amplitude fε (ω → ω ˜ ; E) of Tε = D(Aε , Vε ). The problem is closely related to the resonance state at zero energy of magnetic Schr¨ odinger operators in a natural way. Let R(z; H) denote the resolvent (H − z)−1 of selfadjoint operator H. We write T0 = σ1 ν1 + σ2 ν2 as 0 ν1 − iν2 0 ν− T0 = = , ν1 + iν2 0 ν+ 0 where (ν1 , ν2 ) = −i∇ − A with A = (−∂2 ϕ(x), ∂1 ϕ(x)), ϕ being defined by (1.2). Since ν1 and ν2 satisfies the commutator relation [ν1 , ν2 ] = ν1 ν2 − ν2 ν1 = ib, a simple computation yields ν± ν∓ = ν12 + ν22 ± b = L(A, ±b), so that T02 is diagonalized as T02
=
L(A, −b) 0 0 L(A, b)
.
∗ The two Schr¨odinger operators L(A, ±b) = ν∓ ν∓ ≥ 0 are non-negative, but the spectral structure at zero energy is different. By (1.1), we have
ν+
=
ν1 + iν2 = −i∂1 + ∂2 ϕ + i(−i∂2 − ∂1 ϕ)
=
−i ((∂1 + ∂1 ϕ) + i(∂2 + ∂2 ϕ)) = −ie−ϕ ( ∂1 + i∂2 ) eϕ .
(1.8)
Hence L(A, −b)u = 0 has a bounded solution behaving like ρ(x) = e−ϕ(x) = |x|−α (1 + O(|x|−1 )),
|x| → ∞.
(1.9)
By assumption (1.3), ρ is not in L2 , and hence L(A, −b) has a resonance state at zero energy. On the other hand, L(A, b) does not have a resonance state. The amplitude fε is represented in terms of the boundary values R(E + i0; Tε ) = lim R(E + iδ; Tε ) δ↓0
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to the real axis of resolvent R(E + iδ; Tε ). We now define the unitary operator Jε : [L2 ]2 → [L2 ]2 by (Jε u)(x) = ε−1 u(x/ε), (1.10) then we have Tε = ε−1 Jε T Jε∗ for T = D(A, V ), and hence R(E + i0; Tε ) = εJε R(k + i0; T )Jε∗ ,
k = εE.
(1.11)
Thus the analysis relies on the behavior at low energy of resolvents R(k + i0; T0 ) = (T0 + k)R(k 2 + i0; T02) and R(k + i0; T ), and a basic role is played by the zero energy resonance of the magnetic Schr¨ odinger operator L(A, −b). We note that there is no fear of our confusing the operator Jε with the Bessel function Jν (x) in the argument below. We take the limit ε → 0 in a formal way. It follows from (1.4) that Aε is convergent to the Aharonov-Bohm potential A0α (x), and hence Tε = D(Aε , Vε ) → Dα = D(A0α , 0)
(1.12)
on [C0∞ (R2 \ {0})]2 . However A0α is strongly singular at the origin, and it has the δ-like field 2παδ(x) as a magnetic field. We know ([14, 19, 21]) that Dα is not essentially selfadjoint and it has the deficiency indices (1,1). According to the Krein theory, we can obtain a family of selfadjoint extensions {Hκ } with one real parameter κ, − ∞ < κ ≤ ∞. The element u = t (u1 , u2 ) in the domain D(Hκ ) is specified by the boundary condition u−1 + iκ u−2 = 0
(1.13)
at the origin under assumption (1.3), where u−1 = lim rα u1 (x), r→0
u−2 = lim r1−α e−iθ u2 (x) r→0
(1.14)
in the polar coordinate system (r, θ). If κ = ∞, then u−2 = 0 and the second component u2 (x) has a weak singularity near the origin for u ∈ D(H∞ ), while the first component u1 (x) has a weak singularity for κ = 0. The boundary condition in which both components remain bounded is not in general allowed ([14, 19]). In Section 2, we explicitly calculate the amplitude of Hκ after discussing the problem of selfadjoint extension in some detail. The amplitude fε in question is expected to converge to that of Hκ for some κ. We state the obtained results somewhat loosely. All the main theorems are ˜ ; E) the scattering amplitude of formulated in Section 5. We denote by gκ (ω → ω Hκ . As stated above, gκ can be calculated explicitly. If the scalar potential V (x) vanishes identically, then fε is shown to converge to g∞ (Theorem 5.1). However the situation changes as soon as V is added as a perturbation (Theorem 5.2). It is interesting that this occurs even for small perturbations. We here deal with only
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the simple but generic case that T has neither bound state nor resonance state at zero energy. The definition of resonance state is given in Section 5. Roughly speaking, it means that the equation T u = 0 admits a bounded solution. We note that T does not have a resonance state for V small enough. The obtained result depends on the flux α of field b. The amplitude fε is proved to converge to g∞ for 0 < α < 1/2 and to g0 for 1/2 < α < 1. If α = 1/2, then fε is convergent to gκ for some κ determined from the resonance state ρ = e−ϕ of L(A, −b). A similar problem has been studied by the physical literature [2, Section 7.10] for the scattering outside the small disk {|x| < ε}, and it has shown that the limit takes a different form according as 0 < α < 1/2, α = 1/2 or 1/2 < α < 1. However the argument there is based on the explicit calculation using the Bessel functions, and the connection with zero energy resonance has not been recognized. As stated in the beginning, another motivation of this work comes from the study on the scattering of Dirac particles in the interaction of cosmic string with matter. This problem is mathematically formulated as follows (see [7] for the detail on the physical background). Let Aε , bε = ∇ × Aε and Vε be defined by (1.7). We consider two kinds of particles (for example, lepton and quark) moving in the magnetic field bε and interacting with each other through the scalar potential Vε . If we denote by w = t (u, v) = t (u1 , u2 , v1 , v2 ) the wave function of these two particles, then w obeys the equation Tε w = T0ε w + Vε w = Ew at energy E > 0, where T0ε 0 T0ε = , 0 T0ε
Vε =
0 Vε
Vε 0
(1.15)
,
T0ε = D(Aε , 0).
We assume that the wave function w has only u-wave as an incident wave. Then w behaves like w ∼ t (τ (ω), 0)eiEx·ω + wscat + o(r−1/2 ),
r → ∞,
where τ (ω) is defined by (1.6), and the scattering wave wscat takes the form ˜ ; E)t (τ (˜ ω ), 0) + f2ε (ω → ω ˜ ; E)t (0, τ (˜ ω )) eiEr r−1/2 (1.16) wscat = f1ε (ω → ω along direction ω ˜ . The amplitude f2ε (ω → ω ˜ ; E) describes the v-wave produced by incident u-wave, and it is an important physical quantity in the interaction of cosmic string with matter. We analyze the asymptotic behavior as ε → 0 of ˜ ; E). The asymptotic form is shown to take the form f2ε (ω → ω f2ε (ω → ω ˜ ; E) = Cα ε|2α−1| (1 + o(1)) ,
ε → 0,
for some constant Cα (Theorem 5.3). The constant is independent of incident and final directions ω and ω ˜ , but is different according as 0 < α < 1/2, α = 1/2 or
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1/2 < α < 1. A similar asymptotic form has been derived by the earlier work [7] in the special case that A(x) = A0α (x) is the Aharonov-Bohm potential and V (x) is the characteristic function of the unit disk. However the calculation there is again based on the explicit calculation using the Bessel functions, and the important role of zero energy resonance seems to have been completely hidden behind this explicit calculation. In this work we make clear from a mathematical point of view how the leading coefficient Cα is determined and how it is related to the resonance state ρ of L(A, −b) at zero energy. We confine ourselves to the positive energy case E > 0 for notational brevity, and we fix E > 0 throughout the whole exposition. The dependence on E does not matter. We end the section by noting that the obtained results easily extend to the operator σ1 ν1 + σ2 ν2 + mσ3 + V with mass m > 0.
2 Dirac operators with point-like fields In this section we calculate the scattering amplitude gκ (ω → ω ˜ ; E) of selfadjoint extension Hκ obtained from Dα defined by (1.12) after explaining briefly the Krein theory on the problem of selfadjoint extension. The problem of selfadjoint extension for two-dimensional Dirac operators with singular magnetic fields has already been studied by several authors. We refer to [14, 19, 21] for details, and, in particular, to [21] for the recent references. The argument here follows [23]. The operator Dα = D(A0α , 0) =
0 π+
π− 0
,
π± = π1 ± i π2 ,
(2.1)
2 defined over C0∞ (R2 \ {0}) is symmetric, where (π1 , π2 ) = −i∇ − A0α . The two operators π± are represented as π+ = eiθ −i∂r + r−1 (∂θ − iα) , π− = e−iθ −i∂r − r−1 (∂θ − iα) (2.2) in terms of polar coordinates (r, θ), and we have π+ π− = π12 + π22 = −∂r2 − r−1 ∂r + r−2 (−i∂θ − α)
2
for r = |x| > 0, and similarly for π− π+ . We denote by Dα and Dα∗ the closure and adjoint of Dα respectively, and we set Σ± = {u ∈ [L2 ]2 : (Dα∗ ∓ i) u = 0}. The pair (n+ , n− ), n± = dim Σ± , is called the deficiency indices of Dα . As is well known, Dα has selfadjoint extensions if and only if n+ = n− . (1)
We show that n+ = n− = 1. We denote by Hµ (z) = Hµ (z) the Hankel function of first kind, and all the Hankel functions are understood to be of first
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kind throughout. If u = t (u1 , u2 ) ∈ [L2 ]2 solves (Dα − i) u = 0, then u2 satisfies (π+ π− + 1) u2 = 0 in R2 \ {0}, and u1 is given by u1 = −iπ− u2 . By formula, Hµ (z) satisfies (2.3) (d/dz) z ±µ Hµ (az) = ±az ±µ Hµ∓1 (az). The same formula is still true for Jµ (z). This formula yields π− H1−α (ir)eiθ = H−α (ir) = eiαπ Hα (ir). Hence we see that Σ+ is the one-dimensional space spanned by u+ = Nα t (−ieiαπ Hα (ir), H1−α (ir)eiθ ), where u+ is normalized as u+ L2 = 1. Similarly Σ− is also the one-dimensional space spanned by u− = Nα t (ieiαπ Hα (ir), H1−α (ir)eiθ ),
u− L2 = 1.
All the possible selfadjoint extensions are determined by the Krein theory ([8, 20]). Let U : Σ+ → Σ− be the unitary mapping defined by multiplication U u+ = eiζ u− with −π < ζ ≤ π. Then the selfadjoint extension HU associated with U is realized as the operator HU u = Dα v + icu+ − iceiζ u− acting on the domain D(HU ) = {u ∈ [L2 ]2 : u = v + cu+ + ceiζ u− , v ∈ D(D α ), c ∈ C}. We examine which boundary condition u ∈ D(HU ) satisfies at the origin. The Hankel function Hµ (z) with non-integer µ > 0 is represented as Hµ (z) = (i/ sin µπ) e−iµπ Jµ (z) − J−µ (z) (2.4) in terms of Bessel functions, and it behaves like Hµ (z) = (−i/ sin µπ) (2µ /Γ(1 − µ)) z −µ 1 + O(|z|2µ ) + O(|z|2 )
(2.5)
as |z| → 0. If v = t (v1 , v2 ) ∈ D(D α ), then v obeys v1 = o(|x|−α ) and v2 = o(|x|−(1−α) ) as |x| → 0, so that u = t (u1 , u2 ) ∈ D(HU ) has the limits u−1 and u−2 in (1.14). If we take account of the above asymptotic formula of Hankel functions, then the ratio κ = iu−1 /u−2 = 22α−1 Γ(α)/Γ(1 − α) tan(ζ/2) is calculated as a quantity independent of u. Thus we obtain the family of selfadjoint extensions {Hκ } parameterized by real number κ, − ∞ < κ ≤ ∞, and the operator has the domain D(Hκ ) = {u = (u1 , u2 ) ∈ [L2 ]2 : Dα u ∈ [L2 ]2 , u−1 + iκu−2 = 0},
(2.6)
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where Dα u is understood in the distribution sense, and u−1 and u−2 are defined by (1.14). We move to calculating the scattering amplitude of Hκ . It has already been calculated in the physical articles ([17]) for the special case κ = 0 or κ = ∞. We again note that ω ∈ S is often identified with the azimuth angle from the positive x1 axis. ˜ ; E), ω ˜ = ω, denote the scattering amplitude of Hκ Proposition 2.1 Let gκ (ω → ω for the scattering from initial direction ω into final one ω ˜ at energy E > 0. Then ei(˜ω−ω)/2 2κτα E 2α−1 −1/2 + gκ = − (2πiE) sin απ , (2.7) sin((˜ ω − ω)/2) i(κτα E 2α−1 − eiαπ ) where
τα = 21−2α Γ(1 − α)/Γ(α).
(2.8)
If, in particular, κ = 0 or κ = ∞, then g0 g∞
sin απ
ei(˜ω−ω)/2 , sin((˜ ω − ω)/2)
= − (2πiE)−1/2 sin απ
e−i(˜ω−ω)/2 , sin((˜ ω − ω)/2)
−1/2
= − (2πiE)
and if α = 1/2, then −1/2
gκ = − (2πiE)
ei(˜ω−ω)/2 2κ + sin((˜ ω − ω)/2) 1 + iκ
.
We need two lemmas to prove the proposition. Before stating the lemmas, we briefly discuss the problem of selfadjoint extensions for magnetic Schr¨odinger operator 2 (2.9) Lα = L(A0α , 0) = (−i ∇ − A0α ) with Aharonov-Bohm potential A0α . We know ([1, 13]) that Lα has the deficiency indices (2,2) as a symmetric operator on C0∞ (R2 \{0}), and the Krein theory again yields the family of all possible selfadjoint extensions {LU } parameterized by 2 × 2 unitary mapping U from one deficiency subspace to the other one. The selfadjoint operator LU is realized as a differential operator with some boundary conditions at the origin. If w is in the domain D(LU ), then w behaves like
w = w−0 r−α + w+0 rα + o(rα ) + w−1 r−(1−α) + w+1 r1−α + o(r1−α ) eiθ + o(r) for some coefficients w±k , k = 0, 1, and there exist 2 × 2 matrices B± for which the boundary condition is described as the relation w−0 w+0 B− + B+ =0 w−1 w+1
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between these four coefficients. We distinguish the two operators by the following special notation : D(LAB ) = {w ∈ L2 : Lw ∈ L2 , w−0 = w−1 = 0} D(LZ ) = {w ∈ L2 : Lw ∈ L2 , w+0 = w−1 = 0}
(2.10)
among admissible selfadjoint extensions. The first operator LAB is known as the Aharonov-Bohm Hamiltonian ([3]). We denote by γ(x; ω) the azimuth angle from ω. The operator Lα defined by (2.9) admits the polar coordinate decomposition ⊕ hl , Lα l∈Z
where hl = −(d/dr)2 + (ν 2 − 1/4)r−2 with ν = |l − α|. If we define ϕ± (x; E, ω) = e∓iνπ/2 eilγ(x;∓ω) Jν (Er)
(2.11)
l∈Z
for ν = |l − α|, then ϕ± vanishes at the origin and solves Lα − E 2 ϕ± = 0. Thus ϕ± becomes the generalized eigenfunction of LAB with eigenvalue E 2 . The first lemma is due to [16] (see [3, 10] also). Lemma 2.1 Let ϕ+ (x; E, ω) be as above. Define ϕin (x; E, ω) = eiEx·ω eiα(γ(x;ω)−π)
(2.12)
for x = rθ, θ = ω. Then ϕ+ (x; E, ω) obeys ϕ+ (rθ; E, ω) = ϕin (rθ; E, ω) + g+ (ω → θ; E)eiEr r−1/2 (1 + o(1)) ,
r → ∞,
along direction θ, where −1/2
g+ (ω → θ; E) = − (2πiE)
sin απ
ei(θ−ω)/2 . sin((θ − ω)/2)
(2.13)
This lemma implies that ϕ+ (x; E, ω) is the outgoing eigenfunction of LAB , and g+ (ω → θ; E) defines the scattering amplitude. This is known as the AharonovBohm scattering amplitude ([3]). On the other hand, ϕ− (x; E, ω) is shown to be the incoming eigenfunction, but its asymptotic form is not required in the argument below. We move to the second lemma. The proof of this lemma uses the following formula for the Bessel functions : ±iEJν±1 (Er)ei(l±1)θ (l ≥ 1) ilθ = π± Jν (Er)e (2.14) ∓iEJν∓1 (Er)ei(l±1)θ (l ≤ 0) for ν = |l − α| with 0 < α < 1. This follows from (2.3) after a direct computation. The same formula remains true for the Hankel Hν (Er).
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Lemma 2.2 Let π+ be as in (2.2) and let g+ be as in Lemma 2.1. Then (π+ ϕ+ ) (rθ; E, ω) = Eeiω ϕin (rθ; E, ω) + Eeiθ g+ (ω → θ; E)eiEr r−1/2 (1 + o(1)) as r → ∞ along direction θ, θ = ω. Proof. We calculate I = (π+ ϕ+ )(x; E, ω)/E. Since eilγ(x;−ω) = eilθ eil(π−ω) for x = rθ, we obtain I= ie−iνπ/2 Jν+1 (Er)ei(l+1)θ eil(π−ω) − ie−iνπ/2 Jν−1 (Er)ei(l+1)θ eil(π−ω) l≥1
l≤0
by use of formula (2.14). We use the simple relation ei(l+1)θ eil(π−ω) = −ei(l+1)γ(x;−ω) eiω . If l ≥ 1, then ν + 1 = |l + 1 − α| and ie−iνπ/2 = −e−i|l+1−α|π/2 , and if l ≤ −1, then ν − 1 = |l + 1 − α| and ie−iνπ/2 = e−i|l+1−α|π/2 . If we take account of these relations, then we make a change of variables l + 1 → l to obtain that e−iνπ/2 eilγ(x;−ω) Jν (Er) − e−i(α−1)π/2 Jα−1 (Er)eiθ , I = eiω l =1
so that it equals
I = eiω ϕ+ (x; E, ω) + e−i(1−α)π/2 J1−α (Er) − e−i(α−1)π/2 Jα−1 (Er) eiθ . Hence it follows from (2.4) that I = eiω ϕ+ (x; E, ω) + e−iαπ/2 sin απH1−α (Er)eiθ .
(2.15)
The Hankel function Hµ (z), µ > 0, is known to behave like Hµ (z) = (2/iπ)1/2 e−iµπ/2 eiz z −1/2 1 + O(|z|−1 )
(2.16)
as |z| → ∞. This, together with Lemma 2.1, implies that I = eiω ϕin (x; E, ω) + g˜(ω → θ; E)eiEr r−1/2 (1 + o(1)) , where
−1/2
g˜ = eiω g+ (ω → θ; E) − 2i (2πiE)
sin απeiθ .
A simple computation yields
g˜ = (2πiE)−1/2 sin απ −e−i(θ−ω)/2 / sin((θ − ω)/2) + 2/i eiθ = g+ (ω → θ; E)eiθ . This proves the lemma.
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Proof of Proposition 2.1. Let Dα = D(A0α , 0) be as in (2.1). We look for the solution ψ = (ψ1 , ψ2 ) to equation (Dα − E) ψ = 0 in the form ψ1 = ϕ+ (x; E, ω) + βκ Hα (Er),
ψ2 = (1/E) (π+ ψ1 ) (x; E, ω)
(2.17)
with some constant βκ . If ψ takes the above form, then it is easy to see that ψ solves the equation. The coefficient βκ is determined so as to satisfy the boundary condition (1.13) at the origin. Then ψ = ψ(x; E, ω) becomes the eigenfunction of selfadjoint operator Hκ and the amplitude gκ is determined through the asymptotic form of ψ(x; E, ω). We calculate the limits u−1 and u−2 defined by (1.14). The eigenfunction ϕ+ of LAB vanishes at the origin, so that u−1 = lim rα ψ1 = βκ (−i/ sin απ) (2α /Γ(1 − α)) E −α r→0
by (2.5). Since π+ Hα (Er) = −iEHα−1 (Er)eiθ = iEe−iαπ H1−α (Er)eiθ by (2.14), it follows from (2.15) that
ψ2 = eiω ϕ+ (x; E, ω) + e−iαπ/2 sin απ + ie−iαπ βκ H1−α (Er)eiθ
(2.18)
and hence
u−2 = (−i/ sin απ) e−iαπ/2 sin απ + ie−iαπ βκ 21−α /Γ(α) E −1+α .
Thus βκ is determined as
βκ = ieiαπ/2 sin απ κτα E 2α−1 /(κτα E 2α−1 − eiαπ ) ,
(2.19)
where τα is defined in (2.8). By Lemmas 2.1 and 2.2 and by (2.16), ψ(x; E, ω) behaves like ˜ ; E)τ (˜ ω )eiEr r−1/2 + o(r−1/2 ) ψ = τ (ω)ϕin (x; E, ω) + gκ (ω → ω
(2.20)
as r → ∞ along direction ω ˜ = ω, where τ (ω) is in (1.6), and gκ = g+ (ω → ω ˜ ; E) + 2(2πiE)−1/2 e−iαπ/2 βκ . This determines the desired amplitude and the proof is complete.
We end the section by making some additional comments on the outgoing eigenfunction ψ+ (x; E, ω) and the incoming one ψ− (x; E, ω) of H∞ . These eigenfunctions are used to represent the amplitude f (ω → ω ˜ ; E) of T = D(A, V ) in Section 4. The outgoing eigenfunction ψ+ = t (ψ+1 , ψ+2 ) is defined by (2.17) with β∞ = ieiαπ/2 sin απ, and we have ψ+1 = ϕ+ (x; E, ω) + β∞ Hα (Er),
ψ+2 = eiω ϕ+ (x; E, ω)
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by (2.18). This is expanded as ψ+1 (x; E, ω) =
l =0
ψ+2 (x; E, ω) =
eiω
e−iνπ/2 eilγ(x;−ω) Jν (Er) + eiαπ/2 J−α (Er),
e−iνπ/2 eilγ(x;−ω) Jν (Er).
(2.21)
l∈Z (2)
(2)
The Hankel function Hµ (z) of second kind is related to Hµ (z) through Hµ (z) = (2) (2) Hµ (z) for z ∈ R, and it satisfies H−µ (z) = e−iµπ Hµ (z). If we make use of these relations, a similar argument enables us to construct the incoming eigenfunction ψ− (x; E, ω) = t (ψ−1 , ψ−2 ) as ψ−1 = ϕ− (x; E, ω) + β ∞ Hα (Er),
ψ−2 = eiω ϕ− (x; E, ω)
with ϕ− defined by (2.11), and it admits the expansion ψ−1 (x; E, ω)
=
l =0
ψ−2 (x; E, ω)
= eiω
eiνπ/2 eilγ(x;ω) Jν (Er) + e−iαπ/2 J−α (Er),
eiνπ/2 eilγ(x;ω) Jν (Er).
(2.22)
l∈Z
3 Resolvent of selfadjoint extensions We here establish the relation between the two resolvents R(E +i0; Hκ ) and R(E + i0; H∞ ). We fix several new notation. We denote by ( , ) the scalar product in L2 2 or L2 , and write f ⊗ g = ( · , g)f for the integral operator with kernel f (x) g(y). This acts as (f ⊗ g) u = (u, g)f on u ∈ L2 . We also use a similar notation u ⊗ v = (uj ⊗ vk )1≤j,k≤2 ,
u = t (u1 , u2 ),
v = t (v1 , v2 ),
for a vector version over [L2 ]2 . We further define the two basic functions ξ+ (x; E) = t −ieiαπ Hα (Er), H1−α (Er)eiθ ,
ξ− (x; E) = t −ie−iαπ Hα (Er), H1−α (Er)eiθ
(3.1)
for E > 0. The second function may be written as
(2) ξ− (x; E) = t −ie−iαπ Hα(2) (Er), H1−α (Er)eiθ . If we repeat almost the same argument as in the previous section, then it is easy to see that these two functions solve (Dα − E) u = 0, and form a pair of linearly independent solutions. The aim here is to prove the following proposition.
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Proposition 3.1 Let ξ± = ξ± (x; E) be as above. Then R(E + i0; Hκ ) = R(E + i0; H∞ ) − cκ E (ξ+ ⊗ ξ− ) , where
cκ = sin απ/(4(κτα E 2α−1 − eiαπ ))
with τα defined by (2.8). If, in particular, α = 1/2, then cκ = −1/(4(i − κ)). The proposition is proved at the end of this section. Let LAB and LZ be defined in (2.10), and let Aε and bε = ∇ × Aε be as in (1.9). We again set T0ε = D(Aε , 0), which is convergent to Dα = D(A0α , 0) as ε → 0 on [C0∞ (R2 \ {0})]2 by (1.12). We represent R(E + i0; H∞ ) in terms of resolvents of LAB and LZ . We repeat the same argument as used in Section 1 to obtain R(z 2 ; L−ε ) 0 R(z; T0ε ) = (T0ε + z) , L±ε = L(Aε , ±bε ), 0 R(z 2 ; L+ε ) for z, Im z = 0. According to the results in [23, Section 3], we have R(z; T0ε ) → R(z; H∞ ) and R(z; L+ε ) → R(z; LAB ),
R(z; L−ε ) → R(z; LZ ),
as ε → 0 in norm (in norm resolvent sense). We also have ER(E 2 + i0; LZ ) π− R(E 2 + i0; LAB ) R(E + i0; H∞ ) = . π+ R(E 2 + i0; LZ ) ER(E 2 + i0; LAB )
(3.2)
We now calculate the Green kernels of R(E 2 + i0; LAB ) and R(E 2 + i0; LZ ). To do this, we decompose L2 = L2 (0, ∞) ⊗ L2 (S), and we define the mapping Ul by (Ul f )(r) = (2π)−1/2 r1/2
0
2π
f (rθ)e−ilθ dθ : L2 → L2 (0, ∞)
for l ∈ Z. Then (Ul∗ g)(x) = (2π)−1/2 r−1/2 g(r)eilθ : L2 (0, ∞) → L2 , and R(E 2 + i0; LAB ) admits the decomposition R(E 2 + i0; LAB ) = ⊕ Rl , Rl = Ul∗ R(E 2 + i0; hl )Ul , l∈Z
where the domain of selfadjoint operator hl = −(d/dr)2 + (ν 2 − 1/4)r−2 ,
ν = |l − α|,
(3.3)
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is specified by the boundary condition lim r−(1/2−α) g(r) = 0 at the origin. Simir→0
larly we have ˜0 ⊕ R(E 2 + i0; LZ ) = R
⊕ Rl ,
˜ 0 = U ∗ R(E 2 + i0; ˜h0 )U0 , R 0
(3.4)
l =0
and the domain of selfadjoint operator ˜ 0 = −(d/dr)2 + (α2 − 1/4)r−2 h is specified by the condition lim r−(1/2+α) (g(r) − g0 r1/2−α ) = 0
r→0
with g0 = lim r−(1/2−α) g(r). The two functions r1/2 Jν (Er) and r1/2 Hν (Er) are r→0
linearly independent solutions to (hl − E 2 )g = 0 for E > 0. By formula, we know W (Jµ , J−µ )(z) = −2 sin µπ/(πz) for the Wronskian of Bessel functions, so that W (Hµ , Jµ )(z) = −2i/(πz),
W (Hµ , J−µ )(z) = −2ie−iµπ /(πz)
by (2.4). Thus we can construct the Green kernels Rl (x, y) ˜ 0 (x, y) R
= (i/4) Hν (E(r ∨ ρ))Jν (E(r ∧ ρ))eil(θ−ϕ) , = ieiαπ /4 Hα (E(r ∨ ρ))J−α (E(r ∧ ρ))
(3.5)
in the standard way, where r ∨ ρ = max (r, ρ) and r ∧ ρ = min (r, ρ) for (x, y) = (reiθ , ρeiϕ ). We are now in a position to prove Proposition 3.1. Proof of Proposition 3.1. According to the Krein theory ([8]), the two resolvents are related to each other through the relation in the proposition. We have only to calculate the constant cκ . We set t
(u1 , u2 ) = R(E + i0; Hκ )F
for F = t (f, 0) with f ∈ C0∞ (R2 \ {0}). Then u1 = v1 − ieiαπ cσHα (Er),
u2 = v2 + cσH1−α (Er)eiθ ,
c = −cκ E,
where t
(v1 , v2 ) = R(E + i0; H∞ )F = t (ER(E 2 + i0; LZ )f, π+ R(E 2 + i0; LZ )f )
by (3.2), and σ = (F, ξ− ) is the scalar product between F = t (f, 0) and ξ− . The constant cκ is determined by boundary condition u−1 + iκu−2 = 0, where
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u−1 and u−2 are defined by (1.14). We calculate the limits u−1 and u−2 . Since t (v1 , v2 ) = R(E + i0; H∞ )F , v2 obeys v2 = o(r−(1−α) )eiθ , and hence it follows from (2.5) that u−2 = cσ (−i/ sin απ) 21−α /Γ(α) E α−1 . If we use (3.5) and (3.1), then v1 behaves like ˜ 0 f + o(1) = (σE/4)J−α (Er) + o(1), v1 = E R and hence
r → 0,
u−1 = σ E/4 − ceiαπ / sin απ (2α /Γ(1 − α)) E −α
by (2.5). Then cκ is determined as in the proposition.
4 Scattering amplitudes in the presence of scalar potentials The aim here is to derive the representation (4.6) below for the scattering amplitude f (ω → ω ˜ ; E) of T = D(A, V ) with scalar potential V ∈ C0∞ (R2 → R), where 2 A ∈ C ∞ (R → R2 ) is defined by (1.1). The derivation requires two lemmas. Lemma 4.1 Write ψ− (ω) for the incoming eigenfunction ψ− (x; E, ω), defined by (2.22), of H∞ . Let F (x) = t (f1 (r)eimθ , f2 (r)ei(m+1)θ ),
m ∈ Z,
for f1 , f2 ∈ C0∞ [0, ∞). Then 1/2
(R(E + i0; H∞ )F )(rω ˜ ) = (iE/8π)
(F, ψ− (˜ ω )) τ (˜ ω )eiEr r−1/2 + o(r−1/2 )
as r → ∞ uniformly in ω ˜ ∈ S, where (F, ψ− (ω)) is the scalar product in [L2 ]2 between F and ψ− (ω). Proof. We prove the lemma for the case m = 0 only. A similar argument applies to the other cases. Set t (u1 , u2 ) = R(E + i0; H∞ )F for F as in the lemma. Then u1 = Ev1 + π− v2 ,
u2 = π+ v1 + Ev2
by (3.2), where v1 = R(E 2 + i0; LZ )f1 ,
v2 = R(E 2 + i0; LAB )(f2 eiθ ).
˜ 0 f1 and v2 = R1 (f2 eiθ ). The two It follows from (3.3) and (3.4) that v1 = R ˜ operators R0 and R1 have the kernels (3.5). By assumption, f1 and f2 have compact support. Hence we have v1 = (ieiαπ /4)(f1 , J−α )Hα (Er),
v2 = (i/4)(f2 , J1−α )H1−α (Er)eiθ
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for |x| 1. Since π− H1−α (Er)eiθ = −iEH−α (Er) = −iEeiαπ Hα (Er) by (2.14), it follows from (2.16) that u1
=
(iE/4) eiαπ ((f1 , J−α ) − i(f2 , J1−α )) Hα (Er)
=
(iE/8π)1/2 eiαπ/2 ((f1 , J−α ) − i(f2 , J1−α ))eiEr r−1/2 + o(r−1/2 )
as r → ∞. The eigenfunction ψ− has the expansion (2.22), and we have (F, ψ− (˜ ω )) = (f1 , ψ−1 (˜ ω )) + (f2 eiθ , ψ−2 (˜ ω )) = eiαπ/2 ((f1 , J−α ) − i(f2 , J1−α )) . This yields the desired asymptotic form for u1 . We can show in a similar way that u2 also takes the asymptotic form in the theorem. Thus the proof is complete. We now introduce the Banach spaces B and B ∗ with norms u B =
∞ j=0
2
j
1/2
2
Ωj
|u(x)| dx
,
u B ∗ = sup
R>0
1 R
|x|
1/2 2
|u(x)| dx
,
where Ω0 = {|x| ≤ 1} and Ωj = {2j−1 < |x| ≤ 2j } for j ≥ 1. The two spaces fulfill the inclusion relations L2s ⊂ B ⊂ L21/2 ,
L2−1/2 ⊂ B ∗ ⊂ L2−s
for s > 1/2, where L2s = L2 (R2 ; x2s dx) with x = (1 + |x|2 )1/2 . We use the notation o∗ (r−1/2 ) as r = |x| → ∞ to denote functions u obeying the bound 1 |u(x)|2 dx → 0, R → ∞. R |x|
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(see (2.2) there), we can construct a smooth magnetic potential a(x) : R2 → R2 associated with field b such that a(x) = (a1 , a2 ) = A0α = α −x2 /|x|2 , x1 /|x|2 , |x| > 2. (4.1) We define the auxiliary operator K as K = D(a, V ).
(4.2)
This is selfadjoint with domain D(K) = [H 1 (R2 )]2 , and we know ([11, 15]) that the boundary value R(E + i0; K) to the real axis exists as a bounded operator from [L2s ]2 into [L2−s ]2 for s > 1/2. We further introduce a basic cut-off function χ0 ∈ C0∞ (R2 → R) with the properties supp χ0 ⊂ {|x| < 2},
χ0 = 1 on {|x| < 1}.
(4.3)
We set χ+ (x) = χ0 (x/2) and χ− (x) = χ0 (x/4). We study the behavior at infinity of eigenfunction ψ(x; E, ω) of K. Since K = D(A0α , 0) = Dα over {|x| > 2} by (4.1), we have (1−χ+ ) (K − E) ψ+ = 0 for the outgoing eigenfunction ψ+ (ω) = ψ+ (x; E, ω) of H∞ . Hence the eigenfunction ψ = ψ(x; E, ω) with incident wave ϕin (x; E, ω) as in Lemma 2.2 is written as ψ = (1 − χ+ )ψ+ + R(E + i0; K)Π+ ψ+ ,
(4.4)
where Π+ = [Dα , χ+ ]. Similarly ψ+ (x; E, ω) is represented as ψ+ = (1 − χ− )ψ + R(E + i0; H∞ )Π− ψ with Π− = [Dα , χ− ]. Hence it follows from Lemma 4.2 that ψ = ψ+ − (iE/8π)1/2 (Π− ψ, ψ− (˜ ω )) τ (˜ ω )eiEr r−1/2 + o∗ (r−1/2 ).
(4.5)
We insert (4.4) into ψ on the right side of (4.5). Since Π− (1 − χ+ ) = 0 and Π∗− = −Π− , we obtain (Π− ψ, ψ− (˜ ω )) = −(R(E + i0; K)Π+ ψ+ (ω), Π− ψ− (˜ ω )). We recall that ψ+ obeys (2.20) with κ = ∞. Hence the amplitude f (ω → ω ˜ ; E) of K is given by f = g∞ (ω → ω ˜ ; E) + (iE/8π)1/2 (R(E + i0; K)Π+ ψ+ (ω), Π− ψ− (˜ ω )),
(4.6)
where g∞ is the amplitude of H∞ . The amplitude of T = D(A, V ) is shown to be represented in the same way. Since A and a have the same field b, we have the relation A = a + ∇h (4.7)
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for some function h ∈ C ∞ (R2 → R), and T = eih Ke−ih . The difference obeys A − a = O(|x|−2 ) at infinity, so that h falls off with h = O(|x|−1 ) and eih(x) = 1 + O(|x|−1 ). Thus T has the same scattering operator as K and hence the scattering amplitude of T is also represented as (4.6). To sum up, the amplitude f (ω → ω ˜ ; E) of T = D(A, V ) is defined through the asymptotic form ˜ ; E)τ (˜ ω )eiEr r−1/2 + o∗ (r−1/2 ) ψ = τ (ω)ϕin (x; E, ω) + f (ω → ω as r = |x| → ∞ of solution ψ to equation T ψ = (T0 + V ) ψ = Eψ, and it has the representation (4.6). In the mathematical scattering theory, it is standard to define the scattering amplitudes through integral kernels of scattering matrices after establishing the basic problems such as the existence and completeness of wave operators and the limiting absorption principle [9, 15, 18, 24, 25]. However, K has the special property that it admits the polar coordinate decomposition on {|x| > 2}. If we make use of this property, the Agmon-H¨ ormander theory ([5]) enables us to define directly the scattering amplitude through the asymptotic form of eigenfunction. We can show that these two representations defined in a different way coincide with each other, but we do not go into the details here.
5 Scattering by electromagnetic fields with small support In this section we formulate the results on the asymptotic behavior of amplitudes for the scattering by electromagnetic fields with small support. We obtain the three main theorems and the remaining four sections (Sections 6, 7, 8 and 9) are devoted to the proof of these theorems. ˜ ; E) the scattering Let Aε and Vε be defined by (1.7). We denote by fε (ω → ω amplitude of Tε = D(Aε , Vε ). If we set Kε = D(aε , Vε ),
aε = ε−1 a(x/ε),
(5.1)
then aε (x) = A0α (x) over |x| > 2ε, and the amplitude fε has the representation ˜ ; E) + (iE/8π)1/2 (R(E + i0; Kε )Π+ ψ+ (ω), Π− ψ− (˜ ω )), fε = g∞ (ω → ω
(5.2)
where Π± = [Dα , χ± ] with χ+ = χ0 (x/2) and χ− = χ0 (x/4) again. We have explicitly calculated the scattering amplitude gκ (ω → ω ˜ ; E) of Hκ in Proposition 2.1. It admits the representation ˜ ; E) + (iE/8π)1/2 (R(E + i0; Hκ )Π+ ψ+ (ω), Π− ψ− (˜ ω )) gκ = g∞ (ω → ω
(5.3)
in terms of resolvent R(E + i0; Hκ ). In fact, this is obtained by repeating almost the same argument as used to derive (4.6). We first deal with the case without electric fields.
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Theorem 5.1 Assume that V = 0 identically. Then fε (ω → ω ˜ ; E) → g∞ (ω → ω ˜ ; E),
ε → 0,
for ω = ω ˜. Next we discuss the case when V ∈ C0∞ (R2 → R) does not vanish identically. We assume that V (x) ≥ 0, (5.4) so that the scalar product λ0 = (V ρ, ρ) > 0
(5.5) −ϕ(x)
is strictly positive for the resonance function ρ(x) = e defined by (1.9). The assumption (5.4) does not matter, but λ0 = 0 is important to the future argument. Before stating the second theorem, we define the resonance state of Dirac operator T = D(A, V ) at zero energy. The definition is different according as 0 < α ≤ 1/2 or 1/2 < α < 1. Definition 5.1. (1) Let 0 < α ≤ 1/2. Assume that the equation T v = 0 has a non-trivial solution such that v = t (v1 , v2 ) ∈ L2 × L∞ and v2 (x) = O(|x|−1+α ) at infinity. If v2 ∈ L2 , then T is said to admit a resonance state at zero energy, and if v2 ∈ L2 , then T has an eigenvalue at zero energy. (2) Let 1/2 < α < 1. Assume that T v = 0 has a non-trivial solution such that v = t (v1 , v2 ) ∈ L∞ × L2 and v1 (x) = O(|x|−α ) at infinity. If v1 ∈ L2 , then T is said to admit a resonance state at zero energy, and if v1 ∈ L2 , then T has an eigenvalue at zero energy. In the present work, we deal with only the case that T has neither eigenstates nor resonance states at zero energy. This case is simple but generic. Thus we always assume that T has neither eigenstates nor resonance states at zero energy.
(5.6)
If |V | 1 is small enough, then it can be shown that T fulfills (5.6). The lemma below plays an important role in proving the remaining two main theorems. This basic lemma is proved in Section 7. Lemma 5.1 Assume that (5.6) is fulfilled. Then : (1) Let 0 < α ≤ 1/2. Then there exists a unique solution e ∈ L∞ × L∞ to equation T e = 0 such that e = t (e1 , e2 ) obeys e1 = r−α + O(|x|−1−α ),
e2 = O(|x|−1+α )
(5.7)
at infinity, and e2 (x) behaves like e2 (x) = iλ2 r−1+α eiθ + O(|x|−2+α ), for some real constant λ2 .
|x| → ∞,
(5.8)
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(2) Let 1/2 < α < 1. Then there exists a unique solution e ∈ L∞ × L∞ to T e = 0 such that e = t (e1 , e2 ) obeys e1 = O(|x|−α ),
e2 = ir−1+α eiθ + O(|x|−2+α )
(5.9)
at infinity, and e1 (x) behaves like e1 (x) = λ1 r−α + O(|x|−1−α ),
|x| → ∞,
(5.10)
for some real constant λ1 . We are now in a position to state the second theorem. When the scalar potential V is added as a perturbation, the situation changes even for small perturbation. The limit heavily depends on the values α of fluxes and it changes discontinuously at half-integer flux α = 1/2. Theorem 5.2 Let V ∈ C0∞ (R2 ) satisfy (5.4), and assume that T fulfills (5.6). If ω = ω ˜ for incident and final directions w and ω ˜ , then one has the following asymptotic form as ε → 0 : (1) Let 0 < α < 1/2. Then fε (ω → ω ˜ ; E) → g∞ (ω → ω ˜ ; E). (2) Let α = 1/2 and let λ2 be as in (5.8) of Lemma 5.1. Then fε (ω → ω ˜ ; E) → gκ (ω → ω ˜ ; E) for κ = 1/λ2 (κ = ∞ provided that λ2 = 0). (3) Let 1/2 < α < 1. Then ˜ ; E) → g0 (ω → ω ˜ ; E). fε (ω → ω The third theorem is concerned with the scattering of Dirac particles appearing in the interaction of cosmic string with matter. We now consider the 2 × 2 ˜ ) in question is desystem (1.15) of Dirac equations. The amplitude f2ε (ω → ω fined through the asymptotic form of solution w to equation (1.15). The solution behaves like w
=
t
(τ (ω), 0)ϕin (x; E, ω) + f1ε (ω → ω ˜ ; E)t (τ (˜ ω ), 0)eiEr r−1/2 + f2ε (ω → ω ˜ ; E)t (0, τ (˜ ω ))eiEr r−1/2 + o∗ (r−1/2 ),
r → ∞,
for incident wave t (τ (ω), 0)ϕin (x; E, ω). The aim of the third theorem is to analyze ˜ ; E). the asymptotic behavior as ε → 0 of f2ε (ω → ω Theorem 5.3 Let V ∈ C0∞ (R2 → R) satisfy (5.4), and assume that T fulfills (5.6). Then the amplitude f2ε (ω → ω ˜ ; E) behaves like 1/2 iE Cα ε|2α−1| + o(ε|2α−1| ), ε → 0, f2ε = 8π
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α −α α 2 0 < α < 1/2, (2 E i /Γ(1 − α)) 2πλ2 , α = 1/2, 4E −1 iλ2 (1 + λ22 )−1 , Cα = 2 1−α α−1 1−α 2 E i /Γ(α) 2πλ1 , 1/2 < α < 1.
We end the section by making some comments on Theorems 5.2 and 5.3. (1) As stated in Section 1, a result similar to Theorem 5.2 has been obtained by Afanasiev [2, Section 7.10], where the behavior of amplitude has been analyzed for the scattering by the small obstacle {|x| < ε} under a certain impenetrable boundary condition in the background of the δ-like field 2παδ(x). As ε → 0, the amplitude fε is convergent to g∞ , gκ with κ = −1 or g0 according as 0 < α < 1/2, α = 1/2 or 1/2 < α < 1. (2) The assumption that A(x) and V (x) are smooth is not essential. The two theorems extend to the case of bounded electromagnetic fields, and the extension is possible even for singular magnetic potentials. For example, the theorems apply to the case that A(x) = A0α (x) is the Aharonov-Bohm potential and V (x) is the characteristic function of unit disk {|x| < 1}. If we consider (1.13) with κ = ∞ as the boundary condition at the origin, we can calculate λ1 and λ2 explicitly. In fact, if we set e(x) = t (e1 (r), e2 (r)eiθ ), then it follows from (2.2) that e solves e 1 + α r−1 e1 + iV e2 = 0,
e 2 + (1 − α)r−1 e2 + iV e1 = 0,
where e = (d/dr)e. We use the formula (2.14) to solve the equation above. If we take account of (5.7), then λ2 is determined as λ2 = −J1−α (1)/J−α (1) for 0 < α ≤ 1/2, while (5.9) yields λ1 = −J−α (1)/J1−α (1) for 1/2 < α < 1. (3) As a work related to Theorem 5.3, [7] has dealt with the case that the electric potential is λV (x) and A(x) is the Aharonov-Bohm potential A0α (x) with boundary condition (1.13) with κ = ∞ or κ = 0, where λ > 0 is a small coupling constant and V still denotes the characteristic function of the unit disk.
6 Behavior of resolvent at low energy The proof of all the theorems in the previous section is based on the behavior as ε → 0 of resolvent R(E + i0; Kε ). We first follow the idea from [6, Chapter I.1.2] to derive the basic representation for R(E + i0; Kε ). The derivation is done by repeated use of the resolvent identity. If we set K0ε = D(aε , 0), then Kε = K0ε +Vε , and we have R(E + i0; Kε ) = R(E + i0; K0ε ) − R(E + i0; Kε )Vε R(E + i0; K0ε ) by the resolvent identity. We have assumed that V (x) ≥ 0. If we further define Yε = Vε1/2 R(E + i0; K0ε )Vε1/2 : [L2 ]2 → [L2 ]2 ,
(6.1)
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then the resolvent identity yields the relation R(E + i0; Kε )Vε1/2 (1 + Yε ) = R(E + i0; K0ε )Vε1/2 . The operator 1 + Yε has the bounded inverse (1 + Yε )−1 : [L2 ]2 → [L2 ]2 , which follows from the fact that the outgoing solution to equation (Kε −E)u = 0 identically vanishes. Thus R(E + i0; Kε ) is represented as R(E + i0; K0ε ) − R(E + i0; K0ε )Vε1/2 (1 + Yε )−1 Vε1/2 R(E + i0; K0ε ) by the resolvent identity. Let Jε : [L2 ]2 → [L2 ]2 be again the unitary operator defined by (Jε u) (x) = ε−1 u(x/ε). We set Xε = Jε∗ Yε Jε . Since K0ε = ε−1 Jε K0 Jε∗ for K0 = D(a, 0), we have Xε = Jε∗ Yε Jε = V 1/2 R(k + i0; K0 )V 1/2 ,
k = εE > 0,
(6.2)
and hence R(E + i0; Kε ) = R(E + i0; K0ε ) − ε−1 Γε (E + i0)(1 + Xε )−1 Γε (E − i0)∗ , (6.3) where
Γε (E ± i0) = R(E ± i0; K0ε )Jε V 1/2 .
(6.4)
This is a basic representation. This section is devoted to the analysis on the behavior as ε → 0 of Xε as the first step towards proving the three theorems. By (4.7), the potential a : R2 → R2 takes the form a = (−∂2 ϕ(x), ∂1 ϕ(x)) + ∇h = A + ∇h for some h ∈ C ∞ (R2 → R) falling off like h = O(|x|−1 ) at infinity, and the field b = ∇ × a has support in {|x| < 1}. We set p = (p1 , p2 ) = −i∇ − a and write K0 as 0 p− K0 = σ1 p1 + σ2 p2 = p+ 0 in the matrix form, where p± = p1 ± ip2 . We define the Schr¨ odinger operators L± by (6.5) L± = L(a, ±b) = p21 + p22 ± b = (−i∇ − a)2 ± b. These are selfadjoint with domain D(L± ) = H 2 (R2 ) in L2 . Since i[p1 , p2 ] = i(p1 p2 − p2 p1 ) = −b, we have L± = p± p∓ = p∗∓ p∓ , and R(k + i0; K0 ) is represented as kR(k 2 + i0; L−) p− R(k 2 + i0; L+ ) R(k + i0; K0 ) = . p+ R(k 2 + i0; L− ) kR(k 2 + i0; L+ )
(6.6)
Thus the problem is reduced to the study on the behavior of R(k 2 + i0; L± ) as k → 0.
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The two operators L± = p∗∓ p∓ ≥ 0 are non-negative, and since 0 < α < 1 by assumption (1.3), it follows by the Aharonov-Casher theorem ([4]) that L± have no bound states at zero energy. However, the spectral structure at zero energy is different in the sense that L− has a resonance state. The resonance state is defined as a bounded solution u to equation L− u = p− p+ u = 0. If u is such a solution, then a simple calculation using integral by parts shows that p1 u and p2 u are in L2 , so that p+ u = 0. By (4.7) (see also (1.8)), we have p+ = −ieih e−ϕ ( ∂1 + i∂2 ) eϕ e−ih .
(6.7)
Thus L− has the resonance state behaving like u(x) = e−ϕ eih = |x|−α 1 + O(|x|−1 ) at infinity. On the other hand, L+ = p+ p− does not have a resonance state. We note that if α > 1, L− has bound states at zero energy with multiplicity [α] by the Aharonov-Casher theorem again. We now introduce the following notation : η ∈ C0 (R2 ) is a continuous function with compact support and η0 ∈ C0 (R2 ) is a function compactly supported away from the origin. We further use the notation Op(εσ ) and op(εσ ) to denote the classes of bounded operators obeying the bound O(εσ ) and o(εσ ) in norm respectively. We make a brief review on the behavior at low energy of R(k 2 + i0; L±) obtained by ([23, Propositions 4.2 and 4.3]). We first consider L− . Let h(x) be as in (6.7). Then (6.8) ρ0 (x) = e−ϕ eih , solves L− ρ0 = 0 and behaves like
ρ0 (x) = |x|−α 1 + O(|x|−1 )
(6.9)
at infinity. We know ([23]) that L− has the one-dimensional resonance space spanned by ρ0 at zero energy. Proposition 6.1 Let ρ0 be as above and let γ0 be the constant defined by γ0 = −22(1−α) πΓ(1 − α)/Γ(α). Then
(6.10)
ηR(k 2 + i0; L− )η = γ− (k)i2α k −2α η(ρ0 ⊗ ρ0 )η + Op(ε0 )
for some coefficient γ− (k) obeying γ− (k) = −1/γ0 + o(1) as k → 0. Remark 6.1 (1) The proposition above corresponds to Proposition 4.3 in [23], where the resonance function ρ0 (x) is normalized as ρ0 (x) = (2πα)−1/2 e−ϕ eih , so that the constant γ− (k) undergoes a suitable change. (2) By elliptic estimate, ∇ηR(k 2 + i0; L−)η admits a similar asymptotic form under a natural modification.
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Next we move to L+ which has neither bound states nor resonance states at zero energy. We set L2com = {u ∈ L2 (R2 ) : supp u ⊂ BM },
BM = {|x| < M },
for M 1 fixed arbitrarily but sufficiently large. We have shown in [23] that there exists a limit G+ = lim R(k 2 + i0; L+ ) : L2com → L2−1 (6.11) k→0
as a bounded operator from L2com to L2−1 = L2 (R2 ; x−2 dx). We further know that the equation L+ = p+ p− u = 0 has a unique solution behaving like ω+l = rν eilθ + O(1),
|x| → ∞,
(6.12)
for l = 0, 1, where ν = |l − α| again. Proposition 6.2 Let the notation be as above. Then there exists γ+l (k) such that γ+l (k)i−2ν k 2ν η(ω+l ⊗ ω+l )η + Op(ε2 ), ηR(k 2 + i0; L+)η = ηG+ η + l=0,1
where the two constants γ+l (k), l = 0, 1, are bounded uniformly in k = εE > 0. This proposition has been obtained as Proposition 4.2 in [23]. We can make precise the behavior as k → 0 of the constant γ+l (k), but the argument below does not require such an asymptotic form. By (6.7), p+ = −2i eihe−ϕ ∂eϕ e−ih with ∂ = (1/2) ( ∂1 + i∂2 ). The CauchyRiemann operator ∂ has the fundamental solution (1/π) (x1 + ix2 )−1 . We denote −1 by ∂ the convolution operator ∂ and we define and p−1 −
−1
−1
= (1/π) (x1 + ix2 )
−1 ih −ϕ p−1 e e ∂ + = −(2i)
∗
−1 ϕ −ih
e e −1 ∗ = p+ . By definition, we have p± p−1 ± = 1.
Lemma 6.1 One has the relations p− G+ f = p−1 + f,
G+ p+ f = p−1 − f
for any bounded function f with compact support. Proof. We prove only the first relation. The second one follows by taking the 2 adjoint of both sides. Let f be as in the lemma, and set w1 = p−1 + f . Then w1 ∈ L and it solves p+ w1 = f . If, on the other hand, we set w2 = p− G+ f , then w2 satisfies p+ w2 = p+ p− G+ f = L+ G+ f = f.
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Since w2 ∈ L2−1 by (6.11), it follows that w2 ∈ L2 . In fact, we have p− G+ f L2 < ∞ by a simple use of partial integration. Set w = eϕ e−ih (w1 − w2 ). Then ∂w = 0, so that w is an entire function. Note that eϕ = O(|x|α ) at infinity for 0 < α < 1. Since w1 − w2 ∈ L2 , we can easily show that w = 0, and hence w1 = w2 . Thus the lemma is obtained. Lemma 6.2 Let ω+0 be as in (6.12). Then one has p− ω+0 = 0. Proof. Set v0 = e−ih eϕ . Then p− v0 = 0 and the difference u = ω+0 − v0 is bounded. The function u solves p+ p− u = L+ u = L+ ω+0 − p+ p− v0 = 0. Hence it follows from Lemma 4.3 of [22] (or by the argument used in its proof) that p− u = 0. This implies that p− ω+0 = 0, and the proof is complete. Lemma 6.3 Let ω+1 be also as in (6.12). Then one has p− ω+1 = cρ0 for some c. Proof. Set u = p− ω+1 . Then u obeys the bound u = O(|x|−α ) at infinity, and it solves the equation L− u = p− L+ ω+1 = 0. This implies that u is in the resonance space of L− at zero energy. Since the resonance space is one-dimensional, the lemma follows at once. If we make use of the simple relation p+ R(k 2 ± i0; L− ) = R(k 2 ± i0; L+ )p+ , then we obtain from (6.6) that kR(k 2 + i0, L− ) R(k + i0; K0 ) = R(k 2 + i0; L+ )p+
p− R(k 2 + i0; L+ ) kR(k 2 + i0, L+ )
for k = εE > 0. Thus we combine Propositions 6.1, 6.2 and Lemmas 6.1, 6.2 and 6.3 to get the following proposition. Proposition 6.3 As ε → 0, ηR(k + i0; K0 )η takes the form ηR(k + i0; K0)η = η γ(ε) (˜ ρ0 ⊗ ρ˜0 ) ε1−2α + G0 + O(ε2(1−α) )G1 η + Op(ε), where ρ˜0 = t (ρ0 , 0) and 0 p−1 + G0 = , p−1 0 −
G1 =
0 ω+1 ⊗ cρ0
cρ0 ⊗ ω+1 0
,
c being as in Lemma 6.3, and γ(ε) = i2α E 1−2α γ− (εE) = −i2α E 1−2α (1/γ0 + o(1)) ,
ε → 0.
(6.13)
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In particular, Xε defined by (6.2) takes the form Xε = γ(ε) (q0 ⊗ q0 ) ε1−2α + Z0 + O(ε2(1−α) )Z1 + Op(ε), where and Z0 = V
q0 = V 1/2 ρ˜0 , 1/2
G0 V
1/2
and Z1 = V
1/2
G1 V
ρ˜0 = t (ρ0 , 0), 1/2
(6.14) (6.15)
.
7 Resonance at zero energy: proof of Lemma 5.1 The second step is to analyze the inversion of (1 + Xε )−1 which appears in representation (6.3) for the resolvent R(E + i0; Kε ) under consideration. We also prove Lemma 5.1 at the end of the section. As is easily seen from assumption (5.6), K = D(a, V ) = K0 + V has neither eigenstates nor resonance states at zero energy. Lemma 7.1 Assume that 0 < α ≤ 1/2. Let Z0 be as in Proposition 6.3. If (5.6) is fulfilled, then Z0 : [L2 ]2 → [L2 ]2 has the bounded inverse (1 + Z0 )−1 on [L2 ]2 . Proof. The operator Z0 is compact. Set Φ = ker (1 + Z0). It suffices to show that dim Φ = 0. The proof is done by contradiction. Assume that u = t (u1 , u2 ) ∈ Φ does not vanish identically. If we set v = t (v1 , v2 ) = G0 V 1/2 u for u as above, then V 1/2 v = Z0 u = −u, and v satisfies K0 v = V 1/2 u = −V v, so that v solves Kv = 0. We can easily see that v is notidentically zero. The first −1 ∗ 1/2 component v1 = p−1 u2 is in L2 . Since p−1 is the integral operator + V − = p+ with kernel −(2πi)−1 eϕ eih (x1 − ix2 )−1 ∗ e−ϕ e−ih , 1/2 u1 behaves like the second component v2 = p−1 − V
v2 (x) = −(2πi)−1 (u1 , V 1/2 ρ0 )eϕ eih (x1 −ix2 )−1 +O(|x|−2+α ) = O(|x|−1+α ) (7.1) as |x| → ∞. This implies that K has either eigenstates or resonance states at zero energy. This contradicts the assumption and the proof is complete. By assumption (5.5), λ0 = (V ρ0 , ρ0 ) = 0. This enables us to define P = λ−1 0 (q0 ⊗ q0 ),
q0 = V 1/2 ρ˜0 ,
(7.2)
as a projection on [L2 ]2 . Lemma 7.2 Assume that 1/2 < α < 1. Let Q = 1 − P and Σ = Ran Q. If (5.6) is −1 fulfilled, then QZ0 Q : Σ → Σ has the bounded inverse (1 + QZ0 Q) on Σ.
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Proof. We again show by contradiction that dim Ψ = 0, where Ψ = {u ∈ Σ : QZ0 Qu = −u}. Assume that u not vanishing identically belongs to Ψ. We set v = t (v1 , v2 ) = G0 V 1/2 u − d˜ ρ0 , 1/2 ρ˜0 ) = λ−1 ˜0 = 0 and since where d = λ−1 0 (Z0 u, V 0 (Z0 u, q0 ). Since K0 ρ
V 1/2 v = Z0 u − P Z0 u = QZ0 u = −u, we see that v satisfies K0 v = V 1/2 u = −V v, and hence v solves Kv = 0. We also have that v = 0. The first component v1 behaves like v1 (x) = −dρ0 (x) + O(|x|−1−α ) = O(|x|−α ) at infinity. We claim that v2 ∈ L2 , which follows from (7.1). In fact, we have only to note that (u1 , V 1/2 ρ0 ) = (u, V 1/2 ρ˜0 ) = −(V v, ρ˜0 ) = −(V 1/2 Z0 u − dV ρ˜0 , ρ˜0 ) = 0 by the choice of constant d. Thus v ∈ L∞ × L2 becomes either eigenstate or resonance state. This proves the lemma. Remark 7.1 The converse statements of the two lemmas above are also true, although we do not prove it here. The proof is easy. Hence, if |V | 1 is small enough, then (5.6) is fulfilled. Lemma 7.3 (1)
Let 0 < α ≤ 1/2 and set q = (1 + Z0 )−1 q0 ∈ L2 × L2 .
Then q is represented as q = V 1/2 e with e = t (e1 , e2 ) ∈ L∞ × L∞ , and e uniquely solves Ke = 0 under the condition that e1 = r−α + O(|x|−1−α ), (2)
e2 = O(|x|−1+α ),
|x| → ∞.
(7.3)
Let 1/2 < α < 1 and set q = q0 − (1 + QZ0 Q)−1 QZ0 q0 .
Then q = V 1/2 e for some e = t (e1 , e2 ) ∈ L∞ × L∞ , and e uniquely solves Ke = 0 under the condition that e1 = O(|x|−α ), Proof.
e2 = −i (λ0 /2π) r−1+α eiθ + O(|x|−2+α ),
|x| → ∞.
(7.4)
(1) If we set e = ρ˜0 − G0 V 1/2 q, then it follows that q = q0 − Z0 q = V 1/2 e.
We assert that e has the desired properties. By definition, e satisfies
Ke = −V 1/2 q + V ρ˜0 − G0 V 1/2 q = V 1/2 (q0 − q − Z0 q) = 0 and obeys (7.3). Since K has neither eigenstates nor resonance states, it is easy to see that e uniquely solves Ke = 0. This proves (1).
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(2) This is verified in almost the same way as (1). We set r = −(1 + QZ0 Q)−1 QZ0 q0 . Then we have r = −QZ0 r − QZ0 q0 = −Z0 r − Z0 q0 + P Z0 r + P Z0 q0 and hence q is represented as q = q0 + r = V 1/2 e, where e = d1 ρ˜0 − G0 V 1/2 r − G0 V 1/2 q0
(7.5)
with constant d1 = 1 + (Z0 (r + q0 ), q0 )/λ0 . A simple calculation yields Ke = V 1/2 (d1 q0 − (r + q0 ) − Z0 (r + q0 )) = V 1/2 (d1 q0 − q0 − P Z0 (r + q0 )) = 0. It is easy to see that e1 = O(|x|−α ). We look at the second component e2 . If we note that (V 1/2 r, ρ˜0 ) = (Qr, V 1/2 ρ˜0 ) = (Qr, q0 ) = 0, then it follows from (7.1) that the second component of G0 V 1/2 r obeys O(|x|−2+α ). 1/2 The second component −p−1 q0 of the term −G0 V 1/2 q0 behaves like − V 1/2 −p−1 q0 = (2πi)−1 eϕ eih r−1 eiθ λ0 + O(|x|−2+α ). − V
This yields the coefficient −i(λ0 /2π) in (7.4). Thus we can show that e has the desired properties and the lemma is proved. We end the section by proving Lemma 5.1. Proof of Lemma 5.1. (1) Assume that 0 < α ≤ 1/2. Let q = t (q1 , q2 ) = (1 + Z0 )−1 q0 = V 1/2 e be as in Lemma 7.3, where e = ρ˜0 − G0 V 1/2 q. Then the second component e2 = 1/2 q1 behaves like −p−1 − V e2 = iλ2 r−1+α eiθ + O(|x|−2+α ),
|x| → ∞,
for some constant λ2 . We show that λ2 is real. To to this, we compute ((1 + Z0 )−1 q0 , q0 ) = (q, q0 ) = (V e, ρ˜0 ) = −(K0 e, ρ˜0 ) = −(p− e2 , ρ0 ). Recall the representation (2.2) for π− in terms of the polar coordinates. Since p− = π− on {|x| > 2} and since p+ ρ0 = 0, we have ((1 + Z0 )−1 q0 , q0 ) = i lim e−iθ e2 ρ0 ds = −2πλ2 R→∞
|x|=R
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by partial integration. This yields λ2 = −((1 + Z0 )−1 q0 , q0 )/2π
(7.6)
and λ2 is real. This implies that e has all the desired properties. (2) We proceed to proving (2). Assume that 1/2 < α < 1. Let e be defined by (7.5) in the proof of Lemma 7.3. We calculate the constant d1 in (7.5). According to the argument in the proof of Lemma 7.3, we have d1
=
1 + ((r + q0 ), Z0 q0 )/λ0 = 1 + (V 1/2 e, Z0 q0 )/λ0
=
1 + (q, Z0 q0 )/λ0 = 1 + (q0 − (1 + QZ0 Q)−1 QZ0 q0 , Z0 q0 )/λ0 1 + (q0 , Z0 q0 ) − ((1 + QZ0 Q)−1 QZ0 q0 , QZ0 q0 ) /λ0 .
=
Thus d1 is real, and e1 behaves like e1 (x) = d1 r−α + O(|x|−1−α ). The desired solution is obtained as −(2π/λ0 )e, and then λ1 = −(2π/λ0 )d1 is also determined as a real number. This completes the proof.
(7.7)
8 Convergence of resolvent: proof of Theorems 5.1 and 5.2 In this section we prove Theorems 5.1 and 5.2 through a series lemmas. We recall that η0 ∈ C0 (R2 ) has support away from the origin. We also use the notation o2 (1) to denote remainder terms of which the L2 norm obeys the bound o(1) as ε → 0. We start by the following two lemmas. Lemma 8.1 Let ξ± = ξ± (x; E) be defined by (3.1). Then η0 R(E ± i0; H∞ )Jε η = β± η0 (ξ± ⊗ r˜0 ) ηε1−α + Op(ε), where r˜0 (x) = t (r0 (x), 0) with r0 (x) = |x|−α , and β± = ∓ 2α−2 /Γ(1 − α) E 1−α . Lemma 8.2 Let the notation be as in Lemma 8.1. Then η0 R(E ± i0; K0ε )Jε η = β± ((η0 ξ± + o2 (1)) ⊗ ρ˜0 ) ηε1−α + Op(ε) and, in particular, Γε (E ± i0) defined by (6.4) takes the form η0 Γε (E ± i0) = β± ((η0 ξ± + o2 (1)) ⊗ q0 ) ε1−α + Op(ε), where q0 = t (V 1/2 ρ0 , 0) ∈ [L2 ]2 is defined by (6.15).
(8.1)
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Proof of Lemma 8.1. We prove the lemma for the + case only. For brevity, we write ξ+ = t (ξ1 , ξ2 ), ξ1 = −ieiαπ Hα (Er), ξ2 = H1−α (Er)eiθ . The resolvent R(E+i0; H∞ ) is represented in terms of R(E 2 +i0; LAB ) and R(E 2 + i0; LZ ) by (3.2). We first consider R(E 2 + i0; LZ ). This admits the decomposition ˜0 ⊕ R(E 2 + i0; LZ ) = R ⊕ Rl l =0
˜ 0 and with respect to angular momentum (see (3.4)), and the Green kernels of R Rl are defined by (3.5). Since η0 has support away from the origin, we can take ε ˜ 0 Jε η has so small that |x| > ε|y| when x ∈ supp η0 and y ∈ supp η, and hence η0 R the kernel G(x, y) = ε(ieiαπ /4)η0 (x)Hα (E|x|)J−α (εE|y|)η(y) by a change of variables. This implies that ˜ 0 Jε η = β+ η0 (ξ1 ⊗ r0 )ηε1−α + Op(ε). Eη0 R A similar argument applies to Rl , l = 0, and we obtain η0 Rl Jε η = Op(ε) uniformly in l. Thus we have Eη0 R(E 2 + i0; LZ )Jε η = β+ η0 (ξ1 ⊗ r0 )ηε1−α + Op(ε). Since π+ ξ1 = Eξ2 by (2.14), we make use of this relation to obtain that η0 π+ R(E 2 + i0; LZ )Jε η = β+ η0 (ξ2 ⊗ r0 )ηε1−α + Op(ε). Similarly R(E 2 + i0; LAB ) is shown to obey η0 R(E 2 + i0; LAB )Jε η = Op(ε),
η0 π− R(E 2 + i0; LAB )Jε η = Op(ε).
This proves the lemma. Proof of Lemma 8.2. We again prove the lemma for the + case only. Set ζε (x) = ζ(x/ε),
ζ(x) = 1 − χ0 (x/2),
for the basic cut-off function χ0 (x) with property (4.3). Then we have supp ζε ⊂ {|x| > 2ε},
ζε = 1 on {|x| > 4ε}.
We may assume that ζε η0 = η0 for ε small enough, and we have η0 R(E + i0; K0ε )Jε η
=
η0 R(E + i0; H∞ )ζε Jε η
+
η0 R(E + i0; H∞ )Wε R(E + i0; K0ε )Jε η
(8.2)
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by the resolvent identity, where Wε = H∞ ζε − ζε K0ε . By (4.1), H∞ = K0ε = Dα over |x| > 2ε. If we make use of relations ζε = Jε ζJε∗ and Dα = ε−1 Jε Dα Jε∗ , Wε equals the commutator Wε = [Dα , ζε ] = ε−1 Jε [Dα , ζ]Jε∗ . If we further use the relation Jε∗ R(E + i0; K0ε )Jε = εR(k + i0; K0 ) with k = εE, then we obtain η0 R(E + i0; K0ε )Jε η = η0 R(E + i0; H∞ )Jε ζη + Fε R(k + i0; K0)η,
(8.3)
where Fε = η0 R(E + i0; H∞ )Jε [Dα , ζ]. It follows from Lemma 8.1 that Fε is of the form 0 β+ η0 (ξ1 ⊗ r0 ) [π− , ζ]ε1−α Fε = + Op(ε) 0 β+ η0 (ξ2 ⊗ r0 ) [π− , ζ]ε1−α with ξ+ = t (ξ1 , ξ2 ) as in the proof of Lemma 8.1. Next we evaluate Fε R(k + i0; K0 )η. The operator ηR(k + i0; K0 )η admits the decomposition in Proposition 6.3 for η ∈ C0 (R2 ). We calculate : ρ0 ⊗ ρ˜0 ) ηε1−2α = (o2 (1) ⊗ ρ˜0 ) η, Fε (˜ 1−α Fε G0 η = β+ η0 (ξ+ ⊗ r˜0 )[ π− , ζ]p−1 + Op(ε), − ηε
O(ε2(1−α) )Fε G1 η = (o2 (1) ⊗ ρ˜0 ) η + Op(ε) for G0 and G1 as in Proposition 6.3. We combine these relations with Lemma 8.1. Then η0 (E + i0; K0ε )Jε η = β+ ((η0 ξ+ + o2 (1)) ⊗ r˜1 ) ηε1−α + Op(ε) with r˜1 = t (r1 , 0), where r1 = ζr0 + p−1 + [ζ, π+ ]r0 ,
r0 (x) = |x|−α .
Since ζπ+ r0 = 0, it is easy to see that p+ r1 = 0, and also r1 (x) behaves like r1 (x) = |x|−α + O(|x|−1−α ) at infinity. By uniqueness, this implies that r1 = ρ0 , and the proof is complete. Theorem 5.1 is obtained as an immediate consequence of the lemma below. Lemma 8.3 One has η0 R(E ± i0; K0ε )η0 → η0 R(E ± i0; H∞ )η0 ,
ε → 0,
in norm. Proof. We deal with the + case only. Let ζε be defined by (8.2). Since ζε η0 = η0 for ε small enough, we have η0 R(E + i0; K0ε )η0
=
η0 R(E + i0; H∞ )η0
+
η0 R(E + i0; K0ε )Wε∗ R(E + i0; H∞ )η0
(8.4)
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by the resolvent identity, where ∗
Wε∗ = (H∞ ζε − ζε K0ε ) = ζε H∞ − K0ε ζε = ε−1 Jε [ζ, Dα ]Jε∗ . We decompose the second term on the right side of (8.4) into the product F1ε F0ε F2ε of three operators, where F1ε = η0 R(E + i0; K0ε)Jε η, F2ε = ηJε∗ R(E + i0; H∞)η0 = (η0 R(E − i0; H∞ )Jε η)
∗
for some η ∈ C0 (R2 ), and F0ε = ε−1 [ζ, Dα ]. By Lemmas 8.1 and 8.2, F1ε and F2ε take the form Op(ε1−α ) Op(ε) Op(ε1−α ) Op(ε1−α ) F1ε = , F2ε = Op(ε1−α ) Op(ε) Op(ε) Op(ε) and F0ε equals
F0ε =
0 ε−1 [ζ, π− ] −1 0 ε [ζ, π+ ]
.
A simple computation yields F1ε F0ε F2ε = Op(ε1−α ). This proves the lemma.
Proof of Theorem 5.1. If we recall that fε and g∞ are represented by (5.2) and (5.3) respectively, then the theorem follows from Lemma 8.3 at once. We proceed to the proof of Theorem 5.2. We first accept the lemma below as proved to complete the proof of the theorem. Lemma 8.4 Assume that (5.6) is fulfilled. Recall that P : [L2 ]2 → [L2 ]2 is the projection defined by (7.2), and set Q = 1−P . Then (1+Xε )−1 obeys the following asymptotic form as ε → 0: (1) If 0 < α < 1/2, then (1 + Xε )−1 = (1 + Z0 )−1 + Op(ε1−2α ). (2) If α = 1/2, then (1 + Xε )−1 = (1 + Z0 )−1 + a (q ⊗ q) + op(ε0 ), where a = −i/(2π + iτ ),
τ = (q, q0 ),
q = (1 + Z0 )−1 q0 .
(8.5)
(3) If 1/2 < α < 1, then
(1 + Xε )−1 = δ+ (ε)P 1 + Op(ε2α−1 ) P
− δ+ (ε)Q (Q + QZ0 Q)−1 QZ0 + Op(ε2α−1 ) + Op(ε2(1−α) ) P
− δ+ (ε)P Z0 Q(Q + QZ0 Q)−1 + Op(ε2α−1 ) + Op(ε2(1−α) ) Q + Q (Q + QZ0 Q)−1 + Op(ε2α−1 ) Q,
where δ+ (ε) = 1/µ+ (ε),
µ+ (ε) = 1 + γ− (k)i2α k 1−2α λ0 ,
k = εE.
(8.6)
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Proof of Theorem 5.2. The proof is based on the relation R(E + i0; Kε ) = R(E + i0; K0ε ) − ε−1 Γε (E + i0)(1 + Xε )−1 Γε (E − i0)∗ derived by (6.3). By Lemma 8.3, we have η0 R(E + i0; K0ε )η0 → η0 R(E + i0; H∞ )η0 ,
ε → 0,
in norm for the first operator on the right side. We analyze the second operator R(ε) = ε−1 η0 Γε (E + i0)(1 + Xε )−1 Γε (E − i0)∗ η0 . The behavior as ε → 0 of R(ε) takes a different form according as 0 < α < 1/2, α = 1/2 or 1/2 < α < 1. (1) Let 0 < α < 1/2. Then it follows from Lemmas 8.2 and 8.4 that R(ε) = O(ε−1 )O(ε2(1−α) ) = O(ε1−2α ), so that η0 R(E + i0; K0ε )η0 → η0 R(E + i0; H∞ )η0 ,
ε → 0,
and hence fε → g∞ . This proves (1). (2) If α = 1/2, then β± = ∓2−3/2 E 1/2 /π 1/2 by (8.1), so that β+ β− = −E/8π. By Lemmas 8.2 and 8.4 again, we have R(ε) → a0 η0 (ξ+ ⊗ ξ− )η0 , where
a0 = β+ β− τ + aτ 2 = −(E/4) (i + 2π/τ )−1 .
Since λ2 = −τ /2π by (7.6), it follows from Proposition 3.1 that η0 R(E + i0; K0ε )η0 → η0 R(E + i0; Hκ )η0 ,
κ = 1/λ2 .
This proves (2). (3) The final case is 1/2 < α < 1. Recall that q0 2 = V 1/2 ρ0 2 = (V ρ0 , ρ0 ) = λ0 by (5.5). Since P q0 = q0 and Qq0 = 0, we have by Lemmas 8.2 and 8.4 that R(ε) behaves like R(ε) = a1 (ε)η0 (ξ+ ⊗ ξ− )η0 + op(ε0 ),
a1 (ε) = ε−1 β+ β− ε2(1−α) δ+ (ε)λ0 .
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2 We calculate β+ β− = − 2α−2 E 1−α /Γ(1 − α) by (8.1). Since
γ− (k) → −1/γ0 = Γ(α)/ 22(1−α) πΓ(1 − α) in Proposition 6.1, it follows that ε1−2α δ+ (ε) → −γ0 i−2α E 2α−1 /λ0 and hence a1 (ε)
2 → γ0 i−2α E 2α−1 2α−2 E 1−α /Γ(1 − α) =
−(E/4) (π/Γ(α)Γ(1 − α)) i−2α = −(E/4) sin απ/eiαπ .
This, together with Proposition 3.1, implies that fε → g0 , and (3) is obtained. Thus the proof of the theorem is now complete. Proof of Lemma 8.4. By Proposition 6.3, we have 1 + Xε = 1 + Z0 + γ− (k)i2α k 1−2α (q0 ⊗ q0 ) + O(ε2(1−α) )Z1 + Op(ε) for k = εE > 0, where γ− (k) = −1/γ0 + o(1) as ε → 0. (1) Assume that 0 < α < 1/2. If K = K0 + V has neither bound nor resonance state at zero energy, then 1 + Z0 : [L2 ]2 → [L2 ]2 admits a bounded inverse by Lemma 7.1, and hence (1 + Xε )−1 takes the form as in the lemma. (2) If α = 1/2, we have 1 + Xε = 1 + Z0 + (i/2π) (q0 ⊗ q0 ) + op(ε0 ). Let q = (1 + Z0 )−1 q0 and τ = (q, q0 ) be as in (8.5). Then 1 + Xε = (1 + Z0 ) (1 + (i/2π) (q ⊗ q0 )) + op(ε0 ). A simple computation yields −1
(1 + (i/2π) (q ⊗ q0 )) with a as in the lemma. Hence (1 + Xε )
−1
= 1 + a (q ⊗ q0 )
takes the desired form.
(3) We deal with the case 1/2 < α < 1. We employ the method from [12], which has been applied to the analysis on the behavior at low energy of resolvents of Schr¨ odinger operators −∆ + V in two dimensions. We write µ(ε) and δ(ε) = 1/µ(ε) = O(ε2α−1 ),
ε → 0,
for µ+ (ε) and δ+ (ε) respectively. Then 1 + Xε = µ(ε)P + Q + Z0 + O(ε2(1−α) )Z1 + Op(ε)
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by Proposition 6.3. If we use the two simple relations (µ(ε)P + Q)−1 = δ(ε)P + Q,
(1 + QZ0 P )−1 = 1 − QZ0 P,
then 1 + Xε takes the form 1 + Xε = (µ(ε)P + Q) (1 + QZ0 P ) Gε , and hence (1 + Xε )
−1
= G−1 ε (δ(ε)(P − QZ0 P ) + Q) ,
(8.7)
where Gε is represented in the form Gε = 1 + QZ0 Q + δ(ε)(1 − QZ0 )P Z0 + QOp(ε2(1−α) ) + Op(ε). We now set Σ0 = Ran P and Σ = Ran Q. The second factor on the right side of (8.7) has the matrix representation δ(ε)P 0 Σ0 Σ0 δ(ε)(P − QZ0 P ) + Q = : → , (8.8) −δ(ε)QZ0 P Q Σ Σ while Gε = (Gjk (ε))0≤j,k≤1 has the components G00 = P (1 + Op(ε2α−1 ))P,
G01 = P (δ(ε)Z0 + Op(ε))Q,
G10 = Q(−δ(ε)Z0 P Z0 + Op(ε2(1−α) ))P,
G11 = Q(1 + Z0 + Op(ε2α−1 ))Q.
By Lemma 7.2, Q + QZ0 Q : Σ → Σ has a bounded inverse, so that G−1 11 : Σ → Σ exists for ε small enough. If we take account of this fact, then G−1 ε = Eε = (Ejk (ε))0≤j,k≤1 can be calculated as −1 E00 = G00 − G01 G−1 , 11 G10
−1 E01 = − G00 − G01 G−1 G01 G−1 11 G10 11 ,
−1 E10 = − G11 − G10 G−1 G10 G−1 00 G01 00 , Hence (1 + Xε ) −1
(1 + Xε )
−1
−1 E11 = G11 − G10 G−1 . 00 G01
takes the form
= δ(ε)(E00 P − E01 QZ0 P ) + E01 Q + δ(ε)(E10 P − E11 QZ0 P ) + E11 Q
by use of (8.7) and (8.8). Each component Ejk (ε) behaves like : E00 E01
= P (1 + Op(ε2α−1 ))P, = P (−δ(ε)Z0 Q(Q + QZ0 Q)−1 + Op(ε2(2α−1) ) + Op(ε))Q,
E10 E11
= Q(δ(ε)(Q + QZ0 Q)−1 QZ0 P Z0 + Op(ε2(2α−1) ) + Op(ε2(1−α) ))P, = Q((Q + QZ0 Q)−1 + Op(ε2α−1 ))Q.
If we take account of these relations, (1 + Xε ) in the lemma, and the proof is complete.
−1
can be shown to take the form
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We end the section by making a brief comment on the case when α < 0 and α > 1. Remark 8.1 If we replace the magnetic potential A(x) by −A(x), the argument here extends to the case −1 < α < 0 without any essential change. If |α| > 1, then the magnetic Schr¨ odinger operator L(A, −b) has eigenstates at zero energy besides the resonance state by the Aharonov-Casher theorem [4], so that the norm convergence of resolvent η0 R(E + i0; Kε )η0 can not be expected ([23]). However the strong convergence can be expected, and hence Theorems 5.1 and 5.2 seem to remain true in the case |α| > 1 also.
9 Scattering in the interaction of cosmic string with matter The last section is devoted to proving Theorem 5.3. We begin by representing the ˜ ; E) in question in terms of the resolvent R(E + i0; Kε ) of amplitude f2ε (ω → ω K0ε 0 Vε 0 . Kε = K0ε + Vε = + Vε 0 0 K0ε If we decompose V into the product 1/2 0 V V 0 0 V= = V 0 0 V 1/2 V 1/2
V 1/2 0
= V1 V2 ,
then almost the same argument as used to derive (6.3) enables us to obtain −1
R(E + i0; Kε ) = R(E + i0; K0ε ) − ε−1 Γ1ε (E + i0) (1 + Xε )
Γ2ε (E − i0)∗ , (9.1)
where Xε = V2 R(k + i0; K0 )V1 with k = εE > 0, and Γ1ε (E + i0) = R(E + i0; K0ε )Jε V1 , A direct computation yields Γε (E + i0) 0 Γ1ε = , 0 Γε (E + i0)
Γ2ε (E − i0) = R(E − i0; K0ε )Jε V2 . Γ2ε =
0 Γε (E − i0) Γε (E − i0) 0
where Γε (E ± i0) is defined by (6.4). We further have 0 Xε (1 − Xε2 )−1 −1 Xε = , (1 + Xε ) = Xε 0 −Xε (1 − Xε2 )−1
−Xε (1 − Xε2 )−1 (1 − Xε2 )−1
We divide R(E + i0; Kε ) into the block form R(E + i0; Kε ) = (Rjk (E + i0; Kε ))1≤j,k≤2 , where Rjk (E + i0; Kε ) acts on [L2 ]2 . In particular, we have R21 (E + i0; Kε ) = −ε−1 Γε (E + i0)(1 − Xε2 )−1 Γε (E − i0)∗ .
,
.
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We can represent f2ε (ω → ω ˜ ; E) as ˜ ; E) = (iE/8π)1/2 (R21 (E + i0; Kε )Π+ ψ+ (ω), Π− ψ− (˜ ω )) f2ε (ω → ω by repeating the same argument as in Section 4, and hence we have f2ε = −ε−1 (iE/8π)1/2 (Γε (E + i0)(1 − Xε2 )−1 Γε (E − i0)∗ Π+ ψ+ (ω), Π− ψ− (˜ ω )). (9.2) The argument here is based on this representation. Lemma 9.1 The operator K0 − V has a resonance at zero energy if and only if so does K = K0 + V , and the same statement is also true for an eigenstate. Proof. The lemma is easy to prove. For brevity, we consider the case 0 < α ≤ 1/2 only. A similar argument applies to the case 1/2 < α < 1. Let v+ = (v1 , v2 ) ∈ L2 × L∞ be a resonance state of K0 + V . If we set v− = (v1 , −v2 ), then v− solves (K0 − V ) v− = 0, and it becomes a resonance by Definition 5.1. The case of eigenstate is also shown in the same way. We keep the same notation as in the previous sections. The lemma above implies the existence of bounded inverses (1 − Z0 )−1 : [L2 ]2 → [L2 ]2 and (1 − QZ0 Q)−1 : Σ → Σ. The following lemma is verified in exactly the same way as in the proof of Lemmas 8.4. We skip the proof. Lemma 9.2 If (5.6) is fulfilled, then (1 − Xε )−1 has the following asymptotic form as ε → 0 : (1) If 0 < α < 1/2, then (1 − Xε )−1 = (1 − Z0 )−1 + Op(ε1−2α ). (2) If α = 1/2, then (1 − Xε )−1 = (1 − Z0 )−1 + a (q ⊗ q ) + op(ε0 ), where
a = i/(2π − iτ ),
τ = (q , q0 ),
q = (1 − Z0 )−1 q0 .
(9.3)
(3) If 1/2 < α < 1, then (1 − Xε )−1 = δ− (ε)P 1 + Op(ε2α−1 ) P
+ δ− (ε)Q (Q − QZ0 Q)−1 QZ0 + Op(ε2α−1 ) + Op(ε2(1−α) ) P
+ δ− (ε)P Z0 Q(Q − QZ0 Q)−1 + Op(ε2α−1 ) + Op(ε2(1−α) ) Q + Q (Q − QZ0 Q)−1 + Op(ε2α−1 ) Q, where δ− (ε) = 1/µ− (ε),
µ− (ε) = 1 − γ− (k)i2α k 1−2α λ0 ,
k = εE.
(9.4)
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Lemma 9.3 Let ξ± be defined by (3.1). Set I+ = (ξ+ , Π− ψ− (˜ ω )) , Then
I+ = −4eiαπ/2 /E,
I− = (ξ− , Π+ ψ+ (ω)) . I− = 4e−iαπ/2 /E.
Proof. We calculate I+ only. A similar computation applies to I− . For brevity, we write ξ+ = ξ = t (ξ1 , ξ2 ),
ψ− = ψ = t (ψ1 , ψ2 ),
χ− (x) = χ0 (x/4) = χ(x).
By (4.3), χ has support in {|x| < 8} and χ = 1 on {|x| < 4}. Since Π− ψ = [Dα , χ]ψ = [Dα − E, χ]ψ = (Dα − E) χψ for x = 0, I+ equals I+ = lim δ→0
|x|>δ
ξ1 (π− χψ2 − Eχψ1 ) + ξ2 (π+ χψ1 − Eχψ2 ) dx.
Note that (Dα − E) ξ = 0, and π+ and π− take the form π+ = eiθ (−i∂r + · · · ) ,
π− = e−iθ (−i∂r + · · · )
by (2.2). We integrate by parts to calculate I+ . Since χ = 1 on {|x| = δ}, we have iθ I+ = −i lim e ξ1 ψ 2 + e−iθ ξ2 ψ 1 ds, ds = δ dθ. δ→0
|x|=δ
By (2.5) and (2.22), the first term in the integrand obeys eiθ ξ1 (x)ψ 2 (x) = O(r1−2α ) + O(1),
and hence lim
δ→0
|x|=δ
r = |x| → 0,
eiθ ξ1 ψ 2 ds = 0,
because 0 < α < 1. On the other hand, the second term behaves like −1 iαπ/2
e−iθ ξ2 ψ 1 = (−i/ sin απ) (1/Γ(α)Γ(1 − α)) (Er/2)
e
(1 + o(1))
as |x| → 0. Since Γ(α)Γ(1 − α) = π/ sin απ by formula, we have −i lim e−iθ ξ2 (x)ψ 1 (x) ds = −4eiαπ/2 /E. δ→0
|x|=δ
This yields the desired value. We now define Iε by Iε = (1 − Xε2 )−1 q0 , q0 = (1 + Xε )−1 q0 , (1 − Xε∗ )−1 q0 .
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Lemma 9.4 Let λ1 and λ2 be as in Lemma 5.1. Then one has the following statements: (1) If 0 < α < 1/2, then Iε = −2πλ2 + o(1),
ε → 0.
(2) If α = 1/2, then Iε = −2πλ2 (1 + λ22 )−1 + o(1),
ε → 0.
(3) If 1/2 < α < 1, then Iε = −γ02 (λ1 /2π)i−4α E 2(2α−1) ε2(2α−1) (1 + o(1)) ,
ε → 0.
We complete the proof Theorem 5.3, accepting this lemma as proved. Throughout the proof of the theorem, we use the notation O2 (ε) to denote remainder terms of which the L2 norm obeys O(ε). Proof of Theorem 5.3. We set η± = η0 ξ± + o2 (1) in Lemma 8.2. The amplitude f2ε (ω → ω ˜ ; E) is represented as (9.2). If we use Lemma 8.2, then a simple computation enables us to evaluate the amplitude as follows : f2ε
= + +
−(iE/8π)1/2 β− β+ (Π+ ψ+ (ω), η− )(η+ , Π− ψ− (˜ ω ))Iε ε1−2α −α 2 −1 O(ε )((1 − Xε ) O2 (ε), q0 ) O(ε−α )((1 − Xε2 )−1 q0 , O2 (ε)) + O(ε).
(9.5)
The leading term comes from the first term on the right side of (9.5). We first consider the case 1/2 < α < 1. If 1/2 < α < 1, then it follows from Lemmas 8.4 and 9.2 that (1 − Xε2 )−1 takes the form (1 − Xε2 )−1
= P Op(ε2(2α−1) )P + Q Op(ε0 )Q + P Op(ε2α−1 )Q + Q Op(ε2α−1 )P
and hence we have |((1 − Xε2 )−1 O2 (ε), q0 )| + |((1 − Xε2 )−1 q0 , O2 (ε))| = O(ε2α ), because Qq0 = 0. This implies that the three remainder terms on the right side of (9.5) obey O(εα ) = O(ε2α−1 )O(ε1−α ) = o(ε2α−1 ). Thus we have f2ε = −(iE/8π)1/2 β− β+ (Π+ ψ+ (ω), η− )(η+ , Π− ψ− (˜ ω ))Iε ε1−2α + o(ε2α−1 ). If we combine Lemmas 9.3 and 9.4, the desired asymptotic form is obtained after a little tedious computation of the leading constant Cα . Next we move to the case 0 < α ≤ 1/2. By Lemmas 8.4 and 9.2 again, (1 − Xε2 )−1 is bounded uniformly in ε, so that |((1 − Xε2 )−1 O2 (ε), q0 )| + |((1 − Xε2 )−1 q0 , O2 (ε))| = O(ε).
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Then the remainder terms on the right side of (9.5) obey O(ε1−α ) = o(ε1−2α ). Thus we have ω ))Iε ε1−2α + o(ε1−2α ). f2ε = −(iE/8π)1/2 β− β+ (Π+ ψ+ (ω), η− )(η+ , Π− ψ− (˜ We again combine Lemmas 9.3 and 9.4 to obtain the desired asymptotic form for the case 0 < α ≤ 1/2, and the proof is complete. It remains to prove Lemma 9.4. The proof requires two auxiliary lemmas. The first lemma below is proved in the same way as Lemma 7.3. We skip the proof. Lemma 9.5 (1)
If 0 < α ≤ 1/2, then q = (1 − Z0 )−1 q0 = V 1/2 e
for some e = t (e1 , e2 ) ∈ L∞ × L∞ , and e uniquely solves (K0 − V ) e = 0 under the condition that e1 = r−α + O(|x|−1−α ),
(2)
e2 = O(|x|−1+α ),
|x| → ∞.
If 1/2 < α < 1, then q = q0 + (Q − QZ0 Q)−1 QZ0 q0 = V 1/2 e
for some e = t (e1 , e2 ) ∈ L∞ × L∞ , and e uniquely solves (K0 − V ) e = 0 under the condition that e1 = O(|x|−α ),
e2 = i(λ0 /2π)r−1+α eiθ + O(|x|−2+α ),
|x| → ∞.
Lemma 9.6 Assume that 0 < α ≤ 1/2. Let τ and τ be the real numbers as in (8.5) and (9.3) respectively. Then one has τ = (q, q0 ) = (1 + Z0 )−1 q0 , q0 = −2πλ2 , τ = (q , q0 ) = (1 − Z0 )−1 q0 , q0 = −2πλ2 . Proof. We write e+ = t (e1 , e2 ) for e in Lemma 7.3 and e− for e in Lemma 9.5. Then it follows by uniqueness that e− is given as e− = t (e1 , −e2 ) for 0 < α ≤ 1/2. We prove the first relation only. The second relation is obtained in a similar way. By Lemma 7.3, τ = (V e, ρ˜0 ) and e solves Ke = (K0 + V ) e = 0. Hence τ = −(K0 e, ρ˜0 ) = −(p− e2 , ρ0 ). Note that p∗− ρ0 = p+ ρ0 = 0, and p− takes the form p− = e−iθ (−i∂r . . .). Hence we have τ = i lim e−iθ e2 ρ0 ds, ds = R dθ, R→∞
|x|=R
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by partial integration. Since ρ0 (x) = r−α + O(r−1−α ) as |x| → ∞ and since e2 (x) = iλ2 eiθ r−1+α + O(r−2+α )
by Lemma 5.1, the desired relation follows from (7.6). t
Proof of Lemma 9.4. We again write e+ = (e1 , e2 ) for e in Lemma 7.3 and e− for e in Lemma 9.5. If 0 < α ≤ 1/2, then e− = t (e1 , −e2 ), and if 1/2 < α < 1, then e− = t (−e1 , e2 ). (1) Assume that 0 < α < 1/2. By Lemmas 8.4 and 9.2, it follows that Iε = ((1 + Z0 )−1 q0 , (1 − Z0 )−1 q0 ) + o(1),
ε → 0.
We further obtain Iε = (V e+ , e− ) + o(1) by Lemmas 7.3 and 9.5. The leading term on the right side equals (V e+ , e− ) = −((K0 − V )e+ , e− )/2,
(9.6)
because (K0 ± V )e± = 0. We assert that ((K0 − V )e+ , e− ) = 4πλ2 ,
(9.7)
which implies that Iε = −2πλ2 + o(1). We shall show (9.7). By definition, ((K0 − V )e+ , e− ) = ((p− e2 − V e1 ), e1 ) − ((p+ e1 − V e2 ), e2 ). We recall that p± = e±iθ (−i∂r . . .) for |x| 1. Hence we have −iθ e e2 e1 − eiθ e1 e2 ds, ((K0 − V )e+ , e− ) = −i lim R→∞
ds = R dθ,
|x|=R
by integration by parts. Thus Lemma 5.1 yields (9.7). (2) Assume that α = 1/2. According to Lemmas 8.4 and 9.2, we have −1
(1 + Xε ) Hence
q0 = (1 + aτ )q + o2 (1),
−1
(1 − Xε∗ )
q0 = (1 + a τ )q + o2 (1).
Iε = (1 + aτ )(1 + a τ )(q, q ) + o(1),
ε → 0.
We repeat the same argument as used in proving (1) to obtain that (q, q ) = ((1 + Z0 )−1 q0 , (1 − Z0 )−1 q0 ) = −2πλ2 . On the other hand, Lemma 9.6, together with (8.5), implies that 1 + aτ = 1 − iτ /(2π + iτ ) = 2π/(2π + iτ ) = (1 − iλ2 )−1 , and similarly 1 + a τ = (1 + iλ2 )−1 (see (9.3)). This proves (2).
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(3) Let 1/2 < α < 1. (3) is verified in almost the same way as (1). Since Qq0 = 0 and P q0 = q0 , it follows from Lemmas 8.4 and 9.2 that (1 + Xε )−1 q0 ∼ δ+ (ε) q0 − Q(Q + QZ0 Q)−1 QZ0 q0 , (1 − Xε∗ )−1 q0 ∼ δ− (ε) q0 + Q(Q − QZ0 Q)−1 QZ0 q0 , and hence we have Iε
= δ+ (ε)δ− (ε)(V e+ , e− ) + o(ε2(2α−1) ) = −δ+ (ε)δ− (ε)((K0 − V )e+ , e− )/2 + o(ε2(2α−1) )
by Lemmas 7.3 and 9.5. Note that e1 behaves like e1 (x) = −(λ1 λ0 /2π)r−α + O(|x|−1−α ),
|x| → ∞,
for the real number λ1 as in Lemma 5.1. Hence the scalar product ((K0 −V )e+ , e− ) is calculated as −iθ ((K0 − V )e+ , e− ) = −i lim −e e2 e1 + eiθ e1 e2 ds = −λ1 λ20 /π (9.8) R→∞
|x|=R
by use of partial integration. As is seen from (8.6) and (9.4), δ± (ε) = 1/µ± (ε) = ∓(γ0 /λ0 )i−2α E 2α−1 ε2α−1 (1 + o(1)), because γ− (k) → −1/γ0 as k = εE → 0. This, together with (9.8), yields the desired asymptotic form.
References [1] R. Adami and A. Teta, On the Aharonov-Bohm Hamiltonian, Lett. Math. Phys. 43, 43–53 (1998). [2] G.N. Afanasiev, Topological Effects in Quantum Mechanics, Kluwer Academic Publishers (1999). [3] Y. Aharonov and D. Bohm, Significance of electromagnetic potential in the quantum theory, Phys. Rev. 115, 485–491 (1959). [4] Y. Aharonov and A. Casher, Ground state of a spin-1/2 charged particle in a two-dimensional magnetic field, Phys. Rev. A 19, 2461–2462 (1979). [5] S. Agmon and L. H¨ ormander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Anal. Math. 30, 1–38 (1976). [6] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, Text and Monographs in Physics, Springer (1988).
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[7] M.G. Alford, J. March-Russell and F. Wilczek, Enhanced baryon number violation due to cosmic strings, Nucl. Phys. B 328, 140–158 (1989). [8] N. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert space, Vol. 2, Pitman (1981). [9] E. Balslev and B. Helffer, Limiting absorption and resonances for the Dirac operators, Adv. in Appl. Math. 13, 186–215 (1992). [10] M.V. Berry, R.G. Chambers, M.D. Large, C. Upstill and J.C. Walmsley, Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue, Eur. J. Phys. 1, 154–162 (1980). [11] A. Berthier and V. Georgescu, On the point spectrum of Dirac operators, J. Func. Anal. 71, 309–338 (1987). [12] D. Boll´e, F. Gesztesy and C. Danneels, Threshold scattering in two dimensions, Ann. Inst. H. Poincar´e 48, 175–204 (1988). [13] L. Dabrowski and P. Stovicek, Aharonov-Bohm effect with δ-type interaction, J. Math. Phys. 39, 47–62 (1998). [14] Ph. de Sousa Gerbert, Fermions in an Aharonov-Bohm field and cosmic strings, Phys. Rev. D 40, 1346–1349 (1989). [15] Y. Gˆ atel and D. Yafaev, Scattering theory for the Dirac operator with a long-range electromagnetic potential, J. Func. Anal. 184, 136–176 (2001). [16] C.R. Hagen, Aharonov-Bohm scattering amplitude, Phys. Rev. D 41, 2015– 2017 (1990). [17] C.R. Hagen, Aharonov-Bohm scattering amplitude with spin, Phys. Rev. Lett. 64, 503–506 (1990). [18] H.T. Ito, High-energy behavior of the scattering amplitude for a Dirac operator, Publ. RIMS. Kyoto Univ. 31, 1107–1133 (1995). [19] U. Percoco and V.M. Villalba, Aharonov-Bohm effect for a relativistic Dirac electron, Phys. Lett. A 140, 105–107 (1989). [20] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II, Fourier Analysis, Self-Adjointness, Academic Press (1975). [21] Y.A. Sitenko, Self-adjointness of the two-dimensional massless Dirac Hamiltonian and vacuum polarization effects in the background of a singular magnetic vortex, Ann. Phys. 282, 167–217 (2000). [22] H. Tamura, Norm resolvent convergence to magnetic Schr¨ odinger operators with point interactions, Rev. Math. Phys. 13, 465–512 (2001).
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[23] H. Tamura, Resolvent convergence in norm for Dirac operator with Aharonov-Bohm field, J. Math. Phys. 44, 2967–2993 (2003). [24] B. Thaller, Dirac Equations, Texts and Monographs in Physics, Springer (1992). [25] O. Yamada, On the principle of limiting absorption for the Dirac operator, Publ. RIMS. Kyoto Univ. 8, 557–577 (1972/73). Hideo Tamura Department of Mathematics Okayama University Okayama 700–8530 Japan email:
[email protected] Communicated by Bernard Helffer submitted 05/05/03, accepted 31/07/03
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 119 – 133 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/010119-15 DOI 10.1007/s00023-004-0162-z
Annales Henri Poincar´ e
On the Flux-Across-Surfaces Theorem for Short-Range Potentials Takeyuki Nagao Abstract. The Flux-Across-Surfaces theorem is established for three-dimensional Schr¨ odinger equation with short-range potentials V satisfying the decay condition |∂xα V (x)| ≤ Cα (1 + |x|)−σ , |α| ≤ 2 for some σ > 3. Exceptional cases are treated as well and the required decay index is σ > 5. Explicit conditions for the initial states are found. A stationary representation for the integrated flux is obtained and exploited in the proof. Spatial asymptotic expansion of the resolvent is employed to calculate the limit of the integrated flux at spatial infinity.
1 Introduction Particles moving in the three-dimensional Euclidean space under the influence of the potential are described by the time-dependent Schr¨ odinger equation i∂t u(t, x) = −u(t, x) + V (x)u(t, x). Choosing a scattering state as the initial condition, we consider the scattering for this equation. Namely, u(0, x) = f (x) with f belonging to the absolutely continuous subspace Hac (H) of the Hamiltonian H = − + V . The principal object of this paper is to prove the Flux-Across-Surfaces theorem, which provides us an alternative way of defining the scattering cross section in terms of the integrated flux for the particle. Given a cone K with vertex at the origin, we take a section Σρ = K ∩ {|x| = ρ} and define the integrated flux Iρ (f ) across this surface by the integral ∞ Iρ (f ) = J · n dSdt, (1.1) 0
Σρ
where J = 2Im(u∇u) is the quantum flux density and n is the outward unit normal on the surface Σρ . The integrated flux is interpreted as the expectation of the particle crossing the surface Σρ at sometime in the future and its limit at infinity limρ→∞ Iρ (f ) can be a reasonable substitute for the definition of the scattering cross section as long as the following limit relation is valid with F+ denoting the generalized Fourier transform associated with the Schr¨odinger operator H (cf. [7]). lim Iρ (f ) = F+ f 2L2 (K) .
ρ→∞
(1.2)
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The above identity is called the Flux-Across-Surfaces theorem, or the FAS theorem for short, and proved under various assumptions, including the free (V = 0) [5, 7], short-range [4, 6, 18], and long-range cases [3]. The usual proofs, however, require rather implicit assumptions on the initial state. As is pointed out by Dell’AntonioPanati [6], it is preferable to establish the FAS theorem under explicit decay and smoothness assumptions on the initial state. In this paper, we shall prove the FAS theorem for short-range potentials and formulate the condition on the initial state in terms of the weighted Sobolev spaces H m,s (R3 ) defined by the norm f H m,s (R3 ) = (1 − )m/2 (1 + |x|2 )s/2 f L2 (R3 ) . The potential V = V (x) is supposed to be a bounded real-valued function on R3 and satisfy the following condition with decay index at least σ > 3. |∂xα V (x)| ≤ C(1 + |x|)−σ ,
|α| ≤ 2.
(1.3)
Due to the possible existence of zero eigenvalues or zero resonance and the resulting singular temporal asymptotics of the propagator, we shall separate the problem into two cases. We factor the potential into the product V = vw with v(x) = |V (x)|1/2 and consider v and w as multiplication operators. The integral operator with kernel 1/(4π|x − y|) is denoted by G0 . By our assumption on the potential, the operator vG0 w is compact in L2 (R3 ). Definition 1.1 The case where 1 + vG0 w has a bounded inverse in L2 (R3 ) is called generic and the case where 1 + vG0 w has no bounded inverse in L2 (R3 ) is called exceptional. The main result of this paper is the following theorem. We state the theorem for the generic case and the statement for the exceptional case is enclosed by the parentheses. Theorem 1.1 Suppose that the potential V satisfies the decay condition (1.3) with σ > 3 (σ > 5). Then, for each f ∈ Hac (H) ∩ H 2,s (R3 ) with s > 3/2 (s > 5/2), we have lim Iρ (f ) = F+ f 2L2 (K) . ρ→∞
In our analysis, we will employ a stationary representation for the integrated flux Iρ (f ) and analyse the integrated flux by investigating the trace of the resolvent onto the sphere. We remark that the integrated flux can be written as ∞ 2Im∂r e−itH f |e−itH f L2 (Σρ ) dt, (1.4) Iρ (f ) = 0
where ∂r = |x|−1 x · ∂x is the radial derivative and f |gL2 (Σρ ) = Σρ f g¯dS with dS the surface measure on the surface Σρ . The trace operator τρ onto the sphere
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Sρ = {|x| = ρ} will be omitted in various places. The first step in our approach is to establish the following representation, describing the integrated flux by means of the resolvents. ∞ Iρ (f ) = 2Im−i∂r R+ (λ)f |EH (λ)f L2 (Σρ ) dλ. (1.5) 0
As usual, R± (λ) are the boundary values of the resolvent R(z) = (H − z)−1 at λ > 0 from the upper (+) and the lower (−) half-planes. The spectral density EH is given by the formula EH (λ) = R+ (λ) − R− (λ) /(2πi). (1.6) In the generic case, the expression (1.5) is verified by showing the H-smoothness of the operator τρ Pac , where Pac is the projection onto the absolutely continuous spectrum of H. Because of the singularity of the resolvent at zero energy, τρ Pac may not be H-smooth in the exceptional case and we use a different method to justify the representation in this case. Making use of the spatial asymptotic expansion of the resolvent, we calculate the limit of the integrated flux limρ→∞ Iρ (f ). Specifically, we show that √ (∂r ∓ i λ)R± (λ)f L2 (Sρ ) → 0 as ρ → ∞ and by the asymptotic expansion of the resolvent we obtain the following formulae, relating the limiting behavior of the resolvent at spatial infinity with the generalized Fourier transform. ∞√ 2 lim λ R+ (λ)f L2 (Σρ ) dλ = πF+ f 2L2 (K) , ρ→∞
0
lim
ρ→∞
0
∞
√ λR± (λ)f |R∓ (λ)f L2 (Σρ ) dλ = 0.
The above three limit relations lead to the FAS theorem. Major benefit of our approach is that the above limit relations are valid also for the exceptional case, and the main difference between the generic and exceptional cases resides in establishing the stationary representation for the integrated flux, which permits us to reduce the analysis of the integrated flux into the trace estimates of the resolvent.
2 Generic case In this section, we shall prove the FAS theorem in the generic case. The FAS theorem for the unperturbed system is included in this case. Actually, most of the analysis are devoted to the investigation of the free system and the results for the perturbed system are deduced by simple perturbative arguments.
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Throughout this section, we suppose that the potential V = V (x) satisfies the following decay condition with a constant σ > 3. |∂xα V (x)| ≤ C(1 + |x|)−σ ,
|α| ≤ 2.
(2.1) 2
3
Further, we suppose that the operator 1+vG0 w is invertible in L (R ) (see Section 1 for the definition of this operator). Under these assumptions, H is selfadjoint with the same domain H 2 (R3 ) as the free Hamiltonian H0 = −, its absolutely continuous spectrum is σac (H) = σ(H0 ) = [0, ∞), H has no positive eigenvalues, and the singular continuous spectrum is absent. It is well known (cf. [1, 10, 15]) that as a function of z, the resolvent R(z) = (H − z)−1 is continuous in each closed quadrant Ω± = {z ∈ C : Rez ≥ 0, ±Imz ≥ 0} in the topology of B(L2s (R3 ); H 2,−s (R3 )) with any weight s > 1, where B(X; Y ) denotes the totality of bounded operators from a Banach space X into another Banach space Y with operator topology and L2s (R3 ) = H 0,s (R3 ). On several occasions we denote by R± (λ) = limε↓0 R(λ ± iε) the boundary values of the resolvent at λ > 0 and this notation is also used for the free resolvent R0 (z) = (H0 − z)−1 . Including these boundary values, the resolvent satisfies the estimate R(z)f H m,−s (R3 ) ≤ Cs z(m−1)/2 f L2s (R3 ) ,
z ∈ Ω± , 0 ≤ m ≤ 2
(2.2)
for s > 1 with z = (1 + |z|2 )1/2 and the resolvent identities R(z) − R0 (z) = −R0 (z)V R(z) = −R(z)V R0 (z)
(2.3)
hold for z ∈ Ω± as identities for operators in B(L2s (R3 ); H 2,−s (R3 )) with s > 1. We remark that the estimate (2.2) is, in general short-range scattering, only valid for z away from the origin (we only need s > 1/2 for this. See, e.g., [1]), but for the generic case we are dealing with, (2.2) is valid with all z ∈ Ω± if s > 1 (cf. [10, 17]). It is easy to see from the estimate (2.2) and the differentiability of the potential that V R(z)f H 2,s (R3 ) ≤ Cs f H 2,s (R3 ) ,
z ∈ Ω± , s > 1.
(2.4)
The weighted resolvent estimates (2.2) and (2.4) enable us to reduce the analysis to the unperturbed problem via the resolvent identity (2.3), and to handle the reduced problem we shall use the integral representation of the free resolvent R0 (z) = (H0 − z)−1 . For z from each Ω± , we denote by K ± (x, y) = K ± (z; x, y) the integral kernel of R0 (z), viz. √ e±i z|x−y| ± ± . (2.5) K (z; x, y)f (y)dy, K (z; x, y) = R0 (z)f (x) = 4π|x − y| R3 First of all, we state and prove the most important lemma in this paper, which handles integral operators similar to the free resolvent and gives us the decay ratio of the trace of such operators at spatial infinity.
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kernel K(x, y) of the integral operator T f (x) = Lemma 2.1 Assume that the 3 K(x, y)f (y)dy, x ∈ R satisfies |K(x, y)| ≤ C|x − y|−α for some constant 3 R α with 0 ≤ α < 5/2. Then for any s > 3/2 there exists a constant Cs such that the following inequality holds for any ρ > 1. T f L2(Sρ ) ≤ Cs ρ1−α f L2s(R3 ) . Proof. We define u1 (x), u2 (x) by K(x, y)f (y)dy, u1 (x) = |y|<|x|/2
u2 (x) =
K(x, y)f (y)dy, |y|>|x|/2
and show that they satisfy uj L2 (Sρ ) ≤ Cs ρ1−α f L2s(R3 ) , (j = 1, 2). First, we consider u1 (x). If |y| < |x|/2, then |x − y| > |x|/2, thus |u1 (x)| ≤ C|x|−α f L1 . Taking the square of the both sides and integrating over the spere |x| = ρ, we obtain u1 2L2 (Sρ ) ≤ Cρ2−2α f 2L1 ≤ Cs ρ2−2α f 2L2(R3 ) . Next, we estimate u2 (x). s Choose positive constants µ, ν in such a way that µ + ν = 1, 2αµ < 1, αν < 2. 2
2µ
Since, |x − y|2 = (|x| − |y|) + 2|x||y|(1 − x ˆ · yˆ) ≥ ||x| − |y|| with x ˆ = |x|−1 x, we see that −αν
|u2 (x)| ≤ C|x|
|y|>|x|/2
||x| −
|f (y)| dy. −x ˆ · yˆ)αν/2
|y||αµ (1
By the Schwarz inequality, we have |u2 (x)|2 ≤ C|x|−2αν g1 (x)
y2s |f (y)|2 dy, (1 − x ˆ · yˆ)αν/2
R3
with g1 defined by g1 (x) =
|y|>|x|/2
y−2s dy. ||x| − |y||2αµ (1 − x ˆ · yˆ)αν/2
Using the polar coordinate, g1 can be written as g1 (x) = βg2 (x), with g2 (x) =
∞
|x|/2
τ 2 (1 + τ 2 )−s dτ, ||x| − τ |2αµ
β= |y|=1
(1 − x ˆ · yˆ)−αν/2 dS(y).
By changes of variable, we see that g2 (x) ≤ |x|3−2s−2αµ
∞
1/2
τ 2−2s dτ, |1 − τ |2αµ
1
β = 2π −1
ν
[2|x||y|(1 − x ˆ · yˆ)]
(1 − t)−αν/2 dt
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and these integrals are finite if 2αµ < 1, αν < 2 and s > 3/2. Note that β is independent of x. Combining the above computations, we have y2s |f (y)|2 2 3−2s−2α |u2 (x)| ≤ C|x| dy ˆ · yˆ)αν/2 R3 (1 − x and integrating the both sides over the sphere |x| = ρ, we see that u2 2L2 (Sρ ) ≤ Cρ5−2s−2α f 2L2s (R3 ) ≤ Cρ2−2α f 2L2s (R3 ) as claimed. Applying Lemma 2.1 for the free resolvent, we obtain the following trace estimate which is uniform in the radius of the sphere. τρ R0 (z)f L2(Sρ ) ≤ Cs f L2s (R3 ) ,
s > 3/2, z ∈ Ω± , ρ > 1.
(2.6)
More detailed results will be obtained later in Proposition 2.2 for the generic case and in Proposition 3.1 for the exceptional case. Now, we are in a position to prove the smoothness of τρ . At this point, we remark that the classical trace estimate τρ f L2 (Sρ ) ≤ Cα f H α (R3 ) ,
α > 1/2,
(2.7)
is valid with a uniform constant Cα , where the uniformity refers to ρ > 1. One can deduce this uniform inequality from the estimate (2.2) applied to the free resolvent and the formula below, relating the trace with the spectral density of the free resolvent via the Fourier transform. |τρ f (x)|2 dS(x) = 2ρEH (ρ2 )fˆ|fˆL2 (R3 ) . 0 |x|=ρ
Lemma 2.2 For large γ > 0: (1) τρ is H0 -smooth. (2) τρ Pac is H-smooth. (3) τρ ∂r (H + γ)−1/2 is H0 -smooth. (4) τρ ∂r (H + γ)−1/2 Pac is H-smooth. Proof. (1) It suffices to show that τρ R0 (z)τρ∗ gL2 (Sρ ) ≤ CρgL2 (Sρ ) ,
z ∈ Ω± ,
(2.8)
where the adjoint τρ∗ is taken with respect to the inner product of L2 (Sρ ). This is equivalent to the following estimate. K ± (x, y)g(y)dS(y) ≤ CρgL2 (Sρ ) , z ∈ Ω± . |y|=ρ 2 L (Sρ )
ˆ · yˆ), so If |x| = |y| = ρ, |x − y|2 = 2ρ2 (1 − x (2.8) follows from Young’s inequality.
|y|=ρ
|K ± (x, y)|dS(y) ≤ ρ. Hence,
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(2) In view of (2.8), we only have to prove τρ [R(z) − R0 (z)] τρ∗ gL2 (Sρ ) ≤ CgL2 (Sρ ) ,
z ∈ Ω± , ρ > 1.
(2.9)
This is an easy consequence of the identity R(z) − R0 (z) = −R0 (z)V R0 (z) + R0 (z)V R(z)V R0 (z),
(2.10)
combined with (2.2) and (2.6). (3) Set A = τρ ∂r (H + γ)−1/2 and A0 = τρ ∂r (H0 + γ)−1/2 . It is easy to see from (1) that A0 is H0 -smooth. Since x−s is H0 -smooth if s > 1, we only have to show that A − A0 has the following decomposition. A − A0 = Mρ x−σ ,
sup Mρ B(L2 (R3 );L2 (Sρ )) < ∞, ρ>1
(2.11)
where σ is the decay index for the potential. By our assumption on the potential, B = (H0 + γ) (H + γ)−1/2 − (H0 + γ)−1/2 xσ is a bounded operator in L2 (R3 ), so if we set Mρ = τρ ∂r (H0 + γ)−1 B, we see from the trace theorem (see (2.7) and the remark for it) that Mρ is bounded with norm uniformly bounded by a constant for ρ > 1. (4) We use the same A and A0 as in (3). Notice that (3) is equivalent to |ImAR0 (z)A∗ g|gL2 (Sρ ) | ≤ C(1 + ρ)g2L2 (Sρ ) ,
z ∈ Ω± , ρ > 1.
Hence, from (2.10) and (2.2), it is enough to show that AR0 (z)f L2 (Sρ ) ≤ Cs f L2s(R3 ) ,
s > 3/2, z ∈ Ω± , ρ > 1.
(2.12)
Observe that by (2.6), (2.12) is true if we replace A by A0 . To finish the proof, we use the decomposition (2.11) and the estimate (2.2), which is also valid for the free resolvent. As a consequence of the above lemma, the integrated flux Iρ (f ) is well defined as a function of f ∈ Hac (H) ∩ H 1 (R3 ) and it is continuous in the norm topology of H 1 (R3 ). Indeed, H-smoothness of τρ Pac is equivalent to ∞ τρ e−itH f 2L2 (Sρ ) dt ≤ Cρ f 2L2 (R3 ) , f ∈ Hac (H) −∞
and H-smoothness of τρ ∂r (H + γ)−1/2 Pac is equivalent to ∞ τρ ∂r e−itH f 2L2 (Sρ ) dt ≤ Cρ f 2H 1 (R3 ) , f ∈ Hac (H) ∩ H 1 (R3 ). −∞
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Hence the integral (1.4) is absolutely convergent and the integrated flux satisfies the estimate |Iρ (f )| ≤ Cρ f 2H 1 (R3 ) . The Plancherel theorem applied to Hilbert space-valued functions gives the identity (1.5). We note that the mappings t → τρ e−itH f and t → τρ ∂r e−itH f are L2 functions with values in L2 (Sρ ) if f ∈ Hac (H)∩H 1 (R3 ) and that the mappings λ → τρ R± (λ)f and λ → τρ ∂r R± (λ)f belong to L2 ((0, ∞); L2 (Sρ )) by the Fourier transform with respect to the variable t. To summarize the above consideration, we have obtained the following Proposition 2.1 The integrated flux Iρ (f ) is a well-defined function of f ∈ Hac (H) ∩H 1 (R3 ) and continuous in the norm topology of H 1 (R3 ). Moreover, the following identity holds if f belongs to Hac (H) ∩ H 1 (R3 ). ∞ Iρ (f ) = 2Im−i∂r R+ (λ)f |EH (λ)f L2 (Σρ ) dλ. (2.13) 0
Although the integrated flux satisfies the estimate |Iρ (f )| ≤ Cρ f 2H 1 (R3 ) , this is not satisfactory for our purpose since we are interested in the limiting behavior of the flux Iρ (f ) as ρ → ∞. It is obvious from the proof of Lemma 2.2 that the constant Cρ can be chosen as a linear function of ρ. Scaling argument shows that this is the correct order as ρ → ∞ for the unperturbed problem. In order to gain uniform control over the integrated flux Iρ (f ), we shall estimate the trace of the resolvent by means of Lemma 2.1. Proposition 2.2 For any s > 3/2 there exists a constant Cs > 0 such that for all z ∈ Ω± , ρ > 1, multi-indices α with |α| ≤ 1, and f ∈ H 2,s (R3 ) we have ∂xα R(z)f L2 (Sρ ) ≤ Cs z|α|/2−1 f H 2,s (R3 ) .
(2.14)
Proof. The proof is reduced to the unperturbed case, since by the resolvent identity (2.3) we have the decomposition R(z)f = R0 (z)gz ,
gz = f − V R(z)f,
(2.15)
and from (2.4), gz H 2,s (R3 ) ≤ Cf H 2,s (R3 ) with the constant C independent of z. We only prove the case for α = 0, since the other case |α| = 1 can be handled similarly. For the sake of simplicity, we suppose z ∈ Ω+ . As we have already seen in (2.6), the estimate (2.14) is true if we remove the decay factor z−1 , so we may assume that |z| > 1. In the polar coordinate, the formula (2.5) can be written as ∞ i√zr re f (x − rω)dS(ω)dr. (2.16) R0 (z)f (x) = 4π 0 |ω|=1 Multiplying the both sides by z and integrating by parts twice, we see that
|∂j ∂k f (y)| |∇f (y)| |zR0 (z)f (x)| ≤ C |f (x)| + dy . dy + 2 |x − y| R3 |x − y| R3 j,k=1,2,3
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Taking the · L2 (Sρ ) norm of both sides and applying the trace estimate (2.7) and Lemma 2.1, we have R0 (z)f L2 (Sρ ) ≤ Cz−1 f H 2,s (R3 ) with C independent of ρ > 1 and z. Also as a consequence of Lemma 2.1, we obtain the first one of the three limit relations claimed in the introduction. Proposition 2.3 Assume that f ∈ H 2,s (R3 ) for some constant s > 3/2. Then, we have √ (2.17) lim (∂r ∓ i λ)R± (λ)f L2 (Sρ ) = 0, λ > 0. ρ→∞
Proof. By the same argument as in the proof of Proposition 2.2, we reduce the proof to the free system. We only deal with the case of R0+ . The integral kernel of the operator ∂r R0+ (λ) is given by
√ 1 (x − y) · x . Kr (λ; x, y) = G(x, y) i λ − K + (λ; x, y), G(x, y) = |x − y| |x − y||x| Notice that the function G satisfies the inequalities |y|2µ |y|2 0 ≤ 1 − G(x, y) ≤ min 2, ≤ 21−2µ µ 2|x||x − y| |x| |x − y|µ
(2.18)
for any µ with 0 ≤ µ ≤ 1. We can prove (2.17) by this simple observation. Now, split the kernel Kr into the sum Kr = K1 + K2 with √ K1 (λ; x, y) = i λG(x, y)K + (λ; x, y) and let T be the operator with integral kernel K1 . Since |K2 (λ; x, y)| ≤ C|x− y|−2 , we can neglect K2 in view of Lemma 2.1 and it is enough to show the following identity. √ lim T − i λR0+ (λ) f 2 = 0. (2.19) ρ→∞
L (Sρ )
Choose the constants µ, σ in such a way that σ + 2µ ≤ s (2.20) √ + and decompose the function in (2.19) as [T −i λR0 (λ)]f = Sg, where the integral kernel K3 of the operator S and the function g are defined by √ K3 (λ; x, y) = i λ[1 − G(x, y)]K + (λ; x, y)y−2µ , g(y) = y2µ f (y). 0 < µ < 1,
σ > 3/2,
From (2.18) and (2.20) we have |K3 (λ; x, y)| ≤ C|x|−µ |x − y|−1−µ ,
gL2σ (R3 ) ≤ Cf L2s (R3 ) ,
and thus, by Lemma 2.1, we see that as ρ → ∞, √ [T − i λR0+ (λ)]f L2 (Sρ ) = SgL2(Sρ ) ≤ Cρ−2µ gL2,σ (R3 ) → 0 .
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Computation of the limit limρ→∞ Iρ (f ) is based on the spatial asymptotic expansion of the resolvent. We shall state and prove a version of asymptotic expansion lemma suitable for our purpose. Some notations from scattering theory are required. Let F0 be the Fourier transform and F0 (λ) be its trace onto the unit sphere, namely, F0 f (ξ) = fˆ(ξ) = (2π)−3/2 e−ixξ f (x)dx, R3
√ F0 (λ)f (ω) = 2−1/2 λ1/4 (F0 f )( λω),
|ω| = 1, λ > 0.
We define the generalized Fourier transform F± by setting (F± f )(ξ) = 21/2 |ξ|−1/2 (F± (ξ 2 )f )(ξ/|ξ|), F± (λ)f = F0 (λ)[1 − V R± (λ)].
(2.21)
It is well known (see [1], for example) that for short-range potentials, F± is well defined as a partial isometry on L2 (R3 ) and its trace F± (λ) is a bounded operator from L2s (R3 ) to L2 (S1 ) for s > 1/2. Actually, the mapping f → F± (λ)f can be extended to a bounded operator from L2 (R3 ) to L2 (0, ∞; L2 (S1 )) satisfying the equality ∞
0
F± (λ)f 2L2 (Σ1 ) dλ = F± f 2L2 (K)
(2.22)
for any cone K and its section Σ1 by the unit sphere. Lemma 2.3 Suppose that f ∈ H 2,s (R3 ) for some constant s > 3/2. Then, the function R± (λ)f (x) can be decomposed as R± (λ)f (x) = u± (f ; x, λ) + E± (f ; x, λ), u± (f ; x, λ) =
√
πλ−1/4
e
(2.23)
√ ±i λr
(F± (λ)f ) (±ω), r where r = |x|, ω = |x|−1 x and the remainder term E± (f, x, λ) satisfies ∞√ lim λE± (f ; ·, λ)2L2 (Sρ ) dλ = 0. ρ→∞
(2.24)
(2.25)
0
Proof. By Proposition 2.2, we have √ λR± (λ)f 2L2 (Sρ ) ≤ Cλ−3/2 f 2H 2,s (R3 ) ,
ρ>1
and by the definition of u± we have the identity √ λu± (f ; ·, λ)2L2 (Sρ ) = πF± (λ)f 2L2 (S1 ) .
(2.26)
Hence the integrand in (2.25) is uniformly bounded by an L1 function and the integral is uniformly bounded by f 2H 2,s (R3 ) . Here, the uniformity refers to ρ > 1.
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Therefore, by the dominated convergence, we only have to show the pointwise decay E± (f ; ·, λ)L2 (Sρ ) → 0 as ρ → ∞
(2.27)
for each λ > 0 and smooth rapidly decreasing functions f . In order to prove this, we apply the same reduction argument as in the proof of Proposition 2.2. For the free resolvent, it is well known that the error term E± has the decay E± (f ; x, λ) = O(|x|−2 ) as |x| → ∞, hence (2.27) follows by integration. The remaining two limit relations are direct consequence of Lemma 2.3. Proposition 2.4 Assume that f ∈ H 2,s (R3 ) for some constant s > 3/2. Then the following limit relations hold with K ± = {x ∈ R3 : ±x ∈ K}.
∞
lim
ρ→∞
0
√ ± 2 λ R (λ)f L2 (Σρ ) dλ = πF± f 2L2 (K ± ) ,
lim
ρ→∞
0
∞
√ λR± (λ)f |R∓ (λ)f L2 (Σρ ) dλ = 0.
(2.28)
(2.29)
Proof. By Lemma 2.3, we can replace R± (λ)f by u± (λ) = u± (f ; ·, λ). (2.28) follows from the identity √ λu± (λ)2L2 (Σρ ) = πF± (λ)f (±·)2L2 (Σ1 ) .
(2.30)
(2.29) follows from the identity √ √ λu± (λ)|u∓ (λ)L2 (Σρ ) = πe±2iρ λ F± (λ)f (±·)|F∓ (λ)f (∓·)L2 (Σ1 )
(2.31)
and the Riemann-Lebesgue lemma. Finally, we put all the above propositions together to prove the FAS theorem. Proof of Theorem 1.1. By Proposition 2.2, the integrand in (2.13) is uniformly bounded by Cλ−3/2 f 2H 2,s (R3 ) , which is integrable in λ on the half-line. Therefore, from Proposition 2.3 and the identity (1.6), followed by Proposition 2.4, we conclude that ∞√ λ + R (λ)f 2L2 (Σρ ) dλ = F+ f 2L2 (K) . lim Iρ (f ) = lim ρ→∞ ρ→∞ 0 π
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3 Exceptional case In this section, we shall prove the FAS theorem in the exceptional case. The decay index for the potential is taken to be σ > 5 and we suppose that 1 + vG0 w has no bounded inverse in L2 (R3 ). The operator τρ Pac may not be H-smooth in this case, since the resolvent R(z) has a singularity at z = 0. Hence, we take another approach to prove the representation (1.5). After establishing this identity, we can proceed exactly in the same way as in the generic case except for the increased weight s > 5/2. According to the work of Jensen-Kato [10], we have the following two propositions, describing the low-energy asymptotics of the resolvent and the long-time asymptotics of the propagator. The topology used here is slightly different from that of [10], but one can prove them in a similar manner as in [10]. Proposition 3.1 For any s > 5/2, we have for small z ∈ Ω+ \ {0}, ˜1 (z). R(z) = −z −1 B−2 − iz −1/2 B−1 + B0 + B
(3.1)
The operators B−2 to B0 belong to B(L2,s (R3 ); H 2,−s (R3 )) and the remainder term satisfies ˜ (3.2) B1 (z) 2,s 3 2,−s 3 = O(z 1/2 ), z → 0. B(L
(R );H
(R ))
Moreover, we have the following expressions for the first two coefficients. P0 denotes the projection onto the zero eigenspace of H, G3 is the operator with integral kernel |x − y|2 /(24π), and ψ is the resonant function for H. B−2 = P0 ,
B−1 = P0 V G3 V P0 − ·|ψψ.
(3.3)
A similar result holds for z ∈ Ω− . Proposition 3.2 For any s > 5/2, we have ˜ e−itH Pac = −(πit)−1/2 B−1 + D(t).
(3.4)
B−1 is the same operator as in Proposition 3.1 and the remainder term satisfies −3/2 ˜ D(t) ), B(L2,s (R3 );H 2,−s (R3 )) = O(t
t → ∞.
(3.5)
As we have seen in the proof of Proposition 2.2, the trace estimate (2.14) is a consequence of the inequality (2.4) and the estimate (2.14) for the free resolvent. From Proposition 3.1, we have for f ∈ Hac (H) ∩ H 2,s (R3 ) with s > 5/2, −1/2
V R(z)f H 2,s (R3 ) ≤ Cs z0
f H 2,s (R3 ) ,
z ∈ Ω± \ {0},
(3.6)
where we denote by z0 the function which equals z for |z| ≤ 1 and equals 1 for |z| > 1. This leads to the following trace estimate for the resolvent.
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Proposition 3.3 For any s > 5/2 there exists a constant Cs > 0 such that for all z ∈ Ω± \ {0}, ρ > 1, multi-indices α with |α| ≤ 1, and f ∈ Hac (H) ∩ H 2,s (R3 ) we have −1/2 ∂xα R(z)f L2 (Sρ ) ≤ Cs z0 z|α|/2−1 f H 2,s (R3 ) . (3.7) The stationary representation (1.5) follows from Propositions 3.1 to 3.3 and the spectral decomposition. Proposition 3.4 Let s > 5/2. Then, the integrated flux Iρ (f ) is a well-defined function of f ∈ Hac (H) ∩ H 2,s (R3 ) and continuous in the norm topology of H 2,s (R3 ). Moreover, the following identity holds if f belongs to Hac (H) ∩ H 2,s (R3 ). ∞ 2Im−i∂r R+ (λ)f |EH (λ)f L2 (Σρ ) dλ. (3.8) Iρ (f ) = 0
Proof. First, we show that the integral (1.4) is absolutely convergent. The integrand is bounded by a constant for finite t if f ∈ H 2 (R3 ). Hence, in view of Proposition 3.2, it is enough to show that Im∂r B−1 f |B−1 f L2 (Σρ ) = 0.
(3.9)
This follows from the fact that the resonant function ψ is real-valued and that B−1 f (x) = cψ(x) for a constant c independent of x, if f ∈ Hac (H) ∩ H 2,s (R3 ). Similarly, one can show that the integral (3.8) is absolutely convergent. Now, we prove the identity (3.8). It is easy to see from the spectral decomposition that for f ∈ Hac (H) ∩ H 2,s (R3 ), ∞ −itH −itH 2Im∂r e f |e f L2 (Σρ ) = 2Im∂r e−itH f |EH (λ)f L2 (Σρ ) dλ. (3.10) 0
The integral on the right-hand side is absolutely convergent and its value is bounded by a constant independent of t. Multiplying the both sides of (3.10) by e−εt and integrating over t, we see that ∞ Iρ (f ) = lim 2Im−i∂r R(λ + iε)f |EH (λ)f L2 (Σρ ) dλ. ε↓0
0
From Proposition 3.3 and a similar argument as above about the resonant function, we see that the integrand is bounded by an L1 function of λ independent of ε. Therefore, (3.8) follows by the dominated convergence. One can use the resolvent estimate (3.7) to prove Propositions 2.3 and 2.4 for the exceptional case. All weights in the statements should be replaced by s > 5/2 and the condition f ∈ H 2,s (R3 ) in Proposition 2.4 by f ∈ Hac (H) ∩ H 2,s (R3 ), but the proofs remain unchanged. The proof of Theorem 1.1 is exactly the same as in the generic case.
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Acknowledgments The author is grateful to Prof. Kenji Yajima for helpful suggestions and constructive remarks, and also to Prof. Detlef D¨ urr for interesting discussion on Bohmian mechanics. Part of this work was done during the author’s stay at Mathematisches Institut Ludwig-Maximilians-Universit¨ at M¨ unchen as a program student and the author expresses his sincere gratitude to Prof. Heinz Siedentop for the hospitality.
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[13] C.E. Kenig, A. Ruiz, and C.D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55, 329–347 (1987). [14] T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys. 1, 481–496 (1989). [15] S.T. Kuroda, An introduction to scattering theory, Aarhus Universitet Matematisk Institut, Aarhus (1978). [16] M. Murata, Asymptotic expansions in time for solutions of Schr¨ odinger-type equations, J. Funct. Anal. 49, 10–56 (1982). [17] I. Rodnianski and W. Schlag, Time decay for solutions of Schr¨ odinger equations with rough and time-dependent potentials, preprint, mp arc 01-369 (2001). [18] S. Teufel, D. D¨ urr, and K. M¨ unch-Berndl, The flux-across-surfaces theorem for short range potentials and wave functions without energy cutoffs, J. Math. Phys. 40, 1901–1922 (1999). [19] D. Yafaev, On solutions of the Schr¨ odinger equation with radiation conditions at infinity, Estimates and asymptotics for discrete spectra of integral and differential equations (Leningrad, 1989–1990), Adv. Soviet Math. 7, 179– 204 (1991). odinger operators. [20] K. Yajima, The W k,p -continuity of wave operators for Schr¨ III. Even-dimensional cases m ≥ 4, J. Math. Sci. Univ. Tokyo 2, 311–346 (1995). Takeyuki Nagao Graduate School of Mathematical Sciences University of Tokyo 3-8-1 Komaba, Meguro-ku Tokyo 153-8914 Japan email:
[email protected] Communicated by Gian Michele Graf Submitted 01/03/03, accepted 30/05/03
Ann. Henri Poincar´e 5 (2004) 135 – 168 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/010135-34 DOI 10.1007/s00023-004-0163-y
Annales Henri Poincar´ e
On the Plancherel Formula for the (Discrete) Laplacian in a Weyl Chamber with Repulsive Boundary Conditions at the Walls∗ J.F. van Diejen
Abstract. It is known from early work of Gaudin that the quantum system of n Bosonic particles on the line with a pairwise delta-potential interaction admits a natural generalization in terms of the root systems of simple Lie algebras. The corresponding quantum eigenvalue problem amounts to that of a Laplacian in a convex cone, the Weyl chamber, with linear homogeneous boundary conditions at the walls. In this paper we study a discretization of this eigenvalue problem, which is characterized by a discrete Laplacian on the dominant cone of the weight lattice endowed with suitable linear homogeneous conditions at the boundary. The eigenfunctions of this discrete model are computed by the Bethe Ansatz method. The orthogonality and completeness of the resulting Bethe wave functions (i.e., the Plancherel formula) turn out to follow from an elementary computation performed by Macdonald in his study of the zonal spherical functions on p-adic simple Lie groups. Through a continuum limit, the Plancherel formula for the ordinary Laplacian in the Weyl chamber with linear homogeneous boundary conditions is recovered. Throughout this paper we restrict ourselves to the case of repulsive boundary conditions.
1 Introduction It is well known that the quantum eigenvalue problem for n Bosons on the line that interact pairwise through a delta-potential can be solved by the Bethe Ansatz method [LL, Mc, BZ, Y1, Y2, G1, G2, O]. From a physical point of view, this manybody system describes the n-particle sector of the quantized nonlinear Schr¨ odinger field theory (i.e., the quantum NLS). For an overview of the literature concerning both the mathematical and physical aspects of this model we refer to the collections [M, G4, KBI]. The Hamiltonian of the n-particle system in question is given by the Schr¨ odinger operator H = −∆ + g
δ(xj − xk ),
(1.1)
1≤j=k≤n ∗ Work supported in part by the Fondo Nacional de Desarrollo Cient´ ıfico y Tecnol´ ogico (FONDECYT) Grant # 1010217 and the Programa Formas Cuadr´ aticas of the Universidad de Talca.
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where x1 , . . . , xn denote the position variables, ∆ = ∂x21 + · · · + ∂x2n , δ(·) refers to the delta distribution, and g represents a real coupling parameter determining the strength of the interaction. For g > 0 the pairwise interaction is repulsive and for g < 0 it is attractive. Mathematically, the eigenvalue problem for H (1.1) amounts to that of a free Laplacian −∆ with jump conditions on the normal derivative of the (continuous) wave function at the hyperplanes xj = xk , 1 ≤ j < k ≤ n. (Specifically, the jump of the normal derivative of the wave functions at the hyperplanes should be 2g times the value of the wave function.) By exploiting the permutation and translational symmetry, the eigenvalue problem at issue can be reduced to the form −∆ψ = ξ2 ψ
(where ξ2 := ξ12 + · · · + ξn2 ),
(1.2a)
for a domain of wave functions ψ = ψ(x; ξ) := ψ(x1 , . . . , xn ; ξ1 , . . . , ξn ) supported in the closure of the fundamental convex cone C = {x ∈ Rn | x1 > x2 > · · · > xn , x1 + · · · + xn = 0},
(1.2b)
and subject to linear homogeneous boundary conditions at the walls given by (1.2c) (∂xj − ∂xj+1 − g)ψ xj =xj+1 = 0, j = 1, . . . , n − 1. (Here the variable ξ ∈ RN plays the role of the spectral parameter.) The idea of the Bethe Ansatz method is now to construct the solution of this eigenvalue problem as a permutation-invariant linear combination of plane waves, with suitable coefficients such that the boundary conditions at the walls are satisfied. An important problem is the question of the orthogonality and completeness of the Bethe eigenfunctions in a Hilbert space setting. This problem is commonly referred to in the mathematically oriented literature as the Plancherel Problem. For the repulsive regime g > 0, the spectrum of the Hamiltonian is absolutely continuous; the corresponding Plancherel formula was demonstrated formally by Gaudin [G1, G2, G4]. For the attractive regime g < 0, one has both discrete and continuous spectrum; in this case the Plancherel problem was solved by Oxford [O] by building on work of Yang [Y1] and exploiting ideas from an analysis of a related Plancherel problem for the infinite volume XXX isotropic Heisenberg spin chain by Babbitt and Thomas [T, BT]. Thanks to a fundamental observation by Gaudin, it is known that the nBoson system with delta-potential interaction admits natural generalization in terms of the root systems of simple Lie algebras [G3, G4]. From this perspective, the original n-particle model with pairwise interaction corresponds to a root system of type An−1 (i.e., the Lie algebra sl(n; C)). Other classical root systems appear when restricting the particles to a half-line or by distributing them symmetrically around the origin. It turns out that the eigenfunctions of these generalized deltapotential models related to root systems can again be constructed with the Bethe Ansatz method [G3, GS, G, G4]. The corresponding Plancherel formula was proven
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recently by Heckman and Opdam, who considered both the repulsive and the attractive regime [HO]. The aim of the present paper is to study a discrete version of the spectral problem for the Laplace operator with a delta-potential on root systems. Throughout the paper, we restrict ourselves to the repulsive case. More specifically, we consider a discrete Laplacian acting on lattice functions with support in the dominant cone of the weight lattice of the root system, subject to suitable repulsive boundary conditions. We construct the eigenfunctions of this discrete Laplacian through the Bethe Ansatz method. The resulting eigenfunctions turn out to correspond to (the parameter deformations of) the zonal spherical functions on p-adic Lie groups studied by Macdonald [M1, M3]. In particular, the Plancherel problem reduces in this discrete setting to an elementary calculation already carried out by Macdonald to prove the orthogonality of the spherical functions in question with respect to the Plancherel measure. Finally, we perform a continuum limit as the lattice spacing tends to zero and recover the repulsive case of the Plancherel formula for the Laplace operator with a delta-potential on root systems from [HO]. In this limit the discrete Laplacian converges in the strong resolvent topology to the Laplacian of the continuous model. To give rigorous meaning to our continuum limit in a Hilbert space sense, we employ techniques developed by Ruijsenaars in his study of the continuum limit of the infinite isotropic Heisenberg spin chain [R]. The material is organized as follows. Section 2 serves to prepare some basic definitions and notations from the theory of root systems that are needed to formulate the results. Section 3 recalls the eigenfunctions and exhibits the Plancherel formula for the Laplacian in the Weyl chamber with repulsive boundary conditions at the walls. Section 4 is devoted to the discretization of this Laplacian. Specifically, we introduce our discrete Laplacian on the dominant cone of the weight lattice endowed with linear homogeneous conditions at the boundary. The eigenfunctions of the discrete Laplacian are constructed with the Bethe Ansatz method and the Plancherel problem for the repulsive case is resolved by connecting to Macdonald’s theory of zonal spherical functions on p-adic Lie groups. In Section 5 it is shown how – by passing to the continuum limit – the eigenfunctions and the Plancherel formula for the (continuous) Laplacian in Section 3 can be recovered from the eigenfunctions and the Plancherel formula for the discrete Laplacian in Section 4. A few technical points concerning the proof of the Plancherel inversion formula in the continuous situation have been isolated in Appendix A. Furthermore, some crucial results due to Macdonald – which constitute the backbone of the proof for the Plancherel formula in the discrete situation – have been outlined in Appendix B at the end of the paper.
2 Preliminaries on root systems Throughout the paper we will make extensive use of the language of root systems. For a thorough treatment of the concepts and theory surrounding root systems the
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reader is referred to the standard texts [B, H1, H2, K]. Here we restrict ourselves to recalling just the bare minimum of definitions, notations, and properties needed for our purposes. This section is probably best skipped at first reading and referred back to as needed.
2.1
Roots
Let E be a real finite-dimensional Euclidean vector space with the inner product denoted by ·, ·. For a nonzero vector α ∈ E, the action of the orthogonal reflection rα : E → E in the hyperplane through the origin perpendicular to it is given explicitly by (x ∈ E), (2.1) rα (x) = x − x, α∨ α where α∨ := 2α/α, α. By definition, a (crystallographic) root system is a nonempty subset R ⊂ E \ {0} satisfying the properties (i) rα (R) = R, ∀α ∈ R (reflection invariance), (ii) α, β ∨ ∈ Z, ∀α, β ∈ R (integrality).
(2.2)
A vector in R is referred to as a root. The roots generate an Abelian group Q := SpanZ (R) called the root lattice of R. The dimension of Q is called the rank of the root system. Here we will always assume that the ambient Euclidean space E is chosen minimal in the sense that dim(E) is equal to the rank of the root system (i.e., SpanR (R) = E). If one fixes a choice of normal vector generically, in the sense that the hyperplane through the origin perpendicular to it does not intersect R, then the hyperplane in question divides the root system in two subsets of equal size called the positive and negative roots: R = R+ ∪ R−
with
R− = −R+ .
(2.3)
The positive roots determine a nonnegative semigroup Q+ := SpanN (R+ ) of the root lattice. A positive root α is called simple if α − β ∈ R+ for any β ∈ R+ . Let us denote the simple roots by α1 , . . . , αN . These simple roots form a basis for Q and Q+ , i.e., Q = Z α1 ⊕ · · · ⊕ Z αN
and Q+ = N α1 ⊕ · · · ⊕ N αN .
(2.4)
(Hence, the number of simple roots N is equal to the rank of the root system.) It means that starting from the origin we can reach any vector in the root lattice Q by successive addition or subtraction of simple roots. One defines the height of a vector κ ∈ Q as (2.5) ht(κ) := κ, ρ∨ , with ρ∨ := α∈R+ α∨ /2. In the basis of simple roots the height reads ht(κ) = ht(k1 α1 + · · · + kN αN ) = k1 +· · ·+kN . In particular, for κ ∈ Q+ the height function ht(·) counts the number
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of simple roots in κ. The (unique) positive root α0 such that ht(α) ≤ ht(α0 ) for all α ∈ R+ is called the maximal root of R. A root system is said to be irreducible if it cannot be decomposed as a direct orthogonal sum of two (smaller) root systems. Furthermore, a root system is called reduced if any half-line starting from the origin contains at most one single root α ∈ R. (This amounts to the condition that for any α ∈ R the multiple kα is a root if and only if k = 1 or k = −1.)
2.2
The Weyl group
The group W ⊂ O(E; R) generated by all reflections rα , α ∈ R is called the Weyl group of R. The first defining property (i) of a root system states that R is invariant with respect to the action of the Weyl group; the second defining property (ii) guarantees moreover that the root lattice Q is also invariant with respect to this action. In the case of an irreducible reduced root system, the action of the Weyl group splits up R in at most two orbits. More specifically, there are two possible situations: (i) either all roots have the same length, in which case the action of W on R is transitive, or (ii) the roots come in two different sizes, in which case R splits up in an orbit Rs consisting of the short roots and an orbit Rl consisting of the long roots. The reflections in the simple roots rj := rαj , j = 1, . . . , N are referred to as the simple reflections. They form a minimal set of generators for the Weyl group W . In other words, any Weyl group element w ∈ W can be decomposed (non-uniquely) in terms of simple reflections w = rj1 rj2 · · · rj
(2.6)
(with the indices j1 , . . . , j ∈ {1, . . . , N } not necessarily distinct). The number is referred to as the length of the decomposition. If, for given w ∈ W , the length is minimal then the corresponding decomposition is called reduced. An important property of Weyl groups (used frequently in our analysis below) is that a group element w ∈ W admits a reduced decomposition ending in the simple reflection rj (i.e., with rj in (2.6) equal to rj ) if and only if w(αj ) ∈ R− . Let us – for R both irreducible and reduced – define the following (length) functions on W (w)
:= |{α ∈ R+ | w(α) ∈ R− }|,
(2.7a)
s (w) l (w)
− := |{α ∈ R+ s | w(α) ∈ Rs }|, − := |{α ∈ R+ l | w(α) ∈ Rl }|,
(2.7b) (2.7c)
± ± ± where R± s := Rs ∩ R , Rl := Rl ∩ R , and | · | refers to the cardinality of the set in question. Clearly (w) = s (w) + l (w). (If all roots have the same length, then by convention Rs := R and Rl := ∅, so s (w) = (w) and l (w) = 0.) It turns out that the numbers (w), s (w) and l (w) count, respectively, the number of
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simple reflections, the number of short simple reflections and the number of long simple reflections that appear in a reduced decomposition (2.6) of w into simple reflections. For later use, it will be convenient to split up the height function ht(·) (2.5) as a sum of partial height functions as well hts (κ) := κ, ρ∨ s , htl (κ) :=
κ, ρ∨ l ,
with
ρ∨ s :=
with
ρ∨ l
:=
α∨ /2,
α∈R+ s
α∈R+ l
∨
α /2.
(2.8a) (2.8b)
For κ = k1 α1 + · · · + kN αN , this gives hts (κ) =
1≤j≤N αj short
kj ,
htl (κ) =
kj ,
(2.9)
1≤j≤N, αj long
which for κ ∈ Q+ amounts to a count of, respectively, the number of short and long simple roots in κ.
2.3
Weights
The weight lattice P and its nonnegative dominant cone P + are the duals of the root lattice Q and its nonnegative semigroup Q+ , i.e., P P+
:= :=
{λ ∈ E | λ, α∨ ∈ Z, ∀α ∈ R}, {λ ∈ E | λ, α∨ ∈ N, ∀α ∈ R+ }.
(2.10a) (2.10b)
One has that Q ⊂ P but Q+ ⊂ P + (unless N = 1). A vector in P is called a weight. Furthermore, a weight in P + is called a dominant weight. The special dominant weights ω1 , . . . , ωN that are related to the simple roots via the duality ωj , α∨ k = δj,k are referred to as the fundamental weights. These fundamental weights form a basis for P and P + , i.e., P = Z ω1 ⊕ · · · ⊕ Z ωN
and P + = N ω1 ⊕ · · · ⊕ N ωN .
(2.11)
λ µ ⇐⇒ λ − µ ∈ Q+
(2.12)
The following definition ∀λ, µ ∈ P :
endows the weight lattice with a natural partial order. This partial order is usually referred to as the dominance order. The cone of dominant weights P + constitutes a fundamental domain for P with respect to the action of the Weyl group, in the sense that for any µ ∈ P the Weyl orbit W (µ) := {w(µ) | w ∈ W } (2.13)
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intersects the dominant cone P + precisely once. For µ ∈ P, one defines wµ ∈ W as the unique shortest Weyl group element such that wµ (µ) ∈ P + .
(2.14)
The group element wµ admits a reduced decomposition ending in rj if and only if µ, α∨ j < 0 (i.e., if and only if the hyperplane perpendicular to αj separates µ and wµ (µ)). It is instructive to reformulate this criterion in terms of the partial order in Eq. (2.12): the group element wµ admits a reduced decomposition ending in rj if and only if rj (µ) µ. In particular, it means that any dominant weight λ is maximal in its Weyl orbit W (λ), i.e., ∀λ ∈ P + :
λ w(λ),
∀w ∈ W.
(2.15)
The stabilizer of a weight λ ∈ P is defined as Wλ := {w ∈ W | w(λ) = λ}.
(2.16)
The stabilizer Wλ is a subgroup of the Weyl group W that is generated by the simple reflections rj such that rj (λ) = λ.
3 Laplacian on the Weyl chamber In this section we review the solution of the spectral problem for the Laplacian in a Weyl chamber with repulsive boundary conditions at the walls and formulate the associated Plancherel theorem. Note. From now on it will always be assumed that our root system R is both irreducible and reduced. A helpful list of all irreducible root systems and their concrete properties can be found in Bourbaki’s tables [B].
3.1
Eigenvalue problem
The Weyl chamber is the open convex cone C = {x ∈ E | x, α > 0, ∀α ∈ R+ }.
(3.1)
It is bounded by the walls Cj = {x ∈ E | x, αj = 0 and x, α > 0, ∀α ∈ R+ \ {αj } }
(3.2)
perpendicular to the simple roots αj , j = 1, . . . , N . Let gs , gl be two generic (possibly complex) parameters and let us set gs if α ∈ Rs , gα := (3.3) gl if α ∈ Rl .
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The generalization of the eigenvalue problem in Eqs. (1.2a)–(1.2c) to the case of an arbitrary root system R is given by −∇2x ψ(x; ξ) = ξ2 ψ(x; ξ),
x, ξ ∈ C,
with linear homogeneous boundary conditions at the walls of the form ∇x ψ, αj − gαj ψ = 0, j = 1, . . . , N. x∈Cj
(3.4a)
(3.4b)
Here ∇2x and ∇x denote the Laplacian and gradient on E, respectively, and ξ := ξ, ξ. Theorem 3.1 (Eigenfunction). The wave function Ψ0 (x; ξ) =
α, ξ − igα w eix,ξw , α, ξ w +
(3.5)
w∈W α∈R
with ξ w := w(ξ), solves the eigenvalue problem in Eqs. (3.4a), (3.4b). Theorem 3.1 is due to Gaudin, who constructed the wave function in question by means of the Bethe Ansatz Method [G3, G4]. It is clear that the linear combination of plane waves Ψ0 (x; ξ) (3.5) solves the eigenvalue equation in Eq. (3.4a), since −∇2x eix,ξw = ξ w , ξw eix,ξw = ξ, ξeix,ξw . To infer that the boundary conditions in Eq. (3.4b) are also satisfied it suffices to perform a small computation based on the action of the directional derivative on plane waves: ∇x eix,ξ , αj = iαj , ξeix,ξ . Specifically, the following sequence of elementary manipulations reveals that for x ∈ Cj ∇x Ψ0 , αj α, ξ − igα w iαj , ξ w eix,ξw = α, ξ w w∈W α∈R+ α, ξ − ig α w eix,ξw = (gαj + iαj , ξ w ) α, ξ w + w∈W
(i)
=
gαj
α∈R α=αj
α, ξ − igα w eix,ξw α, ξ w +
w∈W α∈R α=αj (ii)
=
gαj
1−
w∈W
=
gαj
α∈R α=αj
α, ξ − igα w eix,ξw α, ξ w +
w∈W α∈R
=
igαj α, ξ w − igα ix,ξ w e αj , ξw α, ξ w +
gαj Ψ0 .
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In Steps (i) and (ii) one exploits the fact that the expressions under consideration are symmetrized with respect to the action of the Weyl group W . Notice in this connection that the relevant terms on the third and fifth line are built of factors that are (skew-)symmetric with respect to the simple reflection rj . Indeed, we have the skew-symmetry αj , rj (ξ w ) = −αj , ξ w (as rj (αj ) = −αj ) and the symmetries α, ξ − ig α, ξ − igα α w w rj = α, ξ α, ξw w + + α∈R α=αj
α∈R α=αj
(as the simple reflection rj permutes the positive roots other than αj ) and x, rj (ξ w ) = x, ξ w (as x ∈ Cj so rj (x) = x). When symmetrizing with respect to the action of the Weyl group the skew-symmetric parts involving αj , ξ w thus drop out.
3.2
Continuous Plancherel formula
Note. From now on we will restrict attention to the repulsive case of nonnegative parameters gs , gl (and hence gα ). Let H0 = L2 (C, dx) be the Hilbert space of square-integrable functions on the Weyl chamber equipped with the standard inner product (f, g)H0 = f (x)g(x)dx (∀f, g ∈ H0 ), (3.6) C
ˆ 0 (ξ) dξ) be the Hilbert space of square-integrable ˆ 0 = L2 (C, (2π)−N ∆ and let H functions on the Weyl chamber with respect to the positive weight function ˆ 0 (ξ) = ∆
igα −1 , 1+ α, ξ
(3.7)
α∈R
equipped with the normalized inner product 1 ˆ 0 (ξ)dξ (fˆ, gˆ)Hˆ 0 = g (ξ) ∆ fˆ(ξ)ˆ (2π)N C
ˆ 0 ). (∀fˆ, gˆ ∈ H
(3.8)
For f ∈ H0 , we now define the eigenfunction transform fˆ0 = F0 f by means of the pairing f (x)Ψ0 (x; ξ)dx, (3.9a) fˆ0 (ξ) = (F0 f )(ξ) := C
ˆ 0 we define the adjoint with Ψ0 (x; ξ) given by Eq. (3.5). Reversely, for fˆ ∈ H ˆ ˆ eigenfunction transform f0 = F0 f as 1 ˆ ˆ ˆ 0 (ξ)dξ. (3.9b) f0 (x) = (F0 f )(x) := fˆ(ξ)Ψ0 (x; ξ) ∆ (2π)N C
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(So formally: fˆ0 (ξ) = (f, Ψ0 (ξ))H0 and f0 (x) = (fˆ, Ψ0 (x))Hˆ 0 .) For gs , gl = 0, the transformations F0 and Fˆ0 amount to the Fourier and inverse Fourier transformation on C, respectively. The following theorem generalizes this state of affairs to the case of general nonnegative parameter values gs , gl . Theorem 3.2 (Continuous Plancherel Formula). The eigenfunction transform F0 ˆ 0 , with (3.9a) constitutes a unitary Hilbert space isomorphism between H0 and H ˆ the inverse transformation given by F0 (3.9b), i.e., ˆ
F 0 ,F 0 ˆ 0, H0 ←→ H
Fˆ0 F0 = IH0 ,
F0 Fˆ0 = IHˆ 0 .
(3.10)
Below we will show that Theorem 3.2 arises as a degeneration of a more elementary “polynomial” Plancherel formula for a discretization of the eigenvalue problem in Eqs. (3.4a), (3.4b). The Plancherel formula of Theorem 3.2 is in agreement with the previous results due to Gaudin [G1, G2, G4] (for root systems of type A) and HeckmanOpdam [HO] (for arbitrary root systems), who showed that the transformation F0 ˆ 0 with left-inverse Fˆ0 (3.9b). The idea (3.9a) constitutes an isometry of H0 into H of the proof for this inversion formula outlined by Heckman and Opdam [HO] is far from elementary: it hinges on a deep result due to Peetre concerning the abstract characterization of differential operators as support preserving linear operators acting on spaces of smooth functions [P1, P2]. For the reader’s convenience, we have included a completely elementary proof of this inversion formula in Appendix A at the end of the paper. It follows from Theorems 3.1 and 3.2 that the operator −∇2x in the Weyl chamber, with repulsive boundary conditions at the walls of the form in Eq. (3.4b), determines a unique selfadjoint extension in H0 given by the pullback of the mulˆ 0 with respect to the eigenfunction tiplication operator fˆ(ξ) → ξ2 fˆ(ξ) in H transformation F0 . From this observation the following corollary is immediate. Corollary 3.3 (Spectrum and Self-adjointness). The operator −∇2x in the Weyl chamber C (3.1), with repulsive boundary conditions of the form in Eq. (3.4b) at the walls, is essentially selfadjoint in H0 and (its closure) has a purely absolutely continuous spectrum filling the nonnegative real axis.
4 Discrete Laplacian on the cone of dominant weights In this section we introduce a discrete Laplacian with repulsive boundary conditions on the cone of dominant weights and solve the associated spectral problem.
4.1
Action of the discrete Laplacian and boundary conditions
A nonzero dominant weight σ is called minuscule if σ, α∨ ≤ 1 for all α ∈ R+ and it is called quasi-minuscule if σ, α∨ ≤ 1 for all α ∈ R+ \ {σ} (without it being
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minuscule). The number of minuscule weights is one less than the index |P/Q|, which means that there are no minuscule weights iff the root lattice Q fills the whole weight lattice P. A quasi-minuscule weight, on the other hand, always exists and it is moreover unique. Specifically, it is given by the dominant weight σ such that σ ∨ is the maximal root of the dual root system R∨ := {α∨ | α ∈ R}. We will now associate to a (quasi-)minuscule weight σ a discrete Laplace operator Lσ acting on the space C(P + ) of complex functions over the cone of dominant weights P + (2.10b). Definition (Discrete Laplacian). Let σ ∈ P + be (quasi-)minuscule and let ts , tl denote two generic complex parameters. The action of the discrete Laplace operator Lσ : C(P + ) −→ C(P + ) on an arbitrary lattice function ψ ∈ C(P + ) is defined as ψλ+ν (λ ∈ P + ), (4.1a) Lσ ψλ = ν∈W (σ) +
where for λ+ν ∈ P \P the value of ψλ+ν is determined by the boundary condition ψλ+ν
(wλ+ν )
= tss (wλ+ν ) tl l
ψwλ+ν (λ+ν)
s (ν) −htl (ν) + θλ+ν t−ht tl (1 s
−
(4.1b)
t−1 s )ψλ ,
with θµ := ht(wµ (µ) − µ) − (wµ ).
(4.1c)
To appreciate the structure of the above boundary conditions the following proposition is helpful. It exploits the decomposition of Weyl group elements in terms of simple reflections to disentangle the boundary conditions completely in terms of simple reflection relations. In this alternative characterization it turns out to be convenient to work with W invariant parameters tα , α ∈ R upon setting (cf. Eq. (3.3)) ts if α ∈ Rs , tα := (4.2) tl if α ∈ Rl . Proposition 4.1 (Boundary Reflection Relations). Let λ be a dominant weight and let σ ∈ P + be (quasi-)minuscule. Then the boundary conditions in Eqs. (4.1b), (4.1c) are equivalent to the requirement that ∀ν ∈ W (σ) such that λ + ν ∈ P \ P + , and for all simple roots αj such that λ + ν, α∨ j < 0, the following reflection relations are satisfied if ht (rj (λ + ν) − λ − ν) = 1, (I) tαj ψrj (λ+ν) ψλ+ν = tαj ψrj (λ+ν) + (tαj − 1)ψλ if ht (rj (λ + ν) − λ − ν) = 2, (II) or equivalently tαj ψλ+ν+αj ψλ+ν = tαj ψλ tαj ψλ+αj + (tαj − 1)ψλ
∨ if λ, α∨ j = 0 and ν, αj = −1, (Ia ) ∨ if λ, α∨ j = 1 and ν, αj = −2, (Ib ) ∨ if λ, α∨ j = 0 and ν, αj = −2. (II )
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Proof. Let us first check that the reflection relations in (I), (II) and in (Ia ), (Ib ), ∨ (II ) are indeed equivalent. Since λ, α∨ j ≥ 0 (as λ is dominant) and ν, αj ≥ −2 with equality holding only when ν = −αj (as ν ∈ W (σ) with σ (quasi-)minuscule), the condition λ + ν, α∨ j < 0 breaks up in the three cases (Ia ), (Ib ) or (II ). It is readily verified that Cases (Ia ) and (Ib ) correspond to (I) and Case (II ) corresponds to (II). Indeed, we have: rj (λ + ν) = λ + ν + αj in Case (Ia ), ν = −αj and rj (λ + ν) = λ in Case (Ib ), and ν = −αj and rj (λ + ν) = λ + αj in Case (II ). Hence, the corresponding reflection relations match in each case. (Notice also that for σ minuscule we are always in Case (Ia ) (i.e., (I)); the Cases (Ib ) or (II ) (i.e., (II)) can only occur when σ is quasi-minuscule.) Next we verify that the conditions in the proposition amount to the boundary conditions in Eqs. (4.1b), (4.1c). To this end we exploit the decomposition in simple reflections to perform induction on the length of wλ+ν , starting from the trivial induction base (wλ+ν ) = 0. (Notice in this connection that (wλ+ν ) = 0 implies that λ + ν is dominant, which agrees with the fact that formally the r.h.s. of Eq. (4.1b) reduces in this situation to ψλ+ν .) For (wλ+ν ) > 0, there exists a simple reflection rj such that wλ+ν = wrj (λ+ν) rj with (wrj (λ+ν) ) = (wλ+ν ) − 1. One furthermore has that rj (λ + ν) λ + ν, i.e., λ + ν, α∨ j < 0. We thus fall in either one of the three cases (Ia ), (Ib ) or (II ), which are to be analyzed separately below. (Ia ) In this situation rj (λ+ν) = λ+rj (ν), which implies that wλ+ν = wλ+rj (ν) rj . By applying first the reflection relation in (Ia ) and then the induction hypothesis we get ψλ+ν
= tαj ψλ+rj (ν) s (wλ+rj (ν) ) l (wλ+rj (ν) ) tl ψwλ+rj (ν) (λ+rj (ν))
= tαj ts
−htl (rj (ν))
s (rj (ν)) + tαj t−ht tl s
(wλ+ν )
= tss (wλ+ν ) tl l +
θλ+rj (ν) (1 − t−1 s )ψλ
ψwλ+ν (λ+ν)
s (ν) −htl (ν) t−ht tl θλ+ν (1 s
− t−1 s )ψλ ,
which coincides with the expression on the r.h.s. of Eq. (4.1b). (Ib ) In this situation rj (λ + ν) = λ, which implies that wλ+ν = rj and ν = −αj ∈ Rs . We get from the reflection relation in (Ib ) ψλ+ν = tαj ψλ = ts ψλ , which corresponds to Eq. (4.1b) with s (wλ+ν ) = s (rj ) = 1, l (wλ+ν ) = l (rj ) = 0, hts (ν) = hts (−αj ) = −1, htl (ν) = htl (−αj ) = 0, and θλ+ν = θλ−αj = ht(αj ) − (rj ) = 0. (II ) In this situation rj (λ + ν) = λ + rj (ν) = λ + αj , which implies that wλ+ν = wλ+αj rj and ν = −αj ∈ Rs . By applying first the reflection relation in (II )
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and then the induction hypothesis we get ψλ+ν
=
tαj ψλ+αj + (tαj − 1)ψλ
=
tαj ts
=
s (wλ+αj ) l (wλ+αj ) tl ψwλ+αj (λ+αj )
+ (tαj − 1)ψλ
(w ) tss (wλ+ν ) tl l λ+ν ψwλ+ν (λ+ν) s (ν) −htl (ν) + t−ht tl θλ+ν (1 − t−1 s s )ψλ ,
which coincides with the expression on the r.h.s. of Eq. (4.1b). Since all three cases lead to the boundary condition in Eqs. (4.1b), (4.1c), this completes the induction step (and therewith the proof of the proposition). Remark. (i) It is clear from the proof of the proposition that for σ minuscule θλ+ν = ht(wλ+ν (λ + ν) − λ − ν) − (wλ+ν ) = 0 (as we are always in Case (Ia )). Hence, in this situation the boundary condition in Eq. (4.1b) reduces to (wλ+ν )
ψλ+ν = tss (wλ+ν ) tl l
ψwλ+ν (λ+ν) .
(4.4)
Remark. (ii) The parameters ts and tl play the role of coupling parameters that determine the strength of the boundary conditions. There are two special extremal situations worth singling out. For ts , tl → 1 the action of Lσ reduces to that of a (n) free Laplacian Lσ : C(P + ) → C(P + ) with Neumann type boundary conditions:
L(n) σ ψλ =
ψwλ+ν (λ+ν) .
(4.5a)
ν∈W (σ) (d)
For ts , tl → 0 the action of Lσ reduces to that of a free Laplacian Lσ : C(P + ) → C(P + ) with Dirichlet type boundary conditions: L(d) σ ψλ = −Nσ (λ)ψλ +
ψλ+ν ,
(4.5b)
ν∈W (σ) λ+ν∈P +
where Nσ (λ) = 0 if σ is minuscule and Nσ (λ) is equal to the number of short (0) simple roots perpendicular to λ if σ is quasi-minuscule. Let Lσ : C(P) −→ C(P) denote the free Laplacian on the (full) weight lattice characterized by the action L(0) σ ψλ =
ψλ+ν
(λ ∈ P).
(4.6)
ν∈W (σ) (n)
(d)
(0)
The operators Lσ (4.5a) and Lσ (4.5b) can be seen as the reduction of Lσ (4.6) to the space of W invariant functions and W skew-invariant functions on P, respectively (upon restriction to the fundamental domain P + ).
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Bethe Ansatz solution
Let Q∨ denote the dual root lattice SpanZ (R∨ ) and let us write TR for the torus E/(2πQ∨ ). It is evident that the plane waves ψλ (ξ) = exp(iλ, ξ), ξ ∈ (0) TR constitute a (Fourier) basis of eigenfunctions for the free Laplacian Lσ : C(P) → C(P) in Eq. (4.6). The corresponding eigenvalues are given by Eσ (ξ) = ν∈W (σ) exp(iν, ξ), ξ ∈ TR . Following the Bethe Ansatz method, we will now construct suitable linear combination of plane waves that satisfies the boundary conditions in Eqs. (4.1b), (4.1c). By construction, the resulting wave function will thus constitute an eigenfunction of our Laplacian Lσ (4.1a)–(4.1c). Specifically, as Bethe Ansatz wave function we take an arbitrary Weyl group invariant linear combination of plane waves of the form Ψλ (ξ) =
1 (−1)w C(ξ w )eiρ+λ,ξw , δ(ξ)
(4.7a)
w∈W
with (−1)w := det(w) = (−1)(w) , and δ(ξ) = (eiα,ξ/2 − e−iα,ξ/2 ),
(4.7b)
α∈R+
ρ
=
1 α. 2 +
(4.7c)
α∈R
(This wave function is W invariant in the sense that Ψλ (ξ w ) = Ψλ (ξ).) The following theorem matches the coefficients so as to meet the boundary conditions (4.1b), (4.1c). Theorem 4.2 (Bethe Wave Function). Let Lσ : C(P + ) → C(P + ) be the discrete Laplacian with boundary conditions defined in Eqs. (4.1a)–(4.1c). Then the Bethe Ansatz wave function Ψλ (ξ) (4.7a)–(4.7c) solves the eigenvalue equation Lσ ψ(ξ) = Eσ (ξ)ψ(ξ) with Eσ (ξ) = exp(iν, ξ), (4.8) ν∈W (σ)
provided that C(ξ) =
(1 − tα e−iα,ξ )
(4.9)
α∈R+
(or a scalar multiple thereof ). Proof. It suffices to check that the Bethe Ansatz wave function Ψλ (ξ) (4.7a)–(4.7c) satisfies the boundary conditions (4.1b), (4.1c), provided that C(ξ) is of the form stated by the theorem. To this end we compute C(ξ) from the boundary reflection relations of Proposition 4.1. Indeed, upon assuming the technical conditions detailed in the proposition, substitution of the Bethe Ansatz wave function in
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the boundary reflection relations readily leads to the stated expression for the coefficients C(ξ). Specifically, we find in Case (I) that equating Ψλ+ν (ξ)
1 (−1)w C(ξ w )eiρ+λ+ν,ξw δ(ξ) w∈W wρ+λ+ν −1 = (−1) δ (ξ) (−1)wµ eiµ,ξ =
µ∈W (ρ+λ+ν)
(−1)w C(ξ w )
w∈Wρ+λ+ν
to tαj Ψrj (λ+ν) (ξ) = = =
tαj Ψλ+ν+αj (ξ) tαj (−1)w C(ξ w )eiαj ,ξw eiρ+λ+ν,ξw δ(ξ) w∈W tαj (−1)wρ+λ+ν δ −1 (ξ) (−1)wµ eiµ,ξ ×
µ∈W (ρ+λ+ν)
(−1) C(ξw )eiαj ,ξw w
w∈Wρ+λ+ν
leads to the relation
(−1)w C(ξw ) = tαj
w∈Wρ+λ+ν
(−1)w C(ξ w )eiαj ,ξw .
w∈Wρ+λ+ν
Because rj stabilizes ρ + λ + ν (i.e., rj ∈ Wρ+λ+ν ), the latter relation can be rewritten as (−1)w [C(ξw ) − C(rj (ξ w ))] w∈Wρ+λ+ν w −1 (αj )∈R+
= tαj
(−1)w [C(ξw )eiαj ,ξw − C(rj (ξ w ))e−iαj ,ξw ].
w∈Wρ+λ+ν w −1 (αj )∈R+
By induction on the cardinality of the stabilizer Wρ+λ+ν , starting from the smallest value |Wρ+λ+ν | = 2 (as it contains as subgroup the cyclic group of order 2 generated by rj ), one concludes that C(ξ) − C(rj (ξ)) = tαj [C(ξ)eiαj ,ξ − C(rj (ξ))e−iαj ,ξ ], or equivalently (assuming C(ξ) is nontrivial in the sense that it does not vanish identically) 1 − tαj e−iαj ,ξ C(ξ) = . (4.10) C(rj (ξ)) 1 − tαj eiαj ,ξ
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From varying λ and ν, it is clear that the reflection relation in Eq. (4.10) should hold for all simple reflections rj , j = 1, . . . , N . We thus conclude that C(ξ) must in fact be of the form (1 − tα e−iα,ξ ), C(ξ) = c0 (ξ) α∈R+
where c0 (ξ) denotes an arbitrary W invariant overall factor (i.e., c0 (ξ w ) = c0 (ξ), ∀w ∈ W ). It remains to check that this choice for the coefficient C(ξ) is also compatible with the boundary conditions of Case (II). This follows from an analysis similar to that of Case (I). Indeed, we see by equating Ψλ+ν (ξ) = = =
Ψλ−αj (ξ) 1 (−1)w C(ξ w )eiρ+λ−αj ,ξw δ(ξ) w∈W 1 (−1)w [C(ξw )e−iαj ,ξw − C(rj (ξ w ))]eiρ+λ,ξw δ(ξ) w∈W w −1 (αj )∈R+
to the sum of tαj Ψrj (λ+ν) (ξ) = tαj Ψλ+αj tαj = (−1)w C(ξ w )eiρ+λ+αj ,ξw δ(ξ) w∈W tαj = (−1)w [C(ξ w )eiαj ,ξw − C(rj (ξ w ))e−2iαj ,ξw ]eiρ+λ,ξw δ(ξ) w∈W w −1 (αj )∈R+
and (tαj − 1)Ψλ (ξ) =
(tαj − 1) (−1)w C(ξ w )eiρ+λ,ξw δ(ξ) w∈W
=
(tαj − 1) δ(ξ)
(−1)w [C(ξw ) − C(rj (ξ w ))e−iαj ,ξw ]eiρ+λ,ξw ,
w∈W w −1 (αj )∈R+
that it is sufficient to require that C(ξ)e−iαj ,ξ − C(rj (ξ)) =
tαj [C(ξ)eiαj ,ξ − C(rj (ξ))e−2iαj ,ξ ] +(tαj − 1)[C(ξ) − C(rj (ξ))e−iαj ,ξ ].
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Upon collecting the factors of C(ξ) and C(rj (ξ)) the latter relation can be rewritten as C(ξ)(1 − tαj eiαj ,ξ )(1 − e−iαj ,ξ ) = C(rj (ξ))(1 − tαj e−iαj ,ξ )(1 − e−iαj ,ξ ), which leads us back to the reflection relation in Eq. (4.10).
4.3
Discrete Plancherel formula
Next we will address the question of the orthogonality and completeness of the Bethe wave functions given by Theorem 4.2. Note. From now on it will always be assumed that the parameters lie in the (repulsive) domain 0 < ts , tl < 1 (unless explicitly stated otherwise). It is straightforward to rewrite the Bethe wave function of Theorem 4.2 as Ψλ (ξ) =
1 − tα e−iα,ξw eiλ,ξw . −iα,ξw 1 − e + w∈W
(4.11)
α∈R
From this expression it can be seen that the functions Ψλ (ξ) amount in essence to (a parameter deformation of) the zonal spherical functions on p-adic Lie groups computed by Macdonald [M1, M3]. (To make this connection with Macdonald’s work more explicit, the interested reader might want to compare Ψλ (ξ) (4.11) with [M1, Theorem (4.2.1)] and [M3, Eq. (10.1)].) The upshot is that the solution of the Plancherel problem is now a direct consequence of Macdonald’s orthogonality relations for the (deformed) spherical functions in question. To describe the result, some notation is needed. Let H = 2 (P + , ∆λ ) denote the Hilbert space of complex functions on the cone of dominant weights P + (2.10b) that are square-summable with respect to the positive weight function ∆λ =
hts (α) htl (α) tl hts (α) htl (α) tα ts tl
1 − ts
+
α∈R λ,α∨ =0
1−
(4.12)
(λ ∈ P + ). The standard inner product on H is given by (f, g)H =
fλ gλ ∆λ
(∀f, g ∈ H).
(4.13)
λ∈P +
ˆ = L2 (A, |W |−1 Vol(A)−1 ∆(ξ)dξ) ˆ Furthermore, let H denote the Hilbert space of complex functions on the Weyl alcove A = {ξ ∈ E | 0 < ξ, α < 2π, ∀α ∈ R+ }
(4.14)
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that are square-integrable with respect to the positive weight function ˆ ∆(ξ) = =
|δ(ξ)|2 C(ξ)C(−ξ) 1 − eiα,ξ α∈R
(4.15a) (4.15b)
1 − tα eiα,ξ
ˆ reads (ξ ∈ A). The normalized inner product on H 1 ˆ ˆ fˆ(ξ)ˆ g (ξ)∆(ξ)dξ (f , gˆ)Hˆ = |W |Vol(A) A
ˆ (∀fˆ, gˆ ∈ H).
(4.16)
To the Bethe wave function Ψλ (ξ) in Theorem 4.2, we associate the integral ˆ given by the Fourier pairing transformation F : H → H fˆ(ξ) = =
(F f )(ξ) := (f, Ψ(ξ))H fλ Ψλ (ξ)∆λ
(4.17a) (4.17b)
λ∈P +
ˆ → H given by the Fourier (∀f ∈ H), and the adjoint integral transformation Fˆ : H pairing fλ
= =
(Fˆ fˆ)λ := (fˆ, Ψλ )Hˆ 1 ˆ fˆ(ξ)Ψλ (ξ)∆(ξ)dξ |W | Vol(A) A
(4.18a) (4.18b)
ˆ (∀fˆ ∈ H). Theorem 4.3 (Discrete Plancherel Formula). The eigenfunction transformation ˆ in Eqs. (4.17a), (4.17b) constitutes a unitary Hilbert space isomorF :H →H phism with the inverse transformation F −1 given by the adjoint eigenfunction ˆ → H in Eqs. (4.18a), (4.18b). transformation Fˆ : H Proof. The theorem is a direct consequence of the fact that the zonal spherical ˆ satisfying the orthogonality functions Ψλ (ξ), λ ∈ P + form an orthogonal basis of H relations [M1, M3] if λ = µ, ∆−1 λ (Ψλ , Ψµ )Hˆ = (4.19) 0 if λ = µ. To keep our treatment selfcontained, a brief outline of Macdonald’s proof of these orthogonality relations is isolated in Appendix B at the end of the paper. ˆσ : H ˆ→H ˆ be the multiplication operator Let E (Eˆσ fˆ)(ξ) := Eσ (ξ)fˆ(ξ),
(4.20)
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where Eσ (ξ) stands for the eigenvalue in Eq. (4.8). It is a straightforward consequence of Theorem 4.2 and Theorem 4.3 that the discrete Laplace operator Lσ ˆσ with respect to the (4.1a)–(4.1c) is the pullback of the multiplication operator E transformation F . From this observation the following two corollaries are immediate. Corollary 4.4 (Spectrum). The discrete Laplace operator Lσ (4.1a)–(4.1c) has a purely absolutely continuous spectrum in the Hilbert space H given by the compact ¯ ⊂ C. set Spec(Lσ ) = {Eσ (ξ) | ξ ∈ A} Remark. (i) The complex conjugate of the function Eσ (ξ) is given by E−w0 (σ) (ξ), where w0 denotes the longest element in the Weyl group W (i.e., the unique element w0 ∈ W such that w0 (A) = −A). Corollary 4.5 (Adjoint). The adjoint of Lσ in H is given by L−w0 (σ) . In particular, this means that Lσ is selfadjoint if and only if w0 (σ) = −σ. This is for instance the case when σ is quasi-minuscule or when w0 = −Id. If w0 (σ) = −σ, then one can make the eigenvalue problem selfadjoint by passing to the operator (Lσ + L−w0 (σ) ). Remark. (ii) It is instructive to detail the contents of the Plancherel formula in Theorem 3.2 for the special parameter limit cases, corresponding to the free discrete Laplacians over P + with Neumann and Dirichlet type boundary conditions, exhibited in the second remark at the end of Section 4.1. For ts , tl → 1, the Bethe wave function Ψλ (ξ) (4.11) reduces to the monomial symmetric function (n) Ψλ (ξ) = |Wλ | mλ (ξ), with mλ (ξ) = eiµ,ξ , (4.21) µ∈W (λ)
ˆ and ∆(ξ) = 1, ∆λ = 1/|Wλ | (cf. Eq. (B.4) of Appendix B below). The eigenfunction transform F amounts in this situation to the W invariant part of the Fourier transformation on 2 (P): fˆ(ξ) = fλ mλ (ξ), (4.22a) λ∈P +
with the inversion formula fλ =
1 |W (λ)| Vol(A)
A
fˆ(ξ)mλ (ξ)dξ.
(4.22b)
For ts , tl → 0 the Bethe wave function Ψλ (ξ) (4.11) reduces to the Weyl character 1 (d) (−1)w eiρ+λ,ξw , (4.23) Ψλ (ξ) = χλ (ξ), with χλ (ξ) = δ(ξ) w∈W
2
ˆ and ∆(ξ) = |δ(ξ)| , ∆λ = 1. The eigenfunction transform F amounts in this situation to the W skew-invariant part of the Fourier transformation on 2 (P): fλ χλ (ξ), (4.24a) fˆ(ξ) = λ∈P +
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with the inversion formula 1 fλ = |W | Vol(A)
A
fˆ(ξ)χλ (ξ)|δ(ξ)|2 dξ.
Ann. Henri Poincar´e
(4.24b)
5 The continuum limit In this section it is shown that the discrete Plancherel formula of Theorem 4.3 degenerates to continuous Plancherel formula of Theorem 3.2 in the continuum limit as the lattice distance tends to zero. The discrete Laplacian from Eqs. (4.1a)–(4.1c) degenerates in this limit – upon symmetrization and rescaling – in the strong resolvent sense to the continuous Laplacian from Eqs. (3.4a)–(3.4b). The approach in this section is inspired by Ruijsenaars’ proof of the fact that the ground-state representation of the infinite isotropic Heisenberg spin chain converges in the continuum limit to a free Boson gas [R]. Note. Throughout this section we will employ the parametrization tα = e−gα with > 0 and with gα positive (and W invariant, cf Eq. (3.3)).
5.1
Embedding
To perform the continuum limit, we embed the Hilbert space H = 2 (P + , ∆λ ) from 2 Section 4 isometrically in the Hilbert space H0 = L (C, dx) with standard inner product (f, g)H0 = C f (x)g(x)dx. This is done via the one-parameter family of embeddings J : H → H0 , > 0, which associate to a lattice function f ∈ H the staircase function f ∈ H0 defined by −N/2 1/2 ∆[−1 x] f[−1 x] . f (x) = (J f )(x) := det(P)
(5.1a)
Here det(P) := det(ω1 , . . . , ωN ) and for x ∈ C ∨ + [x] := [x, α∨ 1 ] ω1 + · · · + [x, αN ] ωN ∈ P
(where [x] denotes the integral part of a nonnegative real number x obtained via ˆ ˆ = L2 (A, |W |−1 Vol(A)−1 ∆(ξ)dξ) truncation). Similarly, the dual Hilbert space H 2 −N ∨ −1 ˆ = L (A, (2π) det(Q ) ∆(ξ)dξ) from Section 4 is embedded isometrically in ˆ 0 = L2 (C, (2π)−N dξ) with normalized inner product (fˆ, gˆ) ˆ = the Hilbert space H H0 1 ˆ g (ξ)dξ. This is done via the one-parameter family of embeddings (2π)N C f (ξ)ˆ ˆ →H ˆ 0 , > 0, which associate to a function fˆ ∈ H ˆ the rescaled function Jˆ : H ˆ 0 defined by fˆ ∈ H ˆ 1/2 (ξ)fˆ(ξ). fˆ (ξ) = (Jˆ fˆ)(ξ) := ∆ det(Q∨ ) N/2
∨ −N Here det(Q∨ ) := det(α∨ |W |Vol(A). 1 , . . . , αN ) = (2π)
(5.1b)
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ˆ := Jˆ (H) ˆ ⊂H ˆ 0 . The eigenfunction transLet H := J (H) ⊂ H0 and let H ˆ ˆ ˆ (4.18a), (4.18b) form F : H → H (4.17a), (4.17b) and its inverse F : H → H ˆ lift under the embeddings J and J , respectively, to a corresponding transform ˆ and its inverse Fˆ : H ˆ → H of the form F : H → H fˆ (ξ) = (F f )(ξ) := f (x)Φ[−1 x] (ξ)dx (5.2a) C
and f (x) = (Fˆ fˆ)(x) :=
1 (2π)N
C
fˆ(ξ)Φ[−1 x] (ξ)dξ,
(5.2b)
with a kernel given by 1/2
()
ˆ 1/2 (ξ)χ (ξ)Ψ[−1 x] (ξ) Φ[−1 x] (ξ) = ∆[−1 x] ∆ A −1 1/2 () S1/2 (ξ w )ei[ x],ξw , = ∆[−1 x] χA (ξ)
(5.3a)
w∈W
where S (ξ) =
sin (α, ξ − igα ) 2 , sin 2 (α, ξ + igα ) +
(5.3b)
α∈R ()
and with χA (ξ) denoting the characteristic function of the rescaled alcove −1 A ⊂ C. It follows from Theorem 4.3 that the transform F and its inverse Fˆ define a unitary Hilbert space isomorphism between the closed subspaces H ⊂ H0 and ˆ 0 . In other words, we have the following commutative diagram of unitary ˆ ⊂ H H Hilbert space isomorphisms H J H
F ,Fˆ
←→ ˆ F ,F
←→
ˆ H Jˆ ˆ H
Fˆ F = IH
F Fˆ = IHˆ
Fˆ F = IH
F Fˆ = IHˆ
.
(5.4)
ˆ : H ˆ0 → H ˆ on the closed The orthogonal projections Π : H0 → H and Π ˆ ⊂ H ˆ 0 , respectively, are given explicitly by subspaces H ⊂ H0 and H −N (Π f )(x) = f (y) dy, (5.5a) det(P) T() ([−1 x]) ∨ ∨ with T() (λ) := {x ∈ E | λ, α∨ j ≤ x, αj < (λ, αj + 1), j = 1, . . . , N }, and by ˆ fˆ)(ξ) = χ() (ξ)fˆ(ξ). (Π (5.5b) A
If we extend the definitions of F and Fˆ in Eqs. (5.2a) and (5.2b) to arbitrary ˆ 0 , respectively, then clearly f ∈ H0 and fˆ ∈ H F Π = F
ˆ = Fˆ . and Fˆ Π
(5.6)
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This gives rise to the following commutative diagrams of bounded transformations H0 ↑ J H
F
−→
F
−→
ˆ0 H ↑Jˆ ˆ H
H0 ↑ J H
Fˆ
←−
ˆ F
←−
ˆ0 H ↑Jˆ , ˆ H
(5.7)
ˆ 0 and Fˆ : H ˆ 0 → H0 being contractive operators in the sense that with F : H0 → H ˆ ˆ ∀f ∈ H0 and ∀f ∈ H0 and Fˆ fˆH0 ≤ fˆHˆ 0
F f Hˆ 0 ≤ f H0 1/2
(5.8)
1/2
(where · H0 := ·, ·H0 and · Hˆ 0 := ·, ·Hˆ ). 0
5.2
The continuum limit → 0: eigenfunction transform
For x and ξ in the interior of the Weyl chamber C, it is straightforward to check that in the limit → 0 the kernel function Φ[−1 x] (ξ) (5.3a) degenerates pointwise to Φ0 (x; ξ) = =
ˆ 1/2 (ξ) Ψ0 (x; ξ) ∆ 0 1/2 S0 (ξ w )eix,ξw ,
(5.9a)
w∈W
with S0 (ξ) =
α, ξ − igα . α, ξ + igα +
α∈R
So, formally the eigenfunction transform F (5.2a) and its adjoint Fˆ (5.2b) degenerate in this limit to fˆ0 (ξ) = (F0 f )(ξ) := f (x)Φ0 (x; ξ)dx (5.10a) C
and its adjoint f0 (x) = (Fˆ0 fˆ)(x) :=
1 (2π)N
C
fˆ(ξ)Φ0 (x; ξ)dξ,
(5.10b)
respectively. From the fact that |S0 (ξ)| = 1, combined with the Plancherel property of the Fourier transform on L2 (E), it follows that the integral transforms in Eqs. ˆ 0 and Fˆ0 : H ˆ 0 → H0 . (5.10a) and (5.10b) define bounded operators F0 : H0 → H The following two lemmas provide a precise meaning to the intuitive idea that for →0 ˆ → H ˆ0 and F → F0 , Fˆ → Fˆ0 . (5.11) H → H0 , H
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Lemma 5.1 (Continuum Limit: the Hilbert Space). One has that s − lim Π = IH0 →0
ˆ = Iˆ s − lim Π H0
and
(5.12)
→0
(strongly). Proof. Since Π is a projection operator, it is obvious that Π H0 ≤ 1 uniformly ∀ > 0. Hence, for validating the first limit in Eq. (5.12), it suffices to show that lim→0 Π φ = φ for any φ in the dense subspace C0∞ (C) ⊂ H0 . It is obvious from the definition in Eq. (5.5a) that, for any test function φ ∈ C0∞ (C), the staircase approximation (Π φ)(x) converges pointwise to φ(x) when tends to 0. Moreover, since φ has compact support it is clear that the difference |Π φ − φ| admits an L2 upper bound that is uniform in (for ≤ 1 say). The desired convergence lim→0 Π φ − φH0 = 0 thus follows by the dominated convergence theorem of Lebesgue. To demonstrate the second limit in Eq. (5.12), we simply observe that ˆ0 for any fˆ ∈ H () ˆ fˆ − fˆ2ˆ = lim (2π)−N 1 − χA (ξ) |fˆ(ξ)|2 dξ, lim Π H 0
→0
→0
C
which converges to zero (again by Lebesgue’s dominated convergence theorem). Lemma 5.2 (Continuum Limit: the Eigenfunction Transform). One has that i) ∀f ∈ H0 :
lim (F f )(ξ) = (F0 f )(ξ),
ξ∈C
→0
(5.13a)
(pointwise) and that ii) s − lim Fˆ = Fˆ0
(5.13b)
→0
(strongly). Proof. i). The action of F on f ∈ H0 reads () −1 1/2 χA (ξ)S1/2 (−ξw ) f (x)∆[−1 x] e−i[ x],ξw dx. (F f )(ξ) = C
w∈W
For any x, ξ ∈ C, we have that for → 0 ()
χA (ξ) → 1,
S (ξ w ) → S0 (ξ w ), −1
∆[−1 x] → 1,
ei[
−1
x],ξw
→ eix,ξw (5.14)
pointwise. Since |e−i[ x],ξw | = 1 and ∆[−1 x] ≤ 1, the pointwise limit in Eq. (5.13a) follows by Lebesgue’s dominated convergence theorem. ˆ 0 is given by ii). The action of Fˆ on any fˆ ∈ H −1 1 () −1 (Fˆ fˆ)(x) = ∆ fˆ(ξ)S1/2 (ξ w )ei[ x],ξw χA (ξ)dξ. [ x] (2π)N C w∈W
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The pointwise limit lim→0 (Fˆ fˆ)(x) = (Fˆ0 fˆ)(x) thus follows by dominated convergence from the pointwise convergence in Eq. (5.14) combined with the bounds −1 () |S (ξ w )| = 1, |e−i[ x],ξw | = 1 and |χA (ξ)| ≤ 1. It remains to show that the transition Fˆ → Fˆ0 converges in fact strongly. Since Fˆ is uniformly bounded in in view of Eq. (5.8), it suffices to show that lim→0 Fˆ φˆ = Fˆ0 φˆ for any φˆ in the ˆ 0 . The latter limit follows from the estimate dense subspace C0∞ (C) ⊂ H ˆ ≤ C(1 + x2N )−1 |(Fˆ φ)(x)|
(5.15)
uniformly in for sufficiently small. Indeed, the already established pointwise ˆ ˆ convergence lim→0 (Fˆ φ)(x) = (Fˆ0 φ)(x) combined with the L2 -bound in Eq. (5.15) guarantees the convergence of the limit in the Hilbert space H0 by the bounded convergence theorem. In order to verify the estimate in Eq. (5.15), we note that from the explicit formula for the action of Fˆ it is clear that 1/2 1 2N −1 2N ˆ ˆ ˆ S (ξ w )φ(ξ) |∇2N |dξ, (5.16) [ x] |(F φ)(x)| ≤ ξ N (2π) C w∈W
provided is sufficiently small so as to ensure that the support of φˆ is contained in −1 A. Now let ∂ξ1 , . . . , ∂ξN be the partial derivatives associated to an orthonormal basis e1 , . . . , eN of E. Then ∇2ξ = ∂ξ21 + · · · , ∂ξ2N . Hence, to show that the bound (5.15) follows from (5.16) it suffices to check that the partial derivatives m 1/2 ˆ The partial derivatives in ∂ξj j ∂ξmkk S (ξ) are bounded in on the support of φ. question are sums of products of expressions of the form sin (α, ξ − ig ) 1/2 α n 2 ∂ξjj ∂ξnkk . (5.17) sin 2 (α, ξ + igα ) The derivatives in Eq. (5.17) are in turn sums of products built of expressions of the form sin 2 (α,ξ−igα ) 1/2 sin (α,ξ+ig ) α,ej cos 2 (α,ξ−igα ) α,ej cos 2 (α,ξ+igα ) , sin 2 (α,ξ−igαα ) , , , sin (α,ξ+igα ) sin (α,ξ+igα ) sin (α,ξ+igα ) 2
2
2
2
and α, ej , which remain bounded as → 0. With the aid of Lemmas 5.1 and 5.2, we are now in the position to push through the continuum limit → 0 at the level of the Plancherel formula. ˆ 0 → H0 constitutes an Proposition 5.3 (Isometry). The transformation Fˆ0 : H ˆ isometry with left-inverse F0 : H0 → H0 . Proof. The transform Fˆ0 inherits from Fˆ the property that it is an isometry in ˆ0 view of Lemmas 5.1 and 5.2. Indeed, for any fˆ ∈ H Fˆ0 fˆH0
Eq. (5.13b)
=
Eq. (5.4)
=
lim Fˆ fˆH0
→0
ˆ fˆ ˆ lim Π H0
→0
Eq. (5.12)
=
Eq. (5.12)
=
ˆ fˆH0 lim Fˆ Π
→0
fˆHˆ 0 ,
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ˆ 0 → H0 is an isometry. To see that F0 is a left-inverse of Fˆ0 , we whence Fˆ0 : H consider the identity (cf. the commutative diagram in Eq. (5.4)) ˆ φˆ F Fˆ φˆ = Π
(5.18)
for φˆ ∈ C0∞ (C). The l.h.s. of this identity is given by () −1 1/2 ˆ χ (ξ)S 1/2 (−ξ ) (Fˆ φ)(x)∆ e−i[ x],ξw dx. −1 A
w∈W
w
C
[
x]
(5.19)
ˆ We know from the second part of the proof of Lemma 5.2 that (Fˆ φ)(x) admits an L2 -bound that is uniform in for sufficiently small (cf. Eq. (5.15)) and that for ˆ By following the steps in the first part → 0 it converges pointwise to (Fˆ0 φ)(x). of the proof of Lemma 5.2, we readily infer from the expression in Eq. (5.19) that ˆ ˆ = (F0 Fˆ0 φ)(ξ) (pointwise). On the other hand, it follows from lim→0 (F Fˆ φ)(ξ) ˆ ˆ ˆ φ)(ξ) (the proof of) Lemma 5.1 that lim→0 (Π = φ(ξ). We thus conclude that for → 0 the identity in Eq. (5.18) degenerates to ˆ F0 Fˆ0 φˆ = φ, ˆ 0 and the operators whence F0 Fˆ0 = IHˆ 0 (since the subspace C0∞ (C) is dense in H involved are bounded). In other words, Proposition 5.3 states that Fˆ0 is a unitary Hilbert space ˆ 0 ) ⊂ H0 . The following ˆ 0 and the closed subspace Fˆ0 (H isomorphism between H ˆ ˆ proposition ensures that in fact F0 (H0 ) = H0 . ˆ 0 → H0 is surjective, Proposition 5.4 (Completeness). The transformation Fˆ0 : H ˆ 0 ) = H0 . i.e., Fˆ0 (H ˆ 0 → H0 it is enough to show that Proof. For proving the surjectivity of Fˆ0 : H ˆ F0 : H0 → H0 is injective (in view of Proposition 5.3). This injectivity is verified in Appendix A below. Combination of Propositions 5.3 and 5.4 entails that the transformation F0 : ˆ 0 constitutes a unitary Hilbert space isomorphism with inverse Fˆ0 : H ˆ0 → H0 → H H0 : F0 ,Fˆ0 ˆ 0, Fˆ0 F0 = IH0 , F0 Fˆ0 = I ˆ . (5.20) H0 ←→ H H0
The Plancherel formula in Theorem 3.2 is now immediate upon performing the ˆ 1/2 fˆ at the spectral side, so as to trade the uniform gauge transformation fˆ → ∆ 0 ˆ 0 (ξ)dξ. Lebesgue measure dξ for the Plancherel measure ∆
5.3
The continuum limit → 0: Laplacian
ˆσ, and E ˆ0 be multiplication operators in H ˆ 0 of the form Let E ˆ0 (ξ)fˆ(ξ), (Eˆσ, fˆ)(ξ) = Eˆσ, (ξ)fˆ(ξ) and (Eˆ0 fˆ)(ξ) = E
(5.21a)
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with ˆσ, (ξ) = −2 E
Ann. Henri Poincar´e
ˆ0 (ξ) = ξ2 . 1 − cos(ν, ξ) and E
(5.21b)
ν∈W (σ)
ˆσ, and E ˆ0 We introduce the operators Lσ, and L0 in H0 as the pullbacks of E ˆ 0 and F0 : H0 → H ˆ 0: with respect to the eigenfunction transforms F : H0 → H Lσ, L0
:= Fˆ Eˆσ, F , ˆ0 F0 . := Fˆ0 E
(5.22a) (5.22b)
The operator L0 (5.22b) amounts to the Laplacian −∇2x in the Weyl chamber C with boundary conditions at the walls of the form in Eq. (3.4b), and the operator Lσ, (5.22a) corresponds to the lift of −2 |W (σ)| − Lσ /2 − Lw0 (σ) /2 from H to H : 1 (5.23) Lσ, J = 2 2|W (σ)| − Lσ − Lw0 (σ) , 2 where Lσ denotes the discrete Laplacian defined in Eqs. (4.1a)–(4.1c). The following proposition states that, in the continuum limit → 0, the discrete difference operator Lσ, (5.22a) tends (up to a positive factor) to the differential operator L0 (5.22b) in the strong resolvent sense. Proposition 5.5 (Continuum Limit: the Laplacian). Let z ∈ C \ [0, ∞). Then s − lim (Lσ, − zIH0 )−1 = (cσ L0 − zIH0 )−1 →0
for some positive constant cσ . Proof. From the limit 1 1 lim 2 1 − cos(ν, ξ) = →0 2 ν∈W (σ)
|ν, ξ|2 = cσ ξ2
ν∈W (σ)
ˆσ, (ξ) = cσ E ˆ0 (ξ) pointfor some positive constant cσ , one concludes that lim→0 E ˆ ˆ wise. Hence, for any f ∈ H0 and z ∈ C \ [0, ∞) lim (Eˆσ, − zIHˆ 0 )−1 fˆ = (cσ Eˆ0 − zIHˆ 0 )−1 fˆ
→0
strongly, by the dominated convergence theorem. The proposition now follows from the telescope (Lσ, − zIH0 )−1 f − (cσ L0 − zIH0 )−1 f H0 ≤ (Lσ, − zIH0 )−1 (Fˆ0 − Fˆ )F0 f H0 ˆ0 − zI ˆ )−1 ]F0 f H0 + Fˆ [(Eˆσ, − zIHˆ 0 )−1 − (cσ E H0 ˆ0 − zI ˆ )−1 F0 f H0 + (Fˆ − Fˆ0 )(cσ E H0 upon sending to zero (and invoking of Lemma 5.2).
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The inversion formula: Continuous case
In the proof of Proposition 5.4 we needed the fact that the transformation F0 : ˆ 0 in Eq. (5.10a) – or equivalently the transformation F0 : H0 → H ˆ 0 in H0 → H Eq. (3.9a) – is injective. In principle this injectivity follows from the analysis by Heckman and Opdam. Indeed, it was shown in Ref. [HO] that Fˆ0 F0 = IH0
(A.1)
upon restriction to the dense subspace C0∞ (C) ⊂ H0 (cf. also the comments just after Theorem 3.2). Since all operators involved are bounded, the inversion formula in Eq. (A.1) is readily extended from C0∞ (C) to the whole of H0 (by taking the ˆ 0 (and thus the transformation closure), whence the transformation F0 : H0 → H ˆ 0 ) is injective. F0 : H0 → H The proof of Eq. (A.1) indicated in [HO] is quite sophisticated and hinges on a deep result due to Peetre regarding the characterization of differential operators as support preserving operators on smooth test functions [P1, P2]. In this appendix we present an elementary proof for this inversion formula. Let φ ∈ C0∞ (C). Then (F0 φ)(ξ) = φ(x)Ψ0 (x; ξ)dx C = C0 (−ξ w ) φ(x)e−ix,ξw dx C
w∈W
=
˘ C0 (−ξ w ) φ(ξ w ),
(A.2a)
α, ξ − igα α, ξ +
(A.2b)
φ(x)e−ix,ξ dx.
(A.2c)
w∈W
where C0 (ξ) =
α∈R
and
˘ φ(ξ) = E
Substitution of φˆ = F0 φ into ˆ (Fˆ0 φ)(x)
= = =
1 ˆ φ(ξ)Ψ 0 (x; ξ)∆0 (ξ)dξ (2π)N C 1 ix,ξw ˆ ∆0 (ξ)dξ φ(ξ)C 0 (ξ w )e (2π)N C w∈W 1 1 ˆ eix,ξw dξ φ(ξ) (2π)N C (−ξ ) 0 C w w∈W
(A.3)
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yields (Fˆ0 F0 φ)(x)
= =
1 (2π)N 1 (2π)N
w1 ,w2 ∈W
w∈W
E
C
C0 (−ξw1 ) ˘ φ(ξ w1 )eix,ξw2 dξ C0 (−ξw2 )
C0 (−ξ) ˘ φ(ξ)eix,ξw dξ, C0 (−ξw )
(A.4a)
where C0 (−ξ) C0 (−ξw )
=
α, ξ + igα α, ξ w α, ξ α, ξ w + igα + +
α∈R
=
α∈R+ ∩w −1 (R− )
α∈R
α, ξ + igα . α, ξ − igα
(A.4b)
The inversion formula Fˆ0 F0 φ = φ for φ ∈ C0∞ (C) is now immediate from Eqs. (A.4a), (A.4b) combined with the fact that for x ∈ C and w ∈ W φ(x) if w = Id, 1 C0 (−ξ) ˘ ix,ξw φ(ξ)e dξ = (A.5) N (2π) 0 if w = Id. E C0 (−ξ w ) To infer the equality in Eq. (A.5), let us first note that the case w = Id is clear as it amounts to the standard Fourier inversion formula on E. The case w = Id is verified with the aid of the following straightforward observations. ˘ (A.2c) is entire in ξ and rapidly (i) For φ ∈ C0∞ (C) the Fourier transform φ(ξ) decreasing on the tubular domain E − iC∨ , where C∨ denotes the open convex cone dual to C, generated by the positive roots (i.e., C∨ := SpanR+ (R+ )). (ii) The parameter restriction gα > 0 ensures that the quotient C0 (−ξ)/C0 (−ξ w ) (A.4b) is holomorphic and bounded on the tubular domain E − iCw , where Cw := {ξ ∈ E | ξ, α > 0, ∀α ∈ R+ ∩ w−1 (R− )}. ∨ −1 (iii) For all x ∈ C and ϑ ∈ C∨ (−C∨ ) = SpanR+ (R+ ∩ w−1 (R− )), w := C ∩ w one has that x, ϑw < 0.
Indeed, we conclude from (i) and (ii) and the Cauchy integral theorem that for an arbitrary but fixed ϑ ∈ Cw ∩ C∨ w C0 (−ξ) ˘ C0 (−ξ) ˘ ix,ξw φ(ξ)e φ(ξ)eix,ξw dξ dξ = (A.6) C (−ξ ) C E 0 E−isϑ 0 (−ξ w ) w for all s ≥ 0. Furthermore, it follows from (i), (ii) and (iii) that for s → ∞ the r.h.s. of Eq. (A.6) tends to zero, whence the case w = Id of the equality in Eq. (A.5) follows.
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Note. To convince oneself that the cone Cw ∩C∨ w is nonempty for any w ∈ W \{Id}, we observe that it contains the nonzero vector ρ − ρw−1 (where ρ = α∈R+ α/2). Indeed, we have on the one hand that ρ ∈ C and ρw−1 ∈ w−1 (C), so ρ−ρw−1 ∈ Cw , while on the other hand ρ − ρw−1
= =
1 1 1 α− α+ α 2 2 2 + + + + − −1 −1 α∈R α∈R ∩w (R ) α∈R ∩w (R ) α, α∈R+ ∩w −1 (R− )
so ρ − ρw−1 ∈ C∨ w.
B Macdonald’s orthogonality relations In this appendix we outline the proof of Macdonald’s orthogonality relations [M1, M3] if λ = µ, ∆−1 λ (Ψλ , Ψµ )Hˆ = (B.1) 0 if λ = µ, for the Bethe wave functions Ψλ (ξ) of Theorem 4.2. (Here ∆λ is given by Eq. (4.12).) The proof, which follows Macdonald’s treatment in Ref. [M3, §10], hinges on two key lemmas. The first lemma states that the Bethe wave function expands triangularly with respect to the dominance order on the basis of monomial symmetric functions mµ (ξ) = ν∈W (µ) eiν,ξ , µ ∈ P + . Lemma B.1 (Triangularity). For any λ ∈ P + , the Bethe wave function Ψλ (ξ) (4.11) expands as Ψλ (ξ) = ∆−1 λ mµ (ξ) +
aλµ mµ (ξ),
µ∈P + , µ≺λ
for certain coefficients aλµ ∈ C. The second lemma describes a biorthogonality relation between the Bethe wave function Ψλ (ξ) and the monomial symmetric functions mµ (ξ) corresponding to dominant weights µ that are not bigger than λ in the dominance order. Lemma B.2 (Biorthogonality Relations). The Bethe wave functions Ψλ (ξ), λ ∈ P + and the monomial symmetric functions mµ (ξ), µ ∈ P + satisfy the biorthogonality relations 1 if µ = λ, (Ψλ , mµ )Hˆ = 0 if µ λ.
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Before going into the proof of these lemmas, let us first observe that they imply the orthogonality relations in Eq. (B.1). Indeed, it is immediate from Lemmas −1 B.1 and B.2 that (Ψλ , Ψλ )Hˆ = (Ψλ , ∆−1 ˆ = ∆λ and that (Ψλ , Ψµ )H ˆ = 0 if λ mλ )H µ λ. But then (Ψλ , Ψµ )Hˆ must in fact vanish for all dominant weights µ = λ in view of the symmetry (Ψλ , Ψµ )Hˆ = (Ψµ , Ψλ )Hˆ , whence the orthogonality relations follow.
Proof of Lemma B.1 Starting point for the proof of the first lemma is the formula for Ψλ (ξ) in Eqs. (4.7a)–(4.7c) with C(ξ) of the form in Eq. (4.9): Ψλ (ξ) =
1 (−1)w eiρ+λ,ξw (1 − tα e−iα,ξw ). δ(ξ) + w∈W
α∈R
By expanding the product over the positive roots, one arrives at c 1 (−1)w (−1)|X| eiρ(X )−ρ(X)+λ,ξw tα , Ψλ (ξ) = δ(ξ) + w∈W
α∈X
X⊂R
with ρ(X) :=
1 α, 2
ρ(X c ) :=
α∈X
1 2
α.
α∈R+ \X
−1
Next, by exploiting the symmetry eiµ,ξw = eiw (µ),ξ and combining contributions of monomials with weights in the same Weyl orbit into Weyl characters χµ (ξ) = δ −1 (ξ) w∈W (−1)w eiρ+µ,ξw , one finds the expansion (−1)wX (−1)|X| tα χλ(X) (ξ), (B.2) Ψλ (ξ) = X⊂R+
α∈X
where we have introduced the notation wX := wρ(X c )−ρ(X)+λ ,
λ(X) := wX (ρ(X c ) − ρ(X) + λ) − ρ.
In the expansion (B.2) the factor χλ(X) (ξ) vanishes if the weight λ(X) is not dominant. Indeed, from the fact that ρ = ω1 + · · · + ωN (the sum of the fundamental weights) [B, H1], it follows that in such case the dominant weight ρ + λ(X) cannot be regular (i.e., it must have a nontrivial stabilizer), whence the skewsymmetrization with respect to the Weyl-group action in the definition of the Weyl character leads to zero. It follows from Eq. (B.2) that the Bethe function Ψλ (ξ) expands triangularly on the basis of Weyl characters: bλµ χµ (ξ), Ψλ (ξ) = µ∈P + , µ λ
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for certain complex coefficients bλµ . Indeed, one has that λ(X)
= wX (λ) + wX (ρ(X c ) − ρ(X)) − ρ α+ = wX (λ) − α∈R+ ∩wX (X)
α
α∈R− ∩wX (X c )
wX (λ) λ (where in the last step we exploited the well-known fact that any dominant weight is maximal (with respect to the dominance order) in its Weyl orbit, i.e., for all λ ∈ P + and w ∈ W : λw λ [B, H1]). To compute the leading coefficient bλλ , it is needed to collect all terms in Eq. (B.2) for which λ(X) = λ. These terms correspond to those subsets X ⊂ R+ for −1 which wX (λ) = λ and wX (ρ) = ρ(X c ) − ρ(X), or equivalently, to those subsets X for which X = {α ∈ R+ | w(α) ∈ R− } with w ∈ Wλ . By summing the contributions of all such subsets X, one obtains that bλλ is given by the following Poincar´e type series of the stabilizer Wλ : (w) bλλ = tss (w) tl l . (B.3) w∈Wλ
The lemma now follows from the well-known fact that the Weyl characters expand unitriangularly on the monomials χλ = mλ + µ∈P + , µ≺λ cλµ χµ , combined with Macdonald’s celebrated product formula [M2, M3] for the Poincar´e type series in Eq. (B.3) 1 − tα tshts (α) thtl (α) (w) l tss (w) tl l = . (B.4) hts (α) htl (α) 1 − t t s w∈Wλ l α∈R+ λ,α∨ =0
Proof of Lemma B.2 Starting from the formula for the Bethe wave function Ψλ (ξ) in Eq. (4.11), we obtain upon taking the inner product with the monomial symmetric function mµ (ξ): 1 |W | Vol(A) |Wµ | 1 − tα e−iα,ξw1 ˆ ∆(ξ) eiλ,ξw1 × e−iµ,ξw2 dξ −iα,ξw1 1 − e A w1 ∈W w2 ∈W α∈R+ 1 − eiα,ξ 1 (i) dξ = eiλ,ξ−iµ,ξw Vol(T) |Wµ | 1 − tα eiα,ξ w∈W T α∈R+ ∞ 1 (ii) 1+ = eiλ−µw ,ξ (tnαα − tnαα −1 )einα α,ξ dξ, Vol(T) |Wµ | T + n =1
(Ψλ , mµ )Hˆ =
w∈W
α∈R
α
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ˆ where T denotes the torus E/(2πQ∨ ). Here step (i) follows by plugging in ∆(ξ) (4.15a), (4.15b), exploiting the Weyl-group invariance of the integrand, and using the standard fact that the Weyl alcove A (4.14) constitutes a fundamental domain for the action of the Weyl group W on the torus T [B, H2, K]. Furthermore, step (ii) follows by expansion of the denominators with the aid of a geometric series. Clearly, the integral on the last line picks up the constant term of the integrand multiplied by the volume of the torus T. It is immediate that a nonzero constant term can occur only if λ − µw ∈ −Q+ (i.e., µw λ) for some w ∈ W . Now, if µ λ, then for all w ∈ W also µw λ (since µw µ, cf. the proof of Lemma B.1 above). Hence, in this situation the constant term vanishes. On the other hand, if µ = λ then the constant part of the term labelled by w is nonzero (namely equal to 1) if and only if w ∈ Wλ . By summing over all these contributions originating from the stabilizer Wλ the lemma follows.
Acknowledgments This paper was written in large part while the author was visiting the Graduate School of Science and Engineering of the Tokyo Institute of Technology, Tokyo, Japan (January–March, 2003). It is a pleasure to thank the Department of Mathematics, and in particular Professors K. Mimachi and N. Kurokawa, for the warm hospitality. Thanks are also due to S.N.M. Ruijsenaars for several helpful discussions.
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E. Br´ezin and J. Zinn-Justin, Un probl`eme `a N corps soluble, C.R. Acad. Sci. Paris S´er. A-B 263, B670–B673 (1966).
[B]
N. Bourbaki, Groupes et alg`ebres de Lie, Chapitres 4–6, Hermann, Paris, 1968.
[G1]
M. Gaudin, Bose gas in one dimension, I. The closure property of the scattering wave functions, J. Math. Phys. 12, 1674–1676 (1971).
[G2]
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J.F. van Diejen Instituto de Matem´ atica y F´ısica Universidad de Talca Casilla 747 Talca, Chile and Graduate School of Science and Engineering Tokyo Institute of Technology 2-12-1 Oh-okayama Meguro-ku Tokyo, 152-8551 Japan email:
[email protected] Communicated by Rafael D. Benguria Submitted 27/05/03, accepted 14/10/03
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 169 – 188 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/010169-20 DOI 10.1007/s00023-004-0164-x
Annales Henri Poincar´ e
A Singular Expansion of Solution for a Regularized Compressible Stokes System Jae Ryong Kweon Abstract. A compressible Stokes system is considered in a sector of the plane. The continuity equation is regularized by adding the diffusion term −∆p. We give a high-order expansion of corner singularities for the regularized system when the corner singularities for the Laplacian are implemented. A solution formula is constructed in an abstract way, and new associate singular functions are introduced for extracting high-order corner singularities. In the expansion, the smoother parts of the associate singular functions are used.
1 Introduction Many issues for the compressible Stokes or Navier-Stokes systems in domains with singular boundaries are still open. The known results are very limited [10, 11, 14]. A reason is because the system is of mixed type which is neither elliptic nor hyperbolic. The boundary value problems of elliptic type have been exhaustingly investigated in such regions. For instance, see [6, 7, 9] for the Laplace or convection diffusion equations and [4, 6, 8, 9] for the incompressible Stokes equations. Note that there are numerous papers not listed in this paper. The investigation for the incompressible Stokes case is due to the known theory of elliptic problems while in the compressible case, such an adequate approach does not exist. In [10, 11] it is shown that the lowest-order corner singularity of the stationary compressible Stokes or Navier-Stokes system is the same as that of the Laplace equation near the concave vertices. In this case the continuity equation is solved for pressure along the characteristic lines directed by an ambient velocity vector. In [12, 14], when the domain is a convex polygon, the singular functions produced by the Stokes equations were considered in order to obtain a smoother part for the solution of the stationary Stokes or Navier-Stokes system. When it is not convex, the result is not true any more. To understand this difficulty in a different way, we regularize the system by adding − ∆p to the continuity equation. The goal of this paper is to consider the regularized compressible Stokes system in any sector of the plane and to give a high-order expansion of the corner singularities by the Laplacian. The system to be considered is − ∆p + κU · ∇p + λp + divu = −µ ∆u + ∇p
g in Ω,
=
f in Ω,
u, p =
0 on Γ.
(1.1)
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Here the unknowns are the velocity vector u = [u, v] and the pressure p; and λ are positive numbers, κ is the compressibility constant, µ is the viscosity constant; ¯ f and g are U is a given smooth vector function with UΓ = 0, say U ∈ C 1 (Ω); given functions. The region Ω is an infinite sector defined by Ω = {(r cos θ, r sin θ) : r = x2 + y 2 > 0, ω1 < θ < ω2 } where ωl , l = 1, 2, are two numbers satisfying ω1 < ω2 < ω1 + 2π. The angle of the sector is ω = ω2 − ω1 which is assumed 0 < ω ≤ 2π. The two rays that make up the boundary Γ of Ω are denoted Γl , l = 1, 2. Throughout this paper we shall denote by Z = κU · ∇ + λI. Note that there are no inflow and outflow due to UΓ = 0 and (1.1c). So a total mass of the gas is fixed. The equations in (1.1) are derived as follows. The second equation is derived by dropping the terms ν∇divu and ρU · ∇u in the momentum equation. The first equation is derived by the time-dependent continuity equation ρt + div(ρu) = 0. Using the relation ρ = ρ(p) we have ρ (p)pt + ρ(p)divu + ρ (p)u · ∇p = 0. When (U, P ) is a given ambient flow, a linearized continuity equation around the ambient flow is ρ (P )pt + ρ(P )divu + ρ (P )U · ∇p = 0. The derivative pt can be replaced by p(x, t)−p(x, t−∆t) /∆t when ∆t is small. When λ = κ/∆t with κ = ρ (P )/ρ(P ), we have κU · ∇p + λp + divu = g where g = λ P (x, t − ∆t). Adding −∆p, (1.1a) is obtained. One may extend the analysis of sector to a polygon but for simplicity, we will focus on the behavior of the solution near the origin of the sector Ω. Let χ ∈ C0∞ (R2 ) be a cutoff function which is identically 1 in a neighborhood of (0,0), and which satisfies χ(x, y) = 0 for r ≥ 1. Using this, (1.1) becomes − ∆(χp) + Z(χp) + div(χu) = χg − 2 ∇χ · ∇p + p(− ∆χ + U · ∇χ) + u · ∇χ,
(1.2)
−µ ∆(χu) + ∇(χp) = χ f − 2µ∇χ · ∇u − µu∆χ + p∇χ. It is seen that [χu, χp] is a weak solution of (1.2) with zero Dirichlet boundary data. From the property of χ, near r = 1 the solution [χu, χp] is sufficiently smooth and vanishes. In this sense, throughout this paper we assume that the solution [u, p] vanishes outside r = 1. Denote Ωa by a finite sector of Ω, truncated at radius a > 0. us means the norm of Hs (Ω) and us,Ωa the norm of Hs (Ωa ). Set γ0 = λ − |∇(κU)|∞ . We state a main result of this paper, which is shown in Section 4. For j = 1, 2, . . . let sj = jα + 1 with α = π/ω be given. Theorem 1.1 Suppose that the number γ0 is positive, in other words, either λ is sufficiently large or U is close to a constant. Assume that in (1.1), [u, p] vanishes for r ≥ 1. Then, if [f , g] ∈ L2 × L2 , then there is a unique solution [u, p] ∈ H10 × H10 of (1.1) satisfying √ √ √ µu1,Ω1 + ∇p1,Ω1 + γ0 p0,Ω1 ≤ C(f 0 + g0 ).
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If 1 ≤ s < s1 and [f , g] ∈ Hs−2 × Hs−2 , then µ us,Ω1 + ps,Ω1 ≤ C(f s−2 + gs−2 ) for a constant C. For integer l ≥ 1, let sl < s < sl+1 . If [f , g] ∈ Hs−2 × Hs−2 and U ∈ Hs , then the solution [u, p] is split as follows: [u, p] = [us , ps ] + [uR , pR ],
[uR , pR ] = [u, p] − [us , ps ]
with l l−1 ˜ [Cj , Cj3 ]φj + [ηj,1 , τj,1 ]. [us , ps ] = j=1
j=1
Furthermore µuR s,Ω1 + pR s,Ω1 ≤ C(f s−2 +gs−2 ) where C is a constant. Here the coefficient [C˜j , Cj3 ] with C˜j = [Cj1 , Cj2 ] solves the system (4.14) and satisfies (4.16). The function φj is defined in Section 3, and for j ≥ 1, [ηj,1 , τj,1 ] is defined by ηj,1 = ηj − µ−1
j
Λi (χfs,j )φi ,
τj,1 = τj − −1
i=1
j
Λi (χgs,j )φi ,
i=1
where Λi is a bounded linear functional on Hs−2 for s > si (see Theorem A in Section 3), and where fs,1 = −C13 ∇φ1 , gs,1 = C˜1 · ∇φ1 − C13 Zφ1 , ··· fs,j = −(Cj3 ∇φj + ∇τj−1,1 ), gs,j = C˜j · ∇φj − Cj3 Zφj − (Zτj−1,1 + div ηj−1,1 ). The functions ηj and τj are the new associate singular functions which satisfy −µ∆ηj = χ fs,j in Ω, ηj = 0 on Γ and − ∆τj = χ gs,j in Ω, τj = 0 on Γ, respectively. In Theorem 1.1, we note that if s < s1 the solution [u, p] is as regular as permitted in the data [f , g] and that if s > s1 the solution is split with singular and regular parts. In the singular expansion when s > s2 , some new associate singular functions are needed in the expansion while they are not necessary in the Laplacian case [6, 7, 9]. Such singular information of solution might have an application to certain realistic modeling. Consider a high-speed flow over a body where a wall of the body is turned downward at the corner through a deflection angle [2]. If the flow is in high speed, say, supersonic or hypersonic, then at the corner, the flow properties may change drastically. It is thought that a complete understanding of the solution
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near the corner is an essential ingredient in studying such flows. Also, in the solid mechanics, when a solution domain is composed of different materials, the corner singularities occur at the intersections of their internal interfaces. In order to apply to problem (1.1) known results for the Laplacian, we define A : L2 ( or H−1 ) −→ H10 by u := AF where u is the solution of −∆u = F in Ω,
u = 0 on Γ.
(1.3)
In what follows, the following spaces and norms are used [1, 13]. Denote by L2 the space of all measurable functions u defined on Ω for which u0 := 1/2 2 . Let L20 be the set of the functions v in L2 which satisfies Ω |u(x)| dx v(x)dx = 0. Denote by |u|∞ = ess supx∈Ω |u(x)|. For integer l ≥ 0, define Ω Hl =
v ∈ L2 : vl :=
1/2 ∇α v20
<∞ .
|α|≤l 1 l Let Hl0 = Hl ∩ H 0 . Let C be the space of all l times differentiable functions on Ω with ul,∞ := |α|≤l ∇α v∞ < ∞. For s > 0 we denote by Hs the space of all distributions u defined in Ω such that us < ∞ where s = n + σ is nonnegative and is not an integer. The norm is defined by
us =
un + |η|=n
Ω×Ω
|Dη u(x) − Dη u(y)|2 dx dy |x − y|2+2σ
1/2 .
We denote by Hs0 = Hs ∩ H10 and H−s the dual space of Hs0 normed by f −s =
sup
0=v∈Hs0
f, v vs
where , denote the duality pairing. s s s Denote by the bold face H = H ×H . Similar for other spaces. For simplicity, 2 2 we define [v, χ]s = vs + χs . We use C to denote a generic positive constant. Note that C may take different values in different places. The paper is organized as follows. In Section 2 a solution formula is constructed. Using this, existence and regularity are established in a fractional Sobolev space. In Section 3, based on the corner singularities of the Laplace equation, the solution is split into singular and regular parts up to the second-order of singularities. In Section 4 a high-order expansion of corner singularities is derived by introducing new associate singular functions.
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2 Existence We consider some bilinear forms for (1.1) and associate operators with the forms. We define a solution operator and show an unique existence. When no corner singularity is split from the solution of (1.1), a best regularity result that the solution can have is established in a fractional Sobolev space (see Lemma 2.2). Let V = H10 , M = L2 and V = V × V . Define bilinear forms a(v, w) on V × V, b(χ, v) on M × V and c(p, χ) on V × V as follows:
a(u, v) = ∇u · ∇v dx, b(χ, v) = χ divv dx, Ω
Ω c(η, χ) = ∇p · ∇χ + −1 Zp χ dx. Ω
Using the forms, we associate the following problem: for [f , g] ∈ V × V , find [u, p] ∈ V × V such that µ a(u, v) − b(p, v) = f , v , ∀v ∈ V, c(p, χ) + b(χ, u) = g, χ , ∀χ ∈ V,
(2.1) (2.2)
where . , . is the duality pairing between V and V or V and V . We next associate with the forms a, b and c, operators A : V −→ V , defined by
div : V −→ M ,
∇ : M −→ V ,
C : V −→ V
(2.3)
Av, w = a(v, w), v, w ∈ V,
divv, χ = b(v, χ), v ∈ V, χ ∈ M,
∇χ, v = −b(v, χ), v ∈ V, χ ∈ M,
Cp, χ = c(p, χ), p ∈ V, χ ∈ V.
Using the operators, (2.1) and (2.2) can be expressed by µ Au + ∇p = f in V ,
Cp + divu = g in V .
(2.4) (2.5)
Note that C = A + Z. We next give some properties for the operators defined in (2.3) and define a mapping T which is used in constructing the solution formula. Lemma 2.1 (i) There is a constant α∗ > 0 such that α∗ vV ≤ AvV for all v ∈ V. (ii) Assume that U ∈ C 1 (Ω). Define T = C − µ−1 divA∇, where A (= A−1 ) is the solution operator defined in (1.3). If γ0 > 0, then T : V −→ V has a bounded inverse S := T −1 : V −→ V , with a bound S = sup
0=h∈V
ShV ≤ 1/ min{, γ0 }. hV
(2.6)
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(i) is clear. We show (ii). From T η = h in V and using (2.3),
T η, χ = Cη, χ − µ−1 divA∇η, χ = c( η, χ) − µ−1 b(A∇η, χ),
∀χ ∈ V.
Let u = µ−1 A∇η. So µAu = ∇η in V and −b(u, η) = µa(u, u) ≥ µα∗ u2V = µ−1 α∗ A∇η2V . Since A : V −→ V is bounded and f V ≤ AAf V for all f ∈ V , we have
T η, η = c( η, η) − µ−1 b(A∇η, η) 1 ≥ ∇η2M + (λ − |∇(κU)|∞ )η2M + µ−1 α∗ A∇η2V 2 1 ≥ ∇η2M + (λ − |∇(κU)|∞ )η2M + µ−1 α∗ A−2 ∇η2V 2 ≥ ∇η2M + γ0 η2M , ∀η ∈ V. (2.7) Thus T is bounded below and has a bounded inverse S = T −1 . The estimation for S follows from (2.7). We are going to express u and p in terms of f and g. Indeed, let [f , g] ∈ H−1 × H−1 . From (2.4), u = µ−1 A(f − ∇p) ∈ H10 .
(2.8)
Using this, (2.5) and Lemma 2.1, we have T p + µ−1 divAf = g and p = −µ−1 SdivAf + Sg ∈ H10 .
(2.9)
Inserting the formula p of (2.9) into (2.8), u = µ−1 K1 Af − µ−1 A∇Sg ∈ H10 ,
(2.10)
where I is the identity operator and K1 = I + µ−1 A∇Sdiv. In a compact form the solution is given by u f =M p g
where M=
µ−1 K1 A
−µ−1 A∇S
−µ−1 SdivA
S
(2.11) .
For each fixed > 0, M is a solution operator for (1.1) which maps from H−1 ×H−1 into H10 × H10 . We next show an unique existence and establish a best regularity result that the solution can have when no corner singularity is split.
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¯ Suppose that γ0 is positive. Suppose [u, p] vanishes Lemma 2.2 Let U ∈ C 1 (Ω). outside r = 1. If [f , g] ∈ L2 × L2 , then there is a unique solution [u, p] ∈ H10 × H10 of (1.1), satisfying √ √ √ µu1,Ω1 + ∇p1,Ω1 + γ0 p0,Ω1 ≤ C(f 0 + g0 ). (2.12) Furthermore there exists a constant C such that if 1 ≤ s < s1 and [f , g] ∈ Hs−2 × Hs−2 , then µ us,Ω1 + ps,Ω1 ≤ C(f s−2 + gs−2 ). Proof.
(2.13)
First, (2.12) follows from (2.1) and (2.2). Consider the diagram A
S
∇
A
H−1 −→ H10 −→ L2 −→ H10 ⊂ L2 −→ L2 ⊂ H−1 −→ H10 . div
(2.14)
The following mappings are well defined: A∇SdivA,
A∇S,
SdivA.
(2.15)
By the theory of corner singularity, if s < α + 1 and f ∈ Hs−2 , then Af ∈ Hs (see [6, 7, 9]). Hence the sequences of mappings can be considered: f ∈ Hs−2 ∩ H−1 −→Af ∈ Hs ∩ H10 −→divAf ∈ Hs−1 ∩ L2 −→SdivAf ∈ Hs ∩ H10 −→∇SdivAf ∈ H
s−1
(2.16) s
∩ H −→A∇SdivAf ∈ H ∩ 0
H10 ,
and g ∈ Hs−2 ∩ L2 −→Sg ∈ Hs ∩ H10 −→∇Sg ∈ Hs−1 ∩ H−1 −→A∇Sg ∈ Hs ∩ H10 .
(2.17)
Assuming that [f , g] ∈ Hs−2 × Hs−2 , it follows from (2.16) and (2.17) that K1 Af = [I + µ−1 A∇Sdiv]Af ∈ Hs , s
SdivAf ∈ H ,
A∇Sg ∈ Hs ,
s
Sg ∈ H .
By (2.6), S ∼ min{, γ0 }−1 ∼ −1 if γ0 > 0. So K1 A ∼ 1 + (µ)−1 and A∇S ∼ −1 . Using the operator M, µ us,Ω1 ≤ c1 −1 (f s−2 + gs−2 ),
(2.18)
where c1 is a constant, and
ps,Ω1 ≤ S µ−1 f s−2 + gs−2 ≤ c2 −1 (f s−2 + gs−2 ),
where c2 = C(divA, γ0 ). Thus (2.13) follows with c = max{c1 , c2 }.
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3 Decomposition We first cite a basic result on corner singularities for the Laplacian (see [6, 7, 9]). Let α = π/ω and define sj = jα + 1 for j = 1, 2, . . . Theorem A. There are linear functionals Λj and functions ψj , j = 1, 2, . . . , satisfying the following properties. (i) Λj is a bounded linear functional on Hs−2 for s > sj , but not for s ≤ sj . (ii) ψj ∈ Hs ∩ H10 for s < sj but not for s ≥ sj . Also ψj is smooth everywhere in Ω except at r = 0 and ∆ψj is smooth everywhere ¯ (iii) If sj < s < sj+1 , f ∈ Hs−2 , u is a solution of the Laplace problem: in Ω. −∆u = f in Ω, u = 0 on Γ, and u vanishes outside r = 1, then uj := u −
j
Λi (f )ψj ∈ Hs (Ω1 ), with uj s,Ω1 ≤ Cf s−2 .
i=1
There are many sets of singular functions that may be used in Theorem A. One set is given by the formula (see [7]) jα jα = integer, r sin[jα(θ − ω1 )], (ln r) rk sin[k(θ − ω1 )] + (θ − ω1 )rk cos[k(θ − ω1 )] ψj (x, y) = −(−1)j ω csck ω [−x sin ω1 + y cos ω1 ]k , jα = k = integer. Also specific formulas for the corresponding linear functionals Λj can be found in [7]. In this section, based on Theorem A, we split the first and second leading corner singularities of the Laplacian equation from the exact solution of (1.1). We show that the lowest-order corner singularity of (1.1) is the same as that of the Laplacian. We also show that the second leading singularity can be sorted out after some new associate singularity functions are subtracted from the regular solution of the first step. We shall denote by φj = χψj , j = 1, 2, . . . where χ is the smooth cutoff function. Step 1. s1 < s < s2 . We split the solution [u, p] of (1.1) as follows: u = C˜1 φ1 + u1 ,
p = C13 φ1 + p1
(3.1)
where C˜1 = [C11 , C12 ] and C13 are parameters to be constructed later. Inserting (3.1) into (1.1) and setting f1 = f + µ C˜1 ∆φ1 − C13 ∇φ1 , g1 = g + C13 ( ∆φ1 − Zφ1 ) + C˜1 · ∇φ1 , system (1.1) becomes −µ ∆u1 + ∇p1
=
f1 in Ω ,
− ∆p1 + Zp1 + divu1
=
g1 in Ω ,
u1 , p1
=
0
on Γ.
(3.2)
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We next show a required regularity for f1 and g1 so that the linear functional Λ1 given in Theorem A can be applied. Lemma 3.1 If [f , g] ∈ Hs−2 × Hs−2 for s1 < s < s2 , then [f1 , g1 ] ∈ Hs−2 × Hs−2 . Proof. Since φ1 ∈ Hs for s < s1 , we have ∇φ1 ∈ Ht for t < α. Since 2α − 1 < α, we have ∇φ1 ∈ Hs−2 for s < s2 . Since ∆φ1 is sufficiently smooth, the required result follows. When Theorem A is applied to (3.2), it is required that Λ1 (f1 − ∇p1 ) = 0, Λ1 (g1 − Zp1 − divu1 ) = 0.
(3.3) (3.4)
Using (2.11) and letting K∗ = I + µ−1 ∇SdivA, f1 − ∇p1 = K∗ f1 − ∇Sg1 .
(3.5)
Letting ∇1 = ∇x , ∇2 = ∇y and Sij = −∇i S∇j , Mi = −∇i S∆ + −1 (∇i SZ − K∗ ∇i ),
(3.6)
Inserting f1 and g1 into (3.5), the vector equation (3.3) becomes λ1,11 C11 + λ1,12 C12 + λ1,13 C13 = h1,1 , λ1,21 C11 + λ1,22 C12 + λ1,23 C13 = h1,2 ,
(3.7)
where λ1,11 = µ Λ1 (K∗ ∆φ1 ) + Λ1 (S11 φ1 ), λ1,12 = Λ1 (S12 φ1 ), λ1,13 = Λ1 (M1 φ1 ), λ1,22 = µ Λ1 (K∗ ∆φ1 ) + Λ1 (S22 φ1 ), λ1,21 = Λ1 (S21 φ1 ), λ1,23 = Λ1 (M2 φ1 ), h1,1 = Λ1 (∇x Sg − K∗ f1 ), h1,2 = Λ1 (∇y Sg − K∗ f2 ), f = [f1 , f2 ]. We next derive the algebraic equation for (3.4). Set J1 = ZSdivA − divK1 A, J2 = I − ZS + µ−1 divA∇S, −1
J3 = −(J2 Z + µ
J1 ∇),
Ri = (ZS∇i − ∇i K1 )A∆ + J2 ∇i ,
(3.8)
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where K1 is defined in (2.10). Using (2.11) and (3.8), g1 − Zp1 − divu1 = µ−1 J1 f1 + J2 g1 = C11 (R1 φ1 ) + C12 (R2 φ1 ) + C13 (J2 ∆φ1 + J3 φ1 ) +µ−1 J1 f + J2 g.
(3.9)
Using (3.4), λ1,31 C11 + λ1,32 C12 + λ1,33 C13 = h1,3 ,
(3.10)
where λ1,31 = Λ1 (R1 φ1 ), λ1,32 = Λ1 (R2 φ1 ), λ1,33 = Λ1 (J2 ∆φ1 ) + Λ1 (J3 φ1 ), h1,3 = −Λ1 (µ−1 J1 f + J2 g). Before solving (3.6) and (3.10) for C1j , we need to show thatthe coefficients λ1,ij and h1,j are well defined and the determinant of the matrix λ1,ij 1≤i,j≤3 is not zero. Lemma 3.2 Let s1 < s < s2 . The following functions belong to Hs−2 : K∗ ∆φ1 , K∗ ∇φ1 , Sij φ1 , Mi φ1 , Ri φ1 . Proof.
Consider the following diagram A
S
∇
Hs−2 −→ Hs−1 −→ Hs−2 −→ Hs−1 −→ Hs−2 . div
(3.11)
Since ∆φ1 ∈ Hs−2 and K∗ = I + µ−1 ∇SdivA, we have K∗ ∆φ1 ∈ Hs−2 . Since ∇φ1 ∈ Ht for t < α and 2α − 1 < α, we have ∇φ1 ∈ Hs−2 . Using the diagram (3.11), we see that ∇S∇φ1 , ∇SZφ1 and K1 ∇φ1 belong to Hs−2 . Hence the functions Sij φ1 , Mi φ1 , Ri φ1 are in Hs−2 . Lemma 3.3 Suppose γ0 > 0. The mappings K∗ , J2 and J3 are nontrivial. Proof. Suppose K∗ f = 0 for a nonzero f . We have µf + ∇SdivAf = 0. Set p = −µ−1 SdivAf . So p|Γ = 0 and f − ∇p = 0. Using S = T −1 , Cp = µ−1 divA(∇p − f ) = 0. Since C = A + Z, we have ∇p20 + γ0 p20 ≤ 0, where γ0 = λ − |∇(kU)|∞ . So, if γ0 > 0, p ≡ 0 and f = 0. This is a contradiction. Suppose J2 g = 0 for a nonzero g. Without loss of generality, let g = 0 outside r = 1. Let p = Sg ∈ H10 (Ω1 ). Then J2 g = g − Zp + µ−1 divA∇p = 0. Let
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u = −µ−1 A∇p. Then g − Zp − divu = 0. From p = Sg, we have Cp + divu = g. Thus ∆p = 0 in Ω1 . Since p|∂Ω1 = 0, p ≡ 0 on Ω1 , so Sg = 0. Since S is invertible, g = 0, which is also a contradiction. Suppose J3 p = 0 for a nonzero p. Without loss of generality we assume that p = 0 outside r = 1. Let u = −µ−1 A∇p. Since K1 = I + µ−1 A∇Sdiv, we have J3 p = −J2 Zp − (ZSdiv − divK1 )u = Zp + divu − ZS(Zp + divu) + µ−1 divA∇S(Zp + divu). Let q = S(Zp + divu) ∈ H10 (Ω1 ) and w = −µ−1 A∇q. Then J3 p = Z(p − q) + div(u − w) = 0. From q = S(Zp + divu), we have ∆q = 0 in Ω1 . Thus q ≡ 0 on Ω1 and Zp + divu = 0 in Ω1 . From u = −µ−1 A∇p, we have µ∆u = ∇p in Ω1 . So µ∇u20,Ω1 + γ0 p20,Ω1 ≤ 0. We have p ≡ 0, which is a contradiction. Using Lemmas 3.2 and 3.3 we obtain Lemma 3.4 (a) The coefficients λ1,ij are well defined. (b) Suppose that γ0 > 0. The determinant of the matrix (λ1,ij )1≤i,j≤3 is a quadratic equation in µ and if µ is not a root of the quadratic equation, the components of C1 solve (3.7) and (3.10). (c) If [f , g] ∈ Hs−2 × Hs−2 , then |C1 | =
3
|C1i | ≤ C(f s−2 + gs−2 ).
(3.12)
i=1
Proof. (a) From Lemma 3.2, the coefficients λ1,ij are well defined. (b) The determinant for the matrix (λ1,ij ) is expressed by d1 (, µ) = a21 (a2 + a3 )µ2 + (c1 + c2 )µ + c3
(3.13)
where ci are some constants and a1 = Λ1 (K∗ ∆φ1 ), a2 = Λ1 (J2 ∆φ1 ), a3 = Λ1 (J3 φ1 ). From Lemma 3.3, the numbers ai = 0. Let = −a3 /a2 . If µ is large enough or if µ is not a root of d1 (, µ) = 0, then d1 (, µ) = 0. Hence the equations (3.7) and (3.10) are solvable for C1j . Estimating h1,j , (3.12) follows. Using Lemmas 3.2 and 3.4, we obtain Theorem 3.1 Let s1 < s < s2 . Suppose γ0 > 0. Assume that [f , g] ∈ Hs−2 × Hs−2 . Let [u, p] be the solution of (1.1) with [u, p] ≡ [0, 0] for r ≥ 1. Then the solution is split as follows: [u, p] = C1 φ1 + [u1 , p1 ],
[u1 , p1 ] = [u, p] − C1 φ1 .
(3.14)
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The constant C1 = [C˜1 , C13 ] = [C11 , C12 , C13 ] is the solution of the equations (3.7), (3.10) and satisfies (3.12). The regular part satisfies µu1 s,Ω1 + p1 s,Ω1 ≤ C[f s−2 + gs−2 ]. Proof. Using Lemma 3.4, C1 solves (3.7) and (3.10), so (3.14) follows from (3.1). We next apply Theorem A to the equations −∆u1 = F and −∆p1 = G where F = µ−1 (f1 − ∇p1 ), G = −1 (g1 − Zp1 − divu1 ). Using (3.5) and (3.9), F = µ−1 (K∗ f1 − ∇Sg1 ),
G = −1 (µ−1 J1 f1 + J2 g1 ).
So µ u1 s,Ω1 ≤ CFs−2 ≤ C(K∗ f1 s−2 + ∇Sg1 s−2 ) p1 s,Ω1
≤ c1 −1 (f1 s−2 + g1 s−2 ), ≤ CGs−2 ≤ C(µ−1 J1 f1 s−2 + J2 g1 s−2 ) ≤ c2 (f1 s−2 + g1 s−2 ),
where ci are generic constants. The constants c1 and c2 are finite because K∗ and ∇S are bounded mappings from Hs−2 into Hs−2 , and J1 , J2 from Hs−2 into Hs−2 . Computing f1 s−2 and g1 s−2 and using (3.12), the inequality follows. Before stopping this step, using (3.6), (3.8), φj and Λj , we define the following numbers that will be used later: for integer j ≥ 1, λj,11 = µ Λj (K∗ ∆φj ) + Λj (S11 φj ), λj,12 = Λj (S12 φj ), λj,13 = Λj (M1 φj ), λj,22 = µ Λj (K∗ ∆φj ) + Λj (S22 φj ), λj,21 = Λj (S21 φj ), λj,23 = Λj (M2 φj ),
(3.15)
λj,31 = Λj (R1 φj ), λj,32 = Λj (R2 φj ), λj,33 = Λj (J2 ∆φj ) + Λj (J3 φj ). / Hs−2 . To extract the second leading corner Step 2. s2 < s < s3 . Recall that ∇φ1 ∈ singularity from [u1 , p1 ], the following terms −C13 ∇φ1 , C˜1 ·∇φ1 and −C13 Zφ1 must be removed in the functions f1 and g1 in (3.2), respectively. For this, let fs,1 = −C13 ∇φ1 ,
gs,1 = C˜1 · ∇φ1 − C13 Zφ1 .
(3.16)
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Define
An Expansion of Solution for a Regularized Compressible Stokes System
η1,1 = η1 − µ−1 Λ1 (χ fs,1 )φ1 , τ1,1 = τ1 − −1 Λ1 (χ gs,1 )φ1 ,
181
(3.17)
where η1 and τ1 are the solutions of the problems −µ∆η1 = χ fs,1 in Ω, η1 = 0 on Γ, − ∆τ1 = χ gs,1 in Ω, τ1 = 0 on Γ. Since ∇φ1 ∈ Ht−2 for s1 < t < s2 , and from Theorem A, we have [η1,1 , τ1,1 ] ∈ Ht × Ht for s1 < t < s2 . Define ¯ 1 = u1 − η1,1 , u p¯1 = p1 − τ1,1 . Then [¯ u1 , p¯1 ] ∈ Ht × Ht for s1 < t < s2 and solves −µ ∆¯ u1 + ∇¯ p1
= ¯f1 in Ω,
− ∆¯ p1 + Z p¯1 + div¯ u1 ¯ 1 , p¯1 u
= g¯1 in Ω, = 0 on Γ,
(3.18)
where ¯f1 = f + [µ C˜1 − Λ1 (χ fs,1 )]∆φ1 + (1 − χ)fs,1 − ∇τ1,1 ,
(3.19)
g¯1 = g + [ C3 − Λ1 (χ gs,1 )]∆φ1 + (1 − χ)gs,1 − (Zτ1,1 + div η1,1 ). We next show a regularity result for [η1,1 , τ1,1 ] and [¯f1 , g¯1 ]. Lemma 3.5 (i) If s1 < s < s2 , [η1,1 , τ1,1 ]s ≤ C[fs,1 , gs,1 ]s−2 ≤ C|C1 |. (ii) If s2 < s < s3 , [¯f1 , g¯1 ]s−2 ≤ C[f , g]s−2 and [η1,1 , τ1,1 ]s−1 ≤ C|C1 |, provided that [f , g] ∈ Hs−2 × Hs−2 . Proof. (i) Since ∇φ1 ∈ Ht for all t < 2α, and since 3α − 1 < 2α, we have ∇φ1 ∈ Ht for all t < 3α − 1. Using Theorem A, we have, for all t < s2 , [η1,1 , τ1,1 ]t ≤ C[χfs,1 , χgs,1 ]t−2 ≤ C|C1 |.
(3.20)
(ii) For s2 < s < s3 we have, using (3.20), ∇τ1,1 s−2 + Zτ1,1 + divη1,1 s−2 ≤ C[η1,1 , τ1,1 ]s−1 ≤ C|C1 |. Using (3.12), (3.19) and (3.21), [¯f1 , g¯1 ]s−2 ≤ [f , g]s−2 + C(µ + + 2)|C1 | +|C1 |(1 − χ)(|∇φ1 | + Zφ1 )s−2 +∇τ1,1 s−2 + Zτ1,1 + divη1,1 s−2 ≤ C[f , g]s−2 .
(3.21)
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We now subtract the second singular function φ2 from [¯ u1 , p¯1 ]. Define ¯ 1 − C˜2 φ2 , u2 = u
p2 = p¯1 − C23 φ2
(3.22)
where C2 = [C˜2 , C23 ] will be constructed soon. Using (3.22) and (3.18), −µ ∆u2 + ∇p2 − ∆p2 + Zp2 + divu2
= =
f2 in Ω , g2 in Ω ,
u2 , p2
=
0
(3.23)
on Γ,
where f2 = ¯f1 + µ C˜2 ∆φ2 − C23 ∇φ2 , g2 = g¯1 + C23 ( ∆φ2 − Zφ2 ) − C˜2 · ∇φ2 . We are going to construct C2 . In the view of Theorem A, we must require that Λ2 (f2 − ∇p2 ) = 0, Λ2 (g2 − Zp2 − divu2 ) = 0.
(3.24) (3.25)
From(3.24) and (3.25), and using the same procedures as used in Step 1, λ2,11 C21 + λ2,12 C22 + λ2,13 C23 = h2,1 , λ2,21 C21 + λ2,22 C22 + λ2,23 C23 = h2,2 , λ2,31 C31 + λ2,32 C32 + λ2,33 C33 = h2,3 ,
(3.26)
where the coefficients λ2,ij are given in (3.15) and h2,1 = Λ2 (∇x S¯ g1 − K∗ f¯11 ), g1 − K∗ f¯12 ), h2,2 = Λ2 (∇y S¯ h2,3 = −Λ2 (µ−1 J1¯f1 + J2 g¯1 ), where ¯f1 = [f¯11 , f¯12 ] and g¯1 are given in (3.19). We are going to solve the algebraic system (3.26) for C2j and show that λ2,ij and h2,j are well defined and its determinant is not zero. The following lemmas enable us to do them. Lemma 3.6 Let s2 < s < s3 . The following functions belong to Hs−2 : K∗ ∆φ2 , K∗ ∇φ2 , Sij φ2 , Mi φ2 , Ri φ2 . Proof. Since ∆φ2 ∈ Hs−2 and K∗ = I + (µ)−1 ∇SdivA, we have K∗ ∆φ2 ∈ Hs−2 . Since ∇φ2 ∈ Ht for t < 2α, and 3α − 1 < 2α, we have ∇φ2 ∈ Hs−2 . Furthermore ∇S∇φ2 , ∇SZφ2 and K1 ∇φ2 belong to Hs−2 . So Sij φ2 , Mi φ2 , Ri φ2 are in Hs−2 .
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Lemma 3.7 (a) The coefficients λ2,ij are well defined. (b) Suppose γ0 > 0. The determinant of the matrix (λ2,ij )1≤i,j≤3 is not zero and the parameters C2j (j = 1, . . . , 3) is the solution of (3.26). (c) If [f , g] ∈ Hs−2 × Hs−2 , then |C2 | =
3
|C2i | ≤ C(f s−2 + gs−2 ).
(3.27)
i=1
Proof.
The proof is similar to the one of Lemma 3.4.
Using Lemmas 3.6 and 3.7, we obtain Theorem 3.2 Let s2 < s < s3 . Let [f , g] ∈ Hs−2 × Hs−2 . Suppose γ0 > 0. Let [u, p] be the solution of (1.1) with [u, p] ≡ [0, 0] for r ≥ 1. Then u2 = u −
2
C˜i φi − η1,1 ,
p2 = p −
i=1
2
Ci3 φi − τ1,1
(3.28)
i=1
where C1 is defined in Step 1, and C2 = [C˜2 , C23 ] is the solution of (3.26) and satisfies (3.27). Furthermore µu2 s,Ω1 + p2 s,Ω1 ≤ C(f s−2 + gs−2 )
(3.29)
where C is a constant. Proof. The proof is similar to the one of Theorem 3.1. Using (2.11) and (3.23), we have u2 = µ−1 (K∗ f2 − ∇Sg2 ) and p2 = −1 (µ−1 J1 f2 + J2 g2 ), so µu2 s,Ω1 ≤ c1 (f2 s−2 + −1 g2 s−2 ) and p2 s,Ω1 ≤ c2 (µ−1 f2 s−2 + g2 s−2 ). Estimating f2 s−2 , g2 s−2 , (3.29) follows.
4 High-order expansion Based on the corner singular functions φ2 , φ3 , φ4 , . . . , for the Laplace equation (see Theorem A) and also introducing new associate singular functions [ηj , τj ] to be defined below, we derive a high-order singular expansion for (1.1). Setting fs,2 = −(C23 ∇φ2 + ∇τ1,1 ), gs,2 = C˜2 · ∇φ2 − C23 Zφ2 − (Zτ1,1 + divη1,1 ), the functions f2 and g2 in (3.23) are rewritten by f2 = f + µ g2 = g +
2 i=1 2 i=1
C˜i ∆φi − Λ1 (χfs,1 )∆φ1 + (1 − χ)fs,1 + fs,2 ,
(4.1)
Ci3 ∆φi − Λ1 (χgs,1 )∆φ1 + (1 − χ)gs,1 + gs,2 .
(4.2)
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Let χ be a smooth function which vanishes for r > 1 and which is 1 in a neighborhood of the origin. Let j ≥ 2 be an integer. Assume that the parameters Ci = [Ci1 , Ci2 , Ci3 ] (1 ≤ i ≤ j) have been constructed in the previous steps. Define the associate singular functions [ηj , τj ] to satisfy −µ∆ηj = χfs,j in Ω,
ηj = 0 on Γ,
(4.3)
− ∆τj = χgs,j in Ω,
τj = 0 on Γ,
(4.4)
where fs,j = −(Cj3 ∇φj + ∇τj−1,1 ), gs,j = C˜j · ∇φj − Cj3 Zφj − (Zτj−1,1 + div ηj−1,1 ). Define the smoother part of the associate singular function as follows: −1
ηj,1 = ηj − µ
j
Λi (χfs,j )φi ,
(4.5)
Λi (χgs,j )φi .
(4.6)
i=1
τj,1 = τj − −1
j i=1
Next we subtract the smoother part [ηj,1 , τj,1 ] from [uj , pj ] so that the resulted right-hand sides at the (j + 1)th-step have an enough regularity. Define ¯ j = uj − ηj,1 , u
p¯j = pj − τj,1 .
(4.7)
Then [¯ uj , p¯j ] ∈ Ht × Ht for sj < t < sj+1 and solves −µ ∆¯ uj + ∇¯ pj
= ¯fj
in Ω,
− ∆¯ pj + Z p¯j + div¯ uj ¯ j , p¯j u
= g¯j
in Ω,
= 0
on Γ,
(4.8)
where ¯fj = f + µ
j
C˜i ∆φi + (1 − χ)
i=1
g¯j = g +
j
fs,i + Φj (χfs,j , φ) − ∇τj,1 ,
(4.9)
i=1
j
Ci3 ∆φi + (1 − χ)
i=1
j
gs,i + Φj (χgs,j , φ) − (Zτj,1 + divηj,1 ),
i=1
Φj (zj , φ) = − Λ1 (z1 )∆φ1 +
2 i=1
Λi (z2 )∆φi + · · · +
j
Λi (zj )∆φi .
i=1
Next we establish regularities for [fs,j , gs,j ], [ηj,1 , τj,1 ] and [¯fj , g¯j ].
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185
Lemma 4.1 Let j ≥ 2 be an integer. (a) For sj < s < sj+1 , we have [fs,j , gs,j ]s−2 j ≤ C i=1 |Ci | and [ηj,1 , τj,1 ]s ≤ C[fs,j , gs,j ]s−2 . (b) For sj+1 < s < sj+2 , we j have [ηj,1 , τj,1 ]s−1 ≤ C i=1 |Ci | and [¯fj , g¯j ]s−2 ≤ C[f , g]s−2 , provided that [f , g] ∈ Hs−2 × Hs . Proof. The proof follows by an induction argument. The case j = 2 easily follows. Assume that they are true for j − 1. If s < sj , then φj ∈ Hs , and [ηj−1,1 , τj−1,1 ] ∈ Hs × Hs for all s < sj by the assumption, so the functions: ∇φj , Zφj , ∇τj−1,1 , Zτj−1,1 , divηj−1,1 belong to Ht or Ht for t < jα. Hence, since jα − (sj+1 − 1) = 1 − α > 0, we have [fs,j , gs,j ] ∈ Hs−2 × Hs−2 for all s < sj+1 . Thus, when we apply Theorem 3.1 to (4.3)–(4.6), we have [ηj,1 , τj,1 ] ∈ Hs × Hs for s < sj+1 . Since j 2 Φj (χfs,j , φ)s−2 ≤ C |Λ1 (χfs,1 )| + |Λi (χfs,2 )| + · · · + |Λi (χfs,j )| i=1
i=1
≤ C[f , g]s−2 and similarly Φj (χgs,j , φ)s−2 ≤ C[f , g]s−2 , we have [¯fj , g¯j ]s−2 ≤ [f , g]s−2 + C(µ + + 1)
j
|Ci | + C[ηj,1 , τj,1 ]s−1
i=1
+[Φj (χfs,j , φ), Φj (χgs,j , φ)]s−2 ≤ C[f , g]s−2 . Thus the inequality follows. We split the (j + 1)-th singular function φj+1 from the solution [¯ uj , p¯j ] of (4.8): ¯ j = uj+1 + C˜j+1 φj+1 , u
p¯j = pj+1 + Cj+1,3 φj+1
(4.10)
where C˜j+1 = [Cj+1,1 , Cj+1,2 ] and Cj+1,3 are parameters to be determined later. Inserting (4.10) into (4.8), system (4.8) becomes −µ ∆uj+1 + ∇pj+1 − ∆pj+1 + Zpj+1 + divuj+1
= =
fj+1 in Ω, gj+1 in Ω ,
uj+1 , pj+1
=
0
(4.11)
on Γ,
where fj+1 = ¯fj + µ C˜j+1 ∆φj+1 − Cj+1,3 ∇φj+1 , gj+1 = g¯j + C˜j+1 · ∇φj+1 + Cj+1,3 ( ∆φj+1 − Zφj+1 ). As in the previous steps, we construct the parameter Cj+1 and establish the regularity of the solution [uj+1 , pj+1 ] of (4.11).
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Let l = j + 1, for simplicity. Requiring that Λl (fl − ∇pl ) = 0,
(4.12)
Λl (gl − Zpl − divul ) = 0,
(4.13)
one can have, like the previous steps, λl,11 Cl1 + λl,12 Cl2 + λl,13 Cl3 = hl,1 , λl,21 Cl1 + λl,22 Cl2 + λl,23 Cl3 = hl,2 , λl,31 Cl1 + λl,32 Cl2 + λl,33 Cl3 = hl,3 ,
(4.14)
where the coefficients λl,ij are given in (3.15) and gj − K∗ f¯j1 ), hl,1 = Λl (∇x S¯ hl,2 = Λl (∇y S¯ gj − K∗ f¯j2 ), hl,3 = −Λl (µ−1 J1¯fj + J2 g¯j ),
(4.15) ¯fj = [f¯j1 , f¯j2 ].
Using (2.11) and like Lemmas 3.4 and 3.7, one can show that the coefficients λl,ij are well defined. The determinant is of the form dl (µ, ) = a21 (a2 + a3 )µ2 + (c1 + c2 )µ+ c3 , where a1 = Λl (K∗ ∆φl ), a2 = Λl (J2 ∆φl ), a3 = −Λl (J2 Zφl ), which are not zero by Lemma 3.3. So the determinant is not zero by the proof of Lemma 3.4. Thus Cl = [Cl1 , Cl2 , Cl3 ] solves (4.14) and satisfies |Cl | ≤ C(¯fj s−2 + ¯ gj s−2 ) ≤ C[f , g]s−2 .
(4.16)
Since [fl , gl ] ∈ Hs−2 × Hs−2 (s < sl+1 ), and using (2.12) and (4.11), µul = K1 Afl − A∇Sgl , pl = −µ−1 SdivAfl + Sgl , which are in Hs (Ω) and Hs (Ω), respectively. Summarizing all results obtained and the ones of Section 3, we obtain: Theorem 4.1 Suppose that γ0 > 0, in other words, either λ is large enough or U is close to a constant function. Assume that [u, p] vanish for r > 1. For an integer l ≥ 1, if sl < s < sl+1 and [f , g] ∈ Hs−2 × Hs−2 , then [u, p] = [us , ps ] + [uR , pR ] with [uR , pR ] = [u, p] − [us , ps ] and [us , ps ] =
l i=1
[C˜i , Ci3 ]φi +
l−1
[ηi,1 , τi,1 ].
i=1
Here [ηi,1 , τi,1 ] is the smoother part of the new associate singular function [ηi , τi ] defined by (4.3) and (4.4), respectively, the constant [C˜i , Ci3 ] with C˜i = [Ci1 , Ci2 ] is the solution of (4.14) and satisfies (4.16). Moreover, the regular part [uR , pR ] satisfies µuR s,Ω1 + pR s,Ω1 ≤ C(f s−2 + gs−2 ) where C is a constant.
Vol. 5, 2004
Proof.
An Expansion of Solution for a Regularized Compressible Stokes System
The proof is similar to the ones of Theorems 3.2 and 3.3.
187
Thus Theorem 1.1 is obtained, which follows from a combination of Lemma 2.2 and Theorem 4.1.
References [1] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] J.D. Anderson, Jr., Fundamentals of Aerodynamics, 2nd ed., McGraw-Hill, New York, 1991. [3] H. Beir˜ ao da Veiga, An Lp -theory for the n-dimensional, Stationary, Compressible Navier-Stokes Equations, and the Incompressible Limit for Compressible Fluids. The Equilibrium Solutions, Commun. Math. Phys. 109, 229– 248 (1987). [4] M. Dauge, Stationary Stokes and Navier-Stokes systems on two- or threedimensional domains with corners. Part I: Linearized Equations, SIAM J. Math. Anal. 20, 74–97 (1989). [5] V. Girault, P.-A. Raviart, Finite element methods for Navier-Stokes equations: Theory Algorithms, Springer-Verlag, 1986. [6] P. Grisvard, Elliptic problems in non-smooth domains, Pitman Advanced Publishing Program, Boston. London. Melbourne, 1985. [7] R.B. Kellogg, Corner singularities and singular perturbations, Ann. Univ. Ferrara – Sez. VII – Sc. Mat., Vol. XLVII, 177–206 (2001). [8] R.B. Kellogg, J. E. Osborn, A regularity result for the Stokes problem in a convex polygon, J. Funct. Anal. 21, 397–431 (1976). [9] V.A. Kozlov, V.G. Maz’ya, J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, AMS, 2001. [10] J.R. Kweon, R.B. Kellogg, Compressible Stokes problem on non-convex polygon, J. Differential Equations 176, 290–314 (2001). [11] J.R. Kweon, R.B. Kellogg, Regularity of solutions to the Navier-Stokes equations for compressible barotropic flows on a polygon, Arch. Rational Mech. Anal. 163 1, 35–64 (2002). [12] J.R. Kweon, A regularity result of solution to the compressible Stokes equations on a convex polygon, to appear in ZAMP. [13] J.L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications, Springer-Verlag Berlin Heidelberg New York, 1972.
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[14] S.A. Nazarov, A. Novotny, K. Pileckas, On steady compressible Navier-Stokes equations in plane domains with corners, Math. Ann. 304 1, 121–150 (1996). Jae Ryong Kweon1 Department of Mathematics Pohang University of Science and Technology Pohang 790–784 Korea email:
[email protected] Communicated by Rafael D. Benguria submitted 14/11/02, revised 12/08/03, accepted 04/10/03
To access this journal online: http://www.birkhauser.ch
1 This
work was supported by Korea Research Foundation Grant(KRF–2001–015–DS0002)
Ann. Henri Poincar´e 5 (2004) 189 – 201 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/010189-13 DOI 10.1007/s00023-004-0165-9
Annales Henri Poincar´ e
Outgoing Radiation from an Isolated Collisionless Plasma Simone Calogero Abstract. The asymptotic properties at future null infinity of the solutions of the relativistic Vlasov-Maxwell system whose global existence for small data has been established by the author in a previous work are investigated. These solutions describe a collisionless plasma isolated from incoming radiation. It is shown that a non-negative quantity associated to the plasma decreases as a consequence of the dissipation of energy in form of outgoing radiation. This quantity represents the analogue of the Bondi mass in general relativity.
1 Introduction and main results The dynamics of a collisionless plasma in interaction with the mean electromagnetic field generated by the charges is described by the relativistic Vlasov-Maxwell system. In this model the unknowns are the electromagnetic field (E, B) and a set of N non-negative functions fα which give the distributions in phase space of N different species of particles. The system consists of the Vlasov equation ∂t fα + pα · ∂x fα + qα (E + pα ∧ B) · ∂p fα = 0,
∀ α = 1, . . . , N,
coupled to the Maxwell equations with charge density ρ and current density j given by ρ(t, x) = R3
N
qα fα dp,
j(t, x) =
α=1
N
R3
qα fα pα dp.
(1.1)
α=1
In the previous equations, t ∈ R is the time, x ∈ R3 , p ∈ R3 are the position and the momentum of the particles, qα the charge of a particle of species α, p pα = , m2α + p2
p2 ≡ |p|2
denotes the relativistic velocity and mα is the mass of a particle of species α. Units are chosen so that the speed of light is equal to unity. The relativistic Vlasov-Maxwell system has several applications in plasma physics and in astrophysics, where it is used for instance to model the dynamics of the solar wind. Many mathematical problems remain unsolved. For example, existence of global classical solutions is known only under certain restrictions on the size of the initial data (see [2, 5, 6, 10]); global existence for large data has
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been proved for a modified version of the system in which the particle density is forced to have compact support in the momentum (see [4]). An important feature of the dynamics, which is due to its relativistic character, is the presence of radiation fields. The radiation is defined as the part of the electromagnetic field which carries energy to null infinity. It is distinguished in outgoing radiation, which propagates energy to the future null infinity I + , and incoming radiation, which propagates energy to the past null infinity I − . The latter can be interpreted as a flux of energy flowing in onto the system from I − . For an isolated system the incoming radiation should be ruled out by appropriate boundary conditions, which will be now briefly discussed. The amount of energy Ein (v1 , v2 ) carried to I − by the incoming radiation in the interval [v1 , v2 ] of the advanced time, v = t + |x|, can be formally calculated by the limit v2 Ein (v1 , v2 ) = − lim [S · k](s − r, x)dx ds, (1.2) r→+∞
v1
|x|=r
where k = x/|x| and S is the Poynting vector, S = (4π)−1 (E ∧ B). It should be emphasized that (1.2) is only a formal definition, since it is not known in general whether the above limit exists for a solution of the relativistic Vlasov-Maxwell system. Analogously, the limit u2 [S · k](s + r, x)dx ds (1.3) Eout (u1 , u2 ) = lim r→+∞
u1
|x|=r
gives the energy which is propagated to I + by the outgoing radiation in the interval [u1 , u2 ] of the retarded time, u = t − |x|. A solution of the relativistic Vlasov-Maxwell system is isolated from incoming radiation if Ein (v1 , v2 ) = 0, for all v1 , v2 ∈ R. In [2] it was proved that for small data this system admits solutions which satisfy this property. These solutions are defined by replacing the Maxwell equations with the retarded part of the field. The resulting system has been called retarded relativistic Vlasov-Maxwell system and reads ∂t fα + pα · ∂x fα + qα (Eret + pα ∧ Bret ) · ∂p fα = 0, ∀ α = 1, . . . , N, dy , (∂x ρ + ∂t j)(t − |x − y|, y) Eret (t, x) = − |x − y| 3 R dy , Bret (t, x) = (∂x ∧ j)(t − |x − y|, y) |x − y| 3 R
(1.4) (1.5) (1.6)
where ρ and j are given by (1.1). The purpose of this paper is to derive information about the asymptotic behaviour at future null infinity of such isolated solutions, i.e., to study the properties of the outgoing radiation generated by the plasma. Let us first recall, for sake of reference, the global existence result of [2]. Define P = sup{|p| : (x, p) ∈ suppfα (t), t ∈ R, 1 ≤ α ≤ N }
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and denote by BR (0) the sphere in R6 with center in the origin and radius R > 0 and by λ the set of constants {qα , mα }. Theorem 1 Let fαin (x, p) ≥ 0 be given in C02 (R3 × R3 ) such that fαin = 0 for 2 (x, p, α) ∈ BR (0)c × {1, . . . , N }. Define ∆ = α |µ|=0 ∇µ fαin ∞ , where µ ∈ N6 is a multi-index. Then there exists a positive constant ε = ε(R, λ) such that for ∆ ≤ ε the retarded relativistic Vlasov-Maxwell system has a unique global solution {fα } ∈ (C 2 )N which satisfies fα (0, x, p) = fαin (x, p). Moreover Fret = (Eret , Bret ) ∈ C 1 (R × R3 ) and there exists a positive constant C = C(R, λ) such that P ≤ C and the following estimates hold for all (t, x) ∈ R × R3 : |Fret (t, x)| ≤ C∆(1 + |t| + |x|)−1 (1 + |t − |x||)−1 ,
(1.7)
ρ(t, x) ≤ C∆(1 + |t| + |x|)−3 .
(1.8)
The estimate (1.7) shows that the solution of Theorem 1 is isolated from incoming radiation in the sense specified above. We note that the statement of Theorem 1 differs from the main result of [2] in two aspects. Firstly in [2] only the case of a single species of particles is considered. However the restriction to this case has been made only to simplify the notation and the generalization of the result of [2] to the case of a mixture is straightforward. Secondly we claim here that the distribution functions are twice continuously differentiable, whereas in [2] we only proved that they are C 1 . We shall return to this point at the end of the introduction. We state now the main results of this paper. Let us define 1 2 2 |Eret | + |Bret | + e(t, x) = m2α + p2 fα dp, 8π 3 R α 1 (Eret ∧ Bret ) + p(t, x) = 4π α
R3
pfα dp,
the local energy and momentum of a solution of (1.4)–(1.6), respectively. We also set [e − p · k](u + |x|, x) dx (1.9) M∨ (u) = R3
∨
and note that M (u) is non-negative. Theorem 2 Let {fα } ∈ (C 2 )N be a solution of the retarded relativistic VlasovMaxwell system with data as stated in Theorem 1, such that Fret ∈ C 1 (R × R3 ) and fα (t, x, p) = 0, ∀x ∈ R3 : |x| ≥ R + a|t|, α = 1, . . . , N, (1.10) for some a ∈ [0, 1). Then the limit F rad (u, k) =
lim
|x|→+∞
|x|Fret (u + |x|, x)
(1.11)
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exists and is attained uniformly in k = x/|x| ∈ S 2 and u ∈ K, for all K ⊂ R compact. Moreover the radiation field F rad = (E rad , B rad ) satisfies the following algebraic properties: E rad · B rad = 0,
|E rad | = |B rad |,
E rad · k = B rad · k = 0, (E rad ∧ B rad ) · k = |E rad |2 . Theorem 3 Let {fα } ∈ (C 2 )N be a solution of the retarded relativistic VlasovMaxwell system with data as stated in Theorem 1 and such that Fret ∈ C 1 (R× R3 ). Assume the following estimates hold (i) P ≤ C (ii) |Fret (t, x)| ≤ C(1 + |t| + |x|)−1 (1 + |t − |x||)−1 (iii) ρ(t, x) ≤ C(1 + |t| + |x|)−3 for some positive constant C and for all (t, x) ∈ R × R3 . Then M∨ ∈ C 1 and the following equation is satisfied: d 1 M∨ (u) = − |E rad (u, k)|2 dk. (1.12) du 4π S 2 Note that the conclusions of Theorems 2 and 3 apply to the solution of Theorem 1. However the proofs do not require the solution to be small. In fact the proofs of Theorems 2 and 3 do not make use explicitly of the Vlasov equation either. The only tools which enter into play are the continuity equation, ∂t ρ + ∇x · j = 0, and the local energy conservation law, ∂t e+∇x ·p = 0, which are of course satisfied by any “good” matter model. However since the existence of global solutions of the retarded relativistic Vlasov-Maxwell system which satisfy the assumptions of Theorems 2 and 3 is known by Theorem 1, we restrict ourselves to consider this case. Let us now comment the meaning of the results stated above. In Theorem 2 it is claimed that the retarded field generated by the charge distribution is asymptotically null and outwardly directed along the future pointing null geodesics. (We recall that an electromagnetic field (E, B) is said to be null if E · B = 0 and |E| = |B|, cf. [11], page 322.) This result follows essentially from [7]. However, for sake of completeness and to help the reader who is not familiar with the formalism used in [7], we give in Section 2 a complete proof of Theorem 2 adapted to our case. The property (1.10), which for the solution of Theorem 1 follows from the estimate P ≤ C, is in turn a special case of the assumption made in [7] that the matter has to be contained in a timelike world-tube. With regard to Theorem 3, it shows that the function M∨ plays in the context of the retarded relativistic Vlasov-Maxwell system the same role as the Bondi mass in general relativity (cf. [1, 12]). In fact, M∨ is non-increasing and its variation
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on the interval [u1 , u2 ] equals the energy dissipated in form of outgoing radiation in such interval of the retarded time, as it follows from (1.3) and Theorem 2. Although the Bondi mass loss formula in general relativity is extensively studied, it seems the first time that its generalization to plasma physics is considered and that (1.12) appears in the literature. It should be mentioned, however, that a rigorous mathematical derivation of the Bondi formula in general relativity is a much more difficult task (cf. [3, 8]). Let us now deal with the technical point concerning the regularity of the solution. We consider for simplicity the system for a single species of particle and denote by fret the unique global solution for small C 2 data. In [2] we stated that fret ∈ C 1 ; here we claim that fret ∈ C 2 . This gain of regularity can be easily understood by appealing to the smoothing effect which was pointed out in [9]. We recall that the solution of the Vlasov equation can be represented as fret = f in (X(0), P (0)) where X(s), P (s) are the characteristics of (1.4) and are given by s P(τ ) dτ, X(s) = x + P (s) = p + t
t s
Eret (τ, X) + P ∧ Bret (τ, X) dτ.
It was observed in [9] that the time integral of the field evaluated on the characteristics is one derivative smoother than the field itself provided that X(s) is a timelike curve. The latter condition is satisfied by the solution of [2] in virtue of the estimate P ≤ C. Hence the characteristics are C 2 and since f in is also given as a C 2 function, then the solution of the Vlasov equation itself is twice continuously differentiable.
2 Algebraic properties of the radiation field In this section we prove Theorem 2. Let us denote by φ any of the components of the electromagnetic field, i.e., we set φ = Ei or Bi and define
−(∂t ji + ∂xi ρ) for the electric field E, F = for the magnetic field B. (∂x ∧ j)i Then F ∈ C 1 (R × R3 ) and by means of (1.10), F (t, x) = 0 for |x| ≥ R + a|t|. The retarded field defined by (1.5), (1.6) has the form dy , F (t − |x − y|, y) φ(t, x) = |x − y| Ξa (t,x) where Ξa (t, x) = {y ∈ R3 : |y| ≤ R + a|t − |x − y||}, which is a compact set for any fixed t ∈ R, x ∈ R3 . We have the following
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Ann. Henri Poincar´e
lim
|x|→+∞
|x| φ(u + |x|, x) =
F (u + k · y, y) dy, Ωa (u)
uniformly in (u, k) ∈ K × S 2 , for all K ⊂ R compact, where Ωa (u) = {y ∈ R3 : |y| ≤ (1 − a)−1 (R + a|u|)}. Proof. The crucial point to prove this lemma is that for t = u + |x| and for |x| large, i.e., where we need to evaluate the function φ, the domain of integration Ξa (t, x) is contained in Ωa (u), a compact set whose measure depends only on the fixed u. Let ρ = |x|−1 and define g(ρ) = [1 − 2ρk · y + ρ2 |y|2 ]−1/2 =
h(ρ) = =
|x| , |x − y|
ρ−1 [1 − ρk · y − (1 − 2ρk · y + ρ2 |y|2 )1/2 ] |x| − k · y − |x − y|, for ρ = 0
and put h(0) = 0, h (0) = 32 (k · y)2 − 12 |y|2 , so that h ∈ C 1 (R) (here the prime denotes the derivative with respect to ρ). Then we have F (u + k · y + h(ρ), y) g(ρ) dy. lim |x| φ(u + |x|, x) = lim ρ→0
|x|→+∞
Ωa (u)
Setting Gu,k (ρ, y) = F (u + k · y + h(ρ), y)g(ρ), we have to prove that lim Gu,k (ρ, y) dy − Gu,k (0, y) dy = 0, ρ→0
Ωa (u)
Ωa (u)
uniformly in (u, k) ∈ K × S 2 . By the mean value theorem we have Gu,k (ρ, y) = Gu,k (0, y) + ρ Ru,k (ρ, y), where the remainder is bounded as |Ru,k (ρ, y)| ≤ sup{|Gu,k (τ, y)|, 0 ≤ τ ≤ ρ}. Hence
Gu,k (ρ, y) dy − Ωa (u)
Gu,k (0, y) dy
Ωa (u)
≤ ρ sup |Gu,k (τ, y)|, 0 ≤ τ ≤ ρ, y ∈ Ωa (u) Vol[Ωa (u)]. Since G is C 1 and Ωa (u) is compact, the lemma is proved.
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By means of Lemma 1, the radiation field is continuous and is given by Eirad (u, k) = − (∂i ρ + ∂t ji )(u + k · y, y) dy, (2.1) Ωa (u)
(∂x ∧ j)i (u + k · y, y) dy.
Birad (u, k) =
(2.2)
Ωa (u)
It remains to prove the algebraic properties of (E rad , B rad ). In (2.1) we replace the identities ∂t ji (u + k · y, y) = ∂u ji (u + k · y, y), ∂i ρ(u + k · y, y) =
∂yi [ρ(u + k · y, y)] + ∂y · [j(u + k · y, y)]ki −∂u (j · k)(u + k · y, y)ki ,
the second one being a consequence of the continuity equation. After integrating by parts we get rad Ei (u, k) = ∂u [(j · k)ki − ji ](u + k · y, y) dy. Ωa (u)
Let M denote the vector [(j · k)k − j](u + k · y, y) dy.
M (u, k) =
(2.3)
Ωa (u)
Since the integrand function in (2.3) is C 1 and vanishes on the boundary of Ωa (u), then we have E rad = ∂u M . Analogously, from (2.2) we get B rad = ∂u N , where (j ∧ k)(u + k · y, y) dy.
N (u, k) = Ωa (u)
The next lemma describes the algebraic properties of the vectors M , N . Lemma 2 ∀(u, k) ∈ R × S 2 the vector fields defined by (2.3), (2.4) satisfy (1) M (u, k) · k = N (u, k) · k = 0 (2) M (u, k) · N (u, k) = 0 (3) |M (u, k)| = |N (u, k)| (4) (M (u, k) ∧ N (u, k)) · k = |M (u, k)|2 .
(2.4)
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Proof. The proof of (1) is straightforward. For (2) we put j = j(u + k · y, y) and j = j(u + k · y , y ) for short and write M ·N = [j − (j · k)k] · (j ∧ k) dy dy = j · (j ∧ k) dy dy = − j · (j ∧ k) dy dy, where it is understood that the integrals are over the set Ωa (u). Then interchanging y and y we get M · N = −M · N , i.e., M · N = 0. To prove (3) we write |M |2 = (j − (j · k)k) · (j − (j · k)k) dy dy = [j · j − (j · k)(j · k)] dy dy
and |N |2 =
(j ∧ k)(j ∧ k) dy dy.
In the previous equation we use the following rule of vector calculus (a ∧ b) · (c ∧ d) = (a · c)(b · d) − (a · d)(b · c), which is valid for any vectors a, b, c, d and the identity (3) follows at once. To prove (4) we write (M ∧ N ) · k = [((j · k)k − j) ∧ (j ∧ k)] · k dy dy = − [((j · k)k − j) ∧ k] · (j ∧ k) dy dy = (j ∧ k)(j ∧ k) dy dy = |N |2 = |M |2 . The following lemma permits to relate the algebraic properties of the vectors M , N to the ones of the radiation field and concludes the proof of Theorem 2. To simplify the notation we suppress the dependence on k and denote by an upper dot the differentiation with respect to u. Lemma 3 Let M (u), N (u) be C 1 vector fields satisfying the properties (1)–(4) of Lemma 2. Then ∀u ∈ R: (a) N˙ (u) · k = M˙ (u) · k = 0
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(b) M˙ (u) · N˙ (u) = 0 (c) |M˙ (u)| = |N˙ (u)| (d) (M˙ (u) ∧ N˙ (u)) · k = |M˙ (u)|2 . Proof. The identity (a) is proved at once by differentiating (1) with respect to u. To prove the other identities we consider a coordinate system in which the z-axis is parallel to k. In this frame the vectors M and N have the form M (u) = (m1 (u), m2 (u), 0), N (u) = (n1 (u), n2 (u), 0). Now, because of (2), (3) and (4) of Lemma 2: m1 (u)n1 (u) + m2 (u)n2 (u) = 0,
(2.5)
m1 (u)2 + m2 (u)2 = n1 (u)2 + n2 (u)2 , m1 (u)n2 (u) − m2 (u)n1 (u) = m1 (u)2 + m2 (u)2 .
(2.6) (2.7)
After some elementary algebra, (2.5)–(2.7) give m1 = n2 , m2 = −n1 . Hence the vectors M, N can be represented in the following form: M (u) = (q(u), r(u), 0) ⇒ M˙ = (q, ˙ r, ˙ 0), N (u) = (−r(u), q(u), 0) ⇒ N˙ = (−r, ˙ q, ˙ 0), by which the properties (b), (c) and (d) follow at once.
3 Bondi mass of the plasma In this section we prove Theorem 3. Let us introduce m∨ (r, u) = [e − p · k](u + |x|, x) dx.
(3.1)
|x|≤r
Note that m∨ (·, u) is non-decreasing and so its limit as r → +∞ exists. We first prove (1.12) assuming that ( ) m∨ (r, u) converges as r → +∞ for all u ∈ R. Assume ( ) holds. Then M∨ (u) is well defined as improper integral and we have M∨ (u) = lim m∨ (r, u). r→+∞
Let K be a generic compact subset of R. Evaluating the energy conservation law ∂t e = −∂x · p on the future light cone corresponding to the value u of the retarded time we have ∂u e(u + |x|, x) = −∂x · p(u + |x|, x).
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In the previous equation we use the identity ∂x · p(u + |x|, x) = ∂x · [(p(u + |x|, x)] − ∂u (p · k)(u + |x|, x) and so doing we get ∂u (e − p · k)(u + |x|, x) = −∂x · [p(u + |x|, x)].
(3.2)
We now integrate (3.2) on the region |x| ≤ r, and use the Gauss theorem to transform the right-hand side into a surface integral over the sphere of radius r. Since f (t, x, p) is supported on the region |x| ≤ R + a|t|, with a ∈ [0, 1) then for r large enough, f (u + |x|, x, p) vanishes on Sr = {x : |x| = r} and so we get ∂u (e − p · k)(u + |x|, x) dx lim r→+∞ |x|≤r = −(4π)−1 lim (Eret ∧ Bret ) · k (u + r, x) dSr . (3.3) r→+∞
Sr
The integral in the left-hand side of (3.3) is equal to ∂u m∨ (r, u), ∀r > 0. By Theorem 2 and (3.3) , ∂u m converges to the right-hand side of (1.12), uniformly in u ∈ K, as r → +∞. Hence, using ( ) we infer that m∨ (r, u) converges uniformly in u ∈ K and also d d lim m∨ (r, u) = M∨ (u). lim ∂u m∨ (r, u) = r→+∞ du r→+∞ du Thus the proof of (1.12) is complete if we show that the property ( ) above is satisfied. We note that to this purpose, a direct use of the estimate (ii) in Theorem 3 is not enough, since it entails |E(u + |x|, x)| = O(|x|−1 ), as |x| → ∞, which is too weak to bound M∨ . To solve this problem we rewrite M∨ in a way that the decay at past null infinity, which is faster by means of the absence of incoming radiation, enters into the estimates (in other words, we make the advanced time v appear instead of the retarded time u). For this purpose we introduce, besides m∨ (r, u), the following function (e + p · k)(v − |x|, x) dx. (3.4) m∧ (r, v) = |x|≤r
We also set M∧ (v) = lim m∧ (r, v). r→+∞
(3.5)
Lemma 4 Under the assumptions of Theorem 3, M∧ (v) is continuously differentiable and satisfies d M∧ (v) = 0, ∀v ∈ R. (3.6) dv
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Proof. We split M∧ (v) into four parts as follows: M∧ (v) = I1 (v) + I2 (v) + I3 (v) + I4 (v), where
1 (|Eret |2 + |Bret |2 )(v − |x|, x) dx, 8π R3 I2 (v) = m2α + p2 fα (v − |x|, x, p) dp dx, I1 (v) =
R3
α
1 4π
I3 (v) = I4 (v) =
|p|≤C
α
R3
(Eret ∧ Bret ) · k (v − |x|, x) dx,
R3
|p|≤C
p · k fα (v − |x|, x, p) dp dx.
Since I3 (resp. I4 ) is dominated by I1 (resp. I2 ), it suffices to estimate I1 (v) and I2 (v). By means of (ii), I1 (v) is uniformly bounded in v ∈ K. For I2 (v) we use that fα (v − |x|, x, p) = 0 for |x| ≥ R + a|v − |x||, which implies fα (v − |x|, x, p) = 0 for |x| ≥ (1 − a)−1 (R + a|v|). Thus |I4 (v)| ≤ CVol[{x : |x| ≤ (1 − a)−1 (R + a|v|)}] ≤ C,
∀v ∈ K.
Since m∧ (·, v) is non-decreasing, the limit (3.5) exists for all v ∈ R and by the previous estimates, M∧ (v) converges, as improper integral, uniformly in v ∈ K. To prove (3.6) we repeat the argument which led to (3.2), now evaluating on the past light cone. So doing we get ∂v (e + p · k)(v − |x|, x) = −∂x · [p(v − |x|, x)]. Integrating and using the Gauss theorem we obtain, lim ∂v (e + p · k)(v − |x|, x) dx r→+∞ |x|≤r (Eret ∧ Bret ) · k (v − r, x) dSr , = −(4π)−1 lim r→+∞
(3.7)
(3.8)
Sr
where again we used that f (v − r, x, p) vanishes on Sr for large r. The estimate (ii) implies that the right-hand side of (3.8) tends to zero uniformly in v ∈ K and so we get 0 = lim ∂v (e + p · k)(v − |x|, x) dx r→+∞ |x|≤r d d lim M∧ (v), (e + p · k)(v − |x|, x) = = dv r→+∞ |x|≤r dv where the uniform convergence has been used to shift the derivative.
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Note that in the proof of Lemma 4 the estimate (iii) in Theorem 3 has not been used. This will become important for the completion of the proof of ( ). The identity (3.6) represents the counterpart of (1.12) on the backward cones of light. In fact, from one hand the conservation of M∧ (v) is due to the absence of incoming radiation and, on the other hand, the Bondi mass M∨ (u) decreases as a consequence of the emission of outgoing radiation. We are able now to complete the proof of ( ). Integrating (3.7) between u and u + 2|x| we get u+2|x| (e + p · k)(u + |x|, x) − (e + p · k)(u − |x|, x) = − ∂x · [p(v − |x|, x)] dv. u
Integrating in the region |x| ≤ r we get (e + p · k)(u + |x|, x) dx = |x|≤r
|x|≤r
(e + p · k)(u − |x|, x) dx
− Now we use the identity u+2|x| p(v − |x|, x) dv] ∂x · [ u
=
u+2|x|
u
|x|≤r
(3.9)
∂x · [p(v − |x|, x)] dv dx.
2p · k (u + |x|, x)
u+2|x|
+ u
∂x · [p(v − |x|, x)] dv.
Substituting into (3.9) and using the Gauss theorem we obtain m∨ (r, u) = m∧ (r, u) + q(r, u). where
q(r, u) = −
u+2r
Sr
u
α
Sr
(3.10)
p · k (v − r, x) dv dSr
u+2r 1 (Eret ∧ Bret ) · k (v − r, x) dv dSr 4π Sr u u+2r − p · k fα (v − r, x) dv dSr .
= −
u
|p|≤C
Using (ii) in the first term and (iii) in the second term, we conclude that limr→+∞ |q(r, u)| is bounded for all u ∈ R and so the property ( ) follows from (3.10) and Lemma 4. This concludes the proof of Theorem 3. Acknowledgments. The results presented in this paper have been obtained while the author was preparing his PhD thesis at the Albert Einstein Institute in Potsdam, which is thereby acknowledged for the hospitality. Support from the European Network HYKE (contract HPRN-CT-2002-00282) is also acknowledged.
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References [1] H. Bondi, M.G.J. van der Burg, A.W.K. Metzner, Gravitational waves in general relativity VII. Waves from axi-symmetric isolated systems, Proc. R. Soc. London, Ser. A 269, 21–52 (1962). [2] S. Calogero, Global Small Solutions of the Vlasov-Maxwell System in the Absence of Incoming Radiation, Indiana Univ. Math. Journal, (to appear) Preprint: math-ph/0211013. [3] D. Christodoulou, S. Klainerman, The global non-linear stability of the Minkowski space, Princeton Mathematical series 41 (1993). [4] R. Glassey, W. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rat. Mech. Anal. 92, 59–90 (1986). [5] R. Glassey, W. Strauss, Absence of shocks in an initially dilute collisionless plasma, Comm. Math. Phys. 113, 191–208 (1987). [6] R. Glassey, J. Schaeffer, Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data, Comm. Math. Phys. 119, 353–384 (1988). [7] J.N. Goldberg, R.P. Kerr, Asymptotic Properties of the Electromagnetic Field, J. Math. Phys. 5, 172–176 (1964). [8] S. Klainerman, F. Nicol´ o, The Evolution Problem in General Relativity, Birkh¨auser (Basel) (2003). [9] S. Klainerman, G. Staffilani, A new approach to study the Vlasov-Maxwell system, Comm. Pure Appl. Anal. 1, 1, 103–125 (2002). [10] G. Rein, Generic global solutions of the relativistic Vlasov-Maxwell system of plasma physics, Comm. Math. Phys. 135, 41–78 (1990). [11] J.L. Synge, Relativity: The Special Theory, North-Holland, Amsterdam (1965). [12] R.M. Wald, General relativity, (Chicago, IL: The University of Chicago Press) (1984). Simone Calogero Department of Mathematics Chalmers University S-412 96 G¨ oteborg Sweden email:
[email protected] Communicated by Vincent Rivasseau submitted 13/07/03, accepted 11/09/03
Ann. Henri Poincar´e 5 (2004) 203 – 233 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/020203-31 DOI 10.1007/s00023-004-0166-8
Annales Henri Poincar´ e
Proof of the Ergodic Hypothesis for Typical Hard Ball Systems N´andor Sim´ anyi∗
Abstract. We consider the system of N (≥ 2) hard balls with masses m1 , . . . , mN and radius r in the flat torus TνL = Rν /L · Zν of size L, ν ≥ 3. We prove the ergodicity (actually, the Bernoulli mixing property) of such systems for almost every selection (m1 , . . . , mN ; L) of the outer geometric parameters. This theorem complements my earlier result that proved the same, almost sure ergodicity for the case ν = 2. The method of that proof was primarily dynamical-geometric, whereas the present approach is inherently algebraic.
1 Introduction Hard ball systems or, a bit more generally, mathematical billiards constitute an important and quite interesting family of dynamical systems being intensively studied by dynamicists and researchers of mathematical physics, as well. These dynamical systems pose many challenging mathematical questions, most of them concerning the ergodic (mixing) properties of such systems. The introduction of hard ball systems and the first major steps in their investigations date back to the 40’s and 60’s, see Krylov’s paper [K(1942)] and Sinai’s ground-breaking works [Sin(1963)] and [Sin(1970)], in which the author – among other things – formulated the modern version of Boltzmann’s ergodic hypothesis (what we call today the Boltzmann-Sinai ergodic hypothesis) by claiming that every hard ball system in a flat torus is ergodic, of course after fixing the values of the trivial flow-invariant quantities. In the articles [Sin(1970)] and [B-S(1973)] Bunimovich and Sinai proved this hypothesis for two hard disks on the two-dimensional unit torus T2 . The generalization of this result to higher dimensions ν > 2 took fourteen years, and was done by Chernov and Sinai in [S-Ch(1987)]. Although the model of two hard balls in Tν is already rather involved technically, it is still a so-called strictly dispersive billiard system, i.e., such that the smooth components of the boundary ∂Q of the configuration space are strictly concave from outside Q. (They are bending away from Q.) The billiard systems of more than two hard balls in Tν are no longer strictly dispersive, but just semi-dispersive (strict concavity of the smooth components of ∂Q is lost, merely concavity persists), and this circumstance causes a lot of additional technical troubles in their study. In the series of my ∗ Research
supported by the National Science Foundation, grant DMS-0098773.
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joint papers with A. Kr´ amli and D. Sz´ asz [K-S-Sz(1989)], [K-S-Sz(1990)], [K-SSz(1991)], and [K-S-Sz(1992)] we developed several new methods, and proved the ergodicity of more and more complicated semi-dispersive billiards culminating in the proof of ergodicity of four billiard balls in the torus Tν (ν ≥ 3), [K-S-Sz(1992)]. Then, in 1992, Bunimovich, Liverani, Pellegrinotti and Sukhov [B-L-P-S(1992)] were able to prove the ergodicity for some systems with an arbitrarily large number of hard balls. The shortcoming of their model, however, is that, on one hand, they restrict the types of all feasible ball-to-ball collisions, on the other hand they introduce some additional scattering effect with the collisions at the strictly concave wall of the container. The only result with an arbitrarily large number of balls in a flat unit torus Tν was achieved in the twin papers of mine [Sim(1992I-II)], where I managed to prove the ergodicity (actually, the K-mixing property) of N hard balls in Tν , provided that N ≤ ν. The annoying shortcoming of that result is that the larger the number of balls N is, larger and larger dimension ν of the ambient container is required by the method of the proof. On the other hand, if someone considers a hard ball system in an elongated torus which is long in one direction but narrow in the others, so that the balls must keep their cyclic order in the “long direction” (Sinai’s “pen-case” model), then the technical difficulties can be handled, thanks to the fact that the collisions of balls are now restricted to neighboring pairs. The hyperbolicity of such models in three dimensions and the ergodicity in dimension four have been proved in [S-Sz(1995)]. The positivity of the metric entropy for several systems of hard balls can be proven relatively easily, as was shown in the paper [W(1988)]. The articles [L-W(1995)] and [W(1990)] are nice surveys describing a general setup leading to the technical problems treated in a series of research papers. For a comprehensive survey of the results and open problems in this field, see [Sz(1996)]. Pesin’s theory [P(1977)] on the ergodic properties of non-uniformly hyperbolic, smooth dynamical systems has been generalized substantially to dynamical systems with singularities (and with a relatively mild behavior near the singularities) by A. Katok and J-M. Strelcyn [K-S(1986)]. Since then, the so-called Pesin’s and Katok-Strelcyn’s theories have become part of the folklore in the theory of dynamical systems. They claim that – under some mild regularity conditions, particularly near the singularities – every non-uniformly hyperbolic and ergodic flow enjoys the Kolmogorov-mixing property, shortly the K-mixing property. Later on it was discovered and proven in [C-H(1996)] and [O-W(1998)] that the above-mentioned fully hyperbolic and ergodic flows with singularities turn out to be automatically having the Bernoulli mixing (B-mixing) property. It is worth noting here that almost every semi-dispersive billiard system, especially every hard ball system, enjoys those mild regularity conditions imposed on the systems (as axioms) by [K-S(1986)], [C-H(1996)], and [O-W(1998)]. In other words, for a hard ball flow (M, {S t }, µ) the (global) ergodicity of the system actually implies its full hyperbolicity and the B-mixing property, as well. Finally, in our joint venture with D. Sz´ asz [S-Sz(1999)], we prevailed over the difficulty caused by the low value of the dimension ν by developing a brand
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new algebraic approach for the study of hard ball systems. That result, however, only establishes complete hyperbolicity (nonzero Lyapunov exponents almost everywhere) for N balls in Tν . The ergodicity appeared to be a harder task. We note, however, that the algebraic method developed in [S-Sz(1999)] is being further developed in this paper in order to obtain ergodicity, not only full hyperbolicity. Consider the ν-dimensional (ν ≥ 2), standard, flat torus TνL = Rν /L · Zν as the vessel containing N (≥ 2) hard balls (spheres) B1 , . . . , BN with positive masses m1 , . . . , mN and (just for simplicity) common radius r > 0. We always assume that the radius r > 0 is not too big, so that even the interior of the arising configuration space Q is connected. Denote the center of the ball Bi by qi ∈ Tν , and let vi = q˙i be the velocity of the i-th particle. We investigate the uniform motion of the balls B1 , . . . , BN inside the container Tν with half a unit 1 N 1 2 of total kinetic energy: E = . We assume that the collisions i=1 mi ||vi || = 2 2 between balls are perfectly elastic. Since – beside the kinetic energy E – the total ν momentum I = N i=1 mi vi ∈ R is also a trivial first integral of the motion, we make the standard reduction I = 0. Due to the apparent translation invariance of the arising dynamical system, we factorize the configuration space with respect to uniform spatial translations as follows: (q1 , . . . , qN ) ∼ (q1 + a, . . . , qN + a) for all ν translation vectors The configuration space Q of the arising flow is then a∈T . ν N the factor torus (T ) / ∼ ∼ = Tν(N −1) minus the cylinders Ci,j = (q1 , . . . , qN ) ∈ Tν(N −1) : dist(qi , qj ) < 2r (1 ≤ i < j ≤ N ) corresponding to the forbidden overlap between the i-th and j-th spheres. Then it is easy to see that the compound configuration point q = (q1 , . . . , qN ) ∈ Q = Tν(N −1) \
Ci,j
1≤i<j≤N
moves in Q uniformly with unit speed and bounces back from the boundaries ∂Ci,j of the cylinders Ci,j according to the classical law of geometric optics: the angle of reflection equals the angle of incidence. More precisely: the post-collision velocity v + can be obtained from the pre-collision velocity v − by the orthogonal reflection across the tangent hyperplane of the boundary ∂Q at the point of collision. Here we must emphasize that the phrase “orthogonal” should be understood with respect to the natural Riemannian metric (the so-called mass metric) N ||dq||2 = i=1 mi ||dqi ||2 in the configuration space Q. For the normalized Liouville measure µ of the arising flow {S t } we obviously have dµ = const · dq · dv, where dq is the Riemannian volume in Q induced by the above metric and dv is the surface measure (determined by the restriction of the Riemannian metric above) on the
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sphere of compound velocities ν(N −1)−1
S
=
ν N
(v1 , . . . , vN ) ∈ (R )
:
N i=1
mi vi = 0 and
N
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2
mi ||vi || = 1 .
i=1
The phase space M of the flow {S t } is the unit tangent bundle Q × Sd−1 of the configuration space Q. (We will always use the shorthand notation d = ν(N − 1) for the dimension of the billiard table Q.) We must, however, note here that at the boundary ∂Q of Q one has to glue together the pre-collision and post-collision velocities in order to form the phase space M, so M is equal to the unit tangent bundle Q × Sd−1 modulo this identification. A bit more detailed definition of hard ball systems with arbitrary masses, as well as their role in the family of cylindric billiards, can be found in §4 of [SSz(2000)] and in §1 of [S-Sz(1999)]. We denote the arising flow by (M, {S t }t∈R , µ). In the series of articles [K-S-Sz(1989)], [K-S-Sz(1991)], [K-S-Sz(1992)], [Sim(1992-I)], and [Sim(1992-II)] the authors developed a powerful, three-step strategy for proving the (hyperbolic) ergodicity of hard ball systems. First of all, all these proofs are inductions on the number N of balls involved in the problem. Secondly, the induction step itself consists of the following three major steps: Step I. To prove that every non-singular (i.e., smooth) trajectory segment S [a,b] x0 with a “combinatorially rich” (in a well-defined sense) symbolic collision sequence is automatically sufficient (or, in other words, “geometrically hyperbolic”, see below in §2), provided that the phase point x0 does not belong to a countable union J of smooth sub-manifolds with codimension at least two. (Containing the exceptional phase points.) The exceptional set J featuring this result is negligible in our dynamical considerations – it is a so-called slim set. For the basic properties of slim sets, see §2 below. Step II. Assume the induction hypothesis, i.e., that all hard ball systems with N balls (2 ≤ N < N ) are (hyperbolic and) ergodic. Prove that then there exists a slim set S ⊂ M (see §2) with the following property: For every phase point x0 ∈ M \ S the entire trajectory S R x0 contains at most one singularity and its symbolic collision sequence is combinatorially rich, just as required by the result of Step I. Step III. By using again the induction hypothesis, prove that almost every singular trajectory is sufficient in the time interval (t0 , +∞), where t0 is the time moment of the singular reflection. (Here the phrase “almost every” refers to the volume defined by the induced Riemannian metric on the singularity manifolds.) We note here that the almost sure sufficiency of the singular trajectories (featuring Step III) is an essential condition for the proof of the celebrated theorem on local ergodicity for algebraic semi-dispersive billiards proved by B´ alint-ChernovSz´asz-T´oth in [B-Ch-Sz-T (2002)]. Under this assumption the theorem of [B-ChSz-T (2002)] states that in any algebraic semi-dispersive billiard system (i.e., in
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a system such that the smooth components of the boundary ∂Q are algebraic hypersurfaces) a suitable, open neighborhood U0 of any sufficient phase point x0 ∈ M (with at most one singularity on its trajectory) belongs to a single ergodic component of the billiard flow (M, {S t }t∈R , µ). In an inductive proof of ergodicity, steps I and II together ensure that there exists an arc-wise connected set C ⊂ M with full measure, such that every phase point x0 ∈ C is sufficient with at most one singularity on its trajectory. Then the cited theorem on local ergodicity (now taking advantage of the result of Step III) states that for every phase point x0 ∈ C an open neighborhood U0 of x0 belongs to one ergodic component of the flow. Finally, the connectedness of the set C and µ(M \ C) = 0 easily imply that the flow (M, {S t }t∈R , µ) (now with N balls) is indeed ergodic, and actually fully hyperbolic, as well. The main result of this paper is the Theorem. In the case ν ≥ 3 for almost every selection (m1 , . . . , mN ; L) of the outer geometric parameters from the region mi > 0, L > L0 (r, ν), where the interior of t the phase space is connected, it is true that the billiard flow (Mm,L , {S }, µm,L ) of the N -ball system is ergodic and completely hyperbolic. Then, following from the results of Chernov-Haskell [C-H(1996)] and Ornstein-Weiss [O-W(1998)], such a semi-dispersive billiard system actually enjoys the B-mixing property, as well. Remark 1. We note that the main result of this paper and that of [Sim(2003)] nicely complement each other. They precisely assert the same, almost sure ergodicity of hard ball systems in the cases ν ≥ 3 and ν = 2, respectively. It should be noted, however, that the proof of [Sim(2003)] is primarily dynamical-geometric (except the verification of the Chernov-Sinai Ansatz), whereas the novel parts of the present proof are fundamentally algebraic. Remark 2. The above inequality L > L0 (r, ν) corresponds to physically relevant situations. Indeed, in the case L < L0 (r, ν) the particles would not have enough room even to freely exchange positions. The paper is organized as follows: §2 provides all necessary prerequisites and technical tools that will be required by the proof of the theorem. Based on the results obtained in [S-Sz(1999)], the subsequent Section §3 carries out Step I of the inductive strategy outlined above, but for the case when the outer geometric parameters (m1 , . . . , mN ; L) are incorporated in the algebraic process as variables. (Just as the positions and velocities of the particles!) Finally, the closing Section §4 utilizes a “Fubini type argument” by proving Step I for almost every (with respect to the Lebesgue measure of the (m1 , . . . , mN ; L)-space) hard ball system (N ≥ 2, ν ≥ 3). This will finish the inductive proof of the theorem, for Steps II and III of the induction strategy are easy consequences of some earlier results.
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2 Prerequisites 2.1
Cylindric billiards
Consider the d-dimensional (d ≥ 2) flat torus Td = Rd /L supplied with the usual Riemannian inner product . , . inherited from the standard inner product of the universal covering space Rd . Here L ⊂ Rd is assumed to be a lattice, i.e., a discrete subgroup of the additive group Rd with rank(L) = d. The reason why we want to allow general lattices, other than just the integer lattice Zd , is that otherwise the hard ball systems would not be covered. The geometry of the structure lattice L in the case of a hard ball system is significantly different from the geometry of the standard lattice Zd in the standard Euklidean space Rd , see later in this section. The configuration space of a cylindric billiard is Q = Td \ (C1 ∪ · · · ∪ Ck ), where the cylindric scatterers Ci (i = 1, . . . , k) are defined as follows. Let Ai ⊂ Rd be a so-called lattice subspace of Rd , which means that rank(Ai ∩ L) = dimAi . In this case the factor Ai /(Ai ∩ L) is a sub-torus in Td = Rd /L which will be taken as the generator of the cylinder Ci ⊂ Td , i = 1, . . . , k. Denote by d Li = A⊥ i the ortho-complement of Ai in R . Throughout this paper we will always assume that dimLi ≥ 2. Let, furthermore, the numbers ri > 0 (the radii of the spherical cylinders Ci ) and some translation vectors ti ∈ Td = Rd /L be given. The translation vectors ti play a role in positioning the cylinders Ci in the ambient torus Td . Set
Ci = x ∈ Td : dist (x − ti , Ai /(Ai ∩ L)) < ri . In order to avoid further unnecessary complications, we always assume that the interior of the configuration space Q = Td \ (C1 ∪ · · · ∪ Ck ) is connected. The phase space M of our cylindric billiard flow will be the unit tangent bundle of Q (modulo the natural gluing at its boundary), i.e., M = Q × Sd−1 . (Here Sd−1 denotes the unit sphere of Rd .) The dynamical system (M, {S t }t∈R , µ), where S t (t ∈ R) is the dynamics defined by the uniform motion inside the domain Q and specular reflections at its boundary (at the scatterers), and µ is the Liouville measure, is called a cylindric billiard flow. We note that the cylindric billiards – defined above – belong to the wider class of so-called semi-dispersive billiards, which means that the smooth components ∂Qi of the boundary ∂Q of the configuration space Q are (not necessarily strictly) concave from outside of Q, i.e., they are bending away from the interior of Q. As to the notions and notations in connection with semi-dispersive billiards, the reader is kindly referred to the article [K-S-Sz(1990)]. Throughout this paper we will always assume – without explicitly stating – that the considered semi-dispersive billiard system fulfills the following conditions: intQ is connected, and the d-dim spatial angle α(q) subtended by Q at any of its boundary points q ∈ ∂Q is uniformly positive.
(2.1.1) (2.1.2)
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We note, however, that in the case of hard ball systems with a fixed radius r of the balls (see below) the non-degeneracy condition (2.1.2) only excludes countably many values of the size L of the container torus TνL = Rν /L · Zν from the region L > L0 (r, ν) where (2.1.1) is true. Therefore, in the sense of our theorem of “almost sure ergodicity”, the non-degeneracy condition (2.1.2) does not mean a restriction of generality.
2.2
Hard ball systems
Hard ball systems in the flat torus TνL = Rν /L · Zν (ν ≥ 2) with positive masses m1 , . . . , mN are described (for example) in §1 of [S-Sz(1999)]. These are the dynamical systems describing the motion of N (≥ 2) hard balls with a common radius r > 0 and positive masses m1 , . . . , mN in the flat torus of size L, TνL = Rν /L · Zν . (Just for simplicity, we will assume that the radii have the common value r.) The center of the i-th ball is denoted by qi (∈ TνL ), its time derivative is vi= q˙i , i = 1, . . . , N . One uses the standard reduction of kinetic N energy E = 12 i=1 mi ||vi ||2 = 12 . The arising configuration space (still without the removal of the scattering cylinders Ci,j ) is the torus ν TνN L = (TL )
N
= {(q1 , . . . , qN ) : qi ∈ TνL , i = 1, . . . , N }
supplied with the Riemannian inner product (the so-called mass metric) v, v =
N
mi vi , vi
(2.2.1)
i=1 N
in its common tangent space RνN = (Rν ) . Now the Euklidean space RνN with the inner product (2.2.1) plays the role of Rd in the original definition of cylindric billiards, see §2.1 above. The generator subspace Ai,j ⊂ RνN (1 ≤ i < j ≤ N ) of the cylinder Ci,j (describing the collisions between the i-th and j-th balls) is given by the equation (2.2.2) Ai,j = (q1 , . . . , qN ) ∈ (Rν )N : qi = qj , see (4.3) in [S-Sz(2000)]. Its ortho-complement Li,j ⊂ RνN is then defined by the equation N Li,j = (q1 , . . . , qN ) ∈ (Rν ) : qk = 0 for k = i, j, and mi qi + mj qj = 0 , (2.2.3) see (4.4) in [S-Sz(2000)]. Easy calculation shows that the cylinder Ci,j (describing the overlap of the i-th and j-th balls) is indeed spherical and the radius of its base m m sphere is equal to ri,j = 2r mi i+mjj , see §4, especially formula (4.6) in [S-Sz(2000)]. N
The structure lattice L ⊂ RνN is clearly the lattice L = (L · Zν )
= L · ZN ν .
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Due to the presence of an extra invariant quantity I = N i=1 mi vi , one usually N makes the reduction i=1 mi vi = 0 and, correspondingly, factorizes the configuration space with respect to uniform spatial translations: (q1 , . . . , qN ) ∼ (q1 + a, . . . , qN + a),
a ∈ TνL .
(2.2.4)
The natural, common tangent space of this reduced configuration space is then Z=
(v1 , . . . , vN ) ∈ (Rν )N :
N i=1
mi vi = 0
=
⊥ Ai,j = (A)⊥
(2.2.5)
i<j
supplied again with the inner product (2.2.1), see also (4.1) and (4.2) in [SSz(2000)]. The base spaces Li,j of (2.2.3) are obviously subspaces of Z, and we take A˜i,j = Ai,j ∩ Z = PZ (Ai,j ) as the ortho-complement of Li,j in Z. (Here PZ denotes the orthogonal projection onto the space Z.) Note that the configuration space of the reduced system (with the identification (2.2.4)) is naturally the torus RνN /(A + L · ZνN ) = Z/PZ (L · ZνN ).
2.3
Collision graphs
Let S [a,b] x be a nonsingular, finite trajectory segment with the collisions σ1 , . . . , σn listed in time order. (Each σk is an unordered pair (i, j) of different labels i, j ∈ {1, 2, . . . , N }.) The graph G = (V, E) with vertex set V = {1, 2, . . . , N } and set of edges E = {σ1 , . . . , σn } is called the collision graph of the orbit segment S [a,b] x. For a given positive number C, the collision graph G = (V, E) of the orbit segment S [a,b] x will be called C-rich if G contains at least C connected, consecutive (i.e., following one after the other in time, according to the time-ordering given by the trajectory segment S [a,b] x) subgraphs.
2.4
Trajectory branches
We are going to briefly describe the discontinuity of the flow {S t } caused by a multiple collisions at time t0 . Assume first that the pre-collision velocities of the particles are given. What can we say about the possible post-collision velocities? Let us perturb the pre-collision phase point (at time t0 − 0) infinitesimally, so that the collisions at ∼ t0 occur at infinitesimally different moments. By applying the collision laws to the arising finite sequence of collisions, we see that the postcollision velocities are fully determined by the time-ordering of the considered collisions. Therefore, the collection of all possible time-orderings of these collisions gives rise to a finite family of continuations of the trajectory beyond t0 . They are called the trajectory branches. It is quite clear that similar statements can be said regarding the evolution of a trajectory through a multiple collision in reverse time. Furthermore, it is also obvious that for any given phase point x0 ∈ M there are two, ω-high trees T+ and T− such that T+ (T− ) describes all the possible
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continuations of the positive (negative) trajectory S [0,∞) x0 (S (−∞,0] x0 ). (For the definitions of trees and for some of their applications to billiards, cf. the beginning of §5 in [K-S-Sz(1992)].) It is also clear that all possible continuations (branches) of the whole trajectory S (−∞,∞) x0 can be uniquely described by all pairs (B− , B+ ) of ω-high branches of the trees T− and T+ (B− ⊂ T− , B+ ⊂ T+ ). Finally, we note that the trajectory of the phase point x0 has exactly two branches, provided that S t x0 hits a singularity for a single value t = t0 , and the phase point S t0 x0 does not lie on the intersection of more than one singularity manifolds. In this case we say that the trajectory of x0 has a “simple singularity”.
2.5
Neutral subspaces, advance, and sufficiency
Consider a nonsingular trajectory segment S [a,b] x. Suppose that a and b are not moments of collision. Definition 2.5.1 The neutral space N0 (S [a,b] x) of the trajectory segment S [a,b] x at time zero (a < 0 < b) is defined by the following formula:
N0 (S [a,b] x) = W ∈ Z : ∃(δ > 0) such that ∀α ∈ (−δ, δ) V (S a (Q(x) + αW, V (x))) = V (S a x) and V S b (Q(x) + αW, V (x)) = V (S b x) . (Z is the common tangent space Tq Q of the parallelizable manifold Q at any of its points q, while V (x) is the velocity component of the phase point x = (Q(x), V (x)).) It is known (see (3) in §3 of [S-Ch (1987)]) that N0 (S [a,b] x) is a linear subspace of Z indeed, and V (x) ∈ N0 (S [a,b] x). The neutral space Nt (S [a,b] x) of the segment S [a,b] x at time t ∈ [a, b] is defined as follows: Nt (S [a,b] x) = N0 S [a−t,b−t] (S t x) . It is clear that the neutral space Nt (S [a,b] x) can be canonically identified with N0 (S [a,b] x) by the usual identification of the tangent spaces of Q along the trajectory S (−∞,∞) x (see, for instance, §2 of [K-S-Sz(1990)]). Our next definition is that of the advance. Consider a non-singular orbit segment S [a,b] x with the symbolic collision sequence Σ = (σ1 , . . . , σn ) (n ≥ 1), meaning that S [a,b] x has exactly n collisions with ∂Q, and the i-th collision (1 ≤ i ≤ n) takes place at the boundary of the cylinder Cσi . For x = (Q, V ) ∈ M and W ∈ Z, W sufficiently small, denote TW (Q, V ) := (Q + W, V ). Definition 2.5.2 For any 1 ≤ k ≤ n and t ∈ [a, b], the advance α(σk ) : Nt (S [a,b] x) → R of the collision σk is the unique linear extension of the linear functional α(σk ) defined in a sufficiently small neighborhood of the origin of Nt (S [a,b] x) in the following way: α(σk )(W ) := tk (x) − tk (S −t TW S t x).
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Here tk = tk (x) is the time moment of the k-th collision σk on the trajectory of x after time t = a. The above formula and the notion of the advance functional αk = α(σk ) : Nt S [a,b] x → R has two important features: (i) If the spatial translation (Q, V ) → (Q + W, V ) is carried out at time t, then tk changes linearly in W , and it takes place just αk (W ) units of time earlier. (This is why it is called “advance”.) (ii) If the considered reference time t is somewhere between tk−1 and tk , then the neutrality of W with respect to σk precisely means that W − αk (W ) · V (x) ∈ Aσk , i.e., a neutral (with respect to the collision σk ) spatial translation W with the advance αk (W ) = 0 means that the vector W belongs to the generator space Aσk of the cylinder Cσk . It is now time to bring up the basic notion of sufficiency (or, sometimes it is also called geometric hyperbolicity) of a trajectory (segment). This is the utmost important necessary condition for the proof of the fundamental theorem for algebraic semi-dispersive billiards, see Theorem 4.4 in [B-Ch-Sz-T(2002)]. Definition 2.5.3 (i) The nonsingular trajectory segment S [a,b] x (a and b are supposed not to be moments of collision) is said to be sufficient if and only if the dimension of Nt (S [a,b] x) (t ∈ [a, b]) is minimal, i.e., dim Nt (S [a,b] x) = 1. (ii) The trajectory segment S [a,b] x containing exactly one singularity (a so-called “simple singularity”, see 2.4 above) is said to be sufficient if and only if both branches of this trajectory segment are sufficient. Definition 2.5.4 The phase point x ∈ M with at most one (simple) singularity is said to be sufficient if and only if its whole trajectory S (−∞,∞) x is sufficient, which means, by definition, that some of its bounded segments S [a,b] x are sufficient. In the case of an orbit S (−∞,∞) x with a simple singularity, sufficiency means that both branches of S (−∞,∞) x are sufficient.
2.6
No accumulation (of collisions) in finite time
By the results of Vaserstein [V(1979)], Galperin [G(1981)] and Burago-FerlegerKononenko [B-F-K(1998)], in a semi-dispersive billiard flow with the property (2.1.2) there can only be finitely many collisions in finite time intervals, see Theorem 1 in [B-F-K(1998)]. Thus, the dynamics is well defined as long as the trajectory does not hit more than one boundary components at the same time.
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Slim sets
We are going to summarize the basic properties of codimension-two subsets A of a connected, smooth manifold M with a possible boundary. Since these subsets A are just those negligible in our dynamical discussions, we shall call them slim. As to a broader exposition of the issues, see [E(1978)] or §2 of [K-S-Sz(1991)]. Note that the dimension dim A of a separable metric space A is one of the ˇ three classical notions of topological dimension: the covering (Cech-Lebesgue), the ˇ small inductive (Menger-Urysohn), or the large inductive (Brouwer-Cech) dimension. As it is known from general general topology, all of them are the same for separable metric spaces. Definition 2.7.1 A subset A of M is called slim if and only if A can be covered by a countable family of codimension-two (i.e., at least two) closed sets of µ-measure zero, where µ is a smooth measure on M . (Cf. Definition 2.12 of [K-S-Sz(1991)].) Property 2.7.2 The collection of all slim subsets of M is a σ-ideal, that is, countable unions of slim sets and arbitrary subsets of slim sets are also slim. Proposition 2.7.3. (Locality) A subset A ⊂ M is slim if and only if for every x ∈ A there exists an open neighborhood U of x in M such that U ∩ A is slim. (Cf. Lemma 2.14 of [K-S-Sz(1991)].) Property 2.7.4 A closed subset A ⊂ M is slim if and only if µ(A) = 0 and dim A ≤ dim M − 2. Property 2.7.5. (Integrability) If A ⊂ M1 × M2 is a closed subset of the product of two smooth manifolds with possible boundaries, and for every x ∈ M1 the set Ax = {y ∈ M2 : (x, y) ∈ A} is slim in M2 , then A is slim in M1 × M2 . The following propositions characterize the codimension-one and codimension-two sets. Proposition 2.7.6 For any closed subset S ⊂ M the following three conditions are equivalent: (i) dim S ≤ dim M − 2; (ii) intS = ∅ and for every open connected set G ⊂ M the difference set G \ S is also connected; (iii) intS = ∅ and for every point x ∈ M and for any open neighborhood V of x in M there exists a smaller open neighborhood W ⊂ V of the point x such that for every pair of points y, z ∈ W \ S there is a continuous curve γ in the set V \ S connecting the points y and z. (See Theorem 1.8.13 and Problem 1.8.E of [E(1978)].)
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Proposition 2.7.7 For any subset S ⊂ M the condition dim S ≤ dim M − 1 is equivalent to intS = ∅. (See Theorem 1.8.10 of [E(1978)].) We recall an elementary, but important lemma (Lemma 4.15 of [K-S-Sz(1991)]). Let R2 be the set of phase points x ∈ M \ ∂M such that the trajectory S (−∞,∞) x has more than one singularities. Proposition 2.7.8 The set R2 is a countable union of codimension-two smooth sub-manifolds of M and, being such, it is slim. The next lemma establishes the most important property of slim sets which gives us the fundamental geometric tool to connect the open ergodic components of billiard flows. Proposition 2.7.9 If M is connected, then the complement M \ A of a slim Fσ set A ⊂ M is an arc-wise connected (Gδ ) set of full measure. (See Property 3 of §4.1 in [K-S-Sz(1989)]. The Fσ sets are, by definition, the countable unions of closed sets, while the Gδ sets are the countable intersections of open sets.)
2.8
The subsets M0 and M#
Denote by M# the set of all phase points x ∈ M for which the trajectory of x encounters infinitely many non-tangential collisions in both time directions. The trajectories of the points x ∈ M \ M# are lines: the motion is linear and uniform, see the appendix of [Sz(1994)]. It is proven in lemmas A.2.1 and A.2.2 of [Sz(1994)] that the closed set M \ M# is a finite union of hyperplanes. It is also proven in [Sz(1994)] that, locally, the two sides of a hyper-planar component of M \ M# can be connected by a positively measured beam of trajectories, hence, from the point of view of ergodicity, in this paper it is enough to show that the connected components of M# entirely belong to one ergodic component. This is what we are going to do in this paper. Denote by M0 the set of all phase points x ∈ M# the trajectory of which does not hit any singularity, and use the notation M1 for the set of all phase points x ∈ M# whose orbit contains exactly one, simple singularity. According to Proposition 2.7.8, the set M# \ (M0 ∪ M1 ) is a countable union of smooth, codimension-two (≥ 2) submanifolds of M, and, therefore, this set may be discarded in our study of ergodicity, please see also the properties of slim sets above. Thus, we will restrict our attention to the phase points x ∈ M0 ∪ M1 .
2.9
The “Chernov-Sinai Ansatz”
An essential precondition for the theorem on local ergodicity by B´ alint-ChernovSz´asz-T´oth (Theorem 4.4 of [B-Ch-Sz-T(2002)]) is the so-called “Chernov-Sinai Ansatz” which we are going to formulate below. Denote by SR+ ⊂ ∂M the set of all phase points x0 = (q0 , v0 ) ∈ ∂M corresponding to singular reflections (a
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tangential or a double collision at time zero) supplied with the post-collision (outgoing) velocity v0 . It is well known that SR+ is a compact cell complex with dimension 2d − 3 = dimM − 2. It is also known (see Lemma 4.1 in [K-S-Sz(1990)]) that for ν-almost every phase point x0 ∈ SR+ the forward orbit S (0,∞) x0 does not hit any further singularity. (Here ν is the Riemannian volume of SR+ induced by the restriction of the natural Riemannian metric of M.) The Chernov-Sinai Ansatz postulates that for ν-almost every x0 ∈ SR+ the forward orbit S (0,∞) x0 is sufficient (geometrically hyperbolic).
2.10
The theorem on local ergodicity
The theorem on local ergodicity by B´ alint-Chernov-Sz´ asz-T´oth (Theorem 4.4 of [B-Ch-Sz-T(2002)]) claims the following: Let (M, {S t }t∈R , µ) be a semi-dispersive billiard flow with (2.1.1)–(2.1.2) and with the property that the smooth components of the boundary ∂Q of the configuration space are algebraic hyper-surfaces. (The cylindric billiards automatically fulfill this algebraicity condition.) Assume – further – that the Chernov-Sinai Ansatz holds true, and a phase point x0 ∈ (M \ ∂M) ∩ M# is given with the properties (i) S (−∞,∞) x has at most one singularity, and (ii) S (−∞,∞) x is sufficient. Then some open neighborhood U0 ⊂ M of x0 belongs to a single ergodic component of the flow (M, {S t }t∈R , µ). (Modulo the zero sets, of course.)
3 Non-sufficiency occurs on a codimension-two set. The case ν ≥ 3 The opening part of this section contains a slightly modified version of Lemma 4.43 from [S-Sz(1999)]. The reason why we had to modify the recursion for the sequence C(N ) (from C(N ) = (N/2) · max {C(N − 1), 3} to C(N ) = (N/2) · (2C(N − 1) + 1)) is that our Corollary 3.5 (below) requires (2C(N ) + 1)-richness instead of the usual C(N )-richness. In the present paper the sequence C(N ) always denotes the one defined by the recursion in Lemma 3.1 instead of the one defined in Lemma 4.43 of [S-Sz(1999)]. This should not cause any confusion. We note that the upcoming lemma is purely combinatorial. Lemma 3.1 Define the sequence of positive numbers C(N ) recursively by taking C(2) = 1 and C(N ) = (N/2) · (2C(N − 1) + 1) for N ≥ 3. Let N ≥ 3, and suppose that the symbolic collision sequence Σ = (σ1 , . . . , σn ) for N particles is C(N )-rich. Then we can find a particle, say the one with label N , and two indices 1 ≤ p < q ≤ n such that
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N ∈ σp ∩ σq , N∈ / q−1 j=p+1 σj , σp = σq =⇒ (∃j) (p < j < q & σp ∩ σj = ∅), and Σ is (2C(N − 1) + 1)-rich on the vertex set {1, . . . , N − 1}.
(Here, just as in the case of derived schemes, we denote by Σ the symbolic sequence that can be obtained from Σ by discarding all edges containing N .) Proof. The hypothesis on Σ implies that there exist subsequences Σ1 , . . . , Σr of Σ with the following properties: (1) For 1 ≤ i < j ≤ r every collision of Σi precedes every collision of Σj , (2) the graph of Σi (1 ≤ i ≤ r) is a tree (a connected graph without loop) on the vertex set {1, . . . , N }, and (3) r ≥ C(N ). Since every tree contains at least two vertices with degree one and C(N ) = (N/2) · {2C(N − 1) + 1}, there is a vertex, say the one labeled by N , such that N is a degree-one vertex of Σi(1) , . . . , Σi(t) , where 1 ≤ i(1) < · · · < i(t) ≤ r and t ≥ 2C(N − 1) + 1. Thus (iv) obviously holds. Let σp the edge of Σi(1) that contains N and, similarly, let σq be the edge of Σi(t) containing the vertex N . Then the fact t ≥ 3 ensures that the following properties hold: (i) N ∈ σp ∩ σq , (iii) σp = σq =⇒ ∃j p < j < q & σp ∩ σj = ∅, σj = σp . Let σp , σq (1 ≤ p < q ≤ n) be a pair of edges σp , σq (1 ≤ p < q ≤ n) fulfilling (i) and (iii) and having the minimum possible value of q −p . Elementary inspection shows that then (ii) must also hold for σp , σq . Lemma 3.5.1 is now proved. Let us fix a triplet (Σ, A, τ ) of the discrete (combinatorial) orbit structure with Property (A) (just as in [S-Sz(1999)], see Definition 3.31 there), and assume that Σ = (σ1 , . . . , σn ) is C(N )-rich, i.e., it contains at least C(N ) consecutive, connected collision graphs. We also consider the complex analytic manifold Ω (Σ, A, τ ) of all complex (Σ, A, τ )-orbits ω (Definition 3.20 in [S-Sz(1999)]) and the open, dense, connected domain D (Σ, A, τ ) ⊂ C(2ν+1)N +1 of all allowable initial data x = x(ω), see Definition 3.18 in [S-Sz(1999)]. Let, finally, Q(x) be a common irreducible divisor of the polynomials P1 (x), . . . , Ps (x) from (4.3) in [S-Sz(1999)]. (If such a common divisor exists.) In this section we will need several results about such common irreducible divisors Q(x) of the polynomials P1 (x), . . . , Ps (x). The first of them, as it is classically known from algebraic geometry (see, for example, [M(1976)]), is that the solution set V = {Q(x) = 0} of the equation Q(x) = 0 is a so-called irreducible (or, indecomposable) complex algebraic variety of codimension 1 in C(2ν+1)N +1 , which means that V is not the union of two, proper algebraic sub-varieties. Secondly, the smooth part S of V turns out to be a connected complex analytic manifold, while the non-smooth part V \ S of V
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is a complex algebraic variety of dimension strictly less that dimV = (2ν + 1)N , see [M(1976)]. Finally, if a polynomial R(x) vanishes on V , then Q(x) must be a divisor of R(x). (The last statement is a direct consequence of Hilbert’s theorem on Zeroes, see again [M(1976)].) The first result, specific to our current dynamical situation, is Proposition 3.2 The polynomials P1 (x), . . . , Ps (x) of (4.3) in [S-Sz(1999)] are homogeneous in the masses m1 , . . . , mN and, consequently, any common divisor Q(x) of these polynomials is also homogeneous in the masses. Proof. It is clear that the complex dynamics encoded in the orbits ω ∈ Ω (Σ, A, τ ) only depends on the ratios of masses m2 /m1 , m3 /m1 , . . . , mN /m1 . Consequently, all algebraic functions fi (x) (i = 1, . . . , s) featuring the proof of Lemma 4.2 of [SSz(1999)] are homogeneous of degree 0 in the masses. Since the rational function Pi (x) ∈ K0 = C(x) Qi (x) is the product α of all conjugates of fi (x) (see the proof of the lemma just cited), Pi (x) is also homogeneous of degree 0 in the variables m1 , . . . , mN . we get that Qi (x) Then elementary algebra yields that both Pi (x) and Qi (x) are homogeneous (of the same degree) in the masses. Since any factor of a homogeneous polynomial is easily seen to be also homogeneous, we get that the common divisor Q(x) of P1 (x), . . . , Ps (x) is also homogeneous in the variables m1 , . . . , mN . Our next result, specific to our dynamics, that will be needed later is Proposition 3.3 Let ν ≥ 3, (Σ, A, τ ) be a discrete orbit structure with Property (A) and a C(N )-rich symbolic collision sequence Σ = (σ1 , . . . , σn ). Denote by P1 (x), . . . , Ps (x) the polynomials of (4.3) in [S-Sz(1999)] just as before, and let Q(x) be a common irreducible divisor of P1 (x), . . . , Ps (x). (If such a common be an extended discrete orbit structure with divisor exists.) Let, finally, Σ, A, ρ Property (A) and an extended collision sequence Σ = (σ0 , σ1 , . . . , σn ). We claim that the irreducible (indecomposable) solution set V of the equation Q(x) = 0 cannot even locally coincide with any of the following singularity manifolds C defined by one of the following equations: (1) vi00 − vj00 2 = 0, (2) vi00 − vj00 ; q˜i00 − q˜j00 − L · a0 = 0, (3) mi0 + mj0 = 0, i.e., the irreducible polynomial Q(x) is not equal to any of the (irreducible) polynomials on the left-hand sides of (1)–(3). (In [S-Sz(1999)] these equations fea .) Consequently, the open subset ture Definition 3.18 of the domain D Σ, A, ρ V ∩ D Σ, A, ρ of V is connected and dense in V .
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Remark. The first point where we (implicitly) use the condition ν ≥ 3 is the irreducibility of the polynomial vi00 − vj00 2 on the left-hand side of (1). Indeed, in the case ν = 2 this polynomial splits as √ vi00 − vj00 2 ≡ vi00 1 − vj00 1 + −1 vi00 2 − vj00 2 √ · vi00 1 − vj00 1 − −1 vi00 2 − vj00 2 . However, it is easy to see that, in the case ν ≥ 3, all polynomials on the left of (1)–(3) are indeed irreducible. Proof. First of all, we slightly reformulate the negation of the statement of the proposition. Fix one of the three equations of (1)–(3) above, and denote the irreducible polynomial on its left-hand side by R(x). By using the quadratic (or linear) equation R(x) = 0, we eliminate one variable xj out of x by expressing it as an algebraic function xj = g(y ) of the remaining variables y of x, so that the algebraic function g only contains finitely many field operations and (at most one) square root. After this elimination xj = g(y ), the meaning of R(x) ≡ Q(x) (i.e., the negation of the assertion of the proposition) is that all algebraic functions fi (x) ≡ f˜i (y ) in the proof 4.2 of [S-Sz(1999)] (i = 1, . . . , s, constructed for (Σ, A, τ ), not of Lemma for Σ, A, ρ ) are identically zero in terms of y , meaning that every complex orbit segment ω ∈ Ω (Σ, A, τ ), with the initial data x(ω) in the solution set of R(x) = 0, is non-sufficient, see also the “Dichotomy Corollary” 4.7 in [S-Sz(1999)]. Thus, the negation of the proposition means that no orbit segment ω ∈ Ω (Σ, A, τ ) in the considered singularity is sufficient. Now we carry out an induction on the number of balls N quite in the spirit of the proof of Key Lemma 4.1 of [S-Sz(1999)]. Indeed, the statement of the proposition is obviously true in the case N = 2, for in that case there are no non-sufficient (complex) trajectories ω ∈ Ω (Σ, A, τ ), i.e., the greatest common divisor of the polynomials P1 (x), . . . , Ps (x) is 1. Assume now that N ≥ 3, ν ≥ 3, and the statement of Proposition 3.3 has been proven for all values N < N . Suppose, however, that the statement with N balls and Property is false for some (Σ, A, τ ) and extension Σ, A, ρ (A), i.e., that there exists a common irreducible divisor Q(x) of all the polynomials P1 (x), . . . , Ps (x), and Q(x) happens to be one of the irreducible polynomials on the left-hand side of (1), (2), or (3). By using the C(N )-richness of Σ = (σ1 , . . . , σn ), we select a suitable label k0 ∈ {1, 2, . . . , N }, say k0 = N , for the above, substitution mN = 0 along the lines of Lemma by also ensuring the exis 3.1 tence of the derived schemes (Σ , A , τ ) and Σ , A , ρ for the (N −1)-ball-system {1, 2, . . . , N − 1} and properties (i)–(iv) (of Lemma 3.1) for Σ , see Corollary 4.35 ˜ x) the polynomial obtained of [S-Sz(1999)] and Lemma 3.1 above. Denote by Q( from Q(x) after the substitution mN = 0. ˜ x) is not constant. Lemma 3.4 The polynomial Q(
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˜ x) ≡ c ∈ C. The case c = 0 means that mN is a divisor of Proof. Assume that Q( Q(x), thus mN ≡ Q(x) (for Q(x) is irreducible), which is impossible, since Q(x) has to be one of the polynomials on the left-hand side of (1), (2), or (3). If, however, c = 0, then we have that Q(x) ≡ c + mN S(x) with some nonzero polynomial S(x). (S(x) has to be non-zero, otherwise Q(x) would be a constant, not an irreducible polynomial.) However, this contradicts to the proved homogeneity of the polynomial Q(x) in the masses, see Proposition 3.2 above. This finishes the proof of the lemma. Remark. If one takes a look at the equations (1), (2), (3), he/she immediately ˜ x) ≡ Q(x) (when Q(x) is the polynomial on the left-hand realizes that either Q( ˜ x) ≡ mi0 when side of (1) or (2), or Q(x) ≡ mi0 + mj0 and N ∈ {i0 , j0 }), or Q( Q(x) ≡ mi0 + mj0 and N = j0 . In this way one can directly and easily verify Lemma 3.4 without the above “involved” algebraic proof. The reason why we still included the above proof is that later on in this section (in the proof of Sub-lemma 3.7) we will need the idea of the presented proof. The next lemma will use Definition 3.5 Suppose that two indices 1 ≤ p < q ≤ n and two labels of balls i, j ∈ {1, . . . , N } are given with the additional requirement that if i = j, then i ∈ q−1 l=p+1 σl . Following the proof of Lemma 4.2 of [S-Sz(1999)], denote by Q1 (x), Q2 (x), . . . , Qν (x) (x ∈ C(2ν+1)N +1 ) the polynomials with the property that for every vector of initial data x ∈ D (Σ, A, τ ) and for every k, k = 1, . . . , ν, the following equivalence holds true: ∃ ω ∈ Ω such that x(ω) = x & (vip (ω))k = vjq−1 (ω) k
⇐⇒ Qk (x) = 0. Our next lemma is a strengthened version of Lemma 4.39 of [S-Sz(1999)]: Lemma 3.6 Assume that the combinatorial-algebraic scheme (Σ, A, τ ) has Property (A), and use the assumptions and notations of the above definition. We claim that the polynomials Q1 (x), Q2 (x), . . . , Qν (x) do not have any non-constant common divisor. In other words, the equality vip (ω) = vjq−1 (ω) only takes place on an algebraic variety with at least two codimensions. Remark. Lemma 4.39 of [S-Sz(1999)] asserted that at least one of the above polynomials Qk (x) is nonzero. Then, by the permutation symmetry of the components k ∈ {1, . . . , ν}, all of these polynomials are actually nonzero. Proof. Induction on the number N ≥ 2. 1. Base of the induction, N = 2: First of all, by performing the substitution L = 0, we can annihilate all adjustment vectors, see (I), (IV), (VII) of Lemma 4.21 in [S-Sz(1999)], and Remark 4.22 there. Then, an elementary inspection shows that for any selection of positive real masses (m1 , m2 ), indeed, the
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equality vip (ω) = vjq−1 (ω) only occurs on a manifold with ν − 1 (≥ 2) codimensions in the section Ω (Σ, A, τ , m) of Ω (Σ, A, τ ) corresponding to the selected masses, since any trajectory segment of a two-particle system with positive masses and A = 0 has a very nice, totally real (and essentially ν-dimensional) representation in the relative coordinates of the particles: The consecutive, elastic bounces of a point particle moving uniformly inside a ball of radius 2r of Rν . Therefore, the statement of the lemma is true for N = 2. Assume now that N ≥ 3, and the lemma has been successfully proven for all smaller numbers of balls. By re-labeling the particles, if necessary, we can achieve that (i) N = i, N = j and (ii) if i = j, then the ball i has at least one collision between σp and σq with a particle different from N . For the fixed combinatorial scheme (Σ, A, τ ), select a derived scheme (Σ , A , τ ) corresponding to the substitution mN = 0, see Definition 4.11 and Corollary 4.35 in [S-Sz(1999)]. Our induction step is going to be a proof by contradiction. Assume, therefore, that the nonzero polynomials Q1 (x), Q2 (x), . . . , Qν (x) do have a common irreducible divisor R(x). According to Proposition 3.2 above, the (irreducible) ˜ x) polynomial R(x) is homogeneous in the variables m1 , . . . , mN . Denote by R( the polynomial that we obtain from R(x) after the substitution mN = 0. Similarly to Lemma 3.4 above, we claim ˜ x) is not constant. Sub-lemma 3.7. The polynomial R( Remark. The reason why we cannot simply apply Lemma 3.4 is that in the proof of that lemma we used the assumption that the irreducible polynomial Q(x) was one of the polynomials on the left-hand side of (1), (2), or (3) of Proposition 3.3. Right here we do not have such an assumption. ˜ x) ≡ c, where c ∈ C is a constant, i.e., R(x) ≡ c+mN ·S(x). Proof. Suppose that R( In the case c = 0 the polynomial mN ≡ R(x) would be a common divisor of all the polynomials Q1 (x), Q2 (x), . . . , Qν (x), meaning that in the considered N -ball system Ω (Σ, A, τ ) the equation mN (ω) = 0 implies the equality vip (ω) = vjq−1 (ω). This, in turn, means that in the (N − 1)-ball system {1, . . . , N − 1} (with the discrete algebraic scheme (Σ , A , τ )) the equality vip (ω) = vjq−1 (ω) is an identity, thus contradicting to the induction hypothesis. Therefore c = 0, and in the expansion R(x) ≡ c + mN · S(x) of the irreducible polynomial R(x) we certainly have S(x) ≡ 0, and this means that R(x) is not homogeneous in the mass variables, thus contradicting to Proposition 3.2. This finishes the proof of the sub-lemma. ˜ Finishing the proof of Lemma 3.6. Denote by Qk (x) the polynomial that we obtain from Qk (x) after the substitution mN = 0 (k = 1, . . . , ν), and by Tk (x) the polynomial constructed for the (N − 1)-ball system {1, . . . , N − 1} (with the discrete algebraic scheme (Σ , A , τ )) along the lines of Lemma 4.2 of [S-Sz(1999)],
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describing the event (vip (ω))k = (vjq−1 (ω))k in this subsystem (k = 1, . . . , ν). It follows from the induction hypothesis that the zero set Wk of the polyno˜ k (x) (in the phase space of the (N − 1)-ball system (Σ , A , τ )) has a mial Q codimension-two intersection with the singularities of the (Σ , A , τ ) system. In deed, otherwise we would have (vip )k ≡ vjq−1
on some (irreducible) singularity k
manifold of the (Σ , A , τ ) subsystem. Then, by the symmetry with respect to the coordinates k = 1, 2, . . . , ν, we would have vip ≡ vjq−1 on a codimensionone singularity of the subsystem (Σ , A , τ ), contradicting to the induction hypothesis. This means that the polynomial Tk (x) vanishes on the zero set Wk of ˜ k (x), so the non-constant common divisor R( ˜ x) of Q ˜ k (x) is a common divisor of Q T1 (x), . . . , Tν (x), contradicting to the induction hypothesis. This finishes the proof of Lemma 3.6. Continuing the proof of Proposition 3.3. Denote by P˜1 (x), . . . , P˜t (x) the “Pi polynomials” of the N -ball system (Σ, A, τ ) with the constraint mN = 0 constructed the same way as the polynomials P1 (x), . . . , Ps (x) in (4.3) of [S-Sz(1999)] for the general case mN ∈ C, see also the proof of Lemma 4.2 in the cited paper. It follows from the algebraic construction of these polynomials that the irreducible polyno˜ x) is a common divisor of P˜1 (x), . . . , P˜t (x). Recall that, according to our mial Q( indirect assumption made right before Lemma 3.4, Q(x) is a common, irreducible divisor of the polynomials P1 (x), . . . , Ps (x) and, at the same time, Q(x) is one of the polynomials on the left-hand side of (1), (2), or (3) in Proposition 3.3. The ˜ x) was obtained from Q(x) by the substitution mN = 0. polynomial Q( Let us focus now on Lemma 4.9 of [S-Sz(1999)]. The non-sufficiency of the N -ball system (Σ, A, τ ) with the side condition mN = 0 comes from two sources: Either from the parallelity of the relative velocities in (2) of Lemma 4.9, or from the non-sufficiency of the (N − 1)-ball part of the orbit segment ω with the combinatorial scheme (Σ , A , τ ). The first case takes place on a complex algebraic set of (at least) 2 codimensions, thanks to our original assumption ν ≥ 3 and Lemma 3.6 above. Concerning the application of the “non-equality” Lemma 3.6 (ω) are not equal, the above, we note here that once the velocities vipp (ω) and viq−1 q p q−1 p (ω) − vipp (ω) and vN (ω) − viq−1 (ω) = vN (ω) − viq−1 (ω) are relative velocities vN q q
p q−1 not parallel, unless the common velocity vN (ω) = vN (ω) belongs to the complex p q−1 line connecting the different velocities vip (ω) and viq (ω), which is a codimensionp ˜ x) = 0 with (ω). Therefore, the equation Q( (ν − 1) condition on the velocity vN ˜ ˜ the irreducible common divisor Q(x) of the polynomials P1 (x), . . . , P˜t (x) can only ˜ x) should describe the non-sufficiency of the (Σ , A , τ )-part of the system, thus Q( lack the kinetic and mass variables corresponding to the ball with label N , as the following sub-lemma states: ˜ x) of the polynomials Sub-lemma 3.8. The irreducible common divisor Q(
P˜1 (x), . . . , P˜t (x) does not contain the variables with label N .
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Proof. Let D = D Σ, A, τ mN = 0 ⊂ C(2ν+1)N be the open, connected and dense domain in C(2ν+1)N defined analogously to Definition 3.18 (of [S-Sz(1999)]) but incorporating the constraint mN = 0, see also Lemma 3.19 in [S-Sz(1999)]. Let, further, N ⊂ D be a small, complex analytic submanifold of D with complex dimension (2ν + 1)N − 1, holomorphic to the unit open ball of C(2ν+1)N −1 , and ˜ x) (x ∈ D) vanishes on N . (Such a manifold N ⊂ D such that the polynomial Q( exists by the induction hypothesis of Proposition 3.3.) We split the vectors x ∈ D as x = (y , z ), so that the z-part precisely contains the variables bearing the ball label N . We may assume that N ⊂ B1 × B2 , where B1 and B2 are small, open balls in the spaces of the components y and z, respectively. Assume, to the contrary of the statement of the sub-lemma, that the polyno˜ x) ≡ Q( ˜ y , z ) does depend on the component z. Then, for typical but fixed mial Q( y , the “slice” { y0 } × B2 intersects the manifold N in a set of comvalues y0 of plex codimension one. However, this fact clearly contradicts our earlier observation that the non-sufficiency of the orbit segments ω ∈ D = D Σ, A, τ mN = 0 imposes a codimension-2 condition on the coordinates z bearing the label N . This contradiction finishes the proof of the sub-lemma. Finishing the proof of Proposition 3.3. If Q(x) is the left-hand side of (1) or (2) ˜ x) ≡ in 3.3, then we arrive at the conclusion that the irreducible polynomial Q( ˜ ˜ Q(x) divides P1 (x), . . . , Pt (x), and N = i0 , N = j0 by Sub-lemma 3.8. This means, however, that the statement of the proposition is false for the (N − 1)ball system with the discrete algebraic scheme (Σ , A , τ ), contradicting to our induction hypothesis. If, however, the polynomial Q(x) is mi0 + mj0 , then in the case if N ∈ {i0 , j0 } we arrive at a contradiction just the same way as above. If N ∈ {i0 , j0 }, ˜ x) ≡ mi0 , and mi0 is a common divisor of all polynomials say N = j0 , then Q( ˜ ˜ P1 (x), . . . , Pt (x) describing the non-sufficiency of the (Σ , A , τ ) subsystem with the N − 1 balls {1, 2, . . . , N − 1}. This means that the above (Σ , A , τ ) subsystem is always non-sufficient, provided that mi0 = 0. In the case N ≥ 4 it follows from Lemma 4.1 of [S-Sz(1999)] (applied to the (N − 2)-ball system {1, 2, . . . , N } \ {i0 , N }) and from the “non-equality” Lemma 3.6 that almost every (Σ , A , τ )orbit with mi0 = 0 is in fact sufficient. One easily checks by inspection that, in the case N = 3, actually every orbit of the 2-ball system {1, 2} with the side condition mi0 = 0 is hyperbolic (sufficient). The obtained contradiction finishes the inductive proof of Proposition 3.3. Corollary 3.9. Keep all the notations and assumptions of Proposition 3.3, except that we assume now that Σ = (σ1 , . . . , σn ) is (2C(N ) + 1)-rich and the singularity manifold C is defined by one of the following equations: (1) vikk − vjkk 2 = 0, (2) vikk − vjkk ; q˜ikk − q˜jkk − L · ak = 0, (3) mik + mjk = 0
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with some k, 1 ≤ k ≤ n. Let Q(x) be a common irreducible divisor of the polynomials P1 (x), . . . , Ps (x) in (4.3) of [S-Sz(1999)] constructed for the entire discrete structure (Σ, A, τ ) as above. We again claim the same thing: The manifold C and the solution set of Q(x) = 0 cannot locally coincide. Consequently, an open, dense, and connected part of the irreducible variety V = {Q(x) = 0} belongs to the domain D (Σ, A, τ ) of the allowable initial data. Proof. We write Σ = (σ1 , . . . , σn ) in the form Σ = (Σ1 , σk , Σ2 ), where Σ1 = (σ1 , . . . , σk−1 ), Σ2 = (σk+1 , . . . , σn ). Then either Σ1 or Σ2 is C(N )-rich. If Σ2 turns out to be C(N )-rich, then we can directly apply the proposition after a simple time shift 0 −→ k. In the other case, when we only know that Σ1 is C(N )rich, beside the time shift 0 −→ k an additional time-reversal is also necessary to facilitate the applicability of the proposition. Another consequence of Proposition 3.3 is Corollary 3.10. Let ν ≥ 3, (Σ, A, τ ) be a discrete orbit structure with Property (A) and a C(N )-rich symbolic collision sequence Σ = (σ1 , . . . , σn ). Denote by P1 (x), . . . , Ps (x) the polynomials of (4.3) of [S-Sz(1999)] just as before, and let Q(x) be a common irreducible divisor of P1 (x), . . . , Ps (x). (If such a common be an extended discrete orbit structure with divisor exists.) Let, finally, Σ, A, ρ Property (A) and an extended collision sequence Σ = (σ0 , σ1 , . . . , σn ). According to Lemma 3.1, we can find a particle, say the one with label N , and two indices 1 ≤ p < q ≤ n such that (i) (ii) (iii) (iv)
N ∈ σp ∩ σq , q−1 N∈ / j=p+1 σj , σp = σq =⇒ (∃j) (p < j < q & σp ∩ σj = ∅), and Σ is (2C(N − 1) + 1)-rich on the vertex set {1, . . . , N − 1}.
(Here, just as in the case of derived schemes, we denote by Σ the symbolic sequence that can be obtained from Σ by discarding all edges containing N .) Denote by ˜ x) the polynomial that we obtain from Q(x) after the substitution mN = 0. Q( ˜ x) ≡ 0, otherwise there would not be any sufficient orbit segment (Obviously, Q( ω ∈ Ω (Σ, A, τ ) with mN = 0.) We claim that none of the irreducible polynomials on the left-hand side of (1) vi00 − vj00 2 = 0, (2) vi00 − vj00 ; q˜i00 − q˜j00 − L · a0 = 0, (3) mi0 + mj0 = 0 ˜ x). is a divisor of Q( ˜ x). According to SubProof. Consider and fix an irreducible factor R(x) of Q( lemma 3.8, the polynomial R(x) ≡ R(y , z ) does not contain any variable bearing the label N , i.e., R(x) ≡ R(y ).
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Assume, to the contrary of the statement that we want to prove, that the irreducible polynomial R(x) ≡ R(y ) is identical to one of the irreducible polynomials on the left-hand side of (1), (2), or (3). In particular, we have that N = i0 , of Sub-lemma 3.8, the algebraic N = j0 . As we saw in the course of the proof variety V = y ∈ C(2ν+1)(N −1)+1 R(y ) = 0 , defined by one of the equations (1), (2), or (3), describes the non-sufficiency of the derived system Ω (Σ , A , τ ) that one obtains from the original Ω (Σ, A, τ ) by taking mN = 0, i.e., for any point y ∈ D (Σ , A , τ ) it is true that y ∈ V if and only if there is some non-sufficient complex orbit segment ω ∈ Ω (Σ , A , τ ) with y (ω) = y. However, this statement contradicts to the assertion of Proposition 3.3. ˜ x) cannot be a constant c = 0, otherwise Remark. Note that the polynomial Q( the original polynomial Q(x) = c + mN · S(x) would not be homogeneous in the mass variables, see also the proof of Lemma 3.4. The main result of this section is Key Lemma 3.11. Keep all the notations and notions of this section. Assume that ν ≥ 3 and the symbolic collision sequence Σ = (σ1 , . . . , σn ) of the discrete algebraic frame (Σ, A, τ ) (with Property (A)) is C(N )-rich. We claim that all orbit segments ω ∈ Ω (Σ, A, τ ) are sufficient apart from an algebraic variety of codimension-two (at least two, that is), i.e., the polynomials P1 (x), . . . , Ps (x) of (4.3) of [S-Sz(1999)] do not have a non-constant common divisor. Proof. The inductive proof employs many of the ideas of the proof of Proposition 3.3 and it will use the statement of the proposition itself. (More precisely, the statement of Corollary 3.9 is to be used.) Indeed, the assertion of this key lemma is trivially true in the case N = 2, for in that case there are no non-sufficient, complex orbit segments ω ∈ Ω (Σ, A, τ ) at all. Assume that N ≥ 3, and the statement of the key lemma has been successfully proven for all smaller values (2 ≤) N < N . Our induction step is going to be a proof by contradiction. Suppose, therefore, that the polynomials P1 (x), . . . , Ps (x) do have a common irreducible divisor Q(x). Following the assertion of Lemma 3.1, select a suitable label k0 ∈ {1, . . . , N } for the substitution mk0 = 0 so that a derived scheme (Σ , A , τ ) (with Property (A)) exists for the arising (N − 1)-ball system {1, . . . , N } \ {k0 } with a symbolic sequence Σ , possessing the properties (1)–(4) of Lemma 3.1, the same way as we did in the proof of Proposition 3.3. Without loss of generality, we may assume that k0 = N . Consider now the original system Ω (Σ, A, τ ) with the constraint mN = 0. After the substitution mN = 0 the polynomial Q(x) becomes a new, non-constant ˜ x), see the proof of Sub-lemma 3.7 above. Let S(x) be an irreducible polynomial Q( ˜ divisor of Q(x). The (indecomposable) algebraic variety V = {S(x) = 0} has one ˜ = ΩmN =0 of Ω (Σ, A, τ ), and for every x ∈ V codimension, in the submanifold Ω there exists a non-hyperbolic complex orbit segment ω ∈ Ω (Σ, A, τ ) with x(ω) = x
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and mN (ω) = 0. As far as the non-sufficiency of the orbits ω ∈ Ω (Σ, A, τ ) with mN (ω) = 0 is concerned, we again take a close look at Lemma 4.9 of [S-Sz(1999)]. We saw earlier (see the proof of Lemma 3.6, which clearly carries over to the models subjected to the side condition mN = 0) that the parallelity of the relative p q−1 velocities vN (ω) − vipp (ω) and vN (ω) − viq−1 (ω) takes place on a manifold with q codimension at least two in our case of ν ≥ 3. Therefore, according to Lemma 4.9 of [S-Sz(1999)], the “codimension-one event” x(ω) ∈ V (⇐⇒ S(x(ω)) = 0) for orbits with mN (ω) = 0 can only be equivalent to the non-sufficiency of the {1, . . . , N − 1}-part trunc(ω) ∈ Ω (Σ , A , τ ) of the system. In this way it follows from Lemma 4.9 that the irreducible polynomial S(x) lacks all variables bearing the label N , see also the statement and the proof of Sub-lemma 3.8. We conclude that for every x ∈ V (i.e., with S(x) = 0) there exists an orbit segment ω ∈ Ω (Σ, A, τ ) with mN (ω) = 0, x(ω) = x, and a non-sufficient truncated segment ω = trunc(ω) ∈ Ω (Σ , A , τ ). According to Proposition 3.3 above (applied to the (N − 1)-ball system {1, . . . , N − 1} with the algebraic scheme (Σ , A , τ )), the variety {S(x) = 0} does not even locally coincide with the singularities of the complex dynamics Ω (Σ , A , τ ). This means that a codimension-one family of complex orbit segments ω = trunc(ω) ∈ Ω (Σ , A , τ ), x(ω) ∈ V , is not sufficient. This, in turn, contradicts the induction hypothesis of the proof of Key Lemma 3.11 by actually finishing it.
4 Finishing the proof of ergodicity From C back to R First of all, we transfer the main result of the previous section (Key Lemma 3.11) from the complex set-up back to the real case. This result will be an almost immediate consequence of Key Lemma 3.11. Fix a discrete algebraic scheme (Σ, A, τ ) for N balls with Property (A) (see Definition 3.31 in [S-Sz(1999)]) and a C(N )-rich symbolic collision sequence Σ = (σ1 , . . . , σn ). (The definition of the threshold C(N ) can be found in Lemma 3.1.) Denote by ΩR = ΩR (Σ, A, τ ) the set of all elements ω ∈ Ω (Σ, A, τ ) for which (1) all kinetic functions q˜ik (ω) j , vik (ω) j , mi (ω), and L(ω) take real values, i = 1, . . . , N ; k = 0, 1, . . . , n; j = 1, . . . , ν; (2) τk (ω) = tk (ω) − tk−1 (ω) > 0 for k = 1, . . . , n; (3) out of the two real roots of (3.8) of [S-Sz(1999)] the root τk is always selected as the smaller one, k = 1, . . . , n. It is clear that either ΩR = ΩR (Σ, A, τ ) is a ((2ν + 1)N + 1)-dimensional, real analytic submanifold of Ω = Ω (Σ, A, τ ), or ΩR = ∅. Of course, we will never investigate the case ΩR = ∅. Consider the corresponding polynomials P1 (x), . . . , Ps (x) of (4.3) of [S-Sz(1999)] describing the non-sufficiency of the complex orbit segments ω ∈ Ω (Σ, A, τ ), along the lines of Lemma 4.2 of [S-Sz(1999)], in terms of the kinetic
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data x = x(ω). According to the statement in the third paragraph on p. 61 of [S-Sz(1999)], these polynomials Pi (x) admit real coefficients. By Key Lemma 3.11, the greatest common divisor of P1 (x), . . . , Ps (x) is 1, hence the common zero set x ∈ R(2ν+1)N +1 P1 (x) = P2 (x) = · · · = Ps (x) = 0 of these polynomials does not contain any smooth real submanifold of (real) dimension (2ν + 1)N . In this way we obtained Proposition 4.1. Use all the notions, notations and assumptions from above. There exists no smooth, real submanifold M of ΩR with dimR M = dimR ΩR − 1 (= (2ν + 1)N ) and with the property that all orbit segments ω ∈ M are nonsufficient. (For the concept of non-sufficiency, please see §2.) The “Fubini-type” argument Our dynamics Ω (Σ, A, τ ) has the obvious feature that the variables mi = mi (ω) (i = 1, . . . , N ) and L(ω) (the so-called outer geometric parameters) remain unchanged during the time-evolution. Quite naturally, we do not need Proposition 4.1 directly but, rather, we need to use its analog for (almost) every fixed (N +1)-tuple (m1 , . . . , mN ; L) ∈ RN +1 . This will be easily achieved by a classical “Fubini-type” product argument. The result is Proposition 4.2. Use all the notions, notations and assumptions from above. Denote by
N S = N S (Σ, A, τ ) = ω ∈ ΩR (Σ, A, τ ) dimC N (ω) > ν + 1 the set of all non-sufficient, real orbit segments ω ∈ ΩR = ΩR (Σ, A, τ ). (For the definition of the complex neutral space N (ω), please see (3.21) in [S-Sz(1999)].) Finally, we use the notation
= m, and L(ω) = L L) = ω ∈ ΩR m(ω) ΩR (m, for any given (N + 1)-tuple (m, L) = (m1 , . . . , mN , L) ∈ RN +1 . We claim that N +1 for almost every (m, L) ∈ R (for which ΩR (m, L) = ∅) the intersection N S ∩ L) has at least 2 codimensions in ΩR (m, L). ΩR (m, Remark 4.3. As it is always the case with such algebraic systems, the exceptional zero-measure set of the parameters (m, L) turns out to be a countable union of smooth, proper submanifolds of RN +1 . Proof of Proposition 4.2. It is clear that the statement of the proposition is a local one, therefore it is enough to prove that for any small, open subset U0 ⊂ ΩR of ΩR = ΩR (Σ, A, τ ) the set
(m, L) ∈ RN +1 dimR (N S ∩ ΩR (m, L) ∩ U0 ) ≥ 2νN − 1 of the “bad points” (m, L) has zero Lebesgue measure. The points ω ∈ U0 can be identified locally (in U0 ) with the vector x = x(ω) ∈ DR = D (Σ, A, τ )∩ΩR of their
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initial coordinates. After this identification the small open set U0 ⊂ ΩR naturally becomes an open subset U0 ⊂ DR . Furthermore, we split the points x ∈ U0 as x = ((m, L), y ), where y contains all variables other than m1 , . . . , mN , L. In this way we may assume that U0 has a product structure U0 = B0 × B1 of two small open balls, so that B0 ⊂ RN +1 , while the open ball B1 ⊂ R2νN contains the y -parts of the points x = ((m, L), y ) ∈ U0 . Assume that the statement of the proposition is false. Then there exists a small open set U0 = B0 × B1 ⊂ RN +1 × R2νN (with the above splitting) and there is a positive number 0 such that the set
L) × B1 ) ∩ N S contains a A0 = (m, L) ∈ B0 ((m, (2νN − 1)-dimensional, smooth, real submanifold with inner radius > 0 has a positive Lebesgue measure in B0 . Then one can find an orthogonal projection P : R2νN → H onto a hyperplane H of R2νN such that, by taking Π(x) = Π ((m, L), y ) = P (y ), (Π : R(2ν+1)N +1 → H), the set
L) × B1 ) ∩ N S] contains A1 = (m, L) ∈ B0 Π [((m, an open ball of radius > 0 /2 in H has positive Lebesgue measure in B0 . By the Fubini theorem the set ˜ [N S ∩ (B0 × B1 )] Π has positive Lebesgue measure in B0 × H, where ˜ x) = Π ˜ ((m, Π( L), y) = ((m, L), P (y )) ∈ B0 × H for x ∈ B0 × B1 . However, dimR (B0 × H) = (2ν + 1)N , and dimR (N S ∩ ΩR ) ≤ (2ν + 1)N − 1 (according to Proposition 4.1). Thus, we obtained that the real algebraic set ˜ (N S ∩ (B0 × B1 )) ⊂ B0 × H Π has dimension strictly less than dimR (B0 × H) = (2ν + 1)N , yet it has positive Lebesgue measure in the space B0 × H. The obtained contradiction finishes the proof of Proposition 4.2. Finishing the proof of the theorem We will carry out an induction with respect to the number of balls N (≥ 2). For N = 2 the system is well known to be a strictly dispersive billiard flow (after the obvious reductions m1 v1 + m2 v2 = 0, m1 ||v1 ||2 + m2 ||v2 ||2 = 1 (m1 , m2 > 0), and after the factorization with respect to the uniform spatial translations, as usual) and, as such, it is proved to be ergodic by Sinai in [Sin(1970)], see also the paper [S-W(1989)] about the case of different masses.
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Assume now that N ≥ 3, ν ≥ 3, and the theorem has been successfully proven for all smaller numbers of balls N < N . Suppose that a billiard flow
t M, {S t }t∈R , µ = Mm,L , µm,L , Sm,L is given for N balls and outer geometric parameters (m, L) = (m1 , . . . , mN , L) (mi > 0, L > 0) in such a way that, besides the always assumed properties (2.1.1)– (2.1.2), (∗) the vector (m, L) of geometric parameters is such that for any subsystem 1 ≤ i1 < i2 < · · · < iN ≤ N (2 ≤ N ≤ N ) and for any C(N )-rich discrete algebraic scheme (Σ, A, τ ) (with Property (A)),for this subsystem mi1 , . . . , miN , L it is true that the parameter vector mi1 , . . . , miN , L does not belong to the zero-measured exceptional set of parameters featuring Proposition 4.2. According to Lemma 4.1 of [K-S-Sz(1990)], the set R2 ⊂ M of the phase points with at least two singularities on their trajectories is a countable union of smooth submanifolds of M with codimension two, so this set R2 can be safely discarded in the proof, for it is slim, see also §2 about the slim sets. Secondly, by the induction hypothesis and by Theorem 5.1 of [Sim(1992-I)] (adapted to the case of different masses) there is a slim subset S1 ⊂ M such that for every phase point x ∈ M \ S1 (i) S (−∞,∞) x contains at most one singularity, and (ii) S (−∞,∞) x contains an arbitrarily large number of consecutive, connected collision graphs. (In the case of a singular trajectory S (−∞,∞) x we require that both branches contain an arbitrarily large number of consecutive, connected collision graphs.) Then, by Proposition 4.2 just proved, there is another slim subset S2 ⊃ S1 of M such that (H) for every x ∈ M\S2 the trajectory S (−∞,∞) x contains at most one singularity and it is sufficient (or, geometrically hyperbolic). According to Theorem 6.1 of [Sim(1992-I)] (easily adapted to the case of different masses) and Proposition 4.2, the so-called Chernov-Sinai Ansatz (see §2) holds true, i.e., for almost every singular phase point x ∈ SR+ the positive semitrajectory S (0,∞) x is non-singular and sufficient. This is the point where the fundamental theorem for algebraic semidispersive billiards (Theorem 4.4 in [B-Ch-Sz-T(2002)]) comes to play! According to that theorem, by also using the crucial conditions (H) and the Ansatz above, it is true that for every phase point x ∈ (intM) \ S2 some open neighborhood Ux of x in M belongs to a single ergodic component of the considered billiard t flow Sm,L . Since the set (intM) \ S2 contains an arc-wise connected set C with full µ-measure (see Proposition 2.7.9 above), we get that the entire set C belongs to a single ergodic component of the flow ergodicity theorem.
t Sm,L . This finishes the proof of the
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5 Concluding remark: The irrational mass ratio Due to the natural reduction N i=1 mi vi = 0 (which we always assume), in §§1–2 we had to factorize the configuration space with respect to spatial translations: (q1 , . . . , qN ) ∼ (q1 + a, . . . , qN + a) for all a ∈ Tν . It is a remarkable fact, howN ever, that (despite the reduction i=1 mi vi = 0) even without this translation factorization the system still retains the Bernoulli mixing property, provided that the masses m1 , . . . , mN are rationally independent. (We note that dropping the above-mentioned configuration factorization obviously introduces ν zero Lyapunov exponents.) For the case N = 2 (i.e., two disks) this was proven in [S-W(1989)] by successfully applying D. Rudolph’s following theorem on the B-property of isometric group extensions of Bernoulli shifts [R(1978)]: Suppose that we are given a dynamical system (M, T, µ) with a probability measure µ and an automorphism T . Assume that a compact metric group G is also given with the normalized Haar measure λ and left invariant metric ρ. Finally, let ϕ : M → G be a measurable map. Consider the skew product dynamical system (M × G, S, µ × λ) with S(x, g) = (T x, ϕ(x) · g), x ∈ M , g ∈ G. We call the system (M × G, S, µ × λ) an isometric group extension of the base (or factor) (M, T, µ). (The phrase “isometric” comes from the fact that the left translations ϕ(x) · g are isometries of the group G.) Rudolph’s mentioned theorem claims that the isometric group extension (M × G, S, µ × λ) enjoys the B-mixing property as long as it is at least weakly mixing and the factor system (M, T, µ) is a B-mixing system. But how do we apply this theorem to show that the typical system of N hard N balls in Tν with i=1 mi vi = 0 is a Bernoulli flow, even if we do not make the factorization (of the configuration space) with respect to uniform spatial translations? It is simple. The base system (M, T, µ) of the isometric group extension (M ×G, S, µ×λ) will be the time-one map of the factorized (with respect to spatial translations) hard ball system. The group G will be just the container torus Tν with its standard Euklidean metric ρ and normalized Haar measure λ. The second component g of a phase point y = (x, g) ∈ M × G will be just the position of the center of the (say) first ball in Tν . Finally, the governing translation ϕ(x) ∈ Tν is quite naturally the total displacement 0
1
v1 (xt )dt
(mod Zν )
of the first particle while unity of time elapses. In the previous sections the Bmixing property of the factor map (M, T, µ) has been proven successfully for typical geometric parameters (m1 , . . . , mN ; L). Then the key step in proving the Bproperty of the isometric group extension (M × G, S, µ × λ) is to show that the latter system is weakly mixing. This is just the essential contents of the article [S-W(1989)], and it takes advantage of the assumption of rational independence of the masses. Here we are only presenting to the reader the outline of that proof. As a matter of fact, we not only proved the weak mixing property of the extension
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(M × G, S, µ × λ), but we showed that this system has in fact the K-mixing property by proving that the Pinsker partition π of (M × G, S, µ × λ) is trivial. (The Pinsker partition is, by definition, the finest invariant, measurable partition of the dynamical system with respect to which the factor system has zero metric entropy. A dynamical system is K-mixing if and only if its Pinsker partition is trivial, i.e., it consists of only the sets with measure zero and one, see [K-S-F(1980)].) In order to show that the Pinsker partition is trivial, in [S-W(1989)] we constructed a pair of measurable partitions (ξ s , ξ u ) for (M × G, S, µ × λ) made up by open and connected sub-manifolds of the local stable and unstable manifolds, respectively. It followed by standard methods (see [Sin(1968)]) that the partition π is coarser than each of ξ s and ξ u . Due to the S-invariance of π, we have that π is coarser than S nξs ∧ S nξu . (∗) n∈Z
n∈Z
In the final step, by using now the rational independence of the masses, we showed that the partition in (∗) is, indeed, trivial.
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A. Kr´amli, N. Sim´anyi, D. Sz´ asz, The K-Property of Four Billiard Balls, Commun. Math. Phys. 144, 107–148 (1992).
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N. Sim´ anyi, The K-property of N billiard balls I, Invent. Math. 108, 521–548 (1992).
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Ya.G. Sinai, On the Foundation of the Ergodic Hypothesis for a Dynamical System of Statistical Mechanics, Soviet Math. Dokl. 4, 1818–1822 (1963).
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Ya.G. Sinai, Dynamical systems with countably multiple Lebesgue spectrum II, Amer. Math. Soc. Transl. 68 No. 2, 34–38 (1968).
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Ya.G. Sinai, Dynamical Systems with Elastic Reflections, Russian Math. Surveys 25, 2, 137–189 (1970).
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I. Stewart, Galois Theory, Chapman and Hall, London (1973).
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Ya.G. Sinai, N.I. Chernov, Ergodic properties of certain systems of 2-D discs and 3-D balls, Russian Math. Surveys 42 No. 3, 181–207 (1987).
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N. Sim´anyi, D. Sz´ asz, The K-property of Hamiltonian systems with restricted hard ball interactions, Mathematical Research Letters 2 No. 6, 751–770 (1995).
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N. Sim´ anyi, M. Wojtkowski, Two-particle billiard system with arbitrary mass ratio, Ergod. Th. & Dynam. Sys. 9, 165–171 (1989).
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N´ andor Sim´ anyi University of Alabama at Birmingham Department of Mathematics Campbell Hall Birmingham, AL 35294 USA email: simanyimath.uab.edu Communicated by Eduard Zehnder submitted 17/10/02, accepted 01/12/03
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 235 – 244 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/020235-10 DOI 10.1007/s00023-004-0167-7
Annales Henri Poincar´ e
Bel-Robinson Energy and Constant Mean Curvature Foliations Lars Andersson∗
Abstract. An energy estimate is proved for the Bel-Robinson energy along a constant mean curvature foliation in a spatially compact vacuum spacetime, assuming an L∞ bound on the second fundamental form, and a bound on a spacetime version of Bel-Robinson energy.
1 Introduction ¯ , g¯) be a 3 + 1-dimensional C ∞ maximal globally hyperbolic vacuum Let (M ¯ has compact Cauchy sur(MGHV) space-time, which is spatially compact, i.e., M faces. One of the main conjectures (the CMC conjecture, see [5] for background) concerning spatially compact MGHV spacetimes states that if there is a constant ¯ , then there is mean curvature (CMC) Cauchy surface M0 , in such a spacetime M ¯ a foliation in M of CMC Cauchy surfaces with mean curvatures taking on all geometrically allowed values. Specifically, in case the Cauchy surface M0 is of Yamabe type −1 or 0, then the mean curvatures take all values in (−∞, 0), or (0, ∞), depending on the sign of the mean curvature of M0 , while in case M0 is of Yamabe type +1, the mean curvatures take on all values in (−∞, ∞). The only progress towards proving the CMC conjecture so far has been made under conditions of symmetry, cf. [11], or curvature bounds [3], [6]. One approach to the CMC conjecture is to view it as a statement about the global existence problem for the Einstein vacuum field equations ¯ αβ = 0, R
(EFE)
in the CMC time gauge. It is known that in the CMC gauge with zero shift, the (EFE) are non-strictly hyperbolic [8] while in other gauges such as wave coordinates, the (EFE) form a system of quasi-linear wave equations for the metric g¯. In this context, it has been conjectured that the Cauchy problem for the (EFE) is well posed for data in H 2 × H 1 (the H 2 conjecture, see [10]). From this point of view it is interesting to consider continuation principles for the (EFE), in CMC gauge. In this note we will use a scaling argument to prove an energy estimate for CMC foliations. The energy we consider is a version of the ∗ Supported in part by the Swedish Research Council, contract no. R-RA 4873-307, the NSF, contract no. DMS 0104402, and the Erwin Schr¨ odinger Institute, Vienna.
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¯ , the energy expression Bel-Robinson energy. For a spatial hypersurface M in M we consider is defined by (|E|2 + |B|2 )µg , Q(M ) = M
where E, B are the electric and magnetic parts of the Weyl tensor (defined w.r.t. the timelike normal T of M ). Roughly speaking, Q bounds Cauchy data (g, K) on M in H 2 × H 1 . Here, g is the induced metric on M and K is the second fundamental form of K. Therefore, if the H 2 conjecture is true, apriori bounds for the Bel-Robinson energy can be expected to be relevant to the global existence problem for the (EFE). Let H = trK denote the mean curvature, and assume the CMC gauge condition H = t. Define the spacetime Bel-Robinson energy of a CMC foliation FI = {Mt , t ∈ I} by Q(FI ) = dt N (|E|2 + |B|2 )µg , (1.1) I
Mt
where N is the lapse function. We are now ready to state our main result ¯ , g¯) be a MGHV space-time, and let I = (t− , t+ ) with −∞ < Theorem 1.1. Let (M ¯ , g¯). Let t0 ∈ I. t− < t+ < 0, be such that there is a CMC foliation FI in (M Suppose that lim supt→t+ Q(t) = ∞. Then at least one of the following holds: 1. lim suptt+ Q(F[t0 ,t) ) = ∞, 2. lim suptt+
||K(t)||L∞ |H(t)|
= ∞.
The time reversed statement with t+ replaced by t− also holds. Remark 1.1. Let (M, γ) be a compact hyperbolic 3-manifold with sectional cur¯ = (0, ∞) × M is flat. It vature −1. Then the metric γ¯ = −dρ2 + ρ2 γ on M follows from the work of Andersson and Moncrief [7] that for small perturbations ¯ , γ¯), there is a global CMC foliation F[t ,t) in the expanding direction, and of (M 0 for this foliation, the Bel-Robinson energy decays as Q(t) = O(H 2 (t)), which implies that the space-time Bel-Robinson energy Q(F ) is bounded in this case. It is interesting to consider the behavior of Q(F[t0 ,t) ) when t0 decreases. The proof of Theorem 1.1 is based on a scaling argument, which we now sketch. The statement of the theorem is symmetric in time, but here we consider only the future time direction, the argument in the reverse direction is similar. Suppose for a contradiction there is a constant Λ < ∞ so that Q(F[t0 ,t∗ ) ) ≤ Λ, ||K||2L∞ /H 2 ≤ Λ for all t ∈ [t0 , t∗ ), and that lim sup Q(t) = ∞ . tt∗
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An energy estimate shows that rh (t)Q(t) ≤ C, where rh is an L1,p harmonic radius, for some fixed p, 3 < p < 6, and hence if rh is bounded from below there is nothing to prove. Suppose for a contradiction that rh → 0 as t t∗ . The combination rh Q is scale invariant, and hence by rescaling g¯ to g¯ = rh−2 g¯, we get a sequence of metrics g with Q bounded. Q bounds g in L2,2 and hence we may pick out a subsequence of (g , N ), which converges weakly to a solution (g∞ , N∞ ) of the static vacuum Einstein equations, ∆N = 0, 2
∇ N = N Ric.
(1.2a) (1.2b)
It follows from our assumptions that the limit g∞ is complete, and the limiting N∞ is bounded from above and below. Then by [2], g∞ must be flat, with infinite harmonic radius, which contradicts rh = 1, by the weak continuity of rh on L2,2 . We conclude that in fact rh is bounded away from zero. and hence that Q does not blow up, which proves the theorem.
2 Preliminaries ¯ we denote its timelike normal T and induced For a space-like hypersurface M in M metric and second fundamental form (g, K). We assume all fields are C ∞ unless otherwise stated. Let lower case greek indices run over 0, . . . , 3 while lower case latin indices run over 1, . . . , 3. We work in an adapted frame eα , with e0 = ∂t . Our convention for K is Kab = − 21 LT g¯ab , so that if the mean curvature H = trK is negative, T points in the expanding direction. We will sometimes use an index T to denote contraction with T , for example uT = uα T α . In a nonflat spatially compact, globally hyperbolic, vacuum spacetime, the maximum principle implies uniqueness of constant mean curvature (CMC) Cauchy ¯ is contained in at most one CMC Cauchy surfaces. In particular, each x ∈ M surface, and for each t ∈ R, there is at most one Mt with mean curvature t. ¯ is a foliation FI = Let I ⊂ R be an interval. A CMC foliation FI in M {Mt , t ∈ I} such that for each t ∈ I, Mt is a C ∞ CMC Cauchy surface with mean curvature t. When convenient we will write g(t), K(t) for the data induced on Mt . Introducing coordinates xα with x0 = t, the lapse and shift N, X of the foliation are defined by ∂t = N T + X. We may without loss of generality assume X = 0. ¯ if there is no interval I containing We call FI a maximal CMC foliation in M ¯ , we write I as a strict subset with a CMC foliation FI . Given a foliation in M ¯ F for the support of F . M ¯ contains a compact, constant mean curvature (CMC) Cauchy Assume that M surface M0 with mean curvature H 0 < 0. By standard results there is then an interval I = (t− , t+ ) ⊂ R, H 0 ∈ I, such that there is a CMC foliation FI , and by uniqueness, MH 0 = M0 . Hence if FI is a maximal CMC foliation, then I has nonempty interior.
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The Bel-Robinson energy
¯ , g¯) and let ∗ W denote its (left) dual (in vacuum Let W be the Weyl tensor of (M ∗ ∗ ¯ , g¯) is defined by W = W ). The Bel-Robinson tensor Q of (M Qαβγδ = Wαµγν Wβµδ ν + ∗ W αµγν ∗ W β
µ ν δ .
(2.3)
Then Q is totally symmetric and trace-less, and in vacuum, Q has vanishing divergence. Let Eαβ = WαT βT , Bαβ = ∗ W αT βT be the electric and magnetic parts of the Weyl tensor. Then E, B are symmetric, t-tangent (i.e., EαT = BαT = 0) and trace invariant, g ab Eab = g ab Bab = 0. In vacuum, we have Eab = Ricab + HKab − Kac K cb ,
Bab = −curlKab ,
where for a symmetric tensor in dimension 3, 1 st ( ∇t Asb + bst ∇t Asa ). 2 a Recall that for symmetric traceless tensors in dimension 3, the Hodge system A → (divA, curlA) is elliptic. The following identities, see [7], relate Q to E and B, curlAab =
QT T T T = Eab E ab + Bab B ab = |E|2 + |B|2 ,
(2.4a)
QaT T T = 2(E ∧ B)a ,
(2.4b)
1 QabT T = −(E × E)ab − (B × B)ab + (|E|2 + |B|2 )gij , 3 where by definition, for symmetric tensors A, B in dimension 3,
(2.4c)
(A ∧ B)a = abc Abd Bdc , 1 1 (A × B)ab = acd bef Ace Bdf + (A · B)gab − (trA)(trB)gab . 3 3 From equation (2.4a) it follows that QT T T T ≥ 0 with equality if and only if W = 0. ¯ . The Bel-Robinson energy Q(t) of Mt ∈ F w.r.t. the Let F be a foliation in M time-like normal T , is defined by QT T T T µg = (|E|2 + |B|2 )µg . Q(t) = Q(Mt ) = Mt
Mt
An application of the Gauss law gives in vacuum, N QαβT T π αβ µg , ∂t Q(t) = −3 Mt
¯ α Tβ . A computation shows that the only nonzero components of where παβ = ∇ παβ are πab = −Kab , πT a = N −1 ∇a N . Thus N QαβT T π αβ = −N QabT T K ab − QaT T T ∇a N.
(2.5)
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3 Proof of Theorem 1.1 We will assume that the complement of points 1., 2. of Theorem 1.1 holds, and prove from this that Q(t) does not blow up. Assume for a contradiction there is a constant Λ > 1 so that for t ∈ [t0 , t∗ ), Q(F[t0 ,t) ) ≤ Λ,
||K(t)||2L∞ ≤ Λ, H 2 (t)
(3.6)
and that lim suptt∗ Q(t) = ∞. We let Ls,p denote the Lp Sobolev spaces and write H s for Ls,2 . We will sometimes use subindices x or t, x to distinguish function spaces defined w.r.t. space or space-time. For a foliation FI , we may without loss of generality assume that T = N −1 ∂t , where N > 0 is the lapse function of the foliation. Then g¯ is of the form g¯ = −N 2 dt2 + gab dxa dxb . The lapse function satisfies −∆N + |K|2 N = 1,
(3.7)
which using the maximum principle implies the estimate 1/||K||2L∞ ≤ N ≤ 3/H 2 .
(3.8)
¯ . From (2.5) we get Let F be a foliation in M |∂t Q| ≤ C1 (||∇N ||L∞ + ||N ||L∞ ||K||L∞ )Q, ˆ = K − (H/3)g with C1 a universal constant. By assumption, |K|2 /H 2 ≤ Λ. Let K ˆ 2 ≥ H 2 /3, we get be the traceless part of K. Using |K|2 = H 2 /3 + |K| |∂t Q| ≤ C(||∇N ||L∞ +
Λ )Q. |H|
(3.9)
We may assume without loss of generality that ||∇N ||L∞ ≥ Λ/|H|, since otherwise there would be nothing to prove. Therefore we may absorb Λ/|H| in the constant in (3.9) to get (3.10) |∂t Q| ≤ C||∇N ||L∞ Q.
3.1
The blowup
Choose once and for all a fixed p satisfying 3 < p < 6.
(3.11)
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On the Riemannian manifold (Mt , gt ), let rh (x) denote the L1,p harmonic radius of (Mt , gt ) at x ∈ Mt ; thus rh (x) is the radius of the largest geodesic ball about x on which there is a harmonic coordinate chart in which the metric coefficients gab satisfy rh (x)−3/p ||gab − δab ||Lp (Bx (rh (x))) + rh (x)(p−3)/p ||∂gab ||Lp (Bx (rh (x))) ≤ C, (3.12) where C is a fixed constant (say C = 1), cf. [1, 4]. By Sobolev embedding, in Bx (rh (x)), the C β norm of gab is controlled, for β = 1 − 3p . The presence of the factors of rh (x) in (3.12) means the estimate (3.12) is scale invariant. It follows from this that rh (x) scales as a distance. It is well known that the Laplacian in such a local harmonic coordinate chart on Bx (rh (x)) has the form ∆u = g ab ∂a ∂b u. Thus, within Bx (rh (x)), ∆ is given in these local coordinates as a non-divergence form elliptic operator, with uniform C β control on the coefficients g ab , and uniform bounds on the ellipticity constants. We have the following standard (interior) Lp elliptic estimate for this Laplace operator, cf. [9, Thm. 9.11]. Let B = Bx (rh (x)) and B = Bx ( 12 rh (x)). Then ||N ||L2,p (B ) ≤ C(rh (x), p)[||∆N ||Lp (B) + ||N ||Lp (B) ].
(3.13)
We drop the dependence on p, since p is fixed. We need to make explicit the dependence of the constant C on rh (x). This is done by a standard scaling argument. Thus, assume (by rescaling if necessary), that rh (x) = 1. Then (3.13) becomes ||N ||L2,p (B ) ≤ C[||∆N ||Lp (B) + ||N ||Lp (B) ]. By Sobolev embedding, since p > 3 is fixed, and B = B( 12 ), we have ||∇N ||L∞ (B ) ≤ c · ||N ||L2,p (B ) , so that ||∇N ||L∞ (B ) ≤ Co [||∆N ||Lp (B) + ||N ||Lp (B) ]. and in particular, ||∇N ||L∞ (B ) ≤ Co [||∆N ||L∞ (B) + ||N ||L∞ (B) ],
(3.14)
where Co is an absolute constant, (i.e., independent of N , given control on ∆ from definition of rh = 1). Now we put in scale factors to make (3.14) scale invariant and write (3.14) as rh (x)||∇N ||L∞ (B ) ≤ Co [rh (x)2 ||∆N ||L∞ (B) + ||N ||L∞ (B) ].
(3.15)
Note that the function N is itself scale invariant. Each term in (3.15) is invariant under scaling, and thus (3.15) holds in any scale. Therefore, it holds in the metric g(t).
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Let rh = rh (t) = inf inf rh (x). s≤t x∈Ms
From the lapse equation,
∆N = N |K|2 − 1.
ˆ 2 we find Using (3.8) and |K|2 = H 2 /3 + |K| 0 ≤ ∆N ≤ 3 Thus, we have
ˆ 2 |K| |K|2 ≤ 3 ≤ 3Λ. H2 H2
1 rh ||∇N ||L∞ (B ) ≤ 3Co rh2 Λ + 2 . H
(3.16)
In particular, this gives the estimate ||∇N (t)||L∞ ≤ C(Λ, t∗ )/rh (t).
(3.17)
Integrating (3.10), (recall we absorbed the term Λ/|H| in (3.9) into the constant), gives t1 Q(t1 ) ≤ Q(t0 ) + C ds||∇N (s)||L∞ Q(s). t0
We may without loss of generality assume the last term is bigger than 1, so we may absorb Q(t0 ) into C. Multiplying both sides by rh , and using (3.16) we have t1 1 rh Q(t1 ) ≤ C ds rh2 Λ + 2 Q(s). H (s) t0 We may without loss of generality assume rh ≤ 1/|H|, since otherwise there would be nothing to prove, and therefore we can absorb the term rh2 Λ into the constant. Then we have t1 1 rh Q(t1 ) ≤ C ds 2 Q(s). H (s) t0 The inequality (3.8) implies
N ≥ 3Λ−1 H −2 ,
which by the definition of the spacetime Bel-Robinson energy Q(F[t0 ,t1 ) ), see (1.1), gives rh (t1 )Q(t1 ) ≤ CQ(F[t0 ,t1 ) ). (3.18) By assumption, lim suptt∗ Q(t) = ∞, which by the assumed bound on Q(F[t0 ,t1 ) ) implies lim rh (t) = 0. tt∗
We will show that this contradicts (3.6).
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Suppose then that there is an increasing sequence of times ti , ti → t∗ as i → ∞, so that ri = rh (ti ) satisfy limi→∞ ri = 0 (recall that by construction rh (t) is decreasing). Now we have from (3.18) and our assumptions, ri Q(ti ) ≤ C.
(3.19)
Now we introduce the blowup scale. Let g¯i = ri−2 g¯. We will denote the scaled versions of g, K by gi , Ki . We scale the coordinates as ti = ri−1 t, xi = ri−1 x , so that the coordinate components of gi are scale invariant. Then |Ki | ≤ Λri , while the lapse N does not scale, Ni (xi ) = N (x). After translating the time coordinate as ti = ri−1 (t − ti ), we focus our attention on the time interval ti ∈ [−1, 0]. We further translate the space coordinate so that the center of the coordinate system (0,0) is the point where the harmonic radius achieves its minimum value. Since rh Q is scale invariant, we have rh Q = Q , and hence the inequality Q (ti ) ≤ C
(3.20)
holds. This means in view of the definition of the Bel-Robinson energy that at the blowup scale, Rici is bounded in L2 . By construction rh ≥ 1 and by [4], it follows from the Ricci bound that gi is bounded in L2,2 loc . Similarly the Hodge system . relating K to B leads to Ki bounded in L1,2 loc The Einstein vacuum equation is scale invariant, and therefore holds at the blowup scale. We will argue in the next section, that the above bounds on gi , Ki allow us to pick out a weakly convergent subsequence of (gi , Ni ) with limit g∞ , N∞ solving the static vacuum Einstein equation, cf. equation (1.2) below, with g∞ complete.
3.2
Weak convergence
Let g¯i be the sequence of rescaled spacetime metrics. We consider rescaled time ti in the interval [−1, 0]. By construction, the L1,p harmonic radius satisfies ri (0) = 1, and ri (t) ≥ 1 for t ∈ [−1, 0]. Equation (3.8) implies that the rescaled lapse is bounded from above and below, 1 t2− Λ
≤ Ni ≤
3 . t2+
(3.21)
By (3.20), we have Q (t) ≤ C, for t ∈ [−1, 0], and hence we have (gi , Ki ) bounded 1,2 in L∞ ([−1, 0]; L2,2 loc × Lloc ). It follows that there is a subsequence which converges 1,2 weak- to a limit (g∞ , K∞ ) ∈ L∞ ([−1, 0]; L2,2 loc × Lloc ), with corresponding spacetime metric g¯∞ . By passing to a further subsequence if necessary, which we still denote using the index i, we may assume that gi (0) g∞ (0) weakly in L2,2 loc . Let us consider the properties of this limit. First note that |Ki | ≤ Λri → 0 as i → ∞, and hence K∞ ≡ 0. The relation ∂ti gi = −2Ni Ki holds in the limit
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and since K∞ ≡ 0, we conclude that g∞ is time-independent, so that the limiting spacetime metric g¯∞ is static. The lapse equation now implies that the limiting lapse function satisfies (3.22a) ∆∞ N∞ = 0, where ∆∞ is the Laplace operator defined w.r.t. g∞ . The rescaled spacetime metrics g¯i are solutions of the Einstein vacuum equation and the evolution equation for K, ∂t K = −∇2 N + N (Ric + HK − 2K:K), where (K : K)ab = Kac K cb , holds weakly in the limit. In view of the fact that g → Ric is weakly continuous on L2,2 loc and K∞ ≡ 0, we get the equation 0 = −∇2∞ N∞ + N∞ Ric∞ .
(3.22b)
By construction, g∞ is complete, and hence in view of (3.22) we have a complete solution of the static Einstein equations with N∞ > 0. It follows by [2, Theorem 3.2] that g∞ is flat and N∞ is constant. In particular, rh [g∞ ](0) = ∞. Now, since rh is by definition the L1,p harmonic radius, 3 < p < 6, the map g → rh is weakly continuous on L2,2 loc and hence by construction rh [g∞ ](0) = 1. This is a contradiction, and it follows that in fact lim inf i→∞ ri > 0, which by the BR energy estimate (3.19) implies that Q(t) does not blow up. This completes the proof of Theorem 1.1. Acknowledgments. The problem studied in this paper was suggested by Mike Anderson. I am grateful to him for many helpful suggestions, and to Vince Moncrief and Jim Isenberg for useful discussions on the topic of this paper.
References [1] Michael T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 no. 2, 429–445 (1990). [2]
, Scalar curvature, metric degenerations and the static vacuum Einstein equations on 3-manifolds. I, Geom. Funct. Anal. 9 no. 5, 855–967 (1999).
[3]
, On long-time evolution in general relativity and geometrization of 3-manifolds, Comm. Math. Phys. 222 no. 3, 533–567 (2001).
[4] Michael T. Anderson and Jeff Cheeger, C α -compactness for manifolds with Ricci curvature and injectivity radius bounded below, J. Differential Geom. 35 no. 2, 265–281 (1992). [5] Lars Andersson, The global existence problem in general relativity, grqc/9911032, to appear in “50 Years of the Cauchy problem in General Relativity”, eds. Piotr T. Chru´sciel and Helmut Friedrich.
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, Constant mean curvature foliations of flat space-times, Comm. Anal. Geom. 10 no. 5, 1125–1150 (2002).
[7] Lars Andersson and Vincent Moncrief, Future complete vacuum spacetimes, gr-qc/0303045, to appear in “50 Years of the Cauchy problem in General Relativity”, eds. Piotr T. Chru´sciel and Helmut Friedrich. [8] Yvonne Choquet-Bruhat and Tommaso Ruggeri, Hyperbolicity of the 3 + 1 system of Einstein equations, Comm. Math. Phys. 89 no. 2, 269–275 (1983). [9] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, second ed., Springer-Verlag, Berlin, 1983. [10] Sergiu Klainerman, Geometric and Fourier methods in nonlinear wave equations, preprint, Princeton, 2001. [11] Alan D. Rendall, Constant mean curvature foliations in cosmological spacetimes, Helv. Phys. Acta 69 no. 4, 490–500, (1996), Journ´ees Relativistes 96, Part II (Ascona, 1996), gr-qc/9606049. Lars Andersson Department of Mathematics University of Miami Coral Gables, FL 33124 USA email:
[email protected] Communicated by Sergiu Klainerman Submitted 25/07/03, accepted 27/01/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 245 – 260 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/020245-16 DOI 10.1007/s00023-004-0168-6
Annales Henri Poincar´ e
On the Uniqueness of AdS Space-Time in Higher Dimensions Jie Qing
Abstract. In this paper, based on an intrinsic definition of asymptotically AdS spacetimes, we show that the standard anti-de Sitter space-time is the unique strictly stationary asymptotically AdS solution to the vacuum Einstein equations with negative cosmological constant in dimension ≤ 8. Instead of using the positive energy theorem for asymptotically hyperbolic spaces with spin our approach appeals to the classic positive mass theorem for asymptotically flat spaces.
1 Introduction Recently, there has been some interest in the study of space-times that satisfy Einstein equations with negative cosmological constant in association with the so-called AdS/CFT correspondence. With the presence of a negative cosmological constant, the anti-de Sitter space-time replaces the Minkowski space-time as the ground state of the theory. Boucher, Gibbons and Horowitz demonstrated that in 3 + 1 dimensions, the only strictly stationary asymptotically AdS space-time that satisfies the vacuum Einstein equations with negative cosmological constant is the anti-de Sitter space-time in [BGH] (see also [CS]). Another class of globally static asymptotically locally AdS space-times, the AdS solitons, are also important in the theory. In [GSW], Galloway, Surya and Woolgar proved a uniqueness theorem of the AdS solitons. Later, in [ACD], Anderson, Chru´sciel and Delay improved the uniqueness theorem of AdS solitons. In [Wa1], the uniqueness result of [BGH] was generalized to higher dimensions when the space-time is static and the static slice is of spin structure. Proofs in [BGH] and [Wa1] all appeal to the positive energy theorem for asymptotically hyperbolic spaces (see [CH], [Wa3] and some early references therein). Our proof of the uniqueness of the AdS space-time instead appeals to the classic positive mass theorem for asymptotically flat spaces. By this approach we may use the classic positive mass theorem of Schoen and Yau [SY] to drop the spin structure assumption on the static slice in dimension less than 8. The anti-de Sitter space-time in n + 1 dimensions is given by (Rn+1 , gAdS ) where 1 gAdS = −(1 + r2 )dt2 + dr2 + r2 dσ0 (1.1) 1 + r2
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in coordinates (t, r, θ) ∈ R × [0, ∞) × S n−1 and dσ0 is the standard round metric on a unit (n − 1)-sphere. It is a static solution to the vacuum Einstein equation 1 Ric − R g + Λg = 0 2
(1.2)
with negative cosmological constant Λ = − 12 n(n − 1). The staticity implies that √ (Rn+1 , gAdS ) can be constructed by a triple (Rn , gH , 1 + r2 ) where gH is the hyperbolic metric on Rn and (1.3) ∇2 1 + r2 = 1 + r2 gH on the hyperbolic space (Rn , gH ). The simplest examples of space-times that are asymptotically the same as the anti-de Sitter space-time at the infinity are the so-called Schwarzschild-AdS space-times whose metrics are given by + = −(1 + r2 − gM
M 1 )dt2 + 2 rn−2 1+r −
M r n−2
dr2 + r2 dσ0 .
(1.4)
They also satisfy the vacuum Einstein equation (1.2), but the difference is that on ∂ while this is the AdS space-time there is an everywhere time-like Killing field ∂t not so on the Schwarzschild-AdS space-times. In other words, the AdS space-time is strictly stationary, but the Schwarzschild-AdS space-times are not. We always assume in this paper that every orbit of the time-like Killing field is complete for stationary space-times. We will follow the idea in [AM] to give a definition of asymptotically AdS space-times (see Definition 2.1). One can find a good discussion of the comparisons of different definitions of asymptotically AdS space-times in [CS]. Then we show Theorem 1.1 Suppose that (Y n+1 , g) is a strictly stationary asymptotically AdS space-time satisfying the causality axiom. And suppose that g satisfies the vacuum Einstein equation with negative cosmological constant. Then (Y n+1 , g) is static, i.e., n+1 =R×Σ Y (1.5) g = −V dt2 + h where V > 0 on Σ and
√ √ ∆ V =n V √ √ Ric[h] + nh = ( V )−1 ∇2 V
(1.6)
on the Riemannian manifold (Σ, h). We adopt the definition of causality axiom from [Ca], which simply requires no closed time-like curves in the space-time. By the Frobenius Theorem, staticity is locally equivalent to θ = ω ∧ dω = 0 where ω is the dual of the given Killing
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vector field X. Instead of using topological assumptions to write ∗θ = dψ to prove the vanishing of θ in classic Lichnerowicz argument (see [BGH], [Ca]) we observe that 1 1 d( ω) = − 2 iX θ, (1.7) V V which allows us to calculate the boundary integral to show the vanishing of θ from the behavior of X near the infinity. We adopt the method of Fefferman and Graham [FG], [G] to construct a preferable coordinate system near the infinity which allows us to know the asymptotic behavior of both the metric g and the Killing field X near the infinity rather precisely. √ Our next goal is to prove the static solution (Σn , h, V ) must be the same √ as (Rn , gH , 1 + r2 ) for some choice of coordinates. Namely, Theorem 1.2 Suppose that a space-time is asymptotically AdS and space-like geodesically complete. And suppose that the asymptotically AdS space-time √ √ is a static solution satisfying (1.5) and (1.6). Then (Σn , h, V ) = (Rn , gH , 1 + r2 ) for some choice of coordinates in dimension 3 ≤ n ≤ 7. First we would like to remark here that the large class of static solutions in dimension 4 constructed by Anderson, Chru´sciel and Delay in [ACD] have different asymptotic behaviors than the one we imposed here. Our approach is similar to the one used to prove the uniqueness of conformally compact Einstein manifolds in √ [Q]. We use the global defining function ( V + 1)−1 to turn (Σ, h) into a compact ¯ which has the round sphere as its totally umbilical boundary and manifold (Σ, h) whose scalar curvature is non-negative. The nonnegativity of the scalar curvature √ n(n − 1)(V − |∇ V |2 − 1) follows from the application of the strong maximum principle and the following Bochner formula: Lemma 1.3 Suppose that an asymptotically AdS space-time is a static solution satisfying (1.5) and (1.6). Then √ √ √ √ √ ∇ V −∆(V − |∇ V |2 − 1) = 2|∇2 V − V h|2 − √ · ∇(V − |∇ V |2 − 1). (1.8) V One should compare (1.8) with an identity of Lindblom [L] for n = 3 (see also [BGH] [BS] for its applications). Then we appeal to the recent work in [Mi](see also [ST]) to conclude that it has to be scalar flat, which implies √ √ ∇2 V = V h. (1.9) Note that [Mi] relies on the classic positive mass theorem of Schoen and Yau [SY]. Theorem 1.2 then follows from the following lemma similar to a theorem of Obata in [Ob]. Lemma 1.4 Suppose that (M n , g) is a complete Riemannian manifold. And suppose that there is a positive function φ such that ∇2 φ = φg. Then (M, g) is isometric to (Rn , gH ).
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In summary we prove Main Theorem Suppose that an asymptotically AdS space-time (Y n+1 , g) satisfies the causality axiom and is space-like geodesically complete. And suppose that (Y n+1 , g) is a strictly stationary solution to the vacuum Einstein equations with negative cosmological constant. Then (Y n+1 , g) is the standard AdS space-time for 3 ≤ n ≤ 7.
2 Asymptotically AdS space-times We assume through out this paper that a space-time is always orientable and connected as a manifold. In this section we will start with an intrinsic definition of asymptotically AdS space-times and derive some properties of a strictly stationary asymptotically AdS space-time. We will then prove a lemma of Lichnerowicz type similar to the one in [BGH]. We note that, in fact, it was asked whether the uniqueness theorem in their paper [BGH] still holds if one uses the definition of asymptotically AdS space-times proposed by Ashtekar and Magnon in [AM] (see also [Ha]). Let us first introduce the AdS space-time in general dimensions. The anti-de Sitter space-time in (n + 1) dimensions can be given by (Rn+1 , gAdS ) where gAdS = −(1 + r2 )dt2 +
1 dr2 + r2 dσ0 1 + r2
(2.1)
dσ0 is the unit round metric on S n−1 , t ∈ (−∞, +∞), and r ∈ [0, +∞). In the following we will adopt the definition of an asymptotically AdS space-time given by Ashtekar and Magnon in [AM]. To do so, let us first discuss what a conformal completion for a space-time is following the idea of Penrose in [Pe]. Suppose that Y n+1 is a manifold with boundary ∂Y n+1 = X n . Then Ω is said to be a defining function of X n in Y n+1 if a) Ω > 0, in Y n+1 ; b) Ω = 0, on X n ; and c) dΩ = 0 on X n . A space-time (Y n+1 , g) has a C k conformal completion if Y n+1 is a manifold with boundary X n and the metric Ω2 g for a defining function Ω of X n in Y n+1 extends in C k to the closure of Y n+1 . Definition 2.1 A space-time (Y n+1 , g) of dimension (n + 1) is said to be asymptotically AdS if 1) (Y n+1 , g) has a C k conformal completion and its boundary ∂Y n+1 = X n is topologically R × S n−1 ; 2) the space-time (Y n+1 , g) satisfies the Einstein equation with a negative cosmological constant Λ 1 Rab − Rgab + Λgab = 8πGTab 2 where Ω−n Tab admits a C k extension to the closure of Y n+1 ; 3) (X n , Ω2 g|T X n ) is conformal to (R × S n−1 , g0 ) where g0 = −dt2 + dσ0 .
(2.2)
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For the convenience, from now on, we will always assume Λ=−
n(n − 1) . 2
(2.3)
First, by definition, defining functions for X n in Y n+1 are not unique and two different defining functions differ by a positive function on the closure of Y n+1 . Therefore only the class of quadratic forms Ω2 g|T X n up to a conformal factor is determined by g. Second, by requiring the fall-off of the energy-stress tensor, one can compute that the sectional curvature of g would asymptotically go to −|dΩ|2Ω2 g , and conclude |dΩ|2Ω2 g |X n = 1 > 0. Therefore X n is a time-like hypersurface in (Y n+1 , Ω2 g). Finally the conformal flatness of the Lorentz metric Ω2 g|T X n should depend only on the Lorentz metric g. Hawking in [Ha] suggested that local conformal flatness is a necessary boundary condition, see more detailed discussions on this in dimension 4 by Chru´sciel and Simon in [CS]. But we choose to impose the global conformal flatness here to accommodate the existence of a time-like Killing vector field. We next want to choose a coordinate system near the boundary for an asymptotically AdS space-time. What we will do is mostly an analogue to the Euklidean cases which have been established in [FG], [G]. First, we construct a special defining function, at least in a tubular neighborhood of the boundary for each given metric in the class [−dt2 + dσ0 ] on the boundary by solving a first-order PDE. Namely, Lemma 2.1 Suppose (Y n+1 , g) is an asymptotically AdS space-time, and Ω is a defining function. Then, for each metric gˆ = e2φ g0 where g0 = −dt2 + dσ0 , there is a unique defining function s in a tubular neighborhood of the boundary X n in Y n+1 such that a) s2 g|T X n = gˆ; b) |ds|s2 g = 1 in the tubular neighborhood. Proof. Set s = ew Ω. Then ds = ew (dΩ + Ωdw) and
|ds|2s2 g = |ds|2e2w Ω2 g = e−2w |ds|2Ω2 g = |dΩ + Ωdw|2Ω2 g = |dΩ|2Ω2 g + 2Ω(dΩ, dw)Ω2 g + Ω2 |dw|2Ω2 g .
Thus, the requirement |ds|s2 g = 1 is equivalent to solving 2(dΩ, dw)Ω2 g + Ω|dw|2Ω2 g =
1 − |dΩ|2Ω2 g Ω
.
(2.4)
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The boundary condition is determined as follows: if we denote Ω2 g|T X n = e2ψ g0 , then w|X n = φ − ψ. (2.5) It is easily seen that (2.4) and (2.5) is non-characteristic. Notice that both dΩ and dw are space-like. Lemma 2.2 Suppose (Y n+1 , g) is an asymptotically AdS space-time. Suppose that s is the special defining function obtained in Lemma 2.1 for which s2 g|T X n = g0 . Then g = s−2 (ds2 + gs ) (2.6) where gs = −(1 +
s2 2 2 s2 ) dt + (1 − )2 dσ0 + O(sn ). 4 4
(2.7)
Proof. The proof again is adopted from the argument given in [FG], [G]. By the fall-off condition of the energy-momentum tensor Tab one can rewrite the equation (2.2) in coordinates R × S n−1 × [0, ) near the boundary as 1 hab +(1−n)hab −hcdhcd hcd −shcd hac hbd + shcd hcd hab −2sRab [h] = O(sn ), (2.8) 2 where h stands for gs for convenience. The signature of gs here does not make any difference in terms of solving the expansion of gs . Therefore, similar to what is known for Euklidean case, all odd-order terms of order ≤ n−1 vanish and all evenorder terms of order ≤ n − 1 is determined by the metric g0 on X n . Moreover, when n is odd, the nth order term is traceless; when n is even, in general one would need to add one more term in the order of sn log s which is traceless and determined by g0 while the trace part of the nth order is also determined by g0 . By comparing to the AdS space gAdS = s−2 (ds2 − (1 +
s2 2 2 s2 ) dt + (1 − )2 dσ0 ) 4 4
which is of the same boundary metric −dt2 + dσ0 , we may complete the proof. Remark 2.3 In the above argument, it is clear that a weaker fall-off condition of the energy-momentum would imply a weaker control of the asymptotic of the metric g. Next we will follow [BGH] to restrict ourselves to the so-called strictly stationary space-time. That is to assume, for an asymptotically AdS space-time, there ∂ is a global everywhere time-like Killing field which approaches ∂t asymptotically towards the boundary. In [BGH] it was demonstrated that a strictly stationary asymptotically AdS space-time (by their definition in dimension 3) which solves the vacuum Einstein equations with negative cosmological constant must be a
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static one. Before proceeding to prove the staticity we want to study the asymp∂ totic behavior of a Killing field that approaches ∂t at the infinity. We will use the favorable coordinates constructed in Lemma 2.2. Denote the Killing field by X = a(s, t, σ)
∂ ∂ ∂ , + b(s, t, σ) + ci (s, t, σ) ∂s ∂t ∂θi
(2.9)
where (s, t, θ1 , . . . , θn−1 ) ∈ [0, ) × R × S n−1 . For similar computations, please see [Wa2]. First of all, by the boundary condition, we know that b(0, t, θ) = 1,
ci (0, t, θ) = 0, ∀i = 1, . . . , n − 1.
(2.10)
∂ ∂ Computing Xg( ∂s , ∂s ) one gets
a ∂a = . ∂s s
(2.11)
Therefore a(s, t, θ) = sa(t, θ). For convenience we denote by t = θ0 and b = c0 , ∂ , ∂θ∂α ) one gets and use Greek letters to include zero. Computing Xg( ∂s gs (
∂a ∂ ∂ ∂cβ +s , ) = 0. ∂θα ∂θβ ∂s ∂θα
s ∂a b(s, t, θ) = 1 − ugu0β du ∂θβ 0 s ∂a cα (s, t, θ) = − uguαβ du . ∂θβ 0
Therefore
(2.12)
In the other directions one gets gs (
∂ ∂ γ ∂ ∂ γ , )c + gs ( , )c = 0. ∂θα ∂θγ ,β ∂θβ ∂θγ ,α
(2.13)
Notice that, if we denote the Christoffel symbols of metric gs on the slices by Γα βγ ¯ a for the ones of metric g, then and Γ bc
¯α Γα βγ = Γβγ ,
¯ α = 1 hαγ ∂ hγβ − 1 δαβ , Γ sβ 2 ∂s s
(2.14)
where again, for convenience, we use h = gs . Thus hαγ (
∂cγ ∂cγ ∂ − cδ Γγδβ ) + hβγ ( − cδ Γγδα ) = 2ahαβ − sa hαβ . ∂θβ ∂θα ∂s
(2.15)
Taking s = 0 in (2.15) we immediately see that a(t, θ) = 0 and surprisingly get ∂ X = ∂t in this neighborhood. Moreover (2.15) implies gs is independent of t in this neighborhood. Let us summarize what we obtained in a lemma.
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Lemma 2.4 Suppose that (Y n+1 , g) is an asymptotically AdS space-time and that s is the special defining function such that s2 g|T X = −dt2 + dσ0 . If X is a Killing ∂ ∂ field that approaches to ∂t at the infinity, then X = ∂t and gs is independent of t in a tubular neighborhood of the boundary. A strictly stationary asymptotically AdS space-time (Y n+1 , g) comes with a free R1 action. In fact the action is proper if the asymptotically AdS space-time satisfies the causality axiom (cf. [Ca]) since the behavior at the spatial infinity is described in the above Lemma 2.4. The causality axiom excludes any timelike closed curves in the space-time. Therefore, by Theorem 1.11.4 in [DK], we know that Y is a R1 -principle bundle over the smooth orbit space Y /R. Thus topologically, Y = R × M , since Y is orientable and connected, where M is a smooth n-manifold with boundary S n−1 in the light of Lemma 2.4. Now let us discuss the staticity of a space-time. A good reference for this discussion is [Ca]. A space-time is said to be static if there is an everywhere time-like Killing field whose trajectories are everywhere orthogonal to a family of space-like hypersurfaces. We emphasize that our definition of staticity adopted from [Ca] requires not only (2.17) below but also the existence of a global integral hypersurface. Let us introduce some notations. Let {ea } be an orthonormal frame and {wa } be its co-frame. Suppose X = k a ea is an everywhere time-like Killing field and let ω = ka wa . In this notation X is a Killing field if ka,b + kb,a = 0.
(2.16)
To apply Frobenius’ Theorem, we need to ask first that the differential ideal generated by the differential ω be closed under exterior differentiation, i.e., θ = ω ∧ dω = 0.
(2.17)
After (2.17) is established, we would like to say such connected space-time becomes R × Σ where Σ is a static slice (topologically the same as M ). This is easily seen because the following. First, take any global space-like hypersurface, say M of Y , such M exists since Y topologically is R×M where R is the action generated by the given time-like Killing field. Then, any piece of embedded space-like hypersurface is a graph over M and the function of the graph is “reference time” relative to M . For a maximal integral hypersurface Σ which is everywhere orthogonal to the Killing field through a given point in the space-time, we claim that, the “reference time” never reaches infinite in any bounded region. Simply because Σ may be considered at the same time in a “real time”. Hence, relative to Σ, we may read the “real time” at each point on M and it never gets to infinite in any bounded region. Therefore the maximal integral hypersurface Σ has to reach to the space infinity where we have better idea what happens. Thus the maximal integral hypersurface is global static slice. The metric may written as g = −V dt2 + gΣ where V = −k a ka and gΣ is the metric of Euklidean signature on the slice Σ . Now we are ready to prove the following lemma of Lichnerowicz type (cf. [BGH] [Ca]). Our proof is adapted for
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general dimensions and uses no additional topological assumptions. Notice that the definition of an asymptotically AdS space-time in [BGH] is different from ours in this note. Lemma 2.5 Any strictly stationary asymptotically AdS space-time (Y n+1 , g) which satisfies the causality axiom and the vacuum Einstein equations with negative cosmological constant Λ is static. To prove this lemma we observe Lemma 2.6
iX θ ω = −d( ) V2 V
(2.18)
where V = −k a ka . Proof. We simply compute d(
dω 1 1 iX θ ω )= − 2 dV ∧ ω = − 2 (dω(−V ) + dV ∧ ω) = − 2 . V V V V V
Because iX ω = ω(X) = ka k a = −V and
iX dw = iX (ka,b wb ∧ wa ) = ka,b iX wb wa − ka,b wb iX wa = ka,b k b wa − ka,b k a wb = −2k b kb,a wa = dV.
Proof of Lemma 2.5. Let us consider the Hodge dual ∗θ of θ . Since d∗ θ = (k[a,b kc] )c wa ∧ wb =
2 c k Rc[a kb] wa ∧ wb = 0 3
(2.19)
due to the fact that Rab = nηab where ηab is the standard Minkowski metric (please see Chapter 6 in Part II of [Ca]) and iX (∗θ) = ∗(θ ∧ ω) = 0, it follows
(2.20)
iX θ ∧ ∗θ ω iX (θ ∧ ∗θ) ∧ ∗θ) = − =− . (2.21) V V2 V2 The next step is to integrate over a space-like hypersurface Σ whose boundary is a large (n − 1)-sphere S n−1 = {s = , t = c} in the preferable coordinates. We therefore have |θ|2 ω(N ) ω ∧ ∗θ (2.22) dσ = − 2 n−1 V V Σ S d(
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where N is the unit normal of Σ , and dσ is the volume element of Σ in the space-time. Notice that ω(N ) > 0 and θ is space-like since θ ∧ ω = 0. Now let us ∂ recall that the Killing field X is just ∂t in the preferable coordinates. Thus ω|Sn−1 = −V dt + g0k dθk = −V dt + sn−2 τ0k dθk = n−2 τ0k dθk where g = s−2 (ds2 + (1 +
s2 2 2 s2 ) dt + (1 − )2 dσ0 + sn τ ) 4 4
and V = s−2 (1 +
s2 2 ) − sn−2 τ00 . 4
(2.23)
(2.24)
(2.25)
Then dω = −dV ∧ dt + dg0k ∧ dθk , θ = −V dt ∧ dg0k ∧ dθk − sn−2 τ0k dθk ∧ dV ∧ dt + sn−2 τ0k dθk ∧ dg0l ∧ θl , (2.26) and
∂V ∂g0k + g0k ) ∗ (dt ∧ ds ∧ dθk ) ∂s ∂s = Csn−5 ∗ (dt ∧ ds ∧ dθk )
∗θ|Sn−1 = (−V
(2.27)
k ∧ · · · ∧ dθn−1 = Cdθ1 ∧ · · · dθ where C stands for some function on S n−1 . Therefore ω ∧ ∗θ|Sn−1 = O(n )dθ1 ∧ dθ2 ∧ · · · ∧ dθn−1 . (2.28) V This implies that θ = 0 on the hypersurface Σ. But Σ is arbitrary, so θ = 0 on Y n+1 , which finishes the proof in the light of the discussions before the statement of this lemma. Let us conclude this section by making it clear what a static asymptotically AdS space-time which satisfies the vacuum Einstein equations with negative cosmological constant Λ is. We first state an observation in the following lemma. Lemma 2.7 Under the assumption of Lemma 2.5, in the preferable coordinate sys∂ is orthogonal tem at the infinity, indeed, a slice of constant t is a static slice, i.e., ∂t to the slice of constant t. Proof. Consider the conformal completion (Y n+1 , g¯) where g¯ = ds2 +gs . By the construction of the preferable coordinate system, each curve γ(s) = (s, t, θ1 , . . . , θn−1 ) is a geodesic from the point (0, t, θ1 , . . . , θn−1 ) in the space-time (Y n+1 , g¯). On the other hand, a static slice Σ of (Y n+1 , g) is still a maximum integral hypersurface ∂ everywhere with respect to g¯. Because g¯ = ds2 + gs which is orthogonal to ∂t is independent of t, such Σ is totally geodesic in (Y n+1 , g¯). Therefore a geodesic emanating from a boundary point (0, t0 , θ1 , . . . , θn−1 ) with respect to the metric g¯ stays in a static slice. Thus a slice of constant t coincides with a static slice. So the proof is complete. We summarize our result in the following theorem:
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Theorem 2.8 Suppose that (Y n+1 , g) is a space-like geodesically complete spacetime. And suppose (Y n+1 , g) is a strictly stationary asymptotically AdS space-time that satisfies the causality axiom and the vacuum Einstein equations with negative . Then Y n+1 = R × Σn , cosmological constant Λ = − n(n−1) 2 g = −V dt2 + h and, on Σ,
√ √ ∆ V =n V √ √ Ric[h] + nh = ( V )−1 ∇2 V ,
(2.29)
(2.30)
where h is the metric of Euklidean signature induced from g on a static slice Σn . Moreover (Σn , h) is complete and conformally compact of the same regularity as of the conformal completion of (Y n+1 , g) with the conformal infinity (S n−1 , [dσ0 ]) where V −1 h|T S n−1 = dσ0 .
3 Static asymptotically AdS space-times In this section we study static asymptotically AdS space-times. We will prove the uniqueness of static asymptotically AdS space-times. In dimension 3 + 1, with a bit restrictive definition of asymptotically AdS space-times, the uniqueness was first proved in [BGH] (see also [CS]). Then, assuming spin structure for n > 3, the uniqueness of static solutions (M n , g, V ) to the vacuum Einstein equations with negative cosmological constant was established in [Wa1] (also see the definition of a static solution (M, g, V ) in [CS], [Wa1]). Our proof will not use the spin structure in dimensions higher than three, but instead will rely on a recent work of Miao [Mi] (see also [ST]) which in turn depends on the classic positive theorem of Schoen and Yau [SY] for asymptotically flat manifolds. By Theorem 2.8 in the previous section, a static asymptotically AdS spacetime which satisfies the vacuum Einstein equations with negative cosmological √ constant is given by a static solution (Σ, h, V ) in our notation. Therefore, by Lemma 2.2 in the previous section, we know that h = s−2 (ds2 + (1 − V = s−2 ((1 + and
s2 2 ) dσ0 + τ sn + o(sn )), 4
s2 2 ) − αsn + o(sn )), 4
√ V − |∇ V |2 − 1 = nαsn−2 + o(sn−2 )
where α = −Trdσ0 τ (these were known in [Wa1]).
(3.1)
(3.2)
(3.3)
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To motivate our argument in this section we recall the following fact about 2 2 2 2 1+|x| the static solution (B n , ( 1−|x| 2 ) |dx| , 1−|x|2 ) associated with the AdS space-time (Rn+1 , gAdS ). Namely, if one uses the global defining function 1 − |x|2 1 , = u= √ 2 V +1 then
u2 h = |dx|2 . √ Therefore, for a static solution (Σ, h, V ), if we denote u = global defining function for S n−1 in Σ and u2 h = where
√ 1 , V +1
then u is a
1 s2 √ (ds2 + (1 − )2 dσ0 + τ sn + o(sn )) 4 s2 ( V + 1)2
√ √ √ s2 ( V + 1)2 = ( s2 V + s)2 = s2 V + 2s s2 V + s2 = 1 + 2s + O(s2 ).
So
u2 h = (1 + 2s + O(s2 ))ds2 + (1 + 2s + O(s2 ))dσ0 + O(s2 ).
(3.4)
2
Thus (Σ, u h) is a compact manifold with the standard (n−1)-sphere as its boundary and the second fundamental form for ∂Σ in Σ is dσ0 (i.e., the boundary is totally umbilical). In the light of (2.30) one may compute the scalar curvature for u2 h as follows: n−2 n−2 4(n − 1) ∆u 2 + R[h]u 2 ) n−2 √ = n(n − 1)(V − |∇ V |2 − 1),
R[u2 h] = u−
n+2 2
(−
(3.5)
which goes to zero as s → 0 by (3.3). We observe the following lemma which will allow us to apply the Strong maximum principle to conclude that R[u2 h] ≥ 0. Namely, Lemma 3.1
√ √ 2 √ √ √ ∇ V 2 2 −∆(V − |∇ V | − 1) = 2|∇ V − V h| − √ · ∇(V − |∇ V |2 − 1). (3.6) V
Proof. We simply compute √ −∆V = 2(−nV − |∇ V |2 ) and
√ √ √ √ ∆|∇ V |2 = 2(|∇2 V |2 + ( V )i ( V )ijj ) √ √ √ √ √ = 2(|∇2 V |2 + ( V )−1 ( V )i ( V )ij ( V )j )).
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Therefore √ −∆(V − |∇ V |2 − 1) √ √ √ √ √ √ ( V )i √ = 2|∇2 V − V h|2 − √ (2( V )( V )i − 2( V )ij ( V )j ) (3.7) V √ √ √ √ ∇ V = 2|∇2 V − V h|2 − √ · ∇(V − |∇ V |2 − 1). V √ Theorem 3.2 Suppose that (Σn , h, V ) is a geodesically complete and conformally compact static solution to √ the vacuum Einstein equations with negative cosmological constant, i.e., (Σn , h, V ) comes from Theorem 2.8 in the previous section. √ 2 2 2 2 1+|x| Then (Σn , h, V ) = (B n , ( 1−|x| 2 ) |dx| , 1−|x|2 ) for some choice of coordinate in dimension 3 ≤ n ≤ 7. Proof. We consider the defining function u = √V1+1 and the compact manifold (Σ, u2 h). From (3.4) we know that ∂Σ = S n−1 and u2 h|∂Σ = dσ0 . Moreover, from (3.4), we know that the standard round S n−1 is the boundary of (Σ, u2 h) and has the second fundamental form dσ0 (i.e., it is totally umbilical). On the other hand, by (3.5) and (3.3), the scalar curvature R[u2 h] goes to zero as s → 0 when n > 2. Using the above Lemma 3.1 and the strong maximum principle, we therefore conclude that R[u2 h] ≥ 0. Now we appeal to the recent work of Miao [Mi] (see also works of Shi and Tam [ST]). We apply the work in [Mi] to the manifold (M, G) where Ω = Σ and g− = u2 h, and M \ Ω = Rn \ B n and g+ is the Euklidean metric. By Corollary 5.1 in [Mi], for example, √ we then conclude that R[u2 h] ≡ 0 for 3 ≤ n ≤ 7. √ n Back to (Σ , h, V ), in the light of (3.6) in Lemma 3.1 and V −|∇ V |2 −1 = 0, we observe that √ √ ∇2 V = V h. (3.8) Similar to what was proved in [Ob], we prove that (3.8)√implies that (Σn , h) is isometric to the standard hyperbolic space form and V = 1 + r2 for some choice of coordinates in the following lemma. Then the proof of theorem is complete. Lemma 3.3 Suppose that (M n , g) is a complete Riemannian manifold. And suppose that there is a positive function φ such that ∇2 φ = φg. (3.9) √ Then (M n , g) = (Rn , gH ) and φ = c 1 + r2 for some choice of coordinates. Proof. First one observes that, φ has one and only one global minimum point p0 on M , since it is strictly convex. Due to the homogeneity of (3.9), one may assume that φ(p0 ) = 1. Let us consider a geodesic γ(s) emanating from p0 and
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parameterized with its length s. Then, along this geodesic, for φ(γ(s)), we have φ − φ = 0 φ(0) = 1 (3.10) φ (0) = 0. ∂ Thus φ(s) = cosh s. Now take an othonormal base X 0 = ∂s , X 1 , X 2 , . . . , X n−1 at p0 and parallel translate them along γ(s). We want to calculate d(expp0 )(sX 0 ) (X k ). That is, we compute the Jacobi field Y k (s) along γ(s) such that k k 0 0 0 0 ∇X ∇X Y + R(Y , X )X = 0 (3.11) Y k (0) = 0 k k ∇X 0 Y (0) = X (0). k i Let Y (s) = fi (s)X (s). Then (3.11) becomes i j fi X + fi R0i0j X = 0 (3.12) fi = 0 fi = δik .
Notice that, by (3.9) and Ricci identity, φa,bc − φa,cb = φd Rdabc = sinh sR0abc = φc δab − φb δac ,
(3.13)
which gives us R0i0j = Plugging into (3.12), we have
1 (φj δi0 − φ0 δij ) = −δij . sinh s fi − fi = 0 fi (0) = 0 fi (0) = δik .
(3.14)
(3.15)
Thus Y k (s) = sinh sX k (s). To show that (M, g) is a hyperbolic space form, we use the exponential map expp0 which takes the tangent space Tp0 M onto M in the light of completeness, thus gives a nice global coordinate chart. Next we want to calculate the metric g under these coordinates. Let us use spherical coordinates for Tp0 M , that is, (s, v) ∈ [0, ∞)×S n−1 and expp0 (sv) ∈ M . By the above calculations of Jacobi fields, we immediately have g = ds2 + (sinh s)2 dσ0 ,
(3.16)
which is the hyperbolic metric. So (M, g) is a hyperbolic space √ form. Finally let us point out that, if we denote r = sinh s, then φ = cosh s = 1 + r2 . Acknowledgment. The author is deeply indebted to Professor P.T. Chru´sciel and the referee for their careful reading of the paper and many corrections of statements.
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References [ACD] M. Anderson, P.T. Chru´sciel and E. Delay, Non-trivial, static, geodesically complete, vacuum space-times with a negative cosmological constant, arXiv: hep-th/0211006. [AM]
A. Ashtekar and A. Magnon, Asymptotically anti-de Sitter space-times, Class. Quantum Grav. 1, L39–L40 (1984).
[BS]
R. Beig and W. Simon, On the spherical symmetry of a static perfect fluid in general relativity, Lett. Math. Phys. 21 no. 3, 245–250 (1991).
[BGH] W. Boucher, G.W. Gibbons and G.T. Horowitz, Uniqueness theorem for anti-de Sitter spacetime, Phys. Review D. (3) 30, no. 12, 2447–2451 (1984). [Ca]
B. Carter, Black hole equilibrium states, Part II, in “Black Holes”, edited by C. DeWitt and B. DeWitt (New York, 1973).
[CH]
P.T. Chru´sciel and M. Herzlich, The mass of asymptotically hyperbolic Riemannian manifolds, Preprint math.DG/0110035.
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P.T. Chru´sciel and W. Simon, Towards the classification of static vacuum spacetimes with negative cosmological constant, J. Math. Phys. 42, no. 4, 1779–1817 (2001).
[DK]
J.Duistermaat and J. Kolk, “Lie groups”, Springer-Verlag, Berlin, New York, 2000.
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C. Fefferman, and C.R. Graham, Conformal invariants, in The mathematical heritage of Elie Cartan, Ast´erisque, 1985, 95–116.
[GSW] G.J. Galloway, S. Surya and E. Woolgar, On the geometry and mass of static, asymptotically AdS space-times,and uniqueness of the AdS soliton, arXiv: hep-th/0204081. [G]
C.R. Graham, Volume and Area renormalizations for conformally compact Einstein metrics. The Proceedings of the 19th Winter School “Geometry and Physics” (Srn`i, 1999). Rend. Circ. Mat. Palermo (2) Suppl. No. 63 (2000), 31–42.
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S. Hawking, The boundary conditions for gauged supergravity, Phys. Lett. B126, no. 3-4, 175–177 (1983).
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L. Lindblom, Static uniform-density stars must be spherical in general relativity, J. Math. Phys. 29, no. 2, 436–439 (1988).
[Mi]
P. Miao, Positive mass theorem on manifolds admitting corners along a hypersurface, ArXiv: math-ph/0212025.
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[Ob]
M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan. 14, no. 3, 333–340 (1962).
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R. Penrose, Asymptotic properties of fields and space-times, Phys. Rev. Lett. 10, 66–68 (1963).
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J. Qing, On the rigidity for conformally compact Einstein manifolds, International Mathematics Research Notices 21, 1141–1153 (2003), ArXiv: math.DG/0305084.
[SY]
R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. Topics in calculus of variations (Montecatini Terme, 1987), 120–154, Lecture Notes in Math., 1365, Springer, Berlin, 1989.
[ST]
Y. Shi and L. Tam, Positive mass theorem and the boundary behaviors of a compact manifolds with nonnegative scalar curvature, arXiv: math.DG/0301047.
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X. Wang, Uniqueness of AdS space-time in any dimension, arXiv: math.DG/0210165.
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X. Wang, On conformally compact Einstein manifolds, Math. Res. Lett. 8, no. 5–6, 671–688 (2001).
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X. Wang, The mass of Asymptotically hyperbolic manifolds, J. Diff. Geo. 57, no. 2, 273–299 (2001).
Jie Qing Department of Mathematics University of California, Santa Cruz Santa Cruz, CA 95064 USA email:
[email protected] Communicated by Piotr T. Chrusciel Submitted 17/10/03, accepted 07/11/03
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 261 – 287 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/020261-27 DOI 10.1007/s00023-004-0169-5
Annales Henri Poincar´ e
D´eformation Elliptique de la M´etrique de Randall et Sundrum Michel Gaudin
0 Introduction Dans sa version originelle, le mod`ele de Randall et Sundrum1) pour la hi´erarchie de masse consid`ere un univers global V5 , de dimension cinq, a` la topologie d’un cylindre aplati formant comme un ruban avec ses deux faces identifi´ees. La base en est le cercle [−π ϕ +π] modulo la sym´etrie de parit´e. Les deux bords du ruban sont les images de deux parois de dimension quatre, fronti`eres de V5 , l’une V (0) dite “cach´ee”, l’autre V (π) dite “visible” comme support des champs de particules. Les ´equations d’Einstein postul´ees dans V5 , dont les sources ont pour support les parois, font donc intervenir des singularit´es de type δ (ϕ) , δ (ϕ − π) (modulo 2π) dans les composantes du tenseur d’impulsion, ce qui ne pose pas trop de difficult´e ´etant donn´ee la lin´earit´e du premier membre dans les d´eriv´ees secondes de la m´etrique. Cependant il vaut mieux s’assurer dans ces probl`emes non-lin´eaires des relations de conservation sur la zone singuli`ere par une r´egularisation, ce qui revient `a introduire une ´epaisseur de la paroi. Mais alors, le bord de V5 n’est plus mat´erialis´e par un pic de la pression exactement localis´e, et ne subsiste que la notion g´eom´etrique due `a la sym´etrie impos´ee et `a la topologie. Sur le cercle de base la paroi diffuse est seulement une zˆone de variation rapide du tenseur des contraintes au voisinage de 0 et π. Celui-ci est d´ecrit soit par les scalaires de pression et densit´e dans l’hypoth`ese de fluide parfait dans V5 (isotropie locale dans V4 ), soit par un champ scalaire contrˆol´e par un puits de potentiel donn´e pour cr´eer les localisations ad hoc. Le probl`eme de la paroi diffuse V (0) a ´et´e trait´e par Ichinose2) en postulant le potentiel standard biquadratique V (φ) poss´edant un maximum central en φ = 0, et deux minima sym´etriques φ = ±φ0 . Une m´ethode perturbative en fonction d’un petit param`etre d’´epaisseur permet de d´eterminer num´eriquement la m´etrique et de reproduire les traits du mod`ele R.S. a` une paroi. Le r´esultat ne d´epend que de la forme qualitative et des deux param`etres du potentiel entre les deux minima. L’analyse perturbative devient inutile si l’on a la chance d’obtenir une solution exacte pour un potentiel convenable. Or ceci est possible dans le mod`ele d’Ichinose avec le potentiel sinuso¨ıdal le plus simple V (φ) = V0 cos
πφ φ0
V0 > 0
(1)
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o` u l’intervalle −φ0 φ φ0 est suffisant pour couvrir V5 . Il s’agit donc l` a d’une solution exacte, a` une variable, du champ dit de sine-Gordon coupl´e `a la gravit´e. La solution de ce mod`ele dit “ap´eriodique” sugg`ere une extension naturelle au mod`ele p´eriodique `a deux parois d´ependant en outre d’un param`etre continu. C’est au prix de l’introduction d’un second champ dans un second potentiel, mais coupl´e `a la gravit´e de V5 avec le signe oppos´e (M → −M ), ce qui g´en´eralise naturellement le choix des constantes cosmologiques oppos´ees de R.S. Cette hypoth`ese un peu hardie ´etant admise, les r´esultats de R.S. en d´ecoulent, n’´etant qu’un cas limite du mod`ele d´eform´e propos´e. La notion stricte de paroi s’´evanouit par l` a-mˆeme, n’´etant qu’une forme limite de localisation approch´ee dans V5 .
1 Rappels et notations La m´etrique de R.S. est formellement analogue a` une m´etrique cosmologique R.W. o` u le rˆ ole du temps cosmique est tenu par la cinqui`eme variable d’espace: V5 :
ds2
avec x0 ≡
= ≡
dy 2 + e−2σ(y) d s˜2 gµν dxµ dxν
y,
µ = 0, 1, . . . , 4
V4 : d˜ s2 g00
= g˜µν dxµ dxν
(3)
g0µ = 0 (µ = 0)
= 1
= e−2σ g˜µν ,
(µ, ν = 0) √ Le facteur de r´eduction est b = e et le d´eterminant g = b4 . Si V4 est une vari´et´e de pure gravit´e quelconque gµν
(2)
∂0 g˜µν=0
(4)
−σ
V4 :
˜ µν = 0 R
les ´equations de V5 en pr´esence d’une source s’´ecrivent 2 6σ − 3σ gµν = κ Tµν 6σ 2 = κ T00
(5)
(6)
avec la notation du couplage κ−1 = M 3 . La source de V5 peut ˆetre le champ scalaire φ (y) 1 Tµν = ∂µ φ∂ν φ − gµν (∂φ.∂φ + 2V (φ)) 2
(7)
dont la conservation est assur´ee par l’´equation K.G. ∇2 φ −
∂V = 0. ∂φ
(8)
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On peut la consid´erer aussi comme un pseudo-fluide isotrope, de pression p, densit´e ρ Tµν = pgµν + (ρ − p) uµ uν (9) o` u uµ est un courant unitaire de V5 du genre espace, u.u = 1. Si n−1 est le volume sp´ecifique, le courant nuµ est conserv´e, soit √ (10) n g = nb4 = Cste . La conservation ∇µ Tµν = 0 ´equivaut alors a` d ρb4 dρ p=ρ−n = . dn d (b4 )
(11)
La structure (6) impose ´evidement que φ, p, ρ ne d´ependent que de y, avec u0 = 1, et l’on a la correspondance. −p =
1 2 φ + V, 2
2 φ = ρ − p,
1 2 φ − V. 2 −2V = ρ + p ρ=
(12)
et l’´equation K.G. ´equivalente a` (11) φ − 4σ φ −
∂V =0. ∂φ
A l’aide de (6) et (12), les ´equations du probl`eme `a une paroi pos´e par Ichinose s’´ecrivent 2 3σ 2= κφ . (13) 3 σ − 4σ = 2κV (φ) On peut g´en´eraliser au cas d’un espace V4 qui soit un espace d’Einstein ˜ g´en´eral de constante cosmologique Λ ˜ µν = Λ ˜ g˜µν . R
(14)
Les ´equations de Friedman-Lemaˆıtre3) pour V5 s’´ecrivent 2 ˜ Λ 1 ˜ 2σ 3 b κρ = 3σ 2 − Λe − 2 = b b 2
b 6 b
=
κ (2p − ρ)
Pour un VD+1 on aurait
˜ D (D − 1) b2 − Λ b D (D − 1) b
.
(15)
= 6 σ 2 − σ
= 2 κρb2 = κ (Dp − (D − 2) ρ)
(16)
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dont une solution particuli`ere bien connue montre que les espaces d’Einstein sont des “sph`eres g´en´eralis´ees” b = sh y ⇐⇒ ds2 = dy 2 + sh2 y d s˜2 D−1 ˜ Λ = (D − 1) ⇒ Λ = −D κρ = κp = − Λ . 2
(17)
2 Le mod`ele ap´eriodique La solution a` deux parois donn´ee par R.S. pour la fonction continue σ (y), paire et p´eriodique de p´eriode 2yc = 2πrc σ (y) = k |y|
,
−yc y yc
(18)
correspond a` une densit´e ρ (y) constante et positive κρ = 6k 2
(19)
tandis que la pression pr´esente deux pics delta d’intensit´es oppos´ees sur les parois κp (y) = κρ + 6k (−δ (y) + δ (y − yc ))
(20)
(y mod 2yc ) . 3 u la constante cosmologique Λ= Entre les parois, on a κp = κρ = − Λ o` 2 2 −4k < 0 caract´erise V5 (un adS si V4 est un M4 , mais les ´equations restent valides si V4 est un trou noir). La loi de conservation (11), ´ecrite sous la forme 4σ (p − ρ) = −ρ ≡ 0
(21)
implique que la fonction σ soit consid´er´ee comme la limite d’une fonction continue impaire anti-p´eriodique de sorte que l’on ait ,,
lim σ (y)δ(y) = k “ (y)δ(y) = 0
(22)
(idem en yc ). Ceci rappel´e, nous partons maintenant du mod`ele d’Ichinose avec champ u la paroi “physique” scalaire et potentiel, dans la limite ap´eriodique (rc → ∞) o` est envoy´ee `a l’infini de la coordonn´ee y, ce qui ne veut pas dire qu’elle n’existe plus mais qu’elle se situe `a l’horizon. Introduisons une unit´e de longueur y0 (de l’ordre de la longueur de Planck −1 MPl dans la th´eorie de Kaluza4 )) et la coordonn´ee sans dimension u = y/y0 , ainsi que le param`etre sans dimension = ky0 , de sorte que l’on a u ≡ ky .
(23)
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En terme de la coordonn´ee u, les ´eqs. (13) restent formellement inchang´ees dσ en rempla¸cant V par V y02 avec d´esormais σ = . du Inversant la d´emarche d’Ichinose qui cherche `a r´esoudre pour le potentiel standard donn´e, nous proposons la fonction d’essai suivante, qui constitue une r´egularisation tr`es simple, approchant R.S. `a la limite → 0, u → ∞, u fini σ (u) = log ch u
ou b (u) =
⇒ σ = th u → σ =
1
(ch u)
= e−σ
(24)
κ = φ2 2 3 ch u
on introduit le champ sans dimension ϕ
π ϕ κ 1 φ, d’o` u ϕ = , qui s’int`egre en eu = tg + , soit th u = ϕ≡ 3l ch u 4 2 sin ϕ et 1 π π = cos ϕ = ϕ , − < ϕ < + . (25) ch u 2 2 Le champ ϕ ob´eit ` a l’´equation du pendule ap´eriodique et repr´esente le demi angle de rotation a` partir de l’´equilibre stable ϕ = 0, la variable u jouant le rˆ ole du temps m´ecanique. On d´eduit de (13) le potentiel V (φ) v (ϕ)
2κ 2 y V (φ) = σ − 4σ 2 3 0 = cos2 ϕ − 4 2 sin2 ϕ = − (1 + 4 ) sin2 ϕ
≡
ou encore
(26)
4κy02 V = (1 − 4 ) + (1 + 4 ) cos 2ϕ . (27) 3 Nous avons donc le r´esultat annonc´e en (1) pour le potentiel avec 2 φ0 π 3 φ0 = et V = (1 + 4 ) . 0 2 κ πy0 On d´eduit des relations (12) les expressions de la densit´e et de la pression 2κy02 ρ = ϕ2 − v (ϕ) = 4 2 th2 u = 4 2 sin2 ϕ 0 32 . 2κy0 p = − ϕ2 − v (ϕ) = 4 2 2 (2 + 1) = − + (2 + 1) sin2 ϕ 3 ch2 u (28) On voit comment le mod`ele R.S. a` une paroi est atteint comme limite du pr´ec´edent lorsque et y0 tendent vers z´ero de sorte que lim /y0 = k, puisque l’on a 1 1 lim = δ (y) 2 2y0 ch (y/y0 ) 2κp 2κρ = 4k 2 ; lim = 4k 2 − 4kδ (y) . (29) lim 3 3 2v ≡
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Il est int´eressant d’exprimer la m´etrique dans la coordonn´ee angulaire ϕ qui π π ram`ene la base sur l’intervalle d’une p´eriode − < ϕ < + 2 2 ds2
=
y02 du2 +
=
y02
d˜ s2 2
(ch u)
dϕ2 + (cos ϕ)2 d˜ s2 . cos2 ϕ
(30)
π ne sont atteintes par les g´eod´esiques 2 de V5 qu’` a la limite d’un temps propre (de V4 ) infini: si dt2 ≡ −d˜ s2 Les extr´emit´es de l’intervalle ϕ = ±
| u| = |ky| ∝ log | t| .
(31)
Il s’agit donc d’un horizon, infranchissable dans la coordonn´ee ϕ. La m´etrique y est toujours singuli`ere puisque le d´eterminant y est soit nul 1 1 > soit infini < . Cependant si 2 est entier un prolongement π4 4 p´eriodique est possible, qui ne respecte la signature que si est entier. Une seconde forme limite de la m´etrique est a` remarquer pour → +∞ lim
lim → ∞ 2Kr02 3
√ lim ϕ = ϕ0
y02 = r02 ds2 lim V
= =
r02 dϕ20 1
+ −
2
e−ϕ0 4ϕ20
d˜ s2
.
(32)
Ne subsiste du potentiel que le voisinage du maximum central (potentiel parabolique invers´e) tandis que les deux puits sont envoy´es `a l’infini. Ceci montre que sur tout l’intervalle de variation du param`etre = ky0 (0 < < ∞) la d´eformation continue du mod`ele n’a pas d’autre effet que de modifier les largeur et profondeur relatives du pic de pression et du facteur de r´eduction. Ce mod`ele `a une paroi ne permet pas de d´efinir le facteur de r´eduction des masses, sans l’introduction d’une coupure ad hoc, fixant la distance de la paroi visible au voisinage de l’horizon. On aura not´e que c’est le terme “cin´etique” φ2 du champ scalaire qui simule le pic delta de la pression en y = 0. Il est donc impossible de simuler le second pic de signe oppos´e selon R.S. en introduisant un second champ scalaire, sans changer le signe du couplage gravitationnel. Ce fait semble moins choquant lorsqu’il concerne seulement les constantes cosmologiques de signe oppos´e des deux parois V (0) et V (π), mais il est incontournable. Cette hypoth`ese essentielle admise, nous montrons dans la section suivante comment l’introduction de deux champs et de deux potentiels analogues a` (27) permet de construire un mod`ele p´eriodique r´egularis´e qui tend vers celui de R.S. dans la limite consid´er´ee plus haut.
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3 Le mod`ele des deux pendules Le mod`ele exact `a une paroi trait´e dans la section pr´ec´edente est celui du champ scalaire “sine-Gordon” coupl´e `a la gravit´e dans V5 . La variation du champ avec la cinqui`eme coordonn´ee spatiale est r´egie par l’´equation du pendule ap´eriodique dans la variable de temps de l’analogie m´ecanique. Cette ´equation (25), ϕ = cos ϕ, est un cas fronti`ere de celle du pendule p´eriodique ϕ2 + k 2 sin2 ϕ = 1
(33)
d´ependant du module k dans la notation traditionnelle que nous garderons. Pour ´eviter toute confusion avec le param`etre de masse de R.S., on notera d´esormais ce dernier kRS (par ex. l = y0 kRS selon (23)) . • Si 0 < k 2 < 1, le pendule fait un tour complet autour de son axe (2ϕ → 2ϕ + 2π) dans une p´eriode not´ee 2K, o` u K (k) est l’int´egrale elliptique compl`ete (notation de Whittaker et Watson, chap. XXII)
π 2
K (k) = 0
−1/2 dϕ 1 − k 2 sin2 ϕ
(34)
avec le comportement au voisinage de k = 1 K (k) ∝ log
4 , k
k 2 + k 2 = 1 .
(35)
1 • Si k > 1, le pendule oscille avec une amplitude angulaire |ϕ| sin . k π pour les petits mouLa demi-p´eriode est ReK (k) = k −1 K k −1 , a` la limite 2k vements k 1. La solution de (33) est la fonction “amplitude elliptique” de module k (W.W., p. 494) −1
2
ϕ = ϕ+π
=
am (u, k) ⇒ ϕ = dn (u, k)
(36)
am (u + 2K) .
On restaure ainsi dans la m´ethode du champ scalaire d’Ichinose une p´eriode 2K en u, ou 2Ky0 = 2yc en y, qui correspond certes `a la p´eriode π du potentiel, mais a` condition que le prolongement p´eriodique soit possible, c’est-`a-dire pour k 2 < 1. La d´eformation elliptique permet de retomber sur le mod`ele R.S. p´eriodique r´egularis´e. On postule un lagrangien source de V5 d´ependant d’un couple de champs scalaires {φ1 , φ2 } avec la structure suivante 1 −1 5 √ ∂φ1 .∂φ1 + V1 (φ1 ) κ L12 = d x g 2 1 − ∂φ2 .∂φ2 − V2 (φ2 ) + V (φ1 , φ2 ) (37) 2
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qui donne le tenseur d’impulsion Tµν
=
∂µ φ1 ∂ν φ1 − ∂µ φ2 ∂ν φ2 1 − gµν (∂φ1 ∂φ1 − ∂φ2 .∂φ2 + 2 (V1 − V2 + V )) 2
(38)
dont la conservation est assur´ee par les deux relations
2 φ1 −
∂ ∂ (V1 + V ) = 0, 2 φ2 − (V2 − V ) = 0. ∂φ1 ∂φ2
(39)
et qui dans l’hypoth`ese de la seule d´ependance en u s’´ecrivent φ1 − 4σ φ1 − ∂ (V1 + V ) = 0, etc. . . ∂φ1 Les ´equations du probl`eme g´en´eralisant (13) sont 2 κ 2 = σ 3 φ1 − φ2 (40) 2 σ − 4σ 2 = 2κ 3 y0 (V1 (φ1 ) − V2 (φ2 ) + V (φ1 φ2 )) o` u les potentiels vont ˆetre d´etermin´es par la m´ethode inductive pr´ec´edente. La fonction d’essai σ (u) ou b (u) qui ´etend naturellement (24) au cas p´eriodique est l
b (u) = (dn (u, k)) ou σ = −l log dn u
(41)
o` u la fonction dn u = dn (−u) = dn (u + 2K) est positive, oscillant entre le maximum dn (0) = 1, et le minimum dnK = k , avec la propri´et´e dn u dn (u + K) = k qui entraˆıne donc b (u + K) =
(42)
k 2l . b (u) ¯
A la limite ap´eriodique k → 1, k → 0, K → ∞ avec k ∝ 4e−K = e−K , ¯ = K − log 4 et dn (u, 1) = 1 etc. . . K ch u On d´eduit de (41) σ
=
σ
=
dn sn cn = k 2 dn dn sn2 cn2 k 2 2 k 2 − cn2 − sn2 + k 2 ≡ dn dn2 dn2 −
On peut encore ´ecrire = dn2 u − dn2 (u + K) σ σ 2 = 2 1 + k 2 − dn2 u − dn2 (u + K)
.
.
(43)
(44)
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D’o` u les identifications fond´ees sur (40) et (44), dans les notations sans diκ 2κy02 V, φ et v = mension d´ej`a utilis´ees en (25) et (26), ϕ = 3l 3 ϕ1 = dn u ou ϕ1 = am u (45) ϕ2 = dn (u + K) ou ϕ2 = am (u + K) v1 − v2 + v ≡ σ − 4σ 2
≡ −4 2 1 + k 2 + (4 + 1) dn2 u + (4 − 1) dn2 (u + K) . On d´eduit de (45) sin ϕ1 = tg
ϕ1 =
sn u
tg ϕ1 =
sc u
tg ϕ2 =
sc u 1 − cs u k
.
(46)
(47)
D’o` u la relation n´ecessaire entre ϕ1 (u) et ϕ2 (u) k tg ϕ1 tg ϕ2 = −1
(48)
qui entraˆıne encore selon (42) 1 − k 2 sin2 ϕ1 1 − k 2 sin2 ϕ2 = k 2 ou ϕ1 ϕ2 = k . Enfin pour les potentiels 2v1 (ϕ1 ) ≡
(49)
k 2 (1 + 4 ) cos 2ϕ1 = (1 + 4 ) 2dn2 u − 1 − k 2
2v2 (ϕ2 ) ≡
k 2 (1 − 4 ) cos 2ϕ2 = (1 − 4 ) 2dn2 (u + K) − 1 − k 2
(50)
a condition d’avoir la contrainte ` v (ϕ1 , ϕ2 ) = 0
(51)
au cours du mouvement, ce qui justement est r´ealis´e en (48). • Analogie des deux pendules Les angles ou amplitudes 2ϕ1 et 2ϕ2 , fonction du temps u, sont les angles avec la verticale du champ de pesanteur de deux pendules de mˆeme p´eriode 2K, ob´eissant a la mˆeme ´equation horaire (33) ou ` 2ϕ + k 2 sin 2ϕ = 0,
(52)
mais d´ecal´es d’une demi-p´eriode K, ce qui est une question de condition initiale. La relation (48) est alors conserv´ee au cours du temps.
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La liaison entre les deux pendules n’est pas dynamique, mais g´eom´etrique. On peut se repr´esenter deux masses ponctuelles identiques oscillant (ou en rotation) sur deux grands-cercles verticaux de la mˆeme sph`ere, les deux plans formant en di`edre d’angle ψ donn´e par k = cos ψ. La relation (48) peut s’´ecrire cos γ ≡ cos ϕ1 cos ϕ2 + cos ψ sin ϕ1 sin ϕ2 = 0 .
(53)
Or ϕ1 , ϕ2 sont les angles avec la verticale des deux masses vues du z´enith de la sph`ere (2ϕ1 , 2ϕ2 angles vus du centre de rotation). π En vertu de (53), γ = ± , l’angle entre les deux masses vu du z´enith est 2 droit; les masses restent en positions conjugu´ees, ou si l’on veut, en quatrature vues du z´enith. Stabilisons cette liaison par un potentiel entre les deux pendules, minimal en quadrature 1 v (ϕ1 , ϕ2 ) = v0 cos2 γ 0 . (54) 2 Nous avons ∂v = v0 cos γ (− sin ϕ1 cos ϕ2 + k cos ϕ1 cos ϕ2 ) . ∂ϕ1 Utilisant (41) on trouve ∂v k = v0 cos γ = v0 cos γ ∂ϕ1 dn u
ϕ2
(55)
v´erifions les relations de conservation (39) ϕ1 − 4σ ϕ1 = soit 1 2
∂v1 ∂v + ∂ϕ1 ∂ϕ1
= = =
on en d´eduit
∂v =0 ∂ϕ1
1 ∂ (v1 + v) 2 ∂ϕ1
sn cn dn dn 1 − (4 + 1) k 2 sn cn = − k 2 (4 + 1) sin 2ϕ1 2 1 ∂v1 + 2 ρ∂ϕ1
(56)
∂v = 0 sur la trajectoire ∂ϕ2
(57)
−k 2 sn cn − 4 k 2
,
ce qui r´esulte en effet de cos γ = 0, ∀v0 > 0. Le puits stabilisateur v (ϕ1 , ϕ2 ) poss`ede une ´equipotentielle minimale, ligne de points paraboliques qui constitue une trajectoire pour la condition initiale opposition d’une demi-p´eriode, ou quadrature vue du z´enith.
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Prenant la limite v0 → +0, il est permis d’oublier ce potentiel mutuel et de consid`erer les champs scalaires φ1 et φ2 comme ind´ependants, avec des tenseurs s´epar´ement conserv´es, pourvu que les potentiels V1 et V2 soient donn´es par (50). On notera qu’il sont de mˆeme forme sinuso¨ıdale, mais d’intensit´e diff´erente sauf dans la r´egion 1, ou d’intensit´e oppos´ee pour 1.
4 La m´etrique d´eform´ee de V5 3/2 Deux champs scalaires (ϕ1 , ϕ2 ), de dimension M , r´egis par deux potentiels πφ semblables en cos , coupl´es `a la gravit´e d’un V5 avec deux signes oppos´es, φ0 donnent lieu a` la m´etrique suivante dans l’hypoth`ese o` u ils ne d´ependent que de la cinqui`eme coordonn´ee: 2
ds2 = y02 du2 + (dn u) d˜ s2 o` u le param`etre ne d´epend que de φ0, 2 κ 2φ0 κ = M −3 = 3 π
(58)
(59)
tandis que le module elliptique k d´epend de l’intensit´e des potentiels une fois fix´ee πφ1 l’unit´e de longueur y0 . Si l’on ´ecrit V1 (φ1 ) = W1 cos , etc. . . , on a selon (50) φ0 2 2 2 kφ0 k 3 κ 4φ0 W1 = (1 + 4 ) = 1+ 2y0 κ πy0 3 π . (60) 1 + 4 W1 /W2 = 1 − 4 Du module k, on calcule K demi-p´eriode de la coordonn´ee u et yc = Ky0 , demi-p´eriode en y = y0 u. Revenant `a la m´etrique (58) on note qu’une translation u → u + K ´equivaut a changer le signe de et l’´echelle des longueurs dans V4 ` ds2 = y02 du2 +
1 2
(dn u)
k 2 d˜ s2
ce qui d´efinit formellement le facteur de r´eduction ¯ ¯ b(K) −K K ≡ K − log 4 b(0) = k ∼ e si K 1.
(61)
(62)
Il est int´eressant d’exprimer la m´etrique de V5 dans la coordonn´ee angulaire ϕ1 = ϕ 2 dϕ2 ds2 = y02 s . (63) + 1 − k 2 sin2 ϕ d˜ 2 2 1 − k sin ϕ On passe de ϕ1 ` a ϕ2 , comme en (61).
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La m´etrique est p´eriodique et reguli`ere, ∀l, si k 2 < 1. Enfin pour la pression et la densit´e nous avons les formules 2 1 2 2 dn 2 2 κρ = 2 = 2 1 + k 2 − dn2 u − dn2 (u + K) 0 3 y0 dn y0 1 κ (ρ − p) = 2 dn2 u − dn2 (u + K) 3 y0 κp 2 y0 = 2 2 1 + k 2 − (1 + 2 ) dn2 u + (1 − 2 ) dn2 (u + K) . 3 On encore, si l’on veut, sachant que ϕ1 et ϕ2 sont li´es 1 2 k 4 sin2 2ϕ 2 2 k 2 2 sin ϕ1 + sin2 ϕ2 − 1 ≡ κρ = 2 2 2 2 3 2y0 1 − k sin ϕ y0 1 k 2 2 κ (p − ρ) = 2 = sin ϕ1 − sin2 ϕ2 . 3 y0
(64)
(65)
(66)
On v´erifie ais´ement que la pression (65) donne la limite de R.S., formule (20), pour → 0, k → 0 avec lim = kRS , lim Ky0 = yc , lim y0 = y. y0 • Remarque sur l’extension a` un V4 qui soit un espace d’Einstein de constante ˜ cosmologique Λ. Selon les ´eqs. de F.L. (15), il suffit d’effectuer le remplacement ˜ 2σ κρ → κρ + 2Λe κp → ˜ 2σ κp + Λe (67) 1 2 e2σ = dn (u + K) . k 2 ˜ = 0, ne va fonctionner ici que pour = 1, La m´ethode utilis´ee dans le cas Λ 2 la modification ne portant que sur Φ2 et V2 ϕ2 2 v2 Posant
→ ϕ2 2 → v2
− +
˜ 2 2σ Λy 0 3 e 2 2σ ˜ Λy0 e .
(68)
˜ 2 ˜ = Λ . y0 λ 3 k 2
le nouveau champ ϕ2 devient
˜ dn2 (u + K) soit ϕ2 = 1 − λ ˜ am (u + K) . = 1 − λ ϕ2 2
2 D’autre part le coefficient
de dn (u+ K) dans v2 (formule (50)) s’obtient en ˜ =3 λ ˜−1 . rempla¸cant (1 − 4 ) par 1 − 4 + 3λ
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Il est remarquable que le second champ, terme cin´etique et potentiel, dis˜ = 1, c’est-`a-dire pour les valeurs paraisse pour λ 2
˜ = 3k > 0 Λ y02
(69)
˜ assez petit. ce qui d´etermine k pour Λ Pour cette valeur du module, la m´etrique dS 2 = y02 du2 + dn2 ud˜ s2
(70)
est donc celle d’un V5 coupl´e `a un seul champ scalaire de sine-Gordon avec le signe ˜ usuel, pourvu que V4 soit une espace d’Einstein de constante Λ. Si cette constante cosmologique devait ˆetre de l’ordre observ´e R0−2 o` u R0 est le rayon de l’univers V4 , on aurait k ∼
y0 10−33 = = 10−60 R0 1027
ce qui entraˆınerait K ∼ 140, alors que la hi´erarchie de masse exigeant qyc ∼ 35 dans la limite R.S. ce qui nous donnerait plutˆ ot selon la formule (80) pour l = 1, K = 2qyc ∼ 70. Le facteur de r´eduction donn´e par la contrainte cosmologique (69) serait donc de 10−30 au lieu de 10−15 . L’int´erˆet de cette propri´et´e, probablement ind´ependante de la dimension, vient de ce que (70) g´en´eralise celle des espaces d’Einstein (cf. (17)) usuels au cas d’une source de type sine-Gordon. Il serait alors possible d’avoir un scalaire de courbure tr`es grand pour un adS (V5 rempla¸cant le V4 pr´ec´edent) et une constante cosmologique tr`es petite pour une section V4 de rapidit´e convenable. • La constante de gravitation de V4 −2 Si κ = M −3 est la constante de couplage pour V5 , celle de V4 est κ ˜ = MPl et l’on d´efinira la masse q par κ ˜ = κq . (71) Dans le mod`ele de R.S. on a la relation suivante entre q et kRS kRS = 1 − e−2yc kRS ∼ 1 . q Dans le mod`ele d´eform´e nous avons encore yc e−2σ(y) dy q −1 = −yc
(72)
(73)
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M. Gaudin
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c’est-`a-dire 1 2qy0
= =
π 2
− 12 (dn u) du = dϕ 1 − k 2 sin2 ϕ 0 0 2 1+k π − 12 (k ) P− 12 2 2k K
2
(74)
o` u Pν (z) d´esigne la fonction de Legendre d’ordre ν r´eel, holomorphe pour z > −1. Pour k = 0 (mod`ele ap´eriodique), on a ∀ 1 Γ + 12 (75) qy0 = √ π Γ ( ) et notamment pour voisin de z´ero qy0 ∝ , ce qui entraˆıne dans ce mod`ele limite ` une paroi q = kRS . a Pour k petit, la formule (75) donne le terme dominant, la correction ´etant de l’ordre k ∝ 4e−K , a` condition d’exclure le voisinage de = 0. La non-uniformit´e est claire: 1 . (76) ∀k > 0 , lim qy0 = →0 2K La condition de validit´e de la formule (75) est K 1. Enfin, selon Szeg¨o8) , uniform´ement en k pour 1 1 − 12 qy0 ∝ . (76 ) k π Si l’´etude des excitations confirme que le facteur de r´eduction dans ce mod`ele est (k ) , celui-ci sera exponentiellement petit, soit pour assez grand ∀k (0 < k < 1), soit pour k assez petit pour que l’on ait K assez grand. Dans le premier cas 1, la formule (76 ) nous donne K (∀k ou K) . (77) qyc ∝ k π Dans le second cas K 1, l’expression (77) ci-dessus reste encore une bonne estimation de l’ordre de grandeur (approximation de Stirling) pour 1; 1 1 une ´evaluation correcte de l’int´eEnfin pour K fini, c’est-` a-dire = O K grale (74) se fait en coupant l’intervalle en et en notant que l’approximation deux, dn u ∝ ch1u est valide uniferm´ement sur 0, K 2 , pour K assez grand. √ K −K En effet on a dn 2 , k = k ∝ 2e 2 ∝ ch1K on en d´eduit pour K fini, 2 K 1 K2 K2 1 1 −2u −2K 1 − e−2K (78) ∝ e du + e e2u du ∝ 2qy0 2 0 0
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1 1 = 1 − e−2K qyc K
275
(79)
on a donc l’identification K = kRS yc , qui montre l’´equivalence du mod`ele d´eform´e et du mod`ele limite R.S. lorsque est de l’ordre de K −1 . q c Pour 12 , voici quelques valeurs du rapport qy ecroissant en gros K ≈ kRS , d´ 1 comme √ qyc = 12 = π2 K qyc 1 = 1 = 2E(k) K (80) qyc 4 = 32 = 3π(1+k 2 ) K qyc 1 = √kπ . K
5 Excitations et ´echelle de masse Sans entrer dans les justifications physiques, nous consid´erons ici l’´equation de Klein-Gordon pour une onde r´eelle ψ scalaire sur V5 ∇2 ψ = 0 .
(81)
gµν en coordonn´ees harPar exemple, si V4 est plat, la fluctuation de spin 2, δ˜ moniques est r´egie, `a l’approximation lin´eaire, par l’´equation (81). On peut aussi consid´erer ψ comme la fluctuation (scalaire) d’un coefficient m´etrique diagonal relatif a` une sixi`eme dimension. Nous consid´erons plus g´en´eralement l’´equation K.G. massive sur V5
soit
1 √ √ ∂µ ( gg µν ∂ν ψ) = M52 ψ g
(82)
˜ 2 ψ + e4σ ∂0 e−4σ ∂0 ψ = M52 ψ e2σ ∇
(83)
dont nous cherchons une solution factoris´ee de masse m2 dans V4 ˜ 2 ψ = m2 ψ. ∇
(84)
Pour la d´ependance dans la cinqui`eme coordonn´ee x0 = y = y0 u, en notant σ =
dσ , du
nous obtenons l’´equation en ψ (u) ψ − 4σ ψ + y02 m2 e2σ − M52 ψ = 0.
(85)
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M. Gaudin
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Prenant comme fonction inconnue χ = b2 ψ ≡ e−2σ ψ .
(86)
Nous obtenons pour χ (u) −χ + y02 M52 − m2 e2σ + 4σ − 2σ χ = 0,
(87)
et utilisant la formule (12)–(13) pour la pression σ − 2σ 2 −χ + y02 M52 +
2κ 3 p
= − κ3 y02 p − m2 e2σ χ
(88)
= 0.
Puisque l’´equation de KG r´esulte du principe variationnel √ δ ∂ψ · ∂ψ gd5 x = 0
(89)
qui d´erive forc´ement, si elle est valide, du lagrangien d’origine a` l’approximation quadratique, il convient d’exiger la convergence des int´egrales • si m2 = 0
+∞
2
ψ (u) e
−2σ(u)
−∞
du
≡
+∞
χ2 e2σ du
(90)
−∞
• ou si m2 = 0,
+∞
χ2 (u) du.
−∞
Ces conditions ach`event de d´eterminer la fluctuation m´etrique comme une fonction d’onde r´eelle d’´etat li´e en m´ecanique quantique, ce qui illustre en passant l’id´ee primitive de Klein qui voit la relativit´e g´en´erale, notamment en dimension cinq, englobant la m´ecanique ondulatoire de la premi`ere quantification. Dans le cas particulier des masses nulles M52 = 0, m2 = 0, on a la solution triviale ψ = Cste ou “mode z´ero”, ce qui donne l’onde li´ee χ (u) = (ch 1u)2 dans ce mod`ele ap´eriodique. Dans le mod`ele p´eriodique, l’onde apparaˆıt plutˆ ot comme l’´etat fondamental d’une bande (impulsion nulle). Mod`ele ap´eriodique (une paroi) Selon (24) et (28), l’´equation (88) en χ s’´ecrit 2 (2 + 1) 2 2 2 −χ + y02 M52 + 4 2 − − y m (ch u) χ=0. 0 ch2 u
(91)
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Dans la coordonn´ee x = th u = sin ϕ, notre ´equation devient 2 4µ2 dχ y02 m2 2 d χ + 2 (2 + 1) − 1−x − 2x + χ=0 +1 dx2 dx 1 − x2 (1 − x2 ) o` u l’on a pos´e
4µ2 = 4 2 + y02 M52
(µ > 0) .
277
(92)
(93)
Le cas de masse nulle dans V4 , m2 = 0, nous donne l’´equation des fonctions de Legendre associ´ees dont les param`etres 2 et 2µ sont a priori des r´eels positifs quelconques. Avec la d´efinition de Hobson (cf. W.W., p. 326) de la fonction µ 1+x 1 1−x 2µ (x) = F −2 , 2 + 1, 1 − 2µ; P2 Γ (1 − 2µ) 1 − x 2 on a g´en´eriquement un comportement singulier aux extr´emit´es de l’intervalle [−1, +1] −µ 1 − x2 ∼ (ch u)2µ a l’exception bien connue des polynˆ ` omes et fonctions de Legendre associ´ees pour 2 = entier positif, 2µ = entier positif, 0 ≤ µ ≤ . Cette classe tr`es particuli`ere est celle des fonctions p´eriodiques r´eguli`eres dans la 2µ variable angulaire ϕ P2 , int´egrables selon la norme (90) relative `a m2 = 0; Mais, 2 selon la norme m = 0 qui est +∞ +∞ dx χ2 (u) du = P 2 (x) , (94) 1 − x2 −∞ −∞ la fonction relative a` µ = 0 est exclue. Le param`etre 2 n’ayant pas de raison d’ˆetre quantifi´e, la classe des fonctions propres convenable est, pour donn´e, la suite finie index´ee par l’entier r, avec max r = [2 ] − 1 soit
−2+r χr = ar P2 (x)
([ ] = partie enti`ere) ;
r = 0, 1, 2, . . . , [2 ] − 1
et explicitement χr =
ar 22 Γ (2
r − d 2 1 − x2 1 − x2 2 r + 1) dx
(95)
on a aussi χr
−− r2 1 − x2 × (Polynˆ ome de Gegenbauer de degr´e r) 1 ≡ × (polynˆ ome en th u) , (ch u)2−r =
(96)
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ce qui montre nettement le comportement `a l’infini. Pour ≤ 12 , il n’existe qu’une seule solution qui est le mode z´ero s’´ecrivant ∀ χ0 (u) =
a0 (ch u)2
= a0 (cos ϕ)2
(97)
avec la normalisation (94) `a l’unit´e a−2 0 =
√ Γ (2 ) . π Γ 2 + 12
Avec la croissance du param`etre 2 apparaissent des ´etats nouveaux formant une suite finie orthogonale de dimension [2 ], ce qui r´esulte ´evidemment de l’´equation diff´erentielle (91) ou d’un calcul direct partant de (95). Cette propri´et´e n’a pas de rapport avec l’orthogonalit´e de la suite infinie des fonctions de Gegenbauer, ou des fonctions de Legendre r´eguli`eres Pnm appartenant au mˆeme indice m. Il s’agit en fait d’une forme limite des fonctions ellipso¨ıdales qui constituent une d´eformation des harmoniques sph´eriques. Nos fonctions jouent plutˆ ot le rˆ ole de la suite eimϕ pour donn´ee. Selon (99), nous avons 2µ = − + r > 0, d’o` u le spectre en vertu de (93) −y02 M52 = r (4 − r) ≥ 0 .
(98)
En dehors du mode z´ero, on obtient un specre tachyonique dans V5 , ou si l’on veut, de masse nulle dans un V6 de signature adS. On n’a pas calcul´e le coefficient de normalisation ar ( ) de ces fonctions peu usit´ees. La norme usuelle des fonctions de Legendre correspond au choix ar ≡ 1 +∞ −∞
mais notre norme
+∞ −∞
χ2r (u) chdu 2u
=
2 4+1
·
Γ(4−r+1) , r!
(99)
χ2r (u) du semble plus difficile `a obtenir.
Cette ´equation d’onde de masse nulle a ´et´e ´etudi´ee comme KG sur une brane DS par Bertola and all [10]. • Cas m2 = 0 Nous ne nous attarderons pas sur le cas des excitations massives dans ce mod`ele `a une paroi. En effet la force r´epulsive due au potentiel −m2 (ch u)2 les ´eloigne de la paroi, et le probl`eme physique est d´eplac´e vers le voisinage de la seconde paroi, ici rejet´ee `a l’infini. Outre le mode z´ero et les r´esonances tachyoniques peu perturb´ees, les ondes instables sont calculables par la m´ethode W.K.B. Elles sont de carr´e sommable bien que dans le continu, mais les deux courants oppos´es sous-jacents ne le sont pas, puisque constants.
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Pour terminer ce paragraphe, il reste a` montrer que la forme limite de l’´equation (91) des excitations ap´eriodiques est l’´equation donn´ee par Randall et Sundrum. Avec lim y0 = q = kRS , apr`es division de (91) par y02 , nous avons 2 d2 χ 2q (1 + 2 ) y 2 2 2 χ=0 (100) − 2 + M5 + 4q − − m ch dy y0 y0 ch2 yy0 qui, a` la limite y0 → 0, devient l’´equation R.S. −
d2 χ 2 2 2 2q|y| χ=0 + M + 4q − 4qδ (y) − m e 5 dy 2
(101)
en prenant M5 = 0. Mod`ele elliptique (` a deux parois). Selon l’expression (65) pour la pression, l’´equation (88) s’´ecrit y02 m2 k 2 2 2 −χ + 4µ − 2 (1 + 2 ) dn u + 2 (1 − 2 ) 2 − χ=0 dn u (dn u)2 o` u l’on a pos´e
4µ2 = y02 M52 + 4 2 1 + k 2 ,
(102)
(103)
dont la forme limite pour k → 0 est (91). La fonction potentielle v (u) =
−2 (1 + 2 ) dn2 u + 2 (1 − 2 ) dn2 (u + K) −
y02 m2
2
(dn u)
(104)
est 2K-p´eriodique et r´eguli`ere sur une p´eriode. Mode z´ero Pour m2 = 0, M52 = 0, on a ´evidemment le mode z´ero χ = e−2σ = (dn u)2 , dont la norme est donn´ee par la formule (74) en rempla¸cant par 2 , avec la bonne estimation (75) lorsque k est petit. Selon le principe de continuit´e avec le module, il existe sˆ urement une classe de solutions de spectre discret tachyonique correspondant a` (95); nous y reviendrons. Pour m2 = 0, le probl`eme est de d´ecrire la classe de solutions χ qui soient p´eriodiques (r´eelles) et, en cons´equence, r´eguli`eres sur l’axe r´eel puisque v (u) est r´egulier. Une propri´et´e importante de ce potentiel est la suivante (105) v u + K, ; m2 ≡ v u; − , m2q avec la d´efinition m2q =
m2 k 2
(106)
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ou si k est petit, ∀ non nul, ¯
m2 ∝ m2q e−2K .
(107)
Cette propri´et´e traduit la sym´etrie de ce mod`ele dans l’´echange des deux parois (ou des deux potentiels φ1 et φ2 ) u
←→ u + K
κ
←→ −κ
,
←→ −
(108)
(ou M ←→ −M )
a condition de changer l’´echelle des masses par le facteur de r´eduction, selon (106). ` Il faudrait faire une ´etude d´etaill´ee du potentiel en fonction des deux param`etres et k, dans le domaine d’int´erˆet k 10, mais nous reviendrons sur le comportement qualitatif qui est assez simple: deux puits de potentiel s´epar´es par une barri`ere, dont les profondeurs sont dans le rapport du facteur r´eduction. Les ´etats physiques sont li´es dans le plus profond qui est celui de la seconde paroi. • Echelle de masse dans la limite R.S. Nous examinons la limite R.S. dans le voisinage du second puits: il suffit d’effectuer la translation d’une demi-p´eriode K sur l’´equation (102). Posant χ (u + K) = ξ (u), nous avons k 2 2 2 2 2 2 −ξ + 4µ − 2 (1 + 2 ) 2 + 2 (1 − 2 ) dn u − y0 mq (dn u) ξ = 0. (109) dn u nous avons l’approximation uniforme si k 1 1 2 (1 − 2 ) 2 2 2 4µ + ξ=0 (110) − y 0 mq 2 ch2 u (ch u)
Dans la r´egion |u| < −ξ +
K 2,
c’est l’´equation analogue a` (100) avec → − , la limite R.S., lim /y0 = q, nous obtenons −
m → mq . Apr`es division par y02
d2 ξ 2 + M5 + 4q 2 + 4qδ (y) − m2q e−2q|y| ξ = 0 . 2 dy
(111)
Distinguons les deux cas de parit´e, le potentiel δ ´etant inop´erant sur les ´etats impairs. L’onde li´ee d´ecroissant `a l’infini en exp (−2µ |y|) est exactement
m ξ + (y) Jν qq e−q|y|
(112) ξ − (y) (y) Jν mq e−q|y| q o` u l’on a pos´e 2µ = νq, soit ν=
1/2 M2 ≥ 2. 4 + 25 q
(113)
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on a les conditions spectrales mq mq mq Jν + 2Jν =0 q q q mq Jν =0 q
• Parit´e +: • Parit´e −:
(114) (115)
Pour la parit´e +, ce sont les extrema de x2 Jν (x), donc entre les z´eros de Jν . On ´ecrira dans les deux cas m(n) = qxn q
n = 1, 2, 3, . . .
(116)
o` u la parit´e de (n − 1) sera celle de l’´etat. Dans le cas M52 = 0, ν = 2, on aura J1 (xn ) = 0 n impair (parite´ +) n = 1, 3, . . . J2 (xn ) = 0 n pair (parite´ −) n = 2, 4, . . . Asymptotiquement avec n xn ∼
3 n+ 2
x = 3, 83 . . . π 1 x2 = 5, 13 . . . 2 x = 7, 01 . . . 3
(117)
En tout cas, dans la limite R.S., nous avons le r´esultat essentiel de la hi´erarchie de masse avec la d´efinition (106) de mq m2 = m2q k 2 = x2 q 2 k 2
(118)
m2 (n) ∼ x2n q 2 e−2K = x2n m2c
(119)
mc = qe−K ,
(120)
soit encore avec qui serait la masse de r´ef´erence du mod`ele standard. L’´equation (110) et sa forme limite (111) montrent clairement comment l’excitation massive d´ecrite par l’onde ψ dans V5 est li´ee `a la seconde paroi (visible), par un potentiel attractif induit de largeur q −1 qui en mesure l’´epaisseur. Ce m´ecanisme qui d´etermine la paroi physique r´esulte de la dynamique des champs dans les potentiels choisis. Les excitations de masse nulle sont li´ees `a la premi`ere paroi en appelant mc = qe−qyc la masse de r´ef´erence du mod`ele standard. Pour obtenir mc de l’ordre de 100 GeV, il suffit donc d’avoir qyc = K ∼ 40 au voisinage de la limite R-S, mais le truc de la r´eduction exponentielle fonctionne ∀k en vertu de l’estimation (77), pourvu que atteigne 10 ou 100.
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Contribution au propagateur des ´etats massifs La contribution des ´etats massifs – `a l’exclusion du mode z´ero – au propagateur de la fluctuation m´etrique (champ scalaire ψ) s’´ecrit en repr´esentation impulsion p et dans le cas statique p4 = 0 G (p) =
∞
1 ≡ G+ + G− . 2 + m2 x2 p c n n=1
(121)
Dans le cas M5 = 0 (ν = 2), nous avons dx J1 (x) 1 2G+ (p) = 2π (C) J1 (x) p2 + m2c x2
(122)
o` u l’int´egrale de Cauchy porte sur une fonction impaire m´eromorphe dans tout le plan et born´ee uniform´ement si Ju x > 0. Le contour (C) est l’union de (C+ ) et (C− ) , (C+ ) entourant dans Rex > 0 la suite {+xn }, (C− ) dans Rex < 0 la suite {−xn }, toujours positivement. Le contour (C) est ´equivalent a` un circuit n´egatif autour des trois pˆ oles x = 0, x = ±i mpc , et l’on obtient 2G+ (p) =
1 I1 mc p I1
p mc
−
1 I2 1 ≡ 2 p mc p I1
p mc
>0.
De mˆeme pour la parit´e (−) 1 1 I3 p dx J2 2G− (p) = ≡ >0. 2π J2 p2 + m2c x2 mc p I2 mc
(123)
(124)
D’o` u les comportements `a petite et `a grande impulsion • p mc
5 1 p2 G (p) = − +O 24m2c 144 m4c
• p mc 1 +O G (p) = mc p
mc p3
p2 m6c
(125)
(126)
on en d´eduit la correction a` la loi de Newton en 1r due au mode z´ero. Ce type de correction a ´et´e consid´er´e par Tanaka et Montes [3] dans le mod`ele de Randall Sundrum. 1 G (p) e−i p · r d3 p G(r) = 2 2π ∞ 1 −xn mc r e ≡ . (127) r n=1
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• A tr`es courte distance r m−1 c 2 1 G (r) ∼ +O πmc r2
1 r
283
(128)
terme qui n’est plus une correction, mais domine la loi de Newton. • A distance sup´erieure a` quelques longueurs de Compton m−1 c , la loi n’admet qu’une correction n´egligeable en exp − (x1 mc r) due a` la masse la plus basse. • Dans la zˆone de transition π2 mc r de l’ordre de 1, la loi asymptotique (117) donne l’estimation e−mx1 r . (129) G (r) ∼ π 1 − e − 2 mc r Pour des transferts d’impulsion petits devant mc , l’interaction r´esiduelle ´equivaut a` une interaction de contact selon (125) 5π δ r (130) G (r) ∼ 6m2c ce qui ´evoque naturellement l’interaction de Fermi, v´ehicul´ee par un boson de Higgs de masse O (mc ) dans le mod`ele standard, mais ici due `a tout le spectre d’excitation. • Etude qualitative du cas 1 On pourrait ´etudier les choses en d´etail en fonction des deux param`etres. Pour fini, k petit les r´esultats sont essentiellement inchang´es. Examinons seulement le cas de d´eformation extrˆeme 1. Le potentiel est en gros constitu´e de deux puits de profondeurs comparables centr´es en u = 0 et u = K, s´epar´es par une barri`ere k a k1 , avec X = 1 pour u = K en n = K 2 . Posons X = dn2 u , croissant avec u de k ` 2 . 2 + 1 y 2 m2 + (2 − 1) X + 0 X v (X) = − 2 k (131) X k ∂v 2 + 1 y02 m2 −1 = − 2k 2 − 1 − X + . (132) ∂X X2 k = 4 2 1 + k 2 + 2 k 2 + y02 m2 −v (u = 0) − 2 k 2 −v (u = K) = 4 2 1 + k 2 + y02 m2q (133) = 8 2 k −v u = K + y02 m2q k 2 On voit que
admet un unique z´ero pour 1 √ 2 + 1 e Xmax = k + ··· 1+ 2 − 1 4 1 = 1+O 2
∂v ∂X
(134)
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d’o` u le maximum de la barri`ere centr´ee en n = K eriode). Pour que les 2 (quart de p´ profondeurs des puits soient comparables, on doit avoir y02 m2q de l’ordre de 4 2 . Posons y02 m2q = 4 2 ξ 2 on a les termes dominants a` l’approximatiojn exponentielle O k 2 , en supposant k nettement diff´erent de 1 : −v (u = 0) ∼ 4 2 1 + k 2 + 2 k 2 − 2 k 2 −v (u = K) ∼ 4 2 1 + k 2 + ξ 2 (135) ∼ 8 2 k . −v u = K 2 La barri`ere disparaˆıt dans la limite k → 1 (k → 0) o` u le facteur de r´eduction exponentiel est d’autant moins efficace. Supposons encore k assez petit pour que les deux puits d´ecouplent pratiquement les deux ´etats d’´energie commune −4µ2 ≡ −4 2 , si M5 = 0. Dans ce cas extrˆeme, ∀ assez grand, on a les deux probl`emes d´ecoupl´es `a l’ordre dominant (n´egligeant devant 2 ) • Paroi u + 0
−
K 2
u
−X −
K 2
2 (2 + 1) X + 4 2 X = 0 ch2 u
(136)
qui nous donne le mode z´ero X0 . • Paroi u = K. Posant v = K − u, |v| K 2 ξ2 2 X=0. −X + 4 1 − 2 ch v L’approximation WKB donne pour le spectre ξn 1/2 2 ξ 1 2 dv = n + − 1 π. 2 ch2 v Par une approximation assez grossi`ere π 1 = y 0 mq . 2ξn = n + 2 2
(137)
(138)
(139)
Utilisant la valeur de q estim´ee en (76), pour grand et k petit qy0 ∼ π , l’estimation (139) nous donne √ 1 π (n 1) (140) mq = 2q n + 2 2 ´eliminant √ la d´ependance explicite en , formule tr`es analogue au spectre (116) au facteur 2 pr`es.
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6 Conclusion L’introduction de deux champs scalaires φ r´egis par deux potentiels p´eriodiques de mˆeme forme cos π φφ0 , coupl´es `a la gravit´e d’un V5 avec des signes oppos´es, a permis de construire une sorte de d´eformation du mod`ele de Randall et Sundrum d´ependant de deux param`etres. Ce sont la p´eriode 2φ0 des champs (li´ee au param`etre ), et la p´eriode 2Ky0 = 2yc de la cinqui`eme dimension y (K est li´ee au module k), qui sont reli´ees entre elles par les constantes de gravitation κ de V5 et κ ˜ de V4 . (A la limite R.S. on a seulement y0 = 3π4 2 κ ˜ φ20 .) Selon l’id´ee d´evelopp´ee par Ichinose en introduisant un potentiel la paroi est une notion d´eriv´ee ; elle est n´ecessairement diffuse, la paroi-section de R.S. ´etant un concept limite. La p´eriodicit´e, trait essentiel de la th´eorie de Klein, r´esulte habituellement de la des petits mouvements au voisinage d’un minimum du potentiel. 2 dynamique k 1 . Avec l’analogie du pendule circulaire dont l’angle de rotation 2ψ est toujours croissant k 2 < 1 , l’hypoth`ese topologique du cercle pour la cinqui`eme dimension porte de fa¸con ´equivalente sur le champ qui est un angle. L’image des deux pendules en opposition illustre la liaison cin´ematique des deux champs, n´ecessairement li´es puisqu’ils ne d´ependent que d’une seule variable dans cette solution sp´eciale. Cette liaison ´equivaut `a une contrainte g´eom´etrique sans transfert d’´energie. Dans cette analogie m´ecanique o` u la cinqui`eme variable d’espace joue le rˆole de “temps”, le ph´enom`ene des deux parois diffuses, a` la limite R.S. d’´epaisseur nulle, apparaˆıt comme dˆ u a` l’approche du mouvement pendulaire ap´eriodique (limite k → 1, K → +∞). En alternance, les pendules passent quasiment une demi-p´eriode au voisinage de l’´equilibre instable (le z´enith 2ϕ1 = ±π, ϕ1 petit, ϕ2 grand, et vice-versa). D’o` u les pics cin´etiques, ou de pression, caract´erisant les parois. Ce syst`eme pendulaire sert d’horloge de r´ef´erence pour la coordonn´ee u = y/y0 et les champs scalaires φ, tandis que la courbure de V5 ou le facteur de dy r´eduction e−σ = dτ mesure le rapport du temps m´ecanique des pendules dy au temps propre dτ de V4 , sur une g´eod´esique nulle de V5 . La source de cette courbure est le diff´erentiel de densit´e d’´energie et de pression entre les deux champs, qui en fait ne constituent qu’une seule entit´e. Toute particule test massive est de masse rapidement variable sur une p´eriode, de l’ordre de la masse de Planck au voisinage de la paroi cach´ee. Cette masse n’est stabilis´ee `a une valeur physique observable que dans le voisinage de la paroi visible `a laquelle l’excitation massive est comme li´ee. Le couplage avec signes oppos´es apparaˆıt certes comme un artifice. Mais il pr´eserve cependant une sym´etrie d’´echange des deux champs ; le syst`eme global est invariant par la triple op´eration: a) M ←→ −M (ou κ ←→ −κ du fait de la dimension impaire), b) φ1 ←→ φ2 , c) ←→ − , ou inversion du facteur d’´echelle.
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Cette variation math´ematique sur le th`eme de la m´etrique R-S ne change rien sur le fond de l’interpr´etation et semble indiquer une certaine stabilit´e structurelle du mod`ele, tout en accentuant le rˆ ole d’une hypoth`ese artificielle. Concernant la m´etrique, il n’est pas sans int´erˆet de voir s’introduire simplement et exactement une p´eriodisation de la variable de rapidit´e (ou angle hyperbolique) g´en´eralisant les espaces d’Einstein (`a constante cosmologique) qui sont d´ej` a des sph`eres g´en´eralis´ees (ou hyperbolo¨ıdes), au prix d’une source de type sine-Gordon. Si la cinqui`eme coordonn´ee devait ˆetre identifi´ee `a la variable compactifi´ee de K.K., l’unification g´eom´etrique gravit´e-´electromagn´etique serait `a transposer dans ce contexte. La p´eriode 2yc est alors d´etermin´ee. 2π yc = √ y˜ ; α
√ = G = longueur de Planck. ˜ M
y˜ =
√ 2 G yc e2 = Cste de structure fine. ≡ . (141) e π Puisque le syst`eme global est formellement invariant par changement de coordonn´ee, on peut appliquer a` la solution particuli`ere obtenue la transformation sp´eciale qui pr´eserve l’hypoth`ese cylindrique de la th´eorie de Kaluza. Ce qui sugg`ere, en pr´esence d’un champ e.m additionnel A (x), la m´etrique invariante de jauge, d´ependant d’un champ scalaire a (x) α=
2 √ √ ds2 = dy − 2 GAdx + e−2σ(y−2 Ga) d˜ s2
(142)
inchang´ee par √ 2 G S (x) y→y+ e
,
ea → ea + S
,
eA → eA + ∂S
(143)
sur une p´eriode 2yc , l’action S augmente de 2π. Choisir a (x) ≡ 0, fixe la jauge. Mais la projection V5 → V4 , avec le d´ecouplage exact de Kaluza, Thiry, etc. . . qui ´etait dˆ u a` l’hypoth`ese cylindrique, n’est plus qu’approximative et n´ecessite la moyenne pr´ealable sur une p´eriode. Le potentiel Aµ n’est plus un vecteur de Killing de V5 , sinon en moyenne, puisque l’on a dans V5 , 1 ∇µ Aν + ∇ν Aµ = + √ σ gµν G
(144)
ce qui revient en gros a` remplacer dans les formules de Thiry le champ de V4 Fµν par Fµν + √1G σ e−2σ g˜µν . D’o` u une formule analogue a` celle de Thiry avec le terme de pression suppl´ementaire dˆ u a` la source scalaire de sine-Gordon. Le champ F admet comme source un courant qui est proportionnel a` A dans la jauge particuli`ere a (x) ≡ 0, ce qui d´etermine implicitement une action S.
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L’´equation de Klein-Gordon associ´ee en pr´esence du champ A dans la jauge fix´ee sort ainsi sous la forme r´eelle qu’avait not´ee Schr¨odinger ˜ 2Ψ ¯ + e 2 A · A − m2 Ψ ¯ =0 2 ∇ (145) dans la jauge telle que le courant conserv´e soit justement proportionnel a` A 2 ¯ A =0. (146) ∂ Ψ
References [1] L. Randall et R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999), Phys. Rev. Lett. 83, 4690 (1999), preprints hep-ph/9905221–9906064. [2] S. Ichinose, Class. Quantum Grav. 18, 421–432 (2001). Voir aussi S. Kobayashi, K. Koyama et J. Soda, preprint hep-th/0107025 [3] T. Tanaka et X. Montes, Nucl. Phys. B 582, 259 (2000). [4] Pour les ´equations de Friedmann-Lemaˆıtre, r´ef´erences cit´ees dans Essais Cosmologiques J.P. Luminet. L’invention du Big Bang, Le Seuil (1997). [5] Th. Kaluza Sitz. Preuss. Akad., 966 (1921), Modern Kaluza-Klein Theories Appelquist, Chodos and Freund, A.W.P. (1987). [6] O. Klein, Zeitschrift f¨ ur Physik 37, 895 (1926). [7] J.M. Souriau, Five-dimensional Relativity. Nuovo Cimento XXX, 2 (1963). [8] Whittaker and Watson, Modern Analysis, IV`eme ´ed. Cambridge U.P. (1958). [9] G. Szeg¨ o, Orthogonal Polynomials R.I. AMS (1939). [10] M. Bertola, J. Bros, V. Gorini, U. Moschella, R. Schaefer, Decomposing quantum fields on branes, Nuclear Physics B581, 575–603 (2000). Michel Gaudin Service de Physique Th´eorique Orme des Merisiers CEA/Saclay F-91191 Gif sur Yvette France email:
[email protected] Communicated by Vincent Rivasseau Submitted 07/09/01, accepted 25/11/03
Ann. Henri Poincar´e 5 (2004) 289 – 326 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/020289-38 DOI 10.1007/s00023-004-0170-z
Annales Henri Poincar´ e
Conformal Transformations and the SLE Partition Function Martingale Michel Bauer and Denis Bernard∗
Abstract. We present an implementation in conformal field theory (CFT) of local finite conformal transformations fixing a point. We give explicit constructions when the fixed point is either the origin or the point at infinity. Both cases involve the exponentiation of a Borel subalgebra of the Virasoro algebra. We use this to build coherent state representations and to derive a close analog of Wick’s theorem for the Virasoro algebra. This allows to compute the conformal partition function in non trivial geometries obtained by removal of hulls from the upper half-plane. This is then applied to stochastic Loewner evolutions (SLE). We give a rigorous derivation of the equations, obtained previously by the authors, that connect the stochastic Loewner equation to the representation theory of the Virasoro algebra. We give a new proof that this construction enumerates all polynomial SLE martingales. When one of the hulls removed from the upper half-plane is the SLE hull, we show that the partition function reduces to a useful local martingale known to probabilists, thereby unraveling its CFT origin.
1 Introduction Since its very origins, the statistical mechanics of two-dimensional critical systems has seen a deep interplay between physics and mathematics. This was already true for Onsager’s solution of the 2d Ising model and the computation of the magnetization by Yang [19]. In the 80’s, the link between physics and mathematics was mainly through representation theory, affine Lie algebras and the Virasoro algebra playing the most central roles. Two-dimensional conformal field theories [4] have led to an enormous amount of exact results, including the computation of multi-point correlators and partial classifications. The study of multi-fractal properties of conformally invariant critical clusters has been less systematic, but has nevertheless produced a number of remarkable successes (see, e.g., refs.[18, 6, 10] and references therein), the famous Cardy formula giving the probability for the existence of a connected cluster percolating between two opposite sides of a rectangle in two-dimensional critical percolation [5] being one of the highlights. Cardy’s formula is now a theorem [22]. More recently, probability theory, stochastic processes to be precise, have started to play an important role, due to a beautiful connection between Brownian motion and critical clusters discovered by Schramm [21]. This connection is ∗ Member
of the CNRS
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via the Loewner evolution equation, which describes locally growing domains Kt (called hulls) in the upper half-plane implicitly by prescribing the variation of the normalized uniformizing map for the complement. In this way, the growth of the hull is encoded by a real continuous function. Taking this function to be a Brownian sample path leads to stochastic (chordal) Loewner evolutions (SLE) of growing hulls whose properties are those expected for conformally invariant critical clusters. There is a single parameter, denoted κ, which is the time scale for the Brownian motion. This has led to important probabilistic theorems, among which Brownian intersection exponents [15]. Moreover, this framework has opened a way to prove that critical lattice statistical models possess a conformally invariant scaling limit. The link between SLE and standard conformal field theory (CFT) was obscure for several years, but recently we proposed a direct connection [1]. The idea is to couple CFT to SLE via boundary conditions, namely to look at a boundary CFT in the random geometry of the complement of the hull in the upper halfplane. The crucial observation is that if one inserts at the origin (where the hull starts to grow) a primary boundary operator (leading to a boundary state |ω) of appropriate weight in a CFT of appropriate central charge, and then lets the hull grow, the corresponding conformal state is a local martingale in the sense of probability theory, i.e., a quantity whose probabilistic average is time-independent1 . In this way, many quantities computed by probabilistic methods can be shown to be directly related to correlation functions of CFT [2]. Another relation between CFT and chordal SLE which uses the restriction property has been presented in ref. [12]. The purpose of this paper is twofold. The first is SLE independent. We give a rigorous construction of the CFT operator implementing finite local conformal transformations that fix a point. This amounts to show how to go from certain sub-algebras of the Virasoro algebra to a corresponding Lie group via exponentiation. As a first application, we use coordinates on these groups to build coherent state highest weight representations of the Virasoro algebra. We observe a striking similarity with the representations of the Virasoro algebra that appear in matrix models [8]. This is a pedestrian implementation of the geometric ideas ` a la BorelWeil presented in [3]. Under some global conditions, one can multiply operators corresponding to local conformal transformations fixing different points, leading to an embryonic version of the Virasoro group (which is ill defined in the CFT context: the central extension of the group of diffeomorphisms of the circle is not what is needed). As a byproduct, we give a theorem which does for the Virasoro algebra what Wick’s theorem does for oscillator algebras. This kind of computation could have been made right at the early stages of CFT, in the 80’s. It seems that certain analogous 1 Under certain boundedness conditions: technically, nice linear forms applied to this state are time-independent in mean.
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formulæ were derived at that time [25], but we have not been able to trace those back in the published literature. These purely algebraic considerations have applications to SLE. The uniformization of the growing hull Kt is given, close to the point at infinity, by a suitably normalized local conformal transformation kt . This leads immediately to a clean definition of the conformal state Gkt |ω describing the growing hull Kt . The invertible operator Gkt is then shown to satisfy a stochastic differential equation2 which implies that Gkt |ω is a local martingale. We give a brief account of the proof, using the above-mentioned coherent state representations of the Virasoro algebra, that Gkt |ω is the generating function of all SLE martingales in a precise algebraic sense and that these martingales build a certain highest weight representation of the Virasoro algebra with a non trivial character. This is an elaboration of [3]. Finally, we turn to the partition function martingale. If a CFT is coupled via boundary conditions not only to the growing hull Kt but also to a fixed (deterministic) hull A disjoint from Kt , the CFT partition function contains a universal contribution corresponding to some kind of interaction between A and Kt . This is by definition a local martingale. We use Wick’s theorem for the Virasoro algebra to give yet another illustration that the SLE quantities computed by Lawler, Schramm and Werner [16] are in fact deeply rooted in CFT. For κ = 8/3, this martingale computes the probability that Kt never touches A. The previous paragraph is definitely not a claim that mathematicians have rediscovered things that were known to theoretical physicists. Quite the opposite is true: the discoveries of probabilists have motivated us to go back to the foundations of CFT to realize that maybe certain basic construction had not been given enough attention and that some CFT jewels had been left dormant.
2 (Chordal) SLE and CFT The aim of this section is to recall basic definitions of stochastic Loewner evolutions (SLE) and its generalizations that we shall need in the following. Most results that we recall can be found in [20, 14, 15, 16]. See [7] for a nice introduction to SLE for physicists and [23] for pedagogical summer school notes. A hull in the upper half-plane H = {z ∈ C, z > O} is a bounded simply connected subset K ⊂ H (for the usual topology of C) such that H \ K is open, connected and simply connected. The local growth of a family of hulls Kt parameterized by t ∈ [0, T [ with K0 = ∅ is related to complex analysis in the following way. The complement of Kt in H is a domain Ht which is simply connected by hypothesis, so that by the Riemann mapping theorem Ht is conformally equivalent to H via a map ft . This map can be normalized to behave as ft (z) = z + 2t/z + O(1/z 2): 2 In our previous papers, this equation was used as a heuristic definition of G . We had to kt leave aside analytical questions of existence of solutions, relying on physical intuition.
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the P SL2 (R) automorphism group of H allows to impose ft (z) = z + O(1/z) for large z, and then the coefficient of 1/z is fixed to be 2t by a time reparametrization. The crucial condition of local growth leads to the Loewner differential equation ∂t ft (z) =
2 , ft (z) − ξt
ft=0 (z) = z
with ξt a real function. For fixed z, ft (z) is well defined up to the time τz ≤ +∞ for which fτz (z) = ξτz . Then Kt = {z ∈ H : τz ≤ t}. √ (Chordal) stochastic Loewner evolutions is obtained [21] by choosing ξt = κ Bt with Bt a normalized Brownian motion and κ a real positive parameter so that E[ξt ξs ] = κ min(t, s). Here and in the following, E[· · · ] denotes expectation value. The next section, which also contains basic definitions that the reader can refer to, is devoted to a careful discussion of the implementation of finite local conformal transformations in conformal field theory. In the rest of this section, we simply assume that such an implementation is possible, and we derive a direct connection between SLE and CFT. SLE is defined via an ordinary differential equation, but for our reinterpretation in terms of conformal field theories, it is useful to define kt (z) ≡ ft (z) − ξt which satisfies the stochastic differential equation dkt =
2dt − dξt . kt
We observe that the conditions at spatial infinity satisfied by kt imply that its germ there, which determines it uniquely, belongs to the group N− of germs of holomorphic functions at ∞ of the form z + m≤−1 fm z m+1 , the group law being composition. In this way, the Loewner equations describe trajectories on N− in a time-dependent left-invariant vector field, whose value at the identity element is (2/z − ξ˙t )∂z . Due to the fact that ξt is almost surely nowhere differentiable, this observation has to be taken with a grain of salt. We let f ∈ N− act on O∞ , the space of germs of holomorphic functions at infinity, by composition, γf · F ≡ F ◦ f . Observe that γg◦f = γf · γg so this is an anti-representation. Ito’s formula gives 2dt κ − dξt + (γkt · F ) dγkt · F = (γkt · F ) kt 2 from which we derive γk−1 · dγkt = dt t
2 κ ∂z + ∂z2 z 2
− dξt ∂z .
The operators ln = −z n+1 ∂z are represented in conformal field theories by operators Ln which satisfy the Virasoro algebra vir c [Ln , Lm ] = (n − m)Ln+m + (n3 − n)δn+m,0 [c, Ln ] = 0. 12
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The representations of vir are not automatically representations of N− , one of the reasons being that the Lie algebra of N− contains infinite linear combinations of the ln ’s. However, as we shall see in the next section, highest weight representations of vir can be extended in such a way as to become representations of N− . We take this for granted for the moment and associate to γf an operator Gf acting on appropriate representations and satisfying Gg◦f = Gf Gg and κ 2 + dξt L−1 . L dG = dt −2L + G−1 kt −2 kt 2 −1 The basic observation is the following [1]: Let |ω be the highest weight vector in the irreducible highest weight repreand conformal weight hκ = 6−κ sentation of vir of central charge cκ = (6−κ)(8κ−3) 2κ 2κ . Then Gkt |ω is a local martingale. Assuming appropriate boundedness conditions on v|, the scalar v|Gkt |ω is a martingale. In particular E[ v|Gkt |ω] is time-independent. This is a direct consequence of the fact that for this special choice of central charge and weight, the irreducible highest weight representation is degenerate at level 2 and (−2L−2 + κ2 L2−1 )|ω = 0. Then κ dGkt |ω = Gkt dt −2L−2 + L2−1 + dξt L−1 |ω = dξt Gkt |ω . 2 In probabilistic terms, a random variable whose Ito derivative contains only a dξt contribution (no dt) is called a local martingale. We refer the interested reader to the mathematical literature [13]. From the definition of Ito integrals, dξt and Gkt are independent, so that naively dE[ v|Gkt |ω] = 0 for any v|. A word of caution is needed here. Before talking about E[ v|Gkt |ω], we should in principle show that v|Gkt |ω is an integrable random variable. Then v|Gkt |ω is a martingale. This is true for instance if v| is a finite excitation of ω|, but this condition is far too restrictive for probability theory and for conformal field theory as well3 . This result can be interpreted as follows. Take a conformal field theory in Ht . The correlation functions in this geometry can be computed by looking at the same theory in H modulo the insertion of an operator representing the deformation from H to Ht . This operator is Gkt . Suppose that the central charge is cκ and the boundary conditions are such that there is a boundary changing primary operator of weight hκ inserted at the tip of kt (the existence of this tip is more or less a consequence of the local growth condition). Then in average the correlation functions of the conformal field theory in the fluctuating geometry Ht are timeindependent and equal to their value at t = 0. 3 We shall often drop the term local, even if the notion of martingale, though closely related to the notion of local martingale, is more restrictive. In particular, the time-independence of expectations is always true for martingales.
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We call Gkt |ω a generating function for conserved quantities because for any time-independent bra v| satisfying the integrability condition, the scalar E[ v|Gkt |ω] is time-independent. We shall see later that in an algebraic sense, all conserved quantities for chordal SLE are of this form.
3 Conformal transformations in conformal field theory A (rather provocative) definition of (boundary) conformal field theory is that it is the representation theory of the Virasoro algebra vir. The Virasoro algebra has a sub-algebra n− , with generators the Ln ’s n < 0, which is closely related to N− , the group of germs of conformal transformations that fix ∞. This is crucial for the construction of Gkt . Our goal in this section is to show that indeed, N− acts on sufficiently many physically relevant representations of vir to be able to make sense of conformal field theories in the fluctuating geometry Ht . In the same spirit, the group N+ of germs of conformal transformations that fix 0 is closely related to the sub-algebra n+ of vir with generators the Ln ’s n > 0. This group will also play an important role in the forthcoming discussion.
3.1
Background
The theories we shall study will mostly be boundary conformal field theories, and we shall talk of field or operator without making always explicit whether the argument is in the bulk or on the boundary. The basic principles of conformal field theory state that the fields can be classified according to their behavior under (local) conformal transformations. Then the correlation functions in a region U are known once they are known in a region U0 and an explicit conformal map f from U to U0 that preserves boundary conditions is given. Primary fields have a very simple behavior under conformal transformations: for a bulk primary field ϕ of weight (h, h), ϕ(z, z)dz h dz h is invariant, and for a boundary conformal field ψ of weight δ, ψ(x)|dx|δ is invariant. So the statistical averages in U and U0 are related by · · · ϕ(z, z) · · · ψ(x) · · ·U h
= · · · ϕ(f (z), f (z))f (z)h f (z) · · · ψ(f (x))|f (x)|δ · · ·U0 . Such a behavior is described as local conformal covariance. In a local theory, small deformations are generated by the insertion of a local operator, the stress tensor. Local conformal covariance can then be rephrased: the stress tensor of a conformal field theory is not only conserved and symmetric, but also traceless, so that it has only two independent components, one of which, T , is holomorphic (except for singularities when the argument of T approaches the argument of other insertions), and the other one, T , is anti-holomorphic (again
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except for short distance singularities). The field T itself is not a primary field in general, but a projective connection: c · · · T (z) · · ·U = · · · T (f (z))f (z)2 + Sf (z) · · ·U0 . 12 2 (z) (z) In this formula, c is the central charge and Sf (z) = ff (z) − 12 ff (z) is the Schwarzian derivative of f at z. If U is a non empty simply connected region strictly contained in C, the Riemann mapping theorem states that U0 can be chosen to be unit disk D or equivalently the upper half-plane H – then the point at infinity is a boundary point. This second choice will prove most convenient for us in the sequel. In boundary conformal field theory, T and T are not independent: they are related by analytic continuation. The relationship is expressed most simply in the upper half-plane. The vectors fields z n+1 ∂z and z n+1 ∂z are generators of infinitesimal conformal transformations in C but only the combination z n+1 ∂z + z n+1 ∂z ≡ − n preserves the boundary of H, that is, the real axis. Write z = x + iy and for a while write T (x, y) for what we usually write T (z). Choosing boundary conditions such that there is no flow of energy momentum across the boundary x = 0, T (x, y) is real along the real axis, and by the Schwarz reflection principle has an analytic extension to the lower half-plane as T (x, −y) ≡ T (x, y) = T (x, y). Due to this property, most contour integrals involving T and T in the upper half-plane can be seen as contour integrals involving only T but in the full complex plane. Using conformal field theory in H to express correlators in any simply connected region strictly contained in C has another advantage: one can use the formalism of radial quantization in a straightforward way. The statistical averages are replaced by quantum expectation values: ˆ · · · |Ω. ˆ z) · · · ψ(x) · · · T (z) · · · ϕ(z, z) · · · ψ(x) · · ·H = Ω| · · · Tˆ (z) · · · ϕ(z, r
In this formula, |Ω is the vacuum and r denotes radial ordering: the fields are ordered from left to right from the farthest to the closest to the origin. The integral dzz n+1 Tˆ (z) along any contour of index 1 with respect to 0, defines an operator Ln (note again that from the point of view of contour integrals in the upper halfplane, Ln involves T and T ). The fact that the stress tensor is the generator of infinitesimal conformal maps implies that ˆ ˆ [Ln , ψ(x)] = xn+1 ∂x + δ(n + 1)xn ψ(x) n+1 [Ln , ϕ(z, ˆ z) ˆ z)] = z ∂z + h(n + 1)z n + z n+1 ∂z + h(n + 1)z n ϕ(z, n+1 c [Ln , T (z)] = z ∂z + 2(n + 1)z n T (z) + (n3 − n)z n−2 12 c 3 [Ln , Lm ] = (n − m)Ln+m + (n − n)δn+m,0 . 12 Except for the anomalous central c-term, the commutation relations of the Ln ’s are those of the n ’s. Let us take this opportunity to recall that the crucial point
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to implement symmetries in quantum mechanics is to have the symmetries act well on operators, i.e., that the adjoint action represents the symmetries. Hence symmetries in quantum mechanics act projectively, and this leaves room for central terms such as c in vir. The advantage of the operator formulation of conformal field theory is that one can use the powerful methods of representation theory, applied to the Virasoro algebra.
3.2
Some representation theory
Basic references for this standard material are for instance [9]. In the sequel we denote by h the (maximal) Abelian sub-algebra of vir generated by L0 and c, by n− (resp. n+ ) the nilpotent4 Lie sub-algebra of vir generated by the Ln ’s, n < 0 (resp. n > 0) and by b− (resp. b+ ) the Borel Lie sub-algebra of vir generated by the Ln ’s, n ≤ 0 (resp. n ≥ 0) and c. If g is any Lie algebra, we denote by U(g) its universal enveloping algebra. Then a representation of g is the same as a left U(g)-module. Let us describe representations of vir by starting with the simplest ones, which we call positive energy representations. These are representations whose underlying space M splits as a direct sum M = m≥0 Mm of finite-dimensional subspaces such that Ln maps Mm to Mm−n for any m, n ∈ Z (with the convention that Mm ≡ {0} for m < 0) and L0 is diagonalizable on each Mm . If M has positive energy, we can define the contravariant representation of vir whose underlying space is the little graded dual of M , which we define as M ∗ ≡ ∗ ∗ m≥0 Mm , where Mm is the standard algebraic dual of the finite-dimensional space Mm . Observe that one can view Ln acting on M as a collection of linear maps Ln : Mm → Mm−n indexed by m. For each of these maps, one can take the ∗ ∗ algebraic transpose t Ln : Mm−n → Mm , defined (as usual for finite-dimensional t ∗ . We define Ln acting on spaces) by Ln y, x ≡ y, Ln x for (x, y) ∈ Mm × Mm−n ∗ t ∗ ∗ M by the collection L−n : Mm → Mm−n . We decide that c is the multiplication by the same scalar on M ∗ as on M . The representation property is checked by a simple computation. Note that M ∗∗ is canonically isomorphic to M as a virmodule. The most important examples of positive energy representations are highest weight modules and their contravariants. A vir highest weight module M is a representation of the Virasoro algebra which contains a vector v such that (i) Cv is a one-dimensional representation of h and is annihilated by n+ and (ii) the smallest sub-representation of M containing v is M itself, i.e., all states in M can be obtained by linear combinations of strings of generators of vir acting on v. Because Cv is a one-dimensional representation of b+ , all states in M can be obtained by linear combinations of strings of generators of n− acting on v. On such a representation, the generator c acts on M as multiplication 4 Triangular would be more accurate, but we keep this definition by analogy with finitedimensional Lie algebras.
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by a scalar, which we denote by c again and call the central charge. The number h such that L0 v = hv is called the conformal weight of the representation. One can write M = m≥0 Mm where L0 acts on Mm by multiplication by h + m, M0 = Cv and Mm is finite-dimensional with dimension at most p(m), the number of partitions of m. For convenience, we define Mm ≡ {0} for m < 0. Then Ln maps Mm to Mm−n for any m, n ∈ Z. By construction, highest weight cyclic modules have positive energy. The existence of highest weight modules for given c and h is ensured by a universal construction using induced representation. Let R(c, h) denote the onedimensional representation of h, of central charge c and conformal weight h. View R(c, h) as a representation of b+ where n+ act trivially. This turns R(c, h) into a left U(b+ )-module. For any g, U(g) acts on itself on the left and on the right, so by restriction, we can view U(vir)
as a left U(vir)-module and as a right U(b+ )module. Then V (c, h) ≡ U(vir) U (b+ ) R(c, h) is a left U(vir)-module, called the Verma module with parameters (c, h). As a U(n− )-module, V (c, h) is isomorphic to U(n− ) itself, so the number of states in V (c, h)n is exactly p(n). Any highest weight cyclic module M with parameters (c, h) is a quotient of V (c, h). The contravariant M ∗ of a highest weight module is not always highest weight: U(vir)M0∗ is always irreducible, hence is a proper submodule of M ∗ if M is not irreducible.
3.3
Completions
In the following, we shall often need to deal with infinite linear combinations of Virasoro generators. For instance, formally T (z) = n Ln z −n−2 . So we make some new definitions. We denote by n+ the formal completion of n+ which is made of arbitrary (not necessarily finite) linear combinations of Ln ’s, n > 0. The Lie algebra structure on n+ extends to a Lie algebra structure on n+ if we define
m>0
am L m ,
n>0
bn Ln ≡
k>0
(m − n)am bn Lk .
m>0,n>0, m+n=k
As usual with formal power series, this works because for fixed k, the sum m>0,n>0, is a finite sum. m+n=k
We can go one step further and define vir+ as the direct sum n+ ⊕ b− , which is still a Lie algebra with the obvious definition. One can make analogous definitions for n− , b+ , b− , n− ⊕ b+ . All these Lie algebras are contained in n− ⊕ h ⊕ n+ , but we shall not (!) try to put a Lie algebra structure on that space. Note that vir, n− , n+ , b− and b+ are graded Lie algebras, so their universal enveloping algebras are graded too (the grading should not be confused with the
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filtration which exists for any Lie algebra). We denote by U(vir)n , U(n− )n , U(n+ )n , U(b− )n and U(b+ )n the subspace of degree n in eachof the corresponding algebras. Using the grading, one checks that U(n+ ) ≡ n>0 U(n+ )n , the formal completion5 of U(n+ ) has a natural associative algebra structure which extends that of degrees U(n+ ). Actually, U(n+ ) is made of formal series of monomials of arbitrary in the Ln ’s, n > 0. One can make analogous remarks for U(n− ) ≡ n<0 U(n− )n . If M is a positive energy representation, its formal completion M = m Mm is still a vir-module, though not a finite energy one. Any positive energy representation M of vir is also a representation of vir+ = n+ ⊕ b− and a U(n+ )-module, whereas M is a representation of vir− = n− ⊕ b+ and a U(n− )-module.
4 Finite deformations in conformal field theory Suppose now that H is a domain of the type represented on Fig. (1), that is mapped to H by some conformal transformation f .
H Figure 1: A typical hull geometry.
We are going to show that just as an infinitesimal deformation is described by the insertion of an element of the Virasoro algebra, the finite deformation that leads from the conformal field theory on H to that on H can be represented by an operator Gf implementing the map f : ˆ · · · Gf |Ω. · · · ϕ(z, ˆ · · · ϕ(z, z) · · · ψ(x) · · ·H = Ω|G−1 z) · · · ψ(x) f r
This relates correlation functions in H to correlation functions in H where the field arguments are taken at the same point (!) but sandwiched inside a conjugation by Gf .
4.1
Finite deformations around 0
Let N+ be the space of power series of the form z + m≥1 fm z m+1 which have a non-vanishing radius of convergence. With words, N+ is a subset of the space O0 of germs of holomorphic functions at the origin, consisting of the germs which fix the origin and whose derivative at the origin is 1. In physical applications, we 5 indexed by Following standard practice, if I is a set and Ei , i ∈ I a family of vector spaces I, i Ei is the set theoretic product of the Ei , whereas ⊕i Ei is the subspace of i Ei consisting of families with only a finite number of nonzero components.
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shall only need the case when the coefficients are real. But in certain intermediate constructions, it will be useful to consider the fm ’s as independent commuting indeterminates (so that we forget about convergence and deal with formal power series): the following statements can be translated in a straightforward way to deal with this more general situation. As a set, N+ is convex. Moreover, N+ is a group for composition. Our aim is to construct a group (anti)-isomorphism from N+ with composition onto a subset N+ ⊂ U(n+ ) with the associative algebra product. The possibility to do that essentially boils down to the fact that n+ is nilpotent. We let N+ act on O0 by γf · F ≡ F ◦ f for f ∈ N+ and F ∈ O0 . This representation is faithful. Because γg◦f = γf γg , we see by taking g = z + εv(z) for small ε that γf +εv(f ) F = γf · F + εγf · (v · F ) + o(ε), where v · F (z) ≡ v(z)F (z) is the standard action of vector fields on functions. Using the Lagrange inversion formula6 , we compute that for m ≥ 1 f (w) z m+1 = f (z)n+1 dwwm+1 , f (w)n+2 0 n≥m
so that
∂γf f (w) = γf dwwm+1 z n+1 ∂z . ∂fm f (w)n+2 0 n≥m
This system of first-order partial differential equations makes sense in U(n+ ) if we replace z n+1 ∂z by −Ln . We define a connection f (w) Am ≡ Ln dwwm+1 f (w)n+2 0 n≥m
which satisfies the zero curvature condition ∂Al ∂Ak − = [Ak , Al ]. ∂fk ∂fl
(1)
Hence we may construct Gf ∈ U(n+ ) for each f ∈ N+ by solving the system ∂Gf f (w) = −Gf Ln dwwm+1 m ≥ 1. (2) ∂fm f (w)n+2 0 n≥m
This system is guaranteed to be compatible, because the representation of N+ on O0 is well defined for finite deformations f , faithful and solves the analogous system. However, as the argument for zero curvature is instructive, we give a direct proof in Appendix A. the convention that 0 is an integration along a small contour of index 1 around the origin, with the prefactor (2iπ)−1 included, or equivalently that 0 is taking the residue at the origin, a purely algebraic operation which can be performed without a real integration. 6 With
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Once the compatibility conditions are checked, the existence and uniqueness of Gf , with the initial condition Gf =z = 1, is obvious: expansion of Gf using the grading in U(n+ ) leads to a recursive system. The group (anti)-homomorphism property is true because it is true infinitesimally and N+ is convex. As an illustration, Gf = 1 − f1 L1 +
f12 2 (L + 2L2 ) − f2 L2 + · · · . 2 1
Some useful general properties of Gf are collected in Appendix C. Observe that N+ acts by conjugation on vir+ ≡ n+ ⊕ b− . To get orientation, let us consider the action of f ∈ N+ not on functions but on vector fields. First, we extend the action of N+ on O0 by composition to Q0 , the field of fractions of O0 . A vector fields v = v(z)∂z with coefficient in Q0 (i.e., a derivations of Q0 ) acts on Q0 too. Let us consider (γf −1 .v.γ vf ≡ γf −1 .v.γf , a simple compu f )F (z). Defining tation shows that vf F (z) = v ◦ f −1 (z) f ◦ f −1 (z)F (z). So, as expected, vf −1 (z) is still a derivation, and writing v ≡ v (z)∂ , one finds v (z) = v ◦ f f f z f f ◦ f −1 (z). Lagrange inversion shows that f (w)2 vf (z) = z n+1 dwwm+1 for v(z) = z m+1 . n+2 f (w) 0 n≥m
Because of the correspondence between −z m+1 ∂z and Lm , it is not surprising that, for every m ∈ Z: f (w)2 c m+1 L G = dww Sf (w) + L dwwm+1 G−1 m f n f 12 0 f (w)n+2 0 n≥m
≡
Lm (f ).
(3)
The proof of this identity is relegated to appendix B. One can also check directly and painfully that the Lm (f )’s satisfy the Virasoro algebra commutation relation with central term c, but this is guaranteed by the fact that Lm (f ) is obtained from Lm by a conjugation. If we define a truncated stress tensor Tl (z) ≡ m≥l Lm z −m−2 , which belongs to vir+ , we have that = Lm (f )z −m−2 G−1 f Tl (z)Gf m≥l
=
n≥l
Ln
n≥m≥l
z −m−2
dwwm+1 0
f (w)2 f (w)n+2
c −m−2 + z dwwm+1 Sf (w). 12 0 m≥l
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Now let us try to let l → −∞. In the c-term the m summation converges to Sf (z) if z belongs to the disk of convergence of Sf (z). In the same way, for fixed n, the m summation converges to f (z)−n−2 f (z)2 if z belongs to the pointed disk of convergence of f (z)−n−2 f (z)2 . When n varies, this leads only to two constraints, namely that z belong to the pointed disc of convergence of f −1 and to the disk of convergence of f . Thus, for z in a non void pointed disk centered at the origin, the infinite summations appearing for fixed vir degree in G−1 f T (z)Gf are absolutely convergent and c 2 Sf (z). (4) G−1 f T (z)Gf = T (f (z))f (z) + 12 Hence we have an operatorial version of finite deformations that has all the expected properties. The last equation can then be extended by analytic continuation if f (z) allows it. One important lesson to draw from this computation is that, quite naturally in fact, if the Lm ’s are the basic objects and T is constructed from them, changes of coordinates act nicely only if some convergence criteria are fulfilled. Similar consideration would apply if we would consider the action of Gf on other local fields. Now that we have the stress tensor at our disposal, we can rewrite the variations of Gf in a familiar way: if f is changed to f + δf with δf = εv(f ), we find that δGf = −εGf T (z)v(z)dz. 0
If v is not just a formal power series at the origin, but a convergent one in a neighborhood of the origin, we can freely deform contours in this formula.
4.2
Finite deformations around ∞
Now, let us look at the holomorphic functions at ∞ instead of 0. So let N− be the space of power series of the form z + m≤−1 fm z m+1 which have a non-vanishing radius of convergence. We let it act on O∞ , the space of germs of holomorphic functions at infinity, by γf ·F ≡ F ◦f . The adaptation of the previous computations ∂γ f (w) n+1 shows that ∂fmf = γf n≤m ∞ dwwm+1 f (w) ∂z where ∞ is along a small n+2 z contour of index −1 with respect to the point at infinity. We transfer this relation to U(n− ) to define an (anti)-isomorphism from N− to N− ⊂ U(n− ) mapping f to Gf such that f (w) ∂Gf = −Gf Ln dwwm+1 , m ≤ −1. ∂fm f (w)n+2 ∞ n≤m
All the previous considerations can be extended to that case.
4.3
Dilatations and translations
We close this section with a small extensions that, for different reasons, demand to leave the realm of formal power series.
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The first one has to do with dilatations. Up to now, we have been dealing with deformations around 0 and ∞ that did not involve dilatation at the fixed point: f (0) or f (∞) was unity. Hence the operator L0 appears nowhere in the above formulæ. To gain some flexibility in the forthcoming discussion, we decide (this is a convention) to authorize dilatations at the origin. The operator associated to a pure dilatation f (z) = f (0)z isf (0)−L0 . One can view a general f fixing 0 as the composition f (z) = f (0)(z + m fm z m+1 ) of a deformation at 0 with derivative 1 at 0 followed by a dilatation. As before, the operators are multiplied in the opposite order, so that Gf = Gf /f (0) f (0)−L0 . From this formula, one checks that eqs.(3,4) remain valid even when f has f (0) = 1. To keep the group composition property, we demand that f (0) be real and positive. second extension deals with translations. Suppose that f (z) = f (0)(z + Them+1 ) is a generic invertible germ of holomorphic function fixing the origin m fm z (f (0) = 0). If a is in the interior of the disk of convergence of the power series expansion of f and f (a) = 0, we may define a new germ fa (z) ≡ f (a + z) − f (a) with the same properties. What is the relationship between Gf and Gfa ? At the a infinitesimal level, we compute df da |a=0 ≡ v(f ). The use of the Lagrange formula yields f (w)2 n+1 f dw , v(f ) = f (w)n+2 0 n≥0
which implies G−1 fa
dGfa da |a=0
=
−
n≥0
=
Ln
dw 0
f (w)2 f (w)n+2
L−1 f (0) − G−1 f L−1 Gf .
The last equality comes eq. (3) for m = −1. We conclude that for general a, G−1 fa
dGfa = L−1 fa (0) − G−1 fa L−1 Gfa . da
This differential equation is easy to solve formally: Gfa = e−aL−1 Gf ef (a)L−1 .
(5)
This formal solution has an analytic meaning at least as long as a is in the interior of the disk of convergence of the power series expansion of f and f (a) = 0 (extensions will require analytic continuation). This is a special case of the yet to come Wick theorem for the Virasoro algebra.
5 An application to representation theory In this section, we use the above formulæ for finite deformations to make contact with [3]. Our goal is to construct generalized coherent states representations of vir that will allow us to understand the structure of SLE martingales.
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Representations associated to deformations near 0
Suppose that M is a positive energy representation of vir. Then so is its dual ∗ M ∗ . Let f be an element of N+ . For (x, y) ∈ M × M , consider the expectation value Gf y, x or G−1 f y, x . From eq. (15) in Appendix C, these expectations are polynomial in the coefficients of f = z + m≥1 fm z m+1 . Take as M a Verma module V (c, h) and take x= 0 in the highest weight −1 ∗ space of M . Then the space { Gf y, x , y ∈ M } or { Gf y, x , y ∈ M ∗ } is the space of all polynomials in the independent variables f1 , f2 , . . . . Indeed, choose the basis of M indexed by ordered monomials in the Ln ’s with negative n, acting on the highest weight state x, and the dual basis in M ∗ . Then eq. (15) shows that when we take for the elements of the dual basis, the matrix y successively −1 elements Gf y, x or Gf y, x enumerate a basis of the space of polynomials in f1 , f2 , . . . . So we have two linear isomorphisms from M ∗ to q[f1 , f2 , . . . ] where q is the preferred field of the reader (Q is a minimal choice), and we can use these isomorphisms to transport the action of vir. 5.1.1 The case of Gf For y ∈ M ∗ , define Py ≡ Gf y, x. We are going to give formulæ for PLn y as a first-order differential operator acting on Py . The case when n ≥ 1 is simple. Indeed, using formula (2) for the partial derivatives of Gf , one checks that −
m≥n
0
dz
f (z)n+1 ∂ Gf = Gf Ln . z m+2 ∂fm
So for n ≥ 1, PLn y = −
m≥n
0
dz
f (z)n+1 ∂Py . z m+2 ∂fm
(6)
To deal with n < 1, we write Gf Ln = (Gf Ln G−1 f )Gf and use that −1 −1 −1 Gf Ln Gf ∈ n+ ⊕ b− to decompose Gf Ln Gf = (Gf Ln Gf )n+ + (Gf Ln G−1 f )b− . From eq. (3) for the compositional inverse of f , we get after a change of variable c f (w)n+1 n+1 Sf (w) + n ∈ Z. = − dwf (w) L dw m+2 Gf Ln G−1 n f 12 0 f (w) w f (w) 0 m≥n
The b− part contains the central charge term and the sum n ≤ m ≤ 0. For m < 0, Lm Gf y, x = Gf y, L−m x = 0 because x is a highest weight state, and
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L0 Gf y, x = h Gf y, x because x has weight h. So (Gf Ln G−1 ) G y, x b f − f Sf (w) 1 c + h dwf (w)n+1 2 dwf (w)n+1 = − Py . 12 0 f (w) w f (w) 0 −1 To deal with the n+ part, we observe that G−1 f (Gf Ln Gf )n+ Gf belongs to n+ but −1 −1 −1 on the other hand Gf (Gf Ln Gf )n+ Gf = Ln − Gf (Gf Ln G−1 f )b− Gf . Hence −1 −1 −1 G−1 f (Gf Ln Gf )n+ Gf = −(Gf (Gf Ln Gf )b− Gf )n+ ,
n < 1.
For the second conjugation, we use eq. (3) for f itself. This leads to PLn y +
c 12
f (w)n+1 Py w2 f (w) 0 0 0 f (z)2 f (w)n+1 =− dw m+2 dzz m+1 Gf Ll y, x . (7) w f (w) f (z)l+2 0 m=n 0
dwf (w)n+1
Sf (w) −h f (w)
dw
l≥1
One can express the right-hand side of this formula as an explicit differential operator. The details are tedious and best relegated to Appendix D. The final result is that, for n < 1, c f (w)n+1 n+1 Sf (w) − h dw 2 dwf (w) PLn y + Py 12 0 f (w) w f (w) 0 0 f (w)n+1 (fj−m (j − m + 1) = dw m+2 w f (w) j≥1 m=n 0 0 um+1 f (u)2 f (v)k+1 ∂Py − du dv j+2 . (8) f (u)k+2 v ∂fj 0 0 k=m
Eqs. (6,8) give the desired representation of the action of the Virasoro algebra on V ∗ (c, h) as first-order differential operators on the space q[f1 , f2 , . . . ]. To be explicit, we quote the expression for a system of generators of vir: ∂ L2 = − fj fk fl ∂fm m≥2 j+k+l=m−2 ∂ L1 = − fj fk ∂fm m≥1
L0 = h +
m≥1
j+k=m−1
mfm
∂ ∂fm
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L−1 = −2f1 h +
((m + 2)fm+1 − 2f1 (m + 1)fm )
m≥1
305
∂ ∂fm
L−2 = −(f2 /2 − f1 /12 − f12 /3)c − (4f2 − 7f12 )h + the differential part . For the positive generators, the convention f0 = 1, fn = 0 n < 0 is used within the sums. 5.1.2 The case of G−1 f For y ∈ M ∗ , define Qy ≡ G−1 f y, x . We are going to give formulæ for QLn y as a first-order differential operator acting on Qy . Note that Qy is nothing but Py expressed in terms of the coefficients of the inverse (for composition) of f . So in principle, the two constructions are related by a simple changeof variables. −1 We use eq. (3) to work on QLn y = (G−1 f Ln Gf )Gf y, x . Again, we write
−1 −1 G−1 f Ln Gf = (Gf Ln Gf )n+ +(Gf Ln Gf )b− and use the definition of contravariant representation on the b− part to keep only the diagonal action of h. This leads to
QL n y
=
f (w)2 c dwwn+1 Sf (w) + h dwwn+1 12 0 f (w)2 0 f (w)2 −1 L + dwwn+1 G y, x . m f f (w)m+2 0 m≥1
The definition of hm in eq. (17) and its characteristic property eq. (18) are in fact valid for every m ∈ Z. This allows to rewrite the linear combinations of Lm ’s as linear combinations of partial derivatives as: QL n y +
=
c 12
dww
n+1
Sf (w) + h
0
dww 0
fm−n (m − n + 1) −
l,n≤l≤0
m≥max(1,n)
(w)2 f (w)2
n+1 f
du
0
un+1 f (u)2 f (u)l+2
In particular Ln
=
(m + 1)fm
m≥0
L0
=
h+
m≥1
mfm
∂ ∂fn+m
∂ ∂fm
n≥1
(9)
Qy
dv 0
f (v)l+1 ∂Qy . v m+2 ∂fm
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L−1
=
2f1 h +
((m + 2)fm+1 − 2f1 fm )
m≥1
L−2
=
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∂ ∂fm
(f2 /2 − f1 /12 − f12 /3)c + (4f2 − f12 )h + the differential part .
Let us note that the formula for the action of the positive generators Ln , n ≥ 1 is strikingly similar to the one that arises in matrix models [8]. 5.1.3 Representation theoretic remarks By definition, a (non trivial) highest weight vector x of a Verma module V (c, h) generates V (c, h) when acted on by the Virasoro generators. On the other hand, the dual x∗ of x in V ∗ (c, h) generates the irreducible highest weight representation of weight (c, h) when acted on by the Virasoro generators. Hence, if (c, h) is generic, i.e., if the Verma module V (c, h) is irreducible, then so is V ∗ (c, h) and they are equivalent as vir modules. However, if (c, h) is non generic, x∗ generates only a proper subspace of V ∗ (c, h). For instance, suppose that c = (6−κ)(8κ−3) and h = 6−κ 2κ 2κ for some κ. Then κ 2 V (c, h) is not irreducible, (−2L−2 + 2 L−1 )x is a singular vector in V (c, h), annihilated by the Ln ’s, n ≥ 1, so that it does not couple to any descendant of x∗ . How does this show up in the two representations on polynomials that we constructed? To keep consistent notations, denote by Pn (resp. Qn ) the operator differential −1 −1 such that Gf Ln y, x = Pn Gf y, x (resp. Gf Ln y, x = Qn Gf y, x ) for y ∈ V ∗ (c, h). If y is a descendant of x∗ , κ 2 G−1 L y, (−2L + )x = 0. −2 f 2 −1 On the other hand, by copying for Pn , n ≥ 1, the argument leading to the formula −1 one checks that for n ≥ 1 Ln G−1 y, x = −P y, x . We conclude that G n f f all the polynomials in f1 , f2 , . . . obtained by acting repeatedly on the polynomial 1 with the Qm ’s (they build the irreducible representation with highest weight (c, h)) are annihilated by 2P2 + κ2 P12 . For generic κ there is no other singular vector in V (c, h), and this leads to a satisfactory description of the irreducible representation of highest weight (h, c): the representation space is given by the kernel of an explicit differential operator acting on q[f1 , f2 , . . . ], and the states are build by repeated action of explicit differential operators on the highest weight state 1. The same argument would apply to general singular vectors.
5.2
Representations associated to deformations near ∞
The presentation parallels quite closely the case of deformations around 0 so we shall not give all the details. All arguments can be adapted straightforwardly.
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Again, M and its dual M ∗ are supposed to be positive energy representation ∗ of vir. But now we take f in N− . For (x, y) ∈ M × M , consider the expectation value y, Gf x or y, G−1 f x
7
. As for the deformations around 0, these expecta tions are polynomial in the coefficients of f = z + m≤−1 fm z m+1 . As M , take a Verma module V (c, h) and take x = 0 in the highest weight ∗ space of M . The space { y, Gf x , y ∈ M ∗ } or { y, G−1 f x , y ∈ M } is the space of all polynomials in the independent variables f−1 , f−2 , . . . . So we have two linear isomorphisms from M ∗ to q[f−1 , f−2 , . . . ] and we can use these isomorphisms to transport the action of vir.
5.2.1 The case of Gf For y ∈ M ∗ , define Ry ≡ y, Gf x. We give formulæ for RLn y as a first-order differential operator acting on Ry . We write Ln y, Gf x = y, L−n Gf x and conjugate to obtain RLn y
c = 12
dzz ∞
1−n
Sf (z)Ry +
m≤−n
dzz 1−n ∞
f (z)2 y, Gf Lm x , f (z)m+2
where ∞ is around a small contour of index −1 with respect to the point at infinity. Using the highest weight property of x we get RLn y −
c 12
dzz 1−n Sf (z) + h
∞
f (z)2 dzz 1−n Ry f (z)2 ∞ f (z)2 = dzz 1−n y, Gf Lm x . f (z)m+2 ∞
m≤−1
As in the previous sections, we may express the right-hand side as an explicit differential operator. Define, for n ∈ Z, in (z) ≡ z
1−n
f (z) −
1−m
f (z)
m, n≤m≤0
du ∞
u1−n f (u)2 , f (u)2−m
which has the property that in (z) = O(1) and ∞
dzz 1−n
f (z)2 = f (z)m+2
dz ∞
in (z)f (z) f (z)m+2
for
m = −1, −2, . . . .
neither Gf x nor G−1 f x is a finite excitation of x in general, the matrix elements −1 y, Gf x and y, Gf x are well defined because y ∈ M ∗ is by definition a finite excitation. 7 Though
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The z expansion reads in (z) = z m+1 (fm+n (m + n + 1) m≤−1
−
du
u
∞
l, n≤l≤0
1−n
2
f (u) f (u)2−l
1−l
dv ∞
f (v) . v m+2
This leads to the formula 2 c 1−n 1−n f (w) RLn y = dww Sf (w) + h dww Qy 12 ∞ f (w)2 ∞ − (fm+n (m + n + 1) m≤min(−1,−n)
−
du
l, n≤l≤0
u
1−n
∞
2
f (u) f (u)2−l
1−l
dv ∞
f (v) ∂Ry , (10) v m+2 ∂fm
which yields Ln
=
−
(m + 1)fm
m≤0
L0
=
h−
mfm
m≤−1
L−1
=
−2f−1 h −
∂ ∂fm−n
∂ ∂fm mfm−1 −
m≤−1
L−2
=
n≥1
fk fl + 2f−1 fm
k+l=m−1
∂ ∂fm
2 3f−1 )
−cf−2 /2 − h(4f−2 − (m − 1)fm−2 − − m≤−1
fj fk fl
j+k+l=m−2
+ 3f−1
k+l=m−1
5.2.2 The case of G−1 f
fk fl + (4f−2 −
2 3f−1 )fm
∂ . ∂fm
y, G−1 x . We give formulæ for SLn y as a first-order f −1 differential operator acting on Sy . We write Ln y, G−1 x = y, L G x . The −n f f case n ≥ 1 is easy. From f (z)1−n ∂ −1 dz m+2 G−1 f = L−n Gf z ∂f m ∞
For y ∈ M ∗ , define Sy ≡
m≤−n
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we infer that SL n y =
m≤−n
dz
∞
f (z)1−n ∂Sy z m+2 ∂fm
309
n ≥ 1.
In particular SL 1 y
=
SL 2 y
=
∂Sy ∂f−1 m≤−2
dz ∞
1 ∂Sy . m+2 f (z)z ∂fm
The study of the case n < 1 follows closely the discussion in Section 5.1.1. As it plays no role in the application to SLE we leave the computation to the reader. 5.2.3 Application to SLE martingales We assume that c = (6−κ)(8κ−3) and h = 6−κ 2κ 2κ for some κ. Then V (c, h) is not κ 2 irreducible, (−2L−2 + 2 L−1 )x is a singular vector in V (c, h), annihilated by the Ln ’s, n ≥ 1, so that it does not couple to any descendant of x∗ , the dual of x. The descendants of x∗ in V ∗ (c, h) generate the irreducible highest weight representation of weight (c, h). We denote by Rn (resp. Sn ) the differential operator such that −1 ∗ Ln y, Gf x = Rn y, Gf x (resp. Ln y, G−1 f x = Sn y, Gf x ) for y ∈ V (c, h). Now for n ≥ 1, y, Gf L−n x = −Sn y, Gf x. If y is a descendant of x∗ , κ y, Gf (−2L−2 + L2−1 )x = 0 2
All the polynomials in f−1 , f−2 , . . . obtained by acting repeatedly on the polynomial 1 with the Rm ’s (they build the irreducible representation with highest weight (c, h)) are annihilated by 2S2 + κ2 S12 . For generic κ there is no other singular vector in V (c, h), and this leads to a satisfactory description of the irreducible representation of highest weight (h, c): the representation space is given by the kernel of an explicit differential operator acting on q[f−1 , f−2 , . . . ], and the states are build by repeated action of explicit differential operators (the Rm ’s) on the highest weight state 1. We are now in position to rephrase the main results of [3] in the language of this paper. If we take f = kt , the coefficients f−1 , f−2 , . . . of f become random variables (for instance f−1 is simply a Brownian motion of covariance κ). One can show (see [3] for details) that for fixed t the coefficients f−1 , f−2 , . . . seen as functions over the Wiener sample space are algebraically independent. So the above computation can be interpreted as follows: the space of polynomials of the coefficients of the expansion of kt at ∞ for SLEκ can be endowed with a Virasoro module structure isomorphic to V ∗ (cκ , hκ ). Within that space, the subspace of martingales is a submodule isomorphic to the irreducible highest weight representation of weight (cκ , hκ ).
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6 “Wick’s theorem” for the Virasoro algebra Up to now, we have only dealt with finite deformations close to 0 or ∞. These are the most natural points for radial quantization in conformal field theory. However, this is not always convenient. A typical situation is as depicted in Fig. (2).
Figure 2: A typical two hulls geometry.
We want to evaluate correlation of operators in a geometry where the natural series at 0 or at ∞ either do not exist at all, or do not converge at the location of the insertions.
6.1
Basic commutative diagram
In this situation, we may obtain a uniformizing map fA∪B by first removing B by fB , which is regular around ∞ and such that fB (z) = z + O(1) at infinity, then A˜ ≡ fB (A) by fA˜ which is regular around 0 and fixes 0 (as mentioned before, fA˜ (0) = 1 is allowed). Suppose that B is included in an open ball of radius r and A˜ is included in the complement of a closed ball of radius R, both centered at the origin. Now choose z such that |z| > r but |fB (z)| < R 8 . For such z’s, first the composition fA∪B (z) = f ˜ ◦ fB (z) can be computed by inserting the A −1 series expansions, and second G−1 f ˜ GfB T (z)GfB GfA˜ is well defined, given by A
c absolutely convergent series, and is equal to T (fA∪B (z))fA∪B (z)2 + 12 SfA∪B (z). Of course, the roles of A and B could be interchanged, and we could first ˜ ≡ fA (B) by f ˜ remove A by fA which is regular around 0 and fixes 0 and then B B which is regular around ∞ and such that fB˜ (z) = z + O(1). As they uniformize the same domain, we know that fA˜ ◦ fB and fB˜ ◦ fA differ by a (real) linear fractional transformation: there is an h ∈ P SL2 (R) such that fB˜ ◦ fA = h ◦ fA˜ ◦ fB . Suppose that fA and fB are given. There is some freedom in the choice of fA˜ and fB˜ : namely we can replace fA˜ by h0 ◦ fA˜ where h0 is a linear fractional transformation fixing 0, and fB˜ by h∞ ◦ fB˜ where h∞ is a linear
8 Such z’s exist in the above geometry, for instance in a small neighborhood of the segment of the real axis that separates A and B. In such a region, radial ordering is also preserved by the maps.
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fractional transformation such that h∞ (z) = z +O(1) at infinity, i.e., a translation. A simple computation shows that unless there is a z such that fA (z) = ∞ and fB (z) = 0, there is a unique choice of fA˜ and fB˜ such that fB˜ ◦ fA = fA˜ ◦ fB . This commutative diagram was introduced in ref. [16]. In the sequel, we shall concentrate on this generic situation. So we deduce that for z’s in some open
Figure 3: The generic commutative diagram. −1 −1 −1 set, G−1 ˜ . As the modes Ln of T fA˜ GfB T (z)GfB GfA˜ = GfB˜ GfA T (z)GfA GfB generate all states in a highest weight representation, the operators GfB GfA˜ and GfA GfB˜ have to be proportional: they differ at most by a factor involving the central charge c. We write GfB GfA˜ = Z(A, B) GfA GfB˜ , or
−1 G−1 fA GfB = Z(A, B) GfB˜ Gf ˜ . A
(11)
As implicit in the notation, Z(A, B) depends only on A and B: a simple computation shows that it is invariant if fA is replaced by h0 ◦ fA and fB by h∞ ◦ fB . Formula (11) plays for the Virasoro algebra the role that Wick’s theorem plays for collections of harmonic oscillators. We call Z(A, B) a partition function for the following reason: we can write −1 ˆ Ω G−1 GfB GfA˜ Ω fA˜ GfB · · · T (z) · · · r 1 ˆ Ω G−1 G−1 GfB GfA˜ Ω . = fB fA · · · T (z) · · · ˜ Z(A, B) r But |Ω is annihilated by b+ and Ω| is annihilated by n− so 1 ˆ (z) · · · GfB Ω , · · · T (z) · · ·HA∪B = · · · T Ω G−1 fA Z(A, B) r and
Z(A, B) = Ω G−1 G . f BΩ fA
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Computation of the partition function
The computation of Z(A, B) goes along the following lines. If one changes A by a small amount, the variation of fA can be written as δfA = vA (fA ). In order to keep the initial properties of A and B, we impose that vA is a vector field holomorphic in the full plane but for cuts along the real axis with positive discontinuities, satisfies the Schwarz reflexion principle (vA (z) = vA (z)), and is such that the ˜ We may open disk of convergence of its power series expansion at 0 contains B. µA (x)dx with µA (x) positive. write vA in terms of its discontinuities as vA (z) = z−x Similar considerations hold if B is distorted slightly, we write δfB = vB (fB ) and vB satisfies corresponding conditions. Then we know that −1 −1 δ(G−1 G ) = v (u)T (u)duG G − G G vB (v)T (v)dv A fA fB fA fB fA fB 0 ∞ −1 = Z(A, B) vA (u)T (u)duGfB˜ G−1 − G G v (v)T (v)dv . f B ˜ f˜ f˜ B A
0
A
∞
By hypothesis, we can deform the small contour around 0 to a contour in a region where vA and fB˜ have a convergent expansion, and the small contour around ∞ to a contour in a region where vB and fA˜ have a convergent expansion. Then we may conjugate, with the result δ(G−1 fA GfB ) Z(A, B)
= −
c GfB˜ ( vA (u)(T (fB˜ (u))fB˜ (u)2 + SfB˜ (u))du 12 c vB (v)(T (fA˜ (v))fA˜ (v)2 + SfA˜ (v))dv)G−1 fA˜ . 12
Taking the vacuum expectation value yields c vA (u)SfB˜ (u)du − vB (v)SfA˜ (v)dv . δ log Z(A, B) = 12 The explicit value of log Z(A, B) can be computed by means of several formulæ. The most symmetrical ones are obtained if A and B are both described by integrating infinitesimal deformations of H. Consider two families of hulls, As and Bt that interpolate between the trivial hull and A or B respectively. We arrange that fAs and fBt are generic, so that unique fAs,t and fBt,s exist, which satisfy fBt,s ◦ fAs = fAs,t ◦ fBt . ∂f ∂f Define vector fields by vAs and vBt by ∂sAs = vAs (fAs ) and ∂tBt = vBt (fBt ). Now set As,t = fBt (As ) and Bt,s = fAs (Bt ), and define vector fields vAs,t and vBt,s by
∂fAs,t ∂s
= vAs,t (fAs,t ) and σ L(Aσ , Bτ ) ≡ − ds 0
∂fBt,s ∂t
0
τ
= vBt,s (fBt,s ). Set dt dw dz vAs,t (w) Γw
Γz
6 vB (z) (z − w)4 t,s
(12)
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where the contours Γw and Γz are simple contours in C of index 1 with respect to 0, such that the bounded component of C\Γz contains the cuts of fB−1 , the t,s bounded component of C\Γw contains Γz and the unbounded component contains the cuts of fA−1 as described on Fig. (4). We observe that the kernel is a four order s,t pole, i.e., is proportional to the two-point correlation function for the stress energy tensor in the plane geometry. Moreover, by writing vAs,t and vBt,s as integrals of their (positive) discontinuities one sees that L(A, B) is positive. We claim that c Z(Aσ , Bτ ) = exp L(Aσ , Bτ ). 12
Γw
cut for A s,t
cut for B t,s
Γz O
Figure 4: Integration contours intrication. This formula is very symmetrical, but it does not make clear that log Z(Aσ , Bτ ) really depends only on Aσ and Bτ , not on the full trajectories As , s ≤ σ and Bt , t ≤ τ . The following steps are also useful to show that eq. (12) has the correct variational derivative. We start by the change of variable z = fAs,t (ζ), which is valid for z in a simply connected neighborhood of Γw containing the origin, hence on Γz . Taking the t-derivative of fBt,s ◦ fAs = fAs,t ◦ fBt , we obtain vBt,s (fAs,t (ζ)) = ∂fA
(ζ)
∂fAs,t (ζ) + fA s,t (ζ)vBt (ζ). ∂t
s,t is a holomorphic function of ζ in a neighborhood of the origin conBut ∂t taining the ζ integration contour, so in eq. (12) we may replace vBt,s (z)dz by fA s,t (ζ)vBt (ζ)fA s,t (ζ)dζ.
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Hence
L(Aσ , Bτ ) =
−
0
−
dt 0
σ
dw
−1 fA (Γz )
Γw
dt 0
dζ vAs,t (w)
s,t
τ
ds 0
τ
ds
=
σ
−1 fA (Γz )
Ann. Henri Poincar´e
6fA s,t (ζ)2 (fAs,t (ζ) − w)4
dζ vA (fAs,t (ζ))fA s,t (ζ)2 vBt (ζ) . s,t
vBt (ζ) (13)
s,t
In the second line, the w integral has been computed by the residue formula. This is legitimate because, by hypothesis, vAs,t (w) is holomorphic in the bounded component of C\Γw . We can now make use of a useful identity for the variations of the Schwarzian derivative. From its definition one checks that S(f + εv(f ))(f ) = εv (f ) + O(ε2 ). Combined with the cocycle property S(f + εv(f ))(z)dz 2 = S(f + εv(f ))(f )df 2 + S(f )(z)dz 2 this yields d S(f + εv(f ))(z)|ε=0 = v (f (z))f (z)2 . dε Finally
L(Aσ , Bτ )
= −
0
dt 0
dt
τ
τ
ds
= −
σ
0
−1 fA (Γz ) s,t
−1 fA (Γz )
dζ
d SfAs,t (ζ)vBt (ζ) ds
dζ SfAσ,t (ζ)vBt (ζ).
(14)
s,t
The roles of Aσ and Bτ could be interchanged to remove the Γw and t integrations, leading to σ ds dw vAs (w)SfBτ,s (w) L(Aσ , Bτ ) = 0 τ dt dz vBt (z)SfAσ,t (z). = − 0
Using these formulæ, it is apparent that L(Aσ , Bτ ) does not depend on the detailed way the hulls are built: only the final hulls count. It is also clear that setting c L(Aσ , Bτ ) A ≡ Aσ , A ∪ δA = Aσ+dσ , B ≡ Bτ , B ∪ δB = Bτ +dτ , the variation of 12 is exactly the one of log Z(A, B). So we have proved σ c log Z(Aσ , Bτ ) = ds dw vAs (w)SfBτ,s (w) 12 0 τ c dt dz vBt (z)SfAσ,t (z). = − 12 0 We present two examples in Appendix E. The quantity Z(A, B) has a remarkable interpretation [24] in terms of the Brownian loop-soup [17]: it is the probability that no loop of the soup intersects both A and B, with λ = −c the parameter of the soup.
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Factorization of unity and Virasoro vertex operators
Consider a hull A whose closure contains neither 0 nor ∞. There is a one parameter family of maps uniformizing the complement of A in H and which are regular both at the origin and at infinity. Let us pick one of them, which we call fA (z). Since fA (z) is regular at the origin, we may implement it in conformal field theory by GA+ fA (0)−L0 with GA+ in N+ . Alternatively, since it is also regular at infinity, we may implement it by GA− fA (∞)−L0 with GA− ∈ N− . The product VA ≡ GA− fA (∞)−L0 fA (0)L0 G−1 A+ is the Virasoro analogue of what vertex operators of dual or string models are for the Heisenberg or the affine Kac-Moody algebras. It does not depend on the representative one chooses in the one parameter family. This product is well defined and non trivial in positive energy representation. It may be thought of as the factorization of the identity since the conformal transformation it implements is the composition of two inverse conformal maps.
7 The partition function martingale We now come to the application that has motivated most of our investment in the explicit implementation of conformal transformations. For the convenience of the reader, we start with a quick reminder of [1] phrased in a more rigorous setting. Remember that cκ = (6−κ)(8κ−3) and hκ = 6−κ 2κ 2κ . The Verma module V (cκ , hκ ) is κ 2 not irreducible, and (−2L−2 + 2 L−1 ) acting on the highest weight state is another highest weight generating a sub-representation. We quotient V (cκ , hκ ) by this subrepresentation and denote by |ω the highest weight state in the quotient. Then (−2L−2 + κ2 L2−1 )|ω = 0.
7.1
Ito’s formula for Gkt
The maps ft and kt = ft − ξt that uniformize the growing hull Kt fix the point at infinity, so that there are well-defined elements Gft , Gkt ∈ N− ⊂ U(n− ) implementing them in CFT. The maps are related by a change of the constant coefficient in the expansion around ∞, so the operators are related by Gkt = Gft eξt L−1 . 2 , the The map ft satisfies the ordinary differential equation ∂t ft (z) = ft (z)−ξ t 2 corresponding vector field being v(f ) = f −ξt whose expansion at infinity reads v(f ) = 2 m≤−2 f m+1 ξt−m−2 , so that G−1 ft dGft = −2dt
Lm ξt−m−2
m≤−2
= −2e
ξt L−1
L−2 e−ξt L−1 dt.
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ξt L−1 To get G−1 which kt dGkt it remains only to compute the Ito derivative of e κ 2 −ξt L−1 ξt L−1 de = L−1 dξt + 2 L−1 dt. Finally, reads e κ 2 G−1 kt dGkt = (−2L−2 + L−1 )dt + L−1 dξt 2 as announced in Section 2. In particular, dGkt |ω = L−1 dξt Gkt |ω, so that Gkt |ω is a (generating function of) local martingale(s).
7.2
The partition function martingale
We have given an explicit formula for Z(A, B), but motivated by the martingale generating function, we shall sandwich G−1 fA GfB not with the vacuum Ω but with another highest weight state, namely |ω. Using the Virasoro-Wick theorem, one computes that (remember that ω| is annihilated by n− , but |ω is not annihilated by b+ , the L0 part contributes) −1 hκ ω|G−1 ˜ (0) . fA GfB |ω = Z(A, B) ω|Gf ˜ |ω = Z(A, B)fA A
Observe that while the vacuum expectation value depends only on the hulls, the expectation value in a non conformally invariant state depends on the choices of fA and fB . We apply the results of Section 6 to the case when B is the growing hull Kt and A is another disjoint hull. From the previous computation we know that ω|G−1 fA Gkt |ω is a local martingale. We start from fA and ft to build a commutative diagram as before, with maps denoted by fA˜t and f˜t uniformizing respectively ft (A) and fA (Kt ), and satisfying f˜t ◦ fA = fA˜t ◦ ft . Now −1 ξt L−1 ω|G−1 |ω fA Gkt |ω = ω|GfA Gft e ξt L−1 = Z(A, Kt ) ω|G−1 |ω fA˜t e −1 = Z(A, Kt ) ω| e−ξt L−1 GfA˜t efA˜t (ξt )L−1 |ω.
From eq. (5) we know that the operator e−ξt L−1 GfA˜t efA˜t (ξt )L−1 corresponds to the map z → fA˜t (ξt + z) − fA˜t (ξt ), so that −1 ω| e−ξt L−1 GfA˜t efA˜t (ξt )L−1 |ω = fA˜t (ξt )hκ . 2 From the Loewner equation vKt (z) = z−ξ and t t 2 L(A, Kt ) = − dτ dz SfAτ (z) z − ξτ 0 t dτ SfAτ (ξτ ) . = −2 0
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Finally hκ ω|G−1 exp − ˜t (ξt ) fA Gkt |ω = fA
c 6
317
t
0
dτ SfAτ (ξτ ),
were fAτ ◦ fτ uniformizes the two hull geometry corresponding to A ∪ Kτ and fAτ is normalized to ensure the commutativity of the uniformization diagram as explained before. It should be noted that the randomness in the above formula is explicit through the appearance of ξt but also implicit through fA˜τ which is a random function. This local martingale was discovered without any recourse to representation theory by Lawler, Schramm and Werner [16], but we hope to have convinced the reader that it is nevertheless deeply rooted in CFT. For the sake of completeness, we shall give two illustration of how this machinery is used to compute explicit probabilities. The following discussion does not claim originality, as the derivations merely sketch the ones given in [16].
7.3
Restriction
We already know that ω|G−1 fA Gkt |ω is a local martingale. One can show that it is a true martingale for κ ≤ 4, let us just note that the region κ ≤ 4 is also the one for which, almost surely, the SLE hull Kt is a simple curve that avoids the real axis at all positive times. For the rest of this section assume κ ≤ 4. Suppose that A is bounded and choose a very large semi circle CR of radius R in H centered at the origin. Let τR be the first time when Kt touches either A or CR . Then τR is a stopping time. It is crucial to normalize fA correctly, and one does so by imposing that it fixes 0 (as already done) and that moreover fA (z) = z + O(1) close to ∞, which by use of the commutative diagram ensures that ensures that fA˜t (z) = z + O(1) close to ∞ as well. These three conditions fix fA completely. Then we claim that fA˜ (ξτR ) is 0 if the SLE hull hits A at τR and goes to 1 for τR
large R if the SLE hull hits CR at τR . Indeed, when the hull approaches A, one or more points on A˜t approach ξt , and at the hitting time, a bounded connected component is swallowed ξt (this uses the normalization of fA ) indicating that the derivative has to vanish there. On the other hand, if CR is hit first, then A˜τR is dwarfed so that (this uses again the normalization of fA ) fA˜τ is close to the R identity map away from A˜τR and in particular at the point ξτR . The behavior of the other factor in the martingale, Z(A, Kt ), is much harder to control, so we now restrict to κ = 8/3, which is the same as cκ = 0 because κ ≤ 4. So the partition function martingale fA˜ (ξt )h8/3 , at t = τR , is 0 if A is hit before CR and close to t 1 if the opposite is true. But the expectation of a martingale is time-independent, h8/3 = fA (0)5/8 . so that the probability that Kt does not hit A is fA˜ (ξt )|t=0 t
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7.4
Ann. Henri Poincar´e
Locality
Let us consider again the case when B is the SLE hull Kt and A another disjoint hull. We may apply the Virasoro-Wick theorem to G−1 fA Gkt to get −1 G−1 ˜t Gf . fA Gkt = Z(A, Kt ) Gk At
Here k˜t is a uniformizing map of the image of the SLE hull by fA and it defines the SLE growth in H \ A. Its lift Gk˜t in N− depending locally on kt is random. A simple computation shows that its Ito derivative is G−1 ˜t = (−2L−2 + ˜ dGk k t
κ 2 κ−6 L )f (0)2 dt + L−1 fA (0)dt + L−1 fA (0)dξt . t t 2 −1 A t 2
Hence, for κ = 6, Gk˜t is statistically equivalent to Gkt up to a time reparametrisation, dt → ds = fA (0)2 dt. This expresses the locality property of critical percot lation.
A
Proof of identity (1)
We start with the proof of eq. (1): the operators Am ≡
f (w) f (w)n+2
n≥m
Ln
0
dwwm+1
satisfy the zero curvature equation ∂Ak ∂Al − = [Ak , Al ]. ∂fk ∂fl
Integration by parts gives ∂ ∂fl
dww 0
m+1
f (w) m+1 ∂ = f (w)n+2 n + 1 ∂fl
so
1 dww =−(m+1) n+1 f (w) 0
m
∂Al ∂Ak − = (k − l) Lj ∂fk ∂fl j
dw 0
dw 0
wl+m+1 f (w)n+2
wk+l+1 . f (w)j+2
On the other hand, [Ak , Al ] =
(m − n)Lm+n
m,n
duuk+1 0
f (u) f (u)m+2
dvv l+1 0
f (v) . f (v)n+2
Split this sum in two pieces by splitting m − n = (m + 1) − (n + 1). In the sum involving m + 1 use uk k+1 f (u) (m + 1) duu = (k + 1) du . m+2 f (u) f (u)m+1 0 0
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In the sum involving n + 1 use dvv l+1
(n + 1) 0
f (v) = (l + 1) f (v)n+2
dv 0
vl , f (v)n+1
interchange the dummy variables m and n, and also u and v. This leads to [Ak , Al ] =
Lm+n
m,n
du
dv
0
0
f (v)((k + 1)uk v l+1 − (l + 1)ul v k+1 ) . f (u)m+1 f (v)n+2
Up to now, the contours in the u and v planes where independent. But if they are adjusted in such a way that |f (v)| < |f (u)|, we can fix j = m + n and sum over m to obtain f (v)((k + 1)uk v l+1 − (l + 1)ul v k+1 ) Lj du dv . [Ak , Al ] = (f (u) − f (v))f (v)j+2 0 0 j Inside the u-plane contour, the singularities of the u-integrand consist now in a simple pole at u = v, and taking the residue leads to [Ak , Al ] = (k − l)
Lj
j
dw 0
wk+l+1 ∂Al ∂Ak = − . f (w)j+2 ∂fk ∂fl
This concludes the proof.
B Proof of identity (3) We continue with the proof of eq. (3): G−1 f Lm Gf
c = 12
dww
m+1
0
Sf (w) +
n≥m
Ln
dwwm+1 0
f (w)2 f (w)n+2
m ∈ Z.
Observe that if we extend the summation over all n’s, the integrals with n < m vanish anyway. Defining Lm (f ) to be the right-hand side, one way to prove this identity could be the tedious check that both sides have the same variation when f is changed into f + εz k+1 , i.e., ∂Lm (f ) f (w) = Ll dwwk+1 , Lm (f ) k ≥ 1. ∂fk f (w)l+2 0 l≥k
This can be done, but it is simpler to consider the variation of Gf and Lm (f ) when f is changed to f + εv(f ). If v(f ) = l≥1 vl f l+1 , we know that the variation of
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Gf is −Gf
l≥1 vl Ll .
Ann. Henri Poincar´e
Now
2 f (w) = vl Ll , Ln dwwm+1 n+2 f (w) 0 n l≥1
+
vl
l≥1
((l − n)Ll+n
n
c δl+n,0 (l3 − l)) 12
dwwm+1 0
f (w)2 . f (w)n+2
term involving the central charges we sum over n, then l and get For the m+1 dww f (w)2 v (f (w)). For the remaining terms, for fixed l we replace the 0 dummy variable n by n − l, leading to f (w)2 vl (2l − n)Ln dwwm+1 , f (w)n−l+2 0 n c 12
l≥1
which is the same as f (w)2 (f (w)v (f (w)) − (n + 2)v(f (w))) Ln dwwm+1 . f (w)n+3 0 n Finally f (w)2 (f (w)v (f (w)) − (n + 2)v(f (w))) vl Ll , Lm (f ) = Ln dwwm+1 f (w)n+3 0 n l≥1 c + dwwm+1 f (w)2 v (f (w)). 12 0 It is easily seen that this is nothing but dLm (f + εv(f )) , dε |ε=0 which shows that G−1 f Lm Gf
and
c 12
dwwm+1 Sf (w) + 0
n≥m
Ln
dwwm+1 0
f (w)2 , f (w)n+2
which coincide at f (z) = z, have the same tangent map. Convexity ensures that they coincide everywhere.
C
A few properties of Gf
The expansion of Gf in powers of the fm ’s has an important property that is already apparent in the expansion above. Let I = (i1 , i2 , . . . ) be a sequence of
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non-negative integers with finitely many nonzero terms. Let Em be the sequence made of zeroes except for a single 1 in the m-th position, so that I = m im Em . We define |I| ≡ i (which we call degree), d(I) ≡ mi (which we call m m m m im and LI ≡ m Limm (with the convention that grading) I! ≡ im !, fI ≡ m fm L1 factors are on the utmost right, then L2 , and so on). Then we claim that Gf =
(−)|I| I!
I
fI (LI + lower order terms)
where “lower order terms” mean f -independent linear combinations of LJ ’s with |J| < |I| but d(J) = d(I). The same statement would be true if we had chosen the opposite convention to order the Lm ’s in LI . The statement that d(J) = d(I) is simply that a dilation on z multiplies Ll by λl but divides fl by the same factor. Alternatively, one can check that the factor m+1 f (w) f (w)n+2 that appears in eq. (2) is a polynomial in the fl ’s of grading 0 dww n − m. The proof that |J| < |I| is obtained by taking a commuting limit: we set fm ≡ εϕm and Λm ≡ εLm (think of ε as ). Then in the limit ε → 0 keeping the ϕm ’s fixed, on the one hand the Λm ’s commute, and on the other hand f (w) dwwm+1 f (w) n+2 = δn,m so that the differential system defining Gf reduces to 0 ∂Gf ∂ϕm
= −GΛm , with solution Gf = e−
m
ϕm Λm
. This implies that in the ε expan-
(−)|I| I I! ϕI (ΛI
+ O(ε)). But expressed in sion in terms of ϕm ’s and Λm ’s, Gf = terms of fm ’s and Lm ’s the result is ε-independent. This means that the coefficient of ϕI εk involves only ΛJ ’s with |J| = |I| − k. This concludes the proof. An analogous computation would show that G−1 f =
1 fI (LI + lower order terms). I! I
We can rephrase these results as follows: Gf
=
(−)|I| I
G−1 f
=
I!
LI (fI + higher order terms),
1 LI (fI + higher order terms), I!
(15) (16)
I
where “higher order terms” mean L-independent linear combinations of fJ ’s with |J| > |I| but d(J) = d(I). In particular the polynomials in the fm ’s that appear as coefficients of the LI ’s in the above expansions form a basis of the space of all polynomials in the fm ’s. These observations will be useful for the application to representation theory in Section 5.
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We can also write down a general recursive formula. We define PI by Gf ≡ (−)|I| m+1 f (w) I I! fI PI and combinatorial coefficients CJ (m, n) by 0 dww f (w)n+2 ≡ (−)|J| m−n dw J J! fJ CJ (m, n). The integrand can we written as w w times a function l in which each fl is multiplied by z : CJ (m, n) = 0 unless d(J) = n− m. The partial
differential equations for Gf lead to difference equations for the PI ’s. One gets PK+Em =
I+J=K
K! CJ (m, m + d(J))PI Lm+d(J) . I!J!
One finds PEm = Lm , PEm +En = Lm Ln + (n + 1)Lm+n , . . . .
D Final steps for the proof of (8) We start from eq. (7), repeated here for convenience: PLn y +
c 12
0
f (w)n+1 − h dw 2 Py w f (w) 0 0 f (z)2 f (w)n+1 =− dw m+2 dzz m+1 Gf Ll y, x . w f (w) f (z)l+2 0 m=n 0
Sf (w) dwf (w)n+1 f (w)
l≥1
(z)2 Now, fix m and concentrate on l≥1 0 dzz m+1 ff(z) l+2 Gf Ll y, x. From the m+1 Lagrange formula, one can expand z f (z) in powers of f (z) as
z
m+1
f (z) =
k+1
f (z)
0
k≥m
Define hm (z) ≡ z
f (z) −
um+1 f (u)2 . f (u)k+2
m+1
du
k+1
f (z)
du 0
k, m≤k≤0
um+1 f (u)2 . f (u)k+2
(17)
By definition, hm (z) is a O(z 2 ) and its z expansion reads hm (z) =
z
j+1
fj−m (j − m + 1) −
j≥1
0 k=m
um+1 f (u)2 du f (u)k+2 0
f (v)k+1 dv j+2 v 0
.
On the other hand, by construction, hm (z) is such that dzz 0
m+1
f (z)2 = f (z)l+2
dz 0
hm (z)f (z) f (z)l+2
for
l = 1, 2, . . . .
(18)
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so, using again eq. (2), l≥1
dzz m+1
0
=−
f (z)2 Gf Ll f (z)l+2
fj−m (j − m + 1) −
j≥1
E
0 k=m
um+1 f (u)2 du f (u)k+2 0
f (v)k+1 dv j+2 v 0
∂Gf . ∂fj
Two explicit computations
Let a and b be real positive numbers
E.1 Example 1: Two slits We define the hull Bb to be the segment ]i0, ib] and Aa the segment [ia, i∞[ in H. Assuming that 0 ≤ b < a ≤ ∞ we compute L(Aa , Bb ). We interpolate between the empty hull and Bb (resp. Aa ) by Bβ , β ∈]0, b] (resp. Aα , α ∈ [a, ∞[). To uniformize H\Bβ we take the map fBβ (z) = (z 2 +β 2 )1/2 and for H\Aα the map fAα (z) = (z −2 + α−2 )−1/2 . Observe that fBβ maps Aα to αβ Aγ where γ = (α2 − β 2 )1/2 while fAα maps Bβ to Bδ , where δ = (α2 −β 2 )1/2 . One checks that fBδ ◦ fAα = 1−b12 /a2 fAγ ◦ fBβ , so we get a commutative diagram by taking fAα,β = 1−b12 /a2 fAγ and fBβ,α = fBδ . 2
2
2
2
2
3(z +2γ ) 3(z +2(a −β )) Now SfAα,β (z) = SfAγ (z) = − 2(z 2 +γ 2 )2 so SfAa,β (z) = − 2(z 2 +a2 −β 2 )2 . On
the other hand L(Aa , Bb ) = −
d dβ fBβ
=
b
dβ 0
β fBβ
so vBβ (z) =
β z.
To resume,
dzvBβ (z)SfAa,β (z) =
b
dβ
dz
0
β 3(z 2 + 2(a2 − β 2 )) . z 2(z 2 + a2 − β 2 )2
The relevant z-integral encircles the singularity at 0 and no other, so b β L(Aa , Bb ) = 3 0 dβ a2 −β 2 . Finally 3 L(Aa , Bb ) = − log(1 − b2 /a2 ). 2
E.2 Example 2: A slit and a half-disc We keep the definitions above for Aa , Aα , α ∈ [a, ∞[) and fAα . But now Bb is the intersection of the disc of center 0 and radius b with H, and to interpolate between the empty hull and Bb we use the half-discs Bβ , β ∈]0, b]. To uniformize H\Bβ we choose the map fBβ (z) = z+β 2 /z. Observe that fBβ maps Aα to Aγ where now γ = (α2 − β 2 )/α. The Schwarzian derivative is insensitive to the precise normalization 3(z 2 +2(a2 −β 2 )2 /a2 ) of fAα,β , so we can compute it by using fAγ : SfAa,β (z) = − 2(z 2 +(a2 −β 2 )2 /a2 )2 .
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d dβ fBβ
fBβ −
"
fB 2 −4β 2 β
z−
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√
z 2 −4β 2
On the other hand = so vBβ (z) = , where the β β square root is defined to ensure the appropriate properties of vBβ : this vector field is holomorphic in H with negative imaginary part, real on the real axis away from the cut and satisfies the Schwarz reflexion principle. Hence # b z − z 2 − 4β 2 3(z 2 + 2(a2 − β 2 )2 /a2 ) L(Aa , Bb ) = dβ dz . β 2(z 2 + (a2 − β 2 )2 /a2 )2 0 The relevant z-integral encircles the cut [−2β, 2β] and no other singularity. We may compute it with the help of the residue formula, because the integrand is meromorphic in the unbounded component of the complement of the integration contour, regular at infinity but with double poles at z = ±i(a2 − β 2 )/a. The index is −1 for both, and the residue is the same as well. This leads to b dβ β 2 (β 2 + 2a2 ) L(Aa , Bb ) = 3 . a4 − β 4 0 β Finally L(Aa , Bb ) =
1 + b2 /a2 3 log . 4 (1 − b2 /a2 )3
We observe in these two examples that L(A, B) becomes singular when A and B have a contact. As expected L(A, B) is positive. Acknowledgments. We take this opportunity to warmly thank Wendelin Werner for many illuminating explanations on the probabilistic and geometric intuition motivating SLE constructions and Misha Gromov for his questions on finite conformal transformations in conformal field theory. Work supported in part by EC contract number HPRN-CT-2002-00325 of the EUCLID research training network.
References [1] M. Bauer, D. Bernard, SLE growth processes and conformal field theories, Phys. Lett. B543, 135–138 (2002). [2] M. Bauer, D. Bernard, Conformal field theories of stochastic Loewner evolutions, arXiv:hep-th/0210015, Commun. Math. Phys. 239, 493–521 (2003). [3] M. Bauer, D. Bernard, SLE martingales and the Virasoro algebra, arXiv:hepth/0301064, Phys. Lett. B557, 309–316 (2003). [4] A. Belavin, A. Polyakov, A. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B241, 333–380 (1984). [5] J. Cardy, Critical percolation in finite geometry, J. Phys. A25, L201–206 (1992).
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[6] J. Cardy, Conformal invariance and percolation, arXiv:math-ph/0103018. [7] J. Cardy, Conformal invariance in percolation, self-avoiding walks and related problems, arXiv:cond-mat/0209638. [8] F. David, Mod. Phys. Lett. A5, 1019 (1990), R. Dijkgraaf, H. Verlinde, E. Verlinde, Loop equations and Virasoro constraints in nonperturbative 2-d quantum gravity, Nucl. Phys. B348 435 (1991), V. Kazakov, Mod. Phys. Lett. A4 2125 (1989). [9] J. Dixmier, Alg`ebres enveloppantes. Gauthier-Villars, Paris 1974; V.G. Kac and A.K. Reina, Bombay lectures on highest weight representations of infinite dimensional Lie algebras, Adv. Series in Math. Phys., vol. 2, World Scientific, Singapore, 1987. [10] B. Duplantier, Conformally invariant fractals and potential theory, Phys. Rev. Lett. 84, 1363–1367 (2000). [11] B. Duplantier, Higher conformal multifractality, J. Stat. Phys. 110, 691–738 (2003). [12] R. Friedrich and W. Werner, Conformal fields, restriction properties, degenerate representations and SLE, C.R. Acad. Sci. Paris, Ser I Math, arXiv:math.PR/0209382; Conformal restriction, highest weight representations and SLE, arXiv:math-ph/0305061, to appear in Commun. Math. Phys. [13] I. Karatzas, S.E. Shreve, Brownian motion and stochastic calculus, GTM 113, Springer, (1991). [14] G. Lawler, Introduction to the Stochastic Loewner Evolution, URL http://www.math.duke.edu/∼jose/papers.html, and references therein. [15] G. Lawler, O. Schramm, W. Werner, Values of Brownian intersections exponents I: half-plane exponents, Acta Mathematica 187, 237–273 (2001), arXiv:math.PR/9911084; G. Lawler, O. Schramm, W. Werner, Values of Brownian intersections exponents II: plane exponents, Acta Mathematica 187, 275–308 (2001), arXiv:math.PR/0003156; G. Lawler, O. Schramm, W. Werner, Values of Brownian intersections exponents III: two-sided exponents, Ann. Inst. Henri Poincar´e 38, 109–123 (2002), arXiv: math.PR/0005294. [16] G. Lawler, O. Schramm, W. Werner, Conformal restriction: the chordal case, arXiv:math.PR/0209343. [17] G. Lawler, W. Werner, The Brownian loop soup, arXiv: math.PR/0304419.
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[18] B. Nienhuis, Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas, J. Stat. Phys. 34, 731–761 (1983). [19] L. Onsager, Phys. Rev. 65, 117 (1944), L. Onsager, Nuovo Cimento 6, supplement, 261 (1949), C.N. Yang, Phys. Rev. 85, 808 (1952). [20] S. Rhode, O. Schramm, Basic properties of SLE, and references therein, arXiv:math.PR/0106036. [21] O. Schramm, Israel J. Math., 118, 221–288 (2000). [22] S. Smirnov, Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits, C.R. Acad. Sci. Paris 333 239–244 (2001). [23] W. Werner, Lectures notes of the 2002 Saint Flour summer school. [24] W. Werner, private communication. [25] P.A. Wiegmann, private communication. Michel Bauer and Denis Bernard Service de Physique Th´eorique de Saclay CEA/DSM/SPhT Unit´e de recherche associ´ee au CNRS CEA-Saclay F-91191 Gif-sur-Yvette France email:
[email protected] email:
[email protected] Communicated by Vincent Rivasseau Submitted 13/06/03, accepted 21/10/03
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 327 – 346 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/020327-20 DOI 10.1007/s00023-004-0171-y
Annales Henri Poincar´ e
Field-theoretic Weyl Quantization as a Strict and Continuous Deformation Quantization Ernst Binz, Reinhard Honegger and Alfred Rieckers Abstract. For an arbitrary (possibly infinite-dimensional) pre-symplectic test function space (E, σ) the family of Weyl algebras {W(E, σ)}∈R , introduced in a previous work [1], is shown to constitute a continuous field of C*-algebras in the sense of Dixmier. Various Poisson algebras, given as abstract (Fr´echet-) *-algebras which are C*-norm-dense in W(E, 0), are constructed as domains for a Weyl quantization, which maps the classical onto the quantum mechanical Weyl elements. This kind of a quantization map is demonstrated to realize a continuous strict deformation quantization in the sense of Rieffel and Landsman. The quantization is proved to be equivariant under the automorphic actions of the full affine symplectic group. The relationship to formal field quantization in theoretical physics is discussed by suggesting a representation dependent direct field quantization in mathematically concise terms.
1 Introduction The asymptotic correspondence between classical and quantum physics is intimately connected with the name of N. Bohr, but it was P.A.M. Dirac, who first shaped this general principle into a limiting equality of the scaled quantum commutator with the classical Poisson bracket. The various concise mathematical formulations for such a limit → 0 developed much later. Initiated especially by the seminal paper [2] there has been in the last years an extensive study of various forms of the so-called deformation quantization (let us here only mention [3] with its references), which associates a quantum mechanical algebraic structure with a rather arbitrary, finite-dimensional Poisson manifold P. More precisely, one considers in this context often a Poisson algebra P of functions on P, which is not – as in traditional Hilbert space quantization – mapped into a set of selfadjoint operators, but which acquires for itself a deformed non-commutative product, replacing the usual pointwise commutative product of functions. The C*-algebraic version of deformation quantization, the so-called strict deformation quantization, (cf., e.g., [4], [5], [6] [7], [8]), is a combination of the theoretical framework of algebraic quantum theory and the deformation concept. Based on [8] we introduce in Section 2 the concepts of strict deformation quantization and its strengthened version of a continuous quantization in a slightly generalized form. There, the classical Poisson algebra P is mapped, for each value of , into a C*-algebra A . That means, that the basic principles of algebraic quantum theory remain unchanged, whereas the transition between classical and quantum
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theory is formulated in more concise terms. In the continuous (strict) quantization the family {A | ∈ I} is used to construct a continuous field of C*-algebras and especially strong continuity properties are imposed on the corresponding quantization map for the classical limit → 0. Under certain assumptions the inverse of the strict quantization map pulls back the C*-algebraic product and defines a deformed, non-commutative product in the original set of phase space functions. In this manner the connection to the deformation idea is made evident. Most examples and methods of strict deformation quantization treated in the literature are executed on the level of finite degrees of freedom, resp. of locally compact groups, avoiding general C*-algebraic concepts by the use of special Hilbert space representations. The point, in which the various ansatzes differ from each other, is mostly the choice of the classical function algebra. For certain finitedimensional Poisson manifolds, given by locally compact abelian groups, Rieffel has developed a strict deformation quantization, which starts with a rather large function algebra, comprising the almost periodic functions. He develops a mathematically concise version of the Moyal products by means of oscillatory integrals [9], [4], [5]. It is shown, that this coincides with Weyl’s famous quantization of phase space functions, cf., e.g., [8]. The work of Weaver [10] generalizes in some sense Rieffel’s construction from Rn -actions to infinite-dimensional Hilbert space actions. The image of the quantization map is there part of a special, Hilbert space dependent von Neumann algebra. The phase space, and thus the test function space given by the pre-dual, carries a non-degenerate symplectic form, expressed by the imaginary part of the scalar product. Our subsequent developments employ the very flexible and universal construction of the C*-Weyl algebra [11], [1]: To a general pre-symplectic test function space (E, σ) (a real vector space E equipped with an anti-symmetric bilinear form σ), which expresses the degrees of freedom of a physical system with possible superselection rules, we have associated for each ∈ R a C*-algebra W(E, σ). It is generated by abstract Weyl elements W (f ), f ∈ E, satisfying the Weyl relations W (f )W (g) = exp{− 2i σ(f, g)}W (f + g) , W (f )∗ = W (−f ) ,
∀f, g ∈ E .
(1.1)
The requirement that the representations of W(E, σ) reproduce all projective, unitary realizations of the vector group E, determines W(E, σ) uniquely. The appropriate C*-norm, to complete the linear hull of the W (f ), f ∈ E, is obtained by maximizing the values of the projectively positive-definite functions on E, as is shortly recapitulated in the present Section 3. The thus obtained observable algebra is such an extreme generalization of the simple Weyl algebra for non-degenerate σ (and = 0), commonly used in algebraic quantum field theory, that it covers also the case of classical fields (at value = 0 resp. σ = 0). It corresponds to the formal field quantization in so far as it constitutes the smallest mathematically sufficient structure which contains a set of basic elements. In regular Hilbert space representations the Weyl elements
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may, in fact, be replaced by the unbounded field operators in their role as basic elements, provided that additional conventions, concerning the ordering of field operator products, are introduced. For discussing quantization proper, we have to specify the classical Poisson algebra P, considered as a sub-*-algebra of W(E, 0) in our scheme. The elements of the purely algebraically introduced P are infinite series of Weyl elements, which converge in a certain Fr´echet-topology. In the more explicit representation of P in terms of phase space functions a locally convex vector space topology τ is introduced on E in Section 4, and its topological dual Eτ becomes a Poisson manifold. In this manner the Poisson algebra P can be realized as a *-algebra of differentiable functions, densely contained in the continuous almost periodic functions on Eτ . This is, in some sense, a much smaller function algebra than Rieffel’s, but it is used also for infinite-dimensional E and is directly connected with the classical field observables. Only at this stage we have completed the scenario for introducing a (strict) quantization of the Poisson manifold P ≡ Eτ . Since we have now at our disposal a family of C*-Weyl algebras W(E, σ), ∈ R, the most suggestive quantization map relates the classical Weyl elements W 0 (f ) ( = 0) with the quantum mechanical ones W (f ) ( = 0), under preservation of the test function. We investigate at first this so-called symmetrical or Weyl quantization in Section 5 in terms of a global continuous quantization. The proof of the continuity properties with respect to the Planck parameter requires more mathematical technicalities than the verification of the Dirac condition. The fact, that the latter essentially may be reduced to the Dirac condition for Weyl elements, belongs to the merits of the present approach. We find that the concise form of the field-theoretic Weyl quantization enjoys all of the desirable properties: It is not only a strict deformation quantization, but also a continuous quantization, distinguished by the norm-continuity of all products of quantized observables, and is related with a continuous field of C*-Weyl algebras; and it commutes with the affine symplectic actions in the pertinent observable algebras. In the final Section we indicate shortly the relationship to a direct Hilbert space quantization of the (smeared) fields, in order to clarify further the connection with usual quantum field theory.
2 Strict and continuous deformation quantization We introduce here a C*-algebraic version of deformation quantization as it has been developed by Rieffel and Landsman. We take for the values of the Planck parameter a subset I ⊂ R, which contains = 0, and for which I0 := I\{0} accumulates at 0. Definition 2.1 (Strict Deformation Quantization) A strict quantization (A , Q )∈I of the (complex) Poisson algebra (P, {., .}) consists for each value ∈ I of a linear, *-preserving quantization map Q : P → A , where A is a
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C*-algebra with norm . , such that Q0 is the identical embedding of P into A0 , and such that for all A, B ∈ P the following conditions are satisfied: (a) [Dirac’s Condition] The -scaled commutator [X, Y ] := proaches the Poisson bracket as → 0:
i (XY
− Y X) ap-
lim [Q (A), Q (B)] − Q ({A, B}) = 0 .
→0
(b) [von Neumann’s Condition] In the limit → 0 one has the asymptotic product homomorphy: lim Q (A)Q (B) − Q (AB) = 0 .
→0
(c) [Rieffel’s Condition] I → Q (A) is continuous. The strict quantization (A , Q )∈I is called a strict deformation quantization, if Q is injective and Q (P) is closed with respect to the product of A for each ∈ I0 (the latter is equivalent for Q (P) being a sub-*-algebra of A ). The basic condition is Dirac’s condition, which is part of every physically relevant quantization prescription in one form or the other. The C*-algebraic framework has been proved valuable especially for systems with infinitely many degrees of freedom. Von Neumann’s condition is independent from Dirac’s condition in virtue of a different scaling. Rieffel’s condition reinforces the smoothness demand. Note that we did not include the richness condition that the *-algebraic span of Q (P) be dense in A , since the latter is always obtainable by restricting A to the smallest sub-C*-algebra containing Q (P). A strict (deformation) quantization of a Poisson algebra is, of course, nonunique. In the case of a strict deformation quantization one may stay, if one wishes to, in the space of classical observables P equipping it with the deformed, in general non-commutative product A· B := Q−1 (Q (A)Q (B)) for A, B ∈ P. This renders P into a *-algebra with product · , which is *-algebraically isomorphic to the image Q (P), a strategy, which has acquired much attention in the literature. Even stronger continuity conditions, which may be of interest in quantum field theory, are expressible by means of continuous fields of C*-algebras in the sense of J. Dixmier [12, Chapter 10]. We present the pertinent notions and results in a way, adapted to our context of -dependent quantization. For our subset I ⊆ R we denote by ∈I A the cartesian product of the family of C*-algebras A , ∈ I, which may be considered as a bundle over the base manifold I. The elements K of ∈I A are then sections I → K() ∈ A , which we write explicitly as [ → K()] ∈ ∈I A . If the *-algebraic operations (scalar multiplication, addition, product, *-operation) are taken pointwise, then the cartesian product ∈I A becomes a *-algebra. Clearly, for each ∈ I the point evaluation α defined by α (K) := K() is a *-algebraic homomorphism from ∈I A onto A .
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Definition 2.2 (Continuous Field of C*-Algebras) A continuous field of C*algebras ({A }∈I , K) consists of a sub-*-algebra K of ∈I A satisfying: (a) I → K() is continuous for all K ∈ K. This ensure the notion of continuous sections for the elements of K. (b) For each ∈ I the set {K() | K ∈ K} is dense in A . (c) Let K ∈ ∈I A . If for each 0 ∈ I and each ε > 0 there exists a H ∈ K and a neighborhood U0 of 0 so that K() − H() < ε ∀ ∈ U0 , then K ∈ K. If K = [ → K()] ∈ K and u : I → C is continuous, then it follows that [ → u()K()] ∈ K. Moreover, one has always {K() | K ∈ K} = A which strengthens (b). The next result from [12] is essential for the construction of continuous fields of C*-algebras (cf. Subsection 5.1). Lemma 2.3 Let D be a sub-*-algebra of ∈I A such that (a) and (b) of the above Definition are fulfilled (with K replaced by D). Then there exists a unique continuous field of C*-algebras ({A }∈I , K) with K such that D ⊆ K. Moreover, K consists of those K ∈ ∈I A which satisfy: For each 0 ∈ I and each ε > 0 there is an H ∈ D and a neighborhood U0 of 0 so that K() − H() < ε ∀ ∈ U0 . In order to associate a C*-algebra with a given continuous field of C*-algebras K, Dixmier restricts himself to the continuous sections K ∈ K for which I → K() vanishes at infinity. This sub-*-algebra K∞ becomes a C*-algebra if one introduces the C*-norm Ksup := sup∈I K() . There is, however, a larger C*algebra naturally associated with K and more fitting to our Weyl quantization, namely the C*-algebra Kb of the bounded continuous sections K ∈ K, i.e., with Ksup < ∞. In our notion of a continuous quantization we do not, however, specify such a global C*-algebra and work instead directly with the continuous field of C*algebras. (This generalizes slightly [8, Definition II.1.2.5], where K∞ is preselected). Definition 2.4 (Continuous Quantization) Let be given a Poisson algebra (P,{.,.}), a continuous field of C*-algebras ({A }∈I , K), and a linear, *-preserving map Q : P → K. Then ({A }∈I , K; Q) is called a continuous quantization of (P, {., .}), if the following conditions are valid: P ⊆ A0 , and α0 (Q(A)) = A for all A ∈ P, and furthermore, Dirac’s condition is fulfilled for Q := α ◦ Q. Q is denoted a global quantization map. If ({A }∈I , K; Q) is a continuous quantization, then (A , Q )∈I is a strict quantization (but in general not a strict deformation quantization). To prove this it remains only to check the validity of von Neumann’s condition: For A, B ∈ P
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put K := Q(A)Q(B) − Q(AB) ∈ K. Then, when applying the point evaluation *-homomorphisms α , one concludes that K() = Q (A)Q (B) − Q (AB). Now von Neumann’s condition follows immediately from the continuity of → K() in Definition 2.2(a). For the converse reasoning, let us mention without proof: Suppose (A , Q )∈I to be a strict quantization of the Poisson algebra P with fulfilled richness condition, then there exists a continuous quantization ({A }∈I , K; Q) of P with Q = α ◦ Q ∀ ∈ I, if and only if I → P is continuous for each polynomial P of the Q (A), A ∈ P.
3 General C*-Weyl algebras We recapitulate here some results from [1] for later use. Let be given an arbitrary pre-symplectic space (E, σ) and an ∈ R. We start from the linear hull ∆(E, σ) := LH{W (f ) | f ∈ E}
(3.1)
of linearly independent W (f ), f ∈ E, called Weyl elements. Equipped with the twisted product and the *-operation according to the Weyl relations (1.1), the linear hull ∆(E, σ) becomes a *-algebra. Its identity is given by 1 := W (0), and every W (f ) is unitary. Let C(E, σ) be the convex set of the normalized, projectively positivedefinite functions C : E → C. Normalization means C(0) = 1, and projecn tive positive-definiteness is zi zj exp{ 2i σ(fi , fj )}C(fj − fi ) ≥ 0 for arbitrary i,j=1
n ∈ N, fj ∈ E, and zj ∈ C. The latter is a generalization of the notion of positivedefinite functions, familiar from harmonic analysis. We extend each C ∈ C(E, σ) to the unique linear functional ωC on ∆(E, σ) satisfying ωC ; W (f ) = C(f ) for all f ∈ E. By the definition of C(E, σ), every state on the *-algebra ∆(E, σ) is of type ωC with a unique C ∈ C(E, σ). On the *-algebra ∆(E, σ) there exists a unique C*-norm . , given by A = sup{
ωC ; A∗ A | C ∈ C(E, σ)} ,
(3.2)
such that every representation of ∆(E, σ) is . -continuous. Consequently, if . is a further C*-norm on ∆(E, σ) with A ≤ A for all A ∈ ∆(E, σ), then . = . . The Weyl algebra W(E, σ) is the completion of the *-algebra ∆(E, σ). with respect to the C*-norm .. It is simple, if and only if σ is non-degenerate and = 0. Every representation and each state of the *-algebra ∆(E, σ) extends . continuously to the completion W(E, σ), the extension of which are denoted by the same symbol. Thus the mapping C → ωC is an affine bijection from C(E, σ) onto the state space of W(E, σ). C is called the characteristic function of the state ωC .
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Because of the linear independence of the Weyl elements W (f ), f ∈ E, we may define the vector space norm .1 on ∆(E, σ) by n n zk W (fk )1 := |zk | k=1
(3.3)
k=1
for arbitrary n ∈ N, and zk ∈ C, but different fk ’s from E. The inequality A ≤ A1 ,
∀A ∈ ∆(E, σ) ,
(3.4)
1
is immediate. The .1 -completion ∆(E, σ) of ∆(E, σ) together with the .1 continuous extension of the *-algebraic operations from ∆(E, σ) turns out to be a Banach-*-algebra. The estimation (3.4) yields that the identical map from ∆(E, σ) onto itself extends .1 -. -continuously to a mapping from ∆(E, σ)
1
1
into W(E, σ). This mapping is injective, and thus ∆(E, σ) is a sub-*-algebra of W(E, σ) consisting of those A ∈ W(E, σ) which possess the unique decom∞ zk W (fk ) with different fk ’s from E and summable coefficients position A = k=1
zk ∈ C, i.e., A1 =
∞ k=1
|zk | < ∞. 1
The Banach-*-algebra ∆(E, σ) is the twisted group algebra of the discrete vector group E with respect to the multiplier exp{− 2i σ(f, g)} occurring in the Weyl relations (1.1). The Weyl algebra W(E, σ) is its enveloping C*-algebra and thus the twisted group C*-algebra of E (cf. also [13], [11], [14], and references therein). Finally we present two additional facts concerning characteristic functions and states. The first one is proved, e.g., in [15, Theorem 3.4]. The second one is shown in [16], it generalizes the well-known fact from harmonic analysis that the product of two positive-definite functions is positive-definite, too. Proposition 3.1 The following assertions are valid: (a) A Gaussian function E f → exp{− 41 || s(f, f )}, where s is a positive symmetric R-bilinear form on E, is an element of C(E, σ), if and only if σ(f, g)2 ≤ s(f, f ) s(g, g) for all f, g ∈ E. (b) Suppose = 1 + 2 , j ∈ R. If C1 ∈ C(E, 1 σ) and C2 ∈ C(E, 2 σ), then C1 C2 ∈ C(E, σ) (pointwise product), and hence there exists a unique state ω on W(E, σ) with characteristic function ω; W (f ) = C1 (f )C2 (f ) for all f ∈ E.
4 Classical field-theoretic setup For the classical case = 0 the Weyl relations (1.1) imply the product formula W 0 (f + g) = W 0 (f )W 0 (g) = W 0 (g)W 0 (f ) for the Weyl elements W 0 (f ), f ∈ E.
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Thus the *-algebra ∆(E, 0) from Eq. (3.1), the Banach-*-algebra ∆(E, 0) , as well as the C*-Weyl algebra W(E, 0) are commutative. We are going to elaborate, how a given non-trivial pre-symplectic form σ on E gives rise to the construction of a Poisson bracket {., .} on suitable sub-*-algebras P satisfying ∆(E, 0) ⊆ P ⊆ 1
∆(E, 0) . Since the Weyl elements W 0 (f ), f ∈ E are by assumption linearly independent, the ansatz {W 0 (f ), W 0 (g)} := σ(f, g)W 0 (f + g) ,
∀f, g ∈ E ,
(4.1)
leads to a well-defined Poisson bracket {., .} on the commutative *-algebra ∆(E, 0) by means of complex bilinear extension. Let us henceforth write A = f ∈E zf W 0 (f ) instead of A = k zk W 0 (fk ) with different fk ’s. If A ∈ ∆(E, 0), then we have zf = 0 up to finitely many f ∈ E. 1 By Section 3, A = f zf W 0 (f ) ∈ ∆(E, 0) with A1 = f |zf | < ∞ and with at most countably many non-vanishing coefficients zf ∈ C. Let be given a semi-norm κ on E. Then for each n ∈ N0 the definition 1
n
Pκn := {A ∈ ∆(E, 0) | Aκ < ∞} leads to a Banach space with respect to the norm n n zf W 0 (f )κ := κ(f )m |zf | . f ∈E
Consequently, Pκ∞ :=
m=0 f ∈E
Pκn turns out to be a Fr´echet space with respect to the
n∈N
metrizable locally convex Hausdorff topology τκ arising from the increasing system 1 n 0 of norms .κ , n ∈ N. For n = 0 we re-obtain Pκ0 = ∆(E, 0) and .κ = .1 . n n ∞ Obviously, ∆(E, 0) is .κ -dense in Pκ and τκ -dense in Pκ . Conversely, for fixed n ∈ N {∞} the spaces Pκn are in inverse-order-preserving correspondence with the semi-norms κ on E. Lemma 4.1 Pκn constitutes a sub-*-algebra of the commutative Banach-*-algebra 1 ∆(E, 0) for each n ∈ N {∞}. Furthermore: (a) For each n ∈ N it holds Aκ = A∗ κ and ABκ ≤ cn Aκ Bκ for all A, B ∈ Pκn with some constant cn ≥ 1 defined in Eq. (4.2) below. (Since cn > 1 for n ≥ 2, Pκn is only a Banach-*-algebra for a norm equivalent to n .κ .) n
n
n
n
n
(b) Pκ∞ is a Fr´echet-*-algebra (product and *-operation are [jointly] τκ -continuous).
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n Proof. Anκ = A∗ nκ is immediate κ(f ) = κ(−f since ), thus0 Pκ is invariant 0 under the *-operation. Let A = f uf W (f ) and B = g vg W (g) be arbitrary 1 elements of Pκn (where uf , vg ∈ C). Then AB = f,g uf vg W 0 (f + g) ∈ ∆(E, 0) . We show AB ∈ Pκn . From the semi-norm property κ(f + g) ≤ κ(f ) + κ(g) we obtain that n n m n ABκ ≤ κ(f + g)m |uf | |vg | ≤ κ(f ) + κ(g) |uf | |vg | m=0 f,g∈E
m=0 f,g∈E
n m
m m−k k = κ(f ) |u | κ(g) |v | f g f g k m=0 k=0
n
n
(4.2)
which yields AB ∈ Pκn . Now the rest is immediate.
≤ sup{( nk ) | k = 0, 1, . . . , n} Aκ Bκ < ∞ , =: cn
In order to extend our above Poisson bracket {., .} from ∆(E, 0) to suitable *-algebras Pκn , we suppose the existence of a semi-norm ς on E such that |σ(f, g)| ≤ c ς(f ) ς(g) ,
∀f, g ∈ E ,
(4.3)
for some constant c > 0. Especially, ς has to be a norm for non-degenerate σ. Theorem 4.2 With the notations introduced above it holds that n−1
{A, B}ς
n
n
≤ c cn−1 Aς Bς ,
∀A, B ∈ ∆(E, 0) ,
∀n ∈ N .
So the Poisson bracket {., .} extends continuously to the jointly continuous mapping Pςn × Pςn −→ Pςn−1 , (A, B) −→ {A, B} . (4.4) Thus (Pς∞ , {., .}) is a Poisson algebra with jointly τς -continuous Poisson bracket {., .}. Proof. We have {A, B} = f,g σ(f, g)uf vg W 0 (f + g) ∈ ∆(E, 0) for A = f uf 0 W 0 (f ) and B = g vg W (g) from ∆(E, 0) by Eq. (4.1). Estimation (4.3) and proceeding similarly to the proof of Lemma 4.1 ensures that n−1
{A, B}ς
≤
n−1
ς(f + g)m |σ(f, g)| |uf | |vg |
m=0 f,g∈E
n−1 m
m m−k+1 k+1 ≤c ς(f ) |u | ς(g) |v | f g f g k m=0 k=0 n−1 n n ≤ c sup{ k | k = 0, 1, . . . , n − 1} Aς Bς < ∞ . = cn−1 (see Eq. (4.2)) This yields the stated results.
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The Poisson bracket {., .} may be realized indeed in terms of a bivector field. For this let be given a locally convex Hausdorff vector space topology τ on E. The topological dual Eτ is interpreted as the (flat) phase space manifold of our classical field theory. The commutative C*-Weyl algebra W(E, 0) is *-isomorphic to the C*-algebra of the almost periodic, σ(Eτ , E)-continuous functions on Eτ [1]. The Weyl elements W 0 (f ) are realized by the periodic functions W 0 (f ) : Eτ → C ,
F → exp{iF (f )} = W 0 (f )[F ] .
(4.5)
For A : Eτ → R the total differential dF A ∈ TF∗ Eτ = E, where the cotangent space TF∗ Eτ is equipped with the σ(Eτ , E)-topology, is given by dF A[G] := d dt A[F + tG]|t=0 for all G ∈ TF Eτ ≡ Eτ , F ∈ Eτ , provided existence. For a Cvalued function A on Eτ we put dF A := dF A1 + idF A2 with its real and imaginary parts A1 resp. A2 . Our above Poisson bracket {., .} now is given in terms of the constant bivector field F → ΣF arising from σ (more details are found in [17]), {A, B}[F ] = ΣF (dF A, dF B) := −σ(dF A1 , dF B1 ) − iσ(dF A1 , dF B2 ) − iσ(dF A2 , dF B1 ) + σ(dF A2 , dF B2 ) . Especially, Eq. (4.1) is reproduced for the periodic functions W 0 (f ) from Eq. (4.5). The presented Poissonian structure is independent of the chosen locally convex topology τ on E, or equivalently, from the phase space Eτ . Consequently, the only essential ingredients of the algebraized classical field theory are the presymplectic test function space (E, σ) and a semi-norm ς satisfying (4.3), the remaining C*- and Poisson algebraic structure is a functor.
5 Field-theoretic Weyl quantization We select, for given pre-symplectic space (E, σ) and semi-norm ς satisfying (4.3), the two cases P = ∆(E, 0) resp. P = Pς∞ for a Poisson algebra (P, {., .}), both being dense in the commutative, classical Weyl algebra W(E, 0). After having specified the observable algebras of the quantized systems (with possibly intrinsic classical observables, i.e., superselection rules) as the C*-Weyl algebras W(E, σ), = 0, we may now proceed to a quantization proper. Dirac’s original notion as well as its mathematical explication of a quantization in Section 2 indicate for our special case a linear, *-preserving correspondence Q : P → W(E, σ), which should display certain asymptotic properties for → 0. The most suggestive, but certainly not only, choice for Q is the prescription (5.1) Q ( k zk W 0 (fk )) := k zk W (fk ) , zk ∈ C , fk ∈ E , which is well defined because of the Weyl elements being linearly independent. We do not, however, study this quantization map directly but prefer the global point of view of a continuous quantization, which deals with all values of the Planck parameter simultaneously. A specific quantization is then gotten by fixing the value of . The demonstration of the correct → 0 asymptotics comes afterwards.
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Continuous field of C*-Weyl algebras
By means of the Weyl relations it is immediately checked that (5.2) ∆WF (E, σ) := LH{ [ → exp{−is}W (f )] | (s, f ) ∈ R × E} (R × E the cartesian product) constitutes a sub-*-algebra of ∈R W(E, σ). Lemma 5.1 The generating elements [ → exp{−is}W (f )], where (s, f ) ∈ R × E, of the *-algebra ∆WF (E, σ) are linearly independent. Proof. An arbitrary element K ∈ ∆WF (E, σ) possesses the form K=
p m
zj,l [ → exp{−isj,l }W (gj )] ,
(5.3)
j=1 l=1
where m, p ∈ N and zj,l ∈ C, and where the g1 , . . . , gm ∈ E are different, and for each j ∈ {1, . . . , m} the sj,1 , . . . , sj,p ∈ R are different. Since the Weyl elements are linearly independent, we have K = 0, if and only if 0 = l zj,l exp{−isj,l } for all ∈ R and all j = 1, . . . , m. But the maps → exp{−is}, s ∈ R, constitute an orthonormal base of the Hilbert space of almost periodic functions on R [18]. The Weyl relations imply that W (f )∗ W (g)∗ W (f )W (g) = exp{−iσ(f, g)}1 ,
∀f, g ∈ E .
Thus, the *-algebra ∆WF (E, σ) is *-algebraically generated by the sections [ → W (f )], f ∈ E, if and only if σ = 0. Theorem 5.2 (Continuous Weyl C*-Field) There exists a unique continuous field of C*-algebras ({W(E, σ)}∈R , K) such that [ → W (f )] ∈ K for each f ∈ E. Proof. Let σ = 0. We show that D := ∆WF (E, σ), which is algebraically generated by the [ → W (f )], fulfills the assumptions of Lemma 2.3. Since {K() | K ∈ ∆WF (E, σ)} = ∆(E, σ) is dense in W(E, σ) by Section 3, part (b) of Definition 2.2 is already fulfilled. We turn to part (a). We put lim sup G() := lim sup{G() | ∈ [−λ + 0 , λ + 0 ]\{0}}, for a →0
λ0
map R → G() ≥ 0, and analogously for lim inf G(). →0
Suppose to be given an arbitrary element K =
n k=1
zk [ → exp{−isk }
W (fk )] of ∆WF (E, σ) with different tuples (sk , fk ) ∈ R × E. Let EK := LHR {f1 , . . . , fn }. Then W(EK , σ) is a sub-C*-algebra of W(E, σ) [1], and it suffices to evaluate the C*-norm K() with the states on ∆(EK , σ). One easily constructs a positive symmetric R-bilinear form s on EK so that σ(f, g)2 ≤
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s(f, f ) s(g, g) for all f, g ∈ EK (the construction of such a form s may fail for infinite-dimensional E, that is the reason why we go over to EK ). By Proposition 3.1(a), Cs ∈ C(EK , ( − 0 )σ) for all ∈ R, where Cs (f ) := exp{− 41 | − 0 | s(f, f )}. By part (b) of the same Proposition there exists for every C ∈ C(EK , 0 σ) and each ∈ R a unique state ωC on W(EK , σ) with the characteristic function s ωC ; W (f ) = C(f )C (f ). Eq. (3.2) yields ; K()∗ K() = ωC
n zj zk exp{i(sj − sk + 12 σ(fj , fk ))}
j,k=1 2
× C(fk − fj )Cs (fk − fj ) ≤ K() for all ∈ R. Taking the limit → 0 we obtain for every C ∈ C(EK , 0 σ) that 0 ; K(0 )∗ K(0 ) = lim ωC ; K()∗ K() ωC →0
=
lim inf ωC ; K()∗ K() →0
2
≤ lim inf K() , →0
from which with Eq. (3.2) we get the estimation K(0 )0 ≤ lim inf K() . →0
Because of the linear independence of the Weyl elements we conclude that m wj W 0 (gj ) ∈ ∆(E, 0 σ) with wj ∈ C but different gj ’s the for arbitrary A = j=1
inverse image of α0 is given by α−1 0 (A) = {K ∈ ∆WF (E, σ) | K(0 ) = A} p zj,l exp{−i0 sj,l } = wj for j = 1, . . . , m . = K from Eq. (5.3) | l=1
Defining KA :=
m j=1
wj [ → W (gj )] ∈ α−1 0 (A), we obtain an injective linear
mapping A → KA from ∆(E, 0 σ) into ∆WF (E, σ). Thus for K ∈ α−1 0 (A) we get the estimation K() ≤ KA () + K() − KA () ≤ KA () +
p m
|zj,l | |exp{−isj,l } − exp{−i0 sj,l }| .
j=1 l=1
Since lim sup |exp{−is} − exp{−i0 s}| = 0 for any s ∈ R it follows that →0
lim sup K() ≤ lim sup KA () . →0
→0
Interchanging the roles of K and KA finally yields that the expression lim sup K() = lim sup KA () =: A0 , →0
→0
∀K ∈ α−1 0 (A) ,
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depends only on A ∈ ∆(E, 0 σ). One immediately checks that A → A0 defines a further C*-norm on ∆(E, 0 σ). Thus for each K ∈ ∆WF (E, σ) we have the estimation K(0 )0 ≤ lim inf K() ≤ lim sup K() = K(0 )0 . →0
→0
By Section 3 .0 = .0 , and consequently, → K() is continuous at 0 . For σ = 0 the Weyl algebra field is constant, and for D one may take the constant sections [ → A] with A ∈ ∆(E, 0), or A ∈ W(E, 0). It is immediate that [ → exp{−is}W (f )] ∈ Kb for each tuple (s, f ) ∈ R × E for the C*-algebra Kb of bounded continuous sections for our continuous field of C*-Weyl algebras ({W(E, σ)}∈R , K). Consequently, ∆WF (E, σ) is a sub∗ *-algebra of Kb , the .sup -closure of which is denoted by CWF (E, σ). Consider arbitrary tuples (sk , fk ) ∈ R × E and coefficients zk ∈ C satis∞ fying k |zk | < ∞. By Section 3 it holds K() := zk exp{−isk }W (fk ) ∈ k=1
1
∆(E, σ) , even if some of the tuples (sk , fk ) coincide. Hence we obtain the section K = [ → K()] =
∞
zk [ → exp{−isk }W (fk )] ∈
k=1
Put Kn () :=
n
∈R W(E, σ) .
(5.4)
zk exp{−isk }W (fk ) ∈ ∆WF (E, σ) for all n ∈ N. Then for
k=1
each ε > 0 there exists an m ∈ N such that K() − Km () ≤
∞
|zk | < ε
k=m+1
∗ uniformly for ∈ R. Definition 2.2(c) implies K ∈ K. CWF (E, σ) being complete leads to:
Lemma 5.3 K from Eq. (5.4) (with
k
∗ |zk | < ∞) is an element of CWF (E, σ).
Because of the linear independence stated in Lemma 5.1, we may introduce in our *-algebra ∆WF (E, σ) the vector space norm .1 by n n zk [ → exp{−isk }W (fk )]1 := |zk | k=1
(5.5)
k=1
for different tuples (sk , fk ) ∈ R × E and arbitrary zk ∈ C and n ∈ N. The estimation Ksup ≤ K1 , ∀K ∈ ∆WF (E, σ) , already has been established in the above proof. So, similarly as for the Weyl algebra in Section 3 we perform the .1 -completion of ∆WF (E, σ), which is de1
noted by ∆WF (E, σ) . The .1 -continuous extension of the *-algebraic operations
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1
in ∆WF (E, σ) equips ∆WF (E, σ) with the structure of a Banach-*-algebra. We 1
see that ∆WF (E, σ) consists of those sections K from Eq. (5.4), for which the tuples (sk , fk ) ∈ R × E are different, and for which K1 = k |zk | < ∞. From Lemma 5.3 we conclude that we have the inclusions 1
∗ ∆WF (E, σ) ⊆ ∆WF (E, σ) ⊆ CWF (E, σ) ⊆ Kb ,
(5.6)
in the sense of being sub-*-algebras. The identity is given by [ → 1 ], and for every (s, f ) ∈ R × E the continuous section [ → exp{−is}W (f )] is a unitary element. ∗ (E, σ) are deferred to [19]. Further investigations concerning the C*-algebra CWF
5.2
Strict deformation quantization
The continuous field of C*-Weyl algebras ({W(E, σ)}∈R , K) from the previous Subsection 5.1 leads to a definite global quantization mapping Q : ∆(E, 0) → ∆WF (E, σ), which is given by the complex linear extension of Q(W 0 (f )) := [ → W (f )] ,
∀f ∈ E .
Q is well defined and injective, since the Weyl elements W 0 (f ), f ∈ E, as well as the sections [ → exp{−is}W (f )], (s, f ) ∈ R × E, are linearly independent. It is immediately checked that Q is an isometry with respect to the two norms .1 . Thus Q extends .1 -.1 -continously to a linear, *-preserving, surjective isometry 1
1
Q : ∆(E, 0) −→ ∆WF (E, σ) .
(5.7)
By means of the point evaluation α from Section 2 for our continuous field of C*-Weyl algebras let us define for each ∈ R the quantization map 1
1
Q := α ◦ Q : ∆(E, 0) −→ ∆(E, σ) ⊆ W(E, σ) . Then, Q ( k zk W 0 (fk )) = k zk W (fk ) for k |zk | < ∞, which coincides with (5.1). The quantization map Q is a linear, *-preserving .1 -.1 -isometry from 1
1
∆(E, 0) onto ∆(E, σ) . Obviously, Q0 is just the identity for the classical case = 0. 1
Lemma 5.4 The jointly continuous bracket Pς1 × Pς1 → ∆(E, 0) , (A, B) −→ {A, B} from Theorem 4.2 (choose n = 1 in Eq. (4.4)) fulfills the Dirac condition from Definition 2.1(a) with respect to the Banach-*-algebra norms .1 on 1
∆(E, σ) respectively, i.e., lim [Q (A), Q (B)] − Q ({A, B})1 = 0 ,
→0
∀A, B ∈ Pς1 . 1
(5.8)
Furthermore, Q (P) is a .1 -dense sub-*-algebra of ∆(E, σ) and a . dense sub-*-algebra of W(E, σ) for every ∈ R.
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Proof. The Poisson bracket relations (4.1) and the Weyl relations (1.1) yield that [Q (W 0 (f )), Q (W 0 (g))] − Q ({W 0 (f ), W 0 (g)}) 1 = i exp{− 2i σ(f, g)} − exp{ 2i σ(f, g)} W (f + g) − σ(f, g)W (f + g)1 i exp{− i σ(f, g)} − 1 →0 exp{ 2 σ(f, g)} − 1 2 −→ 0 , = i − i − σ(f, g) where we used the differential limits
exp{± 2i σ(f, g)} − 1 d exp{± 2i σ(f, g)} lim = →0 d
=0
i = ± σ(f, g) . 2
The mean value theorem of differential calculus ensures exp{± i σ(f, g)} − 1 1 c 2 ≤ |σ(f, g)| ≤ ς(f ) ς(g) , 2 2 which leads for every f, g ∈ E and all 0 = ∈ R to the domination [Q (W 0 (f )), Q (W 0 (g))] − Q ({W 0 (f ), W 0 (g)}) ≤ 2c ς(f ) ς(g) . 1 0 0 1 Since for A = f uf W (f ) and B = g vg W (g) from Pς the dominant is summable, we may exchange the limit → 0 with f,g . . . by Lebesgue’s dominated convergence theorem in order to obtain Eq. (5.8). 1
∞ It remainsto prove that Q (A)Q (B) ∈ Q (Pς ) ⊆ ∆(E, 0) for A, B ∈ ∞ 0 0 Pς . For A = f uf W (f ) and B = g vg W (g), it is Q (A)Q (B) = Q (C) 1 i 0 with C ∈ ∆(E, 0) given by C := f,g uf vg exp{− 2 σ(f, g)}W (f + g). With i exp{− σ(f, g)} = 1 we conclude as in the proof of Lemma 4.1 that Cn ≤ ς n2 n cn Aς Bς for each n ∈ N, thus C ∈ Pς∞ . 1
The estimation A ≤ A1 for all A ∈ ∆(E, σ) implies the validity of Dirac’s condition for the C*-norms . . As in the above proof one may show that von Neumann’s condition holds with respect to the norms .1 for all A, B ∈ 1
∆(E, 0) , and thus for the C*-norms . . (The latter follows also automatically from our continuous field of C*-Weyl algebras by Section 2.) Summarizing we have shown the following results: Theorem 5.5 (Continuous Quantization) ({W(E, σ)}∈R , K; Q) constitutes a continuous quantization of the Poisson algebra (Pς∞ , {., .}). Theorem 5.6 (Strict Deformation Quantization) (W(E, σ), Q )∈R constitutes a strict deformation quantization of (P, {., .}), where P = ∆(E, 0) or P = Pς∞ . If the quantization maps Q resp. Q are restricted to an arbitrary sub-Poisson algebra P˜ with ∆(E, 0) ⊂ P˜ ⊂ Pς∞ (proper inclusions), then we obtain again a ˜ {., .}) continuous quantization Q, but only a strict quantization (Q )∈R of (P, ˜ (possibly Q (P) is not invariant under products, cf. Definition 2.1).
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Affine symplectic actions
Let symp(E, σ) be the symplectic group on (E, σ) (its elements T are R-linear the character group bijections on E with σ(f, g) = σ(T f, T g) ∀f, g ∈ E), and E (semidirect of the vector group E. The affine symplectic group symp(E, σ) E product) consists of pairs (T, χ) with group multiplication (T1 , χ1 ) · (T2 , χ2 ) := (T1 T2 , χ2 (χ1 ◦ T2 )). For each ∈ R there exists an automorphic action (T, χ) → αT,χ on W(E, σ) with the *-automorphisms αT,χ satisfying αT,χ (W (f )) = χ(f )W (T f ) ∀f ∈ E, [1], a combination of gauge with Bogoliubov transformations [20]. The action is .1 -isometric on the invariant sub-*-algebras ∆(E, σ) 1
and ∆(E, σ) . For the classical case = 0, α0 constitutes even a group of Poisson automorphisms on (∆(E, 0), {., .}), which in general cannot be extended to the larger Poisson algebra Pς∞ . As an immediate consequence of the construction of the quantization maps Q in the previous Subsection we get the equivariance of the strict deformation quantization from Theorem 5.6 with respect to the family of actions (α )∈R , that is, αT,χ (Q (A)) = Q (α0T,χ (A)) ,
1
∀A ∈ ∆(E, 0) .
are τ -continuous for some locally convex If T ∈ symp(E, σ) and χ ∈ E Hausdorff topology τ on E, then the dual operator T to T acts bijectively on Eτ , and there exists a G ∈ Eτ with χ(.) = exp{iG(.)} [21]. One easily deduces that the *-automorphism α0T,χ is the pullback of the affine symplectic diffeomorphism F → T F + G on the phase space manifold Eτ , i.e., α0T,χ (A)[F ] = A[T F + G] ∀F ∈ Eτ , where A ∈ W(E, 0) is considered as an almost periodic function on Eτ as in Section 4.
6 Direct field quantizations For = 0 one defines in a regular representation Π of W(E, σ) the field operators by differentiation with respect to the parameter t ∈ R [20]: d , f ∈E. (6.1) ΦΠ (f ) ≡ ΦΠ (f ) := −i Π (W (tf )) dt t=0 The unbounded, selfadjoint field operators may differ essentially from each other in the various representations of W(E, σ). In the GNS representation over an ordered state (e.g., a condensed boson state [20] or macroscopic coherent state [22]) they may even exhibit a classical part in addition to the quantum mechanical part. This clearly transcends the purely algebraic regime of the theory. Let us indicate, how our algebraic quantization theory provides, nevertheless, also a strategy for a representation dependent Weyl quantization, leading eventually to the quantization of field expressions.
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For a family Π ≡ (Π )∈R of regular, faithful representations Π of the W(E, σ) the Π-dependent quantization mappings may be defined by the linear and .1 -continuous extension of 0 QΠ (W (f )) := Π (W (f )) ,
= 0, ∀f ∈ E .
(6.2)
In virtue of the norm preservation of any faithful representation, we may transcribe the previous abstract results into the language of operator quantizations, as may be shortly indicated. In a completely analogous manner as before we introduce a continuous family ({Π (W(E, σ))}∈R , KΠ ) of represented C*-Weyl algebras and a global quantization map Qw Π : P → KΠ . Corollary 6.1 Let Π ≡ (Π ) be an arbitrary family of regular, non-degenerate, and faithful representations Π of W(E, σ), 0 = ∈ R. Then the following assertions are valid: (a) ({Π (W(E, σ))}∈R , KΠ ; Qw Π ) constitutes a continuous quantization of the Poisson algebra (P, {., .}). (b) (Π (W(E, σ)), Qw Π, )∈R constitutes a strict deformation quantization of (P, {., .}). In order to deal with the unbounded fields it is certainly desirable to make the representations from the family Π compatible with each other, e.g., by setting Π := Π≡1 ◦ β , where β is a *-isomorphism from W(E, σ) onto W(E, σ), = 0. The latter may be realized, e.g., by β (W (f )) = W 1 (T f ) for an R-linear bijection T on E with σ(T f, T g) = σ(f, g), [17]. In the almost periodic function realization one has for the classical field prodn ucts Φ0 (f1 ) . . . Φ0 (fn ) = (−i)n ∂t1∂...∂tn W 0 ( k tk fk )|tk =0 . Our presented version of the Weyl quantization suggests the extension of QΠ , = 0, from (6.2) to unbounded field polynomials by the linear extension of n 0 ∂n Π 0 0 n Π Q W ( tk fk ) Q (Φ (f1 ) . . . Φ (fn )) := (−i) ∂t1 . . . ∂tn k=1 t =...=tn =0 1 (6.3) n n ∂ n = (−i) Π W ( tk fk ) , ∂t1 . . . ∂tn k=1
t1 =...=tn =0
n ∈ N and fk ∈ E arbitrary, with the tk being real parameters. Since there are used the derivatives of the quantized (represented) Weyl operators, the quantization map for field polynomials given in Eq. (6.3) is (at least in quantum optics) still called ‘Weyl quantization’ (or ‘symmetric quantization’). There are in use, however, further quantization prescriptions for fields, here sym˜ Π (Φ0 (f )) = Φ (f ) bolized by a tilde, which satisfy the first quantization step Q Π for every f ∈ E in an unchanged manner. The quantization of a field monomial is, however, of the general form 0 0 ˜Π Q (Φ (f1 ) . . . Φ (fn )) = Pn (f1 , . . . , fn ) ,
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where the choice of the real-multilinear operator-valued Pn , n ≥ 2, determines ˜ Π . A normally ordered monomial in the creation the special kind of quantization Q and annihilation operators, introduced by means of a complex structure, is just a special real-multilinear operator expression of the fields. It is elaborated in [23], how the various operator orderings define direct field quantizations. They all lead back to a decorated Weyl quantization of the form 0 Qw Π, (W (f )) := w(, f )Π (W (f )) ,
∀f ∈ E ,
(6.4)
where the w(, f ) are certain numerical factors. With some modifications of the foregoing arguments it is shown in [23] that these are again strict and continuous deformation quantizations, which all of them refer to the described continuous field of C*-Weyl algebras and which are equivalent, in the sense of [8], to the Weyl quantization. Acknowledgment. This work has been supported by the Deutsche Forschungsgemeinschaft.
References [1] E. Binz, R. Honegger, and A. Rieckers, Construction and uniqueness of the C*-Weyl algebra over a general pre-symplectic form, Preprint Mannheim, T¨ ubingen, 2003. [2] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerovicz, and D. Sternheimer, Deformation theory and quantization, J. Oper. Th. 3, 237–269 (1980). [3] M. DeWilde and P.B.A. Lecompte, Existence of star-products and of formal deformations of a Poisson Lie algebra of arbitrary symplectic manifolds, Lett. Math. Phys. 7, 487–496 (1983). [4] M.A. Rieffel, Deformation quantization for actions of Rd , Mem. Amer. Math. Soc. 106 (1993). [5] M.A. Rieffel, Quantization and C*-algebras, In R.S. Doran, editor, C*Algebras: 1943–1993, pages 67–97. Contemp. Math. 167, Providence, RI, Amer. Math. Soc., 1994. [6] M.A. Rieffel, Questions on quantization, Berkeley, quant-ph/9712009, 1998. [7] N.P. Landsman, Strict quantization of coadjoint orbits, Amsterdam, mathph/9807027, 1998. [8] N.P. Landsman, Mathematical Topics Between Classical and Quantum Mechanics, Springer, New York, 1998.
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[9] J.E. Moyal, Quantum mechanics as a statistical theory, Proc. Cambridge Philos. Soc. 45, 99–124 (1949). [10] N. Weaver, Deformation quantization for Hilbert space actions, Commun. Math. Phys. 188, 217–232 (1997). [11] J. Manuceau, M. Sirugue, D. Testard, and A. Verbeure, The smallest C*algebra for canonical commutation relations, Commun. Math. Phys. 32, 231– 243 (1973). [12] J. Dixmier, C*-Algebras, North-Holland, Amsterdam, 1977. [13] C.M. Edwards and J.T. Lewis, Twisted group algebras I, II, Commun. Math. Phys. 13, 119–141 (1969). [14] H. Grundling, A group algebra for inductive limit groups. Continuity problems of the canonical commutation relations, Acta Appl. Math. 46, 107–145 (1997). [15] D. Petz, An Invitation to the Algebra of Canonical Commutation Relations, volume 2 of Leuven Notes in Mathematical and Theoretical Physics, Leuven University Press, Leuven Belgium, 1990. [16] R. Honegger and A. Rieckers, Partially classical states of a Boson field, Lett. Math. Phys. 64, 31–44 (2003). [17] E. Binz, R. Honegger, and A. Rieckers, Field-theoretic Weyl quantization of large Poisson algebras, Preprint Mannheim, T¨ ubingen, 2003. [18] F. Riesz and B. Sz.-Nagy, Vorlesungen u ¨ber Funktionalanalysis, Deutscher Verlag der Wissenschaften, Berlin, 1982.
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[19] E. Binz, R. Honegger, and A. Rieckers, Infinite dimensional Heisenberg group algebra and field-theoretic deformation quantization, Preprint Mannheim, T¨ ubingen, 2003. [20] O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, volume II, Springer-Verlag, New York, 1981. [21] E. Hewitt and K.A. Ross, Abstract Harmonic Analysis I, II, Springer-Verlag, New York, 1963, 1970. [22] R. Honegger and A. Rieckers, The general form of non-Fock coherent boson states, Publ. RIMS Kyoto Univ. 26, 397–417 (1990). [23] R. Honegger and A. Rieckers, Some continuous field quantizations, equivalent to the C*-Weyl quantization. to appear in Publ. RIMS Kyoto Univ., 2004.
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E. Binz, R. Honegger and A. Rieckers
Ernst Binz Institut f¨ ur Mathematik und Informatik Universit¨ at Mannheim D-68131 Mannheim Germany email:
[email protected] Reinhard Honegger Institut f¨ ur Mathematik und Informatik Universit¨ at Mannheim D-68131 Mannheim Germany and Institut f¨ ur Theoretische Physik Universit¨ at T¨ ubingen D-72076 T¨ ubingen Germany email:
[email protected] Alfred Rieckers Institut f¨ ur Theoretische Physik Universit¨ at T¨ ubingen D-72076 T¨ ubingen Germany email:
[email protected] Communicated by Joel Feldman Submitted 07/10/03, accepted 07/11/03
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e
Ann. Henri Poincar´e 5 (2004) 347 – 379 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/020347-33 DOI 10.1007/s00023-004-0172-x
Annales Henri Poincar´ e
Density of States and Thouless Formula for Random Unitary Band Matrices Alain Joye Abstract.We study the density of states measure for some class of random unitary band matrices and prove a Thouless formula relating it to the associated Lyapunov exponent. This class of random matrices appears in the study of the dynamical stability of certain quantum systems and can be considered as a unitary version of the Anderson model. It is also related with orthogonal polynomials on the unit circle. We further determine the support of the density of states measure and provide a condition ensuring it possesses an analytic density.
1 Introduction The stability of quantum dynamical systems generated by time periodic Hamiltonians is sometimes characterized by means of the spectral properties of the corresponding unitary evolution operator over a period, also called monodromy operator, see [Be, Ho1, Co3]. Unfortunately, even for this relatively simple timedependence, except for certain specific models, e.g., [Co2, DF, Bo], it is rarely the case that one has enough information about the actual monodromy operator so that a complete spectral analysis can be performed. Therefore, one resorts to different approximation techniques in some specific regimes to say something about the spectrum. For example, KAM inspired techniques, see, e.g., [Be, Co1, DS, ADE, DLSV, GY], or adiabatic related approaches, see, e.g., [Ho2, Ho3, Ho4, N1, J, N2], have been used to tackle this problem. In case the complexity of the monodromy operator is important enough to forbid of a complete description of it, one may resort to a statistical modelization. It is the case in particular in the study of the quantum dynamics of electrons confined to a ring threaded by a time-dependent magnetic flux, see, e.g., the paper [BB] and references therein. A modelization of this dynamics by means of an effective random monodromy operator taking into account the details of the metallic structure of the ring is considered and tested numerically in [BB]. We refer the reader to this paper and [BHJ] for a more detailed account of the construction of the monodromy operator. Motivated by this approach, the spectral analysis of a class of random and deterministic unitary operators, which contains the above monodromy operator, is performed in [BHJ]. The main characteristics of these unitaries is that, when expressed as matrices in some basis, they display a band structure: more precisely they are five-diagonal. The coefficients of the matrix are determined by an infinite
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set of triples {rk , αk , θk }k∈Z , where rk ’s are reflection coefficients in ]0, 1[ and αk ’s and θk ’s are phases. For example, in the statistical modelization of the physical situation mentioned above, the phases are considered as random, whereas the reflection coefficients are deterministic. While the construction of the set of unitaries studied in [BHJ] is patterned after the above-mentioned physical model, we believe it contains sufficiently many parameters to be useful for a wider class of problems. Another motivation in favor of the spectral analysis of such unitary operators stems from the recent paper [CMV] where it is shown that certain infinite matrices associated with the construction of orthonormal polynomials on the unit circle display the same five-diagonal structure as our set of monodromy operators. Under certain conditions, these matrices define unitary operators which actually form a subset of those considered in [BHJ]. The authors of [CMV] show that these matrices are to orthogonal polynomials with respect to a measure on the circle what Jacobi matrices are to orthogonal polynomials with respect to a measure on the real line. Orthogonal polynomials on the circle are determined by an infinite set of complex numbers {ak }k∈N such o recurrence that |ak | < 1, called reflection coefficients, through the so-called Szeg¨ relations, see, e.g., [G] or [BGHN]. And indeed, we will see that |ak | = rk , for all k ∈ N. Therefore, once given the expression of the five-diagonal matrix in terms of these reflection coefficients, the orthogonality measure on the circle coincides with the spectral measure of the corresponding unitary operator. These operators are further shown in [CMV] to be unitarily equivalent to unitary operators introduced almost ten years ago in [GT] for the study of the same orthonormal polynomials on the unit circle. The matrix form of the latter operators displays a different structure, namely that of a Hessenberg matrix: it has zero coefficients for indices i, j when i ≥ j − 1 only. Although more complicated, this structure can allow for operator theoretical approaches of orthogonal polynomial on the circle as, e.g., in [GT] or [GNV]. Note in particular that in [GT], properties of random polynomials defined by means of random reflection coefficients ak are investigated through the corresponding random unitary operator, whereas some of the perturbative analyses performed in [GNV] and [BHJ] bear strong resemblance. Nevertheless, we emphasize that the operators under consideration in [BHJ] and the present paper are more general than those constructed in [GT] and [CMV] and therefore their spectral analysis is richer. In particular in the random case, the way randomness appears in the coefficients of the matrix elements may lead to different characteristics of the spectral measure due to the availability of one more random variable. The goal of the present paper is to pursue the analysis of such random unitaries in the random setting considered in the paper [BHJ]: the phases (αk , θk ) are random variables and the reflection coefficients rk are all set to r ∈]0, 1[. This means that the phases of the matrix elements of the five-diagonal operators are random whereas the deterministic moduli depend on the parameter r only. Hence, if the phases are all set to zero, what we will call the “free case”, the unitary
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operator depends on the reflection coefficient r ∈]0, 1[. Note that, specializing to the (random) orthogonal polynomials setting, this means we consider cases with |ak | = r for all k’s whereas the argument of the ak ’s are random. Also the free case is linked to the so-called Geronimus polynomials, constructed by means of constant (complex) reflection coefficients ak = a ∈ C, for all k. However, while the analysis of [BHJ] focused on spectral issues, i.e., proving singularity of the almost sure spectrum by means of a unitary version of the Ishii-Pastur theorem and the positivity of the Lyapunov exponent obtained via Furstenberg’s Theorem, the main object of the present study is the density of states measure and its links with the corresponding Lyapunov exponent. The Lyapunov exponent here is of course characterizing the asymptotic behavior of generalized eigenvectors of the unitary operator. More precisely, expressing the density of states as the density of eigenvalues of a series of unitary operators restricted to “boxes”, we are able to state this relation as what is known as a Thouless formula. This formula allows to compute the Lyapunov exponent by means of the density of states and to recover the a.c. component of the density of states measure by means of a derivative of the Lyapunov exponent. A consequence of our version of Thouless formula is the extension of some results of [BHJ] providing, in particular, an explicit value of the Lyapunov exponent in these cases. We also prove the validity of the Thouless formula for the deterministic free case, by explicit computations of the relevant quantities. When applied to the orthogonal polynomials setting, the existence of the density of states measure can be expressed as the determination of a sequence of random polynomials with a distribution of zeros converging to a measure whose support is the support of the orthogonality measure, almost surely. These polynomials are associated with the random orthogonal polynomials, but they do not coincide with them as the zeros of the former are, by construction, on the unit circle whereas those of the latter lie strictly in the unit disk. Such polynomials are also constructed in [GT] by suitable truncations of the Hessenberg matrix considered. Our Thouless formula relates the potential of the density of states measure, see, e.g., [SaT], [StT] for these notions, with the Lyapunov exponent. Actually, the Lyapunov exponent is essentially the limit of the potentials of the distributions of zero of the random polynomials mentioned above and the density of states is the equilibrium measure in the external field given by the Lyapunov exponent, see below. The existence of the limit almost surely is a consequence of the ergodic properties of the phase distributions. Let us also note here that a Thouless formula is proven for the unitary random operator studied in [GT]. The Lyapunov exponent there characterizes the asymptotics of the difference equation corresponding to the Szeg¨o relations associated with random complex ak ’s. In the second part of the paper, we further assume that some natural linear combination of the original phases {ηk } are i.i.d. random variables, in order to take advantage of the analogy of our unitary matrices with the one-dimensional discrete Schr¨odinger operator. In that case, we characterize the support of the density of states in terms of that of the distribution of the ηk ’s. Finally, we provide
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an effective criterion ensuring analyticity of the integrated density of states in terms of the exponential decay rate of the Fourier coefficients of the distribution of these phases. This result relies on some kind of propagation estimates for the free evolution. The above-mentioned assumption on the phases makes (αk , θk ) correlated random variables. In particular, in the orthogonal polynomials language, this means that when the phase of each reflection coefficient ak (of constant modulus) is given by a sum of k i.i.d. random phases, the almost sure support of the random orthogonality measure can be determined. The plan of the paper is as follows. Section 2 is devoted to the definition of the model and its basic properties. In particular, the link with the constructions of [CMV] to describe orthogonal polynomials on the unit circle is recalled there. The density of states is introduced in the next section and Thouless formula is proven in Section 4. The statements about the support of the density of states and its analyticity properties are made in Section 5, whereas an appendix contains some technical items. The main results will be expressed in the general framework described above. We shall content ourselves with commenting on their translation in the orthogonal polynomial language, where appropriate, except in Section 4 where a little bit more material about potential theory is provided.
2 The model We present here the unitary matrices we will be concerned with and recall some of their basic properties to be used later. The unitary operator we consider has the following explicit form in the canonical basis {ϕk }k∈Z of l2 (Z) Uω ϕ2k
Uω ϕ2k+1
ω
ω
= irte−iη2k ϕ2k−1 + r2 e−iη2k ϕ2k ω ω + irte−iη2k+1 ϕ2k+1 − t2 e−iη2k+1 ϕ2k+2 ω
ω
= −t2 e−iη2k ϕ2k−1 + itre−iη2k ϕ2k ω ω + r2 e−iη2k+1 ϕ2k+1 + irte−iη2k+1 ϕ2k+2 ,
(2.1)
for any k ∈ Z. According to [BHJ], the random phases {ηkω }k∈Z are functions of some physically relevant i.i.d. random variables {(θkω , αω k )}k∈Z on the torus given by ω ω + αω (2.2) ηkω = θkω + θk−1 k − αk−1 , for all k ∈ Z and the coefficients r, t ∈]0, 1[ are interpreted as reflection and transition coefficients linked by r2 + t2 = 1. We will identify the operator and its matrix representation (2.1). Let us recall that these parameters are assumed to be different from their extreme values 0 and 1, because in case r = 1 ⇐⇒ t = 0 the operator Uω is diagonal and if r = 0 ⇐⇒ t = 1, it is unitarily equivalent to the direct sum of two shifts. Let us finally mention that our Uω is a particular case
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of the construction in Section 2 of [BHJ] that we briefly recall below, in order to make contact with the matrices considered in [CMV].
2.1
Link with orthogonal polynomials
Consider the set of 2 × 2 unitary matrices defined for any k ∈ Z by rk e−iαk itk −iθk Sk = e , itk rk eiαk
(2.3)
parameterized by αk , θk in the torus T and the real parameters tk , rk , the reflection and transition coefficients, linked by rk2 + t2k = 1. Then, let Pj be the orthogonal projector on the span of ϕj , ϕj+1 in l2 (Z), and let us introduce Ue , Uo two 2 × 2 block diagonal unitary operators on l2 (Z) defined by Ue = P2k S2k P2k and Uo = P2k+1 S2k+1 P2k+1 . (2.4) k∈Z
k∈Z
In matrix representation in the canonical basis, . .. S−2 Ue = S0 S2
..
(2.5)
.
and similarly for Uo , with S2k+1 in place of S2k . Note that the 2 × 2 blocks in Ue are shifted by one with respect to those of Uo along the diagonal. The unitary operator (2.6) U = Uo Ue coincides with (2.1) in case tk = t ⇐⇒ rk = r, for any k ∈ Z. Actually, a supplementary phase factor appears in the off-diagonal elements of all Sk ’s in the original definition of [BHJ]. We omit it here, as this phase is shown to be irrelevant in the spectral analysis of U , see Lemma 3.2 in [BHJ]. Without entering into the details, orthogonal polynomials on the unit circle with respect to a measure µ are determined
∞by a set of ak ’s such that |ak | < 1 for all k ∈ N, and we shall assume that k=0 |ak | = ∞, which is equivalent to saying that the corresponding Hessenberg matrix is the matrix representation of a unitary operator, [GT], Lemma 2.2. The equivalent five-diagonal matrix F of [CMV] described below is unitary as well. This matrix is constructed in the same way as (2.6) is, by means of blocks of the type (2.3) for k ≥ 0 of the form iγk −i(π/2−γk ) 2 −|a |a |e 1 − |a | |e i 1 − |ak |2 k k k = −i Θk = 1 − |ak |2 |ak |e−iγk i 1 − |ak |2 |ak |ei(π/2−γk ) (2.7)
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where ak = |ak |eiγk , see Section 3 of [CMV]. This corresponds to the particular choices (2.8) θk = π/2, αk = π/2 − γk , rk = |ak |. The definition of F is supplemented by particular “boundary conditions” at zero of the type (3.5) described below, as it is infinite in one direction only. One of the main properties of the matrix F shown in [CMV] is that the determinant of its principal n×n submatrices coincides with the n-th (monic) orthogonal polynomial, as is also true for the corresponding Hessenberg matrix. This property makes the analogy between Jacobi matrices and such F matrices all the more striking. Note that despite the fact that the above matrix is infinite in one direction only whereas ours is infinite in both directions, a “duplication procedure” described in Section 3 of [BHJ] allows to go from the former to the latter case modulo a finite rank perturbation. Hence claims about the spectrum of the doubly infinite matrix also hold for the previous matrix, modulo Birman-Krein’s theorem on finite rank perturbations and multiplicity considerations. From now on, we shall stick to doubly infinite matrices and we further make the choice rk = r ∈]0, 1[, for all k ∈ Z.
2.2
Ergodic properties
More precisely, let us introduce a probabilistic space (Ω, F , P), where Ω is identified with {TZ }, T being the torus, and P = ⊗k∈Z Pk , where P2k = P0 and P2k+1 = P1 for any k ∈ Z are probability distributions on T and F the σ-algebra generated by the cylinders. We introduce a set of random vectors on (Ω, F , P) given by βk = (θk , αk ) : Ω → T2 , k ∈ Z, θkω = ω2k , αω k = ω2k+1 .
(2.9)
The random vectors {βk }k∈Z are thus i.i.d on T2 . We denote by Uω the random unitary operator corresponding to the random infinite matrix (2.1). In analogy with Jacobi matrices describing the discrete Schr¨ odinger equation, we will also denote the vector ϕk by the site k, k ∈ Z. Introducing the shift operator S on Ω by S(ω)k = ωk+2 , k ∈ Z,
(2.10)
we get an ergodic set {S j }j∈Z of translations. With the unitary operator Vj defined on the canonical basis of l2 (Z) by Vj ϕk = ϕk−2j , ∀k ∈ Z,
(2.11)
we observe that for any j ∈ Z US j ω = Vj Uω Vj∗ .
(2.12)
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Therefore, our random operator Uω is a an ergodic unitary operator. Now, general arguments on the properties of the spectral resolution of ergodic operators Eω (∆), where ∆ is a Borel set of the torus T, ensure that this projector is weakly measurable, as well as Eωx (∆) = Pωx Eω (∆), where x = p.p., a.c. and s.c., denote the pure point, absolutely continuous and singular continuous components, see [CL], chapter V. The analysis performed in [BHJ] for the case where {(θkω , αω k )}k∈Z are uniformly distributed on the torus shows that the a.c. component of the spectrum of Uω is almost surely empty.
2.3
Lyapunov exponent
Let us proceed by recalling some facts concerning the Lyapunov exponent. It is shown in [BB] and [BHJ] that generalized eigenvectors defined by Uω ψ = eiλ ψ, ψ= ck ϕk , ck ∈ C, λ ∈ C
(2.13)
k∈Z
in our unitary setting can be computed by means of 2 × 2 transfer matrices due to the structure of the matrix Uω . They are such that for all k ∈ Z, ([BHJ]) c2k−2 c2k = T (k) (2.14) c2k+1 c2k−1 where the randomness lies in the phases ηk (λ) ≡ ηkω (λ) defined by ηk (λ) = ηk + λ,
(2.15)
and T (k)11 T (k)12 T (k)21 T (k)22
= −e−iη2k−1 (λ) (2.16)
r −iη2k−1 (λ) = i e −1 t
r = i ei(η2k (λ)−η2k−1 (λ)) − e−iη2k−1 (λ) t
1 r2 = − 2 eiη2k (λ) + 2 ei(η2k (λ)−η2k−1 (λ)) + 1 − e−iη2k−1 (λ) . t t
Note the properties T (k) ≡ T (η2k (λ), η2k−1 (λ)) i(η2k −η2k−1 )
(2.17)
is independent of λ. whereas det T (k) = e Therefore, knowing, e.g., the coefficients (c0 , c1 ), we compute for any k ∈ N, c0 c0 c2k = T (k) . . . T (2)T (1) ≡ Φ(k) c2k+1 c1 c1 c−2k c0 c = T (−k + 1)−1 . . . T (−1)−1 T (0)−1 ≡ Φ(−k) 0 . (2.18) c−2k+1 c1 c1
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The dynamical system at hand being ergodic and the determinant of the transfer matrices being of modulus one, we get the existence of a deterministic Lyapunov exponent γ(eiλ ), for any λ ∈ C, such that 1 ln Φ(k) = γ(eiλ ) a.s. k→±∞ |k| lim
(2.19)
Writing eiλ = z ∈ C\{0}, we also know from classical arguments, see, e.g., [CFKS], that γ is a subharmonic function of z.
3 Density of states Following the standard approach in the self-adjoint case, we start by a definition of the density of states by averaging over the phases and invoking the Riesz-Markov theorem. Then we relate the density of states with alternative definitions in terms of the density of eigenvalues of truncations of the original operator to l2 ([M, N ]), as N − M → ∞. Definition. The density of states is the (non-random) measure dk on T defined by f (eiλ )dk(λ) := E[ ϕ0 |f (Uω )ϕ0 + ϕ1 |f (Uω )ϕ1 ]/2, (3.1) T
for any continuous function f : S 1 → C. The average over the ϕ0 and ϕ1 matrix elements is motivated by the forms of the matrix (2.1) and shift (2.10). Note also that this definition makes dk a probability measure. Now we turn to the definition of appropriate finite size unitary matrices constructed from (2.1). There are several possible constructions suited to our purpose. Those we use below result from considering Uω provided with boundary conditions at certain sites forbidding transitions through these sites, in the more general definition (2.6) with variable reflection and transition coefficients. More precisely, such a boundary condition at site N corresponds to imposing tN = 0 whereas all other tk ’s are kept equal to their common value t. Therefore, one immediately gets that the matrix takes a block structure which decouples the sites with indices smaller than N from those with indices larger than N . Let us drop temporarily the sub- and super-scripts ω in the notation. Fix N ∈ Z and consider the unitary operator U 2N on l2 (Z) obtained from the original operator U by imposing the following boundary conditions at the sites 2N . Let / {2N, 2N + 1} where U 2N be defined by (2.1) for k ∈ η2N −1 = η2N = η2N +1 = η2N +2 = 0
(3.2)
and, for k ∈ {2N, 2N + 1} U 2N ϕ2N = itϕ2N −1 + rϕ2N U 2N ϕ2N +1 = rϕ2N +1 + itϕ2N +2 .
(3.3)
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Similarly, a boundary condition imposed at site 2N + 1 defines U 2N +1 by (2.1) for k∈ / {2N, 2N + 1, 2N + 2, 2N + 3} where η2N +1 = η2N +2 = 0
(3.4)
and, for k ∈ {2N, 2N + 1, 2N + 2, 2N + 3} U 2N +1 ϕ2N = irte−iη2N ϕ2N −1 + r2 e−iη2N ϕ2N + itϕ2N +1 U 2N +1 ϕ2N +1 = −t2 e−iη2N ϕ2N −1 + irte−iη2N ϕ2N + rϕ2N +1 U 2N +1 ϕ2N +2 = rϕ2N +2 + irte−iη2N +3 ϕ2N +3 − t2 e−iη2N +3 ϕ2N +4 U 2N +1 ϕ2N +3 = +itϕ2N +2 + r2 e−iη2N +3 ϕ2N +3 + irte−iη2N +3 ϕ2N +4 .
(3.5)
For any M ∈ Z, the corresponding operator U M has a the block structure mentioned above and it is unitary. Then, given (M, N ) ∈ Z2 such that M + 4 < N , one defines a unitary matrix U M,N on l2 (Z) by imposing boundary conditions at sites M and N . By construction, U M,N contains an isolated (N − M ) × (N − M ) unitary block on l2 ([M + 1, N ]) we denote by V M,N . Remark. In the definition of the boundary conditions, we put some phases equal to zero around the sites 2N and 2N + 1, in order to avoid having to deal with random boundary conditions later. We could have set them equal to any other value, without changing the main properties of the construction. Introducing the characteristic function χM,N of the set [M + 1, N ] ∈ Z, we denote by the same symbol the projector on the sites [M + 1, N ], corresponding to the multiplication operator by χM,N . Therefore V M,N = χM,N U M,N = U M,N χM,N = χM,N U M,N χM,N . We now consider two measures related to finite matrices as follows. ˜ M,N on T are defined by Definitions. The measures dkM,N and dk f (eiλ )dkM,N (λ) := tr (f (V M,N ))/(N − M ) T ˜ M,N (λ) := tr (χM,N f (U )χM,N )/(N − M ), f (eiλ )dk
(3.6)
(3.7) (3.8)
T
for any continuous function f : S 1 → C. Notice that dkM,N is nothing but the counting measure on T associated with the ˜ M,N is associated with the projection spectrum of the finite block V M,N , and dk of U on [M + 1, N ]. This former operator is unitary whereas the latter is not. We denote the trace norm by · 1 and first show a slight generalization of [GT] allowing to get Lemma 3.1 With the above notations, assume
(U M,N − U )χM,N 1 = o(N − N ), as N − M → ∞,
(3.9)
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then 1 tr (f (V M,N )) − tr (χM,N f (U )χM,N ) = 0. N −M→∞ N − M lim
(3.10)
Remark. The hypothesis is satisfied in particular if Rank(U M,N − U ) < ∞ and uniformly bounded in (N, M ), as is the case with the definitions of U M,N above by means of (3.3, 3.5). Proof. We first note that it is enough to consider functions which are polynomials in z and z¯, z ∈ S 1 . Any f ∈ C(S 1 ) can be approximated by trigonometric polynomials
R PR = j=−R gj eij· in such a way that if > 0 is given, there exists R( ) < ∞ so that (3.11) sup f (θ) − PR() (θ) ≤ . θ∈T
Hence we get using (3.6), tr (f (V M,N ) − χM,N f (U )χM,N ) = tr (χM,N (f (U M,N ) − f (U ))χM,N ) = tr (χM,N (PR() (U M,N ) − PR() (U ))χM,N ) +tr (χM,N ((f − PR() )(U M,N ) − (f − PR() )(U ))χM,N ),
(3.12)
where the trace norm of the last term is bounded by 2 (N −M ), so that it becomes negligeable when divided by (N − M ). We are thus to consider z s and z¯s , with s ∈ N. We can write for any s ≥ 1 U s − (U N,M )s =
s−1
U j (U − U N,M )(U N,M )s−j−1 ,
(3.13)
j=0
so that χM,N (U s − (U N,M )s )χM,N =
s−1
χM,N U j (U − U N,M )χM,N (U N,M )s−j−1 . (3.14)
j=0
Therefore, tr (χM,N (U s − (U N,M )s )χM,N ) s (U − U N,M )χM,N 1 ≤ . N −M N −M
(3.15)
The same result is true if s < 0, with all unitaries replaced by their adjoints. Thus, −R( ) ≤ s ≤ R( ) and the hypothesis on the trace norm of (U − U N,M )χM,N yield the result. Then, restoring the dependence on ω in the notation, we get by the same arguments as in the self adjoint case, that the density of states is almost surely ˜ M,N as N − M → ∞. the limit in the vague sense of the measures dkM,N and dk A proof is provided in the appendix for completeness.
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Proposition 3.1 For any continuous function f : S 1 → C, iλ ˜ ω lim f (e )dk M,N (λ) = f (eiλ )dk(λ) N −M→∞
T
a.s. ,
357
(3.16)
T
and the support of the density of states dk coincides with Σ, the a.s. spectrum of Uω . Remark. The two previous results show that there exists a series of polynomials whose asymptotic distribution of zeros converges to the measure dk, as announced in the introduction. These polynomials are the characteristic polynomial of the unitary matrix V M,N . As we noted earlier, there is some freedom in the definition of the boundary conditions giving rise to these matrices, therefore this series of polynomials is not unique. Observe also that the difference between these polynomials and the orthogonal polynomials only lies in the boundary conditions used to define V M,N , as recalled at the end of Section 2.1.
4 Thouless formula The link between the density of states and the Lyapunov exponent is provided by an analysis of the spectrum of the finite unitary matrices V M,N . It reads Theorem 4.1 [Thouless Formula ] For any z ∈ C \ {0} γ(z) = 2 ln |z − eiλ |dk(λ ) + ln(1/t2 ) − ln |z|.
(4.1)
T
Remarks. 0) The identity γ(1/¯ z) = γ(z) holds. i) It follows from the above formula, as in Theorem 4.6 in [GT], that the integrated density of states is continuous and satisfies λ ln(2/t2 ) , where N (λ) = dk(λ ), (4.2) |N (λ1 ) − N (λ2 )| ≤ | ln |eiλ1 − eiλ2 || −π by an argument of Craig and Simon [CS]. ii) In case z = eiλ ∈ S 1 , the formula can be cast into the form iλ γ(e ) = ln(sin2 ((λ − λ )/2))dk(λ ) + ln(4/t2 ),
(4.3)
T
from which we recover the estimate 0 ≤ γ(eiλ ) ≤ ln(4/t2 ) that follows from the form of the transfer matrices (2.16). The proof of this version of Thouless formula is given at the end of the section and its translation in terms of potentials of measures is given after the proof. We proceed with a Corollary and an application of this formula. The Corollary essentially expresses the radial derivative of the Lyapunov exponent as the Poisson integral of the density of states measure dk, which allows to recover the a.c. component of dk by a limiting procedure.
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Corollary 4.1 For any > 0 and any λ ∈ T,
lim γ(eiλ e− ) = γ(eiλ ), ∂ 1 − |eiλ e± |2 γ(eiλ e± ) = ∓ dk(λ) ≡ ∓P [dk](eiλ e± ). iλ − eiλ e± |2 ∂ |e T
→0+
(4.4) (4.5)
Therefore, if n(λ)dλ/2π denotes the a.c. component of dk(λ), lim
→0+
∂ ∂ γ(eiλ e− ) = n(λ ) = γ(eiλ ), ∂ ∂
(4.6)
where the limit and the derivative exist for Lebesgue almost all λ ∈ T. Remark. As in [CS], it follows also from the subharmonicity of γ(z), that if γ(eiλ0 ) = 0, then γ : S 1 → R+ is continuous at eiλ0 . Proof. Let us first consider the second statement with lower indices only. We compute iλ − 2 (4.7) γ(e e ) = + ln(1/t ) + ln(1 + e−2 − e− 2 cos(λ − λ ))dk(λ), T
which we can differentiate under the integral sign as long as > 0 to get −2e−2 + e− 2 cos(λ − λ ) ∂ iλ − γ(e e ) = 1 + dk(λ) −2 − e− 2 cos(λ − λ ) ∂ T 1+e 1 − e−2 dk(λ) = P [dk](eiλ e− ). (4.8) = −2 − e− 2 cos(λ − λ ) 1 + e T The existence for almost all λ ∈ T of the limit and the first equality in (4.6) is a direct consequence of the above equality. The existence and equality with the derivative at zero for such λ follows from the mean value Theorem. To get the first statement, notice that 1 + e−2 − e− 2 cos(x) > 2e− (1 − cos(x)) in formula (4.7) above yields 0 ≤ − ln((1 + e−2 − e− 2 cos(λ − λ ))/4) < − ln(2e− (1 − cos(λ − λ ))/4) = (4.9)
− ln((1 − cos(λ − λ ))/2), where the last function is in L1 (T, dk) by Thouless formula. Therefore, an application of the dominated convergence Theorem shows we can take the limit → 0 inside the integral to get the result. We consider now the properties of Uω characterized by i.i.d. phases θkω and in the definition (2.2), assuming one set of phases is uniformly distributed on T. In that situation, not only can we prove the transfer matrices have a (positive) Lyapunov behavior, but we can also exactly compute the Lyapunov exponent αω k
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γ(eiλ ). This shows that in this situation, the spectrum of Uω is almost surely singular, in view of the unitary version of the Ishii-Pastur Theorem proven in [BHJ]. This strengthens the corresponding results of [BHJ], Theorem 4.1 and Propositions 5.4. There Furstenberg’s Theorem is applied to prove positivity of the Lyapunov exponent, so that no value for γ(eiλ ) is provided. Theorem 4.2 Let (θkω )k∈Z and (αω k )k∈Z be i.i.d. on T and assume the distribution ’s is uniform on T. Then, for any λ ∈ T, of either the θkω ’s or the αω k dk(λ) = dλ/2π,
and γ(eiλ ) = ln(1/t2 ) > 0,
(4.10)
therefore, σ(Uω )a.c = ∅ and σ(Uω )sing. = S 1 almost surely.
(4.11)
Remark. The assumption on the distribution of the phases actually implies that the ηk ’s are i.i.d. and uniform on T , see Lemma 4.1 below. This explains why the a.s. spectrum coincides with S 1 and why the density of states is flat. Proof of Theorem 4.2. We first use the following lemma of purely probabilistic nature proven in the appendix. Lemma 4.1 Under the hypotheses of Theorem 4.2, the ηkω ’s are i.i.d. and uniform on T . Then we show the density of states is uniform for uniformly distributed phases. Expanding (2.2) in the ηk (ω)’s we can write for any n = 0, ϕj |Uωn ϕj = (Uω )j,k1 (Uω )k1 ,k2 . . . (Uω )kn−1 ,j k=k1 ,k2 ,...,kn−1
=
k
exp −i
pl ηl (ω) (U0 )j,k1 (U0 )k1 ,k2 . . . (U0 )kn−1 ,j , (4.12)
l∈L
where U0 corresponds to Uω when all phases ηk = 0 and where L is a finite set of indices depending on j, k, n and pl are integers. Observing that the variables ηk (ω)’s all appear with the same sign in (2.1), no compensation can take place between contributions of different matrix elements above and one at least among the integers pl , for l ∈ L is strictly positive when n = 0. Using independence and the characterization E(e−imηk (ω) ) = δm,0 of the uniform distribution, we get E( ϕj |Uωn ϕj ) = δn,0 =⇒ einλ dk(λ) = δn,0 (4.13) T
and the first statement follows. The second equality is a consequence of Thouless formula together with the identity 2π ln |1 − eiλ |dλ = 0. (4.14) 0
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The singular nature of the almost sure spectrum of Uω comes from the unitary version of Ishii-Pastur Theorem proven as Theorem 5.3 in [BHJ], which is independent of the properties of the common distributions of the αk ’s and θk ’s and only requires ergodicity. Finally, Proposition 3.1 yields the result about the support of the a.s. singular spectrum. We compute here, for the sake of completeness, the density of states and Lyapunov exponent for the deterministic free operator U0 corresponding to Uω in case ηk = 0, ∀k ∈ Z. In this case, equation (3.16) of Proposition 3.1 becomes a definition of the free density of states dk0 , provided the limit exists. That the limit exists, is the content of the next Lemma 4.2 The free density of states dk0 exists when defined for any f ∈ C(S 1 ) by iλ f (e )dk0 (λ) = lim f (eiλ )dk˜M,N (λ). (4.15) T
N −M→∞
T
As we know essentially everything about the purely a.c. operator U0 , we can also use a direct approach to perform these computations. In particular, the integrated density of states of U0 can be defined as the distribution function on T of the band functions yielding the spectrum Σ0 of U0 . This direct approach of the density of states coincides with the above definition, see the proofs of Proposition 4.1 and Lemma 4.2 in the appendix. We note here that the spectrum of U0 consists in the set 2 2 (4.16) Σ0 = {e±i(arccos(r −t cos(y))) , y ∈ T}. We get in particular that Σ0 is the support of the density of states whereas Σc0 is that of the Lyapunov exponent: Proposition 4.1 If N0 , dk0 and γ0 denote the integrated density of states, the density of states and Lyapunov exponents of U0 , respectively. We have for λ ∈ T ] − π, π], √ | sin(λ)| dλ if |λ| < arccos(r2 − t2 ) 2π t4 −(r 2 −cos(λ))2 (4.17) dk0 (λ) = 0 otherwise
2 1 arccos r −cos(λ) if λ ∈ [− arccos(r2 − t2 ), 0] 2 2π
2t (4.18) N0 (λ) = 1 − 1 arccos r −cos(λ) if λ ∈ [0, arccos(r2 − t2 )] 2π t2 2 2 02
if |λ| ≤ arccos(r − t ) iλ γ0 (e ) = (4.19) cosh−1 r −cos(λ) otherwise. t2 Finally, Thouless formula (4.1) holds true for these quantities with z = eiλ , λ ∈ T. Remarks. Note that the density of dk0 (λ) diverges as 1/ |λ − arccos(r2 − t2 )| at the band edges and behaves as 1/2πt as λ → 0.
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The integrated density of states N0 (λ) tends to its values 0 and 1 as |λ − arccos(r2 − t2 )| at the band edges. Also, in keeping with the fact that U0 becomes a shift if t = 1 and the identity as r = 1, N0 (λ) becomes linear in λ as t → 1 and a step function as r → 1. The Lyapunov exponent, where non zero, is equivalently given by 2 2 − cos(λ) 2 r − cos(λ) r γ0 (eiλ ) = ln + − 1 . (4.20) t2 t2 It is an even C ∞ function of λ on {|λ| > arccos(r2 − t2 )}, strictly increasing on 2 2 iλ [arccos(r − t ), π]. And dγ0 (e )/dλ behaves as 1/ λ − arccos(r2 − t2 ) as λ → arccos(r2 − t2 )+ . Given Lemma 4.2 above, it is clear that Thouless formula holds for the above quantities. A direct proof of this fact is nevertheless given in the appendix. Finally, in terms of orthogonal polynomials, the free case is related to the choice of constant reflection coefficients ak = a ∈ C, for all k, which yields the Geronimus orthogonal polynomials on the circle. For any such choice, the corresponding five diagonal operator equals −U0 , see (2.8), (2.2), and depends on |a| only. The spectral picture corresponds to the one above, rotated by π. This is in agreement with the accounts of this special case given in [G] and [GNV] for example, modulo a point mass or eigenvalue stemming from the boundary condition at the origin which we don’t consider here, see Section 2.1.
4.1
Proof of Thouless formula
We now turn to the proof of Theorem 4.1. Writing down explicitly the effect of the boundary conditions at N > M on the coefficients of the eigenvector (2.13) we obtain the following relations, which depend on the parity of N and M . Let ψ M,N = χM,N ψ and consider V M,N ψ M,N = eiλ ψ M,N
in l2 [M + 1, N ].
(4.21)
We get by inspection, Lemma 4.3 Assume (4.21) is satisfied. Then, if M is even 1 −it(r − e−iλ ) cM+2 iλ = cM+1 b1 (e ) ≡ cM+1 2 . cM+3 (r − eiλ ) + r(r − e−iλ ) t If M is odd,
cM+1 cM+2
1 = cM+1 b2 (e ) ≡ cM+1 it iλ
it eiλ − r
(4.22)
.
Similarly, if N is even, 1 (r − eiλ ) + r(r − e−iλ ) cN −2 = cN b3 (eiλ ) ≡ cN 2 . −it(r − e−iλ ) cN −1 t
(4.23)
(4.24)
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If N is odd,
cN −1 cN
= cN −1 b4 (eiλ ) ≡ cN −1
1 it
eiλ − r it
Ann. Henri Poincar´e
.
(4.25)
These relations together with the formulas (2.18) allow to describe the spectrum of V M,N in a convenient manner. Corollary 4.2 Let M < N be fixed and consider non zero vectors a1 , a2 ∈ C2 such that aj (eiλ ) ∈ (bj+2 (eiλ )C)⊥ , j = 1, 2. Then, eiλ ∈ σ(V M,N ) iff a1 (eiλ )|T (N/2 − 1) . . . T (M/2 + 2)b1 (eiλ ) = 0, a2 (e )|T ((N + 1)/2 − 1) . . . T (M/2 + 2)b1 (e ) = 0, iλ
iλ
M, N even M even , N odd
a1 (e )|T (N/2 − 1) . . . T ((M + 1)/2 + 1)b2 (e ) = 0, M odd , N even a2 (eiλ )|T ((N + 1)/2 − 1) . . . T ((M + 1)/2 + 1)b2 (eiλ ) = 0, M, N odd (4.26) iλ
iλ
Remark. In particular, a possible choice for the aj ’s is a1 (eiλ ) = b1 (e−iλ ), a2 (eiλ ) = b2 (e−iλ ).
(4.27)
Each of the above quantities denotes a matrix element of a product of transfer matrices of the type (2.18), which depend on eiλ , and will be linked in the limit N − M → ∞ to the Lyapunov exponent. Let eiλ = z ∈ C \ {0} and n0 , m0 ∈ Z. Defining Φm0 ,n0 (z) = T (n0 − 1) . . . T (m0 + 2),
(4.28)
one sees that the matrix elements aj (z)|Φm0 ,n0 (z)bk (z) correspond to those in the above corollary for values N = 2n0 , N = 2n0 − 1, M = 2m0 , M = 2m0 + 1, depending on the choice of indices j, k. Lemma 4.4 For any z ∈ C \ S 1 and any indices j, k = 1, 2 lim
n0 −m0 →∞
1 ln | aj (z)|Φm0 ,n0 (z)bk (z) | = 2(n0 − m0 ) ln |z − eiλ |dk(λ ) + ln(1/t) − ln(|z|1/2 ),
(4.29)
T
Proof. We note that for any k ∈ Z, there exist 2 × 2 matrices A(k), B(k), C(k) such that (with z = eiλ ) 0 0 iη2k T (k) = zA(k) + B(k) + C(k)/z, where A(k) = (4.30) 0 − −et2 (k)
(k)
Also, for any j = 1, 2, there exist vectors bj , aj , k = −1, 0, 1 such that ak (z) =
zak + ak + ak
(1)
(0)
bk (z) =
(1) zbk
(0) bk
+
(−1)
+
/z,
(−1) bk /z,
(4.31)
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(1)
where b2 = a2 = 0 are the only zero vectors with the choice (4.27). Thus, taking into account the above property, Pj,k (z) = z n0 −m0 +(1−k) aj (z)|Φm0 ,n0 (z)bk (z)
(4.32)
is a polynomial in z of degree 2(n0 − m0 ) + 2 − (k + j). Let pj,k be the coefficient of the highest power of z of Pj,k . Then, because of corollary 4.2, we can write
degPj,k
Pj,k (z) = pj,k
(z − eiλl ),
(4.33)
l=0
where {eiλl } is the set of eigenvalues of V M,N and we compute (2−j)
|pj,k | = | aj
|
n 0 −1
(1)
A(l)bk | =
l=m0 +2
K0 t2(n0 −m0 )
(n0 −m0 )−2 K1 −it 0 0 0 = 2(n0 −m0 ) (4.34) 0 1 r 1 t
where K0 , K1 are some constants that depend on j, k and t. Therefore, for any z ∈ C \ S1, degPj,k ln |z − eiλl | ln |Pj,k (z)| = ln(1/t2 ) + lim n0 −m0 →∞ (n0 − m0 ) n0 −m0 →∞ (n0 − m0 )
(4.35)
lim
l=0
Introducing the continuous function fz : S 1 → R given by fz (x) = ln |z − x|, the last term can be written deg Pj,k
lim
n0 −m0 →∞
l=0
fz (eiλj ) tr (fz (V M,N )) =2 = 2 lim M−N →∞ n 0 − m0 N −M
T
fz (eiλ )dk(λ )
(4.36) by application of Lemma 3.1 and Proposition 3.1. This ends the proof of the lemma. Then we make use the following easy lemma Lemma 4.5 If Φ : C2 → C2 is linear and aj , bj ∈ C2 , j = 1, 2 are such that span (a1 , a2 ) = span (b1 , b2 ) = C2 , then Φ := maxj,k | aj |Φbk | is a norm for Φ, noting that its hypothesis is satisfied by ak (z), bj (z), for all z = −1, and of the fact that the Lyapunov exponent is defined independently of the norm used in (2.19) to deduce that (4.29) actually equals half the Lyapunov exponent. Finally, the fact that both the Lyapunov exponent and the right-hand side of (4.29) are subharmonic and coincide on C \ S 1 implies the relation (4.1) on C as well, by classical arguments, see [CS]. This ends the proof of the Thouless formula.
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Link with potentials of measures
We now express the Thouless formula as a property of the potential of the density of states measure. Following [SaT], we briefly and informally recall the main definitions. The (logarithmic) potential of a probability measure µ on the circle is defined by (4.37) p(dµ; z) = − ln |z − eiλ |dµ(λ), T
the (logarithmic) energy of such a measure is defined by I(dµ) = − ln |eiθ − eiλ |dµ(λ)dµ(θ),
(4.38)
T2
whereas the energy E of a set Σ ⊆ S 1 is E = inf{I(dµ) | supp dµ ⊆ Σ}.
(4.39)
In case an external field Q coming from a weight w(z) = e−Q(z) , z ∈ S 1
(4.40)
is added, the weighted energy of the measure is defined by ln |eiθ − eiλ |dµ(λ)dµ(θ) + 2 Q(eiλ )dµ(λ) Iw (dµ) = − T2
(4.41)
T
and the weighted energy Ew of a set Σ is defined as above, with Iw in place of I. Now, the equilibrium measure of a set Σ is the unique measure dµΣ realizing the infimum of the energy Ew , when finite. These quantities are defined according to
n the electrostatic analogy. For example, if dµA = j=1 n1 δzj , where zj ∈ S are the zeros (with multiplicity) of some monic polynomial A, µA is the distribution of the zeros of A and its potential equals 1 ln |z − zj | = − ln |A(z)|1/n , n j=1 n
p(dµA ; z) = −
(4.42)
and if Σ = S 1 , the equilibrium measure dµS 1 is the normalized Lebesgue measure so that 0 if |z| ≤ 1 . (4.43) p(dµS 1 ; z) = − ln |z| if |z| > 1 Hence we can cast our Thouless formula for dk under the form p(dk; z) + γ(z)/2 = ln(1/t)
∀z ∈ S 1 ,
(4.44)
which, in view of Theorem I.3.3 of [SaT] and the subharmonicity of γ says that the density of states measure dk is the equilibrium measure on S 1 for the weight given
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by w(z) = e−γ(z)/2 . More generally, we can observe a similarity between the proof or our Thouless formula and Theorem III.4.1 in [SaT]. This theorem essentially says, in a deterministic framework, that if {An }n≥0 is a sequence of asymptotically extremal monic polynomials for a weight w (i.e., such that the asymptotic behavior as n → ∞ of (supz∈S 1 |w(z)n An (z)|)1/n is essentially given by a constant), then we have equivalence between lim
n→∞,n∈N
|An (z0 )|1/n = e−p(dµw ;z0 )
(4.45)
and lim dµAn = dµw ,
(4.46)
n→∞
in the vague sense, where dµw denotes the weighted equilibrium measure corresponding to w and N denotes an infinite subsequence of N. Regarding the definition of dk and the proof of Thouless formula, on the one hand we have that p(dk; z) =
lim
M−N →∞
p(dµ∆V M,N ; z)
(4.47)
where ∆V M,N (z) = det(z − V M,N ) is such that dµ∆V M,N → dk vaguely, and, on the other hand, that this potential is related to the Lyapunov exponent in such a way that dk is the equilibrium measure corresponding to the weight w(z) = e−γ(z)/2 . Hence, in our random setting, we can say our construction selects the asymptotically extremal monic polynomials allowing a discrete approximation of the equilibrium measure associated to the external field given by the Lyapunov exponent.
5 Properties of the density of states We mentioned several times the analogy between our unitary operator Uω and Jacobi matrices corresponding to the self-adjoint case. In this section we slightly drift away from the physical motivations underlying the study of (2.1) and consider more closely the links between these cases. The analogy is made clearer by the following Lemma which will be useful later. Lemma 5.1 Denoting unitary equivalence by , we have U ω D ω S0 , and
S0 =
..
.
rt r2 rt −t2
ω
with Dω = diag {e−iηk } −t2 −rt r2 −tr
(5.1)
rt r2 rt
−t2 −rt r2
−t2
−tr
..
U0 ,
(5.2)
.
where the translation along the diagonal is fixed by ϕ2k−2 |S0 ϕ2k = −t2 , k ∈ Z.
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Remarks. In some sense, the Lemma says that, up to unitary equivalence, Uω is a unitary analog of the one-dimensional discrete random Schr¨ odinger operator where the a.c. unitary S0 plays the role of the discrete Laplacian, the pure point diagonal operator Dω plays the role of the potential on the sites, and the operator sum is replaced by a product. We also recall that tridiagonal unitary matrices are spectrally uninteresting as they either correspond to a shift of to infinite direct sums of blocks of size one or two, see Lemma 3.1 in [BHJ]. The Lemma also shows that our operator Uω is essentially a product of an absolutely continuous unitary and a pure point unitary, whereas it was constructed in Section 2 of [BHJ] as a product of two pure point unitaries. Proof. Let us define a collection of rank two operators by Pj = |ϕj ϕj | + |ϕj+1 ϕj+1 |, j ∈ Z, and the unitary V by the direct sum V =
⊕
P2j−1
j∈Z
ir t −it r
(5.3)
P2j−1 .
(5.4)
It is just a matter of computation to check that we can write Uω = (Uω U0−1 )U0 ≡ V −1 Dω V U0 = V −1 Dω (V U0 V −1 )V ≡ V −1 (Dω S0 )V, (5.5) with the required properties for S0 and Dω .
ηkω
Now, forgetting that the phases are in general correlated random variables, see (2.2), if we consider them as i.i.d., but not necessarily uniformly distributed on T, we get some unitary Anderson-like model. This is where we depart from the physical motivation, as it is recalled in Lemma 4.2 in [BHJ] that independence of the ηk ’s is associated with a uniform distribution.
5.1
Support of the density of states
Nevertheless, assuming the random phases {ηkω }k∈Z are i.i.d. according to the measure dµ on T, we can characterize the almost sure spectrum of Uω in term of the support of µ and of the spectrum Σ0 of U0 . Theorem 5.1 Under the above hypotheses, the almost sure spectrum of Uω consists in the set (5.6) Σ := exp(i suppµ)Σ0 = {eiα Σ0 | α ∈ suppµ}. Remarks. In the case where the ηk (ω) are i.i.d. and uniform on T, we recover the fact that the almost sure spectrum of Uω is S 1 . We recall that in the orthogonal polynomial setting, the hypothesis implies each phase γk of the reflection coefficients is given by a sum of i.i.d. phases, see (2.2), (2.8).
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Proof. To show that Σ belongs to the almost sure spectrum, we simply construct Weyl sequences corresponding to the corresponding quasi-energies, with probability one. We know from Section 6 of [BHJ] that for any eiλ ∈ Σ0 , there exists a generalized eigenvector ψλ such ψλ = cj (λ)ϕj , U0 ψλ = eiλ ψλ , and 0 < K < |cj (λ)| < 1/K, ∀j ∈ Z, (5.7) j∈Z
for some K > 0. The last property can be checked also by means of the transfer matrices (2.16) Let α ∈ suppµ. Then, for all > 0, there exists a set I α such that |I | ≤ , and µ(I ) > 0. With the notation ω(k) = ηk (ω), k ∈ Z, we define for all n ∈ N and k ∈ Z, An (k) = {ω(kn) ∈ I , ω(kn + 1) ∈ I , . . . , ω(kn + n − 1) ∈ I }.
(5.8)
Due to the assumed independence, we have for any k, P(An (k)) = µ(I )n > 0 so that for any n > 0, by Borel-Cantelli, P(∪k∈Z An (k)) = 1. Let ∆n (k) = {kn, kn + 1, . . . , kn + n − 1} denote the set of indices appearing in An (k) and consider now ψn,k (λ) = cj (λ)ϕj = χ(∆n (k))ψ(λ), (5.9) j∈∆n (k)
where χ(∆n (k)) is the projector on the span of {ϕj }j∈∆n (k) Because of (5.7), − + (λ) + Rk(n+1) , U0 ψn,k (λ) = eiλ ψn,k (λ) + Rkn
(5.10)
where the vectors Rj± have at most four components close to the index j and
Rj± ≤ R, where R is uniform in j.
(5.11)
Also, by construction of An (k), U0 and Uω , we have
Uω ψn,k (λ) − eiα U0 ψn,k (λ)
≤
(Uω − eiα U0 )χ(∆n (k))| ψn,k (λ)
=
O( ) ψn,k (λ) ,
(5.12)
where the estimate O( ) is uniform in n and k. Therefore, for all > 0 and all n > 0, there exists, with probability one, a k such that An (k) and the corresponding ψn,k (λ) have the above properties so that
Uω ψn,k (λ) − ei(α+λ) ψn,k (λ) / ψn,k (λ)
= ( (Uω − eiα U0 )ψn,k (λ) + eiα (U0 − eiλ )ψn,k (λ) )/ ψn,k (λ)
≤ O( ) + 2R/ ψn,k (λ) = O( + 1/n).
(5.13)
It remains to chose n = [1/ ] to conclude that ei(α+λ) ∈ σ(Uω ) almost surely.
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Let us now show that S 1 \ Σ belongs to the resolvent set of Uω . In order to do so we use Lemma 5.1 Therefore, we can consider as well the spectrum of the product Dω S0 to which the perturbation theory recalled in Chap.1, §11 of [Yaf] for example, applies. In particular, dropping the ω in the notation as randomness plays no role here, if we know that for all j ∈ Z, ηj ∈ [α, β] ⊂ T, then σ(D) ⊆ (δ1 , δ2 ) where (δ1 , δ2 ) denotes the corresponding arc on the unit circle swept in the positive direction from δ1 ∈ S 1 to δ2 ∈ S 1 . We denote by |(δ1 , δ2 )| the length on the torus of this arc. Since σ(S0 ) = Σ0 corresponds to the 2 2 2 2 symmetric arc (e−i arccos(r −t ) , ei arccos(r −t ) ), perturbation theory tells us that after (multiplicative) perturbation by S0 , the spectrum of U DS0 is a subset of an arc of wider aperture than (δ1 , δ2 ). Quantitatively, Theorem 8, p.65 in [Yaf] tells 2 2 2 2 us that the arc (ei arccos(r −t ) δ2 , e−i arccos(r −t ) δ1 ) belongs to the resolvent set of 2 2 2 2 U , provided |(δ1 , δ2 )| < |(ei arccos(r −t ) , e−i arccos(r −t ) )|. This condition simply insures that the subset of the resolvent set we are talking about is not reduced to the empty set. This is enough to get the result in case the support of µ is such that Σ is connected. In case this set is not connected, as |Σ0 | > 0, it consists of a finite set of connected components, each of which can be associated with the convex hull of sufficiently far apart subsets of the support of µ. Denoting these subsets by mj , j = 1, . . . , N and the associated arcs on S 1 by (M1 (j), M2 (j)), we have that the spectrum of D is the disjoint union of subsets σj satisfying σj ⊆ (M1 (j), M2 (j)). The same argument as above says that the spectrum of DS0 is confined to the 2 2 2 2 finite union of arcs ((ei arccos(r −t ) M1 (j), (e−i arccos(r −t ) M2 (j)), which ends the proof of the Theorem.
5.2
Analyticity of the density of states
Without really entering the delicate analysis of the smoothness of the density of states, we can further exploit the relation (4.12) in order to obtain, at the price of some combinatorics, a condition on the common distribution of the ηk ’s ensuring the analyticity of the density of states. Recall that a function f on T is analytic, if and only if its Fourier coefficients fˆ satisfy an estimate of the form |fˆ(n)| ≤ Ae−B|n| , ∀n ∈ Z,
(5.14)
for some positive constants A, B. We have Theorem 5.2 Assume the ηk ’s are distributed according to a law that has an analytic density f characterized by the estimate (5.14) with A, B > 0. Then, if B > ln(1 + 2rt) + ln A,
(5.15)
the density of states dk admits an analytic density, so that the integrated density of states N is analytic as well. Remarks. As fˆ(0) = T f (η)dη = 1, A ≥ 1. When the Theorem applies, it prevents the Lyapunov exponent from being zero on a set of positive measure.
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This result has to be compared with Proposition VI.3.1. of [CL] stating a similar result for the d-dimensional Anderson model. As an immediate consequence, using r2 + t2 = 1, we get the following Corollary 5.1 If the ηk ’s have an analytic density f , characterized by (5.14) with B > ln A, then there exist r+ (f ) and r− (f ) in ]0, 1[ such that the density of states is analytic provided the reflection coefficient r satisfies 1 > r > r+ (f ) or 0 < r < r− (f ). If B > ln(2A), The density of states is analytic ∀r ∈ [0, 1]. Remark. It is easy to check that in both the extreme cases r = 1 and r = 0, the density of states is analytic. Indeed, if r = 1, dk(λ) = f (λ)dλ, where f is the density of the ηk ’s, whereas if r = 0, dk(λ) = dλ/(2π). Proof of Theorem 5.2. By hypothesis, for any n ∈ Z, |Φη (n)| = eiηn f (η)dη ≤ Ae−B|n| . Then, in (4.12) above,
(5.16)
T
l∈L
|E ϕj |Uωn ϕj | ≤ An e−Bn
pl = n, so that using independence
|(U0 )j,k1 ||(U0 )k1 ,k2 | . . . |(U0 )kn−1 ,j |
(5.17)
k1 ,k2 ,...,kn−1
Here the sum carries over a set of indices that form paths of length n + 1 from index j to index j. The allowed paths are those giving rise to non zero matrix elements (U0 )l,m in the sum above. In order to compute this last sum, we proceed as follows. Let us introduce more general j-dependent subsets Cn−1 (j) of indices of Zn−1 that appear in the computation of the matrix element ϕ0 |Uωn ϕj . This set consists of paths of the form {k0 = 0, k1 , k2 , . . . , kn−1 , kn = j} of length n + 1 in Z from 0 to j with the condition that km+1 − km ∈ {0, +1, −1, +2}
if
km is odd
km+1 − km ∈ {0, +1, −1, −2}
if
km is even,
(5.18)
for all m = 0, 1, . . . , n − 1. Let us define Sn−1 (j) := |(U0 )0,k1 ||(U0 )k1 ,k2 | . . . |(U0 )kn−1 ,j |,
(5.19)
Cn−1 (j)
where the matrix elements |(U0 )l,m | are given by r2 , rt and t2 respectively, when |l − m| equals 0, 1 and 2 respectively. This quantity actually gives a crude upper bound on the probability to go from site 0 to j in n time steps, under the free evolution. It is crude in the sense that it does not take the phases into account during that free evolution. We are actually interested in the computation of Sn−1 (0) and of the similar quantity appearing in the computation of ϕ1 |Uωn ϕ1 , which correspond the sum
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in the right-hand side of (5.17), in the asymptotic regime n → ∞. The case of the matrix element ϕ1 |Uωn ϕ1 being similar, we only consider Sn−1 (0). The plan is to use a transfer matrix formalism to evaluate the generating function associated with Sn−1 (j) and then to compute the asymptotics of Sn−1 (0). In view of (5.17), the following proposition implies the theorem. Proposition 5.1 For some constant c > 0, Sn−1 (0) =
c(r + t)2n √ (1 + o(1)) as n → ∞. n
(5.20)
Proof of Proposition 5.1. Let
Pn (x) =
Sn−1 (j)xj
(5.21)
−2n≤j≤2n
be this generating function which we split into two parts Pn (x) = Pn+ (x) + Pn− (x) where Sn−1 (j)xj . (5.22) Pn± (x) = −2n≤j≤2n j
even odd
Clearly we have for n = 0, 1, P0+ (x) = r2 , P0− (x) = 0, P1+ (x) = r2 + t2 x−2 , P1− (x) = rt(x + x−1 ).
(5.23)
It is readily shown by induction that a transfer matrix allows to compute Pn (x) for any n: Lemma 5.2 For any n ≥ 0, 2 + Pn+1 (x) r + t2 x−2 = − rt(x + x−1 ) Pn+1 (x)
rt(x + x−1 ) r2 + t2 x2
Pn+ (x) Pn− (x)
,
with P0+ (x) = r2 , P0− (x) = 0. Denoting by T (x) the transfer matrix defined in this Lemma, and introducing the parameter τ = t/r ∈]0, ∞[, (5.24)
we rewrite it as T (x) = r2
1 + τ 2 x−2 τ (x + x−1 ) τ (x + x−1 ) 1 + τ 2 x2
.
(5.25)
We will consider first the case t = r ⇐⇒ τ = 1. The case τ = 1, for which more can be said about Sn−1 (j), see Proposition 5.2, is dealt with below.
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5.2.1 Case τ = 1 The eigenvalues of T (x) are given by r2 times λ± (x), where λ± (x) = 1 + τ (x2 + x−2 )/2 ± (1 + τ (x2 + x−2 )/2)2 − (1 − τ 2 )2 ,
so that n
2n
T (x) = r A(x)
with A(x) =
λn+ (x) 0
0 λ− (x)n
A(x)−1
λ+ (x) − (1 + τ 2 x2 ) λ− (x) − (1 + τ 2 x2 ) τ (x + x−1 ) τ (x + x−1 )
(5.26)
(5.27) .
(5.28)
For the moment, x is just book keeping parameter, so that we ignore the potential problems of the definition of A(x) in case the eigenvalues are degenerate and we further compute 2 + r Pn (x) n = T (x) (5.29) Pn− (x) 0 r2n τ (x + x−1 ) = 2 (1 + τ (x2 + x−2 )/2)2 − (1 − τ 2 )2 λ+ (x)n+1 − λ− (x)n+1 − (λ+ (x)n − λ− (x)n )(1 + τ 2 x2 ) × . τ (x + x−1 )(λ+ (x)n − λ− (x)n ) We note at this point that one checks, using the binomial Theorem, that despite the presence of square roots in the expressions for Pn± (x), these quantities actually are given by finite Laurent expansions in x, as they should. Focusing on Pn+ (x) we √ can rewrite with the shorthand · for the square root of the denominator above Pn+ (x) 2n
=
−1
r τ (x + x √ 2 ·
(5.30) √ 2 ) · n n τ −2 2 n n (λ+ (x) + λ− (x) ) . (λ+ (x) − λ− (x) ) (x + x ) + 2 2
The quantity of interest to us is Sn−1 (0), the coefficient of x0 in the expansion of Pn+ (x). Substituting eiθ for x in Pn+ , we get a trigonometric polynomial whose zero’th Fourier coefficient is obtained by integration Sn−1 (0) = Pn+ (eiθ )dθ/(2π). (5.31) T
It remains to perform the asymptotic analysis of the above integral as n → ∞. It is a matter of routine to verify the following properties: The eigenvalues, as functions of θ ∈ T ] − π, π], are continuous. If τ < 1, they are real valued, with discontinuity of the derivative at θ = ±π/2, where they cross and are given by 1 − τ 2 . At all other values of θ, they are C ∞ and they satisfy λ+ (eiθ ) > λ− (eiθ ), with λ+ (eiθ ) > 1 − τ 2 .
(5.32)
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If τ > 1, the eigenvalues become complex conjugate. Let θc = arccos( τ τ−2 2 )/2 be the critical value where the square root becomes zero. If θ ∈ [θc , π − θc ] ∪ [−π + θc , −θc ], the eigenvalues are complex conjugate, of modulus |1−τ 2 |. Otherwise they are real valued, and satisfy (5.32) as well. Therefore, the asymptotics as n → ∞ of (5.31) is determined by λ+ only. Moreover, in both cases, ln(λ+ (eiθ )) admits non degenerate maxima at θ = 0 and π, where λ+ reaches its maximum value (1 + τ 2 ). Therefore, Laplace’s method yields the asymptotics of the proposition. 5.2.2 Case τ = 1 The course of the proof being the same, it is presented in the appendix. However, instead of computing Sn−1 (0) as n → ∞, we can get exact forms for all Sn−1 (j)’s. The proposition we actually show is Proposition 5.2 Sn−1 (j)
=
Sn−1 (j)
=
1 2n − 1 , −2n ≤ j ≤ 2(n − 1), j even 2n j/2 + n 1 2n − 1 , −2n + 1 ≤ j ≤ 2n − 1, j odd (5.33) 2n (j − 1)/2 + n
Remark. √ Of course, Stirling’s formula for n large yields proposition 5.1 with r = t = 1/ 2: 1 2n − 1 2n Sn−1 (0) = n (5.34) √ . n 2 πn
6 Appendix Proof of Proposition 3.1. We have by definition, T
ω
˜ f (eiλ )dk M,N (λ) =
1 N −M
N
ϕj |f (Uω )ϕj ,
(6.1)
j=M+1
where, depending on the parity of M and N and due to the fact that f is uniformly bounded, the right-hand side can be rewritten as N/2 1 1 ) ϕ2k |f (Uω )ϕ2k + ϕ2k+1 |f (Uω )ϕ2k+1 + Of ( N −M N −M k=(M+1)/2 N/2 1 1 = ϕ0 |f (US k (ω) )ϕ0 + ϕ1 |f (US k (ω) )ϕ1 + Of ( ). N −M N −M k=(M+1)/2
(6.2)
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Now, by the Birkhoff theorem, there exists Ωf of measure one such that for all ω ∈ Ωf , 1 N −M→∞ N − M
N/2
lim
ϕj |f (US k (ω) )ϕj =
k=(M+1)/2
1 E( ϕj |f (Uω )ϕj ), ∀j ∈ Z, (6.3) 2
therefore, 1 1 tr (χM,N f (Uω )) → (E( ϕ0 |f (Uω )ϕ0 + ϕ0 |f (Uω )ϕ0 )) . N −M 2
(6.4)
Then, C(S 1 ) being separable, we have the existence of a countable set of {fj }j∈N , dense in C(S 1 ), for which the above is true, on a set of probability one, which proves the almost sure convergence stated in the proposition. f such that Now assume eiλ0 ∈ Σ and take a continuous non-negative f (eiλ0 ) = 1 and f |Σ = 0. Then f (Uω ) = 0 a.s. so that f (eiλ )dk(λ) = 0 and 0 eiλ0 ∈ supp k. Conversely, if eiλ ∈ supp k, there exists a non-negative continuous f with f (eiλ0 ) = 1 and f (eiλ )dk(λ) = 0. Hence, a.s., ϕ0 |f (Uω )ϕ0 + ϕ1 |f (Uω )ϕ1 = 0, therefore, by ergodicity, ϕj |f (Uω )ϕj = 0 a.s. for any j and f (Uω ) = 0. As f is continuous and equals one at eiλ0 , we get that eiλ0 ∈ Σ. Proof of Lemma 4.1. We only deal with the case where the θkω ’s are i.i.d. and ω uniform, the other case being similar. Let Φη (n) = E(einηk ) be the characteristic function of the random variable ηkω , and similarly for αω k , and Φθ (n) = δn,0 . Then, using independence, Φη (n) = Φθ (n)2 Φα (n)Φα (−n) = δn,0 |Φα (n)|2 = δn,0 ,
(6.5)
so that the ηk ’s are uniformly distributed. Consider now Φηk0 ,ηk1 ,...,ηkj (n0 , n1 , . . . , nj ) = E(ei
j
l=0
kl ηl
).
(6.6)
We can assume the kj ’s are ordered and we observe that ηk and ηk+j are independent as soon as j ≥ 2, see (2.2). Therefore, we can consider consecutive indices kl and deal with Φηk ,ηk+1 ,...,ηk+j (n1 , n2 , . . . , nj ) = E(e
(6.7)
in0 θk−1 +i(n0 +n1 )θk +···+i(nj−1 +nj )θk+j−1 +nj θj )
E(f (α, n)),
where the second expectation contains αk ’s only. Then Φηk ,ηk+1 ,...,ηk+j (n1 , n2 , . . . , nj ) = Φθ (n0 )Φθ (n0 + n1 ) . . . Φθ (nj−1 + nj )Φθ (nj )E(f (α)) = δn0 ,0 δn1 ,0 . . . δnj ,0 E(f (α, n)) = δ E(f (α, 0)) = δ , n,0
with the obvious notation, which yields the result.
n,0
(6.8)
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Proof of Proposition 4.1. We first prove this Proposition with the definition of the density of states as the distribution function of the “band functions” of U0 , to be defined below. Then we will see in the course of the proof of Lemma 4.2 below the equivalence with the definition as an average counting measure. The proof of Proposition 6.2 in [BHJ] shows that U0 on l2 (Z) is unitarily equivalent to the operator multiplication by the matrix 2 r − t2 e2ix 2itr cos(x) V (x) = (6.9) on L2 (T) L2+ (T) ⊕ L2− (T), 2itr cos(x) r2 − t2 e−2ix √ by the unitary mapping that sends ϕk → eikx / 2π, and where L2± (T) is the subspace generated by even/odd harmonics {eikx }k∈Z . The eigenvalues of V (x) are (6.10) λ± (x) = e±iα(x) , where α(x) = arccos(r2 − t2 cos(2x)). We note that λ± (x) = λ± (−x) and V (x) = JV (−x)J where J =
0 1
1 0
.
(6.11)
Hence, the corresponding eigenvectors χ± (x) satisfy V (x)χ± (x) = λ± (x)χ± (x) and V (x)Jχ± (−x) = λ± (x)Jχ± (−x),
(6.12)
so that χ± (x) and Jχ± (−x) are linearly dependent. This is in keeping with the fact that the subspace of generalized eigenvectors is of dimension 2, see (2.14). Also, one checks that for any phase β ∈] − arccos(r2 − t2 ), 0[∪]0, arccos(r2 − t2 )[, α−1 (β) = {x1 , x2 , −x2 − x1 } ⊂] − π, π[.
(6.13)
Therefore, due to (6.12), only half these points contribute for the computation of the density of states. We can now compute the integrated density of states N0 (β) as follows: Taking into account the normalization by a factor 1/2π in the definition (3.1), the fact that supp k ⊂ [− arccos(r2 − t2 ), arccos(r2 − t2 )] and the symmetries, we have for any β ∈ [− arccos(r2 − t2 ), 0] N0 (β) =
1 4π
=
1 2π
T
dλχ{−α(λ)<β≤0} =
1 2π
arccos((r 2 −cos(β))/t2 )
= 0
π/2 −π/2
dλχ{cos(2λ)>(r2 −cos(β))/t2 }
1 arccos 2π
r2 − cos(β) t2
(6.14)
.
(6.15)
A similar computation for β ∈ [0, arccos(r2 − t2 )) yields (4.18). Therefore, dk0 is absolutely continuous w.r.t. Lebesgue and, for any |λ| < arccos(r2 − t2 ), dk0 (λ) = N (λ)dλ, from which the result on the density of states follows. In order to obtain the Lyapunov exponent, it is enough to observe that the transfer matrices (2.14) T ,
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now independent of k, are of determinant one and trace equal to 2(r2 − cos(λ))/t2 . Then, an explicit computation of the eigenvalues of T together with definition (2.19) yield γ0 (eiλ ). In order to prove the last statement, we first rewrite the right-hand side of Thouless formula with dk0 (λ ) above as 1 1 ln((x − y)2 ) √ dx + ln 2 (6.16) 2π −1 1 − x2 by means elementary manipulations, changing variables to x = (r2 − cos(λ ))/t2 and introducing y = (r2 − cos(λ))/t2 ∈ [−1, (r2 + 1)/t2 ]. Hence we are to show that (6.16) above equals 0 if y ≤ 1 and ln(y + y 2 − 1) if y > 1. That this is true follows from standard manipulations: differentiation w.r.t. y, deformation of contours of integration in the complex plane and computation of residues. Proof of Lemma 4.2. We use freely the notations above. Let us introduce the eigenprojectors P± (x) associated with λ± (x) such that V (x) = P+ (x)λ+ (x) + P− (x)λ− (x).
(6.17)
These quantities are analytic in x, in a strip including the real axis. Let f ∈ C(S 1 ) and let us compute by means of (6.9) and the definition of L2± (T) tr χM,N |f (U0 )χM,N = ϕj |f (U0 )ϕj
=
1 2π even
j M <j≤N
+
odd
j M <j≤N
M<j≤N
1 2π
T
T
! 1 1 dx (f (λ (x))P (x) + f (λ (x))P (x)) + + − − 0 0 ! 0 0 dx. (f (λ+ (x))P+ (x) + f (λ− (x))P− (x)) 1 1
(6.18)
The summand being independent of j and uniformly bounded, we can rewrite the above trace as N − M gets large as N −M f (λ+ (x)) tr P+ (x) + f (λ− (x)) tr P− (x)dx + O(1) 4π T N −M = f (λ+ (x)) + f (λ− (x))dx + O(1). (6.19) 4π T Hence, with λ± (x) = e±iα(x) as in (6.10), and taking into account the properties of α, we get 1 iλ f (e )dk0 (λ) = f (eiα(x) ) + f (e−iα(x) )dx 4π T T π/2 1 = f (eiα(x) ) + f (e−iα(x) )dx, (6.20) 2π −π/2
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which is easily seen to coincide with the “direct” definition of dk0 in the above proof. Proof of Proposition 5.2. As in that case a common term 21n can be factorized, see (5.17), we compute the generating function of |Cn−1 (j)|, the cardinal of the set of relevant indices. Using the same symbols as above, we consider this time |Cn−1 (j)|xj , (6.21) Pn (x) = −2n≤j≤2n
which we split into two parts Pn (x) = Pn+ (x) + Pn− (x) that satisfy for n = 0, 1, P0+ (x) = 1, P0− (x) = 0, P1+ (x) = 1 + x−2 , P1− (x) = x + x−1 .
(6.22)
As above, Lemma 6.1 For any n ≥ 0, + Pn+1 (x) 1 + x−2 = − x + x−1 Pn+1 (x)
x + x−1 1 + x2
Pn+ (x) Pn− (x)
,
with P0+ (x) = 1, P0− (x) = 0. By diagonalization of the corresponding transfer matrix, we get 0 0 n T (x) = A(x) A(x)−1 0 (x−1 + x)2n
where A(x) =
1 + x2 x + x−1 −1 −(x + x ) 1 + x2
(6.23)
(6.24)
and we compute
Pn+ (x) Pn− (x)
(x2 + 1)2n−1 1 1 = . = T (x) 0 x x2n n
(6.25)
Using the binomial theorem we obtain for Pn± (x) Pn+ (x)
=
n−1
2l
x
l=−n
Pn− (x)
=
n−1 l=−n
hence the end result.
2n − 1 l+n
2l+1
x
2n − 1 l+n
,
(6.26)
Acknowledgments. It is a pleasure to thank O. Bourget and R. Bacher for useful discussions, D. Damanik for pointing out reference [CMV] to me and B. Simon for comments on a previous version of the manuscript.
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Alain Joye Institut Fourier Universit´e de Grenoble 1, BP 74 F-38402 Saint-Martin d’H`eres Cedex France email:
[email protected] Communicated by Eugene Bogomolny Submitted 07/11/03, accepted 15/01/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 381 – 403 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/020381-23 DOI 10.1007/s00023-004-0173-9
Annales Henri Poincar´ e
On the Singularities of the Magnetic Spectral Shift Function at the Landau Levels Claudio Fern´ andez and Georgi Raikov Abstract. We consider the three-dimensional Schr¨ odinger operators H0 and H± where H0 = (i∇+A)2 −b, A is a magnetic potential generating a constant magnetic field of strength b > 0, and H± = H0 ± V where V ≥ 0 decays fast enough at infinity. Then, A. Pushnitski’s representation of the spectral shift function (SSF) for the pair of operators H± , H0 is well defined for energies E = 2qb, q ∈ Z+ . We study the behaviour of the associated representative of the equivalence class determined by the SSF, in a neighbourhood of the Landau levels 2qb, q ∈ Z+ . Reducing our analysis to the study of the eigenvalue asymptotics for a family of compact operators of Toeplitz type, we establish a relation between the type of the singularities of the SSF at the Landau levels and the decay rate of V at infinity. R´esum´e. On consid` ere les op´erateurs de Schr¨ odinger tridimensionnels H0 et H± o` u H0 = (i∇ + A)2 − b, A est un potentiel magn´etique engendrant un champ magn´etique constant d’intensit´e b > 0, et H± = H0 ±V o` u V ≥ 0 d´ ecroˆıt assez vite ` a l’infini. Alors, la repr´esentation obtenue par A. Pushnitski de la fonction du d´ecalage spectral pour les op´erateurs H± , H0 est bien d´efinie pour des ´energies E = 2qb, q ∈ Z+ . On ´ etudie le comportement du repr´esentant associ´e de la classe d’´equivalence d´ etermin´ee par la fonction du d´ecalage spectral, au voisinage des niveaux de Landau 2bq, q ∈ Z+ . En r´ eduisant l’analyse ` a l’investigation de l’asymptotique des valeurs propres d’une famille d’op´erateurs de Toeplitz compacts, on ´etablit une relation entre le type des singularit´es de la fonction du d´ecalage spectral aux niveaux de Landau et la vitesse de la d´ecroissance de V ` a l’infini.
1 Introduction In this paper we analyze the singularities of the spectral shift function (SSF) for the three-dimensional Schr¨ odinger operator with constant magnetic field, perturbed by an electric potential which decays fast enough at infinity. Let us recall the definition of the abstract SSF for a pair of self-adjoint operators. First, let us consider two self-adjoint operators T0 and T acting in the same Hilbert space, such that T − T0 ∈ S1 where S1 denotes the space of trace class operators. Then, there exists a unique function ξ(.; T , T0 ) ∈ L1 (R) such that the Lifshits-Kre˘ın trace formula Tr(φ(T ) − φ(T0 )) = ξ(E; T , T0 )φ (E)dE, φ ∈ C0∞ (R), (1.1) R
holds (see, e.g., [17, Theorem 8.3.3]). Let now H0 and H be two lower-bounded selfadjoint operators acting in the same Hilbert space. Assume that for some γ > 0,
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and λ0 ∈ R lying strictly below the infima of the spectra of H0 and H, we have that (1.2) (H − λ0 )−γ − (H0 − λ0 )−γ ∈ S1 . Set ξ(E; H, H0 ) :=
−ξ((E − λ0 )−γ ; (H − λ0 )−γ , (H0 − λ0 )−γ ) 0 if E ≤ λ0 .
if
E > λ0 ,
Then, similarly to (1.1), Tr(φ(H) − φ(H0 )) =
R
ξ(E; H, H0 )φ (E)dE,
φ ∈ C0∞ (R),
(see [17, Theorem 8.9.1]). The function ξ(.; H, H0 ) is called the SSF for the pair of the operators H and H0 . Evidently, it does not depend on the particular choice of γ and λ0 in (1.2). If E lies below the infimum of the spectrum of H0 , then the spectrum of H below E could be at most discrete, and we have ξ(E; H, H0 ) = −N (E; H)
(1.3)
where N (E; H) denotes the number of eigenvalues of H in the interval (−∞, E), counted with the multiplicities. On the other hand, for almost every E in the absolutely continuous spectrum of H0 , the SSF ξ(E; H, H0 ) is related to the scattering determinant det S(E; H, H0 ) for the pair (H, H0 ) by the Birman-Kre˘ın formula det S(E; H, H0 ) = e−2πiξ(E;H,H0 ) (see [2] or [17, Section 8.4]). A survey of various asymptotic results concerning the SSF for numerous quantum Hamiltonians is contained in [15]. In the present paper the role of H0 is played by the operator H0 := (i∇ + A)2 − b, is essentially self-adjoint on C0∞ (R3 ). Here the magnetic potential A = which bx2 bx1 − 2 , 2 , 0 generates the constant magnetic field B = curl A = (0, 0, b), b > 0. It is well known that σ(H0 ) = σac (H0 ) = [0, ∞) (see [1]), where σ(H0 ) denotes the spectrum of H0 , and σac (H0 ) its absolutely continuous spectrum. Moreover, the so-called Landau levels 2bq, q ∈ Z+ := {0, 1, . . .}, play the role of thresholds in σ(H0 ). For x = (x1 , x2 , x3 ) ∈ R3 we denote by X⊥ = (x1 , x2 ) the variables on the plane perpendicular to the magnetic field. We assume that V satisfies x = (X⊥ , x3 ) ∈ R3 , (1.4) with C0 > 0, m⊥ > 2, m3 > 1, and x := (1 + |x|2 )1/2 , x ∈ Rd , d ≥ 1. Most of our results will hold under a more restrictive assumption than (1.4), namely V ≡ 0,
V ∈ C(R3 ),
V ≡ 0,
0 ≤ V (x) ≤ C0 X⊥ −m⊥ x3 −m3 ,
V ∈ C(R3 ),
0 ≤ V (x) ≤ C0 x−m ,
m > 3,
x ∈ R3 .
(1.5)
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Note that (1.5) implies (1.4) with any m3 ∈ (0, m) and m⊥ = m − m3 . In particular, we can choose m3 ∈ (1, m − 2) so that m⊥ > 2. Set H± := H0 ± V so that the electric potential ±V has a definite sign. Obviously, inf σ(H+ ) = 0, inf σ(H− ) ≥ −C0 . The role of the perturbed operator H is played in this paper by H± . By (1.4) and the diamagnetic inequality (see, e.g., [1]), V 1/2 (H0 − λ0 )−1 with λ0 < 0 is a Hilbert-Schmidt operator. Therefore, the resolvent identity implies (H± − λ0 )−1 − (H0 − λ0 )−1 ∈ S1 for λ0 < inf σ(H± ) ≤ inf σ(H0 ), i.e., (1.2) holds with H = H± , H0 = H0 , and γ = 1, and, hence, the SSF ξ(.; H± , H0 ) exists. A priori the SSF ξ(E; H± , H0 ) is defined only for almost every E ∈ R. In Sec˜ H± , H0 ) of the equivalence class tion 2 below we introduce a representative ξ(.; determined by ξ(.; H± , H0 ), which is well defined and uniformly bounded on each compact subset of the complement of the Landau levels. Moreover, ξ˜ is continuous on R \ 2bZ+ everywhere except at the eigenvalues, isolated, or embedded in the continuous spectrum, of the operator H± . The main goal of the paper is the study of the asymptotic behaviour as λ → 0 ˜ of ξ(2bq + λ; H± , H0 ) with fixed q ∈ Z+ . Our results establish the asymptotic ˜ coincidence of ξ(2bq + λ; H± , H0 ) with the traces of certain functions of compact Toeplitz operators. Many of the spectral properties of those Toeplitz operators are well known, which allows us to describe explicitly the asymptotics as λ → 0 ˜ of ξ(2bq + λ; H± , H0 ) in several generic cases. These asymptotic results admit an interpretation directly in the terms of the SSF, which is independent of the choice of the representative of the equivalence class. In particular, these results reveal the link between the type of the singularities of the SSF at the Landau levels, and the decay rate of V at infinity. The paper is organized as follows. In Section 2 we introduce the representative ξ˜ of the SSF. In Section 3 we formulate our main results, summarize some known spectral properties of compact Toeplitz operators, and obtain as corollaries explicit asymptotic formulas describing the singularities of the SSF at the Landau levels. Section 4 contains preliminary estimates. The proofs of our main results can be found in Section 5. Finally, in Section 6 we prove some of the corollaries of the main results.
2 A. Pushnitski’s representation of the SSF 2.1. In this subsection we introduce some basic notations used throughout the paper. We denote by S∞ the class of linear compact operators acting in a fixed Hilbert space. Let T = T ∗ ∈ S∞ . Denote by PI (T ) the spectral projection of T associated with the interval I ⊂ R. For s > 0 set n± (s; T ) := rank P(s,∞) (±T ).
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For an arbitrary (not necessarily self-adjoint) operator T ∈ S∞ put n∗ (s; T ) := n+ (s2 ; T ∗ T ),
s > 0.
(2.1)
If T = T ∗ , then evidently n∗ (s; T ) = n+ (s, T ) + n− (s; T ),
s > 0.
(2.2)
Moreover, if Tj = Tj∗ ∈ S∞ , j = 1, 2, then the Weyl inequalities n± (s1 + s2 , T1 + T2 ) ≤ n± (s1 , T1 ) + n± (s2 , T2 )
(2.3)
hold for each s1 , s2 > 0. Further, we denote by Sp , p ∈ [1, ∞), the Schatten-von Neumann class of compact ∞ 1/p operators for which the norm T p : = p 0 sp−1 n∗ (s; T ) ds is finite. If T ∈ Sp , p ∈ [1, ∞), then the following elementary inequality n∗ (s; T ) ≤ s−p T pp
(2.4)
holds for every s > 0. If T = T ∗ ∈ Sp , p ∈ [1, ∞), then (2.2) and (2.4) imply n± (s; T ) ≤ s−p T pp,
s > 0.
Finally, we define the self-adjoint operators Re T := 1 ∗ 2i (T − T ). Evidently, n± (s; Re T ) ≤ 2n∗ (s; T ),
(2.5) 1 2 (T
+ T ∗ ) and Im T :=
n± (s; Im T ) ≤ 2n∗ (s; T ).
(2.6)
2.2. In this subsection we summarize several results due to A. Pushnitski on the representation of the SSF for a pair of lower-bounded self-adjoint operators (see [8]–[10]). dt Let I ∈ R be a Lebesgue measurable set. Set µ(I) := π1 I 1+t 2 . Note that µ(R) = 1. Lemma 2.1. [8, Lemma 2.1] Let T1 = T1∗ ∈ S∞ and T2 = T2∗ ∈ S1 . Then 1 n± (s1 + s2 ; T1 + t T2 ) dµ(t) ≤ n± (s1 ; T1 ) + T2 1 , s1 , s2 > 0. πs 2 R
(2.7)
Let H± and H0 be two lower-bounded self-adjoint operators acting in the same Hilbert space. Let λ0 < inf σ(H± ) ∪ σ(H0 ). First of all, assume that (1.2) holds with H = H± for some γ > 0. Further, let
Finally, suppose that
V := ±(H± − H0 ) ≥ 0,
(2.8)
V 1/2 (H0 − λ0 )−1/2 ∈ S∞ .
(2.9)
V 1/2 (H0 − λ0 )−γ ∈ S2
(2.10)
holds for some γ > 0. For z ∈ C with Im z > 0 set T (z): = V 1/2 (H0 − z)−1 V 1/2 .
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Lemma 2.2. (see, e.g., [8, Lemma 4.1]) Let (2.8)–(2.10) hold. Then for almost every E ∈ R the operator-norm limit T (E + i0) := n − limδ↓0 T (E + iδ) exists, and by (2.9) we have T (E + i0) ∈ S∞ . Moreover, Im T (E + i0) ∈ S1 . Theorem 2.1. [8, Theorem 1.2] Let (1.2) with H = H± , and (2.8)–(2.10) hold. Then for almost every E ∈ R we have ξ(E; H± , H0 ) = ± n∓ (1; Re T (E + i0) + t Im T (E + i0)) dµ(t). R
Remark. The representation of the SSF described in the above theorem has been generalized to non-sign-definite perturbations in [6] in the case of trace-class perturbations, and in [10] in the case of relatively trace-class perturbations. These generalizations are based on the concept of the index of orthogonal projections. We will not use them in the present paper. Suppose now that V satisfies (1.4). Then relations (1.2) and (2.8)–(2.10) hold with V = V , H0 = H0 , and γ = γ = 1. For z ∈ C, Im z > 0, set T (z) := V 1/2 (H0 − z)−1 V 1/2 . By Lemma 2.2, for almost every E ∈ R the operator-norm limit T (E + i0) := n − lim T (E + iδ) (2.11) δ↓0
exists, and Im T (E + i0) ∈ S1 .
(2.12)
For trivial reasons the limit in (2.11) exists and (2.12) holds for each E < 0. In Corollary 4.3 below we show that this is also true for each E ∈ [0, ∞) \ 2bZ+ . Hence, by Lemma 2.1, the quantity R n∓ (1; Re T (E + i0) + t Im T (E + i0)) dµ(t) is well defined for every E ∈ R \ 2bZ+ . Set ˜ ξ(E; H± , H0 ) = ± n∓ (1; Re T (E + i0) + t Im T (E + i0)) dµ(t), E ∈ R \ 2bZ+ . R
(2.13)
By Theorem 2.1 we have ˜ ξ(E; H± , H0 ) = ξ(E; H± , H0 )
(2.14)
for almost every E ∈ R. Remark. In [4] it is shown that the function ξ˜ defined on R \ 2bZ+ is continuous away from the eigenvalues of the operator H± . Note that, in contrast to the case b = 0, we cannot rule out the possibility of existence of embedded eigenvalues, by imposing short-range assumptions of the type of (1.4) or (1.5): Theorem 5.1 of [1] shows that there are axisymmetric potentials V of compact support such that below each Landau level 2bq, q ∈ Z+ , there exists an infinite sequence of eigenvalues of H− which converges to 2bq. On the other hand, generically, the only possible accumulation points of the eigenvalues of the operators H± are the Landau levels (see [1, Theorem 4.7], [5, Theorem 3.5.3 (iii)]). Further information of the location of these eigenvalues can be found in [4].
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3 Main results 3.1. In this subsection we formulate our main results. To this end we need some more notations. Introduce the Landau Hamiltonian 2 2 ∂ bx2 ∂ bx1 h(b) := i − + i + − b, (3.1) ∂x1 2 ∂x2 2 i.e., the two-dimensional Schr¨odinger operator with constant scalar magnetic field b > 0, essentially self-adjoint on C0∞ (R2 ). It is well known that σ(h(b)) = ∪∞ q=0 {2bq}, and each eigenvalue 2bq, q ∈ Z+ , has infinite multiplicity (see, e.g., [1]). For x, x ∈ R2 denote by Pq,b (x, x ) the integral kernel of the orthogonal projection pq (b) onto the subspace Ker (h(b) − 2bq), q ∈ Z+ . It is well known that Pq,b (x, x ) =
b Lq 2π
b|x − x |2 2
b (3.2) exp − (|x − x |2 + 2i(x1 x2 − x1 x2 )) 4
(see [7] or [12, Subsection 2.3.2]) where Lq (t) :=
q 1 t dq (tq e−t ) q (−t)k e , = k q! dtq k!
t ∈ R,
q ∈ Z+ ,
k=0
b are the Laguerre polynomials. Note that Pq,b (x, x) = 2π for each q ∈ Z+ and x ∈ R2 . Introduce the orthogonal projections Pq : L2 (R3 ) → L2 (R3 ), q ∈ Z+ , by (Pq u)(X⊥ , x3 ) = Pq,b (X⊥ , X⊥ ) u(X⊥ , x3 ) dX⊥ , u ∈ L2 (R3 ). (3.3) R2
Assume that (1.4) holds. Set W (X⊥ ) :=
R
V (X⊥ , x3 )dx3 ,
X ⊥ ∈ R2 .
(3.4)
X ⊥ ∈ R2 ,
(3.5)
If, moreover, V satisfies (1.5), then 0 ≤ W (X⊥ ) ≤ C0 X⊥ −m+1 , where C0 = C0
R
x−m dx. For q ∈ Z+ and λ > 0 introduce the operator 1 ωq (λ) := √ pq W pq . 2 λ
Evidently, ωq (λ) is self-adjoint and non-negative in L2 (R2 ). Lemma 3.1. Let U ∈ Lr (R2 ), r ≥ 1, and q ∈ Z+ . Then pq U pq ∈ Sr .
(3.6)
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Proof. If U ∈ L∞ (R2 ), then evidently pq U pq ≤ U L∞ . If U ∈ L1 (R2 ), we write pq U pq = pq |U |1/2 ei arg U |U |1/2 pq , check that pq |U |1/2 22 =
b U L1 , 2π
ei arg U |U |1/2 pq 22 =
b U L1 , 2π
b U L1 . Interpolating, we get pq U pq rr ≤ and conclude that pq U pq 1 ≤ 2π b r 2π U Lr which implies the desired result.
Remark. The proof of Lemma 3.1 follows the idea of the proof of [11, Lemma 5.1]. We include it here in order to make the exposition self-contained. If λ > 0, and V satisfies (1.4) with m⊥ > 2 and m3 > 1, then Lemma 3.1 with U = W implies ωq (λ) ∈ S1 . Theorem 3.1. Assume that (1.5) is valid. Let q ∈ Z+ , b > 0. Then the asymptotic estimates ˜ (3.7) ξ(2bq − λ; H+ , H0 ) = O(1), and ˜ − λ; H− , H0 ) ≤ −n+ ((1 + ε); ωq (λ)) + O(1), −n+ ((1 − ε); ωq (λ)) + O(1) ≤ ξ(2bq (3.8) hold as λ ↓ 0 for each ε ∈ (0, 1). Suppose that the potential V satisfies (1.4). For λ > 0 define the matrixvalued function w11 w12 (3.9) , X ⊥ ∈ R2 , Wλ = Wλ (X⊥ ) := w21 w22 where w11 :=
R
√ V (X⊥ , x3 ) cos ( λx3 )dx3 , 2
w12 = w21 :=
R
w22 :=
R
√ V (X⊥ , x3 ) sin2 ( λx3 )dx3 ,
√ √ V (X⊥ , x3 ) cos ( λx3 ) sin ( λx3 )dx3 .
Introduce the operator 1 Ωq := √ pq Wλ pq . (3.10) 2 λ Evidently, Ωq (λ) is self-adjoint and non-negative in L2 (R2 )2 . Moreover, using the fact that ωq (λ) ∈ S1 , it is easy to check that Ωq (λ) ∈ S1 as well. Theorem 3.2. Assume that (1.5) is valid. Let q ∈ Z+ , b > 0. Then the asymptotic estimates 1 ˜ Tr arctan ((1 ± ε)−1 Ωq (λ)) + O(1) ≤ ξ(2bq + λ; H± , H0 ) π 1 ≤ ± Tr arctan ((1 ∓ ε)−1 Ωq (λ)) + O(1) π hold as λ ↓ 0 for each ε ∈ (0, 1). ±
(3.11)
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The proofs of Theorems 3.1 and 3.2 can be found in Section 5. In the following ˜ subsection we will describe explicitly the asymptotics of ξ(2bq − λ; H− , H0 ) and ˜ ξ(2bq + λ; H± , H0 ) as λ ↓ 0 under generic assumptions about the behaviour of W (X⊥ ) as |X⊥ | → ∞. 3.2. Relations (3.8) and (3.11) allow us to reduce the analysis of the behaviour as ˜ λ → 0 of ξ(2bq + λ; H± , H0 ), to the study of the asymptotic distribution of the eigenvalues of Toeplitz-type operators pq U pq . The following three lemmas concern the spectral asymptotics of such operators. Lemma 3.2. [11, Theorem 2.6] Let the function U ∈ C 1 (R2 ) satisfy the estimates 0 ≤ U (X⊥ ) ≤ C1 X⊥ −α ,
|∇U (X⊥ )| ≤ C1 X⊥ −α−1 ,
X ⊥ ∈ R2 ,
for some α > 0 and C1 > 0. Assume, moreover, that U (X⊥ ) = u0 (X⊥ /|X⊥ |)|X⊥ |−α (1 + o(1)),
|X⊥ | → ∞,
where u0 is a continuous function on S1 which is non-negative and does not vanish identically. Then for each q ∈ Z+ we have n+ (s; pq U pq ) =
b
X⊥ ∈ R2 |U (X⊥ ) > s (1 + o(1)) = 2π ψα (s; u0 , b) (1 + o(1)),
where |.| denotes the Lebesgue measure, and ψα (s) = ψα (s; u0 , b) := s
−2/α
b 4π
S1
u0 (t)2/α dt,
s > 0.
s ↓ 0,
(3.12)
Remark. Theorem 2.6 of [11] contains a considerably more general result than Lemma 3.2. For the sake of the simplicity of exposition, here we reproduce only the special case of asymptotically homogeneous U . Lemma 3.3. [13, Theorem 2.1, Proposition 4.1] Let 0 ≤ U ∈ L∞ (R2 ). Assume that ln U (X⊥ ) = −µ|X⊥ |2β (1 + o(1)),
|X⊥ | → ∞,
for some β ∈ (0, ∞), µ ∈ (0, ∞). Then for each q ∈ Z+ we have n+ (s; pq U pq ) = ϕβ (s)(1 + o(1)), where ϕβ (s) = ϕβ (s; µ, b) :=
b | ln s|1/β if 0 < β 2µ1/β 1 β= ln (1+2µ/b) | ln s| if β −1 | ln s| if β−1 (ln | ln s|)
s ↓ 0,
< 1, 1, 1 < β < ∞,
s ∈ (e, ∞). (3.13)
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Lemma 3.4. [13, Theorem 2.2, Proposition 4.1] Let 0 ≤ U ∈ L∞ (R2 ). Assume that the support of U is compact, and that there exists a constant C > 0 such that U ≥ C on an open non-empty subset of R2 . Then for each q ∈ Z+ we have n+ (s; pq U pq ) = ϕ∞ (s) (1 + o(1)), where
ϕ∞ (s) := (ln | ln s|)−1 | ln s|,
s ↓ 0,
s ∈ (e, ∞).
(3.14)
Remark. For each β ∈ (0, ∞] and c > 0 we have ϕβ (cs) = ϕβ (s)(1 + o(1)) as s ↓ 0. Employing Lemmas 3.2, 3.3, 3.4, and the above remark, we find that (3.8) immediately entails the following corollary. Corollary 3.1. Let (1.5) hold with m > 3. i) Assume that the hypotheses of Lemma 3.2 hold with U = W and α = m − 1. Then we have √ b ˜ ξ(2bq − λ; H− , H0 ) = − X⊥ ∈ R2 |W (X⊥ ) > 2 λ (1 + o(1)) 2π √ = −ψm−1 (2 λ; u0 , b) (1 + o(1)), λ ↓ 0, (3.15) the function ψα being defined in (3.12). ii) Assume that the hypotheses of Lemma 3.3 hold with U = W . Then we have √ ˜ ξ(2bq − λ; H− , H0 ) = −ϕβ ( λ; µ, b) (1 + o(1)), λ ↓ 0, β ∈ (0, ∞), the functions ϕβ being defined in (3.13). iii) Assume that the hypotheses of Lemma 3.4 hold with U = W . Then we have √ ˜ ξ(2bq − λ; H− , H0 ) = −ϕ∞ ( λ) (1 + o(1)), λ ↓ 0, the function ϕ∞ being defined in (3.14). ˜ Remark. In the special case q = 0 when −ξ(−λ; H− , H0 ) coincides for almost every λ > 0 with the eigenvalue counting function for the operator H− (see (1.3)), relation (3.15) was established for the first time in [16]. Here we use a different approach related to the one developed in [11]. Similarly, the combination of Theorem 3.2 with Lemmas 3.2–3.4 yields the following corollary. Corollary 3.2. i) Let (1.5) hold with m > 3. Assume that the hypotheses of Lemma 3.2 are fulfilled for U = W and α = m − 1. Then we have √ b ˜ arctan ((2 λ)−1 W (X⊥ ))dX⊥ (1 + o(1)) ξ(2bq + λ; H± , H0 ) = ± 2 2π R2 √ 1 =± ψm−1 (2 λ; u0 , b) (1 + o(1)), λ ↓ 0. 2 cos (π/(m − 1))
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ii) Let (1.5) hold with m > 3. Suppose in addition that V satisfies (1.4) for some m⊥ > 2 and m3 > 2. Finally, assume that the hypotheses of Lemma 3.3 are fulfilled for U = W . Then we have √ 1 ˜ ξ(2bq + λ; H± , H0 ) = ± ϕβ ( λ; µ, b) (1 + o(1)), 2
λ ↓ 0,
β ∈ (0, ∞).
iii) Let the assumptions of the previous part be fulfilled, except that the hypotheses of Lemma 3.3 are replaced by those of Lemma 3.4. Then we have √ 1 ˜ ξ(2bq + λ; H± , H0 ) = ± ϕ∞ ( λ) (1 + o(1)), 2
λ ↓ 0.
The proof of Corollary 3.2 can be found in Section 6. 3.3. In this subsection we present a possible interpretation of our results directly in the terms of the SSF ξ(.; H± , H0 ) which is invariant of the choice of the representative of the equivalence class determined by the SSF. For λ > 0, and q ∈ Z+ , introduce the averaged values of the SSF Ξ± q,< (λ) :=
1 λ
Ξ± q,> (λ) :=
1 λ
2bq
2bq−λ
ξ(s; H± , H0 )ds =
1 λ
ξ(s; H± , H0 )ds =
1 λ
2bq+λ
2bq
λ
0
ξ(2bq − t; H± , H0 )dt,
λ
0
ξ(2bq + t; H± , H0 )dt.
± Since ξ(.; H± , H0 ) ∈ L1loc (R), the quantities Ξ± q,< (λ) and Ξq,> (λ) are well defined for every λ > 0. Applying (2.14), we find that the asymptotic bound Ξ+ q,< (λ) = O(1) as λ ↓ 0 follows from (3.7). Further, Corollary 3.1 i) implies
Ξ− q,< (λ) = −
√ m−1 ψm−1 (2 λ; u0 , b) (1 + o(1)), m−2
λ ↓ 0,
m > 3,
while Corollary 3.1 ii)–iii) entails √ Ξ− q,< (λ) = − ϕβ ( λ; µ, b) (1 + o(1)),
λ ↓ 0,
β ∈ (0, ∞].
Finally, it follows from Corollary 3.2 i) that Ξ± q,> (λ) = ±
√ m−1 1 ψm−1 (2 λ; u0 , b) (1 + o(1)), λ ↓ 0, m > 3, 2 cos (π/(m − 1)) m − 2
while Corollary 3.2 ii)–iii) implies Ξ± q,> (λ) = ±
√ 1 ϕβ ( λ; µ, b) (1 + o(1)), λ ↓ 0, 2
β ∈ (0, ∞].
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4 Preliminary estimates −1 d2 For z ∈ C with Im z > 0, define the operator R(z) := − dx bounded in 2 − z 3
L2 (R), as well as the operators
Tq (z) := V 1/2 Pq (H0 − z)−1 V 1/2 ,
q ∈ Z+ ,
bounded in L2 (R3 ) (see (3.3) for the definition of the orthogonal projection Pq ). Rz (x) = The √ operator √ R(z) admits the integral √ kernel Rz (x3 − x3 ) where √ i z|x| ie /(2 z), x ∈R, the branch of z being chosen so that Im z > 0. Moreover, Tq (z) = V 1/2 pq (b) ⊗ R(z − 2bq) V 1/2 . For λ ∈ R, λ = 0, define R(λ) as the operator with integral kernel Rλ (x3 − x3 ) where √ e− √−λ|x| if λ < 0, 2√ −λ x ∈ R. (4.1) Rλ (x) := lim Rλ+iδ (x) = iei √λ|x| if λ > 0, δ↓0 2 λ
Evidently, if w ∈ L2 (R) and λ = 0, then wR(λ)w¯ ∈ S2 . For E ∈ R, E = 2bq, q ∈ Z+ , set Tq (E) := V 1/2 pq (b) ⊗ R(E − 2bq) V 1/2 . Proposition 4.1. Let E ∈ R, q ∈ Z+ , E = 2bq. Let (1.4) hold. Then Tq (E) ∈ S2 , and Tq (E) 22 ≤ C1 b/|E − 2bq| (4.2) with C1 independent of E, b, and q. Moreover, lim Tq (E + iδ) − Tq (E) 2 = 0.
(4.3)
δ↓0
Proof. The operator Tq (E) admits the representation Tq (E) = M (Gq ⊗ JE−2bq ) M
(4.4)
where M : L2 (R3 ) → L2 (R3 ) is the multiplier by V (X⊥ , x3 )1/2 X⊥ m⊥ /2 x3 m3 /2 , Gq : L2 (R2X ) → L2 (R2X⊥ ) is the operator with integral kernel ⊥
−m⊥ /2 )X⊥ , X⊥ −m⊥ /2 Pq,b (X⊥ , X⊥
X⊥ , X⊥ ∈ R2 ,
(see (3.2) for the definition of Pq,b ), while Jλ : L2 (Rx3 ) → L2 (Rx3 ) is the operator with integral kernel x3 −m3 /2 Rλ (x3 − x3 )x3 −m3 /2 ,
x3 , x3 ∈ R,
λ ∈ R \ {0}.
(4.5)
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Evidently, Tq (E) 22 ≤ M 4 Gq 22 JE−2bq 22 . By (1.4) we have M 4 ≤ C02 . Further, JE−2bq 22
= R
|RE−2bq (x3 − x3 )|2 x3 −m3 x3 −m3 dx3 dx3 1 ≤ 4|E − 2bq|
R
−m3
x3
2 dx3
.
b Finally, since Gq ≤ 1, we have Gq 22 ≤ Gq 1 = 2π X⊥ −m⊥ dX⊥ . Hence, R2 2 2 C (4.2) holds with C1 = 8π0 R x3 −m3 dx3 X⊥ −m⊥ dX⊥ . To prove (4.3), we R2 write 2 2 Tq (E + iδ) − Tq (E) 2 = V (X⊥ , x3 )V (X⊥ , x3 )|Pq,b (X⊥ , X⊥ )| R2
R2
R
R
|RE−2bq+iδ (x3 − x3 ) − RE−2bq (x3 − x3 )|2 dx3 dx3 dX⊥ dX⊥ ,
note that limδ↓0 RE−2bq+iδ (x) = RE−2bq (x) for each x ∈ R, and that the integrand in the above integral is bounded from above for each δ > 0 by the L1 (R6 )-function 1 2 , x3 )|Pq,b (X⊥ , X⊥ )| , V (X⊥ , x3 )V (X⊥ |E − 2bq|
(X⊥ , x3 , X⊥ , x3 ) ∈ R6 .
Therefore, the dominated convergence theorem implies (4.3). Remark. Using more sophisticated tools than those of the proof of Proposition 4.1, it is shown in [4] that for E = 2bq we have not only Tq (E) ∈ S2 , but also Tq (E) ∈ S1 . We will not use this fact here. Corollary 4.1. Assume that (1.4) holds. Let E ∈ R, q ∈ Z+ , E = 2bq. Then Im Tq (E) ≥ 0. Moreover, if E < 2bq, then Im Tq (E) = 0. Proof. The non-negativity of Im Tq (E) follows from the representation Im Tq (E + iδ) = δV 1/2 Pq ((H0 − E)2 + δ 2 )−1 Pq V 1/2 ,
δ > 0,
and the limiting relation Im Tq (E) = n − limδ↓0 Im Tq (E + iδ), E = 2bq, which on its turn is implied by (4.3). Moreover, if E < 2bq, then (4.1) entails Tq (E) = Tq (E)∗ so that Im Tq (E) = 0. Corollary 4.2. Under the assumptions of Corollary 4.1 we have Im Tq (E) ∈ S1 . Furthermore, if E > 2bq, then b (E − 2bq)−1/2 V (x)dx. (4.6) Im Tq (E) 1 = Tr Im Tq (E) = 4π R3
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Proof. Bearing in mind the representation (4.4), we find that the inclusion Im Tq (E) ∈ S1 would be implied by the inclusion Gq ∈ S1 and Im JE−2bq ∈ S1 . The first inclusion follows from Lemma 3.1, and the second one from the obvious fact that rank Im JE−2bq ≤ 2. Further, the first equality in (4.6) follows from the non-negativity of the operator Im Tq (E) which is guaranteed by Corollary 4.1. Since the operator Im Tq (E) with E > 2bq admits the kernel 1 √ V (X⊥ , x3 ) cos E − 2bq(x3 − x3 ) Pq,b (X⊥ , X⊥ ) 2 E − 2bq , x ), (X , x ), (X , x ) ∈ R3 , V (X⊥ ⊥ 3 ⊥ 3 3 the Mercer theorem (see, e.g., the lemma on pp. 65–66 of [14]) implies the second equality in (4.6). Proposition 4.2. Let q ∈ Z+ , λ ∈ R, |λ| ∈ (0, b], and δ > 0. Assume that V satisfies (1.4). Then the operator series Tq+ (2bq + λ + iδ) :=
∞
Tl (2bq + λ + iδ),
(4.7)
Tl (2bq + λ)
(4.8)
l=q+1
Tq+ (2bq + λ) :=
∞ l=q+1
are convergent in S2 . Moreover, Tq+ (2bq + λ) 22 ≤
∞ C0 b (2b(l − q) − λ)−3/2 V (x)dx. 8π R3
(4.9)
l=q+1
Finally,
lim Tq+(2bq + λ + iδ) − Tq+ (2bq + λ) 2 = 0. δ↓0
(4.10)
Proof. For each q , q ∈ Z+ such that q + 1 ≤ q < q < ∞ we have −1 q q √ √ 2 d2 + 2 Tl (2bq + λ) 2 = V pl ⊗ − 2 + 2b(l − q) − λ V 2 dx 3 l=q l=q −2 q d2 ≤ C0 Tr pl ⊗ − 2 + 2b(l − q) − λ V dx3 l=q
q dη C0 b V (x)dx = 2 (2π)2 (η 2 + 2b(l − q) − λ) R3 l=q R q C0 b (2b(l − q) − λ)−3/2 V (x)dx. = 8π R3
l=q
(4.11)
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−3/2 Since the numerical series ∞ is convergent, and S2 is a l=q+1 (2b(l − q) − λ) Hilbert (hence, complete) space, we conclude that (4.11) entails the convergence in S2 of the operator series in (4.8), as well as the validity of (4.9). The convergence of the series in (4.7) is proved in exactly the same manner. Finally, (4.10) follows from the estimate Tq+ (2bq + λ + iδ) − Tq+ (2bq + λ) 22 ∞ dη 2 C0 b ≤δ V (x)dx (2π)2 (η 2 + 2b(l − q) − λ)2 ((η 2 + 2b(l − q) − λ)2 + δ 2 ) R3 l=q+1 R ∞ ∞ C0 b dη ≤ δ2 2 (2b(l − q) − λ)−7/2 V (x)dx. 2π (η 2 + 1)4 R3 0 l=q+1
For E = 2bq + λ with q ∈ Z+ , and λ ∈ R, |λ| ∈ (0, b], set Tq− (E) := T (E) − Tq (E) − Tq+ (E) (see (4.8)). Note that if q = 0, then Tq− (E) = 0, and if q−1 q ≥ 1, then Tq− (E) = l=0 Tl (E). Corollary 4.3. For E = 2bq + λ with q ∈ Z+ and λ ∈ R, |λ| ∈ (0, b] the operatornorm limit (2.11) exists, and T (E + i0) = Tq− (E) + Tq (E) + Tq+ (E).
(4.12)
Re T (E + i0) = Re Tq− (E) + Re Tq (E) + Tq+ (E),
(4.13)
Moreover,
Im T (E + i0) = Im
Tq− (E)
+ Im Tq (E) ∈ S1 .
(4.14)
q Proof. Let δ > 0. Evidently, T (E + iδ) = l=0 Tl (E + iδ) + Tq+ (E + iδ) (see (4.7)). Proposition 4.1 implies that n − limδ↓0 ql=0 Tl (E + iδ) = Tq− (E) + Tq (E), while Proposition 4.2 implies that n − limδ↓0 Tq+ (E) = Tq+ (E). Combining the above two relations, we get (4.12). Relation (4.13) follows immediately from (4.12) and Tq+ (E) = Tq+ (E)∗ , while (4.14) is implied by (4.12) and Corollaries 4.1 and 4.2.
5 Proof of the main results 5.1. This subsection contains a general estimate which will be used in the proofs of all our main results. Informally speaking, we show that we can replace the operator T (E + i0) by Tq (E) in the r.h.s of (2.13) when we deal with the first ˜ asymptotic term of ξ(E; H± , H0 ) as the energy E approaches a given Landau level 2bq, q ∈ Z+ .
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Proposition 5.1. Assume that (1.4) holds. Let E = 2bq+λ with q ∈ Z+ , and λ ∈ R, |λ| ∈ (0, b]. Then the asymptotic estimates n± (1 + ε; Re Tq (E) + t Im Tq (E)) dµ(t) + O(1) R ≤ n± (1; Re T (E + i0) + t Im T (E + i0)) dµ(t) R ≤ n± (1 − ε; Re Tq (E) + t Im Tq (E)) dµ(t) + O(1) (5.1) R
hold as λ → 0 for each ε ∈ (0, 1). Proof. Using (4.13) and (4.14), and applying the Weyl inequalities (2.3), we get n± (1 + ε; Re Tq (E) + t Im Tq (E)) dµ(t) R − n∓ (ε; Re Tq− (E) + Tq+ (E) + t Im Tq− (E)) dµ(t) R ≤ n± (1; Re T (E + i0) + t Im T (E + i0)) dµ(t) R ≤ n± (1 − ε; Re Tq (E) + t Im Tq (E)) dµ(t) R + n± (ε; Re Tq− (E) + Tq+ (E) + t Im Tq− (E)) dµ(t). (5.2) R
In order to conclude that (5.2) implies (5.1), it remains to show that n± (ε; Re Tq− (E) + Tq+ (E) + t Im Tq− (E)) dµ(t) = O(1), λ → 0,
(5.3)
R
for each ε > 0. Employing (2.7) and (2.3), we find that n± (ε; Re Tq− (E) + Tq+ (E) + t Im Tq− (E)) dµ(t) R
2 Im Tq− (E) 1 επ 2 Im Tq− (E) 1 , ≤ n± (ε/4; Re Tq− (E)) + n± (ε/4; Tq+(E)) + επ ≤ n± (ε/2; Re Tq− (E) + Tq+ (E)) +
ε > 0.
(5.4)
If q = 0, and, hence, Tq− (E) = 0, we need only to apply (2.5) with p = 2, and (4.9), in order to get n± (ε/4; Tq+(E)) ≤ 16ε−2 Tq+ (E) 22 ≤
∞ 2C0 b −3/2 (2b(l − q) − λ) V (x)dx, ε2 π R3 l=q+1
(5.5)
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which combined with (5.4) yields (5.3). If q ≥ 1 we should also utilize the estimate n± (ε/4; Re Tq− (E)) ≤ 32ε−2 Tq−(E) 22 ≤ 32ε−2 qC1 b
q−1
(2b(q − l) + λ)−1
(5.6)
l=0
which follows from (2.6), (2.4) with p = 2, and (4.2), as well as the estimate Im Tq− (E) 1 ≤
q−1 b (2b(q − l) + λ)−1/2 V (x)dx, 2π R3
(5.7)
l=0
which follows from (4.6). Thus, in the case q ≥ 1, estimate (5.3) is implied by from the combination of (5.4)–(5.7). 5.2. In this subsection we complete the proof of the first part of Theorem 3.1. Since Im Tq (2bq−λ) = 0 and Re Tq (2bq−λ) = Tq (2bq−λ) ≥ 0 if λ > 0, Proposition 5.1 implies immediately the following corollary. Corollary 5.1. Under the hypotheses of Proposition 5.1 the asymptotic estimates n− (1; Re T (2bq − λ + i0) + t Im T (2bq − λ + i0)) dµ(t) = O(1), (5.8) R
and ≤
R
n+ (1 + ε; Tq (2bq − λ)) + O(1) n+ (1; Re T (2bq − λ + i0) + t Im T (2bq − λ + i0)) dµ(t) ≤ n+ (1 − ε; Tq (2bq − λ)) + O(1)
(5.9)
hold as λ ↓ 0 for each ε ∈ (0, 1). Now the combination of (2.13) and (5.8) yields (3.7). 5.3. In this section we complete the proof of the second part of Theorem 3.1. For q ∈ Z+ and λ > 0 define Oq (λ) : L2 (R3 ) → L2 (R3 ) as the operator with integral kernel 1 , x ), √ V (X⊥ , x3 ) Pq,b (X⊥ , X⊥ ) V (X⊥ (X⊥ , x3 ), (X⊥ , x3 ) ∈ R3 . 3 2 λ Proposition 5.2. Under the hypotheses of Theorem 3.1 the asymptotic estimates n+ ((1 + ε)s; Oq (λ)) + O(1) ≤ n+ (s; Tq (2bq − λ)) ≤ n+ ((1 − ε)s; Oq (λ)) + O(1) (5.10) hold as λ ↓ 0 for each ε ∈ (0, 1) and s > 0.
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Proof. Fix s > 0 and ε ∈ (0, 1). Then the Weyl inequalities entail n+ ((1 + ε)s; Oq (λ)) − n− (εs; Tq (2bq − λ) − Oq (λ)) ≤ n+ (s; Tq (2bq − λ)) ≤ n+ ((1 − ε)s; Oq (λ)) + n+ (εs; Tq (2bq − λ) − Oq (λ)). In order to get (5.10), it suffices to show that n± (t; Tq (2bq − λ) − Oq (λ)) = O(1),
λ ↓ 0,
(5.11)
for every fixed t > 0. Denote by T˜q the operator with integral kernel −
1 , x ), V (X⊥ , x3 ) |x3 − x3 | Pq,b (X⊥ , X⊥ ) V (X⊥ 3 2
(X⊥ , x3 ), (X⊥ , x3 ) ∈ R3 .
(5.12)
Pick m ∈ (3, m), and write ˜ q,m−m ⊗ J˜(0) M ˜ m,m ˜ m,m G T˜q = M m
˜ m,m is the multiplier by the bounded function V (X⊥ , x3 )X⊥ (m−m )/2 where M ˜ q,m−m : L2 (R2 ) → L2 (R2 ) is the operator with integral x3 m /2 , (X⊥ , x3 ) ∈ R3 , G kernel
−(m−m )/2 X⊥ −(m−m )/2 Pq,b (X⊥ , X⊥ )X⊥ ,
X⊥ , X⊥ ∈ R2 ,
(0) and J˜m : L2 (R) → L2 (R) is the operator with integral kernel 1 − x3 −m /2 |x3 − x3 |x3 −m /2 , 2
x3 , x3 ∈ R.
˜ q,m−m is compact, and Since m − m > 0, Lemma 3.1 implies that the operator G (0) ˜ ˜ since m > 3 we have Jm ∈ S2 . Finally, since Mm,m is bounded, we find that the operator T˜q is compact. Further, ˜ q,m−m ⊗ J˜(λ) M ˜ m,m ˜ m,m G Tq (2bq − λ) − Oq (λ) = M m (λ) where J˜m , λ > 0, is the operator with integral kernel √ √ λ|x3 − x3 | 1 −m /2 − 12 λ|x3 −x3 | x3 −m /2 , e sinh − √ x3 2 λ
x3 , x3 ∈ R.
(λ) Applying the dominated convergence theorem, we easily find that limλ↓0 J˜m − (0) J˜m 2 = 0. Therefore, T˜q = n − limλ↓0 (Tq (2bq − λ) − Oq (λ)). Fix t > 0. Choosing
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λ > 0 so small that Tq (2bq − λ) − Oq (λ) − T˜q < t/2, and applying the Weyl inequalities, we get n± (t; Tq (2bq − λ) − Oq (λ)) ≤ n± (t/2; Tq (2bq − λ) − Oq (λ) − T˜q ) + n± (t/2; T˜q ) = n± (t/2; T˜q ). (5.13) Since the r.h.s. of (5.13) is finite and independent of λ, we conclude that (5.13) entails (5.11). Proposition 5.3. Assume that (1.4) holds. Then for each q ∈ Z+ , λ > 0, and s > 0 we have (5.14) n+ (s; Oq (λ)) = n+ (s; ωq (λ)) (see (3.6) for the definition of the operator ωq (λ)). Proof. Define the operator K : L2 (R3 ) → L2 (R2 ) by , x )u(X , x ) dx dX , Pq,b (X⊥ , X⊥ ) V (X⊥ (Ku)(X⊥ ) := ⊥ 3 3 ⊥ 3 R2
R
X ⊥ ∈ R2 ,
where u ∈ L2 (R3 ). The adjoint operator K ∗ : L2 (R2 ) → L2 (R3 ) is given by Pq,b (X⊥ , X⊥ )v(X⊥ ) dX⊥ , (X⊥ , x3 ) ∈ R3 , (K ∗ v)(X⊥ , x3 ) := V (X⊥ , x3 ) R2
2
2
where v ∈ L (R ). Obviously, 1 Oq (λ) = √ K ∗ K, 2 λ
1 ωq (λ) = √ K K ∗ . 2 λ
Since n+ (s; K ∗ K) = n+ (s; K K ∗ ) for each s > 0, we get (5.14). Putting together (2.13), (5.9), (5.10), and (5.14), we get (3.8). Thus, we are done with the proof of Theorem 3.1. 5.4. In this subsection we complete the proof of Theorem 3.2. Proposition 5.4. Let q ∈ Z+ and b > 0. Assume that (1.5) holds. Then the asymptotic estimates (5.15) n± (s; Re Tq (2bq + λ)) = O(1) are valid as λ ↓ 0 for each s > 0. Proof. The operator Re Tq (2bq + λ + i0) admits the integral kernel √ 1 , x ), − √ V (X⊥ , x3 ) sin λ|x3 − x3 | Pq,b (X⊥ , X⊥ ) V (X⊥ 3 2 λ (X⊥ , x3 ), (X⊥ , x3 ) ∈ R3 . Arguing as in the proof of Proposition 5.2, we find that n− limλ↓0 Re Tq (2bq + λ) = T˜q (see (5.12)). Fix s > 0. Choosing λ > 0 so small that Re Tq (2bq+λ)−T˜q < s/2, and applying the Weyl inequalities, we get n± (s; Re Tq (2bq + λ)) ≤ n± (s/2; T˜q ) which implies (5.15).
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Taking into account Propositions 5.1 and 5.4, and applying the Weyl inequalities and the evident identities 1 n± (s; tT )dµ(t) = Tr arctan (s−1 T ), s > 0, π R where T = T ∗ ≥ 0, T ∈ S1 , we obtain the following Corollary 5.2. Let q ∈ Z+ , b > 0. Assume that V satisfies (1.5). Then the asymptotic estimates 1 Tr arctan ((1 + ε)−1 Im Tq (2bq + λ)) + O(1) π ≤ n± (1; Re Tq (2bq + λ) + t Im Tq (2bq + λ))dµ(t) R
1 Tr arctan ((1 − ε)−1 Im Tq (2bq + λ)) + O(1) π are valid as λ ↓ 0 for each ε ∈ (0, 1). ≤
(5.16)
Proposition 5.5. Assume that (1.4) holds. Then for each q ∈ Z+ , λ > 0, and s > 0, we have n+ (s; Im Tq (2bq + λ)) = n+ (s; Ωq (λ)) (5.17) (see (3.10) for the definition of the operator Ωq (λ)). Consequently, Tr arctan (s−1 Im Tq (2bq + λ)) = Tr arctan (s−1 Ωq (λ))
(5.18)
for each q ∈ Z+ , λ > 0. Proof. The proof is quite similar to that of Proposition 5.3. Define the operator K : L2 (R3 ) → L2 (R2 )2 by Ku := v = (v1 , v2 ) ∈ L2 (R2 )2 , where
v1 (X⊥ ) :=
R2
v2 (X⊥ ) :=
R2
R
u ∈ L2 (R3 ),
√ , x )u(X , x ) dx dX , Pq,b (X⊥ , X⊥ ) cos( λx3 ) V (X⊥ ⊥ 3 3 ⊥ 3
√ Pq,b (X⊥ , X⊥ ) sin( λx3 ) R , x )u(X , x ) dx dX , V (X⊥ ⊥ 3 3 ⊥ 3
X ⊥ ∈ R2 .
Evidently, the adjoint operator K∗ : L2 (R2 )2 → L2 (R3 ) is given by √ (K∗ v)(X⊥ , x3 ) := cos( λx3 ) V (X⊥ , x3 ) Pq,b (X⊥ , X⊥ )v1 (X⊥ ) dX⊥ √ + sin( λx3 ) V (X⊥ , x3 )
R2
R2
Pq,b (X⊥ , X⊥ )v2 (X⊥ ) dX⊥ ,
(X⊥ , x3 ) ∈ R3 ,
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where v = (v1 , v2 ) ∈ L2 (R2 )2 . Obviously, 1 Im Tq (2bq + λ) = √ K∗ K, 2 λ
1 Ωq (λ) = √ K K∗ . 2 λ
Since n+ (s; K∗ K) = n+ (s; K K∗ ) for each s > 0, we get (5.17). Now the combination of (2.13), (5.1), (5.16), and (5.18) yields (3.11).
6 Proof of Corollary 2.2 Introduce the matrix-valued functions W 0 (1) W (X⊥ ) := , 0 0 W
(2)
(X⊥ ) = W
(2)
(X⊥ ; λ) := Wλ (X⊥ ) − W
(1)
(X⊥ ) =
−w22 w21
w12 w22
(see (3.4) and (3.9) for the definitions of W and Wλ , respectively), as well as the operators 1 (j) Ω(j) pq , q (λ) : √ pq W 2 λ
λ > 0,
q ∈ Z+ ,
j = 1, 2,
(j)
compact in L2 (R2 )2 . Evidently, Ωq (λ) ∈ S1 , j = 1, 2. Proposition 6.1. (i) Let (1.5) hold with m ∈ (3, 4]. Then for each q ∈ Z+ , s > 0, and δ > 4−m 2 , we have −δ Tr arctan (s−1 Ωq (λ)) − arctan (s−1 Ω(1) ), λ ↓ 0, (6.1) q (λ)) = O(λ (see (3.10) for the definition of the operator Ωq (λ)). (ii) Let (1.4) hold with m⊥ > 2, m3 > 2. Then for each q ∈ Z+ and s > 0 we have Tr arctan (s−1 Ωq (λ)) − arctan (s−1 Ω(1) λ ↓ 0. (6.2) q (λ)) = O(1), Proof. Applying the Lifshits-Kre˘ın trace formula (1.1), we easily get Tr arctan (s−1 Ωq (λ)) − arctan (s−1 Ω(1) q (λ)) 2 −1 = ξ(E; s−1 Ωq (λ); s−1 Ω(1) dE, q (λ))(1 + E ) R
s > 0. (6.3)
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Tr arctan (s−1 Ωq (λ)) − arctan (s−1 Ω(1) q (λ))
|ξ(E; s−1 Ωq (λ); s−1 Ω(1) q (λ))|dE ≤
1 1 Ωq (λ) − Ω(1) Ω(2) q (λ) 1 = q (λ) 1 s s (6.4)
(see [17, Theorem 8.2.1]). Further, b (2) Ωq (λ) 1 ≤ √ w22 (X⊥ )2 + w12 (X⊥ )2 dX⊥ 2π λ R2 √ b b √ ≤ |wj2 (X⊥ )|dX⊥ ≤ √ V (X⊥ , x3 )| sin( λx3 )|dx3 dX⊥ . 2π λ j=1,2 R2 π λ R2 R (6.5) , and m ∈ Assume now that V satisfies (1.5) with m ∈ (3, 4]. Pick δ > 4−m 2 (−2δ + 2, m − 2). Then we have √ V (X⊥ , x3 )| sin ( λx3 )|dx3 dX⊥ λ−1/2 R2 R −δ −(m−m ) ≤ λ C0 X⊥ dX⊥ x3 −m |x3 |−2δ+1 dx3 . (6.6) R2
R
Since m − m > 2 the integral with respect to X⊥ ∈ R2 is convergent, and since m + 2δ − 1 > 1 the integral with respect to x3 is convergent as well. Now the combination of (6.3)–(6.6) entails (6.1). Further, suppose that V satisfies (1.4) with m⊥ > 2 and m3 > 2. Then √ λ−1/2 V (X⊥ , x3 )| sin ( λx3 )|dx3 dX⊥ R2 R ≤ C0 X⊥ −m⊥ dX⊥ x3 −m3 |x3 |dx3 . (6.7) R2
R
Putting together (6.3)–(6.5), and (6.7), we get (6.2). Now note that if V satisfies (1.5) with m ∈ (3, 4], we can choose 4−m <δ< 2 so that in this case Proposition 6.1 entails −1/(m−1) Tr arctan (s−1 Ωq (λ)) − arctan (s−1 Ω(1) ), λ ↓ 0. (6.8) q (λ)) = o(λ
1 m−1
Moreover, if V satisfies (1.5) with m > 4, then it satisfies (1.4) with m⊥ > 2 and m3 > 2, and, hence, (6.2) is valid. Finally, −1 ωq (λ)) Tr arctan (s−1 Ω(1) q (λ)) = Tr arctan (s ∞ n+ (st; ωq (λ)) dt, = 1 + t2 0
s > 0,
λ > 0, (6.9)
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(see (3.6) for the definition of the operator ωq (λ)). Putting together (6.9), (6.8), and (6.2), we conclude that Corollary 3.2 follows easily from Theorem 3.2 and Lemmas 3.2–3.4. Acknowledgments. The authors are grateful to the anonymous referee whose valuable remarks contributed to the improvement of the text. Georgi Raikov was partially supported by the Chilean Science Foundation Fondecyt under Grant 1020737.
References [1] J. Avron, I. Herbst, B. Simon, Schr¨ odinger operators with magnetic fields. I. General interactions, Duke Math. J. 45, 847–883 (1978). ˇ Birman, M.G. Kre˘ın, On the theory of wave operators and scattering [2] M.S. operators, Dokl. Akad. Nauk SSSR 144 (1962), 475–478 [in Russian]; English translation in Soviet Math. Doklady 3 (1962). ˇ Birman, D.R. Yafaev, The spectral shift function. The papers of M.G. [3] M.S. Kre˘ın and their further development, (Russian) Algebra i Analiz 4, 1–44 (1992); English translation in St. Petersburg Math. J. 4, 833–870 (1993). [4] V. Bruneau, A. Pushnitski, G.D. Raikov, Spectral shift function in strong magnetic fields, Algebra i Analiz 16, 207–238 (2004). [5] C. G´erard, I. L aba, Multiparticle Quantum Scattering in Constant Magnetic Fields, Mathematical Surveys and Monographs, 90, AMS, Providence, RI, 2002. [6] F. Gesztesy, K. Makarov, The Ξ operator and its relation to Kre˘ın’s spectral shift function, J. Anal. Math. 81, 139–183 (2000). [7] L. Landau, Diamagnetismus der Metalle, Z. Physik 64, 629–637 (1930). [8] A. Pushnitski˘ı, A representation for the spectral shift function in the case of perturbations of fixed sign, Algebra i Analiz 9, 197–213 (1997) [in Russian]; English translation in St. Petersburg Math. J. 9, 1181–1194 (1998). [9] A. Pushnitski, Estimates for the spectral shift function of the polyharmonic operator, J. Math. Phys. 40, 5578–5592 (1999). [10] A. Pushnitski, The spectral shift function and the invariance principle, J. Funct. Anal. 183, 269–320 (2001). [11] G.D. Raikov, Eigenvalue asymptotics for the Schr¨ odinger operator with homogeneous magnetic potential and decreasing electric potential. I. Behaviour near the essential spectrum tips, Commun. P.D.E. 15, 407–434 (1990); Errata: Commun. P.D.E. 18, 1977–1979 (1993).
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[12] G.D. Raikov, M. Dimassi, Spectral asymptotics for quantum Hamiltonians in strong magnetic fields, Cubo Mat. Educ. 3, 317–391 (2001). [13] G.D. Raikov, S. Warzel, Quasi-classical versus non-classical spectral asymptotics for magnetic Schr¨ odinger operators with decreasing electric potentials, Rev. Math. Phys. 14, 1051–1072 (2002). [14] M. Reed, B. Simon, Methods of Modern Mathematical Physics. III. Scattering Theory, Academic Press, New York, 1979. [15] D. Robert, Semiclassical asymptotics for the spectral shift function, In: Differential Operators and Spectral theory, AMS Translations Ser. 2 189, 187–203, AMS, Providence, RI, 1999. [16] A.V. Sobolev, Asymptotic behavior of the energy levels of a quantum particle in a homogeneous magnetic field, perturbed by a decreasing electric field. I, Probl. Mat. Anal. 9, 67–84 (1984) [in Russian]; English translation in: J. Sov. Math. 35, 2201–2212 (1986). [17] D.R. Yafaev, Mathematical scattering theory. General theory, Translations of Mathematical Monographs, 105 AMS, Providence, RI, 1992. Claudio Fern´ andez Departamento de Matem´aticas Facultad de Matem´ aticas Pontificia Universidad Cat´ olica de Chile Av. Vicu˜ na Mackenna 4860 Santiago Chile email:
[email protected] Georgi D. Raikov Departamento de Matem´aticas Facultad de Ciencias Universidad de Chile Las Palmeras 3425 Santiago Chile email:
[email protected] Communicated by Bernard Helffer submitted 23/09/03, accepted 15/01/04
Ann. Henri Poincar´e 5 (2004) 405 – 434 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/030405-30 DOI 10.1007/s00023-004-0174-8
Annales Henri Poincar´ e
Exact Solution of the AVZ-Hamiltonian in the Grand-Canonical Ensemble Stephan Adams and Jean-Bernard Bru Abstract. The thermodynamic behavior of the Angelescu-Verbeure-Zagrebnov (AVZ) Hamiltonian [1], also called the superstable Bogoliubov model, is solved for any temperature and any chemical potential. It is found that its thermodynamics coincides with one for the Mean-Field Gas for small chemical potential or high temperature. However, for large chemical potential or low temperature, a non-conventional Bose condensation appears with, even at zero-temperature, a (non-zero) particle density outside the condensate. Following [2], the analysis in the present paper corresponds to the main technical step to deduce, in the canonical ensemble, a new microscopic theory of superfluidity at all temperatures explained in [3].
1 Introduction Let an interacting homogeneous gas of n spinless bosons with mass m be enclosed 3
in a cubic box Λ = × L ⊂ R3 . We denote by ϕ (x) = ϕ (x) a (real) two-body α=1
interaction potential and we assume that: (A) ϕ (x) ∈ L1 R3 . (B) Its (real) Fourier transformation λk = d3 xϕ (x) e−ikx , k ∈ R3 , R3
satisfies: λ0 > 0 and 0 ≤ λk = λ−k ≤
lim λk for k ∈ R3 .
k→0+
(C) The interaction potential ϕ (x) satisfies: λ0 λ0 + g00 ≥ 0, or (C2) : + g00 < 0, 2 2 where the (effective coupling) constant g00 equals 2 1 3 λk d k < 0, g00 ≡ − 3 εk 4 (2π) R3 (C1) :
(1.1)
with the one-particle energy spectrum defined by εk ≡ 2 k 2 /2m. The last conditions (C1)–(C2) will be important at the end of this paper and first appeared in the study of the Weakly Imperfect Bose Gas [1, 4–6].
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The (non-diagonal) AVZ-Hamiltonian [1], also called the superstable Bogoliubov Hamiltonian, is defined for λ0 > 0 by SB B HΛ,λ ≡ HΛ,0 + UΛMF , 0 >0
(1.2)
where UΛMF B HΛ,λ 0
≡
λ0 2V
a∗k1 a∗k2 ak2 ak1 =
k1 ,k2 ∈Λ∗
λ0 2 NΛ − NΛ , 2V
≡ TΛ + UΛD + UΛN D + UΛBMF ,
and NΛ
≡
(1.3) (1.4)
a∗k ak ,
k∈Λ∗
TΛ
≡
εk a∗k ak ,
k∈Λ∗
UΛD UΛN D UΛBMF
≡ ≡ ≡
1 2V 1 2V
k∈Λ∗ \{0}
λk a∗0 a0 a∗k ak + a∗−k a−k ,
(1.5)
2 λk a∗k a∗−k a20 + a∗0 ak a−k ,
(1.6)
k∈Λ∗ \{0}
λ0 ∗2 2 λ0 ∗ a a + a0 a0 2V 0 0 V
a∗k ak .
(1.7)
k∈Λ∗ \{0}
SB acts on the boson Fock space The Hamiltonian HΛ,λ 0 +∞
(n)
FΛB ≡ ⊕ HB , n=0
(n)
with HB defined as the symmetrized n-particle Hilbert spaces (n) (0) HB ≡ L2 (Λn ) symm , HB ≡ C, see [7, 8]. Using periodic boundary conditions, let 2πnα ∗ 3 , nα = 0, ±1, ±2, . . . , α = 1, 2, 3 Λ ≡ k ∈ R : kα = L ∗ be the set of wave vectors. Also, note that a# k = {ak or ak } are the usual boson 1 creation / annihilation operators in the one-particle state ψk (x) = V − 2 eikx , k ∈ Λ∗ , x ∈ Λ, acting on the boson Fock space FΛB . Under assumptions (A) and (B) SB on the interaction potential ϕ (x) the Hamiltonian HΛ,λ is superstable [8]. 0 To fix the notations, β > 0 is the inverse temperature, µ the chemical potential, ρ > 0 the fixed full particle density. Before we embark on the rigorous results of this model, it may be useful to give briefly its origin and history.
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B First note that, for λ0 > 0, the Hamiltonian HΛ,λ (1.4) is the Bogoliubov 0 >0 Hamiltonian [9–13], so-called the Weakly Imperfect Bose Gas. It is the starting point of the first microscopic theory of superfluidity proposed in 1947 by Bogoliubov [9–11]. Resuming the observations of [1, 2, 6], in various respects the BoB is not appropriate as the model of superfluidity. The first goliubov model HΛ,λ 0 problem of this theory was highlighted by Angelescu, Verbeure and Zagrebnov in B . 1992 [1]. It concerns the instability1 for positive chemical potential µ > 0 of HΛ,λ 0 In a sense the Bogoliubov theory is a series of recipes, which, after the first ansatz B , give a formula (the second ansatz) saving the theory from instaleading to HΛ,λ 0 bility for µ > 0 and from the gap in the spectrum. For a more detailed discussion of this problem, see the review [6]. Therefore, a “minimal” stabilization of the Bogoliubov Hamiltonian is to add the “forward scattering” interactions between particles above zero-mode. This apSB . Their proach was first developed in papers [1,14,15] and leads to the model HΛ,λ 0 main object was of course to correct the instability for positive chemical potenB but also to find a gapless Bogoliubov tials of the Bogoliubov Hamiltonian HΛ,λ 0 SB , spectrum. In [1], they use a Bogoliubov approximation partially applied on HΛ,λ 0 MF i.e., they save the Mean-Field interaction UΛ (1.3), whereas in [15], the authors use a “generalized” Bogoliubov approximation. This “generalized” ap √ Bogoliubov √
proximation corresponds to partially change the operators a0 / V , a∗0 / V by a
suitable function b (c) , b (c) in (1.2) except in the Mean-Field interaction UΛMF . Then, they prove a Bose condensation in zero-mode via second-order phase transition and a linear asymptotic of the elementary excitation spectrum in condensed phase for k → 0, see also discussions in Section 3.4 of [6]. Here we show that the two procedures [1,15] are inexact, in the sense that they SB . For are equivalent to some drastic modifications of the original Hamiltonian HΛ,λ 0 example, as Bogoliubov did, they were forced in [1] to add some additional assumptions to find a gapless spectrum. As it is explained with more details in [2], it was SB , in the grand-canonical ensemble, had unlikely that the exact solution of HΛ,λ 0 a gapless spectrum even in the presence of Bose-condensation. In fact, we prove that, on the thermodynamic level, the spectrum always has a gap in the grandcanonical ensemble (see below Remark 2.4). The main problem of their methods (Bogoliubov et al.) is to assume, a priori, the Bose condensation by directly doing the Bogoliubov approximation with an arbitrary choice of c or b (c) , without exactly solving it in terms of the thermodynamic behavior. As the review [6] explained in the “outline” section, we should be discouraged “from performing sloppy manipulations with Bose condensations, quantum fluctuations and different kinds of ans¨ atze”. The analysis in the present paper provides another strong warning in doing it. SB correActually, in the grand-canonical ensemble, the Hamiltonian HΛ,λ 0 sponds to a weaker truncation than the Bogoliubov one. This non-diagonal Bose 1 The
corresponding grand-canonicale pressure is infinite
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gas is here rigorously solved at the thermodynamic level in the grand-canonical ensemble. We do not search for a gapless spectrum in the grand-canonical ensemble. However, this result represents a crucial technical step to find a new microscopic theory of superfluidity with a gapless spectrum. The philosophy of this new approach is explained in [2] and comes from a constructive criticism of the Bogoliubov theories. In particular, the paper [2] gives the (new) physical arguments leading to SB as a tool in the grand-canonical ensemble. use the superstable Hamiltonian HΛ,λ 0 Then, using the present paper by choosing λ0 > 0 as an arbitrary parameter, we provide in [3] a new theory of superfluidity with a gapless spectrum at any particle densities and temperatures. It leads us to a deeper understanding of the Bose condensation phenomenon in liquid helium: coexistence in the superfluid liquid of particles inside and outside the Bose condensate (even at zero temperature), Bose/Bogoliubov distribution, “Cooper-type pairs” in the Bose condensate. In the next section, we present the exact thermodynamic behavior of the SB in the grand-canonical ensemble. In particular, the infinite Hamiltonian HΛ,λ 0 volume pressure is explicitly found via variational problems in theorem 2.2. These variational problems are then solved in theorem 2.3 leading to the exact phase diagram. The corresponding phase transition is finally explained by the last theorem of Section 2 (theorem 2.5). It concerns the existence of a non-conventional Bose condensation for large chemical potential µ or high inverse temperatures β. Meantime, even for β → +∞, i.e., for a zero-temperature, only a fraction of the full density is in the condensate: there is a coexistence of particles inside and outside the condensate (see (2.20)). Note that this last phenomenon is already known as the depletion of the condensate. Some discussions corresponding to this problem can be found in [6, 12, 13, 16–21]. To simplify our purpose, the proofs are given in Section 3. They are technically based on two papers [22, 23]. First we use the proof of the exactness of the Bogoliubov approximation in the grand-canonical ensemble for a superstable gas [8], as done by Ginibre [22]. Then, we use the “superstabilization” method [23]. Note that we recall the Bogoliubov u-v transformation in the appendix.
2 Thermodynamics in the grand-canonical ensemble SB In this section we give the thermodynamic behavior of the AVZ-model HΛ,λ in 0 the grand-canonical ensemble. Before entering this study recall the definitions of SB the grand-canonical pressure pSB Λ (β, µ) and particle density ρΛ (β, µ) associated SB with HΛ,λ0 :
pSB Λ (β, µ) ≡ ρSB Λ (β, µ) ≡
SB 1 ln T rFΛB e−β (HΛ,λ0 −µNΛ ) , βV NΛ (β, µ) = ∂µ pSB Λ (β, µ) . V H SB Λ,λ0
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Here − H SB (β, µ) represents the (finite volume) grand-canonical Gibbs state for Λ SB HΛSB . From the superstability of the Hamiltonian HΛ,λ it follows that pSB Λ (β, µ) 0 is defined for every pair (β, µ) ∈ QS ≡ {β > 0} × {µ ∈ R} , even in the thermodynamic limit [8].
2.1
The grand-canonical pressure
The first step is to use the Bogoliubov approximation, i.e., √ √ a0 / V → c ∈ C, a∗0 / V → c ∈ C, SB SB SB (µ) ≡ HΛ,λ −µNΛ . Since the model HΛ,λ is superstable for the Hamiltonian HΛ,λ 0 0 0 [8], Ginibre [22] proves the exactness of the Bogoliubov approximation in the sense that SB SB (β, µ) = sup p (β, µ, c) ≡ lim p (β, µ, c) , pSB (β, µ) = lim pSB sup Λ Λ Λ
Λ
c∈C
c∈C
(2.1) with pSB Λ (β, µ, c) ≡
SB 1 ln T rFB e−βHΛ,λ0 (µ,c) , pSB (β, µ, c) ≡lim pSB Λ (β, µ, c) . (2.2) Λ βV
Here
+∞
(n)
FB ≡ ⊕ HB,k=0 n=0
(n)
is the boson Fock space of the symmetrized n-particle Hilbert spaces HB,k=0 for non-zero momentum bosons. Note that SB B (µ, c) = HΛ,λ (µ, c) + HΛ,λ 0 0
λ0 2 NΛ,k=0 − NΛ,k=0 with λ0 > 0, 2V
(2.3)
B where HΛ,λ (µ, c) is defined by (A.1) in the appendix, and NΛ,k=0 is the operator 0 of the number of particles outside the zero-mode.
Remark 2.1 The applicability of the Ginibre’s proof [22] on the exactness of the Bogoliubov approximation concerns any superstable Bose systems of linear form of order 4 in operators a0 , a∗0 . The main difficulty is to control the upper bound of the pressure in the thermodynamic limit. This is mostly performed by some algebra: Taylor expansion around a0 , a∗0 , and then explicit calculations or estimations in relation with the superstability property. Briefly, this approach leads to an quadratic form in operator δa0 ≡ a0 − a0 and δa∗0 , allowing to use the standard reasoning of the Approximation Hamiltonian Method (see [24, 25]).
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Then, the second step to find the thermodynamic limit pSB (β, µ) of the pressure uses [23] and gives the following result: Theorem 2.2 Let pB 0 (A.9) be the thermodynamic limit of the pressure of the nonB (α, c), then superstable Hamiltonian HΛ,0
p
SB
(β, µ) =sup p c∈C
SB
(β, µ, c) = sup x=|c|2 ≥0
inf
α≤0
pB 0
(µ − α) (β, α, x) + 2λ0
2
, (2.4)
for any (β, µ) ∈ QS . To discuss this theorem, let us consider the Mean-Field Hamiltonian HΛMF = TΛ + UΛMF ≡ TΛ +
λ0 2 NΛ − NΛ , 2V
(2.5)
see [26–32]. Then, by theorem 2.2 and (A.9), we get the following lower bound for the pressure:
2 (µ − α) pSB (β, µ) ≥ inf pB 0 (β, α, 0) + α≤0 2λ0
2 (µ − α) P BG = inf p (β, α) + (2.6) = pMF (β, µ) , α≤0 2λ0 where pP BG (β, α) and pMF (β, µ) are the (infinite volume) pressures respectively for the Perfect Bose Gas and the Mean-Field Bose Gas, see [23, 26–32]. Let α (x) ≡ α (β, µ, x) be the solution of
2 2 (µ − α) (µ − α) (2.7) = pB inf pB 0 (β, α, x) + 0 (β, α, x) + α≤0 2λ0 2λ0 α=α(x) for any fixed x ≥ 0. Thus we have
2 (µ − α) (µ − α) B B ∂α p0 (β, α, x) + = ρ0 (β, α, x) − =0 2λ0 λ0 α=α(x)≤0 α=α(x)≤0 (2.8) for chemical potentials µ ≤ µc (β, x) ≡ λ0 ρB 0 (β, 0, x) ,
(2.9)
whereas for µ ≥ µc (β, x) and α ≤ 0 the corresponding derivative is negative: ρB 0 (β, α, x) −
(µ − α) ≤ 0 which implies α (x) = 0. λ0
(2.10)
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Here, ρB 0
(β, α, x)
≡
∂α pB 0 +
(β, α, x) = x +
1 3
(2π)
R3
1 3
(2π)
x2 λ2k
R3
B B fk,0 + Ek,0 2Ek,0
f k,0B d3 k B eβEk,0 − 1 Ek,0
d3 k.
(2.11)
P BG (β, α ≤ 0) is the critical density of the Perfect Bose Note that ρB 0 (β, α, 0) = ρ Gas. = Since all functions depend only on x = |c|2 , in the following we denote by x x (β, µ) the solution of the first variational problem of Theorem 2.2:
2 (µ − α) pSB (β, µ) = inf pB ) + , (2.12) 0 (β, α, x α≤0 2λ0
and we solve it via the following theorem. Theorem 2.3 For any β > 0, there exists a unique µc (β) such that 2 (µ − α (0)) B = pMF (β, µ) , for µ ≤ µc (β) . p0 (β, α (0) , 0) + 2λ0
SB p (β, µ) = 2 (µ − α (x)) B , for µ > µc (β) . p0 (β, α (x) , x) + 2λ0 x= x>0 The function µc (β) is bijective from [a, +∞) to R+ and we denote by βc (µ) ≥ 0 the inverse function of µc (β), see Figure 2.1. Here a = 0 if (C1) holds whereas if (C2) is satisfied a = µ0 ≡ µc (β = ∞) < 0. The pressure pSB (β, µ) is continuous for µ = µc (β) . Remark 2.4 Since one x) <0 for any µ = µc (β) (Remark 3.1), the spec Bhas α ( B trum Ek,0 (A.5) of HΛ,0 (α ( x) , c) |c|2 =x always has a gap in the grand-canonical ensemble (µ = µc (β)). In other words, on the thermodynamic level, there is a gap SB in the spectrum of HΛ,λ at all chemical potentials µ = µc (β). This rigorous result 0 is different from conclusions of the papers [1, 15], where the corresponding authors find a gapless spectrum using additional hypotheses on the AVZ-Hamiltonian.
2.2
Non-conventional Bose condensation and Bogoliubov distribution
By Lemma 3.5 or by Remark 3.1 for µ = µc (β) or β = βc (µ) the full particle density equals: NΛ µ − α ( x) lim (β, µ) = ρSB (β, µ) ≡ Λ V H SB λ0 Λ,λ0 P BG ρ B (β, α (0)) for µ < µc (β) or β < βc (µ) , = (2.13) ρ0 (β, α ( x) , x ) for µ > µc (β) or β > βc (µ) ,
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λ0 θ c =1/βc (µ)
λ0
(C2) holds (C2) holds (C1) holds
λ0
(C1) holds
+ −0.5φ(0)
µ0
0
µ
Figure 2.1: Illustration of the critical temperature θc = 1/βc as a function of µ. Each curve corresponds to a different value of λ0 (from λ0 = +∞ to 0+ ): when (C1) holds, i.e., λ0 is sufficiently large, the curve starts at µ = 0 in contrast to the cases where (C2) holds. The term λ0 = 0 in this figure corresponds to the model B HΛ,0 where the pressure diverges for µ = −ϕ (0) /2 and not to the Perfect Bose Gas. = x (β, µ) and α (x) = where ρB 0 (β, α (x) , x) is defined by (2.11) and with x α (β, µ, x) the solutions of the variational problems. Now our main results concern the particle densities inside and outside the zero-mode, and are given in the following theorem. Theorem 2.5 Under the assumptions of the previous two theorems it follows for µ = µc (β) or β = βc (µ): (i) A non-conventional Bose condensation induced by the non-diagonal interaction UΛN D for large chemical potentials (high particle densities), or low temperatures: ∗ a0 a0 = 0 for µ < µc (β) or β < βc (µ) . (β, µ) = x (β, µ) = lim > 0 for µ > µc (β) or β > βc (µ) . Λ V SB H Λ,λ0
(ii) No Bose condensation (of any type I, II or III [33–35]) outside the zero-mode: ∗ ak ak ∗ ∀k ∈ Λ \ {0} , lim (β, µ) = 0. Λ V SB HΛ,λ 0 1 ∗ lim lim a a H SB (β, µ) = 0. k k δ→0+ Λ V Λ,λ0 {k∈Λ∗ ,0<k≤δ}
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(iii) A particle density outside the zero-mode equals 1 a∗k ak H SB (β, µ) lim Λ,λ0 Λ V ∗ k∈Λ \{0} fk,0 1 3 = (2π)3 d k βE B B x=x,α=α(x) R3 Ek,0 e k,0 −1
x2 λ2k 1 d3 k + (2π) . 3 B f B 2E +E [ ] k,0 k,0 k,0 R3 x= x,α=α( x) Note that the last limit equals the full particle density ρP BG (β, α (0) ≤ 0) of the Perfect Bose Gas for µ < µc (β) or β < βc (µ). (iv) There is no discontinuity of the particle densities (full density (2.13), density in the zero-mode (i) or outside the zero-mode (iii)) only if condition (C1) is satisfied. Assuming condition (C2), a discontinuity of the three densities appears with a strictly positive jump. (v) For µ < µc (β) or β < βc (µ) one has the Bose distribution for a corresponding chemical potential α (0) < 0: ∀k ∈ Λ∗ : k ≥ δ > 0, lim a∗k ak H SB (β, µ) = Λ
Λ,λ0
1 . eβ(εk −α(0)) − 1
But for µ > µc (β) or β > βc (µ) , i.e., in the presence of a Bose condensation, we get another one, which we call the Bogoliubov distribution, for a corresponding chemical potential α ( x) < 0: lim a∗k ak H SB (β, µ) Λ,λ0 Λ 2 2 f x λk k,0B + = x=x,α=α(x) E B eβEk,0 − 1 B B fk,0 + Ek,0 2Ek,0 k,0 for any k ∈ Λ∗ such that k ≥ δ for all δ > 0. Remark 2.6 (a) Assuming condition (C2), a discontinuity of the densities appears because the direct term of repulsion λ0 2 λ0 ∗2 2 a0 a0 = N0 − N0 , with N0 ≡ a∗0 a0 , (2.14) 2V 2V in (1.2) becomes too weak to beat the attraction induced by UΛN D (1.6). The nondiagonal interaction UΛN D express itself via the effective coupling constant g00 (1.1) [5, 6]. (b) The depletion (Bogoliubov distribution) coincides with the one found for λ0 > 0 [6, 21] at a chemical potential α = 0 (high density regimes) in the thermodynamic B (1.4), so-called the Weakly Imbehavior of the Bogoliubov Hamiltonian HΛ,λ 0 >0 perfect Bose Gas.
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We conclude this section by additional comments about the nontriviality of the Bose condensate at low temperatures, when (C1) holds. Indeed, in this case note that the corresponding kinetic part only turns on the Bose condensation phenomenon via the Bose distribution. Indeed, the solution of the variational problem α ( x) (2.7) for a Bose condensate density x (β, µ) , becomes zero when we reach the critical temperature as for the Mean-Field Bose Gas, but switches again to strictly negative values for x > 0 (Section 3). As soon as the Bose condensate appears, the non-diagonal interaction UΛN D becomes sufficiently important to drastically change all thermodynamic properties of the system by instantly switching the usual Mean-Field Bose Gas to a system of quasi-particles: the Bose-Einstein condensation becomes non-conventional. Whereas the non-diagonal interaction UΛN D is not strong enough to imply alone the Bose-condensation at the critical temperature of the Mean Field Gas when (C1) is satisfied, it strongly dominates all thermodynamics. In the case (C1), the origin of the nontriviality of the Bose condensate is then delicate compare to other Bose systems as for example the Weakly Imperfect Bose Gas [4, 6, 21]. In particular, when (C1) holds, this phenomenon is rather different from the thermodynamic behavior of the Weakly Imperfect Bose Gas [4, 6, 21] ole anymore. where UΛN D plays no rˆ All these previous arguments in the grand-canonical ensemble are valid only for λ0 > 0 and λk > 0 for k < A. Indeed in the limit λ0 0 (and λk 0), notice that (C1) holds but UΛN D does not exist anymore, and we obtain the Perfect Bose Gas, where the Bose condensation is conventional. For more information about this different but expected thermodynamic behavior, see [2].
2.3
The particle density as parameter in the grand-canonical ensemble
Let us consider the fixed particle density ρ in the grand-canonical ensemble which defines a unique chemical potential µβ,ρ satisfying ρSB (β, µβ,ρ ) = ρ.
(2.15)
Actually, at a fixed inverse temperature β the function µβ,ρ is the inverse function of the mean particle density ρSB (β, µ) of the superstable Bogoliubov Hamiltonian. (i) Let
ρc,inf (β) ≡
lim
ρSB (β, µ) .
lim
ρSB (β, µ) .
µ→µ− c (β)
ρc,sup (β) ≡
µ→µ+ c (β)
(2.16)
Recall that µc (β) and βc (µ) are defined in Theorem 2.3 (Figure 2.1). Through (iv) of Theorem 2.5 combined with (2.13) and (3.34), we deduce that ρc (β) ≡ ρc,inf (β) = ρc,sup (β) ≤ ρP BG (β, 0) if condition (C1) is satisfied. At fixed particle density ρ note that we can also define by βc (ρ) the unique critical inverse temperature.
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If condition (C1) is satisfied, an illustration of the behavior of βc (ρ) for a fixed density is performed in Figure 2.2. Because of the interaction note that the critical inverse temperature βc (ρ) is always smaller or equal than the one for the MeanField Bose Gas (which is equal to the one for the Perfect Bose Gas). Unfortunately, at very high densities we are not able to prove an exact equality or the opposite, see Figure 2.2.
βc (ρ)
β 0
ρc (β)
ρ
Figure 2.2: Illustration of the critical inverse temperature βc (ρ) when condition (C1) is satisfied. The dotted line corresponds to the phase digram of the MeanField Bose Gas. The difference with the Mean-Field Bose Gas is always greater or equal to zero. It may be zero for all β > 0 (only at high densities, we are not able to prove an exact equality or the opposite). (ii) By Remark 3.1 and (2.13) we have x) < 0 for ρ ∈ / [ρc,inf (β) , ρc,sup (β)] or β = βc (ρ) . µβ,ρ − λ0 ρ = α ( (iii) Combining Theorem 2.3 with (2.13) and (2.17) we get B λ0 SB + ρ2 p (β, µβ,ρ ) = p0 (β, α (x) , x) 2 x= x
(2.17)
(2.18)
for any ρ > 0, where α ( x) < 0 is the unique solution of the Bogoliubov density equation: ρ = ρB ) for ρ ∈ / [ρc,inf (β) , ρc,sup (β)] or β = βc (ρ) . 0 (β, α, x
(2.19)
(iv) For ρ < ρc,inf (β), one has µβ,ρ < µc (β) , whereas µβ,ρ > µc (β) for ρ > ρc,sup (β), and Theorem 2.5 is still valid for any ρ ∈ / [ρc,inf (β) , ρc,sup (β)], i.e., for
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µβ,ρ = µc (β) or β = βc (ρ). In fact, Theorem 2.5 is verified even for µ = µc (β) or β = βc (µ) but only if (C1) is satisfied (all densities are continuous). Then, for any ρ > ρc,sup (β) there is only one Bose condensation in the zero mode, whereas for ρ < ρc,inf (β) ≤ ρP BG (β, 0) the system behaves as the MeanField Bose Gas (2.5) with no Bose condensations. In particular, note that the result (ii) of Theorem 2.5 excludes any coexistence of non-conventional and conventional Bose condensation, as it appears for high densities in the Weakly Imperfect Bose Gas [6, 21]. Actually, for ρ ∈ [ρc,inf (β) , ρc,sup (β)] , when (C2) holds, the question of Bose condensation is still open in the grand-canonical ensemble. This unsolved problem appears also in the study of the Weakly Imperfect Bose Gas [4, 6, 21] and is quite similar. In this regime, when (C2) is satisfied, it should simply be a coexistence of two phases, see for example Section 4 in [31]. In fact we explain in [3] that this question is not relevant in the canonical ensemble, in the sense that the canonical thermodynamics of the AVZ-Hamiltonian should correspond to the grand-canonical one with the case (C1). Indeed, the superstable interaction UΛMF (1.3) is just a constant in the canonical ensemble. This question is related to the continuity of the particle density which is only satisfied when (C1) holds, see also Remark 2.7. (v) To conclude, note that we have a non-zero particle density outside the zeromode for any ρ > 0 even for zero-temperature: 1 lim lim a∗k ak H SB (β, µβ,ρ ) Λ,λ0 β→+∞ Λ V ∗ k∈Λ \{0} 1 2 2 x λ k 3 d k > 0, = 3 x=x B B (2π) f 2E + E k,0 α=α(x) k,0 k,0 3 R
∀k ∈ Λ∗ , k ≥ δ > 0, lim lim a∗k ak H SB (β, µβ,ρ ) Λ,λ0 β→+∞ Λ 2 2 x λk = > 0. 2E B f + E B x=x k,0 α=α( x) k,0 k,0
(2.20)
In the regime ρ > ρc,sup (β) , the system follows the Bogoliubov distribution (v) of Theorem 2.5, whereas in the absence of the Bose condensation, i.e., for ρ < ρc,inf (β), the (standard) Bose distribution holds. (vi) If we analyze the system as a function of the parameter λ = λ0 in UΛMF (1.3) for a fixed density ρ ∈ / [ρc,inf (β) , ρc,sup (β)] we can find ∂λ0 x = 0 by direct computations (see also [3]). An illustration of the behavior of x for a fixed density is performed in Figure 2.3. This is also true for α ( x). Actually, in [3] we prove that the solutions α ( x) = (λ0 ) of the variational problems (2.7) and (2.12) are also solutions in α ( x, λ0 ) and x the canonical ensemble of other variational problems and they do not depend on the
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λ0
x(β,µ ) 2
417
λ0
(C2) holds (C2) holds (C1) holds
x(β,ρ )
2
λ0
(C1) holds
+ −0.5φ(0)
µ β,ρ
0
µ β,ρ
µ
Figure 2.3: Illustration of x (β, µ) and x (β, µβ,ρ ) = x (β, ρ) for a fixed particle density ρ > 0 in the grand-canonical ensemble. Each curve corresponds to a different value of λ0 in UΛMF (from λ0 = +∞ to 0+ ): when (C1) holds, i.e., λ0 is sufficiently large, there is no jump in contrast with the cases where (C2) holds. B The term λ0 = 0 in this figure corresponds to the model HΛ,0 where the pressure diverges for µ = −ϕ (0) /2 and not to the Perfect Bose Gas. / [ρc,inf (β) , ρc,sup (β)]. Indeed, parameter λ0 in UΛMF for a fixed particle density ρ ∈ the particle density ρSB (β, µ) depends on λ0 for fixed µ, and then µβ,ρ = µβ,ρ (λ0 ) at a fixed density ρ ∈ / [ρc,inf (β) , ρc,sup (β)]. However, as an example, the solution x = x (β, µ, λ0 ) of the variational problem (2.12) is such that x = x (β, ρ) = x (β, µβ,ρ (λ0 ) , λ0 ) does not depend on λ0 ≥ 0 (Figure 2.3). At fixed particle densities ρ > 0, all other densities in the grand-canonical ensemble do not depend on the value λ = λ0 in UΛMF outside the phase transition. Note that the first term in conditions (C1) and (C2) only comes from UΛMF (see Remark 3.4). Remark 2.7 This last phenomenon is expected since for any λ0 > 0 the HamilSB B tonians HΛ,λ (1.2) and HΛ,0 differ only by a constant on the symmetrized n0 (n=[ρV ])
, i.e., in the canonical ensemble. Consequently, in particle Hilbert spaces HB the canonical ensemble all densities do not depend on the value λ = λ0 in UΛMF for a fixed particle density. This phenomenon is true in the grand-canonical ensemble, briefly because of the strong equivalence of ensembles (see [32,36–39] for the notion of strong equivalence). Here n = [ρV ] is defined as the integer of ρV .
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3 Proofs The aim of this section is to give the promised details of the proofs of Theorems 2.2, 2.3 and 2.5.
3.1
Proof of Theorem 2.2
By (2.1) the proof consists of getting the thermodynamic limit pSB (β, µ, c) of the SB pressure pSB Λ (β, µ, c) associated with the Hamiltonian HΛ,λ0 (µ, c). We calculate the pressure pSB Λ (β, µ, c) via another related Hamiltonian defined in part 1. In 2. we consider the specific free energy densities in the canonical ensemble, whereas in 3. we show convexity, such that weak-equivalence of ensembles implies the theorem in part 4. 1. Let
B (γ, c) ≡ H B (γ, c) + γ |c|2 V − λ0 |c|4 V − |c|2 , H Λ,λ0 Λ,λ0 2
(3.1)
B (γ, c) and H SB (µ, c) are well defined on the see (A.1). The Hamiltonians H Λ,λ0 Λ,λ0 boson Fock space FB for any fixed c ∈ C. Here we use two chemical potentials γ B (γ, c) and H SB (µ, c). From the appendix, and µ respectively for the models H Λ,λ0 Λ,λ0 B (γ, c) is diagonalizable by the Bogoliubov canonical u-v the Hamiltonian H Λ,λ0 B transformation, see (A.7), and one gets a perfect Bose gas with a spectrum Ek,λ 0 (A.5). We then have p B Λ,λ0
(β, γ, c) =
pB Λ,λ0
for
λ0 (β, γ, c) − γ |c| + 2 2
2
γ ≤ |c| λ0 +
min
k∈Λ∗ \{0}
2
|c| |c| − V 4
! (3.2)
εk ,
see (A.6) and (A.8) in the appendix. The thermodynamic limit follows as λ0 2 2 B p B x , β, γ, x = |c| ≡lim p B λ0 Λ,λ0 (β, γ, c) = pλ0 (β, γ, x) − γx + Λ 2 cf. (A.9) for γ ≤ xλ0 = lim Λ
2. Note that
B H
" |c|2 λ0 +
# min ∗
k∈Λ \{0}
εk
and λ0 > 0.
$ SB %
0, HΛ,λ (γ, c) , N (µ, c) , NΛ,k=0 = 0. Λ,k=0 = Λ,λ0 0
(3.3)
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B However, for a fixed particle density ρ1 > 0, let f Λ,λ (β, ρ1 , c) and fΛSB (β, ρ1 , c) 0 be the free-energy densities: " #
1 B (0,c) (n,k=0) −β H B Λ,λ0 e ln T r (β, ρ , c) ≡ − f Λ,λ , (n) 1 0 HB,k=0 βV " # (3.4)
(n,k=0) SB 1 e−βHΛ,λ0 (0,c) ln T rH(n) , fΛSB (β, ρ1 , c) ≡ − B,k=0 βV
where
(n)
A(n,k=0) ≡ A HB,k=0 is the restriction of any operator A acting on the boson Fock space FB (n = [ρ1 V ] is defined as the integer of ρV ). Note that
p B Λ,λ0 (β, γ, c) =
+∞ n n 1 B eβV {γ V −fΛ,λ0 (β, V ,c)} . ln βV n=0
(3.5)
B The free-energy density f Λ,λ (β, ρ1 , c) is in fact well defined for any ρ1 > 0 and 0 β > 0 in the thermodynamic limit, i.e., 2 B (β, ρ1 , c) < +∞. f λB0 β, ρ1 , x = |c| ≡ lim f Λ,λ 0 Λ
From (2.3) and (3.1) we then have λ0 2 ρ1 λ0 B fΛSB (β, ρ1 , c) = f Λ,λ ρ1 − − µ |c|2 + (β, ρ , c) + 1 0 2 V 2
|c|2 |c| − V 4
! ,
which gives λ0 λ0 2 f SB β, ρ1 , x = |c| ≡ lim fΛSB (β, ρ1 , c) = f λB0 (β, ρ1 , x) + ρ21 − µx + x2 . Λ 2 2 3. Notice that we do not know if the specific free energy f λB0 (β, ρ1 , x) is convex as a function of ρ1 , which is crucial in order to use [23] for our proof. It is the next step of the proof. By (3.2), there is a unique solution γΛ (ρ1 ) of ∂γ p B Λ,λ0 (β, γΛ (ρ1 ) , c) = ρ1
(3.6)
at all densities ρ1 > 0. By direct computations of (3.6) done via (3.3), the corresponding thermodynamic limit γ (ρ1 ) ≡lim γΛ (ρ1 ) = Λ
< xλ0 for ρ1 < ∂γ p B λ0 (β, λ0 x, x) , xλ0 for ρ1 ≥ ∂γ p B λ0 (β, λ0 x, x) ,
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is an increasing continuous function of ρ1 > 0. By (3.5) we also have p B λ0 (β, γ (ρ1 ) , x)
B ≡ lim p B Λ,λ0 (β, γΛ (ρ1 ) , c) = γ (ρ1 ) ρ1 − fλ0 (β, ρ1 , x) Λ
= sup γ (ρ1 ) t − f λB0 (β, t, x) . (3.7) t>0
Therefore for any ρ1 > 0, ∂ρ1 f λB0 (β, ρ1 , x) = γ (ρ1 ) is an increasing function of ρ1 > 0, i.e., f λB0 (β, ρ1 , x) is a convex function of ρ1 > 0. B (γ, c) for 4. The weak equivalence of ensembles is then verified by the model H Λ,λ0 2 each fixed x = |c| ≥ 0, and using [23] combined with (2.3) and (3.1) we directly find
2 (µ − γ) λ0 2 SB B p (β, µ, c) = inf . (3.8) p λ0 (β, γ, x) + + µx − x γ≤xλ0 2λ0 2 x=|c|2 Therefore the theorem follows by (2.1), (3.3) and the last equality, if we take α = γ − xλ0 ≤ 0 in the expression for the infimum and finally switch from p B λ0 (β, γ, x) B to p0 (β, α, x).
3.2
Proof of Theorem 2.3
From Theorem 2.2 we get pSB (β, µ) =sup {Fβ (α (x) , x)} = {Fβ (α (x) , x)} x≥0
,
x= x
where the function Fβ (α, x) is given by (µ − α)2 (µ − α)2 Fβ (α, x) ≡ pB = ξ0 (β, α, x) + η0 (α, β) + . 0 (β, α, x) + 2λ0 2λ0 2 (µ − α) F∞ (α, x) ≡ lim Fβ (α, x) = η0 (α, x) + . β→∞ 2λ0 (3.9) (β, α, x) = ξ (β, α, x) + η (α, x) is defined by (A.9) in the We recall that pB 0 0 0 appendix. So, we have to evaluate the sign of ∂x {Fβ (α (x) , x)} = {∂x Fβ (α, x)} + {∂x α (x) ∂α Fβ (α, x)} (3.10) α=α(x)
α=α(x)
to obtain x = x maximizing the function Fβ (α (x) , x). The proof is then divided in four parts. First we get in 1. an easier expression of the derivative of the functional Fβ (α(x), x): the second term of (3.10) is in fact zero.
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In the second step 2. we study the solution α(x) and the corresponding critical chemical potential µc (β, x) (2.9). In 3. and 4. we get the first results for β → ∞ and then for arbitrary finite β. 1. Through (3.9) one has ∂α Fβ (α, x) = ρB 0 (β, α, x) −
(µ − α) . λ0
(3.11)
Then, by (2.8)–(2.10), one has {∂α Fβ (α, x)} = 0 for µ < µc (β, x) . α=α(x)<0 {∂α Fβ (α, x)} = 0 for µ = µc (β, x) . α=α(x)=0 < 0 for µ > µc (β, x) . {∂α Fβ (α, x)}
(3.12)
α=α(x)=0
{∂x α (x) ∂α Fβ (α, x)}
Therefore
=0
α=α(x)
for any fixed µ and so (3.10) can be written as
∂x {Fβ (α (x) , x)} = {∂x Fβ (α, x)}
.
(3.13)
α=α(x)
Notice that α (x) = α (β, µ, x) is also a function of the inverse temperature and chemical potential and, in the same way, we get {∂ α (x) ∂ F (α, x)} = 0. β α β α=α(x) (3.14) = 0. {∂µ α (x) ∂α Fβ (α, x)} α=α(x)
2. By (2.9) and (2.11) note that lim µc (β, x) = +∞.
x→+∞
Moreover we have ∂x ρB 0 (β, α, x) =1+ −
1
1 3
2 (2π) 3
4 (2π)
R3
√
xλ2k
" 1+
2 B
εk − α (εk − α + 2xλk )3/2 eβEk,0 − 1 R3 " # εk − α + xλk β d3 k. λk εk − α + 2xλk sinh2 βE B /2 k,0
#
d3 k
(3.15)
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Since the last in term in (3.15) vanishes when β → ∞ for all x ≥ 0 and all α ≤ 0, ∂x ρB 0 (β, α, x) > 0 for sufficiently high β. Thus P BG (β, 0) > 0 inf µc (β, x) = λ0 inf ρB 0 (β, 0, x) = µc (β, 0) = λ0 ρ
x≥0
x≥0
(3.16)
for sufficiently high β > 0 and the critical chemical potential µc (β, x) is an increasing function of x ≥ 0. Consequently there is for µ > µc (β, 0) a solution xµ > 0 of (3.17) µc (β, xµ ) = µ,
such that α (x) =
0 , for 0 ≤ x ≤ xµ , < 0 , for x > xµ > 0,
(3.18)
and for all x2 > x1 > xµ , α (x2 ) < α (x1 ) and
lim α (x) = −∞.
x→+∞
(3.19)
To summarize the behavior of α (x) = α (β, µ, x) : lim α (β, µ, x)
=
−∞ for β, µ fixed,
lim α (β, µ, x)
=
−∞ for β, x fixed,
lim α (β, µ, x)
=
−∞ for µ, x fixed.
x→+∞
µ→−∞ β→0+
(3.20)
3. We consider now the limit β → ∞. To analyze the derivative of the functional F∞ (α(x), x) we only have to consider the partial derivative with respect to x, because we get the same results for F∞ (α(x), x) as in (3.12) and (3.13) for the functional Fβ (α(x), x). Thus by (3.9) we have for any α ≥ 0 ∂x lim Fβ (α, x) = ∂x F∞ (α, x) = α + Ω (α, x) , β→+∞
where Ω (α, x) ≡
1 3
2 (2π)
R3
√ εk − α λk 1 − √ d3 k ≥ 0. εk − α + 2xλk
(3.21)
(3.22)
By direct computations of the partial derivatives with respect to α and x, we find that Ω (α, x) is a strictly increasing concave function of x ≥ 0 for any fixed α ≤ 0 with ϕ (0) , (3.23) Ω (α, 0) = 0 and lim Ω (α, x) = x→+∞ 2 whereas for any fixed x > 0, Ω (α, x) is a strictly increasing convex function of α ≤ 0 with √ εk 1 lim Ω (α, x) = 0 ≤ Ω (0, x) = λk 1 − √ d3 k. (3.24) 3 α→−∞ εk + 2xλk 2 (2π) R3
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Via (3.16) one has
lim
β→+∞
423
inf µc (β, x)
x≥0
= 0,
i.e., we have to consider the cases µ > 0 and µ ≤ 0. 3.1. Let us first discuss the case µ > 0. By (3.17)-(3.19), there is xµ > 0 such that 0 , for 0 ≤ x ≤ xµ . α (x) = (3.25) < 0 , for x > xµ > 0. Combining (3.23)–(3.24) with the previous relation, we get Ω (0, xµ ) ≥ Ω (0, x) > 0 for 0 < x ≤ xµ and µ > 0 and the lower bound sup {F∞ (α (x) , x)} = {F∞ (α (x) , x)} > sup {F∞ (α (x) , x)} = F∞ (0, xµ ) x≥0 0≤x≤x x= x
µ
(3.26) x) < 0 for µ > 0. This first result proves the which implies x > xµ > 0 and α ( theorem for µ > 0 and β → ∞. 3.2. If µ ≤ 0 the condition (2.8) is always satisfied and gives an expression for α = α (x), i.e., α (x) = µ − λ0 ρB 0 (β, α (x) , x). Hence, since the second term in (2.11) vanishes in the limit β → ∞ we can rewrite (3.21): ∂x {F∞ (α (x) , x)} = {∂x F∞ (α, x)} = µ − λ0 ρB 0 (β, α(x), x) + Ω(α(x), x) α=α(x)
2 2 λ0 x λk 3 = µ + Ω (α, x) − λ0 x − k . (3.27) d 3 α=α(x) B B 2 (2π) f E + E k,0 k,0 k,0 3 R
Moreover, notice that ∂x2 F∞
(α, x) = −λ0 +
1 3
2 (2π)
R3
λ2k (εk − α − xλ0 ) d3 k √ 3/2 εk − α (εk − α + 2xλk )
< ∂x2 F∞ (α, x)
, (3.28) x=0
for any α ≤ 0 and x > 0, see (3.15) with β → +∞ for the derivative of the density ρB 0 (β, α, x). We also have λ2k 1 2 3 2 d k ≤ ∂x F∞ (α, x) = −λ0 + ∂x F∞ (α, x) 3 (εk − α) 2 (2π) x=0 x=0,α=0 R3 " # λ0 + g00 , = −2 2
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2 see (1.1). Therefore, by fixing the sign of ∂x F∞ (α, x)
Ann. Henri Poincar´e
the assumptions x=0,α=0
(C1)–(C2) imply two different behaviors for the solution x of the variational problem. (C1) If condition (C1) is satisfied, we find via the two previous expressions that ∂x2 F∞ (α, x) < 0 for all α ≤ 0 and x > 0, which via (3.27) implies µ + Ω (α (x) , x) − λ0 x −
λ0 x2 3
2 (2π)
R3
λ2 k d3 k B B Ek,0 fk,0 + Ek,0
< µ + Ω (α (x) , 0) = µ + Ω (α (0) , 0) = µ ≤ 0, (3.29) for x > 0, i.e., ∂x {F∞ (α (x) , x)} < 0 for all µ ≤ 0. Therefore we get x = 0. x>0
Actually, by combining the results for µ > 0 and µ ≤ 0, notice that > 0 for µ > 0, x = lim x = 0, lim x µ→0− µ→0+ x = 0 for µ ≤ 0,
(3.30)
at infinite inverse temperature (β → ∞). (C2) Assuming now condition (C2), there is a critical value α0 < 0 such that the upper bound of (3.28) becomes positive: ∂x2 F∞ (α, x) > 0, (3.31) x=0
for any α0 < α ≤ 0. Since for µ = 0 one has α (0) = 0 from (2.8), by (3.28) and (3.31) we have ∂ {F (α (x) , x)} = 0, x ∞ µ=0,x=0 (3.32) > 0, ∂x2 {F∞ (α (x) , x)} µ=0,x<δ
for sufficiently small δ > 0, because of continuity. Therefore from the definition of > 0 for µ = 0. Actually, there is a µ0 < 0 such that pSB (β,µ) and (3.9) we have x x > 0 for µ ≥ 0. lim x = lim− x . µ→0 µ→0+ x > 0 for µ0 < µ ≤ 0. (3.33) lim x > lim x = 0. µ→µ+ µ→µ− 0 0 x = 0 for µ < µ0 .
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4. Now we consider the case of finite inverse temperatures β < +∞. By (3.9), (3.13) and (3.14) one has ∂ {F (α (x) , x)} = {∂ ξ (β, α, x) + ∂ F (α, x)} x β x 0 x ∞ α=α(x) (i) < {∂x F∞ (α, x)} , α=α(x) lim ξ0 (β, α (x) , x) = 0, x→+∞ ∂β {Fβ (α (x) , x)} = {∂β ξ0 (β, α, x)} < 0, (ii) α=α(x) lim ξ (β, α, x) = 0, β→+∞
0
for fixed µ ∈ R. By (i) for fixed µ, if x = 0 for β → ∞, then x = 0 for any β ≥ 0. Let µ > 0. By definition of Fβ (α, x) one has Fβ (α, x) > F∞ (α, x) , for µ > 0 and any fixed α ≤ 0. Since by (ii) the function Fβ (α (x) , x) is monotonically decreasing for β ∞, we find that x) , x > 0) < sup {Fβ (α (x) , x)} , Fβ (α (0) , 0) < F∞ (α ( x≥0
for sufficiently high β and large µ > 0, i.e., x > 0. Since one has (3.20), ∂x ξ0 (β, α, x) < 0 (i) and ∂β ∂x ξ0 (β, α, x) =
1 3
(2π)
R3
B
B βEk,0 e Ek,0 B d k 2 ∂x Ek,0 > 0, B βEk,0 1−e 3
for µ > 0, there is an inverse temperature βc (µ) > 0 such that x > 0 for β > βc (µ > 0) and x = 0 for β < βc (µ > 0) . The function βc (µ > 0) is bijective so we define by µc (β) > 0 the inverse function of βc (µ). Note that if µ > µc (β, 0) (2.9) then the arguments done in 2. (cf. (3.15)–(3.19)) and 3.1. still work. So, x > 0 for µ > µc (β, 0) . Consequently µc (β) ≤ µc (β, 0) = λ0 ρB 0 (β, 0, 0) ,
(3.34)
and βc (µ) is a strictly increasing function from [0, +∞) to [0, +∞) . If the condition (C2) is verified, the arguments done here in 4. for µ > 0 work also for µ > µ0 and the function βc (µ > µ0 ) is bijective. In particular the inverse function µc (β) of βc (µ) verifies: lim µc (β) = µ0 < 0,
β→+∞
and (3.33) holds for β > 0. An illustration of the critical temperature θc (µ) = 1/βc (µ) as a function of µ is given by Figure 2.1.
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Remark 3.1 The solution x =x (β, µ) of (2.12) always satisfies ∂
|c|2 p
SB
(β, µ, c)
|c|2 = x
=
∂x pB 0
(β, α, x)
x =x (β, µ) = 0 for µ < µc (β) ,
= 0, for µ > µc (β) ,
x= x,α=α( x)
(3.35) see (3.9) and (3.13). Moreover, from the previous proof we can see that the solution α ( x) = α (β, µ, x ) of (2.7) is always strictly negative for any µ = µc (β) or β = βc (µ). In particular, one always has (2.8) for α = α ( x), x = x (2.12), and µ = µc (β) or β = βc (µ). Remark 3.2 Actually, as an extension for finite β of (3.30) and (3.33) we get two behaviors for x depending on conditions (C1) and (C2): = 0 for µ ≤ µc (β) . x = lim x = 0. lim x (C1) : µ→µ− µ→µ+ c (β) c (β) x > 0 for µ > µc (β) . or
= 0 for µ < µc (β) . x < lim x . 0 = lim x (C2) : µ→µ− µ→µ+ c (β) c (β) x > 0 for µ > µc (β) .
.
Remark 3.3 If condition (C1) holds, using arguments from the proof of Theorem 2.3 (3.2. and 4.) we have µc (β) ≥
inf µc (β, x) .
x≥0
Therefore, for sufficiently high β, i.e., for low temperatures (compare (3.16) and (3.34)), we have µc (β) = µc (β, 0) = λ0 ρP BG (β, 0) . P BG We recall that µc (β, x) = λ0 ρB (β, 0) is the 0 (β, 0, x) is defined by (2.9) and ρ critical density of the Perfect Bose Gas.
Remark 3.4 Notice that the proof of Theorem 2.3 does not depend on the fact that λ0 is the Fourier transformation of the interaction potential for k = 0. Actually, one could have taken as an arbitrary (strictly positive) parameter satisfying either (C1) or (C2). In [3], we explain that λ0 has no physical relevance for a fixed particle density and is then taken arbitrary large enough such that only (C1) holds with a strict inequality.
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Proof of Theorem 2.5
1. Before we prove Theorem 2.5 for µ = µc (β) or β = βc (µ), one useful lemma is shown. Lemma 3.5 For any sequence {EΛ }Λ⊂R3 of subsets EΛ ⊆ Λ∗ such that if k ∈ EΛ then −k ∈ EΛ we have 1 ∗ B lim ak ak H SB (β, µ) = ∂γ p0 (β, α, γ, x) , Λ,λ0 Λ V α( x),γ=0,x= x ∗ k∈EΛ ⊆Λ
with pB 0 (β, α, γ, x) −1 3 B 1 1 −βEk,0 3 B ≡ ln 1 − e d k + fk,0 − Ek,0 d k 3 3 (2π) β 2 (2π) 3 3 R \E R \E 1 −1 B 1 −βEk,0 3 B 3 fk,0 − Ek,0 d k ln 1 − e d k+ + 3 α→α+γ (2π)3 β 2 (2π) E E + α + γ lim χEΛ (0) x, Λ
for any α ≤ 0 and γ ≤ 0. Here χEΛ denotes the characteristic function of EΛ and the set E is given by E ≡lim EΛ ⊆ R3 . Λ
Proof. Let pSB Λ (β, µ, γ) ≡
SB 1 ln T rFΛB e−βHΛ,λ0 ,γ (µ) βV
be the pressure associated with the perturbed (superstable) Hamiltonian SB HΛ,λ (µ) defined by: 0 ,γ SB SB HΛ,λ (µ) ≡ HΛ,λ − µNΛ − γ 0 ,γ 0
a∗k ak .
k∈EΛ ⊆Λ∗ SB Since HΛ,λ (µ) is superstable, its pressure is well defined and convex for any real 0 ,γ µ and γ. Consequently Theorems 2.2–2.3 are still valid for γ ∈ R:
pSB (β, µ, γ) ≡ = =
lim pSB Λ (β, µ, γ) Λ
(µ − α) sup inf (β, α, γ, x) + α≤0 2λ0 x≥0
2 (µ − α) pB 0 (β, α, γ, x) + 2λ0 pB 0
2
α=αγ ( xγ ),x= xγ
,
(3.36)
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with the corresponding pressure pB xγ ) and x γ are the cor0 (β, α, γ, x) . Here αγ ( responding solutions of the variational problems. We also have ∂γ pSB Λ (β, µ, γ) =
1 V
k∈EΛ ⊆Λ∗
a∗k ak H SB
Λ,λ0 ,γ
(β, µ) ,
and, via the Griffiths lemma [40, 41], we get: lim Λ
1 V
k∈EΛ ⊆Λ∗
a∗k ak H SB (β, µ) = ∂γ lim pSB (β, µ, γ) Λ Λ,λ0
Λ
γ=0
= ∂γ pSB (β, µ, γ)
. (3.37) γ=0
From Remark 3.1 for µ = µc (β) or β = βc (µ) combined with (3.36) we get
2 ( x )) (µ − α γ γ ∂γ pSB (β, µ, γ) = ∂γ pB xγ ) , γ, x γ ) + 0 (β, αγ ( 2λ0 = ∂γ pB 0 (β, α, γ, x) α=αγ ( xγ ),x= xγ
for |γ| sufficiently small and µ = µc (β) or β = βc (µ) . Consequently the limit (3.37) combined with the last equation for γ = 0 gives the lemma. 2. Now we are in position to prove the five statements (i)–(v) of Theorem 2.5: (i) Using Lemma 3.5 for EΛ = {0} combined with Remark 3.2 for µ = µc (β), one gets (i). (ii) Let a, b be two arbitrary positive real numbers, with b > a ≥ 0. Lemma 3.5 with EΛ = {k ∈ Λ∗ : k ∈ (a, b)} implies 1 1 lim a∗k ak H SB (β, µ) = ξβ,µ (k) χ(a,b) (k) d3 k 3 Λ,λ0 Λ V (2π) {k∈Λ∗ ,k∈(a,b)} R3
(3.38) where χ(a,b) (k) is the characteristic function of (a, b) , and where ξβ,µ (k) is a continuous function on k ∈ R3 defined by ξβ,µ (k) ≡
1 eβ(εk −α(0))
−1
,
(3.39)
for µ < µc (β) or β < βc (µ) whereas for µ > µc (β) or β > βc (µ) 2 2 f λ x k k,0B + ξβ,µ (k) ≡ βEk,0 B B B Ek,0 e −1 2Ek,0 fk,0 + Ek,0
x= x,α( x)
.
(3.40)
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This last result then implies the first limit in (ii). Moreover for a = 0 and b = δ it also gives the second limit of (ii) by taking the limit δ → 0+ in (3.38). (iii) Since
NΛ = a∗0 a0 +
a∗k ak ,
k∈Λ∗ \{0}
the limit is deduced from (2.13) and (i). (iv) is a direct consequence of Remark 3.2 combined with (i) and (iii). (v) Notice that the mean particle values a∗k ak H SB are defined on the discrete Λ,λ0
set Λ∗ ⊂ R3 . Below we denote by
gβ,µ,Λ (k) ≡ a∗k ak H SB (β, µ)
(3.41)
Λ,λ0
a continuous interpolation of these values from the set Λ∗ to R3 and we define by gβ,µ (k) the corresponding thermodynamic limit: gβ,µ (k) ≡ lim gβ,µ,Λ (k) for k ∈ R3 \ {0} . Λ
(3.42)
This limit exists at least almost surely. In fact, using correlation inequalities [7,42, 43] as it is done for the Weakly Imperfect Bose Gas [21], we can prove the existence of the thermodynamic limit (3.42) for any (β, µ) ∈ QS with an uniformly bound for all k ∈ R3 \ {0}. For any interval (a, b) with 0 < a < b, we have the convergence of the Riemann sums to the integral: lim Λ
1 V
a∗k ak H SB (β, µ) =
{k∈Λ∗ ,k∈(a>0,b)}
Λ,λ0
1 3
(2π)
gβ,µ (k) χ(a,b) (k) d3 k,
R3
which combined with (3.38) implies
1
gβ,µ (k) χ(a,b) (k) d k =
3
(2π)
3
R3
1 3
(2π)
ξβ,µ (k) χ(a,b) (k) d3 k
(3.43)
R3
with the continuous function ξβ,µ (k) defined by (3.39) and (3.40). Since the relation (3.43) is valid for any interval (a, b) ⊂ R with 0 < a < b one gets gβ,µ (k) = ξβ,µ (k) , k ∈ R3 , k ≥ δ > 0. By this and (3.41)–(3.43) combined with (3.39)–(3.40) we finally get the statements in (v) for k ≥ δ > 0.
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Appendix: The Bogoliubov u-v transformation In this subsection we recall the Bogoliubov canonical u-v transformation by applying it on the Bogoliubov approximation [22] B HΛ,λ (α, c) 0 1 2 εk − α + λ0 |c| a∗k ak + = 2 ∗ k∈Λ \{0}
+
1 2
k∈Λ∗ \{0}
2
λk |c|
$
a∗k ak + a∗−k a−k
%
k∈Λ∗ \{0}
$ % λ0 4 2 2 (A.1) |c| V − |c| λk c2 a∗k a∗−k + c2 ak a−k − α |c| V + 2
B B of HΛ,λ (α) ≡ HΛ,λ − αNΛ (1.4) for any λ0 ≥ 0. Then, we compute the corre0 0 sponding pressure
pB Λ,λ0 (β, α, c) =
B 1 ln T rFB e−βHΛ,λ0 (α,c) . βV
(A.2)
After the canonical gauge transformation to boson operators ak e−i arg c , k ∈ Λ∗ \ {0} ,
(A.3)
2
B B (α, c) depends only on x ≡ |c| . Since HΛ,λ (α, c) is a bilinear form note that HΛ,λ 0 0 ∗ in boson operators {ak , ak }k∈Λ∗ \{0} , the Bogoliubov canonical u-v transformation diagonalizes it by using a new set of boson operators {bk , b∗k }k∈Λ∗ \{0} defined by
ak = uk bk − vk b∗−k , a∗k = uk b∗k − vk b−k ,
(A.4)
with real coefficients {uk = u−k }k∈Λ∗ \{0} and {vk = v−k }k∈Λ∗ \{0} satisfying: u2k − vk2 = 1, 2uk vk =
xλk εk , u2k + vk2 = B . B Ek,λ0 Ek,λ0
Here fk,λ0 = ε* k − α + x (λ0 + λk ) , + 2 fk,λ − x2 λ2k = (εk − α + xλ0 ) (εk − α + x (λ0 + 2λk )), 0
B Ek,λ = 0
where we recall that x ≡ |c|2 . Thus u2k
1 = 2
! fk,λ0 1 + 1 , vk2 = B 2 Ek,λ0
(A.5)
! fk,λ0 −1 . B Ek,λ 0
2
Notice that fk,λ0 ≥ xλk and, |c| and α satisfy the inequality: 2
α ≤ |c| λ0 +
min
k∈Λ∗ \{0}
εk .
(A.6)
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The Hamiltonian (A.1) becomes: HB Λ,λ0 (α, c) =
B Ek,λ b∗ b + 0 k k
k∈Λ∗ \{0}
1 2
B 2 Ek,λ0 − fk,λ0 − α |c|
k∈Λ∗ \{0}
λ0 + 2
|c|2 |c| − V 4
! . (A.7)
Therefore, the pressure pB Λ,λ0 (β, α, c) (A.2) equals 2 2 β, α, x ≡ |c| + η α, x ≡ |c| , pB (β, α, c) = ξ Λ,λ Λ,λ 0 0 Λ,λ0 −1 B 1 ξΛ,λ0 (β, α, x) = ln 1 − e−βEk,λ0 , βV k∈Λ∗ \{0} λ0 2 1 x B fk,λ0 − Ek,λ + αx − x − , ηΛ,λ0 (α, x) = 0 2V 2 V ∗
(A.8)
k∈Λ \{0}
and has the following thermodynamic limit: 2 pB ≡ lim pB λ0 β, α, x ≡ |c| Λ,λ0 (β, α, c) = ξλ0 (β, α, x) + ηλ0 (α, x) , Λ −1 B 1 −βEk,λ 0 ln 1 − e d3 k, ξλ0 (β, α, x) ≡ lim ξΛ,λ0 (β, α, x) = 3 Λ (2π) β R3 1 λ0 B fk,λ0 − Ek,λ d3 k + αx − x2 , ηλ0 (α, x) ≡ lim ηΛ,λ0 (α, x) = 3 0 Λ 2 2 (2π) R3
B with Ek,λ , fk,λ0 ≥0 defined by (A.5) and α ≤ xλ0 (cf. (A.6). 0 ≥0
(A.9)
Acknowledgments. The work was supported by DFG grant DE 663/1-3 in the priority research program for interacting stochastic systems of high complexity. Special thanks first go to T. Dorlas and the DIAS for the very nice stay there where this work was finished. J.-B. Bru thanks Institut f¨ ur Mathematik, Technische Universit¨ at Berlin, and its members for their warm hospitality during the academic year 2001-2002 and more precisely S. Adams. J.-B. Bru also wants to express his gratitude to N. Angelescu, A. Verbeure and V.A. Zagrebnov for their useful discussions. And the second author thanks the P. master Dukes and Dido for their help in writing/correcting this article. The authors especially thank the referee for helpful remarks and suggestions.
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References [1] N. Angelescu, A. Verbeure and V.A. Zagrebnov, On Bogoliubov’s model of superfluidity J. Phys. A: Math.Gen. 25, 3473 (1992). [2] S. Adams and J.-B. Bru, Critical Analysis of the Bogoliubov Theory of Superfluidity, Physica A 332, 60–78 (2004). [3] S. Adams and J.-B. Bru, A New Microscopic Theory of Superfluidity at all Temperatures, Annales Henri Poincar´e 5, 437–479 (2004). [4] J.-B. Bru and V.A. Zagrebnov, Exact solution of the Bogoliubov Hamiltonian for weakly imperfect Bose gas, J. Phys. A: Math. Gen. A 31, 9377 (1998). [5] J.-B. Bru and V.A. Zagrebnov, Quantum interpretation of thermodynamic behaviour of the Bogoliubov weakly imperfect Bose gas, Phys. Lett. A 247, 37 (1998). [6] V.A. Zagrebnov and J.-B. Bru, The Bogoliubov Model of Weakly Imperfect Bose Gas, Phys. Rep. 350, 291 (2001). [7] O. Brattelli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol. II, 2nd ed. Springer-Verlag, New York (1996). [8] D. Ruelle, Statistical Mechanics: Rigorous Results, Benjamin-Reading, NewYork (1969). [9] N.N. Bogoliubov, On the theory of superfluidity, J. Phys. (USSR) 11, 23 (1947). [10] N.N. Bogoliubov, About the theory of superfluidity, Izv. Akad. Nauk USSR 11, 77 (1947). [11] N.N. Bogoliubov, Energy levels of the imperfect Bose-Einstein gas, Bull. Moscow State Univ. 7, 43 (1947). [12] N.N. Bogoliubov, Lectures on Quantum Statistics, Vol. 1: Quantum Statistics, Gordon and Breach Science Publishers, New York-London-Paris (1970). [13] N.N. Bogoliubov, Energy levels of the imperfect Bose-Einstein gas, p. 242-257 in: Collection of papers, Vol. 2, Naukova Dumka, Kiev, (1970). [14] N. Angelescu and A. Verbeure, Variational solution of a superfluidity model, Physica A 216, 386 (1995). [15] N. Angelescu, A. Verbeure and V.A. Zagrebnov, Superfluidity III, J. Phys. A: Math.Gen. 30, 4895 (1997). [16] N.N. Bogoliubov and D.N. Zubarev, Wave function of the ground-state of interacting Bose-particles, JETP 28, 129 (1955).
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[17] D.N. Zubarev, Distribution function of non-ideal Bose-gas for zero temperature, JETP 29, 881 (1955). [18] Yu.A. Tserkovnikov, Theory of the imperfect Bose-Gas for non-zero temperature, Doklady Acad. Nauk USSR 143, 832 (1962). [19] V.N. Popov, Functional Integrals and Collective Excitations, Univ. Press, Cambridge (1987). [20] H. Shi and A. Griffin, Finite-temperature excitations in a dilute Bosecondensated gas, Phys. Rep. 304, 1 (1998). [21] J.-B. Bru and V.A. Zagrebnov, On condensations in the Bogoliubov Weakly Imperfect Bose-Gas, J. Stat. Phys. 99, 1297 (2000). [22] J. Ginibre, On the Asymptotic Exactness of the Bogoliubov Approximation for many Bosons Systems, Commun. Math. Phys. 8, 26 (1968). [23] J.-B. Bru, Superstabilization of Bose Systems I: Thermodynamic Study, J. Phys. A: Math.Gen. 35, 8969 (2002). [24] N.N. Bogoliubov (Jr), J.G. Brankov, V.A. Zagrebnov, A.M. Kurbatov and N.S. Tonchev, The Approximating Hamiltonian Method in Statistical Physics (Publ. Bulgarian Akad. Sciences, Sofia, 1981). [25] N.N. Bogoliubov (Jr), J.G. Brankov, V.A. Zagrebnov, A.M. Kurbatov and N.S. Tonchev, Some classes of exactly soluble models of problems in Quantum Statistical Mechanics : the method of the approximating Hamiltonian, Russian Math. Surveys 39, 1 (1984). [26] K. Huang, Statistical Mechanics, Wiley, New York (1963). [27] E.B. Davies, The thermodynamic limit for an imperfect boson gas, Commun. Math. Phys. 28, 69 (1972). [28] M. Fannes and A. Verbeure, The condensed phase of the imperfect Bose gas, J. Math. Phys. 21, 1809 (1980). [29] M. van den Berg, J.T. Lewis and Ph. de Smedt, Condensation in the Imperfect Boson Gas, J. Stat. Phys. 37, 697 (1984). [30] E. Buffet and J.V. Pul`e, Fluctuations Properties of the Imperfect Boson Gas, J. Math. Phys. 24, 1608 (1983). [31] J.T. Lewis, J.V. Pul`e and V.A. Zagrebnov, The Large Deviation Principle for the Kac Distribution, Helv. Phys. Acta 61, 1063 (1988). [32] Vl.V. Papoyan and V.A. Zagrebnov, The ensemble equivalence problem for Bose systems (non-ideal Bose gas), Theor. Math. Phys. 69, 1240 (1986). [33] M. van den Berg and J.T. Lewis, On generalized condensation in the free boson gas, Physica A 110, 550 (1982).
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[34] M. van den Berg, On boson condensation into an infinite number of low-lying levels, J. Math. Phys. 23, 1159 (1982). [35] M. van den Berg, J.T. Lewis and J.V. Pul`e, A general theory of Bose-Einstein condensation, Helv. Phys. Acta 59, 1271 (1986). [36] J.-B. Bru, Superstabilization of Bose systems II: Bose condensations and equivalence of ensembles, J. Phys. A: Math.Gen. 35, 8995 (2002). [37] H.O. Georgii, Large Deviations and the Equivalence of Ensembles for Gibbsian Particle Systems with Superstable Interaction, Probab. Th. Rel. Fields 99, 171 (1994). [38] H.O. Georgii, The equivalence of ensembles for classical systems of particles, Journal of Stat. Phys. Vol. 80, 1341–1378 (1994). [39] S. Adams, Complete Equivalence of the Gibbs ensembles for one-dimensional Markov-systems, Journal of Stat. Phys., Vol. 105, Nos. 5/6, 879–908 (2001). [40] R. Griffiths, A Proof that the Free Energy of a Spin System is extensive, J. Math. Phys. 5, 1215 (1964). [41] K. Hepp E. and H. Lieb, Equilibrium Statistical Mechanics of Matter Interacting with the Quantized Radiation Field, Phys. Rev. A 8, 2517 (1973). [42] M. Fannes and A. Verbeure, Correlation Inequalities and Equilibrium States I, Commun. Math. Phys. 55, 125 (1977). [43] M. Fannes and A. Verbeure, Correlation Inequalities and Equilibrium States II, Commun. Math. Phys. 57, 165 (1977). S. Adams Institut f¨ ur Mathematik Fakult¨ at II, SEK. MA 7-4 Technische Universit¨ at Berlin Strasse des 17. Juni 136 D-10623 Berlin, Germany email:
[email protected] J.-B. Bru School of Theoretical Physics Dublin Institute for Advanced Studies 10 Burlington Rd. Dublin 4, Ireland email:
[email protected] Communicated by Vincent Pasquier Submitted 31/03/03, accepted 01/12/03
Ann. Henri Poincar´e 5 (2004) 435 – 476 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/030435-42 DOI 10.1007/s00023-004-0175-7
Annales Henri Poincar´ e
A New Microscopic Theory of Superfluidity at All Temperatures Stephan Adams and Jean-Bernard Bru
Abstract. Following the program suggested in [1], we propose a new microscopic theory of superfluidity for all temperatures and densities. In particular, the corresponding phase diagram of this theory exhibits: (i) a thermodynamic behavior corresponding to the Perfect Bose Gas for small densities or high temperatures, (ii) the “Landau-type” excitation spectrum in the presence of non-conventional Bose condensation for high densities or small temperatures, (iii) a depletion of the Bose condensate with the formation of “Cooper-type pairs”, even at zero-temperature (experimentally, an estimate of the fraction of condensate in liquid 4 He at T=0 K is 9%, see [2, 3]). In contrast to Bogoliubov’s last approach and while warning that the full interacting Hamiltonian is truncated, the analysis performed here is rigorous by involving a complete thermodynamic analysis of a non-trivial continuous gas in the canonical ensemble.
1 Introduction The first microscopic theory of superfluidity was originally proposed in 1947 by Bogoliubov in [4–8]. A recent analysis of the Bogoliubov theory has already been performed in the review [9], itself containing a summary of [10–15]. The critical analysis performed in [1] leads us to use a truncation of the full Hamiltonian within the framework of the canonical ensemble. The resulting model, different from the Bogoliubov one and defined in Section 2, is here rigorously solved at the thermodynamic level in the canonical ensemble (Section 2.2). In the case of homogeneous systems, this analysis provides a new (canonical) theory of superfluidity with a gapless spectrum at all particle densities and temperatures, leading us to a deeper understanding of the Bose condensation phenomenon in liquid helium explained in Section 3. Actually, at any temperatures T ≥ 0 below a critical temperature Tc , the corresponding Bose gas is a mixture of particles inside and outside the Bose condensate, i.e., there is a depletion of the Bose condensate. Even at zero-temperature, our interpretation is that two Bose subsystems coexist: the Bose condensate and a second system, denoted here as the Bogoliubov system. This comes from a nondiagonal interaction, which, in particular, implies an effective attraction between
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bosons in the zero kinetic energy state, i.e., in the Bose condensate [12]. In contrast with the (conventional) Bose-Einstein condensation, these bosons pair up via the Bogoliubov system to form “Cooper-type pairs” or interacting (virtual) pairs of particles. This Bose condensation constituted by Cooper-type pairs is non-conventional [9, 11, 12, 14, 16, 17], i.e., turned on by the Bose distribution but completely transformed by interaction phenomena. The coherency due to the presence of the Bose condensation is not enough to make the Perfect Bose Gas superfluid, see discussions in [4–6]. The spectrum of elementary excitations has to be collective. In this theory, the particles outside the Bose condensate (the Bogoliubov system, Remark 3.2) follow a new distribution, different from the Bose distribution, which we call the Bogoliubov distribution. The Bogoliubov system coming from the depletion of the Bose condensate is a model of “quasi-particles” or linked pairs of particles with the Landau-type excitation spectrum. Therefore, following Landau’s criterion of superfluidity [18, 19] it is a superfluid gas. The corresponding “quasi-particles” are created from two particles respectively of momenta p and −p (p = 0) through the Bose condensate (p = 0) combined with phenomena of interaction. The theoretical critical temperature where the Landau-type excitation spectrum holds equals Tc ≈ 3.14 K. For the liquid 4 He, the superfluid liquid already disappears at Tλ = 2.17 K, but the Henshaw-Woods spectrum1 [20] does not λ change drastically when the temperature crosses Tλ : there is a temperature T where the Landau-type excitation spectrum persists for Tλ < T < Tλ . For a complete description of this theory in relation with liquid 4 He, see Sections 3.3 to 3.4. The phenomenon of Cooper pairs between two fermions corresponds to the phenomenological explanation given for the existence of superfluidity and Bose condensation in 3 He [21–23]. Therefore, at the end (Section 3.5), we explain how this theory may also be a starting point for a microscopic theory of superfluidity for 3 He within the framework of Fermi systems. Before finishing this short introduction, we recall again that this analysis is based on a truncation of the full interacting Bose gas in the canonical ensemble. This unique truncation hypothesis is still not proven in this paper, but we show that the theory is, at least, self-consistent. In fact, the aim of the present paper is to give the exact solution of a non-diagonal continuous model far from the Perfect Bose Gas in the canonical ensemble at all temperatures and densities. Note that this analysis is technically based on three papers [24–26]. We use the “superstabilization” method [24, 25] to analyze the corresponding model in the canonical ensemble from the grand-canonical one. This study is possible since the exact solution of the (non-diagonal) AVZ-Hamiltonian [27], also called the superstable Bogoliubov Hamiltonian, is found in the grand-canonical ensemble by the paper [26].
1 measure of the excitation spectrum in liquid (1961).
4 He
by a weak-inelastic neutron scattering
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2 Our model for superfluidity We give here our proposal for a model for superfluidity. In particular, we first explain the philosophy of this model and then, we solve it in the canonical ensemble.
2.1
Setup of the appropriate model
Let an interacting homogeneous gas of n spinless bosons with mass m be enclosed 3
in a cubic box Λ = × L ⊂ R3 . We denote by ϕ (x) = ϕ (x) a (real) two-body α=1
interaction potential and we assume that: (A) ϕ (x) ∈ L1 R3 . (B) Its (real) Fourier transformation λk =
d3 xϕ (x) e−ikx , k ∈ R3 ,
R3
satisfies: λ0 > 0 and 0 ≤ λk = λ−k ≤ lim + λk for k ∈ R3 . k→0
The one-particle energy spectrum is εk ≡ 2 k 2 /2m and, using periodic boundary conditions, Λ∗ ≡
2πnα , nα = 0, ±1, ±2, . . . , α = 1, 2, 3 k ∈ R3 : kα = L
∗ is the set of wave vectors. Let a# k = {ak or ak } be the usual boson creation/ 1 annihilation operators in the one-particle state ψk (x) = V − 2 eikx , k ∈ Λ∗ , x ∈ Λ, acting on the boson Fock space +∞
(n)
FΛB ≡ ⊕ HB , n=0
(n)
with HB defined as the symmetrized n-particle Hilbert spaces (n) (0) HB ≡ L2 (Λn ) symm , HB = C, see [28, 29]. Therefore, the corresponding Hamiltonian of the system acting on the boson Fock space FΛB is equal to Λ + UΛMF , HΛ,λ0 >0 = TΛ + U
(2.1)
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with TΛ
≡
εk a∗k ak
k∈Λ∗
Λ U UΛMF
1 2V
≡
λ0 2V
≡
λq a∗k1 +q a∗k2 −q ak1 ak2 ,
k1 ,k2 ,q=0∈Λ∗
a∗k1 a∗k2 ak2 ak1 =
k1 ,k2 ∈Λ∗
Here
NΛ ≡
λ0 2 NΛ − NΛ . 2V
(2.2)
a∗k ak
k∈Λ∗
is the particle number operator. Under assumptions (A) and (B) on the interaction potential ϕ (x), the full Hamiltonian HΛ,λ0 >0 is superstable [29]. Without any Bose condensation, the model should be equal to the MeanField model, i.e., the Perfect Bose gas in the canonical ensemble. Whereas, in Λ should play a crucial role on presence of Bose condensation, the interaction U the thermodynamics. Formally, the Mean-Field interaction UΛMF does not change the “physical properties” of a Bose system (cf. [1, 24, 25]). The “physical” effect of the interaction potential should express itself by the other terms of interaction, Λ . i.e., by U Within the framework of the canonical ensemble, considering the existence of a Bose condensation on the zero-kinetic energy state in liquid 4 He, originally suggested by Fritz London in 1938 [30], one should partially truncate the full interaction without taking into account the Mean-Field interaction since it is a constant in the canonical ensemble. This procedure implies the non-diagonal Hamiltonian: B HΛ,0 ≡ TΛ + UΛD + UΛN D
(2.3)
with UΛD UΛN D
≡ ≡
1 2V 1 2V
k∈Λ∗ \{0}
λk a∗0 a0 a∗k ak + a∗−k a−k ,
(2.4)
2 λk a∗k a∗−k a20 + a∗0 ak a−k .
(2.5)
k∈Λ∗ \{0}
Note that, with the usual Bogoliubov approximation √ √ a0 / V → c, a∗0 / V → c,
(2.6)
B on HΛ,0 , the new Hamiltonian does not commute with the particle number operator NΛ . To solve this problem in the canonical ensemble, Bogoliubov [6] suggests
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a different but similar way corresponding to a “canonical Bogoliubov approximation”. We first use the new set of operators ζk = a∗0 (N0 + I)
−1/2
−1/2
ak , ζk∗ = a∗k (N0 + I)
a0 , k ∈ Λ ∗ .
(2.7)
The set {ζk }k∈Λ∗ \{0} satisfies the Canonical Commutation Relations. Then, for B HΛ,0 the new Bogoliubov approximation corresponds to do the following transformations: 1/2 1/2 (N0 − I) N0 N 2 2 → |c| , 0 → |c| , N0 ≡ a∗0 a0 . (2.8) V V It implies a bilinear form in Bose-operators {ζk }k∈Λ∗ \{0} :
B HΛ,0 (c) =
εk ζk∗ ζk +
k∈Λ∗ \{0}
+
1 2
1 2
2
λk |c|
∗ ζk∗ ζk + ζ−k ζ−k
k∈Λ∗ \{0}
∗ λk c2 ζk∗ ζ−k + c2 ζk ζ−k .
(2.9)
k∈Λ∗ \{0}
This Hamiltonian commutes with the particle number operator NΛ . After the canonical gauge transformation to boson operators ζk e−i arg c , k ∈ Λ∗ \ {0} , 2
B (c) only depends on x ≡ |c| . Then, the Bogoliubov canonithe model HΛ,0 cal u-v transformation diagonalizes it by using a new set of boson operators {bk , b∗k }k∈Λ∗ \{0} defined by
ζk = uk bk − vk b∗−k , ζk∗ = uk b∗k − vk b−k .
(2.10)
The real coefficients {uk = u−k }k∈Λ∗ \{0} and {vk = v−k }k∈Λ∗ \{0} satisfy: xλk εk , u2k + vk2 = . u2k − vk2 = 1, 2uk vk = εk (εk + 2xλk ) εk (εk + 2xλk ) B (c = 0) corresponds to the perfect Bose gas It follows that the Hamiltonian HΛ,0 of quasi-particles defined by 2 BG x ≡ |c| = HP εk (εk + 2xλk )b∗k bk Λ,0 k∈Λ∗ \{0}
+
1 2
εk (εk + 2xλk ) − (εk + xλk ) .
(2.11)
k∈Λ∗ \{0}
In other words, if we consider that this “canonical Bogoliubov approximation” is true, we directly get the well-known Bogoliubov gapless spectrum for a Bose con2 densate density x = |c| > 0.
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The Bogoliubov procedures [4–8] always involved the truncation of the MeanB Field interaction UΛMF . It implied the Bogoliubov Hamiltonian HΛ,λ (A.1), see 0 >0 Appendix A. From the observations of [1,9,27], in various respects the Bogoliubov B is not appropriate as the model of superfluidity. For example, in order model HΛ,λ 0 B , but to deduce its gapless spectrum, Bogoliubov applied (2.6) or (2.8) on HΛ,λ 0 >0 he also needed additional approximations which were shown to be not legitimate. The main problem of the previous attempts (Bogoliubov et al., see for example [4–8, 27, 31, 32]) is to assume, a priori, the Bose condensation by directly doing the Bogoliubov approximation with an arbitrary choice of |c|2 , without exactly solving it in terms of the thermodynamic behavior. In particular, the “canonical B Bogoliubov approximation” (2.8) applied on HΛ,0 has to be proven. For example, Bogoliubov first needed the inexact assumption of 100% of Bose condensate at zero-temperature in the canonical ensemble (Appendix A) and he realized (also with Zubarev) the difficulty with this ansatz: his u-v transformation (2.10) implies a depletion of the Bose condensate due to repulsion between particles. Therefore, 2 what is our value of x = |c| after the approximation (2.8)? Actually, these questions are solved in the next subsection since the canonB is rigorously ical thermodynamic behavior of the non-diagonal Hamiltonian HΛ,0 performed. In particular, it is shown that the “canonical Bogoliubov approximaB corresponds, tion” (2.8) is true in the following sense: the thermodynamics of HΛ,0 at the thermodynamic level, to the perfect Bose gas (2.11) of quasi-particles for k ∈ Λ∗ \ {0} with a Bose condensate density x
(β, ρ) on k = 0 (cf. Theorems 2.2 and 2.3). Note that this new approach is originally explained in [1] and comes from a constructive criticism of the Bogoliubov theories. Moreover, before going further, we want to stress that our approach is different from the Bogoliubov one. In the canonical ensemble, the theory of the present paper is new and distinct from the canonical Bogoliubov theory (Appendix A) in many aspects: B B (2.3) and HΛ,λ = • First, they are based on two separate Bose gases HΛ,0 0 >0 B BMF (A.1). Both of this theories have behind them, in the correHΛ,0 + UΛ sponding truncation, the fundamental hypothesis originally given by Fritz London [30] about existence in the system of a Bose condensation in the zero-mode. • Secondly, the Bogoliubov theories must include other tricks in order to find the Landau-type excitation spectrum (Appendix A). However, his additional ansatz in the canonical ensemble are unnecessary for our approach (see also Section 3). • Finally, the canonical Bogoliubov theory could have been exact only at zerotemperature, whereas our results below conclusion) are valid at any temperatures T ≥ 0 for a fixed particle density ρ > 0. B solves, in the canonical ensemble, In fact we prove in this paper that the model HΛ,0 the problems of the previous Bogoliubov theories and implies a new microscopic theory of superfluidity at all temperatures explained in Section 3.
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To fix the notations, β > 0 is here the inverse temperature, µ the chemical potential, ρ > 0 the fixed full particle density, whereas n = [ρV ], defined as the integer of ρV , is the number of particles in the canonical ensemble. Then, T = (kB β)−1 ≥ 0 is the temperature where kB is the Boltzmann constant. Here − H X (β, ρ) and − H X (β, µ) represent the (finite volume) canonical Λ Λ and grand-canonical Gibbs state respectively for some Hamiltonian HΛX : (n) X −βHΛ (−) e T rH(n) B − H X (β, ρ) ≡ (n) , Λ X e−βHΛ T rH(n) B
− H X Λ
X T rFΛB (−) e−β (HΛ −µNΛ ) (β, µ) ≡ X T rFΛB e−β (HΛ −µNΛ )
(n)
where A(n) ≡ A HB is the restriction of any operator A acting on the boson Fock (n) space FΛB to HB .
2.2
Thermodynamics in the canonical ensemble
B The aim of this section is to examine the Hamiltonian HΛ,0 (2.3) in the canonical B ensemble specified by (β, ρ). The model HΛ,0 turns out to be not sufficient for a microscopic theory of superfluidity in the grand-canonical ensemble because of its instability in presence of Bose condensation (Appendix B). The terms of repulsion B are not strong enough to prevent the system from collapse in the Hamiltonian HΛ,0 B in the grand-canonical ensemble. However, we explain here that the Bose gas HΛ,0 can be solved in the canonical ensemble by superstabilizing it [24,25] in the grandcanonical ensemble. The principle is the following: SB B (i) We denote by HΛ,λ the superstabilization of the model HΛ,0 which is defined by SB B ≡ HΛ,0 + HΛ,λ
λ 2V
k1 ,k2 ∈Λ∗
B a∗k1 a∗k2 ak2 ak1 = HΛ,0 +
λ 2 NΛ − NΛ . 2V
(2.12)
B the Mean-Field interaction UΛMF This procedure adds to the Hamiltonian HΛ,0 SB (2.2) with a sufficiently large parameter λ > 0. The model HΛ,λ is just a technical tool. For λ = λ0 it corresponds to the Angelescu-Verbeure-Zagrebnov Hamiltonian [27], also called the superstable Bogoliubov Hamiltonian. It is rigorously solved in the grand-canonical ensemble (see Appendix C.1) and the corresponding proof does not depend on the fact that λ0 is the Fourier transformation of the interaction potential for k = 0 [26]. The only constraint is to have the superstability of the SB model HΛ,λ , which is verified for large enough λ > 0.
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(ii) Then, we use the notion of strong equivalence of ensembles (see [25, 33–35]) SB for the model HΛ,λ with the arbitrary parameter λ taken such that λ + g00 > 0. 2
(2.13)
Indeed, in the canonical ensemble, note that for a given density ρ, i.e., on the (n=[ρV ]) SB B , the Hamiltonians HΛ,λ and HΛ,0 differ only by a conHilbert space HB stant, i.e., their canonical thermodynamics are equal to each other. In Appendix C, we explain that the strong equivalence between canonical and grand-canonical SB if (2.13) is satisfied. Therefore the canonensembles is verified for the model HΛ,λ B ical thermodynamic properties of HΛ,0 correspond for a fixed particle density ρ SB to the one of HΛ,λ for a fixed density ρ in the grand-canonical ensemble (in [26]: Section 2.3). Remark 2.1 Here the (effective coupling) constant g00 equals 2 1 3 λk g00 ≡ − d k < 0, 3 εk 4 (2π)
(2.14)
R3
with the one-particle energy spectrum εk ≡ 2 k 2 /2m. We explain later in Section 3.2 the quantum interpretation given by [12] of the constant g00 . B Now we give all promised properties of the Hamiltonian HΛ,0 in the canonical ensemble. To simplify our purpose, the proofs are given in Appendix C. B 1. Let fΛ,0 (β, ρ) be the corresponding free-energy density defined for a fixed particle density ρ > 0 by (n=[ρV ]) B 1 B −βHΛ,0 e ln T rH(n) fΛ,0 (β, ρ) ≡ − . (2.15) B βV
Recall that the “canonical Bogoliubov approximation” (2.8) implies the model B B (c) corHΛ,0 (c) (2.9). Here, for technical considerations, we use the operator H Λ,0 B responding in HΛ,0 (c) to replace again the operators {ζk }k∈Λ∗ \{0} by {ak }k∈Λ∗ \{0} . B (c) is well defined on the boson Fock space The Hamiltonian H Λ,0
+∞
(n )
≡ ⊕ HB,k1 =0 FB n1 =0
(n )
of the symmetrized n1 -particle Hilbert spaces HB,k1 =0 for non-zero momentum bosons. The Bogoliubov canonical u-v transformation gives also for k ∈ Λ∗ \ {0} the perfect Bose gas (2.11) of quasi-particles for a Bose condensate (k = 0) density x. We then consider its (infinite volume) free-energy density defined by (n1 =[ρ1 V ],k=0) 1 B e−β HΛ,0 (c) ln T rH(n1 ) f0B (β, ρ1 , x) ≡lim − , (2.16) Λ B,k=0 βV
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for any β > 0, ρ1 > 0 and x = |c|2 ≥ 0, where (n )
A(n1 ,k=0) ≡ A HB,k1 =0 is the restriction of any operator A acting on FB . The (infinite volume) pressure of this gas of quasi-particles is
pB 0 (β, α, x)
≡ = =
B (c)−α −β H Λ,0
a∗ k ak +x
1 k∈Λ∗ \{0} ln T rFB e βV sup α [ρ1 + x] − f0B (β, ρ1 , x) ρ1 >0 −1 1 3 B 1 1 −βEk,0 B ln 1 − e f d k, + − E αx + k,0 k,0 3 β 2 (2π) lim Λ
R3
(2.17) for α ≤ 0 with B fk,0 = εk − α + xλk , Ek,0 =
(εk − α) (εk − α + 2xλk ).
(2.18)
Then we get our first main result: Theorem 2.2 The thermodynamic limit f0B (β, ρ) exists for any β > 0 and ρ > 0. B (i) Moreover, the Hamiltonians HΛ,0 (2.3) is equivalent, at the thermodynamic level, to the perfect Bose gas (2.11) of quasi-particles for k ∈ Λ∗ \ {0} with a density x = x
(β, ρ): B B B . f0 (β, ρ) = inf f0 (β, ρ − x, x) = f0 (β, ρ − x, x) x∈[0,ρ]
x= x<ρ
(ii) More explicitly the free-energy density f0B (β, ρ) equals:
) = α ( x) ρ − pB x) , x
) . f0B (β, ρ) = sup αρ − pB 0 (β, α, x 0 (β, α ( α≤0
Note that f0B (β, ρ1 , x) (2.16) may have been directly defined as the Legendre transformation (C.10) of pB 0 (β, α, x). In this theorem, note that we consider the two systems as equivalent at the thermodynamic level if their corresponding free-energy densities are equal in the thermodynamic limit. In fact the solution x
(β, ρ) = x
(β, µβ,ρ ) of the first variational problem (i) in Theorem 2.2 is originally defined as the solution of (C.2) for a fixed density ρ > 0, i.e., for a chemical potential µβ,ρ (C.5).
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Also the solution α ( x) of (ii) is originally defined as the solution of the variational problem (C.1) for µ = µβ,ρ (C.5). It is the unique solution of the Bogoliubov density equation:
) for ρ > 0. ρ = ρB 0 (β, α, x Here ρB 0 (β, α, x)
≡ ∂α pB 0 (β, α, x) = x +
+
1 3
(2π)
R3
B 2Ek,0
1 3
(2π)
x2 λ2k fk,0 +
R3
B Ek,0
(2.19) f d3 k k,0B βEk,0 B Ek,0 e −1
d3 k.
(2.20)
Moreover, there is a particle density ρc (β) (C.3)-(C.4) such that the solution x
(β, ρ) = 0 for ρ ≤ ρc (β) , whereas for ρ > ρc (β) , 0 < x
(β, ρ) < ρ (even for β → +∞). For a fixed particle density ρ, there is also a critical inverse temperature βc (ρ), see Appendix C.1. An illustration of βc (ρ) is performed in Figure 2.1. Note that ∂ρ f0B (β, ρ) = α ( x) and ∂ρ f0B (β, ρ) < 0 for ρ = ρc (β) or β = βc (ρ) (Remark C.3). B corresponds For ρ ≤ ρc (β), note that the thermodynamic behavior of HΛ,0 to the Perfect Bose Gas (excitation spectrum εk ). B 2. Now we give our main result for the thermodynamic behavior of HΛ,0 in the canonical ensemble (β, ρ).
Theorem 2.3 (i) A non-conventional Bose condensation induced by the non-diagonal interaction UΛN D for high particle densities, or low temperatures: ∗ a0 a0 = 0 for ρ ≤ ρc (β) or β ≤ βc (ρ) . lim (β, ρ) = x
(β, ρ) = > 0 for ρ > ρc (β) or β > βc (ρ) . Λ V B H Λ,0
(ii) No Bose condensation (of any type I, II or III [36–38]) outside the zero-mode for any particle densities or temperatures: ∗ ak ak ∗ ∀k ∈ Λ \ {0} , lim (β, ρ) = 0 Λ V B HΛ,0 lim lim 1 a∗k ak H B (β, ρ) = 0 δ→0+ Λ V Λ,0 ∗ {k∈Λ ,0<k≤δ}
(iii) A particle density outside the zero-mode equals 1 1 fk,0 3 lim a∗k ak H B (β, ρ) = k d B Λ,0 βEk,0 (2π)3 x=x , Λ V B e E − 1 α=α( x) k∈Λ∗ \{0} k,0 R3 1 x2 λ2k 3 + . d k (2π)3 x=x , 2E B f + E B R3
k,0
k,0
k,0
α=α(x)
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βc (ρ)
β ρ
ρc (β)
0
Figure 2.1: Illustration of the critical inverse temperature βc (ρ) as a function of ρ. The dotted line corresponds to the phase diagram of the Perfect Bose Gas. The difference with the Perfect Bose Gas is always greater or equal to zero. It may be zero for all β > 0. B are defined by (2.18) for a chemical potential given by the Here fk,0 and Ek,0 solution α = α ( x) at a fixed particle density ρ > 0. (iv) There is no discontinuity of the particle densities (density in the zero-mode (i) or outside the zero-mode (iii)). (v) For ρ ≤ ρc (β) or β ≤ βc (ρ) one has the Bose distribution for a corresponding chemical potential α (0) < 0:
∀k ∈ Λ∗ : k ≥ δ > 0, lim a∗k ak H B (β, ρ) = Λ
Λ,0
1 eβ(εk −α(0))
−1
.
But for ρ > ρc (β) or β > βc (ρ) , i.e., in the presence of a Bose condensation, we get another one, which we call the Bogoliubov distribution, for a corresponding chemical potential α ( x) < 0: 2 2 λ f x k,0 ∗ k + lim ak ak H B (β, ρ) = B Λ,0 x= x,α=α( x) Λ E B eβEk,0 B B f +E −1 2E k,0
k,0
k,0
k,0
for any k ∈ Λ∗ such that k ≥ δ for any δ > 0. The illustrations of the particle densities inside and outside the zero-mode are given in Figures 2.2 and 2.3 respectively.
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2
Bose Condensation
0
ρc (β)
ρ
Figure 2.2: Illustration of the non-conventional Bose condensate density x
(β, ρ) as a function of ρ. The dashed dotted line corresponds to a zero-temperature, i.e., for β → +∞. The straight line is x = ρ. Remark 2.4 For ρ > ρc (β), there is a non-conventional Bose condensation whereas no Bose condensation (of any type I, II, or III [36–38]) appears outside the zero-mode at all densities ρ > 0 (Theorem 2.3). In contrast to the Bogoliubov theories (see for example [1]), the theory is self-consistent with the corresponding truncation of the full Hamiltonian in the canonical ensemble.
3 A new microscopic theory of superfluidity B The aim of this section is to explain why the model HΛ,0 can imply a new microscopic theory of superfluidity for Bose systems. It is essential here to note that in the canonical ensemble the conditions relating to the interaction potential ϕ (x) may be relaxed as follows. The model is independent of the Fourier transformation of ϕ (x) for k = 0, which may be infinite for some specific interaction potentials. However, the (effective coupling) constant g00 (2.14) and ϕ (0) have to exist.
3.1
Landau-type excitation spectrum in the presence of Bose condensation
In order to obtain a microscopic theory of superfluidity we have to get a Landautype excitation spectrum [18, 19] as Bogoliubov did [4–8] for a suitable choice of c-numbers.
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ρ − x(β,ρ ) 2
0
ρ
ρc (β)
Figure 2.3: Illustration of the particle density outside the zero-mode 2 ρ − | x (β, ρ)| as a function of ρ. Note that for ρ < ρc (β), x
(β, ρ) = 0. The dashed dotted line is the Bogoliubov system density at β → +∞, i.e., at zerotemperature. 1. As Landau’s predictions [18,19], at high densities ρ > ρc (β) (C.3) (or sufficiently B low temperatures) the Bose gas HΛ,0 is equivalent to a “gas of collective elementary excitations” or “quasi-particles” (2.11) for k ∈ Λ∗ \ {0} with a density x
(β, ρ) of Bose condensate on k = 0, cf. Theorems 2.2 and 2.3. Consequently, as stated in Section 2.1, the spectrum of excitations, which is macroscopically relevant, equals the Bogoliubov spectrum at inverse temperatures β > 0 and particle densities ρ > 0: 2 2 ε k = k /2m for β ≤ βc (ρ) or ρ ≤ ρc (β) , (3.1) EkB (β, ρ) = εk (εk + 2 xλk ) for β > βc (ρ) or ρ > ρc (β) , see (2.11). The collective excitation spectrum EkB (β, ρ) has no gap for any densities or temperatures as expected in Section 2.1. The main difficulties are to find B in the canonx
(β, ρ), i.e., the thermodynamic properties of the Hamiltonian HΛ,0 ical ensemble. B , since our Note that we do not rigorously know the exact spectrum of HΛ,0 analysis is only based on its thermodynamic properties. In infinite volume, this B question also implies the problem of definition of lim HΛ,0 ! Λ
2. Now, to find the exact Landau-type excitation spectrum from (3.1), i.e., to get the “phonons” part and the “rotons” one, we can reason along the standard lines of Bogoliubov microscopic theory of superfluidity, see [4–9]. For this approach, we have to assume some specific conditions relating to the two-body interaction potential ϕ (x). In particular, the two-body potential
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ϕ (x) should verify (A)-(B). Here λk is spherically-symmetric, i.e., λk = λk , and additionally, as Bogoliubov did, we assume the absolute integrability of x2 ϕ (x) ∈ 3 1 L R . Actually, we need here the last assumption and the Fourier transformation of ϕ (x) for k = 0 in order to have a Taylor expansion λk = λ0 +
1 2 2 k λ0 + o k , 2
(3.2)
of λk allowing us to analyze EkB (β, ρ) for small k (phonon part). Here λ0 ≤ 0 −1 is the second derivative for k = 0 and |λk | ≤ const. k . Let ρ > ρc (β) or β > βc (ρ), i.e., x
(β, ρ) > 0 (cf. (i) of Theorem 2.3). Then the collective spectrum of excitations EkB (β, ρ) in this domain of (β, ρ) verifies: 1/2 2 λ0 x
k = w k , for k → 0+ . EkB (β, ρ) = (3.3) m εk = 2 k 2 /2m , for k → +∞. The gapless spectrum EkB (β, ρ) is phonon-like for small k (ρ > ρc (β)), whereas for large wave-vectors it behaves like the single-particle excitations εk . Since λk attains its maximum at k = 0, one can choose the potential ϕ (x) in such a way that εk εk + 2 xλk = 0 at k = krot = 0, (3.4) i.e., the spectrum EkB (β, ρ) has a local (“roton”) minimum at krot . On the other hand, one gets: EkB (β, ρ) ≥ k
2 2m
1/2
1/2
≡ k v0 (β, ρ) . min εk + 2λ2k x k
(3.5)
The Bogoliubov spectrum EkB (β, ρ) is a Landau-type excitation spectrum for ρ > ρc (β) or β > βc (ρ) and an illustration is given by Figure 3.1. Remark 3.1 The famous Landau’s criterion of superfluidity of 1941 [18, 19] gives the following critical velocity: B Ek (β, ρ) 0 , for β ≤ βc (ρ) or ρ ≤ ρc (β) . inf = v0 (β, ρ) = > 0, for β > βc (ρ) or ρ > ρc (β) . k k
3.2
Two complementary Bose systems: Cooper-type pairs and gas of quasi-particles
We give here the quantum interpretation of the canonical thermodynamic propB erties of the model HΛ,0 . First note that, in terms of particle densities, we obtain (see Theorem 2.3 and Figures 2.2 and 2.3):
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E Bk (β,ρ)
hw k h2k2 2m h k v 0 (β,ρ) 0
k rot
k
Figure 3.1: The Bogoliubov spectrum EkB (β, ρ) for β > βc (ρ) or ρ > ρc (β) . • A non-conventional Bose condensation appears with the density 0 < x
(β, ρ) < ρ for ρ > ρc (β) (even with β → +∞), whereas at all densities ρ > 0 there is no Bose condensation (of any type I, II, or III [36–38]) outside the zero-mode. • Even for zero-temperature, we have a non-zero particle density outside the zero-mode for any ρ > 0: 1 lim lim a∗k ak H B (β, ρ) = Λ,0 β→+∞ Λ V ∗ k∈Λ \{0} 2 2 1 x λk 3 k > 0, d (2π)3 x=x B B f 2E + E k,0 α=α( x)
k,0 k,0 (3.6) R3 ∗ ∗ ∀k ∈ Λ : k ≥ δ > 0, lim lim a a
(β, ρ) = B k k HΛ,0 β→+∞ Λ 2 2 x λk > 0. 2E B f + E B x=x k,0 α=α(x)
k,0 k,0 In the regime ρ > ρc (β) , the system follows the Bogoliubov distribution (v) of Theorem 2.3, whereas in the absence of the Bose condensation, i.e., for ρ ≤ ρc (β), the (standard) Bose distribution holds. 1. The origin of the Bogoliubov distribution and also of (3.6) is a phenomenon of interaction. Actually, it is known since [12] that the collection of particles outside
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k
λk k’=0
k’=0
−k
Figure 3.2: Non-diagonal-interaction vertices corresponding to UΛN D . the zero-mode imposes, through the non-diagonal interaction UΛN D , a glue-like attraction between particles in the zero-mode. A natural way to see this phenomenon is to remark that the non-diagonal interaction UΛN D (see Figure 3.2) implies an effective interaction term gΛ,00 for bosons with k = 0, see Figure 3.3. Evaluated via a Fr¨ ohlich transformation in the second order [12] (see also the review [9]), gΛ,00 is strictly negative. The corresponding thermodynamic limit lim gΛ,00 = g00 < 0 Λ
remarkably gives (2.14). In particular, this effective attraction term g00 amazingly B plays a crucial rˆ ole in the rigorous thermodynamic analysis of HΛ,0 (see Section 2.2 or in [26]: proof of Theorem 2.3). It is also essential in the rigorous study B , of the Weakly Imperfect Bose Gas, i.e., the Bogoliubov Hamiltonian HΛ,λ 0 >0 see [9–11, 13]. The Bose condensate with the density x
(β, ρ) and the remaining system with the density {ρ − x
> 0}, called here the Bogoliubov system, only exist via this gluelike attraction g00 (Figure 3.3). In fact, the particles inside the condensate pairs up via the Bogoliubov system to form “Cooper-type pairs” or interacting (virtual) pair of particles. This Bose condensation constituted by Cooper-type pairs is then nonconventional [9, 11, 12, 14, 16, 17], i.e., completely transformed by the non-diagonal interaction UΛN D . Remark 3.2 The existence of particles outside the Bose condensate even at zerotemperature is well known as the depletion of the Bose condensate, see for example [7, 8, 39–43]. But we go a step further in our interpretation. We consider this behavior as the coexistence of two subsystems: the Bose condensate and the previously defined Bogoliubov system. We agree that the term “Bogoliubov system” is only a personnel terminology, which stands for the “out-of-condensate Bosons”. 2. As it was claimed by Bogoliubov [4–6], the coherency due to the presence of the Bose condensation is not enough to make the Perfect Bose Gas superfluid. The spectrum of elementary excitations is not collective in this case: it corresponds B , following Landau’s to individual movements of particles. In the Bose gas HΛ,0
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k=0
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k=0
λk
λk −k
k=0
k=0
k=0
k=0
g k=0
Λ,00
k=0
Figure 3.3: Effective interaction for the zero-mode induced by the non-diagonal interaction UΛN D . criterion of superfluidity [18, 19] (Remark 3.1), the Bogoliubov system is here superfluid due to phenomena of interactions, which change, in the presence of the Bose condensate, the behavior of individual particles into an ideal Bose gas of “quasi-particles” with the given spectrum EkB (β, ρ). Indeed, through the Bose condensate, the non-diagonal interaction UΛN D combined with the diagonal interaction UΛD creates quasi-particles from two particles respectively of modes k and −k (k = 0). this can be seen via the Bogoliubov u-v transformation Formally, B (c) |c|2 = x>0 , cf. (2.10). This gas of quasi-particles or linked pairs applied to HΛ,0 of particles, i.e., the Bogoliubov system, exists if and only if the non-conventional Bose condensate exists too. 3. Also for high densities ρ > 0 we have
(β, ρ)} = 0, lim {ρ − x
ρ→+∞
(3.7)
cf. Theorem 2.3, Figure 2.2. Actually, the non-diagonal interaction UΛN D implies an effective repulsion term 1 λp λq 1 x
(β, ρ) gpq ≡ lim gΛ,pq = + ≥ 0, (3.8) Λ 4 εp εq inside each quasi-particle [9, 12], i.e., inside each couple of particles respectively with modes q and −q (q = 0) (Figure 3.4). The larger the Bose condensate den-
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p
λq
λp
−q
k=0
−p
q
p
g −q
Λ ,pq
−p
Figure 3.4: Effective interaction outside the zero-mode induced by the non-diagonal interaction UΛN D . sity x
(β, ρ), the stronger the effective repulsion term gpq . The raise of the nonconventional Bose condensate progressively destroys the Bogoliubov one, see (3.7). The Bose condensate and the Bogoliubov system still remain in competition with each other. Remark 3.3 We could have denoted the Bogoliubov system as the Bogoliubov condensate, for example at zero-temperature. This notion can be confusing since it has nothing to do with a macroscopic occupation of individual particles on some modes k < δ (δ → 0+ ) [36–38]. However, considering this gas as a system of linked pairs of particles, the momentum of all these quasi-particles is always zero. This fact is similar to a usual Bose condensation (seen here as a macroscopic occupation of the zero-momentum) but in a gas of linked pairs of particles and the notion of Bogoliubov condensate may have a sense within this framework.
3.3
Microscopic theory of superfluidity of 4 He?
1. A microscopic interpretation at all temperatures T = (kB β)−1 ≥ 0 of Landau’s theory of superfluidity follows from the Landau-type excitation spectrum EkB (β, ρ) (3.1)–(3.5) (cf. Figure 3.1). Note that Landau’s theory of superfluidity of quantum liquids [4, 6, 7, 44–47] is based on the following principles: • quantum liquid is still fluid even for zero-temperature; • at low temperatures, apart translations (flow), the state of this liquid is entirely described by the spectrum of collective (elementary) excitations;
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• through thermodynamic data [47, 48] (e.g., specific heat capacity) this spectrum for 4 He should be a phonon-like for the long-wave length collective excitations and should be above a straight line with positive slope with (“roo −1
ton”) minimum in the vicinity of krot 2 A
(Figure 3.1).
B HΛ,0
2. The thermodynamic behavior of the Bose gas is also close to the liquid He. This helium liquid is a Bose system with strong interactions. The interaction potential Uth (r) is of Lennard-Jones type [29] and was found by Slater et Kirkwood [49] using the electronic structure of 4 He (see Figure 3.5 with Uth (r) in Kelvin and also [50]). 4
Uth(K)
2.6 3
4.4
o
r(A)
0
−9 K Figure 3.5: The theoretical interaction potential of 4 He. The exact formula for the interaction potential Uth (r) given in [50] is valid only for strictly positive r, whereas close to zero it is given by a polynomial interaction like in Figure 3.5. A caricature of this interaction is the hard sphere interaction potential [51, 52]. This approximation gives surprisingly good estimates of the exB we have perimental condensate fraction: 9% at T = 0 K [2, 3]. In our model HΛ,0 to mimic an interaction potential ϕ (x) close to Uth (r). In particular, in contrast with the hard sphere potential the value of ϕ (x) for x = 0 has to be given and has not to be infinite. A standard way to do it is to cut Uth (r) when r → 0+ as follows: Uth (r) for r > rmin . ϕHe (x) = ϕHe (r = x) = Uth (rmin ) for 0 ≤ r ≤ rmin . This implies a Fourier transformation λ0 (rmin ) of ϕHe (x) for the mode k = 0 which drastically depends on rmin (specially when rmin → 0+ ), i.e., lim λ0 (rmin ) = rmin →0+
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B +∞, but it has no influence on the canonical thermodynamic behavior of HΛ,0 . Moreover, for k = 0 the influence of rmin corresponds only to a small (specially when rmin → 0+ ) perturbation of the Fourier transformation of Uth (r) . In fact one should choose rmin << rmean where rmean ∼ ρ−1/3 is the average length of the inter-particle distance at density ρ > 0. B Then, the thermodynamics of the theoretical Bose gas HΛ,0 is qualitatively 4 quite similar to the one of the liquid He: −1 • for small densities ρ ≤ ρc (β) or high temperatures T ≥ Tc ≡ (kB βc (ρ)) the thermodynamic behavior corresponds to the Perfect Bose Gas, • a non-conventional Bose condensation constituted of Cooper-type pairs appears via a second order transition (no discontinuity of the Bose condensate density) and the spectrum of excitations becomes a Landau-type excitation spectrum in this regime, i.e., for high densities ρ > ρc (β) or small temperatures T < Tc , • a coexistence of particles inside and outside the Bose condensate, even at zero-temperature as it is experimentally found in [2, 3].
As explained above, note that the Bose condensation becomes non-conventional with the formation of Cooper-type pairs via the term of attraction g00 , i.e., because of quantum fluctuations, see Figures 3.3 and 3.4. The importance of quantum fluctuations in helium systems corresponds also to the qualitative explanation for a liquid state at such extreme temperatures [50]. Quantitatively, the critical density ρc (β) is approximately given by ρc (β) ≈ ρP BG (β, 0), cf. (C.3)–(C.4) and Figure 2.1. The theoretical temperature of the BG phase transition Tc verifies Tc ≥ TP = 3.14 K (critical temperature evaluated c for a Perfect Gas of helium particles). The physical reason is that the non-diagonal interaction UΛN D implies an effective attraction in the zero-mode (see Figure 3.3), which helps the formation of the Bose condensation. However, Tc is quite close to BG : TP c Tc ≈ 3.14 K. (In fact we are able to prove an exact equality at small densities but we have no rigorous proof of a such result at very high densities). The experimental transition of the normal liquid 4 He (called 4 He I) to superfluid phase 4 He II (called the “λtransition”, discovered by Kapitza [53] and Allen, Misener [54] in 1938) takes place at a lower temperature Tλ = 2.17 K (along the vapor pressure curve), which is not B . However, note that the Henshaw-Woods so far from the one of the model HΛ,0 spectrum (experimental Landau-type excitation spectrum [20]) does not change drastically when the temperature crosses Tλ , whereas there is no superfluidity for these temperatures. λ > Tλ such that the exRemark 3.4 This means that there is a temperature T λ even if Landau’s criperimental “quasi-particle” system still exists for T < T terion of superfluidity (Remark 3.1) experimentally fails at these temperatures λ. Tλ < T < T
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3. To resume, this analysis is not a complete theory of “real superfluidity”. In particular, the Bogoliubov phonon-maxon-roton dispersion branch is only a part of the spectrum of the full quantum-mechanical Hamiltonian of the helium system. Therefore, this theory fails in being a complete description of all thermodynamics of liquid helium. For example, at temperatures Tλ < T < Tc , a Bose condensation B still exists in HΛ,0 but not for liquid helium even if the system of “quasi-particles” λ (Remark 3.4). However, this theory resists in liquid helium for Tλ < T < T is an interesting mathematical approach to a microscopic theory of many-body interacting boson systems leading to a better understanding of such superfluid systems.
3.4
Additional interpretations of this microscopic theory of superfluidity
B Let us examine other interpretations of the Bose system HΛ,0 in relation with the 4 B liquid He. In fact, we give here two interpretations of the Bose gas HΛ,0 obtained by following or not Landau’s criterion of superfluidity [18, 19] (Remark 3.1). As B is a caricature and may contain only explained above, note that the model HΛ,0 a small part of the physical properties of real liquid helium. The sole purpose of B these discussions is to give some new directions in light of the Bose gas HΛ,0 . 1. It is known [53, 54] that below the critical temperature Tλ of the λ-transition, two fluids (4 He II phase) coexist: the normal fluid and the superfluid liquid. Later justified within the framework of phenomenological Landau’s theory [18,19,47], the picture suggested by Tisza and Landau was to interpret the condensate of frozen in momentum space bosons with p = 0 as a “superfluid component”, and the rest of particles as a “normal component” which is the carrier of the total entropy of the system. Experimentally, a Bose condensate was discovered in 4 He II. The apparition of this Bose condensate transition and the one of the superfluid liquid are strongly correlated to each other. Indeed, from [55–57] if γs is the fraction of superfluid liquid and γ0 the one of the condensate, one has
γs (T) ∼ (Tλ − T) ∼ γ0 (T) , for T → T− λ, η
(3.9)
see Figure 3.6. However, even for zero-temperature the superfluid liquid is not in a full Bose condensate phase which is in contradiction with the assumption of Tisza and Landau. B 2. Following Landau’s criterion of superfluidity [18, 19], the theory based on HΛ,0 might be understood as a microscopic theory of the superfluid liquid. Within this framework, it allows us to understand the close connection between the Bose condensate with density x
(β, ρ) and the Bogoliubov system with density {ρ − x
> 0}. These two systems may compose together the superfluid liquid, which coexists with the normal liquid for non-zero temperature at any positive velocity. Note that Landau’s criterion of superfluidity [18, 19] confronts an initial problem expressed by Remark 3.4 and also a second one: the application of this criterion to the Henshaw-Woods spectrum [20] gives for the critical velocity v0 ≈ 60 m/s
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γ (%)
s
0
100
9
0
Tλ
0
Tλ
Figure 3.6: The fractions, γs of superfluid liquid and γ0 of the Bose condensate, as a function of the temperature T for 4 He
(Remark 3.1), whereas superfluidity in capillaries disappears when velocity is of the order of few cm/s. Moreover, it depends sensitively on the diameter of the channel. The attempts to explain these “misfittings” are concentrated around the idea that the Landau-type spectrum experimentally discovered by Henshaw and Woods [20] is only a part of a plethora of other types of “elementary” excitations not covered by the Bogoliubov theory, see [3, 57]. B , we have seen in Section 3.2 that Within the framework of the model HΛ,0 the Bose condensate has to exist in order to have the superfluidity property via the Bogoliubov system. Indeed, as soon as the non-conventional Bose condensate disappears, the collective phenomenon involved in the formation of the superfluid gas (Bogoliubov system) also vanishes. The introduction of a velocity in an inhomogeneous gas (in capillaries) may change the individual spectrum εk by increasing it. Then, the effective attraction g00 ((2.14), Figure 3.3) becomes smaller, i.e., the non-conventional Bose condensate and the (superfluid) Bogoliubov one may be destroyed for velocities sufficiently large but smaller than v0 (Remark 3.1). Note that the non-conventional Bose condensate constituted of Cooper-type pairs may be changed into a conventional Bose-Einstein condensation as it exists for the Perfect Bose Gas. An experimental study of the spectrum of excitations and also of the Bose condensation phenomenon should be interesting at different velocities. Actually, the collective behavior of this system should be quite delicate to save. A velocity may destroy the Cooper-type pairs and the quasi-particles. The important point is the following: the bigger the density of non-conventional Bose condensate, the stronger the robustness of Cooper-type pairs and quasi-particles to any perturbations. At temperatures T < Tλ even if the Bose condensate exists, its density may be not sufficiently important to keep the collective behavior for any positive velocities: some quasi-particles and Cooper-type pairs may be destroyed and a fraction λ (Remark 3.4) the therof normal fluid appears. At temperatures Tλ < T < T
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mic fluctuations become sufficiently strong to destroy the non-conventional Bose condensate. Consequently, even if the quasi-particle gas resists in liquid helium for λ (Remark 3.4), it is quite unstable and any perturbation of the quasiTλ < T < T particles (like any positive velocity) may quickly destroy the collective system and switch it to a standard liquid where no superfluidity exists. 3. Note that this last conjecture may seem a little naive since the previous discussions are just phenomenological interpretations. Therefore, to conclude we examine B without taking into account Landau’s another interpretation of the Bose gas HΛ,0 criterion of superfluidity [18, 19], which is a phenomenological explanation of superfluidity. ηB −1 If γ0B (T) ∼ (Tc − T) at temperatures T = (kB β) → T− c is the fraction of Bose condensate for a fixed density ρ > 0, then via Theorem 2.3, the fraction γsB (T) = 1 − ρn /ρ satisfies: ηB
where
γsB (T) ∼ (Tc − T) ∼ γ0B (T) , for T → T− c , 1 fk,0 3 ρn (T) = k . d B /T Ek,0 (2π)3 x= x,α=α( x) B e E − 1 k,0 3
(3.10) (3.11)
R
The relation (3.10) is strangely similar to (3.9), see Figure 3.6. The fraction γsB (T) B . Therefore, at a may be considered as the superfluid fraction of the Bose gas HΛ,0 fixed density ρ > 0, the superfluid density ρs equals 2 2 x λk 1 3 ρs (T) = x + k , d 3 x= x,α=α( x) (2π) 2E B f + E B R3
k,0
k,0
k,0
whereas ρn (3.11) is the density of normal fluid which is the carrier of the total entropy of the system. Note that lim ρn = 0 and within this framework there is T→0+
=x
(T). See 100% of superfluid liquid at zero-temperature with a density ρs > x (i) of Theorem 2.3 to see the Bose condensate density at a fixed density ρ > 0. In fact, this conjecture has to be analyzed via the corresponding Hamiltonian with an external velocity field as it has been recently performed with dilute trapped Bose gases at zero-temperature [58].
3.5
Concluding remarks: superfluidity of Fermi systems
The superfluidity of a Fermi system, i.e., the 3 He liquid, was discovered in 1972 for sufficiently low temperatures [59, 60]. All the previous theories concern Bose systems. However, it is remarkable to see that, via the effective coupling constant B implies g00 < 0 (Figure 3.3), the non-diagonal interaction UΛN D of the model HΛ,0 an attraction between particles in the zero-mode. By analogy, it is well known that the phenomenon of superconductivity comes from the effective electron-electron interaction in the BCS theory which results
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from the electron-phonon (non-diagonal) interaction in the second order of perturbation theory, see e.g., [61, 62]. Thus, in a superconductor, electrons can pair up in the metal crystal via phonons to form Cooper pairs which can then condense into a superconducting state. This phenomenon corresponds also to the explanation given for the existence of superfluidity in 3 He [21, 22]. Indeed, by cooling the liquid to a low enough temperature, 3 He atoms can pair up, making it a boson, and therefore superfluidity can be achieved. B , we found exactly the same kind of behavior on bosons: In the Bose gas HΛ,0 a system of linked pair of particles and Cooper-type pairs. Therefore, it should be interesting to study a similar Hamiltonian within the framework of Fermi systems. Of course, the main difference comes from the Fermi distribution. In particular, the critical density ρP BG (β, 0) for the Perfect Bose Gas does not exist for the B Perfect Fermi Gas. For the Bose system HΛ,0 , the corresponding kinetic part only turns on the Bose condensation phenomenon via the Bose distribution. Indeed, the corresponding “chemical potential” α ( x) , as solution of the variational problem (C.1) for a Bose condensate density x
(β, ρ) , becomes zero when we reach the critical density as for the Perfect Bose Gas, but switches again to strictly negative values for x
> 0 (in [26]: proof of Theorem 2.3). As soon as the Bose condensate appears, the non-diagonal interaction UΛN D becomes sufficiently important to drastically change all thermodynamic properties of the system by instantly switching the usual Perfect Bose gas to a gas of quasi-particles: the Bose-Einstein condensation becomes non-conventional in correlation with the creation of the Bogoliubov system and the formation of Cooper-type pairs (Section 3.2). Whereas the non-diagonal interaction UΛN D is not strong enough to imply alone the Bose-condensation at the critical temperature or density of the Perfect Bose Gas, for very small temperatures it strongly dominates all thermodynamics of the system. The non-diagonal interaction UΛN D obviously has a strong impact on the system (see for example the divergence of the grand-canonical pressure of B , Appendix B). It would have implied the non-conventional Bose condensation HΛ,0 without the Bose distribution at sufficiently low temperatures or high densities. In particular, if the Fermi distribution now holds, a similar non-diagonal interaction characterizing by an effective attraction g00 (like in (2.14), Figure 3.3) would drastically oppose the repulsion from the Pauli exclusion principle and would finally become strong enough at sufficiently low temperatures to imply alone the superfluid gas of quasi-particles explained above. This means of course that the critical temperature for the corresponding Fermi system should be quite lower B . Experimentally, the critical temperature of 3 He than that of the Bose model HΛ,0 is very low in comparison with that of 4 He (2.17 K): it is only two milli Kelvin for 3 He [59, 60]. We reserve this analysis on Fermi systems for another paper. To conclude, notice also that the 3 He liquid forms, at sufficiently low temperatures, several superfluid phases (A&B), which are much richer properties than those of the superfluid 4 He. For a complete review of properties of 3 He at low temperatures, see [63, 64].
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Appendix A
459
The canonical Bogoliubov theory of superfluidity [6, 7, 46, 65–67]
Since the atoms of 4 He are bosons, a plausible conjecture links the superfluidity property with the Bose-Einstein condensation phenomenon predicted for the Perfect Bose Gas by Einstein in 1925 [68]. It was originally suggested by Fritz London in 1938 [30], since the transition of the normal liquid 4 He (called He I) to superfluid phase He II takes place at a temperature Tλ (2.17 K) very close to the one of the corresponding Perfect Bose Gas (3.14 K). Experimentally, a fraction of Bose condensate in liquid 4 He was only found in the sixties, almost 30 years after the London’s idea of genius, via deep-inelastic neutron scattering, see [2,3]. In 1941, Landau [18,19] understood for the first time that the properties of quantum liquids like 4 He (or 3 He) can be entirely described by the spectrum of collective excitation, which for liquid 4 He, has two branches : “phonons” for longwavelength excitations and “rotons” for a relatively short-wavelength collective excitations. Guided by the Landau’s idea that (at least) the low energy part of the spectrum in liquid 4 He is defined by coherent collective movements of the system instead of individual ones, Bogoliubov tried to find a physical (or mathematical) mechanism which as in crystals with phonons, favors the collective motions of the “helium jelly”, via some kind of ordering or coherence: the Bose condensation phenomenon suggested by London. Note that the spectrum of the Perfect Bose Gas does not satisfy the Landau criterion of superfluidity and from more recent experiments, recall that the Bose condensate represents at T = 0 K only 9% of the system, whereas there is 100% of Bose-Einstein condensation in the Perfect Bose Gas! In fact, Bogoliubov accepted the crucial role played by the Bose condensation mechanism in superfluid liquid 4 He, but he insisted that “an energy level scheme based on the solution of the quantum mechanical many-body problem with interactions, must be found” (in [7]: Part 3.4). More precisely, his aim was to deduce the Landau-type excitation spectrum from the full interacting gas HΛ,λ0 >0 (2.1). It was a very challenging program, and inspired by all the previous observations, Bogoliubov proposed his microscopic theory of superfluidity in 1947 [4–8]. His Weakly Imperfect Bose Gas was the starting point for this theory in the canonical or grand-canonical ensembles. This model arises from the truncation of the full interacting gas HΛ,λ0 >0 (2.1) by assuming the existence of a Bose condensation on the zero-mode for a weak enough interaction ϕ (x) . The most important terms in (2.1) should be those in which at least two operators a∗0 , a0 appear. This procedure implies his Weakly Imperfect Bose Gas, i.e., the Bogoliubov Hamiltonian (see [7], Part 3.5, eq. (3.81)) defined by B B ≡ HΛ,0 + UΛBMF , HΛ,λ 0 >0
(A.1)
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cf. (2.3). Here the interaction UΛBMF ≡
λ0 ∗2 2 λ0 ∗ a a + a0 a0 2V 0 0 V
a∗k ak , λ0 > 0,
(A.2)
k∈Λ∗ \{0}
comes directly from the Bogoliubov truncation of the Mean-Field interaction UΛMF . Remark A.1 From the beginning, the Bogoliubov theories are not self-consistent. This fact was first highlighted by Angelescu-Verbeure-Zagrebnov [27] in the grandcanonical ensemble. Actually, in the grand-canonical ensemble the Bogoliubov B manifests, for high densities, a conventional Bose-Einstein conmodel HΛ,λ 0 >0 densation [9–11, 13, 14] on modes k = 2π/L = 0, which corresponding terms in (2.1) have been neglected in the truncation of the full interaction. B B seems to be “close” to the Bogoliubov model HΛ,λ , but The model HΛ,0 0 >0 their thermodynamics are in fact very different in the thermodynamic limit. For B is drastically example, in the grand-canonical ensemble, the Bose system HΛ,0 instable at high densities, i.e., the terms of repulsion are not strong enough to prevent the system from collapse for a chemical potential µ > −ϕ (0) /2 (Appendix B exists B), whereas the thermodynamics of the Bogoliubov Hamiltonian HΛ,λ 0 >0 in infinite volume for any µ ≤ 0 [27]. Actually, the interaction UΛBMF has a crucial and, unfortunately, a nasty B , see discussion in impact on the thermodynamics of the Bogoliubov model HΛ,λ 0 >0 [1]. In the canonical ensemble, he then proposed to use the operators {ζk }k∈Λ∗ (2.7) in order to apply his ingenious approximation (2.8). However, a direct application of (2.8) does not give the Landau-type excitation spectrum, again because of the interaction term UΛBMF . In fact, Bogoliubov suggested to eliminate the operator N0 from (A.2) at the cost of further approximations, see [6, 7] and discussion in [65]. He drastiB by using the following cally changed the original Bogoliubov Hamiltonian HΛ,λ 0 >0 approximation
N0 N02 + 2V V
k∈Λ∗ \{0}
ζk∗ ζk =
N0 (NΛ − N0 ) N2 N02 + Λ, 2V V 2V
(A.3)
i.e., the assumption, experimentally inexact for the liquid 4 He [2, 3], that 100% of Bose condensation occurs for T = 0 K. Note that this last ansatz gives a SB B superstabilized version HΛ,λ (2.12) of HΛ,0 with λ = λ0 .2 This model was not explicitly proposed by Bogoliubov since, meantime, he applied (2.8) combined with N0 NΛ 2 ≈ |c| = ρ ≈ V V 2 without
the last term of (2.12)
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B in the canonical ensemble, in order to find HΛ,0 (ρ) (2.9) and the Landau-type SB with λ = λ0 have excitation spectrum. Actually, we recall that the model HΛ,λ ingeniously been proposed in the grand-canonical ensemble in [27]. It has recently been solved in [26]. Therefore, Bogoliubov had to take a completely condensed particle density to get the Landau-type excitation spectrum. This last assumption is not true for B (at least not in the grand-canonical the original Bogoliubov Hamiltonian HΛ,λ 0 >0 ensemble, see [10–14]). The Bogoliubov theory and the approach proposed here are applicable to the weakly interacting gas, but it does not mean that we reach the dilute limit. Following a perturbation theory, the depletion of the condensate is zero in the lowest-order: see for example (2.108)-(2.109) in Section 2.4 of [9]. We could replace N0 by NΛ everywhere it appears apart from the lowest-order Hartree term. This analysis is done after considering the creation/annihilation operators on the ground state as arbitrary chosen complex numbers. However, the crucial point is that, by keeping the creation/annihilation operators on the ground state, the nondiagonal interaction (2.5) of the Bogoliubov Hamiltonian implies a contribution in the second order, see [9, 12] and Section 3.2. This contribution is absolutely not negligible: this term is able to drastically change the thermodynamics of the system [9, 10, 12, 26]. It is the origin of the depletion of the condensate fraction and also of the quasi-particle system with the Landau-type excitation spectrum (Section 3.2). Bogoliubov (and Zubarev) early noticed the difficulty with his ansatz of 100% of Bose condensate. Some discussions corresponding to this problem can be found in [7, 8, 39–43].
Remark A.2 In the grand-canonical ensemble (β, µ) , the operators {ζk }k∈Λ∗ (2.7) are useless and we only need the Bogoliubov approximation (2.6). Then, from the 2 hypothesis µ = λ0 |c| > 0, which is shown to be not legitimate [27], Bogoliubov obtained the Landau-type excitation spectrum. For more details concerning the Bogoliubov theories of superfluidity, see [1, 9].
Appendix B
Thermodynamics in the grand-canonical ensemble
B We explore here the thermodynamic behavior of the Hamiltonian HΛ,0 (2.3) in the grand-canonical ensemble. Even if this study turns out to be useless to explain the superfluidity phenomenon, from the mathematical point of view it highlights the B , see discussion in the last subsection of this appendix. unusual behavior of HΛ,0 The pressure in the grand-canonical ensemble (β, µ) and the grand-canonical particle density are respectively given by B 1 ln T rFΛB e−β (HΛ,0 −µNΛ ) , pB Λ,0 (β, µ) ≡ βV NΛ ρB (β, µ) ≡ (β, µ) = ∂µ pB Λ,0 Λ,0 (β, µ) . V HB Λ,0
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B.1 An upper bound for the grand-canonical pressure Regrouping terms in (2.3) one has B HΛ,0 = HΛI +
1 2V
k∈Λ∗ \{0}
where
HΛI =
∗ ∗ a0 ak + a∗−k a0 ≥ HΛI , λk a∗0 ak + a∗−k a0
k∈Λ∗ \{0}
λk 1 εk − Nk − 2V 2V
λk N0 .
k∈Λ∗ \{0}
Hence we obtain pB Λ,0
(β, µ) ≤
−1 λ 1 −β εk − 2Vk −µ (β, µ) ≡ ln 1 − e βV k∈Λ∗ \{0} −1 1 ln 1 − eβ(µ−µsup,Λ ) + , βV pIΛ
for µ < µsup,Λ ≡ −
1 2V
λk < 0.
(B.1)
k∈Λ∗ \{0}
B.2 A lower bound for the grand-canonical pressure using the Bogoliubov approximation B The corresponding lower bound for the Bogoliubov Hamiltonian HΛ,λ (A.1) 0 >0 found in [27] remains valid even for λ0 = 0 and one gets B pB Λ,0 (β, µ) ≥ sup pΛ,0 (β, µ, c) ,
(B.2)
c∈C
where pB Λ,0 (β, µ, c) is defined by 2 2 β, µ, x ≡ |c| + η µ, x ≡ |c| , (β, µ, c) = ξ pB Λ,0 Λ,0 Λ,0 −1 B 1 ξΛ,0 (β, µ, x) = ln 1 − e−βEk,0 , βV k∈Λ∗ \{0} 1 B fk,0 − Ek,0 + µx, ηΛ,0 (µ, x) = 2V ∗
(B.3)
k∈Λ \{0}
B with Ek,0 , fk,0 defined by (2.18) for α = µ ≤ 0. Therefore one has to analyze the lower bound sup pB Λ,0 (β, µ, c). c∈C
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Lemma B.1 We have sup c∈C
pB Λ,0
(β, µ, c) =
463
P BG pB (β, µ) ; for µ ≤ µsup,Λ < 0 Λ,0 (β, µ, 0) = pΛ +∞ ; for µ > µsup,Λ ,
BG where pP (β, µ) = pB Λ Λ,0 (β, µ, 0) is the grand-canonical pressure for the Perfect Bose Gas.
Proof. Through (2.18) and (B.3), one gets that for µ ≤ 0: −1/2 1 1 2λk (i) ∂x ηΛ,0 (µ, x) = µ + λk − λk 1 + x , 2V 2V εk − µ ∗ ∗ k∈Λ \{0}
k∈Λ \{0}
∂x ηΛ,0 (µ, 0) = µ < 0; (ii)
∂x2 ηΛ,0
1 (µ, x) = 2V
Since lim
x→+∞
1 2V
k∈Λ∗ \{0}
k∈Λ∗ \{0}
√ λ2k εk − µ
3/2
(εk − µ + 2xλk )
> 0.
−1/2 2λk λk 1 + x = 0, εk − µ
even in the thermodynamic limit, (i) implies µ ≤ ∂x ηΛ,0 (µ, x) ≤ µ − µsup,Λ for all x ≥ 0 and lim {∂x ηΛ,0 (µ, x) − µ + µsup,Λ } = 0 .
x→+∞
So, we get with (ii)
sup {ηΛ,0 (µ, x)} = x≥0
ηΛ,0 (µ, x = 0) ; for µ ≤ µsup,Λ +∞ ; for µ > µsup,Λ .
(B.4)
Therefore, for β → ∞ (zero-temperature) the corresponding pressure pB (β, µ, c) (B.3) attains its supremum at c = 0 if µ ≤ µsup,Λ whereas sup Λ,0 pB Λ,0 (β, µ, c) does not exist for any µ > µsup,Λ . By (2.18) and (B.3) note that (i)
∂x ξΛ,0 (β, µ, x) < 0 and
(ii)
∂β ξΛ,0 (β, µ, x) < 0 and
c∈C
lim ξΛ,0 (β, µ, x) = 0,
x→+∞
lim ξΛ,0 (β, µ, x) = 0.
(B.5)
β→+∞
Hence via (B.4) and (B.5) the lemma holds.
Consequently, combining (B.2) with Lemma B.1, we find B pB Λ,0 (β, µ) ≥ pΛ,0 (β, µ, 0) ,
(B.6)
for any µ ≤ µsup,Λ , whereas for µ > µsup,Λ the pressure pB Λ,0 (β, µ) does not exist.
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B.3 Thermodynamic behavior of the model Via the previous upper bound and (B.6) we get B I pB Λ,0 (β, µ, 0) ≤ pΛ,0 (β, µ) ≤ pΛ (β, µ) ,
for µ < µsup,Λ , which gives B P BG pB (β, µ) 0 (β, µ) =lim pΛ,0 (β, µ) = p Λ
(B.7)
in the thermodynamic limit for 1 µ < µsup ≡ lim µsup,Λ = − ϕ (0) < 0, Λ 2
(B.8)
and which can be extended by continuity of the pressure to µ ≤ µsup . Here pP BG (β, µ) is the infinite volume pressure for the Perfect Bose Gas. From (B.7) and Griffiths lemma [69, 70] the infinite volume particle density ρB 0 (β, µ) equals the one for the Perfect Bose Gas ρP BG (β, µ) for µ < µsup and therefore lim
µ→µ− sup
P BG ρB (β, µsup ) < +∞, 0 (β, µ) = ρ
(B.9)
i.e., it is not possible to reach high densities regimes in the grand-canonical ensemble (β, µ). B are, in a way, trivial Hence the thermodynamic properties of the model HΛ,0 for rather negative chemical potential µ ≤ µsup,Λ : they are equivalent to the Perfect Bose Gas. The non-diagonal interaction UΛN D (2.5) is not able to change the system for sufficiently negative chemical potential µ ≤ µsup . This fact is not surprising B since it is exactly the same for the Bogoliubov Hamiltonian HΛ,λ (A.1) for 0 >0 µ ≤ µsup , see the corresponding lower and upper bounds in [27] and discussions in [9, 11]. However, as soon as the non-diagonal interaction UΛN D beats the kinetic part for µ > µsup by attracting particles in the zero-mode (cf. [9, 12], Figures 3.2, 3.3 and 3.4), the system becomes unstable, i.e., all particles collapse in the zero-mode because of the absence of strong enough repulsion terms such as λ ∗2 2 λ 2 a0 a0 = N0 − N0 , with N0 ≡ a∗0 a0 . 2V 2V In fact this term of repulsion is crucial to induce the non-conventional Bose condensation mechanism without any instability in the grand-canonical ensemble (β, µ).
Appendix C
Proofs
The aim of this appendix is to give the promised details of the proofs of Theorems 2.2 and 2.3. But first, we quickly sum up the thermodynamic behavior of the SB (2.12), i.e., of the AVZ-Hamiltonian if superstable Bogoliubov Hamiltonian HΛ,λ λ = λ0 in the grand-canonical ensemble (β, µ).
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C.1 Grand-canonical thermodynamics of the superstable Bogoliubov Hamiltonian [26] Note that the condition (2.13) corresponds to the assumption (C1) in [26] with a strict inequality. In 1. we consider the chemical potential µ as a fixed parameter, whereas in 2. the full particle density ρ is fixed in the grand-canonical ensemble. SB are denoted in 1. The grand-canonical pressure and density associated with HΛ,λ SB the thermodynamic limit as p (β, µ) and NΛ ρSB (β, µ) ≡lim (β, µ) = ∂µ pSB (β, µ) Λ V H SB Λ,λ
respectively. They are defined and explicitly found in [26] for any (β, µ) ∈ {β > 0}× {µ ∈ R}. We have to solve two variational problems. The first one is characterized by α (x) ≡ α (β, µ, x) ≤ 0, i.e., the unique solution of ( ' ( ' (µ − α)2 (µ − α)2 B B inf p0 (β, α, x) + = p0 (β, α, x) + (C.1) α≤0 2λ 2λ α=α(x) for any fixed x ≥ 0, where pB 0 (β, α, x) is defined by (2.17). Whereas the second variational problem directly related to pSB (β, µ) is (( ' ' (µ − α)2 SB B p (β, µ) = sup inf p0 (β, α, x) + α≤0 2λ x≥0 ( ' 2 (µ − α) B = inf p0 (β, α, x , (C.2)
) + α≤0 2λ which solution x
=x
(β, µ) is also unique. Then, via direct calculations, for any β > 0, there is a unique µc (β) such that the pressure pSB (β, µ) equals 2 (µ − α (0)) , for µ ≤ µc (β) . pB 0 (β, α (0) , 0) + 2λ ( ' 2 pSB (β, µ) = (µ − α (x)) B (β, α (x) , x) + , for µ > µc (β) . p 0 2λ x= x>0 The function µc (β) is bijective from R+ to R+ ( lim µc (β) = 0) and we denote β→+∞
by βc (µ) ≥ 0 the inverse function of µc (β). The pressure pSB (β, µ) is continuous for µ = µc (β) .
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Remark C.1 The solution x
=x
(β, µ) of (C.2) always satisfies B = 0, for µ ≥ µc (β) , ∂x p0 (β, α, x) x= x,α=α( x) < 0, x
= 0, for µ ≤ µc (β) , ∂x pB 0 (β, α, x) x>0,α=α(x)
and the solution α ( x) = α (β, µ, x
) of (C.1) is always strictly negative for any µ = µc (β) or β = βc (µ). 2. Let ρc (β) =
lim
µ→µ− c (β)
ρSB (β, µ) =
lim
µ→µ+ c (β)
ρSB (β, µ) ≤ ρP BG (β, 0) ,
(C.3)
SB be the critical density of HΛ,λ , where ρP BG (β, 0) is the critical density of the Perfect Bose Gas (see (2.20) with x = 0). Then, ρc (β) ≈ ρP BG (β, 0) and for sufficiently large β, i.e., for small temperatures, we have
ρc (β) = ρP BG (β, 0) .
(C.4)
By fixing the particle density ρ in the grand-canonical ensemble, we define a unique chemical potential µβ,ρ satisfying ρSB (β, µβ,ρ ) = ρ.
(C.5)
Actually, at a fixed inverse temperature β the function µβ,ρ is the inverse function of the mean particle density ρSB (β, µ) of the AVZ-Hamiltonian. By βc (ρ) we denote the critical inverse temperature for a fixed particle density ρ (Figure 2.1). For ρ ≤ ρc (β), note that µβ,ρ ≤ µc (β) or β ≤ βc (ρ) whereas µβ,ρ > µc (β) , or β > βc (ρ), for ρ > ρc (β). Therefore we get the following properties: (i)
x) < 0, µβ,ρ − λρ = α ( SB (β, α (x) , x) p (β, µβ,ρ ) = pB 0
+ x= x
λ 2 ρ , 2
(C.6)
for any ρ > 0, where α ( x) < 0 is the unique solution of the Bogoliubov density equation ρ = ρB (β, α, x
) (2.19)–(2.20). 0 (ii) All the thermodynamic properties found for a fixed chemical potential µ are also valid for a fixed particle density ρ by using the corresponding chemical potential µβ,ρ . See for example Theorem 2.6 of [26]: for any ρ > ρc (β) there is only one Bose condensation in the zero mode characterized by x
=x
(β, µβ,ρ ) = x
(β, ρ) < ρ, whereas for ρ < ρc (β) the system behaves as the Mean-Field Bose Gas with no Bose condensations. No more Bose condensations outside the zero-mode appears for any ρ > 0.
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C.2 Proof of Theorem 2.2 SB 1. Remark that HΛ,λ commutes with the particle number operator
SB B HΛ,λ , NΛ = HΛ,0 , NΛ = 0.
Let fΛSB (β, ρ) be the free-energy density defined for a fixed particle density ρ > 0 by (n=[ρV ]) SB 1 e−βHΛ,λ ln T rH(n) fΛSB (β, ρ) ≡ − . B βV B Notice that fΛ,0 (β, ρ) (2.15) and fΛSB (β, ρ) are related to each other by B fΛ,0 (β, ρ) = fΛSB (β, ρ) −
λ 2 ρ ρ − . 2 V
(C.7)
B SB The two models HΛ,0 and HΛ,λ are equivalent in the canonical ensemble, in the sense that their (infinite volume) free-energy densities at fixed densities differ only by a constant. Actually their Gibbs states are equal to each other for all (β, ρ). Therefore we only need to compute fΛSB (β, ρ) in the thermodynamic limit to deduce f0B (β, ρ). This is our next step.
2. Since the particle density ρSB (β, µ) as the derivative of the pressure pSB (β, µ) is continuous (as a function of µ) [26], using a Tauberian theorem proven in [71], the existence of pSB (β, µ) already implies the convexity of the thermodynamic limit f SB (β, ρ) for ρ > 0 of fΛSB (β, ρ) and the weak equivalence of the canonical and grand-canonical ensemble: pSB (β,µ) = sup µρ − f SB (β,ρ) = µρSB (β,µ) − f SB β,ρSB (β,µ) ,µ ∈ R, ρ>0 f SB (β,ρ) = sup µρ − pSB (β,µ) = µβ,ρ ρ − pSB (β,µβ,ρ ), ρ > 0 . µ∈R
(C.8) With (C.6) the Legendre transformation (C.8) implies an explicit expression of the corresponding free-energy density: λ SB B f (β, ρ) = α (x) ρ − p0 (β, α (x) , x) + ρ2 , 2 x= x i.e., via (C.7) we also have f0B (β, ρ) = α (x) ρ − pB (β, α (x) , x) 0
(C.9) x= x
with pB 0 (β, α, x) defined by (2.17). 3. Now we give an interpretation of this last equality to show that x
and α ( x) are also solutions of variational problems in the canonical ensemble. Meantime, we explicitly find another, more natural, expression for f0B (β, ρ).
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2 (2.16) is well defined for any 3.1. The free-energy density f0B β, ρ1 , x = |c| ρ1 > 0 and β > 0. Moreover, since the particle density ρB 0 (β, α, x) (2.20) as the derivative of the pressure pB (β, α, x) (2.17) is continuous as a (strictly increasing) 0 function of α ≤ 0, for each fixed x ≥ 0, the function f0B (β, ρ1 , x) is convex for ρ1 > 0. Therefore, via (2.17) we find f0B (β, ρ1 , x) = sup α [ρ1 + x] − pB 0 (β, α, x) α≤0
= α (ρ1 , x) (ρ1 + x) − pB 0 (β, α (ρ1 , x) , x) , (C.10) for ρ1 ≡ ρ − x > 0, with α (ρ1 , x) defined as a solution of the Bogoliubov density equation ρ = ρB 0 (β, α (ρ1 , x) , x) (2.19)-(2.20) for ρ1 ≡ ρ − x > 0. 3.2. Since x
(β, ρ) (C.2) always satisfies x
< ρ even for β → +∞ [26], for x = x
the solution α ( x) < 0 of (C.1) is also the unique solution of the Bogoliubov density equation (2.19), see Appendix C.1. Therefore , (C.11) {α (ρ − x, x) = α (x)} x= x
which by (C.9) and (C.10) implies f0B
(β, ρ) =
f0B
(β, ρ − x, x)
.
(C.12)
x= x
, x
) in (C.12) can be understood as the result The free-energy density f0B (β, ρ − x B of the canonical Bogoliubov approximation applied on HΛ,0 . The equality (C.12) B finally means that the non-diagonal model HΛ,0 is thermodynamically equivalent BG to HP x) (2.11) for k ∈ Λ∗ \ {0} as stated in Section 2. Λ,0 ( 3.3. Moreover, by (C.10)-(C.11), one directly gets B
∂ρ1 f0 (β, ρ1 , x) = α ( x) , ρ1 =ρ− x,x= x
and
B
∂x f0 (β, ρ1 , x)
ρ1 =ρ− x,x= x
= α ( x) −
∂x pB 0
(β, α, x)
. α=α( x),x= x
Therefore, by using Remark C.1 the previous statements imply B B B f0 (β, ρ − x, x) , = inf f0 (β, ρ) = f0 (β, ρ − x, x) x= x<ρ
x∈[0,ρ]
(C.13)
for any ρ > 0. The solution x
=x
(β, µβ,ρ ) = x
(β, ρ) of the variational problem (C.2) is also solution of (C.13) for a fixed density ρ > 0. From (C.11), note that the solution α ( x) of the variational problem (C.1) is the solution in the canonical
> 0. Now we add some remarks to highlight ensemble of (C.10) with ρ1 ≡ ρ − x the important points in order to prepare the discussions of the next subsection.
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Remark C.2 By (C.7), (C.8) and (C.12) we obtain µβ,ρ =
∂ρ f0B
B (β, ρ) + λρ = ∂ρ f0 (β, ρ − x, x)
+ λρ. x= x
Remark C.3 Via (C.6) combined with Remark C.2 it immediately follows that α ( x) = ∂ρ f0B (β, ρ) and ∂λ {α ( x)} = 0. Note that ∂ρ f0B (β, ρ) < 0 for any ρ = ρc (β) (Remark C.1). Remark C.4 For a fixed density ρ we have via (C.13) ∂λ x
= 0. We can see this results using Remark C.1 combined with Remark C.3. An illustration of the behavior of x
for a fixed density is performed in Figure C.1.
x(β,µ ) 2
λ x(β,ρ )
2
0
µ
β,ρ
λ
µ
β,ρ
µ
Figure C.1: Illustration of the density x
(β, µ) as a function of µ for two different parameters λ. The lower curve is for the larger value of λ. At a fixed particle
(β, ρ) is density ρ > 0 in the grand-canonical ensemble, the density x
(β, µβ,ρ ) = x constant as a function of λ.
C.3 Proof of Theorem 2.3 The aim of this subsection is now to deduce the canonical thermodynamic behavior B SB of the model HΛ,0 from the grand-canonical thermodynamic properties of HΛ,λ . This is done by using the notion of strong equivalence [25, 33–35].
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B SB In fact, the two models HΛ,0 and HΛ,λ are equivalent in the canonical ensemB does not ble, see (C.7), i.e., their Gibbs states are equal for all (β, ρ). Since HΛ,0 SB depend on λ, one has to check if the grand-canonical densities for HΛ,λ depends on λ for any fixed particle density. Actually, the solutions α ( x) = α ( x, λ) and x
(λ) of the variational problems (C.1) and (C.2) are the key points of this first study. This is done via Remarks C.3 and C.4 (Figure C.1): the solutions α ( x, λ) and x
(λ) are also solutions in the canonical ensemble of the variational problems (C.10) with
> 0, and (C.13) respectively and they do not depend on λ for a fixed ρ1 ≡ ρ − x particle density ρ > 0. Consequently, all densities in the grand-canonical ensemble do not depend on λ for a fixed full particle density ρ. The parameter λ has no influence on the “physical” thermodynamic behavior of the system for a fixed particle density. Thus in the canonical ensemble the value of λ can be chosen freely as an arbitrary parameter as explained in the beginning SB is the “superstabilization” [24] of the model of Section 2.2. The Hamiltonian HΛ,λ SB such that H for a λ Λ,λ
SB SB = HΛ, HΛ,λ + λ
δ 2 >0 NΛ − NΛ with δ = λ − λ 2V
and
λ λ + g00 > + g00 > 0, 2 2 SB cf. (2.13)–(2.14). Because of the last inequality the model HΛ, satisfies the weak λ equivalence of ensembles for any density ρ > 0, and therefore the Hamiltonian SB satisfies the strong equivalence of ensembles [25] for any ρ > 0 and λ > 0 HΛ,λ sufficiently large. The strong equivalence is understood as follows: Let us consider by AΛ a (positive) quasi-local operator acting on +∞ B FΛB ⊂ F∞ ≡ ⊕ L2 Rnd symm n=0
such that lim AΛ H SB (β, ρ) < +∞ and lim AΛ H SB (β, µ) < +∞, Λ
Λ
Λ,λ
Λ,λ
(C.14)
for any β > 0 and ρ > 0. For β > 0, ρ > 0 and µΛ,β,ρ defined by NΛ (β, µΛ,β,ρ ) = ρ, V H SB Λ,λ
it follows from [25] that lim AΛ H B (β, ρ) =lim AΛ H SB (β, ρ) =lim AΛ H SB (β, µΛ,β,ρ ) , Λ
Λ
Λ,0
Λ,λ
Λ
Λ,λ
(C.15)
SB . Therefore the i.e., the strong equivalence of ensemble is verified by the model HΛ,λ SB B correspond thermodynamic properties in the canonical ensemble of HΛ,λ and HΛ,0 for a fixed particle density ρ to the one described in [26] with a chemical potential given by µ = µβ,ρ =lim µΛ,β,ρ (all densities are continuous). Λ
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C.4 Additional remarks 1. Looking more closely at Theorem 2.6 of [26], the reader may be confused by the problem of the non-continuity of the grand-canonical particle density in the phase transition regime, if λ + g00 < 0, (C.16) 2 holds instead of (2.13) (in [26]: see condition (C2) and (iv) of Theorem 2.6). This in fact appears, because a (direct) coupling constant λ/2 satisfying (C.16) is too small to restore on f SB (β, ρ) the problem of strict convexity of f0B (β, ρ), see (C.7). This comes from the effective attraction g00 on the zero-mode arising from the non-diagonal interaction UΛN D (2.5) (cf. [9, 12], Figures 3.2, 3.3 and 3.4). On the other hand, for λ large enough, i.e., (2.13) is satisfied, the free-energy density f SB (β, ρ) becomes strictly convex. Thus, in this case the grand-canonical density is continuous and the two ensembles are in fact strong equivalent, cf. (C.15). 2. For low dimensions d = 1, 2, the effective coupling constant equals g00 = −∞ and (C.16) is satisfied for any λ > 0. Therefore the effective attraction g00 on the zero-mode should imply the existence of a non-conventional Bose condensation for d = 1, 2 (see [9, 12], Figures 3.2, 3.3 and 3.4). However the method used here to find the canonical thermodynamic properties fails since λ is never large enough to satisfy the condition (2.13). Acknowledgments. The work was supported by DFG grant DE 663/1-3 in the priority research program for interacting stochastic systems of high complexity. Special thanks first go to T. Dorlas and the DIAS for the very nice stay there where this work was finished. J.-B. Bru thanks Institut f¨ ur Mathematik, Technische Universit¨ at Berlin, and its members for their warm hospitality during the academic year 2001–2002 and more precisely S. Adams. J.-B. Bru also wants to express his gratitude to N. Angelescu, A. Verbeure and V.A. Zagrebnov for their useful discussions. And the second author thanks the P. master Dukes and Dido for their help in writing/correcting this article. The authors especially thank the referee for helpful remarks and suggestions.
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[4] N.N. Bogoliubov, On the theory of superfluidity, J. Phys. (USSR) 11, 23 (1947). [5] N.N. Bogoliubov, About the theory of superfluidity, Izv. Akad. Nauk USSR 11, 77 (1947). [6] N.N. Bogoliubov, Energy levels of the imperfect Bose-Einstein gas, Bull. Moscow State Univ. 7, 43 (1947). [7] N.N. Bogoliubov, Lectures on Quantum Statistics, Vol. 1: Quantum Statistics, Gordon and Breach Science Publishers, New York-London-Paris (1970). [8] N.N. Bogoliubov, Energy levels of the imperfect Bose-Einstein gas, p. 242–257 in: Collection of papers, Vol. 2, Naukova Dumka, Kiev, (1970). [9] V.A. Zagrebnov and J.-B. Bru, The Bogoliubov Model of Weakly Imperfect Bose Gas, Phys. Rep. 350, 291 (2001). [10] J.-B. Bru and V.A. Zagrebnov, Exact phase diagram of the Bogoliubov Weakly Imperfect Bose gas, Phys. Lett. A 244, 371 (1998). [11] J.-B. Bru and V.A. Zagrebnov, Exact solution of the Bogoliubov Hamiltonian for weakly imperfect Bose gas, J. Phys. A: Math. Gen. A 31, 9377 (1998). [12] J.-B. Bru and V.A. Zagrebnov, Quantum interpretation of thermodynamic behaviour of the Bogoliubov weakly imperfect Bose gas, Phys. Lett. A 247, 37 (1998). [13] J.-B. Bru and V.A. Zagrebnov, Thermodynamic Behavior of the Bogoliubov Weakly Imperfect Bose Gas, p. 313 in: Mathematical Results in Statistical Mechanics, eds S. Miracle-Sole and al., World Scientific, Singapore (1999). [14] J.-B. Bru and V.A. Zagrebnov, On condensations in the Bogoliubov Weakly Imperfect Bose-Gas, J. Stat. Phys. 99, 1297 (2000). [15] V.A. Zagrebnov, Generalized condensation and the Bogoliubov theory of superfluidity, Cond. Matter Phys. 3, 265 (2000). [16] J.-B. Bru and V.A. Zagrebnov, Exactly soluble model with two kinds of BoseEinstein condensations, Physica A 268, 309 (1999). [17] J.-B. Bru and V.A. Zagrebnov, A model with coexistence of two kinds of Bose condensations, J. Phys. A: Math.Gen. 33, 449 (2000). [18] L.D. Landau, The theory of superfluidity of Helium II, J. Phys. (USSR) 5, 71 (1941). [19] L.D. Landau, On the theory of superfluidity of Helium II, J. Phys. (USSR) 11, 91 (1947).
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[37] M. van den Berg, On boson condensation into an infinite number of low-lying levels, J. Math. Phys. 23, 1159 (1982). [38] M. van den Berg, J.T. Lewis and J.V. Pul`e, A general theory of Bose-Einstein condensation, Helv. Phys. Acta 59, 1271 (1986). [39] N.N. Bogoliubov and D.N. Zubarev, Wave function of the ground-state of interacting Bose-particles, JETP 28, 129 (1955). [40] D.N. Zubarev, Distribution function of non-ideal Bose-gas for zero temperature, JETP 29, 881 (1955). [41] Yu.A. Tserkovnikov, Theory of the imperfect Bose-Gas for non-zero temperature, Doklady Acad. Nauk USSR 143, 832 (1962). [42] V.N. Popov, Functional Integrals and Collective Excitations, Cambridge Univ. Press, Cambridge (1987). [43] H. Shi and A. Griffin, Finite-temperature excitations in a dilute Bosecondensated gas, Phys. Rep. 304, 1 (1998). [44] R.P. Feynman, Application of Quantum Mechanics to Liquid Helium, p. 199 in: Progress in Low Temperature Physics, Vol. 1, Ch. 2. ed. J. Gorter, NorthHolland, Amsterdam (1955). [45] R.P. Feynman and M. Cohen, Energy spectrum of the Excitations in Liquid Helium, Phys. Rev. 102, 1189 (1956). [46] E.H. Lieb, The Bose fluid, in: Lectures in Theoretical Physics, Vol. VII C, ed. W.E. Briffin, University of Colorado, Boulder (1965). [47] I.M. Khalatnikov, An Introduction to the Theory of Superfluidity, BenjaminReading, New York (1965). [48] J. Wilks, An introduction to liquid helium, Claredon, Oxford (1970). [49] J.C. Slater and J.G. Kirkwood, The Van der Waals Forces in Gases, Phys. Rev. 37, 682 (1931). [50] K. Huang, Statistical Mechanics, Wiley, New York (1963). [51] O. Penrose, On the Quantum Mechanics of Helium II, Phil. Mag. 42, 1373 (1951). [52] O. Penrose and L. Onsager, Bose-Einstein condensation and Liquid Helium, Phys. Rev. 104, 576 (1956). [53] P. Kapitza, Nature 141, 74 (1938). [54] J.F. Allen and A.D. Misener, Nature 141, 75 (1938).
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[55] L. Aleksandrov, V.A. Zagrebnov, Zh.A. Kozlov, V.A. Parfenov and V.B. Priezzhev, High energy neutron scattering and the Bose condensate in He II, Sov. Phys.-JETP 41, 915 (1975). [56] E.V. Dokukin, Zh.K. Kozlov, V.A. Parfenov and A.V. Puchkev, Investigation of the temperature dependence of the density of the Bose condensate in helium-4 in connection with the superfluidity phenomenon, Sov. Phys.-JETP 48, 1146 (1978). [57] N.M. Blagoveshchenskii, I.V. Bogoyavlenskii, L.V. Karnatsevich, V.G. Kolobrodov, Zh.A. Kozlov, V.B. Priezzhev, A.V. Puchkov, A.N. Skomorokhov and V.S. Yarunin, Structure of the excitation spectrum of liquid 4 He, Phys. Rev. B 50, 16550 (1994). [58] E.H. Lieb, R. Seiringer and J. Yngvason, Superfluidity in dilute trapped Bose gases, Phys. Rev. B 66, 134529 (2002). [59] D.D. Osheroff, R.C. Richardson, and D.M. Lee, Evidence for a New Phase of Solid He-3, Phys. Rev. Lett. 28, 885 (1972). [60] D.D. Osheroff, W.J. Gully, R.C. Richardson, and D.M. Lee, New Magnetic Phenomena in Liquid He-3 below 3 mK, Phys. Rev. Lett. 29, 920 (1972). [61] C. Kittel, Quantum Theory of Solids, John Wiley and Sons Inc., New York (1963). [62] N.N. Bogoliubov, V.V. Tolmachev and D.V. Shirkov, A New Method in the Theory of Superconductivity, Consultants Bureau Inc., New York (1959). [63] A.J. Leggett, A theoretical description of the new phases of liquid 3He, Rev. Mod. Phys. 47, 331 (1975). [64] D. Vollhardt and P. W¨ olfle, The Superfluid Phases of Helium 3, Taylor and Francis, London (1990). [65] N.N. Bogoliubov, Kinetic equations in the theory of superfluidity, JETP 18, 622 (1948). [66] M. Girardeau, Variational Method for the Quantum Statistics of Interacting Particles, J. Math. Phys. 3, 131 (1962). [67] A.J. Kromminga and M. Bolsterli, Perturbation Theory of Many-Boson Systems, Phys. Rev. 128, 2887 (1962). [68] A. Einstein, Sitzungsberichte der Preussischen Akademie der Wissenschaften I, 3 (1925). [69] R. Griffiths, A Proof that the Free Energy of a Spin System is extensive, J. Math. Phys. 5, 1215 (1964).
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[70] K. Hepp and E.H. Lieb, Equilibrium Statistical Mechanics of Matter Interacting with the Quantized Radiation Field, Phys. Rev. A 8, 2517 (1973). [71] R.A. Minlos and A.Ja. Povzner, Thermodynamic limit for entropy, Trans. Moscow Math. Soc. 17, 269 (1967). S. Adams Institut f¨ ur Mathematik Fakult¨ at II, SEK. MA 7-4 Technische Universit¨ at Berlin Strasse des 17. Juni 136 D-10623 Berlin, Germany email:
[email protected] J.-B. Bru School of Theoretical Physics Dublin Institute for Advanced Studies 10 Burlington Rd. Dublin 4, Ireland email:
[email protected] Communicated by Vincent Pasquier Submitted 31/03/03, accepted 30/01/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 477 – 521 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/030477-45 DOI 10.1007/s00023-004-0176-6
Annales Henri Poincar´ e
A Mountain Pass for Reacting Molecules Mathieu Lewin Abstract. In this paper, we consider a neutral molecule that possesses two distinct stable positions for its nuclei, and look for a mountain pass point between the two minima in the non-relativistic Schr¨ odinger framework. We first prove some properties concerning the spectrum and the eigenstates of a molecule that splits into pieces, a behavior which is observed when the Palais-Smale sequences obtained by the mountain pass method are not compact. This enables us to identify precisely the possible values of the mountain pass energy and the associated “critical points at infinity” (a concept introduced by Bahri [2]) in this non-compact case. We then restrict our study to a simplified (but still relevant) model: a molecule made of two interacting parts, the geometry of each part being frozen. We show that this lack of compactness is impossible under some natural assumptions about the configurations “at infinity”, proving the existence of the mountain pass in these cases. More precisely, we suppose either that the molecules at infinity are charged, or that they are neutral but with dipoles at their ground state.
Introduction In this paper, we study in the non-relativistic quantum Schr¨ odinger framework the case of a molecule that possesses two distinct stable positions for its nuclei, as this is for instance the case for HCN and CNH. Our purpose is somewhat simple: can we obtain a critical point of the energy by using the classical mountain pass method between the two minima? Experiment suggests that this is the case (at least for the HCN↔CNH reaction). Indeed, such mountain pass points are frequently computed by chemists who need to understand the possible behavior of the molecule: it corresponds to a “transition state” during an infinitely slow reaction leading from one minimum to the other. But as far as we know, this problem has never been tackled from the mathematical point of view for the N -body quantum problem, or even in the context of the classical Hartree or Thomas-Fermi type models which are approximations of the exact theory. For a neutral molecule, the proof that there is a minimum with regards to the position of the nuclei can be found in the fundamental work of E.H. Lieb and W.E. Thirring [25] for the Schr¨ odinger model, and in a series of papers by I. Catto and P.-L. Lions [4, 5, 6, 7] for approximate models (Hartree or Thomas-Fermi type), the latter being really more complicated due to the non linearity of these models. In these two works, the authors had to prove that minimizing sequences are compact, the non-compactness behavior being related to the fact that the molecule can split into parts, each moving away from the others. Remark that
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binding does not occur for the Thomas-Fermi model (see the works of E. Teller [31], E.H. Lieb and B. Simon [24], and the references in [4]) and that the result is not known for the Hartree-Fock model, except in very special cases [7]. Let us make some comment on a tool used in these proofs that cannot be simply adapted to our setting. A common idea in these two works is to average over all the possible orientations of each piece in order to simplify the computation of the interaction energy between them, by suppressing the multipoles. To show that the energy can be lower than the energy at infinity, a new term using the correlation between the electrons is then created in [25] to obtain a Van Der Waals term of the form −C/R6 (R is the distance between the molecules), while a very detailed computation of the (exponentially small) combined energy is done in [4, 5, 6, 7] to conclude that the system can bind. Because of the preliminary averaging, the conclusion is that there exists some orientation of the molecules for which this is true, but this position is unknown a priori. In the case of the mountain pass method that we propose here, the noncompactness is obviously also due to a possible splitting of the molecule. However, we want to insist on the fact that we cannot use in this setting the same idea of averaging over all the rotations of the molecules, because we have to pull down the energy along a path. In other words, a comparison between the energies is not sufficient to conclude, and a precise information on the directions on which the energy decreases is needed. This is why we failed to treat the problem in its full generality and we had to add some hypothesis about the configurations “at infinity”. Nevertheless, we wish to ameliorate this first work in the future, and hope that it will stimulate further results. The results proved in this paper are the following. First, we study the spectrum and the eigenfunctions of the Hamiltonian when a molecule splits into parts. We obtain some bounds on the eigenvalues and the bottom of the essential spectrum which allow to show that the “electrons remain in the vicinity of the nuclei” when a fixed excited state is studied. In other words, no electron is lost during the process. This is obtained by a non-isotropic exponential decay of the electronic density, which is shown to be uniform when the distance between the molecules grows. We also specify the behavior of the associated wavefunctions and define the “critical points at infinity”, a concept introduced by A. Bahri [2]. Some parts of this first result are necessary for our min-max problem. Then, we prove a result that enables to identify the possible behavior of the non-compact min-maxing paths. As it is suggested by the intuition, it is shown that the optimal energy of the mountain pass corresponds in this case to a system where the molecule is split into independent parts (the electrons are shared among them), each being at its ground state. This Morse information on the critical points at infinity is rather intuitive. As announced, we were unable to treat the general case and we end this article by showing that this non-compactness behavior is impossible in the special case of
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two interacting molecules with fixed nuclei. This is done under the hypothesis that we are in the easy case of two charged molecules at infinity, or in the more difficult case of two neutral molecules at infinity, but with dipoles at their ground state. This enables to obtain the required result for many practical situations. As explained before, the crucial step is to evaluate the interaction energy between the molecules and we use here a multipole expansion, even for wavefunctions that are not a simple tensor product of two ground states as in [25]. Finally, the expected result is deduced from the fact that the critical points of the dipole/dipole interaction energy which have a nonnegative energy have a Morse index which is at least 2. From a practical point of view, the study of this mountain pass method is really important. As mentioned above, the main idea is that a path leading from one minimum to the other represents an infinitely slow chemical reaction. The mountain pass energy is then interpreted as the lowest energy threshold for the reaction to happen. The numerical computation of this energy and of the optimum (even the whole path) is then a prime necessity for chemists, who have to understand the possible behaviors of the molecule (see for instance [28, 9] for chemical and numerical aspects). However, chemists only consider paths on which the molecule is at its ground state all along it, which leads to obvious problems of smoothness in the case of degeneracy of the first eigenvalue of the Hamiltonian, and can obstruct convergence. For mathematical reasons, we were thus forced to abandon this hypothesis and relax the problem by considering that the wavefunction can vary independently of the nuclear geometry, in order to obtain a critical point with respect to nuclei’s variations. Since we shall show that our min-max energy is in fact the same that the one used in practice, this approach could also be interesting for numerical computations. We conclude with a few words on the mathematical tools used in this paper. As in [4, 5, 6, 7], the proof is guided by P.-L. Lions’ Concentration-Compactness ideas [26], although the localization of the electrons is given by the uniform exponential decay of Theorem 2, and not by this theory. Let us remark that the physical intuition is somewhat often related to the behavior of the electronic density. For instance, when the molecule splits into parts, the latter becomes a sum of functions localized near the nuclei. But this point of view is not sufficient to understand the problem since the main object is not the density, but the wavefunction. The latter will not split into sums, but into sums of tensor products of wavefunctions in lower dimensions (see the work of G. Friesecke [10] for a very clear explanation of this phenomenon). Therefore, we use a variant of N -body geometric methods for Schr¨ odinger operators [30, 29, 18] that enables to relate the behavior of the wavefunction to those of the associated electronic density. This method is used in [10] and enabled G. Friesecke to notice an interesting link between the celebrated HVZ Theorem [18, 32, 33] and the Concentration-Compactness method [26]. Moreover, the HVZ Theorem (which enables to identify the bottom of the spectrum as the ground state energy of the same system but with an electron
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removed), and Zhislin’s Theorem (which states the existence of excited states for positive or neutral molecules) are abundantly used in this article. Finally, we use the results and methods developed by G. Fang and N. Ghoussoub [8, 14] which enable to obtain Morse information on the Palais-Smale sequences, related to the fact that the deformed object are paths (i.e. deformations of [0; 1]). We also use the duality theory developed in [14] which permits to locate critical points. The paper is organized as follows. In the next section, we describe the model in detail and recall known results on the Hamiltonian and its eigenfunctions. Then, in Section 2, we present our results without proof: for the sake of clarity, we have brought all the proofs together in the last section.
1 The model 1.1
Framework
We consider here a positive or neutral molecule with N non relativistic electrons, and M nuclei of charges Z1 +· · ·+ZM ≥ N . The nuclei are supposed to be correctly described by a classical model (Born-Oppenheimer approximation) and are thus represented as pointwise charges at R1 , . . . , RM ∈ R3 . In what follows, we let R = (R1 , . . . , RM ) ∈ Ω := (R3 )M \ (∪i=j {Ri = Rj }) and
Z = (Z1 , . . . , ZM ) ∈ (N∗ )M ,
|Z| = Z1 + · · · + ZM ≥ N.
The system is described by the purely coulombic N -body Hamiltonian H N (R, Z) =
N 1 − ∆xi + VR (xi ) + 2 i=1 VR (u) = −
1≤i<j≤N
M j=1
1 + |xi − xj |
1≤i<j≤M
Zi Zj , |Ri − Rj |
Zj . |u − Rj |
Its operator domain is the Sobolev space Ha2 (R3N , C), and its quadratic form domain is Ha1 (R3N , C). Throughout this paper, the subscript a indicates that we consider wavefunctions Ψ which are antisymmetric under interchanges of variables (expression of the Pauli exclusion principle): ∀σ ∈ SN , Ψ(x1 , . . . , xN ) = (σ)Ψ(xσ(1) , . . . , xσ(N ) ). The quantum energy of the system in a state Ψ ∈ Ha1 (R3N , C) is the associated quadratic form E N (R, Ψ) = Ψ, H N (R, Z)Ψ .
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We refer the reader to [22, 3] for a description of this model and a detailed explanation of the Born-Oppenheimer approximation. The properties of H N (R, Z) and its eigenfunctions are recalled below. For the sake of simplicity, we have neglected the spin and the dynamic of the nuclei, as in [25]. We would like also to mention that all the results in this paper can be adapted to the caseof smeared nuclei, 1 1 is replaced by R3 |x−y−R dµi (y) that is to say when the Coulomb potential |x−R i| i| 1 3 where µi is a probability measure on R . Of course, |Ri −Rj | has to be replaced by 1 R6 |Ri +y−(Rj +z)| dµi (y)dµj (z). Remark that, in contrast to many other papers dealing with minimization, we work here with complex-valued wavefunctions, a hypothesis that plays a role in our results (see for instance Theorem 4 and the associated remarks). Z and N being fixed such that N ≤ |Z|, for each R ∈ Ω, the problem E N (R, Z) = min{E N (R, Ψ), ||Ψ||L2 = 1} has a solution Ψ, which is the ground state of the N electrons interacting with the M nuclei localized at the Ri . For neutral molecules (N = |Z|), it is also known that the problem E N = min E N (R, Z) R∈Ω
admits a solution [25], proving the stability of neutral molecules. We shall assume that (R, Ψ) and (R , Ψ ) are two local minima of E N . We then consider the classical mountain pass method c = inf max E N (γ(t)) γ∈Γ t∈[0;1]
(1)
where Γ = {γ ∈ C 0 ([0; 1], Ω × SHa1 (R3N )), γ(0) = (R, Ψ), γ(1) = (R , Ψ )} SHa1 (R3N ) = {Ψ ∈ Ha1 (R3N ), ||Ψ||L2 = 1} and want to show that c is a critical value of E N . As mentioned in the introduction, the physical interpretation of this minmax method is that paths γ ∈ Γ represent an infinitely slow reaction leading from one minimum to the other. c is thus interpreted as the lowest energy threshold for passing from (R, Ψ) to (R , Ψ ). In practice, the following definition is used c = inf max E N (r(t), Z) r∈R t∈[0;1]
where
R = {r ∈ C 0 ([0; 1], Ω), r(0) = R, r(1) = R }.
As explained in the introduction, the function R → E N (R, Z) is continuous but not necessary differentiable and this is why we shall study the min-max method (1). However, it will be shown that in fact c = c .
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Properties of H N (R, Z)
Let us now recall some well-known facts about the spectrum of H N (R, Z). We introduce inf sup H(R, Z)Ψ, Ψ . (2) λN d (R, Z) = dim(V )=d
Ψ∈V, ||Ψ||L2 =1
In the sequel, we shall denote for all d ≥ 1
E 0 (R, Z) = λ0d (R, Z) :=
1≤i<j≤M
Zi Zj , |Ri − Rj |
Σ0 (R, Z) = +∞.
For a wavefunction Ψ ∈ Ha1 (R3 , C), the electronic density and the electronic kinetic energy density are respectively defined by |Ψ(x, x2 , . . . , xN )|2 dx2 . . . dxN ρΨ (x) = N tΨ (x) = N
R3(N −1)
R3(N −1)
|∇Ψ(x, x2 , . . . , xN )|2 dx2 . . . dxN .
We have brought together the main known results in the following Theorem 1. We assume N ≥ 1. The following results are known: 1. (Self-adjointness [20]) H N (R, Z) is self-adjoint on L2a (R3N ) with operator domain Ha2 (R3N ) and quadratic form domain Ha1 (R3N ). 2. We have [19] σess H N (R, Z) = [ΣN (R, Z); +∞) Zi Zj N ∀d ≥ 1, λN d (R, Z) ≤ Σ (R, Z) ≤ |Ri − Rj | 1≤i<j≤M
3. (HVZ Theorem [18, 32, 33, 19]) We have ΣN (R, Z) = E N −1 (R, Z). N 4. (Compactness below the essential spectrum) If λN d (R, Z) < Σ (R, Z), then N λd (R, Z) is an eigenvalue of finite multiplicity and in particular, there exists a Ψd ∈ Ha2 (R3N ) such that
H N (R, Z)Ψd = λN d (R, Z)Ψd . It is locally lipschitz [21], i.e., Ψd ∈ C 0 (R3N )
and
3N |∇Ψd | ∈ L∞ ), loc (R
and real analytic [13] on U N \ {xi = xj }, where U = R3 \ {Ri }M i=1 . If ρd is the associated electronic density, then [13] ρd ∈ C ω (U ) ∩ C 0,1 (R3 ).
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5. (Zhislin Theorem [33]) For positive and neutral molecules N ≤ |Z|, then N λN d (R, Z) < Σ (R, Z)
for all d ≥ 1, so that N N σ H N (R, Z) = {λN 1 (R, Z) ≤ · · · ≤ λd (R, Z) ≤ · · · } ∪ [Σ (R, Z); +∞). 6. (Negative molecules [19, 23]) For negative molecules N > |Z|, there exists a N δ such that λN δ (R, Z) = Σ (R, Z), and δ = 1 when N ≥ 2|Z| + M . Note that the functions ΣN (R, Z) and λN d (R, Z) (d ≥ 1) are continuous with respect to R.
2 The results In this section, we present the results that we have obtained concerning the mountain pass method defined above. As mentioned, all the proofs are postponed to the next section.
2.1
The spectrum of a molecule that splits into pieces
We begin the study by some general results about the spectrum and the behavior of the eigenstates when the molecule splits into pieces, that is to say when |Ri −Rj | → +∞ for some i and j. As mentioned above, this splitting of the molecule will be shown to be the main reason for the possible lack of compactness of Palais-Smale sequences. Although only the case of ground states will be necessary for the sequel, we tackle here arbitrary excited states. In this section, we consider a positive or neutral molecule (N ≤ |Z|). We fix a 2 ≤ p ≤ M (number of pieces). Let X1 , . . . , Xp : R+ → R3 be p functions that satisfy |Xi (t) − Xj (t)| ≥ t ∗ p for
p all i = j and t large enough. Let be m = (m1 , . . . , mp ) ∈3 (N ) such that some rj,k ∈ R and zj,k ∈ N for j=1 mj = M . We fix a positive constant R0 and
mj j = 1, . . . , p and k = 1, . . . , mj such that |Z| := pj=1 k=1 zj,k ≥ N , |rj,k | ≤ R0 and rj,k = rj,l when k = l. We then let zj = (zj,1 , . . . , zj,mj ),
Z = (z1 , . . . , zp ),
r˜j (t) = (Xj (t) + rj,1 , . . . , Xj (t) + rj,mj ),
rj = (rj,1 , . . . , rj,mj ), R(t) = (˜ r1 (t), . . . , r˜p (t)).
We also introduce ωj = {(rk ) ∈ B(0, R0 )mj , rk1 = rk2 if k1 = k2 }. and
U(R) = R3 \
p
j=1
B(Xj , R) .
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Z3
Z2 ×
X1 Z4
Z1
O ×
Z6
Z5 × X3
t
Y X2 r2,1 ×
*
+
Figure 1: An example with M = 6, p = 3, m1 = 3, m2 = 1 and m3 = 2. 2.1.1 Spectrum and uniform exponential decay We have the following result: Theorem 2 (Spectrum and uniform exponential decay). For all 1 ≤ N ≤ |Z| and all d ≥ 1, we have p 1. lim E N (R(t), Z) = min E Nj (rj , zj ), N1 + · · · + Np = N . t→+∞ j=1 p p Nj 2. limsup λN (R(t),Z) ≤ min λ (r ,z ),N + ···+ N = N, δ = d . j j 1 p j d δj t→+∞ j=1 j=1 3. inf lim inf ΣN (R(t), Z) − λN d (R(t), Z) > 0. rj ∈ωj t→+∞
4. Let ΨR be an eigenfunction associated to the eigenvalue λN d (R, Z), with associated densities ρR and tR . Then there exist positive constants R1 , C and α, depending only on N , d, R0 , p such that ρR (x) ≤ C exp(−αδ(x)) and tR (x) ≤ C exp(−αδ(x))
on U(R1 )
where δ(x) = min{|x − Xj |, j = 1, . . . , d}. The first part 1) identifies the limit of the ground state energy. This type of result is rather intuitive and classical. However, since we do not know a reference in this precise setting, a proof will be given in the next section. The interpretation of the last part 4) is that if a neutral or positively charged molecule splits into parts, then for a fixed excited state, the electrons remain in the vicinity of the nuclei.
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For the sake of simplicity, let us denote, for r = (r1 ,...,rp ) and z = (z1 ,...,zp ) p p N λδjj (rj , zj ), N1 + · · · + Np = N, δj = d . ΛN d (r, z) := min j=1
j=1
2.1.2 Behavior of the wavefunctions, critical points at infinity Now that we have some bounds on the eigenvalues and the bottom of the essential spectrum, we want to prove a result describing the behavior of the eigenfunctions. This will enable us to define the “critical points at infinity” of the model, a concept that was introduced by A. Bahri [2]. The right-hand side of Theorem 2-1), or more generally an equality like c=
p
N
λδjj (rj , zj ),
(3)
j=1
is rather standard from P.-L. Lions’ Concentration-Compactness point of view: when the molecule splits into pieces, then the energy of an electronic excited states becomes a sum of excited states energies of the pieces. In other words, a “critical point at infinity” would be a system constituted by p molecules in some excited state, each being infinitely far from the others, so that the interactions between them vanish. p Let us consider a sequence tn → +∞, some r = (r1 , . . . , rp ) ∈ j=1 ωj , n n and denote by R = R(tn ) := (X1 (tn ) + r1 , . . . , Xp (tn ) + rp ), Xj = Xj (tn ). For the electronic density, such a configuration is then clearly obtained in the case
of dichotomy, that is to say ρn pj=1 ρnj where ρnj is essentially supported in the vicinity of Xjn (see the exponential decay of Theorem 2). But the behavior of the wavefunction Ψn is less simple since these functions will not split into sum of functions, but into sums of antisymmetric tensor products of wavefunctions in lower dimensions. In other words, a simple way to represent these non interacting molecules in terms of the wavefunction is to take Ψn = τX1n · ψ1 ∧ · · · ∧ τXpn · ψp
(4)
where each ψj is an eigenfunction of H Nj (rj , zj ) associated to the eigenvalue N λδjj (rj , zj ). We have used here the notation τv · Ψ(x1 , . . . , xN ) := Ψ(x1 − v, . . . , xN − v) and we recall that the tensor product is defined for ψ ∈ L2a (R3N1 ) and ψ ∈ L2a (R3N2 ) by 1 ε(σ)ψ(x1σ )ψ (x2σ ). ψ ∧ ψ (x1 , . . . , xN1 +N2 ) = √ N !N1 !N2 ! σ∈S N
where x1σ := (xσ(1) , . . . , xσ(N1 ) ) and x2σ = (xσ(N1 +1) , . . . , xσ(N ) ).
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With (4), one easily sees that ρn = pj=1 τXjn · ρj with obvious notations, and that H N (Rn , Z)Ψn − cΨn → 0 in L2 (R3N ) as n → +∞. N When a λδjj (rj , zj ) is degenerated, we can obtain the same behavior by taking a wavefunction which is a sum of such antisymmetric tensor products Ψn ∈
p
N τXjn · ker H Nj (rj , zj ) − λδjj (rj , zj ) .
j=1
To simplify notations, we shall denote τn · (ψ1 ∧ · · · ∧ ψp ) := τX1n · ψ1 ∧ · · · ∧ τXpn · ψp , N so that Ψn = τn · Ψ where Ψ ∈ pj=1 ker H Nj (rj , zj ) − λδjj (rj , zj ) . Suppose now that a molecule splits into two identical pieces: r1 = r2 and z1 = z2 , and that N1 = N2 . At infinity, we shall obtain two molecules with the same configurations of the nuclei, but not the same number of electrons. Since there is no reason to distinguish the two states obtained by inverting the electrons between the two molecules, a wavefunction can be a sum of these two states with the same energies. We are thus led to introduce the following definition. p n Definition 1. Let rn = (r1n , . . . , rpn ) ∈ j=1 ωj be such that rj → rj ∈ ωj , and n n R = R(tn ) := (X1 (tn ) + r1 , . . . , Xp (tn ) + rp ), Xj = Xj (tn ), for some tn → +∞. Let c be such that the set p N AN λδjj (rj , zj ) = c (Nj , δj ) ∈ (Np )2 , N1 + · · · + Np = N, (5) c (r, z) = j=1
is not empty. The sequence (Rn , Ψn ) in Ω × SHa1 (R3N ) converges to a critical point at infinity of energy c if there exists some p N ker H Nj (rj , zj ) − λδjj (rj , zj ) Ψ∈ (6) (Nj ,δj )∈AN c (r,z)
j=1
such that ||Ψn − τn · Ψ||Ha1 (R3N ) → 0. To justify the term critical point, we remark that one can prove Lemma 1. Let be c and Ψ that satisfy (5) and (6). Then (H N (Rn , Z) − c)(τn · Ψ) → 0 in L2 (R3N ). Details will be given later on.
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Saying differently, a critical point at infinity is a class τK (r, Ψ) = τX1 ,...,Xp (r, Ψ), Xj ∈ R3 , |Xi − Xj | ≥ K
Nj p Nj ker H (r , z ) − λ (r , z ) , rj ∈ ωj and where Ψ ∈ (Nj ,δj )∈AN j j j j j=1 (r,z) δ j c p 1 3Nj τX1 ,...,Xp is defined on each Ω × j=1 Ha (R ) by τX1 ,...,Xp · r = (X1 + r1 , · · · , Xp + rp ) τX1 ,...,Xp · (ψ1 ∧ · · · ∧ ψp ) = (τX1 ψ1 ) ∧ · · · ∧ (τXp ψp ). A sequence (Rn , Ψn ) converges to this critical point at infinity when there exists a Kn → +∞ such that lim d[(Rn , Ψn ); τKn (r, Ψ)] = 0.
n→+∞
p Let us now fix a sequence tn → +∞ and some rn = (r1n , . . . , rpn ) ∈ j=1 ωj such that rjn → rj ∈ ωj , and denote Rn = R(tn ), Xjn = Xj (tn ). We then have the following result concerning the behavior of the eigenfunctions Theorem 3. We assume 1 ≤ N ≤ |Z| and d ≥ 1. Let (Ψn ) be a sequence of wavefunctions such that n n H N (Rn , Z) · Ψn = λN d (R , Z) · Ψ . n Then, up to a subsequence, we have limn→+∞ λN d (R , Z) := c with
ΛN 1 (r, z) ≤ c =
p j=1
N
λδ j (rj , zj ) ≤ ΛN d (r, z) j
n n for some (Nj , δj ) ∈ AN c (r, z), and (R , Ψ ) converges to a critical point at infinity of energy c.
2.2
The mountain pass method: a general result
Let us now come back to our mountain pass method, and consider again a neutral molecule (N = |Z|). Recall that (R, Ψ) and (R , Ψ ) are two local minima of (R, Ψ) → E N (R, Ψ), and that c and c are defined by c = inf max E N (γ(t)) γ∈Γ t∈[0;1]
Γ = {γ ∈ C 0 ([0; 1], Ω × SHa1 (R3N )), γ(0) = (R, Ψ), γ(1) = (R , Ψ )}. c = inf max E N (r(t), Z) r∈R t∈[0;1]
R = {r ∈ C 0 ([0; 1], Ω), r(0) = R, r(1) = R }.
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The following result enables to identify c in the case of lack of compactness: Theorem 4. We assume N = |Z|. We have c = c . There exists a min-maxing sequence (Rn , Ψn ) ∈ Ω × Ha1 (R3N ) such that:
If i=j |Rin − Rjn | is bounded then, up to a translation, (Rn , Ψn ) converges strongly in Ω × Ha1 (R3N ) to some critical point (R, Ψ) of E N such that H N (R, Z) · Ψ = c · Ψ,
c = λN 1 (R, Z).
If i=j |Rin − Rjn | is not bounded, there exists a 2 ≤ p ≤ M , some Xjn ∈ R3 with j = 1, . . . , p and a R0 > 0 such that, changing the indices if necessary, Rn = (X1n + r1n , . . . , Xpn + rpn ), ||rjn || ≤ R0 , Z = (z1 , . . . , zp ), and limn→+∞ |Xin − Xjn | = +∞, rjn → rj . Then c = ΛN 1 (r, z) = min
p
E Nj (rj , zj ), N1 + · · · + Np = N
j=1
and (Rn , Ψn ) converges up to a subsequence to a critical point at infinity of energy c = ΛN 1 (r, z). As a consequence, in the non-compact case, the molecule splits into pieces, the electrons being shared among them and at their ground state. We also believe that the rj correspond to positions of the nuclei with a Morse index equal to 0, but this is not necessary for the sequel. This result should be seen as the first step towards concluding the existence of a critical point of energy c, by proving that the second case in Theorem 4 does not happen. Unfortunately, we met with serious difficulties when trying to solve this general problem. This is why the compactness will be shown in the next section for the special case of two interacting molecules with fixed nuclei. Remark. Throughout this paper, we work with complex-valued wavefunctions Ψ. Although in other situations (minimization for instance) one often works with real-valued functions without any change, this is not the case here. In particular, the equality c = c is very easily obtained in this setting, while one can prove that this is also true for real-valued wavefunctions, but for well-chosen ground states Ψ and Ψ only. See the proof for more details.
2.3
Compactness in the case of two interacting molecules
Now that we have identified the critical points at infinity for the mountain pass method, the next step is to show that min-maxing paths cannot approach these critical points. We study here the case of two interacting molecules with fixed nuclei. The parameters are then
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• the distance between the two molecules (denoted by α in the sequel), • the orientation of each molecule (represented by two rotations u and u ), • the electronic wavefunction. u u ... r ... ... . α ..
... r ... ... ... ... ... O
-
Figure 2: Two molecules with fixed nuclei.
m So we consider r = (r1 , . . . , rm ) ∈ B(0, R0 )m and r = (r1 , . . . , rm ) ∈ B(0, R0 ) such that r1 = r1 = 0, ri = rj and ri = rj for i = j, and some z = (z1 , . . . , zm ), z = (z1 , . . . , zm ). We denote by Z = (z, z ), and introduce
R(α, u, u ) = (u · r, α v + u · r ). where v is a fixed vector of norm 1, α ∈ R, and u, u are rotations in R3 . We have used the notation u · r = (u · r1 , . . . , u · rm ). We suppose now that N = |Z| and define E N (α, u, u , Ψ) := E N (R(α, u, u ), Ψ). In [25], it is proved that E N admits a minimum on R × (SO3 (R))2 × SHa1 (R3N ). As in the previous sections, we shall assume that E N possesses two local minima M and M . Up to a rotation of each molecule, we may suppose that α(M ) > 0 and α(M ) > 0. We then consider c = inf max E N (γ(t)) γ∈Γ t∈[0;1]
where Γ is the set of all the continuous functions γ : [0; 1] → X := (0; +∞) × (SO3 (R))2 × SHa1 (R3N ) such that γ(0) = M and γ(1) = M . 2.3.1 The mountain pass method We begin this section by stating a result which is the analogue of Theorem 4 in this special setting. Theorem 5. We have • either there exists a critical point (α, u, u , Ψ) of E N on X, such that H N (R(α, u, u ), Z) · Ψ = c · Ψ, c = λN 1 (R(α, u, u ), Z),
• or
c = min E N1 (r, z) + E N2 (r , z ), N1 + N2 = N .
Roughly speaking, the non compactness of min-maxing sequences is related to the existence of two gradient lines going from a local minimum to some critical point at infinity of index 0. The idea is that an “optimal path” has to follow these
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lines, and then to connect the two critical points at infinity. Since the molecule is split here into two independent parts, the problem is now to find two mountain pass paths connecting each configuration of the two molecules – two similar problems of lower dimension. When the position of the nuclei in each molecule is fixed, these paths can be obtained by only applying some rotations. In other words, the minima always belong to the same connected component and this is why the situation will be simpler in this setting. When the position is not fixed, even if we may assume the existence of such paths (by induction), the situation is much more complicated and we hope to come back to this more general issue in the future. The proof of Theorem 5 is very similar to the one of Theorem 4, and will be omitted. In order to prove that the second case in Theorem 5 does not happen, we need some information on the directions on which the energy decreases near the critical points at infinity. We shall thus need an expansion of the interaction energy between the two molecules when α grows. The terms involving in these expansion are classical. Let us first recall the definitions of the first multipoles. Definition 2. Let be R = (R1 , . . . , RM ) ∈ Ω, Z =(Z1 , . . . , ZM ), and ρ ∈ L1 (R3 ) ∩ S(R3 \ {Rj }) a non-negative function such that R3 ρ = N > 0. Then
M 1. the total density of charge is the measure ρ˜ := ρ − j=1 Zj δRj . The total charge is q := R3 ρ˜ = N − |Z|,
M 2. the dipole moment is the vector P := R3 x˜ ρ(x) dx = R3 xρ(x) dx − j=1 Zj Rj , 3. the quadrupole moment is the matrix Q := R3 xxT − 13 |x|2 I ρ˜(x) dx. When ρ is the electronic density associated to some eigenstate Ψ, we shall use the notations ρΨ , ρ˜Ψ , PΨ and QΨ . This multipoles will be used in the expansion of the interaction energy. To illustrate this point, we give here the following Lemma 2. We assume that N1 and N2 are such that N1 + N2 = N , E N1 (r, z) < ΣN1 (r, z) and E N2 (r , z ) < ΣN2 (r , z ). Let ψ1 and ψ2 be two ground states of respectively H N1 (r, z) and H N2 (r , z ). Denoting Ψ(α, u, u ) = (u · ψ1 ) ∧ (ταv · u · ψ2 ), we have E N (R(α, u, u ), Ψ(α, u, u )) = E N1 (r, z) + E N2 (r , z ) +
q1 q2 uP1 · v u P2 · v (uP1 ) · (u P2 ) − 3(uP1 · v )(u P2 · v ) + q2 − q1 + 2 α α α2 α3 T 1 3(q2 uQ1 uT + q1 u Q2 u )v · v + + O 2α3 α4
for all u, u ∈ SO3 (R) and when α goes to +∞.
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In this result, qk , Pk and Qk are respectively the total charge, the dipole and the quadrupole moment associated to the electronic densities ρk of the states ψk . The terms of this expansion can be interpreted respectively as the energies of the molecules, and the interaction energy between them, which decomposes into the charge/charge (1/α), dipole/charge (1/α2 ), dipole/dipole and charge/ quadrupole (1/α3 ) terms. We are now able to state our main compactness results. As mentioned above, we had to add some hypothesis about the molecules “at infinity”, concerning their multipoles in their ground state. 2.3.2 The case of charged molecules at infinity Our first result will concern the case of monopoles at infinity, that is to say when the molecules are charged. Theorem 6 (Charged molecules at infinity). Let us assume that E N1 (r, z) + E N2 (r , z ) = min {E n1 (r, z) + E n2 (r , z ), n1 + n2 = N }
(9)
for some N1 and N2 with (N1 − |z|)(N2 − |z |) = 0. Then the case 2) in Theorem 5 does not happen. Therefore c is a critical value of E N on X. Remark. By (9), we have for instance
µ := E N1 (r, z) − E |z| (r, z) < E |z | (r , z ) − E N2 (r , z ) := µ for some N1 , N2 such that N1 + N2 = N and N1 < |z|. This can be viewed as a comparison between oxydo-reduction potentials. So (9) will be probably true if one molecule is a oxidant and the other is a reductor. 2.3.3 The case of neutral molecules with dipole moments at infinity If the two molecules at infinity are neutral, the first term involving in the expansion of the interaction energy is the dipole/dipole term. This is why we shall now consider the case of molecules that possess some dipole moment in their ground state (experiment suggests that this is the case for every non symmetric molecule). Let us introduce the following definition Definition 3. Let be R = (R1 , . . . , RM ) ∈ Ω, Z = (Z1 , . . . , ZM ) and N > 0 such N that λN 1 (R, Z) < Σ (R, Z). We shall say that the molecule (R, Z, N ) possesses a dipole moment at its ground state if PΨ = 0 for all ground state Ψ. Since V := ker H N (R, Z) − E N (R, Z) is finite-dimensional, let us notice that this implies min{|PΨ |, Ψ ∈ V, ||Ψ||L2 = 1} > 0.
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We then have the following result: Theorem 7 (Neutral molecules with dipole moments at infinity). Let us assume that
(H1) E |z| (r, z) + E |z | (r , z ) < E N1 (r, z) + E N2 (r , z ) for all N1 , N2 such that N1 + N2 = N and (N1 − |z|)(N2 − |z |) = 0, (H2) the two molecules (r, z, |z|) and (r , z , |z |) possess a dipole moment at their ground state,
(H3) E |z| (r, z) or E |z | (r , z ) is non-degenerated. Then the case 2) in Theorem 5 does not happen. Therefore c is a critical value of E N on X. Remark. (H3) is a purely mathematical restriction that simplifies the proof. Let us explain the general idea of the proof. Recall that the dipole/dipole interaction energy can be written F (P, P )/α3 (see Lemma 2). It is shown in Appendix 2 that the critical points of F which have a non-negative energy have a Morse index which is at least one. If a path approaches a critical point at infinity then, to pull down the energy along the path, one may use either the rotations of the molecules if the dipole/dipole interaction energy is positive (thanks to this Morse index information on F ), or the distance between them if it is negative (because α → F (P, P )/α3 is then increasing). This is why min-maxing paths do not approach the critical points at infinity, and give thus a compact Palais-Smale sequence. Obviously, this general idea does not suffice to lead the proof and there are some other difficulties (essentially due to the complexity of the model) that are explicited in the next section. Remark. This general information on the Morse index is probably true for the others multipoles interaction energies, a fact that could be used to treat the general case.
3 Proofs 3.1
Proof of Theorems 2 and 3
3.1.1 Preliminaries We shall use the following lemma, which is an adaptation of results in [15, 16, 11, 12, 13], and which is proved in Appendix 1. Lemma 3. Let ΨR be an eigenfunction associated to the eigenvalue λN d (R, Z) and ρR be the electronic density. We introduce R = ΣN (R, Z) − λN d (R, Z). Then 1. ρR satisfies the inequation 1 − ∆ρR + VR ρR + R ρR ≤ 0. 2
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2. With R1 () := max R0 + 1, R0 + 2N p and C() := ∪pj=1 {x, |x − Xj | = R1 ()}, and if r > 2R1 (R ), then we have ≤
ρR (x)
||ρR ||L∞ (C(R ))
p
e−
√
R /p(|Xj −x|−R1 (R ))
j=1
≤ ≤
√ p||ρR ||L∞ (C(R )) e− R /p(δ(x)−R1 (R )) √ M e− R /p(δ(x)−R1 (R )) ,
(11)
on U(R1 (R )), where δ(x) = min{|x − Xj |, j = 1, . . . , d}, and M = M (p, N, R1 (R )). The explicit bound (11) has been written in order to show the dependence of all the constants with regard to R . It is clearly not optimal. It shows a nonisotropic exponential decay of the electronic density, which will be uniform if R 0. This type of bounds is studied in the work of Agmon [1] and we do not know if one can use his formalism to obtain the same result. Isotropic exponential bounds for N -body eigenfunctions are frequently seen in the literature, but surprising is the fact that such non-isotropic bounds has not yet been noticed. The next two lemmas will be useful to prove the exponential decay of Theorem 2. Lemma 4. For all α > 0, there exists a constant M = M (α, N, R0 ) such that ρR (y) dy tR (x) ≤ M B(x,α)
on U(R0 + 1/2 + α).
Proof of Lemma 4 – see [16]. Lemma 5. For all j = 1, . . . , p, d ≥ 1 and n ≤ |zj |, we have inf (Σn (rj , zj ) − λnd (rj , zj )) > 0.
rj ∈ωj
Proof. We have ˜ n (rj , zj ) ˜ n (rj , zj ) − λ Σn (rj , zj ) − λnd (rj , zj ) = Σ d ˜n (rj , zj ) and Σ ˜ n (rj , zj ) are the dth eigenvalue and the bottom of the eswhere λ d sential spectrum of the Hamiltonian with the nuclei interaction removed ˜ n (rj , zj ) = H
n 1 − ∆xi + Vrj (xi ) + 2 i=1
1≤i<j≤n
1 . |xi − xj |
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By Zhislin’s Theorem, it is known that ˜ n (rj , zj ) > 0 ˜ n (rj , zj ) − λ Σ d for all rj ∈ ωj and, since this function is continuous with regard to rj , inf (Σn (rj , zj ) − λnd (rj , zj )) > 0.
rj ∈ωj
In the next result, we use both HVZ and Zhislin’s Theorems. This lemma will be useful in the proof of Theorem 2 to construct test functions. Lemma 6. If the minimum p p Nj min λδj (rj , zj ), N1 + · · · + Np = N, δj = d . j=1
j=1
is attained for N1 , . . . , Nj and δ1 , . . . , δp , then necessarily N
N
λδjj (rj , zj ) < Σδjj (rj , zj ) for all j = 1, . . . , p. Proof of Lemma 6. Remark that by definition λ0δj (rj , zj ) < Σ0 (rj , zj ) = +∞ for all δj . We argue by contradiction and suppose that there exists a k such that Nk > 0 k and λN (rk , zk ) = ΣNk (rk , zk ) = E Nk −1 (rk , zk ). Theorem 1 implies Nk ≥ |zk | + 1. δ
k p Since j=1 (Nk − |zk |) = N − |Z| ≤ 0, there exists a l = k such that Nl < |zl |. / {k, l}, δk = 1, Nk = Nk − 1, δl = δk δl , and We then let δj = δj , Nj = Nj for j ∈ Nl = Nl + 1. We obtain p j=1
N λδjj (rj , zj )
−
p j=1
N
λδ j (rj , zj ) = j
Nl +1 l λN (rl , zl ) δl (rl , zl ) − λδ l
≥
E Nl (rl , zl ) − λδN l +1 (rl , zl )
=
ΣNl +1 (rl , zl ) − λδN l +1 (rl , zl )
>
0
l
l
since Nl + 1 ≤ |zl | (Zhislin Theorem), which is a contradiction.
3.1.2 Proof of Theorem 2 We are now able to prove Theorem 2. We first prove 2). Suppose that the right-hand side is pattained for some N1 , . . . , Nj and δ1 , . . . , δp such that N1 + · · · + Np = N and j=1 δj = p. For the
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sake of simplicity, we may assume that Nj > 0 for all j = 1, . . . , p. By Lemma 6 and Theorem 1, there exist eigenfunctions Ψkj ∈ L2a (R3Nj ) satisfying
N
H Nj (rj , zj )Ψkj = λk j (rj , zj )Ψkj ,
R
3Nj
Ψkj Ψlj = δkl
for all j = 1, . . . , p and k = 1, . . . , δj . If Vj = span(Ψkj , k = 1, . . . , δj ) ⊂ L2a (R3Nj ) then we have N
max
Ψ∈Vj , ||Ψ||2L =1
H Nj (rj , zj )Ψ, Ψ = λδjj (rj , zj ).
We now consider a sequence tn → +∞ such that limn→+∞ λN d (R(tn ), Z) = 2 3N (R(t), Z). If Ψ ∈ L (R ), we introduce lim supt→+∞ λN d = τXj (tn ) · Ψkj Ψk,n j 2 3Nj V˜jn = span(Ψk,n ). j , k = 1, . . . , δj ) ⊂ La (R
(we recall that τv is the translation by v). Now, let be Wn = V˜1n ∧ · · · ∧ V˜pn = span(Ψ1k1 ,n ∧ · · · ∧ Ψkpp ,n , 1 ≤ kj ≤ δj ) which is a space of dimension
Ψ=
d
j=1 δj
= d. If
ck1 ,...,kp Ψ1k1 ,n ∧ · · · ∧ Ψkpp ,n ∈ Wn ,
1≤kj ≤δj
and
|ck1 ,...,kp |2 = 1, we have
H N (R, Z)Ψ, Ψ =
p
k ,n
k ,n
|ck1 ,...,kp |2 H Nj (rj , zj )Ψj j , Ψj j + en
j=1 1≤kj ≤δj
where en is the interaction energy between the p molecules. It is the sum of three terms en = e1n + e2n + e3n . e1n is the interaction between electrons in different molecules, and contains terms like kj kj kj kj (Ψj1 1 Ψj1 1 )(x, . . . )Ψj2 2 Ψj2 2 )(y, . . . ) dx dy. |x − y + Xj2 (tn ) − Xj1 (tn )|
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with j1 = j2 . e2n is the interaction between electrons and nuclei of different molecules, and contains terms like
k
k
zj2 ,i (Ψj1j1 Ψj1j1 )(x, . . . ) dx dy |x − rj2 ,i + Xj2 (tn ) − Xj1 (tn )|
with j1 = j2 . Finally, e3n is the interaction between nuclei of different molecules zj1 ,kj1 zj2 ,kj1 . e3n = |r − r j1 ,kj1 j2 ,kj2 + Xj2 (tn ) − Xj1 (tn )| j <j 1
2
1≤kj ≤mj
It is now easy to see that each of this term tends to 0 as n → +∞. By definition, we have λN d (R, Z) ≤ ≤
p
j=1
≤
max
|ck1 ,...,kp |2 =1
p
max
|ck1 ,...,kp
H N (R, Z)Ψ, Ψ
|2 =1
N
1≤kj ≤δj
|ck1 ,...,kp |2 λkjj (rj , zj ) + max en
N
λδjj (rj , zj ) + max en .
j=1
We may now pass to the limit as n → +∞ in this inequality and obtain the bound p p N lim sup λN λδjj (rj , zj ), N1 + · · · + Np = N, δj = d . d (R(t), Z) ≤ min t→+∞ j=1
j=1
We then prove simultaneously 1) 3) 4) by induction on N = 1, . . . , |Z|. For N = 1, it is known that Zi Zj , Σ1 (R, Z) = E 0 (R, Z) = |Ri − Rj | 1≤i<j≤M
and so lim Σ1 (R(t), Z) =
t→+∞
p
j=1 1≤k
zj,k zj,l = E 0 (rj , zj ). |rj,k − rj,l | j=1 p
As a consequence lim inf Σ1 (R(t), Z) − λ1d (R(t), Z) t→+∞ p p ≥ E 0 (rj , zj ) − λ1d (r1 , z1 ) + E 0 (rj , zj ) j=1
j=2
= E (r1 , z1 ) − 0
λ1d (r1 , z1 )
= Σ1 (r1 , z1 ) − λ1d (r1 , z1 )
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and
inf lim inf Σ1 (R(t), Z) − λ1d (R(t), Z) ≥ inf Σ1 (r1 , z1 ) − λ1d (r1 , z1 ) > 0
ri ∈ωi t→+∞
r1 ∈ω1
by Lemma 5. The uniform exponential decay is then a consequence of Lemmas 3 and 4. Let tn → +∞ be such that limn→+∞ E 1 (R(tn ), Z) = lim inf t→+∞ E 1 (R(t), Z), and φn ∈ L2 (R3 ) be such that H 1 (R(tn ), Z)φn = E 1 (R(tn ), Z)φn . By the uniform exponential decay, we may write φn = supp(φnj ) ⊂ B(Xj , rn /3), and ||αn ||H 1 → 0. Then H 1 (R(tn ), Z)φn , φn =
p
˜ 1 (rj , zj )φn , φn + H j j
j=1
p
p j=1
j=1 1≤k
φnj + αn where
zj,k zj,l + en |rj,k − rj,l |
˜ is the Hamiltonian with the nuclei interaction removed, and en → 0. We where H have p ˜ 1 (rj , zj )||φn ||2 2 ˜ 1 (rj , zj )φn , φn ≥ E H j j j L j=1 j=1,...,p ˜ 1 (rj , zj ) ≥ min E ||φnj ||2L2 j=1,...,p
j=1,...,p
so p
zj,k zj,l j=1,...,p |rj,k − rj,l | j=1 1≤k
H 1 (R(tn ), Z)φn , φn ≥ min E˜ 1 (rj , zj ) +
j=1
and finally lim E 1 (R(t), Z) = min
t→+∞
p
E Nj (rj , zj ), N1 + · · · + Np = 1
j=1
.
Let us now assume that 1) 3) 4) have been proved for N − 1 < |Z|. We have −1 (R, Z) so ΣN (R, Z) = λN 1 p lim ΣN (R(t), Z) = min E Nj (rj , zj ), N1 + · · · + Np = N − 1 t→+∞ j=1
=
p j=1
E Nj (rj , zj )
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for some N1 , . . . , Np . But pj=1 (|zj | − Nj ) = |Z| − (N − 1) > 0 so there exists a k such that Nk < |zk |. We then have, for all d ≥ 1, lim inf ΣN (R(t), Z) − λN d (R(t), Z) t→+∞ p k +1 ≥ E Nj (rj , zj ) − λN (rk , zk ) + d j=1
p
E Nj (rj , zj )
j=1, j=k
=E
Nk
(rk , zk ) −
k +1 λN (rk , zk ) d
k +1 = ΣNk +1 (rk , zk ) − λN (rk , zk ) d
and inf lim inf ΣN (R(t), Z) − λN d (R(t), Z)
ri ∈ωi t→+∞
≥
inf (Σn (rj , zj ) − λnd (rj , zj )) > 0
inf
j=1,...,p rj ∈ωj n≤|zj |
by Lemma 5. The uniform exponential decay 4) is then a consequence of Lemmas 3 and 4. We now prove the inequality p lim inf E N (R(t), Z) ≥ min E Nj (rj , zj ), N1 + · · · + Np = N t→+∞ j=1
by using a variant of classical N -body geometric methods for Schr¨ odinger operators [30, 29, 19], which is used in [10]. Let tn → +∞ be such that limn E N (R(tn ), Z) = lim inf t E N (R(t), Z), and Ψn an associated sequence of ground states, with densities ρΨn and tΨn . We denote Rn = R(tn ) and Xjn = Xj (tn ). Due to the uniform exponential decay, one has ρΨn = lim tΨn = 0. (12) lim n→+∞
U (tn /3)
n→+∞
U (tn /3)
Let ξn ∈ C ∞ (R3 , [0; 1]) be a cut-off function such that ξn ≡ 0 on U(tn /3), ξn ≡ 1 on R3 \U(tn /3−1), and ||∇ξn ||∞ ≤ 1, ||∆ξn ||∞ ≤ 2. We then introduce χn (x1 ,...,xN ) = N ˜ ˜ i=1 ξn (xi ) and Ψn = χn Ψn . Using (12), it is then easy to see that ||Ψn − Ψn ||H 1 → 0, and n ˜ n − λN ˜ H N (Rn , Z) · Ψ 1 (R , Z) · Ψn
=
−
N
(2∇xi χn · ∇xi Ψn + Ψn ∆xi χn )
i=1
→ 0 in L2 (R3N ), n ˜ n ) − λN ˜ E (R , Ψ 1 (R , Ψn )||Ψn ||L2 (R3N ) = N
n
N i=1
|Ψn |2 |∇xi χn |2 → 0.
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Now, we may write ξn = pj=1 ξnj where Supp(ξnj ) ⊂ B(Xj , tn /3), and ˜n = Ψ ξnk1 (x1 ) · · · ξnkN (xN )Ψn := Ψkn1 ,...,kN . 1≤kj ≤p
1≤kj ≤p
Since the Ψkn1 ,...,kN have disjoint supports, ˜ n |2 = ˜ n) = |Ψ |Ψkn1 ,...,kN |2 , E N (Rn , Ψ E N (Rn , Ψkn1 ,...,kN ), 1≤kj ≤p
1≤kj ≤p
n k1 ,...,kN →0 H N (Rn , Z) − λN 1 (R , Z) · Ψn
in L2 (R3N ) for all k1 , . . . , kN . To end the proof of Theorem 2, it suffices to bound ˜ n ) from below by the appropriate constant.. We now fix k1 , . . . , kN and E N (Rn , Ψ introduce Cj = {i, ki = j}, Nj = |Cj |. Remark that Ψkn1 ,...,kN is antisymmetric in (xi )i∈Cj for all j = 1, . . . , p. Then p 1 |∇xi Ψkn1 ,...,kN |2 − E N (Rn , Ψkn1 ,...,kN ) = Vr˜j (xi )|Ψkn1 ,...,kN |2 2 j=1 i∈Cj i∈Cj |Ψk1 ,...,kN |2 n + E 0 (rj , zj ) + en + |xk − xl | k,l∈Cj
where en =
1≤j=j ≤p
i∈Cj
|Ψk1 ,...,kN |2 n + en , Vr˜j (xi )|Ψkn1 ,...,kN |2 + |xi − xi | i∈Cj i ∈Cj
en
being the interaction energy between nuclei in different molecules, which easily tends to 0 as n → +∞. Now |Ψk1 ,...,kN |2 3|zj | n + en → 0 |en | ≤ |Ψkn1 ,...,kN |2 + 3 t t n n 1≤j=j ≤p
i∈Cj i ∈Cj
i∈Cj
as n → +∞. Finally, since Ψkn1 ,...,kN is antisymmetric in (xi )i∈Cj for all j = 1, . . . , p and thanks to the translation invariance of the Hamiltonian, p E Nj (rj , zj ) ||Ψkn1 ,...,kN ||2L2 + en . E N (Rn , Ψkn1 ,...,kN ) ≥ j=1
Passing to the limit, we obtain ˜ n ) ≥ min lim E N (Rn , Ψ
n→+∞
p
which ends the proof of Theorem 2.
j=1
E Nj (rj , zj ), N1 + · · · Np = N
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3.1.3 Proof of Theorem 3 The proof uses exactly the same N -body geometric method as the end of the n N n proof of Theorem 2, but with λN 1 (R , Z) replaced by λd (R , Z). If we suppose N n that limn→+∞ λd (R , Z) = c, then passing to the limit and using Theorem 2 n N c ≤ lim sup λN d (R , Z) ≤ Λd (r, z). n→+∞
We have
N n H (R , Z) − c · Ψkn1 ,...,kN → 0
in L2 (R3N ) for all k1 , . . . , kN . Since all the interaction terms tend to 0 (see the proof of Theorem 2), we obtain p H Nj (Xjn + rjn , zj )Cj − c · Ψkn1 ,...,kN → 0 j=1
where the Hamiltonian H Nj (Xjn + rjn , zj )Cj acts on the variables (xi )i∈Cj . Due to the translation invariance, we obtain p ˜ k1 ,...,kN → 0 H Nj (rjn , zj )Cj − c · Ψ n j=1
˜ kn1 ,...,kN (x1 , . . . , xN ) = Ψkn1 ,...,kN (X n + xi ). But due to the exponential dewhere Ψ ki ˜ k1 ,...,kN is precompact in H 1 (R3N ) and converges up to a subsequence cay of Ψn , Ψ n ˜ k1 ,...,kN such that to some Ψ p ˜ k1 ,...,kN = c · Ψ ˜ k1 ,...,kN . H Nj (rj , zj )Cj Ψ j=1
p Nj on the tensor We thus have either c is an eigenvalue of H (r , z ) j j Cj j=1 p ˜ k1 ,...,kN = 0. product j=1 L2a (R3Nj )Cj (with an obvious notation), or Ψ
N
p p Nj j (rj , zj )Cj = (rj , zj )Cj so that Lemma 7. We have σ j=1 H j=1 σ H
p Nj σess H (r , z ) = [Σ; +∞) with j j C j j=1 Σ
= >
for all d ≥ 1.
min
j=j0
ΛN d (r, z)
E Nj (rj , zj ) + ΣNj0 (rj0 , zj0 ), 1 ≤ j0 ≤ p
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Proof of Lemma 7. The fact that the spectrum of pj=1 H Nj (rj , zj )Cj is the sum
p Nj (rj , zj )Cj is standard (see for instance [27], Theorem VIII-33). Supj=1 σ H
Nj (rj , zj ) + ΣNj0 (rj0 , zj0 ) for some 1 ≤ j0 ≤ p. If pose now that Σ = j=j0 E N
N
j0 λd j0 (rj0 , zj0 ) < ΣNj0 (rj0 , zj0 ) then obviously Σ > ΛN d (r, z). If λd (rj0 , zj0 ) = Nj0 N Σ (rj0 , zj0 ) then Σ > Λd (r, z) by Lemma 6.
p Nj As a consequence, if c is an eigenvalue of (rj , zj )Cj , it is necesj=1 H sary below its essential spectrum. It is then easy to see that this implies
˜ k1 ,...,kN ∈ Ψ
p
N ker H Nj (rj , zj ) − λδjj (rj , zj )
j=1
Cj
N
for some λδjj (rj , zj ) < ΣNj (rj , zj ).
Now, we have ||Ψn − τn · Ψ||Ha1 (R3N ) → 0 where Ψ =
3.2
k1 ,...,kN
˜ k1 ,...,kN . Ψ
Proof of Theorem 4
We may suppose c > max(E N (R, Ψ), E N (R , Ψ )). Let us first prove the equality c = c . Indeed, c ≤ c is obvious. Let be rn ∈ R a sequence such that mn := maxt∈[0;1] E N (rn (t), Z) → c as n → +∞. For each n ∈ N, we define cn = inf max E N (rn (t), ψ(t)) ψ∈ΓΨ t∈[0;1]
ΓΨ = ψ ∈ C ([0; 1], SHa1 (R3N )), ψ(0) = Ψ, ψ(1) = Ψ .
0
We may now apply the methods of [8] to obtain some sequences tk ∈ [0; 1] and (Ψk )k≥1 such that 1.
lim E N (rn (tk ), Ψk ) = cn ,
k→+∞
2. H N (rn (tk ), Z) · Ψk − E N (rn (tk ), Ψk ) · Ψk → 0 in L2 (R3N ), 3. E N (rn (tk ), Ψk ) ≤ λN 1 (rn (tk ), Z) + k ,
lim k = 0
k→+∞
1, 2) correspond to the classical fact that one can obtain min-maxing sequences that are almost critical. On the other hand, 3) is a consequence of the less-known fact that one can obtain Palais-Smale sequences with Morse-type information related to the dimension of the homotopy-stable class used in the min-max method, which is 1 here (paths are deformations of [0; 1]). Since we are in C = R2 , eigenvectors N always have an even Morse index and this is why λN 1 appears in 3) and not λ2 . 1 3N The fact that such a sequence (Ψk ) is precompact in Ha (R ) is now a simple consequence of Theorem 1-4). Indeed, the compactness below the essential spectrum is nothing else but the Palais-Smale condition of E N with Morse-type information introduced in [14]. We have the following general lemma, whose proof is postponed until the end of the proof of Theorem 4.
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Lemma 8. We assume that Z = (Z1 , . . . , ZM ) is such that N ≤ |Z|. Let (Rn , Ψn ) be a sequence in Ω × SHa1 (R3N ) such that 1. Rn → R ∈ Ω 2. lim E N (Rn , Ψn ) = c, n→+∞
3. H N (Rn , Z) · Ψn − E N (Rn , Ψn ) · Ψn → 0 in L2 (R3N ), n 4. there exist d0 ≥ 1 and n → 0 such that E N (Rn , Ψn ) ≤ λN d0 (R , Z) + n . Then (Ψn ) is precompact in Ha1 (R3N ) and converges, up to a subsequence, to an eigenfunction Ψ of H N (R, Z) associated to λN d (R, Z) with d ≤ d0 . Applying this result, we obtain, by passing to the limit as k → +∞, c ≤ cn = λN 1 (rn (tn ), Z) ≤ mn for some tn ∈ [0; 1] and so c = c . We now prove the alternative of the Theorem. We introduce Fc (R0 ) = (R, Ψ) ∈ Ω × SHa1 (R3N ), |Ri − Rj | ≤ R0 , E N (R, Ψ) ≥ c i=j
Γ(α) =
! γ ∈ Γ, max E (γ(t)) ≤ c + α . N
t∈[0;1]
We have the following alternative: either there exist R0 > 0 and α > 0 such that, Fc (R0 ) ∩ γ([0; 1]) = ∅ for all γ ∈ Γ(α), or for all R0 > 0 there exists a min-maxing sequence γn ∈ Γ such that Fc (R0 ) ∩ γn ([0; 1]) = ∅. First Case: there exist R0 > 0 and α > 0 such that, Fc (R0 ) ∩ γ([0; 1]) = ∅ for all γ ∈ Γ(α). Since (R, Ψ) and (R , Ψ ) do not belong to Fc (R0 ), we may apply the methods of [14] (Fc (R0 ) is a set which is dual to the homotopy-stable class Γ(α) with boundary B = {(R, Ψ), (R , Ψ )}) to obtain a sequence (Rn , Ψn ) ∈ Ω × SHa1 (R3N ) such that 1. 2. 3.
lim d((Rn , Ψn ), Fc (R0 )) = 0,
n→+∞
lim E N (Rn , Ψn ) = c,
n→+∞
lim ∇R E N (Rn , Ψn ) = 0,
n→+∞
4. H N (Rn , Z) · Ψn − E N (Rn , Ψn ) · Ψn → 0 in L2 (R3N ), n 5. E N (Rn , Ψn ) ≤ λN 1 (R , Z) + n ,
lim n = 0
n→+∞
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Remark that 2, 3, 4, 5) correspond to the fact that one can obtain min-maxing sequences that are almost critical, and with Morse-type information. On the other hand, 1) is the consequence of the duality theory developed in [14] that enables to locate the critical
points. Due to 1), i=j |Rin − Rjn | is bounded. Up to a translation, we may suppose n n Rn → R ∈ Ω (since λN 1 (R , Z) → +∞ when d(R , ∂Ω) → 0 due to the nun clei/nuclei repulsion). Now Ψ converges up to a subsequence to a Ψ in H 1 (R3N ) by Lemma 8. Second Case: for all R0 > 0 there exists a min-maxing sequence γn ∈ Γ such that Fc (R0 ) ∩ γn ([0; 1]) = ∅. Let (rn ) be a sequence in R such that rn → +∞. For each rn , there exists a γn such that, for instance, c ≤ max E N (γn (t)) ≤ c + t∈[0;1]
1 n
˜ n (t)) and fix n. The and Fc (rn ) ∩ γn ([0; 1]) = ∅. We now write γn (t) = (Rn (t), Ψ n n set Kn = {t ∈ [0; 1], |Ri (t) − Rj (t)| ≤ rn } is a compact subset of [0; 1] such that max E N (γ(Kn )) < c. We now introduce # " ˜n ΓΨ = ψ ∈ C 0 ([0; 1], SHa1 (R3N )), ψ|Kn ≡ Ψ |Kn ˜ n (Kn ), and which is an homotopy-stable class of dimension 1 with boundary Ψ cn = inf max E N (Rn (t), ψ(t)) ψ∈ΓΨ t∈[0;1]
(13)
so that
1 . n Applying the methods of [14], we may find a sequence tk ∈ [0; 1] \ Kn and Ψkn ∈ SHa1 (R3N ) such that tk → t¯ and
1. i=j |Rn,i (tk ) − Rn,j (tk )| ≥ rn 2. lim E N (Rn (tk ), Ψkn ) = cn c ≤ cn ≤ c +
k→+∞
3. H N (Rn (tk ), Z) · Ψkn − E N (Rn (tk ), Ψkn ) · Ψkn → 0 in L2 (R3N ) 4. E N (Rn (tk ), Ψkn ) ≤ λN with lim αk = 0 1 (Rn (tk ), Z) + αk k→+∞
By Lemma 8, (Ψkn )k∈N is precompact in Ha1 (R3N ) and converges, up to a subsequence, to some Ψn such that
n n 1. i=j |Ri − Rj | ≥ rn 2. lim E N (Rn , Ψn ) = c n→+∞
n n 3. H N (Rn , Z) · Ψn = λN 1 (R , Z) · Ψ where Rn := Rn (t¯).
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Since i=j |Rin − Rjn | → +∞, there exists a 2 ≤ p ≤ M , some Xjn ∈ R3 with j = 1, . . . , p and a R0 > 0 such that (changing the indices if necessary and up to a subsequence) Rn = (X1n + r1n , . . . , Xpn + rpn ), ||rjn || ≤ R0 , Z = (z1 , . . . , zp ), and limn→+∞ |Xin − Xjn | = +∞, rjn → rj . Passing to the limit, we obtain, by Theorem 2 n N c = lim λN 1 (R , Z) = Λ1 (r, z). n→+∞
We now simply apply Theorem 3 to obtain the convergence to a critical point at infinity of energy c, as defined in the corresponding section. Let us now prove Lemma 8. N Proof of Lemma 8. Let m be an integer such that λN m (R, Z) > λm−1 (R, Z) = N N n N λd0 (R, Z). Due to the fact that λd (R , Z) → λd (R, Z) as n → +∞, we have n N λN m (R , Z) > λd0 (R, Z) ≥ c for n large enough. $m−1 n N n n n Let be Vn = i=1 ker H(R , Z) − λi (R , Z) , and (ψ1 , . . . , ψm−1 ) an n n N n orthonormal basis of V , ψi being eigenfunctions of H (R , Z) (Theorem 1-4)). Due for instance to the uniform exponential decay of Theorem 2, one easily sees that each ψin is precompact and converges up to a subsequence in Ha1 (R3N ) to a $m−1 ψi , with span(ψi ) = i=1 ker H(R, Z) − λN i (R, Z) . Now we can write Ψn = ΨV n + Ψ(V n )⊥ with an obvious definition. Since E N (Rn , Ψn ) is bounded, it is a classical fact that (Ψn ) is bounded in Ha1 (R3N ) and so, up to a subsequence, Ψn Ψ weakly in Ha1 (R3N ). Since dim(Vn ) = m − 1, (ΨVn ) is precompact in H 1 (R3N ) and converges to a ΨV ∈ V . By difference, Ψ(V n )⊥ Ψ(V )⊥ weakly in H 1 (R3N ). Since limn→+∞ E N (Rn , Ψn ) = c, H N (Rn , Z) · Ψn − c · Ψn → 0 in L2 (R3N ), so we obtain (H N (Rn , Z) − c) · ΨVn + (H N (Rn , Z) − c) · Ψ(Vn )⊥ → 0 which implies (H N (Rn , Z) − c) · ΨVn → 0 and (H N (Rn , Z) − c) · Ψ(Vn )⊥ → 0 in L2 (R3N ). Finally, E N (Rn , Ψ(Vn )⊥ ) − c||Ψ(Vn )⊥ ||2L2 (R3N ) → 0. n N n Because min E N (Rn , S(Vn )⊥ ) = λN m (R , Z) > λd0 (R , Z) ≥ c, this implies n ||Ψ(Vn )⊥ ||L2 → 0 and then ||Ψ(Vn )⊥ ||H 1 → 0. Thus Ψ converges in H 1 (R3N ) to a Ψ = ΨV which is an eigenfunction of H N (R, Z) that belongs to V .
3.3
The case of two interacting molecules
3.3.1 Proof of Lemma 2 We shall use the following lemma:
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Lemma 9 (Multipole expansion). There exists a constant C such that, for all R and h ∈ R3 with R + h = 0, % % % 1 1 C |h|3 eR · h 3(eR · h)2 − |h|2 %% % ≤ − − + % |R + h| % |R|3 |R + h| |R| |R|2 2|R|3 with eR = R/|R|. Proof of Lemma 9. It suffices to show % % 3 % % 1 t2 2 % ≤ √ C|t| %√ (3x − 1 + xt + − 1) % % 1 − 2xt + t2 2 1 − 2xt + t2 for all t ∈ R and x ∈ [−1; 1] (take x= −(eR · h)/|h| andt = |h|/|R|). We thus √ 2 introduce f (x, t) = 1 − 1 − 2xt + t2 1 + xt + t2 (3x2 − 1) . One easily computes ∂f ∂x (x, t)
=
3t3 (5x2 −2xt−1) √ , 2 1−2xt+t2
so that
max |f (x, t)| ≤ max{f1 (t), f2 (t), f3 (t), f4 (t)}
x∈[−1;1]
where f1 (t) = |f (x1 (t), t)|11−1≤x1 ≤1 (t),√ f2 (t) = |f (x2 (t), t)|1√ 1−1≤x2 ≤1 (t), f3 (t) = t− t2 +5 t+ t2 +5 |f (−1, t)|, f4 (t) = |f (1, t)|, x1 (t) = and x2 (t) = . It is now easy 5 5 to conclude that |f (x, t)| ≤ C|t|3 for some constant C > 0. We are now able to prove Lemma 2. Proof of Lemma 2. Let ξα ∈ C ∞ (R3 , [0; 1]) be a cut-off function such that ξα ≡ 0 on R3 \ B(O, α/3), ξα ≡ 1 on B(O, α/3 − 1), ||∇ξα ||∞ ≤ 1, ||∆ξα ||∞ ≤ 2. We Nj introduce ψ˜jα (x1 , . . . , xNj ) = k=1 ξα (xk )ψj (x1 , . . . , xNj ), ψjα = ψ˜jα /||ψ˜jα ||L2 and ˜ Ψ(α, u, u ) = (u · ψ1α ) ∧ (ταv · u · ψ2α ). Due to the exponential decay of Theorem 2, one has % % % % N ˜ R(α, u, u ), Ψ(α, u, u ) − E N (R(α, u, u ), Ψ(α, u, u ))% ≤ Ce−aα %E % N % % % %E 1 (r, z) − E N1 (r, ψ1α )% , %E N2 (r , z ) − E N2 (r , ψ2α )% ≤ Ce−aα for some C, a > 0. Let us recall that, by definition, & N1 !N2 ! ψ ∧ ψ (x1 , . . . , xN ) = σ(C)ψ(xC )ψ (xC ) N! |C|=N1
where xC = (xi1 , . . . , xiN1 ) when C = {i1 < · · · < iN1 }, σ(C) = ±1. Applying this ˜ we obtain on the right functions with disjoint supports. We shall equality to Ψ, therefore only study the expansion of E N (R(α, u, u ), (u · ψ1α ) ⊗ (ταv · u · ψ2α )) .
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We have (using the notation x1 = (x1 , . . . , xN1 ) and x2 = (xN1 +1 , . . . , xN )) E N (R(α, u, u ), (uψ1α ) ⊗ (ταv u ψ2α )) = E N1 (ur, uψ1α ) + E N2 (α v + u r , ταv u ψ2α ) N1 N2 m m zi zj |ψ1α (x1 )|2 |ψ2α (x2 )|2 1 2 + dx dx + v + u · x2j − u · x1i | |α v + u · rj − u · ri | 3N |α i=1 j=1 R i=1 j=1 N1 N2 m m α 1 2 α 2 2 α 1 2 α 2 2 z |ψ (x )| |ψ (x )| zi |ψ1 (x )| |ψ2 (x )| 2 j 1 + − dx1 dx2 · x1 − u · r | · r − u · x1 | |α v + u |α v + u 3N i R j j i i=1 j=1
i=1 j=1
so we obtain E N (R(α, u, u ), (uψ1α ) ⊗ (ταv u ψ2α )) = E N1 (r, ψ1α ) + E N2 (r , ψ2α ) m m α zi zj ρα 1 (x)ρ2 (y) + dx dy + v + u · y − u · x| |α v + u · rj − u · ri | R6 |α i=1 j=1 − N1
m i=1
R3
m zj ρα z i ρα 1 (x) 2 (y) dy − N dx 2 |α v + u · y − u · ri | v + u · rj − u · x| 3 |α j=1 R
α where the ρα k are the electronic densities associated to ψk , and finally
E N (R(α, u, u ), (uψ1α ) ⊗ (ταv u ψ2α )) = E N1 (r, ψ1α ) + E N2 (r , ψ2α ) ρ˜α ρα 1 (x)˜ 2 (y) dx dy + |α v + u · y − u · x| 6 R
m
m α α where ρ˜α ˜α 1 (x) = ρ1 (x) − 2 (y) = ρ2 (y) − j=1 zi δri (y) are the i=1 zi δri (x) and ρ α and ρ . Now, by Lemma total densities of charge associated to the distributions ρα 1 2 9, we have R6
−
q1 q2 ρ˜α ρα (uP1α ) · v (u P2α ) · v 1 (x)˜ 2 (y) dx dy = + q2 − q1 2 |α v + u · y − u · x| α α α2
3(uP1α · v )(u P2α · v ) − (uP1α ) · (u P2α ) 3(q2 uQ1 uT + q1 u Q2 u )v · v + α3 2α3 1 |u y − ux|3 |˜ ρα ρα 1 |(x)|˜ 2 |(y) dx dy +O α3 |α v + u · y − u · x| R6 T
But we have 3C |u y − ux|3 |˜ ρα ρα 1 |(x)|˜ 2 |(y) dx dy ≤ |x|3 (|˜ ρ1 | + |˜ ρ2 |)(x)dx. |α v + u · y − u · x| α R3 R6 −aα for some It suffices to notice that |Pkα − Pk | ≤ Ce−aα and |Qα k − Qk | ≤ Ce C, a > 0 to end the proof.
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3.3.2 Proof of Theorem 6 Let us suppose that we are in the second case of Theorem 5, and that c > max{E N (M ), E N (M )}. By the proof of Theorem 4, we obtain a sequence αn → +∞ and paths γn such that N ΛN 1 ≤ c ≤ max E (γn (t)) ≤ c + t∈[0;1]
1 n
and E N (α(t), u(t), u (t), Ψ(t)) < c when α(t) ≤ αn . Let tn1 and tn2 be respectively the minimum and the maximum of {t, α(t) ≥ αn }. By the definition of c, we have 0 < tn1 < tn2 < 1. For the sake of simplicity, we introduce unj = u(tnj ), u nj = u (tnj ), and Ψnj = Ψ(tnj ). n The idea of the proof is now to connect M1n = (αn , un1 , u 1 , Ψn1 ) and M2n = n n n (αn , u2 , u 2 , Ψ2 ) by a path on which α is constant, with a maximum energy that is below c. α *
U
M1n .. × .. .. . × R × M
γn
.. M2n × .. .. . × I ψ1 ∧ ψ2
Figure 3: A schematic representation of the proof We shall use the following lemma Lemma 10. We assume that (α, u, u ) ∈ (0; +∞) × (SO3 (R))2 is fixed, and we abbreviate R = R(α, u, u ). Let Ψ1 and Ψ2 be two wavefunctions of SHa1 (R3N ). Then there exists a continuous path Ψ : [0; 1] → SHa1 (R3N ) such that Ψ(0) = Ψ1 , Ψ(1) = Ψ2 , and for all t ∈ [0; 1] E N (R, Ψ(t)) ≤ max{E N (R, Ψ1 ), E N (R, Ψ2 )}. Proof of Lemma 10. Let us denote by V the finite-dimensional eigenspace asN sociated to the first eigenvalue λN (R, Z), and let PV be the pro1 (R, Z) of H ' jection onto V . We may write Ψ1 = t0 ΨV + 1 − t20 ΨV ⊥ , where t0 ∈ [0; 1], 1) if PV (Ψ1 ) = 0, and ΨV is an arbitrary normalized function ΨV = ||PPV V(Ψ(Ψ1 )|| L2 of V if PV (Ψ1 ) = 0 (then t0 = 0). This enables to define a path connecting Ψ1 to ΨV , on which the energy decreases, by varying t ∈ [t0 ; 1].
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We may find with the same method a path from Ψ2 to some ΨV in V , with a decreasing energy. It remains now to take an arbitrary path in the sphere of V to connect ΨV and ΨV (the sphere of V being pathwise-connected since we are in C), on which E N (R, ·) is constant. Now, let N1 and N2 be such that E N1 (r, z) + E N2 (r , z ) = min {E n1 (r, z) + E n2 (r , z ), n1 + n2 = N } := ΛN 1 and (N1 − |z|)(N2 − |z |) = 0. By Lemma 6, there exist ψ1 and ψ2 , two ground states of respectively ˜ n = (un · ψ1 ) ∧ (τα v · u nk ψ2 ) H N1 (r, z) and H N2 (r , z ). We now introduce Ψ n k k for k = 1, 2. Since SO3 (R) is pathwise connected, one may find two paths Un , Un : n n [0; 1] → SO3 (R) such that Un (0) = un1 , Un (1) = un2 , Un (0) = u 1 and Un (1) = u 2 . n n n ˜ n n n n ˜ n ˜ ˜ This enables to connect M1 := (αn , u1 , u 1 , Ψ1 ) and M2 := (αn , u2 , u 2 , Ψ2 ) by ˜ n , and a path on which α ≡ αn . Applying Lemma 10, we may connect M1n to M 1 n n ˜ M2 to M2 . We finally obtain an admissible path γ˜n connecting M and M , with ˜ n . It is now a consequence of ˜ n and M a maximum that is attained “between” M 1 2 Lemma 2 that 1 (N1 − |z|)(N2 − |z |) max E N (˜ γn (t)) = E N1 (r, z) + E N2 (r , z ) + +O . αn (αn )2 t∈[0;1] Since N1 + N2 = N = |z| + |z |, we have (N1 − |z|)(N2 − |z |) < 0 and so c ≤ max E N (˜ γn (t)) < E N1 (r, z) + E N2 (r , z ) t∈[0;1]
for n large enough, which is a contradiction. 3.3.3 Proof of Theorem 7
As in the previous proofs, we may suppose that c > max{E N (M ), E N (M )}, and that there exists a sequence αn → +∞ and paths γn such that c ≤ max E N (γn (t)) ≤ c + t∈[0;1]
1 n
and E N (α(t), u(t), u (t), Ψ(t)) < c when α(t) ≤ αn . We use the same definitions as n above for tn1 , tn2 , Mkn = (unk , u k , Ψnk ) for k = 1, 2. Applying Lemma 10 if necessary, n we may also assume that Ψnk is a ground state of H N (R(αn , unk , u k ), Z). n Up to a subsequence, we may assume that unk → uk and u k → u k as n goes to +∞ and that each (Ψnk )n converges to some critical point at infinity of energy c (by Theorem 3). But by (H1), (H3) (let us assume for instance that E |z| (r, z) is not degenerated), these points can be written (uψ) ∧ (u ψ ) where ψ is a fixed ground state of H |z| (r, z), and ψ ∈ ker H |z | (r , z ) − E |z | (r , z ) , ||ψ ||L2 = 1.
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So finally, we may assume that each (Ψnk )n converges to a critical point at infinity of the form (uk ψ) ∧ (uk ψk ). Step 1: Rotating the molecules. As above, the idea is now to find a path from each Mkn to some tensor product, with a non-increasing energy. But we cannot apply the method used before since a tensor product may now have an energy which is greater than c if the molecules have a ’bad’ orientation. This is due to the fact that the first term in the expansion of the interaction energy will be the dipole/dipole term. Therefore, our first step will be to change the orientation of the molecules, so that the dipole/dipole interaction energy of a tensor product becomes negative. Let ξ1n ∈ C ∞ (R3 , [0; 1]) be a cut-off function such that ξ1n ≡ 0 on R3 \ B(O, αn /3), ξ1n ≡ 1 on B(O, αn /3 − 1), ||∇ξ1n ||∞ ≤ 1, ||∆ξ1n ||∞ ≤ 2, and let us denote by ξ2n = ταnv ξ1n the translation of ξ1n , and by ξ3n := 1 − ξ1n − ξ2n . We now write as in the proof of Theorem 2 Ψnk =
N l=1
(ξ1n + ξ2n + ξ3n )(xl )Ψnk =
ξkn1 (x1 ) · · · ξknN (xN )Ψnk :=
1≤kl ≤3
(Ψnk )k1 ,...,kN .
1≤kl ≤3
Since k ∈ {1, 2}
and n will be fixed during this step, we shall forget the subscripts and write Ψ = 1≤kl ≤3 Ψk1 ,...,kN . We now introduce for all v1 , v2 ∈ SO3 (R), Ψ(v1 ,v2 ) (x1 , . . . , xN ) :=
Ψk1 ,...,kN (wk1 x1 , . . . , wkN xN ),
1≤kl ≤3
M(v1 ,v2 ) := (αn , v1 u, v2 u , Ψ(v1 ,v2 ) ) where w1 = v1T , w3 = v3 = I and w2 (x) = v2T (x − αn v ) + αn v . We have also forgotten the indices k and n for the sake of simplicity. Obviously, M(I,I) = Mkn . Lemma 11. There exists a path V : [0, 1] → (SO3 (R))2 such that V (0) = (I, I), V (1) is a critical point of (v1 , v2 ) → E N (M(v1 ,v2 ) ), and E N (MV (t) ) ≤ E N (MI,I ) for all t ∈ [0; 1]. Proof of Lemma 11. Since Ψnk is locally Lipschitz by Theorem 1, one easily shows that F : (v1 , v2 ) → E N (M(v1 ,v2 ) ) is C 2 on (SO3 (R))2 . We then simply consider the associated gradient flow to conclude. ˜ connected Applying this lemma, we may obtain some X = (αn , vu, v u , Ψ), n to Mk by a path with a non-increasing energy, and which is a critical point with respect to the rotation of the molecules as defined before. Indeed it is a critical point of the function ˜ (eH1 ,eH2 ) (H1 , H2 ) → E N αn , eH1 v, eH2 v , (Ψ) defined on the space A3 (R) × A3 (R), A3 (R) being the space of 3 × 3 antisymmetric real matrices.
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Step 2: Expression and expansion of the energy and its derivative. Now, we have ˜ 2 H1 H2 = |Ψ| (e ,e )
1≤kl ≤2
˜ k1H,...,kHN |2 + φ |Ψ (e 1 ,e 2 )
with ||φ||H 1 ≤ Ce−aαn (by Theorem 2), so we obtain ˜ (eH1 ,eH2 ) = E N αn , eH1 v, eH2 v , Ψ ˜ k1H,...,kHN + O e−aαn . E N αn , eH1 v, eH2 v , Ψ (e 1 ,e 2 ) 1≤kl ≤2
Let us fix 1 ≤ k1 , . . . , kN ≤ 2 and denote Cl = {i, ki = l}, x1 = (xi )i∈C1 , x = (xi )i∈C2 , Nl = |Cl |. We then have 2
˜ k1H,...,kHN = E N αn , eH1 v, eH2 v , Ψ (e 1 ,e 2 ) H N1 (r, z)C1 Ψk1 ,...,kN , Ψk1 ,...,kN + H N2 (r , z )C2 Ψk1 ,...,kN , Ψk1 ,...,kN
1 ˜ k1 ,...,kN |2 + |(τ−αnv )x2 Ψ H 2 |αn v + e xj − eH1 xi | R3N i∈C1 , j∈C2
m
−
i∈C1 j=1
m zj zi − |αn v + eH2 rj − eH1 xi | i=1 |αn v + eH2 xj − eH1 ri | j∈C2 m m zi zj dx1 dx2 . + H2 r − eH1 r | |α v + e n j i i=1 j=1
Differentiating this expression with regard to H = (H1 , H2 ), we find ˜ k1H,...,kHN (H1 , H2 ) dH=0 E N αn , eH1 v, eH2 v , Ψ 1 2 (e ,e ) eαnv+xj −xi · (H2 xj − H1 xi ) ˜ k1 ,...,kN |2 =− |(τ−αnv )x2 Ψ |αn v + xj − xi |2 3N R i∈C1 , j∈C2
−
m m zj eαnv +rj −xi · (H2 rj − H1 xi ) zi eαnv+xj −ri · (H2 xj − H1 ri ) − 2 |αn v + rj − xi | |αn v + xj − ri |2 i=1 j∈C2 i∈C1 j=1 m m zi zj eαnv+rj −ri · (H2 rj − H1 ri ) dx1 dx2 . + 2 |α v + r − r | n j i i=1 j=1
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Now, we obtain, using Lemma 2, ˜ k1H,...,kHN ∇H1 =0 E N αn , eH1 v, eH2 v , Ψ (e 1 ,e 2 ) ( ) m (N2 − |z |) k1 ,...,kN 2 ˜ =− |(τ−αn v )x2 Ψ | xi − zj rj v T dx1 dx2 (αn )2 R3N i=1 i∈C1 ) ( m (N2 − |z |) k1 ,...,kN 2 T T T ˜ − |(τ−αnv )x2 Ψ | (I −3 v v ) xi xi − zi ri ri dx1 dx2 (αn )3 R3N i=1 i∈C1 1 ˜ k1 ,...,kN |2 (I − 3 v v T ) − |(τ−αnv )x2 Ψ (αn )3 R3N T ( ( ) ) m m ˜ k1 ,...,kN |2 | Ψ 3N R · xi − zi ri xj − zj rj dx1 dx2 + O , (αn )4 i=1 j=1 i∈C1
j∈C2
and a similar expansion for the gradient with regard to H2 . We may now expand the energy with a similar method and obtain ˜ k1 ,...,kN E N αn , v, v , Ψ = H N1 (r, z)C1 Ψk1 ,...,kN , Ψk1 ,...,kN
(N1 − Z1 )(N2 − Z2 ) ˜ k1 ,...,kN |2 |Ψ αn 3N R ( ) m k1 ,...,kN 2 ˜ |(τ−αnv )x2 Ψ | xi − zi ri dx1 dx2
+ H N2 (r , z )C2 Ψk1 ,...,kN , Ψk1 ,...,kN + (N2 − |z |) + ev · (αn )2 −
(N1 − |z|) ev · (αn )2 −
R3N
R3N
(N2 − |z |) 2(αn )3
˜ k1 ,...,kN |2 |(τ−αnv )x2 Ψ
i=1
m
xj −
˜ k1 ,...,kN |2 |(τ−αn v )x2 Ψ
m i=1
R3N
m j=1
(I − 3 v v T )xi · xi
dx1 dx2
˜ k1 ,...,kN |2 |(τ−αn v )x2 Ψ
zj rj dx1 dx2
i∈C1
) zi (I − 3 v v T )ri · ri
j=1
j∈C2
R3N
−
i∈C1
(
− (N1 − |z|) − 2(αn )3
(I − 3 v v T )xj · xj
j∈C2
zi (I − 3 v v T )rj · rj dx1 dx2
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M. Lewin
1 + (αn )3
(
R3N
˜ k1 ,...,kN |2 (I − 3 v v T ) |(τ−αnv )x2 Ψ ·
xj −
xi −
(
zj rj dx1 dx2
+O
R3N
j=1
j∈C2
m
) zi ri
i=1
i∈C1
m
Ann. Henri Poincar´e
˜ k1 ,...,kN |2 |Ψ (αn )4
) .
With a similar computation, it can be proved that ˜ k1 ,...,kN , Ψ ˜ k1 ,...,kN |∇H=0 H N (Rn , Z)Ψ
| = O e−aαn when kl = 3 for some l. As a conclusion, we obtain, by summing these expressions, E
N
αn , e
H1
v, e
1 Bn Cn Dn + En n ˜ v , Ψk = An − + + +O αn (αn )2 (αn )3 (αn )4
H2
where An =
H N1 (r, z)C1 + H N2 (r , z )C2 ΨC1 ,C2 , ΨC1 ,C2
N1 +N2 =N |C1 |=N1
Bn = −
˜ C1 ,C2 ||2 2 ≥ 0 (N1 − |z|)(N2 − |z |)||Ψ L
N1 +N2 =N |C1 |=N1
and with similar definitions for Cn , Dn , En (Dn contains the two terms involving (N1 − |z|) and (N2 − |z |), and En is the third term). Step 3: A boot-strap argument. We have ˜ C1 ,C2 ||2 2 An ≥ E N1 (r, z) + E N2 (r , z ) ||Ψ L |C1 |=N1
N1 +N2 =N
= c
˜ C1 ,C2 ||2 2 ||Ψ L
|C1 |=|z|
+
N1 ˜ C1 ,C2 ||2 2 E (r, z) + E N2 (r , z ) ||Ψ L |C1 |=N1
N1 +N2 =N N1 =|z|
N1 ˜ C1 ,C2 ||2 2 = c + O e−aαn + ||Ψ E (r, z) + E N2 (r , z ) − c L N1 +N2 =N N1 =|z|
≥ c + O e−aαn +
|C1 |=N1
N1 +N2 =N |C1 |=N1 N1 =|z|
≥ c + Bn + O e−aαn
˜ C1 ,C2 ||2 2 ||Ψ L
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where = min{E N1 (r, z) + E N2 (r , z ), N1 + N2 = N, N1 = |z|} − c > 0 (H1). Now, we have by the definition of Mkn and Xkn , ˜ nk < c E N αn , eH1 v, eH2 v , Ψ so we obtain, for n large enough, 1 1 Cn Dn + En 0 ≤ Bn ≤ − − + O Bn ≤ − . 2 αn (αn )2 (αn )3 (αn )4
(14)
˜ n converges to some critical point at infinity, one easily Using the fact that Ψ k sees thatAn , Bn , Cn and Dn are bounded as n goes to +∞, and finally Bn = O (α1n )2 . But we may write for instance, for 1 ≤ kl ≤ 2 with |C1 | := N1 = |z|, by (14) % % ( ) m % % % % ˜ k1 ,...,kN |2 |(τ−αnv )x2 Ψ xi − zi ri dx1 dx2 % %(N2 − |z |)ev · % % R3N i=1 i∈C1 ˜ k1 ,...,kN |2 dx1 dx2 ≤ |N2 − |z ||(N1 + |z|)αn |(τ−αnv )x2 Ψ R3N ˜ k1 ,...,kN ||2 2 = αn |N2 − |z ||O ||Ψ L and so one can easily prove (15) Dn = O (αn )2 Bn . Now Cn = O α1n and by (14), Bn = O (α1n )3 . Using one more time (15), we obtain Cn = O (α1n )2 and Dn = O α1n . As a consequence, Cn = O (αn Bn ) ,
1 ˜ nk = An + En + O E N αn , eH1 v, eH2 v , Ψ . (αn )3 (αn )4 Using the same estimates, we also obtain an expansion of the form 1 ˜ (H ,H ) = En + O =0 ∇H1 =0 E N αn , eH1 v, eH2 v , Ψ 1 2 (αn )3 (αn )4 where En corresponds to the derivative of the dipole/dipole term. But now it is known that * + ˜ (H ,H ) = 0 PA3 (R)⊥ ∇H1 =0 E N αn , eH1 v, eH2 v , Ψ 1 2 where PA3 (R)⊥ is the orthogonal projection onto A3 (R)⊥ . We thus have PA3 (R)⊥ En = O α1n and so PA3 (R)⊥ En → 0 as n goes to +∞. A same result holds for
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M. Lewin
Ann. Henri Poincar´e
the derivative with respect to H2 . Recall now that in fact v = vkn , v = v nk , u = unk n and u = u k . As above we may assume by extracting a sequence if necessary that n n vk and v k converge to some vk and vk as n → +∞. ˜ n, We may now pass to the limit and obtain, by (14) and the convergence of Ψ k (vk uk P ) ,· (vk uk Pk ) − 3(vk uk P · v )(vk uk-Pk · v ) ≤ 0 v v T )(vk uk P )(vk uk Pk )T - = 0 P ⊥ (I − 3 (16) A3 (R) , PA3 (R)⊥ (I − 3 v v T )(vk uk Pk )(vk uk P )T = 0 for k ∈ {1, 2}, and where P := Pψ and Pk := Pψk . Step 4: Study of the critical points of the dipole/dipole interaction. We have the following result, which is proved in Appendix 2. Lemma 12. Let be P, P ∈ R3 \ {0} and e ∈ S 2 . The critical points of the function e T FP,P : (u, u ) → (uP ) · (u P ) − 3(uP · e)(u P · e) = ((I − 3ee )(uP )) · (u P )
defined on SO3 (R)2 are given by 1. u0 P = |P |e, u0 P = |P |e with = ±1. Then (u0 , u0 ) is a minimum of F, e and FP,P (u0 , u0 ) = −2|P | |P |. 2. u4 P = |P |e, u4 P = −|P |e with = ±1. Then (u4 , u4 ) is a maximum of F e with a Morse index equal to 4, and FP,P (u4 , u4 ) = 2|P | |P |. 2 ⊥ 3. u1 P = |P |v, u1 P = −|P |v for some v ∈ S ∩ (e) . Then (u1 , u1 ) has a e Morse index equal to 1, and FP,P (u1 , u1 ) = −|P | |P |. 4. u2 P = |P |v, u2 P = |P |v for some v ∈ S 2 ∩ (e)⊥ . Then (u2 , u2 ) has a Morse e index equal to 2, and FP,P (u1 , u1 ) = |P | |P |. Index
Energy
4
2|P | |P |
2
|P | |P |
6
1
−|P | |P |
6
0
−2|P | |P |
-
-
6
?
-
-
Figure 4: The critical points of the dipole/dipole interaction v By (16), (vk uk , vk uk ) is a critical point of FP,P , with a non-positive energy. k
v Since P = 0 and Pk = 0 (H1), we conclude by Lemma 12 that FP,P (vk uk , vk uk ) ≤ k −|P ||Pk | < 0.
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As a consequence, applying Lemma 2, we obtain n n n n E N αn , vkn unk , v k u k , (vkn unk · ψ) ∧ (ταnv · v k u k · ψk ) = c+
v n n n n FP,P (vk uk , v k u k ) k
(αn )3
+O
1 (αn )4
for n large enough, since v n n v lim FP,P (vk uk , v k u k ) = FP,P (vk uk , v k u k ) < 0. n
n→+∞
n
k
k
Let us now denote by ˜ n := αn , v n un , v n u n , (v n un · ψ) ∧ (τα v · v n u n · ψ ) . M n k k k k k k k k k k ˜ n k ∈ {1, 2}, For n large enough, we may now construct a path connecting Xkn to M k with an energy below c, by Lemma 10. Step 5: Connecting the two tensor products. To end the proof, it remains to connect M1n and M2n by a path with a maximum energy below c. We fix here n and forget this subscript. Let us now introduce the function F defined by v F (ν, ν , φ) = FP,P (ν, ν ) φ
for (ν, ν , φ) ∈ W := SO3 (R)2 × V where V is the unit sphere of the finite di|z | |z | mensional space ker H (r , z ) − E (r , z ) . We now denote by Yk := (uk vk , uk vk , ψk ) for k ∈ {1, 2}. Notice that we have shown max(F (Y1 ), F (Y2 )) ≤ − min{|P ||Pφ |} < 0. φ∈V
Now, let be c˜ := inf max F (Y (t)) Y ∈Γ t∈[0,1]
where
(17)
Γ = {Y ∈ C 0 ([0, 1], W ), Y (0) = Y1 , Y (1) = Y2 }.
We have the following Lemma 13. We have c˜ ≤ − minφ∈V {|P ||Pφ |} := C < 0. Proof of Lemma 13. Suppose that c˜ > max(F (Y1 ), F (Y2 )). By the methods of [14], we may find a sequence yn ∈ W such that F (yn ) → c˜, ∇F (yn ) → 0 and d2(ν,ν ) F ≥ −en on a space of codimension at most 1, with en → 0 as n → +∞. W being compact, up to a subsequence, yn = (νn , νn , φn ) converges to a critical point y = (ν, ν , φ) of energy c˜, and with a Morse index with respect to (ν, ν ) variations at most 1. By Lemma 12, we necessary have F (y) ≤ −|P ||Pφ |.
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Let Yn ∈ Γ be a path connecting Y1 and Y2 such that maxt∈[0;1] F (Y (t)) ≤ We then have c˜ + 1 αn .
E N (αn , νn (t), νn (t), (νn (t) · ψ) ∧ (ταn v · νn (t) · ψn (t))) 1 1 F (Yn (t)) C =c+ +O +O ≤c+
t∈[0;1]
which is a contradiction and ends the proof of Theorem 7.
Appendix 1: Non-isotropic exponential decay of the electronic density We want to prove here Lemma 3, that we recall for the reader’s convenience. Lemma 14. Let ΨR be an eigenfunction associated to the eigenvalue λN d (R, Z) and ρR be the electronic density. We introduce R = ΣN (R, Z) − λN d (R, Z). Then 1. ρR satisfies the inequation 1 (18) − ∆ρR + VR ρR + R ρR ≤ 0. 2 2. With R1 () := max R0 + 1, R0 + 2N p and C() := ∪pj=1 {x, |x − Xj | = R1 ()}, and if r > 2R1 (R ), then we have ρR (x)
≤
||ρR ||L∞ (C(R ))
p
e−
√
R /p(|Xj −x|−R1 (R ))
j=1
≤ ≤
√ p||ρR ||L∞ (C(R )) e− R /p(δ(x)−R1 (R )) , √ M e− R /p(δ(x)−R1 (R ))
(19)
on U(R1 (R )), where δ(x) = min{|x − Xj |, j = 1, . . . , d}, and M = M (p, N, R1 (R )). Proof. The fact that ρR is a solution to (10) is essentially proved in [16] (the proof is written for atoms with no nuclei/nuclei interaction). The proof of (19) follows ideas of [15]. We have VR (x) ≥ −
p j=1
N |x − Xj | − R0
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for all x ∈ U(R1 (R )), so that ρR satisfies p R N 1 − − ∆ρR + ρR ≤ 0. 2 p |x − Xj | − R0 j=1 on U. Let f be the radial positive solution of 1 N R − ∆f + − f =0 2 p |x| − R0 on R3 \B(0, R1 (R )), such that f (x) = 1 if |x| = R1 (R ) (f is a Whittaker function, see [15]). Then fj (x) = f (x − Xj ) is a positive solution of R N 1 − − ∆fj + fj = 0 2 p |x − Xj | − R0
p on R3 \ B(Xj , R1 (R )) and F = j=1 fj is a solution of 1 − ∆F + 2 j=1 p
R N − p |x − Xj | − R0
F = 0.
such that F (x) ≥ 1 if x ∈ C(R ). Since ρR ∈ C ∞ (U(R1 (R ))) and lim|x|→+∞ ρR (x) = 0, the maximum principle implies ρR ≤ ||ρR ||L∞ (C(R )) F (x). But, on R3 \ B(Xj , R1 (R )), we also have R N R − > 0, ≥ p |x − Xj | − R0 2p so that (maximum principle) fj (x) ≤
√ R1 (R ) −√R /p(|x−Xj |−R1 (R )) ≤ e− R /p(|x−Xj |−R1 (R )) e |Xi − x|
on R3 \ B(Xj , R1 (R )) which implies F (x)
≤
p j=1
≤ pe− ≤ pe−
e− √
√
R /p(|x−Xj |−R1 (R ))
1 R /p( p
√
p
j=1
|x−Xj |−R1 (R ))
R /p(δ(x)−R1 (R ))
.
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To end the proof, we now remark that since ρR is real analytic on U(R0 + 1/2), there exists a constant M depending only on R1 (R ) such that ||ρR ||L∞ (C(R )) ≤ M ||ρR ||L1 (U (R0 +1/2))) ≤ M ||ρR ||L1 (R3 ) = M · N and so
ρR ≤ M N pe−
√
R /p(δ(x)−R1 (R ))
.
Appendix 2: Study of the critical points of the dipole/dipole interaction We prove here Lemma 12 that we recall. Lemma 15. Let be P, P ∈ R3 \ {0} and e ∈ S 2 . The critical points of the function e T FP,P : (u, u ) → (uP ) · (u P ) − 3(uP · e)(u P · e) = ((I − 3ee )(uP )) · (u P )
defined on SO3 (R)2 are given by 1. u0 P = |P |e, u0 P = |P |e with = ±1. Then (u0 , u0 ) is a minimum of F, e and FP,P (u0 , u0 ) = −2|P | |P |. 2. u4 P = |P |e, u4 P = −|P |e with = ±1. Then (u4 , u4 ) is a maximum of F e with a Morse index equal to 4, and FP,P (u4 , u4 ) = 2|P | |P |. 2 ⊥ 3. u1 P = |P |v, u1 P = −|P |v for some v ∈ S ∩ (e) . Then (u1 , u1 ) has a e Morse index equal to 1, and FP,P (u1 , u1 ) = −|P | |P |. 4. u2 P = |P |v, u2 P = |P |v for some v ∈ S 2 ∩ (e)⊥ . Then (u2 , u2 ) has a Morse e index equal to 2, and FP,P (u1 , u1 ) = |P | |P |. Proof. It is simpler to study the function (x, y) → |P ||P |x · y − 3(x · e)(y · e) defined on S 2 × S 2 , but we shall not use here this expression, to keep the point of view imposed by the proof of Theorem 7. e If (u, u ) is a critical point of FP,P , then (0, 0) is a critical point of f : e H H (H, H ) → FuP,u P (e , e ), defined on A3 (R) × A3 (R). We may assume u = u = I and |P | = |P | = 1 to simplify notations, and we denote by M := I − 3eeT . Then we have M P P T ∈ A3 (R)⊥ ⇐⇒ M P (P )T ∈ A3 (R)⊥
M P P T = P (P )T M M P (P )T = P P T M
(20)
Multiplying by M , we find for instance M 2 P P T = P P T M 2 and so M 2 P = (P T M 2 P )P , showing that P and P are eigenvectors of M 2 . It is then easy to see that they are eigenvectors of M . Using (20), we obtain P = ±P . The critical points are thus those given in the lemma. The second derivative is given by d2 fP,P (H, H ) = 2(M HP ) · (H P ) + (M H 2 P ) · P + (M (H )2 P ) · P.
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Suppose for instance that P = e. Let (e2 , e3 ) be an orthogonal basis of e⊥ . Then, in the basis (e1 = e, e2 , e3 ), M = diag(−2, 1, 1). We introduce Hij := ei eTj − ej eTi , so that (Hij )1≤i<j≤3 is an orthogonal basis of A3 (R). The matrix of d2P,P f is this basis is then 2 0 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 1 2 ∼ diag(1, 1, 3, 3, 0, 0), dP,P f = 1 0 0 2 0 0 0 0 0 0 0 0 0 0 1 0 0 2 and obviously d2P,−P f ∼ diag(−1, −1, −3, −3, 0, 0). If P ⊥ e, we use the same basis (Hij )1≤i<j≤3 but with e2 = P and find −1 0 0 −2 0 0 0 −1 0 0 1 0 0 0 0 0 0 0 d2P,P f = −2 0 0 −1 0 0 ∼ diag(1, −2, −3, 0, 0, 0), 0 −2 0 0 −1 0 0 0 0 0 0 0 and obviously d2P,−P f ∼ diag(−1, 2, 3, 0, 0, 0).
´ Acknowledgments. I would like to thank Eric S´er´e for his constant attention and helpful remarks.
References [1] S. Agmon, Lectures on exponential decay of solutions of second-order elliptic equations. Princeton University Press (1982). [2] A. Bahri, Critical Points at Infinity in some Variational Problems, Pitman Research Notes in Mathematics Series, 182. Longman Scientific and Technical Ed. (1989). [3] E. Canc`es et al., Computational quantum chemistry: a primer, Handbook of numerical analysis Vol. X, 3–270 (2003). [4] I. Catto, P.-L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. Part I. Comm. Part. Diff. Eq. 17 (7-8), 1051–1110 (1992). [5] I. Catto, P.-L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. Part II, Comm. Part. Diff. Eq. 18 (1-2), 305–354 (1993).
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[6] I. Catto, P.-L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. Part III, Comm. Part. Diff. Eq. 18 (3-4), 381–429 (1993). [7] I. Catto, P.-L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. Part IV, Comm. Part. Diff. Eq. 18 (7-8), 1149–1159 (1993). [8] G. Fang, N. Ghoussoub, Morse-type information on Palais-Smale sequences obtained by min-max principles, Manuscripta Math. 75, 81–95 (1992). [9] J.B. Foresman, A. Frisch, Exploring Chemistry with Electronic Structure Methods, Gaussian Inc, 1996. Chapter 3 and references therein. [10] G. Friesecke, The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions. Mathematical Institute, University of Oxford, U.K. (1999), preprint to appear in Arch. Rat. Mech. Analysis. [11] S. Fournais, M. and T. Hoffmann-Ostenhof, T. Østergaard Sørensen, The Electron Density is Smooth Away from the Nuclei, Commun. Math. Phys. 228, 401–415 (2002). [12] S. Fournais, M. and T. Hoffmann-Ostenhof, T. Østergaard Sørensen, On the regularity of the density of electronic wavefunctions, Mathematical results in quantum mechanics (Taxco, 2001), 143–148, Contemp. Math. 307 (2002). [13] S. Fournais, M. and T. Hoffmann-Ostenhof, T. Østergaard Sørensen, Analyticity of the Density of Electronic Wavefunctions. Preprint (2002). [14] N. Ghoussoub, Duality and perturbation methods in critical point theory, Cambridge University Press (1993). [15] M. and T. Hoffmann-Ostenhof, “Schr¨ odinger inequalities” and asymptotic behavior of the electron density of atoms and molecules, Phys. Rev. A 16 (5), 1782–1785 (1977). [16] M. and T. Hoffmann-Ostenhof, T. Østergaard Sørensen, Electron Wavefunctions and densities for atoms, Ann. Henri Poincar´e 2, 77–100 (2001). [17] M. and T. Hoffmann-Ostenhof, H. Stremnitzer, Local properties of Coulombic wave functions, Commun. Math. Phys. 163, 185–215 (1994). [18] W. Hunziker, On the Spectra of Schr¨ odinger Multiparticle Hamiltonians, Helv. Phys. Acta 39, 451–462 (1966). [19] W. Hunziker, I.M. Sigal, The Quantum N-Body Problem, J. Math. Phys 41(6), 3348–3509 (2000). [20] T. Kato, Fundamental Properties of Hamiltonian Operators of Schr¨ odinger Type, Trans. Am. Math. Soc. 70, 195–221 (1951).
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[21] T. Kato, On the eigenfunctions of many particle systems in quantum mechanics, Commun. Pure and Appl. Math. 10, 151–171 (1957). [22] E.H. Lieb, The Stability of Matter: from Atoms to Stars, Bull. Amer. Math. Soc. 22, 1–49 (1990). [23] E.H. Lieb, Bound on the Maximum Negative Ionization of Atoms and Molecules, Phys. Rev. A. 29 (6), 3018–3028 (1984). [24] E.H. Lieb, B. Simon, The Thomas-Fermi theory of atoms, molecules and solids, Adv. Math. 23, 22–116 (1977). [25] E.H. Lieb, W.E. Thirring, Universal Nature of Van Der Waals Forces for Coulomb Systems, Phys. Rev. A 34, 40–46 (1986). [26] P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part I, Ann. Inst. Henri Poincar´e 1 (2), 109–149 (1984). [27] M. Reed, B. Simon, Methods of Modern Mathematical Physics. I – Functionnal Analysis, Academic Press (1972). [28] H.B. Schlegel, In An Inition Methods in Quantum Chemistry, Lawley Ed, 1987. Vol. 1, p. 249–286 and references cited therein. [29] I.M. Sigal, Geometric methods in the quantum many-body problem. Non existence of very negative ions, Commun. Math. Phys. 85, 309–324 (1982). [30] B. Simon, Geometric methods in multiparticle quantum systems, Commun. Math. Phys. 55, 259–274 (1977). [31] E. Teller, Rev. Mod. Phys. 34, 267 (1962). [32] C. Van Winter, Theory of Finite Systems of Particles. I. The Green function, Mat.-Fys. Skr. Danske Vid. Selsk. 2 (8) (1964). [33] G.M. Zhislin, Discussion of the spectrum of Schr¨ odinger operators for systems of many particles, (In Russian), Trudy Moskovskogo matematiceskogo obscestva 9, 81–120 (1960). Mathieu Lewin CEREMADE, CNRS UMR 7534 Universit´e Paris IX Dauphine Place du Mar´echal de Lattre de Tassigny F-75775 Paris Cedex 16 France email:
[email protected] Communicated by Rafael D. Benguria Submitted 07/07/03, accepted 27/01/04
Ann. Henri Poincar´e 5 (2004) 523 – 577 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/030523-55 DOI 10.1007/s00023-004-0177-5
Annales Henri Poincar´ e
Scattering Theory of Infrared Divergent Pauli-Fierz Hamiltonians Jan Derezi´ nski and Christian G´erard
Abstract. We consider in this paper the scattering theory of infrared divergent massless Pauli-Fierz Hamiltonians. We show that the CCR representations obtained from the asymptotic field contain so-called coherent sectors describing an infinite number of asymptotically free bosons. We formulate some conjectures leading to mathematically well defined notion of inclusive and non-inclusive scattering crosssections for Pauli-Fierz Hamiltonians. Finally we give a general description of the scattering theory of QFT models in the presence of coherent sectors for the asymptotic CCR representations.
1 Introduction The main motivation for this paper is our desire to gain some rigorous understanding of the infrared problem in quantum field theory, in particular in QED. This is not an easy task, since we even do not know how to construct rigorously realistic models of QED. Some authors tried to analyze the infrared problem in the axiomatic framework of local quantum theory. Considerable progress in this direction has been achieved [FMS, Bu]. We will not, however, discuss these results, often deep and interesting. The infrared problem is not restricted, however, to local quantum theory. Some of its aspects persist even in various simplified models derived from QED, which have ultraviolet cutoffs or treat a part of the system in a classical way. In our paper we consider a class of such models. These models are quite far from the “true QED” or from the axioms of local quantum theory. Yet, we will see that their infrared problem is quite nontrivial. Besides, unlike QED, these models can be rigorously defined. Infrared problem, both in “true QED” and in various simplified models appears mostly if we try to compute scattering amplitudes. Thus it is primarily a symptom of a pathological scattering theory. QED is a theory of charged particles interacting with photons. Correspondingly, it has two distinct kinds of the infrared problem: the first kind involves the dynamics of charged particles and the second involves photons. In the following two subsections we would like to make some comments about these two kinds of the infrared problem of QED, focusing mostly on various simplified models.
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Infrared problem for charged particles
Let us shortly discuss the first kind. Scattering of charged particles is made difficult by the long-range nature of their interaction. To partly understand this phenomenon, let us fix the Coulomb gauge in QED, drop photons and use the non-relativistic approximation. Then QED becomes a theory of charged particles whose dynamics is described by the many body Schr¨ odinger Hamiltonian with Coulomb interactions. As is well known, the usual scattering theory breaks down for such systems. Naive rules for computing scattering amplitudes in terms of Feynman diagrams presuppose that we want to construct the usual wave and scattering operators, which do not exist because Coulomb potentials are long-range. Therefore, we get meaningless divergent expressions. It is well understood how to cure this problem, at least in the context of many body Schr¨ odinger Hamiltonians. Two approaches are possible: (1) One can compute only scattering cross-sections, staying away from ill-defined wave and scattering operators. The standard way is to approximate Coulomb interaction by the Yukawa interaction of mass m > 0, which is short range, compute the cross-sections and take the m → 0 limit. This is the approach found in most textbooks on quantum mechanics. (2) One can introduce modified wave and scattering operators. From the conceptual point of view it is a more satisfactory approach – it gives deeper insight into the problem. The mathematics of this approach is very interesting and nowadays well understood (see, e.g., [DG1]). On the other hand, it is more complicated computationally than the first approach and uses non-canonical objects: the modified wave and scattering operators depend on the choice of the so-called modifier. Apart from the remarks above, in our paper we will not touch this aspect of the infrared problem.
1.2
Infrared problem for photons
Let us now discuss the photonic aspect of the infrared problem. In our discussion we will consider both the perturbative QED and various simplified models such as Pauli-Fierz Hamiltonians. If one tries to compute scattering cross-sections involving states with a finite number of asymptotic photons, one often obtains infrared divergent integrals. After an appropriate renormalization, one obtains scattering cross-sections equal to zero. This is usually interpreted by saying that “the vacuum escapes from the physical Hilbert space” and that “all states contain an infinite number of soft photons”. In the literature one can find 4 approaches to cure this problem in QED-like theories that make possible computing physically meaningful cross-sections. (1) One can restrict oneself to the so-called inclusive cross-sections, which take into account all possible “soft photon states” below a certain energy > 0.
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The philosophy behind this prescription is: do not attempt to compute or even ask about the existence of the wave and scattering operators – try to compute scattering cross-sections relevant for realistic experiments. This point of view is most common in standard textbooks [JR] and can be traced back to [BN] (see also [YFS]). (2) Naive rules for computing scattering amplitudes in terms of Feynman diagrams presuppose that the asymptotic fields form a Fock CCR representation. This assumption can be wrong because of the infrared problem. To eliminate this difficulty, one can treat seriously non-Fock representations. One class of non-Fock representations is especially easy to handle – the so-called coherent representations. One can define wave and scattering operators between coherent sectors, and also asymptotic Hamiltonians. Scattering theory is somewhat less intuitive than in the case of Fock representations, but it is naturally defined and not much more difficult. This approach can be traced back to Kibble [Ki]. We regard it as the most satisfactory approach to the infrared problem. It provides an appropriate framework for the infrared problem in the case of exactly solvable van Hove Hamiltonians [De]. In our paper we will argue that this approach works also well in the case of Pauli-Fierz Hamiltonians, although one cannot rule out the appearance of other types of CCR representations besides the coherent ones. In order for this approach to be meaningful, one needs to use a certain version of the so-called LSZ approach, that means, one needs to construct the asymptotic fields. This requires some, usually mild, assumptions on the interaction of the “short range” type. This is the main weakness of this approach. (3) One can keep the formal expression for the Hamiltonian and change the CCR representation. This amounts to a change of the underlying Hilbert space and of the Hamiltonian. The new Hamiltonian is sometimes called the renormalized Hamiltonian. The main requirement for the renormalized Hamiltonian is to have a ground state, which implies that the representation of its asymptotic fields contains a Fock sector. Shifting the asymptotic CCR representations can always be done in the case of exactly solvable van Hove Hamiltonians. In the case of Pauli-Fierz Hamiltonians it seems possible only under some special assumptions on the interaction, such as Assumption 2.D (the possibility to split the interaction into a scalar part and a regular part). One can criticize this approach in two separate points. First of all, as we mentioned above, we need special assumptions to make this approach work. One can argue that Approach (2) is more general and does not need these assumptions.
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Secondly, in general there is a large degree of arbitrariness in how to shift the Hamiltonian. Therefore, the renormalized Hamiltonian is not a canonical object. One can try to give a justification of this approach by using C ∗ -algebras – Approach (4). The passage from the initial to a renormalized Hamiltonian would correspond to a change of a representation of the given C ∗ -algebraic system. If applicable, this approach is very useful. In recent literature it was applied in [Ar] and [HHS]. It will be also an important tool in our paper. (4) Sometimes one can describe a quantum system in terms of a dynamics on a C ∗ -algebra [FNV, BR]. This algebra may have many inequivalent representations. In some of them the dynamics may be generated by a Hamiltonian with a ground state, so that the infrared problem disappears. This approach can be used to justify Approach (3). One can say that the initial Hilbert space is just one of many representations of the C ∗ -algebra and one needs to go to a different representation, where the representation of asymptotic fields has a Fock sector. It seems that this approach is inadequate for Pauli-Fierz systems unless one makes some very special assumptions on the interaction. In general it is difficult (probably impossible) to find a physically motivated C ∗ -algebra which is preserved by the dynamics. In our paper we will discuss in detail Approach (2) to the infrared problem in the context of Pauli-Fierz Hamiltonians. Approach (3) will play an important role, but it will be treated as a tool in the study of Approach (2). We will also discuss Approach (1).
1.3
Scattering theory for Pauli-Fierz Hamiltonians
There exist a number of simplified models that can be used to test some of the photonic aspects of QED. Probably the simplest are quadratic bosonic Hamiltonians with a linear perturbation. In [Sch] such Hamiltonians are called van Hove Hamiltonians, and we will use this name. They are exactly solvable and one can study their scattering theory in full detail [De]. A typical van Hove Hamiltonian can be written in the form: z(k) z(k) ∗ ω(k) a(k) + dk, (1.1) a (k) + ω(k) ω(k) where z(k) is some given function and ω(k) is the dispersion relation, e.g., ω(k) = |k|. Note that if we consider QED with prescribed classical charges, then we obtain a van Hove Hamiltonian In our paper we consider the so-called abstract Pauli-Fierz Hamiltonians. They can also be used to understand interaction of photons with matter, but are
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more difficult and rich than the van Hove Hamiltonians. They are not exactly solvable and their mathematical understanding is far from complete. They are a caricature of QED with charged particles confined in an infinite well. Consider the Hilbert space K ⊗ Γs (L2 (Rd )), where the Hilbert space K describes the confined charged particles and Γs (L2 (Rd )) is a bosonic Fock space. Following the terminology of [DG2, DJ, Ge1], an operator of the form H
:= K ⊗ 1l + 1l ⊗ ω(k)a∗ (k)a(k)dk + v(k) ⊗ a∗ (k)dk + v ∗ (k) ⊗ a(k)dk
will be called a Pauli-Fierz Hamiltonian. For simplicity, in our paper charged particles are described by an abstract Hamiltonian K and their confinement is expressed by the condition that K has a compact resolvent. Let us sketch the main ideas of scattering theory for Pauli-Fierz Hamiltonians. We follow the formalism of [DG2, DG3], which can be traced back to much earlier work, such as [HK]. In the introduction we will not aim at the mathematical precision, for instance we will freely use the operator valued measures a(∗) (k) and we will not precise the type of the limits involved in our statements. All the rigorous details will be provided in subsequent sections. Under appropriate assumptions one can show the existence of the following limits: a∗± (k) := lim eitH e−itω(k) a∗ (k)e−itH , t→∞
a± (k) := lim eitH eitω(k) a(k)e−itH . t→∞
We will call a∗± (k) and a± (k) the asymptotic creation/annihilation operators. (If we want to be more precise, then we will say outgoing/incoming creation/ annihilation operators). Note that they form covariant CCR representations: [a± (k1 ), a± (k2 )] = 0,
[a±∗ (k1 ), a±∗ (k2 )] = 0,
eitH a∗± (k)e−itH = eitω(k) a±∗ (k),
[a± (k1 ), a±∗ (k2 )] = δ(k1 − k2 ),
eitH a± (k)e−itH = e−itω(k) a±∗ (k).
We define K0± to be the space of Ψ ∈ H satisfying a± (k)Ψ = 0,
k ∈ Rd .
Elements of K0± will be called asymptotic vacua. The Fock sectors of the asymptotic space are defined as H0± := K0± ⊗ Γs (L2 (Rd )). ± The wave operators in the Fock sector are defined as linear maps Ω± 0 : H0 → H satisfying ∗ ∗ ±∗ (k1 ) · · · a±∗ (kn )Ψ, Ω± 0 Ψ ⊗ a (k1 ) · · · a (kn )Ω := a
Ψ ∈ K0± ,
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(Ω denotes the vacuum in the Fock space. The same letter decorated by the superscript + or − denotes the appropriate wave operator). We also introduce the Hamiltonian of the asymptotic vacua K0± := H ± , K0
and the full asymptotic Hamiltonian: H0± := K0± ⊗ 1l + 1l ⊗
ω(k)a∗ (k)a(k)dk.
Now the following is true: (1) Ω± 0 are isometric; ± ± (2) Ω± 0 1l ⊗ a(k) = a (k)Ω0 , ± ∗ ±∗ Ω0 1l ⊗ a (k) = a (k)Ω± 0;
(3) K0± contains all eigenvectors of H; ± ± (4) Ω± 0 H0 = HΩ0 .
One can formulate two desirable properties, called sometimes jointly the asymptotic completeness [DG2, DG3]: • The operators Ω± 0 are unitary, in other words, the asymptotic CCR representations are Fock. • All asymptotic vacua are linear combinations of bound states of H. √ For massive bosons, (e.g., if ω(k) = k 2 + m2 with m > 0), under quite weak assumptions one can show that both above properties are true [HK, DG2, DG3]. If m = 0, little is known about these two properties except for the case of van Hove Hamiltonians [De]. Typically, the breakdown of the above properties is closely related to the infrared problem. Note that the conventional scattering theory starts from a given pair of operators: the full Hamiltonian H and the free Hamiltonian H0 and then proceeds to construct wave operators by considering the limit (in appropriate sense) of eitH e−itH0 as t goes to ±∞. The formalism of scattering theory that we described above differs substantially from the conventional one. Instead of the “free Hamiltonian” we have the asymptotic Hamiltonians H0± . The Hamiltonians H0± are simpler than the full Hamiltonian H: they have the form of a “free Pauli-Fierz Hamiltonian”. Nevertheless, they are not given a priori – they are constructed from H. If Ω± 0 is not unitary, then the asymptotic fields have some non-Fock sectors. It may even happen that there are no nonzero asymptotic vacua, so that there are no asymptotic Fock sectors at all. This motivates us to give a description of scattering theory in the presence of non-Fock sectors.
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Among non-Fock sectors the most manageable ones are the so-called coherent sectors. Our paper is to a large extent devoted to the description of scattering theory in their presence. Let Rd k → g(k) be a complex function. We define Kg± to be the space of Ψ ∈ H satisfying √ a± (k)Ψ = 2g(k)Ψ, k ∈ Rd . The elements of Kg± will be called asymptotic g-coherent vectors. The asymptotic g-coherent space is defined as Hg± := Kg± ⊗ Γs (L2 (Rd )). The g-coherent wave operator is the linear map Ω± g : Hg → H defined as ∗ ∗ Ω± g Ψ ⊗ a (k1 ) · · · a (kn )Ω :=
(a±∗ (k1 ) −
√ √ 2g(k1 )) · · · (a±∗ (kn ) − 2g(kn ))Ψ,
Ψ ∈ K0± .
We define the asymptotic Hamiltonian in the g-coherent sector as Hg± := Ω±∗ g HΩ± g . The following can be easily shown: (1) Ω± g are isometric;
√ ± (2) Ω± 2g(k))Ω± g 1l ⊗ a(k) = (a (k) − g , √ ± ∗ ±∗ Ωg 1l ⊗ a (k) = (a (k) − 2g(k))Ω± g ; ± ± (3) Ω± g Hg = HΩg .
(4) There exists a decomposition √ √ Hg± = Kg± ⊗ 1l + 1l ⊗ a∗ (k) + 2g(k) ω(k) a(k) + 2g(k) dk (1.2) (5) If g1 and g2 differ by a square integrable function, then the ranges of Ω± g1 and Ω± g2 coincide. Note that the second term on the right of (1.2) is a van Hove Hamiltonian. If g is not square integrable then the asymptotic CCR representations on the range of Ω± g are non-Fock and the asymptotic Hamiltonians do not have a ground state – nevertheless, we have well defined wave operators that can be used to compute scattering cross-sections. We are not aware of a full description of the above formalism in the literature, although some of its elements may belong to the so-called folklore. In particular, the fact that the asymptotic Hamiltonians have the form given in the equation (1.2) is quite interesting and not obvious.
1.4
Renormalized Hamiltonian and dressing operator
The main new “analytical” result of the paper is the proof of the existence of a nontrivial non-Fock coherent sector for asymptotic fields in a certain nontrivial
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class of Pauli-Fierz Hamiltonians. The most important additional assumption that we need to get this result is the possibility to split the interaction into two parts: an infrared divergent scalar part and an infrared convergent matrix part. Using this assumption we can define the renormalized Hamiltonian Hren . On the formal level the so-called renormalized Hamiltonian is unitarily equivalent to the initial Hamiltonian H: Hren = 1l⊗W (−ig) Hren 1l⊗W (ig),
(1.3)
where W (ig) is formally a Weyl operator. Strictly speaking, however, W (ig) is not well defined, since g is not square integrable. Still, Hren turns out to be a correctly defined Pauli-Fierz operator. Moreover, with an appropriate choice of g, Hren has a mild infrared singularity, so that one can apply the results of [Ge1], which imply that Hren possesses a ground state. Under appropriate assumptions, one can show that for both H and Hren one can define asymptotic fields. Besides, one can define the so-called dressing operators U ± . The dressing operators are some kind of unitary intertwiners between the objects related to Hren and H. They do not intertwine, however, in the usual meaning of this word: it is not true that HU ± = U ± Hren . The action of U ± gives some sort of a translation in phase space by g. In particular, U ± map coherent sectors of the asymptotic fields of Hren onto the coherent asymptotic sectors of H shifting them by g. In particular, they map the Fock sector of the asymptotic CCR representation for Hren onto the g-coherent sector of the CCR representation for H, which is non-Fock. But we know that Hren has a ground state. Hence it has nontrivial Fock asymptotic sectors. Therefore, H has nontrivial g-coherent asymptotic sectors. According to Approach (3) described above one could discard H in favor of Hren , and treat Hren as the physical Hamiltonian. After this replacement, the asymptotic fields have Fock sectors, where the infrared problem is avoided. We, however, prefer the (more canonical and general) Approach (2), which treats H as the basic physical Hamiltonian and Hren as a technical tool used to prove certain properties of scattering for H.
1.5
Comparison with literature
It is difficult to compare our results with the literature, since a large part of it is non-rigorous and a variety of models are studied. Perhaps one of the oldest examples of “infrared renormalization” can be found in a paper of Pauli and Fierz [PF] devoted to non-relativistic QED. In that paper one can find what is nowadays often called “the Pauli-Fierz transformation”, which can be used to make the Hamiltonian of non-relativistic QED less singular. Blanchard considered scattering for the Hamiltonian of QED in the dipole approximation perturbed by a short range potential [Bl]. He showed that it is possible to construct wave operators if one replaces the original Hamiltonian by
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an appropriately renormalized one. Note that Blanchard’s Hamiltonian is different from ours. In particular, in his case one can define usual wave operators and the formalism of asymptotic fields is not necessary, unlike in the case of our Hamiltonian. Faddeev and Kulish made an interesting attempt to define wave and scattering operators for the full QED, taking into account both the long-range nature of the interaction between charged particles and the emergence of non-Fock representations of photons [KF]. Their work was not completely rigorous. The infrared problem for the so-called Nelson model in the one-electron sector was studied by Fr¨ ohlich in [F], and more recently by Pizzo [Pi]. In these papers one ohlich can find an operator essentially equivalent to our dressing operators U ± . Fr¨ and Pizzo consider translation invariant models, which introduces additional complications in their analysis. A complete construction of dressed one electron states is not achieved in [F], (some parts of the construction relied on physically reasonable but conjectural assumptions). A complete construction was recently given by Pizzo [Pi]. Examples of the infrared renormalization, similar to the one in (1.3), can be found in [Ar, HHS]. Our paper can be considered to be a sequel to a number of papers devoted to scattering in quantum field theory [HK, DG2, DG3, Ge2, FGS]. All of these paper, except for [Ge2], are devoted to massive fields, which are not subject to the infrared problem, When comparing the literature on models related to ours one should make a distinction between translation invariant models, such as those considered in [KF, F, Pi, FGS], and the models where the perturbation is localized and thus the translation invariance is broken, such as [HK, DG2, DG3, Ge2] and this paper. Translation invariant models are more difficult and rigorous results about them are scarce. The fact that we restrict ourselves to a confined system without translation invariance enables us to give a more transparent and thorough analysis of the scattering theory in presence of the infrared divergences.
1.6
Organization of the paper
Our paper can be divided into two parts. The first consists of Section 2, where we describe the main results of our paper. We introduce a certain class of abstract Pauli-Fierz Hamiltonians. We recall and partly extend basic results on the existence of asymptotic fields [DG2], [Ge2] and on the existence and non-existence of ground states [Ge1]. The asymptotic fields may have non-Fock sectors. We concentrate our attention on the so-called coherent sectors. We show how to define wave operators, scattering operators and asymptotic Hamiltonians for coherent sectors. We demonstrate that they are not much more difficult than the usual Fock sectors, and thus we explain how one can overcome the conceptual problems caused by the infrared problem. We show the existence of non-Fock sectors for a class of Pauli-Fierz Hamiltonians, that includes a certain class of Nelson Hamiltonians.
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We end Section 2 with a discussion of inclusive cross-sections in our model. Let us stress that, in principle, by using our formalism one can describe predictions for experiments that measure “soft components of the system” and one does not need to restrict oneself to inclusive cross-sections. One can argue, however, that in realistic experiments the soft background should be irrelevant and measurable quantities should depend only on the “hard components”. We discuss how to define such inclusive cross-sections and state some physically motivated conjectures about them. The remaining part of our paper is somewhat more mathematical. It contains a systematic exposition of various elements of mathematical formalism used in Section 2. Some of them are presented in a more general context and proved in bigger generality. Let us stress that Sections 3, 4, 5, 6, 7 and the Appendix can be read independently of Section 2. In Section 3 we study general CCR representations. A particular attention is devoted to the so-called coherent representations. These representations are obtained by translating the Fock representation by an antilinear functional. If the functional is not continuous, then this representation is not unitarily equivalent to the Fock representation. In Section 4 we study the so-called covariant CCR representations. They are CCR representations equipped with a dynamics, which is implemented both on the level of the full space and of the 1-particle space. We show how to describe covariant representations in a coherent sector. It turns out that in every coherent sector the dynamics has a certain natural decomposition, one part of which is given by a quadratic Hamiltonian perturbed by a linear one (a van Hove Hamiltonian). In our opinion this is quite an interesting and hitherto unknown fact. Covariant CCR representations arise naturally in scattering theory of certain quantum systems. Based on the ideas of the LSZ formalism, such representations were constructed and studied, e.g., in [HK], and more recently in [DG2], [DG3] and [Ge2]. In Section 5 we study such representations in an abstract context. One of them describes the observables in the distant past – the incoming representation W − (·), the other describes the observables in the distant future – the outgoing representation W + (·). Collectively, they are called asymptotic representations. We show in particular that eigenvectors of the Hamiltonian are always vacua of both asymptotic representations and thus give rise to nontrivial Fock sectors. Note that the material of Sections 3, 4 and 5 is rather basic and mostly belongs to the folklore (although our presentation has some points which are new). Section 6 is more special: here we introduce the so-called dressing operator between two CCR representations. In Section 7 we introduce a relatively general class of Pauli-Fierz Hamiltonians. For these Hamiltonians, under some relatively mild assumptions on the interaction, asymptotic CCR representations exist and one can apply the formalism developed in the previous sections. One can also introduce the renormalized Hamiltonian and the dressing operators.
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2 Overview of main results and some open problems In this section we describe most of main results of our paper in a somewhat simplified form. We also discuss some aspects of the physical content of our mathematical constructions. We formulate some open mathematical problems inspired by physical considerations. Let us make some remarks about our notation. If A is a closed operator, then DomA, RanA and spA denote its domain, range and spectrum. If A is self-adjoint and Θ a Borel subset of R, then 1lΘ (A) denotes the spectral projection of A onto Θ. We also write x for (1 + x2 )1/2 .
2.1
Pauli-Fierz Hamiltonians
Suppose that K is a separable Hilbert space representing the degrees of freedom of the atomic system. Let K be a positive operator on K – the Hamiltonian of the atomic system. We will sometimes use Assumption 2.A (K + i)−1 is compact on K.
The physical interpretation of this assumption is that the small system is confined. Let h = L2 (Rd , dk) be the 1−particle Hilbert space in the momentum representation and let Γs (h) be the bosonic Fock space over h, representing the field degrees of freedom. Ω will stand for the vacuum in Γs (h). We will denote by k the momentum operator of multiplication by k on L2 (Rd , dk). Let ω := |k| be the dispersion relation. For f ∈ h the operators of creation and annihilation of f are denoted by ∗ f (k)a(k)dk. f (k)a (k)dk, The Hamiltonian describing the field is equal to dΓ(ω) = ω(k)a∗ (k)a(k)dk. (See, e.g., [BR, vol. II] or [DG2, DG3] for basic concepts related to the second quantization.) Assumption 2.B The interaction between the atom and the boson field is described with a coupling function v Rd k → v(k),
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such that for a.e. k ∈ Rd , v(k) is a bounded operator from Dom(K 1/2 ) into K. We will assume: for a.e.
k ∈ Rd ,
1
v(k)(K + 1)− 2 ∈ B(K), 1
∀ Ψ1 , Ψ2 ∈ K, k → (Ψ2 , v(k)(K + 1)− 2 Ψ1 ) is measurable, 1 lim sup (1 + ω(k)−1 ) v(k)(K + R)− 2 2 dk < 1/2. R→∞
1
Note that the functions k → v(k)(K +R)− 2 is measurable (see for example [Ge2, Appendix]), and hence the last condition in Assumption 2.B has a meaning. We set H0 = K⊗1l + 1l⊗ ω(k)a∗ (k)a(k)dk, ∗ H = H0 + v(k) ⊗ a (k)dk + v ∗ (k) ⊗ a(k)dk. H0 is called the free Pauli-Fierz Hamiltonian and H the full Pauli-Fierz Hamiltonian. One can easily show that Theorem 2.1 Under Assumptions 2.A and 2.B, the operator H is self-adjoint and 1/2 bounded from below with the form domain Dom(H0 ).
2.2
The confined massless Nelson model
In this subsection we describe one of the main examples of Pauli-Fierz Hamiltonians. It is a model describing a confined atom interacting with a field of scalar bosons. A similar model (without the ultraviolet cut-off) was studied in a well known paper by Nelson [Ne]. Hence, in a part of the mathematical literature it is called the Nelson model (see [A], [Ar], [LMS]). To be more precise, the model that we will consider can be called the confined massless ultraviolet cut-off Nelson model. We will prove that a large class of such models satisfies all the assumptions of this section. Thus Lemma 2.2 means that all the results presented in in Sections 2 and 7 apply to this class. In particular, their asymptotic CCR representations contain a non-Fock coherent sector. The atom is described with the Hilbert space K := L2 (R3P , dx), where x = (x1 , . . . , xP ), xi is the position of particle i, and the Hamiltonian: K :=
P −1 ∆i + Vij (xi − xj ) + W (x1 , . . . , xP ), 2mi i=1 i<j
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where mi is the mass of particle i, Vij is the interaction potential between particles i and j and W is an external confining potential. We will assume Vij is ∆-bounded with relative bound 0, (H0)
W (x) ≥ c0 |x|2α − c1 ,
W ∈ L2loc (R3P ),
c0 > 0, α > 0.
It follows from (H0) that K is symmetric and bounded below on C0∞ (R3P ). We 1 still denote by K its Friedrich’s extension. Moreover we have Dom((K + b) 2 ) ⊂ H 1 (R3P ) ∩ Dom(|x|α ), which implies that 1
|x|α (K + b)− 2 is bounded.
(2.1)
Note also that (H0) implies that K has compact resolvent on L2 (R3P ). The one-particle space for bosons is h := L2 (R3 , dk), where the observable k is the boson momentum, and the one-particle energy is ω(k) = |k|. The interaction is given by the operator R3 k → v(k) ∈ B(K), where v(k) is a multiplication operator on L2 (R3P , dx) equal to 1 χ(|k|) −ik·xj v(k, x) = √ 1 e 2 j=1 |k| 2 P
where χ ∈ C0∞ (R) is a real, even function such that χ ≡ 1 near 0. The function χ plays the role of an ultraviolet cut-off. Lemma 2.2 If hypothesis (H0) holds for α > 1, the confined Nelson model satisfies assumptions 2.A, 2.B and 2.C, 2.D, 2.E, 2.F below, where in Assumptions 2.D and 2.F we set P χ(|k|) z(k) = √ 1 , 2 |k| 2
1 χ(|k|) −ik·xj vren (k, x) = √ − 1). 1 (e 2 j=1 |k| 2 P
Proof. We already know that Assumption 2.A is true. We have |v(k, x)| ≤ C|k|−1/2 . Therefore, v(k) ∈ L2 (R3 , (1 + |k|−1 )dk), and hence Assumptions 2.B and 2.E are satisfied. We will now show that Assumption 2.C holds with g := C0∞ (R3 \{0}). Let h ∈ g. Define mj,t (x) := h(k)eit|k| v(k, xj )dk + cc =
ei(t|k|−xj ·k) h(k)χ(k) dk + cc. |k|1/2
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(The symbol cc denotes the complex conjugate.) We can write mj,t (x)(1 + K)−1/2
= mj,t (x)1l[0, 2t ] (|x|)(1 + K)−1/2 +mj,t (x)1l] 2t ,∞[ (|x|)x −α x α (1 + K)−1/2 .
(2.2)
1
Since by (2.1) |x|α (K + 1)− 2 is bounded, the second term is O(t−α ). is in C0∞ To deal with the first term, note that the function h(k) χ(|k|) 1 |k| 2
(R3 \{0}). Because of the cutoff function, the phase t|k| − k · xj is smooth without stationary points on |x| < t/2. Using the non-stationary phase method, we obtain that the second term of (2.2) is O(t−∞ ). Hence for α > 1, Assumption 2.C is satisfied. Consider now Assumption 2.D. We note that |e−ik·xj − 1| ≤ |k| |xj |. Hence
(2.3)
vren (k)x −1 ≤ C|k|1/2 ,
which implies vren (k)(1 + K)−1/2 ∈ L2 (R3 , |k|−2 dk). This proves Assumption 2.D. Finally, we prove Assumption 2.F. We set mj,t (xj ) := g(k)eit|k| vren (k, xj )dk + cc √ =P 2
χ(|k|)2 cos(t|k| k2
− xj · k) − cos t|k| dk.
We go to spherical coordinates (r, θ, φ), r ∈ R+ , θ ∈ [0, π], φ ∈ [0, 2π], and get: mj,t (xj ) :=
√ ∞ π 2π
P 2 0 0 0 χ(r)2 cos(tr − |xj | cos θr) − cos tr drd cos θdφ √ sin(tr+|xj |r)−sin(tr−|xj |r)
∞ − 2 cos tr dr = P 22π 0 χ(r)2 |xj |r √ ∞ sin(|xj ||r|)
= P 22π −∞ χ(r)2 cos tr |xj ||r| − 1 dr = O(t−n xj n ).
for any n ∈ N, where in the last step we integrated by parts. By interpolation we actually can replace n with any positive real α. Thus we see that mj,t (xj )(1 + K)−1/2 = mj,t (xj )x −α x α (1 + K)−1/2 = O(t−α ), which ends the proof of Assumption 2.F.
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2.3
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Asymptotic fields
For h ∈ h we define the field and the Weyl operators
1 h(k)a∗ (k) + h(k)a(k) dk, W (f ) := eiφ(h) . φ(h) := √ 2 −1 2 h1 := h ∈ h | (1 + ω(k) )|h(k)| dk < ∞ = Dom(ω −1/2 ),
Let
with the norm h h1 := (1 + ω −1 )1/2 h h . The following assumption can be called the short range condition. Assumption 2.C There exists a dense subspace g ⊂ h1 ∩ Dom(ω 1/2 ) such that for h ∈ g,
∞
itω(k)
∗ −itω(k) −1/2
e
h(k)v(k) + v (k)e h(k) (1+K) dk dt < ∞.
0
B(K)
Theorem 2.3 Suppose that assumptions 2.B and 2.C hold. Then: (1) For all h ∈ h1 , there exist W ± (h) := s− lim eitH 1lK ⊗W (e−itω h)e−itH . t→±∞
(2)
i
W ± (h1 )W ± (h2 ) = e− 2 Im(h1 |h2 ) W ± (h1 + h2 ),
(2.4)
h1 , h2 ∈ h1 ,
R t → W ± (th) is strongly continuous, h ∈ h1 ; in other words,
h1 h → W ± (h)
(2.5)
are regular CCR representations (see Section 3). (3)
eitH W ± (h)e−itH = W ± (eitω h),
h ∈ h1 ,
in other words, (W ± ,ω,H) are covariant CCR representations (see Section 4). (4) If HΨ = EΨ, then 2
(Ψ|W ± (h)Ψ) = e−h
/4
Ψ 2,
(2.6)
in other words, eigenvectors of H are vacua for (2.5) (see Section 3) and Theorem 5.2). The above theorem is a simplified version of Theorem 7.4 proved later in our paper. It is convenient to introduce the following notation. Hp (H) will denote the closure of the span of eigenvectors of H. The set of vacua for (2.5), i.e., the set of
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Φ ∈ H satisfying (2.6) is denoted by K0± . Note that K0± is a closed subspace of H. By Theorem 2.3 (4), K0± contains Hp (H). The closure of the span of vectors W (h)Φ with h ∈ h1 , Φ ∈ K0± will be ± . It is the largest subspace of H on which (2.5) is equivalent to the denoted by H[0] Fock representation. Let us state the following conjecture: Conjecture 2.4 Suppose Assumptions 2.A, 2.B and 2.C hold. Assume also v(k) 2 dk < ∞. k2
(2.7)
Then ± = H, in other words, the asymptotic representations are multiples of the (1) H[0] Fock representation.
(2) K0± = Hp (H), in other words, all the asymptotic vacua are linear combinations of eigenstates of H. There are two situations when we can prove the above conjecture. If dim K = 1, then the Hamiltonian H is the exactly solvable van Hove Hamiltonian and the conjecture follows by explicit computations, see, e.g., [De]. If v(k) = 0 in a neighborhood of zero, then the problem reduces to the case with a positive mass. Conjecture 2.4 (1) follows then from the arguments due to Hoegh-Krohn [HK] described in [DG2], see also a different proof in [DG3]. Conjecture 2.4 (2) follows then from [DG2], see also a somewhat simpler proof given in [DG3]. Note that the power |k|−2 in (2.7) is natural, since it is suggested by the exactly solvable case. However, we do not know how to prove our conjecture even under much stronger assumptions, e.g., if for any N v(k) 2 dk < ∞. kN
2.4
Existence and nonexistence of a ground state
The following assumption will be very important in the sequel: Assumption 2.D v(k) can be split as v(k) = z(k)1lK + vren (k), where 1
z(k) ∈ C, vren (k) ∈ B(Dom(K 2 ), K), (1 + ω(k)−1 )|z(k)|2 dk < ∞, ω(k)−2 vren (k)(K + 1)−1/2 2 dk < ∞.
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In order to use the results of [Ge1] we will also need the following (probably unnecessary) assumption, which is stronger than Assumption 2.B: Assumption 2.E for a.e.
k ∈ Rd ,
1
1
v(k)(K + 1)− 2 , (K + 1)− 2 v(k) ∈ B(K), 1
1
∀Ψ1 , Ψ2 ∈ K, k → (Ψ2 , (K + 1)− 2 v(k)Ψ1 ) and k → (Ψ2 , v(k)(K + 1)− 2 Ψ1 ) are measurable,
1 1 lim (1 + ω(k)−1 ) v(k)(K + R)− 2 2 + (K + R)− 2 v(k) 2 dk = 0. R→∞
Theorem 2.5 Assume Hypotheses 2.D and 2.E. Then: (1) if Assumption 2.A holds and ω(k)−2 |z(k)|2 dk < ∞, then inf sp(H) is an eigenvalue. (2) if inf sp(H) is an eigenvalue, then ω(k)−2 |z(k)|2 dk < ∞. In particular under Assumption 2.A, the existence of a ground state is equivalent to the condition ω(k)−2 |z(k)|2 dk < ∞. Proof. Part (1) has been shown in [Ge1, Thm. 1]. Let us prove part (2) by contradiction. Assume that ω(k)−2 |z(k)|2 dk = ∞, and let Ψ0 ∈ H be a ground state of H. The following pull-through formula is valid (see, e.g., [Ge1, Sect. III.4]): (H + ω(k) − z)−1 a(k)Ψ =
a(k)(H − z)−1 Ψ + (H + ω(k) − z)−1 v(k)(H − z)−1 Ψ, Ψ ∈ H,
(2.8)
as an identity on L2loc (Rd \{0}, dk; H). Applying this identity to Ψ0 , we obtain a(k)Ψ0 = (E − H − ω(k))−1 v(k)Ψ0 , as an identity on L2loc (Rd \{0}, dk; H). Hence a(k)Ψ0 =
z(k) Ψ0 + (E − H − ω(k))−1 vren (k)Ψ0 . ω(k)
Let r(k) := a(k)Ψ0 −
z(k) Ψ0 = (E − H − ω(k))−1 vren (k)Ψ0 . ω(k)
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We have r(k) ≤ c
Ann. Henri Poincar´e
1 1 vren (k)(K + 1)− 2 . ω(k)
Hence, by the last condition of Assumption 2.D, r ∈ L2 (Rd , dk; H). Since applying Lemma 2.6 below we obtain Ψ0 = 0, which is a contradiction.
z ω
Lemma 2.6 Let Ψ ∈ Γs (L2 (Rd )) such that (a(k) − g(k))Ψ 2 dk < ∞,
∈ h,
(2.9)
where k → g(k) ∈ C is measurable and |g(k)|2 dk = ∞. Then Ψ = 0. Proof. We write Ψ = (Ψ0 , Ψ1 , . . . , Ψn , . . . ) where Ψn ∈ ⊗ns h. From (2.9) we obtain since g ∈ L2 (Rd ) and Ψn ∈ ⊗n L2 (Rd ) 1
(n + 1) 2 Ψn+1 (k, k1 , . . . , kn ) − g(k)Ψn (k1 , . . . , kn ) ∈ ⊗n+1 L2 (Rd ),
which implies that Ψn = 0. Hence Ψ = 0.
2.5
Existence of non-Fock sectors for asymptotic fields
Set g(k) :=
√
2ω −1 (k)z(k).
(2.10)
Let us introduce the following assumption: Assumption 2.F
∞
itω(k)
∗ −itω(k) −1/2
e
dt < ∞. g(k)v (k) + v (k)e g(k) (1+K) dk ren ren
0
Theorem 2.7 Assume Hypotheses 2.A, 2.C, 2.D, 2.E and 2.F. Then there exists a nonzero vector Φ ∈ H such that 2
(Φ|W ± (h)Φ) = Φ 2 eiRe(h|g) e−h
/4
.
(2.11)
In particular, if g ∈ L2 , then the CCR representations (2.5) have non-Fock coherent sectors.
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Let us introduce the following notation. The set of vectors Φ satisfying (2.11) will be denoted by Kg± . Such vectors will be called g-coherent vectors for (2.5) (see Section 3). They form a closed subspace of H. The closure of the span of vectors W (h)Φ with h ∈ h1 , Φ ∈ Kg± , will be ± denoted by H[g] . It is the largest subspace of H on which (2.5) is equivalent to the so-called g-coherent representation. Conjecture 2.8 Under the hypotheses of Theorem 2.7, H[g] = H, in other words, the representation of asymptotic fields is equivalent to a multiple of the g-coherent representation. Note that given the methods of the proof of Theorem 2.7, Conjecture 2.8 essentially follows from Conjecture 2.4 (1).
2.6
Renormalized Hamiltonian
In this and the next subsection we will describe the main ideas of the proof of Theorem 2.7. One of them is the use of the so-called renormalized Hamiltonian. It is defined as ∗ (k)a(k))dk, Hren := Kren ⊗ 1l + 1l ⊗ dΓ(ω) + (vren (k)a∗ (k) + vren where Kren
:= K −
|z(k)|2 ω(k)
+
z(k)vren (k) ω(k)
+
∗ vren (k)z(k) ω(k)
dk.
Note that Assumptions 2.A, 2.D, 2.E for H imply Assumptions 2.A, 2.D and 2.E for Hren with zren = 0. Therefore, by the result of [Ge1] quoted in Theorem 2.5 (1), Hren has a ground state. Suppose Assumptions 2.C and 2.F hold as well. Then, by Theorem 7.5, we can define asymptotic fields for Hren ± Wren (h) := s− lim eitHren 1l⊗W (e−itω h)e−itHren . t→±∞
± Clearly, Wren satisfy the obvious analog of Theorem 2.3. The ground state of Hren is a vacuum for the renormalized asymptotic fields.
Remark 2.9 Note that if g ∈ h, then H = W (ig)Hren W (−ig).
(2.12)
If g ∈ h, then W (±ig) is not well defined. Still, we can use (2.12) on a formal level. To make it rigorous we can proceed in a variety of ways. We can choose a sequence of approximations of g gσ := g1l[σ,∞[ (ω), Then it is easy to show that
0 < σ < 1.
−1 (i + Hren )−1 = s− lim i + W (igσ )HW (igσ ) . σ0
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Dressing operators
Clearly, Im(g|e−itω g) is well defined and (1 − e−itω )g ∈ h. Therefore the following definition makes sense: i
U (t) := e 2 Im(g|e
−itω
g) itH
e
W (i(1 − e−itω )g)e−itHren .
Theorem 2.10 Under Assumptions 2.B, 2.D and 2.F, there exists U ± := s− lim U (t). t→±∞
s− limt→±∞ U (t)∗ also exists and equals U ±∗ . The above theorem will be proved under more general conditions later as Theorem 7.5. The operators U ± will be called the dressing operators. They have the following properties: Theorem 2.11 Suppose Assumptions 2.B, 2.C, 2.D and 2.F are true. Then, for h ∈ h1 , we have W ± (h)U ± eitH U ± e−itHren
± = U ± Wren (h)eiRe(h,g) , i
± = U ± Wren (i(1 − eitω )g)e− 2 Im(g|e i
= W ± (i(1 − eitω )g)U ± e 2 Im(g|e
−itω
−itω
g)
g)
.
± Therefore, U ± maps K0,ren onto Kg± .
The above properties of dressing operators are proved in Section 6.
2.8
Wave operators
We define the g-coherent asymptotic space as Hg± := Kg± ⊗ Γs (h). ± It is easy to show that there exists a unique linear operator Ω± g : Hg → H such that −iRe(h|g) Ω± W ± (h)Φ, Φ ∈ Kg± , h ∈ h1 . g Φ⊗W (h)Ω = e ± The operator Ω± g is isometric and its range equals H[g] . It will be called the gcoherent wave operator. (Note that Ω, without any superscripts, still denotes the vacuum in a Fock space.) The g-coherent asymptotic Hamiltonian is defined as ± Hg± := Ω±∗ g HΩg .
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Clearly, Hg± is a self-adjoint operator on Hg± satisfying ± ± Ω± g Hg = HΩg .
What is a little less obvious is the following decomposition of Hg± , proved in Theorem 4.5: z(k) z(k) dk. (2.13) a∗ (k) + Hg± = Kg± ⊗ 1l + 1l ⊗ ω(k) a(k) + ω(k) ω(k) Thus, in particular, the asymptotic Hamiltonians Hg± do not have ground states. ± Note that the subspaces H[g] are invariant with respect to W ± (h) and H. They depend only on the equivalence class [g] of g in h∗1 /h, where h∗1 denotes the space of all antilinear functionals on h1 (see Theorem 3.7 (1)). If one introduces ± ± ±∗ = H ± = Ω± H[g] g Hg Ω g , H[g]
± H[g]
then again depends only on [g]. ± ± and H = H[g] . If Conjecture 2.8 is true then H = H[g]
2.9
Scattering operator
We can define the scattering operator for the g − g channel as − Sgg := Ω+∗ g Ωg . − + = H[g] . It is unitary iff H[g] Suppose that we prepare a state in a distant past inside the incoming gcoherent sector. We can describe it by a density matrix (a positive operator of trace 1) ρ on Hg− . Suppose that we make a measurement in a distant future in the outgoing g-coherent sector. We can describe it by an observable (a self-adjoint operator) A on Hg+ . The expectation value of the experiment is given by the trace ∗ TrSgg ρSgg A.
(2.14)
Note that there is no infrared problem in the formula above. In principle, one has a well defined procedure to compute the expectation value of an experiment involving any initial state and any final observable – there is no need to restrict oneself to “inclusive cross-sections”. The infrared problem manifests itself in the non-canonical choice of the functional g. In fact, g is not determined by the Hamiltonian H itself. One can argue that in a realistic experiment all the quantities depending on the choice of g are unmeasurable (or at least are much more difficult to measure). This is quite similar
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to long-range scattering for Schr¨ odinger operators, where the modified scattering operator depends on a non-canonical modifier and one usually assumes that measurable quantities are independent of its choice. In the remaining part of this section we will analyze scattering of infrared singular Pauli-Fierz Hamiltonians and point out quantities that are likely to be physically relevant. Let us note a certain discrepancy between mathematics and physics of the problem. In the construction of wave and scattering operators the past is treated in the same way as the future. Thus mathematics of scattering theory is in some sense symmetric with respect to time reversal. This is not the case for the formula (2.14), which gives physical interpretation of the scattering operator: the past is represented by a density matrix whereas the future by an arbitrary selfadjoint operator. This asymmetry between past and future will be even more pronounced in the next subsections, where we discuss inclusive cross-sections. It will be clear which observables should be considered in the future, it will be less clear which initial states should be taken into account.
2.10
Soft and hard photons
Let ≥ 0. Define h≤ := Ran1l[0,] (ω),
h> := Ran1l],∞[ (ω),
so that h = h≤ ⊕ h> . Clearly, we can make the identification ± Hg± Hg,≤ ⊗ Γs (h> ),
(2.15)
± where Hg,≤ := Kg± ⊗ Γs (h≤ ). Let us make an additional assumption
1l],+∞[ (ω)g = 0.
(2.16)
Since g is given in terms of z by the equality (2.10), this is equivalent to 1l],+∞[ (ω)z = 0, which we can always assume. By this assumption, the asymptotic Hamiltonian can be written as ∗ z(k) z(k) a (k) + ω(k) ω(k) a(k) + ω(k) dk Hg± = Kg± ⊗ 1l + 1l ⊗ +1l ⊗
ω<
∗
ω(k)a (k)a(k)dk.
ω≥
Therefore, with respect to the decomposition (2.15), the asymptotic Hamiltonians can be written as ± Hg± = Hg,≤ ⊗ 1l + 1l ⊗ dΓ(ω> ), where ω> = ω1l],∞[ (ω).
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One can ask whether the decomposition into soft and hard components is sensitive to the choice of g. Introduce the soft Hamiltonian ± ± ±∗ H[g],≤ := Ω± g Hg,≤ ⊗1l Ωg ,
and the hard Hamiltonian ± ±∗ H[g],> := Ω± g 1l⊗dΓ(ω> ) Ωg .
We have
± ± ± H[g] = H[g],≤ + H[g],>
(2.17)
and the Hamiltonians in (2.17) depend only on [g]. A similar question can be asked concerning the observables. On the level of asymptotic spaces we have clearly ± ± B(Hg± ) B(Hg,≤ ) ⊗ B(Hg,> ).
(2.18)
Denote the range of the homomorphism ±∗ B(Hg± ) A → Ω± g AΩg ∈ B(H) ± by A[g] . A[g] depends only on [g] and is equal to B(H[g] ). Inside A[g] we can distinguish the “algebra of soft observables” ± ± ±∗ A± [g],≤ := Ωg B(Hg,≤ )⊗1l Ωg ,
(2.19)
and the “algebra of hard observables” ± ± ±∗ A± [g],> := Ωg 1l⊗B(Hg,> ) Ωg .
(2.20)
(2.19) and (2.20) depend only on [g]. The hard observables are even more independent of g. The automorphism ± ±∗ B(Γs (h> )) A> → Ω± g 1l⊗A> Ωg ∈ A[g],>
depends only on [g] if we assume (2.16).
2.11
Inclusive cross-sections
To simplify the discussion, we will assume in what follows that + − = H[g] . H = H[g]
(2.21)
In what follows we will drop the subscripts g wherever possible, thus we will write ± ± ± ± Ω± , H± , H ± , H≤ , etc. instead of Ω± g , Hg , Hg , Hg,≤ , etc.
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Set E := inf H. Clearly, − + E = inf H − = inf H + = inf H≤γ = inf H≤γ
for any γ > 0. Note that by the assumption (2.21), the wave operators Ω± are unitary from ± H to H and the scattering operator S = Ω+∗ Ω− is unitary from H− to H+ . Suppose now that the experimentalist can only control the components of the system above the threshold . In particular, since the functional g depends on the soft components, the quantities that depend on g are not measurable. The quantum description of an experiment has two aspects: preparation of the incoming state and measurement of the outgoing observable. It is easy to say which observables can in principle be measured by the experimentalist. They are the observables in the hard algebra A+ > , that means the observables of the form Ω+ 1l⊗A> Ω+∗ , where A> ∈ B(Γs (h> )). It is more difficult to say which incoming states the experimentalist can pre− − ⊗ H> . Thus we can introduce the partial trace wrt pare. Recall that H− = H≤ − H≤ , denoted 1 − l1 (H− ) ρ → Tr− ≤ ρ ∈ l (H> ), where l1 (H) denotes the space of trace class operators on a Hilbert space H. In − particular, if ρ is a density matrix on H− , then Tr− ≤ ρ is a density matrix on H> . We assume that the initial state of the system is described by a density matrix ρ on H− . We also suppose that the experimentalist does not have full information about ρ and is able to control only Tr− ≤ ρ. More precisely, for a given density − matrix ρ> on H> , while preparing his experiment, he can make sure that Tr− ≤ ρ = ρ> .
(2.22)
Of course, there are many density matrices ρ satisfying (2.22). The choice of ρ should be determined by physics. Let us suppose that the experiment is conducted at a low temperature, so that everything tends to have the lowest possible energy. Suppose for a moment that the infrared problem is absent in the sense that ± , has a non-degenerate ground state. the Hamiltonian H, hence also H ± and H≤ Then it is natural to assume that the incoming density matrix equals − 1lE (H≤ ) ⊗ ρ> − − (recall that inf sp(H≤ ) = E, and hence 1lE (H≤ ) denotes the spectral projection − onto the ground state of H≤ ). Thus one can argue that if the experimentalist prepared the hard part of the incoming state as ρ> and measures the observable A, then the expectation value of the measurement (which we will somewhat imprecisely call the cross-section) will be − Tr S 1lE (H≤ )⊗ρ> S ∗ A.
(2.23)
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If we have an infra-red problem – if H has no ground state at all or even if its ground state is degenerate – then it is not clear which ρ satisfying (2.22) should be taken. We can argue that ρ should satisfy − 1l[E,E+δ] (H≤ )⊗1l ρ = ρ
for some small δ > 0. Of course this does not fix the choice of ρ either. − Motivated by these considerations, if δ > 0, ρ> is a density matrix on H> + and A is observable on H , we define Crossδ (ρ> , A) := TrρS ∗ AS : ρ is a density matrix on H− , − ) ⊗ 1l ρ = ρ, Tr− 1l[E,E+δ] (H≤ ≤ ρ = ρ> . This is the set of all possible cross-sections compatible with the pair (ρ> , A) under the assumption that the soft part of the initial state has the excess energy below δ. Clearly, Crossδ (ρ> , A) is a family of nonempty intervals in [− A , A ] decreasing as δ 0. It would be interesting to investigate whether a large class of Pauli-Fierz Hamiltonians has the following property: Property P.a The Pauli-Fierz Hamiltonian H has the property of the continuity of cross-sections at the bottom of spectrum iff for any ρ> and A, Crossδ (ρ> , A)cl (2.24) 0<δ<
is a single point. (The superscript cl denotes the closure of a set.) If Property P.a holds, then the number given by (2.24) can be viewed as the cross-section for the experiment described by ρ> and A. Note that if H has a non-degenerate ground state, then (2.24) contains the number (2.23). Clearly (2.24) depends on the choice of g within its equivalence class, hence one can argue that in such a case it does not correspond to a physical experiment. If one assumes that the observable is of the form A = 1l⊗A> with A> an observable on Γs (h> ), then (2.25) Crossδ (ρ> , 1l ⊗ A> ) does not depend on the choice of g satisfying (2.16), using the covariance properties shown in Subsection 7.5. (2.25) is the set of possible inclusive cross-sections compatible with the pair (ρ> , A> ). One can introduce a property weaker than (P.a): Property P.b The Pauli-Fierz Hamiltonian H has the property of the continuity of inclusive cross-sections at the bottom of spectrum iff for any ρ> and A> , Crossδ (ρ> , 1l⊗A> )cl 0<δ<
is a single point.
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If Property P.a is true then the theory based on the Pauli-Fierz Hamiltonian H has quite a strong predictive power. The experimentalist does not have to worry about preparing precisely the soft part of the initial state; it is enough if its soft part is sufficiently low energetic. Then he can measure all observables he likes, even those involving soft modes. The theory will give well defined cross-sections for his experiments. If the experimentalist measures only hard components of the final state, then it is sufficient that Property P.b holds to have well defined cross-sections for all experiments. Note that the stronger Property P.a is true in the case of the exactly solvable van Hove Hamiltonian, where the scattering operator is equal to identity.
2.12
Insensitivity to soft background
One could argue, however, that Properties P.a and P.b are too modest and do not correspond to realistic physical situations. It may be unjustified to expect that the soft modes of the radiation will dissipate their energy while the experimentalist prepares the experiment. Nevertheless, one can hope that soft modes should not influence the outcome of measurement too much provided that their energy is reasonably bounded. This intuition leads to yet another conjecture. In order to state it, we introduce a new definition. Let δ > 0 and 0 < γ ≤ . Suppose that the experimentalist can control the incoming states up to the modes of energy γ. He can make sure that there are no photons of energy in [γ, ] – the system is in the lowest possible energetic state for the modes of energy in this energy range. This means that Tr− ≤γ ρ = |W (−ig[γ,] )Ω)(W (−ig[γ,] )Ω| ⊗ ρ> .
(2.26)
Here g[γ,] = 1l[γ,] (ω)g, and |W (−ig[γ,])Ω)(W (−ig[γ,] )Ω| denotes the orthogonal projection onto the coherent vector W (−ig[γ,])Ω. Suppose also that the experimentalist can guarantee that the soft modes have the excess of the energy below δ > 0, which however does not have to be very small. This means that − )⊗1l ρ = ρ. 1l[E,E+δ] (H≤γ Note that by (2.26) this is equivalent to − 1l[E,E+δ] (H≤ )⊗1l ρ = ρ.
Cross-sections compatible with this information are given by the set Crossδ,γ (ρ> , A) := TrρS ∗ AS : ρ is a density matrix on H− ,
− )⊗1l ρ = ρ, Tr− 1l[E,E+δ] (H≤γ ≤γ ρ = |W (−ig[γ,] )Ω)(W (−ig[γ,] )Ω|⊗ρ> .
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Clearly Crossδ,γ (ρ> , A) decrease if δ or γ decrease. Moreover if δ < γ, then Crossδ,γ (ρ> , A) = Crossδ (ρ> , A). If A> is as above, then Crossδ,γ (ρ> , 1l ⊗ A> )
(2.27)
does not depend on the choice of g satisfying the condition (2.16). Let [0, ] γ → δ(γ) be a function with values in positive real numbers. One could expect that a large class of Pauli-Fierz Hamiltonians satisfy the following property for δ(γ) such that limγ→0 δ(γ) γ = +∞: Property P.c A Pauli-Fierz Hamiltonian H has the property of δ-insensitivity of inclusive cross-sections to soft background iff the following is true. Let ρ> and A> be as above. Then Crossδ(γ),γ (ρ> , 1l⊗A> )cl 0<γ<
is a single point. Note that the van Hove Hamiltonians have Property P.c with δ(γ) = ∞ – soft modes and hard modes are completely decoupled.
3 Canonical commutation relations Here begins the second part of this paper, consisting of Sections 3–7 and Appendix, which is more mathematical than the previous section. In this part we develop systematically various elements of mathematical formalism useful in the study of infrared problem. In particular we prove most of the statements described in Section 2. Let us stress that this and the following sections can be read independently of Section 2 and of the introduction. In this section we collect basic constructions and facts concerning CCR representations [BR], [BSZ], [DG3], concentrating especially on the so-called coherent representations. The notation that we develop here will be used throughout the paper. Note in particular that in the applications that will start with Section 5, the superscript π will be replaced by the superscript − or + corresponding to the incoming or outgoing representation.
3.1
CCR representations
Let g be a complex vector space with a scalar product (·|·) antilinear wrt the first argument. Let H be a Hilbert space. Let U (H) denote the set of unitary operators on H. Recall that (3.1) g h → W π (h) ∈ U (H)
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is a CCR representation over g in H if i
W π (h1 )W π (h2 ) = e− 2 Im(h1 |h2 ) W π (h1 + h2 ),
h1 , h2 ∈ g.
We say that a vector Ψ ∈ H is regular if R t → W π (th)Ψ, h ∈ g π is continuous. Let Hreg be the set of regular vectors – the regular sector of (3.1). It π is easy to see that Hreg is a closed subspace of H invariant under (3.1). We say that π . The field operator associated to the representation π (3.1) is regular if H = Hreg and h ∈ g is the self-adjoint operator defined as follows: Ψ ∈ Dom(φπ (h)) iff there exists d φπ (h)Ψ = W π (th)Ψ . idt t=0 π . The creation and annihilation Clearly, Dom(φπ (h)) is contained and dense in Hreg operators associated to the representation π are defined as
1 aπ (h) := √ (φπ (h) + iφπ (ih)), 2
1 aπ∗ (h) := √ (φπ (h) − iφπ (ih)). 2
For further reference let us note the identities 1 W π (ig)aπ∗ (h)W π (−ig) = aπ∗ (h) + √ (g|h), 2 1 W π (ig)aπ (h)W π (−ig) = aπ (h) + √ (h|g). 2
3.2
The Fock representation
Let h be a Hilbert space. Γs (h) will denote the symmetric Fock space over h. Ω will denote the corresponding vacuum vector and N the number operator. If h ∈ h, then a∗ (h) denotes the corresponding creation operator, that is the operator defined on finite particle vectors Φ as √ a∗ (h)Φ := h ⊗s N + 1Φ. The same symbol a∗ (h) denotes the closure of this operator. The annihilation operator is defined as a(h) := a∗ (h)∗ and the field and Weyl operators are 1 φ(h) := √ (a∗ (h) + a(h)), 2
W (h) := eiφ(h) .
It is well known that h h → W (h) ∈ U (Γs (h)),
(3.2)
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is a regular CCR representation. It is called the Fock representation. (See [BR], [BSZ].) If f ∈ h, then W (−if )Ω is called the coherent vector centered at f . Note that it satisfies √ 2 a(h)W (−if )Ω = (h|f )W (−if )Ω. This property characterizes coherent vectors, as is seen from Theorem 3.1. In the remaining part of this section, g will be a dense subspace of h and f will be an antilinear functional on g. The action of f on h ∈ g will be denoted by (h|f ), as in the scalar product. The following theorem is well known, for the proof see eg. [De]. Theorem 3.1 Let Ψ ∈ Γs (h). Suppose that for any h ∈ g we have √ 2a(h)Ψ = (h|f )Ψ. Ψ ∈ Dom(a(h)), Then the following is true: (1) If f ∈ h, then Ψ is proportional to W (−if )Ω. (2) If f ∈ h, then Ψ = 0.
3.3
Coherent representations
Note that
g h → W f (h) := W (h)eiRe(f |h) ∈ U (Γs (h))
(3.3)
is a regular CCR representation in Γs (h). We will call (3.3) the f -coherent representation. The corresponding field, creation and annihilation operators will be denoted φf (h), af ∗ (h), af (h). Clearly, φf (h) = φ(h) + Re(h|f ), af ∗ (h) = a∗ (h) + af (h) = a(h) +
√1 (f |h), 2
(3.4)
√1 (h|f ). 2
Note that the vacuum satisfies for h ∈ g: √ f 2 a (h)Ω = (h|f )Ω. Theorem 3.2 (1) If f ∈ h, then W f (h) = W (if )W (h)W (−if ), h ∈ g. (2) If f ∈ h, then there is no operator U such that W f (h) = U W (h)U ∗ , h ∈ g.
(3.5)
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Proof. (1) is immediate. To prove (2), suppose that U satisfies (3.5). Then af (h) = U a(h)U ∗ . Using a(h)Ω = 0 and the last identity of (3.4) we see that √ 2 a(h)U ∗ Ω = (h|f )U ∗ Ω, which means that U ∗ Ω satisfies the assumptions of Theorem 3.1. But U ∗ Ω = 0. Hence f ∈ h.
3.4
Coherent sectors
In this and the following subsection we consider an arbitrary CCR representation g h → W π (h) ∈ U (H).
(3.6)
We are going to describe how to extract f -coherent sub-representations of (3.6). A vector Ψ ∈ H is called an f -coherent vector for (3.6) if for any h ∈ g we have √ π Ψ ∈ Dom(aπ (h)), 2a (h)Ψ = (h|f )Ψ. Let Kfπ be the set of f -coherent vectors for (3.6). Elements of K0π will be called vacua for (3.6). Theorem 3.3 (1) Kfπ is a closed linear subspace. (2) Ψ ∈ Kfπ iff
2
1
(Ψ|W π (h)Ψ) = Ψ 2e− 4 h
+iRe(f |h)
.
(3) All vectors in Kfπ are analytic for φπ (h), h ∈ g. (4) If Ψ1 , Ψ2 ∈ Kfπ , then 2
1
(Ψ1 |W π (h)Ψ2 ) = (Ψ1 |Ψ2 )e− 4 h
+iRe(f |h)
.
Proof. (1) is obvious, since aπ (h) are closed operators. Let us prove (2) ⇐. Let Ψ ∈ H and Ψ = 1. Taking the first two terms of the Taylor expansion of 1 2
t → (Ψ|W π (th)Ψ) = Ψ 2e− 4 t
h2 +itRe(f |h)
,
we obtain (Ψ|φπ (h)Ψ) = Re(f |h),
(Ψ|φπ (h)2 Ψ) =
1 h 2 + (Re(f |h))2 . 2
Similarly, (Ψ|φπ (ih)Ψ) = −Im(f |h),
(Ψ|φπ (ih)2 Ψ) =
1 h 2 + (Im(f |h))2 . 2
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Clearly, [φπ (h), φπ (ih)] = i h 2 . Therefore, √ ( 2 aπ (h) − (h|f ))Ψ 2 = (φπ (h) + iφπ (ih) − (h|f ))Ψ 2 = Ψ| φπ (h)2 + φ(ih)2 + i[φπ (h), φπ (ih)]
−2φπ (h)Re(f |h) − 2φπ (ih)Im(f |h) + |(f |h)|2 Ψ = 0. To prove (2) ⇒, note that Dom(aπ (h)) = Dom(φπ (h)) ∩ Dom(φπ (ih)). Hence if Ψ ∈ Dom(aπ (h)), then the function R t → F (t) := (Ψ|W π (th)Ψ) is C 1 . Now d dt F (t)
=
√i
2
(aπ (h)Ψ|W π (th)Ψ) +
√i (Ψ|W π (th)aπ (h)Ψ) 2
− 2t h 2 F (t)
= (iRe(f |h) − 2t h 2 )F (t). 1 2
2
This implies that F (t) = Ψ 2 e− 4 t h +itRe(f |h) . (3) follows immediately from (2). (4) follows from (2) by polarization.
Set cl π π π H[f ] := Span {W (h)Ψ : Ψ ∈ Kf , h ∈ g},
where Spancl A denotes the closure of the span of the set A ⊂ H. Let P[fπ ] be the π π orthogonal projection onto H[f ] . We will call H[f ] the f -coherent sector of (3.6). Set Hfπ := Kfπ ⊗ Γs (h). π π π will be called the Fock sector of π. If H[f H[0] ] = H (resp. H[0] = H) we will π say that the representation W is of f -coherent type (resp. of Fock type).
Theorem 3.4 π π (1) H[f ] is an invariant subspace of (3.6) contained in Hreg . π (2) There exists a unique operator Ωπf : Hfπ → H[f ] satisfying
Ωπf Ψ⊗W (h)Ω = e−iRe(h|f ) W π (h)Ψ,
Ψ ∈ Kfπ , h ∈ g.
The operator Ωπf is unitary. (3)
Ωπf 1l⊗W (g) = e−iRe(g|f ) W π (g)Ωπf , g ∈ g.
(3.7)
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Proof. (1) is obvious. Let us prove (2). Let Ψ1 , Ψ2 ∈ Kfπ , h1 , h2 ∈ g. Then, by Theorem 3.3 (4), (e−iRe(h1 |f ) W π (h1 )Ψ1 |e−iRe(h2 |f ) W π (h2 )Ψ2 )
i
1
= (Ψ1 |Ψ2 )e 2 Im(h1 |h2 )− 4 h1 −h2
2
= (Ψ1 |Ψ2 )(W (h1 )Ω|W (h2 )Ω) Hence for αj ∈ C, Ψj ∈ Kfπ , hj ∈ g.
2
2
αj e−iRe(hj |f ) W π (hj )Ψj = αj Ψj ⊗W (hj )Ω .
j
j
π Therefore, Ωπf is well defined and isometric. It is obvious that its range equals H[f ]. To show (3), we note:
Ωπf 1l⊗W (g) Ψ⊗W (h)Ω i
= e− 2 Im(g|h) Ωπf Ψ⊗W (g + h)Ω i
= e− 2 Im(g|h) e−iRe(g+h|f ) W π (g + h)Ψ = e−iRe(g+h|f ) W π (g)W π (h)Ψ = e−iRe(g|f ) W π (g) Ωπf Ψ⊗W (h)Ω.
3.5
Comparison of coherent sectors
For h ∈ h we set
Wfπ (h) := Ωπf 1l⊗W (h) Ωπ∗ f .
Theorem 3.5 (1) The map π h h → Wfπ (h) ∈ U (H[f ])
is a regular CCR representation (2) Ωπf Ψ⊗W (h)Ω = Wfπ (h)Ψ,
Ψ ∈ Kfπ , h ∈ h.
(3) For h ∈ g we have Wfπ (h) = e−iRe(f |h) P[fπ ] W π (h), φπf (h) = P[fπ ] (φπ (h) − Re(f |h)), π∗
π √1 aπ∗ f (h) = P[f ] a (h) − 2 (h|f ) ,
aπf (h) = P[fπ ] aπ (h) − √12 (f |h) .
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(1) and (2) are immediate. π π π∗ If we multiply (3.7) from the right by Ωπ∗ f , use P[f ] = Ωf Ωf and the fact that P[fπ ] commutes with W π (h), we obtain the first identity of (3). The other follow immediately.
Remark 3.6 Let us make a comment on the purpose of introducing the operators Wfπ (h). As we see from Theorem 3.5 (3), for various applications, as long as h ∈ g we could use W π (h) instead of Wfπ (h). The advantage of the operators Wfπ (h), however, lies in the fact that they are defined for any h ∈ h. Note also that Wfπ (h) is a different object from the f -coherent representation W f (h) introduced earlier. Theorem 3.7 Let f, g be antilinear functionals on g. (1) Assume that g ∈ h. Then π = Wfπ (−ig)Kfπ . (i) Kg+f π π π π (ii) H[f ] = H[f +g] and P[f ] = P[f +g] . Consequently the f -coherent sector π H[f ] depends only on the class [f ] of f in g∗ /h. π π (iii) Set Wcoh,f (−ig) := Wfπ (−ig) π . Then Wcoh,f (−ig) is a unitary map
from Kfπ to Kfπ+g .
Kf
π (−ig) = Wfπ+g (−ig) (iv) We have Wcoh,f π (−ig)∗ . Wcoh,f
(v) Ωπf
Kπ f
π and Wcoh,f +g (ig) =
π = Ωπf+g Wcoh,f (−ig)⊗W (ig) .
π π (2) If g ∈ h, then H[f ] ⊥ H[f +g] .
Proof. Let us first prove (1.i). Wfπ is a CCR representation, hence for h ∈ g aπf (h)Wfπ (−ig) = Wfπ (−ig)(aπf (h) +
√1 (h|g)). 2
aπ (h)Wfπ (−ig) = Wfπ (−ig)(aπ (h) +
√1 (h|g)). 2
Therefore,
This implies Wfπ (−ig)Kfπ ⊂ Kfπ+g . An analogous reasoning shows the converse inclusion. (1.ii) and (1.iii) follow immediately from (1.i). To prove (1.iv) note that Wfπ (−ig) = Wfπ+g (−ig), which follows from Re(g|ig) = 0.
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Let us prove (1.v). Let Ψ ∈ Kfπ and h ∈ g. π Ωπf+g Wcoh,f (−ig)⊗W (ig) Ψ⊗W (h)Ω
=
Wfπ+g (ig)Ωπf+g Wfπ (−ig)Ψ⊗W (h)Ω
=
Wfπ+g (ig)Wfπ+g (h)Wfπ (−ig)Ψ
=
Wfπ+g (ig)Wfπ+g (h)Wfπ+g (−ig)Ψ
=
Wfπ (h)Ψ
=
Ωπf Ψ ⊗ W (h)Ω.
Let us prove (2). Let us first show that π 0 = Φ ∈ Kfπ+g ∩ H[f ] ⇒ g ∈ h.
(3.8)
In fact, for h ∈ g we have
1 1 π∗ π π∗ 1l⊗a(h) Ωπ∗ f Φ = Ωf a (h) − √ (h|f ) Φ = √ (h|g)Ωf Φ. 2 2 π π∗ But RanΩπf = H[f ] , hence Ωf Φ = 0. By Theorem 3.1, this implies g ∈ h. π π Now suppose that H[f ] is not perpendicular to H[f +g] . Then there exist vecπ π tors Ψ1 ∈ Kf , Ψ2 ∈ Kf +g , h1 , h2 ∈ g such that
(W π (h1 )Ψ1 |W π (h2 )Ψ2 ) = 0.
(3.9)
π π π Set Φ := P[fπ ] Ψ2 . Clearly, Φ ∈ H[f ] . Note that P[f ] commutes with a (h). Hence π π π Φ ∈ Kf +g . Clearly, (W (−h2 )W (h1 )Ψ1 |Φ) equals the left-hand side of (3.9), hence is nonzero. Therefore, Φ = 0. By (3.8), this implies g ∈ h.
4 Covariant CCR representations 4.1
Definition of a covariant CCR representation
In this section we describe properties of a CCR representation equipped with a dynamics. Let h and H be Hilbert spaces. Let g be a dense subspace of h. Let g h → W π (h) ∈ U (H)
(4.1)
be a CCR representation. Let ω be a self-adjoint operator on h and H a selfadjoint operator on H. We say that the triple (W π , ω, H) is a covariant CCR representation iff g is invariant w.r.t. eitω and eitH W π (h)e−itH = W π (eitω h), t ∈ R, h ∈ g.
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Operators dΓ(·)
Let dΓ(ω) be defined in the usual way as a self-adjoint operator on Γs (h). Recall that W (h) denote the Weyl operators on Γs (h). It is well known that eitdΓ(ω) W (h)e−itdΓ(ω) = W (eitω h). Therefore, the triple (W, ω, dΓ(ω)) is a covariant CCR representation (by W we mean the Fock representation over h recalled in Subsection 3.2). For further reference let us note the following identities, where we set z = √1 ωg: 2 W (ig)dΓ(ω)W (−ig) = dΓ(ω) + a∗ (z) + a(z) + (z|ω −1 z), [W (g), dΓ(ω)] = −ia∗ (z)W (g) + iW (g)a(z).
4.3
Van Hove Hamiltonians
Let hn for n ∈ N be the scale of Hilbert spaces associated with the operator ω −1 . This means that for n ≥ 0, hn = Dom(ω −n/2 ), h−n is the space of continuous antilinear functionals on hn . (An alternative notation for h−n is (|ω|−n/2 + 1)h.) Let f ∈ h−1 . Set 1 z := √ ωf ∈ (ω 1/2 + ω)h. 2 It is easy to see that
itω i R t → e 2 Im(f |e f ) W i(1 − eitω )f Γ(eitω ) ∈ U (Γs (h)). (4.2) is a strongly continuous unitary group. Therefore there exists a unique self-adjoint operator dΓf (ω), that we will call the Van Hove Hamiltonian, such that (4.2) equals eitdΓf (ω) . Formally, the van Hove Hamiltonian is given by the following expression: dΓf (ω) := dΓ(ω) + a∗ (z) + a(z) + (z|ω −1 z). (In [De] it is called a van Hove Hamiltonian of the second kind.) Note that the infimum of the spectrum of dΓf (ω) equals 0 and
eitdΓf (ω) W (h)e−itdΓf (ω) = exp iRe(f |(eitω − 1)h) W (eitω h). If f ∈ h, then dΓf (ω) = W (if )dΓ(ω)W (−if ). Theorem 4.1 (1) If f, g ∈ h−1 , then the following identities holds: i
eitdΓf +g (ω) = e 2 Im(g|e
itω
g)+iIm(f |(eitω −1)g)
W (i(1 − eitω )g)eitdΓf (ω) .
(2) If moreover g ∈ h, then: dΓf +g (ω) = W (ig)dΓf (ω)W (−ig)
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Note that if we consider the f -coherent representation h1 h → W f (h) := W (h)eiRe(f |h) ,
(4.3)
then it satisfies
eitdΓf (ω) W f (h)e−itdΓf (ω) = W f (eitω h).
Thus, the triple (W f , ω, dΓf (ω) is a covariant CCR representation.
4.4
Hamiltonian in the Fock sector
In remaining part of this section we consider a covariant representation the
W π , ω, H , as at the beginning of this section. The following facts are immediate [DG3]: Theorem 4.2 (1) The space of vacua K0π is eitH invariant. π (2) The Fock sector H[0] is eitH invariant.
On H0π = K0π ⊗ Γs (h) we define the operator π H0π := Ωπ∗ 0 HΩ0 .
Theorem 4.3 We have where K0π := H
Kπ 0
H0π := K0π ⊗ 1l + 1l ⊗ dΓ(ω), . Moreover, HΩπ0 = Ωπ0 H π .
4.5
Hamiltonian in a coherent sector
One can generalize the constructions described in the previous subsection to the case of coherent sectors. Theorem 4.4 (1) Let g be a dense subspace of h and let f be an antilinear functional on g. Then eitH Kfπ = Keπitω f . π itH -invariant. (2) If in addition f ∈ h−2 , then H[f ] is e
Proof. (1) Let Ψ ∈ Kfπ . Then (eitH Ψ|W (h)eitH Ψ) = (Ψ|W (e−itω h)Ψ) 1
2
+iRe(f |e−itω h)
1
2
+iRe(eitω f |h)
= Ψ 2 e− 4 h = Ψ 2 e− 4 h
.
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π (2) Since f ∈ h−2 , we have (eitω − 1)f ∈ h. Hence by Theorem 3.7, H[f ] = Thus it suffices to apply (1).
π H[e itω f ] .
Set π Hfπ := Ωπ∗ f HΩf .
Theorem 4.5 Suppose that g = h1 and f ∈ h−1 . (1) There exists a unique operator Kfπ on Kfπ such that Hfπ := Kfπ ⊗ 1l + 1l ⊗ dΓf (ω), (2) Ωπf Hfπ = HΩπf . Proof. Let h ∈ h1 . We first check that π
π
eitHf 1l ⊗ W (h) e−itHf =
itH Ωπ∗ W π (h)e−iRe(f |h) e−itH Ωπf f e
=
π itω Ωπ∗ h)e−iRe(f |h) Ωπf f W (e
=
1l ⊗ W (eitω h)e−iRe(f |h)+iRe(f |e
=
1l ⊗ eitdΓf (ω) W (h)e−itdΓf (ω) .
itω
h)
Since linear combinations of W (h), h ∈ h1 , are weakly dense in B(Γs (h)), for B ∈ B(Γs (h)) we have π
π
eitHf 1l⊗B e−itHf = eit1l⊗dΓf (ω) 1l⊗B e−it1l⊗dΓf (ω) . By Lemma A.2, this implies that Hfπ − 1l⊗dΓf (ω) is of the form Kfπ ⊗1l for some self-adjoint operator Kfπ on Kfπ .
4.6
Comparison of coherent sectors of a covariant representation
In this subsection we assume that g = h1 and f ∈ h−1 . Theorem 4.6 Let g ∈ h. Then π π π∗ Hg+f = Wcoh,f (−ig)⊗W (ig) Hfπ Wcoh,f (−ig)⊗W ∗ (ig). π π π∗ = Wcoh,f (−ig)Kfπ Wcoh,f (−ig). Kg+f
Proof. This follows immediately from Theorem 3.7.
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Other natural objects that can be introduced in the context of coherent sectors are the following self-adjoint operators: π π π π∗ K[f ] := Ωf Kf ⊗1l Ωf ,
dΓπ[f ] (ω) := Ωπf 1l⊗dΓf (ω) Ωπ∗ f . π Clearly, they give a natural decomposition of the operator H on the sector H[f ]: π π HP[fπ ] = K[f ] + dΓ[f ] (ω).
(4.4)
The decomposition (4.4) depends only on the class [f ] of f in g∗ /h. π π π π Theorem 4.7 If g ∈ h, then K[f +g] = K[f ] and dΓ[f +g] (ω) = dΓ[f ] (ω).
Theorem 4.7 follows from Theorem 3.7 (iii) and Theorem 4.1 (2).
5 Asymptotic CCR representations 5.1
Construction of asymptotic CCR representations
Suppose that ω is a self-adjoint operator with an absolutely continuous spectrum on a Hilbert space h. Let K be an additional Hilbert space and H a self-adjoint operator on H := K ⊗ Γs (h). Let g be a subspace of h invariant w.r.t. eitω . Throughout this section we make the following assumption: Assumption 5.A For any h ∈ g, there exists s− lim eitH 1l⊗W (e−itω h)e−itH =: W ± (h). t→±∞
It is easy to see that the above assumption implies the following theorem: Theorem 5.1 (1) We have i
W ± (h1 )W ± (h2 ) = e− 2 Im(h1 |h2 ) W ± (h1 + h2 ), In other words,
h1 , h2 ∈ g.
g h → W ± (h) ∈ U (K ⊗ Γs (h)),
(5.1)
are CCR representations. (2)
eitH W ± (h)e−itH = W ± (eitω h), h ∈ g. In other words, (W ± , ω, H) are covariant CCR representations.
We will call (5.1) the asymptotic CCR representations. Let φ± (h), a± (h), a (h), etc, denote the field, annihilation, creation operators, etc. associated with the representations (5.1). All these objects will be called “asymptotic” (or, if there will be a need for a greater precision, “outgoing/incoming”). ±∗
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Wave and scattering operators
For any antilinear functional f on g we can define the space of asymptotic f coherent vectors Kf± , the asymptotic spaces Hf± , the asymptotic Hamiltonian in the f -coherent sector Hf± , etc. The intertwining operators Ω± f will be called the f -coherent wave operators. In the physical interpretation of these concepts an important role is played by the so-called scattering operators: − Sg,f := Ω+∗ g Ωf .
Note that they satisfy Sg,f Hf− = Hg+ Sg,f . Suppose that we prepare a state in the f -coherent sector. It is natural to describe it by a density matrix ρ, which is a positive trace 1 operator on Hf− . Suppose that we measure an observable within the sector g. We can describe it by a self-adjoint operator A ∈ B(Hg+ ). Then according to the standard rules of quantum mechanics, the expectation value of the measurement is given by ∗ A. TrSg,f ρSg,f
5.3
Fock sector of asymptotic representations
Theorem 5.2 Eigenvectors of H are contained in the Fock sector K0± . Proof. We will show first the following property of Weyl operators on the Fock space: w− lim W (eitω h) = exp(− 14 h 2 ). (5.2) t→∞
Let Ψ1 , Ψ2 be vectors with a finite number of particles. Then, by the absolute continuity of ω, a(eitω h)n Ψi → 0 when t → ∞. Hence (Ψ1 |W (eitω h)Ψ2 ) = exp(− 14 h 2 )(e
− √i2 a(eitω h)
Ψ1 |e
√i
2
a(eitω h)
Ψ2 )
→ exp(− 14 h 2 )(Ψ1 |Ψ2 ). Since W (eitω h) is uniformly bounded, this proves (5.2). Assume that HΨ = λΨ. Then (Ψ|W ± (h)Ψ) = lim (Ψ|eitH 1l⊗W (e−itω h)e−itH Ψ) t→±∞
1 = lim (Ψ|1l⊗W (e−itω h)Ψ) = Ψ 2 exp(− h 2 ). t→±∞ 4
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6 Dressing operators 6.1
Dressing operator for a pair of CCR representations
Suppose that h, H are Hilbert spaces and g is a dense subspace of h. Consider two CCR representations (6.1) g h → W π (h) ∈ U (H), π (h) ∈ U (H). g h → Wren
(6.2)
For the representation (6.1) we use the notation described in the previous three sections. All the objects constructed from (6.2) will have an additional subscript ren (for “renormalized”). For instance, φπren , aπren and aπ∗ ren will denote the field, annihilation and creation operators for (6.2). Let g be an antilinear functional on g. We say that U π ∈ U (H) is a g-dressing operator between (6.2) and (6.1) if for h ∈ g, we have π W π (h)U π = U π Wren (h)eiRe(h|g) .
Theorem 6.1 (1) If h ∈ g, then
φπ (h)U π = U π (φπren (h) + Re(g|h)), aπ∗ (h)U π = U π (aπ∗ ren (h) + aπ (h)U π = U π (aπren (h) +
√1 (g|h)), 2
√1 (h|g)). 2
π π = U π Kren,f . (2) Let f be an antilinear functional on g. Then Kg+f π π π (3) Set Ucoh,f := U π π . Then Ucoh,f is a unitary operator from Kren,f to
Kfπ+g .
Kren,f
π π π (4) H[g+f ] = U Hren,[f ] . π∗ (5) Ωπg+f = U π Ωπren,f Ucoh,f ⊗1l
Proof. (1) is immediate. π Consider Ψ ∈ Kren,f . Then 1
2
π (U π Ψ|W π (h)U π Ψ) = eiRe(h|g) (Ψ|Wren (h)Ψ) = e− 4 h
+iRe(h|f +g)
Ψ 2.
This proves (2), which implies (4) and (3). To show (5), we compute for h ∈ g, Ψ ∈ Kren,f , π∗ U π Ωπren,f Ucoh,f ⊗1l Ψ⊗W (h)Ω = U π Ωπren,f U π∗ Ψ⊗W (h)Ω π = e−iRe(h|f ) U π Wren (h)U π∗ Ψ
= e−iRe(h|f +g) W π (h)Ψ = Ωπf+g Ψ⊗W π (h)Ω.
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Dressing operators for a pair of covariant representations
Suppose that H and H ren are self-adjoint operators on H and ω is a self-adjoint operator on h. We assume that g = h1 . Consider two covariant CCR representations π (W π , ω, H) and (Wren , ω, Hren ). Recall that this means that the representations of CCR (6.1) and (6.2) satisfy eitH W π (h)e−itH = W π (eitω h).
(6.3)
π π (h)e−itHren = Wren (eitω h). eitHren Wren
(6.4)
Let g ∈ h−2 and let U ∈ U (H) be a g-dressing operator between (6.1) and (6.2). We say that it is a covariant g-dressing operator between the covariant representations (6.3) and (6.4) if π
eitH U π e−itHren
i
π = U π Wren (i(1 − eitω )g)e− 2 Im(g|e i
= W π (i(1 − eitω )g)U π e 2 Im(g|e
−itω
−itω
g)
g)
,
t ∈ R.
Theorem 6.2 Suppose that g = h1 and f, g ∈ h−1 . Then π π π π∗ = Ucoh,f Kren,f Ucoh,f . Kg+f
Proof. Recall that π Ωπ∗ ren,f Hren Ωren,f
π = Hren,f
π = Kren,f ⊗ 1l + 1l ⊗ dΓf (ω),
π Ωπ∗ g+f HΩg+f
π = Hg+f
π = Kg+f ⊗ 1l + 1l ⊗ dΓg+f (ω).
Hence π∗
π
π
eitUcoh,f Kg+f Ucoh,f ⊗ eitdΓg+f (ω) π
π∗ π = Ucoh,f ⊗1l eitHg+f Ucoh,f ⊗1l π∗ itH π π ⊗1l Ωπ∗ Ωg+f Ucoh,f ⊗1l = Ucoh,f g+f e π∗ itH π π e U Ωren,f = Ωπ∗ ren,f U
= e− 2 Im(g|e
i
−itω
i
−itω
= e− 2 Im(g|e π
g)
π itω Ωπ∗ )g)eitHren Ωπren,f ren,f Wren (i(1 − e
g)+iRe(f |i(1−eitω )g)
i
= eitKren,f ⊗ e 2 Im(g|e
itω
π
1l⊗W (i(1 − eitω )g) eitHren,f
g)+iIm(f |(eitω −1)g)
W (i(1 − eitω )g)eitdΓf (ω)
π
= eitKren,f ⊗ eitdΓg+f (ω) .
6.3
Coherent asymptotic renormalization
Let g ∈ h−1 . Suppose that Hren is a self-adjoint operator on H. Set i
U (t) = e 2 Im(g|e
−itω
g) itH
e
W (i(1 − e−itω )g)e−itHren
= eitH e−it1l⊗dΓg (ω) eit1l⊗dΓ(ω) e−itHren .
(6.5)
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Clearly, Im(g|e−itω g) is well defined and (1 − e−itω )g ∈ h, therefore U (t) is well defined. Moreover, in (6.5) we used the identity from Theorem 4.1. Suppose the following assumption holds: Assumption 6.A s− limt→±∞ U (t) and s− limt→±∞ U ∗ (t) exist. Under Assumption 6.A we set U ± := s− limt→±∞ U (t). Clearly, s− limt→±∞ U (t) = U ±∗ . ∗
Theorem 6.3 Suppose Assumption 5.A holds for the Hamiltonian H and the space g = h1 . Suppose also that Assumption 6.A is satisfied. Then the following is true: (1) Assumption 5.A holds for the operator Hren with g = h1 , that means, for any h ∈ h1 , there exists ± (h). s− lim eitHren 1l⊗W (e−itω h)e−itHren =: Wren t→±∞
(2)
i
± ± ± (h1 )Wren (h2 ) = e− 2 Im(h1 |h2 ) Wren (h1 + h2 ), Wren
h1 , h2 ∈ h1 ,
± ± eitHren Wren (h)e−itHren = Wren (eitω h), h ∈ h1 . ± In other words, the triples (Wren , ω, Hren ) are covariant CCR representations.
(3) For h ∈ h1 , we have W ± (h)U ± eitH U ± e−itHren
± = U ± Wren (h)eiRe(h,g) , i
± = U ± Wren (i(1 − eitω )g)e− 2 Im(g|e i
= W ± (i(1 − eitω )g)U ± e 2 Im(g|e
−itω
−itω
g)
g)
.
Therefore, U ± are covariant g-dressing operators between the covariant CCR ± , ω, Hren) and (W ± , ω, H). representations (Wren Proof. We have eitHren 1l⊗W (e−itω h)e−itHren = eiRe((1−e
−itω
)g|e−itω h) itHren
e
1l⊗W (−i(1 − e−itω )g)W (e−itω h)
W (i(1 − e−itω )g) e−itHren = eiRe((1−e
−itω
)g|e−itω h)
U (t)∗ eitH 1l⊗W (e−itω h)e−itH U (t)
→ e−iRe(g|h) U ±∗ W ± (h)U ± , where we used limt→∞ (g|e−itω h) = 0, which follows from the Riemann-Lebesgue lemma. This proves (1), (2) and the first identity of (3).
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Let us now prove the second identity of (3). We compute: eitH U (s)e−itHren i
−isω
i
−isω
=
e 2 Im(g|e
=
e 2 Im(g|e
g) i(t+s)H
e
g)
i
e 2 Im((e
W (i(1 − e−isω )g)e−i(s+t)Hren
−i(s+t)ω
−e−isω )g|(1−e−i(s+t)ω )g)
×ei(s+t)H W (i(e−i(s+t)ω − e−isω )g)W (i(1 − e−i(s+t)ω )g)e−i(s+t)Hren =
i
e 2 Im(g|e i
−itω
×e 2 Im(g|e i
→ e 2 Im(g|e
g) iIm(g|e−isω (1−e−itω )g) i(s+t)H
e
e
−i(s+t)ω
−itω
g)
g) i(s+t)H
e
W (i(e−i(s+t)ω − e−isω )g)e−i(s+t)H
W (i(1 − e−i(s+t)ω )g)e−i(s+t)Hren
W ± (i(1 − eitω )g)U ± ,
where we used the Riemann-Lebesgue lemma to show that lim Im(g|e−isω (1 − e−itω )g) = 0.
s→∞
7 Pauli-Fierz Hamiltonians In this section we apply the abstract formalism developed in Sections 3–5 to a class of Pauli-Fierz Hamiltonians. We will formulate a set of assumptions that will guarantee a satisfactory scattering theory and the existence of a dressing operator.
7.1
Coupling Fock space ◦
Let h, K be Hilbert spaces. Let h1 and K1 be dense subspaces of h and K. Let ⊗ denote the algebraic tensor product. Let ◦
(K1 ⊗h1 ) × h1 (Ψ1 , Ψ2 ) → (Ψ1 |vΨ2 ) ∈ C be a sesquilinear form. ◦
Let Γs (h1 ) denote the algebraic Fock space over the vector space h1 . We define the annihilation form and creation forms Wick(v ∗ ) and Wick(v) as the forms on ◦ ◦
K ⊗ Γs (h) with the domain K1 ⊗Γs (h1 ) ⊂ K ⊗ Γs (h) as follows: if h1 , h2 ∈ h1 and Ψ1 , Ψ2 ∈ K1 , then √ m(Ψ2 ⊗ h2 |vh1 )(h2 |h1 )n , m = n + 1 ⊗m ⊗n (Ψ2 ⊗h2 |Wick(v)Ψ1 ⊗h1 ) = 0, m = n + 1; ⊗n ∗ (Ψ2 ⊗h⊗m 2 |Wick(v )Ψ1 ⊗h1 ) =
√ n(Ψ2 |v ∗ Ψ1 ⊗h1 )(h2 |h1 )m 0
m = n − 1, m = n − 1
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Note that if v is bounded, then Wick(v) and Wick(v ∗ ) extend to closed operators adjoint to one another. We will write Wick(v1 + v2∗ ) for Wick(v1 ) + Wick(v2∗ ). For a vector z ∈ h the operators |z) ∈ B(C, h) and its adjoint (z| ∈ B(h, C) are defined in the in the usual way: C λ → |z)λ := λz ∈ h,
h h → (z|h := (z|h) ∈ C.
Note that the usual creation and annihilation operators correspond to the case K = C: if z ∈ h, then Wick(|z)) = a∗ (z),
Wick((z|) = a(z).
For further reference let us note the identities W (ig)Wick(v)W (−ig) = Wick(v) + W (ig)Wick(v ∗ )W (−ig) =
√1 1l⊗(g|v, 2 Wick(v ∗ ) + √12 v ∗ 1l⊗|g).
(7.1)
(In the above identities we dropped the factors ⊗1lΓs (h) .) Let us note the following inequalities: Lemma 7.1 For Ψ ∈ Γs (h), R > 0 and a positive operator ω on h we have Wick(v ∗ )Ψ 2 ≤ (Ψ|1l⊗dΓ(ω)Ψ) v ∗ 1l⊗ω −1 v ;
(7.2)
∗
|(Ψ|Wick(v )Ψ)| 1 1 1 1 ≤ 1l ⊗ ω − 2 v(K + R)− 2 B(K,K⊗h) (K + R) 2 ⊗ 1lΨ 1l ⊗ dΓ(ω) 2 Ψ .
(7.3)
Proof. The proof of the first inequality can be found, e.g., in [DJ] and [GGM]. The second inequality is proved, e.g., in [GGM, Corollary 3.10]). For the reader’s convenience we will show how the first inequality implies the second. Set v˜ := v(K + R)−1/2 . Now
|(Ψ|Wick(v ∗ )Ψ)| = | (R + K)1/2 ⊗1l Ψ | Wick(˜ v ∗ )Ψ | ≤ (R + K)1/2 ⊗1l Ψ Wick(˜ v ∗ )Ψ ≤ (R + K)1/2 ⊗1l Ψ 1⊗ω −1/2 v˜ 1l ⊗ dΓ(ω)1/2 Ψ .
7.2
Pauli-Fierz Hamiltonians
Consider a positive operator K on K and a positive operator ω on h. The operator H0 := K ⊗ 1l + 1l ⊗ dΓ(ω), acting on K ⊗ Γs (h) will be called a free Pauli-Fierz Hamiltonian. The following assumption is weaker than Assumption 2.B: Assumption 7.A v is a form on K1 ⊗h1 × K1 such that lim sup ω −1/2 v(K + R)−1/2 < 1/2. R→∞
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From the inequality (7.3) one deduces the following theorem: Theorem 7.2 Under Assumption 7.A, the quadratic form Wick(v + v ∗ ) is form bounded wrt H0 with the bound less than 1. Therefore, by the KLMN theorem, we can define the Pauli-Fierz Hamiltonian as the self-adjoint operator H := H0 + Wick(v + v ∗ ), with the same form domain as H0 .
7.3
Asymptotic CCR representations for Pauli-Fierz Hamiltonians
As before, let hn be the scale of Hilbert spaces associated with ω −1 . The following assumption can be called the short range condition and is the equivalent of Assumption 2.C: Assumption 7.B There exists a subspace g ⊂ h1 ∩ Dom(ω 1/2 ) dense in h1 in the topology of h1 such that for h ∈ g and almost all t ∈ R, the operator
B(t) := 1lK ⊗(e−itω h| v + hc (1+K)−1/2 (7.4)
is bounded and
∞
B(t) dt < ∞.
0
Remark 7.3 Note that in (7.4) 1lK ⊗(e−itω h| v denotes an operator in B(K) and hc stands for its hermitian conjugate, that is the operator v ∗ 1lK ⊗|e−itω h). Theorem 7.4 Suppose Assumptions 7.A and 7.B hold. Then (1) For all h ∈ h1 there exist W ± (h) := s− lim eitH 1l⊗W (e−itω h)e−itH .
(7.5)
h1 h → W ± (h)
(7.6)
t→±∞
(2) The map
is strongly continuous. Consequently (W ± , ω, H) are two regular covariant CCR representations. (3) For all h ∈ h1 W ± (h)(i + H)−1/2 = lim eitH 1l⊗W (e−itω h)(i + H)−1/2 e−itH . t→±∞
(4) The map is norm continuous.
h1 h → W ± (h)(i + H)−1/2
(7.7)
(7.8)
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(5) for all h ∈ h1 , Dom(H + c) 2 ⊂ Dom(φ± (h)) and 1
1
φ± (h)(H + c)− 2 = s− lim eitH φ(e−itω h)eitH (H + c)− 2 . t→±∞
(6) for any > 0 the CCR representations W ± are of Fock type when restricted to 1l[,+∞[ (ω)h. Proof. For shortness, we drop 1lK ⊗ in the formulas below. We have W (e−itω h) = e−itH0 W (h)eitH0 , which implies that t → (1 + H0 )−1 W (e−itω h)(1 + H0 )−1 is C 1 and ∂t (1 + H0 )−1 W (e−itω h)(1 + H0 )−1 = −(1 + H0 )−1 [H0 , iW (e−itω h)](1 + H0 )−1
= − √12 (1 + H0 )−1 a∗ (e−itω h)W (e−itω h) − W (e−itω h)a(e−itω h) (1 + H0 )−1 . (7.9) Using the fact that eitω h ∈ Dom(ω −1/2 ) we see that (1 + H0 )−1/2 a∗ (e−itω h) is bounded. Therefore, from (7.9) and Lemma A.1 we can actually conclude that t → (1 + H0 )−1/2 W (e−itω h)(1 + H0 )−1/2 is C 1 and ∂t (1 + H0 )−1/2 W (e−itω h)(1 + H0 )−1/2
= − √12 (1 + H0 )−1/2 a∗ (e−itω h)W (e−itω h) − W (e−itω h)a(e−itω h) (1 + H0 )−1/2 . (7.10) But (1 + H0 )−1/2 (c + H)1/2 is bounded, so t → (c + H)−1/2 W (e−itω h)(c + H) is C 1 and we can replace (1 + H0 )−1/2 with (c + H)−1/2 in (7.10). Now, t → (c + H)−1 eitH W (e−itω h)e−itH (c + H)−1 is C 1 and we have −1/2
∂t (c + H)−1 eitH W (e−itω h)e−itH (c + H)−1 =
(c + H)−1 eitH i[H, W (e−itω h)]e−itH (c + H)−1
+eitH ∂t (c + H)−1 W (e−itω h)(c + H)−1 e−itH
=
eitH (c + H)−1 i[Wick(v + v ∗ ), W (e−itω h)](c + H)−1 e−itH
√1 (c + H)−1 eitH W (e−itω h) (e−itω h|v − v ∗ |e−itω h) e−itH (c + H)−1 , 2
=
where, in the last step we used the identities (7.1). Eventually, using again Lemma A.1, we can write
=
∂t eitH W (e−itω h)e−itH (c + H)−1/2
√1 eitH W (e−itω h) (e−itω h|v − v ∗ |e−itω h) e−itH (c + H)−1/2 . 2
(7.11)
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The norm of (7.11) can be estimated by
c (e−itω h|v − v ∗ |e−itω h) (1+K)−1/2 . By Assumption 7.B, if h ∈ g, this is integrable. Therefore, by the Cook method there exists lim eitH W (e−itω h)(i + H)−1/2 e−itH . (7.12) t→±∞
If h ∈ h1 , then we will find a sequence (hn ) in g such that hn → h in the norm of h1 . Clearly, hn h1 is uniformly bounded. Now, using Lemma A.1 and estimate (7.3) we get sup eitH W (e−itω h)e−itH (c + H)−1/2 − eitH W (e−itω hn )e−itH (c + H)−1/2 t
≤ sup c (W (e−itω h) − W (e−itω hn ))(1 + dΓ(ω))−1/2 t
≤ c1 ( h − hn + φ(h − hn )(1 + dΓ(ω))−1/2
≤ c2 h − hn + ω −1/2 (h − hn ) . This proves the existence of the norm limit (7.12) for an arbitrary h ∈ h1 , and also shows lim W ± (hn )(c + H)−1/2 = W ± (h)(c + H)−1/2 . n→∞
This proves (3) and (4). Now (1) and (2) follow by a simple density argument. The proof of (5) can be done as in eg [Ge2, Thm. 8.2]. It remains to prove (6). We will use the notion of the number quadratic form associated to a regular CCR representation (see eg [DG3, Sect. 4.2]). Let us fix > 0 and let f be a finite dimensional subspace in h := 1l[,+∞[ (ω)h. Let n± f be the quadratic form equal to: n± f (u, u) =
n
a± (fi )u 2 , with domain
i=1
n
Dom(a± (fi )).
i=1
where (f1 , . . . , fn ) is an orthonormal basis of f. It is easy to see that n± f does not depend on the choice of the o.n.b. of f. One can then define the number quadratic forms n± as: n± := sup n± f . f ⊂h , dimf<∞
Then (see, e.g., [DG3, Thm. 4.3]) the CCR representations W ± are of Fock type iff n± are densely defined. We claim that there exist a constant C, independent of f ⊂ h such that: 1
2 n± f (u, u) ≤ C(u, (H + c)u), u ∈ Dom((H + c) ),
(7.13)
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which implies that Dom((H + c) 2 ) ⊂ Dom(n± ) and hence completes the proof of 1 (6). In fact using (5), we obtain for u ∈ Dom((H + c) 2 ): −itH u, dΓ(eitω πf e−itω )e−itH u), n± f (u, u) = lim (e t→±∞
(7.14)
where πf is the orthogonal projection on f. Next we have eitω πf e−itω ≤ 1l[,+∞[ (ω) ≤ −1 ω, and hence dΓ(eitω πf e−itω ) ≤ −1 dΓ(ω) ≤ C(H + c), uniformly w.r.t. f. By (7.14) this implies (7.13) and completes the proof of the theorem.
7.4
Renormalized Pauli-Fierz Hamiltonian
The following assumption is weaker than Assumption 2.D: Assumption 7.C We assume that v = |z) ⊗ 1lK + vren , z ∈ h,
1
vren ∈ B(Dom(K 2 ), K ⊗ h),
(z|(1 + ω −1 )z) < ∞, ω −1 vren (1+K)−1/2 < ∞. Set g :=
√ −1 2ω z.
Note that g ∈ ω −1/2 h. The assumption below is the equivalent of Assumption 2.F: Assumption 7.D For almost all t ∈ R, the operator
C(t) := 1lK ⊗(e−itω g| vren + hc (1+K)−1/2
is bounded and
∞
C(t) dt < ∞.
0
Introduce the renormalized Hamiltonian ∗ Hren := Kren ⊗ 1l + 1l ⊗ dΓ(ω) + Wick(vren + vren ),
where vren := v − 1lK ⊗ |z), Kren
:= K + (z|ω
−1
z) − 1l⊗(ω −1 z| v − v ∗ 1l⊗|ω −1 z)
∗ 1l⊗ |ω −1 z). = K − (z|ω −1 z) − 1l⊗(ω −1 z |vren − vren
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Scattering Theory of Infrared Divergent Pauli-Fierz Hamiltonians
Note that if g ∈ h, then
571
H = W (ig)Hren W (−ig).
As in (6.5), set i
U (t) = e 2 Im(g|e
−itω
g) itH
e
W (i(1 − e−itω )g)e−itHren .
Theorem 7.5 (1) Suppose Assumptions 7.A, 7.C and 7.D hold. Then there exist U ± := s− lim U (t). t→±∞
Moreover,
U ±∗ = s− lim U ∗ (t). t→±∞
(2) Suppose in addition Assumption 7.B. Then there exist the limits ± s− lim eitHren 1l⊗W (e−itω h)e−itHren =: Wren (h). t→±∞
±
Moreover, U are covariant g-dressing operators between the representations ± , ω, Hren) and (W ± , ω, H) and satisfy all the properties described in Sec(Wren tion 6. Proof. We have d itH −it1l⊗dΓg (ω) e dt e
= =
ieitH (Wick(v + v ∗ ) + K ⊗ 1l − 1l ⊗ a∗ (z) − 1l ⊗ a(z) − (z|ωz)) e−it1l⊗dΓg (ω)
∗ ∗ ieitH Wick(vren + vren ) + Kren ⊗ 1l + (ω −1 z|vren + vren |ω −1 z) e−it1l⊗dΓg (ω)
and d it1l⊗dΓ(ω) −itHren e dt e
Hence, d dt U (t)
=
∗ = −ieit1l⊗dΓ(ω) (Kren ⊗ 1l + Wick(vren + vren )) e−itHren .
d itH −it1l⊗dΓg (ω) e dt e
d eit1l⊗dΓ(ω) e−itHren + eitH e−it1l⊗dΓg (ω) dt
eit1l⊗dΓ(ω) e−itHren i
= ie 2 Im(g|e
−itω
g) itH
e
∗ ∗ Wick(vren + vren ) + (ω −1 z|vren + vren |ω −1 z)
∗ −W (i(1 − e−itω )g)Wick(vren + vren )W (−i(1 − e−itω )g) W (i(1 − e−itω )g)e−itHren i
= ie 2 Im(g|e
−itω
g) itH
e
∗ (e−itω g|vren + vren |e−itω g) W (i(1 − e−itω )g)e−itHren .
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Therefore
∞ 0
Ann. Henri Poincar´e
d dt U (t)(i + Hren )−1/2 dt < ∞.
This means that Assumption 6.A holds and we can apply the results of Section 6.
7.5
Covariance of renormalized objects
The renormalization depends on the splitting of v given in Assumption 7.C into a singular scalar part and the regular part. This splitting is to some extent arbitrary. In this subsection we study how various “renormalized” objects depend on this splitting. We will replace g by g˜ = g + h, for h ∈ h and denote with tildes the new objects obtained with the function g˜. Theorem 7.6 Suppose Assumptions 7.A, 7.B, 7.C and 7.D hold. Let f ∈ h−1 . Then ˜ ren = W (−ih)Hren W ∗ (ih). (1) H ± ± ˜ ren (h1 ) = W (−ih)Wren (h1 )W ∗ (ih), h1 ∈ h1 . (2) W ± ˜± (3) K ren,f = W (−ih)Kren,f . ± ˜± (4) H ren,[f ] = W (−ih)Hren,[f ] . ± ˜± (5) H ren,f = W (−ih)Hren,f .
˜ ± = W (−ih)K ± W (ih). (6) K ren,f ren,f ˜ ± = W (−ih)⊗1l H ± W (ih)⊗1l. (7) H ren,f ren,f ˜ ± and U ˜ ± = W ± (−ih)U ± 1l⊗W (ih). (8) If in addition h ∈ h1 , then there exists U Proof. Direct computation proves (1). To prove (2) we compute for h1 ∈ h1 ˜ ± (h1 ) W ren = s− limt→±∞ W (−ih)eitHren W (ih)W (e−iωt h1 )W (−ih)e−itHren W (ih) = s− limt→±∞ eiRe(h|e
−itω
h1 )
W (−ih)eitHren W (e−iωt h1 )e−itHren W (ih)
± = W (−ih)Wren (h1 )W (ih),
since (h|e−itω h1 ) → 0 when t → ±∞ by the Riemann-Lebesgue lemma.
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(1) and (2) directly imply all the statements but (8), which we prove below: ˜ (t)W (−ih) U ∗ (t)U =
eitHren e−itdΓ(ω) eitdΓg (ω) e−itdΓg˜ (ω) eitdΓ(ω) W (−ih)e−itHren
=
e− 2 Im(h|e
i
itω
h)−iIm(g|(eitω −1)h) itHren −itdΓ(ω)
e
e
W (−i(1 − eitω )h)eitdΓ(ω)
W (−ih)e−itHren
→
i
=
e− 2 Im(h|e
=
e−iIm(˜g|e
strongly
itω
itω
h)−iIm(g|(eitω −1)h) itHren
e
h)+iIm(g|h) itHren
e
W (i(1 − e−itω )h)W (−ih)e−itHren
W (−ie−itω h)e−itHren
± eiIm(g|h) Wren (−ih),
where we used the Riemann-Lebesgue lemma, and the fact that h ∈ h1 , g˜ ∈ h−1 to show that limt→∞ (˜ g |eitω h) = 0. Therefore ± ˜ ± = eiIm(g|h) U ± Wren U (−ih)W (ih) = W ± (−ih)U ± W (ih).
A
Appendix
In the appendix we prove a number of technical lemmas needed in Section 7.
A.1 Differentiability of operator-valued functions Lemma A.1 Consider a function ] − , [ t → C(t) ∈ B(H).
(A.1)
Suppose that for some dense subspaces B, D and Φ ∈ B, Ψ ∈ D the derivative d (Φ|C(t)Ψ) dt
(A.2)
exists. Suppose that ] − , [ t → C (t) ∈ B(H) is a continuous function and (A.2) equals (Φ|C (t)Ψ). Then (A.1) is norm differentiable and its derivative equals C (t), that means C(t + s) − C(t) lim = C (t). (A.3) s→0 s Proof. It suffices to prove (A.3) for t = 0. For Φ ∈ B and Ψ ∈ D, s
= s−1 0 (Φ| C (s1 ) − C (0) Ψ ds1 . Φ| s−1 (C(s) − C(0) − C (0) Ψ
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Hence
Φ| s−1 (C(s) − C(0) − C (0) Ψ ≤ sup{ C (s1 ) − C (0) : |s1 | < |s|} Φ Ψ . Thus
s−1 C(s) − C(0) − C (0) ≤ sup{ C (s1 ) − C (0) : |s1 | < s} → 0.
A.2 1-parameter groups of ∗-automorphisms Let R t → αt
(A.4)
be a group ∗-automorphisms of the ∗-algebra B(H). We say that it is pointwise weakly continuous, if t → (Φ|αt (A)Ψ),
A ∈ B(H), Φ, Ψ ∈ H,
is continuous. It is well known that if (A.4) is a pointwise weakly continuous group of ∗-automorphisms of the ∗-algebra B(H), then there exists a self-adjoint operator H, unique up to an additive constant, such that αt (A) = eitH Ae−itH , see [BR, vol. I, Ex. 3.2.14 and 3.2.35]. Lemma A.2 Suppose that αt and H are as above. Assume that H = H1 ⊗H2 and there exists a self-adjoint operator H2 on H2 such that for any A2 ∈ B(H2 ) we have αt (1l⊗A2 ) = 1l ⊗ eitH2 A2 e−itH2 . Then there exists a unique self-adjoint operator H1 on H1 such that H = H1 ⊗1l + 1l⊗H2 . Proof. For A1 ∈ B(H1 ), αt (A1 ⊗1l) commutes with the operators of the form 1l⊗A2 , A2 ∈ B(H2 ). Hence αt (A1 ⊗1l) is of the form B1 ⊗1l with B1 ∈ B(H1 ). Therefore, αt1 (A1 ) ⊗ 1l := αt (A1 ⊗1l) defines a pointwise weakly continuous group of ∗-automorphisms of the ∗-algebra ˜ 1 on H1 such that B(H1 ). Therefore, there exists a self-adjoint operator H ˜
˜
αt1 (A1 ) = eitH1 A1 e−itH1 . Set ˜ =H ˜ 1 ⊗1l + 1l⊗H2 . H Clearly, ˜
˜
eitH A1 ⊗A2 e−itH = eitH A1 ⊗A2 e−itH . By the weak density, ˜
˜
eitH Ae−itH = eitH Ae−itH , ˜ 1 − c. ˜ − H is a constant. We set H1 := H for all A ∈ B(H). Hence c := H
Vol. 5, 2004
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575
A.3 Continuity of Weyl operators Proposition A.3 For h1 , h2 ∈ h,
(W (h1 ) − W (h2 ))Ψ ≤ 2sin h1 − h2 ( h1 + h2 ) Ψ + 2 sin φ(h12−h2 ) Ψ .
Proof. We have W (h1 ) − W (h2 )
i
= W (h1 )(1l − e− 2 Im(h1 |h2 ) ) i
+e− 2 Im(h1 |h2 ) W (h1 )(1l − W (h2 − h1 )). We note also that Im(h1 |h2 ) = Hence
1 Im ((h1 |h2 − h1 ) + (h1 − h2 |h2 )) . 2 1 ( h1 + h2 ) h1 − h2 . 2 s |eis − 1| = 2 sin . 2
|Im(h1 |h2 )| ≤ Moreover
Acknowledgments. Both authors were partly supported by the NATO Grant PST.CLG.979341 and by the Postdoctoral Training Program HPRN-CT-20020277. The research of J. D. was also partly supported by the Komitet Bada´ n Naukowych (the grants SPUB127 and 2 P03A 027 25). A part of this work was done during his visit to Aarhus University supported by MaPhySto funded by the Danish National Research Foundation.
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[email protected] C. G´erard D´epartement de Math´ematiques Universit´e de Paris Sud F-91405 Orsay Cedex, France email:
[email protected] Communicated by Klaus Fredenhagen Submitted 02/08/03, accepted 30/01/04
Ann. Henri Poincar´e 5 (2004) 579 – 606 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/030579-28 DOI 10.1007/s00023-004-0178-4
Annales Henri Poincar´ e
The Relation between KMS States for Different Temperatures Christian D. J¨ akel∗
Abstract. Given a thermal field theory for some temperature β −1 , we construct the theory at an arbitrary temperature 1/β . Our work is based on a construction invented by Buchholz and Junglas, which we adapt to thermal field theories. In a first step we construct states which closely resemble KMS states for the new temperature in a local region O◦ ⊂ R4 , but coincide with the given KMS state in ˆ By a weak*-compactness the space-like complement of a slightly larger region O. ˆ argument there always exists a convergent subnet of states as the size of O◦ and O tends towards R4 . Whether or not such a limit state is a global KMS state for the new temperature, depends on the surface energy contained in the layer in between ˆ We show that this surface energy can be controlled the boundaries of O◦ and O. by a generalized cluster condition.
1 Introduction A quantum field theory can be specified by a C ∗ -algebra A together with a net O → A(O),
O ⊂ R4 ,
of subalgebras associated with open, bounded space-time regions O in Minkowski space (as described in the monograph by Haag [H]; see also [HK]). The Hermitian elements of A(O) are interpreted as the observables which can be measured at times and locations in O. Technically the algebra A(O) may be thought of as being generated by bounded functions of the underlying smeared quantum fields (see, e.g., [BoY]). For instance, if φ(x) is a hermitian quantum field and if f (x) is a real test function with support region O of space-time, then in a bounded the unitary operator a := exp i dx f (x)φ(x) is a typical element of A(O). In this way the quantum fields provide a “coordinate system” for the algebra A. However, as emphasized by Haag and Kastler, only the algebraic relations between the elements of A are of physical significance. If the time evolution is given by a strongly continuous one-parameter group of automorphisms {τt }t∈R of A, then the pair (A, τ ) forms a C ∗ -dynamical system. Such a description of a QFT fits nicely into the structure of algebraic quantum statistical mechanics (see, e.g., [BR], [E], [R], [Se], [Th]) and we can therefore rely on this well-developed framework. ∗ Partially supported by the IQN network of the DAAD and the IHP network HPRN-CT2002-00277 of the European Union.
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C.D. J¨ akel
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Up till now non-relativistic quantum field theories and spin systems were favored in the framework of algebraic quantum statistical mechanics. In low dimensions the latter have been worked out in great detail (see, e.g., [BR]). Only recently the benefits of formulating thermal field theory in the algebraic framework were emphasized in a series of papers [BJu 86], [BJu 89], [BB 94], [N], [J¨a 98], [J¨ a 99], [J¨ a 04]. Equilibrium states can be characterized by first principles in the algebraic framework: equilibrium states are invariant under the time-evolution and stable against small dynamical (or adiabatic [NT]) perturbations of the time-evolution [HKTP]. Adding a few technical assumptions such a heuristical characterization of an equilibrium state leads to a sharp mathematical criterion [HHW], named for Kubo [K], Martin and Schwinger [MS]: Definition. A state ωβ over A is called a (τ, β)-KMS state for some β ∈ R∪{±∞}, if ωβ aτiβ (b) = ωβ (ba) (1) for all a, b in a norm dense, τ -invariant ∗-subalgebra of Aτ . Here Aτ ⊂ A denotes the set of analytic elements for τ . We note that there are C ∗ -dynamical systems (A, τ ), for which a KMS state exists at one and only one value β ∈ R (see [BR, 5.3.27]). But for a QFT one can specify conditions on the phase-space properties in the vacuum representation, such that KMS states exist for all temperatures β −1 > 0 [BJu 89]. These conditions exclude (see [BJu 86]) the class of models with a countable number of free scalar particles proposed by Hagedorn [Ha]. These models obey all the Wightman and Haag-Kastler axioms but they do not allow equilibrium states above a certain critical temperature. For a generic model one expects that for high temperatures and low densities the set of KMS states contains a unique element1 , whereas at low temperature it should contain many disjoint extremal KMS states and their convex combinations corresponding to various thermodynamic phases and their possible mixtures. The symmetry, or lack of symmetry of the extremal KMS states is automatically determined by this decomposition. Consequently, spontaneous symmetry breaking may occur, when we change the temperature in the sequel. Given a KMS state ωβ over A the GNS-representation (πβ , Hβ , Ωβ ) provides a net of von Neumann algebras: O → Rβ (O) := πβ A(O) , O ∈ R4 . Under fairly general circumstances KMS states for different values of the temperature β −1 lead to unitarily inequivalent GNS-representations (see [T], [BR, 5.3.35]). Hence thermal field theories for different temperatures are frequently treated as completely disjoint objects even if they refer to the same vacuum theory, i.e., even 1 For
non-relativistic fermions with pair-interaction see [J¨ a 95].
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if they show identical interactions on the microscopic level. To understand the relations between these ‘disjoint thermal field theories’ seems to be highly desirable. One simple case is well known ([Pe, 8.12.10]): Assume that the time-evolution τ can be approximated by a net of inner automorphisms such that, for a ∈ A fixed, lim τz (a) − eizhΛ ae−izhΛ = 0,
Λ→∞
hΛ = h∗Λ ∈ A,
uniformly in z on compact subsets of C. If (A, τ ) has a KMS state ωβ at some β = 0, then the net of states Λ → ωΛ , 1 1 ωβ e 2 (β−β )hΛ ae 2 (β−β )hΛ ωΛ (a) = , a ∈ A, ωβ e(β−β )hΛ has convergent subnets and the limit points ωβ := limΛ ωΛ are (τ, β )-KMS states for the new temperature 1/β (0 < β < ∞). But in general, phase transitions may occur while we change the temperature. Consequently “. . . there is no simple prescription for connecting the (τ, β)-KMS states for different β’s” (c.f. [BR, p. 78]). Nevertheless, we will provide a prescription which covers, as far as relativistic systems are concerned, the physically relevant cases. We start form a thermal field theory O → Rβ (O), whose number of local degrees of freedom is restricted in a physically sensible manner. Using a method, which is essentially due to Buchholz and Junglas [BJu 89], we construct a KMS state ωβ and a thermal field theory O → Rβ (O),
β ∈ R+ ,
for a new temperature 1/β > 0. Although we almost repeat their line of arguments, there are some nontrivial deviations due to the mathematical structure we encounter in thermal field theory. ˆ which – up In a first step we construct product states ωΛ , Λ = (O◦ , O), to boundary effects – resemble KMS states for the new temperature 1/β in a local region O◦ ⊂ R4 , but coincide with the given KMS state ωβ in the space-like ˆ complement of a slightly larger region O: ωΛ (ab) = ωΛ (a) · ωβ (b)
∀a ∈ A(O◦ ),
ˆ ). ∀b ∈ A(O
At this point our method is semi-constructive; the product states ωΛ is not uniquely fixed. Intuitively the choice of a particular product state ωΛ corresponds to a choice of the boundary conditions which decouple the local region O◦ , where the state already resembles an equilibrium state for the new temperature, from the spaceˆ Different choices ωΛ , ωΛ should manifest themselves in like complement of O. different expectation values for observables localized in between the two regions ˆ . I.e., we expect O◦ and O ωΛ = ωΛ
⇒
ˆ such that ωΛ (a) = ωΛ (a). ∃a ∈ A(O◦ ∩ O)
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It follows from standard compactness arguments that the net of states Λ → ωΛ has convergent subnets. Whether or not these subnets converge to a global KMS states for the new temperature depends on the surface energy contained in ˆ as their size increases. Introducing an auxiliary between the two regions O◦ and O structure, which can be understood as a local purification, and assuming a cluster condition, we will control these surface energies in all thermal theories which satisfy a certain “nuclearity condition” (see, e.g., [BW], [BD’AL 90a], [BD’AL 90b], [BY] for related work). Consequently, we can single out (generalized) sequences Λi = (i) (i) ˆ such that the limit points2 O◦ , O ωβ (a) := lim ωΛi (a), i→∞
a ∈ A,
are KMS states for the new temperature 1/β (0 ≤ β ≤ ∞). We emphasize that phase transitions are not excluded by our method: by choosing different “boundary conditions” we may encounter disjoint KMS states for the new temperature in the thermodynamic limit. Loosely speaking, we provide a method to heat up or cool down a quantum field theory.
2 Definitions and preliminary results For the Lagrangian formulation of a thermal field theory we refer the reader to the books by Kapusta [Ka], Le Bellac [L] and Umezawa [U], and the excellent review article by Landsman and van Weert [LvW]. Recent work in the Wightman framework can be found in [BB 92], [BB 95], [BB 96], [St]. In this section we will outline the basic structure of a thermal field theory in the algebraic framework.
2.1
List of assumptions
Although it would be more natural – from the viewpoint of algebraic quantum statistical mechanics – to start from a C ∗ -dynamical system (A, τ ) and then characterize equilibrium states ωβ and thermal representations πβ with respect to the dynamics, we will assume here that we are given a thermal field theory O → Rβ (O) acting on some Hilbert space Hβ . How we can reconstruct a C ∗ -dynamical system (A, τ ) from the W ∗ -dynamical system (Rβ , τˆ) is well known and will be indicated in the next subsection (ˆ τ will be defined in (2.1)). We now provide a list of assumptions: i) (Thermal QFT). A thermal QFT is specified by a von Neumann algebra Rβ , acting on a separable Hilbert space Hβ , together with a net (Net structure) 2 We
O → Rβ (O),
O ⊂ R4 ,
have simplified the notation here. In fact, we will have to adjust the relative sizes of a (i) ˆ (i) ) of space-time regions. triple Λi = (O◦ , O (i) , O
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of subalgebras associated with open bounded space-time regions O in Minkowski space. The net O → Rβ (O) satisfies (Isotony)
Rβ (O1 ) ⊂ Rβ (O2 ) if
O1 ⊂ O2
Rβ (O1 ) ⊂ Rβ (O2 )
O1 ⊂ O2 .
and (Locality)
if
As before, O denotes the space-like complement of O. ii) (Dynamical law). The time-evolution τˆ : t → τˆt , τˆt ( . ) = eiHβ t . e−iHβ t , is induced by a strongly continuous one-parameter group of unitaries iH e β t t∈R . It acts geometrically, i.e., τˆt Rβ (O) ⊂ Rβ (O + te) for all t ∈ R. Here e is the unit-vector in the time-direction in the Lorentz-frame distinguished by the KMS state. iii) (Unique KMS vector). There exists a distinguished vector Ωβ , cyclic and separating for Rβ , such that the associated vector state ωβ ( . ) := (Ωβ , . Ωβ ) satisfies the KMS condition (1) w.r.t. the time-evolution τ . Restricting attention to pure phases we assume that Ωβ is the unique – up to a phase – normalized eigenvector with eigenvalue {0} of Hβ . iv) (Reeh-Schlieder property). The KMS vector Ωβ is cyclic and separating for the local algebra Rβ (O), if the space-like complement of O ⊂ R4 is not empty. v) (Nuclearity condition). The thermal field theory O → Rβ (O) has the following phase-space properties: for O bounded the maps Θα,O : Rβ (O) → Hβ given by Θα,O (A) = e−αβHβ AΩβ , 0 ≤ α ≤ 1/2, are nuclear for 0 < α < 1/2 and the nuclear norm (for α 0 or α 1/2 and large diameters r of O) satisfies d −m −m Θα,O ≤ ecr α +(1/2−α) , (2) where c, m, d are positive constants. (We expect that the constant d in this bound can be put equal to the dimension of space in realistic theories, but we do not make such an assumption here. The constant m > 0 may depend on the interaction and the KMS state.) vi) (Regularity from the outside). The net O → Rβ (O) is regular from the outside, i.e., (i) ˆ ˆ (i) O. = Rβ (O), Rβ O O ˆ (i) ⊃O O
(This property can usually be achieved by defining the local algebras in an appropriate way.)
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vii) (Cluster assumption). Let O◦ and O be two space-time regions such that O◦ + te ⊂ O for |t| < δ◦ . Let J denote the modular conjugation (see Subsection 2.3) for the pair (Rβ , Ωβ ). Let Mj ∈ Rβ O◦ ∨ JRβ O◦ J and Nj ∈ Rβ O ∨ JRβ O J . Then, for δ◦ large compared to the thermal wave-length β, N N Mj Nj , (Ωβ , Mj Ωβ )(Ωβ , Nj Ωβ ) − (Ωβ , Mj Nj Ωβ ) ≤ c r◦d δ◦−γ · j=1
j=1
(3) where c , d and γ are positive constants which do not depend on O◦ or O. Here r◦ denotes the diameter of O◦ . Remarks i) The Reeh-Schlieder property is a consequence [J¨a 00] of additivity3 and the relativistic KMS condition proposed by Bros and Buchholz [BB 94]. If the KMS state is locally normal w.r.t. the vacuum representation, then the standard KMS condition (together with additivity of the net in the vacuum representation) is sufficient to ensure the Reeh-Schlieder property of the KMS vector Ωβ [J]. ii) If the KMS state is locally normal w.r.t. the vacuum representation, then it is sufficient to assume that (i) ˆ (i) O, ˆ = R(O), O R O ˆ (i) ⊃O O
holds true in the vacuum representation. For the free scalar field this property was shown by Araki [A 64]. iii) One might try to establish the cluster condition starting from a sharper nuclearity condition. For instance, we might assume that the map Θα,O : Rβ (O) → Hβ given by α > 0, Θα,O (A) = e−α|Hβ | A − (Ωβ , AΩβ ) Ωβ , is nuclear too and satisfies (for αm large in comparison with rd ) the following bound on its nuclear norm Θα,O ≤ c · rd α−m . Formally the bound on the nuclear norm Θα,O follows from taking the limit αm large in comparison with rd in the expression exp(crd α−m ) − 1, where the one is due to the subtraction of the thermal expectation value. (The expression exp(crd α−m ) should provide an upper bound for the nuclear norm of the map A → exp(−α|Hβ |)AΩβ , where α > 0.) 3 The net O → R (O) is called additive if ∪ O = O ⇒ ∨ R (O ) = R (O). Here ∨ R (O ) i i i β i i β i β β denotes the von Neumann algebra generated by the algebras Rβ (Oi ).
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iv) The product state appearing in (3) is induced by a product vectors χi , which satisfies (χi , M N χi ) = (Ωβ , M Ωβ )(Ωβ , N Ωβ ) (i) for M ∈ Mβ O◦ and N ∈ Mβ O(i) . The convergence of the product vector χ → Ωβ follows from ∪O Mβ (O) = 1 (see [D’ADFL]).
2.2
The restricted C ∗ -dynamical system
If the weakly continuous one-parameter group τˆ : t → τˆt fails to be strongly continuous, then we can reconstruct the underlying C ∗ -dynamical system by a suitable smoothening procedure (once again we refer to [S, 1.18]): given a thermal field theory O → Rβ (O) there exists (i) a C ∗ -algebra A and a representation πβ : A → B(Hβ ) such that πβ (A) is a σ-weakly dense C ∗ -subalgebra of Rβ ; (ii) a net O → A(O) of C ∗ -subalgebras of A such that πβ A(O) is a σ-weakly dense C ∗ -subalgebra of Rβ (O) for all O ⊂ R4 ; (iii) a strongly continuous automorphism group t → τt of A such that πβ τt (a) = τˆt (πβ (a) for all a ∈ A. Moreover, the net O → A(O) satisfies isotony and locality and τ respects the local structure of the net O → A(O), i.e., τt A(O) = A(O + te) for t ∈ R. We can now introduce subalgebras Ap of almost local elements in A which are analytic with respect to time-translations [BJu 89]. For the existence of these subalgebras it is crucial that the time-evolution t → τt is strongly continuous, i.e., if we fix some a ∈ A, then limt→0 τt (a) − a = 0. Lemma 2.1. (Buchholz and Junglas). Let p ∈ N be fixed and let Ap ⊂ A be the ∗-algebra generated by all finite sums and products of operators of the form
a(f ) = dt f (t)τt (a), where f is any one of the functions f (t) = const. e−κ(t+w)
2p
(with κ > 0, w ∈ C) and a ∈ ∪O A(O) is any strictly local operator. It follows that (i) each b ∈ Ap is an analytic element with respect to τ , i.e., the operator-valued function t → τt (b) can be extended to a holomorphic function on C; (ii) each b ∈ Ap is almost local in the sense that for any r(i) > 0 there is a local operator b(i) ∈ A(O(i) ) such that b(i) − b ≤ C e−κ(r
(i)
/2)2p
,
κ > 0,
where the constant C > 0 does not depend on r(i) ; (iii) the algebra Ap is invariant under τz , z ∈ C, and norm dense in A.
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The new state ωβ , which we will construct in the sequel, will be a (τ, β )KMS state for the pair (A, τ ). More precisely, it will satisfy the KMS condition (1) for a, b ∈ Ap for some p (p will be specified in Subsection 4.3). As we have just seen, Ap is a norm dense, τ -invariant subalgebra of Aτ . Remark. If the new state ωβ is locally normal w.r.t. πβ , then one might expect that the KMS condition extends to F :=
Rβ (O)
C∗
.
O∈R4
However, the representations πβ and πβ of F will be inequivalent for β = β, and therefore the weak closures πβ (F ) and πβ (F ) will in general be non-isomorphic.
2.3
The opposite net of local algebras
By assumption the KMS vector Ωβ is cyclic and separating for Rβ . Thus TomitaTakesaki theory applies: the polar decomposition S = J∆1/2 of the closeable operator S◦ : AΩβ → A∗ Ωβ , A ∈ Rβ , provides a conjugate-linear isometric mapping J from Hβ onto Hβ and a positive self-adjoint (in general, unbounded, but densely defined and invertible) operator ∆ acting on Hβ . The modular conjugation J satisfies J 2 = 1 and J∆1/2 AΩβ = A∗ Ωβ
∀A ∈ Rβ .
∆ is called the modular operator. J induces a ∗-anti-isomorphism j : A → JA∗ J between the algebra of quasi-local observables Rβ and its commutant (Tomita’s theorem). The opposite net O → j Rβ (O) , O ⊂ R4 , provides a perfect mirror image of the net of local observables. The unitary operators ∆is , s ∈ R, induce a one-parameter group of ∗-automorphism σ : s → σs of Rβ , s ∈ R, A ∈ Rβ . σs (A) = ∆is A∆−is , σ is called the modular automorphism. Takesaki has shown that ωβ is a (σ, −1)KMS state. Moreover, σ is uniquely determined by this condition and consequently ∆is = exp −isβHβ . We conclude that in a thermal field theory the modular automorphism σ coincides – up to a scaling factor – with the time-evolution τˆ. Consequently, the modular automorphism respects the net structure too, i.e., ∀s ∈ R. (4) σs Rβ (O) = Rβ (O + sβ · e) The real parameter β ∈ R+ appearing (until now β was just a dummy index) in (4) distinguishes a length scale, which is called the thermal wave-length. In fact, we
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can turn the argument up side down: given a thermal field theory O → Rβ (O), it is not necessary to provide an explicit expression for the effective Hamiltonian Hβ . It is already uniquely specified by the pair (Rβ , Ωβ ): by Stone’s theorem there exists a unique self-adjoint generator Hβ such that ∆ = exp(−βHβ ). Modular theory implies that for 0 ≤ β < ∞ the operator Hβ is not semi-bounded but its spectrum is symmetric and consists typically of the whole real line [A 72], [tBW].
2.4
Doubling the degrees of freedom
We now present the first step of our construction, which can be understood as a ˆ such local purification. Consider some δ > 0 and two space-time regions O and O ˆ that O + te ⊂ O for |t| < δ. In a forthcoming paper [J¨ a 04] we will show that the so-called split property for the net of von Neumann algebras O → Rβ (O) follows from the nuclearity condition (2). It asserts that there exists a type I factor N such that ˆ (5) Rβ (O) ⊂ N ⊂ Rβ (O). Remark. If the KMS state is locally normal w.r.t. the vacuum representation, then the split property for the vacuum representation automatically implies the split property for the thermal representation. The following result is a consequence of the split inclusion (5). Lemma 2.2. Let O be an open and bounded space-time region. Then the von Neumann algebra Mβ (O) := Rβ (O) ∨ j Rβ (O) is naturally isomorphic to the tensor product of Rβ (O) and j Rβ (O) . I.e., there exists a unitary operator V : Hβ → Hβ ⊗ Hβ such that V Mβ (O)V ∗ = Rβ (O) ⊗ j Rβ (O) . (6) Proof. The split property (5) implies that there exists a product vector Ωp ∈ Hβ , ˆ , such that cyclic and separating for Rβ (O) ∨ Rβ (O) (Ωp , ABΩp ) = (Ωβ , AΩβ )(Ωβ , BΩβ ) ˆ [J¨ for all A ∈ Rβ (O) and B ∈ Rβ (O) a 04]. The product vector Ωp can be utilized to define a linear operator V : Hβ → Hβ ⊗ Hβ by linear extension of V ABΩp = AΩβ ⊗ BΩβ ,
(7)
ˆ . The operator V is unitary. Inspecting (7) we where A ∈ Rβ (O) and B ∈ Rβ (O) find ˆ V ∗ = 1 ⊗ Rβ (O) ˆ . V Rβ (O)V ∗ = Rβ (O) ⊗ 1 and V Rβ (O) (8) ˆ implies that the von Neumann algebra Mβ (O) The inclusion j Rβ (O) ⊂ Rβ (O) is naturally isomorphic to the tensor product of Rβ (O) and j Rβ (O) and the relation (6) is a consequence of (8).
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Remark. The algebras Rβ (O) and j Rβ (O) are weakly statistically independent, i.e., 0 = A ∈ Rβ (O) and 0 = B ∈ j Rβ (O) implies AB = 0 (Schlieder property) [J¨ a 04]. In this sense one can speak of a doubling of degrees of freedom. The elements of Mβ (O) will in general not show analyticity properties with respect to Ωβ . Thus it seems that the essence of a thermal field theory gets lost, when we ‘double the degrees of freedom’ and consider the net O → Mβ (O) instead of the net of observables O → Rβ (O). But, due to the natural tensor product structure of Mβ (O), we can recover certain analyticity properties w.r.t. Ωp and a new auxiliary one-parameter group of unitary operators: Definition. A one-parameter group of unitary operators s → ∆−is p : Hβ → Hβ , s ∈ R, and an anti-unitary operator Jp : Hβ → Hβ are given by linear extension of −is ∗ ∆−is ∆ AΩβ ⊗ ∆is BΩβ , s ∈ R, (9) p ABΩp := V and, respectively,
Jp ABΩp := V ∗ JAΩβ ⊗ JBΩβ ,
ˆ . where A ∈ Rβ (O) and B ∈ Rβ (O) By Stone’s theorem there exists a unique self-adjoint operator Hp such that ∆p = e−βHp
and
Hp Ωp = 0.
ˆ . It follows from The vector Ωp ∈ Hβ is cyclic and separating for Rβ (O) ∨ Rβ (O) the definition (7) of V and the Reeh-Schlieder property of Ωβ that the product vector Ωp is cyclic (and of course separating) for Mβ (O) too. Theorem 2.3. Let O◦ and O be two space-time regions such that O◦ + te ⊂ O for |t| < δ◦ . Then respects the local structure of Mβ (O) for |s| sufficiently small, i.e., (i) ∆−is p is ∆−is p Mβ (O◦ )∆p ⊂ Mβ (O◦ + sβ · e)
∀|sβ| < δ◦ .
(10)
coincides for a ∈ A(O◦ ) and |sβ| < δ◦ – up (ii) the group of unitaries s → ∆−is p to rescaling – with the time-evolution, i.e., is ∆−is p πβ (a)∆p = πβ τsβ (a) for |sβ| < δ◦ . Proof. The inclusion (10) follows from the definition (9) of ∆p and the inclusions τˆt Rβ (O◦ ) ⊂ Rβ (O◦ + te) and τˆt j(Rβ (O◦ )) ⊂ j Rβ (O◦ + te) , which hold for |t| < δ◦ .
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ˆ such that Lemma 2.4. Consider some δ > 0 and two space-time regions O and O ˆ O + te ⊂ O for |t| < δ. Let Ωp and ∆p be the product vector specified in (7) α and the operator defined in (9). Then Mβ (O)Ωp is in the domain D ∆α p of ∆p 1/2
for 0 ≤ α ≤ 1/2. Moreover, the identity Jp ∆p M Ωp = M ∗ Ωp holds true for all M ∈ Mβ (O). Proof. By definition, Jp2 = 1, Jp Ωp = Ωp and ∗ ∗ Jp ∆1/2 A Ωβ ⊗ B ∗ Ωβ = A∗ B ∗ Ωp = (AB)∗ Ωp p ABΩp = V for all A ∈ Rβ (O) and B ∈ j Rβ (O) . Since pα ≤ max(1, p) < 1 + p for 0 ≤ α ≤ 1 1/2 and p > 0, the spectral resolution of the positive operator ∆p implies that α Mβ (O)Ωp ⊂ D ∆p for 0 ≤ α ≤ 1/2 . Nevertheless, Jp and ∆p are not the modular objects associated to Mβ (O), Ωp . Theorem 2.5. Let O◦ and O be two space-time regions such that O◦ + te ⊂ O for |t| < δ◦ . Then the inclusion of von Neumann algebras Mβ (O◦ ) ⊂ Mβ (O) is a standard split inclusion and there exists a unitary operator W : Hβ → Hβ ⊗ Hβ such that W Mβ (O◦ )W ∗ = Mβ (O◦ ) ⊗ 1
and
W Mβ (O) W ∗ = 1 ⊗ Mβ (O) .
(A split inclusion A ⊂ B is called standard (see [DL]), if there exists a vector Ω which is cyclic for A ∧ B as well as for A and B.) Proof. From the split inclusions and j Rβ (O◦ ) ⊂ j N◦ ⊂ j Rβ (O) we infer that there exists a type I factor, namely N◦ ∨ j N◦ , such that Mβ (O◦ ) ⊂ N◦ ∨ j N◦ ⊂ Mβ (O). Rβ (O◦ ) ⊂ N◦ ⊂ Rβ (O)
All infinite type I factors with infinite commutant on the separable Hilbert space Thus Hβ are unitarily equivalent to B(Hβ ) ⊗ 1 ([KR], Chapter 9.3). a there exists unitary operator W : Hβ → Hβ ⊗ Hβ such that N◦ ∨ j N◦ = W ∗ B(Hβ ) ⊗ 1 W . Now consider ωβ ( . ) := (Ωβ , . Ωβ ) and ωp ( . ) := (Ωp , . Ωp ) as two normal states over Mβ (O◦ ) and Mβ (O) , respectively. Set φp (C) := (ωβ ⊗ ωp )(W CW ∗ )
∀C ∈ Mβ (O◦ ) ∨ Mβ (O) .
Then φp is a normal state over Mβ (O◦ ) ∨ Mβ (O) , which satisfies φp (M N ) = ωβ (M ) · ωp (N ) for all M ∈ Mβ (O◦ ) and N ∈ Mβ (O) . In the presence of a separating vector each normal state is a vector state ([KR, 7.2.3]). In fact, there exists a unique vector η in the natural positive cone P Mβ (O◦ ) ∨ Mβ (O) , Ωβ such that (η , M N η) = φp (M N ) = (Ωβ , M Ωβ )(Ωp , N Ωp )
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for all M ∈ Mβ (O◦ ) and N ∈ Mβ (O) ([BR, 2.5.31]). Thus the operator W : Hβ → Hβ ⊗ Hβ can now be specified by linear extension of W M N η = M Ωβ ⊗ N Ωp ,
(11)
where M ∈ Mβ (O◦ ) and N ∈ Mβ (O) . Consequently, W Mβ (O◦ )W ∗ = Mβ (O◦ ) ⊗ 1
and W Mβ (O) W ∗ = 1 ⊗ Mβ (O) .
The vector Ωβ is cyclic and separating for Mβ (O◦ ) and the vector Ωp is cyclic and separating for Mβ (O) . Thus the vector Ωβ ⊗ Ωp is cyclic and separating for Mβ (O◦ ) ⊗ Mβ (O) and the split inclusion Mβ (O◦ ) ⊂ Mβ (O) is standard.
3 Localized excitations of a KMS state Taking the auxiliary structure developed in the previous section into account, we can now adapt the method of Buchholz and Junglas to thermal representations.
3.1
Consequences of the nuclearity condition
Imposing strict localization on an excitation (see Proposition 3.3 (iii) below) does not lead to a convenient notion. The split property provides the key to a more convenient definition of a localized excitation. However, it leaves a lot of freedom, ˆ for instance one could request additional properties for some subregion in O◦ ∩ O. In this sense the following definition only provides one possible choice, fixed by choosing a specific product vector η. ˆ denote three space-time regions such that for some Definition. Let O◦ , O and O δ◦ , δ > 0 ˆ ∀|t| < δ. (12) O◦ + te ⊂ O ∀|t| < δ◦ and O + te ⊂ O ˆ of localized excitations of the KMS The Hilbert space HΛ ⊂ Hβ , Λ := (O◦ , O, O), state ωβ is given by HΛ := Mβ (O◦ )η. (13) The projection onto HΛ is denoted by EΛ . Notation. HereMβ (O◦ ) denotes the von Neumann algebra generated by Rβ (O◦ ) and j Rβ (O◦ ) and η ∈ Hβ denotes the unique4 product vector in the natural positive cone P Mβ (O◦ ) ∨ Mβ (O) , Ωβ satisfying (η , M N η) = (Ωβ , M Ωβ )(Ωp , N Ωp )
(14)
4 Fixing the product vector with respect to some natural positive cone is mathematically convenient, but not necessary. In fact, we expect that different ‘boundary conditions’ are realized by different choices of η. In the thermodynamic limit different choices of the boundary conditions might lead to different phases.
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for all M ∈ Mβ (O◦ ) and N ∈ Mβ (O) . As before, Ωp denotes the unique product ˆ , Ωβ satisfying vector in the natural positive cone P Rβ (O) ∨ Rβ (O) (Ωp , ABΩp ) = (Ωβ , AΩβ )(Ωβ , BΩβ )
(15)
ˆ . for all A ∈ Rβ (O) and B ∈ Rβ (O) Note that W – as specified in (11) – is unitary and W M N W ∗ = M ⊗ N for M ∈ Mβ (O◦ ) and N ∈ Mβ (O) . Using the isometry W we can write HΛ = W ∗ Mβ (O◦ )Ωβ ⊗ Ωp = W ∗ (Hβ ⊗ Ωp ) and EΛ = W ∗ (1 ⊗ PΩp )W . Here PΩp ∈ B(Hβ ) denotes the projection onto C · Ωp . The following proposition summarizes the properties of the Hilbert space HΛ . It justifies the claim stated at the beginning of this subsection. ˆ of space-time regions as specified Proposition 3.1. Given a triple Λ := (O◦ , O, O) in (12) we find: (i) The Hilbert space HΛ is invariant under the action of elements of Mβ (O◦ ), i.e., Mβ (O◦ )HΛ = HΛ . (ii) Vectors in HΛ induce product states for the pair Mβ (O◦ ), Mβ (O) : if Ψ ∈ HΛ , then (Ψ , M N Ψ) = (Ψ , M Ψ)(Ωp , N Ωp ) for all M ∈ Mβ (O◦ ) and N ∈ Mβ (O) . (iii) The vector states associated with HΛ represent strictly localized excitations of the KMS state, i.e., they coincide with the original KMS state ωβ in the ˆ if Ψ ∈ HΛ , then space-like complement of O: (Ψ , πβ (a)Ψ) = ωβ (a)
ˆ ∀a ∈ Ac (O).
ˆ denotes the C ∗ -algebra generated by {a ∈ A : [a, b] = 0 ∀b ∈ Here Ac (O) ˆ ˆ in B(Hβ ). A(O)} and not the commutant of πβ A(O) (iv) HΛ is complete in the following sense: to every normal state φ on Mβ (O◦ ) there exists a Φ ∈ HΛ such that (Φ , M Φ) = φ(M ) for all M ∈ Mβ (O◦ ). Proof. We simply adapt the proof of the corresponding result by Buchholz and Junglas to our situation: (i) follows from the definition; (ii) follows from (13) and (14); (iii) follows from (13), (14) and (15). ˜ ∈ (iv) Since Mβ (O◦ ) has a cyclic and separating vector, there exists a vector Φ Hβ which induces the given normal state φ on Mβ (O◦ ). It follows that the ˜ ⊗ Ωp ) ∈ HΛ satisfies (iv). vector Φ := W ∗ (Φ We need one more lemma, in order to show that the restriction of the operator ∆α p to the subspace HΛ is trace class for 0 < α < 1/2.
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Lemma 3.2. Assume that the nuclearity condition (2) holds true. It follows that (i) the maps ϑα,O : Mβ (O) → Hβ , M → ∆α p M Ωp ,
0 ≤ α ≤ 1/2,
are nuclear for 0 < α < 1/2; (ii) the nuclear norm of ϑα,O is bounded by d −m −m , ϑα,O ≤ e2cr α +(1/2−α)
c, m, d > 0,
where r denotes the diameter of O and c, m, d are the constants appearing in the bound (2) on the nuclear norm of the map Θα,O . Proof. Let A ∈ Rβ (O) and B ∈ j Rβ (O) . By definition, ϑα,O (AB) = V ∗ ∆α AΩβ ⊗ ∆−α BΩβ . The maps A → ∆α AΩβ and B → ∆−α BΩβ are nuclear for 0 < α < 1/2. The tensor product of two nuclear maps itself is a nuclear map and the norm is bounded by the product of the nuclear norms [P]. ˆ be a triple of space-time regions as specified in Proposition 3.3. Let Λ(O◦ , O, O) (12). Assume the nuclearity condition (2) holds true. It follows that the operator ∆α p EΛ , acting on the Hilbert space Hβ , is of trace-class for 0 < α < 1/2, and 2cr d α−m +(1/2−α)−m Tr |∆α , c, m, d > 0, E | ≤ e Λ p where r denotes the diameter of O and c, m, d are the constants appearing in the bound (2) on the nuclear norm of the map Θα,O . Proof. The proof of this proposition is more or less identical to the one given by Buchholz and Junglas [BJu 89] for the vacuum case. We present it for completeness only. i) The first step is to construct a convenient orthonormal basis of HΛ . Let {Ψi }i∈N be an orthonormal basis of Hβ with Ψ1 = Ωβ . Set Ui,j := W ∗ (Mi,j ⊗ 1)W,
(16)
where Mi,j ∈ B(Hβ ) are matrix units given by Mi,j Ψ := (Ψj , Ψ)Ψi ∀Ψ ∈ Hβ . Since W ∗ B(Hβ ) ⊗ 1 W = N◦ ∨ j N◦ , we infer from (16) that Ui,j ∈ N◦ ∨ j N◦ . Furthermore, ∗ = Uj,i , Ui,j
Ui,j Uk,l = δj,k Ui,l ,
and
s − limN →∞
N i=1
Ui,i = 1 .
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Combining (11) and (16) we find Ui,1 η = W ∗ (Ψi ⊗ Ωp ). Thus {Ui,1 η}i∈N is the desired orthonormal basis of HΛ . Note that (11) holds true for all N ∈ N◦ ∨ N η = N Ωp . Consequently, we can introduce an isometry j(N◦ ) . Therefore I ∈ N◦ ∨ j N◦ by setting ∀N ∈ N◦ ∨ j(N◦ ) .
N η = IN Ωp
We can now represent the orthonormal basis {Ui,1 η}i∈N by vectors Γi := Ui,1η = Ui,1 IΩp , where Ui,1 I ∈ N◦ ∨ j N◦ ⊂ Mβ (O). It follows that Γi ∈ D ∆α p for for 0 < α < 1/2. 0 < α < 1/2 and i ∈ N. Especially, η =: Γ1 ∈ D ∆α p α α ii) Polar decomposition of the closeable operator ∆α p EΛ yields ∆p EΛ = F ·|∆p EΛ |, where F is a partial isometry with range in HΛ . Introducing a set of linear functionals φi (which can be chosen to be continuous with respect to the ultra-weakly topology induced by Mβ (O) [BD’AL 90b]) and vectors Φi ∈ Hβ corresponding to the nuclear map ϑα,O we obtain
Tr |∆α p EΛ | =
(Ui,1 IΩp , F ∗ ∆α p Ui,1 IΩp )
i
= Ui,1 IΩp , F ∗ ϑα,O (Ui,1 I) i
= ≤
i
n
i
n
φn (Ui,1 I) · (Ui,1 IΩp , F ∗ Φn ) |φn (Ui,1 I)| · U1,i F ∗ Φn .
Buchholz and Junglas have shown the following inequality [BJu 89]:
|ψ(Ui,1 )| · U1,i Ψ ≤ ψ Ψ,
i
continuous linear functional on Mβ (O). Confor Ψ ∈ Hβ and ψ an ultra-weakly sequently, Tr |∆α p EΛ | ≤ n φn Φn . Taking the infimum with respect to all decompositions of the respective nuclear maps we find Tr |∆α p EΛ | ≤ ϑα,O .
3.2
Local KMS states for a new temperature
Proposition 3.3 allow us to define “local quasi-Gibbs” states, which are local (τ, β )KMS states for the new temperature 1/β in the interior of O◦ and (τ, β)-KMS ˆ Before we do so, we give states for the original temperature 1/β outside of O. a precise meaning to the statement that a local excitation ωΛ of a KMS state ωβ satisfies a local KMS condition for the new temperature 1/β in a bounded region O◦ . Note that any β (0 < β < ∞) can be decomposed into some α (0 < α < 1/2) and some (minimal) n ∈ N such that β = αnβ.
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Definition. Let β > 0 and let n ∈ N be the smallest natural number such that nαβ = β for some α (0 ≤ α ≤ 1/2). A state ωΛ satisfies the local KMS condition at temperature 1/β in some bounded space-time region O◦ ⊂ R4 if for any subregion O◦◦ ⊂ O◦ whose closure is contained in the interior of O◦ there exists some δ◦◦ > 0 and a function Fa,b for every pair of operators a, b ∈ A(O◦◦ ) such that (i) Fa,b is defined on Gn,α := {z ∈ C | 0 < z < nαβ} \ {z ∈ C | |z| ≥ δ◦◦ , z = kαβ, k = 1, . . . , n − 1}; (ii) (iii) (iv) (v)
Fa,b Fa,b Fa,b The
is bounded and analytic in its domain of definition; is continuous for z kαβ and z kαβ, k = 1, . . . , n − 1; is continuous at the boundary for z 0 and z nαβ; respective boundary values are Fa,b (t) = ωΛ aτt (b) and Fa,b (t + inαβ) = ωΛ τt (b)a for |t| < δ◦◦ .
(17)
Remark. To heat up the system locally is quite simple: For β < β/2 we find n = 1, i.e., no cuts appear in G1,α = {z ∈ C | 0 < z < αβ}. To cool down the system locally is more delicate. One needs at least n cuts, where n is the minimal natural number such that β = nαβ (0 < α < 1/2). Whether or not it is useful to operate with more cuts then necessary is unknown to us. ˆ be a triple of space-time regions, as specified Proposition 3.4. Let Λ := (O◦ , O, O) in (12). Let n ∈ N be the minimal natural number such that β = nαβ, 0 < α < 1/2. Set, for n and α fixed, n EΛ ∆α p EΛ n and ωΛ (a) := Tr ρΛ πβ (a) ∀a ∈ A. ρΛ := (18) Tr ∆α p EΛ Then ρΛ is a density matrix, i.e., ρΛ > 0 and Tr ρΛ = 1, and the following statements hold true: (i) The states ωΛ are product states, which coincide with the given KMS state ˆ i.e., ωβ in the space-like complement of O; ωΛ (ab ) = ωΛ (a) ωβ (b ) ˆ As before, Ac (O) ˆ denotes the C ∗ -algebra for all a ∈ A(O◦ ) and b ∈ Ac (O). ˆ generated by {a ∈ A | [a, b] = 0 ∀b ∈ A(O)}. (ii) The states ωΛ are local (τ, nαβ)-KMS states for the space-time region O◦ . ˆ → R4 the denominator in (18) might go to ∞ or 0. In any Remark. For O◦ , O, O case we will leave the representation: we will have no operator convergence, neither in the weak nor in the strong sense and therefore we can only rely on expectation values. After performing the thermodynamic limit, we will use these expectation
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values to construct a new representation and check whether the new state satisfies the KMS condition [Na]. We will see that it will do so, under the assumptions we have imposed on the phase-space properties of our thermal field theory. ˆ and let PΩp denote the projection Proof. (i) Let a ∈ A(O◦ ) and b ∈ Ac (O) onto C · Ωp . Since EΛ ∈ Mβ (O◦ ) ⊂ πβ A(O◦ ) , it follows that [EΛ , πβ (a)] = 0. Moreover, EΛ = W ∗ (1 ⊗ PΩp )W implies EΛ πβ (b )EΛ = ωβ (b )EΛ
ˆ ∀b ∈ Ac (O).
Using the cyclicity of the trace we find n Tr EΛ ∆α πβ (a)EΛ πβ (b )EΛ p EΛ n ωΛ (ab ) = Tr ∆α p EΛ = ωΛ (a)ωβ (b ). (ii) Consider the case n = 2. Let δ◦◦ > 0 and O◦◦ be an open space-time region such that O◦◦ +te ⊂ O◦ for |t| < δ◦◦ . Let a, b ∈ A(O◦◦ ). By assumption, aτt (b) ∈ A(O◦ ) for |t| < δ◦◦ . Set (1) Fa,b (z)
−iz/β
:=
Tr πβ (a)EΛ ∆p
α+iz/β
πβ (b)∆p 2 Tr ∆α p EΛ
EΛ ∆α p EΛ
(1)
for 0 < z < αβ. The function Fa,b (z) is analytic in its domain and continuous −it/β it/β at the boundary. We recall that ∆p πβ (b)∆p = πβ τt (b) ∈ πβ A(O◦ ) for |t| < δ◦◦ . Using once again the cyclicity of the trace and EΛ ∈ πβ A(O◦ ) , we conclude that α Tr πβ (a)EΛ πβ τ z (b) ∆α p EΛ ∆p EΛ (1) lim Fa,b (z) = 2 z 0 Tr ∆α p EΛ 2 Tr πβ aτ z (b) EΛ ∆α p EΛ = ∀|z| < δ◦◦ . 2 Tr ∆α p EΛ Thus
(1) lim Fa,b (z) = ωΛ aτ z (b)
z 0
On the other hand, for |z| < δ◦◦ , (1) lim F (z) z αβ a,b
∀|z| < δ◦◦ .
α Tr πβ (a)EΛ ∆α p πβ τ z (b) EΛ ∆p EΛ = 2 Tr ∆α p EΛ α Tr πβ (a)EΛ ∆p EΛ πβ τ z (b) ∆α p EΛ = . 2 Tr ∆α p EΛ
(19)
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For αβ < z < 2αβ we set (2) Fa,b (z)
−α−iz/β
2α+iz/β
Tr πβ (a)EΛ ∆α πβ (b)∆p p EΛ ∆p := 2 α Tr ∆p EΛ
EΛ
.
(2)
The function Fa,b (z) is analytic in its domain and continuous at the boundary. By definition, α Tr πβ (a)EΛ ∆α (2) p EΛ πβ τ z (b) ∆p EΛ lim F (z) = 2 z αβ a,b Tr ∆α p EΛ (1)
= lim Fa,b (z) ∀|z| < δ◦◦ . z αβ
(2)
Furthermore, Fa,b satisfies (2) lim F (z) z 2αβ a,b
α Tr πβ (a)EΛ ∆α p EΛ ∆p πβ τ z (b) EΛ = 2 Tr ∆α p EΛ 2 Tr πβ (a) EΛ ∆α p EΛ πβ τ z (b) = ∀|z| < δ◦◦ . 2 Tr ∆α p EΛ
Thus lim
z 2αβ
(2) Fa,b (z) = ωΛ τ z (b)a
∀|z| < δ◦◦ .
(20) (1)
(2)
Using the Edge-of-the-Wedge theorem [SW] we conclude that Fa,b and Fa,b are the restrictions to the upper (resp. lower) half of the double cut strip G2,α = {z ∈ C | 0 < z < 2αβ} \ {z ∈ C | |z| ≥ δ◦◦ , z = αβ} of a function
F (2) (z)
αβ < z < 2αβ, Fa,b (z) := for F (1) (z) 0 < z < αβ, a,b
a,b
defined and continuous on the closure of G2,α and analytic for z ∈ G2,α . From (19) and (20) we infer Fa,b (t) = ωΛ aτt (b) and Fa,b (t + i2αβ) = ωΛ τt (b)a for |t| < δ◦◦ . Analogous results for arbitrary n ∈ N can be established by the same line of arguments but with considerable more effort.
4 The thermodynamic limit ˆ → R4 . Since we We will now control the surface energies in the limit O◦ , O, O do not have explicit expressions for the surface energies, our approach is quite involved. The first step is to control the convergence of product vectors.
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Consequences of the cluster condition
(i) ˆ (i) be a sequence of triples Let us introduce some notation: Let Λi = O◦ , O(i) , O (i) (i) of double cones with diameters r◦ , r(i) , rˆ(i) . We consider the product vectors Ωp (i) and ηi , χi ∈ P Mβ (O◦ ) ∨ Mβ (O(i) ) , Ωβ , which satisfy (i) Ωp , AB Ω(i) = (Ωβ , AΩβ )(Ωβ , BΩβ ) p (i) ˆ for A ∈ Rβ O(i) and B ∈ Rβ O , and (i) (ηi , M N ηi ) = Ωβ , M Ωβ Ω(i) p , N Ωp (χi , M N χi ) = (Ωβ , M Ωβ )(Ωβ , N Ωβ ) (i) for M ∈ Mβ O◦ and N ∈ Mβ O(i) . So far there was no restriction on the relative size of the regions O(i) and O . We will now exploit this freedom: If the net of local observables O → Rβ (O) ˆ → C · 1 as O ˆ O. Our aim is regular from the outside, then Mβ (O) ∩ Mβ (O) is to control ηi − Ωβ . The following lemma shows that in order to do so it is sufficient to control χi − Ωβ . (i) Lemma 4.1. Let (O◦ , O(i) ) i∈N be a sequence of pairs of double cones. Then ˆ (i) }i∈N such that (12) holds true and one can find a sequence of double cones {O limi→∞ χi − ηi = 0. (i) Proof. Consider a sequences of pairs of double cones O◦ , O(i) i∈N eventually 4 exhausting each i ∈ N fixed we consider a sequence of double cones all of R . For(i,k) (i,k) ˆ ˆ O such that O O(i) for k → ∞. In order to ease the notation we k∈N set (i) (i,k) ˆ Ai := Mβ O◦ , Bi := Mβ O(i) , Ci,k := Mβ O , ˆ (i)
Di := Ai ∨ Bi , and Ei,k := Ai ∨ Ci,k . For each i ∈ N fixed, the sequence {Ei,k }k∈N of algebras satisfies Ei,k+1 ⊂ Ei,k (this follows from Ci,k+1 ⊂ Ci,k ) and ∩k Ei,k = (i,k) denote the unique product vector in the natural positive cone Di . Now let Ωp (i) ˆ (i,k) ) , Ωβ satisfying P Rβ (O ) ∨ Rβ (O (i,k) Ωp , ABΩ(i,k) = (Ωβ , AΩβ )(Ωβ , BΩβ ) p
(i,k) ˆ . Note that for Ci,k ∈ Ci,k for all A ∈ Rβ O(i) and B ∈ Rβ O (i,k) Ωp , Ci,k Ω(i,k) = (Ωβ , Ci,k Ωβ ). p If we choose product vectors ηi,k and χi in the natural cone P (Di , Ωβ ) such that (ηi,k , M N ηi,k ) = Ωβ , M Ωβ Ω(i,k) , N Ω(i,k) p p
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and (χi , M N χi ) = (Ωβ , M Ωβ )(Ωβ , N Ωβ ) for all M ∈ Ai and N ∈ Bi , then by a result of Araki [A 74] (ηi,k , Di ηi,k ) − (χi , Di χi ). sup ηi,k − χi 2 ≤ Di ∈Di , Di =1
Now assume that for each i ∈ N fixed there exist a sequence {Ei,k ∈ Ei,k | Ei,k = 1}k∈N such that (21) lim (ηi,k , Ei,k ηi,k ) − (χi , Ei,k χi ) ≥ i . k→∞
We demonstrate that this leads to a contradiction. The linear functional (ηi,k , . ηi,k ) − (χi , . χi ) is ultra-weakly continuous on the von Neumann algebra Di . Therefore the sequence {Ei,k ∈ Ei,k | Ei,k = 1}k∈N has a weak limit point w − limk→∞ Ei,k =: Di ∈ Di = ∩k Ei,k such that (ηi,k , Di ηi,k ) − (χi , Di χi ) > 1 i ∀k > ki 2 and some ki ∈ N, in contradiction to (ηi,k , Ei,k ηi,k ) − (χi , Ei,k χi ) = 0 ∀Ei,k ∈ Ei,k , ∀k ∈ N. Therefore, the assumption (21) can not hold true. It follows that there exists some ki ∈ N such that (ηi,k , Dηi,k ) − (χi , Dχi ) < i ∀k ≥ ki . sup D∈Di , D =1
ˆ (i) := O ˆ (i,ki ) , then we can choose i such that limi→∞ χi − ηi = 0. If we set O We will now show that the product vector χ converges to Ωβ if O◦ and O tend to R4 and the relative size of O◦ and O obeys the restrictions imposed by the cluster condition. (i) Lemma 4.2. Let O◦ , O(i) i∈N denote a sequence of pairs of double cones with (i) d (i) −γ (i) δ◦ = 0. It follows diameters r◦ , r(i) , i ∈ N. Assume that limi→∞ r◦ that χi − Ωβ → 0 as i → ∞. (i) Proof. Since χi ∈ P Mβ (O◦ ) ∨ Mβ (O(i) ) , Ωβ , we can again rely on the result of Araki [A 74] concerning the distance of two vectors which belong to the natural (i) positive cone P Mβ (O◦ ) ∨ Mβ (O(i) ) , Ωβ : χi − Ωβ 2 ≤ sup (χi , Di χi ) − (Ωβ , Di Ωβ ); Di =1
(i) where the supremum has to be evaluated over all elements Di ∈ Mβ O◦ ∨ Mβ O(i) . Thus limi→∞ χi − Ωβ = 0 follows from the cluster condition (3) and the assumptions concerning the relative size of O◦ and O stated in the lemma. Combining Lemma 4.1 and Lemma 4.2 we conclude that limi→∞ ηi − Ωβ = (i) ˆ (i) . 0 for an appropriate choice of the relative size of O◦ , O(i) and O
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Bounds on the quasi-partition function
(i) ˆ (i) of triples of double cones Let us consider a sequence {Λi } = O◦ , O(i) , O (i) with diameters (r◦ , r(i) , rˆ(i) ). In order to ensure that (for 0 < α < 1/2 and n ∈ N fixed) the ‘quasi-partition function’ n (i) ˆ (i) , ZΛi (α, n) := Tr EΛi ∆α , Λi = O◦ , O(i) , O p,i EΛi (i)
is bounded from below as i → ∞, it is necessary that O(i) grows rapidely with O◦ . Otherwise the energy contained in the boundary, which is necessary to decouple the local region from the outside, lessens the eigenvalues of EΛi ∆α p,i EΛi so drastically that it outruns the increase in the number of states contributing to the trace by (i) enlarging O◦ . Following once again [BJu 89] we will now demonstrate that the (i) condition on the relative size of r◦ and r(i) which we imposed in order to show that χi converges to Ωβ is already sufficient to exclude this possibility. Lemma 4.3. (Buchholz and Junglas). Let {Λi }i∈N be a sequence of triples of increasing space-time regions such that ηi − Ωβ → 0 for i → ∞. It follows that HΛi tends to the whole Hilbert space Hβ , i.e., s − limi→∞ EΛi = 1. (i) Proof. By assumption ηi = Wi∗ Ωβ ⊗ Ωp converges to Ωβ . Therefore the unitary (i) (11) fulfill Wi∗ Φ ⊗ Ωp → Φ for Φ ∈ Hβ as i → ∞. operators Wi specified in Recall that EΛi = Wi∗ 1 ⊗ PΩ(i) Wi , where PΩ(i) denotes the projection onto p
(i)
p
C · Ωp . Hence ∗ (i) EΛi Φ = Wi∗ 1 ⊗ PΩ(i) Wi Φ − Wi∗ (Φ ⊗ Ω(i) →Φ p ) + Wi Φ ⊗ Ωp p
∀Φ ∈ Hβ ,
as i → ∞. I.e., s − limi→∞ EΛi = 1. (i) (i) denote a sequence of pairs of double cones with Lemma 4.4. Let O◦ , O i∈N (i) d (i) −γ (i) (i) δ◦ = 0. It follows diameters (r◦ , r ), i ∈ N. Assume that limi→∞ r◦ that n >0 ∀n ∈ N. lim inf Tr EΛi ∆α p,i EΛi i
1/2 ∆p,i
(i)
Proof. By definition, is a positive operator. The vector Ωp is the unique eigenvector of Hp for the simple eigenvalue {0}. Let {Ωβ , Ψ1 , Ψ2 , . . .} be an or (i) (i) thonormal basis in Hβ and set Γj = Wi∗ Ψj ⊗ Ωp ∈ Hβ . For 0 < α < 1/2 (i) (i) = Ψj , ∆2α Ψj (Ωp , Ωp ) > 0. Since and j ∈ N this implies that Γj , ∆2α p,i Γj s − limi→∞ EΛi = 1, it follows that ∞ 2 (i) (i) α EΛi Γj , ∆α E ≥ lim inf lim inf Tr EΛi ∆α p,i Λi p,i EΛi ∆p,i EΛi Γj i
i
= lim inf i
∞ j=1
(i)
j=1 (i)
α ∆α p,i Γj , EΛi ∆p,i Γj
= lim inf i
∞ j=1
(Ψj , ∆2α Ψj ) > 0.
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Commutator estimates
ˆ → The unit ball in A∗ is weak∗ -compact. Thus for every net of states Λ(O◦ , O, O) ωΛ there exists a subnet of {ωΛi }i∈I converging to some state ω. Whether or not this state is a (τ, nαβ)-KMS state depends on the energy contained in the (i) ˆ (i) . We show that boundary, i.e., the choice of the relative size of O◦ , O(i) and O the necessary quantitative information restricting the surface energy can be drawn from the bounds on the nuclear norm of the map Θα,O introduced in (2) and the cluster condition (3). (i) ˆ (i) Let Λi = O◦ , O(i) , O be a sequence5 of triples of double cones with i∈N (i) (i) (i) diameters r◦ , r , rˆ , i ∈ N. We will now exploit the fact that the elements of Ap , p ∈ N, introduced at the end of Subsection 2.2, have good localization properties in space-time: we will show that there exists some p ∈ N such that ωΛi aτinαβ (b) − ωΛi (ba) < i ∀a, b ∈ Ap , (22) where i 0 as i → ∞. Thus the surface energy can be controlled by adjusting (i) the relative size of r◦ , r(i) and rˆ(i) . Inspecting the definition (18) of ωΛi we recognize that in order to prove (22) it is sufficient to control Tr ρΛi πβ (a) πβ τikαβ (b) , EΛi , k = 1, . . . , n. Let us consider the case n = 2. Let a, b ∈ Ap , p ∈ N fixed. It follows that τikαβ (b) ∈ Ap for k = 1, 2. Since a and b as well as c := τiαβ (b) and d := τ2iαβ (b) are (i) almost localized in O◦ for i sufficiently large, they almost commute with EΛi . For example, Tr ρΛi πβ (a) πβ τi2αβ (b) , EΛi 2 E π (a) Tr [πβ τi2αβ (b) , EΛi ] · ∆α β p,i Λi = 2 α Tr ∆p,i EΛi 2 [πβ τi2αβ (b) − di , EΛi ] · Tr | ∆α p,i EΛi | · a ≤ 2 Tr ∆α p,i EΛi 2 2a α ≤ 2 τi2αβ (b) − di · Tr |∆p,i EΛi | . α Tr ∆p,i EΛi (i) Here di ∈ A O◦ denotes a local approximation of d := τi2αβ (b) ∈ Ap which satisfies [EΛi , di ] = 0. Thus (i) (i) d c1 −c (r )2p · ec3 (r ) (23) Tr ρΛi πβ (a) πβ τi2αβ (b) , EΛi ≤ 2 · e 2 ◦ α Tr ∆p,i EΛi 5 Note that it is sufficient to work with sequences if the operators a and b appearing in (22) are fixed.
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for certain positive constants c1 = 2C a, c2 = κ/(22p ) and c3 = 2c α−m + (1/2 − α)−m , where m > 0. In the last inequality we made use of Proposition 3.3 and the second part of Lemma 2.1. Inspecting the r.h.s. of (23) closely, we find that (i) the numerator vanishes in the limit i → ∞, if exp −c2 (r◦ )2p · exp c3 (r(i) )d goes to zero as i → ∞. As has been shown in the previous section, the denominator does not vanish as i → ∞, but is bounded from below by some positive constant, (i) (i) if limi→∞ (r◦ )d (δ◦ )−γ = 0. (i)
In other words, the distance δ◦ has to grow sufficiently fast such that ηi → Ωβ , and p has to be chosen sufficiently large such that the elements in Ap are sufficiently well localized to fulfill the boundary condition (17) up to some small error term. We will now establish the KMS property for all weak limit points of {ωΛ }, (i) ˆ (i) tend to the whole space-time in agreeprovided the regions Λi = O◦ , O(i) , O (i) ment with the restrictions imposed on the relative size of r◦ , r(i) and rˆ(i) . Theorem 4.5. Assume that both the nuclearity condition (2) and the cluster condition (3) hold. Then there exists a choice of triples of space-time regions Λi such that every weak limit point of the (generalized) sequence {ωΛi }i∈I is a τ -KMS state at temperature 1/β > 0. Proof. Let n ∈ N and 0 < α < 1/2 be fixed such that β = nαβ. Moreover, let (i) ˆ (i) be a sequence of triples of double cones with diameters r◦(i) , Λi = O◦ , O(i) , O (i) d (i) −γ ˆ (i) O(i) sufficiently fast δ◦ = 0 and O r(i) and rˆ(i) such that limi→∞ r◦ as i → ∞ such that limi→∞ ηi − Ωβ = 0. Let us recall: the nuclearity condition fixes the constants d and m and the cluster condition fixes the constants d and γ. We will now fix p ∈ N. Taking into (i) (i) (i) account the restrictions on the relative size of r◦ and r(i) = r◦ + 2δ◦ imposed (i) d (i) −γ δ◦ goes to zero as by the cluster condition (3) – it is sufficient that r◦ (i) 2p · exp c3 (r(i) )d i goes to infinity – we can now chose p such that exp −c2 (r◦ ) goes to zero as i → ∞. Let a, b ∈ Ap and consider the case n = 2. i) Let ω2αβ denote the limit state of a convergent subnet {ωΛi }i∈I . For every > 0 we can find an index i ∈ I such that ω2αβ aτi2αβ (b) − ba ≤ ωΛi aτi2αβ (b) − ba + . (i) ii) We now approximate τi2αβ (b), τiαβ (b) and b by local elements in A O◦ and apply the commutator estimate (23) several times: for suitable (large) i ∈ N
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we find ω2αβ aτi2αβ (b) − ba 2 α Tr π aτ β i2αβ (b) EΛi ∆p,i EΛi − ω (ba) ≤ + 2 α Λi Tr ∆p,i EΛi α 2 Tr π (a)E π τ β Λi β i2αβ (b) ∆p,i EΛi ≤ − ω (ba) + 2 2 α Λi Tr ∆p,i EΛi Tr π (a)E ∆α π τ (b)E ∆α E β Λi p,i β iαβ Λi p,i Λi = − ωΛi (ba) + 2 2 α Tr ∆p,i EΛi Tr π (a) E ∆α 2 π (b)E β Λi p,i β Λi ≤ − ωΛi (ba) + 3 2 α Tr ∆p,i EΛi Tr π (a)E ∆α E 2 π (b) β Λi p,i Λi β ≤ − ωΛi (ba) + 4 α 2 Tr ∆p,i EΛi = 4. Thus ω2αβ aτ2iαβ (b) = ω2αβ (ba) for all a, b ∈ Ap . Now recall that Ap (for each p ∈ N) is a τ -invariant ∗-subalgebra of the set Aτ of analytic elements of A with respect to τ . Consequently, ω2αβ is a (τ, 2αβ)-KMS state. Similar results for arbitrary n ∈ N can be established by the same line of arguments but with considerable more effort. πβ
Once we have constructed a (τ, β )-KMS state ωβ , the GNS-representation leads to a new thermal field theory O ∈ R4 , O → Rβ (O) := πβ A(O) ,
acting on a new Hilbert space Hβ with GNS-vector Ωβ . If β = β , then the new thermal field theory will not be unitarily equivalent to the old one [T]. In fact, there might even be several extremal (τ, nαβ)-KMS states, which induce unitarily inequivalent representations, i.e., “disjoint thermal field theories”, at the same temperature 1/β = (nαβ)−1 . Acknowledgments. The present work started in collaboration with D. Buchholz. The final formulation is strongly influenced by his constructive criticism and by several substantial hints. Kind hospitality of the II. Institute for theoretical physics, University of Hamburg, the Institute for theoretical physics, University of Vienna, the Erwin Schr¨ odinger Institute (ESI), Vienna, and the Dipartimento di Matematica, Universita di Roma “Tor Vergata” is gratefully acknowledged. This work was financed by the Fond zur F¨ orderung der Wissenschaftlichen Forschung in Austria, Proj. Nr. P10629 PHY and a fellowship of the Operator Algebras Network, EC TMR-Programme.
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Christian D. J¨ akel Math. Inst. d. LMU Theresienstr. 39 D-80333 M¨ unchen, Germany email:
[email protected] Communicated by Klaus Fredenhagen Submitted 28/08/03, accepted 23/02/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 607 – 670 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/040607-64 DOI 10.1007/s00023-004-0179-3
Annales Henri Poincar´ e
Massless Relativistic Wave Equations and Quantum Field Theory Fernando Lled´ o
Dedicated to Rudolf Haag on his 80th birthday.
Abstract. We give a simple and direct construction of a massless quantum field with arbitrary discrete helicity that satisfies Wightman axioms and the corresponding relativistic wave equation in the distributional sense. We underline the mathematical differences to massive models. The construction is based on the notion of massless free net (cf. Section 3) and the detailed analysis of covariant and massless canonical (Wigner) representations of the Poincar´e group. A characteristic feature of massless models with nontrivial helicity is the fact that the fibre degrees of freedom of the covariant and canonical representations do not coincide. We use massless relativistic wave equations as constraint equations reducing the fiber degrees of freedom of the covariant representation. They are characterized by invariant (and in contrast with the massive case non reducing) one-dimensional projections. The definition of one-particle Hilbert space structure that specifies the quantum field uses distinguished elements of the intertwiner space between E(2) (the two-fold cover of the 2-dimensional Euclidean group) and E(2). We conclude with a brief comparison between the free nets constructed in Section 3 and a recent alternative construction that uses the notion of modular localization.
1 Introduction The transformation character of a quantum field involves typically two different types of representations of the corresponding spacetime symmetry group. The first one is a so-called covariant representation which acts reducibly on the test function space of the quantum field. The second one is a unitary and irreducible representation, which is called canonical (and in certain cases also Wigner representation), and which acts on the one-particle Hilbert space associated to the quantum field (see, e.g., [75, Section 2]). In the context of non-scalar free quantum field theory on Minkowski space, one chooses as test function space H-valued Schwartz functions, where H is a fixed finite-dimensional Hilbert space with dim H ≥ 2. Moreover, in order to describe massless models with discrete helicity, the corresponding space carrying the Wigner representations is a C-valued L2 space over the positive light cone. (In contrast with it, massive theories use H-valued L2 functions over the positive mass shell.) The fact that there is a difference between the dimensions of
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the image Hilbert spaces (fibres) of the preceding two function spaces (H vs. Cvalued functions), forces one to introduce some additional set of constraints in the construction of massless models with non-trivial helicity. These constraints guarantee for example that one can embed the test functions into the space carrying the corresponding massless Wigner representation. Physically they express the fact that for massless particles the helicity is parallel to the momentum in all Lorentz frames. In previous papers we have shown that this reduction in the fiber space can be performed in two different ways (see Remark 2.11 below). In this paper we will develop a third point of view in order to explain the reduction, where massless relativistic wave equations will play an essential role. Indeed, one may consider massless relativistic wave equations as constraint equations that reduce the fiber degrees of freedom. Free massless quantum fields with discrete helicity were constructed by Weinberg in [72]. The necessary reduction of degrees of freedom mentioned above has been done in this reference as follows: to define a massless quantum field of helicity j, 2j ∈ Z, one usually constructs first a 2j + 1-component free quantum field. This initial step is a clear reminiscence of massive theories and in fact the unnecessary components are ruled out afterwards by imposing on the quantum field itself a first order constraint equation. This construction procedure has been reproduced almost unchanged several times in the literature (cf. [74, 40, 73]). In the present paper we propose an alternative and direct construction of a free massless quantum field with arbitrary discrete helicity which satisfies the corresponding massless relativistic wave equation in a distributional sense. We will underline the mathematical structures characteristic to massless theories that appear in the group theoretical as well as in the quantum field theoretical context. The construction is naturally suggested by a detailed mathematical analysis of the covariant and canonical representations and, in fact, the reduction of the degrees of freedom can be encoded in suitable one-dimensional invariant projections. In this way the covariant transformation character of the quantum fields becomes completely transparent. They will also satisfy the corresponding Wightman axioms. The main aspects of this paper may be summarized in the following three items: (i) We analyse from a mathematical point of view the role of classical massless relativistic wave equations in the context of induced representations of the Poincar´e group. We show that these equations are characterized by certain invariant (but not reducing) one-dimensional projections. This analysis extends a systematic study of Niederer and O’Rafeartaigh (cf. [54]) concerning massive relativistic wave equations. We will point out the differences w.r.t. the massive case (see Section 2). (ii) We will give an alternative construction of massless free nets using as the essential element an embedding I from the test function space to the space of solution of the corresponding massless relativistic wave equation. This embedding intertwines the covariant and canonical representations mentioned
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above and therefore we can partly interpret the free net construction in the group theoretical context (see Section 3). (iii) We can finally reinterpret the previous embedding I as a one-particle Hilbert space structure and this allows to give a new construction procedure for massless quantum fields with nontrivial helicity. These fields will satisfy directly the relativistic wave equation in the distributional sense as well as the Wightman axioms. We will finally mention some further properties of these fields like, e.g., the conformal covariance (see Section 4). In order to describe induced representations we will consider in the following section the elegant fiber bundle language. In particular the crucial covariant and canonical representations of the Poincar´e group can be described as two different special cases in this framework. (In this context we can even describe the representations of the conformal group that restrict to the massless Wigner representations with discrete helicity (cf. [48]).) From a mathematical point of view, the reason for the need of reducing the fiber degrees of freedom mentioned at the beginning of the introduction lies in the following facts: on the one hand, the canonical or Wigner representations describing massless particles with discrete helicity are induced from non-faithful, one-dimensional representations of the corresponding little group E(2) (the two-fold cover of the 2-dimensional Euclidean group). On the other hand, the covariant representations are induced from at least two-dimensional irreducible representation of the little group SL(2, C) which do not restrict (for non-scalar models) to the inducing representation of the canonical representation when considering E(2) as a subgroup of SL(2, C). In its turn the use of non-faithful representations is due to the fact that the massless little group E(2) is non-compact, solvable and has a semi-direct product structure. (Recall that, in contrast with the previous attributes, the massive little group SU(2) satisfies the complementary properties of being compact and simple.) In the context of massless canonical representations massless relativistic wave equations will naturally appear as constraints performing the mentioned reduction and indeed we may associate with them invariant (but in contrast to massive equations non reducing) projections. It becomes clear that massless relativistic wave equations have a different character than massive ones and are in a certain sense unavoidable for nonscalar models, even if one does not consider discrete transformations of Minkowski space. We will also use in the present paper spinor fields. They can be roughly seen as being “square roots” of null vector fields (cf. [70]) and massless relativistic wave equations can be simply and systematically expressed as differential equations involving these type of fields (cf. [55, 56]). The spinorial language is also particularly well adapted to the general group theoretical framework mentioned before. In the following section we will in addition work out explicitly the Weyl equation as well as Maxwell’s equations (in terms of the field strength, F-eq. for short). These two concrete examples describing models with helicity 12 and 1 will be the base for the construction of massless fermionic and bosonic free nets/fields with nontriv-
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ial helicity. For completeness we will include in our group theoretical context the discussion of the classical Maxwell Equations in terms of the vector potential field (A-eq. for short). For a detailed treatment of quantum electromagnetism in terms of the vector potential (including constraints) we refer to [32]. An alternative and systematic analysis of relativistic particles in the context of geometric quantization is given in [23, 24]. There are other approaches to classical massless relativistic wave equations with different degrees of mathematical rigor, e.g., [76, Chapter II], [69, Chapter 9] or [44, 29, 52, 49, 7, 26, 25]. Concerning item (ii) mentioned previously, we define and prove in Section 3 the main properties of a massless free net. Recall that free nets, as considered in [12, 47] (see also [14]), are the result of a direct and natural way of constructing nets of abstract C*-algebras indexed by open and bounded regions in Minkowski space. They also satisfy the axioms of local quantum physics. The construction is based on group theoretical arguments (in particular on the covariant and canonical representations of the Poincar´e group mentioned before) and standard theory of CAR- and CCR-algebras [5, 51]. No representation of the C*-algebra is used and no quantum fields are initially needed. This agrees with the point of view in local quantum physics that the abstract algebraic structure should be a primary definition of the theory and the corresponding Hilbert space representation a secondary [21, Section 4]. Nevertheless, the free nets will afterwards canonically suggest the construction procedure for massless quantum fields. Therefore the massless free net construction presented here justifies the picture that free nets are the counterpart of free fields at the abstract C*-algebraic level. In addition, the notion of free nets (cf. Part (ii) of Definition 3.3) contains some aspects of Segal’s concept of quantization for bosonic systems (cf. [61, p. 750],[8, p. 106]). Through the requirements on the embedding I we incorporate to this program the axioms of local quantum physics. Note, however, that since Haag-Kastler axioms are stated in terms of abstract C*-algebras, we do not initially require (in contrast with Segal’s approach) that the abstract CCR-algebra is represented in any Hilbert space nor it is necessary to specify any regular state. This point of view is relevant when constraints are present (cf. [31]). Furthermore, the construction presented in Section 3, is in a certain sense complementary to the construction considered in [48], but still will produce isomorphic nets. Instead of using semidefinite sesquilinear forms and the related factor spaces as in [48] we will use in the present paper explicitly the space of solutions of the corresponding massless relativistic wave equation and therefore we need to introduce a different embedding I for characterizing the net. Further in [48] we focused on the covariance of the massless free net under the Poincar´e and conformal group. Here we prove the rest of the properties satisfied by the free net (e.g., additivity, causality etc.). The previous construction of a free net and in particular the group theoretical interpretation of the embedding I particularly pays off in the new construction procedure of massless quantum fields as well as in the verification of the corresponding axioms. Indeed, for the construction procedure of the massless quantum field (cf. item (iii) above) we need to reinterpret I as a one-particle Hilbert space
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structure. In this context we show the continuity of I w.r.t. the corresponding Schwartz space and Hilbert space topologies. Moreover, these fields will directly satisfy the (constraint) massless relativistic wave equations. E.g., in the helicity 12 case the quantum field satisfies in the distributional sense
∂ CC φ(fC ) = 0 , where f is the corresponding vector-valued Schwartz function. The construction presented here is considerably simpler than what is usually done in QFT. In order n to define the fields we will use inducing representations of the type D( 2 , 0) and n D(0, 2 ) since these are enough to construct massless quantum fields with helicity ± n2 (see, e.g., [53]). We conclude this paper commenting on the relations of the free net construction in Section 3 with the construction given in [20] which is based on the notion of modular localization (see also [27]).
2 Representations of the Poincar´e group and relativistic wave equations In the present section we will summarise some results concerning the theory of induced representations in the context of fiber bundles. For details and further generalizations we refer to [6, 64, 65] and [71, Section 5.1]. We will specify these structures in the example of the Poincar´e group to introduce so-called covariant and massive/massless canonical representations. In this group theoretical context we will consider massive and massless relativistic wave equations and analyse their different character. For a study of the conformal group (in 4-dimensions) in this frame and for a proof of the extension of the massless canonical representations with discrete helicity to a corresponding representation of the conformal group see [48] and references cited therein. Let G be a Lie group that acts differentiably and transitively on a C∞ manifold M . For u0 ∈ M denote by K0 := {g ∈ G | gu0 = u0 } the corresponding isotropy group w.r.t. this action. Then, from [38, Theorem II.3.2 and Proposition II.4.3] we have that gK0 → gu0 characterizes the diffeomorphism G/K0 ∼ = D := {gu0 | g ∈ G} . In this context we may consider the following principal K0 -bundle, B1 := (G, pr1 , D).
(1)
pr1 : G → D denotes the canonical projection onto the base space D. Given a representation τ : K0 → GL(H) on the finite-dimensional Hilbert space H, one can construct the associated vector bundle B2 (τ ) := (G ×K0 H, pr2 , D)
(2)
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(see [42, Section I.5] for further details). The action of G on M specifies the following further actions on D and on G ×K0 H: for g, g0 ∈ G, v ∈ H, put G × D −→ D, g0 pr1 (g) := pr1 (g0 g) (3) g0 [g, v] := [g0 g, v] , G × (G ×K0 H) −→ G ×K0 H, where [g, v] = [gk −1 , τ (k)v], k ∈ K0 , denotes the equivalence class characterizing a point in the total space of the associated bundle. Finally we define the (from τ ) induced representation of G on the vector space of sections of the vector bundle B2 , which we denote by Γ(G ×K0 H): let ψ be such a section and for g ∈ G, p ∈ D: (T (g)ψ)(p) := g ψ(g −1 p).
(4)
2.1 Remark. We will now present a way of rewriting the preceding induced representation in (for physicists more usual) terms of vector-valued functions. Indeed, choose a fixed (not necessarily continuous) section s : D → G of the principal K0 -bundle B1 . Now for ψ ∈ Γ(G ×K0 H) we put ψ(p) = [s(p), ϕ(p)], p ∈ D, for a suitable function ϕ : D → H and we may rewrite the induced representation as (T (g)ϕ)(p) = τ (s(p)−1 g s(g −1 p))ϕ(g −1 p),
(5)
where it can be easily seen that s(p)−1 g s(g −1 p) ∈ K0 . Note that till now we have not completely specified the induced representation T . In fact, we have to fix the structure of the representation space Γ(G ×K0 H) (or of the set of H-valued functions). This must be done separately in the three concrete situations considered below: covariant representations as well as massive and massless canonical representations. In these cases we have to specify completely the structure of the corresponding representation spaces. We will also give regularity conditions on the fixed section s of the principal bundle B1 considered before. Typically we will work with fixed Borel or continuous sections of the corresponding principal bundles.
2.1
The Poincar´e group
We will apply next the general scheme of induced representations presented above to specify the so-called covariant and canonical representations of the Poincar´e group. These will play a fundamental role in the definition of the free net in the next section. For additional results concerning induced representations and for the physical interpretation of the canonical (or Wigner) representations we refer to [9, 10, 50, 64, 77] as well as [45, Section 2.1] and [68, Chapters 2 and 3]. Covariant representations. In the general analysis considered above let G := ↑ be the universal covering of the proper orthocronous compoSL(2, C) R4 = P+ nent of the Poincar´e group. It acts on M := R4 in the usual way (A, a) x := ΛA x+a,
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(A, a) ∈ SL(2, C) R4 , x ∈ R4 , where ΛA is the Lorentz transformation associated to ±A ∈ SL(2, C) which describes the action of SL(2, C) on R4 in the semi-direct product. Putting now u0 := 0 gives K0 = SL(2, C) {0}, G/(SL(2, C) {0}) ∼ = R4 , 4 and the principal SL(2, C)-bundle is in this case B1 := (G, pr1 , R ). As inducing representation we use the finite-dimensional irreducible representations of k j j k SL(2, C) acting on the spinor space H( 2 , 2 ) := Sym ⊗ C2 ⊗Sym ⊗ C2 (cf. [66]): j k j k i.e., τ (cov) (A, 0) := D( 2 , 2 ) (A) = ⊗ A ⊗ ⊗ A , (A, 0) ∈ SL(2, C) {0}. From this we have (if no confusion arises we will omit in the following the index ( j2 , k2 ) in D(·) and in H), B2 (τ (cov) ) := (G ×SL(2, C) H, pr2 , R4 ).
(6)
Recalling Remark 2.1 we specify a global continuous section s of B1 (i.e., B1 is a trivial bundle): s : R4 −→ G,
s(x) := (1, x) ∈ SL(2, C) R4 = G .
Note that since τ (cov) is not a unitary representation and since we want to relate the following so-called covariant representation with the irreducible and unitary canonical ones presented below, it is enough to define T on the space of H-valued Schwartz functions S(R4 , H) (T (g)f )(x) := D(A) f (Λ−1 A (x − a)),
f ∈ S(R4 , H) ,
(7)
where we have used that s(x)−1 (A, a) s((A, a)−1 x) = (A, 0), (A, a) ∈ G. T is an algebraically reducible representation even if the inducing representation τ (cov) is irreducible. 2.2 Remark. We will show later that the covariant representation above is related with the covariant transformation character of quantum fields. Thus a further reason for considering this representation space is the fact that in the heuristic picture we want to smear quantum fields with test functions in S(R4 , H). Canonical representations: Next we will consider unitary and irreducible canoni↑ cal representations of P+ and, in particular, we will specify the massive and those massless ones with discrete helicity. We will also state the mathematical differences between these two types of representations. We will mainly apply here Mackey’s theory of induced representations of regular semi-direct products, where each subgroup is locally compact and one of them Abelian [50, 64, 10]. (However, see also Remark 2.18 (b).) First note that in the general context of the beginning of this section if τ is a unitary representation of K0 on H, then Γ(G ×K0 H) turns naturally into a Hilbert space. Indeed, the fibres pr−1 2 (p), p ∈ D, inherit a unique (modulo unitary
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equivalence) Hilbert space structure from H. Assume further that D allows a Ginvariant measure µ. (The following construction goes also through with little modifications if we only require the existence on D of a quasi-invariant measure w.r.t. G.) Then Γ(G ×K0 H) is the Hilbert space of all Borel sections ψ of B2 (τ ) that satisfy, ψ, ψ = ψ(p), ψ(p)p µ(dp) < ∞, D
where ·, ·p denotes the scalar product on the Hilbert space pr−1 2 (p), p ∈ D, and the induced representation given in Eq. (4) is unitary on it. 4 by means of the dual action Put now G := SL(2, C) R4 which acts on R ↑ . It is defined by canonically given by the semi-direct product structure of P+ 4 γ : SL(2, C) R4 → Aut R
and
( γ(A, a) χ)(x) := χ(Λ−1 A (x)) ,
(8)
4 , A ∈ SL(2, C), a, x ∈ R4 . For a fixed character χ ∈ R 4 consider where χ ∈ R M as the orbit generated by the previous action and the corresponding isotropy subgroup is given by (A, a) χ = χ Gχ := (A, a) ∈ SL(2, C) R4 | γ
and recall that
↑ P+ /Gχ ∼ = D.
We have now the principal Gχ -bundle and the associated bundle given respectively by
↑ ↑ , pr1 , D and B2 τ (can) := P+ ×Gχ H, pr2 , D , B1 := P+ where τ (can) is a unitary representation of Gχ on H. If τ (can) is irreducible, then the corresponding induced representation, which is called the canonical representation, is irreducible. Even more, every irreducible representation of G is obtained (modulo unitary equivalence) in this way. Recall also that the canonical representation is unitary iff τ (can) is unitary. 2.3 Remark. In the present context relativistic wave equations appear if one considers reducible representations of the isotropy group Gχ . On the subspace of solutions of these equations (which can be consequently characterized by projections in H) the reducible induced representation will turn irreducible. We will study relativistic wave equations in the following for massive and massless representations. We will show that these have a fundamentally different character as a consequence of the complementary properties of the respective little groups. (Here we use the name little group to denote the subgroup of SL(2, C) appearing in the isotropy group.) Indeed, the representations of the massive little group considered will be unitary and fully decomposable, while the corresponding representations of the massless little group will not satisfy these properties. Massive relativistic wave equations will be characterized by reducing projections, while the massless ones are associated to invariant (but not reducing) projections.
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Massive canonical representations
Choose a character χp˘, with p˘ := (m, 0, 0, 0), m > 0, i.e., χp˘(a) = e−i˘pa , where a ∈ R4 and p˘ a means the Minkowski scalar product. In this case we have Gχp˘ = n SU(2) R4 . As unitary representation of the isotropy subgroup on H( 2 , 0) we take n
τ (can) (U, a) := e−i˘pa D( 2 , 0) (U ) ,
(U, a) ∈ Gχp˘ .
(9)
We can now consider (omitting for simplicity the index ( n2 , 0)) the bundles,
(can) ↑ ↑ (can) + + := P+ , pr1 , Cm and B2 τ := P+ ×Gχp˘ H, pr2 , Cm , B1 ↑ 4 2 /Gχp˘ ∼ where we have used the diffeomorphism P+ = C+ m := {p ∈ R | p = 2 0 + m , p > 0} between the factor space and the positive mass shell Cm . µ(dp) denotes the corresponding invariant measure on C+ m. The principal Gχp˘ -bundle B1(can) is trivial [18]. Indeed, a global continuous section s is ↑ s : C+ m −→ P+ , where Hp := √
1 2m(m+p0 )
↑ s(p) := (Hp , 0) ∈ SL(2, C) R4 = P+ , (10) (m + P ), P = pµ σµ [12, Section A.1] and σµ , µ = µ
0, 1, 2, 3, are the unit and the Pauli matrices. The assignment p → P defines a vector space isomorphism between R4 and the set of self-adjoint elements in Mat2 (C). So, once the section s is fixed and using s(p)−1 (A, a) s(q) = (Hp−1 A Hq , + 2 + ΛHp−1 a) ∈ Gχp˘ , as well as q = Λ−1 A p ∈ Cm , we have on L (Cm , H, µ(dp)) the massive canonical representation (cf. Eq. (5)),
(can) (g)ϕ (p) = e−ipa D(Hp−1 A Hq )ϕ(q), (11) V −1 where g = (A, a) ∈ SL(2, C) R4 and ϕ ∈ L2 (C+ m , H, µ(dp)). The element Hp A (can) is unitary Hq ∈ SU(2) is called the Wigner rotation (e.g., [74, Section 2.3]). V w.r.t. usual L2 -scalar product, ϕ1 , ϕ2 := ϕ1 (p), ϕ2 (p)H µ(dp).
C+ m (can) 2.4 Remark. The representation (L2 (C+ , ·, ·) is equivalent to m , H, µ(dp)), V the representation (hm , V1 , ·, ·β ) used in [12, Section A.1], where we define first for ϕ, ψ a pair of H-valued functions the sesquilinear form ϕ(p), β(p) ψ(p)H µ(dp), ϕ, ψβ :=
C+ m
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with β(p)
:=
P†
:=
n
n
D( 2 , 0) (P † ) = ⊗ P † , 3 1 p0 σ0 − pi σi = m i=1
(12) −1 ∗ −1 Hp . Hp
(13)
hm := {ϕ : C+ m −→ H | ϕ is Borel and ϕ, ϕβ < ∞} .
(14)
Then we put
Finally, the representation n
(V1 (g) ϕ) (p) := e−ipa D( 2 , 0) (A) ϕ(q),
(15)
↑ for g = (A, a) ∈ P+ = SL(2, C) R4 , ϕ ∈ hm and q := Λ−1 A p, is unitary w.r.t. ·, ·β . Roughly, we have removed the Hp -matrices from the definition of V1 at the price of introducing a positive operator-valued function on C+ m , p → β(p), in the definition of the corresponding scalar product. This equivalent definition of canonical representation has been very useful in order to construct massive free nets [12]. We will also adapt this idea to the massless case. The equivalence of the representations mentioned above is given explicitly by the following isometry Ψ : L2 (C+ m , H, µ(dp)) → hm , (Ψϕ)(p) := D(Hp )ϕ(p),
(16)
↑ and it is immediately checked that Ψ V (can) (g) = V1 (g) Ψ, g ∈ P+ . 2.5 Remark. It is now easy to relate the algebraically reducible representation T (cov) in (7) with the canonical representation V (can) given in Eq. (11). Indeed, recalling the definitions introduced in Remark 2.4, we consider the embedding I : S(R4 , H) −→ hm , defined as Fourier transformation, fˆ(p) = R4 e−ipx f (x)d4 x, and restriction to C+ m (recall [58, Section IX.9]). Then the equation I T (cov) (g) = V1 (g) I,
↑ g ∈ P+ ,
(17)
holds. From the preceding Remark we already know that V1 and V (can) are equivalent.
2.3
Massive relativistic wave equations
As already mentioned in Remark 2.3 relativistic wave equations appear when one considers reducible representations of the little group SU(2). They serve to reduce
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the corresponding induced representation and therefore it is natural to associate with relativistic wave equations corresponding projections on H. We will also illustrate the preceding results with two examples that have nontrivial spin, namely the Dirac and the Proca Equations. Niederer and O’Rafeartaigh present Eq. (20.8) in [54] as “. . . the most general covariant wave equation corresponding to a given (nonzero) mass and spin . . . ” (see Definition 2.7 below). It is useful to recognize that in [54] and [10, Chapter 21] the spaces (hm , ·, ·β ) of Remark 2.4 are essentially used. In our context we can equivalently write the mentioned equation also for the space (L2 , V (can) ). Using Eq. (16) the following result is a straightforward consequence of the mentioned equivalence. 2.6 Lemma Let π be a reducing orthoprojection w.r.t. τ (can) , i.e., π τ (can) (g) = τ (can) (g) π, g ∈ SU(2) R4 . Then the following equations are equivalent: (i) π ϕ(p) = ϕ(p), for ϕ ∈ L2 (C+ m , H, µ(dp)). (ii) π(p) ϕ(p) = ϕ(p), where π(p) := D(Hp ) π D(Hp )−1 and ϕ(p) := D(Hp )ϕ(p) ∈ h m , p ∈ C+ . m Following Niederer and O’Rafeartaigh we introduce the notion of massive relativistic wave equation (see [54, § 20 and § 21] for further details and motivation). 2.7 Definition Let π be a reducing orthoprojection w.r.t. τ (can) (SU(2) R4 ). Then we call π(p) ϕ(p) = ϕ(p) , ϕ ∈ hm , p ∈ C+ m, a massive relativistic wave equation associated with π (cf. Lemma 2.6 (ii)). 2.8 Remark. • The equation in the previous definition coincides with [54, Eq. (20.8)] or [10, Eq. (17) of Section 21.1]. Specifying τ (can) and π in this context one obtains the conventional massive relativistic wave equations written in momentum space. For convenience of the reader we mention the examples of Dirac’s and Proca’s equation. More examples of massive relativistic wave equations and the corresponding projections are summarized in [54, Table 2]. • Note further, that the subspace of hm characterized by the equation in Lemma 2.6 (ii) is V1 -invariant. Use for example the relation: D(A)−1 π(p) D(A) = π(Λ−1 A p). Thus we have seen that the reducing subspaces of H are in correspondence with relativistic wave equations. Recall that by compactness the unitary j k representation D( 2 , 2 ) SU(2), (j, k) = (0, 0), is fully decomposable, i.e., it can be decomposed as a direct sum of irreducible subrepresentations.
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A 0 ∈ 0 (A∗ )−1 SL(4, C) for A ∈ SL(2, C). As reducible inducing representation we use τ (U, a) := e−i˘pa τ (U ) , (U, a) ∈ SU(2) R4 . Now, consider the orthoprojection 1 0 1 0 1 0 1 0 1 , π (Dirac) := 2 1 0 1 0 0 1 0 1
The Dirac Equation
[64, Section 9]: Take H := C4 and τ (A) :=
which satisfies π (Dirac) τ (U ) = τ (U ) π (Dirac) , U ∈ SU(2), and using the isometry we can reformulate the Ψ as well as the notation ϕ(p) := (Ψϕ)(p) = τ (Hp )ϕ(p) preceding lemma in the present context as: 2.9 Proposition With the preceding notation we have that ϕ satisfies the Dirac = ϕ(p), p ∈ C+ Equation iff ϕ satisfies π (Dirac) ϕ(p) m. Proof. First of all we note that from Lemma 2.6, π (Dirac) ϕ(p) = ϕ(p) iff π (Dirac) (p) ϕ(p) = ϕ(p) , where π (Dirac) (p) := τ (Hp ) π (Dirac) τ (Hp )−1 . We calculate
m1 P 1 π (Dirac) (p) = , 2m P m1 3 pi σi and P := p0 σ0 − pi σi , so that P P = p2 1 = i=1 i=1
0 P 1 2 we obtain π (Dirac) (p) = 2m (m1 + γ(p)) and thus m 1. Denoting γ(p) := P 0 π (Dirac) (p) ϕ(p) = ϕ(p) iff γ(p) ϕ(p) = m ϕ(p), which is the Dirac equation in momentum space notation.
where P := p0 σ0 +
3
The Proca Equation [54]: In the present case we put H := C2 ⊗ C2 ∼ = C4 and as inducing representation we use 1
1
τ (U, a) := e−i˘pa D( 2 , 2 ) (U ) = e−i˘pa U ⊗ U ,
(U, a) ∈ SU(2) R4 .
As an orthoprojection we take,
π (Proca)
1 1 0 := 2 0 −1
0 2 0 0
0 0 2 0
−1 0 , 0 1
which satisfies π (Proca) τ (U ) = τ (U ) π (Proca) , U ∈ SU(2). Further details, e.g., the explicit relation between the canonical and covariant descriptions in this particular case can be found in [17].
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Massless canonical representations
To specify massless representations with discrete helicity we choose a character χp˘, p˘ := (1, 0, 0, 1) ∈ C+ (the mantle of the forward light cone), i.e., χp˘(a) = e−i˘pa , a ∈ R4 and p˘ a means the Minkowski scalar product. A straightforward computation shows that the isotropy subgroup is given by Gχp˘ = E(2) R4 , where the two-fold cover of the 2-dimensional Euclidean group is i θ i e− 2 θ z e2 E(2) := ∈ SL(2, C) | θ ∈ [0, 4π), z ∈ C . (18) i 0 e− 2 θ The little group E(2) is noncompact and since its commutator subgroup is already Abelian it follows that E(2) is solvable. Further, it has again the structure of a semi-direct product. (In contrast with these facts we have that the massive little group SU(2) is compact and simple.) Since E(2) is a connected and solvable Lie group we know from Lie’s Theorem (cf. [10]) that the only finite-dimensional irreducible representations are 1-dimensional, i.e., H := C. Therefore in order to induce irreducible and unitary representations of the whole group that describe discrete helicity values we define i n (19) τ (can) (L, a) := e−i˘pa e 2 θ , where (L, a) ∈ E(2) R4 = Gχp˘ , n ∈
N. Note that this representation is not 1 z faithful. Indeed, the normal subgroup | z ∈ C is trivially represented 0 1 (see also [72, Section II]). Some authors associate this subgroup to certain gauge degrees of freedom of the system (e.g., [36, 41, 63]). We consider next the bundles,
(can) ↑ ↑ (can) B1 := P+ ×Gχp˘ C, pr2 , C+ , := P+ , pr1 , C+ and B2 τ ↑ ∼ C+ := {p ∈ R4 | p2 = 0 , p0 > where we have used the diffeomorphism P+ /Gχp˘ = 0} between the factor space and the mantle of the forward light cone. We denote by µ0 (dp) the corresponding invariant measure on C+ . In contrast with the massive case the bundle B1(can) has no global continuous section. The reason for this lies in the following topological obstruction: if B1(can) had a global continuous section (hence would be a trivial bundle), then the nth homotopy groups Πn (·), n ∈ N, of the total space would be equal to the direct sum of the homotopy groups of the isotropy subgroup and of the base manifold. We can check in particular for the second homotopy group that on the one hand Π2 (G) = Π2 (S 3 ) = 0 and on the other hand Π2 (Gχp˘ ) ⊕ Π2 (C+ ) = Π2 (S 1 ) ⊕ Π2 (S 2 ) = Z (see [18] for further details).
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Nevertheless, we can specify a Borel section considering a continuous one in a chart that does not include the set {p ∈ C+ | p3 = −p0 } (which is of measure ◦ zero w.r.t. µ0 (dp)). Putting C+ := C+ \ {p ∈ C+ | p3 = −p0 } a (local) continuous section is given explicitly by ↑ ◦ −→ P+ , s : C+
↑ s(p) := (Hp , 0) ∈ SL(2, C) R4 = P+ ,
(20)
p1 − ip2 √ p0 p0 + p3 . − √ p0
(21)
where
√ − p0 (p0 + p3 )
1 Hp := 2p0 (p0 + p3 ) −√p0 (p1 + ip2 ) Recall that the Hp -matrices satisfy the equation
Hp
2 0 Hp∗ = P, 0 0
where
P =
p0 + p3 p1 + ip2
p1 − ip2 p0 − p3
= p0 σ0 +
3
pi σi . (22)
i=1
We use here, as in the massive case, the vector space isomorphism between R4 and the set of self-adjoint elements in Mat2 (C). 2.10 Remark. The representation spaces of the canonical representations are typically L2 -spaces and therefore it is enough to consider fixed Borel sections as above. The fact that we are allowed to choose a continuous section of the corresponding bundle B1 in the massive case is an other pleasant and characteristic feature of these models. From the point of view of quantum fields (to be defined in Section 4) what is crucial is the fact that the singularity of (20) does not affect the continuity (w.r.t. to the Schwartz and L2 -topologies) of the embedding that intertwines the covariant and the massless canonical representations. Indeed, in Theorem 4.2 we give a detailed proof of the mentioned continuity for the embedding associated to the Weyl case and that uses explicitly the section defined in Eq. (20). This ensures that the 2-point distributions defined by the corresponding massless quantum fields are tempered. If we consider the section in Eq. (20) fixed, then we have on L2 (C+ , C, µ0 (dp)) the canonical massless representations (cf. Eq. (5)) i n (U± (g)ϕ)(p) = e−ipa e± 2 θ(A,p) ϕ(q), (23) 4
where g = (A, a) ∈ SL(2, C) R , n ∈ N, q :=
Λ−1 A p
and for A =
a b c d
we compute
i −b(p1 + ip2 ) + d(p0 + p3 ) . e− 2 θ(A,p) := Hp−1 A Hq 22 = | − b(p1 + ip2 ) + d(p0 + p3 )|
∈ SL(2, C)
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U± are unitary w.r.t. usual L2 -scalar product, satisfy the spectrum condition and the helicity of the model carrying one of these representations is ± n2 . 2.11 Remark. In contrast with the massive case, it is now clear that in order to relate the covariant representation with the canonical massless representation it will not be enough to consider the Fourier transformation of suitable test functions and its restriction to C+ (cf. Remark 2.5). Indeed, the fibres of B2 (τ (can) ) are 1dimensional, while the fibres of B2 (τ (cov) ) are at least 2-dimensional if one chooses a nontrivial inducing representation τ (cov) . In other words, if the models describe nontrivial helicity, then some further restriction must be performed on the fibres in order to reduce the covariant representation to the unitary and irreducible canonical one (for a more detailed analysis of this reduction see Subsection 2.6). There are at least three ways to perform the mentioned reduction: (i) One possibility of restricting the dimension of the fibres is to work on the space of solutions of the massless relativistic wave equations (cf. Remark 2.3). We will follow this alternative in the following subsections and present some explicit examples. (ii) A second possibility is to use certain semidefinite sesquilinear forms and the reduction is done by means of the factor spaces that can be naturally constructed from the degeneracy subspaces of the sesquilinear form. This possibility is studied in [12, Part B] and [48] (cf. also Remark 2.13). (iii) Finally one can perform the mentioned reduction for the bosonic models at the C*-level by the so-called T -procedure of Grundling and Hurst [30] (see [46, 47] for details). Imposing ‘quantum constraints’ on the C*-algebra level will show to be equivalent to consider the space of solutions of massless relativistic wave equations as reference space of the corresponding CCR-algebra. In the next section the embedding I intertwining the covariant and massless canonical representations (including the corresponding fiber reduction) will be the fundamental entity. In fact, I specifies completely the corresponding net of C*-algebras satisfying Haag-Kastler’s axioms. Each possibility to carry out the reduction described in (i)–(iii) above, requires its own embedding. Nonetheless the corresponding nets of local C*-algebras turn out to be equivalent (cf. Remark 3.24 (i)).
2.5
Massless relativistic wave equations
In the present subsection we will extend Niederer and O’Rafeartaigh’s systematic analysis of massive relativistic wave equations to the massless case (recall, e.g., Definition 2.7). We will point out the fundamental differences between these two case. We begin showing that, in contrast with the massive case (cf. Subsection 2.3), and due to the mathematical nature of the massless little group one has to give up the central notion of reducing projection for the inducing representation. In
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fact, we will show that, in the context of massless canonical representations, the useful objects are the invariant (but not reducing) projections of the corresponding inducing representations: Motivated by the form of the covariant representation in Eq. (7) (the one we want to reduce) we will consider as inducing representations of the isotropy subgroup j
k
j
k
τ (L, a) := e−i˘pa D( 2 , 2 ) (L) ,
(L, a) ∈ E(2) R4 ,
which act on the Hilbert space H( 2 , 2 ) of dimension (j + 1)(k + 1). (In the rest j k of this subsection we will denote again D( 2 , 2 ) (·) simply by D(·) etc.) These rep1 resentations are nonunitary and reducible for any (j, k) = (0, 0). They are also not fully decomposable [62], [10, p. 607] in contrast to the massive case, i.e., they cannot be decomposed as direct sum of irreducible subrepresentations. Therefore, no nontrivial reducing projection π will exist2 in this context (we will compute explicitly some intertwiner spaces in the following section). Therefore by the general theory, the corresponding induced representation ↑ ◦ of P+ , which is given for g = (A, a) ∈ SL(2, C) R4 , q := Λ−1 A p ∈ C+ , ϕ ∈ 2 L (C+ , H, µ0 (dp)) by (V (g)ϕ)(p) = e−ipa D(Hp−1 A Hq )ϕ(q) ,
Hp−1 A Hq ∈ E(2) ,
(24)
will also be nonunitary and reducible. 2.12 Remark. Before studying in detail the following examples of massless relativistic wave equations we will introduce the massless analogue to the useful space (hm , ·, ·β ) presented in the context of massive representations in Remark 2.4: for ϕ, ψ a pair of H-valued functions the sesquilinear form, ϕ(p), β ϕ, ψ+ := + (p) ψ(p)H µ0 (dp), C+ −1 ∗ −1 ◦ where β + (p) := D(Hp ) D(Hp ) and Hp , p ∈ C+ , is defined in Eq. (21). Then we put (25) h0 := {ϕ : C+ −→ H | ϕ is Borel and ϕ, ϕ+ < ∞} .
Finally, the representation (V (g) ϕ)(p) := e−ipa D(A) ϕ(q),
(26)
1 Wigner already observes in [78, p. 670] using a different formal approach that the massless wave equations cannot be obtained in general from the massive ones by putting m = 0. He also mentions that the irreducible and invariant linear manifold of states corresponding to the massive representations turns reducible for m = 0 and considering the value of the spin bigger than 12 . 2 This is possibly a reason why in [54, Sections 22–25] the authors do not follow the elegant approach used to describe the massive wave equations, when they study the massless case.
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↑ ◦ for g = (A, a) ∈ P+ = SL(2, C) R4 , ϕ ∈ h0 and q := Λ−1 A p ∈ C+ , is equivalent to the representation V defined in (24), i.e., there exists an isometry, Ψ : L2 (C+ , H, µ0 (dp)) −→ h0 , given by (Ψϕ)(p) := D(Hp ) ϕ(p),
(27)
↑ such that, Ψ V (g) = V (g) Ψ, g ∈ P+ . The representation V is also reducible and nonunitary w.r.t. ·, ·+ . 2.13 Remark. If we require the representation V to leave the sesquilinear form ·, ·+ invariant, then we are forced (cf. [9]) to redefine the operator-valued function β + above as β+ (p) := D
Hp−1
∗
D
0 0 0 1
D Hp−1 =: D(P † ) , 1 with P = 2 †
p0 σ0 −
3
pi σi
,
i=1
◦ which is only a semidefinite operator on H for each p ∈ C+ , (compare with the massive case in Remark 2.4). This option is taken in [48] (cf. also [12, Part B]), and has as a consequence that the necessary reduction that must be performed to compare the covariant with the canonical representation (cf. Remark 2.11) is done by means of the factor spaces that can be naturally constructed from the degeneracy subspace of the sesquilinear form ·, ·β+ . This redefinition of the βfunctions is related with the fact of imposing massless relativistic wave equations, which is what we will examine below.
Since in the present context no nontrivial reducing projections exist we will need to introduce the following family of invariant projections: 2.14 Definition From the set of all projections in H, select those orthoprojections π that are invariant w.r.t. D E(2) and satisfy the equation3 π D(L)∗ D(L) π = π ,
L ∈ E(2) .
(28)
2.15 Theorem Let π be as in the preceding definition and put = ϕ(p)} . h := {ϕ ∈ L2 (C+ , H, µ0 (dp)) | π ϕ(p) ↑ , h, g ∈ P+ Then h is a closed V -invariant subspace of L2 (C+ , H, µ0 (dp)) and V (g) is unitary. Further if D(E(2)) is irreducible on πH, then V h is also irreducible. 3 Note that this condition is trivially satisfied by the reducing projections chosen in the massive case.
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↑ Proof. It is obvious that h is a closed subspace and since for g = (A, a) ∈ P+ we have Hp−1 A Hq ∈ E(2), q := Λ−1 p, the invariance follows for ϕ ∈ h from A π(V (g)ϕ)(p)
=
e−ipa π D(Hp−1 A Hq )ϕ(q) = e−ipa π D(Hp−1 A Hq )π ϕ(q)
=
e−ipa D(Hp−1 A Hq )π ϕ(q) = (V (g)ϕ)(p) .
Further, for ϕ, ψ ∈ h we also have V (g)ϕ, V (g)ψ = D(Hp−1 A Hq ) ϕ(q), D(Hp−1 A Hq ) ψ(q)H µ0 (dp) C+ =
ϕ(q), π D(Hp−1 A Hq )∗ D(Hp−1 A Hq ) π ψ(q)H µ0 (dp)
C+ = ϕ, ψ . Here we have used Eq. (28) and the invariance of µ0 . The irreducibility statement follows from the general theory of induced representations stated before in this section. The condition π ϕ(p) = ϕ(p) used before can be rewritten in terms of the equivalent space (h0 , V, ·, ·+ ) of Remark 2.12. This will give the massless relativistic wave equations written in its usual form. 2.16 Lemma Let π be an invariant, orthoprojection w.r.t. τ , i.e., π τ (g) π = τ (g) π for all g ∈ E(2) R4 . Then the following equations are equivalent: (i) π ϕ(p) = ϕ(p), for ϕ ∈ L2 (C+ , H, µ0 (dp)). (ii) π(p) ϕ(p) = ϕ(p), where π(p) := D(Hp ) π D(Hp )−1 ◦ and ϕ(p) := D(Hp )ϕ(p) ∈ h0 , p ∈ C+ . 2.17 Definition Let π be an invariant orthoprojection w.r.t. τ (can) (E(2) R4 ). Then we call ◦ , π(p) ϕ(p) = ϕ(p) , ϕ ∈ h0 , p ∈ C+ a massless relativistic wave equation associated with π (cf. Lemma 2.16 (ii)). The equation in (ii) is the generalization to the massless case of Eq. (ii) in Lemma 2.6 (cf. also [54, Eq. (20.8)] or [10, Eq. (17) of Section 21.1]). In the following we will consider it in different particular cases and show in Corollary 2.25 that it includes the general form of massless relativistic wave equations. 2.18 Remark. (a) As announced in the beginning of this subsection the equation introduced in Definition 2.17 extends neatly to the massless case, the work
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of Niederer and O’Rafeartaigh concerning massive relativistic wave equation [54]. The mentioned equation contains as special cases the conventional massless relativistic wave equations written in momentum space. (See the following Weyl and Maxwell cases below, Corollary 2.25 and the table in Subsection 2.10.) The existence of the corresponding projections π in the massive and massless cases is guaranteed by the fact that SU(2) resp. E(2) are compact resp. solvable Lie groups. Indeed, any finite-dimensional unitary representation τ (can) (SU(2) R4 ) can be decomposed as a direct sum of irreducible ones (cf. [39, Theorem 27.30]). Moreover, since E(2) is solvable and connected, Lie’s Theorem [10, p. 200] guarantees the existence of one-dimensional orthoprojections invariant w.r.t. τ (can) (E(2) R4 ). (b) Notice that in certain steps in this subsection we have make use of nonunitary representations (see, e.g., Eqs. (24) or (26)). Therefore these representations lie outside of Mackey’s theory of induced representations. Nevertheless in these cases we do not use any result of this theory. The justification of this procedure (and the importance of the massless relativistic wave equations) comes from Theorem 2.15. In fact, here we turn back to the description of unitary and irreducible representations and these must be unitarily equivalent to the ones considered by Wigner. The equivalence is given explicitly in the Weyl and Maxwell cases in Propositions 2.21 and 2.24. (For the existence of projections satisfying Eq. (28) recall the previous item.) (c) Note finally the fundamentally different role that massless wave equations play (in contrast to the massive ones) when reducing the covariant representation. Indeed, massive relativistic wave equations appear when we consider reducible representations of SU(2) and therefore will not be present if n we choose, e.g., the irreducible representations given by D( 2 , 0) SU(2). On j k the contrary D( 2 , 2 ) E(2) is always reducible if nontrivial helicity is admitted and therefore the space of solutions of massless relativistic wave equations is unavoidable if we want to work in momentum space with irreducible canonical representations. Indeed, in Sections 2.7 and 2.8, the Weyl and Maxwell’s n equations will naturally appear when considering D( 2 , 0) E(2), n = 1, 2.
2.6
Conditions on the intertwining operator and fiber reduction
It is useful at this point to complete Remark 2.11 on the fiber reduction and study in detail the intertwining operator I between the covariant and massless canonical representation with nontrivial discrete helicity. By Wigner’s analysis and, in particular, due to the dual action defined in Eq. (8), it is clear that I must contain the Fourier transformation . This is also the case in massive models (see Remark 2.5; for scalar models see also [58, Section IX.9]). We can now decompose
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the intertwining operator in its constituents. (The case with helicity H = C2 ) is already typical.)
1 2
(hence
R M 4 , C2 ) −→ S(R4 , C2 ) −→ S(R C ∞ ((R3 \ 0) , C2 ) −→ C ∞ ((R3 \ 0) , C)
⊂ L2 ((R3 \ 0) , C) , where R is the restriction operator onto C+ , (Rf)(p ) := f(|p |, p ), p ∈ (R3 \ 0), and M is the operator performing the fiber reduction mentioned above. The latter operator is characteristic for massless models with nontrial helicity. (In order to keep argument transparent we work here with the massless representation U considered in Eq. (23).) The conditions on the intertwining operator I, which is the composition of the preceding chain of mappings, must satisfy the following conditions: (a) I must be linear (hence M must be linear). (b) I must be continuous between the Schwartz and the Hilbert space topologies (recall Remark 2.10). (c) I must intertwine the covariant and massless canonical representations, i.e., I T (g) = U (g) I ,
↑ . g ∈ P+
Some concrete examples of intertwining operators satisfying the above conditions in the Weyl and Maxwell cases are specified in Eqs. (60) and (62).
2.7
The Weyl Equation
We begin with the simplest representation of E(2) with dimension bigger than
iθ i e2 e− 2 θ z ∈ E(2) we have on one (see, e.g., [37, Section V.A]). For L = i 0 e− 2 θ 1 1 H( 2 , 0) := C2 the representation given by D( 2 , 0) (L) := L. (Notation: In the re1 maining subsection we will denote the representation D( 2 , 0) (·) simply by D(·). We will also skip the index ( 12 , 0) from the objects associated to D(·), e.g., the representation V , the scalar product ·, · etc., in order to keep the notation sim1 ple.) The only nontrivial D(E(2))-invariant subspace is C and we choose the 0
1 0 corresponding invariant orthoprojection π := . In the following lemma we 0 0 will establish the relation between the equation in Lemma 2.16 (ii) and the Weyl Equation. ∈ h0 , for all ϕ ∈ L2 (C+ , C2 , µ0 (dp)) (see 2.19 Lemma Put ϕ(p) := Hp ϕ(p) Eq. (27)). Then we have that ϕ satisfies the equation π ϕ(p) = ϕ(p) iff ϕ satisfies the Weyl Equation.
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Proof. 1. Suppose that ϕ ∈ L2 (C+ , C2 , µ0 (dp)) satisfies π ϕ(p) = ϕ(p), p ∈ C+ . 2 Then there exists a scalar function χ ∈ L (C , C, µ (dp)) such that ϕ(p) = + 0
χ(p) . Next we write the Weyl operator in momentum space as 0 W(p) :=
p0 σ0 −
3
pi σi
i=1 ◦ and notice that we can rewrite W(p), p ∈ C+ , as
∗ W(p) = Hp−1
0 0 Hp−1 = 2P † . 0 2
(29)
χ(p) , satisfies the Weyl Equation W(p) ϕ(p) = W(p) 0
Therefore, ϕ(p) = Hp
χ(p) = 0 . Recall also that in terms of the spinorial components the Weyl Hp 0 equation is usually written as 1
(P † )C C ϕC (p) =
C=0
1
(P † )C C (Hp )C 0 (p) = 0, 0 ϕ
C
∈ {0, 1} .
(30)
C=0
2. Suppose on the other hand that ϕ(p) = Hp ϕ(p) ∈ h0 , with ϕ(p) = ϕ 1 (p) satisfies , satisfies the Weyl Equation W(p) ϕ(p) = 0. Then ϕ 2 = 0 and ϕ ϕ 2 (p) the equation π ϕ(p) = ϕ(p).
The space of solutions of the Weyl equation is therefore given by
χ(p) 2 h+ := Hp | χ ∈ L (C+ , C, µ0 (dp)) , 0 2.20 Lemma Define on the space of solutions of the Weyl Equation h+ the scalar product
∗ ϕ1 , ϕ2 + := ϕ1 (p), Hp−1 Hp−1 ϕ2 (p)C2 µ0 (dp), for ϕi ∈ h+ , i = 1, 2. C+ The representation given for g = (A, a) ∈ SL(2, C) R4 , q := Λ−1 A p and ϕ ∈ h+ by (V1 (g) ϕ)(p) := e−ipa A ϕ(q), is unitary w.r.t. ·, ·+ and irreducible.
(31)
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Ann. Henri Poincar´e
χi (p) , χi ∈ L2 (C+ , C, µ0 (dp)), i = 1, 2, 0
ϕ1 , ϕ2 + =
χ1 (p) χ2 (p) µ0 (dp).
(32)
C+ Since πH is 1-dimensional and π L∗ L π = π, L ∈ E(2), Theorem 2.15 completes the proof. Next we establish the equivalence between the representation (h+ , V1 , ·, ·+ ) defined above and the representation L2 (C+ , C, µ0 (dp)), U+ , ·, ·L2 : The canonical representation for n = 1 (cf. Eq. (23)) is i
(U+ (g) χ)(p) := e−ipa e 2 θ(A,p) χ(q),
χ ∈ L2 (C+ , C, µ0 (dp)) ,
(33)
4 for q := Λ−1 A p and g = (A, a) ∈ SL(2, C) R . U+ (·) is irreducible, strongly con2 tinuous and unitary for the usual L -scalar product, ·, ·L2 . With the preceding notation we have the following equivalence of representations
2.21 Proposition The mapping Φ+ : h+ −→ L2 (C+ , C, µ0 (dp)) defined by,
χ(·) Φ+ H(·) (p) = χ(p) , 0
is an isometric isomorphism between (h+ , ·, ·+ ) and L2 (C+ , C, µ0 (dp)), ·, ·L2 , that commutes with the corresponding representations, i.e., Φ+ V1 (g) = U+ (g) Φ+ , ↑ g ∈ P+ . Proof. That the mapping Φ+ is an isometry follows already from Eq. (32) in the proof of the preceding lemma. The intertwining property is proved by direct computation. Indeed, for g = (A, a) ∈ SL(2, C) R4 and putting q := Λ−1 A p, we have on the one hand (U+ (A, a) Φ+ (ϕ))(p) and on the other hand Φ+ (V1 (A, a) ϕ)(p)
i
= e−ipa e 2 θ(A,p) χ(q)
−1 χ ΛA (·) −i(·)a −1 (p) = Φ+ e AHΛ (·) A 0
−1 χ ΛA (·) −1 = Φ+ H(·) e−i(·)a H(·) (p) AHΛ−1 (·) A 0 i
= e−ipa e 2 θ(A,p) χ(q),
i where for the last equation we have used that Hp−1 AHq 11 = e 2 θ(A,p) .
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With the preceding result we have also proved the equivalence between the representation (h+ , V1 , ·, ·+ ) and the representation (H− , V4 , ·, ·β− ) used in [12, Theorem B.2.17]. Let us finish this subsection defining the space and the representation corresponding to the opposite helicity. They will be denoted by the subindex “–” and the proofs are analogous as before. This representation space associated to the opposite helicity will be necessary in order to construct the reference space of the CAR-algebra (cf. Subsection 3.1). On the space
χ(p) 2 h− := Hp | χ ∈ L (C+ , C, µ0 (dp)) 0 define for g = (A, a) ∈ SL(2, C) R4 , q := Λ−1 A p and ϕ ∈ h− the representation (V4 (g) ϕ)(p) := e−ipa A ϕ(q). This representation is irreducible and unitary w.r.t. the scalar product, −1 ∗ −1 ϕ1 , ϕ2 − := ϕ1 (p), Hp Hp ϕ2 (p) µ0 (dp), for ϕi ∈ h− , i = 1, 2. C2
C+
Finally (h− , V4 , ·, ·− ) is equivalent to L2 (C+ , C, µ0 (dp)), U− , ·, ·L2 , where the latter representation is given by i
(U− (g) χ)(p) := e−ipa e− 2 θ(A,p) χ(q) ,
2.8
χ ∈ L2 (C+ , C, µ0 (dp)) , q := Λ−1 A p.
Maxwell Equations: F-Equation
Now for any L =
i
e2θ 0
i
e− 2 θ z i e− 2 θ
∈ E(2) we have on H(1, 0) := Sym C2 ⊗ C2 the
representation D(1, 0) (L) := L ⊗ L . (Notation: In the remaining subsection we will denote again the representation D(1, 0) (·) simply by D(·) etc.). We select the 1-dimensional sub D(E(2))-invariant 10 10 space characterized by the orthoprojection π := ⊗ . 00 00 ∈ h0 , ϕ ∈ L2 (C+ , H, µ0 (dp)) (see Eq. (27)). 2.22 Lemma Put ϕ(p) := D(Hp ) ϕ(p) Then we have that ϕ satisfies the equation π ϕ(p) = ϕ(p) iff ϕ satisfies the spinorial form of Maxwell Equation (F-Equation for short), which in components is given by 1 (P † )C C ϕCB (p) = 0 , C ∈ {0, 1} , B ∈ {0, 1} . (34) C=0
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Proof. 1. Note first that π ϕ(p) = ϕ(p) iff ϕ 01 (p) = ϕ 10 (p) = 0 = ϕ 11 (p). Therefore from Eq. (30) we get 1
(P † )C C ϕCB (p) =
C=0
1
B (P † )C C (Hp )C 00 (p) = 0 , 0 (Hp )0 ϕ
C=0
C
2. Conversely, suppose that
1
∈ {0, 1} , B ∈ {0, 1} .
(P † )C C ϕCB (p) = 0. From the form of P †
C=0
(see Eq. (29)) and since −1 B CD ϕ EB (p) = (Hp−1 )E (p) , C (Hp )D ϕ
01 (p) = 0. Therefore π ϕ(p) = we obtain ϕ 1B (p) = 0, B ∈ {0, 1}, hence also ϕ ϕ(p). Recall that from a symmetric spinor field ϕCB satisfying the F-Equation, one can construct canonically a real and antisymmetric tensor field Fµν satisfying the source free Maxwell Equations [70, Exercise 13.3], [56, Section 5.1]. The space of solutions of the F-Equation is given by
1 1 2 h+ := D(Hp ) ⊗ χ(p) | χ ∈ L (C+ , C, µ0 (dp)) , (35) 0 0 2.23 Lemma Define on the space of solutions of the F-Equation h+ the scalar product ϕ1 , ϕ2 + := ϕ1 (p), D(Hp−1 )∗ D(Hp−1 )ϕ2 (p)C4 µ0 (dp), for ϕi ∈ h+ , i = 1, 2. C+ The representation given for g = (A, a) ∈ SL(2, C) R4 , q := Λ−1 A p and ϕ ∈ h+ , by (36) (V+ (g) ϕ)(p) := e−ipa D(A) ϕ(q), is unitary w.r.t. ·, ·+ and irreducible. Proof. First note that for ϕi (p) = D(Hp ) i = 1, 2,
1 1 ⊗ χ (p), χi ∈ L2 (C+ , C, µ0 (dp)), 0 0 i
ϕ1 , ϕ2 + =
χ1 (p) χ2 (p) µ0 (dp).
(37)
C+ Since the space πH is 1-dimensional and π D(L)∗ D(L) π = π, L ∈ E(2), Theorem 2.15 completes the proof.
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Next we establish the equivalence between (h+ , V + , ·, ·+ ) defined above and the representation L2 (C+ , C, µ0 (dp)), U+ , ·, ·L2 . Consider on the space L2 (C+ , C, µ0 (dp)) the canonical representation for n = 2 (cf. Eq. (23)), (U+ (g) χ)(p) := e−ipa eiθ(A,p) χ(q),
χ ∈ L2 (C+ , C, µ0 (dp)) ,
(38)
4 for q := Λ−1 A p and g = (A, a) ∈ SL(2, C) R . U+ (·) is irreducible, strongly con2 tinuous and unitary for the usual L -scalar product ·, ·L2 . With the preceding notation we have the following equivalence of representations
2.24 Proposition The mapping Φ+ : h+ −→ L2 (C+ , C, µ0 (dp)) defined by,
1 1 ◦ , Φ+ D H(·) ⊗ χ(·) (p) = χ(p) , p ∈ C+ 0 0
is an isometric isomorphism between (h+ , ·, ·+ ) and L2 (C+ , C, µ0 (dp)), ·, ·L2 that commutes with the corresponding representations, i.e., Φ+ V+ (g) = U+ (g) Φ+ , ↑ . g ∈ P+ Proof. That the mapping Φ+ is an isometry follows already from Eq. (37) in the proof of the preceding lemma. The intertwining property is proved by a direct computation. Indeed, for g = (A, a) ∈ SL(2, C) R4 and putting q := Λ−1 A p, we have on the one hand, (U+ (A, a) Φ+ (ϕ))(p) = e−ipa eiθ(A,p) χ(q)
(39)
and on the other hand computing similarly as in the proof of Proposition 2.21 Φ+ (V+ (A, a) ϕ)(p)
−i(·)a −1 1 1 −1 D H(·) AHΛ−1 (·) ⊗ χ ΛA (·) (p) = Φ+ D H(·) e A 0 0 =
e−ipa eiθ(A,p) χ(q),
where for the last equation we have used that D(Hp−1 AHq )11 = eiθ(A,p) .
Let us finish this subsection defining the space and the representation corresponding to the opposite helicity. They will be denoted by the subindex “–” and the proofs are similar as before. This representation space associated to the opposite helicity will be necessary in order to construct the reference space of the CCR-algebra. On the space
1 1 h− := D Hp (40) ⊗ χ(p) | χ ∈ L2 (C+ , C, µ0 (dp)) , 0 0
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define for g = (A, a) ∈ SL(2, C) R4 , q := Λ−1 A p and ϕ ∈ h− the representation (41) (V− (g) ϕ)(p) := e−ipa D A ϕ(q). This representation is irreducible and unitary w.r.t. the scalar product −1 ∗ −1 ϕ1 (p), D Hp D Hp ϕ2 (p) 4 µ0 (dp), ϕ1 , ϕ2 − := C
C+
for ϕi ∈ h− , i = 1, 2.
2 Finally (h− , V− , ·, ·− ) is equivalent to L (C+ , C, µ0 (dp)), U− , ·, ·L2 , where the latter representation is given by (U− (g) χ)(p) := e−ipa e−iθ(A,p) χ(q) ,
χ ∈ L2 (C+ , C, µ0 (dp)) , q := Λ−1 A p . (42)
Generalization to arbitrary helicity: We collect in this paragraph the obvious generalization of the previous analysis of the Weyl resp. Maxwell equations (which correspond to helicities ± 12 resp. ±1) to the systems carrying arbitrary discrete helicity. We begin characterizing general massless relativistic wave equations corresponding to helicity n2 (cf. [56, p. 375]). The following result contains as special cases Lemmas 2.19 and 2.22. n
2.25 Corollary Consider the D( 2 , 0) (E(2))-invariant subspace characterized by the
n 10 one-dimensional orthoprojection πn := ⊗ , n ∈ N, and put 00 n
ϕ(p) := D( 2 , 0) (Hp ) ϕ(p) ∈ h0 ,
ϕ ∈ L2 (C+ , H, µ0 (dp))
(cf. Eq. (27)).
Then ϕ satisfies the equation π ϕ(p) = ϕ(p) iff ϕ satisfies the massless relativistic wave equation corresponding to helicity n2 . The latter equation is written in momentum space for the spinorial components as 1
(P † )C C ψ CC1 . . . Cn−1 (p) = 0,
C1 , . . . , Cn−1
∈ {0, 1}, C ∈ {0, 1} .
C=0
2.26 Remark. (i) The way of presenting relativistic wave equations written in momentum space is justified by the group theoretical approach which is one of the basic ingredients of the present paper. However, to give a more complete picture of these equations we need to comment on them also as PDEs in position space, since they usually appear in the literature in this form. A general massless relativistic wave equations on position space is given by 1 C=0
∂C C ψ CC1 . . . Cn−1 (x) = 0,
C1 , . . . , Cn−1
∈ {0, 1}, C ∈ {0, 1} ,
(43)
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where ∂C C is the first-order differential operator on spinor fields corresponding to the usual gradient ∂µ , µ = 0, . . . , 3 [70, Eq. 13.1.64]. It can be shown that Eq. (43) is equivalent to the usual wave equation ψ CC1 . . . Cn−1 (x) = 0, which is an hyperbolic equation, together with Eq. (43) holding only as an initial value constraint on a Cauchy surface (e.g., x0 = 0) (for details see [70, pp. 376–377]). This fact confirms the point of view already stated in the introduction (see also Remark 2.11 (i)) that massless relativistic wave equations can be seen as constraint equations restricting the fiber degrees of freedom. The results cited above show that Eq. (43) has a well-posed initial value formulation is relevant if one wants to construct quantum fields on more general (globally hyperbolic) space-times, where the group theoretical approach is not possible due to the lack of symmetry. (ii) We want now complete the generalization to include the corresponding spaces of solutions of the relativistic wave equations, the representations and the associated isometric isomorphisms. For this one needs only to replace the labels (1, 0) resp. (0, 1) by ( n2 , 0) resp. (0, n2 ), n ≥ 3, in the present subsection. In particular we obtain in this way a characterization of the Wigner massless Hilbert spaces with discrete helicity in terms of the space of solutions of the corresponding massless relativistic wave equation (cf. Corollary 2.25).
2.9
Maxwell Equations: A-Equation
For completeness we will include in our group theoretical context the discussion of Maxwell Equations in terms of the vector potential field (A-Equation). We will see that some techniques used in the previous subsection for treating Maxwell Equations in terms of the field strength (F-Equation) will not be applicable anymore (cf. Remark 2.29). Instead we will use a Gupta-Bleuler like procedure will allow to establish an isometric isomorphism to the previous representation space. For a detailed treatment of quantum electromagnetism in terms of the vector potential (including constraints) we refer to [32].
iθ i e− 2 θ z e2 1 1 ∈ E(2) we have on H( 2 , 2 ) := C2 ⊗ C2 ∼ For L = = C4 the − 2i θ 0 e representation 1 eiθ z e−iθ z |z|2 0 eiθ 1 1 0 z . D( 2 , 2 ) (L) := L ⊗ L ∼ = −iθ 0 0 e z 0 0 0 1 (Notation: In the remaining subsection we will denote when no confusion arises 1 1 again the representation D( 2 , 2 ) (·) simply by D(·).) In order to include the nontrivial phases of the diagonal of D(L) (recall that now we want to describe both
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helicity values ±1) one is forced to consider in this context the3-dimensional space 1 0 0 0 0 1 0 0 characterized by the D(E(2))-invariant orthoprojection π := 0 0 1 0 . 0 0 0 0 In the following lemma we will write the divergence equation p0 ψ0 (p) − 3 pi ψi (p) = 0 in an equivalent and for us more convenient form. i=1
ϕ0 (p) ϕ1 (p) ◦ 2.27 Lemma The vector ϕ(p) := ϕ2 (p) , p ∈ C+ , satisfies the equation ϕ3 (p) −(p0 − p3 )ϕ0 (p) + (p1 + ip2 )ϕ1 (p) + (p1 − ip2 )ϕ2 (p) − (p0 + p3 )ϕ3 (p) = 0 (44) 1 0 0 1 0 1 1 0 is a unitary iff the vector ψ(p) := Ws ϕ(p), where Ws := √1 2 0 i −i 0 1 0 0 −1 3 4 pi ψi (p) = 0. matrix acting on C , satisfies the equation p0 ψ0 (p) − i=1
Proof. The proof is straightforward since it uses essentially a unitary transformation acting on H. We write it explicitly down in order to introduce some useful notation for later on. Put 0 0 0 −1 1 0 0 0 0 1 0 0 0 −1 0 0 η := Ws−1 ηMink Ws = 0 0 1 0 , with ηMink := 0 0 −1 0 . −1 0 0 0 0 0 0 −1 (45) Now for p = (p0 , p1 , p2 , p3 ) ∈ C+ and recalling that ·, ·C4 is antilinear in the first argument we have that the equations p, ηMink ψ(p)C4 = p, Ws η Ws−1 ψ(p)C4 = Ws−1 p, η ϕ(p)C4 conclude the proof, since the last term is precisely
√1 2
times the l.h.s. of Eq. (44).
In the following lemma we will establish the relation between the equation in Lemma 2.16 (ii) and the A-Equation (44). ∈ h0 , for all ϕ ∈ L2 (C+ , H, µ0 (dp)) (see 2.28 Lemma Put ϕ(p) := D(Hp ) ϕ(p) Eq. (27)). Then we have that ϕ satisfies the equation π ϕ(p) = ϕ(p) iff ϕ satisfies the A-Equation (44).
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◦ Proof. 1. Suppose that ϕ satisfies the equation π ϕ(p) = ϕ(p), p ∈ C+ . Then there 2 exist three scalar functions χ ∈ L (C , C, µ (dp)), i = 0, 1, 2, such that ϕ(p) = i + 0 χ0 (p) χ1 (p) χ2 (p) . But using Eq. (21) one can explicitly check that ϕ(p) = D(Hp ) ϕ(p) 0 satisfies Eq. (44). 2. Suppose now that ϕ satisfies Eq. (44). Then we can compute:
ϕ 3 (p) = D(Hp )−1 ϕ(p) 3
=
(p0 − p3 )ϕ0 (p) − (p1 + ip2 )ϕ1 (p) − (p1 − ip2 )ϕ2 (p) + (p0 + p3 )ϕ3 (p)
=
0,
◦ . and therefore ϕ satisfies the equation π ϕ(p) = ϕ(p), p ∈ C+
Denote by
h+−
χ0 (p) χ1 (p) 2 | χi ∈ L (C+ , C, µ0 (dp)), i = 0, 1, 2 := D(Hp ) χ2 (p) 0
(46)
the space of solutions of the A-Equation and recall that from the general definition of V given in Eq. (26) we have here, 1
1
(V+− (g) ϕ)(p) := e−ipa D( 2 , 2 ) (A) ϕ(q), where g = (A, a) ∈ SL(2, C) R4 , q := Λ−1 A p and ϕ ∈ h+− . 2.29 Remark. In analogy with the Weyl case or with the F-Equation case, we can try to define on h+− the following sesquilinear form: 2 ϕ, ϕ• := ϕ(p), D(Hp−1 )∗ D(Hp−1 )ϕ• (p)C4 µ0 (dp) = χi (p)χ•i (p) µ0 (dp), C+ where
C+
χ0 (p) χ1 (p) ϕ(p) := D(Hp ) χ2 (p) , 0
i=0
χ•0 (p) χ•1 (p) ϕ• (p) := D(Hp ) χ•2 (p) ∈ h+− . 0
This sesquilinear form is positive definite, but it will not be V+− -invariant, since in the present case we have in general that π D(Hp−1 AHq )∗ D(Hp−1 AHq ) π = π, cf. Eq. (28). Roughly speaking, we have allowed too many degrees of freedom on the fiber in order to apply the arguments used in the Weyl case or in the F-Equation case which are based on Theorem 2.15.
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Motivated nevertheless by the Lorentz-invariance of the Minkowski scalar product we introduce the following sesquilinear form: for ϕ, ϕ• as in the preceding lemma, ϕ, ϕ• +− := ϕ(p), η ϕ• (p)C4 µ0 (dp) (47)
=
C+ 2 C+
χi (p) χ•i (p) µ0 (dp),
(48)
i=1
where η is given in the proof of Lemma 2.27 and can be seen as the spinorial form of the Minkowski metric. Note that since D(A)∗ η D(A) = η
(49)
the above sesquilinear form is V+− -invariant, but now ·, ·+− is only semidefinite and the corresponding (degenerate) space of zero vectors is easily seen to be, χ(p) 0 2 . (50) hd := D(Hp ) (C , C, µ (dp)) | χ ∈ L + 0 0 0 Since hd is a V+− -invariant space denote by VA the natural definition of V+− on the factor space hA := h+− /hd , which is Hilbert space w.r.t. the scalar product, ·, ·A , defined as the lift of ·, ·+− .4 The elements of the factor space are written as [ϕ]0 , where ϕ ∈ h+− and the bracket, [·]0 , specify the corresponding equivalence class. The preceding situation with the appearance of factor spaces is typical when dealing with not fully decomposable representations of a Lie group. This situation is studied in general terms by Araki and our construction above using the space of solutions of the A-Equation is a special case of Theorem 1 in [4]. The following statement justifies the use of the index +− in h+− , since this space carries a representation that contains the irreducible representations describing helicity +1 and −1. In the next result we will show the equivalence of the 4 Notice
that we cannot restrict V+− to the space 0 χ1 (p) | χi ∈ L2 (C+ , C, µ0 (dp)), i = 1, 2 , D(Hp ) χ2 (p) 0
because it is not V+− -invariant. Indeed, this follows from the fact that the space 0 a | a, b ∈ C b 0 is not D(L)-invariant, L ∈ E(2) (cf. [6, Section 5.B.1]).
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representations (VA , hA ) and the direct sum of the canonical representations U+ and U− given in Eqs. (39) and (42) respectively. 2.30 Proposition The mapping, ΦA : hA −→ L2 (C+ , C, µ0 (dp)) ⊕ L2 (C+ , C, µ0 (dp)), defined by χ0 (·)
ΦA D H(·) χ1 (·) (p) := χ1 (p) ⊕ χ2 (p) , χ2 (·) 0 0 is an isometric isomorphism that commutes with the representations VA and U+ ⊕ ↑ . The U− , i.e., the equation, ΦA VA (g) = (U+ (g) ⊕ U− (g)) ΦA holds for all g ∈ P+ representation VA on (hA , ·, ·A ) is unitary, strongly continuous and reducible. Proof. The unitarity of VA follows from the V+− -invariance of the sesquilinear form ·, ·+− and from the construction of the factor space h+− /hd . The isometry property of ΦA follows already from Eq. (48) and the intertwining property can be checked by direct computation as in Proposition 2.24. Use, for instance, the relation χ0 (Λ−1 A ·) −1
·) χ1 (ΛA ΦA D H(·) e−i(·)a D H −1 AH −1 Λ (·) (·) χ2 (Λ−1 ·) (p) A A 0 0 = (U+ (g)χ1 )(p) ⊕ (U− (g)χ2 )(p) .
Recall further that, for g = (A, a) ∈ SL(2, C) R4 , q := Λ−1 A p and χ ∈ L (C+ , C, µ0 (dp)), the canonical representations (cf. Eq. (23) in the case n = 2), 2
(U± (A, a) χ)(p) := e−ipa e±iθ(A,p) χ(q) , are unitary, strongly continuous and irreducible. They correspond to systems with opposite helicity. Finally, we will prove a theorem that relates the pair (hA , VA ) defined above with some combinations of the spaces and representations used in Subsection 2.8. Concretely, using the definitions5 (35), (36), (40) and (41) we consider the following Hilbert space, scalar product and unitary representation w.r.t. it hF := h+ ⊕ h− ·, ·F := ·, ·+ ⊕ ·, ·− (51) VF := V+ ⊕ V− . Then we have the following equivalence between (VF , hF ) and (VA , hA ): 5 Recall that in Subsection 2.8, D(·) means D (1, 0) (·). We have also written D (0, 1) (A) instead of using the notation D (1, 0) (A), A ∈ SL(2, C).
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2.31 Theorem The mapping ΦAF : hF −→ hA , defined by
1 1 1 1 (1, 0) (0, 1) ΦAF D (H(·) ) ⊗ χ (p)⊕D (H(·) ) ⊗ χ (p) (p) 0 0 + 0 0 −
0 (1, 1) χ+ (p) 2 2 (H ) := D p χ− (p) , 0 0
1 1 1 with ϕ+ (p) := D(1, 0) (Hp ) ⊗ χ+ (p) ∈ h+ and ϕ− (p) := D(0, 1) (Hp ) 0 0 0
1 ⊗ χ (p) ∈ h− , is an isometric isomorphism that commutes with the corre0 − ↑ sponding representations, i.e., ΦAF VF (g) = VA (g) ΦAF , g ∈ P+ . Proof. The isometry property follows from the equations,
ΦAF (ϕ+ ⊕ ϕ− ), ΦAF (ϕ+ ⊕ ϕ− )A = |χ+ (p)|2 + |χ− (p)|2 µ0 (dp) C+ = ϕ+ ⊕ ϕ− , ϕ+ ⊕ ϕ− F . The intertwining property is a consequence of Proposition 2.24 (and the corresponding result for the opposite helicity) and of Proposition 2.30. Indeed, note that ΦAF = Φ−1 A ΦF and, therefore, −1 ΦAF VF (g) = Φ−1 A ΦF VF (g) = ΦA (U+ (g) ⊕ U− (g)) ΦF
↑ = VA (g) Φ−1 A ΦF = VA (g) ΦAF , g ∈ P+ ,
and the proof is concluded.
2.32 Remark. Landsman and Wiedemann [45, Theorem 1] interpret the space (hA , ·, ·A , VA ) (written in tensorial language) in the context of MarsdenWeinstein reduction theory. They also mention its equivalence to the triplet (hF , ·, ·F , VF ). For the relation between the two preceding spaces in terms of tensors see also [15] or [37, Section V.B].
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Summary
In the present section we have described covariant and massive/massless canonical representations of the Poincar´e group in the general frame of induced representation theory. Relativistic wave equations appear in the context of canonical representations if one considers reducible representations of the corresponding little groups. Due to the different nature of the massive resp. massless little groups, the corresponding relativistic wave equations play also a different role and are characterized by reducing resp. invariant projections. We will summarize in the following table some of the results concerning massive and massless relativistic wave equations.
massive, m > 0 rel. wave equation
Dirac
Proca
Inducing rep. of SU (2): Unitary and fully decomposable
τ (U ) := U ⊕ U
1 1
D ( 2 , 2 ) (U ) = U ⊗ U
reducing projection
1 1 0 (Dirac) π := 2 1 0 1 0 π (Proca) := 12 0 −1
0 1 0 1
1 0 1 0
0 1 0 1
0 2 0 0
0 0 2 0
−1 0 0 1
massless, m = 0 rel. wave equation
Inducing rep. of E(2): Nonunitary and non fully decomposable
Weyl
D ( 2 ,0) (L) := L
Maxwell: F-Eq.
D (1,0) (L) := L ⊗ L
Maxwell: A-Eq.
1 1 D ( 2 , 2 ) (L) = L ⊗ L
General massless (helicity: n , n ≥ 1) 2
1
n
n
D ( 2 ,0) (L) = ⊗ L
invariant projection
πW :=
πF :=
1 0
1 0 := 0 0
1 0 0 0
0 1 ⊗ 0 0
0 1 πA 0 0
n 1 πn := ⊗ 0
0 0 1 0 0 0
0 0
0 0 0 0
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3 Massless free nets and relativistic wave equations We begin this section stating some core axioms of local quantum physics [34, 35, 11, 14, 59, 28]. Here the point of view is that the correspondence O → A(O) between Minkowski space regions O and local algebras of observables A(O) characterizes intrinsically the theory. Following Haag’s suggestive idea, quantum fields (which are the central objects in other formulations of QFT) can be seen in the present setting just as ‘coordinates’ of the preceding net, in the sense that one may use different quantum fields to describe the same abstract net. In order to avoid any concrete representation of the C*-algebra we will construct the net directly following the strategy presented in [12] (see also [60]), i.e., basing the construction on group-theoretical arguments and standard CAR or CCR-theory (see for the latter algebras [14, Chapter 8] and references cited therein; in the Fermi case we will use Araki’s self-dual approach to the CAR-algebra). We will call the result of this construction a free net and the fundamental object that characterizes it is the so-called embedding that reduces the covariant representation in terms of the corresponding canonical ones (cf. Remark 2.11). As reference spaces of the CARand CCR-algebras we will use in this paper the space of solutions of the Weyland the F-Equation introduced in Subsections 2.7 and 2.8, respectively, and will therefore call the corresponding already typical nets of local C*-algebras Weylresp. F-net. Some relations of the latter with the vector potential will also be mentioned. We will also relate the present construction to the nets specified in [48], since in this reference the nets were given without mentioning explicitly the corresponding relativistic wave equations. Concretely, we will show that the nets associated to the Weyl- and the F-Equation are isomorphic to the nets constructed in [48, cases n = 1 and n = 2], respectively. This isomorphy may then be easily generalized to arbitrary n. Thus producing the same net of local C*-algebras the generalization of the Weyl- and the F-net constructed in the following will present a new methodological aspect w.r.t. [12, 48], namely we will show explicitly the relation to the corresponding relativistic wave equations. In particular the embeddings used here are different from those in [12, 48] and to construct them we will make essential use of distinguished elements of certain intertwiner spaces associated to representations of the little group E(2). By means of these elements the embedding will map any (vector-valued) test function into the space of solutions of the corresponding massless relativistic wave equation. This procedure illuminates another aspect of the reduction of the (fibre) degrees of freedom that is necessary when considering massless representations of nontrivial helicity (cf. Remark 2.11). Denote by B(R4 ) the family of open and bounded regions in Minkowski space partially ordered by inclusion ‘⊂’. (B(R4 ), ⊂) is then a directed index set which is stable under the action of the Poincar´e group [14, Sections 5.1 and 7.1]. 3.1 Definition A correspondence B(R4 ) O → A(O) where the local algebras A(O) are (abstract) unital C*-algebras with common unit 1, is called a HaagKastler net (HK-net for short) if the following conditions are satisfied:
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(i) (Isotony) If O1 ⊆ O2 , then A(O1 ) ⊆ A(O2 ), O1 , O2 ∈ B(R4 ). We denote by A := lim A(O) the corresponding inductive limit which is called the quasi −→ local algebra. (ii) (Additivity) Let {Oλ }λ∈Λ ⊂ B(R4 ) with ∪λ Oλ ∈ B(R4 ). The net O → A(O) satisfies additivity if for any such {Oλ }λ the following equation holds in A: A(∪λ Oλ ) = C∗ (∪λ A(Oλ )). (iii) (Causality) For any O1 ∈ B(R4 ) space-like separated w.r.t. O2 ∈ B(R4 ) (we denote this by O1 ⊥ O2 ), then A(O1 ) commutes elementwise with A(O2 ) in A. ↑ Suppose further that there exists a representation α(·) of the P+ in terms of *↑ automorphisms of A, i.e., P+ g → αg ∈ Aut A. (iv) (Covariance) The net O → A(O) transforms covariantly w.r.t. α, if for every ↑ O ∈ B(R4 ) we have αg (A(O)) = A(gO), g ∈ P+ , where gO := {gx | x ∈ O}. Next we will introduce the notion of isomorphic HK-nets (cf. [22, Section 3]). (i) , 3.2 Definition Two HK-nets (A(i) (O), α(i) (·) )O ∈ B(R4 ) with quasi local algebras A (1) (2) i = 1, 2, are called isomorphic if there exists a *-isomorphism Λ : A → A which preserves localisation, i.e., Λ(A(1) (O)) = A(2) (O), O ∈ B(R4 ), and intertwines between the corresponding actions of the Poincar´e group, i.e., Λ α(1) g = ↑ (2) αg Λ, g ∈ P+ .
Following the strategy suggested in [14, Section 8.3] we will now study a subclass of HK-nets, namely those where the local C*-algebras are certain C*subalgebras of the CAR- resp. CCR-algebras. Due to the nice functorial properties of these algebras it is possible to encode the axioms of isotony, additivity, causality and covariance of the HK-net at the level of the respective reference spaces. We will call the result of this construction a free net. (The indices F/B below denote the Fermi/Bose cases.) ↑ ), where 3.3 Definition We consider the following tuples associated to (R4 , ⊥, P+ ⊥ is the causal disjointness relation given by the Minkowski metric on R4 . (i) In the Fermi or CAR case we have (hF , ·, ·, Γ, VF , TF , TF , IF ), where (hF , ·, ·) is a complex Hilbert space and Γ an anti-unitary involution on ↑ it. VF denotes a unitary representation of P+ on (hF , ·, ·). Further, TF is 4 the set of test functions on R with compact support and TF is a represen↑ tation of P+ on TF satisfying the following support property: if f ∈ TF with ↑ . suppf ⊂ O ∈ B(R4 ), then TF (g)f ∈ TF with supp TF (g)f ⊂ gO, g ∈ P+ Finally, we require for the linear embedding IF : TF −→ hF the following properties:
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(F1) (Γ-invariance.) For an arbitrary f ∈ TF with suppf ⊂ O ∈ B(R4 ), there exists a k ∈ TF such that Γ IF f = IF k and suppk ⊂ O. (F2) (Causality.) For all f, k ∈ TF such that suppf ⊥ suppk, we have IF f , IF k = 0. (F3) (Covariance.) Γ VF (g) = VF (g) Γ and IF TF (g) = VF (g) IF , for all ↑ . g ∈ P+ (ii) In the Bose or CCR case we have (hB , σ, VB , TB , TB , IB ), where VB is a sym↑ plectic representation of P+ on the real symplectic space (hB , σ), i.e., VB (g), ↑ g ∈ P+ , is a bijection of hB that leaves σ invariant. TB is again the set of test ↑ functions on R4 with compact support and TB is a representation of P+ on TB satisfying the same support property as in the fermionic case. We require for the linear embedding IB : TB −→ hB the following properties: (B1) (Causality.) For all f, k ∈ TB such that suppf ⊥suppk, we have σ(IB f, IB k) = 0. ↑ . (B2) (Covariance.) IB TB (g) = VB (g) IB , for all g ∈ P+ Next we will show that the tuples that were specified in the preceding definition characterize in a canonical way a HK-net. 3.4 Theorem Assume the notation given in Definition 3.3 and consider the following nets of local linear submanifolds of the corresponding reference spaces: B(R4 ) O −→ hF (O) 4
B(R ) O −→ hB (O)
:=
{IF f | f ∈ TF
and
suppf ⊂ O} ⊂ hF , (52)
:=
{IB f | f ∈ TB
and
suppf ⊂ O} ⊂ hB . (53)
Then we have (i) The net of local C*-algebras given by B(R4 ) O −→ AF (O) := C∗ ({a(ϕ) | ϕ ∈ hF (O)})
Z2
⊂ CAR(hF , Γ)
is a HK-net. Here a(·) denote the generators of CAR(hF , Γ) and A Z2 means the fixed point subalgebra of the C*-algebra A w.r.t. Bogoljubov automorphism associated to the unitarity −1. The covariance of this net of local C*-algebras is realized by the Bogoljubov automorphisms αg associated to the Bogoljubov ↑ unitaries VF (g), g ∈ P+ . (ii) The net of local C*-algebras given by B(R4 ) O −→ AB (O) := C∗ ({W (ϕ) | ϕ ∈ hB (O)}) ⊂ CCR(hB , σ) is a HK-net. Here W (·) denote the Weyl elements (generators) of CCR(hB ,σ). The covariance of the net is given by the Bogoljubov automorphisms αg as↑ . sociated to VB (g), g ∈ P+ We call the nets of C*-algebras given in (i) and (ii) above free nets.
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Proof. First note that from the Γ-invariance property (F1) we have that Γ hF (O) = hF (O), O ∈ B(R4 ), which implies AF (O)∗ = AF (O) (as a set). The isotony of the local C*-algebras in the CAR and the CCR case follows immediately from the isotony property of the corresponding nets of linear submanifolds O → hF/B (O). To prove additivity we will show first that for {Oλ }λ∈Λ ⊂ B(R4 ) as in Definition 3.1 (ii) we have for the nets of local linear submanifolds hF/B (∪λ Oλ ) = span {hF/B (Oλ ) | λ ∈ Λ}. Indeed, the inclusion ‘⊇’ follows from thelinearity of I and the fact that if supp fλl ⊂ Oλl , l = 1, . . . , L, then supp ( l µλl fλl ) ⊂ ∪λl Oλl for fλl ∈ TF/B and µλl ∈ C. To show the converse inclusion take f ∈ TF/B with supp f ⊂ ∪λ Oλ . By compactness there exists a finite subcovering such that supp f ⊂ ∪L l=1 Oλl and using a subordinate smooth partition of unity (which exists since R4 is paracompact) we can write f = fλ1 + · · · + fλL , where fλl ∈ TF/B and supp fλl ⊂ Oλl , l = 1, . . . , L. Therefore If = Ifλ1 + · · · + IfλL ∈ span {hF/B (Oλ ) | λ ∈ Λ}. Now additivity follows from the properties of the generators of the CAR- and CCR-algebras, cf. [14, Section 8.3]. For the causality property take O1 , O2 ∈ B(R4 ) with O1 ⊥ O2 . Now in the CAR case define the sets Pi of polynomials in the generators a(ϕi ), ϕi ∈ hF (Oi ), i = 1, 2, such that the degree of the corresponding monomials is even. From property (F2) and the CAR’s we have that a(ϕ1 ) a(ϕ2 ) + a(ϕ2 ) a(ϕ1 ) = 0 for all ϕi ∈ hF (Oi ), i = 1, 2, and therefore [P1 , P2 ] = 0. Now, since Pi is dense in AF (Oi ), i = 1, 2, we obtain that [AF (O1 ), AF (O2 )] = 0 in A. In the Bose case note that from (B2) and the Weyl relation we have W (ϕ1 ) W (ϕ2 ) = W (ϕ2 ) W (ϕ1 ) for all ϕi ∈ hB (Oi ) and since span {W (ϕi ) | ϕi ∈ hF (Oi )} is dense in AB (Oi ), i = 1, 2, we also obtain in this case that [AB (O1 ), AB (O2 )] = 0 in A. Finally, to prove the covariance property denote by α(F) resp. α(B) the Bog g ↑ goljubov automorphisms associated to VF (g) resp. VB (g), g ∈ P+ . Now by the support properties of TF (g) and TB (g) as well as by (F3) and (B2) we have VF (g) hF (O) = hF (gO) and VB (g) hB (O) = hB (gO) .
(54)
Now from the way the Bogoljubov automorphisms act on the corresponding generators of the CAR/CCR-algebras it follows from the preceding equations that α(F) g (AF (O)) = AF (gO)
and α(B) g (AB (O)) = AB (gO) ,
↑ g ∈ P+ ,
which concludes the proof. (See further Theorem 3.6 in [48] for a detailed proof of the covariance property.) We will need later on the notion of isomorphic free nets explicitly. The isomorphy can be transcribed in terms of the corresponding reference spaces:
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3.5 Proposition (i) Consider two tuples (hi , Γi , ·, ·i , Vi , T , T, Ii ), i = 1, 2, over the same test function space and satisfying the properties of Definition 3.3 (i). Suppose that there exists a unitary linear mapping λF : h1 → h2 (i.e., λF (ϕ), λF (ψ)2 = ϕ, ψ1 , ϕ, ψ ∈ h1 ) satisfying ↑ λF V1 (g) = V2 (g) λF , λF Γ1 (g) = Γ2 (g) λF , g ∈ P+ ,
and
λF I1 = I2 .
Then the corresponding fermionic free nets are isomorphic. (ii) Consider two tuples (hi , σi , Vi , T , T, Ii ), i = 1, 2, over the same test function space and satisfying the properties of Definition 3.3 (ii). Suppose that there exists a (real) linear symplectic bijection λB : h1 → h2 (i.e., σ2(λB (ϕ), λB (ψ)) = σ1 (ϕ, ψ), ϕ, ψ ∈ h1 ) satisfying ↑ , λB V1 (g) = V2 (g) λB , g ∈ P+
and
λB I1 = I2 .
(55)
Then the corresponding bosonic free nets are isomorphic. Proof. The proof of (ii) is typical: denote by W (ϕi ), ϕi ∈ hi , the Weyl elements of the corresponding C*-algebras CCR(hi , σi ), i = 1, 2. Then the mapping Λ(W (ϕ1 )) := W (λ(ϕ1 )), ϕ1 ∈ h1 , extends uniquely to an isomorphism (also denoted by Λ) of the corresponding CCR-algebras. Further, the equation λB I1 = I2 implies λB h1 (O) = h2 (O), so that for the local C*-subalgebras we have Λ(A1 (O)) = A2 (O), O ∈ B(R4 ). Finally, the intertwining property of λ in Eq. (55) ↑ (2) implies that Λ α(1) g = αg Λ, g ∈ P+ . 3.6 Remark. Part (ii) of Definition 3.3 contains some aspects of Segal’s notion of quantization for bosonic systems (cf. [61, p. 750], [8, p. 106]). With Definition 3.3 and concretely through the requirements on the embedding I we incorporate to this program the axioms of local quantum physics. Note nevertheless that since Haag-Kastler axioms are stated in terms of abstract C*-algebras we do not require initially (in contrast with Segal’s approach) that the abstract CCR-algebra is represented in any Hilbert space nor the specification of any regular state. (For further reasons on this last point see also [31, 32].) We consider next also the spectrality condition, which in the context of free nets can be stated in terms of the tuples considered in Definition 3.3. 3.7 Definition With the notation of Definition 3.3 we require respectively: (F4) There exists a basis projection P on hF (i.e., an orthoprojection satisfying P + ΓP Γ = 1) reducing the representation VF , i.e., P VF (g) = VF (g) P , ↑ g ∈ P+ , and such that the corresponding representation on P hF is strongly continuous and satisfies the spectrality condition (i.e., the spectrum of the corresponding generators of the space-time translations is contained in the forward light cone V + ).
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(B3) There exists a real scalar product s on hB and an internal complexification J satisfying J 2 = −1, σ(ϕ, Jψ) = −σ(Jϕ, ψ), σ(ϕ, Jϕ) = s(ϕ, ϕ) and |σ(ϕ, ψ)|2 ≤ s(ϕ, ϕ)s(ψ, ψ), ϕ, ψ ∈ hB . W.r.t. this complexification VF is a strongly continuous unitary representation on the one particle Hilbert space (hB , kJ ) satisfying the spectrality condition. Here kJ = s + iσ denotes the corresponding complex scalar product. 3.8 Remark. Recall first that the basis projection P resp. the complexification J characterize Fock states of the CAR- resp. CCR-algebras. We will show in this remark that the preceding definition implies the existence of a covariant representation of the C*-dynamical systems (CAR(hF , Γ), R4 , α(·) ) and (CCR(hB , σ), R4 , α(·) ) satisfying the spectrality condition (cf. [16] and [12, Teorems A.4.2 and A.4.5]). Compare also with the notion of covariant representations introduced in [57, 43]. (i) From (F4) and from standard results of the CAR theory [5] it can be shown ↑ , are that the Bogoljubov automorphisms αg , corresponding to VF (g), g ∈ P+ uniquely implemented by unitary operators Qg on Fa (P hF ) (the antisymmetric Fock space over P hF ) that leave the Fock vacuum Ω invariant. Now it is straightforward calculation to show that on the set of finite particle vectors (which is dense in Fa (P hF )) the following equations hold for all ϕ ∈ hF and ↑ g ∈ P+ : πP (αg (a(ϕ))) = Q(P VF (g)) πP (a(ϕ)) Q(P VF (g))−1 and Q(P VF (g)) Ω = Ω , where πP is the Fock representation characterized by P and Q(P VF (g)) denotes the second quantization of the corresponding subrepresentation on Fa (P hF ). This implies that Qg = Q(P VF (g)) and since P VF (g) satisfies the spectrality condition on P hF , Q(P VF (g)) will also satisfy it on Fa (P hF ) [5]. (ii) In the CCR-case we obtain from (B3) and from the definition of the generat1 ing functional hF ϕ → e− 4 s(ϕ,ϕ) that the Bogoljubov automorphisms αg , ↑ corresponding to VB (g), g ∈ P+ , are uniquely implemented by unitary operators Qg on Fs (hB ) (the symmetric Fock space over the one particle Hilbert space hB ) that leave the Fock vacuum Ω invariant (see [14, Section 8.2]). But again a straightforward calculation shows that on the set of coherent vectors (which is total in Fs (hB ) cf. [33, Chapter 2]) the following equations hold for ↑ all ϕ ∈ hB and g ∈ P+ : πJ (αg (W (ϕ))) = Q(VB (g)) πJ (W (ϕ)) Q(VB (g))−1
and Q(VB (g)) Ω = Ω ,
where πJ is the Fock representation characterized by J and Q(VB (g)) denotes the second quantization of VB on Fs (hB ). This shows that Qg = Q(VB (g)). Now by the property of Fock states (cf. [14, Section 8.2.7]) that any positive operator on (h, kJ ) has a positive second quantization on Fs (hB ), we get
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finally that the spectrality condition of VB (g) on the one particle Hilbert space implies the spectrality condition for Q(VB (g)) on Fs (hB ). 3.9 Remark. The existence of the structures given in Definition 3.3 (i) or (ii) satisfying (F1)–(F4) or (B1)–(B3) (and therefore the existence of free nets) is shown in the context of Minkowski space in [48, 12]. In this paper we construct free nets of local C*-algebras associated to massive (massless) systems with arbitrary spin (helicity) [77]. The embedding, which is the central object of the free net construction, is given for example in the massive case of [12] as a direct sum of those mappings that reduce the covariant representation into the irreducible massive canonical representation (cf. also Remark 2.5). In other words the embedding selects from the algebraically reducible covariant representation two irreducible components.6 Summing up, we have transcribed Haag-Kastler’s axioms in terms of the embedding I and given a neat group theoretical interpretation of it in the context of the Poincar´e group. Note finally that the free net construction avoids (in the spirit of local quantum physics) any explicit use of the notion of quantum field. We finish this section adapting Lemma A.1.4 in [12] to the present massless case. This result will be essential for proving the causality property of the following models of free nets. 3.10 Lemma Let x ∈ R4 be a spacelike vector and βn (·) a matrix-valued function on C+ such that at each point p ∈ C+ the matrix elements of βn (p) are homogeneous polynomials of degree n in pµ , µ = 0, 1, 2, 3. Then we have ipx
e − e−ipx βn (p) µ0 (dp) = 0, n even, (56) C+
ipx
e + e−ipx βn (p) µ0 (dp) =
0,
n odd.
(57)
Proof. It is well known that for x2 < 0 the Pauli-Jordan function ∆(x) = e−ipx µ0 (dp)
(58)
C+
C+ ∞
is an even C function, i.e., ∆(x) = ∆(−x) (see [58, pgs. 71 and 107]). Let α be a multi-index and |α| := α0 + α1 + α2 + α3 . Then * ) even function, if |α| is even. ∂ |α| ∆ )) |α| −ipx = (−i) pα e µ0 (dp) is an ) α ∂x x odd function, if |α| is odd. C+ 6 These types of embedding play also an important role in the (rigorous) context of quantized fields, defined mathematically as operator-valued distributions (see, e.g., [58, Theorem X.42] or [8, Appendix B] in the example of the Klein-Gordon field).
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But from hypothesis the matrix elements of βn (p) are homogeneous polynomials in pµ of degree n and therefore the last expression implies Eqs. (56) and (57). We will denote the objects of the constructions in the following Sections with the subindex W or F depending if the net is associated to the Weyl Equation or to the F-Equation, respectively. All the mentioned models will have a particular function β(·) (recall the preceding lemma) that characterizes the corresponding scalar products and symplectic forms.
3.1
Weyl net
The following construction will illustrate the fermionic axioms (F1)–(F4) making explicit use of the Weyl equation. Further it will provide the simplest nontrivial example where certain intertwiner spaces are explicitly introduced in order to define the corresponding embedding that satisfy the conditions already stated in Subsection 2.6. Indeed, making use of the notation and results of Subsection 2.7 and of the particular structure of the intertwiner space associated to the finite-dimensional 1 1 representations D( 2 , 0) and D(0, 2 ) restricted to the massless little group E(2), we will construct the free net associated to the Weyl equation. The free net resulting from this construction is isomorphic to the one given in [48] for n = 1 (cf. Remark 3.17 (i)). We will see later on in the section that this construction procedure can be easily adapted to the bosonic case. Recall that given two representations V, V of a group G on finite dimensional Hilbert spaces H, H the corresponding intertwiner space is defined as (V (G), V (G)) := {Ψ : H → H | ψ is linear and Ψ V (g) = V (g) Ψ , g ∈ G} . 3.11 Lemma With the notion above we compute the following intertwiner spaces:
1 1 0 s = |s∈R , D( 2 , 0) (E(2)), D(0, 2 ) (E(2)) 0 0 1 ( 2 , 0) (1 , 0) D (E(2)), D 2 (E(2)) = C1 . Proof. The first intertwiner space consists of all M ∈ Mat2 (R) such that M L = 0 s L M for all L ∈ E(2). It is now immediate to check that M = , s ∈ R. The 0 0 triviality of the second intertwiner space is a straightforward computation. 3.1.1 CAR-algebra First recall the definitions associated to the Weyl Equation given in Subsection 2.7 ◦ and the form of the Hp -matrices, p ∈ C+ , given in Eq. (21). We consider the complex Hilbert space hW := h+ ⊕ h− ⊕ h+ ⊕ h− with the scalar product given
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by ·, ·W := ·, ·+ ⊕ ·, ·− ⊕ ·, ·+ ⊕ ·, ·− . To define the anti-linear involution on hW consider first the mapping Γ1 : h+ −→ h− given by
χ+ (p) , (Γ1 ϕ+ )(p) := Hp Γ0 Hp−1 ϕ+ (p) = Hp Γ0 0 1
1
where ϕ+ ∈ h+ and Γ0 : H( 2 , 0) → H(0, 2 ) is an anti-unitary involution (conjugation). It can be easily shown that Γ1 is anti-linear and that it satisfies the equation Γ1 ϕ+ , Γ1 ψ+ − = ψ+ , ϕ+ + for all ϕ+ , ψ+ ∈ h+ . Finally, define in terms of Γ1 the anti-unitary involution on hW as −1 ΓW (ϕ+ ⊕ ϕ− ⊕ ψ+ ⊕ ψ− ) := Γ−1 1 ψ− ⊕ Γ1 ψ+ ⊕ Γ1 ϕ− ⊕ Γ1 ϕ+ .
ΓW is anti-linear and it can easily be checked that, Γ2W = 1
and
ΓW ϕ(1) , ΓW ϕ(2) W = ϕ(2) , ϕ(1) W ,
ϕ(i) ∈ hW , i = 1, 2. The C*-algebra CAR(hW , ΓW ) is therefore uniquely given. 3.1.2 Existence theorem for the local algebras 1
We consider here on the test function space spaces TW := S(R4 , H(0, 2 ) ), h+ and ↑ h− the following covariant and canonical representations of P+ = SL(2, C) R4 g = (A, a): for f ∈ TW , ϕ± ∈ h±
(TW (g) f )(x) := A f Λ−1 A (x − a) , (V1 (g) ϕ+ )(p)
:=
(V2 (g) ϕ− )(p)
:=
e−ipa A ϕ+ (Λ−1 A p),
e−ipa A ϕ− (Λ−1 A p),
(V3 (g) ϕ+ )(p)
:=
(V4 (g) ϕ− )(p)
:=
eipa A ϕ+ (Λ−1 A p),
eipa A ϕ− (Λ−1 A p).
Note that the covariant representation TW , satisfies the support property mentioned in part (i) of Definition 3.3. We consider next the following reducible rep↑ over hW : resentation of P+ VW := V1 ⊕ V2 ⊕ V3 ⊕ V4 ↑ 3.12 Lemma The equation ΓW VW (g) = VW (g) ΓW holds for all g ∈ P+ . Proof. The equation is based on the following intertwining properties of Γ1 : + Γ1 V1 (g) = V4 (g) Γ1 (59) Γ1 V3 (g) = V2 (g) Γ1 , which are a direct consequence of the definitions.
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Further we consider the embedding I1,3 : TW −→ h+ and I2,4 : TW −→ h− defined for all f ∈ TW by,
−1 −1 01 01 (I1 f )(p) := Hp Hp f (p) , Hp f(−p) (I3 f )(p) := Hp 00 00
01 01 −1 , , (I2 f )(p) := Hp (I4 f )(p) := Hp Hp Γ0 f (p) , Hp−1 Γ 0 f (−p), 00 00 ◦ where the ‘hat’ f means the Fourier transformation and p is restricted to C+ . Note 0 1 1 1 1 1 that since D(0, 2 ) (E(2)), D( 2 , 0) (E(2)) ∈ D( 2 , 0) (E(2)), D(0, 2 ) (E(2)) 00 the above definitions are consistent. Finally, the embedding that specifies the net structure is given by
IW : TW −→ hW ,
IW f := I1 f ⊕ I2 f ⊕ I3 f ⊕ I4 f .
3.13 Remark. Note for instance that
1 −(p1 − ip2 ) −1 01 Hp = Hp 00 2 −(p0 − p3 )
p0 + p3 p1 + ip2
(60)
.
Therefore, the matrix elements of the previous expression correspond on position space to differential operators and IW will not change the localization properties of f . Further, the embedding (say I1 ) that specify IW can be written in components as −1 , BC E (p) εC B (Hp )B (sum over repeated indices ) , (I1 f )C (p) = (Hp )C B Q E f
0 1 1 0 and (QBC ) := is the matrix corresponding to where (εC B ) := −1 0 0 0 the point 12 (1, 0, 0, 1) in the positive light cone (recall Subsection 2.4). Moreover, Q can be seen as the 1-dimensional projection characterizing the Weyl equation (see Subsection 2.7).
The covariance property of the net characterized by the preceding embedding is guaranteed by the following result ↑ 3.14 Lemma The equation, IW TW (g) = VW (g) IW , holds for all g ∈ P+ . ◦ and q := Λ−1 Proof. First recall that for g = (A, a) ∈ SL(2, C) R4 , p ∈ C+ A p the −1 matrix Hp AHq ∈ E(2). Thus by Lemma 3.11 we have
Hp−1 AHq
01 00
=
01 Hp−1 AHq 00
=
i
e 2 θ(A,a)
01 00
.
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From this, the relations I1,3 TW (g) I2,4 Γ0 TW (g)
= V1,3 (g) I1,3
+
= V2,4 (g) I2,4 Γ0 ,
(61)
can be easily shown and the intertwining equation of the statement is proved. The next result will ensure causality for the net characterized by the embedding IW . 3.15 Lemma If supp f ⊥ supp k for f, k ∈ TW , then the equation IW f , IW kW = 0 holds. −1 ∗ −1 ◦ Proof. First put β and β + (p) := (Hp ) Hp − (p) = Γ0 β+ (p)Γ0 , p ∈ C+ (see also Remark 2.12). Then we compute
IW f , IW kW = (I1 f )(p), β + (p) (I1 k)(p)C2 µ0 (dp) C+ + (I2 Γ0 f )(p), β − (p) (I2 Γ0 k)(p)C2 µ0 (dp)
C+
(I3 f )(p), β + (p) (I3 k)(p)C2 µ0 (dp) C+ + (I4 Γ0 f )(p), β − (p) (I4 Γ0 k)(p)C2 µ0 (dp)
+
= C+ + C+ = C+ + C+ =
0,
C+ f(p), β+ (p) k(p)C2 µ0 (dp) +
, , Γ 0 f (p), β− (p) Γ0 k(p)C2 µ0 (dp)
C+ f(−p), β+ (p) k(−p)C2 µ0 (dp) , , + Γ 0 f (−p), β− (p) Γ0 k(−p)C2 µ0 (dp) C+ f(p), β+ (p) k(p)C2 µ0 (dp) +
C+
f(−p), β+ (p) k(−p)C2 µ0 (dp) +
k(−p), β+ (p) f(−p)C2 µ0 (dp) C+
k(p), β+ (p) f(p)C2 µ0 (dp)
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−1 ∗ 0 0 −1 Hp Hp = P † and β− (p) := Γ0 β+ (p) Γ0 (recall 0 1 Remark 2.13). The last equation follows from Lemma 3.10 and the fact that the matrix elements of P † are homogeneous polynomials of degree 1.
where β+ (p) :=
We can now prove the existence of a free net associated to the Weyl Equation, which we call Weyl net for short. 3.16 Theorem Consider the net of local linear submanifolds of hW given for O ∈ B(R4 ) by . 4 (0, 1 ) 2 ), suppf ⊂ O R4 ⊃ O −→ hW (O) := IW f | f ∈ C∞ . 0 (R , H Then the net of local C*-algebras defined by R4 ⊃ O −→ AW (O) := C∗ ({a(ϕ) | ϕ ∈ hW (O)})Z2 , where the a(·)’s denote the generators of the C*-algebra CAR(hW , ΓW ), is a HKnet. Proof. First note that the local linear submanifolds satisfy the ΓW -invariance property (F1) in Definition 3.3. Indeed, from the relations Γ1 (I1 f ) = I4 (Γ0 f ) and Γ1 (I3 f ) = I2 (Γ0 f ), f ∈ TW , it follows that ΓW IW f = IW f (which for the generators implies a(IW f )∗ = a(IW f )). Now from Lemmas 3.12, 3.14 and 3.15 we have that (hW , ·, ·W , ΓW , VW , TW , TW , IW ) satisfies all conditions stated in Definition 3.3 (i) and by Theorem 3.4 (i) we get that the net of local C*-algebras above is a HK-net. 3.17 Remark. (i) We will show next that the net constructed above is isomorphic to the fermionic net defined in [48, Section 3, case n = 1]. Using the notation and results of the latter reference we specify the unitary λW : hW → h1 (recall Proposition 3.5 (i)): for χ± ,ω± ∈ L2 (C+ , C, µ0 (dp)) put
λF
χ+ (·) χ− (·) ω+ (·) ω− (·) ⊕ H(·) ⊕ H(·) ⊕ H(·) (p) H(·) 0 0 0 0 /
/
0 0 0 0 := Hp ⊕ Hp χ+ (p) + χ− (p) − /
/
0 0 0 0 ⊕ Hp ⊕ Hp ω+ (p) + ω− (p) −
where [·]± denote the classes of the factor spaces H± defined in [48, Section 3]. Using the statements in the proof of [48, Lemma 3.2] it is straightforward to show that λF satisfies the properties required in Proposition 3.5 (i).
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(ii) From the construction given in Remark 3.8 it can be easily shown that 1 0 0 0 0 1 0 0 P := 0 0 0 0 is a basis projection on hW (i.e., P +ΓW P ΓW = 1) that 0 0 0 0 characterizes a Fock state on CAR(hW , ΓW ) satisfying the spectrality condition (recall Definition 3.7 (F4)). Note that V1 (a) ⊕ V2 (a), a ∈ R4 , satisfies the spectrality condition on the one particle Hilbert space P hW = h+ ⊕ h− . (iii) It is also straightforward to generalize the present construction to higher (half-integer) helicity values, just replacing in the preceding construction the indices ( 12 , 0) by ( n2 , 0) and (0, 12 ) by (0, n2 ) with n ≥ 3 and odd. Adapting part (i) above we get the isomorphy to the corresponding nets in [48]. From the isomorphy given in (i) of the previous remark we can assume the structural results of [48, Section 5] (see also [19]). For example we have: 3.18 Corollary The net of von Neumann algebras O → MW (O) obtained from the Weyl net using the canonical Fock space given in Remark 3.17 (ii) transforms in addition covariantly w.r.t. the (fourthfold covering) of the conformal group. Moreover it satisfies essential duality as well as timelike duality for the forward/backward cones. 3.19 Remark. The formulas for the graph of the modular operator and the modular conjugation associated to double cones given for fermionic models in [13, Theorem 5.10] can be also applied to the present construction.
3.2
F-net
The construction bellow will illustrate the bosonic axioms (B1)–(B3) of Section 3, making now use of the F-Equation (34) (recall also the definitions and results in Subsection 2.8). As in the Weyl case the following computation of intertwiner spaces will be essential for the construction of the corresponding embedding. The proof of the following result is similar as in Lemma 3.11. 3.20 Lemma Recalling the notion of intertwiner space in Subsection 3.1 we have: (1, 0)
0 1 0 1 D (E(2)), D(0, 1) (E(2)) = s ⊗ |s∈R , 0 0 0 0
(1, 0) (1, 0) (E(2)), D (E(2)) = C1 . D Next, consider the space hF := h+ ⊕ h− ,
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as a real space with nondegenerate symplectic form given by
1 ϕ, ψF − ψ, ϕF , σF (ϕ, ψ) := Imϕ, ψF = 2i where ·, ·F := ·, ·+ ⊕ ·, ·− and ϕ, ψ ∈ hF . The C*-algebra CCR(hF , σF ) is simple and uniquely given by [51]. The reducible representation VF := V+ ⊕ V− , where for g = (A, a) ∈ SL(2, C) R4 , ϕ ∈ h+ and ψ ∈ h− we define (V+ (g) ϕ)(p) := e−ipa D(1, 0) (A) ϕ(Λ−1 A p) and
(V− (g) ψ)(p) := e−ipa D(0, 1) (A) ψ(Λ−1 A p),
leaves the real-bilinear form ·, ·F invariant and, therefore, the symplectic form σF is also VF -invariant. In the rest of the section we will also write the finitedimensional representation D(1, 0) (A) simply as D(A) and D(0, 1) (A) as D(A), A ∈ SL(2, C). Define also the covariant representation for the present model (which satisfies the support properties required in Definition 3.3 (i)): (TF (g) f )(x) := D(A)f (Λ−1 A (x − a)) , g = (A, a) ∈ SL(2, C) R4 , f ∈ S(R4 , H(0, 1) ) =: TF . In analogy to the Weyl case we introduce the following embedding I1 : S(R4 , H) −→ h+ defined for all f ∈ S(R4 , H) by (I1 f )(p) := (I2 f )(p) :=
and I2 : S(R4 , H) −→ h−
−1 01 ◦ D(Hp ) D D Hp , f(p) , p ∈ C+ 00
01 ◦ , D(Hp )−1 Γ D Hp D p ∈ C+ , 0 f (p) , 00
01 ∈ D(1, 0) (E(2)), D(0, 1) (E(2)) and where the ‘hat’ means 00 ◦ the Fourier transformation which is restricted to C+ as in the Weyl case. Further (0, 1) (1, 0) Γ0 : H →H is again an anti-unitary involution (conjugation). Finally, the embedding that specifies the net structure and satisfies the conditions stated in Subsection 2.6 is given by where D
IF : TF −→ hF , with
IF f := I1 f ⊕ I2 f .
(62)
The covariance property of the net characterized by the preceding embedding is guaranteed by the following result:
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3.21 Lemma Using the notation introduced above the equation IF TF (g) = VF (g) IF ↑ holds for all g ∈ P+ . Proof. The proof is done similarly as in Lemma 3.14. The intertwining equation is now based on the relations, + I1 TF (g) = V+ (g) I1 (63) I2 Γ0 TF (g) = V− (g) I2 Γ0 , ↑ . for any g ∈ P+
The next result will ensure the causality property of the net associated to embedding IF . 3.22 Lemma Suppose that supp f ⊥ supp k for f ,k ∈ TF . Then σF (IF f, IF k) = 0 holds. Proof. First note that I2 (Γ0 f )(p) = Γ0 (I1 f )(−p) for all f ∈ S(R4 , H). Then, −1 ∗ −1 computing similarly as in Lemma 3.15 (putting now β + (p) := D(Hp ) D(Hp ) and β − (p) := Γ0 β+ (p) Γ0 ), we get σF (IF f, IF k) = σF (I1 f ⊕ I2 (Γ0 f ) , I1 k ⊕ I2 (Γ0 k)) 1 = f(p), β+ (p) k(p)C4 µ0 (dp) + k(−p), β+ (p) f(−p)C4 µ0 (dp) 2i C+ C+ k(p), β+ (p) f(p)C4 µ0 (dp) − f(−p), β+ (p) k(−p)C4 µ0 (dp) − C+
C+ = 0,
where the last equation follows from the fact that the matrix elements of the operator-valued function (recall Remark 2.13)
β+ (p) := D
−1 Hp
∗
0 0 0 1
Hp
−1
−(p0 − p3 )(p1 + ip2 ) −(p0 − p3 )(p1 + ip2 ) (p1 + ip2 )2 (p0 − p3 )2 −(p0 − p3 )(p1 + ip2 ) (p0 + p3 )(p0 − p3 ) (p0 + p3 )(p0 − p3 ) −(p0 + p3 )(p1 + ip2 ) 1 ∼ = 4 −(p − p )(p + ip ) (p0 + p3 )(p0 − p3 ) (p0 + p3 )(p0 − p3 ) −(p0 + p3 )(p1 + ip2 ) 0 3 1 2 (p1 − ip2 )2
−(p0 + p3 )(p1 − ip2 ) −(p0 + p3 )(p1 − ip2 )
are homogeneous polynomials of degree 2 (see Lemma 3.10).
(p0 + p3 )2
We will show the existence of a free net associated to the F-Equation, which we call F-net for short.
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3.23 Theorem Consider the net of local linear submanifolds of hF given for O ∈ B(R4 ) by 4 R4 ⊃ O −→ hF (O) := IF f | f ∈ C∞ 0 (R , H), suppf ⊂ O . Then the net of local C*-algebra defined by R4 ⊃ O −→ AF (O) := C∗ ({W (ϕ) | ϕ ∈ hF (O)}) , where the W (·)’s denote the generators of the C*-algebra CCR(hF , σF ), is a HKnet. Proof. By Lemmas 3.21 and 3.22 we have that the tuple (hF , σF , VF , TF , TF , IF ) satisfies all conditions stated in Definition 3.3 (ii) and again by Theorem 3.4 (ii) we get that the net of local C*-algebras above is a HK-net. 3.24 Remark. (i) We will show next that the net constructed above is isomorphic to the bosonic net defined in [48, Section 3, case n = 2]. Using the notation and results of the latter reference we specify the symplectic bijection λB : hF → h2 (recall Proposition 3.5 (ii)): for χ± ∈ L2 (C+ , C, µ0 (dp)) put
1
1 1 1 λB D H(·) ⊗ χ (·) ⊕ D H(·) ⊗ χ (·) (p) 0 0 + 0 0 −
0 /
0 / 0 0 0 0 ⊗ χ (p) ⊕ D(Hp ) ⊗ χ (p) , := D(Hp ) 1 1 + 1 1 − + − where [·]± denote the classes of the factor spaces H± defined in [48, Section 3]. Using again the statements in the proof of [48, Lemma 3.2] it is straightforward to prove that λF satisfies the properties required in Proposition 3.5 (ii). (An isometry to the free net constructed in [12, Part B] is given in [47, Remark 3.3.5].) (ii) The natural complexification of hF given by J(ϕ+ ⊕ ϕ− ) := i ϕ+ ⊕ i ϕ− , ϕ+ ⊕ ϕ− ∈ hF , already defines a Fock state satisfying the spectrum condition (cf. Definition 3.7 (B3) and [14, Subsection 8.2.3]) and where the one-particle Hilbert space carries the representation usually considered in the literature for describing photons with both helicities [40, Section 2]. (iii) It is now obvious that as in the Fermi case we may generalize the preceding construction to arbitrary values of the integer helicity parameter n ∈ N. Replace the index (1, 0) by (n, 0) and (0, 1) by (0, n) etc. Thus we have produced (considering Remark 3.17 (i)) isomorphic nets to the ones given in [48]. Note also that the use of the direct sum of 4 reference spaces in the Fermi case was forced by the self-dual approach to the CAR-algebra. Nevertheless in the Bose and Fermi cases the corresponding one-particle Hilbert spaces given by the canonical Fock states (cf. Remark 3.17 (ii)) are of the form h+ ⊕ h− .
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3.25 Remark. From Theorem 2.31 we can show the equivalence of the C*↑ ↑ A dynamical systems (CCR(hF , σF ), αF g , P+ ) and (CCR(hA , σA ), αg , P+ ), where A σA := Im·, ·A and αg is the Bogoljubov automorphism associated to VA (recall Subsection 2.9). But due to the specific form of the factor space hA and the corresponding covariant representation TA there does not exist a nontrivial embedding IA satisfying the corresponding intertwining property with VA (cf. axiom (B2)). The impossibility of constructing the free net associated to the vector potential is the analogue in our context of the well-known Strocchi no-go theorems, that are formulated in the quantum field theoretical context (cf. [67]). For a detailed treatment of the nets associated to the electromagnetic vector potential (including a general analysis of the localized constraints) see [32].
4 Massless quantum fields In the previous section we have seen that the embedding that characterize the massless free nets naturally reduce the degrees of freedom in the fiber (cf. Re 01 mark 2.11 and Eqs. (60), (62)) by using elements 0 0 of intertwiner space between the little group E(2) and its conjugate E(2). This choice shows explicitly that the embedding map the test functions into the space of solutions of massless relativistic wave equations. Now using the canonical Fock states associated to the CAR- and CCR-algebras (recall Remarks 3.17 and 3.24 (ii)) we will obtain in a natural way quantum fields that satisfy in the distributional sense the Weyl and Maxwell equations. Since these two cases are typical (see Remarks 2.26 (ii), 3.17 and 3.24) the following procedure establishes a neat way to define massless fields for any helicity value. This construction is considerably simpler than what is done usually in QFT, where so-called 2j + 1 quantum fields are introduced (a clear reminiscence of the massive case) and then constrained by imposing suitable equations on them [74], [40, Section 2]. If one considers the canonical Fock states mentioned before, then one can also interpret the embedding I, that where used to completely characterize the free nets in Theorems 3.16 and 3.23, as a one particle Hilbert structure. Indeed, n I can be seen as a real linear map from T := S(R4 , H(0, 2 ) ) into the (complex) one-particle Hilbert space H1 ⊂ h+ ⊕ h− (with scalar product ·, · := ·, ·+ ⊕ ·, ·− ). By Propositions 2.21, 2.24 and Remark 2.26 (ii) the one particle Hilbert space H1 carries representations of the Poincar´e group equivalent to the massless, positive energy, Wigner representations with helicities ± n2 . Thus we can use I to construct canonically massless free quantum fields. We will treat the Weyl (fermionic) and the Maxwell (bosonic) case separately. The fermionic/bosonic fields are defined on the antisymmetric/symmetric Fock space Fa (H1 )/Fs (H1 ) over the corresponding one-particle Hilbert spaces H1 . We denote the corresponding vacua simply by Ω and the scalar products by ·, ·a/s .
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Free Weyl quantum field: Consider the C*-algebraCAR(hW , ΓW ) defined in Sec1 0 0 0 0 1 0 0 tion 3.1 and the basis projection P := 0 0 0 0 specified in Remark 3.17 (ii) 0 0 0 0 (see also Remark 3.8 (i)). Recall that in this context the creation and annihilation operators are given as follows: for ψ, ψ1 , . . . , ψn ∈ H1 := P (IW (TW )) ⊂ h+ ⊕ h− we put c(ψ)Ω
:=
0 n √ n (−1)l−1 ψ, ψl ψ1 ∧ . . . ψˆl · · · ∧ ψn
c(ψ)(ψ1 ∧ · · · ∧ ψn ) :=
l=1
c(ψ)∗ Ω
=
ψ
c(ψ)∗ (ψ1 ∧ · · · ∧ ψn )
=
1 √ ψ ∧ ψ1 ∧ · · · ∧ ψn , n+1
where the wedges mean the antisymmetrised tensor product ψ1 ∧ · · · ∧ ψn := sgn (σ) ψσ(a1 ) ⊗ . . . ⊗ ψσ(an ) . σ∈Pn
The previous creation and annihilation operators are mutually adjoint w.r.t. ·, ·a and satisfy the usual anticommutation relations: for ψ, ψ ∈ H1 one has [c(ψ), c(ψ )∗ ]+ = ψ, ψ a 1 , where [·, ·]+ denotes the anticommutator. In this context we may define the free Weyl quantum field as follows (recall that ΓW IW = IW ): 4.1 Definition Let ωP be the Fock state corresponding to the basis projection P and denote by (Fa (H1 ), ΠP , Ω) the corresponding GNS-data. We define the free Weyl quantum field acting on Fa (H1 ) by ∗ 1 1 φW (f ) := √ ΠP a(IW f ) = √ c P (IW f ) + c P (IW f ) , 2 2 1
where f ∈ TW := S(R4 , H(0, 2 ) ) and a(·) denote the (abstract) generators of CAR(hW , ΓW ). 4.2 Theorem The embedding IW (cf. Eq. (60)) is continuous w.r.t. the corresponding Schwartz and Hilbert space topologies. Proof. It is enough to show the continuity of I1 . Recall that for p ∈ C+ the scalar product is characterized by the positive matrix-valued function β+ (p) =
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−p1 + ip2 . Then we have the estimates p0 + p3 = f(p), β+ (p) f(p)C2 µ0 (dp) C+ ≤
1 C , C = 0
≤
1 C , C = 0
=
C , C = 0
C+ R3 \{0}
1
≤ M
C (p)| · |f C (p)| µ (dp) |β+, C C (p)| · |f, 0
R3 \{0}
1
d 3p C (|p |, p )| · |f C (|p |, p )| |p | · |f, |p | (1 + |p |2 )4 3 C (|p |, p )| · |f C (|p |, p )| · |f, d p (1 + |p |2 )4
C C f, 4,0 · f 4,0 ,
C, C = 0
where M =
R3
1 (1+|p |2 )4
2 2 C C d 3 p and f 4,0 := sup p∈R4 {(1 + |p| ) |f (p)|} is a
particular seminorm corresponding to the Schwartz space topology. Suppose now 1 that fn → 0 in the topology of TW := S(R4 , H(0, 2 ) ). Then by the continuity of Fourier transformation and the previous estimates we conclude that I1 (fn ) → 0 in h+ and the proof is concluded. In the following theorem we will show that the quantum field defined previously satisfies the Wightman axioms as well as the Weyl equation in the distributional sense. 4.3 Theorem The Weyl quantum field φW (f ), f ∈ TW , defined on Fa (H1 ) is a bounded, self-adjoint operator. Moreover we have (i) (Weyl equation) φW (·) satisfies the Weyl equation in the distributional sense: 1 φW ∂ CC fC = 0 , f ∈ S(R4 , H( 2 , 0) ) . (ii) (Poincar´e invariance and spectral condition) φW (·) transforms covariantly under the Poincar´e group: Let TW and Q be the covariant and the second quantization of the canonical representation P VW = V1 ⊕ V2 on Fa (H1 ), respectively (recall Section 3.1). Then Q(g) φW (f ) Q(g)−1 = φW (TW f ) ,
↑ f ∈ TW , g ∈ P+ .
(64)
Further, the representation V1 ⊕ V2 satisfies the spectral condition on H1 and ↑ Q(g)Ω = Ω, g ∈ P+ .
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(iii) (Anticommutation relations) For f, k ∈ TW such that supp f and supp k are space-like separated, the anticommutator of the corresponding smeared fields vanishes: [φW (f ), φW (k)]+ = 0 . (iv) (Regularity) The map TW f → ψ1 , φW (f )ψ2 a ,
ψ1 , ψ2 ∈ Fa (H1 ) ,
is a tempered distribution. Proof. The boundedness and self-adjointness of the field follows from the same properties of the generator a(IW f ) of the CAR-algebra. To show (i) recall that, e.g., the embedding I1 used to specify IW maps into the space of solutions of Weyl equation (cf. (60)). Indeed, we will show that 1 I1 (∂ CC fC ) = 0, f ∈ S(R4 , H( 2 , 0) ): for B ∈ {0, 1} and summing over repeated indices we have
B −1 01 CC CC f (p) ∂ (∂ f )(p) = H H IB C p p C 1 00 C B
−1 01 P CC f Hp = −i Hp C (p) 00 C = 0, where the last equation follows from the fact that
∗ p0 + p3 p1 + ip2 20 CC (P )= Hp . = Hp p1 − ip2 p0 − p3 00
Similarly we obtain I2 (∂ CC fC ) = 0, hence P (IW (∂ CC fC )) = 0 and the field satisfies the Weyl equation as required: ∗ φW ∂ CC fC = c P (IW (∂ CC fC )) + c P (IW (∂ CC fC )) = 0 . The property (ii) follows from Remark 3.8 (i). The anticommutation of the field in (iii) is a consequence of the anticommutation of the generators a(IW f ) and a(IW k) of the CAR-algebra (cf. Lemma 3.15 and Theorem 3.16). The regularity (iv) of the field follows from the continuity of the embedding IW (see Theorem 4.2). Indeed, let fn → 0 in the Schwartz topology of TW . Then by Theorem 4.2 we have IW fn → 0 in the Hilbert space topology. Now for any ψ ∈ Fa (H1 ) φW (fn )ψ ≤ a(IW fn )C∗ ψ ≤ IW fn ψ and therefore s− limn→∞ φW (fn ) = 0. The strong continuity of the field implies finally the temperedness of the distribution. We show in the next theorem some additional properties satisfied by the Weyl quantum field.
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4.4 Theorem The field φW transforms in addition covariantly w.r.t. the (fourfold cover of ) conformal group in Minkowski space SU(2, 2). Proof. The extension of the covariance property to the conformal group follows from the results in [48, Section 5] (recall also the isomorphy between the massless free nets of Section 3.1 and those constructed in [48] given Remark 3.17 (i)). 4.5 Remark. As mentioned above, the Weyl case is typical for fermionic models with nontrivial (half-integer) helicity. Hence replacing for example (0, 12 ) by (0, n2 ) with n ≥ 3 and odd, one can similarly define the massless fermionic free quantum field with helicity n2 by
1 1 ∗ φn (f ) := √ (ΠP (a(In f ))) = √ c (P (IW f )) + c (P (IW f )) , 2 2 n f ∈ Tn := S(R4 , H(0, 0 ) ) . These fields also satisfy Wightman axioms and the adapted version of Theorem 4.4. In particular, it satisfies the corresponding massless relativistic wave equation in the weak sense: C . . . Cn−1 =0 φn ∂ CC fC 1 (cf. Corollary 2.25). Free Maxwell quantum field: Consider the simple C*-algebra CCR(hF , σF ) given in Section 3.2 as well as the canonical Fock state ωJ on CCR(hF , σF ) specified by the internal complexification J(ϕ+ ⊕ ϕ− ) := i ϕ+ ⊕ i ϕ− , ϕ+ ⊕ ϕ− ∈ hF (recall Remarks 3.8 and 3.24 (ii)). Putting H1 := IF (TF ) ⊂ h+ ⊕ h− the generating functional is given by 2 1 H1 ψ → e− 4 ψ . 4.6 Definition Let ωJ be the Fock state associated to the internal complexification J given above and denote by (Fs (H1 ), ΠJ , Ω) the corresponding GNS-data. ΦJ (IF f ), f ∈ TF := S(R4 , H(0, 1) ), is the infinitesimal generator of the strongly continuous unitary group
R t → ΠJ W (t IF f ) = e−it ΦJ (I F f ) , where W (·) are the (abstract) generators of CCR(hF , σF ). Then we define the free Maxwell quantum field acting on Fs (H1 ) by φF (f ) := ΦJ (IF f ) ,
f ∈ TF .
Notice that the free Maxwell quantum field is, as a consequence of the uniqueness of the GNS representation and Nelson’s analytic vector theorem, the closure of the essentially self-adjoint operator 1 ΦJ (IF f ) = √ (a((IF f ))∗ + a((IF f ))) , 2
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on the set Ffin ⊂ Fs (H1 ) of finite particle vectors (cf. [58, Theorem X.41]). The creation and annihilation operators on the symmetric Fock space over the oneparticle Hilbert space H1 are defined as usual: For ϕ, ϕ1 , . . . , ϕn ∈ H1 we put a(ϕ)Ω
:=
a(ϕ)Sn (ϕ1 ⊗ · · · ⊗ ϕn ) :=
0 n √ n ϕ, ϕl Sn−1 (ϕ1 ⊗ · · · ⊗ ϕˆl · · · ⊗ ϕn ) l=1
a(ϕ)∗ Ω
=
ϕ
a(ϕ)∗ Sn (ϕ1 ⊗ · · · ⊗ ϕn )
=
1 √ Sn+1 (ϕ ⊗ ϕ1 ⊗ · · · ⊗ ϕn ) , n+1
where the hat means omission and Sn is the symmetrization operator Sn (ϕ1 ⊗· · ·⊗ ϕn ) := σ∈Pn ϕσ(1) ⊗ · · · ⊗ ϕσ(n) on the n-tensor product space over h+ ⊕ h− . The previous creation and annihilation operators are mutually adjoint w.r.t. ·, ·s and satisfy the usual commutation relations: for ϕ, ϕ ∈ H1 one has [a(ϕ), a(ϕ )∗ ] = ϕ, ϕ s 1 , where [·, ·] denotes the commutator. Similarly as in Theorem 4.2 we can show the following continuity statement for the embedding: 4.7 Theorem The embedding IF (recall Eq. (62)) is continuous w.r.t. the corresponding Schwartz and Hilbert space topologies. We will show next that the Maxwell quantum field also satisfies the Wightman axioms as well as the Maxwell equation in a distributional sense. 4.8 Theorem The Maxwell quantum field φF (f ), f ∈ TF , defined on Fs (H1 ) is an unbounded, self-adjoint operator that leaves the dense subspace Ffin invariant. Moreover we have (i) (Maxwell equation) φF (·) satisfies the following equation in the distributional sense: 1 1 φF ∂ CC fCB = 0 , f ∈ S(R4 , H( 2 , 2 ) ) . (ii) (Poincar´e invariance and spectral condition) φF (·) transforms covariantly under the Poincar´e group: Let TF and Q be the covariant and the second quantization of the canonical representation VF := V1 ⊕ V2 on Fs (H1 ), respectively (recall Section 3.2). Then Q(g) φF (f ) Q(g)−1 = φF (TF f ) ,
↑ f ∈ TF , g ∈ P+ .
(65)
Further, the representation V1 ⊕ V2 satisfies the spectral condition on H1 and ↑ Q(g)Ω = Ω, g ∈ P+ .
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(iii) (Causality) For f, k ∈ TF such that supp f and supp k are space-like separated, the commutator of the corresponding smeared fields vanishes: [φF (f ), φF (k)] = 0
(on Ffin ) .
(iv) (Regularity) The map TF f → ψ1 , φF (f )ψ2 s ,
ψ1 , ψ2 ∈ Ffin ,
is a tempered distribution. Proof. The self-adjointness of the field follows from its definition as generator of a strongly continuous unitary group and the invariance of Ffin is a consequence of the remarks after Definition 4.6. To show (i) recall that, e.g., the embedding I1/2 used to specify IF map into the space of solutions of Maxwell equation (cf. (60)). Reasoning as in the proof of Theorem 4.3 on obtains IF (∂ CC fCB ) = 0 and again this implies ∗ 1 φ(∂ CC fCB ) = √ a (IF (∂ CC fCB )) + a (IF (∂ CC fCB )) = 0 , 2 1
1
where f ∈ S(R4 , H( 2 , 2 ) ). To prove property (ii) note that the Fock state ωJ is invariant w.r.t. the ↑ Bogoljubov automorphism αg generated by VF (g), i.e., ωJ ◦ αg = ωJ , g ∈ P+ , hence by Remark 3.8 (ii) we have Q(g)Ω = Ω. Further, Q(g) also leaves Ffin invariant and for ψ ∈ Ffin we have Q(g) ΦJ (IF f ) Q(g)−1 ψ = ΦJ (IF TF f ) ψ ,
↑ g ∈ P+ .
Since both sides of the previous equation are essentially self-adjoint operators we finally obtain the covariance relation: Q(g) φF (f ) Q(g)−1 = φF (TF f ) ,
f ∈ TF .
The commutation of the field in (iii) is again a consequence of the commutation of the generators W (IF f ) and W (IF k) of the CCR-algebra (cf. Lemma 3.22 and Theorem 3.23). The regularity (iv) of the field follows from the continuity of the embedding IF (see Theorem 4.2). Indeed, let fn → 0 in the Schwartz topology of TF . Then by Theorem 4.2 we have IF fn → 0 in the Hilbert space topology. Now for any k-th particle vector ψ ∈ Ffin we have √ √ φF (fn )ψ ≤ 2 k + 1 IF fn ψ and therefore φF (fn ) → 0 strongly on Ffin . The strong continuity of the field implies finally the temperedness of the distribution. We show in the next theorem some additional properties satisfied by the Maxwell quantum field.
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4.9 Theorem The field φF transforms in addition covariantly w.r.t. the (fourfold cover of ) conformal group in Minkowski space SU(2, 2). Proof. The extension of the covariance property to the conformal group follows from the results in [48, Section 5] (recall also the isomorphy between the massless free nets of Section 3.1 and those constructed in [48] given Remark 3.17 (i)). 4.10 Remark. The present construction can also be generalized to produce massless bosonic free fields with nontrivial (integer) helicity. Hence replacing for example (0, 1) by (0, n) with n ≥ 2 and even, one can similarly define the massless fermionic free quantum field with helicity n2 by 1 n φn (f ) := √ (a((In f ))∗ + a((In f ))) , f ∈ Tn := S(R4 , H(0, 2 ) ) . 2 These fields also satisfy Wightman axioms and the adapted version of Theorem 4.9. In particular, it satisfies the corresponding massless relativistic wave equation in the weak sense: C . . . Cn−1 =0 φn ∂ CC fC 1 (cf. Corollary 2.25).
5 Conclusions In a recent paper Brunetti, Guido and Longo proposed a construction procedure for a bosonic net of von Neumann algebras canonically associated to a positive energy strongly continuous (anti-) unitary Hilbert space representation of the proper Poincar´e group P+ (cf. [20]). They also used the suggestive name of free net as in [14, Example 8.3.1] (see also [12, 48]), since the construction avoids the use of quantum fields as ‘coordinates’ of the corresponding net. The construction of bosonic free nets in Section 3 and the one in [20] are similar in that both use Wigner’s cornerstone analysis of the unitary irreducible representations of the universal cover of the Poincar´e group, as well as the CCR-algebra. Nevertheless, in Section 3 we prefer to work initially with abstract C*-algebras, while in [20] concrete von Neumann algebras in a Fock representation are used. The crucial difference relies in the choice of the localization prescription. We use H-valued Schwartz functions on Minkowski space on which the (algebraically reducible) covariant representation T of the Poincar´e group acts and, in fact, we can also canonically construct the corresponding free massless quantum fields that satisfy Wightman axioms. Brunetti, Guido and Longo use the relatively recent notion of modular localization (see also [27]) which does not need test functions on configuration space. There is also no obvious candidate for covariant representation in this frame. Recall that the covariance of free nets is expressed at the level of local reference spaces h(O) of the CAR resp. CCR-algebras by means of the equation V (g) h(O) = h(gO) ,
↑ , g ∈ P+
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where V is the Wigner representation (see Eq. (54)). The proof of the previous equation is based on the intertwining equation V (g) I = I T (g), where I is the embedding characterizing the free net (for details see the proof of Theorem 3.6 in [48]). The modular localization approach uses, instead, the Bisognano-Wichmann relations as an essential input to introduce modular-like objects at the level of the one-particle Hilbert space H and associated to any wedge W in a suitable family of wedges W. This family is compatible with the action of a one-parameter group of boosts and a time-reversing reflection assigned to each W ∈ W. The ‘Tomita operator’ on H naturally selects a family KW , W ∈ W, of R-linear, closed, standard subspaces of H that transform covariantly under the chosen Wigner representation. By means of suitable intersections of KW ’s one defines a net of subspaces localized in, e.g., causally complete convex regions which also transforms covariantly. A remarkable aspect of the modular localization approach is that one can also naturally associate a free net O → Mcont (O) to the ‘continuous spin’ massless representations. These types of representations are typically excluded by hand from further considerations. It is conjectured in [20, p. 761] that this net should not satisfy the Reeh-Schlieder property for double cones. If so, this would be conceptually a much more satisfactory explanation of the singular character that these representations play in nature. A natural question that arises in this context is the relation of the net O → Mcont (O) with the one associated to discrete helicity representations. In particular, if it is possible to describe at the C*-algebraic level in Mcont the choice of nonfaithful representation of E(2) needed to define discrete helicity. Techniques of local quantum constraints (see [30, 32]) may possibly be applied to O → Mcont (O) in order to consider this question. (Here, the use of abstract C*-algebras in a first step can be relevant.) Recall also that the use of nonfaithful representations of E(2) (hence discrete helicity) is crucial for the extension to the conformal group of the covariance of the corresponding net (see [1, 2, 3, 48] for further points on this subject).
Acknowledgments The present paper is a revised and considerably extended version of some parts of the authors PHD at the University of Potsdam. It is a pleasure to thank Hellmut Baumg¨ artel for supervision and many useful remarks. I would also like to thank Wolfgang Junker for helpful conversations.
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[76] A.S. Wightman, Invariant wave equations, In Proceedings of the Ettore Majorana International School of Mathematical Physics (1977), G. Velo and A.S. Wightman (eds.), Srpinger Verlag, Berlin, 1978. [77] E.P. Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. Math. 40, 149–204 (1939). [78] E.P. Wigner, Relativistische Wellengleichungen, Z. Phys. 124, 665–684 (1947). Fernando Lled´ o Institute for Pure and Applied Mathematics RWTH-Aachen Templergraben 55 D-52056 Aachen Germany email:
[email protected] Communicated by Klaus Fredenhagen Submitted 16/03/03, accepted 11/12/03
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 671 – 741 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/040671-71 DOI 10.1007/s00023-004-0180-x
Annales Henri Poincar´ e
Uniform Lieb-Thirring Inequality for the Three-Dimensional Pauli Operator with a Strong Non-Homogeneous Magnetic Field L´aszl´o Erd˝ os∗ and Jan Philip Solovej†
Abstract. The Pauli operator describes the energy of a nonrelativistic quantum particle with spin 12 in a magnetic field and an external potential. A new LiebThirring type inequality on the sum of the negative eigenvalues is presented. The main feature compared to earlier results is that in the large field regime the present estimate grows with the optimal (first) power of the strength of the magnetic field. As a byproduct of the method, we also obtain an optimal upper bound on the pointwise density of zero energy eigenfunctions of the Dirac operator. The main technical tools are: (i) a new localization scheme for the square of the resolvent of a general class of second order elliptic operators; (ii) a geometric construction of a Dirac operator with a constant magnetic field that approximates the original Dirac operator in a tubular neighborhood of a fixed field line. The errors may depend on the regularity of the magnetic field but they are uniform in the field strength.
1 Introduction 1.1
Notations
Let B ∈ C 4 (R3 ; R3 ) be a magnetic field, div B = 0, and V ∈ L1loc (R3 ) a realvalued potential function. Let A : R3 → R3 be a vector potential generating the magnetic field, i.e., B = ∇ × A. The three-dimensional Pauli operator is the following operator acting on the space of L2 (R3 ; C2 ) of spinor-valued functions: H = H(h, A, V ) := [σ·(−ih∇+A)]2 +V = (−ih∇+A)2 +V (x)+hσ·B(x) , (1.1) where σ = (σ 1 , σ 2 , σ 3 ) is the vector of the Pauli spin matrices, i.e., 0 1 0 −i 1 0 1 2 3 σ = , σ = , σ = . 1 0 i 0 0 −1 ∗ Partially supported by NSF grants DMS-9970323, DMS-0200235, and by the Erwin Schr¨ odinger Institute, Vienna. † Work partially supported by the Danish Natural Science Research Council, by MaPhySto – A network in Mathematical Physics and Stochastics, funded by a grant from The Danish National Research Foundation, and by the EU research network HPRN-CT-2002-00277.
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The spectral properties of H depend only on B and V and do not depend on the specific choice of A. We shall be concerned only with gauge invariant quantities therefore we can always make the Poincar´e gauge choice. In particular, we can always assume that A is at least as regular as B. The operator H = H(h, A, V ) is defined as the Friedrichs’ extension of the corresponding quadratic form from C0∞ (R3 ; C2 ). The Pauli operator describes the motion of a non-relativistic electron, where the electron spin is important because of its interaction with the magnetic field. For simplicity we have not included any physical parameters (i.e., the electron mass, the electron charge, the speed of light, or Planck’s constant ) in the expressions for the operators. In place of Planck’s constant we have the semiclassical parameter h and in most of the paper we also set h = 1. The last identity in (1.1) can easily be checked. If we define the threedimensional Dirac operator D := σ · (−ih∇ + A(x)) ,
(1.2)
then we recognize the last identity in (1.1) as the Lichnerowicz’ formula. The eigenvalues of H below the essential spectrum are of special interest. They determine the possible bound states of a non-relativistic electron subject to the magnetic field B and the external potential V . Under very general conditions on V and B one can show that the bottom of the essential spectrum for the Pauli operator is at zero (see [HNW]). This is in sharp contrast to the case of the spinless magnetic Schr¨ odinger operator, (−ih∇ + A)2 + V (x), whose essential spectrum is not known in general even for decaying potentials. Therefore we shall restrict our attention to the negative eigenvalues, e1 (H) ≤ e2 (H) ≤ · · · ≤ 0 of H. It is known that under very general conditions there are infinitely many negative eigenvalues even for constant magnetic field [Sol], [Sob-86], however their sum is typically finite. We recall that the sum of the eigenvalues below the essential spectrum is equal to the ground state energy of the noninteracting fermionic gas subject to H. The sum of the negative eigenvalues, j ej (H), has been extensively studied recently. In order to find the asymptotic behavior of the ground state energy of a large atom with interacting electrons, one needs, among other things, a semiclas sical asymptotics for j ej (H) as h → 0. The semiclassical formula for the sum of the negative eigenvalues is given as −3 P (h|B(x)|, [V (x)]− )dx (1.3) Escl (h, B, V ) := −h R3
with B P (B, W ) := 2 3π
W
3/2
+2
∞
[2νB −
ν=1
3/2 W ]−
=
∞ 2 3/2 dν B[2νB − W ]− 3π ν=0 (1.4)
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being the pressure of the three-dimensional Landau gas (B, W ≥ 0). Here [x]− = max{0, −x} refers to the negative part of x, d0 := (2π)−1 and dν := π −1 if ν ≥ 1. Observe that if B∞ = o(h−1 ) then Escl reduces to leading order to 5/2 the standard Weyl term, −2(15π 2 )−1 h−3 R3 [V ]− , as h → 0. The main feature of the semiclassical formula is that it behaves linearly with the field strength in the strong field regime. For the proof that j ej (H) is asymptotically equal to Escl as h → 0, first one must establish a non-asymptotic bound on the sum of the negative eigenvalues to control various error terms from the non-semiclassical regions. Such estimates for general Schr¨ odinger type operators are often referred to as Lieb-Thirring (LT) type estimates [LT1]. The bound must behave like the semiclassical formula in all relevant physical parameters; in this case, in particular, it should grow linearly in the field strength. A weaker a priori estimate typically leads to a semiclassical asymptotics that is not uniform in the field strength [Sob-98], [ES-II].
1.2
Summary of previous results
A non-asymptotic LT bound for the Pauli operator has first been established in [LSY-II] for the case of the constant magnetic field, B = const.,
5/2 3/2 (1.5) |ej (H)| ≤ (const.) [V ]− + |B|[V ]− j
with h = 1 and this bound was used to prove that Escl gives the correct asymptotics for the sum of the negative eigenvalues. The first generalizations of such estimates for non-homogeneous magnetic fields The first general bound was of the form (const.) were given in [E-1995].
5/2 3/2 [V ]− + B∞ [V ]− , then the main focus was to study unbounded fields. It was observed, that (1.5) cannot hold in general. There are two problems in connection with (1.5) for non-homogeneous field. Firstly, even when B has constant direction in R3 (1.5) is correct only if |B(x)| is replaced by an effective field strength, Beff (x), obtained by averaging |B| locally on the magnetic lengthscale, |B|−1/2 . Secondly, the existence of the celebrated Loss-Yau zero modes [LY] contradicts (1.5). Indeed, for certain magnetic fields with nonconstant direction the Dirac operator D has a nontrivial L2 -kernel. In this case a small potential perturbation 2 of D shows that j |ej (H)| behaves as n(x)[V (x)]− dx, i.e., it is linear in V . 2 Here n(x) is the density of zero modes, n(x) = j |uj (x)| , where {uj } is an orthonormal basis in Ker D. Thus an extra term linear in V must be added to (1.5). It turns out that in order to estimate n(x) by the magnetic field it is again important to replace |B(x)| by an effective field. The problem of the effective field was first successfully addressed by Sobolev, [Sob-96], [Sob-97] and later by Bugliaro et al. [BFFGS] and Shen [Sh]. In particular, the L2 -norm of the effective field, Beff 2 , is comparable to B2 in [BFFGS], and
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the same holds for any Lp -norm in Shen’s work. In a very general bound proved 3/2 in [LLS] the second term in (1.5) is replaced with B2 V 4 . In the works [E-1995], [Sob-97], [Sh], [LLS], [BFFGS] on three-dimensional magnetic Lieb-Thirring inequalities, the density n(x) is estimated by a function that behaves quantitatively as |B(x)|3/2 . In particular, in the strong field regime these estimates are not sufficient to prove semiclassical asymptotics uniformly in the field strength, they typically give results up to B∞ ≤ (const.)h−1 [Sob-98]. We remark that the bounds in [LLS] and [BFFGS] have nevertheless been very useful in the proof of magnetic stability of matter. In this case the magnetic energy, |B|2 , is also part of the total energy to be minimized, therefore even the second moment of the magnetic field is controlled. We also remark that if the field has a constant direction, then no Loss-Yau zero modes exist, n(x) ≡ 0. In this case Lieb-Thirring type bounds that grow linearly with |B| have been proved in [E-1995] and [Sob-96], [Sob-97]. This problem is technically very similar to the two-dimensional case. Since n(x) scales like (length)−3 and |B(x)| scales like (length)−2 , a simple dimension counting shows that n(x) cannot be estimated in general by the first power of |B(x)| or by any smoothed version Beff (x). However, if an extra lengthscale is introduced, for example certain derivatives of the field are allowed in the estimate, then it is possible to give a bound on the eigenvalue sum that grows slower than |B|3/2 in the large field regime. There are only two results so far in this direction. The work [BFG] uses a lengthscale on which B changes. The estimate eventually scales like b17/12 , if the magnetic field is rescaled as B(x) → bB(x), b 1. 1 is required. However, n(x) As far as local regularity is concerned, only B ∈ Hloc is estimated by a quantity that depends globally on B(x) not just in a neighborhood of x. On physical grounds one expects the following locality property: the zero modes of D are supported near the support of the magnetic field. We prove a stronger locality property, namely that the size of |B| away from a compactly supported negative potential will be irrelevant for the estimate on the sum of the negative eigenvalues. The result of [BFG] does not give such bound for an important technical reason. In order to produce an effective field strength Beff , the |B| is averaged out by a convolution function ϕ that must satisfy |∇ϕ| ≤ (const.)ϕ, i.e., ϕ must have a long tail. For B ∈ L2 the effective magnetic field has a comparable L2 -norm, but it is not true for the localized L2 -norms. Our earlier work [ES-I] had a different approach to reduce the power 3/2 of |B| in the estimate of n(x). We introduced two global lengthscales, L and respectively, to measure the variation scale of the field strength |B| and the unit vector n := B/|B| that determines the geometry of the field lines. This required somewhat more regularity on B than [BFG] and it also involved the unnatural W 1,1 -norm of V . The estimate behaved like b5/4 in the large field regime, if we rescaled B → bB, b 1. For fields with a nearly constant direction, 1, the bound was actually better, it behaved like b + b5/4 −1/2 . This indicates that it is only the variation of n and not that of B that is responsible for the higher b-power.
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Due to the improvement in the b-power from 3/2 to 5/4 in the Lieb-Thirring estimate we could also prove the semiclassical eigenvalue asymptotics in the regime b h−3 for potentials in W 1,1 [ES-II]. This bound turned out to be sufficient to show that the Magnetic Thomas-Fermi theory exactly reproduces the ground state energy of a large atom with nuclear charge Z in the semiclassical regime, i.e., where b Z 3 , Z → ∞ [ES-II]. The condition b Z 3 is optimal as far as the semiclassical theory is applicable as the results of [LSY-I] show for super-strong (b ≥ Z 3 ) constant magnetic fields. Despite the successful application of the bound in [BFG] to the stability of matter with quantized electromagnetic field with an ultraviolet cutoff [BFrG], and despite that the Lieb-Thirring inequality given in [ES-I] fully covered the semiclassical regime of the large atoms, it is still important to establish a uniform Lieb-Thirring type bound with the correct power in the magnetic field. Such bound will likely be the key to generalize the analysis of the super-strong field regime of [LSY-I] to non-homogeneous magnetic fields. In this paper we present a LiebThirring bound that • grows linearly in the field strength; • depends on the potential V in a natural way; • has the locality property in the sense discussed above. We also state the corresponding semiclassical result in Theorem 3.3 but its details, that are similar to [ES-II], will be published separately. A simpler proof of a Lieb-Thirring estimate with both the linear dependence in the field strength and the correct behavior in V is given in [ES-IV]. This approach, however, does not give the locality property.
1.3
Density of zero modes
As a byproduct, we also obtain a bound on the density of the zero modes, n(x), that behaves optimally in the field strength in case of regular fields. Actually, we control the density of all low lying states by giving an estimate for the diagonal element of the spectral projection kernel Π(D2 ≤ c)(x, x) that grows linearly with the strength of the magnetic field for any fixed constant c. We remark that the zero modes of the Dirac operator for particular classes of magnetic fields are well understood. The surprising first examples were due to Loss-Yau [LY] and later the present authors gave a more systematic geometric construction [ES-III]. This construction, in particular, gives examples that show that the density can grow at least linearly in the magnetic field strength. Other generalizations of the original construction of Loss-Yau are also available [AMN], [El-1]. However, there is no complete understanding of all magnetic fields with zero modes yet. It is also known that magnetic fields with zero modes form a slim set in the space of all magnetic fields ([BE], [El-2]) but no quantitative result is available in the general case. Our result (Corollary 3.2) is the first general estimate on the
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density of zero modes that scales optimally (i.e., linearly) in the field strength. This result is formulated as a corollary, since it easily follows from the main theorem, but we shall prove it first on the way to the proof of our main theorem (Theorem 3.1). It is amusing to note that it takes a considerable effort to show that zero modes exist at all, but it is even more difficult to give an optimal upper bound on their densities for strong regular fields. This is actually the main technical achievement of the present paper.
1.4
Organization of the proof
In Section 2 we introduce what we call the combined lengthscale Lc (x) of a given magnetic field B(x). This is a local variation lengthscale on which the magnetic field does not change substantially. More precisely, this is the case only in the regime where the magnetic field is strong; where the field is weak, the combined lengthscale is simply chosen to be of the order of the magnetic lengthscale, |B|−1/2 . In Section 3 we formulate our main result on the new Lieb-Thirring inequality (Theorem 3.1) and its corollary on the density of zero modes. We also state a semiclassical result (Theorem 3.3) whose proof will be published separately. The proof starts in Section 4 with a separation of the contributions from the low and the high energy regimes. The cutoff threshold is space dependent, it is at a level P (x) ∼ Lc (x)−2 . Technically it is done by inserting P (x) into the resolvent in the Birman-Schwinger kernel and using a resolvent expansion. We will call the two regimes the zero mode regime and the positive energy regime, respectively, because the separation is dictated by the need for a special treatment of the zero modes. The basic estimates on the contribution from these regimes are given in Theorem 4.3. We remark that to ensure ultraviolet convergence in the zero mode regime, squares of resolvents need to be estimated as well ([BFFGS]). In both regimes we perform a two-scale localization, like in [ES-I]. For both localizations, however, the approaches used here are substantially improved, as we explain below. The first localization is isotropic and its lengthscale is determined by Lc (x). This is constructed in Section 5. The main difference between the current isotropic localization and the corresponding one in [ES-I] is that in our earlier paper we assumed a universal positive bound on the combined lengthscale, therefore we could use a regular grid of congruent cubes. In order to ensure the locality property, in this paper we need to use a covering argument to select localization domains of different sizes and with a finite overlap. In domains where the magnetic field is relatively weak (|B| ≤ (const.)L−2 c ), we shall neglect all magnetic effects. In Sections 6 and 7 we show how to localize the eigenvalue estimates onto the isotropic domains. In the positive energy regime we apply a version of the IMS localization formula for the resolvent (Proposition 6.1) that was already used in [BFFGS]. However, the same formula does not hold for the square of the resolvent which is needed in the zero mode regime. A new localization scheme is developed in
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Proposition 7.1 to localize the square of the resolvent of a second order elliptic operator. The localized versions of the necessary estimates in the positive energy regime and in the zero mode regime are stated in Propositions 6.2 and 7.2, respectively. Typically it is not hard to localize resolvents of second order elliptic operators onto cubes of size at the expense of an error −2 . However localizing the square (or higher powers) of the resolvent requires off-diagonal estimates on the resolvent kernel (see Proposition 7.1). While these are typically easily available for scalar elliptic operators without spin, we do not know any a priori off-diagonal control on the resolvent of D2 . If the original Pauli operator is estimated by a constant field Pauli operator, then a posteriori we can extract off-diagonal estimates, but without comparison with the constant field problem, we do not have off-diagonal control. This is the main reason why we are unable to extend the elegant and short method of [ES-IV] to give any locality properties. Starting from Section 8 a second localization is performed onto curvilinear cylindrical domains with a transversal lengthscale |B(x)|−1/2 along the field lines. The geometry of the cylindrical domains and the coordinate system are explained in Section 8.1, and a new partition of unity subordinated to the cylindrical domains is constructed in Section 8.2. Within each cylindrical domain the magnetic field is approximated by a field βc , given as a 2-form, that is constant in the appropriate cylindrical coordinates and after a conformal change of the metric (Definition 9.1). The Dirac operator Dc with a magnetic field βc (Definition 9.3) will be used to approximate the original Dirac operator D in the corresponding cylindrical domain. Section 9 is devoted to the construction of Dc and it uses the geometric structure behind the Dirac operator on a non-flat manifold outlined, for example, in [ES-III]. The second main difference between [ES-I] and the current work lies in the cylindrical localization. In [ES-I] we considered straight cylinders to approximate tubular neighborhoods of magnetic field lines and we approximated the field by a constant one within each cylinder. The curving of the magnetic field was not respected by the approximation hence the error was not uniform in the field strength. This is the main reason why the Lieb-Thirring inequality in [ES-I] does not have the optimal |B|-power. In the new construction the cylindrical localization domains are curved in such a way as to follow a field line and we also construct appropriate spinor coordinates. This geometric approach enables us to control D2 with errors that are uniform in the field strength although they depend on the combined lengthscale of B. This eliminates the |B|-dependent error in the large field regime. The nearzero energy states, in particular the zero modes of D, need to be controlled with such a precision in order not to overestimate their contribution to the negative eigenvalues of D2 − V . The proof of the |B|-independent control is quite involved and it relies heavily on the intrinsic geometric properties of the Dirac operator. Section 10 completes the proof of the positive energy regime. In this regime errors that are independent of |B| can be absorbed into the local energy shift
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P (x). Proposition 10.1 contains the necessary spectral estimate localized onto the cylindrical domains in the original coordinates. We first translate the estimate into cylindrical coordinates. In these coordinates the approximating field is constant and we can use the magnetic localization formula (10.18) from [ES-II]. This method yields a |B|-independent cylindrical localization error. Finally, having constructed the approximating local Dirac operators with constant fields, we can use the Lieb-Thirring inequality for the Pauli operator with a constant field obtained in [LSY-II]. In Section 11 we complete the estimate of the zero mode regime. We need to estimate the density of the near-zero energy states of D2 . This is given by the diagonal kernel of the spectral projection operator, Π(D2 ≤ P0 )(x, x) where P0 is the typical value of the regularly varying function P around x. This operator can be bounded by the resolvent, but the diagonal element of the resolvent is infinite because of the ultraviolet divergence. Therefore we need to control Π(D2 ≤ P0 ) by the square of the resolvent, (D2 + P0 )−2 . For regions with weak magnetic fields the magnetic field can be neglected and we can simply use the diamagnetic inequality. The problem thus can be reduced to estimating the resolvent square of the free Laplacian (Section 11.1). For regions with a strong field (Section 11.2) we again use the approximating constant field operators. However, the magnetic localization formula (10.18) is not valid for the square of the Pauli operator, so localizing onto cylindrical domains is more complicated. Fortunately, at this stage we do not need operator inequalities, we need to estimate only the diagonal element of the square of the resolvent at each fixed point u. First we transform the problem into the new coordinates associated with the field line through u (estimates (11.6) and (11.7)). Then we use resolvent expansions extensively to approximate D by Dc . Since we estimate the square of the resolvent, we need to control the offdiagonal elements of the resolvent itself. For a constant field, the offdiagonal decay is Gaussian on a magnetic lengthscale |B|−1/2 . A similar feature is proved for the resolvent with a nonconstant field via the constant field approximation. In the Appendix we collected the proofs of several Propositions and Lemmas which can be skipped at a first reading. It is amusing to note that the most complicated part of the proof (Sections 7 and 11) controls the possible ultraviolet regime of near zero energy states. On physical grounds this regime should be irrelevant if we knew that low energy eigenstates of D2 have transversal momentum of order |B|1/2 and parallel momentum independent of the field strength. The main difficulty is to obtain such information on the low lying states. Convention: Throughout the proof universal constants are denoted by a general c whose value can be different even within the same equation. Constants 3 depending on numbers a, b, . . . are denoted by c(a, b, . . .). Integration over R with respect to the Lebesgue measure, R3 dx, is simply denoted by . We shall say that two positive numbers a, b are comparable if 12 ≤ a/b ≤ 2.
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2 Lengthscales of the magnetic field Let B ∈ C 4 (R3 , R3 ) be a magnetic field and let n := B/|B| be the unit vector field in the direction of the magnetic field at all points where B does not vanish. For any L ≥ 0 and x ∈ R3 we define BL (x) := sup{|B(y)| : |x − y| ≤ L}
(2.1)
bL (x) := inf{|B(y)| : |x − y| ≤ L}
(2.2)
and
to be the supremum and the infimum of the magnetic field strength on the ball of radius L about x. These functions are continuous in both the L and x variables. The Pauli operator will be localized on different lengthscales determined by the magnetic field. We now define these scales. Definition 2.1 (Lengthscales of a magnetic field). Given a C 4 -magnetic field B. We define the magnetic lengthscale of B as Lm (x) := sup{L > 0 : BL (x) ≤ L−2 } .
(2.3)
The variation lengthscale of B is given by
Lv (x) := sup L ≥ 0 : Lγ sup ∇γ B(y) : |x − y| ≤ L ≤ bL (x), γ = 1, 2, 3, 4 . (2.4) Finally we set Lc (x) := max{Lm (x), Lv (x)}
(2.5)
to be the combined lengthscale of B at x. A magnetic field B : R3 → R3 determines two local lengthscales. The magnetic lengthscale, Lm , is comparable with |B|−1/2 . The lengthscale Lv determines the scale on which the field B varies. One may think of Lv as the smaller of the two lenghtscales describing the variation of the field strength, i.e., the variation scale of log |B|, and the variation scale of the field lines n. For weak magnetic fields the magnetic effects can be neglected in our final eigenvalue estimate, so the variational lengthscale becomes irrelevant. This idea is reflected in the definition of Lc ; we will not need to localize on scales shorter than the magnetic scale Lm . Note that for any B ∈ C 4 (R3 , R3 ) we have that 0 < Lc (x) ≤ ∞ for all x ∈ R3 . If Lc (x) = ∞ for some x ∈ R3 , then B is constant on R3 . Moreover the value Lc (x) at any x does not depend on B outside the ball centered at x with radius Lc (x). If follows in particular, that if B vanishes in a ball of radius δ around x, then δ ≤ Lc (x).
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3 Main Theorem We are ready to state our main results. Theorem 3.1 (Uniform Lieb-Thirring inequality). We assume that the magnetic field is B ∈ C 4 (R3 , R3 ). Let A ∈ C 4 (R3 , R3 ) be a vector potential, ∇ × A = B, and let D := σ · (−i∇ + A) be the free Dirac operator with magnetic field B on the trivial spinorbundle over R3 , that can be identified with L2 (R3 , C2 ). Let V be a scalar potential. Then the sum of the negative eigenvalues, ej , of the Pauli operator H := D2 + V = [σ · (−i∇ + A)]2 + V satisfies 5/2 3/2 −1 |Tr H− | = |ej | ≤ c [V ]− + c |B|[V ]− + c (|B| + L−2 c )Lc [V ]− (3.1) j
with universal constants. Notation: For any self-adjoint operator H we let H− := 12 [|H| − H] denote its negative part. Corollary 3.2 (Density of zero modes). Given a magnetic field B ∈ C 4 (R3 , R3 ) with a combined lengthscale Lc , the density of zero modes of the free Dirac operator D with magnetic field B satisfies −1 n(x) := |uj (x)|2 ≤ c(|B(x)| + L−2 (3.2) c (x))Lc (x) j
with a universal constant, where {uj } is an orthonormal basis in the kernel of D. Remarks. (i) The density function n(x) was also estimated in [BFG]. In the strong field limit B → bB, b 1, the estimate behaved as b17/12 . Moreover, unlike in [BFG], our estimate on n(x) uses only local information on B(x) as explained in Section 2. For example, if B vanishes inside a ball centered at x with radius δ, then n(x) ≤ cδ −3 . (ii) The bound (3.2) is optimal as far as the strength of the field |B| is concerned. This fact follows from the construction of Dirac operators with kernels of high multiplicity following the method of [ES-III]. For example, the density of Aharonov-Casher zero modes for a constant magnetic field of strength B 1 on S 2 is of order B. The geometric procedure of [ES-III] allows one to construct a Dirac operator on R3 whose zero energy eigenfunctions are obtained from the eigenfunctions on S 2 by an explicit transformation. The density of these states remain comparable to the strength of the magnetic field at least away from infinity. (iii) Notice that the Lieb-Thirring inequality of [LSY-II] for a constant field is recovered in Theorem 3.1. (iv) The uniform Lieb-Thirring bound for a constant direction field, [Sob-97], [ES-I], does not directly follow from our main theorem as it is stated. On one hand, (3.1) contains a term linear in V that is unnecessary for a constant direction field. On the other hand, we assume high regularity on B. This regularity is
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needed only to construct the appropriate curvilinear cylindrical localization, which is unnecessary for a field with constant direction. However, our present technique to estimate squares of the resolvents can improve these results in another aspect. For example, if the support of B and the support of V are separated, our Lieb-Thirring estimate depends only on the separation distance whereas all previous bounds scale with the magnitude of B. As a byproduct of such a result one can also improve the estimates on the ground state density of the two-dimensional Pauli operator given in [E-93]. Armed with a uniform Lieb-Thirring inequality, the following semiclassical asymptotics may be proved by combining the techniques of the current paper and [ES-II]. The details of the proof will be published separately. Theorem 3.3. We assume that B ∈ C 4 (R3 , R3 ) and V ∈ L5/2 (R3 )∩L1 (R3 ). Then the sum of the negative eigenvalues, ej (b, h), of the Pauli operator [σ · (−ih∇ + bA)]2 + V is asymptotically given by lim
h→0
j
e(b, h)
Escl (h, bB, V )
=1
(3.3)
where the limit is uniform in the field strength b. Remark. This result was obtained for a homogeneous magnetic field in [LSY-II]. Analogous results for d = 2 were obtained in [ES-II] and [Sob-98]. The latter work also extends the two-dimensional analysis to obtain (3.3) for three-dimensional magnetic fields with constant direction. For a general three-dimensional magnetic field the limit (3.3) is proven up to b h−3 for V ∈ W 1,1 in [ES-II]. With a different method Sobolev also obtains (3.3) up to b ≤ (const.)h−1 without assumptions on the derivatives of B and V [Sob-98].
4 Proof of the Main Theorem 3.1 4.1
Tempered lengthscale
Since localization errors decrease with the localization length, we would optimally like to choose the biggest possible scale, i.e., Lc (x), for our localization scale. However, neighboring localization domains must be comparable in size so that the localization errors could be reallocated. This forces us to require a tempered behavior on the localization scales, which may result in choosing a localization scale smaller than Lc . Proposition 4.2 below shows that this technical requirement can be met at the expense of a factor 12 and this justifies the introduction of the tempered lengthscale L := 12 Lc . Before the precise statement we need the following definition:
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Definition 4.1. Let ε > 0 be a positive number. A positive function f (x) on R3 is called ε-tempered if |x − y| ≤ ε−1 f (x) =⇒
f (y) 1 ≤ ≤2 2 f (x)
∀x, y ∈ R3 .
(4.1)
If ε = 1, then a 1-tempered function will be simply called tempered. Proposition 4.2 (Existence of tempered lengthscale). For any not identically constant magnetic field, B ∈ C 4 (R3 , R3 ), L(x) = 12 Lc (x) is finite and defines a tempered function. Moreover, if BL(x) (x) > L(x)−2 , then bL(x) (x) > 0, and for γ = 1, 2, 3, 4
L(x)γ sup ∇γ |B(y)| : |x − y| ≤ L(x) ≤ bL(x) (x), (4.2) and L(x)γ sup{|∇γ n(y)| : |x − y| ≤ L(x)} ≤ 1.
(4.3)
For a constant magnetic field B = const we have Lc = ∞ and we set L(x) := ∞. The proof is given in Section A.1. For a constant magnetic field B the tempered scale has been defined to be infinity for the transparent formulation of our theorem. The estimate (3.1) for this case has been proven in [LSY-II]. It is possible to apply our proof to this case as well, but setting L = ∞ directly may require minor remarks along the proof. In order to avoid this inconvenience, we can choose L to be any fixed real number for which the proof goes through without changes and finally let L → ∞ in the final result (3.1). 1 that has to be chosen small We introduce a universal constant 0 < ε < 1000 enough for the proof to work but we shall not keep track of the exact numerical value needed. We consider it fixed throughout the proof. Let L(x) be the tempered lengthscale of B. Introduce (x) := εL(x), then the properties of L(x) set in Definition 2.1 and Proposition 4.2 are translated into as follows: • (x) is ε-tempered. • For all x ∈ R3 such that sup{|B(y)| : |x − y| ≤ ε−1 } ≥ ε2 −2 and for γ = 1, . . . , 4 we have
(x)γ sup ∇γ |B(y)| : |x − y| ≤ ε−1 (x)
(4.4) ≤ εγ inf |B(y)| : |x − y| ≤ ε−1 (x) and (x)γ sup
γ ∇ n(y) : |x − y| ≤ ε−1 (x) ≤ εγ .
(4.5)
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We define
683
P (x) := ε−5 (x)−2
and for any positive function f > 0 we introduce the notation Rf = R(f ) = (D2 + f )−1 .
4.2
Separation into low and high energy regimes
We shall prove Theorem 3.1 with Lc replaced by L. Since D2 + V ≥ D2 − [V ]− , we may consider only non-positive potentials. For convenience we change the sign and we will work with the operator H := D2 − V with V ≥ 0. By the Birman-Schwinger principle ∞ 1/2 1/2 |Tr H− | = n V RE V , 1 dE (4.6) 0
where n(A, µ) is the number of eigenvalues of the operator A greater than or equal to µ. For any E > 0 we have, by the resolvent identity, that RE = RP +E + RP +E P RE = RP +E + RP +E P RP +E + RP +E P RE P RP +E . Using that P ≤ D2 + P + E and RE ≤ E −1 , we obtain RE ≤ 2RP +E + E −1 RP +E P 2 RP +E . For any positive operators X1 , X2 , n(X1 + X2 , e1 + e2 ) ≤ n(X1 , e1 ) + n(X2 , e2 ) , hence (4.6) is estimated as ∞
|Tr H− | ≤ 0
(4.7)
n V 1/2 RP +E V 1/2 , 14 dE +
∞
n 2V
1/2
2
RP +E P RP +E V
1/2
, E dE . (4.8)
0
The second term carries the contribution of the near zero energy eigenfunctions of the free Pauli operator D2 . This will be called the zero mode regime. For the first term we notice that ∞ 1/2 1/2 1 (4.9) n V RP +E V , 4 dE = Tr(D2 + P − 4V )− 0
by the Birman-Schwinger principle. This term contains the contribution from free eigenfunctions with energy at least O(P ) and it will be called the positive energy regime. The following theorem estimates the two terms in (4.8) and it completes the proof of the Main Theorem by choosing ε sufficiently small.
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Theorem 4.3. For a sufficiently small universal ε and with the notations above we have
2 V 5/2 + |B|V 3/2 , (4.10) Tr(D + P − 4V )− ≤ c(ε)
∞
n 2V
1/2
2
RP +E P RP +E V
1/2
, E dE ≤ c(ε)
V P 1/2 (|B| + P ) .
(4.11)
0
The proof of Theorem 4.3 is given in the rest of the paper. Convention about operator kernels: If A is a Hilbert-Schmidt operator on a Hilbert space of the form L2 (dµ) ⊗ CN , N ∈ N, we denote by A(x, y) its N × N -matrix-valued integral kernel which is L2 on the product space. If, in addition, A is of trace class, we can even define its diagonal kernel which, by a slight abuse of notation, will be denoted by A(x, x). One possible way to define it is to write A as a product of two Hilbert-Schmidt operators, A = HK, and A(x, x) := H(x, y)K(y, x)dµ(y). This is an L1 -matrix-valued function of x and as such it is independent of the choice of H and K. Convention about traces: We shall denote by Tr the trace on L2 (dµ) ⊗ CN and by tr the trace on CN . If A is of trace class on L2 (dµ) ⊗ CN , then trA(x, x) is in L1 (dµ).
5 Isotropic geometry of the first localization In this section we construct the domains for the first localization. The construction is determined by the function (x). We shall construct a discrete set of points {xi }. The localization domains will be balls about xi with radii (xi ) and they will have finite overlap. Moreover, the magnetic field will not change much in each localization ball since (x) determines the local scale of variation of B. Outside of this domain the field will be replaced by a constant field. This procedure will apply to balls with relatively strong fields. On balls where B is small we neglect magnetic effects and replace the Pauli operator by the free Laplacian.
5.1
Regular fields
Definition 5.1. Given , K > 0 and a ball D of radius centered at z0 ∈ R3 . A magnetic field B is called D-strong if |B(z0 )| ≥ ε−2 −2 , otherwise it is called D-weak. A D-strong magnetic field is called (D, K)-regular if for γ = 1, . . . , 4 (i) ∇γ |B| ≤ Kεγ −γ |B(z0 )| on D; (ii) |∇γ n| ≤ Kεγ −γ on D (with n := B/|B|); A (D, K)-regular field B is called extended (D, K)-regular if it is continuous on the whole space and B is constant outside of D. The value of B outside of D is denoted by B∞ and for B∞ = 0 we set n∞ := B∞ /|B∞ |.
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A (D, K)-regular field B clearly has a small total variation on D: |B(x)| − |B(z0 )| ≤ 2Kε|B(z0 )|
(5.1)
for any x ∈ D. For an extended (D, K)-regular field (5.1) is valid for any x ∈ R3 , and n(x) − n∞ ≤ Kε . (5.2) The following statement follows from the definitions above: Lemma 5.2. Let B(x) and (x) satisfy the conditions (4.1), (4.4) and (4.5). Let D 3 be the ball of radius 10(z0 ) about some z0 ∈ R . 1)-regular and |B(x)| ≥ ε−1 (z0 )−2 for any then B is (D, (i) If B is D-strong, x ∈ D. (ii) If B is D-weak, then |B(x)| ≤ ε−2 (z0 )−2 for any x ∈ D. Proof. (i) For ε ≤ ε0 we obtain that for any x ∈ D
|B(z0 )| − |B(x)| ≤ 10ε inf |B(y)| : |x − y| ≤ ε−1 (z0 ) ≤ 10ε|B(z0 )| using (4.4). In particular, |B(x)| ≥ (1 − 10ε)|B(z0 )| ≥ ε−1 (x)−2 for any x ∈ D −2 −2 because B is D-strong, |B(z0 )| ≥ ε (10(z0 )) , and (z0 ) ≤ 2(x). Properties (i) and (ii) in Definition 5.1 follow from (4.4) and (4.5). we have |B(x)| > ε−2 (z0 )−2 . Using that (ii) Suppose that for some x ∈ D 1 −1 |z0 − x| ≤ 10(z0) ≤ 20(x) < ε (x) if ε ≤ 20 , we obtain
|B(z0 )| − |B(x)| ≤ 10ε inf |B(y)| : |y − x| ≤ ε−1 (x) ≤ 10ε|B(z0 )| which contradicts to |B(z0 )| < ε−2 (10(z0 ))−2 .
5.2
Covering lemma and cutoff functions
Let B(x, r) denote the closed ball centered at x with radius r. We introduce the following notations for any x ∈ R3
x := B x, (x) , D 10
Dx := B x, (x) ,
x := B x, 10(x) . D
Definition 5.3. Let (x) be an ε-tempered function and let I be a countable index set. The discrete set of points {xi }i∈I is called an -uniform set of points with intersectionconstant N if xi ; (i) R3 ⊂ i∈I D xi }. xj intersects no more than N other balls from the collection {D (ii) Any ball D The proof of the following covering Lemma is given in Section A.2.
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Lemma 5.4. Let (x) be ε-tempered, then there exists an -uniform set of points {xi }i∈I with some universal intersection constant N . In the rest of the proof we fix such a collection of points {xi }, determined i := D xi , by the magnetic field via (x). For brevity we shall use i := (xi ), D Di := Dxi and Di := Dxi . Definition 5.5. An index i ∈ I, the corresponding point xi and ball Di are called strong (weak) if B is Di -strong (weak). The following Lemma is an application of Lemma 5.2: i , 1)-regular and inf |B| ≥ Lemma 5.6. Let xi be a strong point, then B is (D Di −2 −2 ε−1 −2 i . i |B| ≤ ε i . If xi is a weak point, then supD Given an ε-tempered function (x) and an -uniform set of points {xi }i∈I , for each i ∈ I we choose smooth functions θi , χ i , χi and χ i with values between 0 and 1, such that the following hold: 2 −1 • i θi (x) ≡ 1, supp(θi ) ⊂ Di and ∇θi ∞ ≤ ci ; • χ i ≡ 1 on B(xi , 3i ), supp( χi ) ⊂ B(xi , 4i ), ∇ χi ∞ ≤ 2−1 i ; • χi ≡ 1 on B(xi , 4i ), supp(χi ) ⊂ B(xi , 5i ), ∇χi ∞ ≤ 2−1 i ; • χ i ≡ 1 on B(xi , 6i ), supp( χi ) ⊂ B(xi , 7i ), ∇γ χ i ∞ ≤ (2i )−γ , γ = 1, . . . , 4. i cover. Notice that ∇ Such choice is possible since the balls D χi is supported on the annulus Ai := B(xi , 4i ) \ B(xi , 3i ) . Finally we choose functions {ϕi }i∈I such that ϕi ≡ 1 on Ai , supp(ϕi ) ⊂ B(xi , 5i )\ B(xi , 2i ) and |∇ϕi | ≤ 2−1 i .
5.3
Approximate magnetic fields and Pauli operators
We define approximate vector potentials Ai and magnetic fields Bi := ∇ × Ai , i = 1, 2, . . ., subordinated to the balls Di . The definition is different for weak and strong indices i ∈ I. i be the Poincar´e gauge of B with base point If i ∈ I is a weak index, then let A 3 i | ≤ ci sup |B| ≤ cε−2 −1 holds true xi , in particular ∇ × Ai = B on R and |A i Di i , then Bi = χ i. i by Lemma 5.6. We define Ai := A − (1 − χ i )A i B + ∇ χi × A on D Clearly A(x) = Ai (x) for all x ∈ B(xi , 6i ) and Bi ∞ ≤ c sup |B| ≤ cε−2 −2 i i D
i. with supp Bi ⊂ D If i ∈ I is a strong index, then Ai is given by the following lemma.
(5.3)
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i , 1)Lemma 5.7 (Choice of the local field on strong balls). Assume that B is (D regular, then there exists a vector potential Ai such that Ai ≡ A on B(xi , 6i ) and the magnetic field Bi = ∇ × Ai satisfies Bi (x) ≡ B(x),
x ∈ B(xi , 6i )
x ∈ R3 \ B(xi , 7i ) . (5.4) Moreover, Bi is extended (Di , 100)-regular, in particular for γ = 1, . . . , 4 (5.5) γi ∇γ |Bi | ∞ ≤ 100εγ |Bi (xi )| , γ γ γ ≤ 100ε , (5.6) ∇ ni and
Bi (x) ≡ B(xi )
∞
i
|Bi (x) − B(xi )| ≤ 100ε|B(xi )|
(5.7)
for any x ∈ R3 . Armed with these definitions of Ai , we define Di := σ · (−i∇ + Ai )
(5.8)
i . The operator Di coto be the approximating Dirac operator associated with D incides with D on B(xi , 6i ) because A = Ai in this domain, in particular Dχi = Di χi .
(5.9)
i , 1)-regular, from (5.1) we obtain that Proof of Lemma 5.7. Since B is (D |B(x) − B(xi )| ≤ ε|B(xi )|,
i . x∈D
(5.10)
i of the magnetic field B − B(xi ), then ∇ × Let A# e gauge on D i be the Poincar´ # Ai = B − B(xi ) and (5.11) |A# i (x)| ≤ εi |B(xi )| i . We then define for any x ∈ D χi A# Bi := ∇ × ( i ) + B(xi ) .
(5.12)
i , 100)-regular and (5.5)–(5.7) hold. Easy calculations show that this field is (D # 1 The gauge Ai + 2 B(xi ) ∧ (· − xi ) generates B, hence 1 A = A# i + 2 B(xi ) ∧ (· − xi ) + ∇φi 1 with some φi : R3 → R. Since ( χi A# i ) + 2 B(xi ) ∧ (· − xi ) generates Bi , we define 1 χi A# Ai := ( i ) + 2 B(xi ) ∧ (· − xi ) + ∇φi .
Then ∇ × Ai = Bi .
(5.13)
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6 Positive energy regime: proof of (4.10) in Theorem 4.3 We recall the set {xi } constructed in Section 5 and let i := (xi ),
Pi := P (xi ) = ε−5 −2 i ,
bi := |B(xi )| .
We also recall that for any positive function f we denote the resolvents by Rf := R[f ] := (D2 + f )−1 .
(6.1)
Note that in general Rf and Rg do not commute. For simplicity we also introduce Ri [f ] := (Di2 + f )−1 .
(6.2)
Proposition 6.1 (Pull-up proposition). Let I be a countable index set and let gi , i ∈ I, be a family of nonnegative smooth functions such that 0 < i∈I gi2 (x) < ∞ for every x ∈ R3 . Let Ai , i ∈ I be a family of positive invertible self-adjoint operators on L2 (R3 , C2 ). Then
gi2
1 1 gi2 ≤ gi gi . Ai i∈I gi Ai gi
i∈I
i∈I
(6.3)
i∈I
Proof of Proposition 6.1. This proof is basically given in [BFFGS], we repeat it here for completeness. All positive self-adjoint operators below are interpreted as quadratic forms. We start with the operator inequality J ∗J
1 J ∗ J ≤ J ∗ AJ J ∗ A−1 J
(6.4)
for any positive self-adjoint operator A and any operator J. We define a map J : L2 (R3 , C2 ) → i L2 (R3 , C2 ) =: H as J : ψ → {gi ψ}. on H as A : {ψi } → {Ai ψi }. It is easy to check that We define an operator A = J ∗ AJ
gi Ai gi
on L2 (R3 , C2 ) ,
i
J ∗J =
2 i gi
−1 . Thus −1 = A and that (A)
i
1 1 J ∗J gi2 gi2 = J ∗ J ∗ AJ g A g i i i J i i 1 −1 J = ≤ J ∗ (A) gi gi . Ai i
The following proposition is the localized version of (4.10) for strong balls and its proof is given in Section 10.
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Proposition 6.2 (Positive energy regime). Let D be a ball of radius and let K > 0 be a positive number. Let B be extended (D, K)-regular, let the function 0 ≤ χ ≤ 1 be supported on D. Then for any positive numbers M, µ > 0 there exists a constant ε(M, K, µ) such that for any ε ≤ ε(M, K, µ) we have
2 −5 −2 2 V 5/2 + |B|V 3/2 . (6.5) Tr(D + µε − M χ V )− ≤ c(M, K, ε) D
Armed with these two propositions, we can finish the estimate (4.10) in i ’s (Lemma 5.4), and that Theorem 4.3. Using the finite overlap property of D θi ≤ χi ≤ 1, we see that 1 ≤ Ξ(x) := χ2i (x) ≤ N . i∈I
Moreover, by the localization estimate, 2 2 1 1 |χi Dψ| ≥ 2N |Dχi ψ|2 − |Dψ| ≥ N i∈I
2 N
i∈I
hence D2 ≥
1 2N
χi D 2 χi −
i∈I
ψ, (∇χi )2 ψ ,
i∈I
8 N
−2 i 1(Di ) ,
i∈I
where 1(·) is the characteristic function. Using (5.9) we may simply replace D2 by Di2 on the support of χi . If ε is sufficiently small, we obtain 1 D2 + P + E ≥ 4N χi (Di2 + Pi + E)χi (6.6) i∈I
i. using the finite overlap property and that P is comparable to Pi = ε−5 −2 on D i The resolvent can be estimated by RP +E ≤
2 i χi (Di
4N ≤ 4N Ξ−1 χi Ri [Pi + E]χi Ξ−1 + Pi + E)χi i∈I
using Proposition 6.1. Hence, by the Birman-Schwinger principle ∞
n V 1/2 RP +E V 1/2 , 14 dE Tr(D2 + P − 4V )− = 0 ∞
−1 1/2 1/2 −1 1 ≤ n Ξ V χi Ri [Pi + E]χi V Ξ , 16N dE 0
≤
∞
n 0
i∈I
i∈I
V
1/2
χi Ri [Pi + E]χi V
1/2
,
1 16N
dE .
(6.7)
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Here we used n(ABA, e) = n(B 1/2 A2 B 1/2 , e)
(6.8)
for any nonnegative operator B and arbitrary operator A with the choice A = Ξ−1 and we estimated Ξ−2 ≤ 1. We also use a strengthening of (4.7). If the positive self-adjoint operators A, B are disjointly supported, i.e., there exists an orthogonal projection Π such that ΠAΠ = A and (I − Π)B(I − Π) = B, then n(A + B, e) = n(A, e) + n(B, e) .
(6.9)
The proof is trivial. In order to use Proposition 6.2, we have to pull the summation out in (6.7). We split this sum into a few infinite sums so that each contain disjointly supported i } have uniformly finite overlap with constant N (see (ii) terms. Since the balls {D of Lemma 5.4), there exists a partition of the index set I = I1 ∪ I2 ∪ . . . ∪ IN +1 j ∩ D j = ∅. Such a partition such that if j, j ∈ Ik , for any 1 ≤ k ≤ N + 1, then D can be obtained by a greedy algorithm. We order the index set I in some way and we put each index one by one into one of the sets. We always put the new index into one of the sets where it has no conflict with the indices already put into this set. A new index j is said to be in conflict with a previously placed index i if j = ∅. Since every index can have a conflict with at most N other indices, i ∩ D D each index can be placed somewhere at each step of the placement. Hence, using (4.7) first, then (6.9), we have 1/2 1/2 1 n V χi Ri [Pi + E]χi V , 16N i∈I
= ≤ =
n
N +1
k=1 i∈Ik N +1
n
k=1 N +1
V
1/2
V
1/2
χi Ri [Pi + E]χi V
1/2
χi Ri [Pi + E]χi V
1/2
,
1 16N
,
1 16N (N +1)
i∈Ik
1 n V 1/2 χi Ri [Pi + E]χi V 1/2 , 16N (N +1)
k=1 i∈I k
=
1 n V 1/2 χi Ri [Pi + E]χi V 1/2 , 16N (N +1)
,
i∈I
so combining this estimate with (6.7) and applying the Birman-Schwinger principle in the opposite direction, we obtain (6.10) Tr(D2 + P − 4V )− ≤ Tr(Di2 + Pi − M χ2i V )− i∈I
with M := 16N (N + 1).
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i, We then apply Proposition 6.2 for each strong index i for the ball D = D radius = 10i , and the magnetic field Bi that is extended (Di , K = 100)-regular. For small enough ε we obtain
V 5/2 + |Bi |V 3/2 Tr(Di2 + Pi − M χ2i V )− ≤ c(ε, M ) D i
≤ c(ε, M ) V 5/2 + |B|V 3/2 for i strong, i D
where the last inequality follows from (5.1). For the weak indices i we use Di2 = (−i∇ + Ai )2 + σ · Bi and σ · Bi ≤ cε3 Pi (see (5.3)) and we obtain Tr(Di2 + Pi − M χ2i V )− ≤ Tr((−i∇ + Ai )2 − M χ2i V )− ≤ c(M ) V 5/2 χ5i by the usual Lieb-Thirring inequality for magnetic Schr¨ odinger operators without spin. Summing up these estimates we obtain from (6.10) that
V 5/2 + |B|V 3/2 |Tr(D2 + P − 4V )− | ≤ c(ε) D i∈I i
V 5/2 + |B|V 3/2 , ≤ c(ε) again by the finite overlap property of B. This completes the estimate (4.10).
7 Zero mode regime: proof of (4.11) in Theorem 4.3 The estimate (4.11) essentially involves estimating the square of the resolvent of D2 . However, the analog of Proposition 6.1 does not hold for the square of the resolvent, i.e.,
gi Φ(Ai )gi gi Ai gi gi2 ≤ gi2 Φ with Φ(t) = t−2 is not true in general. Here is a 2 by 2 matrix counterexample with g1 = g2 = 2−1/2 : 1 1 2 1 A1 = , A2 = . 1 2 1 2 Without such an inequality, we have to use a resolvent expansion. In addition to the square of the localized resolvent, we need to control offdiagonal terms. Such an estimate is given in the following proposition, which is a general statement about squares of resolvents of second-order differential operators. The proof is given in Section A.3.
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Proposition 7.1 (Pull-in proposition). Given an ε-tempered function (x) and a function F (x) > 0 satisfying 1 F (x) ≤ ≤2 (7.1) 2 F (y) −5 −2 for all |x − y| ≤ ε−1 (x). Set P (x) . Let A = A · ∇ + B be a first-order = 2ε (x) 3 L (R ), 1 ≤ k < ∞, with smooth coefficients, differential operator acting on k i.e., A(x) is a vector of k × k-matrices, B(x) is a k × k-matrix, all smoothly depending on x. We assume that
sup A(x) ≤ c0 .
(7.2)
x
Let T = A∗ A. Given an -uniform set of points {xi }i∈I as in Lemma 5.4. Let χi , θi , ϕi be chosen as in Section 5.2. We assume that for every i there exists a first-order differential operator Ai = Ai · ∇ + Bi on k L2 (R3 ) such that B = Bi
A = Ai ,
on supp(χi ),
(7.3)
and let Ti = A∗i Ai . Then there exists an ε0 depending only on c0 in (7.2) such that for any ε ≤ ε0 and µ ≥ 0 we have 1 1 F2 T +P +µ T +P +µ Fi2 θi2 Pi−1 ≤c i∈I
1 1 1 θ2 A∗i ϕ2i Ai + Ti + Pi Ti + Pi (Ti + Pi )2 i
(7.4)
i } and Pi = P (xi ). where Fi := sup{F (x) : x ∈ D Remark. If A, B are well behaved, then T looks like an elliptic constant coefficient differential operator on short scales. In this case the estimate localizes the square of the resolvent in such a way that the diagonal element of the operator kernels on the right-hand side of (7.4) remain finite. This is clear for the second term on the RHS since the estimate is integrable in the ultraviolet regime (behaves like p−4 in the momentum p). The first term behaves only as p−2 , but the supports of θi and ϕi are well separated, which makes the diagonal element finite. The diagonal kernels of the localized operators are estimated in the following proposition whose proof is given in Section 11. Proposition 7.2 (Zero mode regime). Let D be a ball of radius > 0 with center z0 ∈ R3 and K > 0. We assume that either = (i) B∞ ≤ cε−2 −2 and B is supported on the ten times bigger ball D B(z, 10); or (ii) B is extended (D, K)-regular.
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Let D be any Dirac operator with magnetic field B. Set P := ε−5 −2 , R[P ] := (D2 + P )−1 , let 0 ≤ ϕ ≤ 1 be a function with dist(z0 , supp ϕ) ≥ 2. If ε ≤ ε(K), then the following estimates hold for any u ∈ D tr R[P ]2 (u, u) ≤
tr R[P ]Dϕ2 DR[P ] (u, u) ≤
c(|B(u)|P −3/2 + P −1/2 )
(7.5)
c(|B(u)|P −1/2 + P 1/2 ) ,
(7.6)
where recall that tr := trC2 stands for the trace in the spin space. Remark. The diagonal elements in (7.5), (7.6) are gauge invariant, i.e, they do not depend on the choice of the vector potential in the Dirac operator. Using these propositions, we can complete the proof of the estimate (4.11) in Theorem 4.3. We use that the function F (x) := P (x) and the operators A := D, Ai := Di satisfy the conditions of Proposition 7.1 by using (5.9) . Setting µ = E in (7.4) we obtain ∞
n 2V 1/2 RP +E P 2 RP +E V 1/2 , E dE
0
≤ 2
∞
n cV 1/2
0
= c
(7.7)
Pi2 θi2 Pi−1 Ri [Pi ]Di ϕ2i Di Ri [Pi ] + Ri2 [Pi ] θi2 V 1/2 , E dE
i∈I
Tr V
θi4
Pi Ri [Pi ]Di ϕ2i Di Ri [Pi ] + Pi2 Ri2 [Pi ]
i∈I
∞ using that 0 n(T, E)dE = Tr T for any positive operator T . This sum of traces can be estimated by
1/2 3/2 4 c dx ≤ c V P 1/2 (|B| + P ) (7.8) V (x)θi (x) |Bi (x)|Pi + Pi i∈I
using Proposition 7.2 with D = Di , = i , B = Bi , K = 100 and for sufficiently small ε. The construction of Bi for both weak and strong indices in Section 5.3 guarantees that either (i) or (ii) holds true in Proposition 7.2. We also used that |Bi | ≤ c(|B| + P i ) and P i ≤ cP (x) on the support of θi (see (5.1), (5.3) and (5.7)), moreover that i θi4 ≤ i θi2 = 1. This completes the proof of (4.11).
8 Cylindrical geometry of the second localization Throughout this section B = (B1 , B2 , B3 ) is an extended (D, K)-regular magnetic field. Let B∞ be the value of B outside D, we set b := |B∞ | and n∞ := B∞ /b. The corresponding magnetic 2-form β is given by β := B3 dx1 ∧ dx2 + B1 dx2 ∧ dx3 + B2 dx3 ∧ dx1 .
(8.1)
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Let z0 be the center of D and define the supporting plane of B, P := {z ∈ R3 : (z − z0 ) · n∞ = −} , to be the plane that is orthogonal to the parallel field lines outside D. We fix an orthonormal basis p1 , p2 in P such that p1 , p2 , n∞ is positively oriented. Any point z in P can be identified with a point zˆ = (ˆ z1 , zˆ2 ) ∈ R2 via z − z0 = zˆ1 p1 + zˆ2 p2 , i.e., zˆi = pi · (z − z0 ), i = 1, 2. We will use these coordinates to parametrize P. For any z ∈ P we denote by ϕz (τ ) the field line through z with arc length parametrization τ , i.e., ϕ˙z (τ ) =
d ϕz (τ ) = n(ϕz (τ )) , dτ
ϕz (0) = z .
Since B is extended (D, K)-regular, ϕ˙ z is constant outside D. If ε is small enough (depending on K), we can assume that the length of the field line within D is at most 4 using (5.2). Therefore ϕ˙ z (τ ) is constant for |τ | ≥ 4. Every field line intersects P since n nowhere vanishes and n · n∞ ≥ 1 − n − n∞ 2 ≥ 12 if ε is sufficiently small. Therefore the field lines {ϕz (τ ) : z ∈ P} form a foliation of R3 and for each x ∈ R3 we denote by π(x) ∈ P the unique point such that x = ϕπ(x) (τ ) for some τ ∈ R.
8.1
Coordinates and conformal factor
The following lemma will be used to introduce coordinates, ξ = (ξ1 , ξ2 , ξ3 ), on R3 associated with the field line ϕz (τ ), z ∈ P, which will be called the central field line. The field line will be characterized by ξ1 = ξ2 = 0. The point z ∈ P will be called the base of the coordinate system. The coordinates are functions of x ∈ R3 and the inverse function will be denoted by x(ξ) : R3 → R3 . We may also use the notation ξ z (x) and xz (ξ) to indicate the dependence on the base. For notational convenience we sometimes use ξ⊥ := (ξ1 , ξ2 ). In order to treat different error terms we introduce a notation similar to the standard “big-oh” notation. Definition 8.1. Let k and α be nonnegative integers and let > 0 be a real number. We say that a complex function f (ξ) is of class Ok (|ξ⊥ |α ) if there exists a constant C such that |ξ | (α−m1 −m2 )+ ⊥ ,1 (8.2) |∂ξm f (ξ)| ≤ C−|m| min for any multiindex m = (m1 , m2 , m3 ) with |m| := m1 + m2 + m3 ≤ k. The definition can be extended to matrix-valued functions and to forms by replacing the absolute value with any matrix or form norm on the left-hand side of (8.2). For = 1 we set Ok (|ξ⊥ |α ) := Ok1 (|ξ⊥ |α ), for k = 0 we set O (|ξ⊥ |α ) := O0 (|ξ⊥ |α ) and for α = 0 we set Ok (1) := Ok (|ξ⊥ |0 ).
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Remark. With a slight abuse of notation Ok (|ξ⊥ |α ) will be used to denote not only the class of these functions but any element of this class, similarly to the way O(1) is used. We also would like the magnetic field to be of constant strength along the central field line which is achieved by a conformal change of metric with a factor Ω. Let ds2 be the standard Euclidean metric and ds2Ω := Ω2 ds2 be a conformally equivalent one. The following lemma describes the necessary information about the new metric and coordinates. The proof is given in Section A.4. Lemma 8.2 (New metric and coordinates). Given positive numbers K, , a ball D of radius , and center z0 , an extended (D, K)-regular magnetic field B, an orthonormal basis p1 , p2 in the supporting plane P such that p1 , p2 , n∞ is positively oriented and the coordinate identification z ∈ P ↔ zˆ = (p1 · (z − z0 ), p2 · (z − z0 )) ∈ R2 . If ε small enough depending on K, then for any z ∈ P there exist coordinate functions ξ = ξ z (x) = (ξ1 , ξ2 , ξ3 ) = (ξ⊥ , ξ3 ), and positive functions Ω(ξ), h(ξ) ∈ C 2 (R3 ) with the following properties: ξ3z (x) = 0
and
z ξ⊥ (x) = x ˆ − zˆ
z ξ⊥ (ϕz (τ )) = 0
for
x∈P ,
∀τ .
(8.3) (8.4)
z , ξ) → xz (ξ) The function (ˆ z , x) ∈ R5 → ξ z (x) ∈ R3 and the inverse function (ˆ 3 5 belong to C (R ). Moreover, if Dξ and Dx denote the Jacobians of these functions, then for γ = 1, 2, 3 we have γ γ D x, D ξ ≤ c(K)εγ −γ+1 . (8.5) The metric ds2Ω := Ω2 ds2 can be expressed as ds2Ω =
2
aij dξi dξj + h2 dξ32 ,
(8.6)
i,j=1
where aij ∈ C 2 (R3 ) satisfies
sup |aij (ξ) − δij | : ≤ |ξ⊥ | ≤ 10 ≤ c(K)ε .
(8.7)
Moreover, aij = δij away from the set ≤ |ξ⊥ | ≤ 10, i.e., ds2Ω = dξ12 + dξ22 + h2 dξ32
on the domain
|ξ⊥ | ≤ or |ξ⊥ | ≥ 10 . (8.8)
The functions Ω, aij and h also satisfy Ω ≡ aij ≡ h ≡ 1
|ξ3 | ≥ 3 ,
(8.9)
h = 1 + εO2 (|ξ⊥ |)
(8.10)
on the domain
Ω = f (ξ3 )(1 + εO2 (|ξ⊥ |)),
and
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1/2 ,
(8.11)
|ξ⊥ | ≥ 10 .
(8.12)
and Ω ≡ 1,
∂ξ⊥ h ≡ 0,
on the domain
Globally, the following bounds hold h − 1∞ , Ω − 1∞ ≤ c(K)ε , ∇γ aij ∞ , ∇γ h∞ , ∇γ Ω∞ ≤ c(K)εγ −γ ,
(8.13) γ = 1, 2 .
(8.14)
Moreover, there exists an orthonormal basis {e1 , e2 , e3 } in the ds2Ω metric such that e3 = h−1 ∂ξ3 everywhere, and ej = ∂ξj , j = 1, 2, apart from the region {ξ : ≤ |ξ⊥ | ≤ 10, |ξ3 | ≤ 4}. Remark. The estimates in Lemma 8.2 actually depend only on the variational lengthscale of the direction of the magnetic field, n, and they are independent of the variational lengthscale of its strength. The proof given in Section A.4 uses a construction that relies only on the field line structure and on the logarithmic gradient of |B| along the field line. However, ∇n |B| = −div n |B| since B = |B|n is divergence-free, therefore derivatives of n alone control the errors. We define the spin-up projection associated with a field line through a given point. Definition 8.3. Given z ∈ P, the field line ϕz (τ ), the associated coordinates ξ z (x) and the inverse function xz (ξ) as defined in Lemma 8.2. Then the spin-up projection associated with ϕz (τ ) is given by a 2 by 2 matrix Pz↑ (x) :=
1 1 + σ · n xz (0, ξ3z (x)) 2
at any point x ∈ R3 . Note that Pz↑ is constant on the level sets of ξ3z .
(8.15)
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Cylindrical partition of unity and grid of field lines
We start with a technical lemma. Lemma 8.4. Given y ∈ P and the associated coordinates {ξky }, k = 1, 2, 3, as constructed in Lemma 8.2, then for any sufficiently small ε ≤ ε(K) and any z ∈ R3 y |ξ⊥ (z)| 1 ≤ ≤2, 2 |y − π(z)|
(8.16)
↑ Py↑ (z) − Pπ(z) (z) ≤ cKε−1 |y − π(z)| ,
(8.17)
where · denotes the standard norm of 2 by 2 matrices. Proof. Denote u = π(z) and set q(τ ) := ξ y (ϕu (τ ))−ξ y (ϕy (τ )) ∈ R3 and let r(τ ) := q⊥ (τ ) = (q1 (τ ), q2 (τ )). We have |q(0)| = |r(0)| = |u − y| and by ϕ(τ ˙ ) = n(ϕ(τ )) and Lemma 8.2 we can estimate |q(τ ˙ )|
≤ ≤
Dx ξ∞ |ϕ˙ u (τ ) − ϕ˙ y (τ )| + Dx2 ξ∞ |ϕu (τ ) − ϕy (τ )| (Dx ξ∞ ∇n∞ + Dx2 ξ∞ )|ϕu (τ ) − ϕy (τ )|
≤ ≤
(Dx ξ∞ ∇n∞ + Dx2 ξ∞ )(Dx ξ)−1 ∞ |q(τ )| c(K)ε−1 |q(τ )|
(8.18)
for |τ | ≤ 4 and q(τ ˙ ) ≡ 0 for |τ | ≥ 4. Therefore supτ |q(τ )| ≤ |u − y|ec(K)ε by Gromwall’s inequality and sup |r(τ ˙ )| ≤ sup |q(τ ˙ )| ≤ cKε−1ec(K)ε |u − y| . τ
τ
Combining this with |r(0)| = |u − y| we obtain 12 |u − y| ≤ supτ |r(τ )| ≤ 2|u − y| if ε is sufficiently small. Note that for some τ y y y |ξ⊥ (z)| = ξ⊥ (ϕπ(z) (τ )) − ξ⊥ (ϕy (τ )) = |r(τ )| , which concludes the proof of (8.16). For the proof of (8.17) we again set u = π(z) and by Definition 8.3 and Lemma 8.2 we estimate Py↑ (z) − Pu↑ (z) ≤ ∇n∞ xy (0, ξ3y (z)) − xu (0, ξ3u (z)) ≤ ∇n∞ xy (0, ξ3y (z)) − xy (0, ξ3u (z))
+ xy (0, ξ3u (z)) − xu (0, ξ3u (z)) ≤
∇n∞ (2Du ξ u ∞ + Du xu ∞ )|y − u|
≤
cKε−1|y − u| ,
using |xy (ξ) − xy (ξ )| ≤ 2|ξ − ξ | that follows from (8.5) if ε is sufficiently small. This completes the proof of (8.17).
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We construct a grid of field lines. Choose a square lattice Y := {yj : j ∈ Z2 } on P with spacing b−1/2 , i.e., |yj − yk | = b−1/2 |j − k|, j, k ∈ Z2 . Applying Lemma 8.2 to each field line ϕyj , we construct conformal factors Ωj , orthonormal bases (j) (j) (j) (j) (j) (j) {e1 , e2 , e3 } and coordinate functions ξ (j) = (ξ1 , ξ2 , ξ3 ). We now construct a set of Gaussian localization functions with a transversal lengthscale of order b−1/2 that are essentially supported around the field lines ϕyj . Let η ≤ 14 be a small positive number to be specified later and we define
ηb (j) vj (x) = exp − [ξ⊥ (x)]2 . 4
(8.19)
We set Pj↑ (x) := Py↑j (x) to be the 2 by 2 spin-up projection matrix associated with the field line through yj (see Definition 8.3). Lemma 8.5. If ε is sufficiently small depending only on K, then for any γ > 0, κ ≥ 0 we have (j) (ηb)κ [ξ⊥ (x)]2κ vj (x)γ ≤ c(γ, κ)η −1 , (8.20) j∈Z2
vj (x)γ
≥
c(γ)η −1 ,
(8.21)
j∈Z2
uniformly in x ∈ R3 . Moreover, there is a universal constant C0 and for any 0 < λ < 1 there exists 0 < η(λ) ≤ 14 such that for any η ≤ η(λ)
(j) ↑ 4 2 2 −2 vj (x) b λ − η b[ξ⊥ (x)] Pj (x) + C0 ≥0. (8.22) j∈Z2 2 κ 2 Proof. Since (ηbξ⊥ ) exp(− γηb 8 ξ⊥ ) ≤ c(γ, κ) uniformly in ξ⊥ , it is sufficient to γ/2 estimate j vj for the proof of (8.20). Using (8.16) we obtain
j∈Z2
γ/2
vj
(x) ≤
j
γηb |yj − π(x)|2 ≤ c(ηγ)−1 exp − 16
since yj runs through a square grid with spacing b−1/2 . The proof of (8.21) is similar. For the proof of (8.22) we define k ∈ Z2 to be an index such that |yk −π(x)| ≤ −1/2 . Then by (8.16) and Schwarz’ inequality b j |yk − yj |2 ≤ 2b−1 + 2|π(x) − yj |2 ≤ 2b−1 + 4|ξ⊥ (x)|2 .
Combining this estimate with (8.17) and using that (P ↑ )2 = P ↑ we have j 2 Pj↑ ≥ 12 Pk↑ − 2Pj↑ − Pk↑ 2 ≥ 12 Pk↑ − −2 |yk − yj |2 ≥ 12 Pk↑ − 4−2 |ξ⊥ | − cb−1 −2
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j 2 Pj↑ ≤ 2Pk↑ + 2(Pj↑ − Pk↑ )2 ≤ 2Pk↑ + 4−2 |ξ⊥ | + cb−1 −2
if ε is sufficiently small. We omitted the x argument for brevity. Therefore we can use (8.20) and (8.21) to estimate j 2 vj4 b(λ − η 2 b|ξ⊥ | )Pj↑ j
≥
b 4 j 2 v (λ − 4η 2 b|ξ⊥ | )Pk↑ 2 j j
j 2 j 4 j 2 −c vj4 λb|ξ⊥ | + cλ + η 2 b2 |ξ⊥ | + η 2 b|ξ⊥ | −2 j
≥ ≥
b −1 (cη λ − c)Pk↑ − cη −1 −2 2 − vj4 (x)c−2 j
if η is sufficiently small. We can choose C0 to be the universal constant c in the last formula and the proof of (8.22) is completed.
9 Dirac operator on R3 with a general metric The following sections summarize basic information about the Dirac operator over a non-flat manifold. More details are found in [ES-III] (the sign of A is chosen to be the opposite in this paper). The presentation here is simplified because the spinor bundle is trivial and we can work in a global orthonormal basis. Throughout this section we shall consider R3 with a general Riemannian metric g = (·, ·) and we shall consider the Dirac operator for this particular Riemannian manifold. The Dirac operator will be an unbounded self-adjoint operator in L2g (R3 ) ⊗ C2 (the subscript g refers to the fact that the measure is the volume form of g). Let {e1 , e2 , e3 } be a global orthonormal basis of vectorfields and let {e1 , e2 , e3 } be the dual basis. If X is a vectorfield on R3 , we denote by PX :=
3 1 1+ (X, ej )σ j 2 j=1
(9.1)
the spin projection in direction X with respect to the basis {e1 , e2 , e3 }. We also introduce a covariant derivative on L2g (R3 ) ⊗ C2 by
Here we define
∇X := ∂X + 2i σ · ω(X) .
(9.2)
ω(X) := (∇X e3 , e2 ), (∇X e1 , e3 ), (∇X e2 , e1 ) ,
(9.3)
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where ∇X refers to the Levi-Civita connection on vector fields for the metric g on R3 . If α is a (real) 1-form we define the corresponding covariant derivative on L2g (R3 ) ⊗ C2 (see Proposition 2.9 in [ES-III]) ∇α X := ∇X + iα(X) .
(9.4)
The magnetic 2-form is β := dα. We define the Dirac operator by Dα :=
3
σ j (−i∇α ej ) .
(9.5)
j=1
It is a symmetric operator in L2g (R3 ) ⊗ C2 (Theorem 3.2 in [ES-III]). Note that Dα also depends on the metric g and the choice of {e1 , e2 , e3 } but this fact will usually be suppressed in the notation. For notational convenience we introduce the following vector of covariant derivatives α α (9.6) Πα := (−i∇α e1 , −i∇e2 , −i∇e3 ) . With this notation we may write Dα = σ · Πα . Note that the components of Πα are not self-adjoint, however the components of the vector Dα = (D1α , D2α , D3α ) := Πα −
i div e1 , div e2 , div e3 2
(9.7)
are self-adjoint operators. For any one form λ = λ1 e1 +λ2 e2 +λ3 e3 we define σ(λ) := λ1 σ 1 +λ2 σ 2 +λ3 σ 3 . The Lichnerowicz’ formula (see, e.g., Theorem 3.4 in [ES-III]) states that 1 [Dα ]2 = [Πα ]∗ · Πα + R + σ(β) , 4
(9.8)
where R is the scalar curvature of g and denotes the Hodge dual. In terms of Djα operators, the Lichnerowicz’ formula reads as 1 1 1 [Dα ]2 = [Dα ]2 + R + [div ej ]2 + ∂e (div ej ) + σ(β) . 4 4 j=1 2 j=1 j 3
3
(9.9)
For the flat Euclidean metric with the standard orthonormal basis we have ω ≡ 0. In this case if a denotes the 1-form dual to the vector field A = (A1 , A2 , A3 ) then Πa = pA where pA is the vector of operators (−i∂1 + A1 , −i∂2 + A2 , −i∂3 + A3 ). Therefore we obtain the usual Dirac operator D = Da defined in Section 1 with β = da given by (8.1). Moreover, PX = Pz↑ if X(x) = n(xz (0, ξ z (x))) from (8.15) and (9.1).
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Gauge transformation
In the previous construction Dα and Πα depend on α and also on {e1 , e2 , e3 }. Up to a unitary equivalent gauge transformation, however, Dα and Πα depend only on the metric g and the magnetic 2-form β. Similarly, the spin projection PX defined in (9.1) is gauge-invariant. More precisely, given another 1-form α with dα = β and another orthonormal basis {e1 , e2 , e3 } with the same orientation, we denote the corresponding op be the spin projection. There exist a real valued erators by D and Π and let PX function φ(x) and a continuous function R(x) ∈ SO(3) on R3 such that α = α+dφ and k wk ek = k (Rw)k ek for any w ∈ R3 . Let UR (x) ∈ SU (2) denote the image of R(x) under the isomorphism SO(3) → SU (2)/{±1}. The requirement that UR (x) be a continuous function of x determines UR uniquely up to a global sign. In particular (9.10) UR (σ · v)UR∗ = σ · (Rv) 2 for any v ∈ R3 , i.e., R(ψ, σψ) = (UR ψ, σUR ψ) for any ψ ∈ C , where (ψ, σψ) denotes the vector (ψ, σ 1 ψ), (ψ, σ 2 ψ), (ψ, σ 3 ψ) ∈ R3 . We define the unitary operator of the form
[UR,φ ψ](x) = eiφ(x) UR (x)ψ(x) ,
(9.11)
∗ = UR,φ PX UR,φ PX
(9.12)
then and
∗ Dα UR,φ , D = UR,φ
∗ and w · Π = (Rw) · UR,φ Πα UR,φ
(9.13)
for any w ∈ R3 . In particular, the spectrum of Dα and the functions
1 1 1 α 2 α D (x, x) tr (x, x) and tr ϕ D ([Dα ]2 + c)2 [Dα ]2 + c [Dα ]2 + c depend only on g and β, where ϕ is any function on R3 and c > 0 is a constant.
9.2
Change of the Dirac operator under a conformal change of the metric
Let Ω be a positive real function on R3 and let gΩ := Ω2 g be a metric which is conformal to g. Consider the (f1 , f2 , f3 ) := (Ω−1 e1 , Ω−1 e2 , Ω−1 e3 ) orthonormal α and DΩ denote the corresponding basis in gΩ . Given a 1-form α we let ∇α,Ω X covariant derivative and Dirac operator. With the notation
α,Ω α,Ω α,Ω (9.14) := − i∇ , −i∇ , −i∇ Πα Ω f1 f2 f3 α we have DΩ = σ · Πα Ω . Then from Section 4 of [ES-III] α DΩ = Ω−2 Dα Ω
(9.15)
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α 1 −1 ∇α,Ω [σ(X ∗ ), σ(dΩ)] X = ∇X + 4 Ω
(9.16)
for any vector X, where X ∗ refers to the 1-form which is dual to the vector X relative to the metric g, and σ(X ∗ ), σ(dΩ) are computed in the {e1 , e2 , e3 } basis. In particular,
−1 α Πα Π − 4i Ω−2 [σ 1 , σ(dΩ)], [σ 2 , σ(dΩ)], [σ 3 , σ(dΩ)] . (9.17) Ω =Ω
9.3
Constant approximation of the magnetic field along a field line
The goal of this section is to express the Dirac operator with a non-homogeneous regular magnetic field as a sum of a constant field Dirac operator and some error terms in a neighborhood of a given field line. This can be done if the original Dirac operator is already written in an appropriate orthonormal basis and with a carefully selected vector potential. The basis and the vector potential are determined by the local magnetic field. Given an extended (D, K)-regular field B. Consider the corresponding 2-form β and a fixed field line. Let the coordinates (ξ1 , ξ2 , ξ3 ), the new metric gΩ = Ω2 ds2 with a conformal factor Ω and the orthonormal basis {e1 , e2 , e3 } be as constructed in Lemma 8.2, associated with the given field line. Let α denote a vector potential, dα = β, to be chosen later. Let Πα be given by (9.6) and let Dα := σ · Πα . On the central line and in the regime |ξ⊥ | ≥ 10 the magnetic field β is constant in the ds2Ω metric: β(e1 , e2 ) = Ω−2 β(Ωe1 , Ωe2 ) = Ω−2 |B| = b,
β(ej , e3 ) = 0,
j = 1, 2 .
This observation gives rise to the following definition. Definition 9.1. Given a field line, the associated coordinate system ξ and the conformal factor Ω as above such that magnetic field β is constant in the ds2Ω metric with strength b = β(e1 , e2 ). Then the magnetic field βc given by βc := b dξ1 ∧ dξ2 is called the approximating constant magnetic field along the field line. 3 The magnetic field βc is clearly constant in the dξ 2 = j=1 dξj2 metric. A convenient gauge is defined as αc := 2b [ξ1 dξ2 − ξ2 dξ1 ], then βc = dαc . In particular β = βc along the central line and in the regime |ξ⊥ | ≥ 10. We compute the norm of the difference field δβ := β −βc and the norm of its derivative in the dξ 2 metric. Using (8.5), (8.7), (8.8), (8.13) and (5.5), (5.6) we obtain δβ(ξ) = εbO2 (|ξ⊥ |) and δβ(ξ) ≡ 0 if |ξ⊥ | ≥ 10.
(9.18)
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Next, we define an appropriate gauge α for the original magnetic field β, dα = β, such that α − αc be small. The following Lemma was given in [ES-I] (Proposition 2.3). Although it was stated in a slightly weaker form, the explicit formula (2.20) of [ES-I] gives the following stronger result with a straightforward computation: Lemma 9.2 (A-formula). Given any C 2 magnetic 2-form β on R3 with Euclidean coordinates (ξ1 , ξ2 , ξ3 ). For k, m ∈ N we define ξ1 ξ2 k m bk,m (ξ⊥ ) := u sup ∇ β(u, z2 , z3 )du + uk sup ∇m β(z1 , u, z3 )du . 0
z2 ,z3
z1 ,z3
0
Then there exists a 1-form α generating β, dα = β, such that α(ξ) ≤ c b0,0 (ξ⊥ ) + b1,1 (ξ⊥ ) , ∇α(ξ)
(9.19)
≤ c sup{β(u) : |u⊥ | ≤ |ξ⊥ |} + b0,1 (ξ⊥ ) + b1,2 (ξ⊥ ) . (9.20)
We apply this lemma to the magnetic 2-form δβ and we denote by δα the generating 1-form. We define α := αc + δα, then α generates the original magnetic field β, dα = β and it is close to the linear gauge αc of the constant field βc using (9.18) and Lemma 9.2: (α − αc )(ξ) = εbO1 (|ξ⊥ |2 ) .
(9.21)
The norm of the left-hand side is computed with respect to the standard metric. Definition 9.3. With the notations above, the Dirac operator := D
3
σ k [−i∂ξk + αc (∂ξk )]
(9.22)
k=1
with a constant field βc in the dξ 2 metric will be called the approximating constant field Dirac operator along the field line. By the properties of the coordinate vector fields ∂ξk and the orthonormal basis {e1 , e2 , e3 } in the gΩ metric from Lemma 8.2 and by the definitions (9.2), (9.4), (9.6) we have, for sufficiently small ε, + D α = σ · Πα = D
3 k=1
σ k [α(ek ) − αc (∂ξk )] +
3
Kk (−i∂ξk ) + M0 ,
(9.23)
k=1
where K = (K1 , K2 , K3 ) and Kk , M0 are 2 by 2 matrix-valued functions. We use the bounds (8.7), (8.10), (8.14) and the estimate (9.21) to obtain, for k = 1, 2, 3, (9.24) [α(ek ) − αc (∂ξk )](ξ) = εbO1 (|ξ⊥ |2 ) .
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We obtain from (9.23) and (9.24) that + Dα = D
3
Kk (−i∂ξk ) + M
(9.25)
k=1
with matrix-valued functions that satisfy M = bO1 (|ξ⊥ |) + −1 O1 (1) ,
K1,2 = O1 (|ξ⊥ |γ ) ,
K3 = O1 (|ξ⊥ |)
(9.26)
for any γ ≥ 0 if ε ≤ ε(K). These estimates follow from Lemma 8.2, especially from the fact that ek = ∂ξk , k = 1, 2, apart from the region ≤ |ξ⊥ | ≤ 10, where (8.6) holds, i.e., K1 , K2 are supported in this region.
10 Positive energy regime: Proof of Proposition 6.2 We first notice that both sides of (6.5) scale as −2 , hence it is sufficient to prove the result for := 1. We can apply the constructions of Section 8 for the magnetic field (j) (j) (j) B to obtain conformal factors Ωj , orthonormal bases {e1 , e2 , e3 }, coordinate (j) (j) (j) functions ξ (j) = (ξ1 , ξ2 , ξ3 ), spin-up projections Pj↑ and Gaussian localization functions vj concentrated along the field line passing through yj . We recall that yj was a lattice with spacing b−1/2 on the supporting plane (see Section 8.2). We first estimate
2 1 1 1 2 D2 + ε−2 2b pA + σ · B ≥ 12 D2 + ε−2 2b D2 = 1 − ε−2 2b pA − ε−2 since b = |B∞ | ≥ ε−2 and sup |B| ≤ 2b if ε is sufficiently small. Using this estimate and (8.20)–(8.21) we have
(10.1) D2 + µε−5 − M χ2 V ≥ 12 D2 + ε−2 1b p2A + µε−5 − 2M χ2 V
≥ cη Dvj4 D + ε−2 1b pA · vj2 pA + µε−5 vj − cM χ2 V vj6 j
if ε is sufficiently small (depending on µ). Notice from the explicit formula (8.19) that 2 (10.2) [pA , vj ] = |∇vj |2 ≤ cηbvj . Therefore by Schwarz’ inequality pA · vj2 pA ≥ 12 vj p2A vj − cηbvj , and using this estimate in (10.1), including the negative error term into µε−5 vj and subtracting the pointwise inequality (8.22) we obtain for any 0 < λ < 1, η ≤ η(λ) (see Lemma 8.5) that D2 + µε−5 − M χ2 V
(j) 1 Dvj4 D + ε−2 2b vj p2A vj − vj4 b(λ − η 2 b[ξ⊥ ]2 )Pj↑ + µ2 ε−5 vj − cM χ2 V vj6 . ≥ cη j
(10.3)
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The error term C0 vj4 in (8.22) has been absorbed into µε−5 vj if ε is small enough depending on η, µ. The inequality (8.22) has been subtracted to prepare for a later step. Hence
Tr D2 + µε−5 − M χ2 V ≤ cη −
j
(j) 1 vj p2A vj − vj4 b(λ − η 2 b[ξ⊥ ]2 )Pj↑ + µ2 ε−5 vj − cM χ2 V vj6 Tr Dvj4 D + ε−2 2b −
(10.4) by |Tr( j Hj )− | ≤ j |Tr(Hj )− | that follows from the variational principle. The following lemma is the cylindrically localized version of Proposition 6.2. Proposition 10.1. With the notations above and setting W := cM χ2 V we have
(j) 1 vj p2A vj − vj4 b(2−7 − η 2 b[ξ⊥ ]2 )Pj↑ + µ2 ε−5 vj − W vj6 Tr Dvj4 D + ε−2 2b −
2 5/2 2 3/2 (vj W ) + b(vj W ) ≤c (10.5) for each j if ε is small enough depending on K, µ, η. Choosing λ := 2−7 and η := η(2−7 ) (see Lemma 8.5), Proposition 6.2 directly follows from this proposition, from (10.4) and (8.20). Proof of Proposition 10.1. The proof contains three transition steps that are performed locally around each field line from the grid constructed in Section 8.2. First we replace D with a Dirac operator DΩ in a metric that is conformal to the Euclidean one. The conformal factor Ω is chosen such that the strength of the magnetic field becomes constant along a field line. Second we replace the volume form dx with the volume form dξ = dξ1 ∧ dξ2 ∧ dξ3 , where ξ is the coordinate system associated with the chosen field line. Then we perform a gauge transformation so := σ · (−i∂ξ + αc (∂ξ )) with a conthat DΩ becomes close to the Dirac operator D stant magnetic field βc = dαc = b dξ1 ∧ξ2 in the linear gauge αc = 2b [ξ1 dξ2 −ξ2 dξ1 ]. can be analyzed explicitly. Finally, the operator D For each fixed j we consider the constructions in Section 9 with the metric gj := gΩj = Ω2j dx2 . We shall apply Section 9.2 to the Euclidean metric with the −1 −1 standard basis vectors {∂1 , ∂2 , ∂3 }. The vectors {Ω−1 j ∂1 , Ωj ∂2 , Ωj ∂3 } form an a a orthonormal basis in gj . Let Dj and Πj denote the corresponding Dirac operator and the vector of derivative operators as defined in (9.15) and (9.14). We recall that a is the 1-form dual to the vector potential A in the standard metric. Using the estimates in Lemma 8.2 to the formula (9.17), we obtain Πaj = Ω−1 j pA + εO2 (1) ,
and Dja = Ω−1 j D + εO2 (1) ,
(10.6)
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where the error terms are functions. By Schwarz’ inequality we obtain the following pointwise bound |pA ψ|2 ≥ 18 |Πaj ψ|2 − c(K)ε2 |ψ|2 since
1 2
and |Dψ|2 ≥ 18 |Dja ψ|2 − c(K)ε2 |ψ|2
(10.7)
≤ Ωj ≤ 2 if ε is sufficiently small. Therefore
and
vj p2A vj ≥ 18 vj [Πaj ]∗ · Πaj vj − c(K)ε2 vj2
(10.8)
Dvj4 D ≥ 18 [Dja ]∗ vj4 Dja − c(K)ε2 vj4 ,
(10.9)
where star denotes the adjoint in the standard L2 -space. By applying the inequalities (10.8) and (10.9) we have
(j) 1 vj p2A vj − vj4 b(2−7 − η 2 b[ξ⊥ ]2 )Pj↑ + µ2 ε−5 vj − W vj6 Tr Dvj4 D + ε−2 2b − 1 ≤ 18 Tr [Dja ]∗ vj4 Dja + ε−2 2b vj [Πaj ]∗ ·
(j) 1 Πaj vj − vj4 b( 16 − 8η 2 b[ξ⊥ ]2 )Pj↑ + µ4 ε−5 vj − 8W vj6 . (10.10) −
The error terms in (10.8) and (10.9) have been absorbed into the µ2 ε−5 vj term using vj ≤ 1 if ε is sufficiently small. The right-hand side of (10.10) is invariant under an SU (2) × U (1) gauge transformation UR,φ as defined in (9.11). We shall choose R to be the rotation −1 −1 from {Ω−1 j ∂1 , Ωj ∂2 , Ωj ∂3 } to the basis {e1 , e2 , e3 } constructed in Lemma 8.2 and φ to be such that α = a + dφ, where α is constructed in Section 9.3. In particular P ↑ becomes σ ↑ := 12 [1 + σ 3 ] according to (9.12) since n = e3 along the central field line. Therefore the right-hand side of (10.10) continues as 1 v[Πα ]∗ (10.10) = 18 Tr [Dα ]∗ v 4 Dα + ε−2 2b
2 1 · Πα v − v 4 b( 16 − 8η 2 bξ⊥ )σ ↑ + µ4 ε−5 v − 8W v 6 , (10.11) −
where we also omitted the j index for brevity, i.e., Dα = Djα , Πα = Πα j , v = vj (j)
and ξ⊥ = ξ⊥ for the rest of this section. Now we translate our problem from the standard L2 (dx) space to the L2ξ := L2 (dξ) space. We change the measure from the volume form dx to dξ = dξ1 ∧ dξ2 ∧ dξ3 . Since these two volume forms are comparable by a factor of at most 4 by (8.13) of Lemma 8.2 if ε is sufficiently small, we can use Lemma A.8 from the appendix to obtain 1 v[Πα ]∗ (10.11) ≤ 18 TrL2ξ [Dα ]∗ v 4 Dα + ε−2 2b
2 · Πα v − v 4 b( 14 − 8η 2 bξ⊥ )σ ↑ + µ4 ε−5 v − 32W v 6 −
2
where the trace and the adjoints are computed in the L (dξ) space.
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We remark that already on the right-hand side of (10.10) it could have been natural to transform the trace on L2 (dx) to the trace on L2 (Ω3j dx), according to Lemma A.8. We introduce the function √ (10.12) G = G(ξ) := 1 + b min{|ξ⊥ |, 1} , and we notice that sup G(ξ)p v(ξ)q ≤ c(p, q)η −p/2 ,
p ≥ 0, q > 0 .
(10.13)
ξ
We recall the definition of Db and the decomposition (9.25) from Section 9.3. Since = 1, the estimates (9.26) are translated into √ M = O(G2 ), ∇M = bO(G), K1,2 = O1 (|ξ⊥ |γ ), K3 = O1 (|ξ⊥ |) . (10.14) for any γ ≥ 0. Here O(Gk ) denotes the class of functions F (ξ) on R3 with supξ |F (ξ)|/Gk (ξ) < ∞. Hence
v 4 |Dα ψ|dξ ≥
1 2
2 dξ − c |v 2 Dψ|
3
|v 2 Kk ∂ξk ψ|2 dξ −
v 4 O(G4 )|ψ|2 dξ .
k=1
(10.15) In the second term on the right-hand side we first commute v through the derivative. Notice that [∂ξk , v] = ηbvO(|ξ⊥ |) for k = 1, 2 and [∂ξ3 , v] = 0. Then we use the estimates (10.14) and (10.13) to obtain 3 |v 2 Kk ∂ξk ψ|2 dξ k=1
≤
2
3
v 2 Kk 2 |∂ξk vψ|2 dξ + 2
k=1
≤
cb−1
3
|∂ξk vψ|2 dξ + c
2
v 2 Kk 2 [ηbO(|ξ⊥ |)]2 |vψ|2 dξ
k=1
v 2 |ψ|2 dξ
(10.16)
k=1
if ε is sufficiently small. From the last term in (10.15) we obtain a similar error term as in (10.16) using (10.13), hence 3 4 α 2 2 −1 2 1 v |D ψ|dξ ≥ 2 |v Dψ| dξ − cb |∂ξk vψ| dξ − c v 2 |ψ|2 dξ . (10.17) k=1
We also define for k = 1, 2, 3 Πη,k := −i∂ξk + (1 + 2η)αc (∂ξk )
and
Dη := σ · Πη =
3 k=1
σ k Πη,k
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which is a Dirac operator with constant field (1 + 2η)b dξ1 ∧ dξ2 in the dξ 2 metric. Notice that + 2iηbv 2 (σ 1 ξ1 + σ 2 ξ2 )σ ↑ . Dη v 2 = v 2 D (10.18) This identity, called the magnetic localization formula, was introduced in [ES-II].
Hence, using (10.18) to continue (10.17), we obtain 2 v 4 |Dα ψ|dξ ≥ 14 |Dη v 2 ψ|2 dξ − 8η 2 b2 v 4 ξ⊥ |σ ↑ ψ|2 dξ −cb
−1
3 k=1
≥
1 8
|∂ξk vψ|
2
|Dη v ψ|2 dξ + 2
−cb−1
3
b 4
(10.19)
dξ − c
v 2 |ψ|2 dξ
|σ ↑ v 2 ψ|2 dξ − 8η 2 b2
2 v 4 ξ⊥ |σ ↑ ψ|2 dξ
|∂ξk vψ|2 dξ − c
v 2 |ψ|2 dξ .
k=1
In the last step we used that Dη2 ≥ 2b(1 + 2η)σ ↑ ≥ 2bσ ↑ , i.e., that on the spin-up subspace {ψ : σ ↑ ψ = ψ} the constant field Pauli operator is bounded from below by twice of the constant field. We shall control the second negative error term on the right-hand side of (10.19) by the term ε−2 (2b)−1 v[Πα ]∗ · Πα v. Notice that the following inequality holds pointwise
1 |∂ek ψ|2 − 4α(ξ)2 |ψ|2 − 4(sup ω)2 |ψ|2 2 3
|Πα ψ|2 ≥
k=1
using (9.2), (9.4) and (9.6). We can estimate α(ξ) ≤ cb1/2 G(ξ) from (9.24) and the explicit choice of αc . We also estimate ω ≤ cKε by Lemma 8.2 and the same lemma is used to estimate the transition from j |∂ej ψ|2 to j |∂ξj ψ|2 . Therefore
1 |∂ξk ψ|2 − cbG2 |ψ|2 , 4 3
|Πα ψ|2 ≥
k=1
if ε is sufficiently small, hence 3
1 1 α 2 2 −1 −2 |Π vψ| dξ ≥ 2 |∂ξk vψ| dξ − cη ε v 2 |ψ|2 dξ 2ε2 b 8ε b
(10.20)
k=1
using (10.13). Combining (10.19) and (10.20) we obtain 2 1 v[Πα ]∗ · Πα v − v 4 b( 14 − 8η 2 bξ⊥ )σ ↑ + νε−5 v 2 − 32W v 6 [Dα ]∗ v 4 Dα + ε−2 2b
≥ 18 v 2 Dη2 v 2 − 32W v 6
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if ε is sufficiently small depending on ν, η and K. Since Tr(X ∗ HX)− ≤ X ∗ XTrH− ,
(10.21) Tr 18 v 2 Dη2 v 2 − 32W v 6 ≤ 18 Tr Dη2 − 256W v 2 − −
≤ c b(1 + 2η)(256W v 2 )3/2 + (256W v 2 )5/2 dξ , where in the last step we used the Lieb-Thirring inequality for a constant magnetic field [LSY-II]. This completes the proof of Proposition 10.1. Remark. The reader may have found it confusing that along the proof of the positive energy regime we used the Birman-Schwinger principle (4.6) back and forth several times. It occurred first in (4.6), (4.9), then in (6.7), (6.10), and finally, implicitly, in (10.21), when we referred to the Lieb-Thirring inequality with a constant magnetic field whose proof also relies on the Birman-Schwinger principle. The frequent changes back to an expression on the sum of the negative eigenvalues were purely for the purpose of compact presentation of the intermediate results. It would have been possible to use only (4.6) and stay with the resolvent language all the time since all estimates done for the operators are equally valid for the resolvents. In this case, of course, we could not have referred directly to the result of [LSY-II] on the constant field case, rather to the details of that proof.
11 Zero mode regime: Proof of Proposition 7.2 First notice that the inequalities in Proposition 7.2 are scale invariant in powers of ; both sides of (7.5) scale like and both sides of (7.6) scale like −1 . Therefore we can set = 1 for the proof. The arguments for weak magnetic fields and for extended (D, K)-regular fields are different.
11.1
Weak magnetic field
For weak fields (7.5) and (7.6) will be estimated by a universal constant c if ε ≤ ε(K). We need the following lemma: Lemma 11.1. Let X, Y be self-adjoint operators such that X ≥ 0, X + Y ≥ 0 and Y ≤ M for some constant M > 0. Then 1 4 4 ≤ ≤ 2 . 2 2 (X + Y + 2M ) (X + M ) X + M2 Proof. Consider the resolvent expansion 1 1 1 1 = − (Y + M ) X + Y + 2M X +M X +M X + Y + 2M
(11.1)
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hence by Schwarz’ inequality 1 (X + Y + 2M )2 1 2 1 1 (Y + M ) ≤ +2 (Y + M ) (X + M )2 X+M (X + Y + 2M )2 X +M 1 2 1 (Y + M )(Y + M ) + 2(2M )−2 ≤ (X + M )2 X +M X +M 4 ≤ (X + M )2 since (Y + M )2 ≤ 2Y 2 + 2M 2 ≤ 4M 2 . By positivity of X we have (X + M )2 ≥ X 2 + M 2 which completes the proof. If B∞ ≤ cε−2 then using Lemma 11.1 we obtain −2 −2 R[P ]2 (u, u) = (−i∇+ A)2 + σ ·B+ ε−5 (u, u) ≤ 4 (−i∇+ A)2 + 12 ε−5 (u, u) if ε is sufficiently small. By the diamagnetic inequality we can continue this estimate as −2 R[P ]2 (u, u) ≤ 4 − ∆ + 12 ε−5 (u, u) ≤ cε5 . This proves (7.5). For the proof of (7.6) we define a smooth function 0 ≤ χ ≤ 1 such that χ(u) = 1, |∇χ| ≤ c, |∇2 χ| ≤ c and supp(χ) ∩ supp(ϕ) = ∅. For brevity we set R := R[P ] with P = ε−5 . Since the inequality (7.6) is gauge invariant, we can choose centered at z0 to generate the magnetic field. In particular, the Poincar´e gauge A −2 and B∞ ≤ cε−2 . A∞ ≤ cε since by assumption B is supported on D Let {X, Y } := XY + Y X denote the anticommutator. Notice that [R, χ] = R{D, [D, χ]}R and σ · ∇χ)} . {D, [D, χ]} = (−i){σ · (−i∇), σ · ∇χ} + (−i){σ · A, We can compactly write {D, [D, χ]} =
3
j + M jK , (−i∇ + A)
j=1
j and M are 2 by 2 matrix-valued functions and from the estimate on where the K j (x), M(x) A and the derivatives of χ we easily obtain that supx K ≤ cε−2 .
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Therefore commuting χ through first, estimating ϕ2 ≤ 1, then using RD2 R ≤ R and applying a Schwarz’ inequality we get χRDϕ2 DRχ
= R{D, [D, χ]}∗ RDϕ2 DR{D, [D, χ]}R
≤ R{D, [D, χ]}∗ R{D, [D, χ]}R 3 j R . (11.2) ∗ RMR +4 ∗ (−i∇ + A) j R(−i∇ + A) jK ≤ 4RM RK j j=1
In the first term we use R ≤ P −1 = ε5 for the middle resolvent then we use the to arrive at the resolvent square, R2 , that was estimated above boundedness of M in the proof of (7.5). 2+σ·B+P ≥ In the second term we use that D2 + P = (−i∇ + A) P P 1 2 −2 −5 + since B ≤ cε ≤ = ε if ε is sufficiently small, therefore (−i∇ + A) 2 2 2 j R(−i∇ + A) j ≤ 1. The estimate of the second term then can be (−i∇ + A) −4 ∗K and referring to the estimate of the square completed by using K j j ≤ cε of the resolvent in (7.5). This finishes the proof of Proposition 7.2 for the case of weak magnetic field (case (i)).
11.2
Strong magnetic field
Here we prove Proposition 7.2 for the case (ii). Throughout the proof we fix u and let z = π(u) be its base point on the supporting plane P. Consider the construction of Lemma 8.2, in particular the coordinate functions ξ = (ξ1 , ξ2 , ξ3 ) and the conformal factor Ω. We know that 12 ≤ Ω ≤ 2 and ∇Ω∞ ≤ 1 if ε is sufficiently small. Recall that we set = 1, therefore the bounds on the right-hand side of (7.5) and (7.6) saturate to c(ε, K)b since P = ε−5 and |B(u)| is comparable with b := |B∞ | ≥ 1. 11.2.1 Transformation into good coordinates Similarly to the positive energy regime in Section 10, we perform three transition a steps. We first change D into DΩ := Ω−2 DΩ and the underlying measure to dx to Ω3 dx, then we change the measure from Ω3 dx to dξ and finally we perform a a gauge transformation. Recall that DΩ is self-adjoint on L2 (ds2Ω ) (see [ES-III]). a a 2 We set RΩ [P ] := ([DΩ ] + P )−1 and we assume that ε is sufficiently small so that P ≥ 29 . Then Lemma A.9 from Appendix A.6 states that a tr R[P ]2 (u, u) ≤ 29 tr (RΩ [P ])2L2 (Ω) (u, u) (11.3)
tr R[P ]Dϕ2 DR[P ] (u, u)
a a 2 a a a ≤ 212 tr RΩ [P ]DΩ ϕ DΩ RΩ [P ] 2 (u, u) + 212 P tr (RΩ [P ])2L2 (Ω) (u, u) (11.4) L (Ω)
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where the operator kernels on the right-hand side are computed in the L2 (Ω) := L2 (Ω3 dx) ⊗ C2 space. Gauge transformation of the form (9.11) leaves the diagonal elements of operator kernels invariant hence we can use the basis {e1 , e2 , e3 } constructed in Lemma 8.2 and vector potential α constructed in Section 9.3 to express the right-hand sides α a of (11.3), (11.4) using DΩ instead of DΩ as in the proof of Proposition 10.1. We recall that in this gauge the decomposition (9.25) holds, i.e., α + DΩ =D
3
Kk ∂ξk + M
(11.5)
k=1
given in (9.22) and Kk , M satisfy the estimates (10.14) with = 1. with D We apply Lemma A.7 from Appendix A.5 to compare operator kernels on the measure spaces with volume forms dµ := Ω3 dx and dν := dξ = dξ1 ∧ dξ2 ∧ dξ3 . Since these two volume forms are comparable at every point by Lemma 8.2, we obtain from (A.33) that α [P ])2L2 (Ω) (u, u) ≤ tr (RΩ
α α 2 α α tr RΩ [P ]DΩ ϕ DΩ RΩ [P ] (u, u) ≤
L2 (Ω)
α α c tr (RΩ [P ]∗ RΩ [P ])L2 (dξ) (u, u) ∗ α α c tr ϕDΩ RΩ [P ]
α α ϕDΩ RΩ [P ] 2 (u, u) , L (dξ)
where the adjoints and the operator kernels on the right-hand sides are computed in the L2 (dξ) ⊗ C2 space. Therefore case (ii) of Proposition 7.2 has been reduced to proving that for ε ≤ ε(K) α α [P ]∗ RΩ [P ] (u, u) ≤ cb , (11.6) tr RΩ α α α α tr (ϕDΩ RΩ [P ])∗ ϕDΩ RΩ [P ] (u, u) ≤ cb , (11.7) where the adjoints and the operator kernels are computed on L2 (dξ) ⊗ C2 . α is. Remark. The operator DΩ in general is not self-adjoint in L2 (dξ) ⊗ C2 , but D
11.2.2 Proof of (11.6) and (11.7) a In this proof we will omit Ω and α from the notation of DΩ and we will simply use D for this operator. This should not be confused with the notation D (see (1.2) with h = 1) used elsewhere in the paper. denotes the Dirac operator with a constant field (see (9.22)). We recall that D = (Π 1, Π 2, Π 3 ) from D = σ · Π, We also need the notations Π = (Π1 , Π2 , Π3 ), Π D = σ · Π. We note that D can be decomposed as
=D ⊥ + N D
(11.8)
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2
j σj Π
and
713
3 = σ 3 (−i∂ξ3 ) . N := σ 3 Π
(11.9)
j=1
These operators are self-adjoint on L2 (dξ 2 ) and ⊥ , Π 3 ] = 0, [D
⊥ , N } = 0, {D
2 2 2 = D ⊥ ⊥ 23 . (11.10) D +N2 = D +Π
and
as in (9.25) We need two decompositions of the error term E := D − D E =Π·K+M
(11.11)
·K +M . E=Π
3
(11.12)
Here K = (K1 , K2 , K3 ) is a vector of 2 by 2 matrices and Π · K := j=1 Πj Kj . The matrices Kj , M satisfy (10.14) with = 1 and the same estimates hold for j , M as well. The following estimates are straightforward from (10.14) and (9.21) K if ε ≤ ε(K) [Πj , Kk ] = O(1),
[Πj , M] = b1/2 O(G),
j, k = 1, 2, 3 ,
(11.13)
and M. In particular the following relations also hold: and the same holds for K E = K · Π + M0 ,
+M ·Π 0 E=K
(11.14)
0 = O(G2 ). with M0 , M The following lemma collects various operator inequalities related to the diamagnetic inequality. The proof is postponed to Section A.7. Lemma 11.2. With the notations above we have in the space L2 (dξ) ⊗ C2 if ε ≤ ε(K): 1 ∗ 1
Πj ≤ Π∗j D2 + P D2 + P D ∗ D
Π∗j Πj ≤ D2 + P D2 + P 1 ∗ 1 Πj Πk ≤ Π∗k Π∗j D2 + P D2 + P 1 ∗ 1 E ≤ E∗ D2 + P D2 + P D ∗ D E∗ E ≤ D2 + P D2 + P
the following operator inequalities
cb ,
(11.15)
cb ,
j = 1, 2, 3
(11.16)
cb2 ,
j, k = 1, 2, 3
(11.17)
O(G4 )
(11.18)
O(G4 ) .
(11.19)
j = 1, 2, 3.
(11.20)
For the constant field operator we have j Π
1 ≤ cb , Π 2 + P j D
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The next lemma estimates the diagonal elements of explicitly computable operators with a constant magnetic field. The proof√is given in Section A.8. We set H(ξ) := min{|ξ⊥ |, 1} and we recall that G = 1 + bH. Lemma 11.3. With the notations above and for any constants P ≥ 1, b ≥ 1 with P ≤ cb we have 1 tr (u, u) ≤ cb . (11.21) 2 (D + P )2 For any k = 1, 2 . . ., m = 1, 2, . . . and for any 2 by 2 matrix-valued function F with F (x) = O(G(x)) we also have 1 1 G2m (u, u) ≤ c(m)b , 2 2 D +P D +P D D 1 tr [F k ]∗ Fk (u, u) ≤ c(k)b , 2 2 2 D +P D +P D +P D D tr H 2m (u, u) ≤ c(m)b1−m , 2 + P 2 + P D D j j Π Π tr H 2m (u, u) ≤ c(m)b2−m , j = 1, 2, 3 . 2 + P 2 + P D D
tr
(11.22) (11.23) (11.24) (11.25)
denote either the identity I, or D, or Π 3 , then Let W tr
D
2 + P D
[F k ]∗
W
2 + P D
G2m
W
2 + P D
Fk
D
2 + P D
(u, u) ≤ c(k, m)b
(11.26)
= 0). Let U denote either the identity I or D or Π j, j = 3 , D] (we recall that [Π 1, 2, 3, and let 0 ≤ ϕ ≤ 1 be a function with dist(u, supp(ϕ)) ≥ 1, then tr
tr
D
2 + P D
[F k ]∗
U
2 + P D
U
2 + P D
ϕ2
U
2 + P D
Gm ϕ2 Gm
U
(u, u) ≤ ce−c
2 + P D
Fk
D
√ b
2 + P D
,
(11.27)
(u, u) ≤ c(k, m)e−c
√ b
. (11.28)
Armed with these lemmas, we complete the proof of (11.6) and (11.7). We start with (11.6). We introduce the notation (· · · )∗ A for A∗ A if A is a long expression. All adjoints are computed in the L2ξ -space. 2 + DE + E D in the following resolvent expansion: We use D2 = D ∗
1 ··· ≤ 3 (A) + (B) + (C) D2 + P
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with (A)
:=
···
(C)
1
2 + P D 1 1 := ··· DE 2 D2 + P D +P
∗ 1 1 ED := ··· . 2 D2 + P D +P
(B)
∗
∗
Term (A) is explicit from (11.21). In term (B) we first use (11.19) in the middle to arrive at (11.22) with m = 2. ·K +M and we expand the resolvent in the middle In term (C) we use E = Π once more. The result is
(C) ≤ 4 (C1) + (C2) + (C3) + (C4) with (C1) :=
···
(C3) := (C4) :=
1
M
D
2 + P 2 + P D D 1 1 D (DE + E D) ··· M 2 2 + P 2 + P D +P D D ∗ 1 ·K D ··· Π 2 2 + P D +P D ∗ 1 1 ·K D (DE + E D) ··· Π . 2 2 + P 2 + P D +P D D
(C2) :=
∗
∗
Term (C1) is explicit from (11.23) after estimating one of the resolvents in the middle by P −1 ≤ 1. Term (C2) is split into two terms,
(C2) ≤ 2 (C21) + (C22) , with
∗
1 1 D M DE 2 + P 2 + P D2 + P D D ∗ D 1 D M E (C22) := ··· . 2 2 + P 2 + P D +P D D
(C21) :=
···
= I, m = k = 2. In (C22) we In (C21) we use (11.19) and finally (11.26) with W = D, m = k = 2. In the term (C3) we use use (11.18) first then (11.26) with W (11.20) then (11.24) with m = 1 together with the estimates (10.14) used for M.
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Finally, for the term (C4) we estimate
(C4) ≤ 2 (C41) + (C42) (C41) := (C42) :=
···
···
∗ D2
∗ D2
1 1 ·K D Π DE 2 2 +P D +P D +P
1 1 ·K D Π ED . 2 + P 2 + P +P D D
In (C41) we first use (11.19) to arrive at ∗ (C41) ≤ c · · · G2
1 2 + P D
·K Π
D
2 + P D
.
(11.29)
For the term (C42) we first observe the following inequality Lemma 11.4. With the notations above ∗ 1 ≤ bO(G8 ) . ED ··· 2 D +P
(11.30)
= Π·N + M, with N , M = O(G2 ). Proof of Lemma 11.4. We can write D Therefore = (Π·K+M)(Π· N + M) = ED
3
Πj Πk b−1/2 O(G4 )+
3
Πj O(G4 )+b1/2 O(G4 )
j=1
j,k=1
after commuting Π through using (11.13) and (10.14). Therefore (11.30) follows from (11.17) and (11.15). Armed with Lemma 11.4, we see that ∗ (C42) ≤ (C43) := cb · · · G4
1 ·K D Π . 2 2 D +P D +P
(11.31)
Since G2 ≤ G4 ≤ G8 and b ≥ 1, it is sufficient to estimate (C43) that will complete the estimate of (C41) from (11.29) as well. ·K into terms containing To estimate (C43), we first separate the term Π Π3 K3 and Π⊥ K⊥ := Π1 K1 + Π2 K2 by Schwarz’ inequality. We arrive at ∗ (C43) ≤ c · · · G4
1 2 + P D
3 (b1/2 K 3) Π
D
2 + P D ∗ + cb · · · G4
1 2 + P D
⊥K ⊥ Π
D
2 + P D
.
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=Π 3 , m = 2, k = 1, using that The first term is explicit from (11.26) with W 1/2 b K3 = O(G). In the second term we estimate G8 ≤ b4 in the middle, then estimate one of the resolvents by P −1 ≤ 1 and first use (11.20) and finally (11.24) This completes the proof of (11.6). with m = 5, together with (10.14) for K. Now we prove (11.7). We need the following lemma. Lemma 11.5. With the notations above, we have ∗ 1 tr · · · ϕE (u, u) ≤ 2 + P D ∗ D 1 tr · · · ϕE M (u, u) ≤ 2 + P 2 + P D D
c e−c c e−c
√ b √ b
,
(11.32)
.
(11.33)
+ M, ·Π then Proof. For the proof of both inequalities (11.32) we write E = K we separate the terms by a Schwarz’ inequality and we use (11.27) and (11.28), respectively, with appropriately chosen U. For the operator on the left-hand side of (11.7) we use a resolvent expansion and a Schwarz’ inequality to obtain
∗ D ≤ 3 (D) + (E) + (F ) ··· ϕ 2 D +P with
(D) (E) (F )
∗
1 2 + P D ∗ 1 D DE := ··· ϕ 2 2 D +P D +P ∗ D D E := ··· ϕ 2 . 2 + P D +P D :=
···
ϕD
+ E applying a Schwarz’ inequality and The estimate of (D) is trivial by D = D using (11.27) and (11.32) for these two terms, respectively. In term (E) we use 1 D D =I −P 2 D2 + P D +P
(11.34)
and separate it by a Schwarz’ inequality: ∗ ∗ 1 1 1 (E) ≤ 2 · · · ϕE E + 2P 2 · · · ϕ 2 . 2 2 D +P D +P D +P For the first term we can use (11.32), for the second one we use ϕ ≤ 1, (11.18) then (11.22) with m = 2.
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Finally for the term (F) we write
(F ) ≤ 2 (F 1) + (F 2) with
(F 1) :=
···
(F 2) :=
∗
···
ϕ
∗
D2
ϕ
D D Π·K 2 + P +P D
D2
D D M . 2 + P +P D
For (F1) we first estimate ϕ ≤ 1, then use (11.16) and (11.24) with m = 1 together with (10.14). For (F2) we need one more resolvent expansion:
(F 2) ≤ 2 (F 21) + (F 22) with (F 21) := (F 22) :=
∗
D 1 M 2 2 D +P D +P ∗ D 1 D (DE + E D) ··· ϕ 2 M . 2 + P 2 + P D +P D D ···
ϕD
+ E: We split (F21) further by using D = D ∗ (F 21) ≤ 2 · · · ϕ
D
2 + P D
M
D
2 + P D
∗ + 2 · · · ϕE
D 1 M . 2 2 D +P D +P
The first term is explicit by (11.23) after ϕ ≤ 1 and estimating D2 by the resolvent. The second term was estimated in (11.33). Finally, to estimate (F22), we use again (11.34) and we split it as follows (F 22) ≤
∗ 3 · · · ϕE
1
M
D
2 + P 2 + P D D
∗ D 1 1 E +3P 2 · · · ϕ 2 M 2 + P 2 + P D +P D D ∗ D D D E +3 · · · ϕ 2 M . 2 + P 2 + P D +P D D
The first term is estimated in (11.33). For the second term we use ϕ ≤ 1 and = 1, m = k = 2. For the third term we use (11.18) first, then (11.26) with W = D, m = k = 2. ϕ ≤ 1 and (11.19) first, then (11.26) with W This completes the proof of (11.7).
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Proof of the technical lemmas
A.1 Proof of Proposition 4.2 on the tempered lengthscale Proof. We recall the definitions of BL (x), bL (x) from (2.1), (2.2) and we notice that BL (x) is increasing in L, while bL (x) is decreasing. Since B and its derivatives are locally bounded and B is not constant everywhere, we easily obtain that the sets appearing on the right-hand sides of (2.3) and (2.4) are non-empty and bounded. Therefore Lm and Lc are positive finite valued functions. We notice that if BL(x) (x) > L−2 (x) then Lm (x) < L(x) < Lc (x), i.e., Lc (x) = Lv (x). We claim that Lc (x) = Lv (x) implies that (4.2) and (4.3) hold even if L(x) is replaced with Lc (x) which is a stronger statement. The validity of this stronger form of (4.2) follows directly from (2.4). This also implies that BLc (x) (x) − bLc (x) (x) ≤ 2Lc (x) · sup{|∇B(y)| : |x − y| ≤ Lc (x)} ≤ 2bLc (x) (x) , therefore bLc (x) (x) ≥ 13 BLc (x) (x), in particular B(y) = 0 and n(y) is well defined for all y with |y − x| ≤ Lc (x). Thus (4.3) with L(x) replaced with Lc (x) follows from (2.4). We also proved that if Lc (x) = Lv (x) then the suprema in (2.3) and (2.4) are actually maxima by the continuity of B. Finally, we have to show that L(x) is tempered. Notice that it is sufficient to show that L(y) 1 (A.1) |x − y| ≤ L(x) =⇒ ≤ 2 L(x) for any x, y ∈ R3 because the inequality L(y)/L(x) ≤ 2 easily follows from this. To see it, we assume that L(y) > 2L(x). Then |x − y| ≤ L(x) implies |x − y| ≤ L(y), so using (A.1) with x, y interchanged we arrive at a contradiction. Now we show that (A.1) holds. Let x, y be two points with |x − y| ≤ L(x) = 1 L (x) and we have to show that L(x) ≤ 2L(y) = Lc (y). This is obvious if c 2 BL(x) (y) ≤ L(x)−2 , since then L(x) ≤ Lm (y) and Lm (y) ≤ Lc (y) by definition. Thus we can assume that BL(x) (y) > L(x)−2 . Since |x − y| ≤ L(x), we know that
z : |y − z| ≤ L(x) ⊂ z : |x − z| ≤ 2L(x) = Lc (x) , (A.2) thus BL(x) (y) ≤ BLc (x) (x), hence BLc (x) (x) > L(x)−2 > Lc (x)−2 , i.e., Lc (x) > Lm (x), so Lc (x) = Lv (x). We will now check that for γ = 1, . . . , 4
(A.3) L(x)γ sup ∇γ |B(z)| : |y − z| ≤ L(x) ≤ bL(x) (y) and
L(x)γ sup ∇γ n(z) : |y − z| ≤ L(x) ≤ 1
(A.4)
hold, which will imply L(x) ≤ Lv (y), hence L(x) ≤ Lc (y). But as we showed above, Lc (x) = Lv (x) implies that (4.2) and (4.3) hold with L(x) replaced by Lc (x). From (A.2) and L(x) < Lc (x) we therefore immediately conclude (A.3), (A.4).
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A.2 Proof of the covering Lemma 5.4
Introduce the notation Dx∗ := B x, 40(x) and Di∗ := Dx∗i . Let S be any compact subset of R3 . First we show how to find a finite set of points within S so that i cover S and they enjoy the finite overlapping property. Let Dx := the balls D B(x, (x)/20) and we cover S by the collection of balls Dx , x ∈ S. By compactness, we can choose points {xα } ⊂ S, with a finite index set α ∈ A, such that the balls {Dα }α∈A cover S. Now we discard certain points from the collection {xα } and relabel the rest by {xi }. Let x1 be the point with the biggest value (x1 ) among all values {(xα ) : α ∈ A}. Then let x2 be the point with the biggest value (x2 ) among all values (xα ) such that xα ∈ R3 \ D1 . Then let x3 be the point with biggest value (x3 ) among all values (xα ) such that xα ∈ R3 \(D1 ∪D2 ), etc. until all xα ’s are covered by D i ’s. This selects a subcollection of the points {xα } and they are relabelled to x1 , x2 , . . .. i ’s cover S. Consider any y ∈ S, then y ∈ Dα We claim that the collection of D for some α. But xα is covered by some Di . We choose the smallest such index i. By the maximality of the radii in the selection procedure, we know that (xα ) ≤ (xi ), i. so |y − xi | ≤ |y − xα | + |xα − xi | ≤ ((xα ) + (xi ))/20 ≤ (xi )/10, hence y ∈ D We claim that the union of the Di∗ balls have the finite covering property with a sufficiently big universal N . From construction, the balls Di# := B(xi , (xi )/40) are disjoint. Fix a point y ∈ R3 and let I be the set of indices i such that y ∈ Di∗ , i ∈ I. Choosing ε < 1/40, we see that y ∈ Di∗ , i.e., |xi − y| ≤ 40(xi ) implies 1/2 ≤ (y)/(xi ) ≤ 2 for all i ∈ I. Hence the balls Di# , i ∈ I, all have radius at least (y)/80 and they are within a ball of radius 81(y) about y. From their disjointness it follows that their number is universally bounded, i.e., the number of Di∗ ’s covering any y is bounded by a universal number N . This completes the construction of the covering balls for any compact set S satisfying (i) with a universal covering property. We denote by P (S) the points {x1 , x2 , . . .} obtained in this procedure and note that P (S) ⊂ S. Let i H(S) := D and H ∗ (S) := Di∗ . i∈P (S)
i∈P (S)
Now we show how to choose points in the whole space. Fix an arbitrary point x, and let Ak := {y : 4k (x) ≤ |y − x| ≤ 4k+1 (x)}, k = 1, 2, . . ., be a sequence 3 x ∪ of annuli. Clearly D k Ak = R . For each annulus we construct the points P (Ak ) defined above and we let x) ∪ P := P (D
∞
P (Ak ).
k=1
This will be our final set of points {x1 , x2 , . . .} after relabeling. It is clear that the i = B(xi , (xi )/10), xi ∈ P , cover the space. balls D
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xi : xi ∈ P }, i.e., Next we prove the finite covering property for the balls {D 3 that the number of balls that cover any given point of R is universally bounded. We need a lemma whose proof is given later. Lemma A.1. Fix any point x ∈ R3 . (i) Let Lk := sup{(u) : u ∈ Ak }, k ≥ 1, then Lk ≤ 4k+1 (x). (ii) If y ∈ Ak , then y ∈ H ∗ (Am ) for any |m − k| ≥ 5. with Recall that for each m every point in R3 is covered by at most N balls D center z ∈ P (Am ) and similarly for balls with center in P (Dx ). Hence (ii) of Lemma A.1 shows that any y is covered by at most 12N balls with center from P . This xi : xi ∈ P }. completes the proof of the finite covering property of the balls {D j = ∅, then i , j i ∩ D Finally we show property (ii) of the Definition 5.3. If D ∗ are comparable by (4.1). Therefore Di covers xj , but any point is covered only by j can be intersected by finitely many D i ’s. finitely many Di∗ ’s, hence D Proof of Lemma A.1. (i) Suppose that there exists u ∈ Ak with (u) > 4k+1 (x). Then |x − u| ≤ 4k+1 (x) < (u), hence (u) ≤ 2(x) by (4.1) which is a contradiction. (ii) Suppose that there is a point z ∈ P (Am ) such that y ∈ Dz∗ , i.e., |y − z| ≤ 40(z). Using (4.1) this implies (z) ≤ 2(y) assuming ε < 1/40. Hence |y − z| ≤ 80(y) ≤ 80 · 4k+1 (x) by (i). But z ∈ Am , so |y − z| ≥ (4m − 4k+1 )(x) which is a contradiction if m ≥ k + 5. Suppose now that m ≤ k − 5. Then |y − z| ≥ (4k − 4m+1 )(x) which contradicts to |y − z| ≤ 40(z) ≤ 40 · 4m+1(x).
A.3 Proof of the localization Proposition 7.1 As a preparation for the proof we define a distance function on the collection {xi }i∈I obtained in Lemma 5.4. Let i ∈ I be a fixed index. We define the following compact sets successively S0 (i) := Di , j , D Sk+1 (i) := j : Dj ∩Sk (i) =∅
and we denote mk := card{j ∈ I : Dj ∩ Sk (i) = ∅} . Lemma A.2. (i) The sets S0 (i), S1 (i), . . . are increasing. (ii) Let u ∈ Sk (i), v ∈ int(Sk+1 (i)), then |u − v| > 4(u). (iii) k Sk (i) = R3 . (iv) mk ≤ N k+1 with the universal constant N from Lemma 5.4. (v) For sufficiently small ε and for any nonnegative function G such that G(x) and G(y) are comparable whenever |x − y| ≤ ε−1 (x), we have sup G ≤ 2k sup G . Sk (i)
S0 (i)
(A.5)
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Proof. For simplicity, we omit i from the arguments since i is fixed. j , we see that Sk ⊂ Sk+1 . (i) Since the balls {Dj } cover R3 and Dj ⊂ D (ii) Let u ∈ Dj for some j. From |u − xj | ≤ (xj ), it follows that (xj ), (u) j ⊂ Sk+1 , we have |v − xj | ≥ 10(xj ), so |v − u| ≥ 9(xj ) > are comparable. Since D 4(u). (iii) Suppose that S := k Sk is not the whole R3 and select a point z ∈ ∂S. Then we can find a sequence of points zk ∈ ∂Sk converging to z such that |zk − z| monotonically decreasing (for example, we can choose the point zk ∈ Sk closest to z). Since (z) > 0, we see that |z − zn | ≤ (z) for some n, hence (zn ) and (z) are comparable. We have |zn+1 − zn | ≤ |zn+1 − z| + |z − zn | ≤ 2(z) ≤ 4(zn ) which contradicts (ii). (iv) It is clear that m0 ≤ N by Definition 5.3 (ii). By induction we show that j in the mk+1 ≤ N mk . This again follows from Definition 5.3 (ii), since each D definition of Sk+1 may intersect at most N balls from the collection {Dj }j∈I . (v) Straightforward by induction on k and by the definition of Sk (i). This lemma gives an integer-valued distance on the collection {xj }j∈I : dij := min{k : xj ∈ Sk (i)} . Clearly dii = 0, and dij + djk ≥ dik , but the distance function is not symmetric. However, we have Lemma A.3. For sufficiently small ε the distance function satisfies dji ≤ 7dij + 1 .
(A.6)
Proof. The proof goes by induction on the value of dij . If dij = 0, xj ∈ Di , then j , i.e., dji ≤ 1. Suppose that (A.6) is proven for all (i, j) pairs with dij ≤ d xi ∈ D and let now dij = d + 1. Then there exists m such that xm ∈ Sd (i), dim ≤ d and a for some index a with |xa − xm | ≤ a + 10m . If ε is sufficiently small, the xj ∈ D radii a , j and m are comparable, and it easily follows that dja ≤ 3 and dam ≤ 3. Therefore dji ≤ dja + dam + dmi ≤ 6 + 7d + 1 < 7dij + 1. For every i ∈ I, k ∈ N we define (i)
uk :=
θj2 .
(A.7)
j : Dj ∩Sk (i) =∅ (i)
(i)
Notice that supp(uk ) ⊂ Sk+1 (i), and uk ≡ 1 on Sk (i). Moreover |∇uk (x)| ≤ cN (x)−1 . (i)
(A.8)
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To see this, we notice that every x is covered by not more than N balls Dj and only these support those θj ’s which do not vanish at x. Moreover j is comparable to (x) for all these j indices, hence ∇θj ∞ ≤ c(x)−1 . In the rest of this section we set Rf = R(f ) := (T + f )−1 ,
Ri [f ] := (Ti + f )−1
for simplicity, in accordance with the notations (6.1), (6.2). Proof of Proposition 7.1. We start with an auxiliary lemma. Lemma A.4. For any number µ ≥ 0 and real function χ on R3 , 1/2 1/2 RP +µ (P + µ)RP +µ ≤ 1 , 1/2 1/2 RP +µ [T, χ]RP +µ ≤ c0 P −1/2 |∇χ| . ∞
(A.9) (A.10)
Proof of Lemma A.4. The first inequality is trivial by inserting P + µ ≤ T + P + µ. For the second inequality we use [T, χ] = A∗ [A, χ] + [A∗ , χ]A ,
(A.11)
and it is sufficient to estimate one of these terms; 1/2 1/2 ∗ 1/2 1/2 RP +µ A [A, χ]RP +µ ≤ RP +µ A∗ [A, χ]RP +µ . Using M = M M ∗ 1/2 , we obtain that the first factor is bounded by 1 (again, using A∗ A = T ≤ T + P + µ). For the second factor we need pointwise commutator bounds (A.12) [A, χ][A, χ]∗ ≤ c0 |∇χ|2 , [A∗ , χ][A∗ , χ]∗ ≤ c0 |∇χ|2 , that follows from (7.2). Hence, we estimate the second factor as 1/2 1/2 1/2 1/2 [A, χ]RP +µ ≤ c0 RP +µ |∇χ|2 RP +µ
1/2 1/2 1/2 1/2 ≤ c0 RP +µ P RP +µ P −1 |∇χ|2 , ∞
and we use (A.9).
The key lemma is the following (recall the definition of χ i from Section 5.2): Lemma A.5. For sufficiently small ε we have the following estimates for any i, j ∈ I χ i RP +µ θj F 2 θj RP +µ χ i 2 i χ i RP +µ F RP +µ χ
≤ c(4c0 ε)2(dij −1)+ Fi2 (Pi + µ)−1 χ i RP +µ χ i ,(A.13) 2 −1 ≤ cFi (Pi + µ) χ i RP +µ χ i , (A.14)
χ i RP2 +µ χ i
≤ (Pi + µ)−2 ,
χi ARP2 +µ A∗ χi
−1
≤ c(Pi + µ)
(A.15) .
(A.16)
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Proof of Lemma A.5. For brevity, we denote R := RP +µ . First we show (A.13). We assume that ε0 is small enough so that F (xi ) and F (xj ) are comparable as i = ∅ (see (7.1)). j ∩ D long as D We first consider the case dij ≤ 1. Then F 2 ≤ cFj2 ≤ cFi2 on the support of θj . We also use θj2 ≤ c(Pj + µ)−1 (P + µ) ≤ c(Pi + µ)−1 (P + µ) .
(A.17)
Hence (A.13) follows from χi ≤ cFi2 χ i Rθj2 R χi ≤ cFi2 (Pi + µ)−1 χ i R(P + µ)R χi χ i Rθj F 2 θj R and finally we use (A.9) to estimate R(P + µ)R ≤ R. (i)
To prove (A.13) for d = dij ≥ 2, we recall the definition of the functions uk (A.7). For brevity, we omit the superscript i. We successively insert the functions u1 , u2 , . . . , ud−1 where d = dij : χ i Rθj
=
χ i u1 Rθj = χ i R[T, u1 ]Rθj = χ i R[T, u1 ]u2 Rθj
=
...χ i R[T, u1 ]R[T, u2 ]Rθj = χ i R[T, u1 ]R[T, u2]R . . . R[T, ud−1]Rθj .
We used that u1 ≡ 1 on the support of χ i , uk+1 ≡ 1 on the support of ∇uk and supp(ud−1 ) ∩ supp(θj ) = ∅. Therefore we can first estimate θj F 2 θj ≤ cFj θj2 , then use the successive insertions to obtain χi ≤ cFj2 χ i R1/2 χ i Rθj F 2 θj R d−1 d−1
∗ R1/2 [T, uk ]R1/2 R1/2 θj2 R1/2 R1/2 [T, uk ]R1/2 × R1/2 χ i . k=1
k=1
(A.18) First we use that R1/2 θj2 R1/2 ≤ c(Pj + µ)−1 R1/2 (P + µ)R1/2 ≤ c(Pj + µ)−1 by (A.17) and (A.9). Then we use (A.10) to estimate the commutator norms and we use (A.8) to get −1/2 |∇uk | ≤ cε5/2 ≤ ε P ∞
for sufficiently small ε. We obtain χ i Rθj F 2 θj R χi ≤ c(c0 ε)2(dij −1) Fj2 (Pj + µ)−1 χ i R χi .
(A.19)
By (A.5), we see that Fj2 (Pj + µ)−1 ≤ 16d Fi2 (Pi + µ)−1 because part (v) of Lemma A.2 applies both to the function G = F and G = (P + µ)1/2 . This completes the proof of (A.13).
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To prove (A.14) we insert a partition of unity χ i RF 2 R χi = χ i Rθj F 2 θj R χi . j∈I
We use (A.13), (iv) of Lemma A.2 and that
(4c0 ε)2(dij −1)+ ≤ 1 + N +
∞
(4c0 ε)p N p ≤ N + 2
(A.20)
p=1
j∈I
if ε ≤ ε0 , where the universal constant N is from Lemma 5.4. This proves (A.14). The proof of (A.15) is straight-forward by applying (A.14) with F ≡ 1, χ i RP2 +µ χ i ≤ c(Pi + µ)−1 χ i RP +µ χ i ,
(A.21)
and then using RP +µ ≤ (P + µ)−1 which is bounded by c(Pi + µ)−1 on the support of χ i since P and Pi are comparable on this set. For the proof of (A.16) we insert χ i that is identically 1 on the support of χi and use (A.21) χi RP2 +µ χ i A∗ χi ≤ c(Pi + µ)−1 χi A χi RP +µ χ i A∗ χi . χi ARP2 +µ A∗ χi = χi A We can remove χ i and use ARP +µ A∗ ≤ 1 to finish the proof.
The next lemma is a strengthening of (A.13) in Lemma A.5. Notice that in (A.13) we lost a resolvent, and the right-hand side is not locally trace class in the high momentum regime. The following lemma remedies this: Lemma A.6. For sufficiently small ε θi2 RP +µ θj F 2 θj RP +µ θi2 ≤
c(4c0 ε)2(dij −1)+ Fi2 θi2 Ri2 [Pi ] + Pi−1 Ri [Pi ]A∗i ϕ2i Ai Ri [Pi ] θi2 . (A.22)
Proof of Lemma A.6. For simplicity, we let R := RP +µ and Ri := Ri [Pi ] in this proof. We start with the identity
i (Pi − P − µ) + A∗i [A, χ i + Ri χ i ] + [A∗ , χ i ]A R , (A.23) χ i R = Ri χ since A and Ai coincide on the support of χ i by (7.3). After a Schwarz’ inequality χ i Rθj F 2 θj R χi ≤ c Ri χ i θj F 2 θj χ i Ri + Ri χ i (Pi − P − µ)Rθj F 2 θj R(Pi − P − µ) χi Ri +
Ri A∗i [A, χ i ]Rθj F 2 θj R[A, χ i ]∗ Ai Ri
∗
∗
∗
∗
+ Ri [A , χ i ]ARθj F θj RA [A , χ i ] Ri 2
.
(A.24)
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The first term is estimated as i θj F 2 θj χ i Ri ≤ cFi2 Ri2 1(dij ≤ 1) , Ri χ
(A.25)
since F ≤ cFi on the support of θj χ i . i , we can freely insert χi in the last three Since χi ≡ 1 on the support of χ terms of (A.24) replacing Rθj F 2 θj R with χi Rθj F 2 θj Rχi everywhere and apply (A.13) after multiplying it by χi from both sides: χi Rθj F 2 θj Rχi ≤ c(4c0 ε)2(dij −1)+ Fi2 (Pi + µ)−1 χi Rχi . In the second term of (A.24) we use R ≤ (P + µ)−1 and that P + µ and Pi + µ are comparable on the support of χ i . Hence Ri χ i (Pi − P − µ)Rθj F 2 θj R(Pi − P − µ) χi Ri ≤ c(4c0 ε)2(dij −1)+ Fi2 Ri2 , (A.26) using (Pi − P − µ)2 ≤ (Pi + µ)2 on the support of χ i . In the third term of (A.24) we again estimate R ≤ (P + µ)−1 ≤ cPi−1 on i ][A, χ i ]∗ | = |∇ χi |2 ≤ cε2 Pi ϕ2i to the support of χ i , we use (A.12) and that | [A, χ obtain Ri A∗i [A, χ i ]Rθj F 2 θj R[A, χ i ]∗ Ai Ri ≤ c(4c0 ε)2dij Fi2 Pi−1 Ri A∗i ϕ2i Ai Ri .
(A.27)
Finally, the fourth term of (A.24) satisfies Ri [A∗ , χ i ]ARθj F 2 θj RA∗ [A∗ , χ i ]∗ Ri ≤
c(4c0 ε)2(dij −1)+ Fi2 Pi−1 Ri [A∗ , χ i ]Aχi Rχi A∗ [A∗ , χ i ]∗ Ri
≤
c(4c0 ε)2dij Fi2 Ri2 ,
(A.28)
since χi ≡ 1 on the support of χ i , we can omit it, and we used ARA∗ ≤ 1 and i = θi . (A.12). Lemma A.6 follows from (A.24)–(A.28) using that θi χ Finally, we complete the proof of Proposition 7.1. We use R = RP +µ for brevity. We insert three partitions of unity and perform a weighted Schwarz’ inequality θi2 Rθj F 2 θj Rθk2 (A.29) RF 2 R = i,j,k∈I
≤
ε(dkj −1)+ −(dij −1)+ θi2 Rθj F 2 θj Rθi2
i,j,k∈I
+ε(dij −1)+ −(dkj −1)+ θk2 Rθj F 2 θj Rθk2
Using (A.6) we see as in (A.20) that ε(dkj −1)+ ≤ ε(djk −8)+ /7 ≤ N 9 k∈I
k∈I
.
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if ε is small enough. Here N is from Lemma 5.4. Therefore (A.29) implies RF 2 R ≤ cN 9 ε−(dij −1)+ θi2 Rθj F 2 θj Rθi2 . i,j
We use (A.22) and sum up the index j similarly to (A.20) with a possible smaller ε
(16c20 ε)(dij −1)+ Fi2 θi2 Ri2 + Pi−1 Ri A∗i ϕ2i Ai Ri θi2 RF 2 R ≤ cN 8 i,j
≤ cN
10
Fi2 θi2 Ri2 + Pi−1 Ri A∗i ϕ2i Ai Ri θi2 .
i
A.4 Proof of Lemma 8.2 on the magnetic coordinates We give the construction of the new coordinates and conformal metric but we do not follow the explicit bounds along the proof as they easily follow by scaling. We consider z ∈ P fixed in the proof and omit the notation z in the sub- and superscripts. We define the function κ(τ ) := −
1 d log |B(ϕ(τ ))| 2 dτ
(A.30)
where κ(τ ) ∈ C 3 (R). At each point ϕ(τ ) we consider a small spherical cap of the sphere S(τ ), going through ϕ(τ ), orthogonal to the field line ϕ and having curvature |κ(τ )|. The different spheres should curve in a direction determined by the sign of κ(τ ): positive curvature means a sphere with outward normal pointing in the direction of ϕ. ˙ We consider now a small cylindrical tubular neighborhood, := {x : inf τ dist(x, ϕ(τ )) ≤ 10}, along the C 5 -curves ϕ, in which the spherical N caps define a C 3 -foliation. Since B is extended (D, K)-regular, such neighborhood exists if ε ≤ ε(K) is sufficiently small. The foliation naturally extends the function such that it is constant on the leaves and τ ∈ C 3 (R3 ). τ onto N This foliation can be extended to the whole of R3 in the following way. Since B = B∞ outside of D = B(z0 , ), for some z0 ∈ R3 , in case of |z −z0 | ≥ 2 we have κ(τ ) ≡ 0 and ϕ is a straight line, so the foliating spherical caps are parallel flat discs and they can be trivially extended to a foliation of R3 with parallel planes. Now we consider the case |z − z0 | < 2, where we only know κ(τ ) ≡ 0 for |τ | ≥ 3. In this region the spherical caps are again parallel flat discs and they can be extended to parallel planes. That leaves a parallel slab unfoliated between the planes passing through ϕ(−3) and ϕ(3). The width of the slab is (6 ± cKε). We consider the smooth function F : R3 → R of the form F (x) = λ(n∞ · x), where λ : R → R is smooth and is chosen such that F (ϕ(τ )) = τ for |τ | ≥ 3, λ − 1∞ ≤ cKε and λ(γ) ∞ ≤ cKε−γ , γ = 2, 3, 4. The level sets of F define a parallel foliation of R3 which coincides with the previous foliation outside of the slab.
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:= B(z0 , 6), Let χ be a smooth cutoff function supported on the ball D −γ ∇ χ∞ ≤ c , γ = 1, . . . , 4, and χ ≡ 1 on B(z0 , 5). Since |z − z0 | ≤ 2, we note ⊂N , so τ is already defined on supp(χ). We define the function that D γ
t := χτ + (1 − χ)F . An easy calculation shows that t ∈ C 3 (R3 ), ∇t − n∞ ≤ cKε and the level sets of t define a regular foliation of R3 . This is clearly an extension of the foliation c . Moreover, if we define a smaller given by τ on D and the leaves are planes on D tubular neighborhood N of the central field line as N := {x : inf τ dist(x, ϕ(τ )) ≤ 2}, then we note that N ⊂ B(z0 , 5) ∪ {|τ | ≥ 3}, therefore t ≡ τ on N . Let N := ∇t−1 ∇t be the unit C 2 -vectorfield orthogonal to the foliation. We remark that the integral curves of N typically do not coincide with the field lines except on the field line ϕ and in the region far away from z0 . Armed with this foliation, we introduce new coordinates on R3 . On the plane P we choose Euclidean orthonormal coordinates ξ1 , ξ2 with origin at z and dual to the basis {p1 , p2 }, i.e., x − z = ξ1 p1 + ξ2 p2 . Clearly (∂ξj , ∂ξk ) = δjk for j, k = 1, 2. For simplicity we set ∂j := ∂ξj and ∇j := ∇∂j in this proof. We extend the coordinate system ξ1 , ξ2 defined on the plane P by setting ξ1 , ξ2 constant on the integral curves of N . It is easy to check that ξ1 , ξ2 ∈ C 3 (R3 ). Together with t they define a regular set of coordinates on R3 . The central line is given by (0, 0, t) in these coordinates. Let b(t) := |B(0, 0, t)| ∈ C 3 (R) be the strength of the magnetic field along the central line, note that b(t) is comparable d ξ3 = [b(t)/b]1/2 with b for all t by (5.1). We define ξ3 = ξ3 (t) to be the solution of dt 4 with ξ3 (0) = 0, clearly ξ3 ∈ C (R). We reparametrize the coordinate t with ξ3 := ξ3 (t). In this way we defined a new coordinate system, {ξ1 , ξ2 , ξ3 }, with origin at z. We shall view the coordinates ξ1 , ξ2 , ξ3 as C 3 functions of x ∈ R3 and whenever the dependence on z is relevant, we use the notation ξ z = (ξ1z , ξ2z , ξ3z ). It is easy to check that the function (ˆ z , ξ) → ξ z belongs to C 3 (R5 ) and the vectorfields 2 ∂i = ∂ξi are C . We note that in the trivial case, |z − z0 | ≥ 2, we simply have ξ z (x) = t R (x − z), where R := [p1 |p2 |n∞ ] is the 3 by 3 matrix with columns p1 , p2 , n∞ , and all statements of Lemma 8.2 are trivial with Ω ≡ h ≡ 1. From now on we shall assume that |z −z0 | < 2. The relations (8.3)–(8.4) and (8.6), i.e., the fact that ds2 has no dξj dξ3 (j = 1, 2) components follow directly from the construction. From the regularity of the magnetic field (Definition 5.1) it easily follows that the Jacobian of the change of coordinates x → ξ z (x) is close to the matrix Rt and it varies regularly in z. This proves that the function (ˆ z , ξ) → xz (ξ) is well defined and C 3 (R5 ), it also proves (8.5) and (8.7) by the inverse function theorem. The metric is diagonal in the ξ coordinate system on the plane P, i.e., for t = 0. The key point is to show that it remains diagonal within the tubular neighborhood N . The diagonal metric elements will define the functions Ω and h. We derive a differential equation for the metric components gjk := (∂j , ∂k )g ∈ C 2 (R3 ) where j, k = 1, 2 within N . We have ∂t gjk = ∂t (∂j , ∂k )g = (∇t ∂j , ∂k )g +
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(∂j , ∇t ∂k )g . Using that ∂t , ∂j , ∂k , are coordinate fields, i.e, have vanishing Lie −1/2 derivatives we have ∇t ∂j = ∇j ∂t . Recall that N = gtt ∂t is the unit normal to 1/2 1/2 the spherical foliation, where gtt := (∂t , ∂t )g . Then ∇j ∂t = gtt ∇j N + ∂j (gtt )N and therefore we have 1/2
1/2
∂t gjk = gtt [(∇j N, ∂k )g + (∂j , ∇k N )g ] = 2gtt Kjk , where Kjk is the second fundamental form of the leaves of the foliation. For a sphere immersed in R3 we have Kjk = κgjk , where κ is the curvature. We recall the choice of κ from (A.30) and that t ≡ τ on N . Thus 1/2
∂t gjk = 2gtt κ(t)gjk .
(A.31)
This proves that since g12 is zero on the supporting plane t = 0, it is zero everywhere in N . It also proves that g11 = g22 everywhere in N since they satisfy the same equation and initial condition. The same relations trivially hold for the −1/2 region |ξ⊥ | ≥ 10, where g11 = g22 = g12 = 1. Moreover, we define Ω := g11 ∈ 2 3 C (R ) and we obtain that within N as well as in the regime |ξ⊥ | ≥ 10 the conformal metric can be written as ds2Ω = dξ12 + dξ22 + Ω2 gtt dt2 = dξ12 + dξ22 + Ω2 gtt f (ξ3 )−2 dξ32 1/2
using the definition of f and ξ3 . This proves (8.8) with h := Ωgtt f (ξ3 ). Since the new coordinates form an orthonormal system for |ξ3 | ≥ 3 and also for |ξ⊥ | ≥ 10 modulo a change of variables in the third direction, the identities (8.9) and (8.12), respectively, follow from the definitions. Along the central line we have gtt (ϕ(t)) = 1. Thus (A.31), (A.30) and g11 ≡ 1 for |τ | ≥ 3 implies that g11 = b/|B|, i.e., Ω = f (ξ3 ) and h ≡ 1 along the central line. Then (8.5) implies (8.10). The global bounds (8.13)–(8.14) also follow from the smoothness of the construction, i.e., from (8.5). The details are left to the reader. Finally, the orthonormal basis {e1 , e2 , e3 } in ds2Ω is defined by first construct ing e1 := ∂1 , e2 := ∂2 , e3 := h−1 ∂3 which are automatically orthonormal apart from the region {ξ : 32 ≤ |ξ⊥ | ≤ 9, |ξ3 | ≤ 3}. On this region we apply a GramSchmidt orthonormalization procedure to obtain {e1 , e2 , e3 } from {e1 , e2 , e3 }.
A.5 Comparison of operators on equivalent L2 -spaces Let dµ and dν be two positive measures on Rd , that are mutually and uniformly absolutely continuous, i.e., dν(x) = F (x)dµ(x) with a a positive bounded function F with bounded inverse F −1 . We let CF := F ∞ F −1 ∞ =
max F . min F
Consider the spaces L2 (dµ) and L2 (dν) = L2 (F dµ) and let A be any operator defined on L2 (dµ). Since these two spaces are the same as sets, we can consider A
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acting on L2 (F dµ) as well. We denote this operator by AF . Let (·, ·) := (·, ·)L2 (dµ) and and (·, ·)F := (·, ·)L2 (F dµ) . Similar convention is used for · and · F and for the traces over these L2 -spaces: Tr := TrL2 (dµ) and TrF := TrL2 (F dµ) . Lemma A.7. Let A be a Hilbert-Schmidt operator on L2 (dµ) with a kernel A(x, y). Then AF is also Hilbert-Schmidt on L2 (dν), and the kernels of these operators satisfy (A.32) A(x, y) = AF (x, y)F (y) . Moreover, the diagonal kernels of A∗ A and A∗F AF are comparable:
∗ ∗ −1 ∞ (A∗ A)(x, x) . CF−1 F −1 ∞ (A A)(x, x) ≤ AF AF (x, x) ≤ CF F
(A.33)
Furthermore, if 0 < α ≤ A∗ A ≤ β for some constants α, β, then CF−1 α ≤ A∗F AF ≤ CF β .
(A.34)
If, in addition, A is of trace class, then so is AF and their diagonal kernels satisfy A(x, x) = AF (x, x)F (x) .
(A.35)
Proof. Recalling the conventions at the end of Section 4, the identities (A.32) and (A.35) are obvious. For (A.33) we estimate A∗F (x, y)AF (y, x)F (y)dµ(y) (A∗F AF )(x, x) = −2 = F (x) |A(x, y)|2 F (y)dµ(y) ≤ CF F −1 ∞ (A∗ A)(x, x) . The lower bound is proven similarly. For (A.34) we notice that (ψ, A∗F AF ψ)F = F 1/2 Aψ2 ≤ (max F )Aψ2 ≤ β(max F )ψ2 and
ψ2 ≤ F −1/2 ψ2F ≤ (max F −1 )ψ2F ,
which proves the upper bound. The proof of the lower bound in (A.34) is similar. Lemma A.8. Let Ak be a finite collection of closed operators on L2 (dµ), let W1 , W2 be nonnegative functions on Rd . Then
(Ak )∗F (Ak )F + W1 − W2 ≤ Tr A∗k Ak + W1 − CF W2 TrF . k
−
k
−
(A.36)
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Proof of Lemma A.8. By the variational principle TrF
(Ak )∗F (Ak )F + W1 − W2
k
−
=
(Ak )∗F (Ak )F + W1 − W2 γ : 0 ≤ γ ≤ 1 inf TrF k
where the infimum is over all finite rank density matrices γ on L2 (F dµ). We can write γ = n λn (fn , ·)F fn with 0 ≤ λn ≤ 1 and {fn } being orthonormal in L2 (F ). Define the operator γ := (min F ) n λn (fn , ·)fn on L2 . Since (φ, γφ) = (min F )
λn |(fn , F −1 φ)F |2 = (min F )(F −1 φ, γF −1 φ)F ≤ φ2 ,
n
γ is a density matrix on L2 . Furthermore, for any A = Ak TrF A∗F AF γ = λn AF fn 2F = λn Afn 2F ≥ (min F ) λn Afn 2 n
=
(min F )
n
n
∗
∗
λn Tr|A Afn fn | = TrA A γ.
n
The potential term is estimated as λn (fn , (W1 − W2 )fn )F TrF (W1 − W2 )γ = n
≥
(min F )
λn (fn , (W1 − CF W2 )fn )
n
=
γ. Tr(W1 − CF W2 )
Therefore
, TrF (Ak )∗F (Ak )F + W1 − W2 γ ≥ Tr A∗k Ak + W1 − CF W2 γ k
k
and (A.36) follows from the variational principle.
A.6 Comparison of Dirac operators under a conformal transformation Let Ω : R3 → R+ be a C 1 -function satisfying
and
1 ≤ Ω(x) ≤ 2 2
(A.37)
∇Ω∞ ≤ −1
(A.38)
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with some constant > 0. We define the metric ds2Ω := Ω2 ds2 that is conformally equivalent to the Euclidean metric ds2 . Let D be a Dirac operator in the ds2 metric, then a Dirac operator in the ds2Ω metric is given by DΩ := Ω−2 DΩ. Notice that DΩ is self-adjoint on L2 (ds2Ω ) ⊗ C2 (see [ES-III]). The following lemma compares certain resolvent kernels of D and DΩ on L2 (ds2 ) ⊗ C2 and on L2 (ds2Ω ) ⊗ C2 , respectively. Lemma A.9. Let P ≥ 29 −2 be a number. Under the conditions (A.37), (A.38) we have 1 1 9 tr (x, x) ≤ 2 tr (x, x) x ∈ R3 . (A.39) 2 + P )2 (D2 + P )2 2 (DΩ 2 L
LΩ
The left-hand side is the diagonal of an operator kernel on L2 (ds2 ) ⊗ C2 , the right-hand side is the diagonal of an operator kernel on L2 (ds2Ω ) ⊗ C2 . Moreover, if 0 ≤ ϕ ≤ 1 is a bounded function then 1 1 Dϕ2 D 2 (x, x) (A.40) tr D2 + P D +P 2 L 1 1 1 12 2 DΩ ϕ DΩ 2 +P 2 ≤ 2 tr 2 (x, x) DΩ + P DΩ + P (DΩ + P )2 2 LΩ
for any x ∈ R3 . Proof of Lemma A.9. Let V : L2 (ds2 ) ⊗ C2 → L2 (ds2Ω ) ⊗ C2 be a unitary map given by Vψ := Ω−3/2 ψ. Notice that D = V ∗ (Ω1/2 DΩ Ω1/2 )V , therefore the unitary operator Ω1/2 DΩ Ω1/2 on L2 (ds2Ω )⊗C2 is unitarily equivalent to D on L2 (ds2 ) ⊗ C2 . In particular, for any real function f f (D) = Ω3/2 f (Ω1/2 DΩ Ω1/2 )Ω−3/2 .
(A.41)
From (A.41) we obtain 1 4 4 ≤ = Ω3/2 Ω−3/2 . (D2 + P )2 (D2 + P )2 + 3P 2 ([Ω1/2 DΩ Ω1/2 ]2 + P )2 + 3P 2 In particular, 1 (x, x) ≤ 2 (D + P )2
4 1/2 (Ω DΩ ΩDΩ Ω1/2 + P )2 + 3P 2
(x, x) . L2
Here the right-hand side is the L2 (ds2 ) ⊗ C2 kernel of the corresponding bounded non self-adjoint operator. However, the same operator can be viewed on L2 (ds2Ω )⊗
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C2 as well, where it is self-adjoint. Using (A.35) from Lemma A.7 we know that the two diagonal kernels differ by a factor Ω3 (x). To conclude (A.39), it is therefore sufficient to show that 1 32 ≤ Ω−1/2 2 Ω−1/2 (DΩ + P )2 (Ω1/2 DΩ ΩDΩ Ω1/2 + P )2 + 3P 2
(A.42)
as self-adjoint operators on L2 (ds2Ω ) ⊗ C2 . Using 2 1/2 Ω1/2 DΩ ΩDΩ Ω1/2 = Ω3/2 DΩ Ω + Ω1/2 [DΩ , Ω]DΩ Ω1/2
and a Schwarz’ inequality, we obtain (Ω1/2 DΩ ΩDΩ Ω1/2 + P )2 + 3P 2 (A.43) 1 3/2 2 1/2 ∗ 3/2 2 1/2 (Ω DΩ Ω ) (Ω DΩ Ω ) ≥ 2
∗
−2 Ω1/2 [DΩ , Ω]DΩ Ω1/2 Ω1/2 [DΩ , Ω]DΩ Ω1/2 + P 2 1 1/2 2 3 2 1/2 1/2 ∗ 1/2 2 Ω DΩ Ω DΩ Ω − 4Ω DΩ [DΩ , Ω] Ω[DΩ , Ω]DΩ Ω + 2P = 2 1 1 1/2 4 1/2 1/2 ∗ 1/2 2 Ω DΩ Ω − 8Ω DΩ [DΩ , Ω] [DΩ , Ω]DΩ Ω + 2P . ≥ 2 8 Using that [DΩ , Ω] = Ω−2 [D, Ω]Ω and (A.38) we can estimate [DΩ , Ω]∗ [DΩ , Ω] ≤ 4∇Ω2∞ ≤ 2−7 P ,
(A.44)
∞
so we can continue
1 1/2 1 4 1 1 1/2 2 2 2 (A.43) ≥ Ω D − P DΩ + P Ω1/2 ≥ Ω (DΩ + P )2 Ω1/2 . 2 8 Ω 16 32
This completes the proof of (A.42) and hence (A.39). For the proof of (A.40) we can use the argument above to reduce the problem to estimating the diagonal element of the self-adjoint operator T := RΩ1/2 DΩ Ω1/2 ϕ2 Ω1/2 DΩ Ω1/2 R with
R :=
1 (Ω1/2 DΩ Ω1/2 )2
+P
viewed on L2 (ds2Ω ) ⊗ C2 , where DΩ is self-adjoint. The resolvent can be written as R = Ω−1/2 R1 Ω−1/2 with R1 :=
1 1 = Ω−1 2 , DΩ ΩDΩ + P Ω−1 DΩ − DΩ [DΩ , Ω]Ω−1 + P Ω−2
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and we can expand R1 = Ω−1
1 1 −1 −2 D + R . [D , Ω]Ω − P (Ω − 1) 1 Ω Ω 2 2 DΩ + P DΩ + P
Therefore, using a Schwarz’ inequality, (A.37) and 0 ≤ ϕ ≤ 1, we can estimate T
≤ 2Ω−1/2 R1 DΩ ϕ2 DΩ R1 Ω−1/2 1 1 Ω−1 DΩ ϕ2 DΩ Ω−1 2 Ω−1/2 ≤ 4Ω−1/2 2 DΩ + P DΩ + P ∗ 1 −1 −1/2 +8 · · · DΩ R1 DΩ [DΩ , Ω]Ω 2 +PΩ DΩ ∗ 1 −2 −1/2 Ω +8 · · · DΩ R1 P (Ω − 1) 2 . DΩ + P
(A.45)
Here we used the shorthand notation (· · · )∗ A for the operator A∗ A where A is a long expression. In the first term on the right-hand side of (A.45) we use DΩ Ω−1 = Ω−1 DΩ + [DΩ , Ω−1 ] and (A.44) to obtain
2 DΩ
1 1 Ω−1 DΩ ϕ2 DΩ Ω−1 2 ≤ +P DΩ + P 8 1 1 DΩ ϕ2 DΩ 2 + 2−8 P 2 , 2 DΩ + P DΩ + P (DΩ + P )2
and both terms explicitly appear on the right-hand side of (A.40). For the other two terms it is sufficient to show that 2 R1 R1 DΩ
≤
4P −1 ,
(A.46)
2 DΩ R1 DΩ R1 DΩ
≤
4,
(A.47)
and then the last two terms in (A.45) can be estimated by the second term on the right-hand side of (A.40) using (A.37), (A.38). 2 For the proof of (A.46) and (A.47) we first use DΩ ≤ 2DΩ ΩDΩ + 2P Ω−1 = −1 2R1 to cancel one of the resolvents. The proof of (A.46) is then completed by estimating the other R1 by 2P −1 . For the proof of (A.47) we notice that DΩ R1 DΩ = DΩ
1 2 DΩ ≤ DΩ 2 DΩ ≤ 2 . −1 DΩ ΩDΩ + P Ω DΩ + 2P Ω−1
This completes the proof of Lemma A.9.
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A.7 Proof of Lemma 11.2: a priori bound on the full resolvent Using (A.34) from Lemma A.7 and since the volume forms dν and dµ = Ω3 dx are comparable at every point, it is sufficient to prove (11.15)–(11.19) in the space α and the components of D = Dα L2 (dµ) = L2 (Ω3 dx). In this space D = DΩ Ω = α i ΠΩ − 2 (divΩ f1 , divΩ f2 , divΩ f3 ) are self-adjoint. We recall that Πα Ω was given in (9.14) and D was defined in general in (9.7). Throughout the proof we will work in α , D = Dα the space L2 (Ω3 dx), and we adapt the notation D = DΩ Ω in this section. 1 We also recall that Πj = Dj + idj with dj := 2 divΩ ej . Using Lichnerowicz’ formula (9.9), supx β(x) = supx B(x) ≤ cb and that all geometric terms are bounded by (8.13) and (8.14), we can estimate D2 ≥ D2 − cb .
(A.48)
We recall that = 1, b ≥ ε−2 ≥ 1 and P = ε−5 ≥ 1. For the proof of (11.15) we start with a Schwarz’ inequality 1 2 1 2 Π∗j Πj ≤ 2Dj Dj + 2 sup |dj | 2 D +P D2 + P and use that |dj | ≤ c. We estimate one of the resolvents trivially and use (A.48) Dj
D2
1 2 b Dj Dj ≤ Dj 2 +P D + Pb ≤ Dj
b b Dj ≤ Dj 2 Dj ≤ b D2 − cb + P b Dj + P b/2
for sufficiently small ε. This completes the proof of (11.15). The proof of (11.16) is identical just we estimate (D/(D2 +P ))2 by (D2 +P )−1 . For the proof of (11.17) we first compute Πj Πk = Dj Dk + i(Dj dk + Dk dj ) − (∂ek dk ) + dj dk . We use a Schwarz’ inequality, the estimate (11.15) and the boundedness of dj ’s together with their derivatives, we obtain 1 2 1 2 Π Π ≤ 2D D Dj Dk + cb . (A.49) Π∗k Π∗j j k k j D2 + P D2 + P We apply Lemma 11.1 to estimate the resolvent square, using that D2 and D2 differ only by an operator bounded by cb ≤ P b/2 if ε is sufficiently small: 1 2 b2 4b2 . (A.50) ≤ ≤ D2 + P (D2 + P b)2 (D2 )2 + (P b)2 /4 We expand (D2 )2 = j Dj4 + j
AB A + AB[A, B] + [B, A]BA + 12 [A, [A, B]]B + B[[B, A], A] + A ↔ B
2
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(the second square bracket contains the same expression as the first one with A and B interchanged). After several Schwarz’s inequalities, we obtain
A2 B 2 + B 2 A2 ≥ 12 AB 2 A + BA2 B − A2 − B 2 − c[A, B][B, A] (A.51) −c[A, [A, B]][[B, A], A] − c[B, [B, A]][[A, B], B] . Using the formula given in Theorem 2.12 [ES-III] for the curvature of the covariant derivative ∇ = ∇α,Ω we obtain [Dj , Dk ] =
−∇[ej ,ek ] − ∂ej dk + ∂ek dj −
3 1 (ea , R(ej , ek )eb )σ a σ b + iβ(ej , ek ) , 4 a,b=1
where R is the Riemannian curvature, j, k = 1, 2, 3. In short, we can write [Dj , Dk ] =
3
a Ujk Da + Wjk ,
a=1 k a a where Ujk , Wjk are 2 by 2 matrix-valued functions with Ujk ∞ ≤ c, ∇Ujk ∞ ≤ c and Wjk ∞ ≤ cb, ∇Wjk ∞ ≤ cb using (8.14). These estimates guarantee bounds on the double commutators as well. From these estimates and (A.51) it follows that
Dj2 Dk2 + Dk2 Dj2 ≥ 12 Dj Dk2 Dj + Dk Dj2 Dk − cbD2 − cb2 .
Therefore
(D ) + (P b) /4 ≥ 2 2
2
1 2
Dj4
Dj Dk2 Dj + Dk Dj2 Dk +
j
+
j
1 4 2 Dj
− cbDj2 + (P b)2 /4 − cb2 .
(A.52)
j
The second line is bigger than (P b)2 /8 if ε is sufficiently small (P = ε−5 ). Every term in the first line is nonnegative, so we can complete the estimate (A.49) using (A.50) and (A.52) Dj Dk
(D2 )2
4b2 8b2 Dk Dj ≤ Dj Dk Dk Dj ≤ 8b2 . 2 2 + (P b) /4 Dk Dj Dk + (P b)2 /8
This completes the proof of (11.17). The proof of (11.18) and (11.19) are straightforward from (11.11), (11.15), (11.16) and (10.14). Finally (11.20) is proven in the same way as (11.15) but now directly on the space L2 (dξ) ⊗ C2 .
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A.8 Proof of Lemma 11.3: Estimates on the resolvent with a constant field The proof of (11.21)–(11.28) may be done by straightforward explicit calculations since the magnetic Schr¨odinger operator with a constant field is exactly solvable. We show below how to obtain these estimates in a reasonably short way. We work in the ξ coordinate system, and use that ξ⊥ (u) = 0. Because of translation invariance in the third direction, we can assume ξ(u) = 0, so in the ξ coordinates we need to estimate the operator kernels at (0, 0). We recall the decompositions (11.8)–(11.10) and let := Π 21 + Π 22 = (−i∂1 − b ξ2 )2 + (−i∂2 + b ξ1 )2 −∆ 2 2 3 . For simplicity, be the two-dimensional magnetic Laplacian that commutes with Π 2 2 + σ 3 b and we denoted ∂j := ∂ξj . By Lichnerowicz’ formula (9.8), D = −∆ + Π 3 3 = −i∂3 . recall that Π has a closed form (see, e.g., Chapter The key idea is that the heat kernel of ∆ 15 in [S79]) b coth(bt) b ib t∆ 2 e (ξ⊥ , ζ⊥ ) = exp − (ξ⊥ − ζ⊥ ) − (ξ2 ζ1 − ξ1 ζ2 ) . 4π sinh(bt) 4 2 (A.53) Then the resolvent can be expressed as ∞ 3 2 1 e−t(P +σ b) et∆ et∂3 dt . (A.54) = 2 + P D 0 We define the following norm on R3 2 |||ξ||| := (bξ⊥ + P ξ32 )1/2 .
Lemma A.10. Let P ≤ cb, then the following bounds hold
1
(ξ, ζ)
2 + P D Π 3 (ξ, ζ) 2 D +P Π j (ξ, ζ) 2 + P D D (ξ, ζ) 2 D +P
≤ cbP −1/2 ≤ cb
e−c|||ξ−ζ||| , |||ξ − ζ|||
(A.55)
e−c|||ξ−ζ||| , |||ξ − ζ|||2
≤ cb3/2 P −1/2 ≤ c e−c|||ξ−ζ|||
e−c|||ξ−ζ||| , |||ξ − ζ|||2
1 +b , 2 |ξ − ζ|
(A.56) j = 1, 2
(A.57) (A.58)
where · refer to the 2 by 2 matrix-norm of the operator kernel as a function from R3 × R3 into the set of 2 by 2 matrices.
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Remark. If b ≤ cP , then the same estimates (A.55)–(A.58) hold with b replaced by P everywhere, including in the definition of ||| · |||. Proof. From (A.53) and (A.54) we estimate
1 2 + P D
(ξ, ζ) ≤
∞
cb 0
e−tP +tb 1 √ bt coth(bt)(ξ⊥ − ζ⊥ )2 + (ξ3 − ζ3 )2 dt . exp − 4t t sinh(bt)
We split the integration into two regimes: bt ≤ 1 and bt ≥ 1. In the first regime we use bt coth(bt) ≥ 1, sin(bt) ≥ bt. In the second regime we estimate coth(bt) ≥ 1 and sinh(bt) ≥ 14 ebt . We obtain 1 1/b −tP (ξ − ζ)2
e dt exp − (ξ, ζ) ≤ c 2 + P 4t t3/2 D 0 b
∞ e−tP (ξ − ζ )2
3 3 2 +b exp − (ξ⊥ − ζ⊥ ) dt exp − 1/2 4 4t 1/b t ≤
c(P 1/2 + bP −1/2 )
e−c|||ξ−ζ||| , |||ξ − ζ|||
after extending both integrations over (0, ∞) and using the resolvent kernels of the one- and three-dimensional free Laplacians. The proofs of (A.56)–(A.57) are similar and left to the reader. For the proof of (A.58), explicit calculation and trivial estimates yield
D
2 + P D
(ξ, ζ)
≤
≤
≤ ≤
∞
e−t(P −b) b2 t(coth(bt) − 1)|ξ⊥ − ζ⊥ | t3/2 sinh(bt) 0 b coth(bt) (ξ3 − ζ3 )2
× exp − |ξ⊥ − ζ⊥ |2 − dt 4t 1/b −t(P +b)4 (ξ − ζ)2
e |ξ⊥ − ζ⊥ | dt c exp − 5/2 4t t 0 (ξ − ζ )2
∞ −tP 2 e |ξ⊥ − ζ⊥ | 3 3 +cb e−cb|ξ⊥ −ζ⊥ | dt exp − 3/2 4t t 1/b 1/b −t(P +b) e (ξ − ζ)2
c dt exp − c t2 0 ∞ −tP t 2 e (ξ3 − ζ3 )2
dt exp − +cb1/2 e−cb|ξ⊥ −ζ⊥ | 3/2 4t 1/b t ∞ √ 1 (ξ − ζ)2
c e− b+P |ξ−ζ| dt exp − c t2 t 0 √ 2 +cb e−cb|ξ⊥−ζ⊥ | −c P |ξ3−ζ3 | 1 c e−c|||ξ−ζ||| +b . |ξ − ζ|2
≤ c
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With the estimates of Lemma A.10 at hand, the proof of Lemma 11.3 is straightforward. For example, the proof of (11.21) is as follows 2 e−c|||ξ||| 1 1 tr (0, 0) = dξ ≤ cbP −3/2 (0, ξ) dξ ≤ cb2 P −1 2 2 + P 2 + P )2 (D R3 D R3 |||ξ||| after a change of variables. The other inequalities are proved similarly.
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A. Balinsky, W.D. Evans, On the zero modes of Pauli operators, J. Funct. Anal. 179(1), 120–135 (2001).
[BFFGS] L. Bugliaro, C. Fefferman, J. Fr¨ ohlich, G.M. Graf and J. Stubbe, A Lieb-Thirring bound for a magnetic Pauli Hamiltonian, Commum. Math. Phys. 187, 567–582 (1997). [BFrG]
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L. Bugliaro, C. Fefferman and G.M. Graf, A Lieb-Thirring bound for a magnetic Pauli Hamiltonian, II, Rev. Mat. Iberoamericana 15, 593–619 (1999).
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D. Elton, New examples of zero modes, J. Phys. A 33 (41), 7297–7303 (2000).
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D. Elton, The local structure of zero mode producing magnetic potentials, Commun. Math. Phys. 229, 121–139 (2002).
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L. Erd˝ os, Ground state density of the Pauli operator in the large field limit, Lett. Math. Phys. 29, 219–240 (1993).
[E-1995] L. Erd˝os, Magnetic Lieb-Thirring inequalities, Commun. Math. Phys. 170, 629–668 (1995). [ES-I]
L. Erd˝ os and J.P. Solovej, Semiclassical eigenvalue estimates for the Pauli operator with strong non-homogeneous magnetic fields. I. Nonasymptotic Lieb-Thirring type estimate, Duke J. Math. 96, 127–173 (1999).
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L. Erd˝ os and J.P. Solovej, Semiclassical eigenvalue estimates for the Pauli operator with strong non-homogeneous magnetic fields. II. Leading order asymptotic estimates, Commun. Math. Phys. 188, 599–656 (1997).
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L. Erd˝ os and J.P. Solovej, Magnetic Lieb-Thirring inequalities with optimal dependence on the field strength, Accepted to J. Statis. Phys. (2003). Available at http://xxx.lanl.gov/pdf/math-ph/0306066.
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B. Helffer, J. Nourrigat and X.P. Wang, Sur le spectre de l’´equation ´ de Dirac (dans R2 ou R3 ) avec champs magn´etique, Ann. scient. Ec. e Norm. Sup. 4 s´erie t. 22, 515–533 (1989).
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E.H. Lieb, M. Loss and J.P. Solovej, Stability of Matter in Magnetic Fields, Phys. Rev. Lett. 75, 985–989 (1995).
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E.H. Lieb, J.P. Solovej and J. Yngvason, Asymptotics of heavy atoms in high magnetic fields: I. Lowest Landau band region, Commun. Pure Appl. Math. 47, 513–591 (1994).
[LSY-II] E.H. Lieb, J.P. Solovej and J. Yngvason, Asymptotics of heavy atoms in high magnetic fields: II. Semiclassical regions, Commun. Math. Phys. 161, 77–124 (1994). [LT1]
E.H. Lieb, W. Thirring, Inequalities for moments of the eigenvalues of the Schr¨ odinger Hamiltonian and their relation to Sobolev inequalities. In: Studies in Mathematical Physics (E. Lieb, B. Simon, A. Wightman eds.) Princeton University Press, 269–330 (1975).
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M. Loss and H.-T. Yau, Stability of Coulomb systems with magnetic fields: III. Zero energy bound states of the Pauli operator, Commun. Math. Phys. 104, 283–290 (1986).
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B. Simon, Functional Integration and Quantum Physics, Academic Press, New York, 1979.
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Z. Shen, On the moments of negative eigenvalues for the Pauli operator, J. Diff. Eq. 149, 292–327 (1998) and 151, 420–455 (1999).
[Sob-86] A. Sobolev, Asymptotic behavior of the energy levels of a quantum particle in a homogeneous magnetic field, perturbed by a decreasing electric field, J. Sov. Math. 35, 2201–2212 (1986). [Sob-96] A. Sobolev, On the Lieb-Thirring estimates for the Pauli operator, Duke Math. J. 82, 607–635 (1996). [Sob-97] A. Sobolev, Lieb-Thirring inequalities for the Pauli operator in three dimensions, IMA Vol. Math. Appl. 95, 155–188 (1997). [Sob-98] A. Sobolev, Quasiclassical asymptotics for the Pauli operator, Commun. Math. Phys. 194, 109–134 (1998).
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S.N. Solnyshkin, The asymptotic behavior of the energy of bound states of the Schr¨ odinger operator in the presence of electric and magnetic fields, Probl. Mat. Fiz. 10, 266–278 (1982).
L´aszl´o Erd˝ os Mathematisches Institut, LMU Theresienstrasse 39 D-80333 Munich Germany email:
[email protected] Jan Philip Solovej Department of Mathematics University of Copenhagen Universitetsparken 5 DK-2100 Copenhagen Denmark email:
[email protected] Communicated by Gian Michele Graf Submitted 31/08/03, accepted 28/01/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 743 – 772 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/040743-30 DOI 10.1007/s00023-004-0181-9
Annales Henri Poincar´ e
Schr¨ odinger Operators on Lattices. The Efimov Effect and Discrete Spectrum Asymptotics Sergio Albeverio, Saidakhmat N. Lakaev and Zahriddin I. Muminov Abstract. The Hamiltonian of a system of three quantum mechanical particles moving on the three-dimensional lattice Z3 and interacting via zero-range attractive potentials is considered. For the two-particle energy operator h(k), with k ∈ T3 = (−π, π]3 the two-particle quasi-momentum, the existence of a unique positive eigenvalue below the bottom of the continuous spectrum of h(k) for k = 0 is proven, provided that h(0) has a zero energy resonance. The location of the essential and discrete spectra of the three-particle discrete Schr¨ odinger operator H(K), K ∈ T3 being the three-particle quasi-momentum, is studied. The existence of infinitely many eigenvalues of H(0) is proven. It is found that for the number N (0, z) of eigenvalues of H(0) lying below z < 0 the following limit exists lim
z→0−
N (0, z) = U0 | log | z ||
with U0 > 0. Moreover, for all sufficiently small nonzero values of the three-particle quasi-momentum K the finiteness of the number N (K, τess (K)) of eigenvalues of H(K) below the essential spectrum is established and the asymptotics for the number N (K, 0) of eigenvalues lying below zero is given.
1
Introduction
One of the remarkable results in the spectral analysis for continuous three-particle Schr¨ odinger operators is the Efimov effect: if in a system of three-particles, interacting by means of short-range pair potentials none of the three two-particle subsystems has bound states with negative energy, but at least two of them have a resonance with zero energy, then this three-particle system has an infinite number of three-particle bound states with negative energy, accumulating at zero. This effect was first discovered by Efimov [5]. Since then this problem has been studied in many physics journals and books [1, 3, 7]. A rigorous mathematical proof of the existence of Efimov’s effect was originally carried out in [25] by Yafaev and then in [20, 22, 23, 24]. Efimov’s effect was further studied in [2, 4, 10, 11, 14, 15, 16, 18, 19]. Denote by N (z), z < 0 the number of eigenvalues of the Hamiltonian below z < 0. The growth of N (z) has been studied by S. Albeverio, R. Høegh-Krohn, and T.T. Wu in [1] for the symmetric case. Namely, the authors of [1] have first found (without proofs) the exponential asymptotics of eigenvalues corresponding to spherically symmetric bound states.
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This result is consistent with the lower bound lim inf
z→0
N (z) >0 | log |z||
established in [23] without any symmetry assumptions. The main result obtained by Sobolev [22] is the limit lim | log |z||−1 N (z) = U0 ,
z→0
(1.1)
where the coefficient U0 does not depend on the potentials vα and is a positive function of the ratios m1 /m2 , m2 /m3 of the masses of the three-particles. In [2] the Fredholm determinant asymptotics of convolution operators on large finite intervals with rational symbols having real zeros are studied, as well as the connection with the Efimov effect. In models of solid state physics [8, 18, 19, 21] and also in lattice field theory [9, 17] discrete Schr¨odinger operators are considered, which are lattice analogs of the continuous three-particle Schr¨ odinger operator. The presence of Efimov’s effect for these operators was demonstrated at the physical level of rigor without a mathematical proof for a system of three identical quantum particles in [18, 19]. Although the energy operator of a system of three-particles on lattice is bounded and the perturbation operator in the pair problem is a compact operator, the study of spectral properties of energy operators of systems of two and three particles on a lattice is more complex than in the continuous case. In the continuous case [6] (see also [7, 21]) the energy of the center-of-mass motion can by separated out from the total Hamiltonian, that is, the energy operator can by split into a sum of a center-of-mass motion and a relative kinetic energy. So that the three-particle “bound states” are eigenvectors of the relative kinetic energy operator. Therefore Efimov’s effect either exists or does not exist for all values of the total momentum simultaneously. In lattice terms the “center-of-mass separation” corresponds to a realization of the Hamiltonians as a “fibered operator”, that is, as the “direct integral of a family of operators” H(K) depending on the values of the total quasi-momentum K∈T3 = (−π, π]3 (see [8, 21]). In this case a “bound state” is an eigenvector of the operator H(K) for some K∈T3 . Typically, this eigenvector depends continuously on K. Therefore, Efimov’s effect may exists only for some values of K∈T3 (see [11]). In [10] was stated the existence infinitely many bound states (Efimov’s effect) for the discrete three-particle Schr¨odinger operators associated with a system of three arbitrary quantum particles moving on three dimensional lattice and interacting via zero-range attractive pairs potentials. In this work only a sketch of proof of results has been given. In [11] the existence of Efimov’s effect for a system of three identical quantum particles (bosons) on a three-dimensional lattice interacting via zero-range attractive pair potentials has been proven, in the case, where all three two-particle subsystems have resonances at the bottom of the three-particle continuum.
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In [12, 13] the finiteness of the number of bound states was proven, in the cases, where either none of the two-particle subsystems or only one of the twoparticle subsystems have a zero energy resonance. In [15] (a detailed proof is in [16]) for the difference operator on a lattice associated with a system of three identical particles interacting via zero-range attractive pair potentials under the assumption that all two-particle subsystems have resonance at the bottom of the three-particle continuum the following results have been established: 1) for the zero value of the total quasi-momentum (K = 0) there are infinitely many eigenvalues lying below the bottom and accumulating at the bottom of essential spectrum (Efimov’s effect). 2) for all K ∈ Uδ0 (0) = {K ∈ T3 : 0 < |K| < δ}, δ > 0 sufficiently small, the three-particle operator has a finite number of eigenvalues below the bottom of essential spectrum. The results are quite surprising and clearly put in evidence the difference between the continuum and discussed cases. In the present work we consider a system of three arbitrary quantum particles on the three-dimensional lattice Z3 interacting via zero-range pair attractive potentials. Let us denote by τess (K) the bottom of essential spectrum of the threeparticle discrete Schr¨odinger operator H(K), K ∈ T3 and by N (K, z) the number of eigenvalues lying below z ≤ τess (K). The main results of the present paper are as follows: (i) for the two-particle energy operator h(k) on the three-dimensional lattice Z3 , k being the two-particle quasi-momentum, we prove the existence of a unique positive eigenvalue below the bottom of the continuous spectrum of h(k), k = 0 for the nontrivial values of the quasi-momentum k, provided that the two-particle Hamiltonian h(0) corresponding to the zero value of k has a zero energy resonance. (ii) we establish the location of the essential spectrum of the discrete threeparticle operator H(K). The infinitely many eigenvalues of the three-particle discrete Schr¨odinger operator arise from the existence of resonances of the two-particle operators at the bottom of three-particle continuum. Therefore we obtain a lower bound for the location of discrete spectrum of H(K) in terms of zero-range interaction potentials. (iii) for the number N (0, z) we obtain the limit result lim
z→−0
N (0, z) = U0 , (0 < U0 < ∞). | log |z||
(iv) for any K ∈ Uδ0 (0) we prove the finiteness of N (K, τess (K)) and establish the following limit result N (K, 0) lim = 2U0 . |K|→0 | log |K||
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We remark that whereas the result (iii) is similar to that of continuous case and, the results (i) and (iv) are surprising and characteristic for the lattice systems, in fact they do not have any analogues in the continuous case. The plan of the paper is as follows: Section 1 is an introduction to the whole paper. In Section 2 the Hamiltonians of systems of two- and three-particles in coordinate and momentum representations are described as bounded self-adjoint operators in the corresponding Hilbert spaces. In Section 3 we introduce the total quasi-momentum and decompose the energy operators into von Neumann direct integrals, choosing relative coordinate systems. In Section 4 we state the main results of the paper. In Section 5 we study spectral properties of the two-particle discrete Schr¨odinger operator h(k), k ∈ T3 on the three-dimensional lattice Z3 . We prove the existence of unique positive eigenvalue below the bottom of the continuous spectrum of h(k) (Theorem 5.4) and obtain an asymptotics for the Fredholm’s determinant associated with h(k). In Section 6 we introduce the ”channel operators” and describe its spectrum by the spectrum of the two-particle discrete Schr¨odinger operators. Applying a Faddeev type system of integral equations we establish the location of the essential spectrum (Theorem 4.3). We obtain a lower bound for the location of discrete spectrum of H(K) lying below the bottom of the essential spectrum (see Theorem 4.4). We prove the finiteness of eigenvalues below the bottom of the essential spectrum of H(K) for K ∈ Uδ0 (0) (Theorem 4.6). In Section 7 we follow closely A. Sobolev method to derive the asymptotics for the number of eigenvalues of H(K) (Theorem 4.7). Throughout the paper we adopt the following conventions: For each δ > 0 the notation Uδ (0) = {K ∈ T3 : |K| < δ} stands for a δ-neighborhood of the origin and Uδ0 (0) = Uδ (0) \ {0} for a punctured δ-neighborhood. The subscript α (and also β and γ) always equal to 1 or 2 or 3 and α = β, β = γ, γ = α.
2 Energy operators for two and three arbitrary particles on a lattice in the coordinate and momentum representations Let Zν − ν-dimensional lattice. 0 of a system of three quantum mechanical particles The free Hamiltonian H on the three-dimensional lattice Z3 in the coordinate representation is usually associated with the following bounded self-adjoint operator on the Hilbert space 2 ((Z3 )3 ): 0 = 1 ∆x1 + 1 ∆x2 + 1 ∆x3 , H (2.1) 2m1 2m2 2m3 with ∆x1 = ∆⊗I⊗I, ∆x2 = I⊗∆⊗I and ∆x3 = I⊗I⊗∆, where mα > 0, α = 1, 2, 3 are different numbers, having the meaning of a mass of the particle α.
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The Laplacian ∆ is a difference operator which describes the transport of a particle from one side to another, i.e., ˆ ˆ ˆ + s)], ψˆ ∈ 2 (Z3 ). (∆ψ)(x) = [ψ(x) − ψ(x |s|=1
of the quantum-mechanical three-particles The three-particle Hamiltonian H systems with two-particle interactions vˆβγ , β, γ = 1, 2, 3 in the coordinate repre0 sentation is a bounded perturbation of the free Hamiltonian H =H 0 − V1 − V2 − V3 , H
(2.2)
where Vα , α = 1, 2, 3 are multiplication operators on 2 ((Z3 )3 ) ˆ 1 , x2 , x3 ) = µα δx x ψ(x ˆ 1 , x2 , x3 ) = vˆβγ (xβ − xγ )ψ(x ˆ 1 , x2 , x3 ), (Vα ψ)(x β γ ψˆ ∈ 2 ((Z3 )3 ). Here µα > 0 interaction energy of particles β and γ, δxβ xγ is the Kronecker delta. It is clear that the three-particle Hamiltonian (2.2) is a bounded self-adjoint operator on the Hilbert space 2 ((Z3 )3 ). we shall introduce the corresponding twoSimilarly as we introduced H, ˆ α , α = 1, 2, 3 as bounded self-adjoint operators on the particle Hamiltonians h Hilbert space 2 ((Z3 )2 ) ˆα = h ˆ 0 − vˆα , h α where ˆ0 = h α
1 1 x β + x γ , 2mβ 2mγ
with xβ = ∆ ⊗ I, xγ = I ⊗ ∆ and ˆ β , xγ ) = µα δxβ xγ ϕ(x ˆ β , xγ ), (ˆ vα ϕ)(x
ϕˆ ∈ 2 ((Z3 )2 ).
Let us rewrite our operators in the momentum representation. Let Fm : L2 ((T3 )m ) → 2 ((Z3 )m ) denote the standard Fourier transform, where (T3 )m , m ∈ N denotes the Cartesian mth power of the set T3 = (−π, π]3 . Remark 2.1 The operations addition and multiplication by real numbers of elements of T3 ⊂ R3 should be regarded as operations on R3 modulo (2πZ1 )3 . For example, let a=(
2π 3π 11π ,− , ) 3 4 12
then a + b = (−
and
b=(
2π 3π π , , − ) ∈ T3 , 3 4 4
2π π 5π ,− , ) ∈ T3 3 2 6
12a = (0, π, π) ∈ T3 .
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The three- resp. two-particle Hamiltonians (in the momentum representation) are given by the bounded self-adjoint operators on the Hilbert spaces L2 ((T3 )3 ) resp. L2 ((T3 )2 ) as follows 3 H = F3−1 HF resp. ˆ α F2 , hα = F2−1 h
α = 1, 2, 3.
One has H = H0 − V1 − V2 − V3 , where ˆ k1 + ∆ ˆ k2 + ∆ ˆ k3 , H0 = ∆ ˆ k1 = ∆ ˆ 1 ⊗ I ⊗ I, ∆ ˆ k2 = I ⊗ ∆ ˆ 2 ⊗ I and ∆ ˆ k3 = I ⊗ I ⊗ ∆ ˆ 3 and ∆ ˆ α , α = 1, 2, 3 with ∆ is the multiplication operator by the function εα (k) ˆ α f )(k) = εα (k)f (k), (∆
f ∈ L2 (T3 ).
The functions εα , α = 1, 2, 3 defined above are of the form εα (p) =
1 ε(p), mα
ε(p) =
3
(1 − cos p(i) ),
p = (p(1) , p(2) , p(3) ) ∈ R3
i=1
and Vα , α = 1, 2, 3 are integral operators of convolution type (Vα f )(k1 , k2 , k3 ) µα δ(kα − kα )δ(kβ + kγ − kβ − kγ )f (k1 , k2 , k3 )dk1 dk2 dk3 , = (2π)3 (T3 )3
f ∈ L2 ((T3 )3 ), where δ(k) denotes the Dirac delta-function. For the two-particle Hamiltonians hα , α = 1, 2, 3 we have: hα = h0α − vα , where ˆk + ˆk , h0α = γ β ˆk = I ⊗∆ ˆk = ∆ ˆ β ⊗ I, ˆ γ and with γ β (vα f )(kβ , kγ ) =
µα (2π)3
(T3 )2
δ(kβ + kγ − kβ − kγ )f (kβ , kγ )dkβ dkγ ,
f ∈ L2 ((T3 )2 ).
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3 Decomposition of the energy operators into von Neumann direct integrals. Quasimomentum and coordinate systems ˆ m , s ∈ Z3 the unitary operators on the Hilbert space Given m ∈ N, denote by U s 3 m 2 ((Z ) ) defined as: ˆsm f )(n1 , n2 , . . . , nm ) = f (n1 + s, n2 + s, . . . , nm + s), (U We easily see that m ˆs+p ˆpm , ˆsm U U =U
f ∈ 2 ((Z3 )m ).
s, p ∈ Z3 ,
ˆsm , s ∈ Z3 is the unitary representation of the Abelian group Z3 . that is, U Via the Fourier transform Fm the unitary representation of Z3 in 2 ((Z3 )m ) induces the representation of the group Z3 in the Hilbert space L2 ((T3 )m ) by −1 ˆ m Us Fm , s ∈ Z3 given by: unitary (multiplication) operators Usm = Fm (Usm f )(k1 , k2 , . . . , km ) = exp − i(s, k1 + k2 + · · · + km ) f (k1 , k2 , . . . , km ), (3.1) f ∈ L2 ((T3 )m ). For any K ∈ T3 we define Fm K as follows 3 m Fm K = {(k1 , . . . , km−1 , K − k1 − · · · − km−1 )∈(T ) :
k1 , k2 , . . . , km−1 ∈ T3 , K − k1 − · · · − km−1 ∈ T3 }. Decomposing the Hilbert space L2 ((T3 )m ) into the direct integral ⊕L2 (Fm L2 ((T3 )m ) = K )dK K∈T3
we obtain the corresponding decomposition of the unitary representation Usm , s ∈ Z3 into the direct integral Usm = ⊕Us (K)dK, K∈T3
where on L2 (Fm K)
Us (K) = exp(−i(s, K))I
and I = IL2 (Fm denotes the identity operator on the Hilbert space L2 (Fm K ). K) ˆ α , α = 1, 2, 3 obviously commute with the and h The above Hamiltonians H ˆs3 and U ˆs2 , s ∈ Z3 , respectively, that is, groups of translations U ˆ 3H =H U ˆ 3, U s s and
ˆα = ˆ ˆs2 , ˆs2 h U hα U
s ∈ Z3
s ∈ Z3 ,
α = 1, 2, 3.
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Correspondingly, the Hamiltonians H and hα , α = 1, 2, 3 (in the momentum representation) commute with the groups Usm , s ∈ Z3 given by (3.1) for m = 3 and m = 2, respectively. Hence, the operators H and hα , α = 1, 2, 3, can be decomposed into the direct integrals ˜ α (k)dk, α = 1, 2, 3, ⊕H(K)dK and hα = ⊕h H= K∈T3
k∈T3
with respect to the decompositions 3 3 ⊕L2 (F3K )dK L2 ((T ) ) =
3 2
and L2 ((T ) ) =
K∈T3
⊕L2 (F2k )dk,
k∈T3
respectively. Given a cyclic permutation αβγ of 123 we introduce the mappings (3)
παβ : (T3 )3 → (T3 )2 ,
(3)
παβ ((kα , kβ , kγ )) = (kα , kβ )
and πα(2) : (T3 )2 → T3 , (3)
πα(2) ((kβ , kγ )) = kβ .
(2)
(3)
(2)
Denote by πK , K ∈ T3 resp. πk , k ∈ T3 the restriction of παβ resp. πα onto F3K ⊂ (T3 )3 resp. F2k ⊂ (T3 )2 , that is, (3)
(3)
πK = παβ |F3K
(2)
and πk = πα(2) |F2k .
(3.2)
At this point it is useful to remark that F3K , K ∈ T3 and F2k , k ∈ T3 are six- and three-dimensional manifolds isomorphic to (T3 )2 and T3 , respectively. (3)
(2)
Lemma 3.1 The mappings πK , K ∈ T3 and πk , k ∈ T3 are bijective from F3K ⊂ (T3 )3 and F2k ⊂ (T3 )2 onto (T3 )2 and T3 with the inverse mappings given by (3) (πK )−1 (kα , kβ ) = (kα , kβ , K − kα − kβ ) and
(2)
(πk )−1 (kβ ) = (kβ , k − kβ ). Let (3)
UK : L2 (F3K ) −→ L2 ((T3 )2 ), UK f = f ◦ (πK )−1 , K ∈ T3 , and
(2)
uk : L2 (F2k ) → L2 (T3 ), uk g = g ◦ (πk )−1 , k ∈ T3 ,
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(2)
where πK and πk are defined by (3.2). Then UK and uk are unitary operators and −1 ˜ α (k)u−1 , α = 1, 2, 3. hα (k) = uk h H(K) = UK H(K)U K , k The operator H(K), K ∈ T3 has form H(K) = H0 (K) − V1 − V2 − V3 . In the coordinates (kα , kβ ) the operators H0 (K) and Vα are defined on the Hilbert space L2 ((T3 )2 ) by f ∈ L2 ((T3 )2 ),
(H0 (K)f )(kα , kβ ) = Eαβ (K; kα , kβ )f (kα , kβ ),
(Vα f )(kα , kβ ) =
µα (2π)3
f (kα , kβ )dkβ ,
f ∈ L2 ((T3 )2 ),
T3
where Eαβ (K; kα , kβ ) = εα (kα ) + εβ (kβ ) + εγ (K − kα − kβ ). The operator hα (k), k ∈ T3 , α = 1, 2, 3 has form hα (k) = h0α (k) − vα , where
(α)
(h0α (k)f )(kβ ) = Ek (kβ )f (kβ ), f ∈ L2 (T3 ), µα f (kβ )dkβ , f ∈ L2 (T3 ) (vα f )(kβ ) = (2π)3 T3
and
(α)
Ek (kβ ) = εβ (kβ ) + εγ (k − kβ ).
(3.3)
4 Statement of the main results For each K ∈ T3 the minimum and the maximum taken over (p, q) of the function Eαβ (K; p, q) are independent of α, β = 1, 2, 3. We set: Emin (K) ≡ min Eαβ (K, p, q), p,q
Emax (K) ≡ max Eαβ (K, p, q). p,q
Definition 4.1 The operator hα (0) is said to have a zero energy resonance if the equation µα mβγ mβ mγ (ε(q ))−1 ϕ(q )dq = ϕ(q), mβγ ≡ 3 (2π) mβ + mγ T3
has a nonzero solution ϕ in the Banach space C(T3 ). Without loss of generality we can always normalize ϕ so that ϕ(0) = 1.
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Let the operator hα (0) have a zero energy resonance. Then the function ψ(q) = (ε(q))−1 is a solution (up to a constant factor) of the Schr¨ odinger equation hα (0)f = 0 and ψ belongs to L1 (T3 ) \ L2 (T3 ). Set ( (ε(q))−1 dq)−1 . (4.1) µ0α = (2π)3 m−1 βγ T3
Hypothesis 4.2 We assume that µα =
µ0α ,
µβ = µ0β and µγ ≤ µ0γ .
The main results of the paper are given in the following theorems. Theorem 4.3 For the essential spectrum σess (H(K)) of H(K) the following equality σess (H(K)) = ∪3α=1 ∪p∈T3 {σd (hα (K − p)) + εα (p)} ∪ [Emin (K), Emax (K)], holds, where σd (hα (k)) is the discrete spectrum of the operator hα (k), k ∈ T3 . Denote by τs (K) the bottom of the spectrum of the self-adjoint bounded operator H(K), that is, τs (K) = inf (H(K)f, f ). ||f ||=1
We set: τsγ (K) ≡ inf [(H0 (K)f, f ) − (Vγ f, f )], γ = 1, 2, 3. ||f ||=1
(4.2)
As in the introduction, let N (K, z) denote the number of eigenvalues of the operator H(K), K ∈ T3 below z ≤ τess (K), where τess (K) ≡ inf σess (H(K)) is the bottom of the essential spectrum of H(K). Theorem 4.4 Assume Hypothesis 4.2. Then for all K ∈ T3 the inequality τsα (K) − µ0β − µγ ≤ τs (K) holds. Theorem 4.4 yields the following Corollary 4.5 Assume Hypothesis 4.2. All eigenvalues of the operator H(K), K ∈ T3 below the bottom of τess (K) belong to the interval [τsα (K) − µ0β − µγ , τess (K)). Theorem 4.6 Assume Hypothesis 4.2. Then for all K ∈ Uδ0 (0), δ > 0 sufficiently small, the operator H(K) has a finite number of eigenvalues below the bottom of the essential spectrum of H(K).
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Theorem 4.7 Assume Hypothesis 4.2. Then the operator H(0) has infinitely many eigenvalues below the bottom of the essential spectrum and the functions N (0, z) and N (K, 0) obey the relations lim
z→−0
N (0, z) N (K, 0) = lim = U0 (0 < U0 < ∞). | log |z|| |K|→0 2| log |K||
(4.3)
Remark 4.8 The constant U0 does not depend on the interaction energies µα , α = m 1, 2, 3 and is given as a positive function depending only on the ratios mαβ , α = β, α, β = 1, 2, 3 between the masses.
5
Spectral properties of the two-particle operator hα (k)
In this section we study the spectral properties of the two-particle discrete Schr¨odinger operator hα (k), k ∈ T3 . We consider the family of the self-adjoint operators hα (k), k ∈ T3 on the Hilbert space L2 (T3 ) hα (k) = h0α (k) − µα v. (5.1) The nonperturbed operator h0α (k) on L2 (T3 ) is multiplication operator by (α) the function Ek (p) (α)
(h0α (k)f )(p) = Ek (p)f (p),
f ∈ L2 (T3 ),
(α)
where Ek (p) is defined in (3.3). The perturbation v is an integral operator of rank one −3 f (q)dq, f ∈ L2 (T3 ). (vf )(p) = (2π) T3
Therefore by the Weyl theorem the continuous spectrum σcont (hα (k)) of the operator hα (k), k ∈ T3 coincides with the spectrum σ(h0α (k)) of h0α (k). More specifically, (α) (α) (k)], σcont (hα (k)) = [Emin (k), Emax where
(α)
(α)
Emin (k) ≡ min3 Ek (p), p∈T
Set lβγ =
mγ , mβ + mγ
(α)
(α) Emax (k) ≡ max3 Ek (p). p∈T
β, γ = 1, 2, 3,
(5.2)
β = γ.
Lemma 5.1 There exist an odd and analytic function pα : T3 → T3 such that for any k ∈ T3 the point pα (k) is a unique nondegenerate minimum of the function (α) Ek (p) and pα (k) = lγβ k + O(|k|3 ) as k → 0. (5.3)
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(α)
Proof. The function Ek (p) can be rewritten in the form (α)
Ek (p) = 3m−1 βγ −
3
(aα (k (j) ) cos p(j) + bα (k (j) ) sin p(j) ),
(5.4)
j=1
where the coefficients aα (k (j) ) and bα (k (j) ) are given by −1 (j) aα (k (j) ) = m−1 , β + mγ cos k
(j) bα (k (j) ) = m−1 . γ sin k
(5.5)
(α)
The equality (5.4) implies the following representation for Ek (p) (α)
Ek (p) = 3m−1 βγ −
3
rα (k (j) ) cos(p(j) − pα (k (j) )),
(5.6)
j=1
where rα (k (j) ) =
a2α (k (j) ) + b2α (k (j) ),
pα (k (j) ) = arcsin
bα (k (j) ) , rα (k (j) )
k (j) ∈ (−π, π].
Taking into account (5.5), we have that the vector function pα : T3 → T3 , pα = pα (k (1) , k (2) , k (3) ) = (pα (k (1) ), pα (k (2) ), pα (k (3) )) ∈ T3 (α)
is odd regular and it is the minimum point of Ek (p). One has, as easily seen from the definition pα (k) = lγβ k + O(|k|3 ) as k → 0. Let C be the complex plane. For any k ∈ T3 and z∈C\σcont (hα (k)) we define a function (the Fredholm determinant associated with the operator hα (k)) (α) ∆α (k, z) = 1 − µα (2π)−3 (Ek (q) − z)−1 dq. T3
Note that the function ∆α (k, z) is real-analytic in T3 × (C\σcont (hα (k))) The following lemma is a simple consequence of the Birman-Schwinger principle and the Fredholm theorem. Lemma 5.2 Let k∈T3 . The point z∈C\σcont (hα (k)) is an eigenvalue of the operator hα (k) if and only if ∆α (k, z) = 0. Lemma 5.3 The following statements are equivalent: (i) the operator hα (0) has a zero energy resonance; (ii) ∆α (0, 0) = 0; (iii) µα = µ0α .
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Proof. Let the operator hα (0) has a zero energy resonance for some µα > 0. Then by Definition 4.1 the equation ϕ(p) = µα mβγ (2π)−3 (ε(q))−1 ϕ(q)dq T3
has a simple solution in C(T3 ) and the solution ϕ(q) is equal to 1 (up to a constant factor). Therefore we see that −3 (ε(q))−1 dq 1 = µα mβγ (2π) T3
and hence ∆α (0, 0) = 1 − µα mβγ (2π)−3
(ε(q))−1 dq = 0
T3
µ0α .
and so µα = Let for some µα > 0 the equality −3
∆α (0, 0) = 1 − µα mβγ (2π)
(ε(q))−1 dq = 0
T3
holds and consequently µα = µ0α . Then only the function ϕ(q) ≡ constant ∈ C(T3 ) is a solution of the equation ϕ(p) = µα mβγ (2π)−3 (ε(q))−1 ϕ(q)dq, T3
that is, the operator hα (0) has a zero energy resonance.
Theorem 5.4 Let the operator hα (0) have a zero energy resonance. Then for all k ∈ T3 , k = 0 the operator hα (k) has a unique simple eigenvalue zα (k) below the bottom of the continuous spectrum of hα (k). Moreover zα (k) is even on T3 and zα (k) > 0 for k = 0. Proof. By Lemma 5.3 ∆α (0, 0) = 1 −
µ0α mβγ (2π)−3
(ε(q))−1 dq = 0
T3
and hence it is easy to see that for any z < 0 the inequality ∆α (0, z) > 0 holds. By Lemma 5.2 the operator hα (0) has no negative eigenvalues. Since p = pα (k) is (α) (α) the nondegenerate minimum of the function Ek (p) we define ∆α (k, Emin (k)) as (α) (α) (α) 0 −3 ∆α (k, Emin (k)) = 1 − µα (2π) (Ek (q) − Emin (k))−1 dq. T3
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By dominated convergence theorem we have (α)
lim
(α)
z→Emin (k)
∆α (k, z) = ∆α (k, Emin (k)).
For all k = 0, q = 0 the inequality (α)
(α)
Ek (q + pα (k)) − Emin (k) < E0 (q) holds and hence we obtain the following inequality (α)
∆α (k, Emin (k)) < ∆α (0, 0) = 0, k = 0.
(5.7) (α)
For each k ∈ T3 the function ∆α (k, ·) is monotone decreasing on (−∞, Emin (k)] and ∆α (k, z) → 1 as z → −∞. Then by virtue of (5.7) there is a number zα (k) ∈ (α) (−∞, Emin (k)) such that ∆α (k, zα (k)) = 0. By Lemma 5.2 for any nonzero k ∈ T3 (α) the operator hα (k) has an eigenvalue below Emin (k). For any k ∈ T3 and z ∈ (α) (−∞, Emin (k)) the equality ∆α (−k, z) = ∆α (k, z) holds and hence zα (k) is even. Let us prove the positivity of the eigenvalue zα (k), k = 0. First we verify, for all k ∈ T3 , k = 0, the inequality ∆α (k, 0) > 0.
(5.8)
Applying the definition of µ0α by (4.1) we have εγ (k − p) − εγ (p) dp. ∆α (k, 0) = µ0α (2π)−3 (α) (α) E0 (p)Ek (p) 3
(5.9)
T
k 2
Making a change of variables q = ∆α (−k, 0) it is easy to show that
∆α (k, 0) = =
µ0α (2π)−3 2
(εγ ( T3
− p in (5.9) and using the equality ∆α (k, 0) = ∆α (k, 0) + ∆α (−k, 0) 2
k k k k + p) − εγ ( − p))(εβ ( + p) − εβ ( − p))F (k, p)dp, 2 2 2 2
where (α)
F (k, p) =
(α)
(α)
E0 ( k2 + p) + E0 ( k2 − p) (α)
(α)
(α)
E0 ( k2 + p)E0 ( k2 − p)Ek ( k2 + p)Ek ( k2 − p)
> 0.
A simple computation shows that 4 k k k k (εγ ( +p)−εγ ( −p))(εβ ( +p)−εβ ( −p)) = 2 2 2 2 mβ mγ Thus the inequality (5.8) is proven.
3
k (i) cos p(i) cos 2 i=1
2 ≥ 0.
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For any k ∈ T3 the function ∆α (k, ·) is monotone decreasing and the inequalities
(α)
∆α (k, 0) > ∆α (k, zα (k)) = 0 > ∆α (k, Emin (k)),
k = 0
hold. Therefore the eigenvalue zα (k) of the operator hα (k) belongs to interval (α) (0, Emin (k)). The following decomposition is important for the proof of the asymptotics (4.3). Lemma 5.5 Let µα = µ0α , α = 1, 2, 3. Then for any k ∈ Uδ (0), δ > 0 sufficiently (α) small, and z ≤ Emin (k) the following decomposition holds: ∆α (k, z) = (20)
3/2 12 µ0α mβγ (α) (α) (02) √ Emin (k) − z + ∆(20) α (Emin (k) − z) + ∆α (k, z), 2π
(α)
(α)
where ∆α (Emin (k) − z) = O(Emin (k) − z) O(|k|2 ) as k → 0. Proof. Let
(α)
as
(α)
(02)
z → Emin (k) and ∆α (k, z) =
(α)
Eα (k, p) = Ek (p + pα (k)) − Emin (k), (α)
(α)
where pα (k) ∈ T3 is the minimum point of the function Ek (p), that is, Emin (k) = (α) Ek (pα (k)). Then using (5.6) we conclude Eα (k, p) =
3
rα (k (j) )(1 − cos p(j) ).
j=1
˜ α (k, w) on T3 × C+ by ∆ ˜ α (k, w) = ∆α (k, E (α) (k) − w2 ), We define the function ∆ min ˜ α (k, w) represented as where C+ = {z ∈ C : Rez > 0}. The function ∆ dp ˜ α (k, w) = 1 − µα (2π)−3 ∆ Eα (k, p) + w2 T3 dp = 1 − µα (2π)−3 . 3 (j) 3 )(1 − cos p(j) ) + w2 T j=1 rα (k Let Vδ (0) be the complex δ-neighborhood of the point w = 0 ∈ C. Denote ˜ α (k, w) to the region T3 × by ∆∗α (k, w) the analytic continuation of the function ∆ 3 (C+ ∪ Vδ (0)). This function is even in k ∈ T . Therefore ˜ (20) (k, w), ∆∗α (k, w) = ∆∗α (k, w) + ∆ α (20)
˜ α (k, w) = O(|k|2 ) uniformly in w ∈ C+ as k → 0. Taylor series expanwhere ∆ sion gives ˜ (01) (0, 0)w + ˜ (02) (0, w)w2 ∆∗α (k, w) = α α
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˜ (02) where α (0, w) = O(1)
Ann. Henri Poincar´e
as w → 0. Simple computation shows that 3/2
µ0 m ∂∆∗α (0, 0) ˜ (01) (0, 0) = α√ βγ = 0. = α ∂w 2π
(5.10)
(α)
Corollary 5.6 The function zα (k) = Emin (k) − wα2 (k) is real-analytic in Uδ (0), ˜ α (k, w) = 0 and wα (k) = where wα (k) is a unique simple solution of the equation ∆ O(|k|2 ) as k → 0. ˜ α (0, 0) = 0 and the inequality (5.10) holds the equation ∆ ˜ α (k, w) = Proof. Since ∆ 0 has a unique simple solution wα (k), k ∈ Uδ (0) and it is real-analytic in Uδ (0). ˜ α (k, w) is even in k ∈ Uδ (0), δ > 0 and Taking into account that the function ∆ (α) 2 wα (0) = 0 we have that wα (k) = O(|k| ). Therefore the function zα (k) = Emin (k)− 2 wα (k) is real-analytic in Uδ (0). Lemma 5.7 Let µα = µ0α for some α = 1, 2, 3. Then for any k ∈ Uδ0 (0) there exists a number δ(k) > 0 such that, for all z ∈ Vδ(k) (zα (k)), where Vδ(k) (zα (k)) is the δ(k)-neighborhood of the point zα (k), the following representation holds ˆ α (k, z). ∆α (k, z) = C1 (k)(z − zα (k))∆ ˆ α (k, z) is continuous in Vδ(k) (zα (k)) and ∆ ˆ α (k, zα (k)) = 0. Here C1 (k) = 0 and ∆ (α)
Proof. Since zα (k) < Emin (k), k = 0 the function ∆α (k, z) can be expanded as follows ∞ Cn (k)(z − zα (k))n , z ∈ Vδ(k) (zα (k)), ∆α (k, z) = n=1
where
3/2
µoα mβγ 1 = 0, C1 (k) = √ 2π 2 E (α) (k) − z (k) α min
k = 0.
ˆ α (k, z) is continuous in Vδ(k) (zα (k)). Since zα (k), k = 0 is a Therefore ∆ (α)
unique simple solution of the equation ∆α (k, z) = 0, z ≤ Emin (k), we have ˆ α (k, zα (k)) = 0. ∆
6 Spectrum of the operator H(K) The “channel operator” Hα (K), K ∈T3 acts in the Hilbert space L2 ((T3 )2 ) as Hα (K) = H0 (K) − Vα .
(6.1)
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The decomposition of the space L2 ((T3 )2 ) into the direct integral 3 2 L2 ((T ) ) = ⊕L2 (T3 )dp p∈T3
yields for the operator Hα (K) the decomposition into the direct integral ⊕Hα (K, p)dp. Hα (K) = p∈T3
The fiber operator Hα (K, p) acts in the Hilbert space L2 (T3 ) and has the form Hα (K, p) = hα (K − p) + εα (p)I, where I is identity operator and hα (k) is the two-particle operator defined by (5.1). The representation of the operator Hα (K, p) implies the equality
(α) (α) σ(Hα (K, p)) = σd (hα (K − p)) ∪ Emin (K − p), Emax (K − p) + εα (p), where σd (hα (k)) is the discrete spectrum of the operator hα (k). The theorem (see, e.g.,[21]) on the spectrum of decomposable operators and above-obtained structure for the spectrum of Hα (K, p) gives Lemma 6.1 The equality holds σ(Hα (K)) = ∪p∈T3 {σd (hα (K − p) + εα (p)} ∪ [Emin (K), Emax (K)]. Lemma 6.2 Let µα = µ0α for some α = 1, 2, 3. Then for any K ∈ Uδ0 (0), δ > 0 sufficiently small, the following inequality τsα (K) < Emin (K) holds, where τsα (K) is defined in (4.2). Proof. By Theorem 5.4 for each K ∈ T3 and p ∈ T3 , p = K the operator hα (K − p) has an unique simple positive eigenvalue zα (K − p) below the bottom of σcont (hα (K − p)). Therefore using Lemma 6.1 for the spectrum σ(Hα (K)) of the operator Hα (K) we conclude that τsα (K) = inf ∪p∈T3 [εα (p) + σ(hα (K − p))] = min3 [εα (p) + zα (K − p)] . p∈T
By Theorem 5.4 for each K ∈ Uδ0 (0) and p = K the inequality (α)
εα (p) + zα (K − p) < Emin (K − p) + εα (p) holds. On the other hand, by computing partial derivatives, it is easy to see that for any K ∈ Uδ0 (0) the point p = K cannot be a minimum point for the function (α) Emin (K − p) + εα (p). Therefore τsα (K) < Emin (K) holds.
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Proof of Theorem 4.4. One can check that sup (Vα f, f ) = µα ,
α = 1, 2, 3.
||f ||=1
We obtain (H(K)f, f ) = (H0 (K)f, f ) − (Vα f, f ) − (Vβ f, f ) − (Vγ f, f ) = (Hα (K)f, f ) − (Vβ f, f ) − (Vγ f, f ) ≥ (Hα (K)f, f ) − (µ0β + µγ )||f ||2 . Thus
inf (H(K)f, f ) ≥ inf (Hα (K)f, f ) − µ0β − µγ .
||f ||=1
||f ||=1
The definition (4.2) of τsα (K) imply that τsα (K) − µ0β − µγ ≤ inf (H(K)f, f ) = τs (K) < τess (K). ||f ||=1
Let Wα (K, z), α = 1, 2, 3 be the operators on L2 ((T3 )2 ) defined as 1
1
Wα (K, z) = I + Vα2 Rα (K, z)Vα2 , where Rα (K, z), α = 1, 2, 3 are the resolvents of Hα (K), α = 1, 2, 3. One can check that 1 1 Wα (K, z) = (I − Vα2 R0 (K, z)Vα2 )−1 , where R0 (K, z) the resolvent of the operator H0 (K). For z < τess (K), τess (K) = inf σess (H(K)) the operators Wα (K, z), α = 1, 2, 3 are positive. (3) Denote by L = L2 ((T3 )2 ) the Hilbert space of vector functions w with 3 2 components wα ∈ L2 ((T ) ), α = 1, 2, 3. Let T(K, z), z < τess (K) be the operator on L with the entries Tαα (K, z) = 0, 1
1
1
1
Tαβ (K, z) = Wα2 (K, z)Vα2 R0 (K, z)Vβ2 Wβ2 (K, z). For any bounded self-adjoint operator A acting in the Hilbert space H not having any essential spectrum on the right of the point z we denote by HA (z) the subspace such that (Af, f ) > z(f, f ) for any f ∈ HA (z) and set n(z, A) = supHA (z) dim HA (z). By the definition of N (K, z) we have N (K, z) = n(−z, −H(K)), −z > −τess (K). The following lemma is a realization of the well-known Birman-Schwinger principle for the three-particle Schr¨ odinger operators on a lattice (see [22, 24] ).
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Lemma 6.3 For z < τess (K) the operator T(K, z) is compact and continuous in z and N (K, z) = n(1, T(K, z)). Proof. First verify the equality 1
1
N (K, z) = n(1, R02 (K, z)V R02 (K, z)),
V = V1 + V2 + V3 .
(6.2)
Assume that u ∈ H−H(K) (−z), that is, ((H0 (K) − z)u, u) < (V u, u). Then 1
1
1
y = (H0 (K) − z) 2 u.
(y, y) < (R02 (K, z)V R02 (K, z)y, y), 1
1
Thus N (K, z) ≤ n(1, R02 (K, z)V R02 (K, z)). Reversing the argument we get the opposite inequality, which proves (6.2). Now we use the following well-known fact. Proposition 6.4 Let T1 , T2 be bounded operators. If z = 0 is an eigenvalue of T1 T2 then z is an eigenvalue for T2 T1 as well of the same algebraic and geometric multiplicities. Using Proposition 6.4 we get 1
1
n(1, R02 (K, z)V R02 (K, z)) = n(1, M(K, z)), where M(K, z) the operator on L with the entries 1
1
Mαβ = Vα2 R0 (K, z)Vβ2 ,
α, β = 1, 2, 3.
Let us check that n(1, M(K, z)) = n(1, T(K, z)). We shall show that for any u ∈ HM(K,z) (1) there exists y ∈ HT(K,z) (1) such that (y, y) < (T(K, z)y, y). Let u ∈ HM(K,z) (1), that is, 3
(uα , uα ) <
α=1
3
1
1
(Vα2 R0 (K, z)Vβ2 uβ , uα )
α,β=1
and hence 3
1
1
((I − Vα2 R0 (K, z)Vα2 )uα , uα ) <
3
1
1
(Vα2 R0 (K, z)Vβ2 uβ , uα ).
β=α=1
α=1 1 2
1 2
1
Denoting by yα = (I − Vα R0 (K, z)Vα ) 2 uα we have 3
(yα , yα ) <
α=1
3
1
1
1
1
(Wα2 (K, z)Vα2 R0 (K, z)Vβ2 Wβ2 (K, z)yβ , yα ),
β=α=1
that is, (y, y) ≤ (T(K, z)y, y). Thus n(1, M(K, z)) ≤ n(1, T(K, z)). By the same way one can check n(1, T(K, z)) ≤ n(1, M(K, z)).
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Σ(K) := ∪3α=1 σ(Hα (K)). (3)
Denote by L2 (T3 ) the space of vector functions w = (w1 , w2 , w3 ), wα ∈ L2 (T3 ), (3) α = 1, 2, 3 and define compact operator T (K, z), z ∈ C \ Σ(K) on L2 (T3 ) with the entries Tαα (K, z) = 0,
√ (Tαβ (K, z)wβ )(kα ) = µα µβ (2π)−3
−1
−1
∆α 2 (K, kα , z)∆β 2 (K, kβ , z) Eαβ (K; kα , kβ ) − z
T3
wβ (kβ )dkβ ,
(3)
w ∈ L2 (T3 ), where ∆α (K, kα , z) := ∆α (K − kα , z − εα (kα )). Now we show that the numbers of eigenvalues greater than 1 of the operators T(K, z) and T (K, z) are coincide. Let (3) (3) Ψ = diag{Ψ1 , Ψ2 , Ψ3 } : L2 ((T3 )2 ) → L2 (T3 ) be the operator with the entries − 32
(Ψα f )(kα ) = (2π)
T3
f (kα , q)dq,
α = 1, 2, 3
(6.3)
and Ψ∗ = diag{Ψ∗1 , Ψ∗2 , Ψ∗3 } its adjoint. Lemma 6.5 The following equalities T(K, z) = Ψ∗ T (K, z)Ψ
and
n(1, T(K, z)) = n(1, T (K, z))
hold. Proof. One can easily check that the equalities 3
−1
Ψα f = (2π) 2 µα 2 Vα1/2 f
−1
and Vα1/2 Wα1/2 f = ∆α 2 (K, kα , z)Vα1/2 f
(6.4)
hold. The equalities (6.4) imply the first equality of Lemma 6.5. By Proposition 6.4 we have n(1, T(K, z)) = n(1, Ψ∗ T (K, z)Ψ) = n(1, T (K, z)ΨΨ∗) = n(1, T (K, z)).
Now we establish a location of the essential spectrum of H(K). For any K ∈ T3 and z ∈ C \ Σ(K) the kernels of the operators Tαβ (K, z), α, β = 1, 2, 3 are continuous functions on (T3 )2 . Therefore the Fredholm determinant DK (z) of the (3) operator I − T (K, z), where I is the identity operator in L2 (T3 ), exists and is a
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real-analytic function on C \ Σ(K). The following theorem is a lattice analog of the well-known Faddeev’s result for the three-particle Schr¨odinger operators with the zero-range interactions and can be proved similarly to that of the identical particle case (see [12]). Theorem 6.6 For any K ∈ T3 the number z ∈ C \ Σ(K) is an eigenvalue of the operator H(K) if and only if the number 1 is eigenvalue of T (K, z). According to Fredholm’s theorem the following lemma holds. Lemma 6.7 The number z ∈ C \ Σ(K) is an eigenvalue of the operator H(K) if and only if DK (z) = 0. Proof of Theorem 4.3. By the definition of the essential spectrum, it is easy to show that Σ(K) ⊂ σess (H(K)). Since the function DK (z) is analytic in C \ Σ(K) by Lemma 6.7 we conclude that the set σ(H(K)) \ Σ(K) = {z : DK (z) = 0} is discrete. Thus σ(H(K)) \ Σ(K) ⊂ σ(H(K)) \ σess (H(K)). Therefore the inclusion σess (H(K)) ⊂ Σ(K) holds.
Now we are going to prove the finiteness of N (K, τess (K)) for K ∈ Uδ0 (0), δ > 0 sufficiently small. First we shall prove that the operator T (K, τess (K)) belongs to the Hilbert-Schmidt class. The point p = 0 is the nondegenerate minimum of the functions εα (p) and zα (p) (see Corollary 5.6) and hence p = 0 is the nondegenerate minimum of Zα (0, p) defined by Zα (K, p) := εα (p) + zα (K − p). By the definition of εα (p) we have εα (p) =
1 2 p + O(|p|4 ) as p → 0. 2mα
(6.5)
(α)
Using the definition of Emin (k), asymptotics (5.3) and (5.4) we obtain that (α)
Emin (k) =
1 k 2 + O(|k|4 ) as k → 0. 2(mβ + mγ )
Corollary 5.6 and simple computations give 100 ∂ 2 Zα (0, 0) 3 nα 0 1 0 , = 2 ∂p(i) ∂p(j) i,j=1 001 where nα ≡
M mα (mβ +mγ ) .
(6.6)
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Therefore for all K ∈ Uδ0 (0) at the nondegenerate minimum point pZ α (K) ∈ of the function Zα (K, p), K ∈ Uδ0 (0) the inequality
Uδ0 (0)
B(K) =
3 ∂ 2 Zα (K, pZ α (K)) i,j=1 > 0 (i) (j) ∂p ∂p
holds. Hence the asymptotics Z Zα (K, p) = τsα (K) + (B(K)(p − pZ α (K)), p − pα (K)) 2 Z + o(|p − pZ α (K)| ) as |p − pα (K)| → 0 (6.7)
is valid, where τsα (K) = Zα (K, pZ α (K)). From Lemma 5.7 we conclude that for all K ∈ Uδ0 (0), p ∈ Uδ(K) (pZ (K)) the equality α ˆ α (K, p, τ α (K)) ∆α (K, p, τsα (K)) = (Zα (K, p) − τsα (K))∆ s
(6.8)
ˆ α (K, pZ (K), τ α (K)) = 0. Putting (6.7) into (6.8) we get the folholds, where ∆ α s lowing Lemma 6.8 Let µα = µ0α , α = 1, 2, 3. Then for any K ∈ Uδ0 (0), δ = δ(K) sufficiently small, there are positive nonzero constants c and C depending on K and Z Uδ(K) (pZ α (K)) such that for all p ∈ Uδ(K) (pα (K)) the following inequalities 2 α Z 2 c|p − pZ α (K)| ≤ ∆α (K, p, τs (K)) ≤ C|p − pα (K)|
(6.9)
hold.
Lemma 6.9 Let µα ≤ µ0α for all α = 1, 2, 3. Then for any K ∈ Uδ0 (0), δ > 0 sufficiently small, the operator T (K, τess (K)) belongs to the Hilbert-Schmidt class. Proof. As we shall see that it is sufficient to prove Lemma 6.9 in the case µα = µ0α for all α = 1, 2, 3. By Lemma 6.2 we have τess (K) = min τsα (K) < Emin (K), K ∈ Uδ0 (0). α
(6.10)
The operator hα (0) has a zero energy resonance. By Theorem 5.4 the operator (α) hα (k), k ∈ T3 , k = 0 has a unique eigenvalue zα (k), zα (k) < Emin (k). Since τsα (K) = minp∈T3 Zα (K, p) the function Zα (K, p) has a unique minimum and hence for all p ∈ T3 \ Uδ (pZ α (K)) we obtain ∆α (K, p, τsα (K)) ≥ C > 0.
(6.11)
According to Lemma 6.2 for all pα , pβ ∈ T3 and K ∈ Uδ0 (0) the inequality Eαβ (K; pα , pβ ) − τsα (K) ≥ Emin (K) − τsα (K) > 0
(6.12)
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holds. Using (6.9), (6.11) and taking into account (6.12) we can make certain that Z for all K ∈ Uδ0 (0) and pα ∈ Uδ (pZ α (K)), pβ ∈ Uδ (pβ (K)) the modules of the kernels Tαβ (K, τess (K); pα , pβ ) of the integral operators Tαβ (K, τess (K)) can be estimated by C0 (K) + C1 , Z |pα − pα (K)||pβ − pZ β (K)| where C0 (K) and C1 some constants. Taking into account (6.10) we conclude that Tαβ (K, τess (K)), α, β = 1, 2, 3 are Hilbert-Schmidt operators. Thus, T (K,τess (K)) belongs to the Hilbert-Schmidt class. Now we shall prove the finiteness of N (K, τess (K)) (Theorem 4.6). Theorem 6.10 Assume Hypothesis 4.2. Then for the number N (K, τess (K)) the relation N (K, τess (K)) ≤ lim n(1 − γ, T (K, τess (K))) γ→0
holds. Proof. By Lemmas 6.3 and 6.5 we have N (K, z) = n(1, T (K, z)) as z < τess (K) and by Lemma 6.9 for any γ ∈ [0, 1) the number n(1 − γ, T (K, τess (K))), K ∈ Uδo (0) is finite. Then according to the Weyl inequality n(λ1 + λ2 , A1 + A2 ) ≤ n(λ1 , A1 ) + n(λ2 , A2 ) for all z < τess (K) and γ ∈ (0, 1) we have N (K, z) = n(1, T (K, z)) ≤ n(1−γ, T (K, τess (K)))+n(γ, T (K, z)−T (K, τess(K))). Since T (K, z) is continuous from the left up to z = τess (K), K ∈ Uδ0 (0), we obtain lim
z→τess (K)
N (K, z) = N (K, τess (K)) ≤ n(1 − γ, T (K, τess (K))) for all γ ∈ (0, 1)
and so N (K, τess (K)) ≤ lim n(1 − γ, T (K, τess (K))). γ→0
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7 Asymptotics for the number of eigenvalues of the operator H(K) We recall that in this section we closely follow A. Sobolev’s method to derive the asymptotics for the number of eigenvalues of H(K) (Theorem 4.7). As we shall see, the discrete spectrum asymptotics of the operator T (K, z) as |K| → 0 and z → −0 is determined by the integral operator Sr , r = 1/2| log( in
|K|2 + |z|)| 2M
L2 ((0, r) × σ (3) ), σ = L2 (S2 ),
with the kernel Sαβ (x − x ; ξ, η), ξ, η ∈ S2 , S2 is the unit sphere in R3 , where Sαα (x; t) = 0, and
Sαβ (x; t) = (2π)−2
uαβ cosh(x + rαβ ) + sαβ t
(7.1)
−1 m−1 mβγ 1 (mαγ mβγ ) 2 βγ mαγ 14 = kαβ , rαβ = log , sαβ = , nα nβ 2 mαγ mγ 1
uαβ
kαβ being such that kαβ = 1 if both subsystems α and β have zero resonances, otherwise kαβ = 0. The eigenvalues asymptotics for the operator Sr have been studied in detail by Sobolev [22], by employing an argument used in the calculation of the canonical distribution of Toeplitz operators. We here summarize some results obtained in [22]. Lemma 7.1 The following equality lim
r→∞
1 −1 r n(λ, Sr ) = U(λ) 2
holds, where the function U(λ) is continuous in λ > 0 and U0 in (4.3) defined as U0 = U(1). Lemma 7.2 Let A(z) = A0 (z) + A1 (z), where A0 (A1 ) is compact and continuous in z < 0 (z ≤ 0). Assume that for some function f (·), f (z) → 0, z → −0 the limit lim f (z)n(λ, A0 (z)) = l(λ),
z→−0
exists and is continuous in λ > 0. Then the same limit exists for A(z) and lim f (z)n(λ, A(z)) = l(λ).
z→−0
Now we are going to reduce the study of the asymptotics for the operator T (K, z) to that of the asymptotics of Sr .
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By (6.5) we obtain Eαβ (K; lα K − p, lβ K − q) (p, q) q2 K2 p2 + O(|K|4 + |p|4 + |q|4 ) as K, p, q → 0, (7.2) + + + 2mαγ mγ 2mβγ 2M
=
where mα is the mass of the particle α and mα , M ≡ m1 + m2 + m3 , lα ≡ M
α = 1, 2, 3.
From Lemma 5.5 and (6.6) we easily receive the following Lemma 7.3 For any K ∈ Uδ (0) and z ∈ [−δ, 0] we have ∆α (K, lα K − p, z) 12 3/2 µ0α mβγ K2 2 − 2z + O(|K|2 + |p|2 + |z|) as K, p, z → 0. (7.3) = nα p + 2π M The following theorem is basic for the proof of the asymptotics (4.3). Theorem 7.4 The equality n(1, T (K, z))
lim
|K|2 M
|K|2 +|z|→0 |log( M
+ |z|)|
= lim
r→∞
1 −1 r n(1, Sr ) 2
holds. Remark 7.5 Since U(.) is continuous in λ, according to Lemma 7.2 a compact and continuous up to z = 0 perturbations of the operator A0 (z), do not contribute to the asymptotics (4.3). During the proof of Theorem 7.4 we use this fact without further comments. First we prove Theorem 7.4 under the condition that all two-particle operators have zero energy resonances, that is, in the case where µ1 = µ01 , µ2 = µ02 and µ3 = µ03 . The case where only two operators hα (0) and hβ (0) have zero energy resonance can be proven similarly. Proof of Theorem 7.4. Let T˜(K, z), z ∈ C \ σess (H(K)), K ∈ Uδ (0) be operator (3) on L2 (T3 ) with the entries T˜αα (K, z) = 0, (T˜αβ (K, z)wβ )(pα ) √ = µα µβ (2π)−3
T3
(3)
w ∈ L2 (T3 ).
−1
−1
∆α 2 (K, lα K − pα , z)∆β 2 (K, lβ K − pβ , z) Eαβ (K; lα K − pα , lβ K − pβ ) − z
wβ (pβ )dpβ ,
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The operators T (K, z), K ∈ Uδ (0) and T˜(K, z), K ∈ Uδ (0) are unitarily equality and the equivalence is performed by the unitary dilation (1)
(2)
(3)
(3)
(3)
YK = diag{YK , YK , YK } : L2 ((T3 )2 ) → L2 ((T3 )2 ), (α)
(YK f )(pα ) = f (lα K − pα ). (3)
2
K + |z|) be an operator on L2 (T3 ) with the entries Let T (δ, 2M
K2 + |z|) = 0, 2M K2 (Tαβ (δ; + |z|)w)(p) 2M K2 K2 χδ (p)χδ (q)(nα p2 + 2( 2M + |z|))−1/4 (nβ q 2 + 2( 2M + |z|))−1/4 = Dαβ w(q)dq, 2 2 2 q p 2 K mβγ + mγ (p, q) + mαγ + 2( 2M + |z|) 3 Tαα (δ;
T
where
−3
Dαβ =
−3
mαγ4 mβγ4 2π 2
, α, β, γ = 1, 2, 3, α = β = γ
and χδ (·) is the characteristic function of Uδ (0) = {p : |p| < δ}. 2
K Lemma 7.6 The operator T˜(K, z) − T (δ; 2M + |z|) belongs to the Hilbert-Schmidt class and is continuous in K ∈ Uδ (0) and z ≤ 0.
Proof. Applying asymptotics (7.2) and (7.3) one can estimate the kernel of the K2 + |z|) by operator T˜αβ (K, z) − Tαβ (δ; 2M 1
1
C[(p2 + q 2 )−1 + |p|− 2 (p2 + q 2 )−1 + (|q|− 2 (p2 + q 2 )−1 + 1] K2 and hence the operator T˜αβ (K, z)−Tαβ (δ; 2M +|z|) belongs to the Hilbert-Schmidt class for all K ∈ Uδ (0) and z ≤ 0. In combination with the continuity of the kernel of the operator in K ∈ Uδ (0) and z < 0 this gives the continuity of T˜ (K, z) − K2 + |z|) in K ∈ Uδ (0) and z ≤ 0. T (δ; 2M
The space of vector functions w = (w1 , w2 , w3 ) with coordinates having supK2 port in Uδ (0) is an invariant subspace for the operator T (δ, 2M + |z|). Denote by Lδ the space of vector functions w = (w1 , w2 , w3 ), wα ∈L2 (Uδ (0)), that is, Lδ = ⊕3α=1 L2 (Uδ (0)). 2
2
K K Let T0 (δ, 2M + |z|) be the restriction of the operator T (δ, 2M + |z|) to the invariant 2 K subspace Lδ . One verifies that the operator T0 (δ, 2M + |z|) is unitarily equivalent
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2
K to the operator T1 (δ, 2M + |z|) with entries
K2 + |z|) = 0, 2M K2 (1) (Tαβ (δ; + |z|)w)(p) = Dαβ 2M
(1) Tαα (δ;
(nα p2 + 2)−1/4 (nβ q 2 + 2)−1/4 q2 mβγ
Ur (0)
2
(3)
+
2 mγ (p, q)
+
p2 mαγ
+2
w(q)dq
−2 acting in L2 (Ur (0)), r = ( |K| . The equivalence is performed by the 2M + |z|) unitary dilation 1
(3)
(3)
Br = diag{Br , Br , Br } : L2 (Uδ (0)) → L2 (Ur (0)),
r δ (Br f )(p) = ( )−3/2 f ( p). δ r
Further, we may replace (nα p2 + 2)−1/4 , (nβ q 2 + 2)−1/4
and
q2 2 p2 + (p, q) + +2 mβγ mγ mαγ
by (nα p2 )−1/4 (1 − χ1 (p)), (nβ q 2 )−1/4 (1 − χ1 (q))
and
q2 2 p2 + (p, q) + , mβγ mγ mαγ
respectively, since the error will be a Hilbert-Schmidt operator continuous up to (3) K = 0 and z = 0. Then we get the operator T (2) (r) in L2 (Ur (0) \ U1 (0)) with entries (2) (r) = 0, Tαα (2) (Tαβ (r)w)(p)
− 14
= (nα nβ )
Dαβ
|p|−1/2 |q|−1/2 q2
Ur (0)\U1 (0) mβγ
+
2 mγ (p, q)
+
p2 mαγ
w(q)dq.
This operator T (2) (r) is unitarily equivalent to the integral operator Sr with entries (7.1). The equivalence is performed by the unitary operator M = (3) diag{M, M, M } : L2 (Ur (0) \ U1 (0)) −→ L2 ((0, r) × σ (3) ), where (M f )(x, w) = e3x/2 f (ex w), x ∈ (0, r), w ∈ S2 .
Acknowledgment The authors are grateful to Prof. R.A. Minlos, Prof. K.A. Makarov and Dr. J.I. Abdullaev for useful discussions and to the referee for useful critical remaks. This work was supported by the DFG 436 USB 113/3 and DFG 436 USB 113/4 projects and the Fundamental Science Foundation of Uzbekistan. The last two named authors gratefully acknowledge the hospitality of the Institute of Applied Mathematics and of the IZKS of the University Bonn.
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References [1] S. Albeverio, R. Høegh-Krohn, and T.T. Wu, A class of exactly solvable threebody quantum mechanical problems and universal low energy behavior, Phys. Lett. A 83, 105–109 (1971). [2] S. Albeverio, S.N. Lakaev, K.A. Makarov, The Efimov Effect and an Extended Szeg˝o-Kac Limit Theorem, Letters in Math. Phys. V. 43, 73–85 (1998). [3] R.D. Amado and J.V. Noble, Efimov effect; A new pathology of three-particle Systems. II, Phys. Lett. B 35 No.1, 25–27(1971); Phys. Lett. D 5 No.8, 1992– 2002 (1972). [4] G.F. Dell’Antonio, R. Figari, A. Teta, Hamiltonian for systems of N particles interacting through point interactions, Ann. Inst. H. Poincar´e Phys. Theor. 60, no. 3, 253–290 (1994). [5] V. Efimov, Energy levels of three resonantly interacting particles, Nucl. Phys. A 210, 157–158 (1973). [6] L.D. Faddeev, Mathematical aspects of the three-body problem in quantum mechanics, Israel Program for Scientific Translations, Jerusalem, 1965. [7] L.D. Faddeev and S.P. Merkuriev, Quantum scattering theory for several particle systems, Kluwer Academic Publishers, 1993. [8] G.M. Graf, D. Schenker, 2-magnon scattering in the Heisenberg model, Ann. Inst. H. Poincar´e Phys. Theor. 67, no. 1, 91–107 (1997). [9] S.N. Lakaev and R.A. Minlos, On bound states of the cluster operator, Theor. and Math. Phys. 39, No.1, 336–342 (1979). [10] S.N. Lakaev, On an infinite number of three-particle bound states of a system of quantum lattice particles, Theor. and Math. Phys. 89, No.1, 1079–1086 (1991). [11] S.N. Lakaev, The Efimov’s Effect of a system of Three Identical Quantum lattice Particles, Funkcionalnii analiz i ego priloj. 27, No.3, 15–28 (1993), translation in Funct. Anal.Appl. [12] S.N. Lakaev, J.I. Abdullaev, Finiteness of the discrete spectrum of the threeparticle Schr¨odinger operator on a lattice, Theor. Math. Phys. 111, 467–479 (1997). [13] S.N. Lakaev and S.M. Samatov, On the finiteness of the discrete spectrum of the Hamiltonian of a system of three arbitrary particles on a lattice, Teoret. Mat. Fiz. 129, No. 3, 415–431 (2001) (Russian).
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[14] S.N. Lakaev and M.Kh. Shermatov, On infiniteness of the discrete spectrum of Hamiltonian of system of three-particles (two fermion and other), Uspexi matem. nauk 54, 165–166 (1999). [15] S.N. Lakaev and J.I. Abdullaev, The spectral properties of the three-particle difference Schr¨odinger operator, Funct. Anal. Appl. 33, No. 2, 84–88 (1999). [16] S.N. Lakaev and Zh.I. Abdullaev, The spectrum of the three-particle difference Schr¨odinger operator on a lattice. Math. Notes 71, No. 5–6, 624–633 (2002). [17] V.A. Malishev and R.A. Minlos, Linear infinite-particle operators. Translations of Mathematical Monographs, 143. American Mathematical Society, Providence, RI, 1995. [18] D.C. Mattis, The few-body problem on lattice, Rev. Modern Phys. 58, No. 2, 361–379 (1986). [19] A.I. Mogilner, The problem of a quasi-particles in solidstate physics I n; Application of Self-adjoint Extensions in Quantum Physics (P. Exner and P. Seba eds.) Lect. Notes Phys. 324 (1998), Springer-Verlag, Berlin. [20] Yu.N. Ovchinnikov and I.M. Sigal, Number of bound states of three-particle systems and Efimov’s effect, Ann. Physics 123, 274–295 (1989). [21] M. Reed and B. Simon, Methods of modern mathematical physics. III: Scattering theory. Academic Press, N.Y., 1979. [22] A.V. Sobolev, The Efimov effect. Discrete spectrum asymptotics, Commun. Math. Phys. 156, 127–168 (1993). [23] H. Tamura, The Efimov effect of three-body Schr¨ odinger operator, J. Funct. Anal. 95, 433–459 (1991). [24] H. Tamura, Asymptotics for the number of negative eigenvalues of three-body Schr¨ odinger operators with Efimov effect. Spectral and scattering theory and applications, 311–322, Adv. Stud. Pure Math., 23, Math. Soc. Japan, Tokyo (1994). [25] D.R. Yafaev, On the theory of the discrete spectrum of the three-particle Schr¨ odinger operator, Math. USSR–Sb. 23, 535–559 (1974).
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S. Albeverio, S.N. Lakaev and Z.I. Muminov
Sergio Albeverio Institut f¨ ur Angewandte Mathematik Universit¨ at Bonn Wegelerstr. 6 D-53115 Bonn Germany SFB 256, Bonn, BiBoS, Bielefeld – Bonn CERFIM, Locarno and Acc. ARch, USI (Switzerland) email:
[email protected] Saidakhmat N. Lakaev Samarkand State University Samarkand Uzbekistan Academy of Sciences of Uzbekistan email:
[email protected] Zahriddin I. Muminov Samarkand State University Samarkand Uzbekistan email:
[email protected] Communicated by Gian Michele Graf submitted 19/11/03, accepted 08/03/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e
Ann. Henri Poincar´e 5 (2004) 773 – 808 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/040773-36 DOI 10.1007/s00023-004-0182-8
Annales Henri Poincar´ e
Rotating Singular Perturbations of the Laplacian Michele Correggi and Gianfausto Dell’Antonio Abstract. We study a system of a quantum particle interacting with a singular timedependent uniformly rotating potential in 2 and 3 dimensions: in particular we consider an interaction with support on a point (rotating point interaction) and on a set of codimension 1 (rotating blade). We prove the existence of the Hamiltonians of such systems as suitable self-adjoint operators and we give an explicit expression for the unitary dynamics. Moreover we analyze the asymptotic limit of large angular velocity and we prove strong convergence of the time-dependent propagator to some one-parameter unitary group as ω → ∞.
1 Introduction In this paper we shall study systems defined by formal time-dependent Schr¨odinger operators on L2 (Rn ), n = 2, 3 H(t) = H0 + Vt = −∆ + Vt
(1.1)
with uniformly rotating potentials Vt (x) = V (R−1 (t) x)
(1.2)
where V is a singular potential (e.g., V (x) = δ(x − y0 )) and R(t) a rotation on the x, y-plane with period 2π/ω: cos(ωt) − sin(ωt) 0 R(t) = sin(ωt) cos(ωt) 0 . 0 1 Regular rotating potentials were studied by Enss et al. [6] in order to extract information about the scattering of a quantum particle: indeed they considered a class of potentials such that the kinetic energy of the system remains bounded on the range of wave operators and they proved existence and completeness of the wave operators. Our purpose is to define in a rigorous way the time-dependent Hamiltonians (1.1) when the potential has a more singular behavior: we shall study rotating point perturbations1 of the Laplacian in 2 and 3 dimensions and rotating blades, namely 1 Point interactions were introduces for the first time in a rigorous way by Berezin and Faddeev in 1961 [3]. For general references about fixed and time-dependent point interactions see [2, 4, 5, 8].
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rotating singular potentials supported over a set of codimension 1 (a segment in 2 dimensions and an half-disk in 3 dimensions respectively). As pointed out by Enss et al., the uniformly rotating Hamiltonians can be studied in a simpler way than general time-dependent operators, indeed, considering the time evolution Urot (t, s) of the system in a uniformly rotating frame around the z−axis, it is easy to see that the following relation with the time evolution in the inertial frame Uinert (t, s) holds Uinert (t, s) = R(t) Urot (t − s) R† (s)
(1.3)
where R(t)Ψ(x) = Ψ(R(t)−1 x) and Urot (t, s) = Urot (t − s) is the one-parameter unitary group (1.4) Urot (t − s) = e−iK(t−s) with a time-independent generator K, formally defined in the following way K = H0 − ωJ + V .
(1.5)
Here J stands for the third component of the angular momentum and V is the time-independent potential (1.2). Using this trick we shall define the previous time-dependent Hamiltonians considering the corresponding formal time-independent generators in the rotating frame and studying their self-adjoint extensions. The last goal of this work will be the analysis of the asymptotic limit of the systems when the angular velocity ω → ∞: by means of the explicit expression of resolvents of singular perturbations of the Laplacian, we shall prove convergence in strong sense of Uinert (t, s) to some one-parameter unitary group Uasympt (t − s) with time-independent generator Hasympt . Moreover we shall see that, for point interactions, Hasympt is the Laplacian with singular perturbation on a circle, while the asymptotic limit of the rotating blade is simply a regular potential supported on a compact set. The same study was performed by Enss et al. [7] for regular rotating potentials.
2 The rotating point interaction in 3D 2.1
The Hamiltonian
The system we shall study is defined by the formal time-dependent Hamiltonian H(t) = H0 + a δ (3) (x − y (t))
(2.1)
where y (t) = R(t)y0 . According to the previous scheme, the formal generator of time evolution in the uniformly rotating frame (with angular velocity ω) is given by K = H0 − ωJ + a δ (3) (x − y0 ) .
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Therefore the Hamiltonian of the system is a self-adjoint extension of the operator K y 0 = Hω D(Ky0 ) = C0∞ (R3 − {y0 }) . The operator Ky0 is symmetric and then closable; let K˙ y0 be its closure, with domain D(K˙ y0 ). The function ∞ l ∞ 1 ϕ∗ (y ) ϕklm (x) Gz (x, y0 ) = dk (2.2) 2 − mω − z klm 0 k 0 l=0 m=−l
3
for x ∈ R − {y0 } and z ∈ C − R, is the unique solution of K˙ y∗0 Ψz (x) = zΨz (x) with Ψ ∈ D(K˙ y∗0 ) (see Proposition A.1). The operator K˙ y0 has then deficiency indices (1, 1) and its self-adjoint extensions are given by the one-parameter family of operators Kα,y0 , α ∈ [0, 2π): D(Kα,y0 ) = {f + cG+ + ceiα G− | g ∈ D(K˙ y0 ), c ∈ C}
(2.3)
Kα,y0 (f + cG+ + ceiα G− ) = K˙ y0 g + icG+ − iceiα G−
(2.4)
where G± (x) = G±i (x, y0 ) =
∞
dk 0
l ∞ l=0 m=−l
k2
1 ϕ∗ (y0 ) ϕklm (x) − mω ∓ i klm
3
for x ∈ R − {y0 }. Moreover the self-adjoint extension Kπ,y0 corresponds to the “free” Hamiltonian H˙ ω : indeed, if Ψ ∈ D(Kπ,y0 ), Ψ = f + c(G+ − G− ) and the difference G+ − G− is a continuous function at x = y0 , which belongs to the domain of Hω , so that Kπ,y0 becomes exactly the operator H˙ ω . Using this result and applying the Krein’s theory of self-adjoint extensions, it is easy to obtain the following Theorem 2.1 The resolvent of Kα,y0 has integral kernel given by (Kα,y0 − z)−1 (x, x ) = Gz (x, x ) + λ(z, α)Gz¯∗ (x , y0 )Gz (x, y0 ) with z ∈ (Kα,y0 ), x, x ∈ R3 , x = x , x, x = y0 and 1 1 = − (z + i) Gz¯(x), G− (x) λ(z, α) λ(−i, α) λ(−i, α) =
1 + eiα . 2iG− (x)2
(2.5)
(2.6) (2.7)
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Proof. Since K˙ y0 is a densely defined, closed, symmetric operator with deficiency indexes (1, 1), we can apply Krein’s theory (cf. [2, 11]) to classify all its self-adjoint extensions: from Krein’s formula we immediately obtain (Kα,y0 − z)−1 − (Kπ,y0 − z)−1 = λ(z, α) Gz¯(x), · Gz (x) for z ∈ (Kα,y0 ) ∩ (Hω ). It follows that (Kα,y0 − z)−1 has integral kernel given by (Kα,y0 − z)−1 (x, x ) = (H˙ ω − z)−1 (x, x ) + λ(z, α)Gz¯∗ (x , y0 )Gz (x, y0 ) Moreover λ(z, α) satisfies the following equation 1 1 = − (z − z ) Gz¯(x), Gz (x) . λ(z, α) λ(z , α) The explicit expression of the factor λ(−i, α) is given in the following Theorem. Theorem 2.2 The domain D(Kα,y0 ), α ∈ [0, 2π), consists of all elements Ψ ∈ R3 which can be decomposed in the following way Ψ(x) = Φz (x) + λ(z, α)Φz (y0 )Gz (x, y0 ) for x = y0 , Φz ∈ D(H˙ ω ) and z ∈ (Kα,y0 ). The previous decomposition is unique and on every Ψ of this form (Kα,y0 − z)Ψ = (Hω − z)Φz . Proof. First of all we observe that functions belonging to D(H˙ ω ) are H¨older continuous with exponent smaller than 1/2 in every compact subset of R3 . Indeed the 2 (R3 ): on every compact domain of self-adjointness of H˙ ω contains functions in Hloc 3 2 S set S ⊂ R , the domain of H0 is strictly contained on the domain of J S , since J S is a bounded operator on D(H˙ 0S ) = H 2 (S), therefore D(H˙ ωS ) = D(H˙ 0S ) = H 2 (S). Hence it makes sense to write Φ(y0 ) for every Φ ∈ D(H˙ ω ) and y0 ∈ R3 . Moreover D(Kα,y0 ) = (Kα,y0 − z)−1 (H˙ ω − z)D(H˙ ω ) and the claim follows from the expression of the resolvent given in the previous Theorem 2.2. To prove the uniqueness of the decomposition let Ψ = 0, so that Φz (x) = −
1 + eiα Φz (y0 )Gz (x) 2iG− (x)2
but Φz (x) must be continuous at x = y0 : it follows that Φz (y0 ) = 0 and then Φz = 0. 2 The
notation AS denotes the restriction of the operator A to the Hilbert space L2 (S).
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Finally the last equality of the theorem easily follows from (Kα,y0 − z)−1 (H˙ ω − z)Φz = Φz + λ(z, α) Gz¯(x), (H˙ ω − z)Φz (x) Gz = Ψ . To find the explicit expression of λ(−i, α) it is sufficient to study the behavior of y0 . Let Ψ(x) ∈ D(Kα,y0 ), functions in D(Kα,y0 ) at Ψ(x) = f (x) + cG+ (x) + ceiα G− (x) with f ∈ D(H˙ y0 ) and c ∈ C. Since ∞ ∞ l G+ (x) = dk 0
l=0 m=−l
1 2i + ϕ∗klm (y0 ) ϕklm (x) = k 2 − mω + i |k 2 − mω − i|2 = G− (x) + 2ig(x, y0 )
where
g(x, y0 ) =
∞
dk 0
l ∞ l=0 m=−l
1 ϕ∗ (y0 ) ϕklm (x) |k 2 − mω − i|2 klm
belongs to D(H˙ ω ), ∀y0 ∈ R3 , we obtain Ψ(x) = f (x) + 2ic g(x, y0 ) + c(1 + eiα )G− (x) and
lim Ψ(x) − c(1 + eiα )G− (x) = 2icG− (x)2L2 .
x→ y0
Thus Ψ can be uniquely decomposed in Ψ(x) = Φ(x) + λ(−i, α)Φ(y0 )G− (x) with Φ ∈ D(H˙ ω ) and boundary condition lim Ψ(x) − λ(−i, α)Φ(y0 )G− (x) = Φ(y0 ) . x→ y0
Comparing the two boundary conditions we obtain Φ(y0 ) = 2icG− (x)2L2 c(1 + eiα ) = λ(−i, α)Φ(y0 ) and then c=
Φ(y0 ) 2iG− (x)2L2
λ(−i, α) =
1 + eiα . 2iG− (x)2L2
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Theorem 2.3 The spectrum σ(Kα,y0 ) is purely absolutely continuous and σ(Kα,y0 ) = σac (Kα,y0 ) = σ(Hω ) = R .
(2.8)
Proof. Considering the explicit expression of the resolvent given in Theorem 2.1, we immediately see that σ(Kα,y0 ) = σ(Hω ) = R: indeed, since (Kα,y0 − z)−1 − (Hω − z)−1 is of rank 1 for each z ∈ R and α ∈ [0, 2π), Weyl’s Theorem (see for example Theorem XIII.14 in [13]) implies σess (Kα,y0 ) = σess (Hω ). In order to prove absence of pure point and singular spectrum, we are going to apply the limiting absorption principle (see Theorem XIII.19 in [13]): to this purpose we need to prove that the following inequality is satisfied for every interval [a, b] ⊂ R, b
−1
p
dx Ψ , Kα,y0 − x − iε Ψ <∞ sup 0<ε<1
a
with Ψ in a dense subset of L2 (R3 ) and p > 1. Since the operator Hω has no singular spectrum, the inequality is easily satisfied if α = π. So, let α = π, from Theorem 2.1 one has −1 −1 Ψ , Kα,y0 − x − iε Ψ = Ψ , Hω − x − iε Ψ +λ(α, x + iε) Gx−iε , Ψ Ψ , Gx+iε and again the inequality holds for the first term. It is very easy to see that the second term is a bounded function of x if ε > 0, so that we have only to control the limit when ε → 0. Since the singular spectrum of Hω is empty, we can choose the dense subset of L2 (R3 ) given by functions of the form (Hω −x)ϕ where ϕ ∈ D(Hω ): −1 Gx−iε , Ψ Ψ , Gx+iε = Hω − x − iε Hω − x ϕ (y0 )
2
−1 · Hω − x − iε Hω − x ϕ∗ (y0 ) −→ ϕ(y0 ) < ∞ ε→0
since functions in D(Hω ) are continuous and because −1 −1 Hω − x − iε Hω − x ϕ (y0 ) = ϕ(y0 ) + iε Hω − x − iε ϕ (y0 ) and
−1
ϕ (y0 ) ≤ lim ε Gx−iε ϕ = 0 . lim ε Hω − x − iε
ε→0
ε→0
Indeed from Proposition A.1 we can easily extract the following upper bound for Gx−iε , C Gx−iε ≤ √ . ε Finally from equation (2.6) it follows that
λ(α, x + iε) −→ 0 . ε→0
Since the previous argument applies for each interval [a, b] ⊂ R, the proof is completed.
Vol. 5, 2004
2.2
Rotating Singular Perturbations of the Laplacian
779
Asymptotic limit of rapid rotation
Let Urot (t−s) the unitary group generated by Kα,y0 for some α ∈ [0, 2π), according to [6], Uinert (t, s) = R(t) Urot (t − s) R† (s) . In the following, we shall prove that s− lim Uinert(t, s) = e−iHγ,C (t−s) ω→∞
where Hγ,C is an appropriate self-adjoint extension of HC , a singular perturbation of the Laplacian supported over a circle of radius y0 in the x, y−plane: let C the curve y(ϕ) = (y0 , π2 , ϕ), ϕ ∈ [0, 2π], and H˙ C the closure of the operator HC = H0 D(HC ) = C0∞ (R3 − C) . We first classify the self-adjoint extensions of H˙ C : Proposition 2.1 The self-adjoint extensions of the operator H˙ C , that are invariant under rotations around the z-axis, are given by the one-parameter family Hγ,C , γ ∈ R, with domain ˜ z ξΨ ∈ H 2 (R3 ), D(Hγ,C ) = {Ψ ∈ L2 (R3 ) | ∃ ξΨ ∈ D(Γγ,C (z)), Ψ − G ˜ z ξΨ = Γγ,C (z)ξΨ } (2.9) Ψ−G C ˜ z ξΨ (2.10) Hγ,C − z Ψ = H0 − z Ψ − G where z ∈ C, (z) > 0, D(Γγ,C (z)) = {ξ ∈ L2 ([0, 2π]) | Γγ,C (z)m ξm ∈ l2 } 1 ξm ≡ √ 2π
and
˜ z ξ (x) ≡ G
2π
0
∞
dk 0
0
dφ ξ(φ)e−imφ
0
Γγ,C (z)ξ (φ) = γξ(φ) − Γγ,C (z)m = γ − 2π
2π
2π
(2.11)
√
ei z|y(φ)−y(φ )| ξ(φ ) dφ 4π|y(φ) − y(φ )|
∞ l=|m|
k2
2 1
ϕklm (y0 ) −z
(2.12) (2.13)
√
ei z|x−y(φ)| ξ(φ) . dφ 4π|x − y(φ)|
Proof. See [14, 15]. The formula for Γα,C (λ)m is obtained expressing the free resolvent in terms of spherical waves.
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Proposition 2.2 For every Ψ ∈ L2 (R3 ), z ∈ (Hγ,C ), (z) > 0 and y0 = (0, y0 , 0), −1 −1 Hγ,C − z Ψ(x) = H0 − z Ψ(x) +
2π m∗ Gm ( x , y ) G ( x , y ) , Ψ( x ) 0 0 z Γ (z)m z L2 (R3 ) m=−∞ γ,C +∞
where
Gm x, y0 ) ≡ z (
∞
dk 0
∞ l=|m|
k2
1 ϕ∗ (y0 ) ϕklm (x) . − z klm
Proof. The expression for the resolvent of Hγ,C for a generic curve C is given in [14, 15]:
−1 −1 ˜ z Γ−1 (z) H0 − z −1 Ψ
Hγ,C − z Ψ(x) = H0 − z Ψ(x) + G . γ,C C
Since Γγ,C (z) is diagonal in the basis em (φ) =
√1 eimφ 2π
of L2 ([0, 2π], dφ),
∞ −1 Γγ,C (z)ξ (φ) =
1 ξm em (φ) Γ (z)m m=−∞ γ,C
and therefore ∞
−1 Γ−1 Ψ = γ,C (z) H0 − z) C
H0 − z)−1 Ψ
C m
Γγ,C (z)m
m=−∞
em (φ) ,
where
H0 − z)−1 Ψ
C m
·
ϕ∗klm (x )
ϕ
klm
1 =√ 2π
2π
dφ e−imφ
R3
0
d3 x
(y (φ)) Ψ(x ) =
√
2π e
im π 2
R3
∞
dk 0
∞ l l=0 m =−l
1 k2 − z
d3 x Gm x , y0 ) Ψ(x ) . z (
Finally ˜ z em )(x) = (G
0
2π
dφ √ 2π
∞
dk 0
=
l ∞ l=0 m =−l
1 ϕ∗ (y (φ)) ϕklm (x) eimφ k 2 − z klm
√ π 2π e−im 2 Gm x, y0 ) . z (
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781
0 Corollary 2.1 If Ψ(x) ∈ L2 (R3 ), Ψ(x) = χ(r)Ylm (θ, φ) and z ∈ (Hγ,C ), 0
(z) > 0, ∞ −1 2 0 Hγ,C − z Ψ (x) = dr r gzl0 (r, r )χ(r ) Ylm (θ, φ) 0
0
+
2π
0 (π/2, 0) Ylm 0
Γγ,C (z)m0
where gzl0 (r, r ) ≡
2 π
∞
dk 0
0 Gm x, y0 ) z (
0
∞
2
dr r gzl0 (y0 , r )χ(r )
−1 k2
m0 (r, r ) jl0 (kr)jl0 (kr ) = H0 − z Hl −z 0
k2
0 and Hlm0 0 is the subspace of L2 (R3 ) spanned by χ(r)Ylm (θ, φ). 0
Proof. The result follows from a straightforward calculation: indeed, if Ψ(x) = 0 (θ, φ), χ(r)Ylm 0 ∞ 2 ∗ m0 ( x , y ) , Ψ( x ) = δ Y (π/2, 0) dr r gzl0 (y0 , r )χ(r ) Gm 0 m,m 0 z l0 0
and
−1 Ψ (x) = H0 − z
∞ 0
2
0 dr r gzl0 (r, r )χ(r ) Ylm (θ, φ) . 0
Now we can state the main result: Theorem 2.4 For every t, s ∈ R, s− lim Uinert(t, s) = e−iHγ,C (t−s) ω→∞
where γ(α, y0 ) ∈ R and ∞ ∞ dk γ(α, y0 ) = 2π 0
l=0
2i 1
ϕkl0 (y0 ) 2 . + iα 2 2 2 (1 + e )|k + i| k +i
Proof. First we observe that (see Lemma 2.1 below) 0 −1 ∗ s− lim dt e−izt Uinert (t, 0) = −i Hγ,C − z = ω→∞
−∞
0
dt e−izt eiHγ,C t
−∞
and, since the previous equality holds for every z ∈ C, (z) > 0, we obtain ∗ (t, 0) = eiHγ,C t , s− lim Uinert ω→∞
and therefore
s− lim Uinert(t, 0) = e−iHγ,C t . ω→∞
The result then follows from the property of the 2-parameters unitary group Uinert (t, s):
∗ s− lim Uinert (t, s) = s− lim Uinert (t, 0) Uinert (s, 0) = e−iHγ,C (t−s) . ω→∞
ω→∞
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The explicit expression of the parameter γ(α, y0 ) is proved in the following Lemma 2.1. Lemma 2.1 For every z ∈ C, (z) > 0, 0 −1 ∗ s− lim dt e−izt Uinert (t, 0) = −i Hγ,C − z . ω→∞
−∞
Proof. We shall verify the equality on the dense subset of L2 (R3 ) given by functions 0 (θ, φ), with l0 = 0, . . . , ∞ and m0 = −l0 , . . . , l0 , of the form Ψ(x) = χ(r)Ylm 0 ∗ Uinert (t, 0)Ψ(x) = eiKα,y0 t R∗ (t)Ψ(x) = ei(Kα,y0 +m0 ω)t Ψ(x) .
Therefore
0
−∞
0
= −∞
∗ dt e−izt Uinert (t, 0)Ψ(x) =
0
dt e−izt ei(Kα,y0 +m0 ω)t Ψ(x)
−∞
−1 dt e−i(z−m0 ω)t eiKα,y0 t Ψ(x) = −i Kα,y0 + m0 ω − z Ψ(x) .
Hence we have now to prove that −1 −1 lim Kα,y0 + m0 ω − z Ψ(x) = Hγ,C − z Ψ(x) . ω→∞
First of all we observe that, for each z ∈ C, (z) > 0, m0 ∈ Z and y0 = (0, y0 , 0), 0 lim Gz−m0 ω (x, y0 ) = Gm x, y0 ) z (
ω→∞
in the norm topology of L2 (R3 ): indeed, since 0 Gz−m0 ω (x, y0 ) = Gm x, y0 ) + Rzm0 (x, y0 ) , z (
with Rzm0 (x, y0 ) =
∞
dk
l ∞
0
l=0 m=−l m=m0
1 ϕ∗ (y0 ) ϕklm (x) , k 2 − (m − m0 )ω − z klm
it is sufficient to prove that lim Rzm0 (x, y0 )L2 (R3 ) = 0 ,
ω→∞
but m R 0 (x, y0 )2 2 3 = z L (R )
∞
dk 0
l ∞ l=0 m=−l m=m0
|k 2
1 |ϕklm (y0 )|2 , − (m − m0 )ω − z|2
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Rotating Singular Perturbations of the Laplacian
783
and the right-hand side is bounded for each ω ∈ R (see Proposition A.1), so that we can exchange the limit with the integration
∞
lim
ω→∞
=
∞
dk 0
dk
∞ l
0
l=0 m=−l m=m0
l ∞
|k 2
1 |ϕklm (y0 )|2 − (m − m0 )ω − z|2
|ϕklm (y0 )|2 lim
ω→∞ |k 2
l=0 m=−l m=m0
1 =0. − (m − m0 )ω − z|2
Now, since (see Theorem 2.1) −1 ∗ Ψ (x) = Gz−m ( x , x ), Ψ( x ) Kα,y0 + m0 ω − z ω 0 +λ(z − m0 ω, α) Gz¯−m0 ω (x , y0 ), Ψ(x ) and lim
ω→∞
∗ Gz−m ( x , x ), Ψ( x ) 0ω
∞
= lim
ω→∞
0
L2 (R3 )
L2 (R3 )
L2 (R3 )
Gz−m0 ω (x, y0 )
π 0∗ = eim0 2 Gm ( x , x ), Ψ( x ) z
L2 (R3 )
2
0 dr r gzl0 (y0 , r ) χ(r ) Ylm (π/2, π/2) 0
Gz¯−m0 ω (x , y0 ), Ψ(x )
L2 (R3 )
π 0 = e−im0 2 Gm x , y0 ), Ψ(x ) z¯ (
lim Gz−m0 ω (x, y0 ) = e
im0 π 2
ω→∞
L2 (R3 )
0 Gm x, y0 ) , z (
we obtain −1 Ψ(x) = lim Kα,y0 + m0 ω − z
ω→∞
0 + β(z, α) Gm x, y0 ) z (
0
∞
0
∞
2
0 dr r gzl0 (r, r )χ(r ) Ylm (θ, φ) 0
−1 2 dr r gzl0 (y0 , r )χ(r ) = Hγ,C − z Ψ(x) ,
3
with
β(z, α) = lim λ(z − m0 ω, α) ω→∞
and
Γγ,C (z)m0 1 = . 2π β(z, α)
3 Actually λ is a function separately of z − m ω and ω, since the Green’s function G ( 0 − x) depends on ω.
784
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Ann. Henri Poincar´e
It remains to find the explicit expression of γ(α, y0 ): using the relation (see Theorem 2.1) 1 1 = − (z − m0 ω + i) G−m0 ω+¯z (x), G− (x) λ(z − m0 ω, α) λ(−i, α) we obtain
1 1 = lim − (z − m0 ω + i) G−m0 ω+¯z (x), G− (x) β(z, α) ω→∞ λ(−i, α) ∞ ∞ ∞ ∞
2 1 1
2i
ϕkl0 (y0 ) 2 + ϕkl0 (y0 ) dk dk = iα 2 2 2 1+e |k + i| k +i 0 0 l=0 l=0 ∞ ∞
2 1
ϕklm0 (y0 ) − dk 2−z k 0 l=|m0 |
and hence the result. We want to stress that, as it was expected, γ ∈ R:
2i 1 2i 1
+ 2
−1 = 2 (1 + eiα )|k 2 + i|2 k +i |k + i|2 1 + eiα
2i + 2ie−iα 1 −1 =0 . = 2 |k + i|2 2 + 2 cos α
3 The rotating point interaction in 2D 3.1
The Hamiltonian
The system we shall study is defined by the formal time-dependent Hamiltonian H(t) = H0 + a δ (2) (x − y (t))
(3.1)
where y (t) = R(t)y0 . The formal generator of time evolution in the uniformly rotating frame (with angular velocity ω) is given by K = H0 − ωJ + a δ (2) (x − y0 ) . Therefore the Hamiltonian of the system is a self-adjoint extension of the operator K y 0 = Hω D(Ky0 ) = C0∞ (R2 − {y0 }) . According to the discussion of Section 2, the Hamiltonian is given by the selfadjoint operator D(Kα,y0 ) = {f + cG+ + ceiα G− |g ∈ D(K˙ y0 ), c ∈ C}
(3.2)
Kα,y0 (f + cG+ + ceiα G− ) = K˙ y0 g + icG+ − iceiα G−
(3.3)
Vol. 5, 2004
Rotating Singular Perturbations of the Laplacian
785
with α ∈ [0, 2π) and where G± (x) = G±i (x, y0 ) Gz (x, y0 ) =
∞
dk 0
∞ n=−∞
k2
1 ϕ∗ (y0 ) ϕkn (x) − nω − z kn
(3.4)
for x ∈ R2 − {y0 }. As in the 3D case, the self-adjoint extension Kπ,y0 corresponds to the “free” Hamiltonian H˙ ω and Theorem 3.1 The resolvent of Kα,y0 has integral kernel given by (Kα,y0 − z)−1 (x, x ) = Gz (x, x ) + λ(z, α)Gz¯∗ (x , y0 )Gz (x, y0 )
(3.5)
with z ∈ (Kα,y0 ), x, x ∈ R2 , x = x , x, x = y0 and 1 1 = − (z + i) Gz¯(x), G− (x) λ(z, α) λ(−i, α)
λ(−i, α) =
1 + eiα . 2iG− (x)2
Proof. See the proof of Theorem 2.1 and Proposition A.2.
(3.6)
(3.7)
Theorem 3.2 The domain D(Kα,y0 ), α ∈ [0, 2π), consists of all elements Ψ ∈ R3 which can be decomposed in the following way Ψ(x) = Φz (x) + λ(z, α)Φz (y0 )Gz (x, y0 ) for x = y0 , Φz ∈ D(H˙ ω ) and z ∈ (Kα,y0 ). The previous decomposition is unique and on every Ψ of this form we obtain (Kα,y0 − z)Ψ = (Hω − z)Φz . Proof. See the proof of Theorem 2.2.
Theorem 3.3 The spectrum σ(Kα,y0 ) is purely absolutely continuous and σ(Kα,y0 ) = σac (Kα,y0 ) = σ(Hω ) = R . Proof. See the proof of Theorem 2.3, Theorem 2.1 and Proposition A.2.
(3.8)
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Ann. Henri Poincar´e
Asymptotic limit of rapid rotation
As in the 3D case, we shall prove that s− lim Uinert(t, s) = e−iHγ,C (t−s) ω→∞
where Hγ,C is an appropriate self adjoint extension of HC , a singular perturbation of the Laplacian supported over a circle of radius y0 : let C the curve y(θ) = (y0 , θ), θ ∈ [0, 2π], and H˙ C the closure of the operator H C = H0 D(HC ) = C0∞ (R2 − C) . Proposition 3.1 The self-adjoint extensions of the operator H˙ C , that are invariant under rotations around the z-axis, are given by the one-parameter family of operators Hγ,C , γ ∈ R, with domain ˜ z ξΨ ∈ H 2 (R2 ), D(Hγ,C ) = {Ψ ∈ L2 (R2 ) | ∃ ξΨ ∈ D(Γγ,C (z)), Ψ − G ˜ z ξΨ = Γγ,C (z)ξΨ } Ψ−G (3.9) C ˜ z ξΨ Hγ,C − z Ψ = H0 − z Ψ − G (3.10) where z ∈ C, (z) > 0, D(Γγ,C (z)) = {ξ ∈ L2 ([0, 2π]) | Γγ,C (z)n ξn ∈ l2 } 2π 1 √ dθ ξ(θ)e−inθ = en , ξΨ ξn ≡ L2 ([0,2π],dθ) 2π 0 √ 2π ei z|y(θ)−y(θ )| ξ(θ) − ξ(θ ) Γγ,C (z)ξ (θ) ≡ dθ γ 4π|y(θ) − y(θ )| 0 ∞
2 1 1
ϕkn (y0 ) Γγ,C (z)n = − 2π dk 2 γ k −z 0 and
˜ z ξ (x) ≡ G
0
2π
(3.11)
(3.12) (3.13)
√
ei z|x−y(θ)| ξ(θ) . dθ 4π|x − y(θ)|
Proof. Singular perturbations of the Laplacian supported on a curve in R2 are analogous to singular perturbations supported on a surface in R3 : indeed the quadratic form
2
2
2
F (Ψ, Ψ) ≡ d x ∇Ψ − dθ γ(θ) Ψ(y (θ)) R2
C
is easily seen to be a closed semibounded quadratic form (see for example [14, 15] and the discussion of Section 5) on ˜ z ξΨ ∈ H 1 (R2 ) D(F ) = Ψ ∈ L2 (R2 ) | ∃ ξΨ ∈ L2 (C), Ψ − G and it can be proved that it is associated to the self-adjoint operator Hγ,C .
Vol. 5, 2004
Rotating Singular Perturbations of the Laplacian
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Proposition 3.2 If Ψ(x) ∈ L2 (R2 ), Ψ(x) = χ(r)en0 (θ) and z ∈ (Hγ,C ), (z) > 0, ∞ −1 Hγ,C − z Ψ (x) = dr r gzn0 (r, r )χ(r ) + where gzn0 (r, r )
2π Gn0 (x, y0 ) Γγ,C (z)n0 z
≡
∞
dk 0
0
∞ 0
dr r gzn0 (y0 , r )χ(r )
−1 k
J (kr)J (kr ) = H − z (r, r ) 0 |n | |n | 0 0 Hn 0 k2 − z
and Gnz (x, y0 )
≡
∞
dk 0
1 ϕ∗ (y0 ) ϕkn (x) . k 2 − z kn
Proof. See the proof of Proposition 2.2 and Corollary 2.1.
Theorem 3.4 For every t, s ∈ R, s− lim Uinert(t, s) = e−iHγ,C (t−s) ω→∞
where γ(α, y0 ) ∈ R and γ(α, y0 ) =
∞
dk k
0
2i 1 + J 2 (ky0 ) . (1 + eiα )|k 2 + i|2 k2 + i 0
Proof. See the proof of Theorem 2.4 and the following Lemma 3.1.
Lemma 3.1 For every z ∈ C, (z) > 0, 0 −1 ∗ s− lim dt e−izt Uinert (t, 0) = −i Hγ,C − z . ω→∞
−∞
Proof. The first part of the proof is analogous to the proof of Lemma 2.1 (the only difference is the dense subset of L2 (R2 ) given by functions of the form Ψ(x) = χ(r)en0 (θ), with n0 ∈ Z). Hence it remains to prove that −1 −1 Ψ(x) = Hγ,C − z Ψ(x) . lim Kα,y0 + n0 ω − z ω→∞
Now, for each z ∈ C, (z) > 0, n0 ∈ Z and y0 = (0, y0 ), lim Gz−n0 ω (x, y0 ) = Gnz 0 (x, y0 )
ω→∞
in the norm topology of L2 (R2 ): since Gz−n0 ω (x, y0 ) = Gnz 0 (x, y0 ) + Rzn0 (x, y0 ) ,
788
M. Correggi and G. Dell’Antonio
with
Rzn0 (x, y0 ) =
∞
dk 0
Ann. Henri Poincar´e
∞
1 ϕ∗ (y ) ϕkn (x) , 2 − (n − n )ω − z kn 0 k 0 n=−∞ n=n0
it is sufficient to prove that lim Rzn0 (x, y0 )L2 (R2 ) = 0 .
ω→∞
But n R 0 (x, y0 )2 2 2 = z L (R )
∞
dk 0
∞
1 |ϕkn (y0 )|2 2 − (n − n )ω − z|2 |k 0 n=−∞ n=n0
and the right-hand side is bounded (see Proposition A.2) for each ω ∈ R, so that exchanging the limit with the integration, we obtain the result. Now, substituting in the expression of the resolvent (see Theorem 3.1), −1 ∗ Kα,y0 + m0 ω − z Ψ (x) = Gz−m ( x , x ), Ψ( x ) ω 0 L2 (R3 )
+λ(z − m0 ω, α) Gz¯−m0 ω (x , y0 ), Ψ(x )
L2 (R3 )
Gz−m0 ω (x, y0 )
the result follows from a straightforward calculation. Moreover we obtain the same relation between γ and α: 1 Γγ,C (z)n0 = 2π β(z, α) where β(z, α) = lim λ(z − n0 ω, α) ω→∞
but
1 1 = − (z − n0 ω + i) G−n0 ω+¯z (x), G− (x) λ(z − n0 ω, α) λ(−i, α)
and then
1 1 = lim − (z − n0 ω + i) G−n0 ω+¯z (x), G− (x) β(z, α) ω→∞ λ(−i, α) ∞ ∞
2
2 1 1
2i
ϕk0 (y0 ) + ϕk0 (y0 ) dk 2 dk 2 = iα 2 1+e |k + i| k +i 0 0 ∞
2 1
ϕkn0 (y0 ) . − dk 2 k −z 0
Vol. 5, 2004
Rotating Singular Perturbations of the Laplacian
789
4 The rotating blade in 3D 4.1
The Hamiltonian
Let D be the half-disk D ≡ {(r, θ, φ) ∈ R3 | 0 ≤ r ≤ A, 0 ≤ θ ≤ π, φ = 0} and ΘD (x, z) its characteristic function. The formal time-dependent Hamiltonian of the system is given by H(t) = H0 + α(x, z) R(t) ΘD (x, z) δ(y)
(4.1)
where R(t)Ψ(x) = Ψ(R(t)−1 x) and α∞ < ∞. Therefore in the rotating frame the formal generator of time evolution is K = H0 − ωJ + α ΘD (x, z) δ(y) or more rigorously a self-adjoint extension of the symmetric operator K D = Hω D(KD ) = C0∞ (R3 − D) . The Hamiltonian cannot be easily defined with the method of quadratic form, because of its unboundedness from below. Hence we shall pursue a different strategy: we shall define a sequence of cut-off Hamiltonians which converge to the operator Hω in the strong resolvent sense and that are self-adjoint and bounded from below; then we shall add the singular perturbation and prove that the so obtained operators are self-adjoint. Finally we shall prove that the limit (in the strong resolvent sense) of the sequence of cut-off perturbed Hamiltonians is a self-adjoint operator that we shall identify with the Hamiltonian of the system. So let (4.2) HωL = Hω ΠL where ΠL is the projector on the subspace of L2 (R3 ) generated by functions of the form χ(r)Ylm (θ, φ), with l ≤ L. It is very easy to prove that the operator HωL is self-adjoint on the domain H 2 (R3 ): the operator J is bounded on the domain of the projector ΠL and therefore it is an infinitesimally bounded perturbation of H0 , so that we can apply the Kato Theorem [9]. Moreover for each z ∈ (HωL ) the resolvent (HωL − z)−1 is given by an integral operator with kernel GzL (x, x ) =
∞
dk 0
l L ϕ∗klm (x ) ϕklm (x) . k 2 − mω − z
(4.3)
l=0 m=−l
Proposition 4.1 The sequence of cut-off Hamiltonians converge as L → ∞ in the strong resolvent sense to the self-adjoint operator Hω .
790
M. Correggi and G. Dell’Antonio
Ann. Henri Poincar´e
Proof. For each L ∈ N and z ∈ C−R, the function GzL (x, x ) belongs to L2 (R3 , d3 x): L Gz (x, x )2 ≤ Gz (x, x )2 < ∞ and then the result is a straightforward consequence of Proposition A.1. The operator Hω was studied in [6, 16]. Now we can define the perturbed cut-off Hamiltonians with the method of quadratic form: let4
2 Fα,L (Ψ, Ψ) = Fω,L (Ψ, Ψ) − dµD (r) α(r) Ψ D (r) (4.4) D
where Fω,L is the closed5 semibounded quadratic form associated to HωL . The form Fα,L is well defined if Ψ ∈ D(Fω,L ) and α is a smooth real function on D bounded away from 0. Proposition 4.2 Let z ∈ C − R, the form Fα,L can be written in the following way, z Fα,L (Ψ, Ψ) = Fω,L (Ψ, Ψ) + Φzα,L (ξΨ , ξΨ ) − 2 (z) Ψ , G˜zL ξΨ (4.5) where z Fω,L (Ψ, Ψ) = Fω,L (Ψ − G˜zL ξΨ , Ψ − G˜zL ξΨ ) − (z)Ψ − G˜zL ξΨ 2 + (z)Ψ2 (4.6)
(z) ξ Φzα,L (ξΨ , ξΨ ) = ξΨ , ΓL Ψ α L2 (D,dµ ) D
and
ξΨ (r) ΓL ( r ) = − (z) ξ dµD (r ) GzL (x, x ) x,x ∈D ξΨ (r ) Ψ α α(r) D
L G˜z ξ (x) ≡ dµD (r ) GzL (x, x ) x ∈D ξ(r ) .
(4.7)
(4.8)
D
Proof. The result follows from a simple calculation: setting
ξΨ (r) = α(r ) Ψ D (r)
(4.9)
one has Fα,L (Ψ, Ψ) − Fω,L (Ψ − G˜zL ξΨ , Ψ − G˜zL ξΨ ) = G˜zL ξ , HωL (Ψ − G˜zL ξ) + Ψ , HωL G˜zL ξ − 4 Here
dµD D
|ξΨ |2 α
dµD ( r ) stands for the restriction of the Lebesgue measure to D, namely dµD ( r) ≡ r 2 dr d cos θ for r = (r, θ) ∈ D; r denotes the restriction of x ∈ R3 to D, i.e. r ≡ (r, θ). 5 The form F 1 3 ω,L is closed on the domain D(Fω,L ) = H (R ).
Vol. 5, 2004
Rotating Singular Perturbations of the Laplacian
=
dµD D
791
2 |ξΨ |2 ˜L − Gz ξ , (HωL − z ∗ )G˜zL ξ − z ∗ G˜zL ξ + 2 z Ψ , G˜zL ξ α 2 = Φzα,L (ξΨ , ξΨ ) − (z)G˜zL ξ + 2 z Ψ , G˜zL ξ ,
since
2
(z) G˜zL ξ = G˜zL ξ , (HωL − z ∗ )G˜zL ξ ,
but
L 2 G˜ ξ = Ψ − G˜L ξ 2 − Ψ2 + 2 Ψ , G˜L ξ z z z
so that we obtain the result.
Of course the form Fα,L is independent on z and the decomposition Ψ = / D(Fω,L ) if ξΨ ∈ L2 (D, dµD ). Moreover the ϕz + G˜zL ξΨ is unique, since G˜zL ξΨ ∈ form Φzα,L (ξ, ξ) is bounded and one can choose z ∈ C such that the form satisfies another useful inequality: Proposition 4.3 The form Φzα,L (ξ, ξ) is bounded for each ξ ∈ L2 (D, dµD ). Proof. The first term of the form is of course bounded if ξ ∈ L2 (D, dµD ) and
L L ∗
≤ ξ 2
˜
G˜z ξ 2 G dµ ξ ξ D z
D L (D,dµD ) D L (D,dµD ) D
but we are going to prove that the function (G˜zL ξ)|D (r) is bounded ∀ r ∈ D, so that L G˜z ξ 2 < C(A) ξ2L2 (D,dµD ) D L (D,dµ ) D
and hence the result. Indeed
2
˜L
Gz ξ D (r) = GzL (x , x) x,x ∈D , ξ(r )
L2 (D,dµD )
2
2 2 2 ≤ GzL (x , x)L2 (D,dµD (r )) ξ L2 (D,dµD ) ≤ C ξ L2 (D,dµD ) since the Green’s function GzL (x, y0 ) belongs to L2 (R3 ), for each z ∈ C − R and y0 ∈ R3 . Proposition 4.4 For each smooth real function α on D bounded away from 0, there exists ζ ∈ R, ζ < 0 such that, for each z ∈ C−R, (z) < ζ, the following inequality holds (4.10) Φzα,L (ξ, ξ) − 2 (z) Ψ , G˜zL ξΨ − (z) + ωL Ψ − G˜zL ξΨ 2 > 0
792
M. Correggi and G. Dell’Antonio
Ann. Henri Poincar´e
Proof. We first point out that (see Proposition A.1) lim GzL (x, y0 ) ≤ C( (z)) < ∞ . (z)→∞
Thus, since the form Φzα,L (ξ, ξ) remains bounded for each z ∈ C − R, (z) = 0, and lim (z)Ψ − G˜zL ξΨ 2 = ∞ (z)→∞
2
(z) Ψ , G˜zL ξΨ ≤ C( (z)) ξ
we can always found a ζ satisfying the requirement.
But now we can prove that the complete form Fα,L is closed and bounded from below: Theorem 4.1 The form Fα,L is bounded from below and closed on the domain (4.11) D(Fα,L ) = Ψ ∈ L2 (R3 ) | ∃ξΨ ∈ L2 (D, dµD ), Ψ − G˜zL ξΨ ∈ H 1 (R3 ) where z ∈ C − R. Proof. Semiboundedness is trivial thanks to Proposition 4.4: since the form Fα,L does not depend on z, we can choose z ∈ C − R, (z) < ζ, so that the inequality (4.10) applies and Fα,L (Ψ, Ψ) ≥ Fω,L (Ψ − G˜zL ξΨ , Ψ − G˜zL ξΨ ) + ωL Ψ − G˜zL ξΨ 2 + (z) Ψ2 ≥ F0 (Ψ − G˜zL ξΨ , Ψ − G˜zL ξΨ ) + (z) Ψ2 ≥ (z) Ψ2 . So it remains to prove closure. Let Ψn = ϕn + G˜z ξn be a sequence in D(Fα,L ) converging to Ψ in the norm topology of L2 (R3 ), such that6 lim Fα,L − (z) (Ψn − Ψm ) = 0 n,m→∞
lim Fα,L − (z) (Ψn − Ψm ) ≥
n,m→∞
lim F0 (ϕn − ϕm ) ≥ 0
n,m→∞
so that lim F0 (ϕn − ϕm ) = 0
n,m→∞
and lim Φzα,L (ξn − ξm ) = 0 .
n,m→∞
The result easily follows, because F0 and Φzα,L are closed forms (see Proposition 4.3). Thus the form Fα,L defines a semibounded self-adjoint operator: 6F
0
is simply the form associated to the free Hamiltonian, i.e. F0 (Ψ, Ψ) =
|∇Ψ|2 .
Vol. 5, 2004
Rotating Singular Perturbations of the Laplacian
793
Proposition 4.5 The operators KαL defined below are self-adjoint: D(KαL ) = Ψ ∈ L2 (R3 ) | ∃ξΨ ∈ L2 (D, dµD ), Ψ − G˜zL ξΨ ∈ D(HωL ), Ψ − G˜zL ξΨ D = ΓL α (z)ξΨ L Kα − z Ψ = HωL − z Ψ − G˜zL ξΨ where α ∈ C(D), α(r ) = 0, for each r ∈ D. Moreover −1 −1 KαL − z Ψ (x) = HωL − z Ψ (x)
−1 L −1
H + d2r ΓL (z) − z Ψ (r ) GzL (x, x ) x ∈D α ω
(4.12) (4.13)
(4.14)
D
D
for each z ∈ (Kα ). Proof. The result easily follows from Theorem 4.1. The explicit expression of the resolvent is a direct consequence of the equation (4.13). We want only to remark z that the operator ΓL α (z) is invertible if (z) = 0: the form Φα,L can be written in the following way 2 |ξ|2 z − (z)G˜zL ξ . dµD Φα,L (ξ, ξ) ≡ α D 2 Since G˜zL ξ is bounded by C( (z)) ξ2 , if (z) = 0, we can always choose the real part of z is such a way that the form is positive. At last we can remove the cut-off in the angular momentum and define the Hamiltonian of the system: Theorem 4.2 For each α ∈ C(D), α(r ) = 0, ∀ r ∈ D, the sequence of semibounded self-adjoint operators KαL converge as L → ∞ in the strong resolvent sense to the self-adjoint (unbounded from below) operator Kα : D(Kα ) = Ψ ∈ L2 (R3 ) | ∃ξΨ ∈ L2 (D, dµD ), Ψ − G˜z ξΨ ∈ D(Hω ), Ψ − G˜z ξΨ D = Γα (z)ξΨ Kα − z Ψ = Hω − z Ψ − G˜z ξΨ ,
(4.15) (4.16)
where
ξΨ (r) Γα (z) ξΨ (r) = − dµD (r ) Gz (x, x ) x,x ∈D ξΨ (r ) α(r) D
G˜z ξ (x) ≡ dµD (r ) Gz (x, x ) x ∈D ξ(r ) . D
(4.17)
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M. Correggi and G. Dell’Antonio
Ann. Henri Poincar´e
Moreover the resolvent of Kα is −1 −1 Kα − z Ψ (x) = Hω − z Ψ (x) + D
−1
d2r Γ−1 Ψ (r ) Gz (x, x ) x ∈D α (z) Hω − z
(4.18)
D
for each z ∈ (Kα ). Proof. The key point of the proof is the application of the Trotter-Kato Theorem (see Theorem VIII.22 in [12]) to the sequence of self-adjoint operators KαL : we shall prove that (KαL − z)−1 converge in the strong sense for all z ∈ C − R to the operator (Kα − z)−1 , then the Trotter-Kato Theorem guarantees that there exists a self-adjoint operator T such that KαL converges in the strong resolvent sense to T . The identification of T with Kα is then trivial. So we shall start with the analysis of the sequence of bounded operators (Kα − z)−1 , z ∈ C − R, defined in (4.14): thanks to Proposition 4.1, the first part of the resolvent converges in the strong sense to (Hω − z)−1 , so that, in order to prove convergence of the whole operator, we need to consider the second part,
−1 L −1
Hω − z d2r ΓL Ψ (r ) GzL (x, x ) x ∈D α (z) D
D
but, for the same reason,
lim GzL (x, x ) x ∈D = Gz (x, x ) x ∈D
L→∞ 2
3
in L (R ) and lim
L→∞
−1
−1
HωL − z Ψ (r ) = Hω − z Ψ (r ) D
D
in L2 (D, dµD ), for all Ψ ∈ L2 (R3 ). Hence, to complete the first part of the proof, it is sufficient to show that −1 = Γ−1 lim ΓL α (z) α (z) L→∞
in the norm topology of L2 (D, dµD ), but this is again a consequence of Proposition 4.1: for each L the operator ΓL α (z) is invertible (see the proof of Proposition 4.5) and, in the same way, we can prove that Γ−1 α (z) is bounded and well defined, if
(z) = 0; moreover it is easy to see that lim ΓL α (z) = Γα (z) .
L→∞
We have then proved that, for each z ∈ C − R, −1 −1 s− lim KαL − z = Kα − z L→∞
Vol. 5, 2004
Rotating Singular Perturbations of the Laplacian
795
and the operator (Kα − z)−1 has of course a dense range. Thus the Trotter-Kato Theorem applies and the limiting self-adjoint operator T is immediately identified with Kα : the domain of Kα is given by functions of the form (Kα − z)−1 Ψ, Ψ ∈ L2 (R3 ), and the action of the operator on its domain follows from (4.18). Theorem 4.3 The spectrum of Kα is purely absolutely continuous and σ(Kα ) = σac (Kα ) = σ(Hω ) = R . Proof. First of all we shall prove that the operator −1 −1 Rzα ≡ Kα − z − Hω − z is a compact operator ∀ z ∈ C − R. Let Ψn be a weakly convergent sequence in L2 (R3 ), namely (ϕ , Ψn − Ψm ) → 0 when n, m → ∞ for each ϕ ∈ L2 (R3 ),
−1
Rzα (Ψn − Ψm ) = d2r Γ−1 (Ψn − Ψm ) Gz (x, x ) x ∈D α (z) Hω − z D
D
z
∗
Rα (Ψn − Ψm ) ≤ Gz Γ−1 α (z) Gz ∗ , Ψn − Ψm
≤ C Gz∗ , Ψn − Ψm −→ 0
and
n,m→∞
Γ−1 α (z)
since the operator is bounded (see the proof of Theorem 4.2). Therefore we can apply Weyl’s theorem and thus σess (Kα ) = σess (Hω ) = R . To prove that the singular and pure points spectrum of Kα are empty, we refer again to the limiting absorption principle. To show that the condition of the principle is satisfied, we have to consider the scalar product (where z = x + iε)
−1
z 2 −1 Ψ , Gz (x, x ) x ∈D
d r Γα (z) Hω − z Ψ
Ψ , Rα Ψ = D D
−1
2
≤ Γα (z) d r Gz∗ (x, x ) x ∈D , Ψ Ψ , Gz (x, x ) x ∈D
. D
The operator Γ−1 α (z) remains bounded when ε → 0 and, applying the same trick used in the proof of Theorem 2.3, one has
2 lim Gx−iε (x, x ) x ∈D , Ψ Ψ , Gx+iε (x, x ) x ∈D = ϕ(r ) < ∞ ε→0
where Ψ = (Hω − x)ϕ and ϕ ∈ D(Hω ), so that b
p
sup dx Ψ , Rx+iε Ψ
<∞ α 0<ε<1
a
for some p > 1 and for each interval [a, b] ⊂ R.
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Asymptotic limit of rapid rotation
In this section we shall study the asymptotic limit of rapid rotation of the unitary group Uinert (t, s) = R(t) Urot (t − s) R† (s) which represents the time evolution in the inertial frame associated to the formal time-dependent Hamiltonian defined in (4.1), while Urot (t− s) is the unitary group associated to the self-adjoint generator Kα : our main goal will be the proof of the following result, s− lim Uinert (t, s) = e−iHα (t−s) ω→∞
where Hα is the self-adjoint generator7 Hα = H0 − α(r ) ΘS (r)
(4.19)
and ΘS (r) is the characteristic function of a sphere S of radius A centered at the origin. Theorem 4.4 For every t, s ∈ R, s− lim Uinert (t, s) = e−iHα (t−s) ω→∞
where Hα = H0 − α(r) ΘS (r) .
Proof. See the proof of Theorem 2.4 and the following Lemma 4.1. Lemma 4.1 For every z ∈ C, (z) > 0, 0 −1 ∗ dt e−izt Uinert (t, 0) = −i Hα − z . s− lim ω→∞
−∞
Proof. Like in the proof of Lemma 2.1, we shall prove the result on the dense subset 0 (θ, φ), with l0 = 0, . . . , ∞ of L2 (R3 ) given by functions of the form Ψ(x) = χ(r)Ylm 0 and m0 = −l0 , . . . , l0 . The first part of the proof of Lemma 2.1 still applies, so that it is sufficient to prove that −1 −1 Ψ(x) = Hα − z Ψ(x) . lim Kα + m0 ω − z ω→∞
First of all we observe that −1 −1 Ψ = Hω + m 0 ω − z Ψ K α + m0 ω − z
∗ ∗ −1 + Γ−1 (z − m ω) H + m ω − z Ψ , G ( x , x )
0 ω 0 z−m ω 0 α x ∈D D
L2 (D,dµD )
7 The operator H is easily defined with the method of quadratic form (see for example [12]): α since the potential α(r) is bounded, it is associated to a form infinitesimally bounded w.r.t. the free Hamiltonian H0 . Hence the operator H0 + α(r) ΘD ( r ) is self-adjoint on the domain of H0 .
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and
797
−1 −1 Ψ = H0 − z Ψ lim Hω + m0 ω − z
ω→∞
as we have proved in Lemma 2.1. Therefore we need only to study the second part of the resolvent: it is easy to see that −1
−1
Ψ = H0 − z Ψ lim Hω + m0 ω − z ω→∞
D
D
−1 in L2 (D, dµD ). Moreover, since H0 − z Ψ D (r) is a function of the form m0 χ(r)Yl0 (θ, 0), we can apply the result found in the following Lemma 4.2: −1
lim Γ−1 Ψ α (z − m0 ω) Hω + m0 ω − z
ω→∞
−1
= Ξα (z) H0 − z Ψ
D
D
−1 = α(r)ΘD (r) H0 − α(r) ΘS (r) − z Ψ. In conclusion we obtain −1 −1 −1 1 + αΘD H0 − α ΘS − z Ψ lim Kα + m0 ω − z Ψ = H0 − z ω→∞
−1 = H0 − α ΘS − z Ψ.
Lemma 4.2 Let Γα (z) the operator defined in (4.17) and Ψ(x) ∈ L2 (R3 ) of the 0 form Ψ(x) = χ(r)Ylm (θ, φ), 0 lim Γ−1 α (z − m0 ω) Ψ|D = Ξα (z) Ψ|D
ω→∞
in L2 (D, dµD ), where −1 H0 − z Ψ|D (r) . Ξα (z)Ψ|D (r) ≡ α(r) H0 − α(r ) ΘS (r) − z Proof. First of all we are going to prove that norm− lim Γα (z − m0 ω) = Λα (z) ω→∞
where
ξ Λα (z) ξ = − α
for the definition of Indeed
0 Gm z
D
0 dµD (r ) Gm x, x ) x,x ∈D ξ(r ) z (
see Proposition 2.2. Γα (z − m0 ω) = Λα (z) + Rzm0
(4.20)
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where Rzm0 is a bounded integral operator on L2 (D, dµD ) with kernel Rzm0 (r, r ) ≡
∞ ∞ 0
l
l=0 m=−l m=m0
ϕklm (r) ϕklm (r ) k 2 − (m − m0 )ω − z
that goes to 0 when ω → ∞ (see the proof of Lemma 2.1). Moreover ∀ ω ∈ R+ the operator Γα (z) is invertible if (z) = 0 (see the proof of Theorem 4.2) and, for each l0 ∈ N, m0 = −l0 , . . . , l0 , z ∈ C − R it can be seen that the operator Λα is also invertible: indeed, let Ψ is the dense subset of 0 L2 (D, dµD ) given by functions of the form χ(r)Ylm (θ, 0), 0 Y m0 (θ, 0) Ψ|D (r, θ) − l0 Λα (z) Ψ|D (r, θ) = α 2π and
0
A
2
dr r gzl0 (r, r ) χ(r )
Ψ|D H0 − z Λα (z) Ψ|D (r) = H0 − z (r) − ΘD (r) Ψ|D (r) α
so that Λ−1 α (z)Ψ|D = Ξα (z)Ψ|D .
5 The rotating blade in 2D 5.1
The Hamiltonian
The formal time-dependent Hamiltonian of the system is given by the operator H(t) = H0 + α(x) R(t) ΘA (x) δ(y)
(5.1)
where ΘA (x) is the characteristic function of the segment 0 ≤ x ≤ A. In the rotating frame the generator of time evolution is a self-adjoint extension of the symmetric operator K S = Hω D(KS ) = C0∞ (R2 − S) where S is the segment S ≡ {(x, 0) ∈ R2 | 0 ≤ x ≤ A}. In order to rigorously define the self-adjoint extensions of the operator KS , we shall proceed like in the 3D case, namely we shall introduce a sequence of cut-off perturbed Hamiltonians and then we shall identify their limit with the Hamiltonian of the system. So let HωN = Hω ΠN (5.2) where ΠN is the projector on the subspace of L2 (R2 ) generated by functions of the form χ(r)en (θ), with |n| ≤ N . The operator HωN is self-adjoint on the domain
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H 2 (R2 ) (see the discussion at the beginning of Section 4) and, for each z ∈ (HωN ), the resolvent (HωN − z)−1 is given by an integral operator with kernel GzN (x, x ) =
∞
dk 0
N ϕ∗kn (x ) ϕkn (x) . k 2 − ωn − z
(5.3)
n=−N
Proposition 5.1 The sequence of cut-off Hamiltonians converge as N → ∞ in the strong resolvent sense to the self-adjoint operator Hω . Proof. See the proof of Proposition 4.1 and Proposition A.2. The perturbed cut-off Hamiltonian is associated to the form
2 Fα,N (Ψ, Ψ) = Fω,N (Ψ, Ψ) − dµS α(r) Ψ S (r)
(5.4)
S
which is well defined8 if Ψ ∈ D(Fω,N ), Fω,N being the closed semibounded form associated to the self-adjoint operator HωN , and α ∈ C(S), α(r) = 0, ∀r ∈ S. Proposition 5.2 Let z ∈ C − R, the form Fα,N can be written in the following way, z (Ψ, Ψ) + Φzα,N (ξΨ , ξΨ ) − 2 (z) Ψ , G˜zN ξΨ Fα,N (Ψ, Ψ) = Fω,N (5.5) where z Fω,N (Ψ, Ψ) = Fω,N (Ψ − G˜zN ξΨ , Ψ − G˜zN ξΨ ) − (z)Ψ − G˜zN ξΨ 2 + (z)Ψ2 (5.6)
Φzα,N (ξΨ , ξΨ ) = ξΨ , ΓN α (z) ξΨ L2 (S,dµS )
ξΨ (r) N Γα (z) ξΨ (r) = − dµS (r ) GzN (x, x ) x,x ∈S ξΨ (r ) α(r) S
N G˜z ξ (x) ≡ dµS (r ) GzN (x, x ) x ∈S ξ(r ) .
and
(5.7)
(5.8)
S
Proof. See the proof of Proposition 4.2.
Now we shall prove that the properties of the form Φzα,N still hold: Proposition 5.3 The form Φzα,N (ξ, ξ) is bounded for each ξ ∈ L2 (S, dµS ). Proof. Using the result proved in Proposition A.2, we can follow the proof of Proposition 4.3. 8 In
the 2D case, the measure dµS is given by r dr.
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Proposition 5.4 For each smooth real function α on S bounded away from 0, there exists ζ ∈ R, ζ < 0 such that, for each z ∈ C−R, (z) < ζ, the following inequality holds Φzα,N (ξ, ξ) − 2 (z) Ψ , G˜zN ξΨ − (z) + ωN Ψ − G˜zN ξΨ 2 > 0 . Proof. See the proof of Proposition 4.4 and Proposition A.2.
We can now state the following theorem: Theorem 5.1 The form Fα,N is bounded from below and closed on the domain D(Fα,N ) = Ψ ∈ L2 (R2 ) | ∃ξΨ ∈ L2 (S, rdr), Ψ − G˜zN ξΨ ∈ H 1 (R2 ) . (5.9)
Proof. See the proof of Theorem 4.1. Proposition 5.5 The operators KαN defined below are self-adjoint: D(KαN ) = Ψ ∈ L2 (R2 ) | ∃ξΨ ∈ L2 (S, dµS ), Ψ − G˜zN ξΨ ∈ D(HωN ), Ψ − G˜zN ξΨ D = ΓN α (z)ξΨ N Kα − z Ψ = HωN − z Ψ − G˜zN ξΨ
(5.11)
where α ∈ C(D), α(r ) = 0, for each r ∈ D. Moreover −1 −1 Ψ (x) = HωN − z Ψ (x) KαN − z
−1 N −1 + Hω − z d2r ΓN Ψ (r ) GzN (x, x ) x ∈D α (z)
(5.12)
(5.10)
D
D
for each z ∈ (Kα ). Proof. The result follows from Theorem 5.1. Like in the 3D case it is possible to prove that the operator ΓN α (z) is invertible if (z) = 0. Theorem 5.2 For each α ∈ C(S), α(r) = 0, ∀ r ∈ S, the sequence of semibounded self-adjoint operators KαN converge as N → ∞ in the strong resolvent sense to the self-adjoint (unbounded from below) operator Kα : D(Kα ) = Ψ ∈ L2 (R2 ) | ∃ξΨ ∈ L2 (S, dµS ), Ψ − G˜z ξΨ ∈ D(Hω ), (5.13) Ψ − G˜z ξΨ S = Γα (z)ξΨ Kα − z Ψ = Hω − z Ψ − G˜z ξΨ (5.14) where
ξΨ (r) Γα (z) ξΨ (r) = − dµS (r ) Gz (x, x ) x,x ∈S ξΨ (r ) α(r) S
G˜z ξ (x) ≡ dµS (r ) Gz (x, x ) x ∈D ξ(r ) . S
(5.15)
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Moreover the resolvent of Kα is −1 −1 Kα − z Ψ (x) = Hω − z Ψ (x)
−1
(z) H − z Ψ (r ) Gz (x, x ) x ∈S + dr r Γ−1 ω α
801
(5.16)
S
S
for each z ∈ (Kα ). Proof. See the proof of Theorem 4.2.
Theorem 5.3 The spectrum of Kα is purely absolutely continuous and σ(Kα ) = σac (Kα ) = σ(Hω ) = R . Proof. See the proof of Theorem 4.3, Theorem 5.2 and Proposition A.2.
Remark. An interesting application of previous results is the study of the 3D rotating needle, i.e., a singular rotating perturbation of the Laplacian supported on a (finite) segment. Indeed the system can be easily reduced to a 2D rotating blade on the plane of rotation and a free motion on its perpendicular axis: the Hamiltonian is formally given by H = H0x,y + α(x) R(t) ΘA (x) δ(y) + H0z where ΘA (x) is the characteristic function of the segment 0 ≤ x ≤ A. According to the previous discussion, the self-adjoint extensions of H are given by the family of operators Kαx,y + H0z , where Kαx,y denotes the Hamiltonians of the 2D rotating blade defined in (5.14). Moreover the domain of self-adjointness can be identified with the set of functions Ψ(x) = f (x, y) g(z) such that f ∈ D(Kα ) and g ∈ H 2 (R).
5.2
Asymptotic limit of rapid rotation
In this section, we shall prove that s− lim Uinert (t, s) = e−iHα (t−s) ω→∞
where Hα is the self-adjoint generator Hα = H0 − α(r) ΘC (r)
(5.17)
and ΘC (r) is the characteristic function of a circle C of radius A centered at the origin. Theorem 5.4 For every t, s ∈ R, s− lim Uinert (t, s) = e−iHα (t−s) ω→∞
where Hα = H0 − α(r) ΘC (r) . Proof. See the proof of Theorem 2.4 and the following Lemma 5.1.
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Lemma 5.1 For every z ∈ C, (z) > 0,
0
s− lim
ω→∞
−∞
−1 ∗ dt e−izt Uinert (t, 0) = −i Hα − z .
Proof. Like in the proof of Lemma 3.1, we shall prove the result on the dense subset of L2 (R2 ) given by functions of the form Ψ(x) = χ(r)en0 (θ), n0 ∈ Z. Following the proof of Lemma 3.1, it remains to prove that −1 −1 lim Kα + n0 ω − z Ψ(x) = Hα − z Ψ(x) ω→∞
but
−1 −1 K α + n0 ω − z Ψ = Hω + n 0 ω − z Ψ
∗ ∗ −1 + Γ−1 (z − n ω) H + n ω − z Ψ , G ( x , x )
0 ω 0 z−n ω 0 α x ∈S
L2 (S,dµS )
S
and
−1 −1 lim Hω + n0 ω − z Ψ = H0 − z Ψ
ω→∞
as we have proved in Lemma 3.1. Moreover −1
−1
Ψ = H0 − z Ψ lim Hω + n0 ω − z ω→∞
S
S
2
in L (S, dµS ) and, applying the result found in the following Lemma 5.2, −1
lim Γ−1 Ψ α (z − n0 ω) Hω + n0 ω − z ω→∞
−1
= Ξα (z) H0 − z Ψ
S
S
−1 = α(r)ΘS (r) H0 − α(r) ΘC (r) − z Ψ. In conclusion we obtain −1 −1 −1 1 + αΘS H0 − α ΘC − z Ψ Ψ = H0 − z lim Kα + n0 ω − z ω→∞
−1 = H0 − α ΘC − z Ψ.
Lemma 5.2 Let Γα (z) the operator defined in (5.15), lim Γ−1 α (z − n0 ω) = Ξα (z)
ω→∞
in L2 (S, dµS ), where
−1 Ξα (z)ξ (r) ≡ α(r) H0 − α(r) ΘC (r) − z H0 − z ξ (r) .
(5.18)
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803
Proof. First of all we are going to prove that norm− lim Γα (z − n0 ω) = Λα (z) , ω→∞
where
1 ξ Λα (z) ξ = − α 2π
S
dµS (r ) gzn0 (r, r )ξ(r ) .
For the definition of gzn0 see Proposition 3.2. Indeed Γα (z − n0 ω) = Λα (z) + Rzn0 where Rzn0 is a bounded integral operator on L2 (S, dµS ) with kernel Rzn0 (r, r )
≡
0
∞
∞
ϕkn (r) ϕkn (r ) −→ 0 k 2 − (n − n0 )ω − z n=−∞ n=n0
as ω → ∞ (see the proof of Lemma 3.1). Moreover for each n0 ∈ Z and z ∈ C − R it can be seen that the operator Λα is invertible: indeed
ξ H0 − z Λα (z) ξ (r) = H0 − z (r) − ΘS (r) ξ(r) , α so that Λ−1 α (z) = Ξα (z).
Remark. As in Section 5.1, we can apply the previous results to analyze the asymptotic limit of rapid rotation of the 3D rotating needle: the time-dependent propagator in the inertial frame factorizes in the product of the time-dependent propagator associated to a 2D rotating blade on the x, y-plane and a the free propagator on the z-axis. Therefore Theorem 5.4 implies convergence of the propagator in the inertial frame to the one-parameter unitary group generated by the time-independent self-adjoint operator Hαx,y + H0z , where Hαx,y is defined in (5.17).
6 Conclusions and perspectives The operators studied in Section 2 and 3 could be viewed as the Hamiltonians of quantum systems given by a particle interacting with a rotating δ-type potential. In this context the results proved about the asymptotic limit of rapid rotation have an heuristic physical meaning: if the angular velocity of the potential is very large with respect to the velocity of the particle, we expect that the particle feels a time-independent potential, which is the mean of the true potential over a period. This result was already proved by Enss et al. [7] for regular potential, and, from this point of view, our work is an extension of their results to singular potentials.
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A future application of that study would be the analysis of the scattering of a particle by a rotating point interaction. Indeed it would be an example of time-dependent scattering that can be reduced to a stationary problem: passing to the rotating frame, we could prove in simpler way, for example, existence and completeness of the wave operators. In Section 3 and 4 we have studied the rotating blade, namely a singular potential with codimension 1. That kind of rotating singular perturbations of the Laplacian are more interesting and could open many suggestive problems. For example in the 3D case we could investigate the dependence of the results on the shape of the blade. While all the properties of the form and the self-adjoint extensions still hold for a blade with a general shape, because the key point is the good behavior of the Green’s function on a compact subset of R3 , the analysis of the asymptotic limit is harder. In fact a semi-spherical shape is very useful to perform the calculation with the Green’s function of Hω expressed in terms of functions with spherical symmetry (the spherical waves), but the same goal can be reached for a blade of different form: if we take a square shaped blade and we express the resolvent of Hω in terms of functions with cylindrical symmetry (essentially the Bessel functions), all the results still hold. On the other hand, if the blade has no symmetry, we could expect the same behavior but it is not clear at all how it can be proved. Finally we want to mention another feature of the problem which can be investigated: the blades we have considered are finite, so it would be interesting to study an infinite blade, for example an half-line in 2D and an half-plane in 3D, but, in that case, many problems arise in the definition of the operator. In particular the form Φzα should not be bounded, unless we impose some condition on the behavior at ∞ of the parameter α.
Acknowledgments M.C. is very grateful to Prof. Ludwik Dabrowski and the INTAS Research Project nr. 00-257 of European Community, “Spectral Problems for Schr¨ odinger-Type Operators”, for the support.
Appendix A
The Green’s function of Hω
In this appendix we shall study the Green’s function Gz (x, y0 ) of Hω and we shall prove that it belongs to L2 (Rn , dn x), ∀y0 ∈ Rn with n = 2, 3. We shall start from the 3D case: Proposition A.1 The resolvent (Hω − z)−1 , z ∈ C − R, has the following integral representation −1 d3 x Gz (x, x )Ψ(x ) (Hω − z) Ψ(x) = R3
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805
with Ψ(x) ∈ L2 (R3 , d3 x) and
Gz (x, x ) =
∞
dk 0
l ∞ l=0 m=−l
1 ϕ∗ (x ) ϕklm (x) . k 2 − mω − z klm
(A.1)
The functions ϕklm (x) are the spherical waves9 : 2k 2 ϕklm (x) = jl (kr)Ylm (θ, ϕ) . π Moreover, for every y0 ∈ R3 and z ∈ C − R, Gz (x, y0 ) ∈ L2 (R3 , d3 x). Proof. The integral representation of the Green’s function of Hω is a straightforward consequence of the eigenvectors decomposition of Hω . Moreover in the following we shall prove that, for each Ψ ∈ L2 (R3 ), z ∈ C − R and y0 ∈ R3 ,
Gz (x, y0 ) , Ψ(x) 2 3 3 < ∞ . L (R ,d x)
2
3
Every function Ψ ∈ L (R ) can be decomposed in terms of spherical harmonics: Ψ(x) =
l ∞
Ψlm (r) Ylm (θ, φ)
l=0 m=−l
with the L2 -condition l ∞ Ψlm (r)2 2 + 2 < ∞ . L (R ,r dr) l=0 m=−l
Thus ∞ l
2 2
Gz (x, y0 ) , Ψ(x) ≤
Gz+mω (x, y0 ) , Ψlm (r)Ylm (θ, φ) l=0 m=−l
≤
l ∞ Gz+mω (x, y0 )2 2 3 3 Ψlm (r)Ylm (θ, φ)2 L (R ,d x) l=0 m=−l
≤ C( (z))
l ∞ Ψlm (r)2 2 + 2 <∞ L (R ,r dr) l=0 m=−l
because the Green’s function of the free Hamiltonian √ z+mω| x− y0 |
Gz+mω (x, x) =
ei
4π|x − x|
9 Here j (r) denotes the spherical Bessel function of order l (see [10, 17]) and Y m (θ, φ), with l l l ∈ N and m = −l, . . . , l, the spherical harmonics.
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belongs to L2 (R3 , d3 x) for each z ∈ C − R and y0 ∈ R3 : we have to choose the root of z + mω with imaginary part 12 2 2 − (z) − mω ((z) + mω) + (z) √ | (z)| ≥ >0
z + mω = 2 2 so that Gz+mω ∈ L2 independently on m ∈ Z.
An analogous result can be proved in the 2D case: Proposition A.2 The resolvent (Hω − z)−1 , z ∈ C − R, has the following integral representation d2 x Gz (x, x )Ψ(x ) (Hω − z)−1 Ψ(x) = R2
with Ψ(x) ∈ L2 (R2 , d2 x) and10 Gz (x, x ) ≡
∞
dk 0
∞ n=−∞
ϕkn (x) =
k2
1 ϕ∗ (x ) ϕkn (x) − ωn − z kn
(A.2)
k J|n| (kr) einθ . 2π
Moreover, for every y0 ∈ R2 and z ∈ C − R, Gz (x, y0 ) ∈ L2 (R2 , d2 x). Proof. Following the proof of Proposition A.1, we shall consider the scalar product Gz (x, y0 ) , Ψ(x) L2 (R3 ,d3 x)
with Ψ(x) =
∞ n=−∞
Ψn (r)
einθ 2π
and we obtain ∞
2
Gz+mω (x, y0 )2 2 3 3 Ψn (r)2 2 + 2 < ∞
Gz (x, y0 ) , Ψ(x) ≤ L (R ,d L (R ,r dr) x) n=−∞
since11
i (1) √ H ( z + nω |x − y0 |) 4 0 √ belongs to L2 (R2 , d2 x), for each z ∈ C − R and ( z + nω) > 0. Gz+nω (x, y0 ) =
10 J
n (r) stands 11 H (1) denotes 0
for the Bessel function of order n ∈ N. the Hankel function of first kind and order zero (see [1]).
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References [1] M. Abramovitz, I.A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1965. [2] S.A. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, Solvable Models in Quantum Mechanics, Springer-Verlag, New York, 1988. [3] F.A. Berezin, L.D. Faddeev, A Remark on Schr¨ odinger Equation with a Singular Potential, Soviet Mathematics Doklady 2, 372–375 (1961). [4] G.F. Dell’Antonio, R. Figari, A. Teta, Schr¨ odinger Equation with Moving Point Interactions in Three Dimensions, in Stochastic Processes, Physics and Geometry: New Interplays, Leipzig, 1999, CMS Conference Proceedings 28, 99–113, American Mathematical Society, Providence, 2000. [5] G.F. Dell’Antonio, Point Interactions, in Mathematical Physics in Mathematics and Physics, Siena, 2000, Fields Institute Communications 30, 139–150, American Mathematical Society, Providence, 2001. [6] V. Enss, V. Konstrykin, R. Schrader, Energy Transfer In Scattering By Rotating Potential, in Proceedings of the Workshop on Spectral and Inverse Spectral Problems for Schr¨ odinger Operators, Goa, India, Proceedings of the Indian Academic of Science, Mathematical Science 112, no. 1, 55–70 (2002). [7] V. Enss, V. Konstrykin, R. Schrader, Perturbation Theory for the Quantum Time-Evolution in Rotating Potentials, in Proceedings of the Conference QMath-8 “Mathematics Results in Quantum Mechanics”, Taxco, Mexico, Contemporary Mathematics 307, 113–127, American Mathematical Society, Providence, 2002. [8] R. Figari, Time-Dependent and Non-Linear Point Interactions, in Proceedings of Mathematical Physics and Stochastic Analysis, Lisbon, 1998, 184–197, World Scientific Publisher, New York, 2000. [9] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1976. [10] A.F. Nikiforov, V.B. Uvarov, Special Functions of Mathematical Physics, Birkh¨auser, Basel, 1988. [11] A. Posilicano, A Krein-like Formula for Singular Perturbation of Self-Adjoint Operators and Applications, Journal of Functional Analysis 183, 109–147 (2001). [12] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, Academic Press, San Diego, 1975.
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[13] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators, Academic Press, San Diego, 1975. [14] A. Teta, Singular Perturbations of the Laplacian and Connections with Models of Random Media, Ph.D Thesis SISSA/ISAS, 1989. [15] A. Teta, Quadratic Forms for Singular Perturbations of the Laplacian, Publications of RIMS, Kyoto University, 26, 803–817 (1990). [16] A. Tip, Atoms in Circularly Polarised Fields: the Dilatation-Analytic Approach, Journal of Physics A: Mathematical and General 16, 3237–3259 (1983). [17] G.N. Watson, A Treatise on the Theory of Bessel Functions, University Press, Cambridge, 1944. Michele Correggi International School for Advanced Studies SISSA/ISAS Trieste Italy email:
[email protected] Gianfausto Dell’Antonio Centro Linceo Interdisciplinare∗ Roma Italy email:
[email protected] Communicated by Gian Michele Graf submitted 07/10/03, accepted 09/12/03
To access this journal online: http://www.birkhauser.ch
* On leave from Dipartimento di Matematica, Universit` a di Roma, “La Sapienza”, Italy.
Ann. Henri Poincar´e 5 (2004) 809 – 870 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/050809-62 DOI 10.1007/s00023-004-0183-7
Annales Henri Poincar´ e
Scaling Algebras for Charged Fields and Short-Distance Analysis for Localizable and Topological Charges Claudio D’Antoni∗, Gerardo Morsella and Rainer Verch
Abstract. The method of scaling algebras, which has been introduced earlier as a means for analyzing the short-distance behavior of quantum field theories in the setting of the model-independent, operator algebraic approach, is extended to the case of fields carrying superselection charges. In doing so, consideration will be given to strictly localizable charges (“DHR-type” superselection charges) as well as to charges which can only be localized in regions extending to spacelike infinity (“BF-type” superselection charges). A criterion for the preservance of superselection charges in the short-distance scaling limit is proposed. Consequences of this preservance of superselection charges are studied. The conjugate charge of a preserved charge is also preserved, and for charges of DHR-type, the preservance of all charges of a quantum field theory in the scaling limit leads to equivalence of local and global intertwiners between superselection sectors.
1 Introduction In an attempt to analyze the short-distance behavior of quantum field theories in a completely model-independent manner, and to have a counterpart of renormalization group analysis at short length scales in the setting of general quantum field theory, so-called “scaling algebras” have been introduced some time ago [11]. The idea of this approach is to associate to a given quantum field theory described in terms of local observable algebras [22, 21] a “scaling algebra” of functions depending on a scaling parameter λ > 0 and taking values in the local observable algebras. These functions are required to have certain properties regarding their localization and energy behavior as λ tends to zero; roughly speaking, the values of the functions at scale parameter λ should be observables localized in spacetime regions of extension proportional to λ, and having energy-momentum transfer proportional to λ−1 . The collection of all these functions, i.e., of all the members of the scaling algebra, may hence be viewed as “orbits” of elements in the local observable algebras under all possible renormalization group transformations. By studying the vacuum expectation values of these functions in the limit λ → 0 (the “scaling limit”), one can then analyze the extreme short distance properties of the given quantum field theory. ∗ supported
by MIUR, INDAM-GNAMPA, and the EU
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This programme, initiated in [11], has been further developed in [9, 12, 8, 25]. It leads to a general classification of the short distance behavior of the given theory which corresponds to the one known in perturbation theory where one distinguishes theories with stable ultraviolet fixed points under renormalization group transformations, as opposed to others with unstable fixed points or no fixed points at all [11]. Moreover, it permits to give a criterion as to when a given quantum field theory possesses “confined charges” which are only visible in the extreme short distance limit while they are absent at finite scale, like the color charge in QCD [7, 9]. According to this criterion a charge is confined if it arises as a superselection charge in the scaling limit theory of the observables which is not a scaling limit of the superselection charges of the original theory at scale λ = 1 (see Sec. 5 for discussion). The effectiveness of this criterion has been illustrated in the example of the two-dimensional Schwinger model [9, 12]. However, with the exception of Ref. [26], the scaling algebra method has up to now only been applied in the setting of local observable algebras, not in the context of local field algebras containing charge-carrying local field algebras. In other words, this method has not yet been applied to studying the short-distance behavior of superselection charges (see [31, 21] and references cited there) and their corresponding charge-carrying fields. In the present work, we generalize the “scaling algebra” framework in the setting of algebraic quantum field theory in the presence of field operators transforming non-trivially under the action of a (global) compact gauge group. We consider separately two cases where the field operators can be localized (1) in bounded spacetime regions and (2) in regions extending to spacelike infinity (so-called “spacelike cones”). The first case corresponds to superselection charges which can be localized in arbitrary bounded regions of spacetime (“DHR-charges”) while the second case corresponds to superselection sectors carrying so-called topological charges (“BF-charges”).1 In both cases, we will assume that the translations act covariantly on the algebras of field operators, and that there is a translationinvariant vacuum. Our principal interest lies in the behavior of the superselection charges in the scaling limit. We propose a criterion specifying what it means that a charge superselection sector of the given quantum field theory is “preserved” in the scaling limit. Then we will show that under quite general conditions, a superselection charge is preserved in the scaling limit exactly if this is also the case for the corresponding conjugate charge. As a further application, we extend an earlier result by Roberts [30] (which was obtained for dilation covariant quantum field theories) by showing that in a quantum field theory where all charges of DHR-type are preserved in the scaling limit, the sets of local and global intertwiners for the superselection charges coincide (see the first part of Sec. 4 for explanation of this terminology). This 1 The acronyms DHR and BF refer to Doplicher-Haag-Roberts [13, 14] and to BuchholzFredenhagen [10], respectively, who have introduced and analyzed the corresponding types of superselection sectors.
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amounts to saying that part of the superselection structure is determined locally if the superselection charges are ultraviolet stable in the sense of being preserved in the scaling limit. Such a property is of some relevance in the construction of superselection theory in a generally covariant setting as recently developed in [37]. This article is organized as follows. In Sec. 2 we define the quantum field theories corresponding to case (1), with strictly localizable field operators, more precisely. We introduce a class of theories which we call “quantum field theories with gauge group action”, abbreviated QFTGA, in the operator-algebraic setting. This class of theories is slightly more general than the class of theories obtained via the Doplicher-Roberts reconstruction from DHR-type superselection charges (which will be considered in Sec. 4). We introduce the scaling algebra for such QFTGAs, and, in close analogy to [11], we introduce scaling limit states and scaling limit theories and study their basic properties. Then, in Sec. 3, we consider QFTGAs with more structure, mainly with additional Poincar´e covariance and clustering properties, and study what additional properties ensue in the scaling limit. In Sec. 4 we introduce “quantum field systems with gauge symmetry” (QFSGSs) according to [16]. These are more special QFTGAs which arise by the Doplicher-Roberts reconstruction theorem from the covariant, strictly localizable (i.e., DHR-type) superselection sectors with finite statistics belonging to a quantum field theory of local observables (cf. again [16]). Charges of this kind would, e.g., correspond to the flavor charges of strong interactions. The reason why we make a distinction between QFTGA and QFSGS is that the scaling limit theories of a QFTGA are again of this type, i.e., are QFTGAs. But scaling limit theories of a QFSGS have in general only the structure of a QFTGA. We summarize parts of the terminology of the theory of superselection sectors and the result on the existence of a corresponding QFSGS, emphasizing the role played by the “field multiplets” in the local field algebras corresponding to each superselection charge. We will make use of this in Sec. 5, where we will state our criterion of preservance of a charge in the scaling limit in terms of such field multiplets: Our criterion demands that a charge of DHR-type is preserved in the scaling limit if scaled families of such multiplets (“scaled multiplets”) have a certain limiting behavior in the scaling limit. Then we briefly discuss mechanisms for the disappearance of charges in the scaling limit. Quite generally, a charge may disappear in the scaling limit if it takes typically more energy than proportional to λ−1 to create the charge within a spacetime region of extension proportional to λ. As will be explained, this may happen if the interaction between the charges of a quantum field theory is, at extremely short distances, either strongly binding or strongly repellent. Moreover, we present some further results on the structure of superselection charges preserved in the scaling limit, like the preservance of the conjugate charge. In Sec. 6 we state and prove our result on the equivalence of local and global intertwiners if all DHR-charges are preserved in the scaling limit. Having up to this point discussed the case (1) of field operators localizable in arbitrary bounded regions, we will turn in Sec. 7 to the discussion of QFS-
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GSs where the field operators are only localizable in infinitely extended spacelike cones (corresponding to case (2) alluded to above). Because of the much weaker localization properties of field operators in this case, the scaling algebra method has to be appropriately adapted, and this is done by defining the scaling algebra functions of field operators such that the functions asymptotically (as the scaling parameter λ tends to 0) commute with the scaling algebra functions of the observables (which are strictly localized) from which the QFSGS is constructed by the Doplicher-Roberts theorem for superselection charges of the BF-type. It will turn out that, for the corresponding definition of the scaling algebra of fields, one obtains scaling limit QFTGAs where the field operators are strictly localizable. In Sec. 8 we will then generalize the criterion for the preservance of charges in the scaling limit to the case of BF-type superselection charges, and discuss conditions sufficient to conclude that such preserved charges induce superselection sectors of the scaling limit theories of the observables. It will be assumed in this discussion that the underlying QFSGS of fields localized in spacelike cones is Poincar´e covariant. There will be two more technical Appendices. In Appendix A, we present the example of the Majorana-Dirac field which possesses a Z2 gauge symmetry, and show that the corresponding charges are preserved in the scaling limit according to our criterion. In Appendix B, we sketch the proof of a Reeh-Schlieder property for an extended scaling limit field algebra used in the construction of the scaling limit QFTGA in Sec. 7. The article is completed by some concluding remarks.
2 Quantum field theories with gauge group action and their scaling algebras and scaling limits In the present section we investigate an extension of the “scaling algebra” approach of [11] to quantum field theories that include a structure which we will call a normal, covariant quantum field theory with gauge group action (QFTGA) since we will see that this structure has a counterpart in the scaling limit. In the next section we add a few more assumptions, such as Lorentz covariance, geometric modular action, and clustering, but it is not before Section 4 that we introduce a normal, covariant quantum field system with gauge symmetry according to [16] which connects quantum field algebras and superselection sectors, and explore some properties of the scaling limits for such theories. Notation. In the following, we consider quantum field theories on n-dimensional Minkowski-spacetime (n ≥ 2), which will be identified with Rn , equipped with the Lorentzian metric η = (ηµν ) = diag(1, −1, −1, . . . , −1). We recall that the set V+ := {(y 0 , . . . , y n−1 ) ∈ Rn : (y 0 )2 > (y 1 )2 + · · · + (y n−1 )2 , y 0 > 0} denotes the open forward lightcone and V+ its closure. A double cone is any set in Rn of the form O = x + V+ ∩ y − V+ for any pair of x, y ∈ Rn so that y ∈ x + V+ . The set of all double cones in Rn will be denoted generically by K.
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Definition 2.1 A quintuple (F, U (Rn ), U (G), Ω, k) is called a normal, covariant quantum field theory with gauge group action (QFTGA) if the following properties are fulfilled: There is a Hilbert-space H and a family {F(O)}O∈K of von Neumann algebras on H which is indexed by the members O of the set K of all double cones in n-dimensional Minkowski spacetime. It will be assumed that isotony holds, i.e., O1 ⊂ O ⇒ F(O1 ) ⊂ F(O) .
(QFTGA.1)
C∗ in B(H) Hence, one may form the smallest C ∗ -algebra F := O F(O) containing all local field algebras F(O). (In the above quintuple, F is short for the family {F(O)}O∈K .) There is a strongly continuous unitary representation Rn a → U (a) ∈ B(H) of the group of translations Rn on H whose action on {F(O)}O∈K is covariant, i.e.,
(QFTGA.2)
U (a)F(O)U (a)∗ = F(O + a) ,
a ∈ Rn , O ∈ K .
Moreover, it will be assumed that the relativistic spectrum condition holds: The joint spectrum of the selfadjoint generators of U (Rn ) is contained in the closed forward lightcone V + . There is a compact group G, and a strongly continuous,2 faithful representation G g → U (g) ∈ B(H) of the group G on H. It is assumed that the action of this unitary representation on {F(O)}O∈K preserves localization, i.e., U (g)F(O)U (g)∗ = F(O) , g ∈ G , O ∈ K ,
(QFTGA.3)
and also that this group representation commutes with the translations: U (g)U (a) = U (a)U (g) ,
g ∈ G , a ∈ Rn .
G will be called the gauge group. There is a unit vector Ω ∈ H which is invariant under all U (a), a ∈ Rn , and under all U (g), g ∈ G, and which moreover has the cyclicity property FΩ = H. This vector is called the vacuum vector.
(QFTGA.4)
There is an element k contained in the center of G and fulfilling k 2 = 1G (the unit group element) so that, upon setting
(QFTGA.5)
F± := 2 whenever
1 (F ± U (k)F U (k)∗ ) , 2
this makes sense, i.e., when G possesses continuous parts
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the following relations hold whenever F ∈ F(O1 ), F ∈ F(O2 ), and the double cones O1 and O2 are spacelike separated: F+ F+ = F+ F+ ,
F+ F− = F− F+ ,
F− F− = −F− F− .
(2.1)
These properties are referred to as normal commutation relations. Remark. It was already mentioned in the introduction that the definition of a QFTGA is slightly more general than that of a quantum field system with gauge symmetry (see Sec. 4) which is more directly related to the theory of superselection charges; however, the differences are minute and mainly of technical nature. The advantage of working with QFTGAs is that their structure is stable with respect to passing to scaling limit theories, as will become clear in the present section. The next task is to introduce the counterpart of the scaling algebra for a QFTGA which was defined in [11] for quantum field theories formulated in terms of local observable algebras. To that end, we assume that we are given an arbitrary normal, covariant quantum field theory with gauge group action (F, U (Rn ), U (G), Ω, k) (henceforth called the “underlying QFTGA”) and keep it fixed. It will be convenient to introduce the following notation for the adjoint actions of translations and gauge group: αa (F ) := U (a)F U (a)∗ ,
βg (F ) := U (g)F U (g)∗ ,
for all F ∈ F, a ∈ Rn , g ∈ G. Definition 2.2 For each O ∈ K, we define F(O) as the set of all functions F : R+ → F, λ → F λ , having the following properties: (a) F λ ∈ F(λO), (b) || F || := supλ ||F λ || < ∞, (c) || αa (F ) − F || → 0 as a → 0, where (αa (F ))λ := αλa (F λ ) ,
(2.2)
(d) || β g (F ) − F || → 0 as g → 1G , where (β g (F ))λ := βg (F λ ) .
(2.3)
In [11] the case was considered that F is an observable algebra. In that case, the action of the gauge group U (G) on H is trivial, and spacelike commutativity holds for the local algebras F(O), meaning that F(O1 ) ⊂ F(O2 ) if O1 and O2 are spacelike separated. The motivation for imposing the conditions (a-d) above is similar as for the scaling algebra in the case that F is an observable algebra discussed in [11]. The idea is to view the F λ as the image of an element F ∈ F
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under the action of any “renormalization group transformation” Rλ (so one should think of F λ as Rλ (F )). In other words, the collection of all functions λ → F λ with the above stated properties corresponds to all possible orbits of elements in F under all (abstract) renormalization group transformations. The general properties of renormalization group transformations in the present, model-independent setting are hence encoded by the conditions (a-d). We point out that (c) ensures that the energy-momentum transferred by F λ scales like const.·1/λ, see [11] for further discussion. As has been indicated to us by D. Buchholz, it should be noted that there may actually be situations where the lifted action of the gauge transformations ought to be defined differently than in (2.3). This occurs for example if the charges of the theory have a dimension which isn’t independent of length or energy (in this sense, they are “dimensionful” charges), and this can happen in two- and threedimensional models. For the time being, we neglect this possibility, but we point out that it deserves attention. There are some simple consequences of Def. 2.2 which we briefly put on record here, see [11] for more details. First, it is easy to see that each F(O), O ∈ K, is a C ∗ -algebra with respect to the C ∗ -norm introduced in (b) when the algebraic operations are defined pointwise for each λ. Clearly one also has isotony, O1 ⊂ O ⇒ F(O1 ) ⊂ F(O) . C∗ One can thus form the C ∗ -algebra F = O F(O) . The “lifted” actions αRn and β G of translations and gauge group, defined in (2.2) and (2.3), respectively, act by automorphisms on F under preservation of the corresponding covariance properties, i.e., αa (F(O)) = F(O + a) ,
β g (F(O)) = F(O) .
(2.4)
Moreover, we may define 1 (F ± β k (F )) 2 and hence obtain relations similar to (2.1) for F ∈ F(O1 ), F ∈ F(O2 ) and O1 and O2 spacelike separated. Finally we note that one may demonstrate the existence of a wealth of elements in F as follows. Let µ be a left-invariant Borel-measure on G and let h be any continuous, compactly supported function on Rd × G. Pick any uniformly bounded function R+ λ → Xλ ∈ F so that Xλ ∈ F(λO) for each λ and some O ∈ K, and define F λ := dn a dµ(g) h(a, g) αλa (βg (Xλ )) (2.5) F± =
where the integral is to be understood in the weak sense. Then it is easily checked that R+ λ → F λ is contained in F(O× ) whenever O× is any open neighborhood of O + g∈G supp h( . , g).
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Having defined the scaling field algebra F of the underlying QFTGA, we may associate with any locally normal state ω on F 3 a parametrized family (ω λ )λ>0 of states on F, where ω λ (F ) := ω (F λ ) , F ∈ F . As in [11], we adopt the following definition of scaling limit states. Definition 2.3 For each locally normal state ω on F, we regard the family (ω λ )λ>0 as a generalized sequence directed towards λ = 0. Hence, by the Banach-Alaoglu theorem [28], the family (ω λ )λ>0 on the C ∗ -algebra F possesses weak-* limit points. This set of weak-* limit points will be denoted by {ω0,ι : ι ∈ I} where I is a suitable F F index set, or simply by SL (ω ). Each ω0,ι ∈ SL (ω ) is a state on F, and is called a scaling limit state of ω . We note that the definition of weak-* limit points means that there exists for (ι) each label ι a directed set Kι together with a generalized sequence (λκ )κ∈Kι of positive numbers converging to 0 so that (F ) = lim ω λ(ι) (F ) , ω0,ι κ
κ
F ∈ F.
∈ SLF (ω ) its Again following [11], we introduce for each scaling limit state ω0,ι GNS-representation (π0,ι , H0,ι , Ω0,ι ) and define
F0,ι (O) := π0,ι (F(O)) ,
F0,ι :=
F0,ι (O)
C∗
.
O
Many of the following results (containing also some new definitions) concerning the structure of scaling limit states and their associated GNS-representations in the present setting are generalizations of similar statements in [11]. Proposition 2.4 1. For each pair of locally normal states ω and ω on F it holds that SLF (ω ) = SLF (ω ) . 2. Let ω be a locally normal state on F. Then each ω0,ι ∈ SLF (ω ) is invariant n under the actions of αa , a ∈ R , and β g , g ∈ G: ◦ αa = ω0,ι , ω0,ι
ω0,ι ◦ β g = ω0,ι .
Hence, there are unitary group representations of the translation group and the gauge group on H0,ι which are, respectively, defined by U0,ι (a)π0,ι (F )Ω0,ι U0,ι (g)π0,ι (F )Ω0,ι
:= π0,ι (αa (F ))Ω0,ι , := π0,ι (β g (F ))Ω0,ι
for all a ∈ Rn , g ∈ G, and F ∈ F. 3a
state ω on F is called locally normal if ω F(O) is normal for each O ∈ K
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3. The unitary group representations U0,ι (a), a ∈ Rn , and U0,ι (g), g ∈ G, are continuous and have the properties U0,ι (a)F0,ι (O)U0,ι (a)∗ = F0,ι (O + a) ,
U0,ι (g)F0,ι (O)U0,ι (g)∗ = F0,ι (O)
for all a ∈ Rn , g ∈ G and O ∈ K. Moreover, the unitary translation group U0,ι (a), a ∈ Rn , fulfills the relativistic spectrum condition. 4. The set N0,ι of all g ∈ G so that U0,ι (g)ψ = ψ holds for all ψ ∈ H0,ι is a closed normal subgroup of G. Therefore, • U0,ι : G•0,ι g • → U0,ι (g)
(2.6)
is a continuous faithful representation of the factor group G•0,ι = G/N0,ι . • Here, g → g • ≡ g0,ι is the quotient map, and in (2.6), g is any element in the pre-image of g • with respect to the quotient map. 5. Define for f ∈ F0,ι , f ± :=
1 • • (f ± U0,ι (k • )f U0,ι (k • )∗ ) 2
• . Then the following holds: If O1 and O2 are spacelike sepawhere k • = k0,ι rated double cones and f ∈ F0,ι (O1 ), f ∈ F0,ι (O2 ), one has the relations
f + f + = f + f + ,
f + f − = f − f + ,
f − f − = −f − f − .
(2.7)
6. The previous statements yield the following corollary: Let ω be a locally nor mal state on F (of the underlying QFTGA) and ω0,ι ∈ SLF (ω ) an arbitrary scaling limit state, then the corresponding scaling limit objects • • (G•0,ι ), Ω0,ι , k0,ι ) (F0,ι , U0,ι (Rn ), U0,ι
form again a normal, covariant quantum field theory with gauge group action (which will be called a scaling limit QFTGA of the underlying QFTGA ). corresponding to ω0,ι Proof. Ad 1. The proof is analogous to that in [11], which uses an argument due to Roberts [30] showing that ||(ω − ω ) F(λO)|| → 0 as λ → 0
(2.8)
holds for any pair of locally normal states ω and ω on F and O ∈ K as a consequence of F(O) = C · 1 . O0
This latter property holds also for the local field algebras owing to the spectrum condition for the translation group and normal commutation relations (2.1), see [11] for details.
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Ad 2. The invariance property is obvious for the case that ω coincides with the vacuum state ω(F ) = Ω, F Ω on F. Then (2.8) implies the analogous property for any other locally normal state. Ad 3. The continuity follows simply from assumptions (c) and (d) of Def. 2.2. The covariance properties are implied by (2.4). The spectrum condition for the translations may be proved as in [11]. • Ad 4. By construction, U0,ι is a faithful unitary representation of G•0,ι on H0,ι . Continuity follows since the quotient map g → g • is open. Ad 5. As indicated above, the relations (2.1) carry over to the scaling algebra F by setting F ± = 12 (F ± β k (F )). The corresponding relations for the scaling limit theories follow directly. (It may however happen that k ∈ N0,ι ; in this case, the last, “fermionic” relation of (2.7) is absent, and spacelike commutativity holds for the local scaling limit algebras F0,ι (O), O ∈ K.) Henceforth, we will (without restriction of generality in view of 1. of Prop. 2.4) always consider scaling limit states ω0,ι ∈ SLF (ω) where ω( . ) = Ω, . Ω denotes the vacuum state. As was done in [11], we will identify scaling limit theories which are isomorphic in a sense that we will describe next. Definition 2.5 Let • • • • (G•0,ι ), Ω0,ι , k0,ι ) and (F0,γ , U0,γ (Rn ), U0,γ (G•0,γ ), Ω0,γ , k0,γ ) (F0,ι , U0,ι (Rn ), U0,ι
be two scaling limit theories of an underlying QFTGA. These two scaling limit theories will be called isomorphic if there exists a C ∗ -algebraic isomorphism φ : F0,ι → F0,γ so that the following properties hold: φ(F0,ι (O)) φ ◦ Ad U0,ι (a)
= F0,γ (O) , O ∈ K , = Ad U0,γ (a) ◦ φ , a ∈ Rn ,
φ ◦ Ad U0,ι (g)
= Ad U0,γ (g) ◦ φ ,
g ∈ G.
Note that the last property induces a natural identification between N0,ι and • • → g0,γ ∈ G•0,γ , so that one N0,γ and hence a natural identification G•0,ι g0,ι obtains, in consequence, • • • • φ ◦ Ad U0,ι (g0,ι ) = Ad U0,γ (g0,γ ) ◦ φ • • • • which holds in particular with k0,ι and k0,γ inserted for g0,ι and g0,γ , respectively. We will moreover say that two isomorphic scaling limit theories have a unique vacuum structure if the connecting isomorphism also has the property
ω0,γ ◦ φ = ω0,ι . Following once more [11], one may now classify a given underlying QFTGA according to the following (mutually exclusive) possibilities:
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(1) All scaling limit QFTGAs are isomorphic, and F0,ι is non-abelian. Then the underlying QFTGA is said to have a unique quantum scaling limit. (2) All scaling limit QFTGAs are isomorphic, and F0,ι is abelian. In this case one says that the underlying QFTGA has a classical scaling limit. (3) There are scaling limit QFTGAs which are non-isomorphic. One then says that the underlying QFTGA has a degenerate scaling limit. The interpretation of these cases is as in the case of observable algebras [11]; see this reference for further discussion. The first case would correspond to an underlying theory which has a single, stable ultraviolet fixed point. The second case is thought to correspond to an underlying theory which has no ultraviolet fixed point. The third case is in a sense intermediate, the underlying theory has a very irregular behavior at small scales and has various, most likely unstable, ultraviolet fixed points. We next put on record a result from [11] connecting the uniqueness of the scaling limit with the existence of a dilation symmetry in the scaling limit theories. The proof proceeds exactly as in the cited reference. Proposition 2.6 [11] Assume that all the scaling limit QFTGAs • • (G•0,ι ), Ω0,ι , k0,ι ), (F0,ι , U0,ι (Rn ), U0,ι
ι ∈ I,
of the underlying QFTGA are isomorphic, i.e., that we are in case (1) or (2) of (0,ι) the just given classification. Then for each ι ∈ I there exists a family (δµ )µ>0 of automorphisms of F0,ι acting as dilations in the corresponding scaling limit theory, which means that the following relations hold: δµ(0,ι) (π0,ι (F(O)) δµ(0,ι) ◦
=
Ad U0,ι (a) =
δµ(0,ι) ◦ Ad U0,ι (g) =
π0,ι (F(µO)) ,
µ > 0, O ∈ K,
Ad U0,ι (µa) ◦ δµ(0,ι) , Ad U0,ι (g) ◦ δµ(0,ι) ,
a ∈ Rn , µ > 0 , g ∈ G, µ > 0.
Furthermore, if the underlying QFTGA also has a unique vacuum structure in the scaling limit, then it follows that the family of dilations leaves the scaling limit (0,ι) states invariant: ω0,ι ◦ δµ = ω0,ι , ι ∈ I, µ > 0.
3 Scaling limits for QFTGAs with additional properties In the present section we consider an underlying QFTGA with additional properties, such as Lorentz-covariance, spacelike clustering and geometric modular action, and we will investigate which further properties for the scaling limit theories ensue. More precisely, let (F, U (Rn ), U (G), Ω, k) be the underlying QFTGA, assumed to satisfy the conditions (QFTGA.1-5) of Def. 2.1. We will consider the following additional properties:
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(Lorentz covariance) There is a strongly continuous unitary represen˜ ↑ L → U˜ (L) ∈ B(H) of the covering group of the proper, ortation L + thochronous Lorentz group L↑+ (in d dimensions) on H so that the following relations are fulfilled:
(QFTGA.6)
U˜(L)U (a) = U˜(L)U (g) = U˜ (L)F(O)U˜ (L)∗ =
U (Λ(L)a)U˜ (L) , U (g)U˜ (L) , F(Λ(L)O) ,
U˜ (L)Ω = Ω
˜ ↑ L → Λ(L) ∈ L↑ ˜ ↑ , a ∈ Rn , g ∈ G and O ∈ K, where L for all L ∈ L + + + denotes the covering projection. (QFTGA.7)
(Irreducibility) F = C · 1 .
(Spacelike clustering) We will assume that a uniform clustering bound holds on the vacuum (for spacetime dimension d ≥ 3). To formulate this, we use the following notation. Elements in the x0 = 0 hyperplane will be denoted by x ∈ Rn−1 and identified with (0, x) ∈ Rn . We define the derivation d α(x0 ,0) (F ) ∂0 (F ) := −i 0 dx x0 =0
(QFTGA.8)
on the domain D(∂0 ) of all F ∈ F so that the (weak) derivative on the righthand side exists as an element in F. Note that D(∂0 ) is a weakly dense subset of F. Then our assumption on the existence of a uniform spacelike clustering bound is: There exists, for the given underlying QFTGA, a constant c > 0 so that for each double cone Or having spherical base of radius r in the x0 = 0 hyperplane there holds the bound |ω(F1 αx (F2 )) − ω(F1 )ω(F2 )| ≤
crn−1 (||F1 || ||∂0 (F2 )|| + ||∂0 (F1 )|| ||F2 ||) |x|n−2
for all F1 , F2 ∈ F(Or ) ∩ D(∂0 ) as soon as |x| > 3r. (Geometric modular action) A wedge region is any Poincar´e-transformed copy of the so-called right wedge WR := {(x0 , . . . , xn−1 ) : |x1 | < x0 , x0 > 0}. For this right wedge, we define the wedge-reflection map rR : Rn → Rn by
(QFTGA.9)
rR (x0 , x1 , x2 , . . . , xn−1 ) := (−x0 , −x1 , x2 , . . . , xn−1 ) , and the Lorentz-boosts ΛR (t)(x0 , x1 , x2 , . . . , xn−1 ) :=
(cosh(t)x0 + sinh(t)x1 , sinh(t)x0 + cosh(t)x1 , x2 , . . . , xn−1 ) .
For any other wedge-region W = LWR with a suitable Poincar´e-transformation L, we define rW := LrR L−1 and ΛW (t) := LΛR (t)L−1 .
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For each wedge region W in Rn , the vacuum vector Ω of the underlying QFTGA is cyclic and separating for the von Neumann algebra F(W ) = {F(O) : O ⊂ W, O ∈ K} . Hence, there correspond to each wedge region W the Tomita-Takesaki modular objects JW , ∆W associated with F(W ), Ω [35]. It will then be assumed that, in the presence of (QFTGA.6), these modular objects act geometrically in the following way: JW U˜ (L)JW = U˜ (Ad˜ rW L),
˜↑ , L∈L +
(3.1a)
JW U (a)JW = U (rW a), ˜ W (2πt)), ∆it = U˜ (Λ
a∈R ,
(3.1b)
t ∈ R,
(3.2)
JW F(O)JW = F (rW O),
O ∈ K.
(3.3)
n
W
t
In these equations, we have denoted by Ad˜ rW the lift of the adjoint action ˜ ↑ , and by Λ ˜ ↑ for which ˜ W (t) the continuous lift of ΛW (t) to L of rW to L + + ˜ W (0) = 1 ˜ ↑ (both of which exist, cf. [4]). Moreover, we have introduced the Λ L+ so-called “twisted” local von Neumann algebras Ft (O) := V F(O)V ∗ ,
O ∈ K,
(3.4)
where the twisting operator V is a unitary on H defined by V := (1 + i)−1 (1 + iU (k)) .
(3.5)
Note that the algebras F(O1 ) and Ft (O2 ) commute for spacelike separated O1 and O2 on account of the assumed normal commutation relations. We shall continue our investigation of the scaling limit theories of an underlying QFTGA satisfying some, or all, of the just stated additional conditions. In order to do that, we have to slightly re-define the scaling algebras F(O) when the underlying QFTGA satisfies Lorentz-covariance. For the remaining part of this article we adopt the following Convention. Suppose that the underlying QFTGA satisfies also the condition of Lorentz-covariance (QFTGA.6). In this case, the local scaling algebras F(O), O ∈ K, are defined as in Def. 2.2 but demanding in addition that the elements F ∈ F(O) fulfill the also the condition (e) where
|| α ˜ L (F ) − F || → 0 as L → 1L˜ ↑
+
(˜ αL (F ))λ := U˜ (L)F λ U˜ (L)∗ .
Again, it is not difficult to demonstrate that, with that convention, the F(O) are C ∗ -algebras containing plenty of elements, and αRn , β G and α ˜ L˜ ↑ act as strongly +
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continuous groups of automorphisms on F with the covariance properties (2.4) and, in addition, α ˜ L (F(O)) = F(Λ(L)O) ,
˜↑ , O ∈ K . L∈L +
The following statement is again essentially a transcription of analogous results established for observable algebras in [11]. Proposition 3.1 Suppose that the underlying QFTGA fulfills the conditions of Def. 2.2. 1. If the underlying QFTGA fulfills also Lorentz-covariance (QFTGA.6), then this property holds also for all scaling limit QFTGAs. 2. If the underlying QFTGA fulfills (QFTGA.6 & 7) and n ≥ 3, then all scaling limit QFTGAs fulfill (QFTGA.6 & 7). 3. If the underlying QFTGA fulfills (QFTGA.8) and n ≥ 3, then all scaling limit QFTGAs fulfill (QFTGA.7). 4. If the underlying QFTGA fulfills (QFTGA.6 & 9), then all scaling limit QFTGAs fulfill (QFTGA.6 & 9), too. Proof. Ad 1. This statement is proved in complete analogy to the corresponding statement in [11]; we note that for any scaling limit state ω0,ι ∈ SLF (ω) (where ω is any locally normal state on F) there holds ω0,ι ◦ α ˜ L = ω0,ι and hence one ˜ ↑ on H0,ι via setting obtains a unitary representation of L + αL (F ))Ω0,ι , U˜0,ι (L)π0,ι (F )Ω0,ι := π0,ι (˜
˜↑ , F ∈ F . L∈L +
It is also easily checked that this unitary representation has all the properties analogous to those listed in (QFTGA.6) with respect to the scaling limit theory. Ad 2. If the underlying theory has the additional properties (QFTGA.6 & 7), then this entails that the underlying theory also has the property (QFTGA.8) according to a result by Araki, Hepp and Ruelle [3]; cf. also the proof of Lemma 4.3 in [11]. The statement then follows from 1. and 3. Ad 3. Let F (1) , F (2) ∈ F(Or ) and define, for some h ∈ C0∞ (Rn ), F (j) := dn a h(a)αa (F (j) ) , j = 1, 2 . Then there is some r > r so that F (j) ∈ F(Or ), and clearly F λ ∈ F(λO) ∩ D(∂0 ). We apply the uniform clustering bound to obtain, for each λ > 0 and |x| > 3r, (j)
|ω λ (F (1) αx (F (2) )) − ω λ (F (1) )ω λ (F (2) )| (1)
(2)
(1)
(2)
= |ω(F λ αλx (F λ )) − ω Ω (F λ )ω(F λ )| ≤ ≤
c(λr)n−1 (1) (2) (1) (2) (||F λ || ||∂0 (F λ )|| + ||∂0 (F λ )|| ||F λ ||) |λx|n−2 crn−1 (|| F (1) || || ∂ 0 (F (2) ) || + || ∂ 0 (F (1) ) || || F (2) ||) , |x|n−2
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(j) ||∂0 (F λ )||
−1
(j) d ) and used the dx0 x0 =0 α(x0 ,0) (F (j) || ∂ 0 (F ) || < ∞ by the definition of
where we have defined ∂ 0 (F (j) ) := −i
823
fact that
≤ λ || ∂ 0 (F ) ||. Now the F (j) , and taking the lim supλ on the left-hand side of the last inequality, one concludes that asymptotic spacelike clustering holds on the vacuum of each scaling limit theory since F (j) approaches F (j) in the scaling algebra norm for h → δ. Because of normal commutation relations in each scaling limit QFTGA, this entails that F0,ι = C · 1 holds in all scaling limit theories. Ad 4. The proof proceeds analogously to the proof of Lemma 4.3 in [11]. (j)
There is another result worth mentioning here which also generalizes a corresponding result established for observable algebras in [11] and connects a duality condition in scaling limit theories with the type of the local von Neumann algebras of the underlying QFTGA. Theorem 3.2 Suppose that the underlying QFTGA fulfills the assumptions of Def. 2.1. Moreover, suppose that there exists a scaling limit QFTGA • • (G•0,ι ), Ω0,ι , k0,ι ) (F0,ι , U0,ι (Rn ), U0,ι
having the property of “twisted wedge duality”, t F0,ι (W ) = F0,ι (rW (W ))
for some wedge region W in Rd (with the definition of the twisted local von Neumann algebras analogous to (3.4) and (3.5) with respect to the corresponding objects in the scaling limit QFTGA); moreover, suppose that F0,ι = C · 1. In this case it holds that the local von Neumann algebras F(O) are of type III1 for each double cone O ⊂ W whose boundary intersects W ∩ rW (W ), and for all translates of such double cones O. If twisted wedge duality holds for all wedge regions in some scaling limit QFTGA, then one concludes that F(O) is of type III1 for all double cones. We refer to Prop. 6.4 in [11] for a proof of this statement. We note also that according to the previous proposition, the validity of conditions (QFTGA.6 & 7 & 9) in the underlying theory implies that the assumptions of Thm. 3.2 are fulfilled.
4 Quantum field systems with gauge symmetry We now wish to investigate the scaling limits of QFTGAs that really correspond to superselection charges of a system of observables. Such QFTGAs are, more specifically, quantum field systems with gauge symmetry in the terminology of Doplicher and Roberts [16]. In order to summarize their definition here, and also for later reference, we first recapitulate some concepts of the Doplicher-Haag-Roberts approach to superselection theory, mainly from the sources [21, 31, 16]. This approach starts from the assumption that one is given an observable quantum system in a vacuum representation together with a further, distinguished
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set of representations modeling localized charges. The structure of an observable quantum system in a vacuum representation is described in terms of a collection of objects (Avac , Uvac (Rn ), Ωvac ) whose properties are assumed to be as follows. (a) Avac symbolizes a family {Avac (O)}O∈K of von Neumann algebras in a separable Hilbert space Hvac , subject to conditions of isotony (see above) and duality, Avac (O) = Avac (O ) := {Avac (O1 ) : O1 ⊂ O , O1 ∈ K} , where O denotes the open causal complement of O. Setting moreover Avac := C∗ , it is assumed that Avac = C · 1. O Avac (O) (b) Uvac (a), a ∈ Rn , is a strongly continuous unitary representation of the translation group on Hvac , acting covariantly on the family {Avac (O)}O∈K , and fulfilling the spectrum condition (see above). Furthermore, Ωvac ∈ Hvac is a unit vector which is let invariant by the action of Uvac (a), a ∈ Rn . Remark. Usually, also the assumption is made that the family {Avac (O)}O∈K has the Borchers property (“Property B”). This property says that given O, O1 ∈ K with O ⊂ O1 and a non-zero projection E ∈ A(O), then there is V ∈ A(O1 ) with V V ∗ = E and V ∗ V = 1. However, Roberts has shown [32] that this property can already be deduced from the other assumptions (essential being separability of Hvac and the spectrum condition). Given an observable quantum system (Avac , Uvac (Rn ), Ωvac ), one may look for representations of Avac describing the presence of charges. Following Doplicher, Haag and Roberts, one may consider the set PDHR of representations π of Avac which are unitarily equivalent to the vacuum representation in restriction to the causal complement of any double cone. That means, if Avac (O ) is defined as the C ∗ -algebra generated by all Avac (O1 ) where O1 ⊂ O , then π is in PDHR if π Avac (O ) is unitarily equivalent to the identical representation of Avac (O ) on B(Hvac ) for each O ∈ K. Such representations describe superselection charges which are strictly localizable, see [21, 31] for further discussion. We shall be interDHR which are translation-covariant, ested only in the subset PDHR cov of those π in P meaning that there is a strongly continuous representation Uπ (a), a ∈ Rn , of the translation group on the representation-Hilbert space of π fulfilling the spectrum condition and the intertwining property Ad Uπ (a)(π(A)) = π(Ad Uvac (a)(A)) ,
a ∈ Rn , A ∈ Avac .
(4.1)
By identifying the representation-Hilbert space Hπ with Hvac , the set PDHR cov may of covariant, alternatively (and equivalently) be described in terms of the set ∆cov t localized and transportable endomorphisms of Avac . Here, an endomorphism ρ : Avac → Avac is called localized in O ∈ K if ρ(A) = A holds for all A ∈ Avac (O ). It is called transportable if, given an arbitrary region O1 ∈ K, there exists a unitary V so that V ρ( . )V ∗ is an endomorphism of A localized in O1 ; one can show that V may be chosen as an element of A.
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An element ρ ∈ ∆cov is called irreducible if ρ(Avac ) = C · 1, and the set t Sect of all equivalence classes cov
[ρ] := {V ρ( . )V ∗ : V ∗ = V −1 ∈ Avac } for irreducible ρ ∈ ∆cov is called the set of translation-covariant superselection t sectors of the given observable quantum system (Avac , Uvac (Rn ), Ωvac ). If ρ, ρ ∈ ∆cov t , one defines by I(ρ, ρ ) the set of intertwiners between ρ and ρ as the set of all T ∈ Avac which satisfy T ρ(A) = ρ (A)T ,
A ∈ Avac .
Strictly speaking, one should refer to I(ρ, ρ ) as the set of global intertwiners between ρ and ρ . Given O1 ∈ K and ρ, ρ ∈ ∆cov localized in O1 , one can introduce t I(ρ, ρ )O , the set of local intertwiners with respect to the localization region O ⊃ O1 , as consisting of all T ∈ Avac fulfilling T ρ(A) = ρ (A)T ,
A ∈ Avac (O) .
Hence it is obvious that I(ρ, ρ )O ⊃ I(ρ, ρ ) for all O ∈ K, and in Sec. 6 we will link the question if local and global intertwiners are equivalent, i.e., if I(ρ, ρ )O = I(ρ, ρ ) holds for all O ∈ K, to the preservance of charges in the scaling limit. Presently, we need to very briefly summarize some further concepts of charge superselection theory (see, e.g. [31] for a more detailed account). First, one can introduce for T1 ∈ I(ρ1 , ρ1 ) and T2 ∈ I(ρ2 , ρ2 ) a product operation T1 × T2 yielding an element in I(ρ1 ρ2 , ρ1 ρ2 ). There is then a distinguished family of intertwiners
(ρ1 , ρ2 ) ∈ I(ρ1 ρ2 , ρ2 ρ1 ), for irreducible ρ1 , ρ2 ∈ ∆cov t , characterized by the property that it describes the exchange in the intertwiner product according to (T2 × T1 ) (ρ1 , ρ2 ) = (ρ1 , ρ2 )(T1 × T2 ) ,
Tj ∈ I(ρj , ρj ) ,
together with the properties (ρ1 , ρ2 ) = 1ρ1 ρ2 if the localization regions of ρ1 and ρ2 are spacelike separated, and (ρ2 , ρ1 ) (ρ1 , ρ2 ) = 1ρ1 ρ2 . Moreover, one can show that each irreducible ρ ∈ ∆cov possesses a left inverse ϕρ , i.e., a positive linear map t on Avac which preserves the unit and fulfills ϕρ (Aρ(B)) = ϕρ (A)B. Then there is for ρ a number λρ so that ϕρ ( (ρ, ρ)) = λρ 1. The number λρ depends only on the equivalence class [ρ] of ρ and is called the statistics parameter of the corresponding superselection sector. If λρ = 0, then the superselection sector is said to have finite statistics. We define by Sectcov fin the set of all translation-covariant superselection sectors of the underlying observable quantum system which have finite statistics, and by ∆cov fin the set of all endomorphisms ρ with [ρ] ∈ Sectcov fin . Finally, we need to recollect the notion of a conjugate charge. One can show cov (cf., e.g., [31]) that for each ρ ∈ ∆cov fin localized in O ∈ K there is some ρ ∈ ∆fin ,
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also localized in O, together with isometries R and R in A(O) which intertwine the endomorphisms ρρ and ρρ, respectively, with the identical endomorphism of Avac , that is, ρ(ρ(A))R = RA and ρ(ρ(A))R = RA ,
A ∈ Avac .
In this case, one calls [ρ] the conjugate superselection sector of [ρ] or, synonymously, the conjugate charge of [ρ]. Doplicher and Roberts [16] have shown that one can construct from ∆cov fin and the interwiners a system of local field algebras, acted upon by a faithful unitary representation of a compact group – called the gauge group – such that the local algebras of the initially given observable quantum system are embedded in the local field algebras as exactly containing the invariant elements under the gauge group action. In more precise terms, they have shown that one can associate with (Avac , Uvac , Ωvac ) a quantum field system with gauge symmetry (QFSGS), defined as follows: Definition 4.1 (F, U (Rn ), U (G), Ω, k) is a QFSGS for (Avac , Uvac , Ωvac ) and ∆cov fin if the following conditions hold: (F, U (Rn ), U (G), Ω, k) is a QFTGA; the Hilbert space on which the von Neumann algebras F(O) of F = {F(O)}O∈K act will be denoted by H. Moreover, F = C1.
(QFSGS.1)
(QFSGS.2)
There is a C ∗ -algebraic monomorphism π : Avac → F
containing the vacuum representation (i.e., the identical representation of Avac on Hvac ) as a sub-representation, and such that π(Avac (O)) consists exactly of all A ∈ F(O) having the property U (g)AU (g)∗ = A for all g ∈ G. We will use the shorter notation A(O) := π(Avac (O)) . Moreover, the sub-Hilbert-space H0 of H which is generated by all vectors A(O)Ω, as O ranges over the double cones, is cyclic for the algebras F(O). Let [ρ] ∈ Sectcov fin be a superselection sector. Then there exists a finitedimensional, irreducible, unitary representation
(QFSGS.3)
v[ρ] = (v[ρ]ji )di,j=1 of G (acting as a matrix representation for some suitable d = d[ρ] ) so that, for each O ∈ K, there is a multiplet ψ1 , . . . , ψd of elements in F(O) having
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the following properties: ∗
U (g)ψi U (g) =
d
ψj v[ρ]ji (g) ,
(4.2)
j=1 d
ψi∗ ψj = δij 1 ,
ψj ψj∗ = 1 ,
(4.3)
j=1
π ◦ ρO (A) =
d
ψj π(A)ψj∗ ,
A ∈ Avac ,
(4.4)
j=1
for some representer ρO of [ρ] localized in O . These properties fix v[ρ] to within unitary equivalence. F(O) is generated by A(O) and all multiplets ψj , j = 1, . . . , d[ρ] , with the properties (4.2),(4.3), (4.4), as [ρ] ranges over all superselection sectors in Sectcov fin . For each finite-dimensional, irreducible, unitary representation v of G there is some superselection sector [ρ] ∈ Sectcov fin so that v = v[ρ] where v[ρ] has the properties of (QFSGS.3).
(QFSGS.4)
The conditions for a QFSGS associated with (Avac , Uvac (Rn ), Ωvac ) and ∆cov fin are given here in a form slightly different from the statement in [16]; however, the present formulation is convenient for our purposes. It is plain that a QFSGS is a QFTGA fulfilling additional properties. Condition (QFSGS.4) states, in particular, that Sectcov fin can be identified with the dual, of the gauge group G. The connection between field algebra and superselection G, sectors is essentially expressed through the multiplet operators ψ1 , . . . , ψd with the properties listed in (QFSGS.3). In fact, the occurrence of such “charge multiplets” associated with the superselection sector [ρ] is equivalent to the presence of the corresponding charge in the QFSGS (F, U (Rn ), U (G), Ω, k). This will, basically, be our starting point for formulating criteria that express “preservation of a charge” in the scaling limit.
5 Preservance of charges in the scaling limit Let us now discuss the problem of characterizing “preservation of charges in the scaling limit” in greater detail. To this end, let (F, U (Rn ), U (G), Ω, k) be a QFSGS n associated with (Avac , Uvac (Rn ), Ωvac ) and ∆cov fin . Since (F, U (R ), U (G), Ω, k) is a QFTGA, we can form the corresponding scaling algebra F as in Sec. 2. We may then define A(O) = {A ∈ F(O) : Aλ ∈ A(λO)} , and it is not difficult to see that A(O) consists exactly of the A ∈ F(O) so that β g (A) = A for all g ∈ G.
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Now let ω0,ι ∈ SLF (ω) be a scaling limit state on F, and denote by • • (F0,ι , U0,ι (Rn ), U0,ι (G•0,ι ), Ω0,ι , k0,ι )
the corresponding scaling limit QFTGA. Let us also denote by A0,ι (O) = π0,ι (A(O)) ,
O ∈ K,
the von Neumann algebra formed by the scaling limits of the observables of the underlying QFSGS, and define by •
• • (g • )f = f U0,ι (g • ) ∀ g • ∈ G•0,ι } F0,ι (O)G0,ι = {f ∈ F0,ι (O) : U0,ι
the fixed point algebra of the gauge group action in the scaling limit. With this notation, and recalling that H0,ι = F0,ι Ω0,ι , we find: Lemma 5.1 •
(i) A0,ι (O) = F0,ι (O)G0,ι , O ∈ K. (ii) Suppose that Ω0,ι is the unique (up to a phase) unit vector in H0,ι which is C∗ invariant under U0,ι (Rn ) (equivalently, F0,ι = C · 1). If A0,ι = O A0,ι (O) is abelian, then F0,ι = C · 1 and hence, H0,ι = CΩ0,ι . •
Proof. (i) Clearly, one has A0,ι (O) ⊂ F0,ι (O)G0,ι . To show that the reverse in • clusion holds, let f ∈ F0,ι (O)G0,ι . Denote by m0,ι (h) = G dµ(g) U0,ι (g)hU0,ι (g)∗ , h ∈ F0,ι , the mean over the action of G on F0,ι . We have m0,ι (f ) = f . Let F (n) , n ∈ N, be a sequence of elements in F(O) so that w-limn→∞ π0,ι (F (n) ) = f . Such a sequence exists because, by a Reeh-Schlieder argument, Ω0,ι is separating for F0,ι (O). Using this separating property of Ω0,ι once more, also m0,ι (π0,ι (F (n) )) approximates f weakly. On the other hand, dµ(g) π0,ι (β g (F (n) )) = π0,ι ( dµ(g) β g (F (n) )) , m0,ι (π0,ι (F (n) )) = G
G
where we made use of the continuity of β G in norm on the scaling algebra to interchange representation and integration. Since G dµ(g) β g (F (n) ) is contained in A(O), we see that f is weakly approximated by elements in A0,ι (O) and hence is itself contained in A0,ι (O). (ii) Under the given hypotheses, a result by Buchholz (Lemma 3.1 in [8]) shows (0,ι) that A0,ι = C · 1. Hence, the strongly continuous group βg = Ad U0,ι (g), g ∈ G, (0,ι) of automorphisms on F0,ι acts ergodically, meaning that βg (f ) = f for all g ∈ G implies f ∈ C · 1. Using Thm. 4.1 in [23], it follows that the unique ergodic state (0,ι) (0,ι) for βG on F0,ι is a trace. The scaling limit vacuum Ω0,ι , . Ω0,ι is a pure βG invariant state on F0,ι and hence is a trace. (Purity of this state holds since the
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space of translation-invariant vectors in H0,ι is one-dimensional.) This implies Ω0,ι , f ∗ U0,ι (x)f Ω0,ι = Ω0,ι , f U0,ι (−x)f ∗ Ω0,ι for each f ∈ F0,ι (O), O ∈ K, and all x ∈ Rn . Arguing with spectrum condition and clustering (as a consequence of the assumption that every translation-invariant vector in H0,ι is a multiple of Ω0,ι ) in the same manner as in the proof of Lemma 3.1 in [8], one concludes that f ∈ C · 1. Hence F0,ι = C · 1. The lemma shows that all charges of the underlying QFSGS disappear in a scaling limit theory once the scaling limit theory is known to be classical for the observables, provided the underlying theory satisfies very general conditions such as clustering (QFTGA.8) or (for n ≥ 3) Lorentz-covariance (QFTGA.6). At this point, we should emphasize the distinction between charges in the scaling limit QFTGA which are “scaling limits of charges of the underlying QFSGS”, and “charges arising as superselection sectors of the scaling limit theory”, as was first discussed by D. Buchholz [7]. Charges of the first mentioned type correspond • (G•0,ι ) acts non-trivially to the situation that G•0,ι is non-trivial and hence U0,ι • (and faithfully) on F0,ι . In this case, the action of U0,ι (G•0,ι ) on the elements of F0,ι may be seen as a short-distance remnant of the action of U (G) on F so that, • of G• may be viewed as repcorrespondingly, the members of the spectrum G 0,ι 0,ι of the underlying QFSGS. resenting short-distance remnants of the charges in G It is important to note that, to some extent, these charges of the scaling limit theory have been present in the underlying QFSGS. We will discuss this case in more detail below. The second type of charges in the scaling limit arises in a different way. One may consider the scaling limit theory (induced by ω0,ι ∈ SLF (ω)) (A0,ι , U0,ι (Rn ), Ω0,ι ) which is gained from the observables of the underlying QFSGS as a new observable quantum system in its own right (provided it fulfills the assumptions of irreducibilcov ity). Then one can assign a set of superselection sectors Sectcov fin = Sectfin (A0,ι ) to this observable quantum system, and by the Doplicher-Roberts reconstruction theorem4 , we can now associate to these data a QFSGS, which we may denote by (F(0,ι) , U (0,ι) (Rn ), U (0,ι) (G(0,ι) ), Ω(0,ι) , k (0,ι) ) . Thus, this QFSGS contains the superselection charges which arise in the scaling limit theory of the observables of the underlying QFSGS. In general, it may occur that F0,ι is properly contained in F(0,ι) and that G•0,ι is a factor group of G(0,ι) by some non-trivial normal subgroup, so that the QFTGA associated with F0,ι may be viewed as a proper subtheory (in the sense of [16]) of the QFSGS associated with 4 Provided that (A , U n 0,ι 0,ι (R ), Ω0,ι ) fulfills all conditions for an observable quantum system in vacuum representation. See our discussion after Prop. 5.6, and Prop. 5.7.
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F(0,ι) . Buchholz [7] proposed to consider such a case as a criterion for confinement, since it models the situation where charges appear as superselection sectors of the (observables’) scaling limit theory which do not arise as scaling limits of charges that occur as superselection sectors in the underlying QFSGS. We refer to [7, 9] for further discussion, and we note that examples for superselection charges of this second type have been constructed for the Schwinger model in two spacetime dimensions [9, 12]. In the present work, we shall restrict attention solely to charges in the scaling limit QFTGAs of an underlying QFSGS of the first mentioned type, i.e., which arise as “scaling limits” of charges present in the underlying QFSGS. Having clarified this basic point, we must find criteria which express that a charge of the underlying QFSGS has a non-trivial scaling limit. There are some prefatory observations which may be helpful as a guideline. We have already seen that the gauge group G•0,ι = G0,ι /N0,ι of a scaling limit QFTGA is a factor group of G0,ι which is itself a copy of G, the gauge group of the underlying QFSGS. It may in general happen that the normal subgroup N0,ι is non-trivial, and hence that G•0,ι is “smaller” than G. In this situation, certainly not all the charges of the underlying QFSGS will have counterparts in the scaling limit QFTGA. Thus, we will in general be confronted with a situation which is in a sense complementary to that of F0,ι ⊂ F(0,ι) mentioned just before and where, morally, the scaling limit QFTGA associated with F0,ι corresponds to a subtheory of the underlying QFSGS, at least as far as the charge structure is concerned.5 However, since there is no inclusion of F0,ι into F, we need to establish a correspondence between elements in F0,ι and in F which allows to decide if charges present in the underlying QFSGS are also present in the scaling limit. As we have mentioned above, the presence of a superselection charge in the underlying QFSGS manifests itself through the presence of charge multiplets ψ1 , . . . , ψd ∈ F which transform under a finite dimensional, irreducible, unitary representation v[ρ] as described in (QFSGS.3). This will be the starting point for our criterion of charge preservance in the scaling limit. To fix ideas, let (F, U (Rn ), U (G), Ω, k) denote the underlying QFSGS, and let [ρ] ∈ Sectcov fin be one of its superselection sectors, and pick some arbitrary O ∈ K. Then there is a finite-dimensional, irreducible, unitary representation v[ρ] of G and, for each λ > 0, a multiplet of elements ψ1 (λ), . . . , ψd (λ) in F(λO) having the properties of (QFSGS.3) with respect to the localization region λO. We will refer to any such multiplet family {ψ1 (λ), . . . , ψd (λ)}λ>0 as a scaled multiplet for [ρ]. The principal idea is now to view the functions λ → ψj (λ) as “would-be” elements of F(O) and to follow their fate as λ approaches 0. However, these functions won’t satisfy the “phase-space constraint” condition (c) of Def. 2.2 which is essential in order to interpret them as orbits of field algebra elements under (abstract) renormalization group transformations. Hence, if ω0,ι is a scaling limit state, in general one cannot 5 The dynamics of the theories corresponding to F 0,ι and F are expected to be different and so the former cannot be a subtheory of the latter in the full sense of the definition.
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form π0,ι (ψj ( . )) since ψj ( . ) won’t belong to the scaling algebra F. But one can still check if, in the scaling limit, scaled multiplets become close to elements of π0,ι (F) so that they can effectively be regarded as representing elements in the scaling limit von Neumann algebras F0,ι (O) = π0,ι (F(O)) . We will introduce some new terminology which gives this idea a more precise shape. Definition 5.2 Let ω0,ι ∈ SLF (ω) be a scaling limit state of the underlying QFSGS. Then we say that a family F = {F(λ)}λ>0 fulfilling (i) F(λ) ∈ F(λO1 ) for some O1 ∈ K, (ii) supλ>0 ||F(λ)|| < ∞, and (iii) supλ ||βg (F(λ))−F(λ)|| → 0 for g → 1G , is asymptotically contained in F0,ι (O) if the following holds: For each given > 0 there are elements F and F in F(O) such that
lim sup ||(F(λκ ) − F λκ )Ω|| + ||(F(λκ ) − F λκ )∗ Ω|| < ,
(5.1)
κ
where the net {λκ }κ∈K of positive numbers converges to 0, with ω0,ι = limκ ω λκ on F. We will next collect some immediate consequences of this definition; this requires yet some further notation. Given a finite-dimensional Lie group X endowed with a Borel measure µ which is invariant under group transformations (for ↑ ), we call a sequence of functions {hν }ν∈N of our purposes, X = Rn or X = P + class L1 (X, µ) ∩ C0∞ (X) a δ-sequence if supp hν+1 ⊂ supp hν , ν supp hν = 1X , supν ||hν ||L1 < ∞, and if X hν χ dµ converges to χ(1X ) as ν → ∞ for all continuous functions χ on X. Here, 1X is the group unit element; note that 1Rn = 0. Lemma 5.3 Let ω0,ι be a scaling limit state of the underlying QFSGS, and suppose that F = {F(λ)}λ>0 is a family of elements in F with the properties as in the previous definition. Then the following statements are equivalent: (a) F is asymptotically contained in F0,ι (O) for all O ⊃ O1 , (b) In the scaling limit, F is approached in the ∗-strong topology by elements in π0,ι (F(O)) in the following sense: Whenever O ⊃ O1 , > 0 and finitely many F (1) , . . . , F (N ) ∈ F are given, then there is an F ∈ F(O) fulfilling ||F || ≤ supλ ||F(λ)|| and
(j) (j) lim sup ||(F(λκ ) − F λκ )F λκ Ω|| + ||(F(λκ ) − F λκ )∗ F λκ Ω|| < , κ
j = 1, . . . , N , where {λκ }κ∈K is as in the previous definition, (c) Given any δ-sequence {hν } on Rn , there holds
lim || ( (αhν F)(λκ ) − F(λκ ) )Ω || + || ( (αhν F)(λκ ) − F(λκ ) )∗ Ω || = 0 , (κ,ν)
(5.2)
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where the limit is taken with respect to the partial ordering on K × N given by (κ, ν) > (κ , ν ) :⇔ κ > κ and ν > ν , and (αh F)(λ) = dn x h(x)αλx (F(λ)) , λ > 0 , h ∈ L1 (Rn ) . (The latter integral is to be interpreted in the weak topology on F; {λκ }κ∈K is as before.) Proof. (a) ⇒ (c). Let > 0 be arbitrary. Then we must show that there exist κ ∈ K and ν ∈ N so that (5.3) || ( (αhν F)(λκ ) − F(λκ ) )Ω || + || ( (αhν F)(λκ ) − F(λκ ) )∗ Ω || < n holds for all (κ, ν) > (κ , ν ). Writing (αh F )λ = d x h(x)αλx (F λ ), we consider the estimate ||( (αhν F)(λκ ) − F(λκ ) ) Ω|| ≤ ||( (αhν F)(λκ ) − (αhν F )λκ ) Ω|| + ||( (αhν F )λκ − F λκ ) Ω|| + ||(F λκ − F(λκ )) Ω|| , (5.4) where (. . . ) stands for either (. . . ) or (. . .)∗ . Now we use the fact that, owing to the definition of asymptotic containment, one may choose F , F in such a way that there is some κ with (supν ||hν ||L1 ||(f (λκ ) − F λκ ) Ω|| < /6 for all κ > κ , where F = F or F = F according if (. . .) = (. . .) or (. . .) = (. . .)∗ . Denoting ˆ the Fourier transform of h and by P = (Pν )n−1 the selfadjoint generators by h ν=0 of the unitary translation group of the underlying QFSGS, the first term on the right-hand side of (5.4) is seen to equal ˆ ν (P )(F(λκ ) − F ) Ω|| ≤ sup ||hν ||L1 ||(F(λκ ) − F ) Ω|| < /6 ||h λκ λκ ν
for all κ > κ . The second term on the right-hand side of (5.4) can be estimated by sup ||hν ||L1 sup ||αx (F ) − F || ν
x∈supp hν
and using the continuity of F with respect to αx , this quantity may be made smaller than /6 for all ν smaller than some suitable ν . Summing up, we obtain (5.3) for all (κ, ν) > (κ , ν ). (c) ⇒ (b). It holds that λ → Φλ = (αhν F)(λ) is contained in F(O× ) where O× is any double cone containing O1 + supp hν . A standard Reeh-Schlieder argument shows that, if W is any wedge region in the causal complement of O× , then t F0,ι (W )Ω0,ι is dense in H0,ι hence F0,ι (W )Ω0,ι is dense in H0,ι . As a consequence, (j) there is for given F ∈ F and given η > 0 some B (j) ∈ F(W ) so that, if V0,ι is the natural twist on F0,ι , (j)
(j)
||(π0,ι (F (j) ) − V0,ι π0,ι (B (j) ))Ω0,ι || = lim ||(F λκ − V B λκ )Ω|| < η. κ
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Thus, making first η and then ν −1 small enough, one can arrange that
(j) (j) lim sup ||(Φλκ − F(λκ ))F λκ Ω|| + ||(Φλκ − F(λκ ))∗ F λκ Ω|| κ
(j) (j) ≤ lim sup ||(Φλκ − F(λκ ))V B λκ Ω|| + ||(Φλκ − F(λκ ))∗ V B λκ Ω|| κ
+4η sup ||F(λκ )|| κ
=
(j) (j) lim sup ||B λκ V ∗ (Φλκ − F(λκ ))Ω|| + ||B λκ V ∗ (Φλκ − F(λκ ))∗ Ω|| κ
≤
+4η sup ||F(λκ )|| κ
(j) lim sup ||B || ||(Φλκ − F(λκ ))Ω|| + ||(Φλκ − F(λκ ))∗ Ω|| κ
+4η sup ||F(λκ )|| κ
can be made smaller than any given > 0; then, for a sufficiently large ν, Φ can be taken as the F required in (b). Note that in passing from the second line to (j) the third we have used that V B λ V ∗ commutes with (Φλ − F(λ)) and its adjoint, because of the localization properties of the operators involved. The implication (b) ⇒ (a) is obvious. Remark. In view of statement (b) of the previous Lemma, one might refer to our notion of asymptotic containment more precisely as ∗-strong asymptotic containment. It should then be obvious how to introduce, e.g., the notion of strong or weak asymptotic containment in F0,ι (O) for families F = {F(λ)}λ>0 fulfilling the properties as in 5.2. One could also drop condition (iii) on F in the definition of asymptotic containment, then having to define in Lemma 5.3 αh F differently, cf. (2.5). After these preparations, we can now present our criterion for preservance of charges in the scaling limit. Definition 5.4 Let ω0,ι ∈ SLF (ω) be a scaling limit state of the underlying QFSGS, and let [ρ] ∈ Sectcov fin be a superselection sector. Then we say that the charge [ρ] is preserved in the scaling limit QFTGA of ω0,ι if, for each O1 ∈ K, there is some scaled multiplet {ψ1 (λ), . . . , ψd (λ)}λ>0 for [ρ] with ψj (λ) ∈ F(λO1 ) such that all families {ψj (λ)}λ>0 , j = 1, . . . , d, are asymptotically contained in F0,ι (O) if O ⊃ O1 . Let us briefly convince ourselves that each family {ψj (λ)}λ>0 of a scaled multiplet satisfies the assumptions (i)–(iii) of Def. 5.2. Clearly, only condition (iii) need be checked, and one has sup ||βg (ψj (λ)) − ψj (λ)|| = sup || λ
λ
d i=1
ψi (λ)(v[ρ]ij (g) − δij )|| ≤ d max |v[ρ]ij (g) − δij | i,j
where the last term tends to 0 if g → 1G if G is a continuous group.
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We remark that, in view of part (c) of Lemma 5.3, a similar criterion has been used recently by Morsella [26]. Part (c) of Lemma 5.3 also provides some insight into the basic mechanism which might cause charges to disappear in the scaling limit. To elaborate on that, we consider a scaled multiplet {ψ1 (λ), . . . , ψd (λ)}λ>0 1 n for nthe charge [ρ]. Moreover, for h ∈ L (R ) with compact support and h ≥ 0, d x h(x) = 1, we define (h,j)
Φλ
= (αh ψj )(λ) , λ > 0 .
Now by Lemma 5.3 it follows that for the charge [ρ] to be preserved in the scaling limit QFTGA of ω0,ι , one must be able to choose a scaled multiplet and h in such a way that ||π0,ι (Φ(h,j) )Ω0,ι || comes arbitrarily close to 1. It could however happen that for all scaled multiplets and any choice of h one ends up with ||π0,ι (Φ(h,j)∗ )Ω0,ι || = 0 , which also implies π0,ι (Φ(h,j) ) = 0 since Ω0,ι is separating for the local field algebras of the scaling limit QFTGA. We can interpret this as follows. The convolution of the scaled charge multiplets ψj (λ) with respect to the scaled action of the translations, which produces elements Φ(h,j) in F, results in an energy damping of the (h,j)∗ to the vacuum vector Ω. charged states that are obtained by applying the Φλ This energy damping scales inversely, that is, proportional to λ−1 , to the localiza(h,j) tion scale of the Φλ . Depending on the dynamics of the underlying QFSGS, it may happen that the amount of energy-momentum required to create the charged vectors ψj (λ)∗ Ω from the vacuum in a small region of scale λ is typically larger than ∼ λ−1 , e.g. of the type ∼ λ−q with some q > 1. In this case, the energy damp(h,j)∗ ing leads to a “blotting out” of the charged contributions of Φλ Ω, resulting in the vanishing of the norm of these vectors as λ approaches 0. In other words, our preservance criterion amounts essentially to requiring that the energy-momentum needed to localize the considered charges is only restricted by Heisenberg’s principle, and, also in view of the specific phase space properties of renormalization group orbits encoded in the scaling algebra construction, it is evident that a condition of this kind is needed in order to single out “elementary, pointlike” charges, which survive the scaling limit (cf. also mechanism (B) below). Let us sketch two – quite distinct – physical mechanisms that may account for the disappearance of charges in the scaling limit. (A) There is a strongly attractive force between the charges at short distances. This may have the effect that certain “compounds” of charges are dynamically more favorable than single charges. That is to say, it may cost far less energy to create a compound of several charges at small scales than the single charges contained in the compound. In this case, the compound charges could survive the scaling limit (i.e., be preserved), while certain single charges disappear since their creation costs too much energy at small
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scales. The compound charges preserved in the scaling limit could then well be invariant under some normal subgroup of the gauge group of the underlying quantum field theory. In a sense, this mechanism is complementary to that of confinement at finite distances of charges which would be viewed as “free” charges in the short-distance scaling limit (asymptotic freedom) as in QCD. There, one expects that the color charges correspond to charges which are present as superselection charges of a scaling limit quantum field theory (corresponding to field multiplets in F(0,ι) , not in F0,ι ), while in the underlying quantum field theory, at finite scale, only color-neutral compounds of the color-charges appear. (B) The charges are strongly repellent at short distances.6 In this situation, in order to localize two or more charges in a small spacetime region of scale λ, one requires more energy than of the order of λ−1 . This will typically lead to disappearance of certain charges in the scaling limit in the following way. If {ψ1 (λ), . . . , ψd (λ)}λ>0 is a scaled multiplet for some charge [ρ], then there acts the tensor product representation v [ρ] ⊗ v[ρ] of the gauge group on the space of vectors spanned by ψj∗ (λ)ψk∗ (λ)Ω, for each λ > 0. This tensor product representation can be decomposed into a sum of irreducibles, so that ∗ a ,m (λ)ψ ,m (λ)Ω ψj∗ (λ)ψk∗ (λ)Ω =
,m
with suitable coefficients a ,m (λ), where the {ψ1,m (λ), . . . , ψdm ,m (λ)}λ>0 are scaled multiplets corresponding to the charges [ρm ] labelled by m which appear in the decomposition of v [ρ] ⊗ v[ρ] . Assuming now that the interaction between charges of type [ρ] is strongly repellent at short distances this will, in keeping with our discussion on the energy damping caused by applying αh , typically result in π0,ι (αh (ψj∗ ψk∗ ))Ω0,ι = 0 for some indices j, k, and therefore in ∗ ))Ω0,ι = 0 π0,ι (αh (ψ ,m
for some indices , m. Concerning the question whether our criterion for preservance of charges is fulfilled in certain quantum field models, we note that the charges of the Majorana-Dirac field satisfy indeed this criterion in all scaling limit states (see Appendix A). We also remark that all charges in a dilation covariant theory complying with the hypotheses of [30] are preserved in all scaling limit QFTGAs: if for a multiplet ψj ∈ F(O), j = 1, . . . , d, associated to a given sector [ρ] ∈ Sectcov fin , we define ψj (λ) = D(λ)ψj D(λ)−1 , j = 1, . . . , d, D(λ) being the unitary implementation 6 This
mechanism has been pointed out to us by D. Buchholz
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of dilations, we get a scaled multiplet {ψ1 (λ), . . . , ψd (λ)}λ>0 for [ρ] such that ψj (·) ∈ F(O), as can be easily derived from the commutation relations between dilations and translations, and then in particular ψj (·) is asymptotically contained in every F0,ι (O). (This proves incidentally that a dilation covariant theory admits no classical scaling limit as soon as it has multiplets with dimension d > 1.) Our criterion of charge preservance not only bars the situation of charge disappearance, but it even implies that the limits of π0,ι (Φ(h,j) ), j = 1, . . . , d, as h tends to the δ-measure, yield charge multiplets corresponding to the charge [ρ] with respect to their transformation behavior under the scaling limit gauge group. This is the content of the following statement. Proposition 5.5 Suppose that the charge [ρ] is preserved in the scaling limit QFTGA of ω0,ι . Let {ψ1 (λ), . . . , ψd (λ)}λ>0 be a scaled multiplet for [ρ] which is asymptotically contained in F0,ι (O), and let Φ(h,j) be defined as before with respect to the {ψj (λ)}λ>0 . Then for any δ-sequence {hν } on Rn , the limit operators ψ j = s- lim π0,ι (Φ(hµ ,j) ) ν→∞
and
ψ ∗j = s- lim π0,ι (Φ(hµ ,j) )∗ ν→∞
(5.5)
ˆ whenexist, are independent of the chosen δ-sequence and are contained in F0,ι (O) ˆ ever O ⊃ O. Furthermore, ψ 1 , . . . , ψ d forms a multiplet transforming under the • adjoint action of U0,ι (G•0,ι ) according to the irreducible, unitary representation v[ρ] . More precisely, denoting by G g → g • ∈ G•0,ι the quotient map, there is a • • of G•0,ι so that v[ρ] (g • ) = finite-dimensional, irreducible, unitary representation v[ρ] v[ρ] (g) for all g ∈ G and • • U0,ι (g • )ψ j U0,ι (g • )∗ =
d
• ψ i v[ρ]ij (g • ) ,
g • ∈ G•0,ι .
i=1
˜ ν˜ } be Proof. First we need to establish existence of the limit. Let {hν } and {h n δ-sequences on R . Choose any > 0. Then one can find ν0 > 0 so that ˜
(h ,j)
˜ ,j) (h
||( π0,ι (Φ(hν ,j) ) − π0,ι (Φ(hν˜ ,j) )Ω0,ι || = lim ||(Φλκν − Φλκν˜ )Ω|| κ
˜ ,j) (hν ,j) (h ≤ lim sup ||(Φλκ − ψj (λκ ))Ω|| + ||(Φλκν˜ − ψj (λκ ))Ω|| < κ
if ν, ν˜ > ν0 . This shows that π0,ι (Φ(hν ,j) )Ω0,ι is a Cauchy sequence in ν → ∞ and hence has a limit in H0,ι ; it shows also that the limit is independent of the chosen δ-sequence. Since Ω0,ι is separating for the local scaling limit field algebras and ||Φ(hν ,j) || is bounded uniformly in ν, one can thus conclude that π0,ι (Φ(hν ,j) ) ˆ if O ˆ ⊃ O. Similarly converges strongly to some ψ j which is contained in F0,ι (O) ∗ one argues that π0,ι (Φ(hν ,j) )∗ converges strongly to ψ j .
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Next we demonstrate ψ ∗j ψ k = δjk 1. To this end, we observe that for any F ∈ F there holds the following chain of equations, π0,ι (F )Ω0,ι , (ψ ∗j ψ k − δjk 1)Ω0,ι = =
(h ,j)
lim lim [ Φλκν
ν→∞
κ
(h ,k)
F λκ Ω, Φλκν
Ω − δjk F λκ Ω, Ω ]
lim lim [ ψj (λκ )F λκ Ω, ψk (λκ )Ω − δjk F λκ Ω, Ω
ν→∞
κ
(h ,j)
+ (Φλκν
(h ,j)
+ Φλκν
− ψj (λκ ))F λκ Ω, ψk (λκ )Ω (h ,k)
Ω, (Φλκν
− ψk (λκ ))Ω ] .
The expression on the third to last line is equal to 0 since ψj (λ)∗ ψk (λ) = δjk 1 by assumption, and the limits of the expressions on the last two lines vanish by the argument having led to the conclusion (c) ⇒ (b) in the proof of Lemma 5.3. This proves ψ ∗j ψ k = δjk 1 by the separating property of Ω0,ι for the local field algebras in the scaling limit. d The proof of j=1 ψ j ψ ∗j = 1 is completely analogous. For the last part of the statement, we observe that U0,ι (g)ψ j U0,ι (g)∗ =
d
ψ k v[ρ]kj (g) ,
g ∈ G,
k=1
is simply a consequence of β g (Φ
(h,j)
)=
d
Φ(h,k) v[ρ]kj (g) ,
g ∈ G;
k=1
this, in turn, can be seen from Φ(h,j) = αh ψj and the commutativity of β g and αx . On the other hand, from the definition of N0,ι one obtains ψ j Ω0,ι = U0,ι (n)ψ j Ω0,ι =
d
ψ k v[ρ]kj (n)Ω0,ι
k=1
for all n ∈ N0,ι , and multiplying by ψ ∗i from the left yields δij Ω0,ι = v[ρ]ij (n)Ω0,ι for all n ∈ N0,ι . This shows v[ρ] (n) = 1 (the unit matrix) for all n ∈ N0,ι and hence • • there is an irreducible, unitary representation v[ρ] of G•0,ι so that v[ρ] (g • ) = v[ρ] (g) for all g ∈ G, proving the last part of the statement. There is an obvious connection between the scaling limits of scaled multiplets for a charge [ρ] and the scaling limits of endomorphisms induced by scaled multiplets in case that [ρ] is preserved in a scaling limit state. While fairly immediate, we put the corresponding result on record here.
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Proposition 5.6 Let ω0,ι ∈ SLF (ω) and let [ρ] ∈ Sectcov fin be a charge of the underlying QFSGS which is preserved in the scaling limit QFTGA of ω0,ι . Moreover, let {ψ1 (λ), . . . , ψd (λ)}λ>0 be a scaled multiplet for [ρ] asymptotically contained in F0,ι (O) and let, with respect to this scaled multiplet, ψ 1 , . . . , ψ d be defined as in (5.5). Then for each A ∈ A the family {ρ(A)(λ)}λ>0 defined by ρ(A)(λ) =
d
ψj (λ)Aλ ψj (λ)∗
j=1
is asymptotically contained in A0,ι . Furthermore, for each δ-sequence {hν } on Rn there holds s- lim π0,ι (αhν ρ(A)) = ν→∞
d
ψ j π0,ι (A)ψ ∗j ,
A ∈ A;
(5.6)
j=1
and ρ defined by ρ(a) =
d
ψ j aψ ∗j ,
a ∈ A0,ι ,
(5.7)
j=1
is a localized, transportable, irreducible endomorphism of A0,ι which is moreover covariant and has finite statistics. Proof. The asymptotic containment in A0,ι of {ρ(A)(λ)}λ>0 is simply a consequence of the asymptotic containment of each {ψj (λ)}λ>0 in F0,ι (O) and the fact that ρ(A)(λ) ∈ A(λ(O1 ∩ O2 )) for A ∈ A(O2 ), with the conventional assumption that ψj (λ) ∈ F(λO1 ). Owing to (5.7) and the properties of a multiplet, ρ is clearly a localized, irreducible endomorphism of A0,ι . The transportability may be seen as follows. According to the definition of preserved charge, there is for any double cone O× different from O a scaled multiplet for [ρ], {ψ˜1 (λ), . . . , ψ˜d (λ)}λ>0 , which is asymptotically contained in F0,ι (O× ). In the same way as the {ψj (λ)}λ>0 lead ˆ for all O ˆ ⊃ O, the {ψ˜j (λ)}λ>0 lead to multito multiplet operators ψ d in F0,ι (O) ˜ contained in F0,ι (O ˆ× ) for all O ˆ × ⊃ O× . For the corresponding plet operators ψ j ˜ it then holds that T ρ ˜ ( . ) = ρ( . )T with the unitary intertwiner endomorphism ρ d ˜∗ T = j=1 ψ j ψ j . Now it is easy to see that the family {T (λ)}λ>0 defined by d T (λ) = j=1 ψj (λ)ψ˜j∗ is asymptotically contained in A0,ι (O∗ ) for some double cone O∗ , and by an argument by now familiar, T = s- limν→∞ π0,ι (αhν T ) showing ˆ∗ ) for O ˆ ∗ ⊃ O∗ . Covariance follows from a general that T is contained in A0,ι (O argument: Given a multiplet ψ 1 , . . . , ψ d , it holds that ρ(U aU ∗ ) = W ρ(a)W ∗ for d ∗ each unitary U with W = j=1 ψ j U ψ j which is itself unitary. Moreover, if a continuous unitary group a → U (a), a ∈ Rn , fulfills the spectrum condition, then a → W (a) = dj=1 ψ j U (a)ψ ∗j is clearly also a continuous unitary group fulfilling the spectrum condition. That ρ has finite statistics follows from the finiteness of the dimension d of the multiplet.
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Thus we have now seen that a preserved charge gives rise to localized, transportable endomorphisms of the observable quantum system in the scaling limit. However, it is not clear if the scaling limit theories (A0,ι , U0,ι (Rn ), Ω0,ι ) gained from the observables of the underlying QFSGS satisfy all the technical properties assumed to hold for an observable quantum system in vacuum representation. Namely, the following two conditions cannot be asserted for (A0,ι , U0,ι (Rn ), Ω0,ι ) from the (0) general assumptions made so far: (1) separability of the Hilbert-space H0,ι = A0,ι Ω0,ι and (2) Haag-duality. Without these two conditions, one is not really in the situation where standard superselection theory applies, and so it is not really clear if one can construct a QFSGS for (A0,ι , U0,ι (Rn ), Ω0,ι ) that could be compared to the scaling limit QFTGA of the underlying theory. Separability of the vacuum Hilbert-space of an observable quantum system is expected to hold quite generally for physically realistic theories, and moreover can be concluded for scaling limit theories from a decent behavior of the energylevel density of the states of the underlying quantum field theory at short scales [8]. On the other hand, the difficulty with the possible failure of Haag-duality can be overcome by passing to the “dual net” to A0,ι . This is the family Ad0,ι = {Ad0,ι (O)}O∈K of von Neumann algebras defined by Ad0,ι (O) := A0,ι (O ) where O denotes the causal complement of O and A0,ι (O ) is the von Neumann algebra generated by the A0,ι (O1 ) with O1 ⊂ O . Note that with this definition, since the family A0,ι satisfies the condition of locality, it holds that A0,ι (O) ⊂ Ad0,ι (O). Quite obviously, U0,ι (Rn ) extends to a covariant action of the translations on Ad0,ι fulfilling the spectrum condition. Now it is known [29] that, if the family A0,ι fulfills the condition of geometric modular action analogous to condition (QFTGA.9), then the condition of Haag duality is fulfilled for the dual net Ad0,ι . In turn, if the observable system (Avac , Uvac (Rn ), Ωvac ) of the underlying QFSGS is Lorentz covariant (QFTGA.6) and fulfills geometric modular action (QFTGA.9), then – observing the convention stated in Sec. 3 – also each scaling limit theory (A0,ι , U0,ι (Rn ), Ω0,ι ) fulfills QFTGA.9 (cf. Prop. 3.1,4). Moreover, it was shown in Sec. 3.4 of [31] that any localized, transportable endomorphism of A0,ι extends to a localized, transportable endomorphism of Ad0,ι provided the latter fulfills the condition of Haag-duality. Therefore we have deduced the following result: Proposition 5.7 Let the dimension of spacetime n be 3 or higher. Assume that the observable quantum system of the underlying QFSGS fulfills Lorentz covariance (QFTGA.6) and geometric modular action (QFTGA.9) and, moreover, that the (0) scaling limit Hilbert-space H0,ι = A0,ι Ω0,ι is separable. Then (Ad0,ι , U0,ι (Rn ), Ω0,ι ) is an observable quantum system in vacuum representation fulfilling the conditions (a) and (b) at the beginning of Sec. 4. If a superselection charge [ρ] of the underlying QFSGS is preserved in this scaling limit, then the corresponding ρ defined in (5.7) extends to a localized, transportable endomorphism of Ad0,ι which is covariant and has finite statistics.
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The last result presented in this section concerns the preservance of the conjugate charge of a preserved charge. To this end, let us assume for the remainder of this section that the underlying QFSGS fulfills also the condition of geometric modular action as formulated in (QFTGA.9) in Sec. 3. (We note that this can be deduced already if a similar form of geometric modular action is initially only assumed to hold for the underlying observable quantum system provided it fulfills some mild additional conditions. We refer to [18, 19, 24] for discussion of this issue.) In this case, let ψ1 , . . . , ψd be a multiplet for the charge [ρ] ∈ Sectcov fin , with all ψj contained in F(O) for some O ∈ K, and assume that W is a wedge region containing O. Let JW denote the Tomita-Takesaki modular conjugation associated with F(W ) and the vacuum vector Ω. Then one can take an arbitrary multiplet ψ1 , . . . , ψd for [ρ] with all ψj ∈ F(rW O), and define a new multiplet of operators ψ j ∈ F(O), j = 1, . . . , d, by ψ j = JW V ψj V ∗ JW where V is the “twist” operator defined in (3.5). It is easy to check that the ψ j indeed form a multiplet, i.e., dj=1 ψ j ψ ∗j = 1 and ψ ∗j ψ k = δjk 1; however, since JW is antilinear, this multiplet transforms under the gauge group action according to the conjugate representation v [ρ] of v[ρ] , U (g)ψ j U (g)∗ =
d
g ∈ G,
ψ i v [ρ]ij (g) ,
i=1
if ψ1 , . . . , ψd transforms under the gauge group according to v[ρ] . This indicates that ψ 1 , . . . , ψ d is a multiplet of the conjugate sector [ρ] of [ρ]. Indeed, writing ρ(A) =
d j=1
ψj Aψj∗ , ρ(A) =
d
1 1 ψ j Aψ ∗j , R = √ ψ j ψj , R = √ ψj ψ j , d j=1 d j=1 j=1 d
d
one can easily check that R and R are isometries in A(O) and moreover, there holds ρ(ρ(A))R = RA and ρ(ρ(A))R = RA , A ∈ A . (This can actually also be deduced from a rather more general argument of [18].) Equipped with these observations, we can now state the result. Theorem 5.8 Suppose that the underlying QFSGS fulfills the conditions of Poincar´e covariance (QFTGA.6) and of geometric modular action (QFTGA.9), and let ω0,ι ∈ SLF (ω) be one of its scaling limit states. Then a charge [ρ] ∈ Sectcov fin is preserved in the scaling limit state ω0,ι if and only if also the conjugate charge [ρ] is preserved. Proof. Assume that [ρ] is preserved in the scaling limit state ω0,ι and let O ∈ K. Then for O1 ∈ K with O1 ⊂ O there is a scaled multiplet {ψ1 (λ), . . . , ψd (λ)}λ>0 for [ρ], contained in F(λrW O1 ) and asymptotically contained in F0,ι (rW O1 ). Let
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ψ j be defined by
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ψ j (λ) = JW V ψj (λ)V ∗ JW
where V is the “twist” operator (cf. eq. (3.5)) and JW is the modular conjugation associated with F(W ) and the vacuum vector Ω. Then {ψ1 (λ), . . . , ψ d (λ)}λ>0 is a scaled multiplet for the conjugate charge [ρ] and each ψ j (λ) is contained in F(λO1 ). Moreover, for compactly supported h ∈ L1 (Rn ) it holds that (αh ψ j )(λ) = JW V (αh◦rW ψj )(λ)V ∗ JW and this shows that the {ψj (λ)}λ>0 are asymptotically contained in F0,ι (O).
Remarks. (i) Note that under the conditions of Thm. 5.8 one also obtains asymptotic scaling limit versions of the isometries which intertwine ρ and ρ. More precisely, suppose that a charge [ρ] is preserved in the scaling limit state ω0,ι , and let ψ 1 , . . . , ψ d be a corresponding multiplet contained in F0,ι (O) induced by a scaled multiplet {ψ1 (λ), . . . , ψd (λ)}λ>0 . As the previous Theorem shows, there is then a conjugate multiplet ψ 1 , . . . , ψ d in F0,ι (O) induced by a scaled multiplet {ψ1 (λ), . . . , ψ d (λ)}λ>0 , and it is straightforward to show that R = d d d−1/2 j=1 ψ j ψ j and R = d−1/2 j=1 ψ j ψ j are given by 1 1 π0,ι (αhν ψ j ψj ) and R = s- lim √ π0,ι (αhµ ψj ψ j ) R = s- lim √ ν→∞ ν→∞ d j=1 d j=1 d
d
where {hν } is any δ-sequence on Rn . Using this, one deduces ρ(ρ(a))R = Ra
and ρ(ρ(a))R = Ra ,
a ∈ A0,ι ,
where ρ and ρ relate to the ψ j and ψ j , respectively, as in (5.7). (ii) In addition, one obtains also the following: Let JW 0,ι and V0,ι denote the analogous objects to JW and V in the scaling limit theory of ω0,ι , then a conjugate charge multiplet ψ 1 , . . . , ψ d to ψ 1 , . . . , ψ d is obtained by ∗ ψ j = JW 0,ι V0,ι ψ j V0,ι JW 0,ι
whenever ψ 1 , . . . , ψ d is a multiplet equivalent to ψ 1 , . . . , ψ d localized in rW O.
6 On equivalence of local and global intertwiners In the present section we will address the question of equivalence of local and global intertwiners of superselection sectors. We shall extend an argument of Roberts [30] who considered the setting of dilation covariant quantum field theories, showing that the preservance of all charges in some scaling limit theories7 is, together with 7 As already remarked in sec. 5, all charges are preserved in all scaling limits of dilation covariant theories.
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the assumption that the local field algebras F(O) are factors, sufficient for the equivalence of local and global intertwiners. Our main technical result is stated in the following Lemma. Lemma 6.1 Let [ρ] ∈ Sectcov fin be a superselection sector of the underlying QFSGS, let O ∈ K, and suppose that there are (i) a scaling limit state ω0,ι ∈ SLF (ω), (ii) a scaled multiplet {ψ1 (λ), . . . , ψd (λ)}λ>0 for [ρ] with ψj (λ) ∈ F(λO), (iii) some compactly supported, non-negative h ∈ L1 (Rn ), such that ||π0,ι (Φ(h,j) )Ω0,ι || > 0 ,
j = 1, . . . , d ,
where Φλ = (αh ψj )(λ). Then for all unitaries U ∈ A(O) ∩ F(O) and all mul˜ tiplets ψ1 , . . . , ψ˜d ∈ F(O) for [ρ] (O ∈ K) there holds (h,j)
ω(ψ˜j∗ U ∗ ψ˜k U ) = δjk ω(ψ˜∗ βk (U ∗ )ψ˜k U ) = δjk j
if βk (ψ˜i ) = ψ˜i if βk (ψ˜i ) = −ψ˜i
(6.1)
Proof. We will treat explicitly the “even” case in (6.1), the “odd” case being completely analogous. First we note that ||π0,ι (Φ(h,j) )Ω0,ι || > 0 for any of the j = 1, . . . , d implies that the π0,ι (Φ(h,j) )Ω0,ι , j = 1, . . . , d, are linearly independent. To see this, note that the contrary assumptionof linear dependence implies that there is an invertible d × d matrix (uj ) so that j π0,ι (Φ(h,j) )Ω0,ι uj = 0 for some . But this implies π0,ι (Φ(h,j) )Ω0,ι uj = π0,ι (Φ(h,k) )Ω0,ι v[ρ]kj (g)uj
0 = U0,ι (g) j
j,k
for all g ∈ G and hence, since v[ρ] is irreducible, π0,ι (Φ(h,j) )Ω0,ι = 0 for all j. We further observe that it constitutes no restriction of generality to prove the statement of the theorem only for O ∈ K which contain the origin 0 ∈ Rn in their spacelike boundary (i.e., the origin is contained both in the boundary of O and in the boundary of its spacelike complement) since the underlying QFSGS is translation covariant. Thus we continue to prove the statement for an arbitrary O of this type. We begin by noting that from our observation above, the π0,ι (Φ(h,j) )Ω0,ι , j = 1, . . . , d, span a d-dimensional subspace of H0,ι . Now let W ⊃ O be a wedge region containing the origin in its spacelike boundary. Then let W be the wedge which is the causal complement of W , and let Wh be a copy of W shifted by some suitable spacelike vector into the interior of W such that Wh lies in the causal complement of O + supp h. By a standard Reeh-Schlieder argument F0,ι (Wh )Ω0,ι is dense in H0,ι and hence, choosing some > 0 arbitrarily, there will be some ˆ ⊂ W and F (1) , . . . , F (d) ∈ F(O) ˆ such that double cone O h | π0,ι (F (j) )∗ Ω0,ι , π0,ι (Φ(h,k) )Ω0,ι − δjk | = |ω0,ι (F (j) Φ(h,k) ) − δjk | < .
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Now let (λκ )κ∈K be a subnet of the positive reals, converging to 0, with ω0,ι = (j) (h,k) limκ ω λκ on F. Since F λκ Φλκ converges weakly to a multiple of 1 owing to O0 F(O) = C1 (see [30]), we obtain ω0,ι (F (j) Φλκ ) = lim ω(F λκ Φλκ ) = lim ω (F λκ Φλκ ) (h,k)
(j)
(h,k)
(j)
κ
(h,k)
κ
for each locally normal state ω on F, and this implies | lim ω (F λκ Φλκ ) − δjk | < (j)
(h,k)
κ
whenever ω is locally normal. On the other hand, since ||(αλx (U ) − U )Ω|| → 0 as λ → 0 uniformly for x ranging over compact sets, it follows that ω(U ∗ F λκ Φλκ U ) (j) = dn x h(x)ω(αλκ x (U ∗ )F λκ αλκ x (ψk (λκ ))U ) + o(λκ ) = dn x h(x)ω(U ∗ α−λκ x (F (j) )λκ ψk (λκ )α−λκ x (U )) + o(λκ ) , (j)
(h,k)
where o(λ) tends to 0 for λ → 0, and we have used invariance of the vacuum state ω under the action of the translations αx . We have also inserted the definition of the Φ(h,k) , so that the scaled multiplets ψk (λ) appear here. Next we write ψj (λ = 1) = ψj , and we notice that ψj (λ) = Tλ ψj where Tλ = d ∗ j=1 ψj (λ)ψj is contained in A(O), and thus commutes with U ∈ A(O) ∩ F(O). We note also that for every B ∈ F we have, ω(U ∗ α−λκ x (F λκ )B) = ω(α−λκ x (F λκ )U ∗ B) (j)
(j)
for λκ ≤ 1 and x ∈ supp h since then α−λκ x (F λκ ) ∈ F(W ) and U ∗ ∈ A(O) ∩ F(O) ⊂ F(W ). Hence we get for λκ ≤ 1, dn x h(x)ω(U ∗ α−λκ x (F (j) )λκ ψk (λκ )α−λκ x (U )) (j)
=
=
d (j) ψi ψi∗ )U ∗ ψk α−λκ x (U )) dn x h(x)ω(α−λκ x (F λκ )Tλκ (
d
i=1
dn x h(x)ω(α−λκ x (F λκ )Tλκ ψi α−λκ x (ψi∗ U ∗ ψk U )) + p(λκ ) (j)
i=1
=
d
ω(F λκ Φλκ ψi∗ U ∗ ψk U ) + p(λκ ) (j)
(h,i)
i=1
with some function p(λ) tending to 0 as λ → 0, where we used that lim ||(ψi∗ U ∗ ψk α−λx (U ) − α−λx (ψi∗ U ∗ ψk U ))Ω|| = 0
λ→0
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uniformly for x ranging over compact sets. Also we used the translational invariance of ω again. Summing up these findings we have for λκ ≤ 1, ω(U ∗ F λκ Φλκ U ) = (j)
(h,k)
d
ω(F λκ Φλκ ψi∗ U ∗ ψk U ) + o(λκ ) + p(λκ ) . (j)
(h,i)
i=1
Making now use of the fact that for all normal states ω it holds that lim |ω (F λκ Φλκ ) − δji | < , (j)
(h,i)
κ
the previous equation yields, upon taking the limit over κ, |ω(ψj∗ U ∗ ψk U ) − δjk | < (d + 1) . Here > 0 was arbitrary, and hence we conclude that ω(ψj∗ U ∗ ψk U ) = δjk holds for all unitary U ∈ A(O) ∩ F(O) and the special multiplet ψj = ψj (λ = 1). However, given any other multiplet ψ˜j in F(O) for the charge [ρ], there is the unitary T = dj=1 ψ˜j ψj∗ in A(O) so that ψ˜j = T ψj , and thus we obtain, for each unitary U ∈ A(O) ∩ F(O), ω(ψ˜j∗ U ∗ ψ˜k U ) = ω(ψj∗ T ∗ U ∗ T ψk U ) = ω(ψj∗ U ∗ ψk U ) = δjk
since U and T commute.
Now we make use of the following result which has been proved in [30] (using also [15]): If, for some O ∈ K, there holds (6.1) for all charge multiplets ψ˜j (of all superselection sectors) contained in F(O) and for all unitaries U contained in A(O) ∩ F(O), then A(O) ∩ F(O) = F(O) ∩ F(O) . If moreover the local field algebras of the underlying QFSGS are factors, i.e., if F(O) ∩ F(O) = C1 ,
O ∈ K,
(6.2)
then equivalence of local and global intertwiners ensues: Given [ρ] and [ρ ] in Sectcov fin it holds that I(ρ, ρ )O = I(ρ, ρ ) for ρ, ρ localized in O .
(6.3)
(Cf. Sec. 4 for the definition of I(ρ, ρ )O and I(ρ, ρ ).) Corollary 6.2 Suppose that all local field algebras of the underlying QFSGS are factors, i.e., that (6.2) holds for all O ∈ K. Moreover, suppose that for each charge F [ρ] ∈ Sectcov fin there is some scaling limit state ω0,ι ∈ SL (ω) (which may depend on [ρ]) such that [ρ] is preserved in that scaling limit state. Then in the underlying QFSGS there holds the equivalence of local and global intertwiners (6.3).
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The factorial property of the local field algebras has been checked in free field models. Assuming that this is a general feature of quantum field theories, the assertion of the Corollary shows that part of the charge superselection structure is determined entirely locally if all charges are preserved in suitable scaling limit states; in other words, if the charges are, in this (somewhat generalized) sense, ultraviolet stable. For further discussion as to how much else of the superselection structure may be determined locally, we refer to [30].
7 Scaling algebras for quantum field systems localized in spacelike cones Up to this point, we have considered quantum fields localizable in arbitrary bounded open regions, corresponding by the Doplicher-Roberts reconstruction theorem to superselection charges of the DHR-type. There are more general types of superselection charges whose localization properties with respect to the vacuum representation of the observables are weaker. Before we enter into a discussion of this fact, let us first introduce the relevant terminology. As before, we identify n-dimensional Minkowski spacetime with Rn . Following [16], the timelike hyperbolic submanifold D = {s ∈ Rn : ηµν sµ sν = −1} will be taken to represent all points at spacelike infinity of n-dimensional Minkowski spacetime, since each s ∈ D represents a spacelike direction of unit Minkowskian length. Let a pair of points s+ , s− in D be given, where s+ ∈ (s− + V+ ), then we call D(s+ , s− ) = (s+ + V− ) ∩ (s− + V+ ) ∩ D a double cone at spacelike infinity with future direction s+ and past direction s− . Then a spacelike cone is a set of the form C = a + {λD(s+ , s− ) : λ > 0} where a is any element of Rn and D(s+ , s− ) is any double cone at spacelike infinity. A spacelike cone is thus a conic set extending to spacelike infinity having its apex at a; it can be viewed as the set of points lying in a certain opening angle around a spacelike direction. With this definition, each spacelike cone is causally complete, i.e., taking its double causal complement reproduces each spacelike cone. For further properties of spacelike cones, we refer the reader to the discussion in the appendix of [16]. We will denote by S the set of all spacelike cones. A profound analysis by Buchholz and Fredenhagen [10] has shown that in a theory with no massless excitations a general superselection charge is generically localized in spacelike cones. Namely, they have proven that for any massive single particle representation π of Avac (i.e., π is a translation covariant representation having no translation invariant vector and the single particle states are separated from the continuum by a gap in the spectrum of the corresponding translations ˜vac (i.e., a representation Uπ ), there exists an irreducible vacuum representation π translation covariant representation with a translation invariant vector) such that, ˜vac Avac (C ), for any spacelike cone C, π Avac (C ) is unitarily equivalent to π ∗ Avac (C ) being the C -algebra generated by all Avac (O) with O ⊂ C .
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Then, even in absence of massless particles, as in massive non-abelian gauge theories, DHR localization could be too strong a requirement. That such sectors really should arise in this kind of theories is suggested by the fact that spacelike cones can be thought as idealizations of flux tubes joining pairs of infinitely separated opposite gauge charges. While in the case of strictly localizable field operators it was meaningful to define the scaling algebra of a QFTGA and to study the corresponding scaling limit theories even for the case that the QFTGA has not all the features of a QFSGS, the situation is somewhat different for the case of field operators which are only localizable in spacelike cones. To illustrate this, consider the case that one is given a collection of von Neumann algebras {F(C)}C∈S indexed by spacelike cones (to be viewed as localization regions). Then one may fix, say, some spacelike cone C having its apex at the origin, and consider uniformly norm-bounded functions F : R+ → F(C). In order to take up the ideas that led to the definition of the scaling algebra for strictly localizable fields, however, one would now have to further restrict these functions in a manner expressing that F λ becomes localized near the origin as λ tends to 0. But since λC = C for all λ > 0, it is obvious that imposing F λ ∈ F(λC) leads to no restriction in localization at all, and hence the localization constraint has to be implemented by making use of additional structure. And this can be achieved if it is assumed that the collection of von Neumann algebras {F(C)}C∈S belongs to a QFSGS corresponding to BF-type superselection charges, where one can exploit the strict localization properties of the quantum system of observables. Therefore, let us now sketch, following [16], the Doplicher-Roberts reconstruction theorem for the case of charges of BF-type, which is very much in parallel to the discussion of the DHR case in Sec. 4. The starting point is again an observable quantum system (Avac , Uvac (Rn ), Ωvac ) fulfilling the properties listed at the beginning of Sec. 4. Then one may, as in [10], consider the set PBF of representations which, upon restriction to the spacelike complements of arbitrary spacelike cones, are unitarily equivalent to some fixed vacuum representation, which may be assumed to be the identical one. That means, in view of the above discussion, π is in PBF iff π Avac (C ) is unitarily equivalent to the identical representation of Avac (C ) on B(Hvac ). Then we restrict to the subset PBF cov of such representations which are translation-covariant, where the definition of covariance is exactly as in Sec. 4 (cf. eqn. (4.1)). It can again be shown that the set PBF cov can be described BF,cov by the set ∆t of covariant, localized, transportable morphisms of Avac , now taking values in B(Hvac ), which are defined similarly as in Sec. 4 with the difference that these morphisms are no longer localized in double cones but in spacelike cones. The concepts of translation-covariant superselection sectors and of (global) intertwiners carry over literally from the situation of Sec. 4. This is likewise the case for the intertwiner product, the notion of statistics and of conjugate charge, keeping in mind that all morphisms are now localized in spacelike cones and that all properties referring to localization and spacelike separation must take this into
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account. This understood, one is led to defining the set of transportable, irreducible, covariant morphisms ∆BF,cov of BF-type that have finite statistics, and fin representing their corresponding unitary equivalence classes collected in SectBF,cov fin the superselection charges of BF-type of the theory. Definition 7.1 One says that a collection of objects (F, U (Rn ), U (G), Ω, k) is a QFSGS associated with (Avac , Uvac (Rn ), Ωvac ) and ∆BF,cov if the following holds: fin (∗)
F denotes a family of von Neumann algebras (called field algebras) {F(C)}C∈S on a separable Hilbert space indexed by spacelike cones. The family satisfies the analogues of conditions (QFTGA.1-5) when changing double cone localization regions of the field algebras to spacelike cones, and observing the following alterations: Since the set S is not directed with respect to inclusion, there is no counterpart to the quasilocal algebra F, and the assumption of cyclicity has to be altered to demanding that the space generated by F(C)Ω, where C ranges over all of S, is dense in H.
(∗∗)
The analogue of (QFSGS.2) holds when replacing double cone localization regions by spacelike cones, more precisely, there is a C ∗ -algebraic monomorphism π : Avac → B(H) so that A(O) := π(Avac (O)) ⊂ F(C) holds for all double cones O and all spacelike cones C ⊃ O. Moreover, π contains the vacuum representation of Avac on Hvac as a sub-representation, and A(O) = F(C)G . A(C) := O⊂C
Furthermore, if one denotes, for D(s+ , s− ) any double cone at infinity, the C ∗ -algebra generated by all F(a + R+ D(s+ , s− )), a ∈ Rn , by F[D(s+ , s− )], then there holds π(Avac ) ∩ F[D(s+ , s− )] = C1 . ∗ (∗∗ ) The analogue of (QFSGS.3) holds, with the difference that the multiplets ψ1 , . . . , ψd are now elements of the F(C), and correspondingly one has to change ρO to ρC , a representer of [ρ] localized in C, in (4.3).
(∗∗ ∗∗ ) The analogue of (QFSGS.4) holds upon replacing double cones O by spacelike cones C. As Doplicher and Roberts [16] have shown, there is for each observable quantum system (Avac , Uvac (Rn ), Ωvac ) together with ∆BF,cov an associated QFSGS. fin We will now assume that we are given such a QFSGS corresponding to BF-type superselection charges, and construct a scaling algebra for it. In order to do so, we have to introduce some notation. First of all, we recall that αa stands for the adjoint action of U (a), and βg stands for the adjoint action
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of U (g). Then, let us denote by Bβ (R+ , B(H)) the C ∗ -algebra of all bounded functions F : R+ → B(H) with C ∗ -norm given by || F || := sup || F(λ) || λ>0
and with pointwise defined algebraic operations, where the functions are also assumed to be continuous with respect to the standard lift of the gauge group action, meaning that || β g (F) − F || = sup || βg (F(λ)) − F(λ) || → 0
for g → 1G .
λ>0
Then we define Cα (R+ , B(H)) as the C ∗ -subalgebra of Bβ (R+ , B(H)) whose elements fulfill || αa (F) − F || = sup || αλa (F(λ)) − F(λ) || → 0
for a → 0 .
λ>0
With this notation, we are ready to define scaling algebras for field operators localizable in spacelike cones. Definition 7.2 Let O be a double cone and C ⊃ O a spacelike cone. We define the scaling algebra A(O) as in Def. 2.2, i.e., consisting of all A ∈ Cα (R+ , B(H)) where Aλ ∈ A(λO), λ > 0. Then we denote by A(O)1 the subset of all elements of A(O) whose norm is bounded by unity, and we define: (I) F(C, O) is the C ∗ -subalgebra of all F in Bβ (R+ , B(H)) having the properties ∼
F(λ) ∈ F(λC)
and
lim sup (sup{ [F(λ), Aλ ] : A ∈ A(O )1 }) = 0 , λ→0
(7.1) where [A, B] = AB − BA denotes the commutator. (II) F(C, O) := F(C, O) ∩ Cα (R+ , B(H)). ∼
Some remarks about this definition are in order. The second condition in 7.1 expresses the fact that the field operators F(λ) that we are considering are asymptotically localized, as λ → 0, in the double cone λO, in the sense that their effect on measurements performed in the spacelike complement of this bounded region vanishes in the limit. Through this requirement, then, we implement in this more general case the basic idea of the scaling algebra approach. As a physical motivation for such a condition, we may note that a behavior of this kind is expected to show up at least in nonabelian asymptotically free gauge theories: as remarked by Buchholz and Fredenhagen, the spacelike cone in which BF charges are localized has to be thought of as a fattened version of a gauge flux string between two opposite charges, one of which has been shifted at spacelike infinity, and it is then natural to expect that such string should become weaker and weaker at
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small scales if the theory is asymptotically free, leaving in the limit a compactly localizable charge. We define the auxiliary C∗ -algebra F× as the C∗ -subalgebra of Cα (R+ , B(H)) generated by all the algebras F(C, O), and we note that for this system of algebras we have the obvious covariance properties αa (F(C, O)) = F(C + a, O + a),
β g (F(C, O)) = F(C, O),
for the actions αRn and β G of the translations and gauge group defined above, so that these restrict to automorphic actions on F× , denoted by the same symbols. For each normal state ω on B(H) we define the family of states (ωλ )λ>0 on F× in analogy to the case of localized fields, ωλ (F ) := ω (F λ ),
F ∈ F× ,
×
and SLF (ω ) = {ω0,ι : ι ∈ I} will be the set of weak* limit points of the net (ω λ )λ>0 (this is non-void, as in the localizable case), and will be called the set of scaling limit states of ω .
Lemma 7.3 Assume that the net of observable algebras in the vacuum representation satisfies the following condition: for each double cone O containing the origin, there holds Avac (O ∩ O0 ), (7.2) Avac (O) = O0 0 ×
where O0 runs through all double cones containing the origin. Then SLF (ω ) is independent of the normal state ω . Remark. The above condition (7.2) is suggested by the fact that O0 0 Avac (O0 ) = C1, by Haag duality and by the time-slice axiom. Its validity can also be proven in free field models. Proof of Lemma 7.3. As the union of all C∗ -algebras F(C, O) is norm dense in F× , it is sufficient to show that, for any two normal states ω 1 , ω 2 on B(H), and for any choice of C, O with O ⊂ C and any F ∈ F(C, O), there holds lim ω 1λ (F ) − ω2λ (F ) = 0.
λ→0
To this end, we adapt Roberts’ argument [30] and assume that this is not true. Then we can find a subnet (F λν )ν of (F λ )λ>0 weakly convergent to some F0 ∈ B(H) and such that |ω 1 (F0 ) − ω 2 (F0 )| = lim|ω1λν (F ) − ω 2λν (F )| > 0. ν
Now we intend to show that F0 is a multiple of 1 which leads to a contradiction, and hence shows validity of the statement of the lemma. We first observe that, if
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D is the double cone at spacelike infinity defined by the spacelike cone C, then from F λν ∈ F(λν C) ⊂ F[D] for each ν, F0 ∈ F[D] follows. Let then O1 be a double cone containing the origin, take A ∈ A(O1 ) such that A ≤ 1 and x → αx (A) is norm continuous, and define for each µ > 0 an element A(µ) ∈ A( µ1 O1 ) by (µ) Aλ
:=
A 0
if λ = µ, if λ = µ.
Then since A(µ) ∈ A(O )1 for µ sufficiently small, from the asymptotic localizability of F ∈ F(C, O) it follows that (λ)
lim sup [F λ , A] = lim sup [F λ , Aλ ] ≤ lim λ→0
sup
λ→0 A∈A(O )1
λ→0
[F λ , Aλ ] = 0,
which implies [F0 , A] = 0, so that, as the multiples of the A’s satisfying the stated requirements form a weakly dense set in each algebra A(O0 ) := A(O0 ) with O0 ⊃ O 1 , we get F0 ∈ A(O0 ) . O0 0
But from the assumption (7.2), together with local normality of the representation π, we have π(Avac ) ⊆ O0 0 A(O0 ), so that, by (∗∗) of Definition 7.1, F0 ∈ π(Avac ) ∩ F[D] = C1. In view of the above lemma, from now on we will only consider the scal× ing limit states SLF (ω), with ω := Ω, (·)Ω the underlying vacuum state. For × × × ω0,ι ∈ SLF (ω), let (π0,ι , H0,ι , Ω0,ι ) be the associated GNS representation and let × × n U0,ι (a), a ∈ R , U0,ι (g), g ∈ G, be respectively the translations group and gauge group representations, obtained as in the case of localized fields (Part 2 of Proposition 2.4), thanks to αRn - and β G -invariance of ω0,ι . We define then for each double cone O the von Neumann algebra × × F0,ι (O) := π0,ι (F(C, O)) , (7.3) C⊃O
× (O)Ω0,ι , a net of von and correspondingly a cyclic Hilbert space H0,ι := O F0,ι × Neumann algebras over it given by F0,ι (O) := F0,ι (O) H0,ι , a translation group × representation U0,ι (a) := U0,ι (a) H0,ι , a ∈ Rn , and a gauge group representation × • • • U0,ι (g ) := U0,ι (g) H0,ι , g ∈ G•0,ι , where, in analogy to the localizable case, G•0,ι := G/N0,ι with N0,ι the closed normal subgroup of G of the elements g ∈ G × such that U0,ι (g) H0,ι = 1H0,ι , and g ∈ G → g • ∈ G•0,ι the quotient map, so that • U0,ι is a faithful unitary representation of G•0,ι on H0,ι . We also denote by π0,ι the × subrepresentation of π0,ι determined by H0,ι .
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• Proposition 7.4 The quintuple (F0,ι , U0,ι (Rn ), U0,ι (G•0,ι ), Ω0,ι , k0,ι ) defined above is a normal, covariant quantum field theory with gauge group action, which will be called a scaling limit QFTGA, corresponding to ω0,ι , of the QFSGS determined by (Avac , Uvac (Rn ), Ωvac ) and ∆BF,cov . fin
Proof. The proof is completely analogous to the one of Proposition 2.4, so we do not repeat it here. The only thing that deserves a comment is the normality of commutation relations within F0,ι . We first note that they hold for the system of algebras F(C, O), in the sense that, by defining F ± := 12 (F ± β k (F )), relations analogous to (2.1) are satisfied for F i ∈ F(Ci , Oi ), with spacelike separated Ci , × i = 1, 2. Clearly this also carries over to the system of algebras π0,ι (F(C, O)) ,
× (Oi ) with spacelike with respect to the grading defined by βk . If then f i ∈ F0,ι × separated Oi , i = 1, 2, we can assume that f i ∈ π0,ι (F (Ci , Oi )) with Ci ⊃ Oi , i = 1, 2, spacelike separated to each other, so that normal commutation relations × , and hence for F0,ι . also hold for the net F0,ι (0,ι)
The classification of the underlying theory in terms of the resulting structure of the scaling limit theories given in Section 2 can be clearly applied also here, the isomorphism notion being again the one of Definition 2.5. We can also show a result analogous to Proposition 2.6, stating that if all the scaling limit QFTGAs are isomorphic, then they are dilation covariant. Since the formulation of this result and its proof are straightforward, we omit them. It is also straightforward to show that if the underlying QFSGS is also Lorentz covariant, where Lorentz covariance is defined as in (QFTGA.6), understanding that spacelike cones substitute double cones,8 then the same is true for each scaling limit QFTGA, provided that in this case one considers the scaling algebras F(C, O) obtained by redefining the C∗ -algebra Cα (R+ , B(H)) appearing in Definition 7.2 as the C∗ -subalgebra of Bβ (R+ , B(H)) whose elements F fulfill αs (F) − F = sup αsλ (F(λ)) − F(λ) → 0 for λ>0
s → 1P˜ ↑ , +
˜ ↑ is a generic element of the covering of the (proper orwhere s := (L, a) ∈ P + ˜↑ = L ˜ ↑ Rn , sλ := (L, λa), and αs := AdU (s), thochronous) Poincar´e group P + + U (L, a) := U (a)U˜(L) (slightly abusing notation). We will denote by U0,ι (s), ˜ ↑ , the corresponding unitary representation of the Poincar´e group on H0,ι , s∈P + with respect to which F0,ι is covariant. We also note, for future use, that U0,ι (s) = × × ˜ ↑ on H× defined U0,ι (s) H0,ι , where U0,ι (s) is the unitary representation of P + 0,ι × × × (s)π0,ι (F )Ω0,ι = π0,ι (αs (F ))Ω0,ι . by U0,ι We introduce here the standard notations × αs(0,ι)× := AdU0,ι (s),
αs(0,ι) := AdU0,ι (s),
˜↑ , s∈P +
8 This happens for instance if one considers the QFSGS determined through the DoplicherRoberts reconstruction theorem from the set of Poincar´e covariant BF sectors.
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˜ ↑ and a ∈ Rn with (L, 0), (1 ˜ ↑ , a) ∈ P ˜↑ and, for simplicity, we will identify L ∈ L + + L+ respectively. We close this section by putting on record a result generalizing Part 4 of Proposition 3.1. Proposition 7.5 Assume that the underlying QFSGS determined by BF sectors is Poincar´e covariant and satisfies the condition of geometric modular action (QFTGA.9), where in this case F(W ) := C⊂W F(C), and where equation (3.1) is substituted by JW F(C)JW = Ft (rW C),
C ∈ S.
(7.4)
Then each scaling limit QFTGA satisfies the condition (QFTGA.9) of geometric modular action. ˜ × (W ) be the C∗ -algebra generated by all π × (F(C, O)) with O ⊂ C Proof. Let F 0,ι 0,ι ˜ × (W ) := F ˜ × (W ) . In Appendix B we prove that the and C ⊂ W , and let F 0,ι 0,ι ˜ × (W ) and that ω0,ι is a scaling limit vacuum Ω0,ι is cyclic and separating for F 0,ι (0,ι)× × ∗ ˜ ), and then also (−2π)-KMS state for the C -dynamical system (F (W ), α 0,ι
˜ W (·) Λ
˜ × (W ), α(0,ι)× ). Then, arguing as in the proof of for the W -dynamical system (F 0,ι ˜ (·) Λ ∗
W
× Lemma 6.2 in [11] and denoting by (∆× 0,ι , J0,ι ) the modular objects determined by ˜ × (W ), Ω0,ι ), it is easy to verify the relations (F 0,ι × × × × U0,ι (L, a)J0,ι = U0,ι (Ad˜ rW L, rW a), J0,ι × it × ˜ (∆ ) = U (ΛW (2πt)),
0,ι × × × J0,ι π0,ι (F(C, O)) J0,ι × But since F0,ι (W ) :=
O⊂W
=
(7.5) (7.6)
0,ι × π0,ι (F(rW C, rW O)) .
(7.7)
× ˜ × (W ), and F0,ι (W ) = F0,ι (O) is contained in F 0,ι
× F0,ι (W ) H0,ι , it follows easily that ω0,ι is also (−2π)-KMS for (F0,ι (W ), αΛ˜ (·) ), W so that we also get ˜ (7.8) ∆it 0,ι = U0,ι (ΛW (2πt)). (0,ι)
This, together with (7.6) and standard arguments of Tomita-Takesaki theory, im(0,ι) plies that if f ∈ F0,ι (W ) is analytic for αΛ˜ (·) there holds W
˜ W (−iπ))f ∗ Ω0,ι = U × (Λ ˜ W (−iπ))f ∗ Ω0,ι = J × f Ω0,ι , J0,ι f Ω0,ι = U0,ι (Λ 0,ι 0,ι × so that one obtains J0,ι H0,ι = J0,ι (since the analytic elements for αΛ˜ (·) are W weakly dense in F0,ι (W )). The equations (7.5), (7.7) then imply geometric modular action for the scaling limit QFTGA. (0,ι)
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8 Preservance of BF-type charges in the scaling limit We will now generalize the notion of preservance of charges given in Section 5 so as to encompass the more general situation of BF-type charges considered in the previous section. In particular, in view of the physical picture of asymptotically free theories discussed after Definition 7.2, and of the ensuing construction of the scaling algebras and scaling limit, we will formulate a criterion implying that a given BF charge of the underlying theory gives rise to a localizable charge in the scaling limit. We consider then a Poincar´e covariant observable net (Avac , Uvac (P↑+ ), Ωvac ) ˜ ↑ ), U (G), Ω, k) deterand the corresponding Poincar´e covariant QFSGS (F, U (P + BF,cov of finite statistics, Poincar´e covariant BF sectors. In mined by the set ∆fin ˜ ↑ ), U (G), Ω, k) satisfies addition, we assume throughout this section that (F, U (P + the condition of geometric modular action (as formulated in Proposition 7.5). We note that also in this case, as for localizable charges, this can be deduced from geometric modular action of the observable net, supplemented by mild additional assumptions [19, 24]. The considerations about the possible phase space behaviors of localizable charges discussed in Section 5 as a motivation for the preservance criterion for DHR-charges clearly apply also to the present case, as we are considering asymptotically localized field operators, for which an asymptotic phase space notion can be recovered. By this, we mean that if ψ1 (λ), . . . , ψd (λ) ∈ F(λC) is some scaled multiplet associated to a fixed BF sector [ρ] of the underlying theory, and if this multiplet is asymptotically localized in some O ∈ K, then we can still think of the states ψj (λ)Ω as describing a charge [ρ] which is, for small λ, essentially localized in λO so that, by looking at the energy content of these states, we can define their phase-space occupation. Furthermore, the direction of the cone C in which this multiplet is localized is irrelevant, in the sense that if Cˆ ⊃ O is another spacelike ˆ still implementing cone, we can find another multiplet ψˆ1 (λ), . . . , ψˆd (λ) ∈ F(λC) the sector [ρ]. But from the picture of spacelike cones as strings which tend to vanish at small scales, we expect that, if also the multiplet ψˆj (λ) is asymptotically localized in O, then it should be possible to choose it in such a way that the charged states ψj (λ)∗ Ω, ψˆj (λ)∗ Ω should become close to each other as λ → 0. This motivates the following generalization to the present setting of the notion of asymptotic containment, for whose formulation we introduce the notation R1 R2 for two arbitrary spacetime regions R1 , R2 , to mean that there exists some neighborhood of the identity N ⊂ P↑+ such that N · R1 ⊆ R2 . Definition 8.1 Let O1 ∈ K, C1 ∈ S be such that O1 ⊂ C1 , and let F ∈ F(C1 , O1 ). F ∼ is said to be asymptotically contained in F0,ι (O) with O ⊃ O1 , if for each spacelike ˆ ∈ F(Cˆ1 , O1 ), with F ˆ = F for Cˆ1 = C1 , fulfilling cone Cˆ1 ⊃ O1 there exist some F ∼ the following properties:
ˆ κ ) − F(λκ ))Ω + (F(λ ˆ κ ) − F(λκ ))∗ Ω = 0; (A) limκ (F(λ
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(B) for any C ⊃ O such that Cˆ1 C, and for any > 0, there exist elements F , F ∈ F(C, O) (depending on C and ) so that
ˆ κ ) − F )∗ Ω < ; ˆ κ ) − F )Ω + (F(λ lim sup (F(λ λκ λκ κ
where {λκ }κ∈K ⊂ R+ is such that ω0,ι = limκ ω λκ . Lemma 8.2 Let F ∈ F(C1 , O1 ) and let O ⊃ O1 . The following statements are ∼ equivalent: (a) F is asymptotically contained in F0,ι (O); ˆ ∈ F(Cˆ1 , O1 ), with F ˆ = F for Cˆ1 = C1 , (b) for each Cˆ1 ⊃ O1 there exist some F ∼ fulfilling property (A) of Definition 8.1 and the following property (B ) for any C ⊃ O such that Cˆ1 C, and for any given > 0 and finitely many F (1) , . . . , F (N ) ∈ F× , there is F ∈ F(C, O) such that for j = 1, . . . , N ;
ˆ κ ) − F )F (j) Ω + (F(λ ˆ κ ) − F )∗ F (j) Ω < ; lim sup (F(λ λκ λ λκ λκ κ κ
ˆ ∈ F(Cˆ1 , O1 ), with F ˆ = F for Cˆ1 = C1 , (c) for each Cˆ1 ⊃ O1 there exist some F ∼ such that
ˆ κ ) − F(λκ ) )Ω + ( (α F)(λ ˆ κ ) − F(λκ ) )∗ Ω = 0 (8.1) lim ( (αhν F)(λ hν (κ,ν)
˜ ↑ , where the limit is taken with respect whenever {hν } is a δ-sequence on P + to the product partial ordering on K × N, and where ˜↑ ) ˆ ˆ ds h(s)αsλ (F(λ)) , λ > 0 , h ∈ L1 (P (αh F)(λ) := + ˜↑ P +
˜ ↑ ). (integral in the weak sense, using the standard invariant measure on P + Proof. (a) ⇒ (c). The proof proceeds analogously to the proof of the corresponding implication in Lemma 5.3 using the estimate (with the notation introduced there) ˆ κ )) Ω ˆ κ ) − F(λκ )) Ω ≤ 2 (F − F(λ ((αhν F)(λ λκ +
sup s∈supp hν
ˆ κ ) − F(λκ )) Ω. αs (F ) − F + (F(λ
(c) ⇒ (b). From the estimate ˆ κ ) − F(λκ )) Ω ≤ (F(λ
sup s∈supp hν
ˆ κ )] Ω [(U (sλκ ) − 1)F(λ
ˆ κ ) − F(λκ )) Ω, + ((αhν F)(λ
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ˆ follows at once, by first and from strong continuity of s → U (s) property (A) for F choosing κ and then ν, depending on κ, sufficiently large. Property (B’) is proven by the same argument as the one in the proof of the corresponding implication in × ˜× ˜ × t (W )Ω0,ι = V × F Lemma 5.3, using here the fact that F 0,ι 0,ι 0,ι (W )Ω0,ι , with V0,ι the × × natural twisting operator on H0,ι , is dense in H0,ι , Theorem B.1. (b) ⇒ (a). Obvious. ˆ Remark. The field net on double cones defined by F(O) := C⊃O F(C) is essentially the net of the QFSGS determined by the localizable sectors of Avac (see [16] ˆ and a scalfor precise statements), and we can associate to it scaling algebras F(O) ˆ ing limit net F0,ι (O) in the way discussed in previous sections. It is then clear that ˆ ˆ complying with F(O) ⊂ F(C, O) for each C ⊃ O and that a function F : R+ → F ˆ properties (i)–(iii) in definition 5.2 is an element of F(C1 , O1 ) for each Cˆ1 ⊃ O1 , so ˆ 0,ι (O), it is∼also asymptotically contained that if F is asymptotically contained in F ˆ = F for each Cˆ1 in Definition 8.1. Conversely, if in F0,ι (O) as it suffices to take F ˆ F with F(λ) ∈ F(λO1 ) is asymptotically contained in F0,ι (O), then thanks to the characterizations (c) in Lemmas 5.3 and 8.2, it is also asymptotically contained ˆ 0,ι (O). in F As in the case of localizable charges, a collection of multiplets {ψ1 (λ), . . . , ψd (λ)}λ>0 inducing a fixed BF sector [ρ] and with ψj (λ) ∈ F(λC) for some C ∈ S, will be called a scaled multiplet for [ρ]. ×
Definition 8.3 Let ω0,ι ∈ SLF (ω) be a scaling limit state of the underlying QFSGS, and let [ρ] ∈ SectBF,cov be a BF superselection sector. Then we say that fin the charge [ρ] is preserved in the scaling limit QFTGA of ω0,ι if, for each O1 ∈ K, C1 ∈ S with C1 ⊃ O1 , there is some scaled multiplet {ψ1 (λ), . . . , ψd (λ)}λ>0 for [ρ] such that all functions λ → ψj (λ), j = 1, . . . , d, are elements of F(C1 , O1 ) and are ∼ asymptotically contained in F0,ι (O) if O ⊃ O1 . Proposition 8.4 Suppose that the charge [ρ] is preserved in the scaling limit QFTGA of ω0,ι . Let {ψ1 (λ), . . . , ψd (λ)}λ>0 be a scaled multiplet for [ρ] such that ψj (·) ∈ F(C1 , O1 ) is asymptotically contained in F0,ι (O). Let, for j = 1, . . . , d and ∼ ˆ Cˆ1 ⊃ O1 , ψjC1 ∈ F(Cˆ1 , O1 ) be as in Lemma 8.2(c). Then the limit operators ∼
ˆ
ψ j = s- lim π0,ι (αhν ψjC1 ) ν→+∞
and
ˆ
ψ ∗j = s- lim π0,ι (αhν ψjC1 )∗ ν→+∞
(8.2)
exist for any δ-sequence {hν }, are independent of Cˆ1 and of the chosen δ-sequence, and are contained in F0,ι (O) whenever O ⊃ O1 . Furthermore, ψ 1 , . . . , ψ d forms a • (G•0,ι ) according to the irremultiplet transforming under the adjoint action of U0,ι ducible, unitary representation v[ρ] . More precisely, denoting by G g → g • ∈ G•0,ι the quotient map, there is a finite-dimensional, irreducible, unitary representation
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• • v[ρ] of G•0,ι so that v[ρ] (g • ) = v[ρ] (g) for all g ∈ G and
• • (g • )ψ j U0,ι (g • )∗ = U0,ι
d
• ψ i v[ρ]ij (g • ) ,
g • ∈ G•0,ι .
i=1
Proof. The proof is similar to the proof of Proposition 5.5, so we will only indicate the major differences. From the inequality ˆ
˜
× × (π0,ι (αhν ψjC1 ) − π0,ι (αh˜ ν˜ ψjC1 ))Ω0,ι
ˆ ˜ ≤ lim sup ((αhν ψjC1 )(λκ ) − ψj (λκ ))Ω + ((αh˜ ν˜ ψjC1 )(λκ ) − ψj (λκ ))Ω , κ
˜ ν˜ } and spacelike cones Cˆ1 , C˜1 ⊃ O1 , tovalid for arbitrary δ-sequences {hν }, {h ˆ × gether with Lemma 8.2(c), it follows that limν→+∞ π0,ι (αhν ψjC1 )Ω0,ι =: Φj exists ˆ and is independent of Cˆ1 and of the chosen δ-sequence. As αhν ψjC1 is uniformly ˜ × (W ) with W Cˆ1 , this implies bounded in ν, and since Ω0,ι is separating for F 0,ι
ˆ
× × that s- limν→+∞ π0,ι (αhν ψjC1 ) =: ψ × j exists and is defined by ψ j f Ω0,ι = f Φj ˜ × (W ) . Then, since for any two spacelike cones Cˆ1 , C˜1 we can find for f ∈ F 0,ι spacelike cones Cˆ2 , . . . , Cˆn with Cˆn = C˜1 , and wedges W1 , . . . , Wn−1 such that ˆ Cˆi ∪ Cˆi+1 Wi , i = 1, . . . , n − 1, we conclude that ψ × j is independent of C1 , as ˆ well as of {hν }. Thus, since for any spacelike cone C ⊃ O there is a C1 ⊃ O1 such × × that Cˆ1 C, we have ψ × j ∈ π0,ι (F(C, O)) , and then ψ j := ψ j H0,ι ∈ F0,ι (O). ˆ
The same argument shows that π0,ι (αhν ψjC1 )∗ converges strongly to ψ ∗j . The rest of the proof is essentially identical to the corresponding part of the proof of Prop. 5.5.
If, for F ∈ F(C1 , O1 ), we define F(λ) := JW V F(λ)V ∗ JW , where W ⊃ C1 is a ∼ wedge and JW is the associated modular conjugation, and recalling that we assume that the underlying QFSGS satisfies geometric modular action, it is easily checked that F ∈ F(rW C1 , rW O1 ) and that (αh F)(λ) = JW V (αh◦Ad˜rW F)(λ)JW V ∗ (where ∼ Ad˜ rW (L, a) := (Ad˜ rW L, rW a)), so that if F is asymptotically contained in F0,ι (O) then F is asymptotically contained in F0,ι (rW O), and it is then straightforward to verify that the following generalization of Theorem 5.8 holds. ×
Theorem 8.5 Let ω0,ι ∈ SLF (ω) be a scaling limit state. Then a BF charge [ρ] ∈ SectBF,cov is preserved in the scaling limit state ω0,ι if and only if also the conjugate fin charge [ρ] is preserved. We would then like to obtain a result corresponding to Proposition 5.6. However, at the present stage of our work, this can be achieved only at the price of some • additional assumptions on the net F0,ι , namely that A0,ι (O) = F0,ι (O)G0,ι . We will comment on this assumption below. Nevertheless, without making this assumption,
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we can at any rate show that the fields ψ j constructed above give rise, in a sense made precise in the following theorem, to positive energy representations of A0,ι . Theorem 8.6 Let ξ = [ρ] be a BF charge which is preserved in the scaling limit QFTGA of a given ω0,ι , and let ψ j ∈ F0,ι (O), j = 1, . . . , d, a multiplet for ξ arising as in Proposition 8.4. Then the state ωξ on A0,ι defined by ωξ (a) :=
d
Ω0,ι , ψ j aψ ∗j Ω0,ι ,
a ∈ A0,ι ,
(8.3)
j=1
obeys ωξ A0,ι (O ) = ω0,ι A0,ι (O ) and induces, via the GNS construction, a representation πξ of A0,ι which is locally normal and translation covariant. Proof. It is evident that ωξ A0,ι (O) is a normal state, so that πξ is a locally normal representation of A0,ι . According to a theorem of Borchers [5, Thm. II.6.6], in order to show that πξ is translation covariant, it is necessary and sufficient to show that the set of vector states of πξ is contained in the norm closure of the set A∗0,ι (V+ ) of functionals φ ∈ A∗0,ι with the following property. For each pair (0,ι)
a, b ∈ A0,ι , the function x ∈ R4 → φ(aαx (b)) is continuous and is the boundary value of a function W which is analytic in the forward tube T := R4 + iV+ and satisfies the bound |W (z)| ≤ abem|Imz| ,
z ∈ T,
for some constant m > 0 which may depend on φ but not on a, b; furthermore the same conditions must be satisfied by φ∗ (φ∗ (a) := φ(a∗ )). Now, the set of operators c ∈ A0,ι with compact support in momentum (0,ι) space, i.e., where there exists a compact ∆ ⊂ R4 such that αh (c) = 0 for h ⊂ R4 \ ∆, is strongly dense in A0,ι . To see this, each h ∈ L1 (R4 ) with supp ˆ (0,ι) take c = αf (c1 ) with c1 ∈ O π0,ι (A(O)) and compact supp fˆ. Then c has compact momentum space support. The set of L1 functions f with compact supp fˆ 1 (0,ι) is is strongly continuous on dense in L . Owing to the fact that the action of α π (A(O)), this implies that there exists a sequence of L1 -functions fn with O 0,ι (0,ι) ˆ compact supp fn so that αfn (c1 ) approaches c1 in norm. One concludes then by noting that O π0,ι (A(O)) is strongly dense in A0,ι . Then, using πξ (c)Ωξ − πξ (d)Ωξ = 2
d
(c − d)ψ ∗j Ω0,ι 2 ,
j=1
where Ωξ is a cyclic vector for πξ , it is sufficient to show that the functionals φc (a) := ωξ (c∗ ac), with c ∈ A0,ι having compact momentum space support, are contained in the norm closure of A∗0,ι (V+ ).
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To this end let c ∈ A0,ι have momentum support in a compact set ∆, and let ∆n be the closed double cone in momentum space with vertices 0 and (n, 0), n > 0. Then the functional φc,n ∈ A∗0,ι defined by φc,n (a) :=
d
Ω0,ι , ψ j E(∆n )c∗ acE(∆n )ψ ∗j Ω0,ι ,
j=1
where by E we denote the spectral measure associated to translations of F0,ι , is (0,ι) such that the function x → φc,n (aαx (b)) is continuous and has distributional Fourier transform with support in −(∆ + ∆n ) + V + . Then, if p = (−m, 0) is such that −(∆ + ∆n ) + V + ⊂ p + V + , one concludes by [5, Thm. II.1.7] that this function is a boundary value of a function W analytic in T and which satisfies, for suitable constants k, M , N > 0, the bound |W (z)| ≤ k(1 + |x|)N (1 + dist(y, ∂V+ )−1 )M em|y| ,
z = x + iy ∈ T.
But there also holds |W (x)| ≤ φc,n ab for x ∈ R4 , and this implies, by a standard analytic function argument,9 the desired estimate |W (z)| ≤ φc,n abem|y| , showing that φc,n ∈ A∗0,ι (V+ ). Then by the inequality |φc (a) − φc,n (a)| ≤
d
2c2 a[E(∆n ) − 1]ψ ∗j Ω0,ι ,
j=1
we get the statement.
vac vac We denote by π0,ι the vacuum representation of A0,ι , defined by π0,ι (a) := vac vac a H0,ι , where H0,ι = A0,ι Ω0,ι is the scaling limit vacuum Hilbert space. Thanks vac to the separating property of Ω0,ι for local algebras, π0,ι A0,ι (O) is an isomorphism of von Neumann algebras. vac is separable, then for Corollary 8.7 If the scaling limit vacuum Hilbert space H0,ι each x ∈ R4 vac πξ A0,ι (O + x) ∼ A0,ι (O + x), (8.4) = π0,ι
i.e., πξ has the DHR property for the class of all translates of the given double cone O. Proof. By the argument in the appendix of [14], the fact that ωξ A0,ι (O ) = ω0,ι A0,ι (O ), together with translation covariance of πξ , imply (8.4) if it is known that vac is separable, then each property B holds in the representation πξ . But if H0,ι vac local algebra π0,ι (A0,ι (O)) has a separable predual, and being ωξ locally normal, from [36, Corollary 3.2] it follows that the Hilbert space Hξ of πξ is separable, and then, by the already recalled argument of Roberts [32], property B holds in the representation πξ . 9 We
are indebted to J. Bros for helpful remarks on this point.
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vac We recall that separability of H0,ι follows from suitable nuclearity properties of the underlying observable net [8]. We now turn to discussing the conditions under which it is possible to generalize Proposition 5.6. At the technical level, the main obstruction is represented • by the fact that, in general, A0,ι (O) F0,ι (O)G0,ι , as it is easy to construct gauge invariant combinations of the αh ψj ’s which need not belong to some scaling algebra A(O) but are only localized in spacelike cones. However, thanks to the fact that these functions are asymptotically localizable in O, it may well happen that, at least in favorable cases, their scaling limits do belong to A0,ι (O). Another problem comes from the fact that we do not know if the scaling limit field algebra is irreducible, as the estimates in [3], when applied to cone-like localized fields, are not sufficient to conclude that the clustering rate in the vacuum state is uniform in λ.10 But in this case also it may be expected that the asymptotically localizable character of the scaling algebra functions under consideration helps in improving the situation. Restricting then ourselves to the consideration of cases in which these two properties are indeed present, we get a quite satisfactory picture of the scaling limit of morphisms. ×
•
Proposition 8.8 Let ω0,ι ∈ SLF (ω) and assume that A0,ι (O) = F0,ι (O)G0,ι and that F0,ι acts irreducibly on H0,ι . Moreover, let [ρ] ∈ SectBF,cov be a charge of fin the underlying QFSGS which is preserved in the scaling limit QFTGA of ω0,ι , let {ψ1 (λ), . . . , ψd (λ)}λ>0 be a scaled multiplet for [ρ] asymptotically contained in F0,ι (O) and let, with respect to this scaled multiplet, ψ 1 , . . . , ψ d be defined as in (8.2). If we define, for each A ∈ A, the family {ρ(A)(λ)}λ>0 as ρ(A)(λ) =
d
ψj (λ)Aλ ψj (λ)∗ ,
j=1
then there holds s- lim π0,ι (αhν ρ(A)) = ν→+∞
d
ψ j π0,ι (A)ψ ∗j ,
A ∈ A;
(8.5)
j=1
and ρ defined by ρ(a) =
d
ψ j aψ ∗j ,
a ∈ A0,ι ,
(8.6)
j=1
is a localized, transportable, irreducible endomorphism of A0,ι which is moreover covariant and has finite statistics. 10 On the contrary, as indicated by D. Buchholz, it can be shown that, if one defines the scaling algebra without requiring asymptotic localizability, then the scaling limit algebra has a nontrivial center.
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The proof of these statements is completely parallel to that of Proposition 5.6. As a final comment, we would like to remark that the condition A0,ι (O) = • F0,ι (O)G0,ι , introduced here as a technical assumption in order to get a well defined scaling limit of morphisms, may turn out to have a sensible physical interpreta• tion. By the above remarks, we see that F0,ι (O)G0,ι contains, apart from the scaling limit observables localized in O, the scaling limit of functions λ → Aλ ∈ A(λC), for every spacelike cone C ⊃ O, i.e., there are gauge invariant families of operators, with localization regions extending to spacelike infinity, which give rise to objects in the scaling limit which are charged with respect to the intrinsic gauge group of A0,ι , so that new charges appear at small scales. This situation, which must not be confused with confinement where the fields carrying the new charges cannot be approximated at all at finite scales, is instead reminiscent of the phenomenon of charge screening,11 much discussed in the physical literature (cf. for instance [33, 34] and references quoted). In this scenario, a charge which is described by an asymptotically free theory at small scales disappears at finite scales because, due to nonvanishing interactions, it is always accompanied by a cloud, extending to spacelike infinity, of charge-anticharge pairs, so that one can expect that the corresponding “charge carrying fields” are neutral and non-compactly localized at finite scales, and become instead charged and localized in the scal• ing limit. Then the condition A0,ι (O) = F0,ι (O)G0,ι could be interpreted as the requirement that in the theory under consideration, no charges are screened.
Concluding remarks A generalization of the scaling algebra framework to the situation where the operator algebras describing the underlying quantum field theory contain chargecarrying fields has been developed in this work, together with a proposal as to what it means that a charge present in the underlying theory is preserved in the scaling limit. A natural concept of confined charge arises as a charge in the scaling limit theory which is not obtainable as a charge of the underlying theory which is preserved in the scaling limit process [7, 9]. We have indicated two basic physical mechanisms for the disappearance of charges in the scaling limit. Moreover, we have seen that the preservance of all charges in the scaling limit leads to the equivalence of local and global intertwiners for the superselection sectors in the underlying theory. We hope that in the future it will be possible to illustrate the mechanisms for charge disappearance in the scaling limit by instructive examples, possibly in lower spacetime dimensions. This should also shed light on the very important issue if the lifted action of the gauge transformations shouldn’t be defined differently than in (2.3) in the case where the physical dimension of charge is related to the dimension of length. Furthermore, it would also appear desirable to develop, based on the method of scaling algebras for the observables of the underlying theory, an 11 This
connection was pointed out to us by Detlev Buchholz.
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abstract renormalization group analysis for superselection charges without using the Doplicher-Roberts reconstruction theorem. This would eventually make superselection charges with braid group statistics and with infinite statistics accessible to short-distance analysis.
Acknowledgments It is a pleasure to thank D. Buchholz for many helpful discussions and instructive comments. Similar thanks are also extended to S. Doplicher, who was PhD supervisor of one of us (G.M.), and to J.E. Roberts and K.-H. Rehren. R.V. gratefully acknowledges financial support by INDAM-GNAMPA. G.M. gratefully acknowledges financial support by DAAD and MPI-MIS.
A
An example of a preserved localizable charge
In this Appendix we shall show that the localizable charge described by the Majorana field in n = 1 + 3 spacetime dimensions with Z2 gauge group satisfies the preservance condition, Definition 5.4, in all scaling limit states. For the definition of the Majorana field, we will mainly follow [20], where also a discussion of the superselection structure is given. We begin with some notational conventions. Let 0 1 0 σj 0 j , γ = , j = 1, 2, 3, (A.1) γ = 1 0 −σj 0 be the Dirac matrices in Chiral representation, where σj are the Pauli matrices. A vector u ∈ C4 (also called a spinor) will be thought as a column matrix and correspondingly its adjoint u† will be a row matrix, so that the standard scalar product on C4 is given by (u, v) → u† v (rows by columns product of matrices). We adopt the notation v := vµ γ µ for any (covariant) vector v ∈ R4 . By Ω± m we 4 := {p ∈ R : p2 = shall indicate the upper and lower mass m > 0 hyperboloid, Ω± m 2 m , ±p0 > 0}. For a given mass m > 0, the Dirac operator is D := γ µ ∂µ +im, and, denoting as usual by D(R4 ; C4 ) the space of spinor valued, compactly supported smooth functions on Minkowski space, we endow the space H0,m := D(R4 ; C4 )/ Im D with the scalar product f, g m := d3 p g (±ωm (p), p), (A.2) fˆ(±ωm (p), p)† P± (p)ˆ R3
where
±
γ 0 ( p + m) P± (p) = , 2p0 p0 =±ωm (p)
ωm (p) =
|p|2 + m2 ,
(A.3)
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and where we made no notational distinction between elements in H0,m and their representatives in D(R4 ; C4 ). Let Hm be the completion of H0,m in this scalar product. The action of the universal covering of the Poincar´e group on Hm is defined, ˜ ↑ , by for (A, a) ∈ P +
A 0 −1 S(A) := u(A, a)f (x) := S(A)f Λ(A) (x − a) , , (A.4) 0 (A† )−1 A ∈ SL(2, C) → Λ(A) ∈ L↑+ being the covering homomorphism. Let C be the antilinear operator on C4 defined by Cu := iγ 2 u, where the bar denotes complex conjugation, which satisfies C 2 = 1, C † = C and Cγ µ C = −γ µ , and define then an antilinear involution Γ on H0,m by (Γf )(x) := Cf (x), which is antiunitary, Γf, Γg m = g, f m (so that it extends to Hm ) and commutes with the action of the Poincar´e group. Let B(Hm ) be the self-dual CAR algebra over Hm [1], generated as a C∗ algebra by elements B(f ), f ∈ Hm , such that f → B(f ) is antilinear, and {B(f ), B(g)} = g, Γf m 1,
B(f )∗ = B(Γf ).
(A.5)
˜ ↑ on Hm induces an automorphic action By CAR unicity, the representation u of P + ↑ ˜ on B(Hm ), defined by α of P + α(A,a) (B(f )) := B(u(A, a)f ),
˜ ↑ , f ∈ Hm , (A, a) ∈ P +
and, by the fact that B(f ) ≤ 2 f m and strong continuity of u, it follows that this action is strongly continuous, i.e., (A, a) → α(A,a) (B) is norm continuous for each B ∈ B(Hm ). We consider on B(Hm ) the quasifree state ω defined, according to [2], by the 2-point function
(A.6) ω B(f )B(g) := Γf, P+ g m , where P+ is the projection on the positive energy states in Hm , defined by ˆ P + f (p0 , p) = P+ (p)f (p0 , p). ˜ ↑ leaves ω invariant, so that if we consider the GNS repThe action α of P + resentation (π, H, Ω) induced by ω, we get on H a unitary strongly continuous ˜ ↑ leaving Ω invariant and such that (π, U ) is a covariant representation U of P + representation of (B(Hm ), α). Definition A.1 The free Majorana field of mass m > 0 is the operator
valued distribution f ∈ D(R4 ; C4 ) → ψ(f ) ∈ B(H) given by ψ(f ) := π B(f ) , f ∈ D(R4 ; C4 ), where on the right-hand side f is identified with its image in Hm . It is straightforward to verify that ψ is covariant with respect to U , U (A, a)ψ(f )U (A, a)∗ = ψ(u(A, a)f ),
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that the translations a → U (1, a) satisfy the spectrum condition with Ω as the unique (up to a phase) translation invariant unit vector in H, and that ψ(f ), ψ(g) anticommute for spacelike separated suppf , suppg. We now turn to the consideration of the net of local von Neumann algebras associated to the free Majorana field, and defined by F(O) := {ψ(f ) : suppf ⊆ O} ,
(A.7)
for O ⊂ R4 open and bounded. On this net the group Z2 acts by an automorphism βk induced by the automorphism of B(Hm ) defined by B(f ) → −B(f ), which leaves the vacuum state ω invariant, and is therefore implemented by a unitary operator U (k) on H such that U (k)2 = U (k 2 ) = 1, so that it induces a direct sum decomposition H = H+ ⊕ H− according to its eigenspaces, i.e., U (k) H± = ±1H± , which is Poincar´e invariant. Define then Uvac (A, a) := U (A, a) H+ , and the net of observable von Neumann algebras associated to the free Majorana field as Avac (O) := F(O)Z2 H+ . (A.8) That in this way we get an example satisfying the assumptions made in Sections 2 and 4 is the content of the following proposition, the proof of which, being standard, is omitted. Proposition A.2 With the above notations, and with Hvac := H+ , let π be the representation of the quasi-local algebra Avac defined by π(A Hvac ) := A. Then ˜ ↑ ), U (Z2 ), (A, Uvac (P↑+ ), Ω) is a Poincar´e covariant observable net, and (F, U (P + Ω, k) is a QFSGS on it. In the next proposition, the very simple superselection structure of Avac described by the field net F is analyzed, cf. [20]. Proposition A.3 The representation π− of Avac given by π− := π(·) H− satisfies the DHR criterion, is covariant, irreducible and with finite statistics, and any irreducible representation of Avac appearing in H is equivalent either to ι, the 4 identity representation, √ or to π− . Moreover, if for f ∈ D(O, C ) with suppf ⊆ O, Γf = f and f m = 2, ρf is the automorphism of Avac induced by the unitary ∼ operator ψ(f ) ∈ F(O), then ρf ∈ ∆cov fin (O) and ρf = π− . Proof. The irreducibility of π− follows from the arguments in [13], taking into account that H− is the subspace associated to the irreducible representation k → −1 of Z2 in the factorial decomposition of U . This also implies that any other irreducible representation of Avac in H is equivalent to ι or π− . To show that π− 4 satisfies the DHR √ criterion, fix a double cone O and an f ∈ D(O; C ) with Γf = f and f m = 2. Then ψ(f ) is unitary by the CARs, and ψ(f )H± = H∓ . Let then Vf := ψ(f ) H− , and g, h ∈ D(O , C4 ). We have Vf π− ψ(g)ψ(h) = ψ(g)ψ(h)Vf , so that Vf intertwines between π− Avac (O ) and ι Avac (O ). Covariance of π− follows by U− := U (·) H− . Finally if πρf (A) = ψ(f )π(A)ψ(f )∗ then ρf is
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localized in O and has finite statistics, and Vf intertwines between π− and ρf , and if suppf1 ⊆ O1 , V := ψ(f )ψ(f1 )∗ H+ intertwines between ρf1 and ρf , which is therefore transportable. Finally we come to the proof of the fact that the charge ξ := [π− ] in the above proposition is preserved in any scaling limit theory, in the sense of Definition 5.4. To this end it is sufficient to find, for every double cone O1 , a family (fλ )λ∈(0,1] of √ functions such that suppfλ ⊆ λO1 , fλ m = 2, and such that condition (5.1) is satisfied for ψ(fλ ). Proposition A.4 For every double cone O1 , there exists f ∈ D(O1 ; C4 ) such that, if fλ ∈ D(λO1 ; C4 ) is defined by 3
fλ (x) := λ 2 −4 f (λ−1 x),
λ ∈ (0, 1],
√ then fλ m = 2 and Γfλ = fλ for λ ∈ (0, 1], and λ → ψ(λ) := ψ(fλ ) is asymptotically contained in F0,ι (O) for each O ⊃ O 1 and each scaling limit state ω0,ι . In the course of the proof of this proposition, we will need the following simple result concerning the action of the Lorentz group on Minkowski space. Lemma A.5 Fix a mass m > 0. For any sufficiently large R > 0, there exists a neighborhood of the identity N in L↑+ such that, for any p ∈ V + with 0 ≤ p2 ≤ m2 √ and |p| > R, and for any Λ ∈ N, it holds, for p := Λp, |p | > |p| / 2. Proof. To simplify the notation, we will write Λ · p for the spatial part, in a given Lorentz frame, of the 4-vector Λp, Λ ∈ L↑+ , p ∈ R4 . Let Λ1 (s), s ∈ R, denote the 1-parameter group of boosts in the p1 direction. 2 2 2 If p ∈ R4 is such that |p1 | ≤ |p2 | + |p3 | , since Λ1 (s) leaves the components 2 2 2 2 p2 , p3 unaffected, we have, for any s ∈ R, |Λ1 (s) · p| ≥ |p2 | + |p3 | ≥ |p| /2. √ 2 2 2 Assume now that |p1 | > |p2 | + |p3 | . This implies |p1 | ≥ |p| / 2 and, since for any sufficiently large R > 0, inf 2
0≤p ≤m |p|>R
2
|p| |p| ≥ inf > 0, |p0 | |p|>R 2 |p| + m2
we can √ find a δ > 0 such that, if |s| < δ, |(Λ1 (s)p)1 | = |sinh s p0 + cosh s p12| ≥ |p1 | / 2 for any p ∈ V + with 0 ≤ p2 ≤ m2 and |p| > R, so that |Λ1 (s) · p| ≥ 2 p21 /2 + p22 + p23 ≥ |p| /2. Then, if we identify in the canonical way SO(3) with a subgroup of L↑+ , we conclude with N := {R1 Λ1 (s)R2 : |s| < δ, R1 , R2 ∈ SO(3)}. Proof of Proposition A.4. In order to shorten formulae, we will use the notation pλ,± := (±ωλm (p), p), as well as the notation Λ · p introduced in the proof of the
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above lemma. Also, |·| will denote the norm of a vector both in R3 and in C4 . A calculation shows d3 p 2 2 γ 0 ( pλ,± + λm)fˆ(pλ,± )2 , fλ m = f λm = 2 R3 4ωλm (p) ±
√ and then, in order to show that there is an f ∈ D(O, C4 ) such that fλ m = 2 and Γfλ = fλ for each λ ∈ (0, 1], it is sufficient to exhibit an f ∈ D(O, C4 ) such that Γf = f , and for which ( p + µ)fˆ(p) is not identically zero on each hyperboloid − Ωµ := Ω+ µ ∪ Ωµ , µ > 0. A direct check shows that these conditions are met by
t f (x) := g(x)(1 + iγ 2 ) 1 0 0 0 where g ∈ D(O; R). We now show that actually λ → ψ(λ) is an element of the scaling algebra itself, i.e., that
(A.9) sup α(A,λa) ψ(fλ ) − ψ(fλ ) = 0, lim (A,a)→(1,0) λ∈(0,1]
which clearly implies the statement. We have
α(A,λa) ψ(fλ ) − ψ(fλ )2 ≤ 4 u(A, a)f − f 2 λm
2 d3 p 0 ipλ,± ·a =4 S(A)fˆ(Λ(A)−1 pλ,± )−fˆ(pλ,± ) , γ ( pλ,± +λm) e 2 4ω (p) 3 λm R ± and, considering only the + term in the sum inside the integral (the other one is estimated in the same way), and writing pλ := pλ,+ ,
2 d3 p 0 γ ( pλ + λm) eipλ ·a S(A)fˆ(Λ(A)−1 pλ ) − fˆ(pλ ) 2 4ω (p) 3 λm R 2 d3 p ipλ ·a 5 e ≤ S(A)fˆ(Λ(A)−1 pλ ) − fˆ(pλ ) 4 R3 |p| 5 S(A) fˆ(Λ(A)−1 pλ ) − fˆ(pλ )2 + (eipλ ·a − 1)fˆ(pλ )2 ≤ 4 2 + S(A) − 1 fˆ(pλ )2 , (A.10) where ·2 denotes the standard norm in L2 (R3 , d3 p/ |p|) ⊗ C4 , and where, for more clarity, we indicated explicitly the variable of integration inside the norms. The last term of the last line in this equation can be estimated uniformly in λ by the fact that, being fˆ ∈ S(R4 ; C4 ), there are constants C > 0, n > 1, such that 2 d3 p ˆ d3 p d3 p 1 1 ≤ C , f (pλ ) ≤ C 2 n 2 2 |p| |p| |p| (1 + ωλm (p) + |p| ) (1 + 2 |p| )n R3 R3 R3 so that it can be made arbitrarily small, as A → 1, uniformly in λ ∈ (0, 1]. For the second term in square brackets at the end of (A.10), we have, by an application
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of Lagrange’s theorem to the exponential, R3
2 d3 p ipλ ·a (e − 1)fˆ(pλ ) ≤ C(|a0 |2 + |a|2 ) |p|
R3
2
2
d3 p |ωm (p)| + |p| , |p| (1 + 2 |p|2 )n
and then, if n > 2, this term is also uniformly small in the relevant limit. Finally, we use the above lemma to estimate the first term in square bracket at the end of (A.10). For each sufficiently large R > 0 let NR be a neighborhood of the identity in SL(2, C), such that Λ(NR ) ⊆ L↑+ is as in the lemma. Then, for |p| > R, A ∈ NR , 1 1 fˆ(Λ(A)−1 pλ ) − fˆ(pλ ) ≤ C + (1 + 2 |Λ(A)−1 · pλ |2 )n (1 + 2 |pλ |2 )n 1 1 ≤C + . 2 2 (1 + |p| )n (1 + 2 |p| )n Thus, again by Lagrange theorem, we have, for A ∈ NR , R3
2 2 1 d3 p ˆ d3 p 1 f (Λ(A)−1 pλ ) − fˆ(pλ ) ≤ C + 2 2 |p| (1 + 2 |p| )n |p|>R |p| (1 + |p| )n 2 d3 p 2 2 (|ωm (p)| + |p| ). + ∂ fˆ2∞ Λ(A)−1 − 1 |p|
and the λ independent right-hand side can be made arbitrarily small by taking R ˜ R ⊆ NR . sufficiently large, and A in a corresponding neighborhood N
˜ × (W ) B Reeh-Schlieder property for F 0,ι ˜ × (C) be the C∗ We employ the notations introduced in Section 7. Let also F 0,ι × (F(C, O)) as O ⊂ C. algebra generated by π0,ι Theorem B.1 The vacuum Ω0,ι is a cyclic and separating vector for the algebras ˜ × (W ). F 0,ι We will give a sketch of the proof of this theorem, which uses in an essential way analyticity of both translations and Lorentz boosts, consequence of geometric modular action and Tomita-Takesaki theory. Similar results can be found in [6, 17], to which we refer the interested readers for the details, which can also be found in [27]. We need some preparations. We recall that we denote by ΛW (t) ∈ P↑+ , t ∈ R, the one parameter group of Poincar´e transformations leaving the wedge W invariant. In order to simplify notations, we will identify ΛW (t) with its unique ˜ ↑ which is the identity for t = 0. smooth lift to P +
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× ˜ ↑ , and Lemma B.2 Let U0,ι be a strongly continuous unitary representation of P + ↑ ↑ ˜ ) is the strong closure ˜ an open neighborhood of the identity. Then U × (P N⊆P + + 0,ι × of the group UN generated by the elements U0,ι (sΛW (t)s−1 ), t ∈ R, s ∈ N.
Proof. We can assume that W = WR , and that N = N1 ×N2 ⊆ SL(2, C)×R4 . Then × by [6, Lemma 2.1], U0,ι (SL(2, C)) is the strong closure of the subgroup of UN gen× erated by U0,ι (ΛΛWR (t)Λ−1 ), t ∈ R, Λ ∈ N1 ,12 so that it is sufficient to show that × × × (x) ∈ UN− . Furthermore, as U0,ι ((Λ, x)ΛWR (t)(Λ, x)−1 )U0,ι (ΛΛWR (t)Λ−1 )∗ = U0,ι × × × −1 n 4 U0,ι (Λ(1 − ΛWR (t))Λ x), and U0,ι (x) = U0,ι (x/n) , x ∈ R , n ∈ N, we reduce the problem to showing that the set E := { i Λi (1 − ΛWR (ti ))Λ−1 i xi : (Λi , xi ) ∈ N, ti ∈ R} is a neighborhood of zero in R4 . Then with eµ , µ = 0, . . . , 3 the canonical basis of R4 and e± := e1 ± e0 , it is easily verified, by first choosing the Λi in the definition of E to be 1 and then to be a small rotation around the e3 axis, that seα ∈ E for |s| sufficiently small and α = +, −, 2, 3, so that E contains a neighborhood of 0. Lemma B.3 The state ω0,ι = Ω0,ι , (·)Ω0,ι is a (−2π)-KMS state for the C∗ ˜ × (W ), α(0,ι)× ). dynamical system (F 0,ι ΛW The proof is completely analogous to the one of the first part of Lemma 6.2 in [11]. For any finite set of spacelike cones C1 , . . . , Cn , we introduce the C∗ -algebra ˜ × (C1 , . . . , Cn ) as the one generated by the algebras F ˜ × (C1 ), . . . , F ˜ × (Cn ). We F 0,ι 0,ι 0,ι ˜× also define G× 0,ι (C1 , . . . , Cn ) to be the set of operators G ∈ F0,ι (C1 , . . . , Cn ) for ˜ ↑ such that αs(0,ι)× (G) ∈ which there exists a neighborhood N of the identity in P + ˜ × (C1 , . . . , Cn ) for any s ∈ N. It is clear that G× (C1 , . . . , Cn ) is a ∗-algebra and F 0,ι 0,ι ˜ × (C˜1 , . . . , C˜n ) ⊆ that for any n-tuple C˜1 , . . . , C˜n with C˜i Ci , i = 1, . . . , n, F 0,ι
G× 0,ι (C1 , . . . , Cn ).
Lemma B.4 Let W be a wedge in Minkowski space and let Ci W , i = 1, . . . , n, ⊥ be spacelike cones. If Φ ∈ (G× 0,ι (C1 , . . . , Cn )Ω0,ι ) , then Φ, αs(0,ι)× (G1 ) . . . αs(0,ι)× (Gm )Ω0,ι = 0 1 m
(B.11)
˜ ↑ , Gi ∈ G× (C1 , . . . , Cn ), i = 1, . . . , m. for any si ∈ P + 0,ι ⊥ Proof. We begin by showing that Φ ∈ (G× 0,ι (C1 , . . . , Cn )Ω0,ι ) implies × ⊥ U0,ι (s)Φ ∈ (G× 0,ι (C1 , . . . , Cn )Ω0,ι ) ,
˜↑ . s∈P +
˜ ↑ such that N−1 · Ci ⊂ W , i = Let N be a neighborhood of the identity in P + × 1, . . . , n, and let G ∈ G0,ι (C1 , . . . , Cn ). Then there exists ε > 0, depending on cited result refers actually to representations of L↑+ , but since the proof uses only properties of its Lie algebra, it can be also applied to the present case. 12 The
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(0,ι)×
s ∈ N, such that Φ, αsΛW (t)s−1 (G)Ω0,ι = 0 for |t| < ε. But by the above lemma (0,ι)×
t → αsΛW (t)s−1 (G)Ω0,ι has an analytic continuation to a function on the strip × ⊥ {−π < Im z < 0}, so that U0,ι (sΛW (t)s−1 )Φ ∈ (G× for any 0,ι (C1 , . . . , Cn )Ω0,ι ) × (s)Φ ∈ t ∈ R, s ∈ N. Then, iterating the argument, and using Lemma B.2, U0,ι ↑ × ⊥ ˜ (G0,ι (C1 , . . . , Cn )Ω0,ι ) for each s ∈ P+ . ˜ ↑ , i = 1, . . . , m, we If we now show that, for any Gi ∈ G× (C1 , . . . , Cn ), si ∈ P +
0,ι (0,ι)× (0,ι)× × (G1 )...αsm (Gm )Φ ∈ (G0,ι (C1 ,...,Cn )Ω0,ι )⊥ , αs1
since G× have 0,ι (C1 , . . . , Cn ) is a ∗-algebra containing the identity operator, the conclusion of the lemma will follow, but this is proven easily by induction, using the first part of the proof. Proof of Theorem B.1. By normal commutation relations, it is sufficient to ˜ × (W ), i.e., (F ˜ × (W )Ω0,ι )⊥ = {0}. Let then show that Ω0,ι is cyclic for F 0,ι 0,ι ˜ × (W )Ω0,ι )⊥ and Fi ∈ F ˜ × (C˜i ), i = 1, . . . , n, be arbitrary operators. For any Φ ∈ (F 0,ι 0,ι ˜ ↑ and a spacelike cone Ci such that s−1 · C˜i Ci i = 1, . . . , n there exists si ∈ P + i (0,ι)× × (C , . . . , C ) and, being Φ ∈ (G (C , . . . , Cn )Ω0,ι )⊥ , W . Then αs−1 (Fi ) ∈ G× 1 n 1 0,ι 0,ι i
(0,ι)×
(0,ι)×
(αs−1 (F1 )) . . . αs(0,ι)× (αs−1 (Fn ))Ω0,ι = 0 Φ, F1 . . . Fn Ω0,ι = Φ, α(0,ι)× s1 n 1
n
× by Lemma B.4, thus Φ is orthogonal to a total set of vectors in H0,ι , and then vanishes.
References [1] H. Araki, On the diagonalization of a bilinear Hamiltonian by a Bogoliubov transformation, Publ. RIMS A 4, 387 (1968). [2] H. Araki, On quasifree states of CAR and Bogoliubov automorphisms, Publ. RIMS 6, 383 (1970/71). [3] H. Araki, K. Hepp, D. Ruelle, On the asymptotic behaviour of Wightman functions in spacelike directions, Helv. Phys. Acta 35, 164 (1962). [4] J.J. Bisognano, E.H. Wichmann, On the duality condition for quantum fields, J. Math. Phys. 17, 303 (1976). [5] H.J. Borchers, Translation group and particle representations in quantum field theory, Lecture Notes in Physics m40, Springer-Verlag, Berlin, 1996. [6] H.J. Borchers, D. Buchholz, Global properties of vacuum states in de Sitter space, Ann. Inst. Henri Poincar´e 70, 23 (1999). [7] D. Buchholz, On the manifestations of particles, in: Mathematical physics towards the 21st century, ed. R. Sen (Proceedings, Beer-Sheva 1993), Ben Gurion University Press, 1994.
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[8] D. Buchholz, Phase space properties of local observables and structure of scaling limits, Ann. Inst. H. Poincar´e 64, 433 (1996). [9] D. Buchholz, Quarks, gluons, colour: Facts or fiction?, Nucl. Phys. B469, 333 (1996). [10] D. Buchholz, K. Fredenhagen, Locality and the structure of particle states, Commun. Math. Phys. 84, 1 (1982). [11] D. Buchholz, R. Verch, Scaling algebras and renormalization group in algebraic quantum field theory, Rev. Math. Phys. 7, 1195 (1995). [12] D. Buchholz, R. Verch, Scaling algebras and renormalization group in algebraic quantum field theory. II. Instructive examples, Rev. Math. Phys. 10, 775 (1998). [13] S. Doplicher, R. Haag, J.E. Roberts, Fields, observables and gauge transformations I, Commun. Math. Phys. 13, 1 (1969). [14] S. Doplicher, R. Haag, J.E. Roberts, Local observables and particle statistics I, Commun. Math. Phys. 23, 199 (1971). [15] S. Doplicher, J.E. Roberts, Fields, statistics and non-abelian gauge groups, Commun. Math. Phys. 28, 331 (1972). [16] S. Doplicher, J.E. Roberts, Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics, Commun. Math. Phys. 131, 51 (1990). [17] W. Driessler, S.J. Summers, E.H. Wichmann, On the connection between quantum fields and von Neumann algebras of local operators, Commun. Math. Phys. 105, 49 (1986). [18] D. Guido, R. Longo, Relativistic invariance and charge conjugation in quantum field theory, Commun. Math. Phys. 148, 521 (1992). [19] D. Guido, R. Longo, An algebraic spin and statistics theorem, Commun. Math. Phys. 172, 517 (1995). [20] K. Fredenhagen, Superselection sectors, notes from lectures held at Hamburg University in the Winter Term 1994/95. [21] R. Haag, Local quantum physics, 2nd edn., Springer-Verlag, Berlin, 1996. [22] R. Haag, D. Kastler, An algebraic approach to quantum field theory, J. Math. Phys. 5, 848 (1964). [23] R. Ho /gh-Krohn, M.B. Landstad, E. Sto /rmer, Compact ergodic groups of automorphisms, Ann. Math. 114, 75 (1981). [24] B. Kuckert, A new approach to spin and statistics, Lett. Math. Phys. 35, 319 (1995). [25] S. Mohrdieck, Phase space structure and short distance behaviour of local quantum field theories, J. Math. Phys. 43, 3565 (2002).
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[26] G. Morsella, The structure of charges in the ultraviolet and an intrinsic notion of confinement, pp. 315–324, in: Operator algebras and mathematical physics, eds. J.M. Combes et al., Theta Foundation, Bucharest, 2003. [27] G. Morsella, Ph.D. thesis, University of Rome I, 2003. [28] M. Reed, B. Simon, Methods of modern mathematical physics, Vol. 1, Academic Press, New York, 1972. [29] C. Rigotti, Remarks on the modular operator and local observables, Commun. Math. Phys. 61, 267 (1978). [30] J.E. Roberts, Some applications of dilation invariance to structural questions in the theory of local observables, Commun. Math. Phys. 37, 273 (1974). [31] J.E. Roberts, “Lectures on algebraic quantum field theory”, pp. 1–112, in: The algebraic theory of superselection sectors. Introduction and recent results, ed. D. Kastler, World Scientific, 1990. [32] J.E. Roberts, Localization in algebraic field theory, Commun. Math. Phys. 85, 87 (1982). [33] H.J. Rothe, K.D. Rothe, J.A. Swieca, Screening versus confinement, Phys. Rev. D19, 3020 (1979). [34] J.A. Swieca, Charge screening and mass spectrum, Phys. Rev. D13, 312 (1976). [35] M. Takesaki, Tomita’s theory of modular Hilbert algebras and its applications, Lecture Notes in Mathematics Vol. 128, Springer-Verlag, New York, 1970. [36] M. Takesaki, M. Winnink, Local normality in quantum statistical mechanics, Commun. Math. Phys. 30, 129 (1973). [37] R. Verch, “Stability properties of quantum fields in curved spacetime”, Habilitation thesis, University of G¨ ottingen, 2002. Claudio D’Antoni Dipartimento di Matematica Universit` a di Roma “Tor Vergata” Via della Ricerca Scientifica I-00133 Roma, Italy email:
[email protected]
Gerardo Morsella Dipartimento di Matematica Universit` a di Roma “La Sapienza” P.le Aldo Moro 2 I-00185 Roma, Italy email:
[email protected]
Rainer Verch Max-Planck-Institut for Mathematics in the Sciences Inselstr. 22 D-04103 Leipzig, Germany email:
[email protected] Communicated by Klaus Fredenhagen submitted 21/01/03, accepted 26/02/04
Ann. Henri Poincar´e 5 (2004) 871 – 914 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/050871-44 DOI 10.1007/s00023-004-0184-6
Annales Henri Poincar´ e
Stability of Interfaces and Stochastic Dynamics in the Regime of Partial Wetting Thierry Bodineau and Dmitry Ioffe∗
Abstract. The goal of this paper is twofold. First, assuming strict convexity of the surface tension, we derive a stability property with respect to the Hausdorff distance of a coarse grained representation of the interface between the two pure phases of the Ising model. This improves the L1 description of phase segregation. Using this result and an additional assumption on mixing properties of the underlying FK measures, we are then able to derive bounds on the decay of the spectral gap of the Glauber dynamics in dimensions larger or equal to three. These bounds are related to previous results by Martinelli [Ma] in the two-dimensional case. Our assumptions can be easily verified for low enough temperatures and, presumably, hold true in the whole of the phase coexistence region.
1 Introduction During the last decade, a series of studies enabled to derive rigorously the occurrence of phase segregation starting from a model with microscopic interactions. The phase separation phenomenon has been established for a fairly general class of models, but the correspondence between the microscopic models and the equilibrium crystal shapes (solution of the Wulff variational problem) is extremely loose. Thus, important questions remain and a complete theory of phase coexistence is far from being achieved. A thorough description of the phase coexistence phenomena should include a characterization of the structure of the interface (thickness, fluctuation, detailed structure. . . ) as well as an understanding of the relaxation of the system to the pure phases away from the interface. So far such complete program has been achieved only in the case of two-dimensional nearest neighbor Ising model [DKS, DH, ISc, Pf, PV, BCK]. The strategy developed in this context, relies on the one-dimensional structure of the interface; this enabled to derive not only the Wulff construction, but as well quantitative statements on the microscopic configurations: existence of a unique large droplet, localization of the interface wrt the Hausdorff distance. . . ∗ Partly
supported by the EU Network Postdoctoral Training Program in Mathematical Analysis of Large Quantum Systems under the contract HPRN-CT-2002-00277 and by the Technion V.P.R fund – E. and J. Bishop research fund. We would like to thank warmly N. Yoshida for numerous discussions on the topic of Glauber dynamics.
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For systems in three or more dimensions the interface is a more complicated geometrical object and a different approach of phase coexistence, the L1 -theory, was initiated in order to bypass the complexity of the microscopic configurations. In this new framework, a weaker characterization of the phase segregation is obtained in terms of local averages of the magnetization. In this way, the occurrence of macroscopic equilibrium crystals whose shapes are solutions of a variational problem can be predicted, but unlike the two-dimensional case, nothing can be inferred on the interface. In fact one cannot even conclude from these results that the equilibrium crystal contains only a pure phase: as the statements are formulated in terms of averages and interfaces are understood only in L1 -sense, one could not rule out the situation when equilibrium crystals contain minority phase impurities or even are made of a collection of small crystals glued together. The first step is to propose a relevant interpretation of the interface. Let us, as an example, consider a three-dimensional Ising model with Dobrushin boundary conditions, i.e., mixed boundary conditions which enforce an horizontal interface. In this case, the interface can be unambiguously defined as the unique open contour in the system. At low temperature, the interface is a rigid two-dimensional hyperplane with some protuberances attached to it (e.g., one-dimensional filaments). The statistic of these excitations is known and the open contour which forms this interface is localized wrt the Hausdorff distance. On the one hand, as the temperature increases above the roughening temperature the interface is expected to be macroscopically flat but with some logarithmic fluctuations. However, as the temperature approaches the critical temperature, the behavior of the microscopic contour becomes irregular and in particular one-dimensional filaments are conjectured to percolate through the whole system [ABL]. Thus a microscopic representation of the interface is then irrelevant since the microscopic contour might be completely delocalized (see [CePi] for a discussion on this phenomenon). The way out is to renormalize the system at a proper mesoscopic scale for which the interface becomes regular. This is characteristic of the physicists heuristics which says that the complex microscopic configurations can be reduced to an effective interface model and should share the same properties on a suitable mesoscopic scale. As mentioned previously, the L1 -theory sheds little light on the statistical properties of random interfaces. The goal of this paper is to show that, nevertheless, on a mesoscopic level some smoothness properties of the interface are restored. Though much more modest than the heuristic picture described above, our results show that the low-dimensional excitations of the coarse grained interface disappear and we recover a macroscopic stability with respect to the Hausdorff distance of the random interface. The exact statement of this stability result is given in Subsection 3.1 along with some comments on the implications for the statistical Hausdorff stability of higher-dimensional mesoscopic Wulff shapes. The second part of this paper deals with dynamical properties, we derive bounds on the logarithmic asymptotics of the spectral gap of the Glauber dynam-
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ics. Such asymptotics are non-trivial whenever the energy level of the bottleneck between two pure, though possibly metastable, phases is strictly higher than the free energy of each of the respective phases. For a two-dimensional Ising model with free boundary conditions, Martinelli and coauthors derived in [CGMS, Ma] the precise logarithmic asymptotics of the spectral gap. He proved that the dominant time scale to reach one equilibrium state starting from the other one is the creation of an interface; once created the interface moves in a much shorter amount of time until the other equilibrium state is reached. Martinelli devised very ingenious techniques in order to control the occurrence of the interface and its motion, in particular the dynamical estimates were reduced to some statements on the equilibrium measure. The analysis of [CGMS, Ma] has been based on very specific facts about the Hausdorff stability of the 2D nearest neighbor Ising model interfaces, on the closely related exponential mixing properties of finite volume pure state and on exact surface order large deviation asymptotics for the magnetization inside the phase coexistence region. In higher dimensions, we are going to use the large deviation estimates of the L1 -theory and the Hausdorff stability of the random interface on the macroscopic scale in order to extend the results of [CGMS, Ma]. The upper and lower bounds derived in higher dimensions on the spectral gap are expressed in terms of variational principles for which (contrary to the twodimensional) the solution is not explicit. For this reason, we cannot prove that the lower and upper bound coincide as it was shown in the two-dimensional case [CGMS, Ma]. A more complete discussion on the interplay between the metastability and the wetting is postponed to Subsection 3.2. Apart from being dependent on the validity of Pisztora coarse graining (c.f. Subsection 2.2) our proof of the interface stability relies on strict convexity of surface tension. The analysis of the spectral gap asymptotics for the Glauber dynamics requires an additional assumption on exponential mixing properties of the underlying FK measures. Both assumptions are described and discussed in Subsection 2.5 and are expected to hold for a wide range of sub-critical temperatures. While completing this paper, we learnt about the recent work by N. Sugimine [Su1, Su2] on upper and lower bounds for spectral gap for the three-dimensional low temperature Ising model with mixed +/∅ boundary conditions.
2 Notations and assumptions 2.1
The microscopic model
We consider the nearest neighbor ferromagnetic Ising model in dimension d 2. For any domain ∆ ⊂ Zd and boundary conditions η outside ∆, the Gibbs measure on {±1}∆ at inverse temperature β will be denoted by µηβ,∆ . Thus, given σ ∈ {±1}∆, 1 µηβ,∆ (σ) = η exp { −βHη∆ (σ)} , Zβ,∆
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where the nearest neighbor Hamiltonian Hη∆ is given by Hη∆ (σ) = −
1 σi σj − 2 i∼j∈∆
ηj σi .
i∈∆ j∈Zd \∆
There exists a critical value βc such for any β > βc a phase transition, characterized by symmetry breaking, occurs. Throughout the paper, we always consider an inverse temperature β for which the system is in a phase coexistence regime and, we denote by m∗ (β) the spontaneous magnetization in the + phase. It will be convenient to work with an alternative representation of the mid be the croscopic system, namely the FK representation. Given ∆ ⊂ Zd let E∆ d set of bonds, i.e., the pairs of nearest neighbor vertices in Z , with at least one d end-point belonging to ∆. The percolation configuration ξ on E∆ , or with an abuse ∆
of notation on ∆, is an element ξ ∈ Ξ∆ = {0, 1}E∆ . We shall suppress the domain sub-index and write Ξ whenever ∆ = Zd . Given ξ ∈ Ξ and a bond b ∈ E d , we say that b is open if ξ(b) = 1. Two sites of Zd are said to be connected if one can be reached from another via a chain of open bonds. Thus, each ξ ∈ Ξ splits Zd into the disjoint union of maximal connected components, which are called the open clusters of Ξ. Given a finite subset B ⊂ Zd we use cB (ξ) to denote the number of different open finite clusters of ξ which have a non-empty intersection with B. Below we give a general definition of FK measures which are related to the finite volume spin Gibbs states on ∆ ⊂ Zd . We use a provisional notation intended to illustrate the connection between the Gibbs states and the FK measures. A more precise notation will be introduced later for particular cases which show up in the main body of the paper. The set of bonds connecting ∆ to ∆c will be denoted by E∂∆ . The boundary conditions are specified by a frozen percolation configuration π ∈ Ξ \ Ξ∆ and by the collection p ∈ [0, 1 − e−2β ]E∂∆ , which describes the “activity” of the bounds on the boundary of ∆. We write ξ ∨ π for the joint configuration in Ξ and define the finite volume FK measure on ∆ with the boundary conditions π and p as: 1−ξb ξb 1 ∆ π,p Φ∆ (ξ) = π 1 − p(b) p(b) (2.1) 2c∆ (ξ∨π) (ξ) , Z∆ d
b
where, for a bond b = (i , j) with (i , j) ∈ ∆, we set p(b) = p(β, b) = 1 − exp(−2β); otherwise if b is in E∂∆ the corresponding percolation probability p(b) = p(b). The two extreme configurations such that p(b) ≡ 1 − exp(−2β) and π ≡ 1 and, respectively, p(b) ≡ 0 or π ≡ 0 for all b in E∂∆ lead to the FK measures with wired Φw ∆ or free Φf∆ boundary conditions respectively. The intermediate values of p have a natural interpretation as a magnetic boundary field. This enables to represent
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the Gibbs measures for which the boundary conditions η have only non-negative components. If η has only non-negative coordinates, the Gibbs state µηβ,∆ can be reconstructed as follows (see [ES]): First for any b = (i , j) ∈ E∂∆ with j ∈ ∆c set p(b) = p(β, η, b) = 1 − e−2ηj β . Next sample a bond configuration ξ ∈ Ξ∆ from the FK measure Φw,p β,∆ , and paint with 1 all the clusters of ξ connected to the regions of the boundary where ηi > 0, whereas all the remaining clusters of ξ are to be painted into ±1 with probability 1/2 each. The corresponding joint bond-spin probability measure is denoted by Pηβ,∆. η,ξ Pηβ,∆ (σ, ξ) = P∆ (σ) Φw,p β,∆ (ξ) , η,ξ denotes the painting measure. The Gibbs state µηβ,∆ is then recovered where P∆ as the σ-marginal of Pηβ,∆. The representation for more general boundary conditions which correspond to sign changing η will be discussed later.
2.2
Scales and coarse graining
All scales are binary. The running microscopic scale will be N = 2n and the associated renormalization scale K = 2k . We shall work either with fixed finite scales K or else we shall explicitly relate K to N as K = N a , where the fixed positive number a = a(n) (the dependence on n is only in order to be compatible with the binary notation) satisfies 0 < a1 a a2 < 1/d.
(2.2)
All our computations go through if instead of K = N a we choose the mesoscopic scale K = C log N for C large enough. We introduce now the mesoscopic partitions of DN = {1, . . . , N }d . At each fixed mesoscopic scale K = 2k we split the microscopic domain DN into the dis∆ joint union of shifts of the mesoscopic box BK = {− 21 K + 1, . . . , 12 K}d. These ∆
shifted boxes are centered at the lattice points from the rescaled set DN,K = K DN/K − (1/2, . . . , 1/2) : BK (i), (2.3) DN = i∈DN,K ∆
where BK (i) = i + BK = (i + BK ) ∩ Zd . As explained in the introduction a key tool to understand the interface behavior is a renormalization procedure. In this paper we will use a coarse graining implemented by Pizstora [Pi] by means of the FK representation. We recall below the main features of this coarse graining.
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First of all we shall set up the notion of good box on the K-scale which characterizes a local equilibrium in a pure phase Definition 2.1 Let us say that a K-box BK (i) ⊂ DN , centered in i, is good with respect to the percolation configuration ξ ∈ Ξ if (1) There exists a crossing cluster C ∗ = C ∗ (B2K (i)) connected to all the faces of the inner vertex boundary of the 2K-box B2K (i). (2) Any FK-connected cluster of vertices of B2K (i) which has a diameter larger than K/10 is necessarily connected to C ∗ . Fundamental techniques developed by Pizstora in [Pi] imply that there exists B a subset of ]βc , ∞[ for which the following holds: for any β ∈ B, there is a constant c > 0, such that for all scales K K0 large enough (and, in particular, for our basic scale K = N a ), inf Φπ,p B2K (i) (ξ is a good configuration in BK (i)) 1 − exp(−cK) , π
(2.4)
uniformly in the boundary conditions π, boundary bond activities p and in i ∈ DN,K . The conjecture B =]βc , +∞[ has been recently derived in [B].
2.3
Equilibrium setting
In equilibrium, our result on the localization concerns primarily interfaces imposed by mixed boundary conditions. We also investigate the consequences of the localization on the structure of the mesoscopic droplet when two phases coexist. We define below the two corresponding frameworks. 2.3.1 Pure boundary conditions The Gibbs measure on the set DN = {1, . . . , N }d with + boundary conditions will w be denoted by µ+ N and the corresponding FK measure by ΦN . An important quantity to study phase coexistence is the surface tension d−1 which we now introduce. Let √ n ∈ S + be a ±unit normal and assume for the definiteness that (n, ed ) > 1/ d. Let ZN and ZN (n) be the partition functions on {−N, . . . , N }d with respectively “+” and mixed boundary conditions, the latter being defined by σi = sign((n, i)), with sign(0) = 1. The bulk surface tension in the direction orthogonal to n is ∆
τ (n) =
lim −
N →∞
± ZN (n) (n, ed ) log + . d−1 (2N ) ZN
(2.5)
The equilibrium crystal shape of volume a > 0 is the Wulff shape Ka defined by Ka =
a |K|
1/d
K,
where K =
x ∈ Rd , (x · n) τ (n) . n
(2.6)
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For our purpose, it will be more convenient to recall the phase coexistence Theorem directly in the FK coarse grained setting. For any configuration ξ in ΞDN , we partition the set DN,K into 3 sets (not necessarily connected) + (ξ) DN,K = CN
0 CN (ξ)
− CN (ξ) ,
(2.7)
+ where CN (ξ) denotes the set of good boxes BK for which the unique crossing − cluster (see Definition 2.1) is connected to the boundary of DN , CN (ξ) denotes the set of good boxes BK for which the crossing cluster is not connected to the 0 boundary of DN and CN (ξ) the boxes which are not good. The phase coexistence will be imposed by a volume constraint at the mesoscopic level defined by
− Nd VN,a = ξ; CN (ξ) a d . K
This is the set of configurations for which there is a density at least a > 0 of good blocks detached from the boundary (i.e., of the − phase in the spin language). The L1 -approach (see [CePi, BIV1]) implies that for any β ∈ B there is a sequence (δN ) vanishing to 0 such that d N − < δN d VN,a = 1 , lim Φw ξ; Zd ∩ (i + Ka(N/K)d ) ∆CN N N →∞ K i∈DN,K
where ∆ denotes the symmetric difference of the sets in DN,K and | · | the cardinal of a set. 2.3.2 Mixed boundary conditions Let Ld be the lattice half-space {i ∈ Zd : id > 0}. The exterior boundary of DN = {1, . . . N }d in Ld will be denoted by ∂ ext DN . The bottom face of DN is denoted by ∂bint DN = {1, . . . , N }d−1 ×{1}. We consider mixed boundary conditions equal to 1 in ∂ ext DN and to −1 outside Ld . The corresponding Ising measure will be denoted µ± N accordingly. The FK representation of the mixed boundary conditions requires some care. Consider the graph Ld , Ld , where the edge set Ld consists of all (unoriented) pairs of nearest neighbor vertices (i, j) ⊂ Ld . Let LdN denote the set of bonds of Ld which have a non-empty intersection with DN . It happens to be convenient to augment the graph (Ld , Ld ) with a ghost site g connected to all the sites in the bottom layer ∂bint DN . In this way the edge set for the model is given by ∆
LdN,± = LdN
(i, g) i ∈ ∂bint DN .
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The sample space for finite volume FK states on DN is given by ∆
ΞN,± = {0, 1}LN,± . d
Define the FK percolation event JN ⊂ ΞN,± as ∆ JN = ξ ∈ ΞN,± g ↔ ∂ ext DN , and set
2.4
(2.8)
w Φ± N ( · ) = ΦN · J N .
Dynamical setting: boundary fields
In the second part of the paper, we are going to study the slow relaxation of the Glauber dynamics which occurs when magnetic fields are applied on the faces of DN . Let h = (h1 , . . . , h2d ) be a vector with non-negative coordinates. The Gibbs measure with each boundary magnetic field hi applied on the ith -face of the cube DN is denoted by µh N . In this way free boundary conditions correspond to h = (0, . . . , 0), whereas pure + boundary conditions correspond to h = (1, . . . , 1). As we shall explain below our results on the relaxation speed are non-trivial only when the boundary magnetic fields h1 , . . . , h2d are in the partial wetting regime, which is the case for example for free boundary conditions, but not for the pure + ones. In a metastable regime, the rescaled evolution of the system can be described by an energy landscape which is related to equilibrium macroscopic quantities. Therefore we first proceed in recalling the basic framework of equilibrium phase coexistence (we refer to [BIV1] for a detailed review). A heuristic discussion of the interplay between the equilibrium properties and the dynamics is postponed to subsection 3.2. The basic macroscopic quantities in this context are the bulk surface tension (2.5) and the boundary free energy. The influence of a magnetic field h ∈ R applied along the boundary leads −,h +,h and ZN with to a specific surface energy. Consider the partition functions ZN “−” and “+” boundary conditions on Ld \ DN and h outside Ld . The boundary free energy ∆h is defined as the difference between the interfacial free energies of the coexisting phases: ∆
∆h =
lim
N →∞
1 N d−1
log
+,h ZN
−,h ZN
.
(2.9)
We refer the reader to [FP1] and [FP2] for a detailed study of the boundary surface tension as well as related phenomena. On the macroscopic level the equilibrium phase coexistence is governed by a variational principle involving the bulk surface tension and the wall free energies ∆hi . As it has been realized in [ABCP] the appropriate macroscopic setting is that
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of the functions of bounded variations and we shall repeatedly refer to [EG] for the necessary background. = [0, 1]d . The microscopic system is embedded in the continuous domain D is represented by a signed indicator A macroscopic distribution of phases in D function u taking values {±1} according to the local equilibrium. Let O be an For non-negative boundary fields, it is enough open smooth neighborhood of D. to consider functions u taking values in the set of bounded variation functions BV (O, {±1}) and equal to 1 outside D. th The boundary of D is denoted by Let Pi be the i -face of the cube D. P = ∪i Pi . The interfacial energy associated to u is defined by 2d (d−1) Wh (u) = τ (nx )dHx + ∆hi dHx(d−1) , (2.10) ∂ ∗ u\P
i=1
∂ ∗ u∩Pi
where ∂ ∗ u is the reduced boundary [EG] of {u = −1}. In the particular case of boundary magnetic field acting only on one of the faces of DN the probability of observing spin configurations which are close (in the L1 sense) to some macroscopic configuration u was proven to decay exponentially fast with the order N d−1 Wh (u) (see [BIV2]). Finally, the optimal interfacial energy under a volume constraint is defined as
m h u(x) dx = ∗ . F (m) = inf Wh (u) (2.11) u∈BV m D
2.5
The assumptions
There are two main assumptions. The first one is of geometric nature and will play a crucial role in the localization of the interface. The second assumption is a mixing property for the FK measure and will be only used in the estimation of the spectral gap. 2.5.1 Strict convexity of the surface tension Recall that a d-dimensional simplex is the convex envelop S = S(u1 , . . . , ud+1 ) of (d + 1) points u1 , . . . ud+1 ∈ Rd in general position. The latter means that S has a non-empty interior. Given such a d-dimensional simplex S let F1 , . . . , Fd+1 be its faces and n1 , . . . , nd+1 the corresponding outer normals. By the Gauss-Green theorem [EG], d+1 |Fk |nk = 0. (2.12) k=1
Given an axis direction ed let us say that a simplex S = S(u1 , . . . , ud+1 ) is ed oriented if: (i) u1 , . . . , ud ∈ {x ∈ Rd : xd = 0}, (ii) ud+1 ∈ {x ∈ Rd : xd > 0}.
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We shall always number the faces of ed -oriented simplices S in such a way that F1 ⊂ {x ∈ Rd : xd = 0} or, equivalently, n1 = −ed . Thus, for a given ed -oriented simplex S, (2.12) yields a representation of ed as a non-trivial linear combination d+1 |Fk |nk . |F1 |ed = k=2
We say that the surface tension τβ is strictly convex at ed if the following strict inequality d+1 |Fk |τ (nk ) (2.13) |F1 |τ (ed ) < k=2
holds for any ed -oriented simplex S = S(u1 , . . . , ud+1 ). In [DS2] (2.13) is called Strong Simplex Inequality. It is shown to be equivalent to the following fact (Lemma 3.5 in [DS2]): Assume that v1 , . . . , vd are d vectors in general position, and assume that ed lies in the interior of the positive cone spanned d by v1 , . . . , vd , that is there exist positive numbers λ1 , . . . , λd , such that ed = k=1 λk vk , then, τ (ed ) <
d
λk τ (vk ).
(2.14)
k=1
Assumption (SC): τ is strictly convex at ed . This assumption is of course true at low enough temperatures when the Wulff shape exhibits a flat facet in the ed direction. Presumably it is true for every β > βc since, at least on the heuristic level, kinks on the boundary of the Wulff shape would correspond to pathological large interface fluctuations. We could have avoided this assumption by simply considering a direction which is orthogonal to smooth portions of the Wulff shape. This would not mend the situation and we prefer this assumption for the notational simplicity and in order to stress the existing flaws in the theory. This assumption (SC) is related to the stability properties of the associated variational problem. In Section 5, we are going to consider more general boundary conditions which would lead to a different variational problem. We proceed now in discussing this new framework and show how assumption (SC) enables to control the stability of the new variational problem. This will be useful only for Section 5, thus the reader is invited to skip this discussion on a preliminary run-through. are denoted = [0, 1]d . The faces of D We consider the macroscopic domain D ext int as follows; the top face ∂t D = {xd = 1}, the bottom face ∂b D = {xd = 0} and A boundary magnetic field equal to 1 the remainder which are the side faces ∂sext D. ext and an ε > 0 is applied on ∂t D, a boundary magnetic field equal to −1 on ∂bint D boundary field on the sides. This last field leads to a boundary surface tension denoted by ∆ε . We refer to Subsection 5.1, for the explicit microscopic definition.
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We are going to check that this modification of the boundary fields has no impact on the stability of the variational problem. Define the modified Wulff shape ∆
Kε = K ∩ {x : |xi | ∆ε ∀ i = 1, . . . , d − 1},
(2.15)
and let τ ε be the support function of Kε . = [0, 1]d . We consider the Let O be an open smooth neighborhood of D {±1} specified by boundary condition g ∈ BV O \ D, g(x) =
1, if xd > 0 , −1, if xd 0 .
define Given a ±1-valued function u on D u(x) u ∨ g(x) = g(x)
, if x ∈ intD . if x ∈ O \ D
(2.16)
It is well known [EG] that u ∨ g ∈ BV(O, {±1}) whenever the phase function {±1}). For any v in BV(O, {±1}), there exists a generalized notion u ∈ BV(intD, of the boundary of {v = −1} called reduced boundary [EG] and denoted by ∂ ∗ v. If {v = −1} is a regular set, ∂ ∗ v coincides with the usual boundary ∂v. Given a {±1}) we use ∂ ∗ u to denote the reduced boundary phase function u ∈ BV(intD, g of u in the presence of the b.c. g: = ∂ ∗ (u ∨ g) \ ∂ ∗ g. ∂g∗ u = ∂ ∗ (u ∨ g) ∩ D
(2.17)
{±1}): ε (·|g) on BV(intD, Finally define the functional W ε u g = W τ ε (nx )dHx(d−1) , ∂g∗ u
where τ ε is the support function of Kε (see (2.15)). Proposition 2.1 Assume that (SC) holds, that is τ is assumed to be strictly convex {±1}). ε (·|g) on BV(intD, at ed . Then, u = ½(·) is the unique minimum of W The proof of Proposition 2.1 is relegated to the Appendix. {±1}) in the aboveCorollary 2.1 Define the functional Wε (·|g) on BV(intD, mentioned notation via τ (nx )dHx(d−1) + ∆ε dHx(d−1) . Wε u g = ∂g∗ u\∂s D
∂g∗ u∩∂s D
Then u = ½(·) is the stable minimum of Wε (·|g) in the following sense: For every ν > 0 there exists c2 = c2 (ν) > 0 such that Wε u g Wε ½ g + c2 (ν) =⇒ u − ½1 ν. (2.18)
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Proof. Proposition 2.1 and standard compactness considerations imply that the ε · g is stable in the above sense. On the other hand functional W ε · g Wε · g , W and, of course, both functionals attain the same value on u = ½. In particular, any function u which satisfies the left-hand side of (2.18) automatically satisfies the ε · g instead of Wε · g . very same inequality with W 2.5.2 Mixing property The localization of the interface will be derived on a coarse grained level. Throughout the paper we will assume that β belongs to B so that Pisztora’s coarse graining holds. The analysis of the dynamics will require an assumption on the exponential mixing of a pure phase. It is well known (see [Gri]) that for all β (except possibly for a countable number) there is no phase transition in the FK representation, i.e., that the limiting FK measures ΦfZd and Φw Zd coincide. We will need an enhanced property of uniqueness and will suppose that the boundary effect vanishes exponentially fast. We introduce ΛN,M = {−N, . . . , N }d−1 × {−M, . . . , M } and consider two types of FK measures on this set with different boundary conditions. We denote w,f,w by Φw,f,f ΛN,M (resp. ΦΛN,M ) the measure with wired boundary conditions on the face {xd = M } (resp. on the faces {xd = ±M }) and free elsewhere. Definition 2.2 Let B1 be the subset of B containing the inverse temperatures β for which there exists c1 = c1 (β), c2 = c2 (β) > 0 such that w,f,f w,f,w ∀(N, M ), ∀b ∈ ΞΛN,M/2 , ΦΛN,M (ξb ) − ΦΛN,M (ξb ) c2 exp − c1 M . Assumption (MP): We will suppose that β ∈ B1 , i.e., that the mixing property holds. The previous assumption holds for β large enough. This can be easily derived along the lines of the proof of Theorem 5.3 (c) of [Gri]. We conjecture that the mixing property should be valid on ]βc , ∞[. In fact (MP) can be related to the notion of strong mixing which was introduced in the context of Ising model by Dobrushin and Shlosman [DS3] (see also Martinelli and Olivieri [MO] for the regular strong mixing property). The counterpart of this notion for the FK model can be stated as follows: there exists c1 (β), c2 (β) > 0 such that for any cube ∆ of Zd , any pair of boundary conditions π, π π ∀b ∈ Ξ∆ , Φ∆ (ξb ) − Φπ∆ (ξb ) c2 exp − c1 dist(b, π ∧ π ) , where π ∧ π refers to the region where π and π differ. This property implies (MP) and we conjecture that it holds for the parameters β for which the FK measure is unique in the thermodynamic limit.
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Finally, we stress the fact that (MP) does not apply directly to the Ising model, nevertheless combined with the localization of the interface, it will have useful implications on the mixing of the spin system. This will be discussed in Section 5.
3 The results Throughout the paper, the dimension d is fixed larger or equal to 3 and β belongs to B, the domain of validity of Pisztora’s coarse graining [Pi]. Theorem 3.3 and Theorem 3.5 hold for every β ∈ B. Results on Hausdorff stability with respect to axis directions (Theorem 3.1 and Theorem 3.2) require an additional assumption (SC), namely we need to assume that the surface tension τβ is strictly convex at ed . These stability results play an important role in our proof of the lower bound on the spectral gap (Theorem 3.4) which also relies on the mixing property (MP).
3.1
The Hausdorff stability
As we have already mentioned the conjectured percolation of minority spins at moderately low temperatures [ABL] suggests that microscopic interfaces are not the appropriate objects to describe stability properties of phase boundaries. In any case, however, the phases are characterized by the order parameter (spontaneous magnetization) ±m∗ (β) in the sense that local spin averages, or local magnetization profiles, inside what is expected to be “+” or “−” phases should converge, as the averaging scale grows, to m∗ (β) or −m∗ (β) respectively. Our main stability result below is formulated in terms of phase boundaries induced by local magnetization profiles on large finite scales. Consider the decomposition (2.3). Given a small number ρ > 0 let us define phase labels u˜ρN,K ∈ {0, ±1}DN,K as follows:
u˜ρN,K (i) =
1, −1 , 0,
1 if d K 1 if d K
∗ σj − m (β) ρ , j∈BK (i) ∗ σj + m (β) ρ , j∈BK (i)
(3.19)
otherwise .
Thus, u˜ρN,K uses the resolution ρ to label the proximity of the local magnetization profile to the order parameter ±m∗ (β) on the renormalization scale K = 2k . For the spin model on DN with mixed boundary conditions described in Subsec∆ tion 2.3.2 we shall (by abuse of notation) extend u˜ρN,K to the whole of ZdK =
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K Zd − (1/2, . . . , 1/2) as follows: 1 if i ∈ ZdK \ DN,K and id > 0 , ρ u ˜N,K (i) = −1 if i ∈ ZdK \ DN,K and id < 0 . It happens to be more convenient to work with the adjusted phase labels uρN,K : uρN,K (i) = 1 (respectively −1) if u˜ρN,K (i) = 1 (respectively to −1) both at i and at all ∗-neighbors of i in ZdK . Otherwise, uρN,K (i) is set to be equal to zero. The advantage of such adjustment is that any nearest neighbor path of vertices of DN,K which connects regions with different phase labels is forced to contain a site with zero phase label. Accordingly let us define the collection of phase boundaries induced by uρN,K (i) as ∆
∂ ρN,K = {i ∈ DN,K : uρN,K (i) = 0}. The set ∂ ρN,K is in general disconnected and for fixed finite values of the renormalization scale K contains (for entropic reasons) many small components even in the case of pure boundary conditions. In the case of mixed boundary conditions, however ∂ ρN,K contains a unique unbounded connected component which we shall ρ ρ denote as ∂N,K . By the construction ∂N,K contains an infinite flat double layer ρ is confined to DN,K . outside DN,K and, in fact, all the non-trivial geometry of ∂N,K Here is our Hausdorff stability result in terms of phase labels uρN,K : Theorem 3.1 Assume that β ∈ B and that the Assumption (SC) holds, that is the surface tension τβ is strictly convex at ed . Then for any ν > 0 and ρ > 0 there exists a finite scale K0 = K0 (ν, ρ) and a positive constant c = c(ν, ρ) such that for every K K0 and for all N sufficiently large, ρ ∂ µ± ∩ {i : i > νN } = ∅ e−c(ν,ρ)N . (3.20) d N N,K The above statement asserts that on large enough, though still finite, renormalization scales K the interface is macroscopically stable in the sense that the order of its fluctuations is smaller than the linear size of the system N . Since the fluctuations in question are expected to be of the log N -size for moderately low temperatures and, at least for axis oriented interfaces, are known to be bounded for sufficiently low ones [DS1], the result is far from being optimal and, in a way, it illustrates limitations of the L1 -approach. The proof of Theorem 3.1 is based on the following result on the stability of the FK interfaces. In the sequel we employ the notation introduced in Subsections 2.2 and 2.3. Theorem 3.2 Assume that β ∈ B is such that (SC) holds. Let K = N a , where a is chosen according to (2.2). Then for any ν > 0, lim Φ± N (There is a bad block in {i : id > νN }) = 0 .
N →∞
(3.21)
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More precisely, there exists c1 = c1 (ν) > 0, such that uniformly in N large enough, −c1 (ν)N . Φ± N (g ↔ {i : id νN }) e
(3.22)
Notice that the second part of the statement implies that in the FK representation the interface is localized even on a microscopic level without any additional renormalization procedures. Let us now prove Theorem 3.1 as a consequence of inequality (3.22). Proof of Theorem 3.1. Let us decompose the percolation event ∆
TN,ν = {ξ ∈ JN : g ↔ {i : id νN }} according to the realization of the maximal connected component C(g) of the ghost site g. By the very definition, ∆
C(g) ⊆ DνN = DN ∩ {i : id < νN }, on TN,ν . Consequently, for every percolation event A which depends only on the bonds connected to the upper sub-box DN \ DνN , the following decoupling bound holds: Φ± (3.23) max ΦπDN \Dν (A). N A TN,ν π
N
We stress the fact that in this Theorem the coarse graining scale K is independent of N , unlike in (3.21). As in [BIV1, BIV2] define the joint spin-bond label ρ ρ = vN,K [σ, ξ] ∈ {0, 1}DN,K via: vN,K ρ (i) vN,K
=
1, if |uρN,K | = 1 and B2K (i) is ξ good , 0, otherwise .
(3.24)
Clearly, ρ ∂ ρN,K ∩ DN,K ⊆ {i ∈ DN,K : vN,K (i) = 0}. ρ On the other hand, it follows from (3.23) that the distribution of the field vN,K on ν ± DN,K \DN,K {0, 1} under Pβ,DN ( · |TN,ν ) stochastically dominates the Bernoulli site , where the probability p = p(β, ρ, K) of a particular site percolation process PBern p to be occupied satisfies limK→∞ p(β, ρ, K) = 1, see Section 3.2 of [BIV2] for more details and references. As a result, (3.20) follows from the exponential decay of connectivities for the sub-critical site percolation once (1 − p) is sufficiently small (or, equivalently, once K is sufficiently large).
We turn now to the case of the Wulff shape for which the phase coexistence is imposed in a more indirect way via a volume constraint. A straightforward modification of the techniques which we shall employ for the proof of Theorem 3.2 yields (see (2.7)):
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Theorem 3.3 Assume that β ∈ B. Let K = N a , where a is chosen according to (2.2). Then for any ν > 0, − w (i + Ka(1−ν)(N/K)d ) ⊂ CN VN,a = 1 . lim ΦN N →∞
i∈DN
Notice that we made no additional assumptions on the strict convexity of τβ . Indeed, directions v at which τβ is not strictly convex correspond to non-smooth portions of the boundary ∂K which have zero surface measure. Theorem 3.3 implies that there is no percolation of the + phase inside the interior of the − droplet. In this way usual conclusion of the L1 -theory is clearly upgraded. On the other hand we are not able to establish a complete statement on the Hausdorff localization, i.e., that there always exists i in DN such that − (i + Ka(1−ν)(N/K)d ) ⊂ CN ⊂ (i + Ka(1+ν)(N/K)d ) . − + This would imply that the interface between CN and CN is always localized close to the boundary of the Wulff shape. This limitation is due to our method of proof: we are able to prove that large protuberances of the interface are not statistically favorable and therefore can be chopped. However, the volume constraint VN,a prevents us to control the percolation of the − phase inside the + phase because erasing a filament of − blocks might be in conflict with the volume constraint.
3.2
Spectral gap
We study the relaxation of the Glauber dynamics for the Ising model in a finite domain with a boundary magnetic field. The metastable behavior of the dynamics will be related to the equilibrium wetting phenomenon which occurs for a certain range of the magnetic field. The evolution of the system is given by the Glauber dynamics. The Dirichlet form associated to the dynamics is h x 2 , ∀f ∈ L2 (µh EN (f, f ) = µh N ), N |f (σ ) − f (σ)| x∈DN
where σ x is the spin configuration deduced from σ ∈ {±1}DN by flipping the spin at site x. The reader is referred to the lecture notes by Martinelli [Ma] and Guionnet, Zegarlinski [GZ] for a precise definition and related results on the Glauber dynamics. In the phase transition regime, the two phases segregate and the relaxation of the system is related to the slow motion of the interfaces. A convenient parameter to capture the signature of this slowing down is the spectral gap of the dynamics defined as follows SG(N, h) = inf f
h EN (f, f ) 2 . h µN f − µh N (f )
(3.25)
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We first consider the case where a positive magnetic field h = (0, . . . , 0, hd , 0, . . . , 0) is applied only on the face {i ∈ DN : id = 1} of the cube. Theorem 3.4 Let β ∈ B is such that both the strict convexity Assumption (SC) and the mixing Assumption (MP) hold. Then, lim inf N →∞
1 log SG(N, h) − τ (ed ) + ∆hd , N d−1
(3.26)
where the wall free energy corresponding to the field hd is denoted by ∆hd . For general fields h = (h1 , . . . , h2d ) with non-negative components, we introduce the functional h
h
G (m) = F (m) −
2d
∆hi ,
(3.27)
i=1
where F h was introduced in (2.11). We get Theorem 3.5 For any β ∈ B, the following asymptotic hold lim sup N →∞
1 log SG(N, h) − sup G h (m) . N d−1 ∗ ∗ m∈]−m ,m [
(3.28)
Remark 3.1 Notice that the statement of Theorem 3.5 does not depend on Assumptions (SG) or (MP). In [Ma] only the free boundary conditions have been considered, but, in view of the results of [PV], the proof pertains to the case of boundary magnetic fields in the partially wetting regime. In dimension 2, Pisztora’s coarse graining is not valid, but an alternative coarse graining for which the L1 -approach holds has been devised in [BoMa]. In particular, the proof of Theorem 3.5 goes through in two dimensions. Furthermore, since in 2 dimensions the Wulff shape is always strictly convex and the mixing property (MP) is known to be valid up to the critical temperature, the conclusion of Theorem 3.4 is also valid in dimension 2 for any β > βc . The functional G h should be interpreted as an energy landscape parametrized by the averaged magnetization. The time for a configuration starting in the − phase to relax to the + phase provides an estimation of the spectral gap. This explains why the supremum is taken over the values of m in ] − m∗ , m∗ [. The supremum of G h is related to the energy of the bottleneck and Theorem 3.5 asserts that if it is positive then the system has a metastable behavior and evolves extremely slowly. In this case the system has time to equilibrate and equilibrium parameters should be relevant as well to describe the influence of the boundary field on the dynamics. We expect that inequality (3.28) is, in fact, an equality. For appropriate choices of h, an explicit upper bound can be obtained. In particular for nearest neighbor Ising model in two dimensions and h = (0, h2 , 0, 0), one can check that the RHS
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of (3.26) and the LHS of (3.28) coincide. In general, estimating the supremum of G h boils down to solve a difficult variational problem involving subtle boundary effects. For general values of h = (h1 , . . . , h2d ), this seems to be out reach for the moment. Moreover the situation is far from being understood even in the case of a cube with free boundary conditions: in dimension d 3, the solutions of the isoperimetric problem (2.11) are not known for general volume constraints. In the particular case of isotropic surface tension and for a volume constraint which is the half of the volume of the cube, it was only recently proven by Barthe and Maurey [BaMa] that the solution is the half cube. We refer the reader to the review by Ros [Ro] for further discussions on this issue. This explains why, contrary to the two-dimensional case, we cannot derive precise asymptotics of the logarithm of the spectral gap. We turn now to a more physical interpretation of our results. The behavior of the dynamics is very sensitive to the boundary conditions. In dimension two, it was establish in [CGMS, Ma] that for free boundary conditions or, equivalently, in the case of zero boundary magnetic fields h ≡ 0, log SG(N, h) scales like −N τ (ed ). Instead, when at least one side of the square has all + boundary conditions and the other sides free boundary conditions (i.e., h = (0, . . . , 0, 1)), then for any ε > 0 and N large enough log SG(N, h) − N 1/2+ε ( −N τ (ed )) . In this case the spectral gap is conjectured to decay polynomially fast. An appealing interpretation of this result would be to relate the dynamics of the Ising model to an effective model evolving in a one well potential (+ bc) or a two well potential (free bc): the positive magnetic field h enforces a unique ground state whereas the transitions between two symmetric wells on the energy landscape in the case of the free boundary conditions have to cross the saddle point whose height scales like the logarithm of the inverse spectral gap. Following the seminal work of Martinelli, various other types of boundary conditions have been investigated to understand better the crossover between the two regimes. Alexander [A] showed that small (at least logarithmic) modifications of the boundary conditions at the corners of a two-dimensional cube leads to drastic changes in the scaling of the spectral gap. Alexander and Yoshida [AY] investigated the influence of an alteration of the + boundary conditions by an arbitrary small density of spins. Roughly speaking, they showed that in two dimensions if the boundary conditions have an average magnetization less than 1, there exists some inverse temperature β0 large enough above which the dynamics exhibits a metastable phase. Our result was originally motivated by [AY]; the magnetic field h < 1 can be interpreted as an effective boundary condition after averaging the spins. For simplicity, let us focus on the case h = (0, . . . , 0, hd , 0, . . . , 0). Extrapolating the results of [AY] to this setting, one can state that in two dimensions and for any hd in [0, 1[, there exists β large enough such that the spectral gap decays
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exponentially fast. Theorems 3.4 and 3.5 will enable us to interpret these results in a more qualitative way. In order to do so, we first recall some statements on the wetting transition. It was derived in [FP2] that ∆hd ∈ [−τ (ed ), τ (ed )]. In fact there exists a critical value hc 1 such that if hd < hc then ∆hd < τ (ed ). The critical value hc characterizes the influence of the boundary field on the thermodynamic properties of the Gibbs measure. More precisely, one should also distinguish the partial drying regime (0 hd < hc ) from the partial wetting regime (0 hd > −hc ). We refer to Pfister, Velenik [PV] or to [BIV1] for further discussions on the equilibrium issues. The previous Theorem shows that hc is also related to the metastable behavior of the system and thus it also plays the role of a critical value in the dynamical setting. Nevertheless, for any hd > 0, the Gibbs measure is unique in the thermodynamic limit. This confirms the fact that the dynamical properties cannot be deduced simply from the bulk properties, but that the metastability is related to surface properties (the picture of the effective magnetization evolving in a one well potential was too simplistic).
4 Hausdorff localization: Proof of Theorem 3.2 4.1
FK phase labels
We use the notation introduced in Subsections 2.2 and 2.3. Define the following dependent percolation process on DN,K : 1, if B2K (i) is FK good and C ∗ (B2K (i)) is connected to ∂ ext DN FK (i) = −1, if B2K (i) is FK good and C ∗ (B2K (i)) is not connected XN,K to ∂ ext DN 0, if B2K (i) is FK bad. (4.29) We recall the choice K = N a for some a ∈]0, 1/d[. Exactly as in [BIV1] the stability assumption implies: Lemma 4.1 For every α > 0 there exists a positive constant c4 = c4 (α), such that Φ± N
FK XN,K (i)
i∈DN,K
(1 − α)
N K
d
d−1 exp −c4 N , K d−1
for all N large enough. Recall that the total number of mesoscopic boxes in DN is
Nd Kd .
(4.30)
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Logic of the proof
Back to Theorem 3.2 we shall use Lemma 4.1 in the regime 0 < α ν 1. We argue that on the event
ξ ∈ ΞN,± :
½{XN,K FK (i) =1} (ξ) α
i∈DN,K
N K
d
(4.31)
νN -long fingers of the minority phase are improbable in the sense that one is always able to find a horizontal layer where such finger can be amputated at a substantial energetic gain. The argument is just a careful computation along the lines of the minimal section method introduced in [BBBP].
4.3
Fingers and finger labels
The excitations of the interface on the coarse grained level will be named fingers. We stress the fact that this terminology does not refer only to the low-dimensional excitations. Given a configuration ξ ∈ ΞN,± , let us define the associated finger FN,K ⊂ DN,K as FK FN,K = i ∈ DN,K : XN,K (i) = 0 or − 1 . For every mesoscopic layer l = 1, 2, . . . define the finger label 1 l = #{i : id = (l − )K, fN,K 2
i ∈ FN,K }.
With ν fixed set R = [νN/K] + 1. To simplify the notation we shall assume that R = 2r . Given a collection 1 R fN,K = fN,K , . . . , fN,K of strictly positive finger labels we shall use F (fN,K ) to denote the set of all those l percolation configurations ξ ∈ ΞN,± which are compatible with fN,K in each of the mesoscopic layers l = 1, 2, . . . , R. Evidently, {ξ ∈ ΞN,± : g ↔ {i : id > νN }} ⊆
F [fN,K ],
(4.32)
fN,K
where the disjoint union is, of course, over all strictly positive finger labels. A large fluctuation of the interface occurs when bad blocks percolate on a distance larger than νN .
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The target estimate on the probability of F(fN,K )
Our proof of Theorem 3.2 relies on the following uniform upper bound: Theorem 4.1 There exists c5 = c5 (β) > 0 such that Φ± e−c5 νN N F [fN,K ]
(4.33)
uniformly in strictly positive finger labels fN,K and in N sufficiently large. Since the total number of different finger labels (recall the choice of scales K = N a ) is bounded above as (d−1) R 2N exp c5 (d)νN 1−a log N , (4.34) K Theorem 3.2 instantly follows.
4.5
Splitting of LN,± with respect to lth mesoscopic layer
Given a mesoscopic layer l = 1, 2, . . . , R define the following mesoscopic sets: 1 ∆ H−,l N,K = {i ∈ DN,K : id (l − )K}, 2 1 ∆ l HN,K = {i ∈ DN,K : id = (l − )K}, 2 1 ∆ +,l HN,K = {i ∈ DN,K : id > (l − )K}. 2 −,l l Their microscopic counterparts are denoted by HN,K , HN,K and, respectively, +,l HN,K , where −,l = HN,K
l B2K (i), HN,K =
i∈H−,l N,K
+,l B2K (i) and HN,K =
B2K (i).
i∈H+,l N,K
i∈HlN,K
Accordingly, we split the set of all edges LdN,± into the disjoint union +,l −,l LdN,± = EN,K EN,K , where
∆
−,l = EN,K
−,l (i , j) : either i or j belong to HN,K ∪g ,
+,l −,l = LdN,± \ EN,K . and EN,K The induced notation for the splitting of the percolation configurations ξ ∈ l l ΞN,± is ξ = ξ− ∨ ξ+ with ∆
−,l
EN,K l ∈ Ξ−,l ξ− N,K = {0, 1}
∆
+,l
EN,K l and ξ+ ∈ Ξ+,l . N,K = {0, 1}
−,l l ∈ Ξ−,l Given a percolation configuration ξ− N,K , we use CN,K (g) to denote the con-
−,l nected cluster of the ghost site g inside HN,K .
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The event Al (fN,K )
Let fN,K be a collection of strictly positive finger labels. For each mesoscopic layer l, we define the associated event Al (fN,K ) ⊂ Ξ−,l N,K : % (i) g ↔ ∂ ext DN inside H−,l N,K −,l l ξ− ∈ ΞN,K Al (fN,K ) = (ii) #{∂ int H−,l ∩ C −,l (g)} f l (2K)d−1 t N,K N,K N,K where
∆
−,l −,l = {u ∈ HN,K : ud = (l + 1/2)K} ∂tint HN,K
−,l . is the top layer of the box HN,K l l l Notice that if ξ = ξ− ∨ ξ+ ∈ F(fN,K ) then, necessarily, ξ− ∈ Al (fN,K ). −,l −,l Indeed, the connected cluster CN,K (g) is capable of hitting the top layer ∂tint HN,K only from within the set B2K (i). i∈FN,K ∩HlN,K
4.7
Surgery in lth mesoscopic layer
l ¯l Given a percolation configuration ξ− ∈ Ξ−,l N,K define a new configuration ξ− as −,l −,l if b = (u, v) with u ∈ ∂tint HN,K ∩ CN,K (g) 0, l and v = u + 1, ¯ d d ξ− (b) = (4.35) l otherwise. ξ− (b), l l → ξ¯− cuts off all the vertical bonds which emerge In other words, the surgery ξ− −,l int −,l l from those points of the top layer ∂t HN,K which are ξ− -connected (inside HN,K ) to the ghost site g. l l → ξ¯− leaves Al (fN,K ) For every l = 1, 2, . . . , R and for each fN,K the map ξ− l fN,K (2K)d−1 invariant and is at most 2 to 1 on the latter set. Furthermore, given fN,K , a mesoscopic layer l and a percolation configuration l l ∈ Al (fN,K ), the concatenation ξ = ξ¯− ∨ η belongs to JN = {ξ ∈ ΞN,± : g ↔ ξ− +,l ∂DN } for every η ∈ ΞN,K . In particular, given any event B ⊂ Ξ+,l N,K and any l ξ− ∈ Al (fN,K ), (4.36) Φ± B ξ¯l max Φπ+,l ( B ) , N
−
π
EN,K
where ΦπE +,l is the FK-measure on Ξ+,l N,K with wired boundary conditions on the N,K
+,l and, respectively, with π boundary condilateral sides and on the top of HN,K +,l tions on the bottom side of HN,K . Thus, the surgery operation (4.35) effectively +,l l on the edges EN,K of the upper box decouples the percolation configuration ξ+ +,l HN,K from the global constraint JN .
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Upper bound in one mesoscopic layer
Given a mesoscopic layer Hm N,K and a percolation configuration ξ ∈ ΞN,± , let us m on Hm define the site percolation process YN,K N,K as: 1, m YN,K (i) = −1 , 0,
m if B2K (i) is good and C ∗ ↔ ∂ ext DN in HN,K , m , if B2K (i) is good and C ∗ ↔ ∂ ext DN in HN,K
if B2K (i) is bad ,
where C ∗ denotes the crossing cluster of B2K (i) (see Subsection 2.2). m FK ) dominates the restriction of (XN,K ) Clearly, the percolation process (YN,K m to HN,K m (i) =1} FK (i) =1} . ½{YN,K ½{XN,K (4.37) i∈Hm N,K
i∈Hm N,K
Lemma 4.2 There exists a positive constant c6 > 0, such that d−2 ½{Y m (i) =1} f exp −c6 min f K, f d−1 K d−1 , max ΦπE m π
N,K
N,K
i∈Hm N,K
(4.38) uniformly in positive integers f and N sufficiently large. A section of a finger on the layer m leads to an exponential decay of order d−2 f K if it contains at least f2 blocks with label 0 or of order f d−1 K d−1 if it contains at least f2 blocks with label −1. The claim of the lemma follows from (2.4) via a straightforward modification of the argument presented in the Appendix A in [BIV1]. We stress the fact that the scaling K = N a will be used only in the derivation of (4.38).
4.9
Upper bound on F(fN,K )
We shall consider only even mesoscopic layers m = 2, 4, . . . , R. For every such m and every positive collection fN,K of finger labels, define the event Bm , which m depends only on the percolation configuration inside the slab HN,K : Bm = Bm (fN,K ) = {ξ :
m m (i) =1} f ½{YN,K N,K }.
i∈Hm N,K
Given an even mesoscopic layer l = 2, 4, . . . , R, F (fN,K ) ⊆
R
m=l+2
Bm (fN,K )
Al (fN,K ).
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We will compute the balance between the energetic cost of chopping a finger at the section l and the gain of erasing this finger above the level l. Consequently, the estimates (4.36) and (4.38) imply: R
± ± F (fN,K ) Φ Φ Bm (fN,K ) Al (fN,K ) N
N
=
Φ± N
l ∈A ξ− l
e
m=l+2 R
l Bm (fN,K ) ; ξ−
m=l+2
l c7 fN,K K d−1
Φ± N
l : ξ l ∈A ξ¯− l −
e
l c7 fN,K K d−1
max ΦπE +,l π N,K
exp
l c7 fN,K K d−1
− c6
R
l Bm (fN,K ) ; ξ¯−
m=l+2 R
Bm (fN,K )
m=l+2
d−2 m m d−1 min fN,K K, (fN,K ) d−1 K .
R m=l+2
(4.39)
4.10
Proof of Theorem 4.1
We claim that there exists a positive constant c8 > 0, such that % R d−2 l d−1 m m d−1 d−1 min − c6 min fN,K K, (fN,K ) K c7 fN,K K − c8 N , 2 l R/2
m=l+2
(4.40) uniformly over all strictly positive collections fN,K of finger labels which comply with the volume bound (4.31), which we rewrite in terms of fN,K as: R
l fN,K α
l=2
N K
d .
(4.41)
In view of (4.30) a substitution of (4.40) to (4.39) yields the target bound (4.33). Let us turn to the proof of (4.40). The percolation of bad blocks between the layers R/2 and R produces an energy cost of an order at least N . Therefore, it is enough to examine the layers below R/2 and to prove that R/2 d−2 c 6 l m m min K d−1 − min fN,K K, (fN,K ) d−1 K d−1 fN,K 0. c8 2 l R/2 m=l+2
(4.42)
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Set n = R/8. Suppose that (4.42) is violated then either there exists a sequence of even numbers 2 l1 < l2 < · · · < ln R/2 such that for every i = 1, . . . , n − 1; li K d−1 fN,K >
n c6 lj KfN,K , c7 j=i+1
(4.43)
or there exists a sequence of even numbers 2 m1 < m2 < · · · < mn R/2 such that for every i = 1, . . . , n − 1; li fN,K >
n d−2 c6 mj d−1 fN,K + 1. c7 j=i+1
(4.44)
The constants c6 and c7 do not depend on the values of ν in R = [νN/K] + 1 and α in (4.41). We claim that under an appropriate choice of 0 < ν α 1 (see (4.49) below) both (4.43) and (4.44) contradict the volume constraint in (4.41). The latter is a consequence of the following two elementary numeric lemmas: Lemma 4.3 Fix χ > 0. Assume that a sequence of positive integers a1 , . . . , an satisfies: n ai χ aj (4.45) j=i+1
for every i = 1, . . . , n − 1. Then n
ai (1 + χ)n−1 −
i=1
1−χ . χ
(4.46)
Lemma 4.4 Fix χ > 0. Set γ = (d − 2)/(d − 1). Assume that a sequence of positive integers a1 , . . . , an satisfies: ai χ
n
aγj + 1
(4.47)
j=i+1
for every i = 1, . . . , n − 1. Then there exists c9 = c(χ, d) > 0, such that n
a i c9 n d .
(4.48)
i=1
The inequality (4.43) corresponds to the choice of χ = K 2−d c6 /c7 in (4.45) and, consequently, the estimate (4.46) yields in this case: R/2 l=2
N l fN,K exp c10 ν d−1 , K
which, by the choice of K = N a ; a < 1/d, is clearly incompatible with (4.41).
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On the other hand, the inequality (4.44) corresponds to the choice of χ = c6 /c8 in (4.47) and, consequently, the estimate (4.48) yields: R/2
l fN,K
c11
l=2
νN K
d .
Thus, the volume constraint (4.41) is violated whenever α νd > , c12
(4.49)
which defines the appropriated choice of α and ν.
4.11
Proof of Lemma 4.3
Consider the sequence a ¯0 , . . . , a ¯n−1 given by: ¯i = χ a ¯0 = 1 and a
i−1
a ¯j
for i > 1.
(4.50)
j=1
The system (4.50) is exactly solvable: a ¯i = χ(1 + χ)i−1
for i 1.
¯n−i for every i = 1, . . . , n. A look at the conditions of Lemma 4.3 reveals that ai a Hence (4.46).
4.12
Proof of Lemma 4.4
For some c > 0, consider the sequence a ¯0 , . . . , a ¯n−1 given by a ¯i = (1 + ic)1/(1−γ) = (1 + ic)d−1 . By convexity: d−2
¯i (d − 1)c (1 + (i + 1)c) a ¯i+1 − a
d−2
(d − 1)c(1 + c)d−2 (1 + ic)
= (d − 1)c(1 + c)d−2 a ¯γi .
Let us choose c = c(χ, d) according to (d − 1)c(1 + c)d−2 = χ. Then, a ¯i χ
i−1
a ¯γj + 1,
j=0
for all j = 1, . . . , n − 1. Comparing with (4.47) we readily infer that (1 + ic)d−1 = a ¯i an−i and, consequently, (4.48) follows with, for example, c9 =
1 (1 + c(n − 1))d . min dc n>1 nd
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5 Exponential mixing The analysis of the spectral gap will rely on two properties of the equilibrium measure: the localization of the interface and the mixing of the system to a pure phase away from the interface. These estimates will be used in Subsection 6.1 in a specific framework, slightly different from the one of Theorem 3.2. In this section we establish the technical estimates which will be necessary for the derivation of the lower bound of the spectral gap. First, we formulate the localization property in the appropriate setting (see Subsection 6.1). Then the mixing property (MP) of the FK measure (see Definition 2.2) is combined with the localization in order to derive a control on the Gibbs measure.
5.1
A new setting
Let δ = 2−m be the relative height of the new domain ∆
DδN = {i ∈ Ld : 1 ik N, k = 1, . . . , d − 1,
0 id δN } .
(5.51)
We use ∂ ext DδN to denote the exterior of DδN in Ld . This boundary & boundary ext δ ext δ ext δ consists of two parts: ∂ DN = ∂s DN ∂t DN , where t stands for top and s stands for sides. The bottom face of DδN is denoted by ∂bint DδN . We are going to established the Hausdorff stability of the interface when a negative (respectively positive) magnetic field is applied on ∂bint DδN (resp. ∂text DδN ) and a small positive field ε > 0 is applied on the faces of ∂sext DδN . This amounts δ to consider the Hamiltonian on {±1}DN which is given by ∆
HN,ε (σ) = −
1 2
+
σi σj −
σi − ε
(i , j) i∈DδN ,j∈∂text DδN
(i , j)⊂DδN
(i , j) i∈DδN ,j∈∂sext DδN
σi (5.52)
σi ,
i∈∂bint DδN
where the first three sums are over (subsets of ) nearest neighbor bonds (i , j). the correFollowing notation introduced in Subsection 2.3, we denote by µ+,ε,− N sponding Gibbs measure and by Φ± N,ε the FK measure. The counterpart of Theorem 3.2 in this context relies also on the strict convexity assumption of the surface tension (SC). Theorem 5.1 Assume (SC) and let δ > 0, ε > 0 be fixed. For any β ∈ B and any ν > 0 there exists c1 = c1 (ν) > 0, such that uniformly in N large enough, −c1 (ν)N Φ± . N,ε (g ↔ {i : id > νδN }) e
The proof is similar to the one of Theorem 3.2.
(5.53)
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Screening
Combining the localization of the interface (Theorem 5.1) and the mixing assumption (MP), we obtain a screening property for the Gibbs measure. that for any function g with Proposition 5.1 Fix β ∈ B1 . Then there is cβ > 0 such support included in SN,δ = i ∈ DδN ; id = 12 δN , the following holds uniformly over the boundary conditions in ∂bint DδN int δ +,ε +,ε,η ∀η ∈ {±1}∂b DN , µN (g) − µN (g) 2N d−1 g∞ exp(−cβ δN ) , (5.54) where µ+,ε,η is the Gibbs measure with η boundary conditions on ∂bint DδN , + boundN ary conditions on ∂text DδN and ε > 0 boundary conditions elsewhere. µ+,ε stands N for the measure with {ηi = 1}i . If ηi = 0 for all i, the FK counterpart of the Ising measure will be denoted by In the proof of the proposition, we are going to show that assumption (MP) implies that for any β in B1 , the probabilities that a site in SN,δ is connected to w/f the wired boundary conditions under Φw N,ε or ΦN,ε are almost identical, i.e., there exists c = c(β) > 0 such that w w/f ∀j ∈ SN,δ , ΦN,ε (j ↔ ∂DδN ) − ΦN,ε (j ↔ ∂ ext DδN ) exp(−cδN ) . (5.55)
w/f ΦN,ε .
We stress the fact that in general the screening property (5.54) for the Ising measure is stronger than (5.55) since for β ∈ B1 a phase transition occurs for the Gibbs measure instead the FK measure is unique in the thermodynamic limit. In particular, if the + boundary conditions on ∂text DδN are replaced by the magnetic field ε then (5.54) does not hold for small values of ε instead (5.55) remains valid uniformly in ε (at least for large enough β). The behavior wrt a magnetic field will be investigated in details in Subsection 6.1. Proof. By definition of the total variation distance +,ε +,ε,η µ+,ε ˜+,ε,η tv g∞ , µN (g) − µN (g) ˜ N −µ N where µ ˜ denotes the projection of the measure on SN,δ . Furthermore, the total variation can be rewritten as +,ε,η ˜ µ+,ε − µ ˜ = inf dΠ(σ, σ ) 1σ =σ , tv N N Π
2 where the infimum is taken over the joint probability measure Π on {±1}SN,δ with marginals µ ˜+,ε and µ ˜+,ε,η . As the measures are ordered wrt the boundary N N
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conditions, there is a coupling Π which preserves this order. dΠ(σ, σ ) ˜ µ+,ε ˜+,ε,η tv inf |σj − σj | , N −µ N Π j∈SN,δ +,ε +,ε µN (σj ) − µ+,ε,η (σj ) µN (σj ) − µ+,ε,− (σj ) . N N j∈SN,δ
(5.56)
j∈SN,δ
In terms of FK representation, this leads to ext δ ˜+,ε,η tv Φw DN ∪ ∂bint DδN ) ˜ µ+,ε N,ε (j ↔ ∂ N −µ N j∈SN,δ ext δ D N ) + Φ± − Φ± N,ε (j ↔ ∂ N,ε (j ↔ g) ,
where ∂ ext DδN ∪ ∂bint DδN is simply the boundary ∂DδN of DδN . Let us fix ν = 14 . δ ˜ µ+,ε ˜+,ε,η tv Φw N,ε (j ↔ ∂DN ) N −µ N j∈SN,δ
− j ↔ ∂ ext DδN ; g ↔ {i : id 1 d−1 ± +N ΦN,ε g ↔ {i : id δN } 4 Φ± N,ε
1 δN } 4
(5.57)
.
We are going to use now the fact that the interface is localized. Conditioning wrt the bond configuration ξ below {i : id = 14 δN }, we get Φ± N,ε
1 ext δ j ↔ ∂ DN ; g ↔ {i : id δN } 4 Φw g ↔ {i : id 1 δN }, JN N,ε w,ξ 4 ext ˜ δ inf ΦD˜ δ ,ε j ↔ ∂ DN , ξ Φw N N,ε (JN )
1 δ ˜δ where Φw,ξ ˜ δ ,ε denotes the FK measure on DN = DN ∩ {i : id 4 δN } with D N ˜ δ . As a consequence of Theorem 5.1, boundary conditions ξ on the lower face of D N we get 1 ± ext δ ΦN,ε j ↔ ∂ DN ; g ↔ {i : id δN } 4 w/f ˜ δ , (5.58) (1 − exp(−c1 δN )) ΦD˜ δ ,ε j ↔ ∂ ext D N N
w/f where ΦD˜ δ ,ε denotes the FK measure N ˜ δ and wired otherwise. face of D N
with free boundary conditions on the bottom
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Combining (5.53), (5.57) and (5.58) we finally derive w/f δ ext ˜ δ ˜ µ+,ε ˜+,ε,η tv Φw DN ) N,ε (j ↔ ∂DN ) − ΦD ˜ δ ,ε (j ↔ ∂ N −µ N N
j∈SN,δ
+N d−1 exp(−c2 δN ) . By the FKG property of the random cluster measures, δ w ˜δ Φw ˜ δ ,ε (j ↔ ∂ DN ) . N,ε (j ↔ ∂DN ) ΦD N
At this stage, it will be enough to apply the strong mixing inequality (5.55) to conclude. Finally, it remains to derive (5.55) from the mixing property (MP). First of all, one has to modify the boundary conditions and to replace ε by 0. This rests on the GHS ferromagnetic inequalities which are available only for the Ising measure (see eg. [El]). Using the correspondence between the Ising and the FK measure, we define ∀ε > 0,
w/f
δ ext δ (σj ) − µ+,ε,0 (σj ) = Φw DN ) , Ψ(ε) = µ+,ε,+ N,ε (j ↔ ∂DN ) − ΦN,ε (j ↔ ∂ Dδ Dδ N
N
where µ+,ε,+ (resp. µ+,ε,0 (σj )) denotes the Gibbs measure on the set DδN with Dδ Dδ N
N
boundary conditions + on the top face ∂text DδN , ε on the sides ∂sext DδN and + on the bottom face ∂bint DδN (resp. 0 on the bottom face). By FKG inequality, the function ε → Ψ(ε) is non-negative and we are going to check that it is non-increasing. Deriving wrt the parameter ε, we get +,ε,+ Ψ (ε) = µDδ (σj ; σi ) − µ+,ε,0 (σj ; σi ) , Dδ i
N
N
where the sum is restricted to the sites i which interact with the boundary field on the sides ∂sext DδN of the box. The GHS inequality ensures that the two point truncated correlation function is a decreasing function of the field (for non-negative fields), i.e., Ψ (ε) 0. Thus, the derivation of (5.55) can be reduce to the case ε = 0 and it is enough to prove that w/f
ext δ int δ ext δ Ψ(0) = Φw N,ε=0 (j ↔ ∂t DN ∪ ∂b DN ) − ΦN,ε=0 (j ↔ ∂t DN ) exp(−cδN ) . (5.59) As ε = 0, the magnetization of σj is simply related to the FK connection of j to the top (and possibly to the bottom) face of DδN . The mixing property (MP) enables us to compare only the probability of events which are locally supported, this is not the case in the previous inequality, thus we need more work to reduce to events with supports independent of N .
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Let BδN/10 (j) be the box centered at j, then Pisztora coarse graining implies that if j is connected to the boundary of BδN/10 (j) then with probability at least 1 − exp(−cδN ) the site j is connected also to the top face of DδN . Define the set of bond configurations A = ξ, j ↔ ∂BδN/10 (j) . Then ext δ int δ Φw N,ε=0 (j ↔ ∂t DN ∪ ∂b DN ) −
=
w/f
ΦN,ε=0 (j ↔ ∂text DδN )
w/f Φw N,ε=0 (A) − ΦN,ε=0 (A) + o exp(−cδN ) .
The previous FK measures are ordered (in the FKG sense). We can consider the joint measure νN (ξ, ξ ) such that the first marginal is Φw N,ε=0 , the second marginal w/f
is ΦN,ε=0 and the measure is supported by the configurations ξ ξ . By construction w/f
Φw N,ε=0 (A) − ΦN,ε=0 (A) = νN 1A (ξ) − 1A (ξ )
=
νN ξb = ξb
b∈BδN/10 (j)
νN ξb − ξb =
b∈BδN/10 (j)
w/f
Φw N,ε=0 (ξb ) − ΦN,ε=0 (ξb ) .
b∈BδN/10 (j)
For any bound b, the probability on the LHS can be estimated thanks to the mixing property (see Definition 2.2) w/f
d Φw N,ε=0 (A) − ΦN,ε=0 (A) N exp(−cδN ) ,
for some c > 0. This completes the derivation of (5.55).
6 Spectral gap estimates 6.1
Lower bound
We turn now to the derivation of the lower bound (3.26) on the spectral gap. The proof follows closely the strategy developed by Martinelli and coauthors [CGMS, Ma] in the two-dimensional case. We will briefly recall the main steps of the proof as they are exposed in the Chapter 6 of [Ma] and focus only on the changes. This comprises a more careful analysis of the boundary effects to take into account the boundary surface tension and a repeated use of Proposition 5.1, whose proof is based on the localization of the interface. Step 1. The first step is to reduce to a block dynamics in order to estimate the spectrum of the single site Glauber dynamics in DN = {1, . . . , N }d .
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For a given δ > 0, we consider the following covering of DN by the overlapping slabs Ri =
i i [δN ] xd < ( + 1)[δN ] . 2 2
x ∈ DN ,
The total number of sets {Ri }i is independent of N and denoted by L = L(δ). The sets Ri are simply shifts of the set DδN introduced in (5.51). The block dynamics is defined in terms of the generator h,σ h µ ∀f ∈ L2 (µh ), L f (σ) = (f ) − f (σ) . N N,δ Ri i
We recall that the single site dynamics on each Ri has a spectral gap larger than exp(−cβ δN d−1 ) (for some cβ > 0). Therefore, according to Proposition 3.4 of [Ma], the following bound holds for some Cβ > 0 d−1 SG(Lh ) SG(Lh N ) exp(−Cβ δN N,δ ) ,
(6.60)
where SG(Lh N,δ ) denotes the spectral gap of the block dynamics. Step 2. Thanks to (6.60), it is enough to derive Lemma 6.1 Let h = (0, . . . , 0, hd , 0, . . . , 0). Then, for any δ > 0, 1
lim inf
N →∞ N d−1
log SG(Lh ed ) + ∆hd . N,δ ) − τ (
The proof boils down to check that the semi-group associated to Lh N,δ is a contraction for some time T , i.e., that there is rN > 0 such that for all N large enough sup E f (σTη ) (1 − rN )f ∞ , (6.61) η
for any f such that µh N (f ) = 0. In our context rN will be such that lim
1
N →∞ N d−1
log rN = −τ (ed ) + ∆hd .
(6.62)
Iterating (6.61), we get for any f ∀t 0,
' ' 'E f (σtη ) − µh (f )' N
∞
( ) t f ∞ exp −rN . T
(6.63)
This L∞ contraction and (6.62) imply Lemma 6.1. We turn now to the derivation of (6.61). For technical reasons, it will be convenient to replace the free boundary conditions by a small coupling ε > 0 and
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to consider the evolution of the Glauber dynamics associated to the generator which takes into account the new boundary conditions. Let us denote by µh,ε N the corresponding Gibbs measure and by Lh,ε N,δ the new generator. The effect of ε is to select the + phase. The two block dynamics are comparable by using the Radon Nykodim derivative; thus as ε vanishes, we recover the result for the original dynamics lim inf N →∞
1 N d−1
log SG(Lh N,δ ) = lim lim inf ε→0 N →∞
1 N d−1
log SG(Lh,ε N,δ ) .
Fix a function f such that µh,ε N (f ) = 0. Let f0 be the image of f at time t = 1 if only the block R0 has been updated at the random time t0 η f0 (η) = E f (σt=1 ) ; 0 < t0 1 < t1 = pµh,ε,η R0 (f ) , where p = L1 P(t0 < 1 t1 ) is the probability that R0 is the only update. Furdenotes the Gibbs measure on R0 with boundary conditions hd thermore, µh,ε,η R0 on the bottom face {i ∈ DN : id = 0} of R0 , ε on the sides and η at the top face. Notice that p depends on L but not on N . By construction f0 satisfies 3 important properties: 1. f0 depends only on the spins in DN \ R0 . 2. f0 ∞ pf ∞ . h,ε 3. µh,ε N (f0 ) = pµN (f ) = 0. Using the Markov property at time t = 1 (see [Ma] page 162), we get η η sup E f (σt=2 ) (1 − p)f ∞ + sup E f0 (σt=1 ) . η
η
Thus (6.61) will follow if one can derive that for any ψ which does not depend on the spins in R0 and has zero mean under µh,ε N , η sup E ψ(σt=1 ) (1 − rN,δ )ψ∞ , (6.64) η
where rN,δ satisfies the asymptotic similar to (6.62) lim lim
δ→0 N →∞
1 log rN,δ = −τ (ed ) + ∆hd . N d−1
Replacing ψ by f0 , we complete Lemma 6.1. Step 3. We turn now to the derivation of (6.64) for any function ψ which does not depend on the spins in R0 . We consider a specific evolution up to time t = 1 with exactly L + 1 updates occurring at the random times (ti )0 i L (see [Ma] p. 159). During the time interval [0, 1], the blocks R0 , R1 , . . . , RL are successively updated at times (ti )0 i L
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such that 0 < t0 t1 . . . tL 1 < tL+1 . The k th -update amounts to modify the spin configuration in the slab Rk , thus we introduce the following mappings on the space of configurations σk (x), x ∈ Rk , σk Tk (σ) = σ(x), x ∈ Rk , To quantify the successive updates, one has to bound η ηL−1 sup dµηR0 (σ0 ) . . . dµRj−1 (σ ) . . . dµ (σ ) ψ(η ) j L L , RL j η
where ηk = Tkσk ◦ · · ·◦ T0σ0 (η). In words, this means that at the j th update the configuration in σj is chosen wrt the Gibbs measure on Rj with boundary conditions equal to σj−1 in Rj−1 and η in Rj+1 . We define η η dµRj (σj ) . . . dµRL−1 (σL ) ψ(ηL ) , ∀j L, gj (η) = L σ
where this time ηk = Tkσk ◦ · · · ◦ Tj j (η) for k j. Thus it is enough to estimate int supη |µh,ε,η R0 (g1 )|, where the boundary conditions are hd on ∂b R0 , η on ∂t R0 and ε on the sides. The influence of the boundary condition η will be related to the stability property of the interface and, unlike [Ma], we resort to the FK representation. be the joint FK measure associated to µh,ε,η Let Ph,ε,η R0 R0 ξ,h,η Ph,ε,η (σ) Φh,w R0 (σ, ξ) = PR0 R0 ,ε (ξ | Cη ) .
The previous formula reads as follows. First a bond configuration is chosen wrt the conditional FK measure; the conditioning Cη imposed by the boundary conditions η is such that ξ cannot connect regions of the boundary with different signs. For a given bond configuration ξ, the spin configuration σ is obtained by a random coloring compatible with the bond configuration ξ and the boundary conditions. . The random coloring is chosen according to the measure PRξ,h,η 0 As ψ does not depend on the spins in R , the support of g1 is included in 0 SN,δ ∪ (R0 ∪ R1 )c , where SN,δ = i ∈ R0 ; id = 12 δN . We consider the event Aη which decouples the spins in SN,δ from the boundary conditions η outside R0 3 Aη = ξ {η = −1} ↔ {i : id δN } . 4 The domain R0 is the analog of DδN viewed upside down and {η = −1} = {i; −1} replaces g (see Section 5). For any η, we write h,w h,ε,η µh,ε,η g1 | Aη + Ph,ε,η g1 1Acη . R0 (g1 ) = ΦR0 ,ε (Aη | Cη ) PR0 R0
ηi =
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This leads to the following decomposition h,ε,η h,ε,η h,w +,ε,η µR0 (g1 ) ΦR0 ,ε (Aη | Cη ) PR0 g1 | Aη − µR0 (g1 ) +,ε,η h,w c +Φh,w R0 ,ε (Aη | Cη ) µR0 (g1 ) + ΦR0 ,ε (Aη | Cη )ψ∞ ,(6.65) denotes the Gibbs measure on R0 where the boundary magnetic field hd and µ+,ε,η R0 on ∂bint R0 has been replaced by +1. In order to complete the evaluation of (6.65), we have to derive the following inequalities: • A bound involving the surface tension d−1 (τ (ed ) − ∆hd + O(ε)) = rN,ε . (6.66) inf Φh,w R0 ,ε (Aη | Cη ) exp − N η
• A characterization of the screening d−1 g1 | Aη − µ+,ε,η exp(−cδN )ψ∞ . sup Ph,ε,η R0 R0 (g1 ) N
(6.67)
η
• A proof of the much “faster” relaxation of the dynamics in the + phase. This boils down to check that d−1 exp(−cδN )ψ∞ . (6.68) sup µ+,ε,η R0 (g1 ) N η
Combining the 3 previous estimates, there is c > 0 such that h,w d−1 c (g ) exp(−cδN )ψ∞ + Φh,w sup µh,ε,η 1 ΦR0 ,ε (Aη | Cη ) N R0 R0 ,ε (Aη | Cη )ψ∞ , η 1 − rN,ε (1 − N d−1 exp(−cδN )) ψ∞ . This concludes the proof of (6.64). 6.1.1 Derivation of inequality (6.66)
The event Aη ∩ Cη is supported by the set of bonds E∆ generated by ∆ = i ∈ R0 , id > 3δ i ∈ ∆ . Since Aη ∩ Cη is decreasing, we 4 N , i.e., E∆ = (i, j) ∈ E, have Φh,w R0 ,ε (Aη | Cη ) =
Φh,w R0 ,ε (Aη ∩ Cη ) Φh,w R0 ,ε (Cη )
Φw ∆,ε (Aη ∩ Cη ) Φh,w R0 ,ε (Cη )
.
In the spin language, it can be rewritten as Φw ∆,ε (Aη ∩ Cη ) Φh,w R0 ,ε (Cη )
h,ε,+ h,ε,+ +,ε,η +,ε,− ZR ZR Z∆ Z∆ 0 0 = +,ε,+ h,ε,η +,ε,+ h,ε,− , Z∆ Z∆ ZR0 ZR0
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+,ε,η * h,ε,η ZR0 is an increasing where we used in the last inequality that the ratio Z∆ function of η. As in, e.g., Lemma 2.1 and Lemma 2.2 in [BIV2] we, taking the thermodynamic limit, recover the surface tension and the surface energy (recall that δN is the height of the box R0 and that the magnetic filed ε is applied on the lateral sides only):
lim lim inf
δ→0 N →∞
1 inf log Φh,w ed ) + ∆hd . R0 ,ε (Aη | Cη ) − τ ( N d−1 η
(6.69)
This concludes (6.66). 6.1.2 Derivation of inequality (6.67) ˜ 0 is the counterpart of ˜ 0 be the set R0 ∩ {i : id 3 δN }. The domain R Let R 4 ˜ δ introduced in the proof of Proposition 5.1. An intermediate step the domain D N ˜ h,ε,η (·|Aη ) and is to estimate the total variation distance between the measures P R0 +,ε h,ε,η µ ˜R˜ which are the projections of the measures PR0 (·|Aη ) and µ+,ε ˜ 0 on the spin R 0 variables in the domain SN,δ . Before applying (5.56), we need to check that the ˜ h,ε,η (·|Aη ) is stochastically dominated by µ measure P ˜+,ε ˜ . R0 R 0
Let ψ be an increasing function supported by {±1}SN,δ ˜ h,ε,η (ψ|Aη ) = P R0
1 Φh,w R0 ,ε (Aη
| Cη )
PRξ,h,η (σ) Φh,w R0 ,ε (ξ | Cη )1Aη (ξ)ψ(σ) . 0
σ,ξ
˜ 0 . Conditioning Let us decompose ξ into (ξ , ξ ), where ξ is the restriction of ξ to R wrt ξ , we get ˜ h,ε,η (ψ|Aη ) = P R0
1
Φh,w R0 ,ε (Aη | Cη )
Φh,w R0 ,ε
1Aη
σ,ξ
PRξ,h,η (σ) 0
Φh,w R0 ,ε (ξ
| ξ )ψ(σ) Cη .
does not take into account the As ξ belongs to Aη , the coloring measure PRξ,h,η 0 constraint imposed by η. Thus one can write ξ,h,η PR0 (σ) Φh,w (ξ | ξ )ψ(σ) = mξ (d ω)µh,ε,ω (ψ) . ˜ R0 ,ε R σ,ξ
0
˜ 0 . As the RHS where mξ is a measure on the boundary conditions ω outside R of the previous inequality is always smaller than µ ˜+,ε the stochastic domination ˜0 R holds.
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Using the property that the measures are ordered, we have according to (5.56) +,ε ˜ h,ε,η µR˜ (σj ) − Ph,ε,η ˜ µ+,ε ˜ − PR0 (·|Aη )tv R0 (σj |Aη ) R 0
j∈SN,δ
0
h,w int ˜ ext ˜ Φw R0 | Aη ∩ Cη ) . ˜ 0 ,ε j ↔ ∂ R0 ∪ ∂b R0 − ΦR0 ,ε (j ↔ ∂ R
j∈SN,δ
By FKG inequality, this leads to ext ˜ ˜0 ˜ h,ε,η R0 ∪ ∂bint R ˜ µ+,ε Φw ˜ 0 ,ε j ↔ ∂ ˜ − PR0 (·|Aη )tv R R 0
j∈SN,δ h,w/f (j 0 ,ε
− ΦR˜
↔ ∂ ext R0 ) .
As β is in B1 , the strong mixing property implies that for some c > 0 d−1 ˜ h,ε,η ˜ µ+,ε exp(−cδN ) . ˜ − PR0 (·|Aη )tv N R 0
(6.70)
and By Proposition 5.1, the total variation distance between the measures µ ˜+,ε ˜ R µ ˜+,ε,η is exponentially small, thus (6.67) is proven. R0
0
6.1.3 Derivation of inequality (6.68) The proof is based on a repeated use of the screening property obtained in Proposition 5.1. +,ε,η +,ε,η +,ε,η µ+,ε,η R0 (g1 ) = µR0 (g1 ) − µR0 ∪R1 (g1 ) + µR0 ∪R1 (g1 ) .
As g1 is supported by SN,δ ∪ (R0 ∪ R1 )c , we can apply Proposition 5.1 to get +,ε,η +,ε,η µ (6.71) R0 (g1 ) − µR0 ∪R1 (g1 ) exp(−cδN )f ∞ . +,ε,η Since µ+,ε,η R0 ∪R1 (g1 ) = µR0 ∪R1 (g2 ), the previous argument can be iterated
µ+,ε,η R0 (g1 ) =
L−1
+,ε,η µ+,ε,η (g ) − µ (g ) + µ+,ε i+1 i+1 R0 ∪···∪Ri R0 ∪···∪Ri+1 N (f ) ,
i=0
where we used that f = gL+1 and DN = R0 ∪ · · · ∪ RL . Using the fact that for i 1, the restriction of gi+1 to R0 ∪ · · · ∪ Ri+1 is measurable wrt R0 ∪ · · · ∪ Ri−1 an estimate similar to (6.71) holds. For ε > 0, an argument similar to the one used in Proposition 5.1 implies +,ε µ (f ) − µh,ε (f ) exp(−cδN )f ∞ . N N By construction µh,ε N (f ) = 0. Summarizing the previous estimates, there exists c > 0 such that d−1 exp(−cδN )f ∞ . sup µ+,ε,η R0 (g1 ) LN η
Thus (6.68) holds.
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Upper bound on the spectral gap
We turn now to the derivation of Theorem 3.5. For any m ∈] − m∗ (β), m∗ (β)[, we set Am = σ ∈ {−1, 1}DN | MN m , where MN denotes the averaged magnetization MN = 1/N d i∈DN σi . Applying formula (3.25) to the test function f (σ) = 1{σ∈Am } , we get the following upper bound on the spectral gap µh N ∂Am SG(N, h) (2N )d h , (6.72) µN (Am )(1 − µh N (Am )) where ∂Am is the boundary of the set Am ∂Am = σ ∈ {−1, 1}DN | ∃x ∈ DN ,
σ ∈ Am , σ x ∈ Am ,
or σ ∈ Am , σ x ∈ Am .
Optimizing this inequality over m will enables us to bound the spectral gap in terms of equilibrium quantities. + For any β in B, the measure µh in the N converges to the pure phase µ thermodynamic limit as soon as one of the coordinates of h is positive. As a ∗ ∗ consequence µh N (Am ) vanishes as N goes to infinity for m in [−m (β), m (β)[. The set Am contains the configurations in the − phase which can be associated, on the macroscopic level, to the function u uniformly equal to −1. In this case there is no interface in the bulk and the interfacial energy is concentrated along the boundary. A straightforward adaptation of Proposition 4.1 of [BIV2] implies
1
lim inf d−1 log µh N (Am ) N →∞ N
h
∗
− F (−m (β)) = −
2d
∆hi .
(6.73)
i=1
Proposition 4.2 of [BIV2] implies the following upper bound lim sup lim sup δ→0
N →∞
1 N d−1
h log µh N {MN ∈ [m − δ, m + δ]} − F (m) .
The previous inequality rests upon the lower semi-continuity of the functional Wh which, for the sake of completeness, is proven in the Appendix. For any δ > 0, ∂Am is included in the set {MN ∈ [m − δ, m + δ]}, thus ∀m ∈] − m∗ (β), m∗ (β)[,
lim sup N →∞
1 h log µh N (∂Am ) − F (m) . N d−1
Combining estimates (6.73) and (6.74), we conclude Theorem 3.5.
(6.74)
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7 Appendix 7.1
Lower semi-continuity
By considering appropriate boundary conditions, we are going to reduce W h to a functional which does not explicitly take into account the boundary field. We set |∆hi | if xi − 1/2 max{|xj − 1/2|} , 2τ (e ) , j i h g (x) = | |∆ h2i , if xi − 1/2 min{−|xj − 1/2|} . j 2τ (ei ) To any function u of bounded variation, we associate Du the vector measure of its first partial derivatives and |Du| the positive measure obtained by taking the total variation of Du. Finally, we denote by n the vector function obtained dDu {±1}), . For any function u in BV(intD, as the Radon Nykodim derivative of d|Du| one sets 1 h W (u) = τβ (nx ) d|Du ∨ g h |(x). (7.75) 2 O |∆
|
This functional is lower semi-continuous wrt L1 -convergence. Let si = 2τβ h(eii ) and recall that P = ∪2d i=1 Pi denotes the faces of the cube D. This functional can be rewritten as follows τβ (nx )dHx(d−1) Wh (u) = ∂ ∗h u\P g
+
(|Pi | − Si )(1 − si )τβ (ei ) + Si (1 + si )τβ (ei ) + C(O),
i
where Si stands for the Hausdorff measure of ∂ ∗ {{u = −1} ∨ g h } ∩ Pi and C(O) We recover W h (u) up to a constant is the variation of g in O \ D. W h (u) = Wh (u) − |Pi | (τβ (ei ) − ∆hi /2) − C(O). (7.76) i
This implies that the functional W h (u) is lower semi-continuous.
7.2
Proof of Proposition 2.1
We split the proof into several steps: Step 1. If τβ is strictly convex at ed , then also τβ is strictly convex at ed . Indeed, define x = (0, . . . , 0, τβ (ed )). Thus, x belongs to ∂K, it is just a point where the ed -orthogonal hyperplane touches ∂K. Of course, τβ (ed ) = (x, ed ) for every > 0. The inequality (2.14) can be equivalently reformulated as follows: at least for one of the vectors vk , τβ (vk ) > (x, vk ), or, in other words, {x ∈ Rd : (x − x, vk ) = 0} ∩ int(K) = ∅.
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Since the Wulff shape K is convex and has a non-empty interior, the latter is equivalent to {x ∈ Rd : (x − x, vk ) = 0} ∩ int(K ) = ∅ for every > 0. Hence τβ (vk ) > (x, vk ) as well. Step 2. Below we use a simplified notation τ ≡ τβ . Let E ⊂ {x : xd 0} be a bounded set of finite perimeter and positive volume (d-dimensional Hausdorff measure). Let ∂ ∗ E be the reduced boundary [EG] of E. Let us split it as ∂ ∗ E = A ∪ Σ, where Σ = ∂ ∗ E \ {x ∈ Rd : xd = 0} = ∂ ∗ E \ A. We claim that
Σ
τ (nx )dHxd−1 > Hd−1 (A)τ (ed ).
(7.77)
By the Gauss-Green formula [EG], H
d−1
(A)ed =
Σ
nx dHxd−1 .
(7.78)
In view of (2.14) it is enough to show that one can find a decomposition of Σ into a disjoint union Σ = Σ1 ∪ · · · ∪ Σd , such that the vectors ∆ nx dHxd−1 k = 1, . . . , d, (7.79) vk = Σk
are in the general position. At this stage the positivity of the volume of E enters the picture. By the continuity one can pick positive numbers 0 = a0 < a1 < · · · < ad−1 < ad = ∞ such that (i) mink d Hd (E ∩ {x : ak−1 < xd < ak }) > 0. (ii) mink d−1 Hd−1 (E ∩ {x : xd = ak }) > 0. Of course, E∩{x : ak−1 < xd < ak } is just the part of E which is chopped out ∆
∆
by ed -orthogonal hyperplanes through the points xk−1 = (0, . . . , 0, ak−1 ) and xk = (0, . . . , 0, ak ) respectively. Since E is a set of finite perimeter we may in addition assume that small perturbations of these hyperplanes retain both properties above. Specifically, there exist positive numbers δ1 , . . . , δd−1 > 0, such that the sets (k = 2, . . . , d − 1) ∆
Sk = E ∩ {x : (x − xk , ed + δkek ) < 0 < (x − xk−1 , ed + δk−1ek−1 )}, ∆
S1 = {x : (x − x1 , ed + δke1 ) < 0 < (x, ed )} and Sd = E \ ∪d−1 k=1 Sk , are disjoint and, furthermore, each and everyone of the corresponding portions of their boundaries, which we denote as Σk = Σ ∩ Sk ; k = 1, . . . , d and Ak = E ∩ {x : (x − xk , ed + δkek ) = 0}; k = 1, . . . , d − 1, has a positive (d − 1)-dimensional Hausdorff measure.
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Subsequent application of the Gauss-Green formula on each of the sets S1 , . . . , Sd leads now to the following chain of equalities for the vectors v1 , . . . , vd defined in (7.79): 0 = vd
= (ed + δd−1ed−1 ) Hd−1 (Ad−1 )
0 = vd−1 = − (ed + δd−1ed−1 ) Hd−1 (Ad−1 ) + (ed + δd−2ed−2 ) Hd−1 (Ad−2 ) ··· 0 = v1 = − (ed + δ1e1 ) Hd−1 (A1 ) + ed Hd−1 (A). Recall that by (7.78) (7.77) follows.
d
vk 1
= ed Hd−1 (A). Consequently, v1 , . . . , vd span Rd and
Step 3. Finally we turn to the proof of Proposition 2.1 proper. Let u ∈ BV int D, {±1} . Set E = {x : u(x) = −1}. As in Step 2, split ∂ ∗ E = Σ ∪ A. The β, (u|g) can be then written (in the notation τ ≡ τ ) as functional W β
β, u g = W By (7.77),
Σ
τ (nx )dHxd−1 + 1 − Hd−1 (A) τ (ed ).
β, ½(·) g , β, u g > W W
as soon as Hd (E) > 0. But this is precisely the claim of the Proposition.
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Thierry Bodineau Laboratoire de Probabilit´es et Mod`eles al´eatoires CNRS-UMR 7599 Universit´es Paris VI & VII, Case 188 F-75252 Paris, Cedex 05, France email: Thierry.Bodineau@@math.jussieu.fr Dmitry Ioffe Faculty of Industrial Engineering Technion Haifa 32000, Israel email: ieioff
[email protected] Communicated by Jennifer Chayes submitted 04/11/03, accepted 05/03/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 915 – 928 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/050915-14 DOI 10.1007/s00023-004-0185-5
Annales Henri Poincar´ e
Minimal Escape Velocities for Unitary Evolution Groups Serge Richard∗
Abstract. Starting from a strict Mourre inequality, the minimal escape velocity for a unitary evolution group in a Hilbert space is derived under some minimal conditions. If the self-adjoint generator H of this evolution is a Schr¨ odinger operator and if the conjugate operator is the generator of dilations, then it follows that H has very good and easily understandable propagation properties. The striking fact is that no proof of the absence of singularly continuous spectrum of H is available yet under such weak conditions.
1 Introduction This paper is a natural sequel of [12] on the minimal escape velocity for the evolution group generated by a self-adjoint operator in a Hilbert space. By improving part of the mentioned work (as suggested in [8]) and by applying these new results to some Schr¨odinger operators in L2 (Rn ), we deduce some sharp propagation estimates. The minimal escape velocity is one variant of the generically called minimal velocity estimates, which are a key ingredient in the proof of asymptotic completeness for various models in quantum mechanics. We refer for example to [19], [21], [9] and [3] for their importance in the N-body problem, and to [11] and [17] for their use in some other anisotropic situations. Let us first concentrate on Schr¨ odinger operators and explain the interest of our estimates. We consider the Hilbert space H := L2 (Rn ), the usual Sobolev space H2 of order two on Rn and the generator A of the dilation group in H. Let V (Q) be a ∆-bounded operator with relative bound less than one, and let H := −∆ + V be the corresponding Schr¨ odinger operator in H with domain H2 . 1 Assume that H is of class Cu (A). The conditions of regularity of H with respect to A are explained in Section 2, but let us already mention that this requirement is very weak in the setting of the conjugate operator theory. Assume moreover that there exists an open interval J of R such that A is strictly conjugate to H on J. We show then that there exist a strictly positive constant υmin and a dense set of vectors ϕ in the spectral subspace EH (J)H of H such that for each υ < υmin , χ(|Q| ≤ υt)e−iHt ϕ → 0 as t → ∞, (1) where χ(|Q| ≤ υt) is the characteristic function of the ball in Rn centered at the origin and of radius equal to υt. ∗ This
work is partially supported by the Swiss National Science Foundation.
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The physical interpretation of this result is that the probability of finding the state e−iHt ϕ in the growing ball goes to zero as the time t goes to infinity. In other words, the state e−iHt ϕ propagates to infinity or “flees the origin” [4] with a velocity at least equal to υmin . Let us point out that the hypotheses of the previous estimate are easily fulfilled by Schr¨odinger operators with very general N-body potentials or cartesian potentials [17]. In the case where V is a two-body potential, the relation (1) is similar to some results obtained in [6]. The natural question which arises is about the nature of the spectrum of H on J. Do such propagation estimates imply the absence of singularly continuous spectrum on J? We do not know the answer but two related works could corroborate a positive one. We mention first the paper [20] in which a connection is drawn between the time of sojourn of the state e−iHt ϕ in any finite region of the space and the absolutely continuous subspace of H with respect to H. Secondly, let us assume for a while that V is a bounded function on Rn satisfying lim|x|→∞ |x|V (x) = 0. In that case, one shows that the relation (1) holds for the corresponding Schr¨ odinger operator on any open interval J of R+ with 0 not in the closure of J (cf. Remark 3). But then, it has been proved in [16] that any Schr¨ odinger operator −∆ + V in L2 [0, ∞) , with a bounded function V satisfying limx→∞ xV (x) = 0, has purely absolutely continuous spectrum on (0, ∞). Anyway, any proof (based on the method of the conjugate operator) of the absolute continuity of the spectrum of H on J requires a stronger condition than the Cu1 (A)-condition needed above. We refer to Chapter 7 of [1] for the most refined version of such results. Let us now develop the abstract side of the minimal escape velocity. We consider two self-adjoint operators H and A in a Hilbert space H (with norm · and scalar product ·, ·). The starting point is a strict Mourre inequality, i.e., the existence of an open interval J of R and of a strictly positive constant θ such that η(H)[iH, A]η(H) ≥ θη 2 (H) for all smooth real functions η with support in J. In order to give an unambiguous meaning to that expression, a regularity condition on H with respect to A must be imposed: H has to be of class C 1 (A). But if H is only slightly more regular we are able to state our first main result. Let us denote by Cc∞ (J) the set of all smooth complex functions defined on J which have a compact support in J. We use the notations χ(A ≤ c) and χ(A ≥ c) for the spectral projections of the operator A on the intervals (−∞, c] and [c, ∞) respectively. Theorem 1. Let H and A be self-adjoint operators in H with H of class Cu1 (A). Assume that there exist an open interval J of R and θ > 0 such that η(H)[iH, A]η(H) ≥ θη 2 (H) for all real η ∈ Cc∞ (J). Let a and t be real numbers. Then for each real η ∈ Cc∞ (J) and for each υ < θ one has χ(A ≤ a + υt)e−iHt η(H)χ(A ≥ a) → 0 uniformly in a.
as
t→∞
(2)
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The localization of the evolution in the spectrum of a conjugate operator has already a long history. We refer for example to [5], [14], [13] or more recently to [2] or [18] for different but related results. It is worth mentioning that in all those references, the operator H has to be more regular with respect to A than in Theorem 1. However, by requiring more regularity of H one may obtain a better control on the decrease of the norm in (2), cf. [12] for this kind of results. We point out that in this reference, H is at least of class C 2 (A). Let us finally describe the content of this paper. In Section 2 we introduce some notations and definitions. The proof of Theorem 1 is given in Section 3. In certain situations, one has some interest in localizing the evolution in the spectrum of another self-adjoint operator B rather than in the spectrum of A. Section 4 is devoted to that question and Proposition 3 contains some sufficient conditions between A and B to that purpose. The last section is the application to Schr¨ odinger operators. Our second main statement, Theorem 2, is exposed and proved. The relation (1) previously discussed is then a straightforward corollary of this theorem.
2 Some notations Almost all the notations and definitions are borrowed from [1], to which we refer for details. For any positive integer k let C k (R) be the algebra of complex functions on R that are k times We also consider various subalgebras continuously differentiable. ∞ (R), the functions whose derivatives have at of C ∞ (R) := C k (R), namely: Cpol most polynomial growth at infinity, S µ (R) with µ ≤ 0, the symbols of degree µ, and Cc∞ (R), the functions with compact support. Let us recall that f ∈ C ∞ (R) is a symbol of degree µ if for each k there exists a constant ck such that |f (k) (x)| ≤ µ−k ck (1 + x2 ) 2 for all x ∈ R. We collect some definitions related with the conjugate operator theory. H is a Hilbert space, B(H) denotes the set of bounded operators in H and {Wt }t∈R is the unitary group in H generated by a self-adjoint operator A. For any T ∈ B(H), we write T ∈ Cu (A), T ∈ C k (A) or T ∈ Cuk (A) if the mapping R t → W−t T Wt ∈ B(H) is continuous in norm, strongly C k or C k in norm respectively. By assuming that T ∈ C 1 (A), the commutator [iT, A], defined in form sense on the domain D(A) of A, extends continuously to a bounded operator in H. Let us mention that T ∈ Cu1 (A) if and only if T ∈ C 1 (A) and [iT, A] belongs to Cu (A). A selfadjoint operator H in H is of class C k (A), resp. Cuk (A), if (H − z)−1 ∈ C k (A), resp. (H − z)−1 ∈ Cuk (A), for some, and then for all, z ∈ C \ σ(H). We have used the notation σ(H) for the spectrum of H. Let Φ : [1, ∞) t → Φ(t) ∈ B(H) be an operator-valued mapping. We say that Φ or by a slight abuse of notation Φ(t) belongs to o(t−k ) if Φ(t) ∈ o(t−k ) or to O(t−k ) if Φ(t) ∈ O(t−k ), i.e., if limt→∞ tk Φ(t) = 0 or if tk Φ(t) ≤ c < ∞ for all t ≥ 1. We shall use on R the Fourier measure dx := (2π)−1/2 dx, where dx is the usual Lebesgue measure. Then a function f : R → C belongs to L1 (R) if f L1 :=
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|f (x)|dx < ∞. For such a function, its Fourier transform F f ≡ fˆ is defined by fˆ(x) := R e−ixy f (y)dy. We recall that F extends canonically to an isomorphism of the space of tempered distributions S ∗ (R) onto itself. For any complex Radon measure on R (simply called measure), we use the notation ν(x)dx for ν(dx). With such a measure ν we associate its variation |ν|, i.e., the smallest positive measure such that |ν(Ω)| ≤ |ν|(Ω) for each bounded and closed subset Ω of R. The measure ν is integrable on R if |ν|(R) < ∞. The space of ∗ all integrable measures on R is identified with a subspace of S (R) by the formula f, ν := R f (x)ν(x)dx, where f is any element of the space S (R) of tempered test functions on R and ·, · is the duality between S (R) and S ∗ (R). We are finally in position to recall a functional calculus. Let A be a selfadjoint operator in H and f ∈ S ∗ (R) such that fˆ is an integrable measure on R. Then for any ϕ, ψ ∈ H, one has (cf. Definition 3.2.7 of [1]): (3) ϕ, f (A)ψ := ϕ, eiAx ψfˆ(x)dx. R
R
3 The abstract theory We first consider a self-adjoint operator A in H and prove estimates for operators which have a certain regularity with respect to A. In the sequel, it is assumed that a and s are real numbers with s ≥ 1 and that f, h, η, . . . are real functions. ∞ (R)Lemma 1. Consider a bounded operator T ∈ C 1 (A) and let h be a bounded Cpol function such that h is an integrable 1measure on R. The norm of the commutator T, h A−a is then less or equal to s [T, A]h L1 . s
In the following proofs, we write As for the operator
A−a s .
Proof. By using the commutator expansions given in Theorem 5.5.3 of [1], one has the following equality in form sense on any core for A: 1 1 [T, h(As )] = dτ eiAs τ x [T, A]eiAs (1−τ )x h (x)dx. s 0 R Since T ∈ C 1 (A) the commutator [T, A] extends continuously to a bounded operator in H, and the estimate on the norm follows straightforwardly. Corollary 1. Assume that T and h satisfy the hypotheses of Lemma 1 and h that is has support in (−∞, 0]. Then the norm of the operator χ(A − a ≥ 0)T h A−a s 1 less or equal to s [T, A]h L1 . Proof. Since χ(x ≥ 0)h xs = 0 for any s ≥ 1 and all x ∈ R, one has the equality: χ(A − a ≥ 0)T h(As ) = χ(A − a ≥ 0)[T, h(As )]. The conclusion is then implied by Lemma 1.
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Lemma 2. Consider a bounded operator B ∈ Cu (A) and let h be a L∞ (R)-function such that h is an integrable measure on R. Then the commutator B, h A−a s belongs to o(s0 ), uniformly in a. Proof. By using the functional calculus introduced in equation (3), one has: i i h(x)|dx [B, h(As )] ≤ R e s Ax Be− s Ax − B | i i h(x)|dx. h(x)|dx + |x|<s1/2 e s Ax Be− s Ax − B | ≤ 2B |x|≥s1/2 |
(4)
The first term of (4) goes to 0 ass increases, and the second term is less or equal hL1 . By the regularity of B with respect to to sup|y|<s−1/2 eiAy Be−iAy − B 0 A, this belongs to o(s ). The next proposition imposes some apparently strong conditions on a certain function. But we shall show in a subsequent remark the existence of a class of functions satisfying all those requirements. Proposition 1. Let H be a self-adjoint operator in H of class Cu1 (A). Assume that there exist an open interval J of R and θ > 0 such that η(H)[iH, A]η(H) ≥ θη 2 (H) ∞ for all η ∈ Cc∞ (J). Let f be a bounded Cpol (R)-function such that f = −g 2 for ∞ some bounded Cpol (R)-function g. Moreover assume that f , g and g are integrable η(H) measures on R. Then for each η ∈ Cc∞ (J) the operator η(H) iH, f A−a s satisfies the estimate
A−a A−a θ η(H) iH, f (5) η(H) ≤ η(H)f η(H) + o(s−1 ), s s s where o(s−1 ) is uniform in a. Proof. a) Let η˜ ∈ Cc∞ (J) be such that η˜η = η. We set T := H η˜(H) (which belongs to Cu1 (A) by Corollary 6.2.6 (b) of [1]) and extension denote by B the continuous of the operator formally given by [iT, A] B belongs to Cu (A) . One observes that η(H)[iH, f (As )]η(H) = η(H)[iT, f (As )]η(H) and that the strict Mourre inequality can be rewritten as η(H)Bη(H) ≥ θη 2 (H) for all η ∈ Cc∞ (J). Through the use of the commutator expansions of Theorem 5.5.3 of [1], one obtains the following equality: 1 1 Rs + Bf (As ) s s 1 i i with Rs = 0 dτ R e s Aτ x Be− s Aτ x − B eiAs x f (x)dx. [iT, f (As )] =
(6)
Since the terms on are bounded, the l.h.s. term of (5) extends the r.h.s. of (6) continuously to η(H) 1s Rs + 1s Bf (As ) η(H).
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b) Let us now observe that 1s Bf (As ) = − 1s g(As )Bg(As ) + o(s−1 ), where o(s ) is independent of a (we have used Lemma 2). Moreover, since η(H) ∈ C 1 (A), some commutator calculations based on Lemma 1 show that −1
− 1s η(H)g(As )Bg(As )η(H) = − 1s g(As )η(H)Bη(H)g(As ) + O(s−2 ) ≤ − θs g(As )η 2 (H)g(As ) + O(s−2 ) = θs η(H)f (As )η(H) + O(s−2 ),
where O(s−2 ) is independent of a. c) It only remains to show that Rs belongs to o(s0 ) uniformly in a. One has that its norm is less or equal to 2B |x|≥s1/2
|f (x)|dx +
1
dτ 0
|x|<s1/2
i s Aτ x − si Aτ x Be − B |f (x)|dx. e
(7)
The first term of s increases. The second term of (7) is less or equal (7) goes to 0 as to sup|y|<s−1/2 eiAy Be−iAy − B f L1 , which belongs to o(s0 ) by the regularity of B with respect to A. One observes that both convergences are uniform in a. Remark 1. Consider g ∈ S µ (R) for some µ < −1. Since g 2 ∈ S 2µ (R) and g ∈ S µ−1 (R), then g, g2 and g are integrable measures on R (Proposition 5.4.5 of [1]). x ∞ Moreover if f (x) := − 0 g 2 (y)dy, then f belongs to Cpol (R) and satisfies all the assumptions of Proposition 1. Proposition 2. Let H be a self-adjoint operator in H of class Cu1 (A). Assume that there exist an open interval J of R and θ > 0 such that η(H)[iH, A]η(H) ≥ θη 2 (H) for all η ∈ Cc∞ (J). Let t be a real number with t ≥ 1. Then for each η ∈ Cc∞ (J), for each f ∈ L∞ (R) with support in (−∞, 0] and for each υ < θ, one has A−a − υ η(H)e−iHt χ(A − a ≥ 0) ∈ o(t0 ) f t uniformly in a. The following proof is inspired from that of Theorem 1.1 of [12], but is considerably simpler in our situation. a) Let g be a Cc∞ (R)-function with support in (υ − θ, 0) and such that Proof. ∞ x 2 2 −∞ g (y)dy = 1. We set h(x) = − 0 g (y)dy and observe that h satisfies all conditions imposed on f in Proposition 1. Furthermore, since h1/2 (x−θ)f (x−υ) = f (x − υ) for all x ∈ R, it is enough to prove that A−a 1/2 − θ η(H)e−iHt χ(A − a ≥ 0) ∈ o(t0 ) (8) h t uniformly in a.
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b) Let us set Φs (t) := η(H)h(At,s )η(H), with At,s equal to A−a−θt . For s each ψ ∈ H, we define ψt := e−iHt χ(A − a ≥ 0)ψ. Then (8) is equivalent to the statement that for all ψ ∈ H, ψt , Φt (t)ψt ≤ o(t0 )ψ2 , with o(t0 ) independent of ψ and a. One observes that t d ψτ , Φs (τ )ψτ dτ ψt , Φs (t)ψt = ψ0 , Φs (0)ψ0 + dτ 0 θ t = ψ0 , Φs (0)ψ0 − ψτ , η(H)h (Aτ,s )η(H)ψτ dτ s 0 t ψτ , η(H)[iH, h(Aτ,s )]η(H)ψτ dτ. +
(9)
(10)
0
By inserting (5) into (10) with a replaced by a + θτ , we find that t o(s−1 )ψ2 dτ ψt , Φs (t)ψt ≤ ψ0 , Φs (0)ψ0 + 0
−1
with o(s ) independent of a, τ and ψ. Moreover, with the help of Corollary 1, one gets that ψ0 , Φs (0)ψ0 ≤
1 ηL∞ [η(H), A]h L1 ψ2 . s
Hence, one has obtained that c ψt , Φs (t)ψt ≤ ψ2 + t o(s−1 )ψ2 s with o(s−1 ) and c independent of a, t and ψ. By setting s = t, this implies (9). Proof of Theorem 1. Since χ 1t x − υ ≤ 0 = χ (x ≤ υt) for any t ≥ 1 and all x ∈ R, the statement of the theorem is a special case of Proposition 2 with f (·) = χ(· ≤ 0).
4 From one localization to another The content of this section is inspired from Section 4.4.1 of [10]. The main difference is that the parameter a is not considered in that monograph. Let us recall from Lemma 7.2.15 of [1] that if T is a bounded operator be∗ longing to C 1 (A), the closure of the symmetric, densely defined operator T∗ AT ∗ D(T AT ) ⊃ D(A) is a self-adjoint operator which we still denote by T AT . Moreover, D(A) is a core for this operator. Therefore, if H is of class C 1 (A) and η˜ ∈ Cc∞ (R), the operator η˜(H)A˜ η (H), defined on D(A), admits a unique self-adjoint extension cf. Theorem 6.2.5 of [1] for the proof that η˜(H) belongs 7.2.16 of the same reference) that if to C 1 (A) . We also mention (Proposition ∞ 1 η ∈ Cc (R), then η(H) belongs to C η˜(H)A˜ η (H) .
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Lemma 3. Let H and A be self-adjoint operators in H with H of class C 1 (A). Let η, η˜ be Cc∞ (R)-functions such that η˜η = η, and let f be a C ∞ (R)-function such that f = 0 in a neighborhood of −∞ and f = 1 in a neighborhood of +∞. Then one has A−a η˜(H)A˜ η (H) − a f −f η(H) ∈ O(s−1 ) s s uniformly in a. η(H)A˜ ˜ η (H)−a ˜ ˜ , R(z) := (As − z)−1 and R(z) := Proof. Let us set As := A−a s , As := s −1 (A˜s − z) for z ∈ C \ R. For space reasons, we write η for η(H) and η˜ for η˜(H). Let ϕ, ψ be elements of H. By using Theorem 6.1.4 (b) of [1] for any integer r ≥ 1, one has that ϕ, {f (As ) − f (A˜s )}ηψ is equal to r−1 1 (k) ˜ + i)}ηψdλ (λ) ϕ, ik {R(λ + i) − R(λ k=0 πk! R f 1 1 ˜ + iµ)}ηψdλ. dµ R µr−1 f (r) (λ) ϕ, ir {R(λ + iµ) − R(λ + π(r−1)! 0
Moreover, one observes that there exist two constants c1 and c2 , independent of a and s, such that for any z ∈ C \ R: ˜ ˜ |ϕ, {R(z) − R(z)}ηψ| = |{A˜s − As }R(z)ϕ, R(z)ηψ| 1 ˜ ˜ ˜ + {R(z) − R(z)}ϕ, [η, η˜A˜ η ]R(z)ψ| = |η{A˜s − As }R(z)ϕ, R(z)ψ s 1 c1 ˜ ˜ ˜ ≤ |η[A, η˜]R(z)ϕ, R(z)ψ| + {R(z)ϕ + R(z)ϕ} R(z)ψ s s c1 ˜ c2 ˜ ˜ + R(z)ϕ R(z)ψ, ≤ R(z)ϕR(z)ψ s s ˜ ˜ ˜ η˜A˜ η ]R(z). By using then the H¨older where we have used that [R(z), η] = 1s R(z)[η, inequality and the identity (cf. Chap. XIII.7, Example 2 of [15]) valid for any self-adjoint operator K: π ϕ2 , (K − λ − iµ)−1 ϕ2 dλ = |µ| R we find that for µ = 0, f (k) (λ) ϕ, ik {R(λ + iµ) − R(λ ˜ + iµ)}ηψdλ ≤ d ϕψ. s|µ| R 1 1 with d = π(c1 + c2 )f (k) L∞ . By choosing r ≥ 2, one has 0 µr−1 |µ| dµ < ∞, and we have therefore obtained that c |ϕ, {f (As ) − f (A˜s )}η(H)ψ| ≤ ϕψ s for some c independent of a and s.
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Let us recall from Theorem 6.2.10 of [1] that if A and B are self-adjoint operators in H with B of class C 1 (A), then D(A) ∩ D(B) is a core for B. Lemma 4. Let A and B be self-adjoint operators in H. Assume that i) B is of class C 1 (A) and A ≤ B on D(A) ∩ D(B), ii) h Bs , A ∈ O(s0 ) for each h ∈ Cc∞ (R). Let f and g be C ∞ (R)-functions such that max(supp g) < min(supp f ). Moreover assume that f = 1 in a neighborhood of +∞ and that g has compact support. Then there exists c < ∞ independent of a and s such that g B f A − a ≤ c (1 + |a|). (11) s s s Proof. Let g˜ be in Cc∞ (R) such that max(supp g˜) < min(supp f ), g˜g = g and 0 ≤ g˜ ≤ 1. Then the operator g˜ Bs belongs to C 1 (A) and the operator g˜ Bs As g˜ Bs , defined on D(A), admits a unique self-adjoint extension which we denote by A˜s (cf. the observations made before Lemma 3). It follows from hypothesis i) that A˜s ≤ g˜ Bs Bs g˜ Bs < min(supp f ) on D(A), and therefore that f (A˜s ) = 0 for any s ≥ 1. In order to obtain (11), it is hence enough to show that there exists c < ∞ independent of a and s such that g B {f (As ) − f (A˜s )} ≤ c (1 + |a|). s s The rest of the proof is now analogous to that given in Lemma 3 and we shall only point out the minor difference. Let us set R(z) := (As − z)−1 and ˜ R(z) := (A˜s − z)−1 for z ∈ C \ R. One has, in form sense on H, that B B ˜ ˜ g {R(z) − R(z)} = R(z)g (A˜s − As )R(z) s s
B 1˜ B B ˜ + R(z) g˜ A˜ g ,g {R(z) − R(z)}, s s s s and that ˜ R(z)g
B s
(A˜s − As )R(z) =
1˜ R(z)g s
B s
A, g˜
B s
+ a R(z).
Hypothesis ii) is now used in order to obtain a uniform bound for the commutators. We now refine Lemma 4 to the case where B dominates only a localized version of A.
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Lemma 5. Let H, A and B be self-adjoint operators in H and let η, η˜ be Cc∞ (R)functions such that η˜η = η. Assume that i) H is of class C 1 (A) and of class C 1 (B), B is of class C 1 (A), η (H) defined on D(A) extend continuously to ii) the operators (B ± i)−1 A˜ bounded operators in H, iii) η˜(H)A˜ η (H) ≤ B on D η˜(H)A˜ η (H) ∩ D(B), iv) h Bs , A ∈ O(s0 ) for each h ∈ Cc∞ (R). Let f and g be C ∞ (R)-functions such that max(supp g) < min(supp f ). Moreover assume that f = 1 in a neighborhood of +∞ and that g has a compact support. Then there exists c < ∞ independent of a and s such that g B f A − a η(H) ≤ c (1 + |a|). s s s Proof. For space reasons, we write η˜ for η˜(H). One has A−a η˜A˜ η−a η˜A˜ η−a g B f − f + f η(H) s s s s B η˜A˜ η−a + O(s−1 ), ≤ ηL∞ g s f s
(12)
where we have used Lemma 3 and thus obtained that O(s−1 ) is independent of a. In order to deal with the first term of (12) we shall use Lemma 4 with η˜A˜ η instead of A. It follows from hypotheses i) and ii) that B is of class C 1 (˜ η A˜ η ) (the proof 4.4.7 of [10]). Thus we only have to prove that to that of Lemma B is similar 0 , η ˜ A˜ η belongs to O(s ) for each h ∈ Cc∞ (R). This commutator is equal to h s in form sense on D(A) :
B B B h , η˜ A˜ η + η˜ h , A η˜ + η˜A h , η˜ . (13) s s s By hypothesis iv) the second term of (13) is bounded uniformly in s. So let us concentrate on the first term (the third one being similar). Let ϕ, ψ belong to D(A) and let r bea strictly positive integer. By using Theorem 6.1.4 (b) of [1], η ψ is equal to the term ϕ, h Bs , η˜ A˜ −1 r−1 1 (k) B k h (λ) ϕ, i − λ − i , η˜ A˜ η ψdλ k=0 πk! R s −1 1 1 dµ R µr−1 h(r) (λ) ϕ, ir Bs − λ − iµ , η˜ A˜ η ψdλ. (14) + π(r−1)! 0 Let us observe that for z ∈ C \ R, −1 −1 B B −z −z , η˜ = [˜ η , B](B − sz)−1 , s s
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where [˜ η , B] extends continuously to a bounded operator. By inserting the first resolvent equation (B − sλ − isµ)−1 = I + (sλ + isµ + i)(B − sλ − isµ)−1 (B + i)−1 and by taking into account hypothesis ii), one obtains that for any s ≥ 1, −1 B c − λ − iµ , η˜ A˜ ηψ ≤ {|λ| + |µ| + 1)}ϕψ ϕ, |µ|2 s where c is independent of s. Finally, by using this estimate in (14) with r ≥ 3 one finds that ϕ, h B , η˜ A˜ η ψ ≤ c ϕψ s for some constant c independent of s.
Proposition 3. Let H, A and B be self-adjoint operators in H such that H is of class Cu1 (A) and of class C 1 (B), B of class C 1 (A) and B ≥ 0. Assume that there exist an open interval J of R and θ > 0 such that η(H)[iH, A]η(H) ≥ θη 2 (H) for all η ∈ Cc∞ (J). Let t be a real number with t ≥ 1 and let η, η˜ be Cc∞ (J)-functions such that η˜η = η. Assume moreover that i) the operators (B ± i)−1 A˜ η (H) defined on D(A) extend continuously to bounded operators in H, ii) η˜(H)A˜ η (H) ≤ B on D η˜(H)A˜ η (H) ∩ D(B), iii) h Bt , A ∈ O(t0 ) for each h ∈ Cc∞ (R). Then for each positive υ < θ, there exists c < ∞ independent of a and t such that χ(B ≤ υt)e−iHt η(H)χ(A ≥ a) ≤ o(t0 ) + c |a|, t where o(t0 ) is uniform in a. Proof. Let υ ∈ (υ, θ) and let g be a Cc∞ (R, [0, 1])-function such that supp g ⊂ (−∞, υ ) and g = 1 on [0, υ]. Let f be a C ∞ (R, [0, 1])-function such that max(supp B g) < min(supp fB) and f (x) = 1 for all x ≥ υ . Since χ(B ≤ υt) = B χ t ≤ υ = χ t ≤ υ g t , one has χ(B ≤ υt)e−iHt η(H)χ(A ≥ a) ≤ g B f A − a η(H) t t A−a + e−iHt η(H)χ(A ≥ a) 1−f . t By Lemma 5, there exists a constant c < ∞ independent of a and t such that the first term on the r.h.s. is less or equal to ct (1 + |a|). Since {1 − f (· + υ )} has support in (−∞, 0], one obtains from Proposition 2 that the second term on the r.h.s belongs to o(t0 ) uniformly in a.
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Remark 2. Since χ(A ≥ a)χ(A ≥ 0) = χ(A ≥ a) for any a ≥ 0, the statement of Proposition 3 can be rewritten in such a situation: for each a ≥ 0 and each υ < θ, one has χ(B ≤ υt)e−iHt η(H)χ(A ≥ a) ∈ o(t0 ), where o(t0 ) is uniform in a.
5 Application to Schr¨ odinger operators We consider the Hilbert space L2 (Rn ) and the Sobolev spaces of order s on Rn denoted by Hs . We recall that for j ∈ {1, . . . , n}, Qj is the operator of multiplication by the variable xj , Pj := −i∇j is a component of the momentum operator and −∆ is equal to P 2 . For any real number a, let us define a− which is equal to max{−a, 0}. Theorem 2. Let V (Q) be a ∆-bounded operator with relative bound less than one, and let H := −∆ + V be the corresponding Schr¨ odinger operator in L2 (Rn ) with 2 1 domain H . Assume that H is of class Cu (A), with A := 12 (P · Q + Q · P ) the generator of dilation, and that there exist an open interval J of R and θ > 0 such that η(H)[iH, A]η(H) ≥ θη 2 (H) for all η ∈ Cc∞ (J). Let a and t be real numbers with t ≥ 1. Then there exists υmin > 0 such that for each η ∈ Cc∞ (J) and each υ < υmin one has χ(|Q| ≤ υt)e−iHt η(H)χ(A ≥ a) ≤ o(t0 ) + c a− , t where o(t0 ) is uniform in a, and where c is a positive constant independent of a and t. Remark 3. Theorem 9.4.10 of [1] contains some sufficient conditions on the potential V such that −∆ + V is a N-body Hamiltonian of class Cu1 (A). In particular, if V (Q), [iV (Q), A] are compact operators from H2 to H, from H2 to H−2 respectively, then H is a two-body Hamiltonian of class Cu1 (A). For example if V is a bounded real function on Rn satisfying lim|x|→∞ |x|V (x) = 0, then the corresponding two-body Hamiltonian H is of class Cu1 (A). Its essential spectrum is equal to [0, ∞) and all its eigenvalues are negative and can accumulate only on 0; moreover, the operator A is strictly conjugate to H on any open interval J of R+ with 0 not in the closure of J (cf. Corollary 1.4 of [7] and Theorem 7.2.9 and Corollary 7.2.11 of [1]). Hence Theorem 2 applies and H has very good propagation properties on J. Remark 4. We also mention that if the operator V (Q) : H2 → H is compact and of the usual short-range or long-range type (cf. for example Definition 9.4.15 of [1]), then the corresponding two-body Hamiltonian H is of class Cu1 (A). In fact, in that situation H satisfies even a slightly stronger regularity condition, the one required in order to prove a limiting absorption principle.
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Proof of Theorem 2. This theorem is an application of Proposition 3. Let b be any strictly positive number and let η˜ be a Cc∞ (J)-function such that η˜η = η. The first step consists in verifying that the positive operator B := bQ ≡ b(1 + Q2)1/2 is of class C 1 (A), and that H is of class C 1 (B). This can be easily obtained with the help of Theorem 6.3.4 (a) of [1]. Secondly, let us observe that hypothesis ii) of Proposition 3 is fulfilled if the operator Q−1/2 η˜(H)A˜ η (H)Q−1/2 is bounded and if the value of b is chosen equal to its norm. But this new condition is quite standard and can be easily proved with some commutators calculations (statement i) of Lemma 6.2 of [17] may help). The other requirements of Proposition 3 are then also easily checked. One finishes the proof by setting υmin := θb and by taking υ into account Remark 2 and the fact that if υ < then χ bQ ≤ υ t χ |Q| ≤ b υt = χ |Q| ≤ υt for t large enough. One obtains the estimate (1) by observing that the set of vectors of the form η(H)χ(A ≥ a)ψ with η ∈ Cc∞ (J), a ∈ R and ψ ∈ H is dense in the subspace EH (J)H of H.
Acknowledgments The author thanks W.O. Amrein for critical reading of the text and for his constant support.
References [1] W.O. Amrein, A. Boutet de Monvel, V. Georgescu, C0 -Groups, Commutator Methods and Spectral Theory of N -Body Hamiltonians, Birkh¨ auser Verlag (1996). [2] A. Boutet de Monvel, V. Georgescu, J. Sahbani, Boundary values of resolvent families and propagation properties, C. R. Acad. Sci. Paris 322 (S´erie 1), 289–294 (1996). [3] J. Derezi´ nski, C. G´erard, Scattering Theory of Classical and Quantum N Particle Systems, Springer-Verlag Berlin Heidelberg (1997). [4] J.D. Dollard, On the definition of scattering subspaces in nonrelativistic quantum mechanics, J. Math. Phys. 18 (2), 229–232 (1977). [5] V. Enss, Asymptotic Completeness for Quantum Mechanical Potential Scattering, Commun. Math. Phys. 61, 285–291 (1978). [6] V. Enss, Asymptotic Observables on Scattering States, Commun. Math. Phys. 89, 245–268 (1983). [7] R. Froese, I. Herbst, Exponential Bounds and Absence of Positive Eigenvalues for N -Body Schr¨ odinger Operators, Commun. Math. Phys. 87, 429–447 (1982).
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[8] V. Georgescu, Review on the paper of Hunziker, Sigal and Soffer [12], Results from MathSciNet: MR1720738 (2001g:47129). [9] C. G´erard, Sharp Propagation Estimates for N -Particle Systems, Duke Math. Journal 67 (3), 483–515 (1992). [10] C. G´erard, I. Laba, Multiparticle Quantum Scattering in Constant Magnetic Fields, Mathematical Surveys and Monographs, Volume 90, American Mathematical Society (2002). [11] C. G´erard, F. Nier, Scattering theory for the perturbations of periodic Schr¨ odinger operators, J. Math. Kyoto Univ. 38 (4), 595–634 (1998). [12] W. Hunziker, I.M. Sigal, A. Soffer, Minimal Escape Velocities, Comm. Partial Differential Equations 24 (11&12), 2279–2295 (1999). [13] A. Jensen, Propagation Estimates for Schr¨ odinger-type Operators, Transactions of the American Mathematical Society 291 (1), 129–144 (1985). [14] E. Mourre, Op´erateurs conjugu´es et propri´et´es de propagation, Commun. Math. Phys. 91, 279–300 (1983). [15] M. Reed, B. Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press (1978). [16] C. Remling, The Absolutely Continuous Spectrum of One-Dimensional Schr¨ odinger Operators with Decaying Potentials, Commun. Math. Phys. 193, 151– 170 (1998). [17] S. Richard, Spectral and Scattering Theory for Schr¨ odinger Operators with Cartesian Anisotropy, to appear in Publ. RIMS, Kyoto Univ. [18] J. Sahbani, Th´eor`emes de Propagation, Hamiltoniens Localement R´eguliers et Applications, Th`ese de Doctorat de l’Universit´e Paris VII (1996). [19] I.M. Sigal, A. Soffer, Local decay and velocity bounds, Preprint, Princeton University (1988). [20] K.B. Sinha, On the absolutely and singularly continuous subspaces in scattering theory, Ann. Inst. Henri Poincar´e XXVI (3), 263–277 (1977). [21] E. Skibsted, Propagation Estimates for N -body Schr¨ odinger Operators, Commun. Math. Phys. 142, 67–98 (1991). Serge Richard Department of Theoretical Physics University of Geneva CH-1211 Geneva 4 Switzerland email:
[email protected]. Communicated by Gian Michele Graf submitted 15/10/03, accepted 19/02/04
Ann. Henri Poincar´e 5 (2004) 929 – 978 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/050929-50 DOI 10.1007/s00023-004-0186-4
Annales Henri Poincar´ e
On the Singular Spectrum for Adiabatic Quasi-Periodic Schr¨ odinger Operators on the Real Line Alexander Fedotov and Fr´ed´eric Klopp Abstract. In this paper, we study spectral properties of a family of quasi-periodic Schr¨ odinger operators on the real line in the adiabatic limit. We assume that the adiabatic iso-energetic curves are extended along the momentum direction. In the energy intervals where this happens, we obtain an asymptotic formula for the Lyapunov exponent, and show that the spectrum is purely singular. R´esum´e. Cet article est consacr´e ` a l’´ etude du spectre d’une famille d’op´erateurs de Schr¨ odinger quasi-p´eriodiques sur l’axe r´eel lorsque les courbes iso-´energ´etiques adiabatiques sont non born´ees dans la direction des moments. Dans des intervalles d’´ energies o` u cette propri´et´ e est v´ erifi´ ee, nous obtenons une formule asymptotique pour l’exposant de Lyapounoff, et nous d´emontrons que le spectre est purement singulier.
0 Introduction In this paper, we continue our analysis of the spectrum of the family of differential operators d2 Hz,ε = − 2 + V (x − z) + W (εx) (0.1) dx acting on L2 (R), where x → V (x) and x → W (x) are periodic, sufficiently regular and real valued, z ∈ R indexes the family of operators, and ε > 0 is chosen so that the potential x → V (x − z) + W (εx) be quasi-periodic. Recall that this implies that the family of Schr¨ odinger operators (Hz,ε )z is ergodic; consequently, its spectrum, its absolutely continuous spectrum and, hence, its singular spectrum, are independent of z (see, e.g., [14, 13]). Moreover, it guarantees the existence of the Lyapunov exponent ([14]). We study the spectral properties of the operator Hz,ε for ε positive small. In [8], we studied this operator near the bottom of the spectrum when W was the cosine. In [7], for a general analytic, periodic potential W , we studied the spectrum located in the “middle” of a spectral band of the “unperturbed” periodic differential operator d2 (0.2) H0 = − 2 + V (x) dx acting on L2 (R). In the present paper, we again consider a rather general analytic potential W ; we only assume that it has exactly one maximum and one minimum in a period,
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and that these are non-degenerate. As for V , it can be rather singular; for the sake of simplicity, we assume that it us locally square integrable. We study the spectrum in an energy interval J such that, for all E in J, the interval E − W (R) contains one or more isolated spectral bands of the periodic operator 0.2 whereas the ends of the interval E − W (R) are in the gaps, see Fig. 1. So, we are interested in the spectrum close to and inside relatively small bands of the unperturbed periodic operator H0 . E − W (R) E
E σ(H0 ) E2n−1 E2n+1
E2n−2
E2n−1 E2n
E2n+1
E2n−2
E2n
E2n+2m−1 E2n+2m−2
E2n+2m+1
E2n+2m
Figure 1. The isolated band: two possible cases. As in [8, 7], our main tool is the monodromy matrix. Most of the present paper is devoted to the asymptotic study of the monodromy matrix for the family of operators (0.1). In the adiabatic limit ε → 0, the monodromy matrix is asymptotic to a trigonometric polynomial; if the interval E − W (R) contains only one isolated spectral band, this is a trigonometric polynomial of the first order. In result, the analysis of (0.1) reduces to the analysis of a “simple” model finite difference operator. Using the monodromy matrix asymptotics, we obtain asymptotic formulae for the Lyapunov exponent for the operator family (0.1). They show that, in J, the considered energy region, the Lyapunov exponent is positive. This implies that the spectrum of (0.1) in J is singular by the Ishii-Pastur-Kotani Theorem ([14]). The spectral results admit a natural semi-classical interpretation. Let E(κ) be the dispersion relation of H0 (see Section 3.2). Consider the real and the complex iso-energy curves ΓR and Γ defined by ΓR = {(κ, ζ) ∈ R2 ; E(κ) + W (ζ) = E}, 2
Γ = {(κ, ζ) ∈ C ; E(κ) + W (ζ) = E}.
(0.3) (0.4)
These curves are 2π-periodic both in ζ and κ. Under our assumptions, the real branches of Γ (the connected components of ΓR ) are isolated continuous curves periodic in κ. In the case when the interval E − W (R) contains only one spectral band, the iso-energy curve is shown in Fig. 2. The real branches are represented by full lines. They are connected by complex loops (closed curves) lying on Γ; the loops are represented by dashed lines. The adiabatic limit can be regarded as a semi-classical limit, and the expression E(κ)+ W (ζ) can be interpreted as a “classical” Hamiltonian corresponding to
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κ
0
2π
ζ
Figure 2. The phase space picture. the operator (0.1). Then, from the quantum physicist point of view (see [17, 18]), a semi-classical particle should “live” near the real branches of the iso-energy curve. In our case, these curves are “extended” in the momentum variable and “localized” in the position variable. Therefore, they have to correspond to localized states. The decay of these states in the position variable is characterized by the complex tunneling between the real branches along the complex loops. So, the Lyapunov exponent is naturally related to the tunneling coefficients. Our results justify these heuristics. This naturally leads to the following conjecture: in a given energy interval, if the iso-energy curve has a real branch that is an unbounded vertical curve, then, in the adiabatic limit, in this interval, the Lyapunov exponent is positive and the spectrum is singular. Note that, in [7], we have proved a dual result for the absolutely continuous spectrum: we have proved that, if, in some energy region, the branches of the real iso-energy curve are unbounded horizontal curves, then, this energy region, except for a set of exponentially small measure, is in the absolutely continuous spectrum.
1 The results We now state our assumptions and results.
1.1
Assumptions on the potentials
On the functions V and W , we assume that (H1) V and W are periodic, V (x + 1) = V (x),
W (x + 2π) = W (x),
x ∈ R;
(1.1)
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(H2) V is real valued and locally square integrable; (H3) W is real analytic in a neighborhood of R, say, in the strip {|Im z| < Y }; (H4) W has exactly one maximum and one minimum in [0, 2π); they are nondegenerate. To fix the notations, assume that, on the interval [0, 2π), the maximum of W is 0 and the minimum is ζ ∗ .
1.2
The assumption on the energy region
To describe the energy regions where we study the spectral properties of the family 0.1, we consider the periodic Schr¨ odinger operator H0 acting on L2 (R) and defined by 0.2. 1.2.1 The periodic operator The spectrum of 0.2 is absolutely continuous and consists of intervals of the real axis [E1 , E2 ], [E3 , E4 ], . . . , [E2n+1 , E2n+2 ], . . . , such that E1 < E2 ≤ E3 < E4 ≤ · · · < E2n ≤ E2n+1 < E2n+2 ≤ . . . , En → +∞, n → +∞. 2
d The points (Ej )j∈N\{0} are the eigenvalues of the differential operator − dx 2 +V (x) 2 acting on L ([0, 2]) with periodic boundary conditions. The intervals defined above are called the spectral bands, and the intervals (E2 , E3 ), . . . , (E2n , E2n+1 ), . . . , are called the spectral gaps. If E2n < E2n+1 , we say that the nth gap is open, and, if [E2n−1 , E2n ] is separated from the remaining part of the spectrum by open gaps, we say that the nth band is isolated.
1.2.2 The “geometric” assumption Let us now describe the energy region where we study (0.1). The spectral window centered at E is defined as the interval W(E) = E − W (R). If W+ = maxx∈R W (x) and W− = minx∈R W (x), then, W(E) = [E − W+ , E − W− ]. We assume that there exists a real compact interval J, such that, for all E in J, the window W(E) contains exactly m + 1 isolated bands of the periodic operator (see Fig. 1). That is, we fix two integers n > 0 and m and assume that (A1) the bands [E2(n+j)−1 , E2(n+j)) ], j = 0, 1, . . . , m, are isolated; (A2) for E in J, these bands are contained in the interior of W(E); (A3) for E in J, the remaining part of the spectrum of the periodic operator is outside W(E).
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Note that energies E satisfying (A1) – (A3) exist only if W+ −W− , the “amplitude” of the adiabatic perturbation, is large enough; e.g., if m = 0, such energies exist if and only if W+ − W− is larger than the size of the nth spectral band, but smaller than the distance between the (n − 1)st and (n + 1)st bands. From now on, unless stated otherwise, we assume that assumptions (H1)– (H4) and (A1)–(A3) are satisfied.
1.3
Iso-energy curve
Our results are formulated in terms of the iso-energy curve Γ defined by (0.4). The iso-energy curve is 2π periodic both in the ζ- and κ- directions (see Lemma 10.1). 1.3.1 The real branches By definition, the real branches of Γ are the connected components of the real iso-energy curve ΓR . To describe them, we define the following collection of subintervals of [0, 2π]. Consider the mapping E : ζ → E − W (ζ). It is monotonous on the intervals I− = [0, ζ ∗ ] and I+ = [ζ ∗ , 2π] and maps each of them onto the spectral window W(E). For n ≤ j ≤ n + m, let z+ j ⊂ I+ − (resp. zj ⊂ I− ) be the pre-image of the jth spectral band in W(E). Let Z be the collection of these intervals. A “period” of the real iso-energy curve is described by Proposition 1.1 Let E ∈ J. The set ΓR ∩ {0 ≤ κ ≤ 2π} consists of 2(m + 1) curves {γ(z), z ∈ Z}. Fix z ∈ Z. The curve γ(z) is the graph {(κ, ζ) ∈ R2 ; ζ = Zz (κ), κ ∈ R} of a function Zz which is continuous, 2π-periodic and even in κ, monotonous on the interval [0, π], and maps [0, π] onto z. The curves γ(z) continuously depend on E ∈ J. Proposition 1.1 is proved in Section 10.1. For m = 0, the real iso-energy curve is shown in Fig. 2. 1.3.2 Complex loops Now, we discuss loops, i.e., closed curves, located on the iso-energy curve Γ and connecting its real branches. − For j = n − 1, . . . , n + m, let g+ j (resp. gj ) be the subinterval of I+ (resp. I− ) that is the pre-image of the part of jth spectral gap located inside W(E). Let − gn−1 = (g+ n−1 − 2π) ∪ gn−1
− and gn+m = g+ n+m ∪ gn+m .
Then, gn−1 is an open interval containing 0, and gn+m is an open interval containing ζ ∗ . Let G be the set consisting of gn−1 , gn+m and the intervals gj± with j = n, . . . , n + m − 1.
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For g ∈ G, let V (g) ⊂ C be a sufficiently small complex neighborhood of the interval g. Let G(g) be a smooth closed curve in V (g) \ g that goes once around the interval g. In Figure 3, we depicted the curves G(g) when m = 0. Consider the projection (κ, ζ) ∈ Γ → ζ ∈ C. In Section 10.2, we show that ˆ each of the curves G(g) is the projection of G(g), a closed curve on Γ. This curve connects the real branches projecting onto the intervals adjacent to g. 1.3.3 Tunneling coefficients To Γ, we associate the tunneling coefficients 1
t(g) = e− 2ε S(g) , where S(g) are the tunneling actions given by S(g) = i κd ζ, ˆ G(g)
g ∈ G,
(1.2)
g ∈ G.
(1.3)
In Section 10.3, we show that, for E ∈ J, each of these actions is real and non-zero. By definition, we choose the direction of the integration so that all the tunneling actions be positive.
+ − 2π ζ2n
G(gn−1 )
G(gn ) − 0 ζ2n−1
+ ζ2n−1 − 2π
ζ∗
− ζ2n
+ ζ2n
+ ζ2n−1
Figure 3. The curves G(g) for m = 0.
1.4
Spectral results
One of the main objects of the spectral theory of quasi-periodic operators is the Lyapunov exponent (for a definition and additional information, see, for example, [14]). Our main spectral result is Theorem 1.1 Let W and V satisfy the hypotheses (H1)–(H4). Let J be an interval satisfying the assumptions (A1)–(A3) for some n and m. Then, on the interval J, for sufficiently small irrational ε/2π, the Lyapunov exponent Θ(E, ε) of (0.1) is positive and it satisfies the asymptotics Θ(E, ε) =
1 1 ε + o(1) = ln S(g) + o(1). 2π t(g) 4π g∈G
g∈G
(1.4)
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Note that, this theorem implies that, if ε/2π is sufficiently small and irrational, then, the Lyapunov exponent is positive for all E ∈ J. Recall that, if ε/2π is irrational, then Hz,ε is quasi-periodic. In this case, its spectrum σ(Hz,ε ) does not depend on z (see [1]). In [8], we have proved Theorem 1.2 ([8]) Let Σ = σ(H0 ) + W (R) = σ(H0 ) + [W− , W+ ]. Then, • for ε ≥ 0, σ(Hz,ε ) ⊂ Σ. • for any compact K ⊂ Σ, there exists C > 0 such that for all ε sufficiently small and all E ∈ K, one has σ(Hz,ε ) ∩ (E − Cε1/2 , E + Cε1/2 ) = ∅. By the Ishii-Pastur-Kotani Theorem [4, 14] and Theorem 1.5 in [13], Theorems 1.1 and 1.2 imply Corollary 1.1 In the case of Theorem 1.1, for ε sufficiently small, for all z ∈ R, one has σ(Hz,ε ) ∩ J = ∅ and σac (Hz,ε ) ∩ J = ∅, where σac (Hz,ε ) is the absolutely continuous spectrum of the family of equations (0.1).
1.5
The monodromy matrix and the Lyapunov exponents
The main object of our study is the monodromy matrix for the family of operators (0.1); we define it briefly (we refer to [10, 8] for more details). The central result of the paper is its asymptotics in the adiabatic limit. 1.5.1 Definition of the monodromy matrix Fix E ∈ R. Consider the family of differential equations indexed by z ∈ R d2 − 2 + V (x − z) + W (εx) ψ(x) = Eψ(x). dx
(1.5)
Definition 1.1 We say that (ψi )i∈{1,2} is a consistent basis of solutions to (1.5) if the two functions ((x, z) → ψi (x, z, E))i∈{1,2} are a basis of solutions to (1.5) whose Wronskian is independent of z and that are 1-periodic in z, i.e., that satisfy ∀x ∈ R, ∀z ∈ R, ∀i ∈ {1, 2},
ψi (x, z + 1, E) = ψi (x, z, E).
(1.6)
The existence of consistent bases as well as more details are provided in [10, 8]. The functions ((x, z) → ψi (x + 2π/ε, z + 2π/ε, E))i∈{1,2} being also solutions to equation (1.5), one can write Ψ (x + 2π/ε, z + 2π/ε, E) = M (z, E) Ψ (x, z, E),
(1.7)
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where
ψ1 (x, z, E) , • Ψ(x, z, E) = ψ2 (x, z, E)
• M (z, E) is a 2 × 2 matrix with coefficients independent of x. The matrix M is called the monodromy matrix associated to the consistent basis (ψ1,2 ). Note that det M (z, E) ≡ 1,
M (z + 1, E) = M (z, E),
∀z.
(1.8)
1.5.2 Lyapunov exponents Consider now a 1-periodic, SL(2, C)-valued function, say, z → N (z), and h > 0 irrational. Consider the finite difference equation ∀n ∈ Z,
Fn+1 = N (z + nh)Fn
Fn ∈ C2 .
(1.9)
The Lyapunov exponent of this finite difference equation is θ(M, h) =
lim
N →+∞
1 log PN (z, h), N
(1.10)
where the matrix cocycle (PN (z, h))N ∈N is defined as PN (z) = N (z + N h) · N (z + (N − 1)h) · · · N (z + h) · N (z). It is well known (see [3, 14, 15] and references therein) that, if h is irrational, and N (z) is sufficiently regular in z, then the limit (1.10) exists for almost all z and is independent of z. 1.5.3 The monodromy equation Set h = 2π ε mod 1. Let M be the monodromy matrix associated to a consistent basis (ψi )i∈{1,2} . Consider the monodromy equation Fn+1 = M (z + nh, E)Fn
∀n ∈ Z Fn ∈ C2 .
(1.11)
The Lyapunov exponent of the monodromy equation (1.11) is defined by θ(E, ε) = θ(M (z, E), h) . There are several deep relations between equation (1.5) and the monodromy equation (1.11) (see [7, 8]). We describe only one of them. Recall that Θ(E, ε) is the Lyapunov exponent of the operator (0.1). One proves Theorem 1.3 ([8]) Assume ε/2π is irrational. The Lyapunov exponents Θ(E, ε) and θ(E, ε) satisfy the relation ε θ(E, ε). (1.12) Θ(E, ε) = 2π The reduction to the monodromy equation is close to the monodromization idea developed in [2] for difference equations with periodic coefficients.
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1.5.4 The asymptotics of the monodromy matrix As W and V are real on the real line, we construct a monodromy matrix of the form a(z, E) b(z, E) (1.13) ¯ a(¯ ¯ . b(¯ z , E) z , E) In the adiabatic case, the asymptotics of a and b have a very simple, model form. We first assume that the integer n in (A1)–(A3) is odd. Then, one has Theorem 1.4 Pick E0 ∈ J. There exists Y > 0 and V0 , a neighborhood of E0 , such that, for sufficiently small ε, the family of equations (1.5) has a consistent basis of solutions for which the corresponding monodromy matrix M is analytic in (z, E) ∈ {z ∈ C; |Im z| < Y /ε} × V0 and has the form (1.13). When ε tends to 0, the coefficients a and b admit the asymptotic representations a = a−m e−2πimz (1 + o(1)),
b = b−m e−2πimz (1 + o(1)),
0 < Im z < Y /ε, (1.14)
and a = am+1 e2πi(m+1)z (1 + o(1)), b = bm+1 e2πi(m+1)z (1 + o(1)), −Y /ε < Im z < 0. (1.15) The coefficients a−m , b−m , am+1 and bm+1 are independent of z. Moreover, there exists a constant C > 1 such that, for ε > 0 sufficiently small, E ∈ V0 ∩ R and j ∈ {−m, m + 1}, one has 1 ≤ T · |aj | ≤ C, C
1 ≤ T · |bj | ≤ C, C
where
T = T (E) =
t(g). (1.16)
g∈G
For Y1 and Y2 such that 0 < Y1 < Y2 < Y , there exists V = V (Y1 , Y2 ), a neighborhood of E0 such that the asymptotics (1.14) and (1.15) for a and b are uniform in (z, E) ∈ {z ∈ C; Y1 < |Im z| < Y2 } × V . In Sections 9.1 and 9.2, we give asymptotic formulae for am+1 , a−m and bm+1 , b−m . The asymptotics (1.14) and (1.15) are obtained by means of the new asymptotic method developed in [10, 9]. In the case n even, one has a similar result. The only novelty is that, in this case, the formulae (1.14) and (1.15) describe the asymptotics of the coefficients of ˜ (z, E) related to the monodromy matrix M (z, E) by the following the matrix M transformation iπz 0 ˜ (z, E) := S −1 (z + h)M (z, E)S(z), S(z) = e M . (1.17) 0 e−iπz
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1.5.5 Fourier coefficients The coefficients am+1 , a−m and bm+1 , b−m are the leading terms of the asymptotics of the (m + 1)st and (−m)th Fourier coefficients of the monodromy matrix coefficients. Theorem 1.4 implies that, in the strip {|Im ζ| < Y }, only a few Fourier series terms of the monodromy matrix dominate. 1.5.6 The case m = 0 When m = 0 (and n odd), Theorem 1.4 imply that, in the whole strip {|Im z| < Y }, the monodromy matrix coefficients a and b admit the asymptotics: a = a0 (1 + o(1)) + a1 e2πiz (1 + o(1)),
b = b0 (1 + o(1)) + b1 e2πiz (1 + o(1)). (1.18)
So, up to the error terms, the entries of monodromy matrix become first order trigonometric polynomials: a0 + a1 u b 0 + b 1 u M ∼ M0 = , u = e2πiz , (1.19) b0 + b1 /u a0 + a1 /u with constant coefficients a0 , a1 , b0 , b1 of order O(1/T (E)) (for real E). We see that, for m = 0, the monodromy equation becomes a “simple” model equation. 1.5.7 Relation to the spectral results In this paper, we use the asymptotics of the monodromy matrix only to prove Theorem 1.1. However, we believe that these asymptotics can be used to get quite detailed information on the spectrum of the operator (0.1) in the adiabatic limit. Therefore, in a subsequent paper [6], we study the monodromy equation (1.11) where the monodromy matrix is replaced with the model matrix M0 . In particular, we hope to prove that, under a Diophantine condition on 2π/ε, the spectrum of the operator (0.1) is pure point and the eigenvalues can be described by quantization conditions of Bohr-Sommerfeld type. 1.5.8 Organization of the paper Section 2 is devoted to the proof of Theorem 1.1 using Theorem 1.4. In Section 3, we recall some well-known facts from the theory of periodic Schr¨ odinger operators on the real line. In Section 4, we recall the main construction of the asymptotic method we use to compute the monodromy matrix. In Sections 5 and 6, using this method, we construct a consistent basis of solutions having a simple “standard” asymptotic behavior in the complex plane of ζ = εz. In Section 7, we discuss the properties of the monodromy matrix for this basis. This is the monodromy matrix the asymptotics of which are described in Theorem 1.4. Sections 8 and 9 are devoted to the computation of the asymptotics of the monodromy matrix. In Section 10, we study the geometry of the iso-energy curve Γ and prove estimates (1.16).
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2 The asymptotics for the Lyapunov exponent In this section, we prove the asymptotics (1.4). We deduce these asymptotics from the asymptotics of the monodromy matrix coefficients described by Theorem 1.4. First, we use a statement of [8] and obtain a lower bound for the Lyapunov exponent. This statement is based on the ideas of [15] generalizing Herman’s argument [12]. Then, using the asymptotics of the coefficients of the monodromy matrix in the complex plane, we estimate them on the real line. This yields an upper bound for the Lyapunov exponent. Comparing the upper and the lower bounds, we obtain (1.4).
2.1
The lower bound
We first recall a result giving a lower bound on the Lyapunov exponent for the finite difference equation (1.9) when N admits some special form; then, we apply this to the monodromy equation (1.11) using the asymptotics for monodromy matrix provided by Theorem 1.4. 2.1.1 Preliminaries Let (M (z, ε))0<ε<1 be a family of SL(2, C)-valued, 1-periodic functions of z ∈ C. Let h be an irrational number. One has Proposition 2.1 ([8]) Pick ε0 > 0. Assume that there exists y0 and y1 such that 0 < y0 < y1 < ∞ and such that, for any ε ∈ (0, ε0 ) one has • the function z → M (z, ε) is analytic in the strip S = {z ∈ C; 0 ≤ Im z ≤ y1 /ε}; • in the strip S1 = {z ∈ C; y0 /ε ≤ Im z ≤ y1 /ε} ⊂ S, M (z, ε) admits the representation M (z, ε) = λ(ε)ei2πn0 z · (M0 (ε) + M1 (z, ε)) , for some constant λ(ε), some integer n0 and a matrix M0 (ε), all of them independent of z; 1 β(ε) ; • M0 (ε) = 0 α(ε) • there exists constants β > 0 and α ∈ (0, 1) independent of ε and such that |α(ε)| ≤ α and |β(ε)| ≤ β; • m(ε) := sup M1 (z, ε) → 0 as ε → 0. z∈S1
Then, there exists C > 0 and ε1 > 0 (both depending only on y0 , y1 , α, β and ε → m(ε)) such that, if 0 < ε < ε1 , one has θ(M (·, ε), h) > log |λ(ε)| − Cm(ε). The basic ideas behind this result go back to [12] and [15].
(2.1)
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In [8], we have assumed that n0 is a positive integer, but the proof remains the same for n0 ∈ Z. We use this result and Theorem 1.4 to get the lower bound for the Lyapunov exponent. In the sequel, n is the index introduced in assumptions (A1)–(A3). The cases n odd and n even are treated separately. 2.1.2 Obtaining the lower bound for n odd In the sequel, we suppose that the assumptions of Theorem 1.4 are satisfied; we use the notations and results from this theorem without referring to it anymore. We assume that E ∈ V0 ∩ R. 0 1 Let σ = . Let us show that the matrix σM (z, E)σ satisfies the assumptions 1 0 of Proposition 2.1. Fix y0 and y1 so that 0 < y0 < y1 < Y . The asymptotics of the monodromy matrix coefficients are uniform in z in the strip S = {z ∈ C; y0 /ε ≤ Im z ≤ y1 /ε} and in E ∈ V0 (reducing V0 if necessary). For E ∈ V0 ∩ R and z ∈ S, formulae (1.14) and (1.15), and estimates (1.16) imply that a(z, E) = am+1 e−2πi(m+1)z (1 + o(1)), b(z, E) a(z, E)
= c(E) (1 + o(1)),
a(z, E) a(z, E)
b(z, E) a(z, E)
= o(1),
= o(1)
where c(E) is independent of z and bounded by a constant uniformly in ε and E. So, we have σ · M (z, E) · σ = am+1 e
−2πi(m+1)z
1 c(E) + o(1) . 0 0
We see that the matrix-valued function z → σM (z, E)σ satisfies the assumptions of Proposition 2.1. Clearly, the Lyapunov exponents of the matrix cocycles associated to the pairs (M (·, E), h) and (σ · M (·, E) · σ, h) coincide. So, Proposition 2.1 implies that θ(E, ε), the Lyapunov exponents of the matrix cocycle associated to (M (·, E), h), satisfies the estimate (2.2) θ(E, ε) ≥ log |am+1 | + o(1). The Lyapunov exponent Θ(E, ε) for (0.1) is related to θ(E, ε) by Theorem 1.3. ε log |a−m | + o(ε). Hence, (1.16) and (2.2) clearly imply Therefore, Θ(E, ε) ≥ 2π Θ(E, ε) ≥
ε log T (E)−1 + O(ε). 2π
(2.3)
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2.1.3 The lower bound when n is even If n is even, then, formulae (1.14) and (1.15) give the asymptotics of the coefficients of the matrix (1.17). Obviously, the Lyapunov exponents for the matrix cocycles generated by M (z, E) and by S −1 (z + h)M (z, E)S(z) coincide. Arguing exactly as in Subsection 2.1.2, we again obtain (2.3).
2.2
The upper bound
Let us first assume that n is odd. Let E ∈ V0 ∩ R. Fix 0 < y0 < Y . The asymptotics (1.14) and (1.15) and estimates (1.16) imply the following estimates for the coefficients of M (z, E), the monodromy matrix: |a|, |b| ≤ C(y0 ) T (E)−1 e2πmy0 /ε , Im z = y0 /ε, |a|, |b| ≤ C(y0 ) T (E)−1 e2π(m+1)y0 /ε , Im z = −y0 /ε.
(2.4)
Here, C(y0 ) is a positive constant independent of ε, Re z, and E. The estimates are valid for sufficiently small ε. Recall that M is analytic and 1-periodic in z. Therefore, (2.4) and the Maximum Principle imply that |a|, |b| ≤ 2C(y0 ) T (E)−1 exp(2π(m + 1)y0 /ε),
z ∈ R.
(2.5)
This leads to the following upper bound for the Lyapunov exponent for the matrix cocycle generated by M (z, E) θ(E, ε) ≤ log T (E)−1 + C + 2π(m + 1)y0 /ε where C is a constant independent of E and ε. In view of Theorem 1.3, we finally get ε log T (E)−1 + εC + 2π(m + 1)y0 . Θ(E, ε) ≤ (2.6) 2π The upper bound (2.6) remains true when n is even as the Lyapunov exponents for the matrix cocycles generated by M (z, E) and by S −1 (z +h)M (z, E)S(z) coincide.
2.3
Completing the proof
Recall that, in (2.6), y0 is an arbitrarily fixed positive number. So, comparing (2.3) ε and (2.6), we get Θ(E, ε) = log T (E)−1 + o(1). This and the formula (1.16) 2π for T (E) imply (1.4) for all E ∈ V0 ∩ R. Recall that V0 ∩ R is an open interval containing E0 ∈ J. The above construction can be carried out for any E0 ∈ J. As the interval J is compact, this completes the proof of Theorem 1.1.
3 Periodic Schr¨ odinger operators We now discuss the periodic Schr¨ odinger operator (0.2) where V is a 1-periodic, real valued, L2loc -function. We collect known results needed in the present paper (see [5, 16, 9, 11]).
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Bloch solutions
Let ψ be a solution of the equation −
d2 ψ(x, E) + V (x)ψ(x, E) = Eψ(x, E), dx2
x ∈ R,
(3.1)
satisfying the relation ψ(x + 1, E) = λ(E)ψ(x, E)
(3.2)
for all x ∈ R and some non-vanishing complex number λ(E) independent of x. Such a solution exists and is called a Bloch solution, and the number λ(E) is called the Floquet multiplier (see [16]). We consider the Bloch solution only at energies E where Ψ(0, E) = 0, and normalize it by the condition Ψ(0, E) = 1. Discuss its analytic properties as a function of E. As in Section 1.2, we denote the spectral bands of the periodic Schr¨ odinger equation by [E2n+1 , E2n+2 ], n ∈ N. Consider S± , two copies of the complex plane E ∈ C cut along the spectral bands. Paste them together to get a Riemann surface with square root branch points. We denote this Riemann surface by S. One can construct a Bloch solution ψ(x, E) of equation (3.1) meromorphic on S. The poles of this solution are located in the open spectral gaps or at their edges; the closure of each spectral gap contains exactly one pole that, moreover, is simple. It is located either on S+ or on S− . The position of the pole is independent of x (see [16, 11]). For E ∈ S, we denote by Eˆ the point on S different from E and having the same projection on C as E. We let ˆ E) = ψ(x, E), ˆ ψ(x,
E ∈ S.
ˆ E) is another Bloch solution of (3.1). Except at the edges of The function ψ(x, the spectrum (i.e., the branch points of S), the functions ψ and ψˆ are linearly independent solutions of (3.1). In the spectral gaps, ψ and ψˆ are real valued functions of x, and, on the spectral bands, they differ only by complex conjugation (see [16, 11]).
3.2
The Bloch quasi-momentum
Consider the Bloch solution ψ(x, E). The corresponding Floquet multiplier λ (E) is analytic on S. Represent it in the form λ(E) = exp(ik(E)). The function E → k(E) is the Bloch quasi-momentum of H0 . Its inverse denoted by k → E(k) is the dispersion relation of H0 . The Bloch quasi-momentum is an analytic multi-valued function of E. It has the same branch points as E → ψ(x, E) (see [11]). Let D be a simply connected domain containing no branch point of the Bloch quasi-momentum. In D, one can fix an analytic single-valued branch of k, say k0 .
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All the other single-valued branches of k that are analytic in D are related to k0 by the formulae k±,l (E) = ±k0 (E) + 2πl, l ∈ Z (3.3) (see [11]). Consider C+ , the upper half-plane of the complex plane. On C+ , one can fix a single-valued analytic branch of the quasi-momentum continuous up to the real line. It can be determined uniquely by the conditions Re k(E + i0) = 0 and Im k(E + i0) > 0 for E < E1 . We call this branch the main branch of the Bloch quasi-momentum and denote it by kp . The function kp conformally maps C+ onto the first quadrant of the complex plane cut at compact vertical slits starting at the points πl, l ∈ N. It is monotonically increasing along the spectral zones so that [E2n−1 , E2n ], the nth spectral band, is mapped on the interval [π(n−1), πn]. Along any open gap, Re kp (E +i0) is constant, and Im kp (E + i0) is positive and has only one non-degenerate maximum. All the branch point of kp are of square root type. Let El be a branch √ point. In a sufficiently small neighborhood of El , the function kp is analytic in E − El , and
kp (E) − kp (El ) = cl E − El + O(E − El ), cl = 0. (3.4) Finally, we note that the main branch can be continued analytically to the complex plane cut along (−∞, E1 ] and the spectral gaps ]E2n , E2n+1 [, n ∈ N∗ , of the periodic operator H0 .
3.3
A meromorphic function
Here, we discuss a function playing an important role in the adiabatic constructions. In [9], we have seen that, on S, there is a meromorphic function ω having the following properties: • the differential Ω = ω dE is meromorphic; its poles are the points of P ∪ Q, where P is the set of poles of E → ψ(x, E), and Q is the set of zeros of k , the derivative of k; • all the poles of Ω are simple; • if the residue of Ω at a point p is denoted by res p Ω, one has res p Ω = 1, ∀p ∈ P \ Q, res q Ω = −1/2, ∀q ∈ Q \ P, res r Ω = 1/2, ∀r ∈ P ∩ Q; (3.5) • if E ∈ S projects into a gap, then ω(E) ∈ R; ˆ • if E ∈ S projects inside a band, then ω(E) = ω(E).
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4 The complex WKB method for adiabatic problems In this section, following [10, 9], we briefly describe the complex WKB method for adiabatically perturbed periodic Schr¨ odinger equations −
d2 ψ(x) + [V (x) + W (εx + ζ)] ψ(x) = Eψ(x), dx2
x ∈ R.
(4.1)
Here, V is 1-periodic and real valued, ε is a small positive parameter, and the energy E is complex; one assumes that V ∈ L2loc and that W is analytic in D(W ), a neighborhood of the real line (W is not necessarily periodic in the present section). The parameter ζ is an auxiliary complex parameter used to decouple the “slow variable” ξ = εx and the “fast variable” x. The idea of the method is to study solutions of (4.1) in some domains of the complex plane of ζ and, then to recover information on their behavior in x ∈ R. Therefore, for D a complex domain, one studies solutions satisfying the condition: ψ(x + 1, ζ) = ψ(x, ζ + ε) ∀ζ ∈ D.
(4.2)
The aim of the WKB method is to construct solutions to (4.1) satisfying (4.2) and that have a simple asymptotic behavior when ε tends to 0. As we will see, this is possible in certain special domains of the complex plane of ζ. These domains will depend continuously on V , W and on the value of E (but not on ε). As ultimately we want to use these solutions to construct the monodromy matrix for Hz,ε , we will consider V and W as fixed, and pick an energy, say E, and, then, construct the WKB objects and solutions in a uniform way for energies in a neighborhood of E. So we essentially consider E to be fixed and in many cases forget all reference to it to alleviate the notations.
4.1
Standard behavior of consistent solutions
We first define two analytic objects central to the complex WKB method, the complex momentum and the canonical Bloch solutions. Then, we describe the standard behavior of the solutions studied in the framework of the complex WKB method. 4.1.1 The complex momentum The complex momentum κ is the main analytic object of the complex WKB method. For ζ ∈ D(W ), the domain of analyticity of the function W , it is defined by the formula κ(ζ) = k(E(ζ)),
E(ζ) = E − W (ζ),
(4.3)
Here, k is the Bloch quasi-momentum defined in Section 3.2. Though κ depends on E, for the reasons explained above, we will focus mainly on its behavior as a function of ζ, and thus not underline the E-dependence.
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Relation (4.3) “translates” properties of k into properties of κ. Hence, ζ → κ(ζ) is a multi-valued analytic function, and its branch points are related to the branch points of the quasi-momentum by the relations El + W (ζ) = E,
l = 1, 2, 3 . . .
(4.4)
Let ζ0 be a branch point of κ. If W (ζ0 ) = 0, then ζ0 is a branch point of square root type. If D ⊂ D(W ) is a simply connected set containing no branch points of κ, we call it regular. Let κp be a branch of the complex momentum analytic in a regular domain D. By (3.3), all the other branches that are analytic in D, are described by the formulae: κ± (4.5) m = ±κp + 2πm, where ± and m ∈ Z are indexing the branches. 4.1.2 Canonical Bloch solutions To describe the asymptotic formulae of the complex WKB method, one needs to construct Bloch solutions to the equation −
d2 ψ(x) + V (x)ψ(x) = E(ζ)ψ(x), dx2
E(ζ) = E − W (ζ),
x ∈ R,
(4.6)
that are moreover analytic in ζ on a given regular domain. Pick ζ0 a regular point (i.e., ζ is not a branch point of κ). Let E0 = E(ζ0 ). Assume that E0 ∈ P ∪ Q. Let U0 be a sufficiently small neighborhood of E0 , and let V0 be a neighborhood of ζ0 such that E(V0 ) ⊂ U0 . In U0 , we fix a branch of
the function k (E) and consider ψ± (x, E), the two branches of the Bloch solution ψ(x, E), and Ω± , the corresponding branches of Ω (see Sections 3.1 and 3.3). For ζ ∈ V0 , put
Ψ± (x, ζ) = q(E) e
E E0
Ω±
ψ± (x, E),
q(E) =
k (E),
E = E(ζ).
(4.7)
The functions Ψ± are called the canonical Bloch solutions normalized at the point ζ0 . The properties of the differential Ω imply that the solutions Ψ± can be analytically continued from V0 to any regular domain D containing V0 . The Wronskian w(Ψ+ (·, ζ), Ψ− (·, ζ)) of the canonical Bloch solutions satisfies (see [9]) w(Ψ+ (·, ζ), Ψ− (·, ζ)) = w(Ψ+ (·, ζ0 ), Ψ− (·, ζ0 )) = k (E0 )w(ψ+ (x, E0 ), ψ− (x, E0 )) (4.8) For E0 ∈ Q ∪ {El }, the Wronskian w(Ψ+ (·, ζ), Ψ− (·, ζ)) is non-zero.
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Solutions having standard asymptotic behavior
Fix E = E0 . Let D be a regular domain. Fix ζ0 ∈ D so that E(ζ0 ) ∈ P ∪ Q. Let κ be a continuous branch of the complex momentum in D, and let Ψ± be the canonical Bloch solutions normalized at ζ0 defined on D and indexed so that κ be the quasi-momentum for Ψ+ . Definition 4.1 Let σ be either + or −. We say that, in D, a consistent solution f has standard behavior (or standard asymptotics) if • there exists V0 , a complex neighborhood of E0 , and X > 0 such that f is defined and satisfies (4.1) and (4.2) for any (x, ζ, E) ∈ (−X, X) × D × V0 ; • f is analytic in ζ ∈ D and in E ∈ V0 ; • for any compact set K ⊂ D, there exists V ⊂ V0 , a neighborhood of E0 , such that, for (x, ζ, E) ∈ (−X, X) × K × V , f has the uniform asymptotic f =e
σ
i ε
ζ ζ0
κ dζ
(Ψσ + o (1))
as ε tends to 0;
(4.9)
• this asymptotic can be differentiated once in x without loosing its uniformity properties. We call ζ0 the normalization point for f . To say that a consistent solution f has standard behavior, we will use the following notation i ζ i ζ f ∼ exp σ κ dζ · Ψσ or f ∼ exp σ κ dζ · Ψσ ε ζ0 ε when the normalization point is of no importance.
4.3
Canonical domains
Canonical domains are important examples of domains where one can construct solutions with standard asymptotic behavior. They are defined using canonical lines. 4.3.1 Canonical lines A curve is called vertical if it is connected, piecewise C 1 , and if it intersects the lines {Im ζ = Const} transversally. Vertical curves are naturally parameterized by Im ζ, and we will always consider them as oriented upward, i.e., for a curve t → ζ(t), Imζ(t) is increasing with t. Let us note here that, to simplify notation in the sequel, we will not distinguish between complex curves (i.e., continuous mappings from an interval in R to C)
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and the set they define (i.e., the range of the mapping); which object is meant will be clear from the context. Let γ be a regular compact curve. On γ, fix κ, a continuous branch of the complex momentum. Definition 4.2 The curve γ is called canonical if it is vertical and if, along γ, ζ 1. Im ζ0 κdζ is strictly monotonously increasing with Im ζ, ζ 2. Im ζ0 (κ − π)dζ is strictly monotonously decreasing with Im ζ. Note that canonical lines are stable under small C 1 -perturbations. 4.3.2 Canonical domains Let K be a regular domain. On K, fix a continuous branch of the complex momentum, say κ. The domain K is called canonical if there exists two points ζ1 and ζ2 located on ∂K such that K is the union of curves that are canonical with respect to κ and that connect ζ1 and ζ2 . One has Theorem 4.1 ([9]) Let K be a bounded domain that is canonical with respect to κ. For sufficiently small positive ε, there exists two solutions of (4.1), say, (f± ) having standard behavior in K:
i ζ κdζ Ψ± . f± ∼ exp ± ε ζ0 For any fixed x ∈ R, the functions ζ → f± (x, ζ) are analytic in the smallest strip {Y1 < Im ζ < Y2 } containing K. One easily calculates the Wronskian of the solutions f± (x, ζ) to get w(f+ , f− ) = w(Ψ+ , Ψ− ) + o(1).
(4.10)
By (4.8), for ζ in any fixed compact subset of K and ε sufficiently small, the solutions f± are linearly independent.
4.4
The strategy of the WKB method
Our strategy to apply the complex WKB method is explained in great detail in [9]; we recall it briefly. First, we find a canonical line. Roughly, we build it out of segments of some “elementary” curves that are described in Section 4.6. Then, we find a “local” canonical domain K “enclosing” this line, see Section 4.5. For this domain, we construct the solutions f± using Theorem 4.1. Second, we describe the asymptotic behavior of f± outside the domain K. Therefore, we use three general principles that we state in Section 4.7.
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Having investigated the behavior of f± for x ∈ (−X, X) and a sufficiently large set of ζ, we recover the behavior of f± on the real line of x by means of condition (4.2). Below, we assume that D is a regular domain, that ζ0 is a fixed point in D, and that κ is a analytic branch of the complex momentum in D. A segment of a curve is a connected, compact subset of that curve.
4.5
Local canonical domains
Let γ ⊂ D be a canonical line (with respect to κ). Denote its ends by ζ1 and ζ2 . Let a domain K ⊂ D be a canonical domain corresponding to the triple κ, ζ1 and ζ2 . If γ ∈ K, then, K is called a canonical domain enclosing γ. One has Lemma 4.1 ([8]) One can construct a canonical domain enclosing any given canonical curve. Canonical domains whose existence is a consequence of this lemma are called local.
4.6
Pre-canonical lines
To construct a local canonical domain we need a canonical line. To construct such a line, we first build a pre-canonical line made of some “elementary” curves. Let γ be a vertical curve in D. We call γ pre-canonical if it is a finite union of segments of canonical lines and/or lines of Stokes type, i.e., of level curves of ζ ζ the harmonic functions ζ → Im ζ0 κdζ or ζ → Im ζ0 (κ − π)dζ. One has Proposition 4.1 ([8]) Let γ be a pre-canonical curve. Denote the ends of γ by ζa and ζb . For V ⊂ D, a neighborhood of γ and Va ⊂ D, a neighborhood of ζa , there exists a canonical line γ ⊂ V connecting the point ζb to a point in Va . When constructing pre-canonical lines, we identify complex number with vectors in R2 in the standard way and interpret the function ζ → κ(ζ) (as well as other functions constructed from it) as vector fields on complex domains. In particular, we will use the following simple result. ζ Lemma 4.2 The lines of Stokes type of the family Im ζ0 κ dζ = Const are tangent ζ to the vector field κ(ζ); those of the family Im ζ0 (κ − π) dζ = Const are tangent to the vector field κ(ζ) − π.
4.7
Tools for computing global asymptotics
A set is said to be constant if it is independent of the parameter ε.
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4.7.1 The Rectangle Lemma: asymptotics of increasing solutions The Rectangle Lemma roughly says that a solution f preserve the standard behavior along a line Im ζ = Const as long as the leading term of the standard asymptotics increases. Fix ηm and ηM positive such that ηm < ηM . Define S = {ζ ∈ C : ηm ≤ Im ζ ≤ ηM }. Let γ1 and γ2 be two vertical lines such that γ1 ∩ γ2 = ∅. Assume that both lines intersect the strip S at the lines Im ζ = ηm and Im ζ = ηM , and that γ1 is located to the left of γ2 . Consider the compact R bounded by γ1 , γ2 and the boundaries of S. Let D=R \ (γ1 ∪ γ2 ). One has Lemma 4.3 (The Rectangle Lemma [7]) Assume that the “rectangle” R is contained in a regular domain. Let f be a solution to (4.1) satisfying (4.2). Then, for sufficiently small ε, one has i
ζ
i
ζ
κdζ
Ψ+ in a neighbor1 If Im κ < 0 in D, and if f has standard behavior f ∼ e ε ζ0 hood of γ1 , then, it has standard behavior in a constant domain containing the “rectangle” R. κdζ
Ψ+ in a neighbor2 If Im κ > 0 in D, and if f has standard behavior f ∼ e ε ζ0 hood of γ2 , then, it has standard behavior in a constant domain containing the “rectangle” R. Lemma 4.3 was proved in [7] where one can find more details and references. 4.7.2 The Adjacent Canonical Domain Principle This principle complements the Rectangle Lemma; it allows us to obtain the asymptotics of decreasing solutions. Let γ be a compact vertical curve. Let S be the minimal strip of the form {C1 ≤ Im ζ ≤ C2 } containing γ. Let U ⊂ S be a regular domain such that γ ⊂ ∂U . We say that U is adjacent to γ. One has Proposition 4.2 (The Adjacent Canonical Domain Principle [7]) Let γ be a canonical line. Assume that f , a solution to (4.1) satisfying (4.2), has standard behavior in a domain adjacent to γ. Then, f has the standard behavior in any bounded canonical domain enclosing γ. 4.7.3 Adjacent canonical domains To apply the Adjacent Canonical Domain Principle, one needs to describe canonical domains enclosing a given canonical line. It can be quite difficult to find a “maximal” canonical domain enclosing a given canonical line. In practice, one uses “simple” canonical domains described in
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Lemma 4.4 (The Trapezium Lemma [9]) Let γ0 be a canonical line, and γ be a canonical line containing γ0 as an internal segment. Let U be a domain adjacent to γ. Assume that, in U , Im κ = 0. Let σu (resp. σd ) be the line of Stokes type beginning at the upper (resp. lower) end of γ0 and going in U downward (resp. upward) from this point (see Fig. 4(a)). One has: Trapezium case Let T ⊂ U be a regular domain bounded by σu , σd , γ0 and γ˜ , one more canonical line not intersecting γ0 . Then, T is part of a canonical domain enclosing γ0 . Triangle case Assume that σu intersects σd . Let T ⊂ U be a regular domain bounded by σu , σd and γ0 . Then, T is part of a canonical domain enclosing γ0 . There always exists a canonical line containing γ0 ; moreover, if Im κ = 0 in U , then, the lines σd and σu described in the Trapezium Lemma always exist (see Lemma 5.3 in [9]). σ3
Trapezium Triangle σu γ0 γ0 σu γ˜ σd σd (a) The Trapezium Lemma
S2
S3 ζ0
σ2
S1 : lines of Stokes type : Stokes lines
σ1
(b) The Stokes lines near a branch point
Figure 4.
4.7.4 The Stokes Lemma The domains where one justifies the standard behavior using the Adjacent Canonical Domain Principle are often bounded by Stokes lines (see definition below) beginning at branch points of the complex momentum. The Stokes Lemma allows us to justify the standard behavior beyond these lines by “going around” the branch points. Notations and assumptions. Assume that ζ0 is a branch point of the complex momentum such that W (ζ0 ) = 0. Definition 4.3 The Stokes lines starting at ζ0 are the curves γ defined by
ζ
(κ(ζ) − κ(ζ0 ))dζ = 0,
Im ζ0
ζ ∈ γ.
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The angles between the Stokes lines at ζ0 are equal to 2π/3. We denote them by σ1 , σ2 and σ3 so that σ1 is vertical at ζ0 (see Fig. 4(b)). Let σ ˜1 be a (compact) segment of σ1 which begins at ζ0 , is vertical and contains only one branch point, i.e., the point ζ0 . Let V be a neighborhood of σ ˜1 . Assume that V is so small that the Stokes lines σ1 , σ2 and σ3 divide it into three sectors. We denote them by S1 , S2 and S3 so that S1 be located between σ1 and σ2 , and the sector S2 be between σ2 and σ3 (see Fig. 4(b)). The statement. In [9], we have proved Lemma 4.5 (Stokes Lemma) Let V be sufficiently small. Let f be a solution to i
ζ
κdζ
(4.1) satisfying (4.2) that has standard behavior f ∼ e ε ζ0 Ψ+ inside the sector S1 ∪ σ2 ∪ S2 of V . Moreover, assume that, in S1 near σ1 , one has Im κ(ζ) > 0 if S1 is to the left of σ1 and Im κ(ζ) < 0 otherwise. Then, f has standard behavior inside V \ σ1 , the leading term of the asymptotics being obtained by analytic continuation from S1 ∪ σ2 ∪ S2 into V . Comments and details on this lemma can be found in [9].
5 A consistent solution We now begin with the construction of the consistent basis the monodromy matrix of which we compute. Recall that V and W satisfy the hypotheses (H1)–(H4), and that J is a compact real interval satisfying the hypotheses (A1)–(A3). We fix E = E0 ∈ J. In the present section, by means of the complex WKB method, we construct and study f a solution of (1.5) satisfying the consistency condition (1.6). To use the complex WKB method, we perform the following change of variable in (1.5) x−z →x
and εz → ζ.
(5.1)
One easily check that (1.5) then takes the form (4.1). In the new variables, the consistency condition (1.6) becomes (4.2). Note also that, in the new variables (5.1), for two solutions to (4.1) to form a consistent basis, in addition to being a basis of consistent solutions, their Wronskian has to be independent of ζ (see Definition 1.1). We now first describe the complex momentum and Stokes lines under our assumptions on V , W , and J. Then, we construct a local canonical domain, hence, a consistent solution to (4.1) by Theorem 4.1. Finally, using the continuation tools, we describe global asymptotics of this solution.
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SY − ζ2n
0 − ζ2n−1
+ ζ2n−1
ζ∗
+ ζ2n
2π
Figure 5. The branch points when m = 0.
5.1
The complex momentum
We begin with the analysis of the mapping E : ζ → E − W (ζ) (where W and E satisfy assumptions (H1)–(H4) and (A1)–(A3)). 5.1.1 The set W −1 (R) As E ∈ R, the set E −1 (R) coincides with W −1 (R). The set W −1 (R) is 2π-periodic. It consists of the real line and of complex branches (curves) symmetric with respect to the real line. There are complex branches separated from the real line, and complex branches beginning at the real extrema of W . The complex branches do not return to the real line. Consider an extremum of W on the real line, say ζ0 . By assumption (H4), it is non-degenerate. This implies that, near ζ0 , the set W −1 (R) consists of a real segment, and of a “complex” curve symmetric with respect to the real axis, intersecting the real axis at ζ0 only. This curve is orthogonal to the real line at ζ0 . For Y > 0, we let SY = {−Y ≤ Im ζ ≤ Y }. We assume that Y is so small that • SY is contained in the domain of analyticity of W ; • the set W −1 (R)∩SY consists of the real line and of the complex lines passing through the real extrema of W ; For such Y , and if m = 0, the set W −1 (R) ∩ SY is shown in Fig. 5. 5.1.2 Branch points The branch points of the complex momentum are related to the branch points of the Bloch quasi-momentum by equation (4.4). So, they lie on W −1 (R), and form a 2π-periodic set.
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As E and W satisfy (H1), (H3), (A2) and (A3), there exists C > 0 such that non real branch points have either an imaginary part larger than C or less that −C. Reducing Y , at no loss of generality, we from now on assume that the strip SY contains only the real branch points of the complex momentum. Consider the branch points located in the interval [0, 2π) of the real line. Recall that, by assumption (H4), the function ζ → W (ζ) has two extrema in [0, 2π]: a maximum at ζ = 0 and a minimum at ζ ∗ , 0 < ζ ∗ < 2π. The mapping E is monotonous on each of the intervals I− = [0, ζ∗ ] and I+ = [ζ ∗ , 2π] and maps both I± onto the interval [E − W+ , E − W− ]. Under the hypotheses (A1) – (A3), the interval [E − W+ , E − W− ] contains 2m + 2 branch points of the Bloch quasi-momentum, namely the points Ej for j = 2n − 1, 2n, . . . , 2(n + m). Therefore, on [0, 2π], one has 4m + 4 branch points of the complex momentum ζj± , j = 2n − 1, 2n, . . . 2(n + m), such that ζj± ∈ I± and E(ζj± ) = Ej . They satisfy the inequalities − − − 0 < ζ2n−1 < ζ2n < · · · < ζ2(n+m) + + + < ζ ∗ < ζ2(n+m) < ζ2(n+m)−1 < · · · < ζ2n−1 < 2π. (5.2)
5.1.3 The sets Z and G Consider the gaps and bands of the periodic operator (0.2). Let Z and G respectively be the pre-image (with respect to E) of the bands and the pre-image of the union of the interval (−∞, E1 ) and the spectral gaps. Clearly, Z ∪ G = W −1 (R), and Z ∩ G = ∅. Connected components of Z and G are separated by branch points of the complex momentum. The set Z is 2π-periodic. Consider the part of Z located in the interval [0, 2π]. Consider Z, the collection of subintervals of [0, 2π] defined in Section 1.3.1. One has − − + + z+ j = n, n + 1, . . . , n + m. (5.3) z− j = [ζ2j−1 , ζ2j ], j = [ζ2j , ζ2j−1 ], Under our assumptions on Y , the set Z ∩ SY consists of the intervals obtained by translating those of Z by 2πm, m ∈ Z. Define the finite collection of intervals G as in Section 1.3.2. One has − − + + + g− j = (ζ2j , ζ2j+1 ), gj = (ζ2j+1 , ζ2j ),
gn−1 =
+ (ζ2n−1
−
j = n, n + 1, . . . n + m − 1,
− 2π, ζ2n−1 ),
− + gn+m = (ζ2(n+m) , ζ2(n+m) ),
(5.4)
The set G ∩ SY consists of the following connected components: 1. the intervals g± j , j = n, . . . n + m − 1; 2. the connected component containing ζ = 0 (it is the union of the interval gn−1 and the complex branch of W −1 (R) ∩ SY passing through 0);
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3. the connected component containing ζ = ζ ∗ (it is the union of the interval gn+m and the complex branch of W −1 (R) ∩ SY passing through ζ ∗ ); 4. all the curves obtained by translating the curves obtained in the points (1), (2) and (3) by 2πm, m ∈ Z. 5.1.4 The main branch of the complex momentum Introduce the main branch of the complex momentum. In the strip {0 < Im ζ < Y }, consider the domain Dp between the complex branches of W −1 (R) beginning at 0 and at ζ ∗ . It is regular and E = E − W (ζ) conformally maps it onto a domain in the upper half of the complex plane. We define the main branch of κ by the formula κp (ζ) = kp (E − W (ζ)), ζ ∈ Dp , (5.5) where kp is the main branch of the Bloch quasi-momentum of the periodic operator (0.2) (see Section 3.2). Clearly, Im κp > 0 in Dp and the function κp is continuous up to the boundary of Dp . Its behavior at the boundary of Dp reflects the behavior of kp along the real line. In particular, for each j = n, n+ 1 . . . , n+ m, it is monotonously increasing on z− j and maps it onto [π(j − 1), πj]. 5.1.5 The Stokes lines − − Consider the Stokes lines beginning at ζ2n−1 and ζ2n . As κp is real on z− n = − − − − [ζ2n−1 , ζ2n ], this interval is a Stokes line both for ζ2n−1 and ζ2n . Pick one of these points. As W = 0 at this point, the angles between the Stokes lines beginning at it are equal to 2π/3. So, one of the Stokes lines is going upward, one is going downward. These two Stokes lines are symmetric with respect to the real line. − going downward, and denote Denote by σ1 the Stokes line starting from ζ2n−1 − by σ2 the Stokes line beginning at ζ2n and going upward (see Fig. 6).
σ1 σ2 − ζ2n−1
R − ζ2n
σ2 σ1 Figure 6. Stokes lines.
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The lines σ1 and σ2 are vertical in SY . Indeed, a Stokes line stays vertical as long as Im κ = 0; the imaginary part of the complex momentum vanishes only on Z, and Z ∩ SY ⊂ R. Reducing Y if necessary, we can assume that σ1 and σ2 intersect the boundaries of SY .
5.2
Local construction of the solution f
We construct f on a local canonical domain. To construct a local canonical domain, we need a canonical line. To find a canonical line, we first build a pre-canonical line. 5.2.1 Pre-canonical line − − Consider the curve β which is the union of the Stokes lines σ1 , [ζ2n−1 , ζ2n ] and σ2 . Let us construct α, a pre-canonical line close to the line β. It goes around the branch points of the complex momentum as shown in Fig. 7.
ζ Im (κ − π)dζ = C
σ1
ζ0
Im
ζ − ζ2n−1
D+ σu
κdζ = 0
σ2
− − ζ2n (κ(ζ2n ) = π) − ζ2n−1
− (κ(ζ2n−1 )
= 0)
ζ Im (κ − π)dζ = 0
ζb
ζa
W −1 (R) ∩ R σd
+ ζ2n
ζ Im κdζ = C
σ1
: α : β
ζ0
Figure 7. The construction of the pre-canonical curve. When speaking of κp along α, we mean the branch of the complex momentum obtained of κp by analytic continuation along this line (the analytic continuation can be done using formula (4.3)). Actually, the line α will be pre-canonical with respect to the branch of the complex momentum related to κp by the formula: for ζ ∈ Dp ,
κ(ζ) =
κp (ζ) − π(n − 1) if n is odd, if n is even. πn − κp (ζ)
In view of (4.5), κ is indeed a branch of the complex momentum. We prove
(5.6)
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Lemma 5.1 Fix δ > 0. In the δ-neighborhood of β, to the left of β, there exists a line α that is pre-canonical with respect to the branch κ such that, at its upper end, Im ζ > Y , and, at its lower end, Im ζ < −Y . Proof. We consider only the case n odd. The analysis of the case of n even − is similar. Note that, for n odd, formula (5.6) implies that κ(ζ2n−1 ) = 0, and ζ − κ(ζ2n ) = π. So, the Stokes lines σ1 and σ2 satisfy the equations Im ζ − κ dζ = 0 2n−1 ζ and Im ζ − (κ − π) dζ = 0. 2n
The pre-canonical line is constructed of lu and ld , two “elementary lines” that are segments of σu and σd , two lines of Stokes type shown in Fig. 7. Let us describe them more precisely. Pick ζu , a point of the line Im ζ = Y , to the left of β close enough to it. The ζ line σu is the line of Stokes type Im ζu (κ − π)dζ = Const containing ζu . Note − − that three curves, σu , σ2 and [ζ2n−1 , ζ2n ] ⊂ R, belong to the same family of curves ζ Im ζu (κ−π)dζ = Const. This implies that, if ζu is close enough to σ2 , then, below − − ζu , σu goes arbitrarily close to σ2 ∪ [ζ2n−1 , ζ2n ] staying to the left of σ2 and above − − [ζ2n−1 , ζ2n ]. We omit elementary details. Pick ζd , a point of the line Im ζ = −Y , to the left of β close enough to it. The ζ line σd is a line of Stokes type Im ζd κdζ = Const containing ζd . Note that the ζ three curves σd , σ1 and σ1 belong to the same family of curves Im ζd κdζ = Const. This implies that, if ζd is close enough to σ1 , then, above ζd , σd goes arbitrarily close to the line σ1 ∪ σ1 and stays to the left of it. We omit the details. Note that, by Lemma 4.2, the curves σu and σd are tangent to the vector fields κ − π and κ respectively. This implies in particular that each of the curves σu and σd stays vertical as long as it does not intersect the set Z. If σu and σd are chosen close enough to β, they intersect one another in a − . Indeed, consider first σu . It goes from ζu downward staying neighborhood of ζ2n−1 − − to the left of σ2 and above [ζ2n−1 , ζ2n ]. So, staying vertical, it has to intersect the − Stokes line σ1 above ζ2n−1 , the origin of σ1 . The intersection is transversal (as, first, σ1 is tangent to the vector field κ, second, σu is tangent to the vector field κ − π, and, third, at the intersection point, Im κ = 0). If σd is sufficiently close to σ1 , then σu also intersects σd transversally. We choose the intersection point as the lower end of lu and the upper end of ld . As σu and σd are defined and are vertical somewhat outside SY , we can assume that the upper end of lu is above the line Im ζ = Y , that the lower end of ld is below the line Im ζ = −Y , and that both ld and lu are vertical. The lines lu and ld then form a pre-canonical line α; it can be chosen as close to the line β as desired and, in particular, inside the δ-neighborhood of β. This completes the proof of Lemma 5.1.
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5.2.2 Local canonical domain and a solution f By Proposition 4.1, one can find a canonical line γ as close to α as desired. We construct γ in the left-hand side of the δ-neighborhood of β. By Lemma 4.1, there is K a canonical domain enclosing γ. We can assume that it is located in an arbitrarily small neighborhood of γ. So, we construct K in (the left part of) the δ-neighborhood of β. By Theorem 4.1, on the canonical domain we construct f , a solution K, i ζ to (4.1) that has standard behavior f ∼ exp ε κdζ Ψ+ in K. We fix the normalization of this solution later.
5.3
Asymptotics of f outside K
Recall that ζ → f (x, ζ) is analytic in {Y1 < Im ζ < Y2 }, the smallest strip containing K. 5.3.1 The results − Denote by ζa (resp. ζb ) the real branch point that is the closest to ζ2n−1 and to − the left of that point (resp. the closest to ζ2n and to the right of that point), see Fig. 7. Let D be a regular domain obtained by cutting SY along the Stokes lines σ1 , and σ2 and along the real intervals (−∞, ζa ] and [ζb , ∞), see Fig. 8.
− ζ2n
ζa 0
ζb
− ζ2n−1
Figure 8. How to “extend” the asymptotics of f . We prove Proposition 5.1 If δ is sufficiently small, then, f has standard behavior i
f = eε
ζ ζ0
κdζ
(Ψ+ (x, ζ, ζ0 ) + o(1))
(5.7)
in the whole domain D. The remainder of this section is devoted to the proof of Proposition 5.1. The proof is naturally divided into “elementary” steps. At each step, applying
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just one of the three continuation tools (i.e., the Rectangle Lemma, the Adjacent domain principle and the Stokes Lemma), we justify the standard behavior of f on one more subdomain of D. Fig. 8 shows where we use each of the continuation principles. Full straight arrows indicate the use of the Rectangle Lemma, circular arrows indicate the use of the Stokes Lemma, and, in the hatched domains, we use the Adjacent Canonical Domain Principle. Again, the analysis of the cases n odd and n even are analogous. For sake of ˜ = D ∪ (∂D ∩ definiteness, we assume that n odd and consider only this case. Let D {Im ζ = ±Y }). We shall use ˜ \ [ζ − , ζ − ], Lemma 5.2 If n is odd, then, in D 2n−1 2n • to the left of the Stokes lines σ1 and σ2 , one has Im κ > 0; • to the right of these Stokes lines, one has Im κ < 0. Proof. The sign of Im κ remains the same in any regular domain not intersecting Z. Moreover, the sign of Im κ changes to the opposite one as ζ intersects a connected component of Z at a point where W = 0. So, in the connected component of ˜ − , ζ − ] to the left of σ1 ∪[ζ − , ζ − ]∪σ2 , one has Im κ = Im (κp −π(n−1)) = D\[ζ 2n−1 2n 2n−1 2n Im κp > 0 (here, we have used (5.6) for n odd). To come from this subdomain to ˜ \ [ζ − , ζ − ] to the right of σ1 ∪ [ζ − , ζ − ] ∪ σ2 , the connected component of D 2n−1 2n 2n−1 2n − − one has to intersect the interval [ζ2n−1 , ζ2n ] which is a connected component of Z. So, in the right subdomain, one has Im κ < 0. 5.3.2 Behavior of f between the lines γ and β To justify the standard asymptotics of f in D between the lines γ and β, we use the Adjacent Canonical Domain Principle. Therefore, we need to describe a canonical domain enclosing γ (more precisely, the part located between γ and β). We do this by means of the Trapezium Lemma 4.4 (first statement). Let us describe the domain U and the curves γ0 , σd and σu used to apply the Trapezium Lemma 4.4. The domain U . It is the domain bounded by β, γ and the lines Im ζ = C containing the ends of γ. In view of Lemma 5.2, choosing the ends of γ closer to the lines Im ζ = ±Y if necessary, we can assume that Im κ > 0 in the domain U . The line σu . As the line σu , we take the line which belongs to the family ζ Im ζ0 κdζ = Const and intersects β at a point ζ˜u satisfying Im ζ˜u = Y . Recall that γ is constructed in the δ-neighborhood of β where δ can be fixed arbitrarily small. One has Lemma 5.3 The line σu enters U at ζ˜u and goes upward. If δ is sufficiently small, then, σu intersects γ at an internal point of γ.
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Proof. The main tool of this proof is Lemma 4.2. Below, we use it without further notice. Recall that, above the real line, β coincides with the Stokes line σ2 . So, it is tangent to the vector field κ(ζ) − π. The line σd is tangent to the vector field κ(ζ). In U , and in particular at ζ˜u , one has Im κ > 0. Therefore, at ζ˜u , the tangent vector to β oriented upward is directed to the right with respect to the tangent vector to σu oriented upward. So, σu enters U at ζ˜u going upward. Note that σu intersect β transversally and is independent of δ. So, if δ is small enough, then σu intersects γ. ζ The line σd . It is the line of Stokes type Im ζ0 κdζ = Const that intersects β at a point ζ˜d satisfying Im ζ˜d = −Y . One has Lemma 5.4 The line σd enters U at ζ˜d and goes downward. If δ is sufficiently small, then σd intersects γ at an internal point. The proof of this lemma being similar to the proof of Lemma 5.3, we omit it. The line γ0 . We choose δ so that both σu and σd intersect γ. Then, γ0 is the segment of γ between its intersection points with σd and σu . 5.3.3 A description of the curve γ˜ Let us describe the canonical line γ˜ needed to apply the first variant of the Trapezium Lemma. As for γ, using Lemma 5.1, we can construct γ˜ so that it be arbitrarily close to β and strictly between γ0 and β. As σu and σd intersect γ and β, they also intersect γ˜ . 5.3.4 The completion of the analysis By the Trapezium Lemma, the domain bounded by γ0 , σu , σd and γ˜ is a part of the canonical domain enclosing γ0 . So, by the Adjacent Canonical Domain Principle, f has the standard behavior here. As γ˜ can be constructed arbitrarily close to β, we conclude that f has the standard behavior in the whole domain bounded by β, γ and the lines |Im ζ| = Y .
5.4
Behavior of f to the left of γ
We justify the standard behavior of f in D to the left of γ by means of the Rectangle Lemma. ˜ Let us describe R, the rectangle used to apply Lemma 4.3: it is the part of D ˜ between the canonical line γ1 = γ and a vertical line in D, say γ2 , staying to the left of γ and going from Im ζ = −Y to Im ζ = Y . By Lemma 5.2, in R, the imaginary part of κ is positive. Moreover, f has standard behavior in a neighborhood of γ. So, by the Rectangle Lemma, f has standard behavior in R.
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˜ can be taken so that Pick ζ ∈ D to the left of γ. As the vertical curve γ2 ⊂ D ζ ∈ R, we see that f has standard behavior in the whole part of D located to the left of γ.
5.5
Behavior of f to the right of σ1
First, by means of the Stokes Lemma, Lemma 4.5, we show that f has standard behavior to the right of σ1 in its neighborhood. Let V˜1 be a sufficiently small constant neighborhood of σ1 and set V1 = ˜ V1 ∩ SY . Let us show that f has standard behavior in V1 \ σ1 . The Stokes lines − − σ1 , σ1 and [ζ2n−1 , ζ2n ] divide V1 into three sectors. By the previous steps, we know that f has the standard behavior in V1 outside the smallest sector bounded − − , ζ2n ], say, S. The Stokes line σ1 is vertical. By by the Stokes lines σ1 and [ζ2n−1 Lemma 5.2, in V1 , to the left of σ1 , the imaginary part of the complex momentum is positive; thus, the Stokes Lemma implies that f has standard behavior inside V1 \ σ1 . Recall that the leading term of the asymptotics of f in S is obtained by − analytic continuation from the remainder of V1 around the branch point ζ2n−1 avoiding σ1 . Having justified the standard behavior of f to the right σ1 in a small neighborhood of σ1 , we justify it in in the remainder of the subdomain of D ∩ {−Y < Im ζ < 0} located to the right of σ1 by means of the Rectangle Lemma. The argument is similar to that carried out in Subsection 5.4. So, we omit the details.
5.6
− − Behavior of f along the interval (ζ2n−1 , ζ2n )
In the previous steps, we have justified the standard behavior of f both above and − − , ζ2n ). We show now that f has standard behavior below the real interval (ζ2n−1 also in this interval. − By Section 5.5, we know that f has standard behavior in a neighborhood of ζ2n−1 cut along σ1 . Moreover, f cannot have the standard behavior in a neighborhood − − − − ζ2n (as ζ2n is a branch point). Hence, there exists a ∈ (ζ2n−1 , ζ2n ] such that f has − the standard behavior in a neighborhood of any point in (ζ2n−1 , a), but not at a. − Assume that a < ζ2n . Let α be a segment of the line Re ζ = a connecting a point a1 ∈ C− to a point a2 ∈ C+ . One has 0 < κ(a) < π. This implies that, if the length of α is sufficiently small, then, α is a canonical line. The solution f has the standard behavior to the left of α. By the Adjacent Canonical Domain Principle, f has standard behavior in any local canonical domain enclosing α, thus, in a − . constant neighborhood of a. So, we obtain a contradiction, and, thus, a = ζ2n − − This completes the analysis of f along (ζ2n−1 , ζ2n ).
5.7
Behavior of f to the right of σ2
One studies f to the right of σ2 in the same way as we have studied it to the right of σ1 : first, using the Stokes Lemma, one justifies the standard behavior to
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the right of σ2 , in a small neighborhood of σ2 , and, then, applying the Rectangle Lemma, one proves that f has the standard behavior in the remaining part of the subdomain of D to the right of this neighborhood. We omit further details. The analysis of f to the right of σ2 completes the proof of Proposition 5.1.
5.8
Normalization of f
To fix the normalization of the leading term of the asymptotics of f , we choose the normalization
point ζ0 in D and, in a neighborhood of ζ0 , we choose a branch of the function k (E(ζ)) in the definition of Ψ+ . As the normalization point, we take ζ0 such that − − < ζ0 < ζ2n . ζ2n−1
(5.8)
Inside any spectral band of the periodic operator, k (E) does not vanish, and there are no poles of the Bloch solution ψ(x, E). So, E(ζ0 ) ∈ P ∪ Q, and the solution Ψ+ is well defined. √ To fix the branch of k , we note that, inside any spectral band of the periodic operator, the main√branch of the Bloch quasi-momentum, kp , is real and satisfies kp > 0. So, we fix k so that
k (E(ζ)) > 0,
− − ζ2n−1 < ζ < ζ2n .
(5.9)
6 The consistent basis Up to now, we have constructed f , one consistent solution to (4.1) (i.e., a solution to (4.1) satisfying (4.2)) with explicit asymptotic behavior in the domain D. We now construct another consistent solution f ∗ so that (f, f ∗ ) form a consistent basis.
6.1
Preliminaries
For ζ ∈ D∗ , the symmetric to D with respect to the real line, we define f ∗ (x, ζ, E) = f (x, ζ, E).
(6.1)
As W is real analytic, the function f ∗ is also a solution of (4.1); it satisfies the consistency condition as f does. In the next subsections, we first study its asymptotic behavior; then, we compute the Wronskian w(f, f ∗ ). We show that, as ε tends to 0, in SY , w(f, f ∗ ) has the form C (1 + o(1)), where C is a non-vanishing constant, and o(1) is a function which can depend on ζ. Finally, we modify the solution f so that it still have the standard behavior in D, and w(f, f ∗ ) be constant. Recall that we study the standard behavior of f ∗ at E = E0 ∈ J.
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The asymptotics of f ∗
− − Note that D ∩ D∗ contains the interval z = (ζ2n−1 , ζ2n ) ⊂ R. One has
Lemma 6.1 In D∗ , the solution f ∗ has the standard behavior f ∼e
− εi
ζ ζ0
κ∗ dζ
Ψ−,∗ (x, ζ, ζ0 ),
(6.2)
where, for E = E0 , • κ∗ is the branch of the complex momentum which coincides with κ on z and is analytic in D∗ , • Ψ−,∗ is the canonical Bloch solution which coincides with Ψ− (corresponding to Ψ+ from the asymptotics of f ) on z and is analytic in D∗ . Proof. Recall that, by Proposition 5.1, f has the standard behavior (5.7) in the domain D. The statement of Lemma 6.1 follows from Proposition 5.1, the definition of f ∗ and the relation ¯
ζ i ζ i ∗ ¯ ζ0 ) = exp − ∀ζ ∈ D , exp κdζ Ψ+ (x, ζ, κ∗ dζ Ψ−,∗ (x, ζ, ζ0 ). ε ζ0 ε ζ0 (6.3) Let us prove this relation. As both the right- and left-hand sides of (6.3) are analytic in ζ, it suffices to check (6.3) along the interval z. Recall that the interval − − [ζ2n−1 , ζ2n ] is a connected component of Z. This implies that ∀ζ ∈ z,
κ(ζ) = κ(ζ),
ψ+ (x, E(ζ)) = ψ− (x, E(ζ)),
(6.4)
where ψ± (x, E) are two different branches of the Bloch solution ψ(x, E). As ζ0 satisfies (5.8), relation (6.3) follows from the first relation in (6.4) and the relation (6.5) ∀ζ ∈ z, Ψ+ (x, ζ, ζ0 ) = Ψ− (x, ζ, ζ0 ). To check (6.5), we recall that Ψ± are defined in Section 4.1.2 by formula (4.7). Therefore, relation (6.5) follows from (5.9), the second relation in (6.4) and the last property of ω listed in Section 3.3. This completes the proof of Lemma 6.1.
6.3
The Wronskian of f and f ∗
The solutions f and f ∗ are analytic in the strip SY . Here, we study their Wronskian. As both f and f ∗ satisfy condition (4.2), the Wronskian is ε-periodic in ζ. One has Lemma 6.2 The Wronskian of f and f ∗ admits the asymptotic representation: ∀ζ ∈ SY ,
w(f, f ∗ ) = w(f, f ∗ )(ζ, E, ε) = w(Ψ+ , Ψ− )(ζ, E)|ζ=ζ0 + g(E, ζ, ε), (6.6)
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where the function (ζ, E) → g(ζ, E, ε) satisfies • ζ → g(ζ, E, ε) is analytic and ε-periodic in SY ; • for ζ ∈ R and E ∈ V0 ∩ R, one has Re g(ζ, E, ε) = 0; • for any compact K ⊂ SY , there exists a neighborhood of E0 , say, V0 , such that sup |g(ζ, E, ε)| = o(1) when ε → 0. (6.7) ζ∈K E∈V0
Remark 6.1 Note that 1. w(Ψ+ , Ψ− )|ζ=ζ0 = 0 (as E(ζ0 ) ∈ P ∪ Q, see (5.8) and the comments to it); 2. w(Ψ+ , Ψ− )|ζ=ζ0 ∈ iR (due to (6.5)). Proof. The domain D ∩ D∗ contains the “rectangle” R bounded by the lines σ1 ∪ σ1 , σ2 ∪ σ2 and Im ζ = ±Y . So, in R, the solutions f and f ∗ have the standard behavior (5.7) and (6.2). Consider the functions κ∗ and Ψ−,∗ from (6.2). Their definitions (see Lemma 6.1) imply that, for E = E0 , ∀ζ ∈ R,
κ∗ (ζ) = κ(ζ),
Ψ−,∗ (ζ, E) = Ψ− (ζ, E).
Using this information and (5.7) and (6.2), one obtains ∀ζ ∈ R,
w(f, f ∗ ) = w(Ψ+ (·, ζ), Ψ− (·, ζ)) + g,
g = o(1).
(6.8)
Being obtained using standard behavior, this estimate is uniform in ζ in any compact of R provided E be in a sufficiently small neighborhood of E0 . By (4.8), the first term in the left-hand side of (6.8) coincides with the first term in (6.6). So, we only have to check that g has all the properties announced in Lemma 6.2. As w(f, f ∗ ) is ε-periodic, so is g. Furthermore, ig is real analytic as iw(f, f ∗ ) and iw(Ψ+ , Ψ− )|ζ=ζ0 are. This completes the proof of Lemma 6.2.
6.4
Modifying f
To recover a consistent basis in the sense of Definition 1.1 by reverting the change of variables (5.1), we need to guaranty that the Wronskian of (f, f ∗ ) be independent of ζ. As Lemma 6.2 shows, this need not be the case. Therefore, we redefine the solution f (and thus f ∗ ) in the following way f := f /ν, ν = 1 + g/w(Ψ+ , Ψ− )|ζ=ζ0 . In terms of this new solution f , we define the new f ∗ by (6.1). The functions (f, f ∗ ) are a consistent basis ; we shall study the monodromy matrix obtained from this basis. For these “new” functions f and f ∗ , we have
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Theorem 6.1 For ε sufficiently small, the solutions f and f ∗ satisfy the condition (4.2) and w(f, f ∗ ) = w(Ψ+ , Ψ− )|ζ=ζ0 . (6.9) Moreover, f has the standard behavior (5.7) in D, and f ∗ has the standard behavior (6.2) in D∗ . Proof. We use Lemma 6.2 and Remark 6.1. Let ζ be in a fixed strip {y1 < Im ζ < y2 } ⊂ SY , and let ε be sufficiently small. Then, g is uniformly small, and f is correctly defined. Recall that ζ → g(ζ) is ε-periodic. So, ν is ε-periodic, and f and f ∗ remain consistent. Furthermore, note that ν is real analytic. This implies (6.9). Finally, as ν = 1 + o(1), the new solutions f and f ∗ still have the “old” standard asymptotic behavior in D and D∗ respectively.
7 General properties of the monodromy matrix for the basis {f, f ∗ } In the previous section (see Theorem 6.1), we have constructed a consistent basis (f, f ∗ ) of solutions of (4.1). If we return to the variables of the initial equation (1.5), we get a consistent basis of (0.1). The matrix discussed in Theorem 1.4 is the monodromy matrix obtained for this basis. In this short section, we check some of its properties. Instead of coming back to the initial variables, we continue to work in the variables (5.1). The definition of the monodromy matrix (1.7) takes the form f (x, ζ) F (x, ζ + 2π) = M (ζ)F (x, ζ), F = , (7.1) f ∗ (x, ζ) and the matrix M becomes ε-periodic in ζ. As the basis solutions f and f ∗ are related by (6.1), the monodromy matrix has the form (1.13). The definition of the monodromy matrix (7.1) implies that a(ζ) ≡ M11 (ζ)
=
b(ζ) ≡ M12 (ζ)
=
w(f (x + 2π, ζ), f ∗ (x, ζ)) , w(f (x, ζ), f ∗ (x, ζ)) w(f (x, ζ), f (x, ζ + 2π)) . w(f (x, ζ), f ∗ (x, ζ))
(7.2)
Finally, we note that the monodromy matrix is analytic in ζ in the strip SY and in E in a constant neighborhood of E0 . Indeed, as the solutions f and f ∗ are analytic functions of both variables, so are the Wronskians in (7.2). Moreover, by (6.9), the Wronskian appearing in the denominators in (7.2) does not vanish. Hence, we have proved the Lemma 7.1 The monodromy matrix corresponding to the basis {f, f ∗ }, say, M satisfies (7.1) and has the form (1.13). The function ζ → M (ζ, E) is analytic and ε-periodic in SY ; the function E → M (ζ, E) is analytic in a constant neighborhood of E0 . Its coefficients are given by (7.2).
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8 General asymptotic formulas To compute the asymptotics of the monodromy matrix defined above, we only need to compute the Wronskians appearing in (7.2). These Wronskians depend on ζ and have different asymptotics in the lower and upper half planes. Rather than repeating similar computations many times, in the present section, we obtain a general asymptotic formula for the Wronskian of two solutions having standard behavior.
8.1
General setting
In this subsection, we do not suppose that W be periodic. Fix E = E0 . Assume that h and g are two solutions of (4.1) having the standard asymptotic behavior in regular domains Dh and Dg : i
h ∼ eε
ζ ζh
κh dζ
i
Ψh (x, ζ),
g ∼ eε
ζ ζg
κg dζ
Ψg (x, ζ).
(8.1)
Here, κh (resp. κg ) is an analytic branch of the complex momentum in Dh (resp. Dg ), Ψh (resp. Ψg ) is the canonical Bloch solution Ψ+ defined on Dh (resp. Dg ), and ζh (resp. ζg ) is the normalization point for h and g. As the solutions h and g satisfy the consistency condition (4.2), their Wronskian is ε-periodic in ζ. We now describe the asymptotics of this Wronskian and of its Fourier coefficients. We first introduce several simple useful objects. ˜ Below, we assume that Dg ∩ Dh contains a simply connected domain, say D. 8.1.1 Arcs Let γ be a regular curve going from ζg to ζh in the following way: staying in Dg , ˜ then, staying in Dh , it goes to ζh . We say that it goes from ζg to some point in D, ˜ γ is an arc associated to the triple h, g and D. ˜ As D is simply connected, all the arcs associated to one and the same triple can ˜ naturally be considered as equivalent; we denote them by γ(h, g, D). ˜ The analysis perLet us continue κh and κg analytically along γ(h, g, D). formed in Section 4.1.1, see (4.5), yields, that, for V a small neighborhood of γ, one has κg (ζ) = σκh (ζ) + 2πm, ∀ζ ∈ V
for some
m ∈ Z, σ ∈ {−1, +1}.
(8.2)
˜ the signature of γ(h, g, D), ˜ and m = m(h, g, D) ˜ the index We call σ = σ(h, g, D) ˜ of γ(h, g, D). 8.1.2 Meeting domains ˜ be as above. We call D ˜ a meeting domain, if, in D, ˜ the functions Im κh and Let D Im κg do not vanish and are of opposite sign.
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Note that, for small values of ε, whether ζ → g(x, ζ) and ζ → h(x, ζ) increase or de i
ζ
κdζ
i
ζ
κdζ
crease is essentially determined by the exponential factors e ε ζg and e ε ζh . So, roughly, in a meeting domain, along the lines Im ζ = Const, the solutions h and g increase in opposite directions. 8.1.3 The amplitude and the action of an arc We call the integral
˜ = S(h, g, D)
κg dζ γ(h,g,d)
˜ Clearly, the action takes the same value for equivthe action of the arc γ(h, g, D). alent arcs. ˜ Consider the function Assume that E(ζ) ∈ P ∪ Q along γ = γ(h, g, D).
qg = k (E(ζ)) and the 1-form Ωg (E(ζ)) in the definition of Ψg . Continue them analytically along γ. Put
A(h, g, γ) = (qg /qh )|ζ=ζh e
γ
Ωg
.
(8.3)
We call A the amplitude of the arc γ. The first three properties of Ω listed in Section 3.3 imply the following result. ˜ coincide. Lemma 8.1 The amplitudes of two equivalent arcs γ(h, g, D) 8.1.4 Fourier coefficients ˜ be the smallest strip of the form {C1 < Im ζ < C2 } containing the Let S(D) ˜ One has domain D. ˜ = D(h, ˜ Proposition 8.1 Let D g) be a meeting domain for h and g, and m = ˜ be the corresponding index. Then, m(h, g, D) ˜ ∀ζ ∈ S(D),
w(h, g) = wm e
2πim (ζ−ζh ) ε
(1 + o(1))
as ε → 0,
(8.4)
and wm is the constant given by i
˜
˜ e ε S(h,g,D) w(Ψ+ (·, ζh ), Ψ− (·, ζh )), wm = A(h, g, D)
(8.5)
˜ where Ψ+ = Ψh and Ψ− is “complementary” to Ψ+ . For any compact K ⊂ S(D), there exists V0 , a neighborhood of E0 such that the asymptotics (8.4) is uniform for (ζ, E) ∈ K × V0 . The factor wm is the leading term of the asymptotics of the mth Fourier coefficient of w(h, g).
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˜ g , let γg (ζ) be a curve in Dg from ζg to ζ. Similarly, define γh (ζ). Proof. For ζ ∈ D ˜ one has First, we check that, for ζ ∈ D, i i i 2πim ˜ κ dζ − κ dζ , (8.6) e ε γg (ζ) g = e ε S(h,g,D) e ε (ζ−ζh ) e ε γh (ζ) h ˜ Ψ− (x, ζ), Ψg (x, ζ) = A(h, g, D)
(8.7)
where Ψ− is the canonical Bloch solution “complementary” to Ψ+ = Ψh in a neighborhood of γh (ζ). ˜ is a meeting domain, in a neighborhood of γ(h, g, D), ˜ one has As D κg = −κh + 2π m .
(8.8)
This implies relation (8.6). ˜ be an arc such that E(γ) ∩ (P ∪ Q) = ∅. Let us check (8.7). Let γ = γ(h, g, D) ˜ Note that qg and qh the arc γ(h, g, D). Continue qg , Ωg and ψg analytically along
are two different branches of the function k (E − W (ζ)). So, they differ at most by a constant factor. Therefore, in a neighborhood of ζh , we get ˜ qh (E(ζ))e Ψg (x, ζ) = A(h, g, D)
γh (ζ)
Ωg
ψg (x, E(ζ)).
(8.9)
Now, recall that, in a neighborhood of ζh , there are only two branches of Ω and ψ. Denote them by ψ± and Ω± so that ψ+ = ψh and Ω+ = Ωh . Then, either ψg = ψ− and Ωg = Ω− or ψg = ψ+ and Ωg = Ω+ . To choose between these two variants, we recall that the Bloch quasi-momentum of a Bloch solution is defined modulo 2π. Note that κh is the Bloch quasi-momentum of ψ+ , and κg is the Bloch quasi-momentum of ψg . By (8.8), we get κg = −κh mod 2π. So, κg must be the Bloch quasi-momentum of ψ− . Thus, we have ψg = ψ− and Ωg = Ω− , and (8.9) ˜ implies relation (8.7) in a neighborhood of ζh . By analyticity, it stays valid in D. ˜ ⊂ Dh ∩ Dg , both h and g have standard behavior in D. ˜ Substituting the As D asymptotics of f and g into w(f, g), and using (8.6) and (8.7), one easily obtains ˜ ∀ζ ∈ D,
i
˜
˜ e ε S(h,g,D) w(h, g) = A(h, g, D) e
2πim ε
(ζ−ζh )
(w(Ψ+ (·, ζ), Ψ− (·, ζ)) + o(1)).
(8.10)
As w(Ψ+ (·, ζ), Ψ− (·, ζ)) is independent of ζ and ε and is non-zero (see (4.8) and comments to it), we get (8.4). As this asymptotic was obtained using the standard behavior of h and g, it has all the announced uniformity properties. This completes the proof of Proposition 8.1.
8.2
The index m and the periods when W is periodic
Here, we only assume that W is 2π-periodic and real analytic in ζ (i.e., we do not assume anything on the critical points of W ), and that E is fixed. We describe the computation of the index m in the special case that one encounters when computing monodromy matrices.
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8.2.1 Periods Pick ζ0 , a regular point. Consider a regular curve γ going from ζ0 to ζ0 + 2π. Fix κ, a branch of the complex momentum that is continuous on γ. We call the couple (γ, κ) a period. Let (γ1 , κ1 ) and (γ2 , κ2 ) be two periods. Assume that one can continuously deform γ1 into γ2 without intersecting any branching point. This defines an analytic continuation of κ1 to γ2 . If the analytic continuation coincides with κ2 , we say that the periods are equivalent. Consider the branch κ along the curve γ of a period (γ, κ). In a neighborhood of ζ0 , the origin of γ, one has κ(ζ + 2π) = σκ(ζ) + 2πm,
σ ∈ {±1},
for some m ∈ Z.
(8.11)
The numbers σ = σ(γ, κ) and m = m(γ, κ) are called the signature and the index of the period (γ, κ). The numbers m (resp. σ) coincide for equivalent periods. Recall that G is the pre-image with respect to E of the union of the interval (−∞, E1 ) and the spectral gaps of H0 . One has Lemma 8.2 Let (γ, κ) be a period such that γ starts at a point ζ0 ∈ G. Assume that γ intersects G exactly N times (N ∈ N) and that, at all intersection points, W = 0. Let r1 , . . . , rN be the values that Re κ takes consecutively at these intersection points as ζ moves along γ from ζ0 to ζ0 + 2π. Then, σ(γ, κ) = (−1)N ,
m(γ, κ) =
1 (rN − rN −1 + rN −2 − · · · + (−1)N −1 r1 ). (8.12) π
Proof. The image E(γ) of γ by E : ζ → E − W (ζ) is a closed curve that starts and ends at E0 = E(ζ0 ). We consider the curve E(γ) as open at E0 . Along γ, we can write κ(ζ) = k(E − W (ζ)) where k is a fixed analytic branch of the quasimomentum. Let ko and ke be the values of k at the origin and at the end of the curve E(γ0 ). Then, κ(ζ0 ) = ko and κ(ζ0 + 2π) = ke . We now study the relationship between ko and ke . Since W = 0 at the points of intersection of γ0 and G, E(γ0 ) intersects exactly N times spectral gaps of the periodic operator. As the values for both m and σ coincide for equivalent periods, it suffices to construct ζ0 such that Im E0 = 0. Assume that a continuous curve begins at E0 , goes along a strait line to one of the ends of a gap, then goes around this gap end along an infinitesimally small circle, and returns to E0 along the same strait line. We call such a curve a simple loop. We distinguish the origin and the end of the loop considering the loop as open at its endpoints. As Im E0 = 0, any simple loop intersects only one gap, namely, the gap around the end of which it goes. Recall that the ends of the gaps coincide with the branching points of the Bloch quasi-momentum, and that these branching points are of square root type. So,
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in a neighborhood of a branching point, the corresponding branches of the Bloch quasi-momentum satisfy the relation k1 (E) + k2 (E) = 2r,
(8.13)
where r is the common value of these branches at the branching point. Note that r is equal to the value of the real part of any of these branches on the spectral gap beginning at the branching point. On a simple loop, fix a continuous branch of the quasi-momentum. Clearly, formula (8.13) also relates the values of the quasi-momentum at the ends of the loop when r is the value of the quasi-momentum at the branching point inside the loop. Recall that k can be analytically continued onto the whole complex plane cut along the spectral gaps of H0 . Therefore, the value of k at the end of E(γ0 ) is equal to the value of k at the end of the curve consisting of N simple loops and going successively around the branch points of k with k = r1 , r2 , . . . rN . So, taking (8.13) into account, we get ke = (−1)N kb + 2(rN − rN −1 + rN2 − . . . r1 ). This implies (8.12) and completes the proof of Lemma 8.2.
8.2.2 The index of arcs that are periods ˜ One has Let us come back to the computation of the index m(h, g, D). ˜ be an arc such that ζh = ζg + 2π. If, in a neighborLemma 8.3 Let γ = γ(h, g, D) hood of ζg , κg (ζ) = s · κh (ζ + 2π), (8.14) where s is either “+” or “-”, then, ˜ = s · σ(γ, κg ), σ(h, g, D)
˜ = m(γ, κg ). m(h, g, D)
(8.15)
˜ κg ) is a period. So, in a neighborhood of ζg , one has Proof. The pair (γ(h, g, D), κg (ζ + 2π) = σ(γ, κg )κg (ζ) + 2πm(γ, κg ). This and (8.14) imply that κg (ζ) = s σ(γ, κg )κh (ζ) + 2πm(γ, κg ) in a neighborhood of ζh . This implies the relations (8.15).
9 Asymptotics of the monodromy matrix We now compute the asymptotics of the coefficients a and b of the monodromy matrix for the basis {f, f ∗ }; in particular, we prove formulae (1.14) and (1.15).
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We concentrate on the case n odd. The computations for n even being similar, we omit them. Recall that a and b are expressed via the Wronskians appearing in formulae (7.2). We compute these Wronskians (and, thus, a and b) using the construction from Section 8.
9.1
The asymptotics of the coefficient b
By (7.2), we have to compute w(f (·, ζ), f (·, ζ + 2π)). One applies the constructions of Section 8 for h(x, ζ) = f (x, ζ),
g(x, ζ) = (T f )(x, ζ) Dh = D,
∀ζ ∈ Dh ,
ζh = ζ0 , κh (ζ) = κ(ζ);
where (T f )(x, ζ) = f (x, ζ + 2π); (9.1)
Dg = D − 2π;
(9.2)
ζg = ζ0 − 2π; ∀ζ ∈ Dg , κg (ζ) = κ(ζ + 2π).
(9.3) (9.4)
9.1.1 The asymptotics in the strip 0 < Im ζ < Y ˜ 0 , the meeting domain, and γ0 = γ(f, T f, D ˜ 0), the arcs used to Let us describe D compute w(f, T f ) in the strip {0 < Im ζ < Y }. ˜ 0 is the subdomain of the strip 0 < Im ζ < Y between the The meeting domain. D Stokes lines σ2 − 2π and σ2 . Indeed, it follows from Lemma 5.2 and (9.4) that, in ˜ 0 , one has Im κg = −Im κh < 0. D The arc. γ0 connects the point ζg to ζh . In view of (9.3), it defines the period (γ0 , κg ). ˜ 0 ). In view of (9.4), the arc γ0 satisfies the assumption of The index m(f, T f, D ˜ 0 ) = m(γ0 , κg ). Due to (9.4), m(f, T f, D ˜ 0 ) = m(γ0 + Lemma 8.3. So, m(f, T f, D 2π, κ). To compute this integer, we use Lemma 8.2. Therefore, we have to compute κ at the intersections of γ0 + 2π and G, the pre-image of the spectral gaps of H0 . The set G ∩ SY is described in Section 5.1.3 where we have listed all its connected components. Recall that the index takes one and the same value for all equivalent periods. We deform the period γ0 + 2π to γ, an equivalent one shown in Fig. 9 (for sake of simplicity, we drew this figure for the case when m = 0 in the hypothesis (A1)). The new period has the following properties • (γ, κ) is a period equivalent to (γ0 + 2π, κ), • γ stays to the left of the complex branch of W −1 (R) starting at 2π and staying in the upper half-plane, • γ has exactly three intersection points with G: it once intersects the complex − , ζb ) (the point branch of W −1 (R) going upward from 0, once the interval (ζ2n
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γ0 + 2π ζb 2π
− ζ2n
ζa 0 − ζ2n−1
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ζ∗ γ1 + 2π
W −1 (R)
Figure 9. Curves equivalent to γ0 + 2π and γ1 + 2π (when m = 0). ζb is defined in Section 5.3.1), and, once the complex branch of W −1 (R) going upward from ζ ∗ . Recall that ζ → Re κ(ζ) is constant on any connected component of G. Therefore, 1 m(γ0 + 2π, κ) = m(γ, κ) = Re κ(ζ)|ζ − − Re κ(ζ)|ζ − + Re κ(ζ)|ζ∈gn+m +i0 2n 2n−1 π 1 = 0 − π + Re κ(ζ)|ζ∈gn+m +i0 . π To compute the last term in this formula, we recall that, in Dp,left , the part of the domain Dp (see Section 5.1.4) located to the left of the Stokes line σ2 , one has κ = κp − π(n − 1). In the domain Dp,right , the part of Dp located to the right of σ2 , κ is obtained by analytic continuation from Dp,left around the branch − − passing below this branch point. As κp (ζ2n ) = πn, for ζ ∈ Dp,right , one point ζ2n has κ(ζ) = (2πn − κp (ζ)) − π(n − 1) = π(n + 1) − κp (ζ). Along gn+m , one has Re κp = π(n + m); hence, we get m(γ0 + 2π, κ) =
1 (0 − π + [π(n + 1) − π(n + m)]) = −m. π
(9.5)
The result. Now, Proposition 8.1, formula (7.2) for b and formula (6.9) imply formula (1.14) for b with ˜ 0 ) e εi S(f,T f,D˜ 0 )+ b−m = A(f, T f, D
2πimζ0 ε
,
(T f )(x, ζ) = f (x, ζ + 2π).
(9.6)
9.1.2 Asymptotics of b below the real line ˜ 1 ) needed to ˜ 1 , the meeting domain, and compute the index m(f, T f, D Describe D get the asymptotics of w(f, T f ) in the strip {−Y < Im ζ < 0}.
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˜ 1 is the subdomain of the strip −Y < Im ζ < 0 located The meeting domain. D between the Stokes lines σ1 − 2π and σ1 . ˜ 1 ) again defines a period, and m(f, T f, D ˜ 1) = The index. The arc γ1 = γ(f, T f, D m(γ1 + 2π, κ). The curve defining a period equivalent to (γ1 + 2π, κ) is shown in Fig. 9. As in the sequel of this computation we only use this curve, we call it γ1 + 2π. To compute the index of this period, we compute Re κ at the intersections of γ1 + 2π and G. We can assume that γ1 + 2π satisfies: • it is located to the left of the complex branch of W −1 (R) in C− starting at 2π, • it has exactly three intersection points with G: it once intersects the complex branch of W −1 (R) going downward from 0, once the interval gn−1 and once the complex branch of W −1 (R) going downward from ζ ∗ . We get 1 Re κ(ζ)|ζ∈gn+m −i0 . π We have used the fact that the interval gn−1 and the complex branch of W −1 (R) going downward from 0 belong to the same connected component of W −1 (R). To finish the computation, we introduce the domain Dp∗ , the symmetric of Dp ∗ with respect to the real line. In Dp,right , the part of this domain located to the right of σ1 , κ can be viewed as the analytic continuation of κp − π(n − 1) from − ∗ Dp across the interval z− n . Along the interval zn , κp is real. So, for ζ ∈ Dp,right , m(γ1 + 2π, κ) =
κ(ζ) = κp (ζ) − π(n − 1), and m(γ1 + 2π, κ) =
1 Re κp (ζ)|ζ∈gn+m +i0 − π(n − 1) = π 1 (π(n + m) − π(n − 1)) = m + 1 . (9.7) π
The result. Now, Proposition 8.1, formula (7.2) for b and (6.9) imply formula (1.15) for b with ˜ 1 ) e εi S(f,T f,D˜ 1 )− bm+1 = A(f, T f, D
9.2
2πi(m+1)ζ0 ε
,
(T f )(x, ζ) = f (x, ζ + 2π). (9.8)
The asymptotics of the coefficient a
The computations of the coefficient a following the same scheme as those of b, we only outline them. Now, h = f ∗, ∀ζ ∈ Dh ,
g = T f; Dh = D∗ , Dg = D − 2π; ζh = ζ0 , ζg = ζ0 − 2π; ¯ κh (ζ) = −κ(ζ); ∀ζ ∈ Dg , κg (ζ) = κ(ζ + 2π).
(9.9) (9.10) (9.11)
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− Recall that the complex momentum is real on z− n , and that ζ0 ∈ zn . This and relations (9.11) imply that
κg (ζ) = −κh (ζ + 2π),
ζ ∼ ζg .
(9.12)
9.2.1 The asymptotics of a above the real line ˜ 0 , the meeting domain, is the subdomain of the strip {0 < Im ζ < Y } In this case, D located between the lines σ2 − 2π and σ1 (which is symmetric to σ1 with respect ˜ 0 ) defines a period (˜ to R). The arc γ(f ∗ , T f, D γ0 , κg ); the curve γ˜0 + 2π is shown in Fig. 10. σ1 + 2π
σ2 γ˜0 + 2π ζa 0
ζ0
ζb
2π
ζ0 + 2π
ζ∗ γ˜1 + 2π σ2 + 2π
σ1
Figure 10. Curves γ˜0 + 2π and γ˜1 + 2π (when m = 0). In view of (9.12), one is again in the case of Lemma 8.14, and, by means ˜ 0 ) = m(˜ of Lemma 8.2, one obtains m(f ∗ , T f, D γ0 + 2π, κ) = −m. This yields formula (1.14) for a with ˜ 0) e εi S(f ∗ ,T f,D˜ 0 )+ a−m = A(f ∗ , T f, D
2πimζ0 ε
.
(9.13)
9.2.2 The asymptotics of a below the real line ˜ 1 , the meeting domain, is the subdomain of the strip {−Y < Im ζ < In this case, D 0} located between the lines σ2 (symmetric to σ2 with respect to R) and σ1 − 2π. ˜ 1 ) defines a period (˜ The arc γ(h, g, D γ1 , κg ); the curve γ˜1 + 2π is shown in Fig. 10. ˜ 1 ) = m + 1. This yields formula (1.15) for a with One obtains m(f ∗ , T f, D ˜ 1 ) e εi S(f ∗ ,T f,D˜ 1 )− am+1 = A(f ∗ , T f, D
2πi(m+1)ζ0 ε
.
(9.14)
10 Iso-energy curve The iso-energy curve Γ is defined by (0.4). In this formula, E(·) is the dispersion law for the periodic operator (0.2), i.e., the function inverse to the Bloch quasi-
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momentum (E = E(k) if and only if k is the value of one of the branches of k when the spectral parameter is equal to E). We begin with a simple general observation: Lemma 10.1 The iso-energy curve Γ is 2π-periodic in ζ- and κ-directions; it is symmetric with respect to any of the lines κ = πm, m ∈ Z. Proof. The periodicity in ζ follows from the periodicity of W . Fix k0 ∈ C. The list (3.3) shows that E(k) takes the same value for all k = σk0 + 2πm, where σ ∈ {±1}, and m ∈ Z. This implies the periodicity and the symmetries in κ. Now, for W satisfying (H1)–(H4) and for E in J, an interval satisfying (A1)–(A3), we discuss the iso-energy curves (0.3) and (0.4) and obtain the estimates (1.16).
10.1
Real iso-energy curve: the proof of Proposition 1.1
A point (ζ, κ) ∈ R2 belongs to ΓR if and only if κ is the value of one of the branches of the complex momentum at ζ. Recall that the intervals z ∈ Z are pre-images (with respect to E) of spectral bands. The complement of these intervals in (0, 2π) is mapped by E into spectral gaps. So, on (0, 2π), κ takes real values only on the intervals of Z. Therefore, in the strip {0 ≤ κ ≤ 2π}, the connected components of ΓR are located above the intervals z ∈ Z (“above” refers to the projection Π : (ζ, κ) ∈ R2 → ζ ∈ R). Pick j ∈ {n, n + 1 . . . n + m}, and σ ∈ {±}. Consider the part of ΓR above the interval z := zσj . Recall that E defines a bijection from z onto the jth spectral band of H0 . So, there exists κ0 , a branch of the complex momentum, continuous on the interval z, σ and mapping it monotonously onto the interval [π(j − 1), πj] so that κ0 (ζ2j−1 )= σ π(j − 1) and κ0 (ζ2j ) = πj. On the interval [π(j − 1), πj], let Zz be the inverse of ζ → κ0 (ζ). We continue κ → Zz (κ) to the real line so that it be 2π periodic and even. Recall that all the values of all the branches of the complex momentum at ζ ∈ z are given by the list (4.5). This and the definition of Zz imply that the points of ΓR above z are points of the graph of Zz and, reciprocally, all the points in the graph of Zz lie above points of z. All the properties of the function Zz announced in Proposition 1.1 follow directly from this construction. To prove that the connected components of ΓR depend continuously on E, it suffices to check that each of the functions Zz depends continuously on E ∈ J. Pick z ∈ Z. As W (ζ) = 0 for all ζ ∈ z, the continuity of E → Zz immediately follows from the Local Inversion Theorem and the definition of the iso-energy curve (0.3). This completes the proof of Proposition 1.1.
10.2
Loops on the complex iso-energy curve
Here, we discuss closed curves in Γ.
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10.2.1 An observation We define the intervals g ∈ G as in Section 1.3.2. We shall use Lemma 10.2 Pick g ∈ G. Let V (g) be complex neighborhood of g sufficiently small so that it contains only two branch points of κ, namely, the ends of g. Let κ be an analytic branch of the complex momentum in a sufficiently small neighborhood of a point of V (g) \ g. Then, κ can be analytically continued to the domain V (g) \ g to a single-valued function. The analytic continuation satisfies κ(ζ) = κ(ζ),
ζ ∈ V (g) \ g.
(10.1)
Proof. We can continue κ to a branch of the complex momentum analytic in the simply connected domain V (g) obtained from V (g) by cutting it, say, along R from the right end of g to +∞. It suffices to check that the values of κ at the edges of the cut coincide. The set (R ∩ V (g)) \ g consists of two intervals. Each of them belongs to Z (the pre-image of the spectral bands with respect to E). So, κ is real both on the left of these two intervals and at the edges of the cut. As κ is real on the left interval, one has (10.1) in V . So, the values of κ on the edges of the cut satisfy κ(ζ+i0) = κ(ζ − i0), and, therefore, being real, coincide. This implies Lemma 10.2. 10.2.2 The loops Pick g ∈ G. On V (g) \ g, fix κ0 , a single-valued analytic branch of the complex momentum. Consider G(g) ⊂ V (g) \ g, a curve turning once around the interval g. One has Lemma 10.3 For each σ ∈ {±1} and m ∈ Z, the curve ˆ (m,σ) (g) = {(ζ, κ) : κ = σκ0 (ζ) + 2πm, ζ ∈ G(g)}, G is a closed curve on Γ. It connects the two connected components of ΓR that project onto the intervals of Z ∩ R adjacent to g. ˆ (0,0) is a closed curve on Γ. This and Proof. As κ0 is univalent on G(g), the curve G ˆ (σ,m) are loops in Γ. As G(g) intersects Lemma 10.1 imply that all the curves G ˆ the intervals of Z ∩ R adjacent to g, G connects the two connected components of ΓR that project onto these intervals.
10.3
Tunneling coefficients
Pick g ∈ G. Fix an analytic branch κ of the complex momentum on V (g)\g. Define the action S(g) = i G(g) κdζ. To study its properties, we use
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Lemma 10.4 Let E ∈ J. If G(g) is positively oriented, then
Im κ dζ, S(g) = ±2
(10.2)
g±i0
where, in the left-hand side, one integrates in the increasing direction on the real axis. Proof. Deform the integration contour G(g) so that it go around g just next to g. Then, relation (10.2) follows directly from (10.1). This lemma immediately implies Corollary 10.1 Let E ∈ J. Then, 1. S(g) is real and non-zero; 2. as a functional of the branch κ, it takes only two values that are of opposite sign. Proof. Inside any spectral gap, the imaginary part of no branch of the Bloch quasi-momentum vanishes. Hence, the first statement follows from (10.2). The second one follows from (4.5) listing all the branches continuous on the integration contour. In the sequel, we choose the branch κ so that, on J, S(g) be positive. S(g) is called the tunneling action.
10.4
Obtaining estimates (1.16)
All the estimates in (1.16) are obtained in the same way. So, we prove only the estimate for b−m in the case of n odd. Recall that we work in a small constant neighborhood of a point E0 ∈ J, say, V0 (where “constant” means independent of ε). The coefficient b−m is given by (9.6). The definition of the amplitude of an arc, formula (8.3), implies that A(f, T f, d1 ) is independent of ε, continuous in E and does not vanish. So, there are two positive constants C1 and C2 such that C1 ≤ |A(f, T f, d0 )| ≤ C2 , E ∈ V0 . (10.3) Let us estimate the factor exp εi S(f, T f, d0 ) for E ∈ V0 ∩R. Therefore, we choose the arc γ = γ(f, T f, d0 ) stretched along the real line and turning around the branch points (between ζ0 − 2π and ζ0 , the origin and the end of γ) along infinitesimally small circles. We compute
i 1 1 exp S(f, T f, d0 ) = exp − Im κg dζ = exp − Im κg dζ , ε ε γ ε ˜ g g∈G
(10.4) where G˜ consists of all the connected components of G ∩ R between ζ0 − 2π and ζ0 , i.e., of all the intervals g± j − 2π, the interval gn+m − 2π and the interval gn−1 .
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Using (9.4), one easily checks that, in the right-hand side of (10.4), Im κg < 0 inside each of the intervals of integration (which are segments of the arc γ). Due to the periodicity of W , we can write
exp i S(f, T f, d0 ) = exp − 1 Im κ dζ , (10.5) ε ε g g∈G
where, on each interval of integration, κ is any continuous branch of the complex momentum such that Im κ < 0. By means of Lemma 10.4, we check that, up to the sign, the expression −2 g Im κdζ is equal to S(g), the tunneling action. As i S(f, T f, d0 ) = both are positive, they coincide. Therefore, exp (t(g))−1 . ε g∈G
This and (10.3) imply the estimate for b−m announced in (1.16). Acknowledgments. A.F. thanks the Universit¨ at Potsdam where part of this work was done. F.K.’s research was partially supported by the program RIAC 160 at Universit´e Paris 13 and by the FNS 2000 “Programme Jeunes Chercheurs”. Both authors thank the Mittag-Leffler Institute where part of this work was done.
References [1] J. Avron and B. Simon, Almost periodic Schr¨ odinger operators, II. The integrated density of states, Duke Mathematical Journal 50, 369–391 (1983). [2] V.S. Buslaev and A.A. Fedotov, Bloch solutions for difference equations, Algebra i Analiz 7(4), 74–122 (1995). [3] R. Carmona and J. Lacroix, Spectral Theory of Random Schr¨ odinger Operators, Birkh¨ auser, Basel, 1990. [4] H.L. Cycon, R.G. Froese, W. Kirsch, and B. Simon, Schr¨ odinger Operators, Springer Verlag, Berlin, 1987. [5] M. Eastham, The spectral theory of periodic differential operators, Scottish Academic Press, Edinburgh, 1973. [6] A. Fedotov and F. Klopp, In progress. [7] A. Fedotov and F. Klopp, On the absolutely continuous spectrum of one dimensional quasi-periodic Schr¨ odinger operators in the adiabatic limit, Preprint, Universit´e Paris-Nord, 2001. Mathematical Physics Preprint Archive, preprint 01-224. [8] A. Fedotov and F. Klopp, Anderson transitions for a family of almost periodic Schr¨ odinger equations in the adiabatic case, Comm. Math. Phys. 227(1), 1–92 (2002).
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[9] A. Fedotov and F. Klopp, Geometric tools of the adiabatic complex WKB method, To appear in Asymp. Anal (2004). [10] A. Fedotov and F. Klopp, A complex WKB method for adiabatic problems, Asymptot. Anal. 27(3-4), 219–264 (2001). [11] N.E. Firsova, On the global quasimomentum in solid state physics. In Mathematical methods in physics (Londrina, 1999), pages 98–141. World Sci. Publishing, River Edge, NJ, 2000. [12] M.-R. Herman, Une m´ethode pour minorer les exposants de Lyapounov et quelques exemples montrant le caract`ere local d’un th´eor`eme d’Arnol’d et de Moser sur le tore de dimension 2. Comment. Math. Helv. 58(3), 453–502 (1983). [13] Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schr¨ odinger operators, Invent. Math. 135(2), 329–367 (1999). [14] L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators. Springer Verlag, Berlin, 1992. [15] E. Sorets and T. Spencer, Positive Lyapunov exponents for Schr¨ odinger operators with quasi-periodic potentials, Comm. Math. Phys. 142(3), 543–566 (1991). [16] E.C. Titschmarch, Eigenfunction expansions associated with second-order differential equations. Part II. Clarendon Press, Oxford, 1958. [17] M. Wilkinson, Critical properties of electron eigenstates in incommensurate systems. Proc. Roy. Soc. London Ser. A 391(1801), 305–350 (1984). [18] M. Wilkinson, Tunnelling between tori in phase space. Phys. D 21(2-3), 341–354 (1986). Alexander Fedotov Department of Mathematical Physics St Petersburg State University 1, Ulianovskaja 198904 St Petersburg-Petrodvorets, Russia email:
[email protected]
Communicated by Bernard Helffer submitted 17/06/03, accepted 05/03/04
Fr´ed´eric Klopp D´epartement de Math´ematique Institut Galil´ee U.R.A 7539 C.N.R.S Universit´e de Paris-Nord Avenue J.-B. Cl´ement F-93430 Villetaneuse, France email:
[email protected]
Ann. Henri Poincar´e 5 (2004) 979 – 1012 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/050979-34 DOI 10.1007/s00023-004-0187-3
Annales Henri Poincar´ e
Negative Discrete Spectrum of Perturbed Multivortex Aharonov-Bohm Hamiltonians∗ M. Melgaard, E.-M. Ouhabaz and G. Rozenblum
Abstract. The diamagnetic inequality is established for the Schr¨ odinger operator (d) H0 in L2 (Rd ), d = 2, 3, describing a particle moving in a magnetic field generated by finitely or infinitely many Aharonov-Bohm solenoids located at the points of a discrete set in R2 , e.g., a lattice. This fact is used to prove the Lieb-Thirring inequality as well as CLR-type eigenvalue estimates for the perturbed Schr¨ odinger (d) operator H0 − V , using new Hardy type inequalities. Large coupling constant eigenvalue asymptotic formulas for the perturbed operators are also proved.
1 Introduction and main results Consider a non-relativistic, spinless quantum particle in Rd , d = 2, 3, interacting with a magnetic field B associated with finitely or infinitely many thin solenoids aligned along the x3 -axis which pass through the points λ of some discrete subset Λ of the x1 x2 plane. The magnetic flux through each solenoid is a noninteger αλ . If, moreover, the radii of the solenoids tend to zero, whilst the flux αλ through each solenoid remains constant then one obtains a particle moving in Rd subject to a finite or an infinite sum of δ-type magnetic fields, the so-called Aharonov-Bohm fields or magnetic vortices, located at the points of Λ which may be interpreted as infinitely thin impurities within a superconductor. Setting Λd = Λ × Rd−2 , the multiply-connected region Rd \ Λd , in which the field B equals zero, represents the configuration space. In the case of a lattice (defined by λkl = kω 1 + lω2 , where ω 1 , ω2 are vectors in R2 and (k, l) runs over the whole of Z2 or a subset of Z2 ) such a situation occurs experimentally in GaAs/AlGaAs heterostructures coated with a film of type-II superconductors [5, 11]. The vector potential A(x1 , x2 ) = (A1 (x1 , x2 ), A2 (x1 , x2 ), 0) associated with B is chosen such that A1 (x1 , x2 ) = Im A(x1 , x2 ), and A2 (x1 , x2 ) = Re A(x1 , x2 ),
(1.1)
where A(z) = A(x1 , x2 ), z = x1 + ix2 , is a meromorphic function having simple poles at λ ∈ Λ with residues αλ ; existence (and examples) of such functions A(z) ∗ The main part of this work was carried out while the first author was an individual Marie Curie Post-Doc Fellow, supported by the European Union under grant no. HPMF-CT-200000973.
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are discussed in Section 2. One easily verifies that ∂x1 A2 − ∂x2 A1 = αλ δ(z − λ) = B λ∈Λ
in the sense of distributions; as usual, it suffices to consider αλ ∈ (0, 1) due to gauge invariance. The dynamics of a spinless particle moving in any of the above-mentioned configurations of Aharonov-Bohm (abbrev. A-B) solenoids in Rd is described by the Schr¨ odinger operator (d) H0 = −(∇ + iA)2 (1.2) acting in L2 (Rd ), where ∇ is the gradient on Rd . Since the singularities of the A-B magnetic potential are very strong, the operator defined initially on functions with support away from the singularities is not essentially self-adjoint. In Section 2 we (d) define the Friedrichs extension of H0 by means of quadratic forms. In the case of a single A-B solenoid the corresponding standard A-B Schr¨odinger operator has been studied intensively in two dimensions and there is an ongoing discussion on the mathematical and physical reasonability of different self-adjoint extensions [31, 1, 10, 15, 35]. The Friedrichs extension considered herein corresponds to the model of solenoids being non-penetrable for electrons, and, moreover, with interaction preserving circular symmetry [1]. Within the theory of Schr¨ odinger operators with magnetic fields L(A) = −(∇ + iA)2 associated with a vector potential A = (A1 , . . . , Ad ) satisfying Aj ∈ L2loc (Rd ), one of the fundamental facts is the diamagnetic inequality [2], viz., |e−tL(A) u| ≤ e−tL0 |u| for all t ≥ 0 and all u ∈ L2 (Rd ); here L0 denotes the negative Laplacian in L2 (Rd ). In Section 4 we show that this inequality is valid also for the Schr¨ odinger (d) operator H0 in L2 (Rd ) for any of the afore-mentioned A-B configurations. Theorem 1.1 The inequality (d)
|e−tH0 u| ≤ e−tL0 |u| holds for all t ≥ 0 and all u ∈ L2 (Rd ) This result does not follow directly from the known diamagnetic inequality since the components (1.1) of the vector potential do not belong to L2loc (Rd ); this latter condition is crucial in all existing proofs of the diamagnetic inequality for Schr¨ odinger operators with magnetic fields. Our proof of Theorem 1.1 uses a recent criterion (see Section 3) for the domination of semigroups due to Ouhabaz [25]1 . This criterion is a generalization (from operators to forms) of the Simon-Hess-Schrader-Uhlenbrock test for domination of semigroups [13]. 1 It might be possible to prove Theorem 1.1 from general results in [22] which, in their turn, are based on [25], but we prefer to give a direct proof.
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As the first application of the diamagnetic inequality we establish the Lieb(d) Thirring inequality for the perturbed Schr¨ odinger operator H0 − V in Section 6. Here the electrostatic potential V is a nonnegative, measurable function on Rd (d) belonging to an appropriate class, which guarantees that the form sum H0 − V generates a semi-bounded, self-adjoint operator in L2 (Rd ) with discrete spectrum below zero. The classic Lieb-Thirring inequality [20] for a d-dimensional Schr¨ odinger operator L0 − V in L2 (Rd ), with L0 = −∆ as above and d ≥ 1, says that d |νj (L0 − V )|γ ≤ bγ,d V (x)γ+ 2 dx, (1.3) Rd
j
where νj (L0 − V ) denote the negative eigenvalues of L0 − V , γ > 0 (γ ≥ 1/2 for d d = 1) and V ∈ Lγ+ 2 . The constant bγ,d is expressible in terms of Γ-functions. The Lieb-Thirring inequality plays a crucial role in the problem of stability of matter (see, e.g., [21]), where the exact value of the constant is important (see [18], [14] for recent developments in obtaining sharp constants). One way of establishing (1.3) is to use the Cwikel-Lieb-Rozenblum (abbrev. CLR) estimate (see, e.g., [27]) which, in its original form, reads d V (x) 2 dx, d ≥ 3. (1.4) N− (L0 − V ) ≤ Cd Rd
Here N− denotes the number of negative eigenvalues of a self-adjoint operator, provided its negative spectrum is discrete. The single assumption, under which (1.4) is valid, is the finiteness of the integral on its right-hand side. In [33, p. 99100] it is shown how one can obtain (1.3) provided (1.4) holds. This, however, does not produce the optimal constant in the Lieb-Thirring inequality. The Lieb-Thirring inequality for d-dimensional Schr¨ odinger operators with magnetic fields L(A) − V , with d ≥ 3 and Aj ∈ L2loc (Rd ), takes the same form and can be obtained from the CLR-estimate for L(A) − V which is shown by means of the diamagnetic inequality (see, e.g., [33, p 168]). In two dimensions there exist certain CLR-type estimates both for L0 − V [34, 6] and L(A) − V [29], provided Aj ∈ L2loc (R2 ) for the latter operator. However, unlike in higher dimensions, these estimates, having a different form, do not produce Lieb-Thirring inequalities. Moreover, in our case, the components (1.1) of the magnetic potential do not belong to L2loc (R2 ). Therefore the question (d) on Lieb-Thirring inequalities for the perturbed Schr¨ odinger operator H0 − V was up to now open. In the present paper we establish the following Lieb-Thirring (abbrev. LT) (d) inequality for the perturbed Schr¨ odinger operator H0 − V in L2 (Rd ), d = 2, 3, for any of the afore-mentioned configurations of A-B solenoids, with constants not depending on the configuration or the strength of magnetic fields.
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M. Melgaard, E.-M. Ouhabaz and G. Rozenblum
Ann. Henri Poincar´e
(d)
Theorem 1.2 Let νj denote the negative eigenvalues of H0 − V , d = 2, 3. If, moreover, γ > 0 and V ∈ Lγ+(d/2) (Rd ) then d γ LTγ,d := |νj | ≤ Cγ,d V (x)γ+ 2 dx, j
Rd
where the constant Cγ,d fulfills the following upper bounds for the most interesting values of γ: 0.5300 for γ = 1/2, 0.3088 for γ = 1, Cγ,2 ≤ 0.2275 for γ = 3/2 ; and Cγ,3
0.1542 0.0483 ≤ 0.0270
for γ = 1/2, for γ = 1, for γ = 3/2.
We note that the expression we obtain for the best constant in Theorem 1.2 is implicit; see (6.3). Using Hardy-type inequalities enables one to further improve the LT estimates; see Section 7. The diamagnetic inequality is one out of the two crucial ingredients in the proof of Theorem 1.2. The other is an abstract CLR estimate for generators of semigroups dominated by positive semigroups. To make the paper self-explanatory we formulate this rather recent result, obtained by Rozenblum and Solomyak, in Section 5. An important application of eigenvalue estimates for Schr¨ odinger operators is to deduce asymptotic formulas for the eigenvalues when the coupling constant q is present and it tends to infinity. The technology of getting the asymptotic formulas from the estimates is well-established nowadays (see, e.g., [27] and [7, 8]), and what is required from the estimates is that they have correct order in the coupling constant. For weakly singular magnetic fields such estimates were obtained by Lieb (see [33]) and Melgaard-Rozenblum [23] in dimensions d ≥ 3, and by Rozenblum-Solomyak [29] in dimension d = 2 (see also [30]). In the case of a single A-B solenoid, the only existing estimate for the corresponding A-B (2) Schr¨ odinger operator HAB − qV , by Balinsky, Evans and Lewis [4], deals with a rather special case of a radially symmetric potential (or with one majorized by a radially symmetric potential). There were no preceding results concerning eigenvalue estimates for many solenoids. Based upon the diamagnetic inequality we establish CLR-type estimates (i.e., (d) estimates having correct order in coupling constant q) for H0 − qV for any of the A-B configurations mentioned above. To achieve this, we derive Hardy-type inequalities for each configuration, which allows us to carry over recent CLR-type estimates for the negative eigenvalues of two-dimensional Schr¨ odinger operators,
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with a regularizing positive (Hardy) term added, to the operators H0 − qV . The Hardy-type inequalities are of interest by themselves and complement the recent result by Balinsky [3]. For finitely many A-B solenoids we prove the Hardy-type inequality by using a conformal mapping. This idea belongs to Balinsky but we use another, more explicit realization, which gives a better control over the weight function in the Hardy-type inequality. In the field of CLR-type estimates, sharp constants are unknown, and at present the known values of constants lie far above the expected ones. In applications to finding eigenvalue asymptotics, the values of these constants are of no importance, and thus we do not try to obtain the best values for our case either. Instead we demonstrate that the presence of the magnetic field and its particular configuration may improve, via Hardy inequalities, eigenvalue estimates, by compensating possible strong singularities or insufficient decay of the electric potential. Similar effect takes place for LT estimates as well. After obtaining CLR-type estimates, to deduce the large coupling constant (d) asymptotics for the eigenvalues of H0 − qV is a standard job. Only a few remarks are needed. The magnetic flux parameters αλ are nonintegers throughout the paper. If αλ are integers, the resulting operator is gauge equivalent to the negative Laplacian in L2 (Rd ). This, however, does not reflect itself in the LT inequality but the eigenvalue estimates in Section 8 are no longer valid, as one can see, e.g., from the factor β −2 in formula (8.7). (d)
2 The unperturbed Hamiltonian H0 Choice of vector potential
As mentioned in the introduction, the vector potential A(x1 , x2 ) = (A1 (x1 , x2 ), A2 (x1 , x2 ), 0) associated with B is chosen such that A1 (x1 , x2 ) = Im A(x1 , x2 ) and A2 (x1 , x2 ) = Re A(x1 , x2 ),
(2.1)
where A(z) = A(x1 , x2 ), z = x1 + ix2 , is a meromorphic function having (only) simple poles at λ ∈ Λ with residues αλ . In the case where Λ is a finite set, say, Λ = {λ1 , λ2 , . . . , λN }, the function A(z) =
N αλj z − λj j=1
has the desired properties. In the general case, where Λ is a discrete set with infinitely many points, without finite limit points, Mittag-Leffler’s theorem guarantees the existence of a meromorphic function with the afore-mentioned properties, unique, up to an entire summand. For an infinite regular lattice where all fluxes are equal to a noninteger α, we can construct such a function A(z) explicitly. Indeed let Φ(z) be an entire function
984
M. Melgaard, E.-M. Ouhabaz and G. Rozenblum
Ann. Henri Poincar´e
such that its set of (only simple) zeros coincide with Λ. Then one can take A(z) = αΦ (z)Φ(z)−1 . In particular, the Weierstrass function σ(z) corresponding to the lattice can serve as Φ(z), and then Φ (z)Φ(z)−1 is the Weierstrass function ζ(z). Magnetic quadratic forms For A = (A1 , A2 ) in (1.1) we observe that d d A1 , A2 ∈ L∞ loc (R \ Λ ).
Let
Ωn = B(0, n) × (−n, n)d−2 \ ∪λ∈Λ B(λ, 1/n) × Rd−2 , n ≥ 2,
where B(λ, r) denotes the disk with center λ and radius r. We define on L2 (Ωn ) (for each n ≥ 2) the form h(d) n [u, v]
=
d j=1
Ωn
∂u + iAj u ∂xj
∂v + iAj v dx ∂xj
(2.2)
(d)
on the domain D(hn ) = H01 (Ωn ). The form is closed since A1 , A2 ∈ L∞ (Ωn ). (d) The associated self-adjoint, nonnegative operators are denoted by Hn . These are 2 2 operators in L (Ωn ). It is convenient to extend them to zero in L (Rd \ Ωn ), thus getting operators in L2 (Rd ); keeping the same notation for extended operators does not create misunderstanding. (d) Define, in addition, the (closed) form ln with the same form expression and (d) domain as hn but with A1 = A2 = 0. The associated self-adjoint, nonnegative (d) operators are denoted by Ln . Define now the form h(d) by (d) h(d) [u, v] = h(d) n [u, v] if u, v ∈ D(hn ),
D(h(d) )
1 = ∪n D(h(d) n ) = ∪n H0 (Ωn ).
Lemma 2.1 The form h(d) is closable. Proof. According to the definition, the form h(d) is closable if and only if any sequence {un }, un ∈ D(h(d) ), for which lim un L2 = 0 and
n→∞
lim h(d) [un − um ] = 0,
n,m→∞
(2.3)
satisfies limn→∞ h(d) [un ] = 0. First observe that (2.3) implies C := sup h(d) [un ]1/2 < ∞.
(2.4)
n
Take > 0 and choose n0 such that h(d) [un − um ] ≤ when n, m ≥ n0 .
(2.5)
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Set, moreover, K = Ωn0 ⊂ Rd \ Λd such that supp un0 ⊂ K. In view of (2.3), |(∇ + iA)(un − um )|2 dx ≤ h(d) [un − um ] −→ 0 as n, m → ∞, (2.6) K
|un |2 dx −→ 0 as n → ∞,
(2.7)
K
and, since A is bounded on K, |Aun |2 dx −→ 0 as n → ∞.
(2.8)
K
Now,
1/2 1/2
2 2 |A(un − um )| dx − |∇(un − um | dx
K K 1/2 ≤ |(∇ + iA)(un − um )|2 dx .
(2.9)
K
According to (2.8), the first term on the left-hand side of the latter inequality tends to zero as n, m → ∞ and, due to (2.6), the same holds for the right-hand side. Thus, |un − um |2 + |∇(un − um )|2 dx −→ 0 as n, m → ∞. K
Since the form of the classical Dirichlet Laplacian is closable it follows from the latter relation, in conjunction with (2.7) that |∇un |2 dx → 0, |un |2 dx → 0, as n → ∞. (2.10) K
K
Now, h(d) [un ] = ≤
h(d) [un , un − un0 ] + h(d) [un , un0 ] h(d) [un ]1/2 h(d) [un − un0 ]1/2 + |h(d) [un , un0 ]|.
(2.11)
It follows from (2.4) and (2.5) that h(d) [un ]1/2 h(d) [un − un0 ]1/2 ≤ C 1/2 when n ≥ n0 . Since A is bounded on K we infer from (2.10) and (2.8) that (d) (∇ + iA)un (∇ + iA)un0 dx → 0 as n → ∞. h [un , un0 ] =
(2.12)
(2.13)
K
Substitution of (2.12)-(2.13) into (2.11) shows that limn→∞ h(d) [un ] = 0 as desired.
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M. Melgaard, E.-M. Ouhabaz and G. Rozenblum
Ann. Henri Poincar´e
(d)
We denote the closure of h(d) by h and the associated semi-bounded (from (d) below), self-adjoint operator by H0 . This is, in fact, the Friedrichs extension of the symmetric operator (1.2) defined initially on C0∞ (Rd \ Λd ). The introduction of ‘approximating’ forms, however, is required for proving the diamagnetic inequality later. We define l(d) in a similar way, viz. l(d) [u, v] = D(l(d) ) =
(d) l(d) n [u, v] if u, v ∈ D(ln ),
1 ∪n D(l(d) n ) = ∪n≥2 H0 (Ωn ).
(d)
(d)
Then l(d) is closable. The closure l has domain D(l ) = H 1 (Rd ). The associated nonnegative, self-adjoint operator is just the negative Laplacian in L2 (Rd ); we suppress d and denote it by L0 .
3 Semigroup criterion Throughout this section H denotes our Hilbert space L2 (Rd ). For a given u ∈ H we denote by u := Re u − iIm u the conjugate function of u. By |u| we denote the
absolute value of u (i.e., the function x → |u(x)| := u(x) · u(x)) and by sign u the function defined by u(x) if u(x) = 0, |u(x)| sign u(x) = 0 if u(x) = 0. Let s be a sesquilinear form which satisfies D(s) is dense in H,
(3.1)
Re s[u, u] ≥ 0, ∀u ∈ D(s),
(3.2)
|s[u, v]| ≤ Cus vs , ∀u, v ∈ D(s),
where C is a constant and us = Re s[u, u] + u2 , and, moreover, D(s), · s is a complete space.
(3.3)
(3.4)
Definition 3.1 Let K and L be two subspaces of H. We shall say that K is an ideal of L if the following two assertions are fulfilled: 1) u ∈ K implies |u| ∈ L. 2) If u ∈ K and v ∈ L such that |v| ≤ |u| then v · sign u ∈ K. Let s and t be two sesquilinear forms both of which satisfy (3.1)-(3.4). The semigroups associated to corresponding self-adjoint operators S, T will be denoted by e−tS and e−tT , respectively. The following result was established by Ouhabaz [25, Theorem 3.3 and its Corollary].
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Theorem 3.2 (Ouhabaz’96) Assume that the semigroup e−tT is positive. The following assertions are equivalent: 1) |e−tS f | ≤ e−tT |f | for all t ≥ 0 and all f ∈ H. 2) D(s) is an ideal of D(t) and Re s[u, |v|sign u] ≥ t[|u|, |v|]
(3.5)
for all (u, v) ∈ D(s) × D(t) such that |v| ≤ |u|. 3) D(s) is an ideal of D(t) and Re s[u, v] ≥ t[|u|, |v|]
(3.6)
for all u, v ∈ D(s) such that u · v ≥ 0. The following lemma is useful when one wishes to apply the criteria in Theorem 3.2. Lemma 3.3 Let Ω ⊂ Rd be an open set and u, v ∈ H 1 (Ω) be functions satisfying u(x) · v(x) ≥ 0 for a.e. x in Ω. Then ∂u ∂u · v = |v| Im · sign u . 1. Im ∂x ∂xj j ∂u ∂v 2. |v| Im ∂xj · sign u = |u| Im ∂xj · sign u . Proof. Let χ{u=0} denote the characteristic function of the set { x | u(x) = 0}. Since (∂u/∂xj ) · χ{u=0} = 0, we have that ∂u ∂u v·u ∂u u χ{u=0} χ{v=0} = |v| · · χ{u=0} . ·v = ·v· · ∂xj ∂xj |u| · |v| ∂xj |u| By taking the imaginary part on both sides of the latter equality, we obtain that ∂u ∂u · v = |v| Im · sign u , Im ∂xj ∂xj which verifies the first assertion. To prove the second assertion we start from |v| · u = |v| · u ·
v·u χ{u=0} χ{v=0} = |u| · v. |u| · |v|
Hence, ∂|v| ∂u ∂|u| ∂v · u + |v| · = · v + |u| · . ∂xj ∂xj ∂xj ∂xj We multiply both sides by sign u = (u/|u|)χ{u=0} and take the imaginary parts on both sides to obtain ∂v ∂v ∂u · sign u = Im · uχ{u=0} = Im ·u . |v| Im ∂xj ∂xj ∂xj The latter in combination with the first assertion (with u substituted by v and vice-versa) shows the second assertion.
988
M. Melgaard, E.-M. Ouhabaz and G. Rozenblum
Ann. Henri Poincar´e
(d)
4 Diamagnetic inequality for H0
The usual diamagnetic inequality is established for vector potentials which belong to L2loc (see, e.g., [2]). In this section we establish the diamagnetic inequality for (d) the Schr¨ odinger operator H0 , i.e. when Aj ∈ L2loc , j = 1, 2. (d) (d) (d) Denote by e−tHn (resp. e−tLn ) the semigroup associated with Hn (resp. (d) Ln ) introduced in Section 2. For each n the diamagnetic inequality holds for these pairs of semigroups. Proposition 4.1 The inequality (d)
(d)
|e−tHn f | ≤ e−tLn |f | holds for all t ≥ 0 and all f ∈ L2 (Ωn ) (n ≥ 2). (2)
Proof. We give the proof for d = 2 and suppress the upper index in hn . With a few obvious modifications the proof for d = 3 is the same. By the domination criterion in Theorem 3.2, assertion 3, it suffices to prove that Re hn [u, v] ≥ ln [|u|, |v|] for all u, v ∈ D(hn ) = H01 (Ωn ) obeying u · v ≥ 0. Let u, v ∈ H01 (Ωn ) be such that u · v ≥ 0. We have that ∂u ∂v ∂u ∂v I1 := Re · + · dx ∂x1 ∂x1 ∂x2 ∂x2 Ωn ∂v ∂u · sign u Re · sign v = Re ∂x1 ∂x1 Ωn ∂u ∂v + Re · sign u Re · sign v dx ∂x2 ∂x2 ∂v ∂u · sign u Im · sign v + Im ∂x1 ∂x1 Ωn ∂u ∂v + Im · sign u Im · sign v dx ∂x2 ∂x2 ∂v ∂u · sign u Re · sign v = Re ∂x1 ∂x1 Ωn ∂u ∂v + Re · sign u Re · sign v ∂x2 ∂x2 ∂u ∂u |v| + Im χ{u=0} · sign u Im · sign u ∂x1 ∂x1 |u| ∂u ∂u |v| χ{u=0} dx, + Im · sign u Im · sign u ∂x2 ∂x2 |u|
(4.1)
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where we applied Lemma 3.3, part 2, in the last equality. From [24, Lemma 4.1] we have that ∂u ∂|u| = Re sign u , ∀u ∈ H 1 (Ωn ) ⊃ H01 (Ωn ). ∂x1 ∂x1 Using this, we find that 2 ∂u ∂|u| ∂|v| ∂|u| ∂|v| |v| χ{u=0} · + · + Im sign u I1 = ∂x ∂x ∂x ∂x ∂x |u| 1 1 2 2 1 Ωn 2 ∂u |v| χ{u=0} dx. + Im sign u ∂x1 |u|
(4.2)
∂v ∂v = Re u ∂x we have that Next, let u, v ∈ H01 (Ωn ) with u · v ≥ 0. Using Re u ∂x 1 1
∂u ∂u ∂v ∂v v − iA2 v + iA1 u + iA2 u dx ∂x1 ∂x2 ∂x1 ∂x2 Ωn ∂u ∂u v − Im (−iA2 ) Im v − Im (−iA1 ) Im ∂x1 ∂x2 Ωn ∂v ∂v − Im (iA1 ) Im u − Im (iA2 ) Im u dx. ∂x1 ∂x2
I2
:= =
Re
−iA1
Using the first part of Lemma 3.3 we may rewrite I2 as ∂u I2 = sign u |v| − Im (−iA2 ) − Im (−iA1 ) Im ∂x1 Ωn ∂v ∂u sign u |v| − Im (iA1 ) Im sign v |u| × Im ∂x2 ∂x1 ∂v sign v|u| dx. − Im (iA2 ) Im ∂x2 Next we apply the second part of Lemma 3.3 to the last two terms in I2 . It follows that ∂u sign u |v| I2 = (A1 − A1 ) Im ∂x1 Ωn ∂u sign u |v| dx = 0. (4.3) +(A2 − A2 ) Im ∂x2 For the last term in hn , we have that 2 2 (A1 + A2 )u · vdx = I3 := Re Ωn
for all u, v ∈ H01 (Ωn ) such that u · v ≥ 0.
Ωn
(A21 + A22 )|u| |v|dx
(4.4)
990
M. Melgaard, E.-M. Ouhabaz and G. Rozenblum
Since Re hn [u, v] =
3
j=1 Ij ,
Ann. Henri Poincar´e
we obtain from (4.2), (4.3), and (4.4) that
2 ∂u ∂|u| ∂|v| ∂|u| ∂|v| Re hn [u, v] = · + · + Im sign u ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 Ωn 2 ∂u |v| |v| 2 2 χ{u=0} + (A1 + A2 )|u| |v| dx. × χ{u=0} + Im sign u |u| ∂x1 |u|
In this expression, the sum of the last three terms is nonnegative, so we infer that ∂|u| ∂|v| ∂|u| ∂|v| Re hn [u, v] ≥ · + · + (A21 + A22 )|u| |v| ∂x1 ∂x1 ∂x2 ∂x2 Ω n ∂|u| ∂|v| ∂|u| ∂|v| ≥ · + · = ln [|u|, |v|] ∂x1 ∂x1 ∂x2 ∂x2 Ωn for all u, v ∈ H01 (Ωn ) obeying u · v ≥ 0. This verifies (4.1).
(d)
The semigroups associated with H0 and L0 , introduced in Section 2, are (d) denoted by e−tH0 and e−tL0 , resp. By means of Proposition 4.1 we are ready to (d) prove Theorem 1.1, i.e., the diamagnetic inequality for the operator H0 . Proof of Theorem 1.1. Bear in mind that when s1 and s2 are closed forms bounded from below then s1 ≥ s2 means that D(s1 ) ⊂ D(s2 ) and s1 [u, u] ≥ s2 [u, u] for u ∈ D(s1 ). A sequence {sn } of closed forms bounded from below is nonincreasing if sn ≥ sn+1 for all n. (d) (d) The forms {hn } defined in (2.2) on the domains D(hn ) = H01 (Ωn ) in L2 (Ωn ), n ≥ 2, compose a nonincreasing sequence (d)
(d)
· · · ≤ hn+1 ≤ h(d) n ≤ hn−1 ≤ . . . , of closed, non-densely defined forms in L2 (Rd ). The monotone convergence theorem for closed forms is also valid for non-densely defined forms [32, Theorem 4.1]. Hence, the latter theorem in conjunction with Lemma 2.1 yields that operators (d) (d) Hn corresponding to forms hn (extended by zero outside Ωn ) converge in strong (d) resolvent sense to H0 or, equivalently, (d)
(d)
e−tH0 = s− lim e−tHn . n→∞
(4.5)
A similar argument yields (d)
(d)
e−tL0 = s− lim e−tLn . n→∞
(4.6)
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Thus we can pass to the limit n → ∞ in the diamagnetic inequality for operators (d) (d) Hn , Ln , Proposition 4.1 and therefore
(d)
(d)
−tH0(d)
f = lim e−tHn f ≤ lim e−tLn |f | = e−tL0 |f |
e n→∞
n→∞
which proves the assertion.
Remark 4.2 Our proof of the diamagnetic inequality also applies to the case where we have another metric, that is, the result holds also for operators of the type (∇ + iA)∗ M (x)(∇ + iA) where M (x) = (mjk (x)) is a symmetric matrix with real-valued and bounded measurable coefficients (satisfying the classical ellipticity condition). The semigroup generated by this operator is dominated by the semigroup generated by the elliptic operator ∇∗ M (x)∇.
5 Abstract CLR eigenvalue estimates and semigroup domination In this section we recall Rozenblum’s and Solomyak’s abstract CLR estimate for generators of positively dominated semigroup. Let Ω be a space with σ-finite measure µ, L2 = L2 (Ω, µ). Let T be a nonnegative, self-adjoint operator in L2 , generating a positivity preserving semigroup Q(t) = e−tT . We suppose also that Q(t) is an integral operator with bounded kernel Q(t; x, y) subject to MT (t) := ess sup x Q(t; x, x), MT (t) = O(t−β ) as t → 0 for some β > 0.
(5.1)
We will write T ∈ P if T satisfies the afore-mentioned assumptions2 . If T ∈ P, the operator Tµ = T + µ also belongs to P. The corresponding semigroup is QTµ (t) = e−µt QT (t) and thus MTµ (t) = e−µt MT (t). We say that the semigroup P (t) = e−tS is dominated by Q(t) if the diamagnetic inequality holds, i.e., if any u ∈ L2 satisfies |P (t)u| ≤ Q(t)|u| a.e. on Ω.
(5.2)
In the latter case we write S ∈ PD(T ). Let now G be a nonnegative, continuous, convex function on [0, ∞). To such a function we associate ∞ g(λ) = L(G)(λ) := z −1 G(z)e−z/λ dz, λ > 0, (5.3) 0
provided the latter integral converges. In other words, g(1/λ) is the Laplace transform of z −1 G(z). 2 Although the diagonal in Ω × Ω may be a set with measure zero in Ω × Ω, the semigroup property defines Q(t, ·, ·) as a function in L∞ (Ω), see [28].
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For a nonnegative, measurable function V such that the operator of multiplication by V is form-bounded with respect to T with a bound less than one, we associate the operators T − V , S − V by means of quadratic forms. The number of negative eigenvalues (counting multiplicity) of T − V is denoted by N− (T − V ); if there is some essential spectrum below zero, we set N− (T − V ) = ∞. Rozenblum and Solomyak [28, Theorem 2.4] have established the following abstract CLR estimate . ∞ Theorem 5.1 Let G, g and T ∈ P be as above and suppose that a MT (t) dt < ∞ for some a > 0. If S ∈ PD(T ) then ∞ dt 1 MT (t)G(tV (x)) dx, (5.4) N− (S − V ) ≤ g(1) 0 t Ω as long as the expression on the right-hand side is finite. The assumption that V is form-bounded with respect to T with a bound smaller than one in conjunction with S ∈ PD(T ) implies that V is form-bounded with respect to S with a bound less than one, thus N− (S − V ) is well defined. In (d) Section 6 we shall apply Theorem 5.1 to prove the LT inequality for H0 − V . Rozenblum has also developed an abstract machinery which, in our situation, allows us to carry over any, sufficiently regular, bound for N− (T − qV ) to N− (S − qV ), as soon as the diamagnetic inequality (5.2) is valid for S, T [30, Theorem 4]. We customize it to our situation. Theorem 5.2 Assume that T ∈ P, S ∈ PD(T ) and V ≥ 0 is a measurable function infinitesimally form-bounded with respect to T . Suppose that, for some p > 0, N− (T − qV ) ≤ Kq p
(5.5)
for all q > 0 and some positive constant K. Then N− (S − qV ) ≤ eCp Kq p ,
(5.6)
with a constant Cp which depends only on p. (d)
6 Lieb-Thirring inequality for H0 − V Having the diamagnetic inequality in Theorem 1.1 as well as the abstract CLR estimate in Theorem 5.1 at our disposal, we are ready to prove Theorem 1.2. Before proceeding with the proof, observe that the assumption V ∈ Lp (Rd ), p > 1 for d = 2 and p ≥ 3/2 for d = 3, in Theorem 1.2 implies that V is infinitesi(d) mally L0 - form-bounded; Theorem 1.1 then implies that V is infinitesimally H0 (d) form-bounded. Thus, according to the KLMN Theorem, the form sum H0 − V generates a lower semi-bounded, self-adjoint operator in L2 (Rd ).
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Proof of Theorem 1.2. As usual, it suffices to prove the estimate for V ∈ L1 ∩ L∞ d and then approximate V ∈ Lγ+ 2 by such functions. It is well known that L0 ∈ P and the kernel of its semigroup e−tL0 on the diagonal is given by Q(t; x, x) = (d) (4π)−d/2 t−d/2 . From Theorem 1.1 we have that H0 ∈ PD(L0 ) and the kernel P of its semigroup obeys P (t; x, x) ≤ (4π)−d/2 t−d/2 . (d) Let µ > 0 and define the auxiliary operators Sµ = H0 + µ and Tµ = L0 + µ. (d) Now L0 ∈ P and H0 ∈ PD(L0 ) imply that Tµ ∈ P and Sµ ∈ PD(Tµ ). For the kernel Pµ (t; x, x) = e−µt P (t; x, x) of the semigroup generated by Sµ we have therefore that |Pµ (t; x, x)| ≤ Qµ (t; x, x) = e−µt Q(t; x, x) = (4π)−d/2 t−d/2 e−µt . Thus we may apply Theorem 5.1 which yields ∞ 1 dt 1 N− (Sµ − V ) ≤ t−d/2 e−µt G(tV (x)) dx (6.1) (4π)d/2 g(1) 0 t Rd for any nonnegative convex function G(s) of subexponential growth, vanishing near zero (which ensures that the integral in (6.1) converges). We will not evaluate the integral in (6.1) as one might be inclined to do. Instead, for γ > 0, we recall that (see, e.g., [21]) (d) γ |νj (H0 − V )| = − µγ dNµ LTγ,d = j
=
γ 0
∞
µγ−1 N− (Sµ − V ) dµ.
(6.2)
We substitute (6.1) into (6.2) and get that ∞ ∞ γ 1 dt γ−1 dx µ dµ t−d/2 e−µt G(tV (x)) . LTγ,d ≤ d/2 g(1) Rd t (4π) 0 0 Making first the change of variables s = V (x)t and then the change of variables τ = µ/V (x) we obtain that d ˜ γ,d LTγ,d ≤ L V (x)γ+ 2 dx, Rd
where ˜ γ,d = L
γ 1 (4π)d/2 g(1)
0
∞
∞
s− 2 −1 e−τ s G(s)τ γ−1 ds dτ. d
0
∞ Now, 0 τ γ−1 e−τ s dτ = s−γ Γ(γ), where Γ(γ) is the Gamma-function evaluated at γ. Choose G(s) = (s − k)+ for some k > 0; this is Lieb’s original choice. Then ∞ d 1 s−γ s− 2 −1 (s − k)+ ds = d. d−2 γ + γ + d2 k γ 2 0 2 Moreover,
g(1) = 1
∞
e−ks s−2 ds ≥
2 e−k − g(1), k k
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i.e., 1/g(1) ≤ ek (k + 2). Thus ˜ γ,d ≤ Cγ,d := L
Γ(γ)ek (k + 2) (4π)d/2 (γ +
d−2 2 )(γ
+ d2 )k γ+
d−2 2
.
(6.3)
The optimization problem for the expression in (6.3) does not admit an exact solution. For the three most interesting values of γ, namely 1, 1/2 and 3/2, one easily finds the numerical values of Cγ,d given in the Theorem. Remark 6.1 In the case of a single A-B solenoid, A. Laptev pointed out to the authors that the LT inequality can be derived without using the diamagnetic inequality [19]. His argument goes as follows. When A = α(−x2 /|x|2 , x1 /|x|2 ) we may use the decomposition L2 (R2 ) = L2 (R+ , r dr)⊗L2 (S1 ) = ⊕n∈Z {L2 (R+ , r dr)[einθ /2π]} ([·] denotes the linear span) to express the A-B Schr¨odinger operator as (2)
HAB = ⊕n∈Z {Hn ⊗ In }, where Hn is the Friedrichs operator in L2 (R+ , r dr) associated with the quadratic form ∞ (n + α)2 2 2 hn [un ] = |un (r)| r dr. |un (r)| + r2 0 Thus, with a slight abuse of notation, the quadratic form associated with HAB is given by h[u] = n∈Z hn [un ]. Taking α ∈ (0, 1/2), we note that |n + α|2 ≥ |1 − α|2 provided n = 0. As a consequence, we have that ∞ n2 2 2 2 hn [un ] ≥ |1 − α| |un (r)| + 2 |un (r)| r dr = |1 − α|2 ln [un ], r 0 where l[u] = n∈Z ln [un ] is the quadratic form of the negative Laplacian in L2 (R2 ). In conclusion, h[u] ≥ |1 − α|2 l[u]. The latter inequality immediately implies that the usual LT inequalities for −∆ − V carry over to the A-B Schr¨odinger operator (2) HAB − V with a constant L2,γ /|1 − α|2 , where L2,γ is the usual Lieb-Thirring constant. A similar reasoning was used in [4]. This argument, however, does not work for many A-B solenoids.
7 Hardy-type inequalities In order to establish eigenvalue estimates in the two-dimensional case for various configurations of A-B solenoids (or magnetic vortices), we require certain Hardytype inequalities which we will obtain in this section. Generally, a Hardy-type inequality is an estimate where the integral involving the gradient of the function majorizes the weighted integral of the square of the function itself. The classical Hardy inequality |u(x)|2 dx ≤ const. |∇u(x)|2 dx, u ∈ C0∞ (Rd \ {0}), 2 |x| d d R R
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does not hold for d = 2. It was discovered by Laptev and Weidl [17], however, that the presence of a magnetic field can improve this situation. In particular, if the gradient ∇ is replaced by the “magnetic gradient” ∇ + iA, where A is the 1 A dx is standard A-B vector potential (see (7.5) below), and the flux α = 2π S1 noninteger then ([17, Theorem 3]) |u(x)|2 −2 dx ≤ ρ(α) |(∇ + iA)u(x)|2 dx, u ∈ C0∞ (R2 \ {0}), (7.1) 2 |x| 2 2 R R where ρ(α) = mink∈Z |k − α|. We are going to find analogies of this fact for configurations of magnetic solenoids considered in Section 2. In what follows, we will freely interchange real and complex pictures in description of our magnetic object. Thus x = (x1 , x2 ), z = x1 + ix2 , A = (A1 , A2 ), A = (A2 + iA1 ), dx = 12 dzdz etc. Finitely many solenoids Let Λ = {λ1 , λ2 , . . . , λJ } with λj = (λ1,j , λ2,j ). For finitely many A-B solenoids located at the points of Λ the corresponding A-B vector potential is given by A(x) =
J j=1
αj (−x2 + λ2,j , x1 − λ1,j ) |x − λj |2
(7.2)
for x = (x1 , x2 ) ∈ R2 \ Λ and αj being the flux through the jth solenoid. The aim is to establish the following Hardy-type inequality. J Proposition 7.1 Suppose that αj ∈ Z, j = 1, 2, . . . , J, and that αs := j=1 αj ∈ Z. Define J |x − λj |−2 . (7.3) W (x) = min{ρ(αj )2 , ρ(αs )2 } j=1
Then there exists a constant C such that 2 W (x)|u(x)| dx ≤ C R2
R2
|(∇ + iA)u(x)|2 dx
(7.4)
is valid for all u ∈ C ∞ (R2 \ Λ) as long as the integral on the right-hand side is finite. Note that the constant C above may depend on the configuration of the solenoids. We will mostly use this and other similar inequalities for u ∈ C0∞ (R2 \ Λ). Sometimes, however, we need the compact support condition dropped. Moreover, we want to stress that, unlike the nonmagnetic case, the magnetic Hardy inequalities hold without compactness of support; this simple observation was somehow overlooked in [17, 3]. We begin by showing a slightly modified version of [17, Theorem 3].
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Lemma 7.2 Assume that α0 ∈ Z and let A0 (x) = Then 2
ρ(α0 )
B(0,R)
α0 (−x2 , x1 ) |x|2
|u(x)|2 dx ≤ |x|2
|(∇ + iA0 )u(x)|2 dx
(7.5)
(7.6)
B(0,R)
holds for any u ∈ C ∞ (B(0, R) \ {0}) as long as the integral on the right-hand side is finite. Proof. In polar co-ordinates (r, θ), we have that ∇ + iA0 = −er (∂/∂r ) + (1/r)eθ [(−∂/∂θ ) + iα0 ].
(7.7)
Therefore, for any function f (r)einθ , n ∈ Z, we have that |(∇ + iA0 )f (r)einθ |2 r drdθ B(0,R) |fr |2 + (1/r2 )|f (r)|2 (n + α0 )2 r drdθ = B(0,R) 1 ≥ |f (r)|2 (n + α0 )2 r drdθ 2 r B(0,R) |f (r)einθ |2 ≥ ρ(α0 )2 r drdθ. r2 B(0,R) This proves (7.6) for spherical functions and therefore for any u ∈ C ∞ (B(0, R) \ {0}) since the left-hand side and the right-hand side of (7.6) are both sums of contributions of spherical functions. In a similar way we establish the following result. αj ∈ Z. Then, provided R > 0 is sufficiently Lemma 7.3 Suppose that αs := large, ΩR = {|x| > R}, the inequality |u(x)|2 |(∇ + iA)u(x)|2 dx ≥ ρ(αs )2 dx (7.8) |x|2 ΩR ΩR holds for any u ∈ C ∞ (ΩR ) as long as the integral on the left-hand side is finite. Proof. First we note that there exists a function ϕ such that A(x) − As (x) = (∇ϕ)(x), αs As (x) = (−x2 , x1 ), (7.9) |x|2 for any x ∈ ΩR provided R > 0 is large enough. Since the right-hand side of (7.8) is gauge invariant, it suffices to show (7.8) for the vector potential As . Now we switch to polar co-ordinates and repeat the reasoning in Lemma 7.2.
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Lemma 7.4 (Local Hardy inequality) Let D be a bounded, simply-connected domain in C with smooth boundary and let z0 ∈ D. Let A(z) = A2 (z)+iA1 (z), z = x1 +ix2 , be a (complex) magnetic vector potential such that A(z) is analytic in D \ {z0 } and has a simple pole at z0 with residue equal to µ0 , and let A = (A1 , A2 ). Then, there exists a constant C > 0 such that for any u ∈ C ∞ (D \ {z0 }), |u(z)|2 −2 dx ≤ ρ(µ ) C |(∇ + iA)u(z)|2 dx, (7.10) 0 2 |z − z | 0 D D as long as the integral on the right-hand side is finite. Proof. Let w = y1 + iy2 = F (z), F : D → B(0, 1) be a conformal mapping of D onto the unit disk B(0, 1) so that z0 is mapped to zero, and let u ˜(y) = u˜(w) = u(F −1 (w)). Since D has a smooth boundary, F is smooth up to the boundary [26, p. 49], together with its inverse. For later purpose we note that there exists c such that
F (z)
c
z
≤
(7.11)
. |z − z0 | F (z)
Indeed, since F is smoothly invertible, F is bounded away from 0. Therefore F /F has the order of 1/F near z0 . Since F has a simple zero at z0 , it has the order of |z − z0 |, which verifies (7.11). Let ωA denote the differential 1-form A1 (z)dx1 + A2 (z)dx2 and let AF (w) = F (A1 (w), AF 2 (w)) be the transformed magnetic vector potential in B(0, 1) such that F ∗ (ωAF ) = ωA (F ∗ denotes the pull-back), i.e., F AF 1 (w)dy1 + A2 (w)dy2 = A1 (z)dx1 + A2 (z)dx2 .
In particular, AF has a simple pole at the origin with residue equal to µ0 . Since F is a conformal mapping it follows that 2 |(∇x + iA)u(x)| dx = |(∇y + iAF )˜ u(y)|2 dy (7.12) D
B(0,1)
∞
for any u ∈ C (D). F Next we gauge away the regular part of AF = AF 2 + iA1 (as we did in the proof of Lemma 7.3). From Lemma 7.2 we immediately get that |˜ u(y)|2 −2 dy ≤ ρ(µ ) |(∇y + iA0 )˜ u(y)|2 dy, (7.13) 0 2 |y| B(0,1) B(0,1) where A0 is the pure A-B vector potential given in (7.5). Finally, we return to the domain D by making the inverse transform F −1 : B(0, 1) → D. Clearly,
F (z) 2 |˜ u(w)|2
dy = |u(z)|2 z dx. (7.14) 2
|w| F (z) B(0,1) D Using (7.11) in conjunction with (7.12) and (7.13) we arrive at (7.10).
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We are ready to give the proof of Proposition 7.1 Proof of Proposition 7.1. Let B(0, R) be a disk centered at the origin with a radius R > 0 so large that all the points of Λ are in B(0, R). Cover the disk B(0, R) with simply connected domains Ωj having smooth boundaries in such a way that Ωj contains λj but no other point from Λ.3 Let κ be the multiplicity of the covering of B(0, R) and let ΩR be the exterior of B(0, R). We clearly have that |(∇ + iA)u(x)|2 dx R2 J ≥ (1 + κ)−1 |(∇ + iA)u(x)|2 dx + |(∇ + iA)u(x)|2 dx . ΩR
j=1
Ωj
The first term on the right-hand side is estimated by the inequality in Lemma 7.3 and each of the terms in the sum on the right-hand side is estimated by the local Hardy inequality in Lemma 7.4. In this way, we obtain that |(∇ + iA)u(x)|2 dx R2
2
≥ ρ(αs )
ΩR
J |u(x)|2 |u(x)|2 −1 2 dx + (1 + κ) c ρ(α ) dx. j j 2 |x|2 Ωj |x − λj | j=1
J Since, inside Ωj , j > 0, we have |x − λj |−2 ≥ C k=1 |x − λk |−2 , and inside ΩR , J we have |x|−2 ≥ C k=1 |x − λk |−2 , this proves (7.4). Remark 7.5 Using a conformal mapping was inspired by A. Balinsky [3]. He has recently derived a Hardy-type inequality for an A-B Schr¨ odinger operator on general punctured domains. His result, however, does not give sufficient control over the Hardy weight, in particular, it does not guarantee strict positivity of the weight everywhere. This does not fit our purpose and, consequently, we have derived a slightly modified Hardy-type inequality. The inequality (7.4) has a shortcoming: if just one of the fluxes is very close to an integer, the weight on the left-hand side deteriorates. The following version of the Hardy inequality takes care of this situation: if the sum of fluxes is non-integer, we can exclude any solenoids we wish from the expression in (7.3). 3 To
construct such a covering, take, e.g., a direction not parallel to any of the straight lines passing through pairs of the points λj . Then one can draw straight lines parallel to this direction, which cut B(0, R) into pieces, each of which contains only one of the points λj . Extending these pieces slightly to domains with smooth boundaries we obtain the desired covering, with multiplicity κ = 2.
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Proposition 7.6 Suppose that αj ∈ Z, j = 1, 2, . . . , J, and that αs := Let J0 be a subset in {1, . . . , J}. Set W (x) = min {ρ(αj )2 , ρ(αs )2 } {j∈J0 }
|x − λj |−2 ,
999
J j=1
αj ∈ Z.
(7.15)
j∈J0
if J0 is nonempty, and W (x) = ρ(αs )2 (1 + |x|2 )−1 .
(7.16)
otherwise. Then there exists a constant C such that (7.4) is satisfied for any u ∈ C ∞ (R2 \ Λ). Proof. We consider the case of empty J0 first. Let, as in the proof of Proposition 7.1, the ball B(0, R) contain all points zj . Let ϕ, ψ ≥ 0 be smooth functions, ϕ2 + ψ 2 = 1, ϕ ∈ C0∞ (B(0, 3R)), 1 − ϕ = 0 outside B(0, R), |∇ϕ|, |∇ψ| < 2/R. Then for any u, |(∇ + iA)u(x)|2 dx JA(u) := R2 = JA(ϕu) + JA(ψu) − (|∇ϕ|2 + |∇ψ|2 )|u|2 dx R<|x|<3R −2 ≥ JA(ϕu) − 8R |u|2 dx. (7.17) R<|x|<3R
Now we use the well-known fact (see, e.g., [33, page 2]) that (for any magnetic potential A), JA(v) ≥ |∇|v||2 dx. (7.18) Applying (7.18) to the function v = ϕu and substituting the result into (7.17), we obtain 2 −2 JA(u) ≥ |∇|ϕu|| dx − 8R |u|2 dx. (7.19) R<|x|<2R
It follows from (7.6) that u ∈ L2loc as soon as JA(u) is finite. At the same time, (7.18)–(7.19) imply that ∇|ϕu| ∈ L2 . Thus |ϕu| belongs to the Sobolev space H01 (B(0, 3R)) and we can apply the Friedrichs inequality to the first integral in (7.19). The second term can be estimated from both sides by R<|x|<3R |x|−2 |u|2 dx. Thus we have JA(u) ≥ C1 |ϕu|2 dx − C2 |x|−2 |u|2 dx R<|x|<3R ≥ C1 |u|2 dx − C2 |x|−2 |u|2 dx. BR
ΩR
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For some > 0, multiply the latter inequality by and add to (7.8), multiplied by 1 − . We obtain 2 2 JA(u) ≥ C1
|u| dx + (ρ(αs ) (1 − ) − C2 ) |x|−2 |u|2 dx. (7.20) ΩR
B(0,R)
Choosing small enough (say, = ρ(αs )2 /(4C2 )), we arrange that the constant before the last integral in (7.20) is greater than C3 ρ(αs )2 and we obtain the required inequality. In the case of nonempty J0 , we split JA(u) = 12 JA(u) + 12 JA(u). For first term here we use the inequality we have just established. To estimate from below the second term, we act as in the proof of Proposition 7.1, i.e., consider the covering of the disk B(0, R) by domains Ωj but we write the local Hardy inequalities only for j ∈ J0 , thus getting 1 JA(u) ≥ C ρ(αj )2 |x − λj |−2 |u|2 dx. 2 Ωj j∈J0
Summing this with the estimate for the case of empty J0 , we arrive at (7.4).
Regular lattice of solenoids For a regular lattice of A-B solenoids we establish the following Hardy-type inequality. Proposition 7.7 Let A(z) = A(x1 + ix2 ) = A2 + iA1 be a magnetic potential such that A is analytical in C with exception of the points zkl = kω1 + lω2 , k, l ∈ Z, and in these points A has simple poles with residue equal to some non-integer α. Then, for any u ∈ C ∞ (C \ ∪zjk ), 2 2 |u|2 W (z) dx1 dx2 , JA(u) = |(∇ + iA)u| dx1 dx2 ≥ Cρ(α) where C > 0, ρ(α) = mink∈Z |k−α| and W (z)−1/2 is the distance from z = x1 +ix2 to the nearest lattice point. Proof. We consider first the case of a lattice Λ with ω1 = 1, ω2 = i. Write JA(u) = 4 × 14 JA(u). Split the lattice Λ into four sublattices, Λ = Λ1 ∪ Λ2 ∪ Λ3 ∪ Λ4 , where Λ1 consists of the points (2k, 2l) and, Λ2 = {(2k + 1, 2l)}, Λ3 = {(2k, 2l + 1)} and Λ4 = {(2k + 1, 2l + 1)}. Around each point zkl ∈ Λj , draw a disk Dkl with radius 0.8. Such a disk does not contain other points in the lattice. For this disk Dkl and any u ∈ C0∞ (C\∪zjk ), we can apply the inequality (7.1), 1 1 |(∇ + iA)u|2 dx1 dx2 ≥ ρ(α)2 |u|2 |z − zkl |−2 dx1 dx2 , (7.21) 4 Dkl 4 Dkl
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since, in the punctured disk Dkl \ {zkl }, the vector potential A is gauge equivalent to the potential α/|z|. For j fixed (j = 1, 2, 3, 4) we now sum (7.21) over all zkl ∈ Λj . The size of the disks is selected in such way that for j fixed, the corresponding disks are disjoint and therefore one can sum (7.21) termwise and get 1 1 JA(u) ≥ ρ(α)2 |u|2 |z − zkl |−2 dx1 dx2 . 4 4 Dkl zkl ∈Λj
Next we sum the latter inequality over j = 1, 2, 3, 4, which yields 1 2 JA(u) ≥ ρ(α) |u|2 |z − zkl |−2 dx1 dx2 4 Dkl
(7.22)
zkl ∈Λ
Now we note that
|z − zkl |−2 ≥ CW (z) for z ∈ Dkl
for some C > 0 and, moreover, the disks Dkl cover the plane. Therefore the expression in (7.22) majorizes ρ(α)2 |u|2 W (z) dx1 dx2 . For an arbitrary lattice we perform the same reasoning, just with disks with radius 0.8 being replaced by equal ellipses of proper size, covering the plane, and with the local Hardy inequality in the ellipse used instead for the one in the disk. The weight function W (z) in Proposition 7.7 is positive and separated from zero, W (z) ≥ W0 > 0. This implies, in particular, that the spectrum of the operator (2) H0 is separated from zero, i.e., the magnetic field produces a spectral gap. It is remarkable to compare this with the result of Geyler-Grishanov [12] who have shown that for another self-adjoint realization of the A-B operator corresponding to an infinite regular lattice of solenoids, the lowest point of the spectrum is zero and, moreover, an eigenvalue with infinite multiplicity.4 Hardy inequalities in dimension d = 3 Although in dimension three one does not need Hardy-type inequalities to establish CLR estimates (the latter is our main reason to study these inequalities), such three-dimensional versions are of certain interest. In order to join all cases, we will denote, for a fixed configuration of vortices, by W (x⊥ ), x⊥ = (x1 , x2 ), the weight which, according to Propositions 7.1, 7.6 or 7.7, enters into the Hardy inequality in R2 , viz. |(∇ + iA)u(x⊥ )|2 dx1 dx2 ≥ W (x⊥ )|u(x⊥ )| dx1 dx2 . (7.23) R2
R2
4 It is an interesting question, whether the lowest point of the spectrum of our operator is an eigenvalue.
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Hence W (x) incorporates the weight W and coefficients depending on the configuration of solenoids and on fluxes αs . Proposition 7.8 Under the assumptions of Propositions 7.1, 7.6 or 7.7, the following Hardy-type inequality holds: |(∇ + iA)u(x⊥ , x3 )|2 d3 x ≥ W (x⊥ )|u(x⊥ , x3 )| d3 x. (7.24) R3
R3
Of course, (7.24) is obtained from (7.23) by integration in x3 . Improvement of LT estimates by means of Hardy inequalities The LT-type inequality obtained in Section 6, does not show any dependence on the configuration of the magnetic fields. This is quite natural for the method of proving, since it was based upon comparing with the nonmagnetic case. By means of Hardy inequalities established above one can get certain improvements of the LT estimate. We explain, first of all, that one should not expect an improvement in the constant Cγ,d in (1.3), at least for a finite system of A-B solenoids. In fact, the magnetic potential decays at infinity, and if the electric potential V is supported far away from the sources of the field, then the influence of the magnetic field on the eigenvalues must be negligible (one can give exact meaning to this statement). On the other hand, the Hardy term can compensate local singularities of V or insufficient decay and thus estimate LTγ,d even in the case when the right-hand side in (1.3) is infinite. Proposition 7.9 Let A be a magnetic potential in Rd , d = 2 or d = 3, and let κ ∈ (0, 1) be an arbitrary number. Then, for the sum LTγ,d of powers of the eigenvalues of the operator H0d − V , the following inequality holds: d d +γ (V (x) − κW (x1 , x2 ))+2 dx, (7.25) LTγ,d ≤ Cγ,d (1 − κ)− 2 −γ Rd
where γ > 0 and Cγ,d is the constant in (1.3). The proof follows immediately from the operator inequality H0d − V
=
(1 − κ)H0d + κH0d − V (x)
≥
(1 − κ)H0d + κW (x1 , x2 ) − V (x)
≥
(1 − κ)(H0d − (1 − κ)−1 (V (x) − κW (x1 , x2 )))
and (1.3) applied to the electric potential (V (x) − κW (x1 , x2 ))+ . The inequality (7.25) includes certain situations which are worth being picked out, e.g., the case of an electric potential having singularities at the magnetic vortices, so that singularities are (partially) compensated by the magnetic field. In the case of the infinite lattice of solenoids, moreover, the Hardy weight W is
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separated from zero, W ≥ c0 , and thus an insufficiently slow decay of V , preventing the right-hand side of (1.3) from being finite, can be compensated; in view of (7.25), d +γ −d −γ 2 LTγ,d ≤ Cγ,d (1 − κ) (V (x) − κc0 )+2 dx. (7.26) Rd
Further versions of the LT-type inequalities, taking into account possible cancelling of the magnetic and electric field, can be obtained by averaging (7.25) in κ. Having chosen some nonnegative function ζ(κ) on (0, 1) which vanishes near one and has integral 1, we can multiply (7.25) by ζ(κ) and integrate. This produces new estimates, first for ‘nice’ V , but then, as usual, since the constants there do not depend on the potential, they extend by continuity to all V for which the quantity on the right-hand side is finite. We give here just one, simple example of this approach, where we do not care much about getting sharp constants but are interested in possible cancellation of singularities. Take ζ(κ) = cσ κσ for κ < 1/2 and ζ(κ) = 0 otherwise; with cσ = (σ + −1 −σ−1 , σ > −1. Then integrating of (7.25) with weight ζ(κ) gives 1) 2 1/2 d +γ −d −γ σ 2 (1 − κ) κ (V (x) − κW (x1 , x2 ))+2 dxdκ. (7.27) LTγ,d ≤ cσ Cγ,d 0
Rd
The integral in (7.27) is estimated as follows. Change the order of integration and d d majorate (1 − κ)− 2 −γ by 2 2 +γ . For the remaining integral in κ we have 1/2 d +γ κσ (V (x) − κW (x1 , x2 ))+2 dκ 0
≤ W (x1 , x2 )
d 2 +γ
0
∞
V (x) −κ W (x1 , x2 )
d2 +γ
κσ dκ.
+
The latter integral is evaluated by the usual change of variables and equals B(σ + d 1, d2 + γ + 1)V (x) 2 +γ+σ+1 W (x1 , x2 )−σ−1 ; B(·, ·) being the Beta function. Thus we come to the estimate d d LTγ,d ≤ Cγ,d cσ B(σ + 1, + γ + 1) V (x) 2 +γ+σ+1 W (x1 , x2 )−σ−1 dx. (7.28) 2 d R In the inequality (7.28) one can choose the exponent σ depending on the particular potential V . The approach above has, in fact, a flavor of interpolation; similar results, based on Hardy inequalities, were obtained for N− (H0 − V ) in [9] (for the magnetic potential in L2loc , d ≥ 3).
8 Eigenvalue estimates and large coupling constant asymptotics (3)
In three dimensions the CLR estimate for H0 − V, V ≥ 0, takes its standard form for any of the configurations of A-B solenoids considered in Section 2, viz. 3 (3) V (x) 2 dx. (8.1) N− (H0 − V ) ≤ C3 R3
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This follows automatically from the non-magnetic inequality and domination, Theorem 5.2; the constant C3 > 0 is absolute. In the two-dimensional case there are a number of inequalities for the nonmagnetic Schr¨ odinger operator, fairly cumbersome, see [34, 6]. Due to domination, they carry over to the A-B Hamiltonian. However, the presence of the magnetic field improves the non-magnetic estimates considerably. The aim of this section is not to obtain the most general nor the best possible bounds for the number of negative eigenvalues for the two-dimensional perturbed A-B Schr¨ odinger operator; rather we demonstrate that any CLR-type eigenvalue estimate (existing or obtained in the future) for the (nonmagnetic) two-dimensional perturbed Schr¨ odinger operator, with a proper Hardy term added, automatically produces a similar estimate for the A-B Schr¨ odinger operator and, moreover, to show that further improvements, using the Hardy inequalities are possible. A single solenoid The (closed) quadratic form h(2) of the unperturbed A-B Schr¨ odinger operator (2) H0 can be written as h(2) [u] =
h(2) [u] h(2) [u] + . 2 2
(8.2)
Let β = ρ(α). To one of the two terms in (8.2), we apply the Hardy type inequality (7.1). This yields |u(x)|2 h(2) [u] β 2 h(2) [u] ≥ + dx. (8.3) 2 2 R2 |x|2 (2)
Let H0 (β 2 r−2 ), r = |x|−2 , denote the operator generated by the form on the (2) right-hand side of (8.3). Since H0 obeys the diamagnetic inequality, it follows (2) from, e.g., the Trotter-Kato formula that H0 (β 2 r−2 ) fulfills the diamagnetic in2 (2) equality as well, in shorthand, H0 (β 2 r−2 ) ∈ PD( L20 + β2 r−2 ). The latter fact in conjunction with Theorem 5.2 allows us to carry over all power in q bounds for odinger the two-dimensional Schr¨ odinger operator L0 + Cr−2 − qV to the A-B Schr¨ (2) operator H0 − qV , with a coupling constant q > 0. In order to keep track of the influence of the value of β, we can write L0 β 2 −2 β2 β 2 −2 β2 + r − qV ≥ L0 + r − qV = (L0 + r−2 − 2β −2 qV ). 2 2 2 2 2 Therefore, to estimate the number of eigenvalues for the operator with given β, we may use the existing estimates for the operator L0 + r−2 with potential 2β −2 qV . Such estimates for the Schr¨odinger operator L(r−2 , qV ) := L0 +r−2 −qV in L2 (R2 ) have been studied in [34, 6, 16]. We present here only the results from [16], not the most general ones, but, probably, the most transparent.
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Proposition 8.1 Let p > 1, V ≥ 0. Denote by Sp (V ) the expression
1/p
1
p
V (rω) dω
r dr
(8.4)
S1
0
in polar co-ordinates (r, ω). Then for a constant Cp , N− (L(r−2 , qV )) ≤ Cp qSp (V ) ;
(8.5)
in particular, if V is radial, V = V (r), then N− (L(r
−2
, qV )) ≤ Cq
V (x) dx.
(8.6)
R2
(2)
Due to domination, this estimate is carried over to H0 − qV , with p chosen arbitrary: Theorem 8.2 (A single solenoid) For some Cp , depending only on p, (2)
N− (H0 − qV ) ≤ Cp qβ −2 Sp (V )
(8.7)
and, for a radial potential, (2) N− (H0
− qV ) ≤ Cqβ
−2
V (x) dx.
(8.8)
R2
(2)
In a similar way, all estimates obtained in [6, 34] carry over to H0 − qV . Of course, the factor β −2 must arise in the estimates, as it was explained above. Finitely many solenoids Let h(2) be the (closed) quadratic form generating the unperturbed magnetic (2) Schr¨ odinger operator H0 associated with finitely many A-B solenoids. Again, write (8.2) and apply the Hardy type inequality established in Proposition 7.1 to one of the two terms in (8.2). We get h(2) [u] ≥
h(2) [u] + 2
R2
CW (x)|u(x)|2 dx, u ∈ C0∞ (R2 \ Λ). (2)
(8.9)
where W (x) is given in (7.3). Let H0 (W (x)) denote the operator generated by the form on the right-hand side of (8.9). As above, (8.9) together with Theorem 5.2 (2) reduce the problem of estimating N− (H0 − qV ) to the similar task for N− (L0 + W (x) − qV ).
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Ann. Henri Poincar´e
˜ j , 1 ≤ j ≤ J, such that λj ∈ Ω ˜ j and Take some splitting of R2 into sets Ω ˜ j and vanishing outside (of course, denote by Vj the function coinciding with V in Ω at least one of these sets must be unbounded). Then L0 + W − qV splits into the sum (in the sense of quadratic forms) of operators J −1 (L0 + W ) − qVj , and from the Weyl inequality we infer that N− (L0 + W (x) − qV ) ≤
J
N− (J −1 L0 + K|x − λj |−2 − qVj ),
(8.10)
j
where the constant K, depends on the positions of the solenoids λj and their fluxes αj , according to the Hardy type inequality (7.4). Separate terms in (8.10) have just been discussed above. For each j we define by Sp (Vj ) the expression of the form (8.4), with V replaced by Vj and integration performed in polar coordinates centered at λj . Proposition 8.1, applied to each Vj , leads to the following result. Theorem 8.3 (Finitely many solenoids) One has (2) N− (H0 − qV ) ≤ Cq Sp (Vj ),
(8.11)
1≤j≤J
where the constant C depends on the positions of the solenoids and their fluxes. For fixed positions of solenoids, the constant C in (8.7) is determined by the flux which is the one closest to integers and it deteriorates if this flux approaches an integer. Using Proposition 7.6 one may exclude arbitrary solenoids from (8.7), and leaving only fluxes which are sufficiently far away from integers, one can get an improved estimate of the form (8.7), with summation performed only over j ∈ J0 . Here the constant will depend only on the remaining fluxes αj , j ∈ J0 and the sum of all fluxes. Regular lattice of solenoids We consider the case with infinitely many solenoids located at the points of the lattice Λ = {λkl = (k, l) ∈ R2 : k, l ∈ Z }. As it is typical for the two-dimensional case, one can here, as for the previous configurations, give different types of CLR estimates. We restrict ourselves to the two, most simple versions. Theorem 8.4 Let V ∈ Lp (R2 ) locally, p ≥ 1. Consider a partition of R2 into unit cubes Qj . Then, for some constant C, (2)
N− (H0 − qV ) ≤ Cq
V Lp(Qj ) ,
j
where the norms involved are the Lp norms over the cubes.
(8.12)
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Proof. We use the inequality, already mentioned, W (x) ≥ W0 > 0, where W (x) is the weight function in Proposition 7.7. Thus, for the (nonmagnetic) two-dimensional Schr¨ odinger operator L0 + W0 − qV , the estimate of the type (8.12) is contained, for example, in [6]. Then the Hardy inequality in Proposition 7.7 and the diamagnetic inequality, together with Theorem 5.2, imply that the same kind (2) of estimate, with some other constant, holds for H0 − qV . Another estimate we give here is in flavor of Theorem 8.3. Theorem 8.5 Let V (x) be split into the sum of nonnegative functions Vkl , viz. Vkl . V (x) = kl
Then, for some constant Cp , (2) N− (H0
− qV ) ≤ Cp q
2 Sp (Vkl )
1 2
,
(8.13)
kl
as long as the quantity on the right-hand side is finite. Proof. Similar to the reasoning in the previous proof, the diamagnetic inequality, the Hardy inequality, and Theorem 5.2 reduce our task to establishing (8.13) for the two-dimensional Schr¨ odinger operator L0 + W − qV . 1/2 Denote Sp (Vkl ) by Rkl and suppose that the series Rkl converges to some 1/2 number M . Set τkl = Rkl M −1 , kl τkl = 1. Then the series ˜ (z) = W τkl |x − λkl |−2 kl
converges for any x ∈ Λ. Moreover, for some constant C, not depending on V , ˜ (x) ≤ CW (x). Thus, due to the max-min principle, it suffices to prove the W ˜ (z) − qV . From Weyl’s inequality it estimate (8.13) for the operator L0 + C W follows that −2 ˜ N− (L0 + C W (z) − qV ) = N− (τkl L0 + Cτkl |x − λkl | − qVkl ) ≤
kl
N− (τkl L0 + Cτkl |x − λkl |−2 − qVkl (|x − λkl |))
kl
=
−1 N− (L0 + C|x − λkl |−2 − qτkl Vkl (|x − λkl |)).
kl
To each term in the latter sum we apply the estimate in (8.5), getting 1/2 −1 −1 ˜ (z) − qV ) ≤ qC N− (L0 + C W τkl Sp (Vkl ) = qC τkl Rkl = qCM Rkl , kl
which coincides with the expression in (8.13).
kl
kl
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Ann. Henri Poincar´e
Improving the estimates Exactly as in the case of LT-type estimates, considered in Section 7, one can further improve CLR-type estimates by means of additional use of Hardy inequalities, which enables one to trace possible cancellations of singularities of the potential V and the Hardy weight. We present only one, simplest result in this direction, a substructure over Theorem 8.2; all other formulations can be obtained following this pattern, and the proofs just repeat the tricks used in Section 7. Proposition 8.6 Consider the case of a single solenoid, with radial potential V (r). Then for any κ ∈ (0, 1) 2 κβ (2) N− (H0 − qV ) ≤ Cq(1 − κ)−1 Sp V − . (8.14) 2|x|2 + (d)
Large coupling constant asymptotics for H0 − qV The task of establishing large coupling constant asymptotics for Schr¨ odinger-type operators is nowadays a routine matter as soon as correct estimates are obtained, see, e.g., [7, 27, 29], where this routine is described in details. Therefore, in the case of a singular magnetic field we just indicate those (minor) modifications one has to make. (d) Having the Schr¨ odinger operator H0 and a potential 0 ≤ V ∈ L1loc , we denote by Σ(V ) the quantity entering in the eigenvalue estimate for the particular of A-B solenoids above. Thus, in dimension three, we set Σ(V ) = configuration 3 V (x) 2 dx. In dimension two, Σ(V ) is the right-hand side of (8.7) or (8.8) under the conditions of Theorem 8.2, and it equals the right-hand side of (8.11) under the conditions of Theorem 8.3. Finally, for the case of a lattice, Σ(V ) is the right-hand side of (8.12), respectively (8.13), under the conditions of Theorem 8.4, respectively Theorem 8.5. (d)
Theorem 8.7 For d = 2, 3, let H0 be the (multivortex) Aharonov-Bohm Schr¨ odinger operator for any of the solenoid configurations in Section 2 and sup(d) pose Σ(V ) is finite. Then, for the negative eigenvalues of HqV = H0 − qV , the following asymptotic formula holds: d d V (x) 2 as q → ∞, (8.15) N− (HqV ) ∼ cd q 2 Rd
where cd is the standard coefficient, cd = (2π)−d ωd and ωd is the volume, resp. area, of the unit ball, resp. disk, in Rd . Note that, similar to the nonmagnetic case, the asymptotic formula in dimension d = 2 requires some additional restrictions compared with just finiteness of the asymptotic coefficient in (8.15).
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To prove the asymptotic formula, one has just to split, for given , the potential V into V , supported away from the solenoids, and V , Σ(V ) < . For the operator with potential V one finds the asymptotics, for example, by means of Dirichlet-Neumann bracketing, and for the operator with potential V the results of this section give estimate with a small coefficient. The matter is completed by applying the asymptotic perturbation lemma from [7], see also the expositions in [27, 29].
Acknowledgment M.M. thanks Professor J. Brasche for enlightening discussions and G.R. is grateful to Professor M. Solomyak for useful consultations. Moreover, the authors thank the Referee for constructive remarks and suggestions.
References [1] R. Adami, A. Teta, On the Aharonov-Bohm Hamiltonian, Lett. Math. Phys. 43, no. 1, 43–53 (1998). [2] J. Avron, I. Herbst, B. Simon, Schr¨ odinger operators with magnetic fields. I. General interactions. Duke Math. J. 45, no. 4, 847–883 (1978). [3] A.A. Balinsky, Hardy type inequality for Aharonov-Bohm magnetic potentials with multiple singularities, Math. Res. Lett. 10, no. 2–3, 169–176 (2003). [4] A.A. Balinsky, W.D. Evans, R.T. Lewis, On the number of negative eigenvalues of Schr¨ odinger operators with an Aharonov-Bohm magnetic field, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457, no. 2014, 2481–2489 (2001). [5] S.J. Bending, K. von Klitzing, K. Ploog, Weak localization in a distribution of magnetic flux tubes Phys. Rev. Lett. 65, 1060–1063 (1990). [6] M.Sh. Birman, A. Laptev, M. Solomyak, The negative discrete spectrum of the operator (−∆)l − αV in L2 (Rd ) for 2l ≥ d, d even, Ark. Mat. 35, 87–126 (1997). [7] M.Sh. Birman, M. Solomyak, The principal term of the spectral asymptotics for “non-smooth” elliptic problems, (Russian) Funkcional. Anal. i Priloˇzen 4, no. 4, 1–13 (1970). Translated in Functional Anal. Appl. 4, 265–275 (1970). [8] M.Sh. Birman, M. Solomyak, Negative discrete spectrum of the Schr¨ odinger operator with large coupling constant: a qualitative discussion; in Order, disorder and chaos in quantum systems (Dubna, 1989), pp. 3–16, Oper. Theory Adv. Appl., 46, Birkh¨ auser, Basel, 1990.
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[9] M.Sh. Birman, M. Solomyak, Estimates for the number of negative eigenvalues of the Schr¨ odinger operator and its generalizations, in: Estimates and Asymptotics of Discrete Spectra. Advances in Soviet Mathematics, 7, Amer. Math. Soc., 1991. ˇ ovˇıˇcek, Aharonov-Bohm effect with δ-type interaction, J. [10] L. D¸abrowski, P. St´ Math. Phys. 39, no. 1, 47–62 (1998). [11] A.K. Geim, V.I. Falko, S.V. Dubonos, I.V. Grigorieva, Single magnetic flux tube in a mesoscopic two-dimensional electron gas conductor, Solid State Commun. 82, no. 10, 831–836 (1992). [12] V.A. Geyler, E.N. Grishanov, Zero modes in a periodic system of AharonovBohm solenoids, JETP Letters 75, 354–356 (2002). [13] H. Hess, R. Schrader, D.A. Uhlenbrock, Domination of semigroups and generalization of Kato’s inequality, Duke Math. J. 44, no. 4, 893–904 (1977). [14] D. Hundertmark, A. Laptev, T. Weidl, New bounds on the Lieb-Thirring constants, Invent. Math. 140, no. 3, 693–704 (2000). [15] H.T. Ito, H. Tamura, Aharonov-Bohm effect in scattering by point-like magnetic fields at large separation, Ann. Henri Poincar´e 2, no. 2, 309–359 (2001). [16] A. Laptev, Yu. Netrusov, On the negative eigenvalues of a class of Schr¨ odinger operators; in Differential operators and spectral theory, pp. 173–186, Amer. Math. Soc. Transl. Ser. 2 189, Amer. Math. Soc., Providence, RI, 1999. [17] A. Laptev, T. Weidl, Hardy inequalities for magnetic Dirichlet forms; in Mathematical Results in Quantum Mechanics (Prague, 1998), pp. 299–305, Oper. Theory Adv. Appl., 108, Birkh¨ auser, Basel, 1999. [18] A. Laptev, T. Weidl, Sharp Lieb-Thirring inequalities in high dimensions, Acta Math. 184, 87–111 (2000). [19] A. Laptev, private communication. [20] E.H. Lieb, W. Thirring, Bound for the kinetic energy of fermions which proves the stability of matter, Phys. Rev. Lett. 35, 687–689 (1975). [21] E.H. Lieb, Kinetic energy bounds and their application to the stability of matter, in Schr¨ odinger operators, pp. 371–382, Lecture Notes in Phys. 345, Springer, Berlin, 1989. [22] V. Liskevich, A. Manavi, Dominated semigroups with singular complex potentials, J. Funct. Anal. 151, no. 2, 281–305 (1997). [23] M. Melgaard, G. Rozenblum, Spectral estimates for magnetic operators, Math. Scand. 79, no. 2, 237–254 (1996).
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[24] E.-M. Ouhabaz, L∞ -contractivity of semigroups generated by sectorial forms J. London Math. Soc. (2) 46, no. 3, 529–542 (1992). [25] E.-M. Ouhabaz, Invariance of closed convex sets and domination criteria for semigroups, Potential Anal. 5, no. 6, 611–625 (1996). [26] Ch. Pommerenke, Boundary behaviour of conformal maps, Springer-Verlag, Berlin, 1992. [27] M. Reed, B. Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press, New York-London, 1978. [28] G. Rozenblum, M. Solomyak, The Cwikel-Lieb-Rozenblum estimator for generators of positive semigroups and semigroups dominated by positive semigroups, (Russian) Algebra i Analiz 9, no. 6, 214–236 (1997); translation in St. Petersburg Math. J. 9, no. 6, 1195–1211 (1998). [29] G. Rozenblum, M. Solomyak, On the number of negative eigenvalues for the two-dimensional magnetic Schr¨odinger operator, in Differential operators and spectral theory, pp. 205–217, Amer. Math. Soc. Transl. Ser. 2, 189, Amer. Math. Soc., Providence, RI, 1999. [30] G. Rozenblum, Domination of semigroups and estimates for eigenvalues, (Russian) Algebra i Analiz 12, no. 5, 158–177 (2000); translation in St. Petersburg Math. J. 12, no. 5, 831–845 (2001). [31] S.N.M. Ruijsenaars, The Aharonov-Bohm effect and scattering theory, Ann. Physics 146, no. 1, 1–34 (1983). [32] B. Simon, A canonical decomposition for quadratic forms with applications to monotone convergence theorems, J. Funct. Anal. 28, no. 3, 377–385 (1978). [33] B. Simon, Functional integration and quantum physics, Academic Press, New York-London, 1979. [34] M. Solomyak, Piecewise-polynomial approximation of functions from odinger H l ((0, 1)d ), 2l = d, and applications to the spectral theory of the Schr¨ operator, Israel J. Math. 86, no. 1–3, 253–275 (1994). [35] H. Tamura, Norm resolvent convergence to magnetic Schr¨ odinger operators with point interactions, Rev. Math. Phys. 13, no. 4, 465–511 (2001). M. Melgaard Department of Mathematics Uppsala University Polacksbacken S-751 06 Uppsala, Sweden email:
[email protected]
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E.-M. Ouhabaz Laboratoire Bordelais d’Analyse et G´eom´etrie Universit´e de Bordeaux 1 351, Cours de la Lib´eration F-33405 Talence cedex, France email:
[email protected] G. Rozenblum Department of Mathematics Chalmers University of Technology and University of Gothenburg Eklandagatan 86 S-412 96 Gothenburg, Sweden email:
[email protected] Communicated by Bernard Helffer submitted 02/12/03, accepted 12/03/04
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Ann. Henri Poincar´e
Ann. Henri Poincar´e 5 (2004) 1013 – 1039 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/061013-27 DOI 10.1007/s00023-004-0188-2
Annales Henri Poincar´ e
A Direct Proof of the Nekhoroshev Theorem for Nearly Integrable Symplectic Maps Massimiliano Guzzo
Abstract. We provide the direct proof of the Nekhoroshev theorem on the stability of nearly integrable analytic symplectic maps. Specifically, we prove the stability of the actions for a number of iterations which grows exponentially with an inverse power of the norm of the perturbation by conjugating the generating function of the map to suitable normal forms with exponentially small remainder.
1 Introduction The stability of the actions of nearly integrable analytic symplectic maps for a number of iterations which grows exponentially with an inverse power of the norm of the perturbation was conjectured by N.N. Nekhoroshev already in his 1977 article (see [7], Section 2.2). Up to now, direct proofs exist only for specific situations (isochronous systems and neighborhoods of elliptic equilibrium points; see [1], [2] and [10]). For the general situation of nearly integrable symplectic maps, with hypotheses which extend in a natural way those of Nekhoroshev theorem, an indirect proof has been provided by Kuksin in [5], where the exponential stability is proved by showing the existence of a quasi-integrable non-autonomous analytic Hamiltonian system interpolating the map (see also [4], [6]). Kuksin’s article is based essentially on a constructive version of Grauert analytic embedding theorem, nevertheless it seems not so straightforward to recover (even up to a small order) the interpolating Hamiltonian of a given map, so we think that a direct proof which provides also explicit algorithms for the construction of the normal forms would be welcome. In this paper we provide such a direct proof, obtaining an exponential stability result which is independent of the one obtained by Kuksin in [5] and by Kuksin– Posch¨el in [6] (see points i–ii and iii below). Precisely, we consider symplectic maps which are generated by a function: S(I, ϕ) = I · ϕ + h(I) + εf (I, ϕ)
(1)
defined for I in an open set B ⊂ Rn and ϕ ∈ Tn ; h and f are analytic functions; ε is a small parameter. The function S generates (implicitly) the map C : (I, ϕ) →
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M. Guzzo
Ann. Henri Poincar´e
(I , ϕ ) through the equations: I ϕ
∂f (I , ϕ) ∂ϕ ∂h ∂f (I ) + ε (I , ϕ). = ϕ+ ∂I ∂I
= I + ε
(2)
For ε = 0 the map is integrable: the actions do not change, while the angles at any iteration rotate by an angle ω(I), with ω = ∂h ∂I . For ε = 0, in general the problem of the stability of the actions arises. The Nekhoroshev theorem for maps then can be stated as follows (conjectured in [7]; first proof by interpolation in [5]): Theorem 1 If h is convex, there exist positive constants ε0 , a, b, t0 , d0 such that for any ε < ε0 , and for any I(0), ϕ(0) ∈ B × Tn , with B = {I ∈ B : dist(I, ∂B) > 2d0 εa }, denoting: (I(t), ϕ(t)) = Ct (I(0), ϕ(0)), it is: |I(t) − I(0)| ≤ d0 εa for any t ∈ Z satisfying: |t| ≤ t0 exp
ε b 0
ε
(3)
.
(4)
The above theorem is different from the one conjectured in [7], Section 2.2 because the ‘P-steepness’ hypothesis on h used in Nekhoroshev’s paper is here replaced by the stronger convexity hypothesis. Here, we refer to the convex situation which is a non-trivial case of P-steepness sufficient to illustrate Nekhoroshev theorem (see [3]). Nevertheless, the result extends also to the P-steepness case. We remark however that the convexity hypothesis cannot be replaced by the weaker quasi-convexity hypothesis, which is commonly used in the Hamiltonian case. Indeed, quasi-convexity of h is not allowed (for the generic steep case, Nekhoroshev replaced the steepness condition, valid for the Hamiltonian case, with the ‘P-steepness’ condition), as can be easily seen by the trivial counter-example S(I, ϕ) = Iϕ + 2πI + ε cos(ϕ). The exponential stability result which is proved in this paper is independent from the one stated in [5] (and also in [6]) for the following reasons: i) the critical parameter ε0 appearing in Kuksin’s result is necessarily smaller than the critical ε∗ for which the analytic Hamiltonian interpolation can be proved to exist. Such a problem here does not exist. ii) within Kuksin’s technique it is necessarily: ε0 → 0 for supI∈B |ω(I)| → ∞, because it can be shown that the analyticity radius with respect to time of the interpolating Hamiltonian goes to 0 as a positive power of 1/ |ω(I)|. A similar limitation (which is not natural in the Nekhoroshev theorem) is absent in our proof.
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iii) Kuksin’s technique, using the Nekhoroshev theorem for quasi-convex Hamiltonian systems, allows to provide the stability exponents a = b = 1/(2(n + 1)) as well as a = 1/4 + b, b = 1/(4(n+ 1)) (for other possible exponents see [8], ‘Theorem 1∗ ’). Instead, with the direct proof, using the original geometric construction by Nekhoroshev, we will obtain a worse stability exponent b ∼ 1/n2 . More precisely, for any n ≥ 2, we obtain: 2 − 6a 1 b= , a ∈ 0 , , 3n2 + 3n − 2 3 which include: b=
6n2
1 1 , a= . + 6n − 4 4
The article is structured as follows: in Section 2 we illustrate the basic idea allowing the construction of resonant normal forms for symplectic maps. In Section 3 we define the geometry of resonances in the action domain. In Sections 4, 5 and 6 we provide the technical details for the construction of the normal forms. In Section 7 we conclude the proof.
2 Normal forms for symplectic maps via generating functions Following [1] and [10], we look for near to the identity symplectic transformations ˜ ϕ) = I · ϕ + εχ(I, ϕ) through the Φ : (I, ϕ) → (I , ϕ ) generated by a function χ(I, equations: I
=
ϕ
=
∂χ (I , ϕ) ∂ϕ ∂χ ϕ + ε (I , ϕ) , ∂I
I + ε
(5)
such that the conjugate map C = Φ−1 ◦ C ◦ Φ is ‘more integrable’ than C. This means that C is generated by a function S (I, ϕ) = I · ϕ + h(I) + εf (I, ϕ), with f smaller than f (except for a possible resonant term). We remark that if ε is suitably small, then C can be also generated by some function S . The specific form of S is obtained by suitably choosing the function χ. The composition of these symplectic maps satisfies the following lemma: Lemma 1 Let S be as in (1), let χ be defined and analytic in B × Tn , and denote by C and Φ the symplectic maps defined implicitly by (2) and (5) respectively. If ε is small the transformation C = Φ−1 ◦ C ◦ Φ is generated by: S = I · ϕ + h(I) + εf (I, ϕ) + ε[χ(I, ϕ) − χ(I, ϕ + ω(I))] + ε2 f (I, ϕ)
(6)
where ω(I) = ∂h ∂I (I), and the remainder f is real analytic and bounded uniformly n in ε in B × T .
The lemma is proved, with detailed estimates, in Section 5.
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Lemma 1 concerns the composition of the symplectic map generated by S, with any near to the identity symplectic map, written in the form (5). The normal forms of S are instead obtained by means of specific choices of χ, depending on the resonance properties of the domain. For example, the non-resonant normal form is obtained when the term of order ε of S does not depend on the angles. This happens if the function χ is chosen as f (I, ϕ) + [χ(I, ϕ) − χ(I, ϕ + ω(I))] = g(I) ,
(7)
where g(I) is analytic in B. The above equation is the ‘homological equation’ for symplectic maps, which can be solved by Fourier series. Denoting: f= fk (I)eik·ϕ , χ = χk (I)eik·ϕ , k∈Zn
k∈Zn
χ and g satisfy eq. (7) if it is: f0 (I) = g(I) fk (I) + χk (I)[1 − eik·ω(I) ] = 0
, if k = 0 .
(8)
A formal solution is therefore given by: g = f0 , χ = −
k∈Zn \0
fk (I) eik·ϕ . 1 − eik·ω(I)
(9)
In general, as for the Hamiltonian case, the above solution is only formal, in the sense that χ is not defined in any open subset of B, because the set of I where for some k ∈ Zn \0 it is: 1 − eik·ω(I) = 0 is dense in B. However, an analytic solution can be found by restricting suitably the spectrum of f and the domain of definition of χ (as it is done in the proof of the Nekhoroshev theorem for Hamiltonian systems). Remark. In the Hamiltonian case, an integrable Hamiltonian h(I) produces the small denominators k · ω(I), with k ∈ Zn \0, and therefore the resonances are given by the equations k · ω(I) = 0, with k ∈ Zn \0. Instead, in the case of symplectic maps, the small denominators are: 1 − eik·ω(I) , with k ∈ Zn \0, and therefore the resonances are given by the equations: k · ω(I) = 2k0 π ,
(10)
with k ∈ Zn \0 and k0 ∈ Z. These kinds of denominators, which appeared already in the proof of the isochronous case (see [1], [10]) are consistent with the representation of the n degrees of freedom integrable map generated by I · ϕ + h(I) as the 2π-time flow of the n + 1 degrees of freedom Hamiltonian system with Hamilton function h(I) 2π + In+1 , whose resonances are given by eq. (10).
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3 The geometry of resonances As explained in the previous section, the geometry of resonances refers to the frequency vector Ω = (Ω0 , . . . , Ωn ) ∈ Rn+1 defined by Ω0 = 2π, and Ωi = ωi (I) for any i = 1, . . . , n. We adapt to maps the original geometric construction of the Nekhoroshev paper [7] (see also [3] for the convex case), introducing some differences with respect to the Hamiltonian case (see the remarks below). As it is usual, the construction of resonant domains is parameterized by a positive number K > 0 and by n positive parameters: 0 < α1 ≤ α2 ≤ · · · ≤ αn < π .
(11)
More precisely, for any choice of these parameters, we define the following structures: • K-lattices: for any K > 0, we first consider the set of all integer lattices (i) Λ ⊆ Zn which are generated by d ≤ n independent integer vectors k , (i) (i) n i ≤ d, with order k = j=1 kj ≤ K; then, among these lattices, we do not consider those which are properly contained in other of these lattices of the same dimension. The remaining lattices will be called K-lattices. • resonant manifolds: for any K-lattice Λ we define its resonant manifold: RΛ
=
{I ∈ B : ∀k ∈ Λ there exists k0 ∈ Z : k · ω(I) + 2πk0 = 0} .
(12)
• resonant zones: For any d-dimensional K-lattice Λ, 1 ≤ d ≤ n, its resonant zone is ZΛ
=
{I ∈ B : ∀k ∈ Λ with |k| ≤ K there exists k0 ∈ Z : (13) |k · ω(I) + 2πk0 | ≤ k αd } ,
where denotes the Euclidean norm of a vector. • resonant blocks: for any d-dimensional K-lattice Λ, 1 ≤ d ≤ n−1, its resonant block is / ZΛ for any K−lattice Λ , with dimΛ = d + 1} DΛ = {I ∈ ZΛ such that I ∈ (14) while the non-resonant block, corresponding to Λ = {0}, is: / ZΛ for any K−lattice Λ , with dimΛ = 1} D0 = {I ∈ B : I ∈
(15)
and the completely resonant block is: DZn = Z Zn .
(16)
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• cylinders: For any I ∈ Rn and δ > 0 denote Ξ(I, δ) = {I ∈ Rn : I − I ≤ δ} . For any Λ with 1 ≤ dimΛ ≤ n, we denote by I + Λ the plane through I spanned by the lattice Λ. We associate to each I ∈ DΛ the cylinder of radius δ, which we denote by CΛ,δ (I), defined as the connected component containing I of the set: Ξ(I , δ) ∩ ZΛ . I ∈I+Λ
The intersection of CΛ,δ (I) with the border of ZΛ gives the cylinder lateral walls. • the extended block is
ext = DΛ,δ
CΛ,δ (I) .
I∈DΛ
Remarks. I) In the definition of the resonant manifolds and zones we have considered resonances k · ω(I) + 2πk0 = 0, with k0 ∈ Z and k ∈ Zn , with the norm of k bounded by K, while in the analogous Hamiltonian construction the norm of the complete resonant vectors (k0 , k) is bounded. In the case of maps any harmonic of the perturbation labeled by some k ∈ Zn produces resonances of the same width with any vector k0 ∈ Z, and therefore it is necessary to consider all these resonances. Nevertheless, the restriction on the norm of vector k is sufficient to control the density and avoid the overlapping of all these resonances. II) Though the resonances are defined with respect to the n + 1-dimensional frequency vector Ω, we need to define only n parameters αi (and not n + 1), because the n + 1-dimensional resonance does not exist. The original confinement argument of the Nekhoroshev theorem resides mainly on two facts: first, the parameters entering the geometric construction are such that the extended block of a given lattice Λ does not intersect the resonant zones of any lattice of the same dimension of Λ; second, in an exponentially long time any motion with initial action in a given cylinder can leave it only through its lateral walls. In the rest of the section we focus our attention on the first point, while the second will be considered in Section 7. Through this paper, we will use the following notations. We denote by I , ϕ the analyticity radii such that h and f are analytic in BI × Tnϕ ; we denote by M a Lipschitz constant of ω(I) = ∂h/∂I in the set BI and by M0 a positive constant such that: 2 ∂ h ≥ M0 u · u (I)u · u (17) ∂I 2
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for any u ∈ Rn and any I ∈ B. The hypotheses of Theorem 1 (analyticity of h and f and convexity of h) guarantee that such constants are strictly positive, and we take the freedom to choose M0 and M in such a way that: M0 < M . We now prove the following technical lemmas about the resonant domains: Lemma 2 Let h : B → R convex, and let M, M0 be defined as above. Consider any K-lattices Λ, Λ with d = dimΛ = dimΛ . If δΛ and the αi satisfy: δΛ ≤
2 M dK d−1 αd , αd+1 ≥ 6K dK d−1 αd M M0
(18)
ext ext ext Ξ(I, δΛ ). ) ∩ (ZΛ ) = ∅, where (DΛ,δ ) = ∪I∈DΛ,δ then it is (DΛ,δ Λ δΛ Λ δΛ Λ
Before proving Lemma 2 we estimate the small denominators in any resonant block and the diameters of the cylinders. Lemma 3 For any K-lattice Λ with dim Λ = d ∈ [1, n − 1] and for any I ∈ DΛ it is: (19) |k · ω(I) + 2πk0 | > k αd+1 for any k ∈ Zn \Λ with |k| ≤ K and for any k0 ∈ Z. For any I ∈ D0 it is: |k · ω + 2πk0 | > α1
(20)
for any k ∈ Z \0 with |k| ≤ K and for any k0 ∈ Z. n
The proof of Lemma 3 follows directly from the definition of the resonant zones and blocks. Lemma 4 For any K-lattice Λ with dim Λ = d ∈ [1, n] and δ ≤ I ∈ DΛ and I ∈ CΛ,δ (I) then it is:
I − I ≤
2 d−1 αd , 5M dK
3 dK d−1 αd . M0
if
(21)
Proof of Lemma 4. Let Iˆ ∈ I + Λ (we recall that I + Λ denotes the plane ˆ δ). The following inequalities hold: through I spanned by Λ) such that I ∈ Ξ(I, M0 I − I 2 ≤ |(ω(I) − ω(I )) · (I − I )| ˆ + |(ω(I) − ω(I )) · (Iˆ − I )| ≤ |(ω(I) − ω(I )) · (I − I)| ˆ + M I − I δ , ≤ |PΛ (ω(I) − ω(I )) · (I − I)|
(22)
where we denote by PΛ v the orthogonal projection of a vector v ∈ Rn over the real space spanned by the lattice Λ. Let k (1) , . . . , k (d) ∈ Znbe a K-base of Λ. Therefore, any i there exist ni , ni such that: ω(I) · k (i) + 2πni ≤ αd and since I, I (i)∈ ZΛ , for ω(I ) · k + 2πn ≤ αd . Moreover, one can choose ni = n : if we consider any arc i i
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I(s) ∈ CΛ,δ (I), s ∈ [0, 1], with I(0) = I, I(1) = I , it is ω(I(s)) · k (i) + 2πni ≤ αd for any s ∈ [0, 1]. In fact, let us suppose that there exists s0 ∈ (0, 1), and a neighborhood U of s0 such that for s1 ≤ s0 < s2 it is any s1 , s2 ∈ U with ω(I(s1 )) · k (i) + 2πni ≤ αd and ω(I(s2 )) · k (i) + 2πni > αd . In such a case, there exists also an integer n ˆ = ni such that ω(I(s2 )) · k (i) + 2πˆ n ≤ αd . It would be also: n − ni | − KM I(s1 ) − I(s2 ) − αd , (23) αd ≥ ω(I(s1 )) · k (i) + 2πni ≥ 2π |ˆ which is in contrast with αd < π as soon as |s1 − s2 | tends to zero. Therefore, for any k (i) it is: ω(I ) · k (i) + 2πni ≤ αd , from which it follows also: PΛ (ω(I) − ω(I )) · k (i) = (ω(I) − ω(I )) · k (i) = ω(I) · k (i) + 2πni − ω(I ) · k (i) − 2πni ≤ 2αd
(24)
From “technical Lemma 1” of [3] it follows:
PΛ (ω(I) − ω(I )) ≤ 2dK d−1 αd .
(25)
M0 I − I 2 ≤ 2dK d−1αd ( I − I + δ) + M I − I δ
(26)
From (22) follows also:
from which follows (21). ext Proof of Lemma 2. Since I ∈ (DΛ,δ ) , there exists I ∈ DΛ with I − I ≤ Λ δΛ 3 d−1 αd + δΛ . The lattice Λ contains at least a vector k ∈ Zn \Λ with |k| ≤ K. M0 dK From Lemma 3, for any n ∈ Z it is
|ω(I) · k + 2πn| ≥ k αd+1 − k M
3 dK d−1 αd − k M δΛ ≥ k αd . M0
(27)
This inequality implies that I ∈ / ZΛ . In fact, if I ∈ ZΛ , for any k ∈ Λ with |k| ≤ K there exists n such that: |ω(I) · k + 2πn| ≤ k αd , and this is in contrast with (27).
4 The resonant normal forms In this section we construct the resonant normal form for any resonant domain. The construction follows closely that of Hamiltonian systems, except that the homological equation is replaced by equation (7), as explained in Section 2. We first fix some notation. For any real domain D ⊆ Rn and σ > 0 we denote by {x ∈ Cn : x − x ≤ σ} (28) D σ = x∈D
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its complex extension of radius σ; we denote by Tnσ = {ϕ ∈ (C/2πZ)n : |Im ϕi | ≤ σ for any i ≤ k}
(29)
the complex extension of Tn of radius σ. We will handle real domains of the form D × Tn with D ⊆ Rn ; for any function u : D × Tn → C, for we consider the Fourier decomposition u= uk (I)eik·ϕ , (30) k∈Zn
and, for any Λ ⊆ Zn , the Fourier projection ΠΛ u = u (I)eik·ϕ and for k∈Λ k ik·ϕ . For any any K > 0 we define the cut-off projection: TK u = |k|≤K uk (I)e ‘extension vector’ σ = (σI , σϕ ) (extension vectors will be always considered with positive entries, and inequalities on extension vectors are intended as inequalities on the entries) we denote by |u|σ the sup-norm in the domain D σI × Tnσϕ . Lemma 5 Let K ≥ 2 be such that:
N= Let the function
32n ϕ ,
≥
K ln K
and define
1 K ϕ . 32n ln K
W = I · ϕ + h(I ) + εf (I , ϕ) Tnϕ .
(31)
(32)
Let Λ ⊆ Z be a K-lattice and let α ∈ (0, 1), rI ∈ be analytic in BI × (0, 2I ], and D ⊆ B be such that for any I ∈ Dr I it is: α (33) 1 − eik·ω(I ) ≥ 2 n+1
for any k ∈ Λ\0 and |k| ≤ K. If r, N and K satisfy: rϕ = K≥ and it is:
ϕ 2 6 ϕ
ln
α2 rI rϕ , nN 2 ϕ 2 Γ ≤ min 1, , , M I n αN εn |f | ≤ Γ
with:
ϕ 27 MN 22n+7 nn−1 n ϕ
, rI ≤
1
22n+18
(34)
(35)
(36)
there exists a symplectic map: Φ : Dr I /2 × Tnrϕ /2 → Dr I × Tnrϕ , which is a diffeomorphism on its image, such that, denoting by C the symplectic map generated by W , its pull-back C = Φ−1 ◦ C ◦ Φ : DrI × Tnrϕ → Dr I × Tnrϕ is generated by the 2 2 function: W (I , ϕ) = I · ϕ + h(I ) + εu(I , ϕ) + εR(I , ϕ) , (37)
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where R and u are analytic in DrI × Tnrϕ , u = ΠΛ u, |u − ΠΛ TK f | r ≤ 2 |f | and 2
2
2
|R| r ≤ 2
1 |f | . 4N N
(38)
Denoting (I , ϕ ) = Φ(I, ϕ), with (I, ϕ) ∈ Dr I /2 × Tnrϕ /2 , it is: |Ii − Ii | ≤
rI 211
(39)
for any i ≤ n.
5 On the conjugation of nearly integrable symplectic maps with near to the identity symplectic map In this section we compute the pull-back of nearly integrable symplectic maps with respect to near to the identity symplectic maps, so as to prove Lemma 1. Consider the generating function: S(I , ϕ) = I · ϕ + k(I ) + εw(I , ϕ) ,
(40)
where k and w are analytic in B × Tn , with B ⊆ Rn open set, ε ∈ R (in the following it will be useful to consider both choices k = 0 and k = h). If ε is suitably small, the following equations: I ϕ
∂w (I , ϕ) ∂ϕ ∂k ∂w (I , ϕ) , = ϕ + (I ) + ε ∂I ∂I = I + ε
(41)
define implicitly a symplectic diffeomorphism: C(I, ϕ) = (I , ϕ ). We consider then a near to the identity symplectic map Φ : (I0 , ϕ0 ) → (I, ϕ) generated by χ(I, ˜ ϕ0 ) = I ·ϕ0 + εχ(I, ϕ0 ), where χ is analytic in a domain D × Tn , with D ⊆ B . We want to compute the generating function for the map C = Φ−1 ◦ C ◦ Φ. It is convenient to refer to the following diagram: C
(I, ϕ) −−−−→ (I , ϕ )
Φ Φ
(42)
C
(I0 , ϕ0 ) −−−−→ (I0 , ϕ0 ). Lemma 6 Let S,χ, C, C and Φ as above. If ε is suitably small, the transformation C is generated by: S
= I · ϕ0 − I · ϕ + I0 · ϕ0 − I · ϕ0 + I · ϕ+ k(I ) + εw(I , ϕ) + ε[χ(I, ϕ0 ) − χ(I , ϕ0 )] ,
where I, ϕ, I , ϕ , I0 , ϕ0 are functions of the independent variables I0 , ϕ0 .
(43)
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Proof of Lemma 6. Because of equations (41) the standard Lie condition reads: (I − I ) · dϕ + (ϕ − ϕ) · dI = d(k(I ) + εw(I , ϕ)) ,
(44)
and it is defined globally on B × Tn . Similarly, to compute the generating function S it is sufficient to compute the differential form (I0 − I0 ) · dϕ0 + (ϕ0 − ϕ0 ) · dI0 , because it is: dS (I0 , ϕ0 ) = d(I0 · ϕ0 ) + (I0 − I0 ) · dϕ0 + (ϕ0 − ϕ0 ) · dI0 .
(45)
With reference to diagram (42), it is: dS(I , ϕ) = dχ(I, ˜ ϕ0 ) = dχ(I ˜ , ϕ0 ) =
d(I · ϕ) + (I − I ) · dϕ + (ϕ − ϕ) · dI d(I · ϕ0 ) + (I0 − I) · dϕ0 + (ϕ − ϕ0 ) · dI d(I · ϕ0 ) + (I0 − I ) · dϕ0 + (ϕ − ϕ0 ) · dI ,
(46)
d(S(I , ϕ) + χ(I, ˜ ϕ0 ) − χ(I ˜ , ϕ0 )) = d(I0 · ϕ0 + I · ϕ − I0 · ϕ0 ) +(I0 − I0 ) · dϕ0 + (ϕ0 − ϕ0 ) · dI0 ,
(47)
and therefore:
so that the new generating function is: S
= I0 · ϕ0 − I · ϕ + S(I , ϕ) + χ(I, ˜ ϕ0 ) − χ(I ˜ , ϕ0 ) = I0 · ϕ0 − I · ϕ + I · ϕ + I · ϕ0 − I · ϕ0 + k(I ) +εw(I , ϕ) + ε[χ(I, ϕ0 ) − χ(I , ϕ0 )] ,
(48)
where I, ϕ, I , ϕ , I0 , ϕ0 are functions of the independent variables I0 , ϕ0 . Lemma 6 provides the analytic expression of the generating function of C . In the following lemma we provide estimates for C on convenient domains. To fix notations, for any set V ⊆ Rn and any σ = (σI , σϕ ) we will denote Vσ = VσI ×Tnσϕ . Lemma 7 Consider the generating function: W (I , ϕ) = I · ϕ + k(I ) + εw(I , ϕ) ,
(49)
where w : B˜I × Tn˜ϕ → C, and k : B˜I → C are analytic. Let ∆I , ∆ϕ defined by: ∂w ∂w εn εn max max , ∆ϕ = ∆I = 4 i ∂ϕi ˜ 4 i ∂Ii ˜
(50)
satisfy the following inequalities: ∆ϕ + M ˜I < ˜ϕ , ∆I < ˜I 2
(51)
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on B˜I . If: ∂2w 1 εn max , ≤ i,j ∂ϕi ∂Ij 2
where M is a Lipschitz constant for
∂k ∂I
(52)
˜
then the equations: ∂w (I , ϕ) ∂ϕ ∂k ∂w = ϕ + (I ) + ε (I , ϕ) ∂I ∂I = I + ε
I ϕ
(53)
define implicitly the analytic symplectic diffeomorphism: C(I, ϕ) = (I , ϕ ) such that: • for any (I, ϕ) ∈ B˜I −∆I × Tn˜ϕ there exists I ∈ B˜I such that: I = I + ε
∂w (I , ϕ) , ∂ϕ
(54)
and therefore B˜I −∆I × Tn˜ϕ is in the domain of C; • for any (I, ϕ) ∈ B˜I −∆I × Tn˜ϕ −∆ϕ it is: ∆I maxi |Ii − Ii | ≤ 2 ∂k maxi ϕi − ϕi − ∂I (I ) ≤ i
maxi |Im ϕi − Im ϕi | ≤ so that C(B−∆ ) ⊆ B ˜
˜I −
∆I 2
× Tn
˜ϕ −
∆ϕ 2
+M˜I
∆ϕ 2
∆ϕ 2
+ M ˜I
,
(55)
;
• for any (I , ϕ ) ∈ B˜I × Tn˜ϕ −∆ϕ there exists ϕ ∈ Tn˜ϕ such that: ϕ = ϕ +
∂k ∂w (I ) + ε (I , ϕ) , ∂I ∂I
(56)
and therefore B˜I × Tn˜ϕ −∆ϕ is in the domain of C−1 ; • for any (I , ϕ ) ∈ B˜I −∆I × Tn˜ϕ −∆ϕ , denoting (I, ϕ) = C−1 (I , ϕ ) it is: ∆I maxi |Ii − Ii | ≤ 2 ∂k maxi ϕi − ϕi − ∂I (I ) ≤ i
maxi |Im ϕi − Im ϕi | ≤ so that C−1 (B−∆ ) ⊆ B ˜
˜I −
∆I 2
× Tn
˜ϕ −
∆ϕ 2
∆ϕ 2
+M˜I
∆ϕ 2
+ M ˜I .
,
(57)
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Proof of Lemma 7. We first prove that equations (53) can be inverted to define C in B˜I −∆I × Tn˜ϕ . Let us fix ϕ ∈ Tn˜ϕ , and denote Bϕ = ∪I ∈B˜ {I + ε ∂w ∂ϕ (I , ϕ)}. I The map: B˜I I
−→ Bϕ
−→ I = I + ε
∂w (I , ϕ) ∂ϕ
(58)
is injective. In fact, if there exist I1 , I2 ∈ B˜I such that: I1 + ε
∂w ∂w (I , ϕ) = I2 + ε (I , ϕ) , ∂ϕ 1 ∂ϕ 2
(59)
then it is trivially:
I1
−
I2
∂2w ≤ εn max I1 − I2 . i,j ∂ϕi ∂Ij
(60)
˜
I1
I2 ,
= the above inequality is in contrast with eq. (52). This means that we If can define a map u : B → B˜I , where B = {(ϕ, I) such that ϕ ∈ Tn˜ϕ and I ∈ Bϕ } , such that: I = u(I, ϕ) + ε
∂w (u(I, ϕ), ϕ) . ∂ϕ
(61)
The map u is analytic. To prove this we use the local inversion theorem for holomorphic maps. Indeed, for any (I0 , ϕ0 ) ∈ B˜I × Tn˜ϕ , the Jacobian matrix of I + ε ∂w ∂ϕ (I , ϕ) is: ∂2w Jik = δik + ε (I , ϕ) , (62) ∂Ik ϕi 2 w which is non-singular because by eq. (52) it follows: εn ∂ϕ∂ i ∂I < 1. This proves j
˜
that u is analytic in a suitable neighborhood of (I0 + ε ∂w ∂ϕ (I0 , ϕ0 ), ϕ0 ), for any n (I0 , ϕ0 ) ∈ B˜I × T˜ϕ , and therefore u is analytic in B. The second equation of (53) defines ϕ as a function of I , ϕ, and therefore it is possible to define ϕ as a function of I, ϕ, analytic in B. This allows one to define the analytic map C : B → B˜I × Tn˜ϕ :
I
ϕ
= =
u(I, ϕ) ∂k ∂w ϕ + (u(I, ϕ)) + ε (u(I, ϕ), ϕ) . ∂I ∂I
(63)
Now, we prove: B˜I −∆I × Tn˜ϕ ⊆ B, so that B˜I −∆I × Tn˜ϕ is in the domain of C. Specifically, for any (I, ϕ) ∈ B˜I −∆I × Tn˜ϕ we prove that there exists I ∈ B˜I
1026
M. Guzzo
Ann. Henri Poincar´e
n such that: I = I + ε ∂w ∂ϕ (I , ϕ). Let us consider the map y : B˜I → C such that y(I ) = I − ε ∂w ∂ϕ (I , ϕ). We prove that y has a fixed point I ∈ B˜I . We consider j the sequence Ij = y (I). Using (52) and (50) one easily proves: Ij − I ≤ ∆I /2 for any j ∈ N, so that all Ij are in the domain of y, and moreover Ij is a Cauchy sequence, so that its limit I exists, it is I ∈ B˜I , and it is a fixed point of y. This proves B˜I −∆I × Tn˜ϕ ⊆ B. In a very similar way we prove that: B˜I × Tn˜ϕ −∆ϕ is in the domain of C−1 . Specifically, for any (I , ϕ ) ∈ B˜I × Tn˜ϕ −∆ϕ we prove that there exists ϕ ∈ Tn˜ϕ ∂k ∂w n n such that: ϕ = ϕ + ∂I (I ) + ε ∂I (I , ϕ). Let us consider the map: z : T ˜ϕ → C , ∂k ∂w such that z(ϕ) = ϕ − ∂I (I ) − ε ∂I (I , ϕ). We prove that z has a fixed point ∂k ϕ ∈ Tn˜ϕ −∆ϕ . We consider the sequence ϕj = z j (ϕ − ∂I (I )). Using (52) and (50) one proves: ϕj − ϕ0 ≤ ∆ϕ /2 for any j ∈ N, so that all ϕj are in the domain of z, and moreover ϕj is a Cauchy sequence, so that its limit ϕ exists, it is ϕ ∈ Tn˜ϕ , and it is a fixed point of z. eqs. (55), (57) immediately follow from eq. (50).
Lemma 8 Consider the generating functions: W (I , ϕ) χ(I ˜ , ϕ)
= I · ϕ + h(I ) + εw(I , ϕ) = I · ϕ + εχ(I , ϕ)
(64)
where w : B˜I × Tn˜ϕ → C, χ : B˜I × Tn˜ϕ → C and h : B˜I → C are analytic and satisfy the estimates: ∂2w ∂2χ 1 1 , εn max (65) εn max ≤ ≤ i,j ∂ϕi ∂Ij i,j 2 ∂ϕi ∂Ij 2 ˜
−δ ˜
with some δ < ˜. Let now ∆ and ζ be defined by: ∂w ∂w εn εn ∆I = max max , ∆ = ϕ 4 i ∂ϕi ˜ 4 i ∂Ii ˜ ∂χ ∂χ εn εn max max ζI = , ζϕ = i i 4 ∂ϕi −δ 4 ∂Ii −δ ˜ ˜
(66)
satisfying: ζ < ˜ − δ ,
∆ϕ + ζϕ + M ˜I ≤ δϕ , ∆I + ζI ≤ δI 2
(67)
∂h where M is a Lipschitz constant for ∂I on B ˜I . Then, denoting by C the canonical transformation generated by W , with Φ the canonical transformation generated by χ ˜ and with C = Φ−1 ◦ C ◦ Φ, the maps C and C−1 are analytic and symplectic diffeomorphisms from B−∆ on their image; Φ and Φ−1 are analytic and symplec˜ −1 tic diffeomorphisms from B−δ−ζ on their image; C and C are analytic and ˜ symplectic diffeomorphisms from B−2δ−∆−2ζ on their image and it is: ˜
) ⊆ B−δ−∆−2ζ , C C (B−2δ−∆−2ζ ˜ ˜
−1
(B−2δ−∆−2ζ ) ⊆ B−δ−∆−2ζ . ˜ ˜
(68)
Vol. 5, 2004
Nekhoroshev Theorem for Nearly Integrable Symplectic Maps
1027
Let now η satisfying: 2δ + ∆ + 2ζ + 2η < ˜ , ∆I + ζI ≤
ηI . 2n
(69)
The map C can be generated by the function: W (I , ϕ) = +
I · ϕ + h(I ) + ε(w(I0 , ϕ0 ) + χ(I0 , ϕ0 ) − χ(I0 , ϕ0 + ω(I0 ))) εw (I , ϕ) (70)
where w is analytic in B−2δ−∆−2ζ−2η and satisfies the following estimate: ˜ ≤ ε−1 |w |−2δ−∆−2ζ−2η ˜
9 4
MnζI2 + 4∆I ζϕ + 4∆ϕ ζI + 8ζI ζϕ
.
(71)
Proof of Lemma 8. It is convenient to refer to the following diagram: C
(I, ϕ) −−−−→
Φ
(I , ϕ )
Φ
(72)
C
(I0 , ϕ0 ) −−−−→ (I0 , ϕ0 ) . First, we prove that C can be generated by a function W (I0 , ϕ0 ). We denote: I0 = I0 + u(I0 , ϕ0 ) .
(73)
Because:
∆ϕ ∆I + ζϕ + M ˜I < δϕ , + ζI ≤ δI , 2 2 for any (I0 , ϕ0 ) ∈ B−δ−∆−2ζ it is (I, ϕ), (I , ϕ ), (I0 , ϕ0 ) ∈ B ˜ Tn , ∆ ˜ϕ −δϕ − 2ϕ −ζϕ +M˜I
˜I −δI −
(74) ∆I 2
−ζI
×
and in particular the functions uj are analytic in B−δ−∆−2ζ . ˜
For Lemma 7, they satisfy the estimates: |uj |−2δ−∆−2ζ ≤ ˜
∆I + ζI , 2
and therefore from Cauchy estimates it follows: ∂uj ∆I ζI ≤ + , ∂I0 2η η I I k −2δ−∆−2ζ−η ˜
(75)
(76)
ηI and because ∆I + ζI ≤ 2n the inversion theorem for holomorphic functions allows to define functions u˜j such that:
˜(I0 , ϕ0 ) , I0 = I0 + u
(77)
1028
M. Guzzo
which are analytic in: A=
{
Ann. Henri Poincar´e
(I0 + u(I0 , ϕ0 ), ϕ0 )} .
(78)
ϕ0 ∈Tn ˜ϕ −2δϕ −∆ϕ −2ζϕ −ηϕ I0 ∈B ˜
I −2δI −∆I −2ζI −ηI
Moreover, it is B−2δ−∆−2ζ−2η ⊆ A: for any (I0 , ϕ0 ) ∈ B˜I −2δI −∆I −2ζI −2ηI × ˜ n T˜ϕ −2δϕ −∆ϕ −2ζϕ −ηϕ we prove that there exists I0 ∈ B˜I −2δI −∆I −2ζI −ηI such that: I0 = I0 + εu(I0 , ϕ0 ). Let us consider the map: y : B˜I −2δI −∆I −2ζI −ηI → Cn , such that y(I) = I0 − εu(I, ϕ0 ). We prove that y has a fixed point I ∈ B˜I −2δI −∆I −2ζI −ηI , and consequently I0 = I ∈ B˜I −2δI −∆I −2ζI −ηI . We consider the sequence Ij = y j (I0 ). Using the inequality: ∆I + ζI ≤
ηI , 2n
(79)
one easily proves that Ij is a Cauchy sequence and Ij − I0 ≤ ηI for any j ∈ N. Therefore, its limit I exists, it is in B˜I −2δI −∆I −2ζI −ηI and it is a fixed point of y. Because I0 is an analytic function of I0 , ϕ0 , we can find a function W (I0 , ϕ0 ) satisfying: dW (I0 , ϕ0 ) = I0 · dϕ0 + ϕ0 · dI0 = d(I0 · ϕ0 ) + (I0 − I0 ) · dϕ0 + (ϕ0 − ϕ0 ) · dI0 (80) which defines implicitly the map C on a suitable domain. We work out a more explicit expression for W . , it is I0 = I0 + u ˜(I0 , ϕ0 ) ∈ B−2δ−∆−2ζ−η , For any (I0 , ϕ0 ) ∈ B−2δ−∆−2ζ−2η ˜ ˜ and then (I, ϕ), (I , ϕ ), ϕ0 are analytic functions of I0 , ϕ0 . Applying Lemma 6 to the generating maps W and χ ˜ we obtain the new generating function: W
= I0 · ϕ0 + h(I ) + ε(w(I , ϕ) + χ(I, ϕ0 ) − χ(I , ϕ0 )) +(I0 · ϕ0 − I · ϕ + I · ϕ + I · ϕ0 − I · ϕ0 − I0 · ϕ0 ) ,
(81)
where I, ϕ, I , ϕ , I0 , ϕ0 are functions of the independent variables (I0 , ϕ0 ) ∈ B−2δ−∆−2ζ−2η . We need to give more explicit expression to (81). We observe ˜ that it is: I0 · ϕ0 − I · ϕ + I · ϕ + I · ϕ0 − I · ϕ0 − I0 · ϕ0 = I0 · (ϕ0 − ϕ0 ) +I · (ϕ0 − ϕ) + I · (ϕ −ϕ0 ) = (I0 − I ) · (ϕ0 − ϕ) + (I0 − I) · (ϕ − ϕ0 ) ∂χ ∂χ ∂w (I , ϕ0 ) · − ε ∂I = ε ∂ϕ (I , ϕ0 ) + ω(I ) + ε ∂I (I , ϕ) ∂χ ∂χ (I , ϕ0 ) − ε ∂w (82) + ε ∂ϕ ∂ϕ (I , ϕ) · ε ∂I (I, ϕ0 ) and therefore it is: W
= I0 · ϕ0 + h(I ) + ε(w(I , ϕ) + χ(I, ϕ0 ) − χ(I , ϕ0 )) ∂χ ∂χ ∂χ ∂w +εω(I ) · (I , ϕ0 ) + ε2 (I , ϕ0 ) · − (I , ϕ0 ) + (I , ϕ) ∂ϕ ∂ϕ ∂I ∂I ∂χ ∂χ ∂w (I , ϕ0 ) + (I , ϕ) · (I, ϕ0 ) − − ∂ϕ ∂ϕ ∂I
Vol. 5, 2004
Nekhoroshev Theorem for Nearly Integrable Symplectic Maps
= I0 · ϕ0 + h(I0 ) + ε[w(I0 , ϕ0 ) + χ(I0 , ϕ0 ) − χ(I0 , ϕ0 + ω(I0 ))] + εw (I0 , ϕ0 ) , where: w
=
1029
(83)
∂χ (I , ϕ0 ) + [w(I , ϕ) − w(I0 , ϕ0 )] ε−1 h(I ) − h(I0 ) + εω(I ) · ∂ϕ +[χ(I, ϕ0 ) − χ(I0 , ϕ0 )] + [χ(I0 , ϕ0 + ω(I0 )) − χ(I , ϕ0 )] ∂χ ∂χ ∂w (I , ϕ0 ) · − (I , ϕ0 ) + (I , ϕ) +ε ∂ϕ ∂I ∂I ∂χ ∂χ ∂w (I , ϕ0 ) + (I , ϕ) · (I, ϕ0 ) (84) − − ∂ϕ ∂ϕ ∂I
We now provide the estimates for the different contributions to w on the set . First of all, we recall that the functions u˜j defined in I0 , ϕ0 ∈ B−2δ−∆−2ζ−2η ˜ (77) are analytic in B−2δ−∆−2ζ . Therefore, for any I0 , ϕ0 ∈ B−2δ−∆−2ζ−2η it ˜ ˜ is (I0 , ϕ0 ) ∈ B−2δ−∆−2ζ−η , and then, by analyticity of C in B−2δ−∆−2ζ−η , all ˜ ˜ . Moreover, it I, ϕ, I , ϕ , I0 , ϕ0 are analytic functions of I0 , ϕ0 ∈ B−2δ−∆−2ζ−2η ˜ is: I0 , I , I ϕ0 , ϕ , ϕ
∈ B˜I −δI −∆I −2ζI −ηI ∈ Tn˜ϕ −δϕ −∆ϕ −2ζϕ −ηϕ .
(85)
The estimate of the Taylor remainder of h around I provides: h(I ) − h(I0 ) + εω(I ) · ∂χ (I , ϕ0 ) = |h(I0 ) − h(I ) − ω(I ) · (I − I0 )| ∂ϕ 1 2 (86) ≤ M I − I0 ≤ MnζI2 , 4 while the other terms are estimated by: |w(I , ϕ) − w(I0 , ϕ0 )| |χ(I, ϕ0 ) − χ(I0 , ϕ0 )| |χ(I0 , ϕ0 ) − χ(I , ϕ0 )| χ(I , ϕ ) − χ(I , ϕ0 + ∂h (I )) 0 0 0 0 ∂I and finally it is:
≤ ≤ ≤ ≤
2ε−1 (∆I ζϕ + ∆ϕ ζI ) 2ε−1 (∆I + ζI )ζϕ 2ε−1 ζI ζϕ ζI ∆ϕ 4ε−1 (ζϕ + + Mn )ζI , 2 2
(87)
∂χ ∂(u+v) ∂χ ε ∂ϕ (I , ϕ0 ) · − ∂I (I , ϕ0 ) + ∂I (I , ϕ) ∂χ ∂χ − − ∂ϕ (I , ϕ0 ) + ∂(u+v) (I , ϕ) · (I, ϕ ) 0 ∂ϕ ∂I ≤ (εn)−1 (2ζI ζϕ + ζI ∆ϕ + ζϕ ∆I ) .
Therefore, collecting all estimates, w satisfies the following estimate: 9 |w |−2δ−∆−2ζ−2η ≤ ε−1 MnζI2 + 4∆I ζϕ + 4∆ϕ ζI + 8ζI ζϕ . ˜ 4
(88)
(89)
1030
M. Guzzo
Ann. Henri Poincar´e
6 Proof of Lemma 5 In this section, for any δ = (δI , δϕ ) we denote Dδ = Dδ I × Tnδϕ . As usual in perturbation theory, we construct the transformation Φ as the composition of many near to the identity symplectic maps, each of which reduces the norm of the remainder by a suitable factor. More precisely, we construct the canonical transformations Φ1 , . . . ΦN , with N defined as in (31): N=
1 K rϕ , 16n ln K
(we remark that because K > 2 it is N ≤ r0 , r1 , . . . , rN with r0 = r and ri =
1 8n rϕ K)
(90) and the extension vectors:
3 r − (i − 1)σ , i = 1, . . . , N 4
(91)
where:
3r , (92) 16N such that: Φi (Dri ) ⊆ Dri−1 is a symplectic diffeomorphism on its image, and it is −1 also Φ−1 i (Dr i ) ⊆ Dr i−1 ; denoting Ci = Ψi ◦ C ◦ Ψi , where Ψi = Φ1 ◦ . . . Φi , the map: σ=
Ci : Dri −→ Dr ,
(93)
is a diffeomorphism on its image and it is generated by a function Wi : Dri → C of the form: Wi (I , ϕ) = I · ϕ + h(I ) + εui (I , ϕ) + εv i (I , ϕ)
(94)
where u0 = 0, v 0 = f , ui = ui−1 + ΠΛ TK v i−1 and v i satisfies the estimate: i |f | v i ≤ 1 r i 4 N
(95)
for any i ≥ 1. If Φ1 , . . . , ΦN exist, then the map Φ of Lemma 5 is Φ = Φ1 ◦ . . .◦ ΦN . Indeed, it is: Φ : D r2 → Dr ; Φ is a symplectic diffeomorphism on its image; C = Φ−1 ◦ C ◦ Φ is generated by W of the form (37) where u=
N
uj , R = v N ,
(96)
j=0
and therefore it is: |u − ΠΛ TK f | r
≤
|R| r
≤
2
2
N N j |f | v j ≤ ≤ 2 |f | r 4j j=1 j=1
1 |f | . 4N N
(97)
Vol. 5, 2004
6.1
Nekhoroshev Theorem for Nearly Integrable Symplectic Maps
1031
Existence of Φ1
We apply Lemma 8 to the generating functions W and I · ϕ + εχ1 where χ1 is defined by the Fourier expansion:
χ1 = −
k∈Zn \0,|k|≤K
fk (I ) eik·ϕ . 1 − eik·ω(I )
(98)
r With the notations of Lemma 8, we set ˜ = r0 = r, and δ = η = 32 . χ1 is analytic in Dr because of (33), and its norm can be estimated following R¨ ussmann ([9]):
|χ1 |r ≤
2n+2 |f |r , α
and therefore by Cauchy estimates we get: 2 f 4εn εn ∂ϕ∂i ∂I |f | ≤ I ϕ 2 j r n+12 ∂ χ1 εn ∂ϕ ≤ εn2 |f | αr r i ∂Ij r−δ I ϕ ∂f εn ∆I = εn maxi ∂ϕ |f | ≤ 2 ϕ i r ∂f εn ∆ϕ = εn maxi ∂I |f | ≤ 2 I i r n+5 ∂χ1 ζI = εn maxi ∂ϕi ≤ εn2rϕ |f | r−δ n+5 1 ζϕ = εn maxi ∂χ ≤ εn2rI |f | ∂Ii
(99)
.
(100)
r−δ
The hypotheses of Lemma 5 allow to apply Lemma 8 which proves that the canonical transformation Φ1 generated by I ·ϕ+εχ1 (I , ϕ) maps Dr1 , with r1 = 34 r, into Dr , it is a symplectic diffeomorphism on its image, and it is also Φ−1 i (Dr 1 ) ⊆ Dr 0 ; denoting C1 = Φ−1 ◦ C ◦ Φ , the map: 1 1 C1 : Dr1 −→ Dr ,
(101)
is well defined, is a diffeomorphism on its image and it is generated by a function W1 : Dr1 → C of the form:
where: and:
W1 (I , ϕ) = I · ϕ + h(I ) + ε˜ u1 (I , ϕ) + εw1 (I , ϕ)
(102)
u ˜1 (I , ϕ) = f (I , ϕ) + χ1 (I , ϕ) − χ1 (I , ϕ + ω(I ))
(103)
1 w
r1
≤ ε−1
9 4
M nζI2 + 4∆I ζϕ + 4∆ϕ ζI + 8ζI ζϕ
.
(104)
|f | Using the hypotheses of Lemma 5 one proves: w1 r1 ≤ (1/8 + 1/32) N . From the definition of χ1 we obtain: u ˜1 (I , ϕ) = ΠΛ TK f (I , ϕ) + (1 − TK )f (I , ϕ) ,
(105)
1032
M. Guzzo
Ann. Henri Poincar´e
and therefore, if we define: u1 = ΠΛ TK f (I , ϕ) and v 1 = w1 + (1 − TK )f (I , ϕ), by estimating the term (1 − TK )f (I , ϕ) as in [3], also using (34), we obtain: |(1 − TK )f |r1 ≤ |(1 − TK )f | ≤ 2
so that:
6.2
1 v
|f | nn−1 22n+2 −K ϕ 4 |f | ≤ , e n ϕ 32N
r1
≤
|f | 4N
.
(106)
(107)
Iteration
We now suppose that Φ1 , . . . , Φi exist and apply Lemma 8 to the generating functions Wi and I · ϕ + εχi+1 where χi+1 is defined by the Fourier expansion: χi+1 = −
k∈Zn \0,|k|≤K
vki (I ) eik·ϕ . 1 − eik·ω(I )
(108)
Setting ξ = r/(32N ), we apply Lemma 8 with ˜ = ri − ξ and δ = η = ξ. The function χi+1 is analytic in Dri because of (33), and its norm can be estimated following R¨ ussmann ([9]): |χi+1 |ri ≤
2n+2 |vi |ri , α
and therefore by Cauchy estimates we get: 2 i ∂ v εn i v ri εn ∂ϕ ≤ ∂I ξ ξ i I ϕ j r i −ξ 2 i+1 εn2n i v ri εn ∂∂ϕχi ∂I ≤ αξ I ξϕ j r i −2ξ i ∂v 8εnN i v ri ∆I = εn 4 maxi ∂ϕi r i −ξ ≤ rϕ i ∂v v i i ∆ϕ = εn ≤ 8εnN 4 maxi ∂Ii i rI r r −ξ ∂χi+1 2n+5 εnN i v ri ζI = εn maxi ∂ϕi i ≤ αrϕ r −2ξ 2n+5 εnN i v ζϕ = εn maxi ∂χ∂Ii+1 ≤ αr ri i I i r −2ξ
(109)
.
(110)
The hypotheses of Lemma 5 allow to apply Lemma 8 which proves that the canonical transformation Φi+1 generated by I ·ϕ+εχi+1 (I , ϕ) maps Dri −6ξ ⊆ Dri+1 into Dri , it is a symplectic diffeomorphism on its image, and it is also Φ−1 i+1 (Dr i+1 ) ⊆ −1 Dri ; denoting Ci+1 = Φi+1 ◦ Ci ◦ Φi+1 , the map: Ci+1 : Dri+1 −→ Dr ,
(111)
Vol. 5, 2004
Nekhoroshev Theorem for Nearly Integrable Symplectic Maps
1033
is well defined, is a diffeomorphism on its image and it is generated by a function Wi+1 : Dri+1 → C of the form:
where:
Wi+1 (I , ϕ) = I · ϕ + h(I ) + ε˜ ui+1 (I , ϕ) + εwi+1 (I , ϕ)
(112)
u˜i+1 (I , ϕ) = v i (I , ϕ) + χi+1 (I , ϕ) − χi+1 (I , ϕ + ω(I ))
(113)
and:
i+1 w i+1 ≤ ε−1 9 M nζ 2 + 4∆I ζϕ + 4∆ϕ ζI + 8ζI ζϕ . (114) I r 4 |vi | Using the hypotheses of Lemma 5 one proves: wi+1 ri+1 ≤ 8 ri . From the definition of χi+1 we obtain: u˜i+1 (I , ϕ) = ΠΛ TK v i (I , ϕ) + (1 − TK )v i (I , ϕ) .
(115)
Therefore, if we define: ui+1 = ui + ΠΛ TK v i (I , ϕ) and v i+1 = wi+1 + (1 − TK )v i (I , ϕ), estimating the term (1 − TK )v i as in [3] and using (34), we obtain: (1 − TK )v i
r i+1
so that:
≤
nn−1 2n+2 −3Kξϕ i 1 v ri ≤ v i ri , e n n 6 ξϕ 8
i+1 v
r i+1
≤
|f | i+1 4 N
.
(116)
(117)
7 Proof of the theorem Through this section, we complete the proof of Theorem 1 and we compute the constants a, b, ε0 , d0 , t0 appearing in the statement as a function of n, I , ϕ , M, M0 , |f | , diamB (we recall that I , ϕ denote analyticity radii such that h and f are analytic in BI × Tnϕ ; M denotes a Lipschitz constant of ω(I) = ∂h/∂I in the set BI and M0 denotes a convexity constant for h in B (see equation 17)). j For any positive K and α1 we fix the parameters αj , δj , rj = (rIj , rϕ ) as follows: M j−1 1 j(j−1) αj = j! 24 K2 α1 M0 1 jK j−1 αj δj = 4M αj rIj = 2M K ϕ j (118) rϕ = 2 for any j = 1, . . . , n, and moreover: rI0 =
α1 ϕ j , rϕ . = 8M K 2
(119)
1034
M. Guzzo
Ann. Henri Poincar´e
We consider only those K, α1 such that it is also: αn < π ,
(120)
and let N be defined as in (31). With such choices, the hypotheses of Lemma 2, Lemma 3 and Lemma 4 are satisfied, so that we deduce that for any K-lattice Λ with dimΛ = j ∈ {1, . . . , n}, and for any I ∈ DΛ and I ∈ CΛ,δj (I), it is:
I − I ≤
3 jK j−1 αj . M0
(121)
In view of the construction of the normal forms described in Lemma 5 in any of the resonant extended blocks, except for the completely resonant one, we remark that ext using (120) and (121) in the case j ∈ {1, . . . , n−1} one proves that if I ∈ (DΛ,δ ) j, j rI ik·ω(I) n ≥ αj+1 /8 for any k ∈ Z \Λ and |k| ≤ K; if I ∈ (D0 )rI0 , then it is: 1 − e ik·ω(I) ≥ α1 /8 for any k ∈ Zn \0 and |k| ≤ K. then it is: 1 − e ext , Then, for any Λ with j =dimΛ ≤ n − 1 we apply Lemma 5 with D = DΛ,δ j j r = r and α = αj+1 /4; while we apply Lemma 5 in the non-resonant block D0 with r = r0 and α = α1 /4. It is possible to apply Lemma 5 to all these sets if the parameters satisfy the following inequalities: 2n+7 n−1 K ≥ max 2, K∗ , 6ϕ ln 2 nn ϕ 2 n+1 3 M , 2 n M0 αn ≤ min 4, 24+n n M M0 I εn |f | ≤ C
K
α3n 3n2 +4−3n 2
(122)
where K∗ , C are defined by: 1
C= 22n+10 Mn!3
M 24 M
3(n−1)
K∗ 1 32n ϕ ln K∗ = 1 n−1 n2 −n−2 2ϕ 16n 4 M min 1, nM , K 2 2 ϕ αn M0 I
0
To compute constants, we assume n ≥ 2, so that C ≥ C0 with: 1 2 ϕ 4n 4 M n−1 2 . min 1, , C0 = 3(n−1) nM I ϕ M0 M 22n+10 M n!3 24 M 0
7.1
.(123)
(124)
Stability of D0
We now consider a motion (It , ϕt ) = Ct (I0 , ϕ0 ), t ∈ Z, with I0 ∈ D0 . Because of Lemma 5, there exists a symplectic map Φ : D0,rI0 /2 × Trϕ0 /2 (I, ϕ)
−→ D0,rI0 × Trϕ0 −→ (I , ϕ )
(125)
Vol. 5, 2004
Nekhoroshev Theorem for Nearly Integrable Symplectic Maps
1035
which conjugates C to the map C generated by the analytic function: W (I , ϕ) = I · ϕ + h(I ) + εu(I ) + εR(I , ϕ)
(126)
|f |
with |R|r0 /2 ≤ 41N N . Therefore, denoting (It , ϕt ) = Φ(It , ϕt ), it is: |It − I0 | ≤ |It − It | + |It − I0 | + |I0 − I0 | with: |I0 − I0 | ≤ |It − It | ≤
rI0 211 rI0 211
|It − I0 | ≤ ε |t|
2 4 1 |f | , |R| r0 ≤ ε |t| 2 rϕ,0 ϕ 4 N N
for those t such that It ∈ Dr0 , and therefore it is |It − I0 | ≤ I 2
ε |t|
7.2
rI0 4
(127)
for any t satisfying:
rI0 4 1 |f | ≤ . ϕ 4 N N 8
(128)
Stability of DΛ
We now consider a K-lattice Λ with j =dimΛ ∈ {1, . . . , n − 1}, and a motion ext (It , ϕt ) = Ct (I0 , ϕ0 ) with I0 ∈ DΛ . We set D = DΛ,δ . Because of Lemma 5 there j exists a symplectic map Φ : Dr j /2 × Trϕj /2 I
(I, ϕ)
−→ Dr j × Trϕj I
−→ (I , ϕ )
(129)
which conjugates C to the map C generated by the analytic function: W (I , ϕ) = I · ϕ + h(I ) + εu(I , ϕ) + εR(I , ϕ)
(130)
|f |
with |u|rj /2 ≤ 2 |f | , ΠΛ u = u and |R|rj /2 ≤ 41N N . To be definite we consider positive t (a very similar argument apply to negative t), and we prove that It satisfies one of the two following statements: • It ∈ CΛ,δj (I0 ) for any t satisfying εn |t|
rIj 4 1 |f | ≤ ; ϕ 4 N N 16
(131)
• there exists t0 satisfying (131) such that It ∈ CΛ,δj (I0 ) for any t with |t| ≤ t0 , and It0 +1 ∈ DΛ , with Λ a suitable K-lattice of dimension strictly smaller than j.
1036
M. Guzzo
Ann. Henri Poincar´e
Let t0 satisfying (131) and also It ∈ CΛ,δj (I0 ) for any positive t ≤ t0 . Actually, it is also: It0 ∈ CΛ,δj /4 (I0 ). In fact, it is It0 − I0 = (It0 − It0 ) + (I0 − I0 ) + (It0 − I0 ), where It0 − It0 ≤ rIj /211 ≤ δj /16, |I0 − I0 | ≤ rIj /16 ≤ δj /16, and: It0 − I0 = ε
∂R ∂u +ε =λ+x ∂ϕ ∂ϕ
(132)
∂u where λ = ε ∂ϕ ∈< Λ >, and satisfies:
∂u 4εn rIj ≤ |f | ≤
λ = ε ∂ϕ ϕ 16 while x can be in any direction but satisfies: ∂R 4εn |f | rIj δj ≤
x = ε ≤ . |t | ≤ 0 N ∂ϕ ϕ N4 16 16
(133)
(134)
But It0 +1 is in a ball of radius rIj /4 + δj /16 from It0 , and therefore it is in CΛ,δj (I0 ), otherwise, because of Lemma 2, it is not in a resonant zone related to a j-dimensional K-lattice. Therefore, it is in a resonant block DΛ with dimΛ ≤ j−1.
7.3
Stability of all motions
From the two previous subsections, it is clear that for a generic initial condition, which is in any of the resonant or non resonant blocks, the actions cannot move more than n times the dimension of the completely resonant cylinder, plus the stability radius of the non resonant domain, estimated by the quantity: d≤
4 2 n−1 rI,0 3 2 n−1 ≤ n K αn + n K αn , M0 4 M0
(135)
in a number of iterations t satisfying: |t| ≤
N N rI0 ϕ 4 , εn 27 |f |
(136)
provided that in the meanwhile they do not leave the action domain B. But such an escape cannot occur if the initial datum is chosen at a distance from the border of B strictly larger than d.
7.4
Choice of the parameters
The stability arguments shown above work if the parameters K, αn are suitably chosen. First of all, we set: π 1 (137) αn = ε 2 −γ 2
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Nekhoroshev Theorem for Nearly Integrable Symplectic Maps
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with 0 ≤ γ ≤ 1/2, so that αi < π for any i = 1, . . . , n, when ε < 2, and in particular it is: 1 π −γ α1 = . (138) n−1 n(n−1) ε 2 M 4 2 2n! 2 M0 K Then, the third estimate of (122), assuming n ≥ 2, is satisfied if: 22 C0 π 3 3n −3n+4 1 K≤ . 6γ−1 8n |f | 2 ε 3n −3n+4 Therefore, we can set: K=
ε b ∗
(140)
ε
where: 6γ − 1 , ε∗ = b= 2 3n − 3n + 4
(139)
C0 π 3 8n |f |
2 6γ−1
.
(141)
The first two equations of (122) and the condition ε < 2 are therefore satisfied by imposing ε ≤ ε˜0 , with 22n+7 nn−1 1b 6 ε˜0 = min 2 , ε∗ max 2, K∗ , ln , ϕ nϕ 2 1−2γ 2 2 4+n M n+1 3 M . (142) min 4, 2 n I , 2 n π M0 M0 Then, up to a number of iterations smaller than: T∗ =
b ϕ ϕ 2ϕ 2ϕ ε∗ b ( εε∗ ) ≥ 26 n e e 26 n ( ε ) 16 2 18 2 2 εn M |f | 2 n M |f |
(143)
the actions cannot move by a quantity larger than: 2n2 ε∗ M0
b(n−1)
∆=
π
εa
(144)
where a = 12 − γ − b(n − 1). The theorem is proved if we can find a γ ∈ (1/6, 1/2) such that a, b > 0. It is sufficient to choose: b=
2 − 6a 3n2 + 3n − 2
(145)
for any a ∈ 0, 13 . In fact, for any choice of a in this interval it is a, b > 0 and also: 1 1 n2 (3 − 6a) + n(6a − 1) + 2 − 8a γ= ∈ , (146) 6n2 + 6n − 4 6 2 for any n ≥ 2.
1038
M. Guzzo
Ann. Henri Poincar´e
Therefore, for any a ∈ 0, 13 , let b be defined as in (145), ε∗ be defined as in (141) (in formula 141 the constants γ, C0 are as in 146,124), ε˜0 be defined as in (142) (in formula 142 the constants γ, K∗ are are as in 146,123 first line), and finally let d0 , t0 be defined by: 2n2 ε∗ M0
b(n−1)
d0 =
π
2ϕ . 218 n2 M |f |
, t0 =
(147)
All the constants a, b, ε∗ , ε˜0 , d0 , t0 turn out to be defined as functions of n, I , ϕ , M, M0 , |f | . Then, for any initial datum (I0 , ϕ0 ) ∈ B × Tn with dist(I0 , ∂B) ≥ 2d0 εa and for any ε ≤ ε˜0 it is: |It − I0 | ≤ d0 εa
(148)
up to a number of iterations t ∈ Z such that: b
ϕ ε∗ |t| ≤ t0 e 26 n ( ε )
.
(149)
Therefore, if the diameter of the action domain B is large that 2d0 , the theorem is proved on a non-empty set of initial conditions setting: 1b ϕ ε∗ , ε0 = min ε˜0 , 6 2 n otherwise the theorem is proved by adding the additional constraint on ε: 2d0 ε ≤ diamB, so that, in any case, the theorem is proved on a non-empty set of initial conditions setting: ε0 = min
diamB 2d0
, ε˜0 ,
1b ϕ ε∗ . 26 n
References [1] A. Bazzani, Normal forms for symplectic maps of R2n . Unpublished note, (1987). [2] A. Bazzani, S. Marmi and S. Turchetti, Nekhoroshev estimate for isochronous nonresonant symplectic maps. Celest. Mech. and Dynamical Astronomy 4, 333–359 (1990). [3] G. Benettin, L. Galgani and A. Giorgilli, A proof of Nekhoroshev’s theorem for the stability times in nearly integrable Hamiltonian systems. Cel. Mech. 37, 1 (1985). [4] R. Douady, Application du th´eor`eme de tores invariantes, Th`ese 3`eme cycle, Universit´e Paris VII (1982).
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Nekhoroshev Theorem for Nearly Integrable Symplectic Maps
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[5] S.B. Kuksin, On the inclusion of an almost integrable analytic symplectomorphism into a Hamiltonian flow, Russian journal of Mathematical Physics 1, 2, 191–207 (1993). [6] S. Kuksin and J. P¨ oschel, On the inclusion of analytic symplectic maps in analytic Hamiltonian flows and its applications, Seminar on Dynamical Systems (St. Petersburg, 1991), 96–116, Progr. Nonlinear Differential Equations Appl., 12, Birkh¨ auser, Basel (1994). [7] N.N. Nekhoroshev, Exponential estimates of the stability time of nearintegrable Hamiltonian systems, Russ. Math. Surveys 32, 1–65 (1977). [8] J. P¨ oschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z. 213, 187 (1993). [9] H. R¨ ussmann, On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus, Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), pp. 598–624. Lecture Notes in Phys. 38 (1975). [10] G. Turchetti and A. Bazzani, Normal forms for symplectic maps. Advances in nonlinear dynamics and stochastic processes, II (Rome, 1986), 25–36, World Sci. Publishing, Singapore, (1987). Massimiliano Guzzo Universit` a degli Studi di Padova Dipartimento di Matematica Pura ed Applicata Via Belzoni 7 I-35131 Padova Italy email:
[email protected] Communicated by Eduard Zehnder submitted 16/06/03, accepted 31/03/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 1041 – 1064 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/061041-24 DOI 10.1007/s00023-004-0189-1
Annales Henri Poincar´ e
Asymptotics of Solutions of the Einstein Equations with Positive Cosmological Constant Alan D. Rendall Abstract. A positive cosmological constant simplifies the asymptotics of forever expanding cosmological solutions of the Einstein equations. In this paper a general mathematical analysis on the level of formal power series is carried out for vacuum spacetimes of any dimension and perfect fluid spacetimes with linear equation of state in spacetime dimension four. For equations of state stiffer than radiation evidence for development of large gradients, analogous to spikes in Gowdy spacetimes, is found. It is shown that any vacuum solution satisfying minimal asymptotic conditions has a full asymptotic expansion given by the formal series. In four spacetime dimensions, and for spatially homogeneous spacetimes of any dimension, these minimal conditions can be derived for appropriate initial data. Using Fuchsian methods the existence of vacuum spacetimes with the given formal asymptotics depending on the maximal number of free functions is shown without symmetry assumptions.
1 Introduction Spacetimes with accelerated expansion have come to play an important role in cosmology. The accelerating phase may be in the early universe (inflation) or at the present epoch (quintessence). The simplest way to produce a model with accelerated expansion which solves the Einstein equations is to introduce a positive cosmological constant. A good survey article on this topic is [19]. The fact that a positive cosmological constant leads to solutions of the Einstein equations with exponential expansion is associated with the term ‘cosmic no hair theorem’. In the following we investigate possibilities of proving theorems related to these ideas. In the setting of formal series a satisfactory answer is obtained for the Einstein equations in vacuum or in the presence of a perfect fluid with linear equation of state. There are formal series solutions which have the expected asymptotic behavior and which depend on the maximum number of free functions. This also holds for vacuum spacetimes in higher dimensions. In the case of even space dimensions it is in general necessary to allow terms with logarithmic dependence on the expansion parameter. This throws some light on what is special about three space dimensions. These results are proved in Section 2. While most of the results in three space dimensions obtained in Section 2 confirm the results of [18], one new phenomenon was observed. Evidence is obtained that for fluids with an equation of state stiffer than that of a radiation fluid inhomogeneous structures can be formed. This is reminiscent of the formation of spikes near the initial singularity in Gowdy spacetimes [17].
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In Section 3 it is shown that in the vacuum case minimal assumptions on the asymptotics in the expanding phase imply that the spacetime has an asymptotic expansion of the form already exhibited as formal series. Unfortunately we do not know in general how to obtain these minimal assumptions starting from conditions on initial data. An exception to this is the case of three space dimensions where it is shown in Section 4 that the minimal assumptions can be deduced from results of Friedrich [4], [5] based on the conformal method. The minimal assumptions can also be verified in the case of certain spatially homogeneous vacuum spacetimes of any dimension, as shown in Section 5. In particular, there are genuine solutions of the Einstein equations whose asymptotics contain non-vanishing logarithmic terms. In Section 6 Fuchsian methods are applied to show the existence of vacuum spacetimes of any dimension with the asymptotics given in Section 2 and depending on the maximum number of free functions. Finally, Section 7 shows that in a model problem, the wave equation on de Sitter space, full information on asymptotics of solutions with arbitrary initial data can be obtained.
2 Perturbative solutions A perturbative treatment of four-dimensional vacuum spacetimes with positive cosmological constant can be found in [18]. In that paper formal solutions are written down without any mathematical derivation being given. In this section a careful discussion of these formal power series solutions is presented. The analysis is generalized to vacuum spacetimes with positive cosmological constant in all dimensions. The expansion for perfect fluid spacetimes given in [18] is also revisited. Consider the vacuum Einstein equations with cosmological constant Λ for a spacetime of dimension n + 1 with n ≥ 2. An n + 1 decomposition with lapse equal to one and vanishing shift results in the constraint equations R − k ab kab + (trk)2 = 2Λ ∇a k a b − ∇b (trk) = 0
(1) (2)
and the evolution equation ∂t k a b = Ra b + (trk)k a b −
2Λ a δ b. n−1
(3)
Here gab is the spatial metric with Ricci tensor Rab and scalar curvature R, kab is the second fundamental form and indices are raised and lowered using gab and its ˜ ab inverse. Let σ a b be the trace-free part of the second fundamental form and R the trace-free part of the spatial Ricci tensor and define the following quantities: ˜ ab E
=
E
=
˜ a b + (trk)σ a b ] ∂t σ a b − [R 2nΛ ∂t (trk) − R + (trk)2 − n−1
(4) (5)
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Asymptotics of Solutions of the Einstein Equations
C Ca
= =
1043
R − k ab kab + (trk)2 − 2Λ ∇b k
b
a
(6)
− ∇a (trk).
(7)
The Einstein equations are equivalent to the vanishing of the evolution quantities ˜ a b and E and the constraint quantities C and Ca . These quantities are linked by E the following consistency conditions ∂t C ∂t Ca
= =
2(trk)C − 2∇a Ca − 2σ a b E˜ b a + 2(1 − 1/n)(trk)E ˜ b a − (1 − 1/n)∇a E. (trk)Ca − (1/2)∇a C + ∇b E
(8) (9)
To do a perturbative analysis of equations (1)–(3) consider the formal power series 0 1 −Ht 2 −2Ht 3 −3Ht gab = e2Ht (gab + gab e + gab e + gab e + ···) (10) where H is a constant. The n+1 form of the Einstein equations are imposed on this expression ina suitable sense. It turns out that for consistency it is necessary to choose H = 2Λ/(n(n − 1)) and so, in particular, Λ must be positive. Products of formal series are defined in the obvious way that the terms in the individual series are multiplied and the resulting terms with the same power of e−Ht collected. The derivatives of a formal series with respect to the space and time variables are defined via term by term differentiation. In order to impose the Einstein equations it is also necessary to have a definition of the inverse g ab of the formal power series metric gab . This can be done uniquely by requiring that the relation gab g bc = δac holds. This allows the coefficient of order m in the series for g ab to be expressed 0 in terms of the coefficients in the series for gab up to order m. Setting gab = δab m and gab = 0 for m > 0 gives an exact solution of the Einstein equations. In the case n = 3 it is the de Sitter solution ([7], p. 125). Given any tensor T , let (T )m denote the coefficient of e−mHt in the expansion m+2 . It follows from (10) that (Ra b )m = 0 of T . With this notation (gab )m = gab for m = 0 and m = 1. It also follows directly from (10) that (trk)0 = −nH and (σ a b )0 = 0. This is consistent with the vanishing of the coefficients of all evolution and constraint quantities for m = 0. The vanishing of (E)1 and (E˜ a b )1 implies that (trk)1 = 0 and (σ a b )1 = 0. It follows that (C)1 = 0 and (Ca )1 = 0 and this ensures the consistency of the series up to order m = 1. Using the relation 1 = 0 and this in turn implies that (Ra b )3 = 0. ∂t gab = −2gack c b shows that gab The relations between coefficients for m ≥ 2 will now be written down. The summation indices p and q in the following formulae are assumed to be no less than two. The evolution equations (3) imply the recursion relations a −1 a a ˜ b )m (σ b )p (trk)q + (R (11) (n − m)(σ b )m = H p+q=m
and (2n − m)(trk)m = H
−1
p+q=m
(trk)p (trk)q + (R)m .
(12)
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A.D. Rendall
Ann. Henri Poincar´e
The Hamiltonian constraint (1) gives (2n − 2)(trk)m = H −1 [−(k a b )p (k b a )q + (trk)p (trk)q ] + (R)m
(13)
p+q=m
and the momentum constraint (2) gives [(Γcab )p (k a c )q − (Γaac )p (k c b )q ]. ∇0a (k a b )m − ∇0b (trk)m =
(14)
p+q=m
0 Here ∇0 is the covariant derivative associated to gab . The consistency conditions ˜ a b )k , (8) and (9) relating evolution and constraint quantities imply that if (E)k , (E (C)k and (Ca )k all vanish for k ≤ m − 1 then
(2n − m)(C)m = −2(n − 1)(E)m
(15)
(n − m)H(Ca )m = −(1/2)∇a (C)m + ∇0b (E˜ b a )m − (1 − 1/n)∇a (E)m .
(16)
Consider first the case n = 3. The form (10) of the series for gab , taking 1 account of the vanishing of the coefficient gab , is contained in [18]. The following theorem formalizes some of the statements in [18]. Here smooth means C ∞ . Theorem 1. Let Aab be a smooth three-dimensional Riemannian metric and Bab a smooth symmetric tensor which satisfies Aab Bab = 0 and ∇a Bab = 0, where the covariant derivative is that associated to Aab . Then there exists a unique formal power series solution of the vacuum Einstein equations with cosmological constant 0 3 m = Aab and gab = Bab . The coefficients gab are Λ > 0 of the form (10) with gab smooth. Proof. The coefficients (k a b )m determine the coefficients (gab )m recursively. For substituting (10) into the relation ∂t gab = −2gac k c b gives m 0 p = 2gac (k c b )m + 2 gac (k c b )q (17) mHgab p+q=m
and hence an equation which expresses (gab )m−2 in terms of (k a b )m and lower order terms for any m ≥ 2. Thus in order to prove the theorem it is enough to show that equations (11)–(14) determine the coefficients (k a b )m uniquely and that when the coefficients have been fixed in this way all the equations (11)–(14) are satisfied. The coefficient (k a b )m is determined by (11) and (12) for all m ≥ 2 except m = 3 and m = 6. The coefficient (k a b )3 is determined by using the condition that 3 gab = Bab . The coefficient (σ a b )6 is determined by (11) while (trk)6 is determined by (13). By construction the evolution equation (12) is satisfied for all values of m except possibly m = 3 and m = 6 while (11) is satisfied except possibly for m = 3. The fact that Bab has zero trace ensures that (12) is satisfied while (11) is automatic for m = 3. It will now be shown by induction that (11)–(14) hold
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for all m. Except for m = 6 only (13) and (14) need to be verified. The equations (11)–(14) hold for m = 1. For 2 ≤ m ≤ 5 the inductive step from m − 1 to m can be carried out as follows. When m = 3 the consistency condition (15) shows that (C)m = 0 and then the consistency condition (16) shows that (Ca )m = 0. That (C)3 and (Ca )3 are zero follows from the conditions on Bab in the hypotheses of the theorem. Knowing that the equations (11)–(14) hold for m ≤ 5 implies using (15) that (E)6 = 0. By construction (C)6 = 0 and then it is straightforward to obtain (Ca )6 = 0 from (16). For m ≥ 7 we can proceed as for 2 ≤ m ≤ 5. This completes the proof. 0 Remark. If Pab and P denote the Ricci tensor and Ricci scalar of gab respectively, 2 −2 0 then gab = H (Pab − (1/4)P gab ), a relation given in [18]. The theorem just proved can be generalized directly to all larger odd values of n, as will now be shown.
Theorem 2. Let Aab be a smooth n-dimensional Riemannian metric with n odd and Bab a smooth symmetric tensor which satisfies Aab Bab = 0 and ∇a Bab = 0, where the covariant derivative is that associated to Aab . Then there exists a unique formal power series solution of the vacuum Einstein equations with cosmological 0 n = Aab and gab = Bab . The coefficients constant Λ > 0 of the form (10) with gab m gab are smooth. Proof. Let s be an integer such that 2s + 1 < n and (k a b )m = 0 for all odd m m = 0 for all odd m with m ≤ 2s − 1 and that with m ≤ 2s − 1. If follows that gab ab (g )m = 0 for all odd m with m ≤ 2s + 1. Putting this information into the Ricci tensor shows that (Ra b )m = 0 for all odd m with m ≤ 2s + 1. Then (11) and (12) imply that (k a b )2s+1 = 0. It can then be proved by induction that (k a b )m = 0 vanishes for all odd m with m < n. From this point on the proof is very similar to that of the previous theorem. The coefficients (k a b )m are uniquely determined for all values of m except m = n and m = 2n. The coefficient (k a b )n is determined n by using the condition that gab = Bab . The coefficient (σ a b )2n is determined by (11) while (trk)2n is determined by (13). By construction the evolution equation (12) is satisfied for all values of m except possibly m = n and m = 2n while (11) is satisfied except possibly for m = n. The fact Bab has zero trace ensures that (12) is satisfied while (11) is automatic for m = n. These statements make use of the fact that the odd order coefficients of k a b of order less than n vanish. It will now be shown by induction that (11)–(14) hold for all m. Except for m = 2n only (13) and (14) need to be verified. The equations (11)–(14) hold for m = 1. For 2 ≤ m ≤ 2n − 1 the inductive step from m − 1 to m can be carried out as follows. When m = n the consistency condition (15) shows that (C)m = 0 and then the consistency condition (16) shows that (Ca )m = 0. That (C)n and (Ca )n are zero follows from the conditions on Bab in the hypotheses of the theorem. Knowing that the equations (11)–(14) hold for m ≤ 2n − 1 implies using (15) that (E)2n = 0. By construction (C)2n = 0 and then it is straightforward to obtain (Ca )2n = 0. For m ≥ 2n + 1 we can proceed as for 2 ≤ m ≤ 2n − 1. This completes the proof.
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The case where n is even is more complicated. The form (10) of the metric must be generalized to 0 gab = e2Ht gab +
Lm ∞
(gab )m,l tl e−mHt
(18)
m=0 l=0
where Lm is a non-negative integer for each m and Lm = 0 for m < n. Given any tensor T with an expansion of the above type let (T )m,l denote the coefficient of tl e−mHt . As before when manipulating series they are differentiated term by term. The recursion relations for the expansion coefficients coming from the evolution equations generalize as follows, where the terms not written out explicitly are lower order in the sense that they can be expressed in terms of the coefficients of k a b with m smaller: (n − m)(σ a b )m,l + H −1 (l + 1)(σ a b )m,l+1 = · · · (2n − m)(trk)m,l + H
−1
(l + 1)(trk)m,l+1 = · · · .
(19) (20)
The terms on the right-hand side not written out are obtained from the terms on the right-hand side of equations (11) and (12) if the indices m, p and q are replaced by the pairs (m, l), (p, l1 ) and (q, l2 ), summing over l1 + l2 = l. The recursion relations implied by the constraints are identical except for the addition of an extra index l. In a similar way, the consistency conditions lead to (2n − m)(C)m,l + H −1 (l + 1)(C)m,l+1 (n − m)H(Ca )m,l + H
−1
(l + 1)(Ca )m,l+1 +∇0 (E˜ b a )m,l b
=
−2(n − 1)(E)m,l
=
−(1/2)∇a (C)m,l
−
(1 − 1/n)∇a (E)m,l
(21) (22)
˜ a b )k,l , (C)k,l and (Ca )k,l vanish whenever k ≤ m − 1. assuming that (E)k,l . (E By using the above relations it is possible to express g0ab (gab )n−2,0 as a func0 tion of gab and its spatial derivatives. We denote this schematically by g0ab (gab )n−2,0 = Z(g 0 ). Similarly, it is possible to write 0 ∇a (gab )n−2,0 = Z˜b (g 0 ). Theorem 3. Let Aab be a smooth n-dimensional Riemannian metric and Bab a smooth symmetric tensor which satisfies Aab Bab = Z(A) and ∇a Bab = Z˜b (A), where the covariant derivative is that associated to Aab . Then there exists a unique formal series solution of the vacuum Einstein equations with cosmological constant 0 = Aab and (gab )n−2,0 = Bab . The coefficients Λ > 0 of the form (18) with gab (gab )m,l are smooth. Proof. As a general principle, when determining coefficients for fixed m we start from l = Lm and proceed to successively lower values of l. If n is odd then the existence follows from Theorem 2. Uniqueness for n odd in the wider class being considered in this theorem in comparison with Theorem 2 is obtained by a straightforward extension of the argument given in the proof of the latter. Consider now
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Asymptotics of Solutions of the Einstein Equations
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the case where n is even. By analogy with the proof of Theorem 2 it can be shown that (k a b )m,l = 0 for m odd. For m < n the coefficients are uniquely determined. The assumption that Lm = 0 in this range is thus unavoidable.The coefficients (trk)n,l are uniquely determined by (20) and vanish for l > 0. The coefficients (σ a b )n,l are uniquely determined for l ≥ 1 and vanish for l > 1. The choice of Bab determines (σ a b )n,0 . For n < m < 2n equations (19) and (20) determine (k a b )m,l . Equation (19) is used to determine (σ a b )2n,l while the analogue of (13) is used to determine (trk)2n,l . For m > 2n (19) and (20) can be used again. That all field equations are satisfied at all orders can be proved much as in the proof of Theorem 2, always proceeding in the direction of decreasing l for each fixed m. A question left open by Theorem 3 is whether it can ever happen that any of the coefficients with l > 0 are non-zero. This is equivalent to the question whether (σ a b )n,1 is ever non-zero. It follows from the proof of the theorem that 0 . In the case n = 2 the coefficient this coefficient is uniquely determined by gab of interest vanishes due to the fact that the Ricci tensor of a two-dimensional metric is automatically traceless. For all even dimensions greater than two there 0 for which (σ a b )n,1 does not vanish. In fact this is the generic are choices of gab case. The coefficient of interest can be written as a polynomial expression in H −1 . 0 If there were no logarithmic terms for a given choice of gab then all terms in this −n+1 , which is the most polynomial would have to vanish. The coefficient of H negative power of H occurring, is a non-zero constant times P˜ a b (trP )k−1 , where k = n/2 and P˜ a b and trP are the tracefree part and trace of the Ricci tensor 0 of gab . There are only two ways in which this coefficient can vanish. Either the 0 0 scalar curvature of gab vanishes identically or gab is an Einstein metric. A necessary condition for the absence of logarithmic terms has now been given but it is unlikely to be sufficient. The coefficients of other powers of H have to be taken into account in order to decide this issue. In [18] the expansions obtained for vacuum spacetimes were extended to the case of a perfect fluid with pressure proportional to energy density. Formalizing these considerations leads to a theorem generalizing Theorem 1 above. The notation here is as follows: ρ = T 00 , j a = T 0a and S ab = T ab . The proper energy density and pressure of the fluid are denoted by µ and p respectively, so that T αβ = (µ + p)uα uβ + pg αβ .
(23)
The equation of state is taken to be p = (γ − 1)µ with 1 ≤ γ < 2. In the case with matter evolution and constraint quantities can be defined by E˜ a b E
= =
˜ a b + (trk)σ a b − 8π S˜a b ] ∂t σ a b − [R ∂t (trk) − [R + (trk)2 + 4πtrS − 12πρ − 3Λ]
(24) (25)
C
=
R − k ab kab + (trk)2 − 16πρ − 2Λ
(26)
Ca
=
∇b k
b
a
− ∇a (trk) − 8πja
(27)
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A.D. Rendall
Ann. Henri Poincar´e
so that their vanishing is equivalent to the Einstein equations. These satisfy the consistency conditions (8) and (9) as in the vacuum case. The components of the energy-momentum tensor can be expressed in terms of the fundamental fluid variables as follows: ρ j
a
Sab
= =
µ(1 + γ|u|2 ) γµ(1 + |u|2 )1/2 ua
(28) (29)
=
µ[γua ub + (γ − 1)δba ]
(30)
where |u|2 = gab ua ub . The following relations will be useful: 1 ∂t ρ − (trk)ρ − (trk)trS 3 5 a ∂t j − (trk)j a 3
=
−∇a j a + σ a b S b a
(31)
=
−∇b S a b + 2σ a b j b .
(32)
It is possible to express µ and ua in terms of ρ and j a . To see this note first that |j| = γµ(1 + |u|2 )1/2 |u| and that as a consequence: |j|2 /ρ2 = γ 2 (1 + |u|2 )|u|2 /(1 + γ|u|2 )2
(33)
If f (x) = γ 2 x2 (1+x2 )(1+γx2 )−2 then f (x) = 2γ 2 x(1+γx2 )−3 (1+(2−γ)x2 ) > 0. It follows that the mapping from the interval [0, ∞) to the interval [0, 1) defined by f is invertible and |u|2 can be expressed as a smooth function of |j|2 /ρ2 for ρ > 0. Since µ can be expressed as a smooth function of ρ and |u|2 it follows that it is a smooth function of ρ and j a . Similarly the fact that ua can be expressed as a smooth function of µ, |u|2 and j a implies that ua is a smooth function of ρ and j a . Next Theorem 1 will be generalized to the case with perfect fluid. The solution is sought as a formal series where each tensor occurring is written as a sum of exponentials. The exponents are taken from an increasing sequence of real numbers M = {mi } which tends to infinity as i → ∞. The solution is of the form (gab )mi e−mi Ht (34) gab = mi ∈M
µ =
(µ)mi e−mi Ht
mi ∈M
u
a
=
(ua )mi e−mi Ht .
mi ∈M
The quantities ρ and j a have similar expansions. The sequence M occurring depends on which quantity is being expanded and on the value of the parameter γ in the equation of state of the fluid. In order to organize this information let real numbers k1 , k2 , k3 and k4 depending on γ be defined as follows. For γ ≤ 4/3 we have k1 = 3γ, k2 = 5 − 3γ, k3 = 3γ, k4 = 5 while for γ ≥ 4/3 we have k1 = 2γ/(2 − γ),
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k2 = (6 − 4γ)/(2 − γ), k3 = 4 and k4 = 5. It is also useful to define the relative order m ˜ i of a coefficient of order mi in the expansion of a particular quantity. For the quantities gab , k a b , µ, ua , ρ and j a these are defined by m ˜ i = mi + 2, m ˜ i = mi , ˜ i = mi − k2 , m ˜ i = mi − k3 and m ˜ i = mi − k4 respectively. m ˜ i = mi − k1 , m Theorem 4. Let Aab be a smooth three-dimensional Riemannian metric and Bab a smooth symmetric tensor, µ0 a smooth positive real-valued function and ua0 a smooth vector field. Suppose that Aab Bab = −(8π/3H 2 )µ0 for γ = 1, Aab Bab = 0 for γ > 1 and ∇a Bab = ∇b (Aac Bac ) + (16πγ/3H)µ0 Abc uc0 where the covariant derivative is that associated to Aab . If γ > 4/3 suppose furthermore that ua is nowhere vanishing. Then there exists a unique formal power series solution of the Einstein-Euler equations with cosmological constant Λ > 0 and equation of state p = (γ − 1)ρ, 1 ≤ γ < 2, of the form (34) with (gab )−2 = Aab , (gab )1 = Bab , (µ)k1 = µ0 and (ua )k2 = ua0 . The coefficients of the series are smooth. They satisfy (µ)mi = 0 for mi < k1 , (ua )mi = 0 for mi < k2 and, except for mi = −2, the coefficient (gab )mi vanishes for mi < 0. Proof. Consider a formal series solution whose coefficients vanish in the ranges indicated in the statement of the theorem. With the given values for k1 and k2 the matter terms do not contribute to the equations for the coefficients of k a b below order three and thus all statements made about these coefficients in the vacuum case can be taken over without change. This follows from the fact that ρ, S a b and ja are all O(e−3Ht ). The exponent in this estimate can be improved except in the case of ρ with γ = 1 and in the case of ja with general γ. The proof splits into several cases. Suppose first that 1 ≤ γ < 1/3. Then |u| = o(1) and so in leading order ρ = µ, j a = γµ|u|ua and S a b = (γ − 1)µδba . Thus the following relations are obtained: (mi − 3γ)(ρ)mi (mi − 5)(j a )mi
= ··· = ··· .
(35) (36)
The terms not written out explicitly are of lower order in the sense that they are combinations of terms of lower relative order than m ˜ i . There is one subtlety involved in showing this. In the case γ = 1 the expression ∇b S a b gives rise to a term which, looking at the exponents, is not of lower order. However the coefficient of this term contains a factor γ −1 and so the term vanishes for γ = 1. The Einstein equations give: (3 − mi )(σ a b )mi = · · · (37) and (6 − mi )(trk)mi = −12πH −1(ρ)mi + · · · .
(38)
The terms on the right-hand side of the last two equations not written out explicitly are lower order. The one explicit term on the right-hand side of the last equation is also of lower order except in the case γ = 1. The energy-momentum quantities
1050
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ρ and j a are linked to the matter quantities µ and ua by the relations (µ)mi a
(u )mi
ρmi + · · ·
= =
γ
−1
(ρ
(39)
−1 a
j )mi + · · · .
(40)
Fix a value of m ˜ i and suppose that all coefficients with lower relative order have been determined. Consider the equations for (ρ)mi and (ua )mi . These coefficients are determined uniquely unless m ˜ i = 0 and if m ˜ i < 0 they vanish. When m ˜i = 0 they are determined by the conditions on (µ)k1 and (ua )k2 in the hypotheses of the theorem and the equations relating ρ and j a to µ and ua . The latter relations also fix the coefficients of µ and ua of the given relative order when m ˜ i > 0. Next consider the equations for (trk)mi and (k a b )mi . By what has been said above we may assume that mi ≥ 3. The unique determination of the coefficients of the given relative order can be shown using the same procedure as in the vacuum case. The additional terms are either already of lower relative order, and hence known, or have been determined in the preceding discussion of the matter equations. By induction on i it can be concluded that all coefficients are uniquely determined. The fact that all field equations are satisfied can be shown much as in the vacuum case since the compatibility conditions are identical. Now consider the case 4/3 < γ < 2. The assumption that ua0 is nowhere vanishing implies in this case that |u|−1 = o(1) and in leading order ρ = γµ|u|2 , j a = γµ|u|ua and S a b = γµua ub . The following relations are obtained: (mi − 4)(ρ)mi a
(mi − 5)(j )mi
= ···
(41)
= ··· .
(42)
The Einstein equations give the same relations as in the previous case. For 4/3 < γ < 2 the energy-momentum quantities ρ and j a are linked to the matter quantities µ and ua by the relations (µ)mi a
(u )mi
= =
(2 − γ)−1 (ρ(1 − |j|2 /ρ2 ))mi + · · · 1/2
((2 − γ)/γ)
(ρ
−1
2
(43)
2 −1/2 a
(1 − |j| /ρ )
j )mi + · · · .
(44)
Using these facts we can proceed as in the case 1 ≤ γ < 4/3. Consider finally the case γ = 4/3 where |u| tends to a finite limit, in general non-zero, as t → ∞. The difference in comparison to the cases already treated is that the relations between ρ and j a on the one hand and µ and ua on the other hand cannot be inverted explicitly in leading order. However the fact, shown above, that the relevant mappings are invertible has an equivalent on the level of formal power series. For if f is a smooth function between open subsets of Euclidean spaces then f (x + y) can be written formally in terms of a Taylor series about x. The resulting expression contains the derivatives of f evaluated at x multiplied by powers of y. If y is replaced by a formal power series without constant term then a well-defined formal power series for f (x + y) is obtained. Thus the same method can be applied as in the previous cases, allowing the proof of the theorem to be completed.
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A case which has been excluded in the above theorem is that where γ > 4/3 and ua may vanish somewhere. In that case the kind of series which has been assumed in the theorem is not consistent. For if it is assumed that an expansion of this kind is possible this leads to different rates of decay for certain quantities, for instance µ, depending on whether |u| does or does not vanish. As a consequence ∇a µ/µ will be unbounded as t tends to infinity although µ is nowhere zero. This contradicts the assumptions which have been made. The situation is reminiscent of the spikes observed near the initial singularity in Gowdy spacetimes [17] and so we may speculate that in reality inhomogeneous features develop in µ so that the density contrast blows up as t → ∞. This behavior for γ > 4/3 is not consistent with the usual picture in inflationary models where the density contrast remains bounded at late times. The issue deserves to be investigated further. It is interesting to ask whether the expansions for a fluid presented here can be extended to the case of collisionless matter. If they can then the result probably resembles that for dust. Limited expansions in some special cases are already known [13], [20]. Note that the analysis of vacuum spacetimes in this section has a close analogue for Riemannian (i.e., positive definite) metrics. A solution of the Einstein equations with positive cosmological constant in the Lorentzian case corresponds to an Einstein metric with negative Einstein constant in the Riemannian case. The equations obtained for a positive definite metric are −R − k ab kab + (trk)2 ∇a k
a
b
=
− ∇b (trk) = ∂t k a b
=
−(n − 1)K
(45)
0
(46)
−Ra b + (trk)k a b + 2Kδ a b
(47)
where K is the Einstein constant, i.e., the n+ 1-dimensional metric satisfies Rαβ = Kgαβ . Asymptotic expansions for this case have been investigated in the literature on Riemannian geometry [3] and string theory [6].
3 From minimal to full asymptotics In the last section consistent formal asymptotic expansions were exhibited for a number of problems. In this section it is shown that minimal information about the asymptotics implies the full expansions given in the last section. For simplicity we restrict consideration to the vacuum case. The following lemma will be used: Lemma 1. Consider an equation of the form ∂t u + ku =
vm,l tl e−mt + O(e−jt )
m,l
for a vector-valued function u(t), where j = k and m < j in the sum.
(48)
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Then there are coefficients um,l , m < j, such that um,l tl e−mt + O(e−jt ). u=
(49)
m,l
If (48) may be differentiated term by term with respect to t as often as desired the same is true of (49). Proof. Note first that ∂t (ekt u) is equal to a sum of explicit terms with a term of order e(k−j)t . Each of the explicit terms has an explicit primitive which is a sum of terms of the same general form and the same value of m but in general several values of l. Thus we can absorb these terms into the time derivative and write um,l tl e−mt )) = O(e(k−j)t ) (50) ∂t (ekt (u − m,l
with m < j in the sum. If j < k we can integrate this relation directly to get the desired result. If j > k then the expression which is differentiated with respect to time converges to a limit as t → ∞, which can be called uk,0 . This gives the desired result in the latter case. If the assumption on time derivatives is satisfied then ∂t u satisfies an equation of the same form as that satisfied by u. Hence ∂t u has an asymptotic expansion wm,l tl e−mt + O(e−jt ). (51) ∂t u = m,l
Integrating this from t0 to t and using (49) gives t wm,l sl e−ms ds = C + um,l tl e−mt + O(e−jt ) m,l
t0
(52)
m,l
for a constant C. It follows that the coefficients wm,l are obtained from um,l by term by term differentiation. This process can be repeated for higher-order derivatives with respect to t. Remark. If the quantities in (48) depend smoothly on a parameter and the equation may be differentiated term by term with respect to the parameter then the same is true for the solution. Theorem 5. Let a solution of the vacuum Einstein equations with cosmological constant Λ > 0 in n + 1 dimensions be given in Gauss coordinates. Suppose that e−2Ht gab , e2Ht g ab , e2Ht σ a b and their spatial derivatives of all orders are bounded. Then the solution has an asymptotic expansion of the form given in Theorem 3. The expansion remains valid when differentiated term by term to any order. Proof. The Hamiltonian constraint can be used to express trk in terms of the scalar curvature R, σ a b and Λ, giving 1/2 n
trk = − −R + σ a b σ b a + n2 H 2 . (53) n−1
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It follows from the assumptions of the theorem that trk = −nH + O(e−2Ht ) and that this relation may be differentiated term by term with respect to the spatial variables. Now (54) ∂t (e−2Ht gab ) = −2e−2Ht gac (k c b + Hδbc ). 0 The right-hand side of this expression is O(e−2Ht ) and so there is some gab such that 0 + O(e−2Ht ) (55) e−2Ht gab = gab
and corresponding relations hold for spatial derivatives of all orders. Using the evolution equations it can be seen that these relations can also be differentiated repeatedly with respect to time. The proof now proceeds by induction. The inductive hypothesis is as follows. There exist coefficients (gab )m,l and (k a b )m,l ,0 ≤ m ≤ M such that gab
=
e2Ht (
Lm M
(gab )m−2,l tl e−mHt + g¯ab ) = [gab ]M + e2Ht g¯ab
(56)
m=0 l=0
ka b
=
Lm M
(k a b )m,l tl e−mHt + k¯ a b = [k a b ]M + k¯ a b
(57)
m=0 l=0
where g¯ab and k¯a b are O(e−(M+)Ht ) and similar asymptotic expansions hold for all derivatives of these quantities. Here is a constant belonging to the interval (0, 1). The inductive hypothesis is satisfied for M = 1. If these expressions are substituted into the Einstein equations then the expansion coefficients written explicitly satisfy the same relations as in the analysis of formal power series solutions carried out above. It is convenient to write the evolution equations in the following form: = −2ˆ gac σ c b − (2/n)(trk + nH)ˆ gab (58) a a a a ˜ ∂t σ b + nHσ b = (trk + nH)σ b + R b (59) 2 (60) ∂t (trk + nH) + 2nH(trk + nH) = (trk + nH) + R ∂t gˆab
where gˆab = e−2Ht gab . Using the inductive hypothesis it follows that if each quantity Q in these equations is replaced by the corresponding quantity [Q]M+1 then equality holds up to a remainder of order e−(M+1+)Ht in (59) and (60). Using this information shows that the corresponding statement holds in (58) with a remainder of order e−(M−1+)Ht . Thus the quantities [Q]M+1 satisfy a system of the type occurring in Lemma 1. It follows from that lemma that the inductive hypothesis is satisfied with M replaced by M + 1. In [14] results similar to those of this section were obtained using a different coordinate system. The time coordinate used there satisfies the condition that that the lapse function is proportional to the inverse of the mean curvature of its level surfaces. This means that the foliation of level surfaces is a solution of the inverse mean curvature flow, a fact which raises serious doubts whether such coordinates
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exist in forever expanding cosmological spacetimes, as will now be explained. The inverse mean curvature flow for hypersurfaces is defined by the condition that a hypersurface flows with a speed equal to the inverse of its mean curvature in the normal direction. In the case of a Riemannian manifold it was used in the work of Huisken and Ilmanen [10] on the Penrose inequality. For spacelike hypersurfaces in a Lorentzian manifold it was studied in [9]. If a spacelike hypersurface with positive expansion (i.e., in the convention used here, with trk < 0) is given then there is a local solution of the inverse mean curvature flow in the contracting direction. Moreover, under reasonable assumptions on the nature of the singularity, there is a global solution. In the expanding direction, in contrast, the equation is backward parabolic and it is to be expected that there is no local solution for general initial data, i.e., for a general starting hypersurface. This is an analogue of the fact that the heat equation cannot be solved backwards in time.
4 Relations to conformal infinity There is a relation between the expansions discussed in the last two sections and the concept of conformal infinity. In this section only the Einstein vacuum equations are considered. Define T = H −1 e−Ht . Then spacetime metric corresponding to (18) becomes −2
(HT )
2
[−dT +
0 (gab
+
Lm ∞
(gab )m,l (−1)l H −l (log(HT ))l (HT )m )].
(61)
m=0 l=0
It is conformal to a metric which is non-degenerate at T = 0 and is written in Gauss coordinates. If there are no non-vanishing coefficients with l > 0 the conformal metric (or unphysical metric) is smooth at T = 0. This is for instance the case when n is odd. In the case n = 3 Friedrich [4], [5] has used conformal techniques to prove results which, as shown in the following, imply that spacetimes evolving from initial data close to standard initial data for de Sitter space indeed have asymptotic expansions of the type presented in the last section. The method used, based on the conformal method, is only known to work in the case n = 3. The occurrence of logarithms in the expansions for even values of n cast doubt on the possibility of implementing an analogous procedure in that case. There are also problems for n = 3 if matter is present. For conformally invariant matter fields the method can be used but for other types of matter, e.g. a perfect fluid with linear equation of state, there is no straightforward way of doing this. The non-integer powers occurring in the formal expansions for this case make the application of the method problematic. Note, however, that a similar problem has been overcome in the study of isotropic singularities [2]. Consider the de Sitter solution with a slicing by intrinsically flat hypersurfaces, as described in Section 2, with the slicing being given by the hypersurfaces
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of constant t. We may assume for convenience that the solution has been identified in a way which is periodic in the spatial coordinates. Consider initial data which is a small perturbation of the initial data induced by this model solution on a hypersurface of constant time. The smallness can be measured in the sense of uniform convergence of a function and its derivatives of all orders. Then, according to Section 9 of [5], the perturbed solution has a Cauchy development which is asymptotically simple in the future. This means that the solution gαβ is conformal to a metric g˜αβ = Ω2 gαβ with Ω > 0 in such a way that gαβ and Ω have smooth extensions through a hypersurface where Ω vanishes. We may choose coordinates in the unphysical metric in the following way. Set T˜ = Ω and choose the spatial coordinates to be constant along the curves orthogonal to the hypersurfaces of constant Ω. In these coordinates the conformal metric takes the form −H −2 α2 dT˜ 2 + g˜ab dX a dX b
(62)
where α is a function of T˜ and X a . The condition −3∇α Ω∇α Ω = Λ (see Lemma 9.2 of [5]) implies that α = 1 for T˜ = 0. In order to compare this with the expansions in the previous sections we need to transform to Gauss coordinates with respect to the physical metric gαβ . As a first step let T˜ = e−HT . Then the physical metric becomes −α2 dT 2 + gab dX a dX b
(63)
with gab = e2HT g˜ab . The following lemma shows that Gauss coordinates of a suitable kind can be introduced. The hypotheses make use of the following inequalities |gab | −1 |α | + |Γabc | |α − 1| + |∂T α| + |∂a α| ˜a
ab
|k b | + |trk + 3H| + |g |
≤ Ce2HT ≤ C
(64) (65)
≤ Ce−HT
(66)
≤ Ce
−2HT
.
(67)
The metric (63) above satisfies inequalities of this type together with corresponding inequalities for spatial derivatives of all orders. In fact the estimates for k˜a b and trk + H are only obviously satisfied with the bound Ce−HT . However this can be improved by using equations (59) and (60) at the end of the last section, or rather their equivalents in the presence of a non-trivial lapse function. Lemma 2. Consider a metric of the form (63) on a time interval [T0 , ∞) and assume that there is a constant C > 0 such that the inequalities (64)–(67) are satisfied, together with the corresponding inequalities for spatial derivatives of all orders. Then for T0 sufficiently large there exists a Gaussian coordinate system based on the hypersurface T = T0 which is global in the future. The transformed metric satisfies the hypotheses of Theorem 5.
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Proof. To construct Gaussian coordinates it is necessary to analyse the equations of timelike geodesics. In 3 + 1 form these are 2 dT d2 T dX a dX b dX a dT −1 −1 −1 + α = 0(68) + α ∂ α + 2α ∇ α k T a ab dτ 2 dτ dτ dτ dτ dτ 2 dT d2 X a dX a dT 2 a + α∇ α − α(trk) 2 dτ dτ 3 dτ dτ −
2αk˜a b
dX b dT dX b dX c + Γabc = 0. dτ dτ dτ dτ
It is helpful for the following analysis to rewrite one of the terms:
dX a dT dX a dX a dT 2 = 2H + 2H −1 − α(trk) 3 dτ dτ dτ dτ dτ 2 dX a dT 2 dX a dT − (trk + 3H) − (α − 1)(trk) . 3 dτ dτ 3 dτ dτ
(69)
(70)
These equations are to be solved for functions T (τ, xb ) and X a (τ, xb ) with initial values T = T0 , dT /dτ = 1, X a = xa and dX a /dτ = 0 at τ = T0 . Strictly speaking Gaussian coordinates based on T = T0 would differ from this by a time translation by T0 but it is convenient here to work with this slight modification. Consider now a solution of these equations on an interval [T0 , τ ∗ ) and suppose for later convenience that T0 ≥ 0. There is a τ ∗ > T0 for which a solution does exist. We assume that on this interval |dX a /dτ | ≤ Ce−2Hτ and that |dT /dτ − 1| < for some ∈ (0, 1/3). For given C and there exists an interval of this kind. On this interval e−T ≤ e−τ0 e−(1−)τ , eT −τ ≤ e(τ −τ0 ) and inequalities of the following form hold, where C is a positive constant depending only on C and . d2 T dτ 2 2 a d X dX a + 2H 2 dτ dτ
=
f (τ ), |f (τ )| ≤ C e−(1−)Hτ
(71)
=
g(τ ), |g(τ )| ≤ C e−2Hτ .
(72)
It follows from the first of these that |dT /dτ − 1| ≤ C e−(1−)HT0 .
(73)
For T0 large enough this strictly improves on the estimate originally assumed for dT /dτ − 1. For small the quantities dX a /dτ can be seen to decay exponentially with an exponent which is as close as desired to −2. The fact that we are dealing with timelike geodesics parametrized by proper time leads to the relation 2 dT dX a dX b . (74) −1 = −α2 + gab dτ dτ dτ This implies that |dT /dτ − 1| = O(e−HT ). Putting this back into the evolution equation for dX a /dτ shows that it is O(e−2Hτ ). By choosing T0 large enough the
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decay estimate for this quantity is recovered and in fact strengthened. Consideration of the longest time interval on which the original inequality holds shows that τ ∗ = ∞. The estimates we have derived up to now hold globally. The estimate for dT /dτ obtained above implies that there are positive constants C1 and C2 such that C1 τ ≤ T ≤ C2 τ . Hence in estimates we can replace e−T by e−τ if desired. Next we would like to obtain corresponding estimates for the spatial derivatives of dT /dτ and dX a /dτ of all orders. Consider the result of differentiating the geodesic equations with respect to the spatial variables. This leads to estimates of the form 2 d ∂ T (75) = fa (τ ), |fa (τ )| ≤ C e−Hτ dτ ∂τ ∂xa 2 a ∂2X a ∂ X d + 2H = gca (τ ), |gca (τ )| ≤ C e−2Hτ . (76) c dτ ∂τ ∂x ∂τ ∂xc This allows us to show that the first order spatial derivatives of the key quantities satisfy the estimates analogous to those satisfied by the quantities themselves. The same argument can be applied to estimate spatial derivatives of any order inductively. Now all the desired information about existence and decay of T and X a has been obtained. It remains to show that they form a coordinate system. This follows from the fact that the initial values of ∂T /∂τ , ∂X a /∂τ and ∂X a /∂xb are one, zero and δba respectively and the exponential decay of their time derivatives which has already been proved. This completes the proof of the lemma. Combining Lemma 2 and Theorem 5 shows that the spacetimes constructed by Friedrich admit global Gaussian coordinates in which they have an asymptotic expansion of the form of (10). Hence any initial data close to that for de Sitter on a flat hypersurface evolves into a solution having an asymptotic expansion of the form given by Starobinsky.
5 The spatially homogeneous case This section is concerned with spatially homogeneous solutions of the vacuum Einstein equations in n + 1 dimensions with positive cosmological constant. It is assumed that the spatial homogeneity is defined by a Lie group G, supposed simply connected, which acts simply transitively. We restrict to spacetimes such that all left invariant Riemannian metrics on G have non-positive scalar curvature. In the case n = 3 this corresponds to Bianchi types I to VIII. Information on the case n = 4 can be found in [8]. It will be shown that the spacetimes of the type just specified have asymptotic expansions with all the properties of the formal expansions in Theorem 3. A spatially homogeneous spacetime of the type being considered can be written in the form (77) −dt2 + gij (t)ei ⊗ ej
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where ei is a left invariant frame on the Lie group G. Basic information about the asymptotics of these spacetimes in 3+1 dimensions are given by Wald’s theorem [21], which provides information on the behavior of the second fundamental form as t → ∞. This can easily be generalized to the present situation. Using the condition on the sign of R, the Hamiltonian constraint implies that on any interval where 2nΛ = (nH)2 . Combining the Hamiltonian constraint a solution exists (trk)2 ≥ n−1 with the evolution equation for trk gives ∂t (trk) ≥
1 2 (trk)2 − Λ. n n−1
(78)
In particular trk is non-decreasing. These facts together show that trk is bounded. Now it will be shown that trk → −nH as t → ∞. For ∂t (trk + nH) ≥ ≥
1 (−trk + nH)(−trk − nH) n −2H(trk + nH).
(79) (80)
It follows that trk = −nH + O(e−2Ht ). Using the Hamiltonian constraint then gives σ i j σ j i = O(e−2Ht ). This bound can be used to get information on σij as in [15]. Then it is possible to proceed exactly as in the proof of Proposition 2 in [13] to show that e−2Ht gij , e2Ht g ij and eHt σ i j are bounded. Then equation (58) can be used as in the previous section to improve the last statement to the boundedness of e2Ht σ i j . The fact that gij , g ij and kij are bounded on any finite time interval implies that the solution exists globally in time. We are now in a situation very similar to that of Theorem 5. However the estimates we have are expressed in term of frame components. Choosing a coordinate system on some subset of the Lie group G with compact closure will give us uniform asymptotic expansions for the components in that coordinate system. Conversely uniform asymptotic expansions for the components in a coordinate system of this type give corresponding asymptotic expansions for the frame components. In this case we will say that the asymptotic expansions are locally uniform. If an expansion of this type holds for a given quantity it also holds for all spatial derivatives in the coordinate representation. The proof of Theorem 5 uses only arguments which are pointwise in space and so it generalizes immediately to the case of locally uniform asymptotic expansions. It can be concluded that for a spatially homogeneous spacetime of the type under consideration locally uniform asymptotic expansions of generalized Starobinsky type are obtained. Restricting to a coordinate domain with compact closure uniform asymptotic expansions are obtained. In general these expansions will contain logarithmic terms. Consider for instance the case of the Lie group H × R where H is a three-dimensional Lie group of Bianchi type other than IX. Let the spacetime be such that the spatial metric at each time is the product of a metric on H with one on R. This is consistent with the constraint equations. For instance the initial data can be chosen to be the product of data on H with trivial data on R. The
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data are invariant under reflection in R and this property is inherited by the solutions. Suppose that H admits no metric of vanishing scalar curvature. This is the 0 case for every Bianchi type except I and VII0 . Then the metric gab corresponding to this solution has non-vanishing scalar curvature and is not an Einstein metric. Hence logarithmic terms are unavoidable.
6 Fuchsian analysis Fuchsian systems are a class of singular equations which can be used to prove the existence of solutions of certain partial differential equations with given asymptotics [11], [12], [16]. It will be shown that Fuchsian methods allow the construction of solutions of the vacuum Einstein equations with positive cosmological constant in any number of dimensions which have asymptotic expansions of the type given in Section 2 and depend on the same number of free functions as the general solution. Before coming to the specific problem of interest here some general facts about Fuchsian equations will be recalled. The form of the equations is t∂u/∂t + N (x)u = tf (t, x, u, ux ).
(81)
Here x denotes the spatial coordinates collectively and ux the spatial derivatives of the unknown u(t, x). The matrix N and the function f are required to satisfy certain regularity conditions and N is required to satisfy a positivity condition. There are forms of the regularity condition adapted to smooth and to analytic functions. The version adapted to analytic functions will be used in the following since it is the one where the most powerful theorems are available. For the precise definition of regularity see [1], where a corresponding definition of regularity of solutions is also given. Roughly speaking, regularity means that the functions concerned are continuous in t and analytic in x and vanish in a suitable way as t → 0. Consider now an ansatz of the form u(t, x) =
Lm ∞
um,l tm (log t)l .
(82)
m=0 l=0
By analogy with what was done in Section 2 we can ask whether the equation (81) has a formal series solution of this kind. Suppose that this is the case. Fix M Lm m l M ≥ 0. Then there exist coefficients um,l such that u ¯ = u− 0 0 um,l t (log t) satisfies t∂ u ¯/∂t + N (x)¯ u = tM+ fM (¯ u) (83) for a regular function fM and a constant > 0 together with the corresponding relations obtained by differentiating term by term with respect to the spatial ¯. Then coordinates any number of times. Let v = t−M u t∂v/∂t + (N (x) + M I)v = t g(t, x, v, vx )
(84)
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for a regular function g. Introducing t as a new time variable, we obtain an equation of the form (81). If we assume that N (x) is bounded then by choosing M large enough it can be ensured that the matrix N (x) + M I is positive definite. Assuming that f and N are regular in the analytic sense the existence theorem of [1] implies the existence of a unique regular solution v vanishing at t = 0. Expressing u in terms of v gives a solution of the original equation which has the given asymptotic expansion up to order M . Consider now the slightly more general equation t∂u/∂t + N (x)u = tf (t, x, u, ux) + h(u).
(85)
In order to have a consistent formal power series solution suppose that for some functions u0,0 and u1,0 we have N u0,0 = h(u0,0 ) and (N + I − Dh(u0,0 ))u1,0 = f (0, u0,0).
(86)
Here Dh denotes the derivative of h as a map between Euclidean spaces. Suppose further that the equation admits a formal power series solution with coefficients u0,0 and u1,0 and L0 = L1 = 0. If these conditions hold then u satisfies the original equation if v = u − u0,0 − u1,0 t satisfies a Fuchsian system and vanishes at the origin. Thus an existence theorem is obtained. To make contact with the Einstein equations we start with the equations (58)–(60) and set τ = e−Ht . Then an equation of the form (85) is obtained, with u = (ˆ gab , σ a b , trk + nH). If it is assumed that the variables σ a b and (trk + nH) vanish at τ = 0 then the consistency conditions on u0,0 and u1,0 are satisfied. The fact that consistent formal expansions were shown to exist in Section 2 allows the above procedure to be carried through. If the data Aab and Bab are chosen to be analytic then this gives an existence theorem for the Einstein evolution equations with Aab and Bab prescribed as in Theorem 3. In this context it is important to note that if Aab and Bab are analytic all the coefficients in the formal expansions whose existence is asserted in Theorem 3 are also analytic. In order to see that a solution of the Einstein equations is obtained it suffices to show that the constraint equations are satisfied. Note that it follows from the results of Section 2 that the constraint quantities vanish to all orders at τ = 0 but since the solution is not analytic at τ = 0 this does not suffice to conclude that the constraint quantities vanish everywhere. To see that they do we need to write the consistency conditions (8) and (9) in Fuchsian form. Using the fact that the Einstein evolution equations are satisfied, and introducing C˜a = e−tH Ca , these equations can be written as ∂t C + 2HC ∂t C˜a + 2H C˜a
= =
2(trk + H)C − 2eHt ∇a Ca ˜ (trk + H)C˜a − (1/2)e−Ht ∇a C.
(87) (88)
Setting τ = e−Ht gives a system of the form (81) and since C and C˜a tend to zero as τ → 0 both of these quantities vanish as a consequence of the uniqueness theorem for Fuchsian systems and the constraints are satisfied. The solution of the
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Einstein equations has an asymptotic expansion of the form given in Theorem 3 truncated at any given finite order. Applying Theorem 5 shows that this solution has an asymptotic expansion of this form to all orders. The results obtained can be summed up as follows: Theorem 6. Let Aab be an analytic n-dimensional Riemannian metric and Bab an analytic symmetric tensor which satisfies Aab Bab = Z(A) and ∇a Bab = Z˜b (A), where the covariant derivative is that associated to A. Then there exists an analytic solution of the vacuum Einstein equations with an asymptotic expansion 0 = Aab and (gab )n−2,0 = Bab . The expansion may be of the form (18) with gab differentiated term by term with respect to the spatial variables as often as desired.
7 The wave equation on de Sitter spacetime In Section 4 it was shown that initial data for the vacuum Einstein equations in 3+1 dimensions close to that for de Sitter space evolve to give a spacetime with asymptotics of Starobinsky type. It has not yet proved possible to obtain the analogous statement in higher dimensions. What is missing are suitable energy estimates. In this section it will be shown how a simpler model problem can be treated. This is the case of the wave equation ∇α ∇α φ = 0 on (the higher-dimensional analogue of) de Sitter space. The spacetime metric in this case is ds2 = −dt2 + e2Ht ((dx1 )2 + · · · + (dxn )2 ).
(89)
Written out explicitly in coordinates the wave equation takes the form: ∂t2 φ + nH∂t φ = e−2Ht ∆φ
(90)
where ∆ is the Laplacian of the flat metric. Consider the ansatz for formal solutions of the equations ∞
(Am (x)e−mHt + Bm (x)te−mHt ).
(91)
m=0
Substituting this into the equation and comparing coefficients gives m(m − n)H 2 Am − (2m − n)HBm
=
∆Am−2
(92)
m(m − n)H 2 Bm
=
∆Bm−2 .
(93)
For any n it is true that B0 = A1 = B1 = 0. In the case that n is odd assume that the coefficients Bm vanish. Then ∆Am−2 = H 2 m(m − n)Am for all m ≥ 2. Then it follows from A1 = 0 that A2k+1 = 0 for all integers k with 2k + 1 < n. The coefficients A2k+1 with 2k + 1 > n are determined by An . The coefficients A2k are determined by A0 . There are no further relations to be satisfied and so A0 and An parametrize the general solution. If the coefficients Bm are not assumed
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to be zero it can be shown that they must vanish for n odd. For n even we have A2k+1 = B2k+1 = 0 for every positive integer k. If 2k < n then B2k = 0. For 2k < n the coefficients A2k are determined successively by A0 . Also Bn is determined by A0 . Then A2k and B2k are determined for all k with 2k > n in terms of the coefficients already determined. Thus the general solution can be parametrized by A0 and An , just as in the case n odd. The difference is that the series obtained contains terms which are multiples of te−mHt . Let E = (∂t φ)2 +e−2Ht |∇φ|2 . Differentiating with respect to t and integrating by parts gives the relation dE/dt ≤ −2HE. We can differentiate the equation through with respect to a spatial coordinate and repeat the argument. This shows that all Sobolev norms of eHt ∂t φ and ∇φ are bounded. By the Sobolev embedding theorem they and all their spatial derivatives satisfy corresponding pointwise bounds. Thus the spatial derivatives of φ are bounded while its time derivative decays like e−Ht . As a consequence φ(t, x) = φ0 (x) + O(e−Ht ) for some function φ0 . Comparing with the formal solutions already obtained we see that these estimates are not likely to be sharp. The equation for φ is equivalent to the system ∂t φ ∂t ψ + nHψ
= ψ = e
−2Ht
(94) ∆φ.
(95)
Starting with the basic information on the asymptotic behavior of φ we already have the method of proof of Theorem 5 can be applied to this system. The result is that any solution has an asymptotic expansion of the type derived on a formal level above. Acknowledgments. I thank Hans Ringstr¨om for discussions on the subject of this paper and Gary Gibbons for pointing me to relevant work on positive definite metrics.
References [1] L. Andersson and A.D. Rendall, Quiescent cosmological singularities, Commun. Math. Phys. 218, 479–511 (2001). [2] K. Anguige and K.P. Tod, Isotropic cosmological singularities 1: Polytropic perfect fluid spacetimes, Ann. Phys. (NY) 276, 257–293 (1999). [3] C. Fefferman and C.R. Graham, Conformal invariants, In: Elie Cartan et les math´ematiques d’aujourd’hui. Ast´erisque (hors s´erie), 95–116 (1985). [4] H. Friedrich, Existence and structure of past asymptotically simple solutions of Einstein’s field equations with positive cosmological constant, J. Geom. Phys. 3, 101–117 (1986). [5] H. Friedrich, On the global existence and the asymptotic behaviour of solutions to the Einstein-Yang-Mills equations, J. Diff. Geom. 34, 275–345 (1991).
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[6] S. de Haro, K. Skenderis and S.N. Solodukhin, Holographic reconstruction of spacetime and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217, 595–622 (2001). [7] S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, Cambridge University Press, Cambridge 1973. [8] S. Hervik, Multidimensional cosmology: spatially homogeneous models of dimension 4+1, Class. Quantum Grav. 19, 5409–5428 (2002). [9] M. Holder, Geometrische Evolutionsgleichungen in kosmologischen Raumzeiten, PhD thesis, University of T¨ ubingen, 1999. [10] G. Huisken and T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Diff. Geom. 59, 353–437 (2001). [11] S. Kichenassamy, Nonlinear Wave Equations, Marcel Dekker, New York 1996. [12] S. Kichenassamy, A.D. Rendall, Analytic description of singularities in Gowdy spacetimes, Class. Quantum Grav. 15, 1339–1355 (1998). [13] H. Lee, Asymptotic behaviour of the Einstein-Vlasov system with a positive cosmological constant. Preprint gr-qc/0308035, To appear in Math. Proc. Camb. Phil. Soc. 2003. [14] W.C. Lim., H. van Elst, C. Uggla and J. Wainwright, Asymptotic isotropization in inhomogeneous cosmology, Preprint gr-qc/0306118 (2003). [15] A.D. Rendall, Cosmic censorship for some spatially homogeneous cosmological models, Ann. Phys. (NY) 233, 82–96 (1994). [16] A.D. Rendall, Fuchsian methods and spacetime singularities, Class. Quantum Grav. 21, S295–S304 (2004). [17] A.D. Rendall and M. Weaver, Manufacture of Gowdy spacetimes with spikes, Class. Quantum Grav. 18, 2959–2975 (2001). [18] A.A. Starobinsky, Isotropization of arbitrary cosmological expansion given an effective cosmological constant, JETP Lett. 37, 66–69 (1983). [19] N. Straumann, On the cosmological constant problems and the astronomical evidence for a homogeneous energy density with negative pressure, Preprint astro-ph/0203330 (2002). [20] S.B. Tchapnda and A.D. Rendall, Global existence and asymptotic behaviour in the future for the Einstein-Vlasov system with positive cosmological constant, Class. Quantum Grav. 20, 3037–3049 (2003).
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[21] R.M. Wald, Asymptotic behaviour of homogeneous cosmological models in the presence of a positive cosmological constant, Phys. Rev. D 28, 2118–2120 (1983). Alan D. Rendall Max Planck Institute for Gravitational Physics Am M¨ uhlenberg 1 D-14476 Golm Germany email:
[email protected] Communicated by Sergiu Klainerman submitted 08/12/03, accepted 30/04/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 1065 – 1080 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/061065-16 DOI 10.1007/s00023-004-0190-8
Annales Henri Poincar´ e
Modular Nuclearity and Localization Detlev Buchholz and Gandalf Lechner Abstract. Within the algebraic setting of quantum field theory, a condition is given which implies that the intersection of algebras generated by field operators localized in wedge-shaped regions of the two-dimensional Minkowski space is non-trivial; in particular, there exist compactly localized operators in such theories which can be interpreted as local observables. The condition is based on spectral (nuclearity) properties of the modular operators affiliated with wedge algebras and the vacuum state and is of interest in the algebraic approach to the formfactor program, initiated by Schroer. It is illustrated here in a simple class of examples.
1 Introduction There is growing evidence that algebraic quantum field theory [23] not only is useful in structural analysis but provides also a framework for the construction of models. Basic ingredients in this context are, on the one hand, the algebras affiliated with wedge shaped regions in the Minkowski space, called wedge algebras for short. On the other hand there enter the modular groups corresponding to these algebras and the vacuum state by the Tomita–Takesaki theory. The wedge algebras are distinguished by the fact that the associated modular groups can be interpreted as unitary representations of specific Poincar´e transformations. This fact was established first by Bisognano and Wichmann in the Wightman framework of quantum field theory [4] and, more recently, by Borchers in the algebraic setting [6], cf. also [22, 33]. It triggered attempts to construct families of such algebras directly within the algebraic framework [9, 32]. A particularly interesting development was initiated by Schroer [36] who, starting from a given factorizing scattering matrix in two spacetime dimensions, recognized how one may reconstruct from these data a family of wedge algebras satisfying locality. A complete construction of these algebras for a simple class of scattering matrices was given in [27]. These results are a first important step in an algebraic approach to the formfactor program, i.e., the reconstruction of quantum fields from a scattering matrix [3,26,39]; for more recent progress on this issue see also [1, 2, 17]. The second step in this approach consists in showing that, besides field operators localized in wedges, there appear also local observables, i.e., operators which are localized in compact spacetime regions, such as double cones. As any double cone in two dimensions is the intersection of two opposite wedges, local observables ought to be elements of the intersection of wedge algebras. The question of whether these intersections are non-trivial turned out to be a difficult one, however, and
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has not yet been settled. Some ideas as to how this problem may be tackled in models are discussed in [38]. It is the aim of the present letter to point out an alternative strategy for the proof of the non-triviality of the intersections of wedge algebras. By combining results scattered in the literature and casting them into a simple condition, we will show that the non-triviality of these intersections can be deduced from spectral (nuclearity) properties of the modular operators on certain specific subspaces of the Hilbert space. Thus the algebraic problem of determining intersections of wedge algebras amounts to a problem in spectral analysis which seems to be better tractable. The subsequent section contains an abstract version of our nuclearity condition and a discussion of its consequences in a general algebraic setting. In Section 3 these results are carried over to a family of theories with factorizing S-matrix in two-dimensional Minkowski space. It is shown that compactly localized operators exist in any theory complying with our condition. Section 4 illustrates the type of computations needed to verify this condition in a simple example. The article closes with a brief outlook.
2 Modular nuclearity and its consequences In this section we present our nuclearity condition in a general setting, extracted from the more concrete structures in field theoretic models, and discuss its implications. We begin by introducing our notation and listing our assumptions. (a) Let H be a Hilbert space and let U be a continuous unitary representation of R2 acting on H. Choosing proper coordinates on R2 , x = (x0 , x1 ), the joint spectrum of the corresponding generators (P0 , P1 ) of U is contained in the cone . V+ = {(p0 , p1 ) ∈ R2 : p0 ≥ |p1 |} and there is an (up to a phase unique) unit vector Ω ∈ H which is invariant under the action of U . (b) There is a von Neumann algebra M ⊂ B(H) such that for each element x of . the wedge W = {y ∈ R2 : |y0 | + y1 < 0} the adjoint action of the unitarities U (x) induces endomorphisms of M, . (2.1) M(x) = U (x)MU (x)−1 ⊂ M, x ∈ W. Moreover, Ω is cyclic and separating for M. It is well known that, under these circumstances, the algebraic properties of M are strongly restricted. As a matter of fact, disregarding the trivial possibility that H is one-dimensional and M = C, the following result has been established in [29, Thm. 3]. Lemma 2.1. Under the preceding two conditions the algebra M is a factor of type III1 according to the classification of Connes. It immediately follows from this result that the algebras M(x) are factors of type III1 as well. Little is known, however, about the algebraic structure of
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the relative commutants M(x) ∩ M of M(x) in M, x ∈ W . Even the question of whether these relative commutants are non-trivial has not been settled in this general setting. Yet this question turns out to have an affirmative answer and, as a matter of fact, the algebraic structures are completely fixed if the inclusions (2.1) are split, i.e., if for each x ∈ W there is a factor N of type I∞ such that M(x) ⊂ N ⊂ M.
(2.2)
First, the split property implies that M is isomorphic to the unique [24] hyperfinite factor of type III1 . We briefly recall here the argument: As Ω is cyclic and separating for M, and hence for M(x), this is also true for N . It follows that N , being of type I∞ , is separable in the ultraweak topology and consequently H is separable, cf. [21, Prop. 1.2]. Now, as U is continuous, M is continuous from the inside, M = x∈W M(x). The split property thus implies that M can be approximated from the inside by separable type I∞ factors and therefore is hyperfinite, cf. [11, Prop. 3.1]. Knowing also that it is of type III1 , the assertion follows. Secondly, the split property implies that M(x) ∩ M, x ∈ W , is isomorphic to the hyperfinite factor of type III1 as well. This can be seen as follows [21]. On a separable Hilbert space H, any factor of type III has cyclic and separating vectors [35, Cor. 2.9.28]. Moreover, for any von Neumann algebra on H with a cyclic and a separating vector there exists a dense Gδ set of vectors which are both, cyclic and separating [20]. Now, taking into account that N is isomorphic to B(H), the relative commutant M(x) ∩ N of the type III factor M(x) in N is (anti)isomorphic to M(x) by Tomita–Takesaki theory. It is therefore of type III and has cyclic vectors in H. This holds a fortiori for M(x) ∩M ⊃ M(x) ∩N and, as Ω is separating for M, the relative commutant M(x) ∩M has a dense Gδ set of cyclic and separating vectors. But the intersection of a finite number of dense Gδ sets is non-empty. So we conclude that the triple M, M(x) and M(x) ∩ M has a joint cyclic and separating vector in H. The inclusion (2.2) is thus a standard split inclusion according to theterminology in [21]. In particular, there is a spatial isomorphism mapping M(x) M on H onto M(x) ⊗ M on H ⊗ H [19]. By taking commutants, we conclude that M(x) ∩ M is isomorphic to M(x) ⊗ M, x ∈ W . The statement about the algebraic structure of the relative commutant then follows. It seems difficult, however, to establish the existence of intermediate type I∞ factors N in the inclusions (2.2) for concretely given {M, U, H}, and this may be the reason why this strategy of establishing the non-triviality of relative commutants has been discarded in [38]. Yet the situation is actually not hopeless, the interesting point being that the existence of the desired factors can be derived from spectral properties of the modular operator ∆ affiliated with the pair (M, Ω). Recalling that a linear map from a Banach space into another one is said to be nuclear if it can be decomposed into a series of maps of rank one whose norms are summable, we extract the following pertinent condition from [12].
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(c) Modular Nuclearity Condition: For any given x ∈ W the map M → ∆1/4 M Ω,
M ∈ M(x),
(2.3)
is nuclear. Equivalently, since ∆1/4 is invertible, the image of the unit ball in M(x) under this map is a nuclear subset of H. Since Ω is cyclic and separating for M and the algebras M(x), M both are factors, it follows from the modular nuclearity condition (c) that the inclusions M(x) ⊂ M, x ∈ W , are split [12, Thm. 3.3]. Conversely, if these inclusions are split, the map (2.3) has to be compact, at least. Thus a proof of the split property (2.2) amounts to a spectral analysis of the operator ∆1/4 on the subspaces M(x) Ω ⊂ H. This task is, as we shall see, manageable in concrete applications. We summarize the results of the preceding discussion in the following proposition. Proposition 2.2. Let {M, U, H} be a triple satisfying conditions (a), (b) and (c), stated above. Then, for x ∈ W , (i) the inclusion M(x) ⊂ M is split; (ii) the relative commutant M(x) ∩ M is isomorphic to the unique hyperfinite type III1 factor. In particular, it has cyclic and separating vectors. We conclude this section by noting that any triple {M, U, H} as in the preceding proposition can be used to construct a non-trivial Poincar´e covariant net of local algebras on the two-dimensional Minkowski space R2 . Following closely the discussion in [5, 6], we first note that the modular group ∆is , s ∈ R, and the modular conjugation J affiliated with (M, Ω) can be interpreted as representations of proper Lorentz transformations Λ (having determinant one). More specifically, if Λ is any such transformation and Λ = (−1)σ B(θ) its polar decomposition, where σ ∈ {0, 1} and B(θ) is a boost with rapidity θ ∈ R, one can show that . U (x, Λ) = U (x) J σ ∆iθ/2π (2.4) defines a continuous (anti)unitary representation of the proper Poincar´e group [5]. Moreover, Ω is invariant under the action of these operators and may thus be . interpreted as a vacuum state. Setting R(ΛW + x) = U (x, Λ)MU (x, Λ)−1 , one obtains a local (as a matter of fact, Haag-dual) Poincar´e covariant net of wedge . algebras on R2 . Denoting the double cones in R2 by Cx,y = (−W + x) ∩ (W + y), x − y ∈ W , the corresponding algebras . (2.5) R(Cx,y ) = R(W + x) ∩ R(−W + y) = M(x) ∩ M(y) are non-trivial according to the preceding proposition. As was shown in [5], they form a local net on R2 which is relatively local to the wedge algebras and transforms covariantly under the adjoint action of U (x, Λ). It may thus be interpreted as a net of local observables. The vacuum vector Ω need not be cyclic for the local algebras, however. In fact, thinking of theories exhibiting solitonic excitations of Ω which are localized in wedge regions, this may also not be expected in general.
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3 Applications to field theoretic models We carry over now the results of the preceding section to the framework of twodimensional models and indicate their significance for the formfactor program, i.e., the reconstruction of local observables and fields from a given factorizing scattering matrix. For the sake of concreteness, we restrict attention here to the theory of a single massive particle with given two-particle scattering function S2 , as considered in [36, 37] and described in more detail in [27]. The Hilbert space of the ∞theory is conveniently represented as the S2 -symmetrized Fock space H = n=0 Hn . Here the subspace H0 consists of multiples of the vacuum vector Ω and, using the parameterization of the mass shell by the rapidity θ, p(θ) = m ch(θ), sh(θ) ,
θ ∈ R,
(3.1)
the single particle space H1 can be identified with the space of square integrable functions θ → Ψ1 (θ) with norm given by (3.2) Ψ1 2 = dθ |Ψ1 (θ)|2 . The elements of the n-particle space Hn are represented by square integrable functions θ1 . . . θn → Ψn (θ1 , . . . , θn ) which are S2 -symmetric, Ψn (θ1 , . . . , θi+1 , θi , . . . , θn ) = S2 (θi − θi+1 ) Ψn (θ1 , . . . , θi , θi+1 , . . . , θn ).
(3.3)
Here ζ → S2 (ζ) is the scattering function which is continuous and bounded on the strip {ζ ∈ C : 0 ≤ Im ζ ≤ π}, analytic in its interior and satisfies, for θ ∈ R, the unitarity and crossing relations S2 (θ)−1 = S2 (θ) = S2 (−θ) = S2 (θ + iπ).
(3.4)
On H there acts a continuous unitary representation U of the proper orthochronous Poincar´e group, given by n . U (x, B(θ)) Ψ n (θ1 , . . . , θn ) = eix j=1 p(θj ) Ψn (θ1 − θ, . . . , θn − θ).
(3.5)
It satisfies the relativistic spectrum condition, i.e., the joint spectrum of the generators P = (P0 , P1 ) of the translations U (R2 , 1) is contained in V+ . Moreover, there is an antiunitary operator J on H, representing the PCT symmetry. It is given by . (3.6) J Ψ n (θ1 , . . . , θn ) = Ψn (θn , . . . , θ1 ). As in the case of the bosonic and fermionic Fock spaces, one can define creation and annihilation operators z † (θ), z(θ) (in the sense of operator-valued
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distributions) on the dense subspace D ⊂ H of vectors with a finite particle number. They are hermitian conjugates with respect to each other and satisfy the Fadeev–Zamolodchikov relations z † (θ)z † (θ ) = S2 (θ − θ ) z † (θ )z † (θ),
z(θ)z(θ ) = S2 (θ − θ ) z(θ )z(θ),
z(θ)z † (θ ) = S2 (θ − θ) z † (θ )z(θ) + δ(θ − θ ) 1.
(3.7)
Their action on D is fixed by the equations (z † (θ1 ) . . . z † (θn ) Ω, Ψ) = (n!)1/2 Ψn (θ1 , . . . , θn ),
z(θ) Ω = 0.
(3.8)
With the help of these creation and annihilation operators one can define on D a field φ, setting . φ(f ) = z † (f+ ) + z(f− ), f ∈ S(R2 ), (3.9) where
. f± (θ) = (2π)−1
dxf (x) e±ip(θ)x
(3.10)
and we adopt the convention that, both, z † ( · ) and z( · ) are complex linear on the space of test functions. It has been shown in [27] that φ transforms covariantly under the adjoint action of the proper orthochronous Poincar´e group, U (x, B) φ(f ) U (x, B)−1 = φ(fx,B ),
(3.11)
. where fx,B (y) = f (B −1 (y − x)), y ∈ R2 . Moreover, φ is real, φ(f )∗ ⊃ φ(f ), and each vector in D is entire analytic for the operators φ(f ). Since D is stable under their action, these operators are essentially selfadjoint on this domain for real f . We mention as an aside that the fields φ(f ) are polarization-free generators in the sense of [7]. Denoting the selfadjoint extensions of φ(f ), f real, by the same symbol, one can define the von Neumann algebras . R(W + x) = {eiφ(f ) : suppf ⊂ W + x} ,
x ∈ R2 ,
(3.12)
. where W denotes, as before, the wedge W = {y ∈ R2 : |y0 | + y1 < 0}. With the help of the PCT operator J one can also define algebras corresponding to the opposite wedges, . R(−W − x) = J R(W + x) J,
x ∈ R2 .
(3.13)
Now, given an arbitrary proper Lorentz transformation Λ with polar decomposition Λ = (−1)σ B, σ ∈ {0, 1}, one obtains a representation of the proper . Poincar´e group, setting U (x, Λ) = U (x, B)J σ . It then follows from the covariance properties (3.11) of the field that U (x, Λ) R(±W + y)U (x, Λ)−1 = R(±(−1)σ W + Λy + x),
(3.14)
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taking into account that the wedge W is stable under the action of boosts. So, by this construction, one arrives at a Poincar´e covariant net of wedge algebras on the two-dimensional Minkowski space. It has been shown in [27] that this net is local, R(±W + x) ⊂ R(∓W + x) ,
(3.15)
and that Ω is cyclic and separating for the wedge algebras (and hence for their commutants). The triple {R(W ), U (R2 , 1), H} satisfies conditions (a) and (b) given in the preceding section. More can be said by making use of modular theory and certain specific domain properties of the field φ. Proposition 3.1. Let R(W ) be the algebra defined above. Then (i) the modular group and conjugation affiliated with (R(W ), Ω) are given by R λ → U (0, B(2πλ)) and J, respectively; (ii) R(W ) = R(−W ) (Haag duality). Proof. Let ∆W , JW be the modular operator and conjugation, respectively, affiliated with (R(W ), Ω). It follows from modular theory that any boost U (0, B) commutes with ∆W and JW since Ω is invariant and R(W ) is stable under its . (adjoint) action. Hence λ → V (λ) = U (0, B(2πλ))∆−iλ is a continuous unitary W representation of R with the latter properties. Moreover, V (λ) commutes with all boosts U (0, B) and, by a theorem of Borchers [5], also with all translations U (x, 1). Since the restriction of U to the proper orthochronous Poincar´e group acts irreducibly on H1 , one concludes that V (λ) H1 = eiλc 1 for fixed real c and any λ ∈ R. Now, for real f with suppf ⊂ W , φ(f ) is a selfadjoint operator affiliated with . R(W ), and the same holds for φλ (f ) = V (λ)φ(f )V (λ)−1 , λ ∈ R, because of the stability of R(W ) under the adjoint action of V (λ). So both operators commute with all elements of R(W ) . Since Ω is invariant under the action of V (λ)−1 and since φ(f )Ω ∈ H1 , the preceding result implies (3.16) φλ (f ) − eiλc φ(f ) A Ω = 0, A ∈ R(W ) . It will be shown below that the dense set of vectors R(W ) Ω is a core, both, for φ(f ) and φλ (f ). Hence φλ (f ) = eiλc φ(f ) which, in view of the selfadjointness of the field operators, is only possible if c = 0. This holds for any choice of f within the above limitations, so V (λ) acts trivially on R(W ). Taking also into account that Ω is cyclic for R(W ), one arrives at V (λ) = 1, λ ∈ R, from which the first part of statement (i) follows. Similarly, modular theory and the theorem of Borchers mentioned above im. ply that the unitary operator I = JW J commutes with all Poincar´e transformations U (x, B), and taking into account relation (3.15), one also has IR(W )I −1 ⊂ . R(W ). Hence, putting φI (f ) = Iφ(f )I −1 , one finds by the same reasoning as in
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the preceding step that φI (f ) = φ(f ). Thus I = 1, proving the second part of statement (i). The statement about Haag duality then follows from the equalities R(W ) = JW R(W )JW = JR(W )J = R(−W ).
(3.17)
It remains to prove the assertion that R(W ) Ω is a core for the selfadjoint operators φ(f ), φλ (f ) and φI (f ), respectively. To this end one makes use of bounds, given in [27], on the action of the field operators on n-particle states Ψn . One has φ(f )Ψn ≤ cf (n + 1)1/2 Ψn , where cf is some constant depending only on f . Since the field operators change the particle number at most by ±1, one can proceed from this estimate to corresponding bounds for Ψ ∈ D, given by φ(f )Ψ ≤ 2cf (N + 1)1/2 Ψ , where N is the particle number operator. Recalling that P0 denotes the (positive) generator of the time translations, it is also clear that m (N + 1) ≤ (P0 + m1). So for Ψ ∈ D ∩ D0 , where D0 is the domain of P0 , one arrives at the inequalities φ(f )Ψ ≤ 2cf (N + 1)1/2 Ψ ≤ 2m−1/2 cf (P0 + m1)1/2 Ψ .
(3.18)
It follows from this estimate by standard arguments that any core for P0 is also a core for the field operators φ(f ). Since the unitary operators V (λ) and I in the preceding steps were shown to commute with the time translations, this domain property is also shared by the transformed field operators φλ (f ) and φI (f ), respectively. In order to complete the proof, one has only to show that R(W ) Ω ∩ D0 is a core for P0 . Now R(W ) Ω is mapped into itself by all translations U (x), x ∈ −W . Hence, taking into account the invariance of Ω under translations, one finds that f(P )R(W ) Ω ⊂ R(W ) Ω ∩ D0 for any test function f with suppf ⊂ −W . But this space of functions contains elements f such that f(P ) is invertible. Hence )R(W ) Ω ⊂ (P0 ± i1)(R(W ) Ω ∩ D0 ) both are dense subspaces of H, (P0 ± i1)f(P proving the statement. In view of the covariance properties of the net, it is apparent that analogous statements hold for all wedge algebras. Thus the only point left open in this reconstruction of a relativistic quantum field theory from scattering data is the question of whether the wedge algebras contain operators which can be interpreted as observables localized in finite spacetime regions, such as the double cones . Cx,y = (W + y) ∩ (−W + x), x − y ∈ W . By the Einstein causality, observables localized in Cx,y have to commute with all operators localized in the adjacent wedges W + x and −W + y. They are therefore elements of the algebra . R(Cx,y ) = R(W + x) ∩ R(−W + y) = R(−W + x) ∩ R(W + y). (3.19) It follows from the properties of the wedge algebras established thus far that the resulting map C → R(C) from double cones to von Neumann algebras defines a local and Poincar´e covariant net on the Minkowski space. So if the theory describes local observables, the algebras R(C) are to be non-trivial.
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At this point the nuclearity condition formulated in Section 2 comes in. Knowing by the preceding proposition the explicit form of the modular operator affiliated with (R(W ), Ω) and taking into account the invariance of Ω under spacetime translations, we are led to consider, for given x ∈ W , the maps A → U (0, B(−iπ/2)) U (x, 1) AΩ,
A ∈ R(W ).
(3.20)
Within the present context one then has the following more concrete version of Proposition 2.2. Proposition 3.2. Let the maps (3.20) be nuclear, x ∈ W . Then (i) the net of wedge algebras has the split property; (ii) for any open double cone C ⊂ R2 the corresponding algebra R(C) is isomorphic to the unique hyperfinite factor of type III1 . In particular it has cyclic and separating vectors. So in order to establish the existence of local operators in the theory, one needs an estimate of the size of the set of vectors U (0, B(−iπ/2)) U (x, 1) AΩ,
A ∈ R(W )1 ,
(3.21)
i.e., the image of the unit ball R(W )1 under the action of the map (3.20). We briefly indicate here the steps required in such an analysis which are similar to those carried out in [8] in an investigation of the Haag–Swieca compactness condition; a more detailed account of these results will be presented elsewhere. Making use of the localization properties of the operators A ∈ R(W ) and the analyticity properties of the scattering function S2 , one can show that the n-particle wave functions θ1 , . . . , θn → (AΩ)n (θ1 , . . . , θn )
(3.22)
extend, in the sense of distributions, to analytic functions in the domain 0 < Im θi < π, −δ < Im (θi − θk ) < δ, where i, k = 1, . . . , n and δ depends on the domain of analyticity of the scattering function S2 . Thus the functions θ1 , . . . ,θn → (U (0, B(−iπ/2)) AΩ)n (θ1 , . . . , θn ) = (AΩ)n (θ1 + iπ/2, . . . , θn + iπ/2)
(3.23)
are analytic in the domain −δ/n < Im θi < δ/n, i = 1, . . . , n. As a matter of fact, if A ∈ R(W )1 , the family of these functions turns out to be uniformly bounded (normal) on this domain. Taking also into account that U is a representation of the Poincar´e group, one obtains for x ∈ W the equality U (0, B(−iπ/2)) U (x, 1) AΩ = ex1 P0 −x0 P1 U (0, B(−iπ/2)) AΩ,
(3.24)
so the n-particle components of the vectors (3.21) have wave functions of the form θ1 , . . . , θn → (U (0, B(−iπ/2)) U (x, 1) AΩ)n (θ1 , . . . , θn ) = em
n
k=1 (x1
ch(θk ) − x0 sh(θk ))
(AΩ)n (θ1 + iπ/2, . . . , θn + iπ/2).
(3.25)
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Since, for x ∈ W , the exponential factor gives rise to a strong damping of large rapidities, it follows from the preceding results that the wave functions (3.25) form, for A ∈ R(W )1 , a bounded subset of the space of test functions S(Rn ) and hence a nuclear subset of Hn ⊂ L2 (Rn ). Moreover, taking into account the spectral properties of (P0 , P1 ), relation (3.24) combined with the estimate U (0, B(−iπ/2)) AΩ ≤ A following from modular theory implies (U (0, B(−iπ/2)) U (x, 1) AΩ)n ≤ en m(x1 +|x0 |) ,
A ∈ R(W )1 .
(3.26)
So these norms tend rapidly to 0 for large n ∈ N if x ∈ W . Combining these facts, one finds after a moments reflection that the sets (3.21) are relatively compact in H, implying that the maps (3.20) are compact. So they can be approximated with arbitrary precision by finite sums of maps of rank one. In order to prove that they are also nuclear, one needs more refined estimates, however.
4 An instructive example In order to illustrate the quantitative estimates needed for the proof that the map (3.20) is nuclear, we consider here the case of trivial scattering, S2 = 1, i.e., the theory of a free massive Bose field φ. There the combinatorial problems appearing in the analysis of the size of sets of the type (3.21) have been settled in [15] and we shall make use of these results here. We begin by recalling some well-known facts: The restrictions of the field φ and of its time derivative φ˙ to the time zero plane are operator-valued distributions on the domain D. These time zero fields, commonly denoted by ϕ and π, satisfy canonical equal time commutation relations. If smeared with test functions h having support in the interval (−∞, 0), they generate the von Neumann algebra R(W ) and, applying them to the vacuum vector Ω, they create closed subspaces Lϕ (W ), Lπ (W ) of the single particle space H1 given by Lϕ (W ) = {θ → h(m sh(θ)) : supp h ⊂ (−∞, 0)}− , Lπ (W ) = {θ → ch(θ) h(m sh(θ)) : supp h ⊂ (−∞, 0)}− ,
(4.1)
where the tilde denotes Fourier transformation. We also consider the shifted sub. . spaces Lϕ (W + x) = U (x, 1) Lϕ (W ) and Lπ (W + x) = U (x, 1) Lπ (W ) and denote the corresponding orthogonal projections by Eϕ (W + x) and Eπ (W + x), respectively. After these preparations we are in a position to apply the results in [15, Thm. 2.1] which we recall here for the convenience of the reader in a form appropriate for the present investigation. Lemma 4.1. Consider the theory with scattering function S2 = 1 and let both, Eϕ (W +x) U (0, B(−iπ/2)) and Eπ (W +x) U (0, B(−iπ/2)) be trace class operators with operator norms less than 1, x ∈ W . Then the sets (3.21) are nuclear.
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Thus the proof that the modular nuclearity condition is satisfied in the present theory reduces to a problem of spectral analysis in the single particle space H1 . We first turn to the task of providing estimates of the norms of the operators appearing in the lemma. h(m sh(θ)), Let Φh ∈ Lϕ (W ) be a vector with wave function θ → Φh (θ) = where h is, as before, a test function with support in (−∞, 0). Because of these support properties, Φh lies in the domain of all boost operators U (0, B(θ)) for complex θ with −π ≤ Im θ ≤ 0. Furthermore, as sh(θ + iπ) = sh(−θ), one has (U (0, B(−iπ))Φh )(θ) = h(m sh(−θ)) = Φh (−θ). Thus U (0, B(−iπ)) Φh = Φh and hence U (0, B(−iπ/2)) Φh ≤ Φh . Making use now of the properties of the representation U , one obtains the estimate, x ∈ W , U (0, B(−iπ/2)) U (x, 1) Φh = ex1 P0 −x0 P1 U (0, B(−iπ/2)) Φh ≤ em(x1 +|x0 |) U (0, B(−iπ/2)) Φh ≤ em(x1 +|x0 |) U (x, 1) Φh .
(4.2)
Since Φh was arbitrary within the above limitations and (x1 + |x0 |) is negative, this yields the norm estimate U (0, B(−iπ/2)) Eϕ(W + x) < 1, x ∈ W . But the adjoint operator Eϕ (W + x) U (0, B(−iπ/2)) has the same norm, so the desired bound follows. In a similar manner one can show that Eπ (W + x) U (0, B(−iπ/2)) also has norm less than 1. It remains to establish the trace class property of these operators. To this end we consider the restriction of the operator U (0, B(−iπ/2)) U (x, 1), x ∈ W , to the subspaces Lϕ (W ) and Lπ (W ), respectively. Let, as before, Φh ∈ Lϕ (W ), then (U (0, B(−iπ/2)) U (x, 1) Φh )(θ) = ex1 p0 (θ) − x0 p1 (θ) Φh (θ + iπ/2).
(4.3)
Making use of the analyticity and boundedness properties of θ → Φh (θ) and the fact that Φh (θ + iπ) = Φh (−θ), one can represent Φh (θ + iπ/2) by a Cauchy integral, 1 1 1 + Φh (θ ). (4.4) Φh (θ + iπ/2) = dθ 2πi θ − θ − iπ/2 θ + θ − iπ/2 Next, for x ∈ W , let Xϕ be the operator on H1 with kernel 1 1 1 x1 p0 (θ) − x0 p1 (θ) Xϕ (θ, θ ) = e + . 2πi θ − θ − iπ/2 θ + θ − iπ/2
(4.5)
Being the sum of products of multiplication operators in rapidity space, respectively its dual space, which are bounded and rapidly decreasing, it is apparent that Xϕ is of trace class. Moreover, U (0, B(−iπ/2)) Eϕ(W + x) = Xϕ Eϕ (W ) U (x, 1)−1 by the preceding results. Since the trace class operators form a *-ideal in B(H1 ), it follows that U (0, B(−iπ/2)) Eϕ (W + x) and its adjoint Eϕ (W + x)U (0, B(−iπ/2)) are of trace class. By a similar argument one can also establish the trace class property of Eπ (W + x) U (0, B(−iπ/2)), the only difference being that for vectors Φh ∈ Lπ (W )
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with wave functions θ → Φh (θ) = ch(θ) h(m sh(θ)) one has Φh (θ +iπ) = −Φh (−θ). As a consequence, the sum in relation (4.4) turns into a difference, but this does not affect the conclusions. So the following statement has been proven. Proposition 4.2. In the theory with scattering function S2 = 1, the sets (3.21) and corresponding maps A → U (0, B(−iπ/2)) U (x, 1) AΩ,
A ∈ R(W ).
(4.6)
are nuclear for any x ∈ W . We thus have verified in the present model the modular nuclearity condition for wedge algebras with all of its consequences. In particular, the wedge algebras have the split property. Although the latter fact was known before [30], there did not yet exist a proof in the literature. By similar arguments one can also treat the theory with scattering function S2 = −1 (related to the Ising model [26]). In fact, there the sets (3.21) turn out to be somewhat smaller since the underlying auxiliary Fock space is antisymmetric and the Pauli principle becomes effective. The details of these computations are presented in [28]. Quantitative estimates of the size of the sets (3.21) in theories with generic scattering function require, however, further analysis.
5 Conclusions Within the algebraic setting of quantum field theory, we have presented a method which allows one to decide whether algebras affiliated with wedge shaped regions in two-dimensional Minkowski space contain compactly localized operators. This method seems to be particularly useful for proving the existence of local operators in theories with factorizing S-matrix. It is thus complementary to the formfactor program, where one tries to exhibit such operators explicitly by solving an infinite system of equations. The upshot of the present investigation is the insight that the basic algebraic problem of checking locality, which amounts to computing relative commutants, can be replaced by an analysis of spectral properties of representations of the Poincar´e group. There exist other methods by which the crucial intermediate step in our argument, the proof of the split property of wedge algebras, could be accomplished [10, 13, 16, 18, 19, 31]. But the present approach requires less a priori information about the underlying theory and also seems better managable in concrete applications. Moreover, in view of the fact that it relies only on the modular structure, it is applicable to theories on arbitrary spacetime manifolds. It is apparent, however, that the split property of wedge algebras is in general an unnecessarily strong requirement if one is merely interested in the existence of compactly localized operators. As a matter of fact, it follows from an argument of Araki that it cannot hold in more than two spacetime dimensions, cf. [10, Sec. 2]. It would therefore be desirable to establish less stringent conditions which still
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imply that the relative commutant fixed by a given inclusion of von Neumann algebras is non-trivial. The present results seem to suggest that this information is encoded in spectral properties of the corresponding modular operators, but a clarification of this point requires some further analysis. An appropriately weakened condition which would allow one to establish the existence of local operators in non-local algebras also in higher dimensions would have several interesting applications. This existence problem was recently met in the context of theories of massless particles with infinite spin [34], for example. It also appears in the algebraic approach to the construction of theories of particles with anyonic statistics [33] and the construction of nets of wedge algebras from information on the modular data [6,14,25,40,41]. A solution of this problem would thus be a major step in the algebraic approach to constructive problems in local quantum physics.
Acknowledgments. We would like to thank the Deutsche Forschungsgemeinschaft for financial support.
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[9] R. Brunetti, D. Guido and R. Longo, Modular localization and Wigner particles, Rev. Math. Phys. 14, 759 (2002) [arXiv:math-ph/0203021]. [10] D. Buchholz, Product states for local algebras, Commun. Math. Phys. 36 (1974) 287. [11] D. Buchholz, C. D’Antoni and K. Fredenhagen, The universal structure of local algebras, Commun. Math. Phys. 111, 123 (1987). [12] D. Buchholz, C. D’Antoni and R. Longo, Nuclear maps and modular structures. 1: General properties, J. Funct. Anal. 88, 233 (1990). [13] D. Buchholz, C. D’Antoni and R. Longo, Nuclear maps and modular structures. 2. Applications to quantum field theory, Commun. Math. Phys. 129, 115 (1990). [14] D. Buchholz, O. Dreyer, M. Florig and S.J. Summers, Geometric modular action and spacetime symmetry groups, Rev. Math. Phys. 12, 475 (2000) [arXiv:math-ph/9805026]. [15] D. Buchholz and P. Jacobi, On the nuclearity condition for massless fields, Lett. Math. Phys. 13, 313 (1987). [16] D. Buchholz and E.H. Wichmann, Causal independence and the energy level density of states in local quantum field theory, Commun. Math. Phys. 106, 321 (1986). [17] O.A. Castro-Alvaredo and A. Fring, Mutually local fields from form factors, Int. J. Mod. Phys. B 16 1915 (2002) [arXiv:hep-th/0112097]. [18] C. D’Antoni and K. Fredenhagen, Charges in space – like cones, Commun. Math. Phys. 94, 537 (1984). [19] C. D’Antoni and R. Longo, Interpolation by type I infinity factors and the flip automorphism, J. Funct. Anal. 51, 361 (1983). [20] J. Dixmier and O. Marechal, Vecteurs totalisateurs d’une alg`ebre de von Neumann, Commun. Math. Phys. 22, 44 (1971). [21] S. Doplicher and R. Longo, Standard and split inclusions of von Neumann algebras, Commun. Math. Phys. 75, 493 (1984). [22] M. Florig, On Borchers’ Theorem, Lett. Math. Phys. 46, 289 (1998). [23] R. Haag, Local Quantum Physics, Springer 1996. [24] U. Haagerup, Connes’ bicentralizer problem and uniqueness of the injective factor of type III1 , Acta Math. 158, 95 (1987).
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[25] R. K¨ ahler and H.W. Wiesbrock, Modular theory and the reconstruction of four-dimensional quantum field theories, J. Math. Phys. 42, 74 (2001). [26] M. Karowski and P. Weisz, Exact form-factors in (1 + 1)-dimensional field theoretic models with soliton behavior, Nucl. Phys. B 139, 455 (1978). [27] G. Lechner, Polarization-free quantum fields and interaction, Lett. Math. Phys. 64, 137 (2003) [arXiv:hep-th/0303062]. [28] G. Lechner, On the existence of local observables in theories with a factorizing S-matrix, [arXiv:math-ph/0405062]. [29] R. Longo, Notes On Algebraic Invariants For Noncommutative Dynamical Systems, Commun. Math. Phys. 69, 195 (1979). [30] M. M¨ uger, Superselection structure of massive quantum field theories in 1 + 1 dimensions, Rev. Math. Phys. 10, 1147 (1998) [arXiv:hep-th/9705019]. [31] M. M¨ uger, Superselection structure of quantum field theories in 1 + 1 dimensions. Doctoral dissertation. University of Hamburg, 1997. DESY-preprint 97-073. [32] J. Mund, The Bisognano-Wichmann theorem for massive theories, Annales Henri Poincar´e 2, 907 (2001) [arXiv:hep-th/0101227]. [33] J. Mund, Modular localization of massive particles with any spin in D = 2+1, J. Math. Phys. 44, 2037 (2003) [arXiv:hep-th/0208195]. [34] J. Mund, B. Schroer and J. Yngvason, String-localized quantum fields from Wigner representations, [arXiv:math-ph/0402043]. [35] S. Sakai, C*-Algebras and W*-Algebras, Springer 1971. [36] B. Schroer, Modular localization and the bootstrap-formfactor program, Nuclear Phys. B 499, 547 (1997). [arXiv:hep-th/9702145]. [37] B. Schroer, Modular wedge localization and the d = 1+1 formfactor program, Annals Phys. 275, 190 (1999) [arXiv:hep-th/9712124]. [38] B. Schroer and H.W. Wiesbrock, Modular constructions of quantum field theories with interactions, Rev. Math. Phys. 12, 301 (2000) [arXiv:hepth/9812251]. [39] F.A. Smirnov, Formfactors in completely integrable models of quantum field theory, Advanced Series in Mathematical Physics, 14. World Scientific 1992. [40] H.W. Wiesbrock, Modular inclusions and intersections of algebras in QFT. Operator algebras and quantum field theory (Rome, 1996), 609–620, Internat. Press, Cambridge, MA, 1997.
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[41] H.W. Wiesbrock, Modular intersections of von Neumann algebras in quantum field theory, Commun. Math. Phys. 193, 269 (1998). Detlev Buchholz and Gandalf Lechner Institut f¨ ur Theoretische Physik Universit¨ at G¨ ottingen D-37077 G¨ ottingen Germany email:
[email protected] email:
[email protected] Communicated by Klaus Fredenhagen submitted 19/04/04, accepted 21/04/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 1081 – 1095 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/061081-15 DOI 10.1007/s00023-004-0191-7
Annales Henri Poincar´ e
C ∗-Independence, Product States and Commutation L.J. Bunce and J. Hamhalter Abstract. Let D be a unital C ∗ -algebra generated by C ∗ -subalgebras A and B possessing the unit of D. Motivated by the commutation problem of C ∗ -independent algebras arising in quantum field theory, the interplay between commutation phenomena, product type extensions of pairs of states and tensor product structure is studied. Roos’s theorem [11] is generalized in showing that the following conditions are equivalent: (i) every pair of states on A and B extends to an uncoupled product state on D; (ii) there is a representation π of D such that π(A) and π(B) commute and π is faithful on both A and B; (iii) A ⊗min B is canonically isomorphic to a quotient of D. The main results involve unique common extensions of pairs of states. One consequence of a general theorem proved is that, in conjunction with the unique product state extension property, the existence of a faithful family of product states forces commutation. Another is that if D is simple and has the unique product extension property across A and B then the latter C ∗ -algebras must commute and D be their minimal tensor product.
1 Introduction One approach to the concept of independence in quantum field theory, in general terms, has that two subsystems of observables may be deemed independent if each can be prepared in any of its states without having regard to the other. In a formulation originating in [5], when observables are realized as hermitian elements in a C ∗ -algebra this translates into the notion of C ∗ -independence. Commutation of the arising observable algebras is not a precondition of C ∗ -independence nor it is a mathematical necessity, as many counterexamples show (see [12, p. 205]). The commutation question was first considered in [9, 10] and more lately in [3] where a natural commutation conjecture involving product states was negated and a request for appropriate sufficient conditions issued. We investigate commutation phenomena occurring in C ∗ -independence. At the same time we attempt to throw light upon the role of product states. Forewarned by [3, III] that faithful product states can exist alongside absence of nontrivial commutation across C ∗ -independent algebras, we proceed indirectly. The introduction of the notion of uncoupled product states allows us access to tensor products as likewise (and more immediately) does a condition which we call faithful independent commutation. Roos’s theorem [11] is generalized and minimal tensor product of C ∗ -algebras is characterized in terms of state extension conditions alone, without assuming mutual commutativity. Our main results are gathered in the third and final section of the paper where we study unique common
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extension properties. Amongst other things we deduce that if a simple C ∗ -algebra has the unique product state extension property across a pair of generating C ∗ subalgebras A and B, then A and B commute and D is canonically isomorphic to the minimal tensor product of A and B. In general, the unique product state extension property together with the existence of a faithful family of product states is sufficient to compel commutation. We use [2] and [13] as general references for C ∗ -algebras, the latter for tensor products particularly. Let A be a C ∗ -algebra, let S(A) be its state space and P (A) its set of pure states. Given ϕ ∈ S(A) and a ∈ A such that ϕ(a a∗ ) = 1 we use ϕa to denote the transformed state of A given by ϕa (x) = ϕ(a x a∗ ) ,
x ∈ A.
We recall [2, 2.4.10] that if ϕ ∈ S(A) then ker πϕ is the largest ideal in ker ϕ, where πϕ is the associated GNS representation. If J is an ideal of A, J 0 denotes its annihilator in A∗ . Let A and B be unital C ∗ -algebras and let β be a C ∗ -norm on the algebraic tensor product A ⊗ B. The completion with respect to β is written A ⊗β B. We may regard A and B as C ∗ -subalgebras of A ⊗β B. If ∈ S(A) and τ ∈ S(B) the tensor product ⊗ τ ∈ A∗ ⊗ B ∗ has unique extension to a state on A ⊗min B, and on A ⊗β B via the canonical ∗-homomorphism onto A ⊗min B. By custom we continue to denote the resulting (tensor product) extension in S(A ⊗β B) by ⊗ τ , regardless of β. Now let A and B be C ∗ -subalgebras of a unital C ∗ -algebra D such that A and B generate D as a C ∗ -algebra and both contain the identity of D. 1.1 Definition. A and B are said to be C ∗ -independent in D if and τ have a common extension in S(D) for all (, τ ) ∈ S(A) × S(B). Thus (when identified with their canonical images) A and B are C ∗ -independent in A ⊗β B for any C ∗ -norm β on A ⊗ B. Conversely [11], in a result referred to as Roos’s theorem, if A and B commute (i.e., a b = b a, for all a ∈ A and b ∈ B) and are C ∗ -independent in D then the natural map is a ∗-isomorphism from A ⊗ B onto the ∗-subalgebra of D generated by A and B. Recent papers that discuss various forms of independence in operator algebras include [3, 4] and [6, 7]. The comprehensive survey [12] is a main source of information on both mathematical and physical aspects. For a more recent development we refer the reader to [8, Chapter 11]. The reader is directed to the references contained in these works for an extensive literature. The related question of the existence and uniqueness of common extensions of a given pair of states is investigated in [1] in the significant case of subsystems of a Fermion system, with decisive results [1, Theorem 4, Theorem 5].
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2 Uncoupled product states Throughout this section D denotes a unital C ∗ -algebra and A and B denote C ∗ subalgebras of D containing the identity of D the union of which generates D. It is not assumed that A and B commute. If S ⊂ S(D) we shall employ the contraction S|A = {ϕ|A | ϕ ∈ S(D)} (with the corresponding meaning for S|B). 2.1 Definition. A state ϕ of D is said to be (a) a product state across A, B if ϕ(a b) = ϕ(a) ϕ(b) whenever a ∈ A and b ∈ B; (b) an uncoupled product state across A, B if n n n ϕ ai b i = ϕ ai ϕ bi , i=1
i=1
i=1
whenever a1 , . . . , an ∈ A , b1 , . . . , bn ∈ B . The definition (a) is symmetric about A and B (states are hermitian). Since 1’s can be ninserted willy-nilly without effect, the value of an uncoupled product state at i=1 ai bi (ai ∈ A, bi ∈ B) is the same at any rearrangement of the product that leaves unaltered the relative order of the ai ’s, and the relative order of the bi ’s. As is clear, product states and uncoupled product states are one and the same if A and B commute, and they coincide with the tensor product states when D is a C ∗ -tensor product of A and B. Let ∆(A, B) and ∆u (A, B), respectively, be the (possibly empty) sets of all product states across A, B and uncoupled product states across A, B. Given ∈ S(A) we shall write E() = {ϕ ∈ S(D) | ϕ|A = } . Similarly, E(τ ) shall denote the set of extensions of τ in S(D) whenever τ ∈ S(B). Routine verifications show that ∆(A, B) and ∆u (A, B) are weak∗ -compact and that E(), E() ∩ ∆(A, B) and E() ∩ ∆u (A, B) are convex and weak∗ -compact. (These facts concerning ∆u (A, B) are also visibly true from those of ∆(A, B) via Lemma 2.5 below.) Let ∈ S(A) and τ ∈ S(B). If and τ have a common extension in ∆u (A, B) we shall denote it ∧τ (by the definition there can be at most one such extension). Thus, τ ∈ E() ∩ ∆u (A, B)|B ⇐⇒ ∈ E(τ ) ∩ ∆u (A, B)|A ⇐⇒ ∧ τ exists. n Further, if τ = i=1 λi τi (convex sum)where τi ∈ S(B) such that ∧ τi exists n for each i, then ∧ τ exists and equals i=1 λi ( ∧ τi ) .
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2.2 Lemma. (a) Let ∈ S(A). Then (i) the restriction map, E() ∩ ∆(A, B) → E() ∩ ∆(A, B)|B is affine and weak∗ -continuous; (ii) the restriction map, E() ∩ ∆u (A, B) → E() ∩ ∆u (A, B)|B is an affine weak∗ -homeomorphism. (b) The map ∆u (A, B)|A × ∆u (A, B)|B → ∆u (A, B), given by (, τ ) → ∧ τ , is a weak∗ -homeomorphism and is affine in each variable. Proof. (a) Affineness and (weak∗ -) continuity are apparent in both (i) and (ii). Further, in (ii), the map is a bijection and so a homeomorphism by compactness. (b) To see continuity let (α , τα ) → (, τ ) in the product weak∗ -topology on the domain. By compactness of ∆u (A, B), some subnet β ∧ τβ of α ∧ τα has weak∗ limit ϕ in ∆u (A, B). Since ϕ must agree with on A and τ on B we have β ∧τβ → ∧ τ , establishing continuity. The remainder is clear. 2.3 Definition. D is said to have the C ∗ -product property (respectively, the C ∗ uncoupled product property) across A, B if and τ have a common product state extension (respectively a common uncoupled product state extension) for all (, τ ) ∈ S(A) × S(B). The C ∗ -product property was introduced in [4]. 2.4 Lemma. Let and τ have common extension in ∆(A, B) (respectively, in ∆u (A, B)) for all (, τ ) ∈ P (A) × P (B). Then D has the C ∗ -product property property (respectively, the C ∗ -uncoupled product property) across A, B. Proof. Let ∈ P (A). The assumption implies that E()∩∆(A, B)|B contains P (B) and so contains S(B) by the Krein-Milman theorem. For any τ ∈ S(B) it follows that E(τ ) ∩ ∆(A, B)|B contains P (A) and therefore contains S(A) whence D has the C ∗ -product property across A, B. A similar argument proves the statement in parentheses. In the next result and later J(A, B) denotes the norm closed ideal of D generated by the set of A-B commutators, {a b −b a | a ∈ A, b ∈ B}. n We shall refer to the elements of the form i=1 ai bi , where a1 , . . . , an ∈ A, b1 , . . . , bn ∈ B as the A-B generators of D. 2.5 Lemma. ∆u (A, B) = ∆(A, B) ∩ J(A, B)0 . Proof. Suppose that ϕ is a product state across A, B vanishing on J where J = J(A, B), and let ϕ be the induced state on D/J. Let a1 , . . . , an ∈ A, b1 , . . . , bn ∈ B. Since the images of A and B pairwise commute in D/J we see that n n n n n ϕ ai b i = ϕ ai · bi + J = ϕ ai · ϕ bi i=1
i=1
i=1
i=1
i=1
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so that ϕ is uncoupled. Conversely, suppose that ϕ is an uncoupled product state across A, B. If a1 , . . . , an ∈ A and b1 , . . . , bn ∈ B where n ≥ 2 we have, by definition ϕ(a1 b1 · · · ai−1 bi−1 (ai bi ) ai+1 bi+1 · · · an bn ) = ϕ(a1 b1 · · · , ai−1 bi−1 (bi ai ) ai+1 bi+1 · · · an bn ) . Thus, if a ∈ A, b ∈ B and x, y are A-B generators, we have ϕ(x a b y) = ϕ(x b a y) , an equation that continues to hold for all x and y in D (since D is the norm closed linear span of its A-B generators and ϕ is linear and continuous). Therefore, ϕ vanishes on all elements of the form x (a b − b a) y , where x, y ∈ D and a ∈ A, b ∈ B . Since the norm closed linear span of all such elements is exactly the ideal J(A, B), ϕ must vanish on J(A, B). We shall now introduce a modified form of commutation. 2.6 Definition. A and B are said to faithfully independently commute if there is a ∗-homomorphism π on D for which the following conditions hold: (a) π is faithful on A and B; (b) π(A) and π(B) commute and are C ∗ -independent in π(D). The above considerations combine to give the following generalized and extended form of Roos’s theorem. 2.7 Theorem. The following conditions are equivalent: (a) and τ have common uncoupled state extension across A, B for all (, τ ) ∈ P (A) × P (B). (b) D has the C ∗ -uncoupled product property across A, B. (c) A and B faithfully independently commute. (d) There is a (unique) C ∗ -norm β on A ⊗ B and a ∗-isomorphism A ⊗β B → D/J(A, B) sending a ⊗ b → a b + J(A, B). (e) There is a (unique) norm closed ideal J of D and a ∗-isomorphism A ⊗min B → D/J, sending a ⊗ b → a b + J.
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Proof. (a) ⇒ (b) This was given in Lemma 2.4. (b)⇒ (c), (d) Assume (b). Let ∈ S(A) and τ ∈ S(B). By assumption and Lemma 2.5 and τ have a common extension in D vanishing on J(A, B), implying that and τ vanish on A ∩ J(A, B) and B ∩ J(A, B), respectively. It follows that A ∩ J(A, B) = B ∩ J(A, B) = {0} . So the quotient map, π : D → D/J(A, B), is faithful on A and B. Since, visibly, π(A) and π(B) are C ∗ -independent in π(D), this proves (c). Moreover, since π ⊗ π is a ∗-algebra isomorphism from A ⊗ B onto π(A) ⊗ π(B) and the latter, by Roos’s theorem, embeds as a ∗-subalgebra of π(D) via π(a) ⊗ π(b) → π(a b), the norm on π(D) pulls back to a C ∗ -norm on A ⊗ B giving (d). (d)⇒ (e) This is immediate from the fact that, canonically, A ⊗min B is a quotient of A ⊗β B. (e)⇒ (a). This is clear.
In the sequel we shall denote by π the GNS representation associated with a given state . 2.8 Lemma. Let and τ have common extension in ∆u (A, B). Then ker π = A ∩ ker πϕ and ker πτ = B ∩ ker πϕ , where ϕ = ∧ τ . Proof. Since ϕ extends the ideal A∩ker πϕ is contained in ker and so is contained in ker π . On the other hand, the norm closed ideal J of D generated by ker π is the norm closed linear span of all elements of the form y = a1 b1 · · · an bn a c1 d1 · · · cm dm where the ai and cj belong to A, the bi and dj belong to B and a lies in ker π . For any such element y we have ϕ(y) = (a1 · · · an a c1 · · · cm ) τ (b1 · · · bn d1 · · · dm ) = 0 . Hence, J is contained in ker πϕ so that ker π is contained in A ∩ ker πϕ as required. 2.9 Proposition. There is a set S of uncoupled product states across A, B such that {πϕ | ϕ ∈ S} is faithful on D if, and only if, there is a ∗-isomorphism A⊗min B → D sending a ⊗ b → a b. Proof. Suppose that {πϕ | ϕ ∈ S} is faithful on D for some subset S of ∆u (A, B). It follows from Lemma 2.5 that J(A, B) ⊂ ker ϕ for each ϕ ∈ S. Hence, A and B pairwise commute. Now let (σ, ω) ∈ P (A) × P (B). We have S = { ∧ τ | ∈ S|A , τ ∈ S|B} .
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Put S1
=
{a | ∈ S|A , (a a∗ ) = 1, a ∈ A}
S2
=
{τb | τ ∈ S|B , τ (b b∗ ) = 1, b ∈ B} .
and
By Lemma 2.8 {π | ∈ S|A} and {πτ | τ ∈ S|B} are faithful on A and B, respectively. It follows from [2, 2.4.8 (ii) and 3.4.2 (ii)] that σ and ω lie in the respective weak∗ -closures of S1 and S2 . Further, given typical elements a and τb of S1 and S2 , we see that a ∧ τb exists and equals ( ∧ τ )a b . Now Lemma 2.2 implies that σ ∧ ω exists. Hence, by the faithfulness assumption together with Theorem 2.7 (a)⇒ (e) (or Roos’s theorem, itself) A ⊗min B is ∗-isomorphic to D as claimed. If D is isomorphic to A ⊗min B in the above way, then the family {π⊗τ | ∈ P (A), τ ∈ P (B)} is well known to be faithful on A ⊗min B. 2.10 Corollary. Let ∈ S(A) and τ ∈ S(B). Then and τ have a common extension to an uncoupled product state across A, B if, and only if, there is a ∗isomorphism π (A) ⊗min πτ (B) → πϕ (D) sending π (a) ⊗ πτ (b) → πϕ (a b), where ϕ = ∧ τ. Proof. Suppose that ϕ = ∧ τ exists. Via Lemma 2.8 there is a ∗-homomorphism between π (A) ⊗min πτ (B) and πϕ (A) ⊗min πϕ (B) sending π (a) ⊗ πτ (b) to πϕ (a) ⊗ πϕ (b). But the induced state ϕ on πϕ (D), given by ϕ(π(x)) = ϕ(x), is an uncoupled product state across πϕ (A), πϕ (B) and πϕ is faithful on πϕ (D). Hence, the required ∗-isomorphism is produced by Proposition 2.9 together with the above remark. The converse is clear. 2.11 Remarks. (a) The C ∗ -uncoupled product property across A, B is implemented by conditional expectations onto A and B as follows. Given in S(A) and τ in S(B) consider the composition, Q , D → D/J → A ⊗min B → B where the first map is the quotient homomorphism, the second is the inverse of the one given by Theorem 2.7 (e) and the third is the projection that sends a⊗ b → (a) b [13, IV, 4.25]; and let Qτ be the corresponding composition finishing at A. Chasing the maps, we see that Q and Qτ are norm one projections onto B and A, respectively, and that τ Q = Qτ = ∧ τ . (b) If D has a faithful uncoupled product state across A, B then, as soon as it is observed, by Lemma 2.5, that A and B must pairwise commute the conclusion
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that D A ⊗min B is seen directly from [12, Corollary 3.5] and [3, Proposition 12]. The same conclusion cannot be drawn if it is known only that there is a faithful product state even when D is finite-dimensional and both A and B are abelian and C ∗ -independent [3, III]. Evidently, ∆u (A, B) = ∆(A, B) in that case. Nevertheless, if A and B are abelian and C ∗ -independent in D, then D always has the C ∗ -uncoupled product property across A, B (see Proposition 2.13 below). (c) The commutator ideal of D, the norm closed ideal of D generated by all commutators x y−y x as x and y range over D, clearly contains J(A, B). If there be any at all, the pure states of D vanishing on the commutator ideal are precisely the multiplicative states of D. Trivially, the latter are examples of uncoupled product states. 2.12 Lemma. Let ϕ ∈ S(D). (a) If ϕ|A is multiplicative, then ϕ(a x) = ϕ(a) ϕ(x) for all a ∈ A and x ∈ D. (b) If ϕ|A and ϕ|B are multiplicative then ϕ is multiplicative and is the unique common extension of ϕ|A and ϕ|B in S(D). Proof. (a) Let ϕ be multiplicative on A and let a and x be self-adjoint elements of A and D, respectively. With α = ϕ(a) the Cauchy-Schwarz inequality gives |ϕ((a − α 1) x)|2 ≤ ϕ((a − α 1)2 ) ϕ(x2 ) = (ϕ(a − α 1))2 ϕ(x2 ) = 0 so that ϕ(a x) = ϕ(a) ϕ(x), from which the general case follows. (b) Let ϕ be multiplicative on both A and B and let a1 , . . . , an ∈ A and b1 , . . . , bn ∈ B where n ≥ 2. Iterating part (a) (symmetrized), n n n ϕ ai b i ai bi = ϕ(a1 ) ϕ(b1 ) ϕ ai b i = ϕ(a1 ) ϕ b1 i=1
= ··· =
n
i=2
i=2
ϕ(ai ) ϕ(bi ) .
i=1
ϕ(xi ) ϕ(yj ) So for any A-B generators x1 , . . . , xn , y1 , . . . ,ym we have ϕ(x i yj ) = n n m m and therefore that ϕ =ϕ implying i=1 xi j=1 yj i=1 xi ϕ j=1 yj that ϕ is multiplicative on D.
2.13 Proposition. (a) Let A be abelian. Then A and B are C ∗ -independent in D if, and only if, D has the C ∗ -product property across A, B. (b) Let A and B be abelian. Then the following conditions are equivalent. (i) A and B are C ∗ -independent in D. (ii) D has the C ∗ -uncoupled product property across A, B.
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(iii) A and B faithfully independently commute. (iv) There is a ∗-isomorphism A ⊗min B → D/J sending a ⊗ b → a b + J, where J is the commutator ideal of D. Proof. (a) If ∈ P (A) and τ ∈ S(B) have common extension ϕ in S(D) then, since is multiplicative, ϕ is a product state across A, B by Lemma 2.12 (a). Hence, if A and B are C ∗ -independent in D, then the latter has the required C ∗ -product property by Lemma 2.4. The converse is clear. (b) Since A and B are abelian, D/J(A, B) is abelian. Therefore J(A, B) contains, and so equals, the commutator ideal of D. By Theorem 2.7 it is enough to prove (i)⇒(ii). Assume (i) and let (, τ ) ∈ P (A) × P (B). Since and τ are multiplicative, the assumption together with Lemma 2.12 (b) implies that and τ have common extension to a multiplicative state on D, and (b) is now immediate from Theorem 2.7. We remark that in the light of Remark 2.11 (b) neither of the conditions in Proposition 2.13 (b) imply that A and B commute.
3 Unique common extensions If K is a compact convex set let A(K) denote the continuous (real) affine functions on K. If K is the state space of a unital C ∗ -algebra A the evaluation map Asa → A(K) (a → a ˆ) is an order isomorphism and linear isometry. The same is true of the evaluation map Wsa → Ab (S) when W is a von Neumann algebra, S is the normal state space of W and Ab (S) denotes the bounded (real) affine functions on S. In all that follows D continues to denote a unital C ∗ -algebra and A and B denote C ∗ -subalgebras the union of which generates D and such that 1 ∈ A ∩ B. As before, we do not assume that A and B commute. In order to formulate solutions to the the commutation problem we study two natural unique common extension properties. The first characterizes faithful independent commutation and determines the set of corresponding uncoupled product states amongst likely weak∗ -closed sets of states. The underlying idea of the proof below is to employ convexity to construct implementing conditional expectations (see Remarks 2.11 (a)). 3.1 Theorem. Let ∆ be a weak∗ -closed subset of S(A) × S(B). Then the following statements are equivalent: (a) and τ have unique common extension in ∆ and ∆ ∩ E(), ∆ ∩ E(τ ) are convex, for all (, τ ) ∈ S(A) × S(B). (b) ∆ = ∆u (A, B) and A and B faithfully independently commute.
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Proof. (b)⇒ (a) This is immediate from Theorem 2.7 and remarks following Definition 2.1. (a)⇒ (b). Assume (a). By the uniqueness condition it is enough to show that ∆ is contained in ∆u (A, B). If ∈ S(A) and τ ∈ S(B), let ϕ,τ be their unique common extension in ∆. Now fix ∈ S(A) and consider the set K = E() ∩ ∆ . We have K = {ϕ,τ | τ ∈ S(B)} and K|B = S(B) . The restriction map, r : K → S(B), is an affine homeomorphism with inverse α : S(B) → K given by α(τ ) = ϕ,τ . Consider now the complex linear extension, Q : D → B, of the composition β
α∗
Dsa → A(S(D)) → A(K) → A(S(B)) → Bsa , where the first and last isometries are the appropriate evaluation map and inverse, respectively, and β is the restriction map, f → f |K. Letting x ∈ Dsa and τ ∈ S(B) we see that Q (x)ˆ= (ˆ x|K) ◦ α and deduce that τ Q = ϕ,τ . In particular, for b ∈ B we have τ Q (b) = τ (b) for all τ ∈ S(B) so that Q (b) = b . Hence, Q is a norm one projection onto B. In addition, for a ∈ A we have Q (a) = (a) 1, since τ Q (a) = (a) = τ ((a) 1) for all τ ∈ S(B). The upshot is that for all ∈ S(A) there is a surjective norm one projection, Q : D → B, such that τ Q = ϕ,τ
for all τ ∈ S(B),
and Q (a) = (a) 1
for all a ∈ A .
By symmetry, for all τ ∈ S(B) there is a surjective norm one projection, Qτ : D → A, such that Qτ = ϕ,τ
for all ∈ S(A)
and Qτ (b) = τ (b) 1
for all b ∈ B .
We claim that for all ∈ S(A) and all a1 , . . . , an ∈ A and b1 , . . . , bn−1 ∈ B, where n ≥ 2, we have Q (a1 b1 · · · an−1 bn−1 an ) = (a1 · · · an ) b1 · · · bn−1 . Observe first that if ∈ S(A), τ ∈ S(B) and a1 , a2 ∈ A and b1 ∈ B then, since Qτ : D → A is a conditional expectation, we have Qτ (a1 b1 a2 ) = a1 Qτ (b1 ) a2 = τ (b1 ) a1 a2 so that τ Q (a1 b1 a2 ) = Qτ (a1 b1 , a2 ) = τ [(a1 a2 ) b1 ]
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which, being valid for all τ in S(B), proves that Q (a1 b1 a2 ) = (a1 a2 ) b1 . Now fix n ≥ 2 and suppose that for all ∈ S(A) we have Q (a1 b1 · · · ak−1 bk−1 ak ) = (a1 · · · ak ) b1 · · · bk−1 whenever the ai are in A, the bj are in B and 2 ≤ k ≤ n. Take any a1 , a2 . . . in A and b1 , b2 . . . in B. For ∈ S(A), τ ∈ S(B) and n ≥ 2 we have Qτ (b1 a1 · · · bn−1 an−1 bn )
= τ Q (b1 a1 · · · bn−1 an−1 bn ) = τ [b1 Q (a1 b2 · · · bn−1 an−1 ) bn ] = τ [b1 ((a1 · · · an−1 ) b2 · · · bn−1 ) bn ] = [τ (b1 · · · bn ) a1 · · · an−1 ] ,
giving Qτ (b1 a1 · · · bn−1 an−1 bn ) = τ (b1 · · · bn ) a1 · · · an−1 and, in turn, τ Q (a1 b1 · · · an bn an+1 )
= Qτ (a1 b1 · · · an bn an+1 ) = [a1 Qτ (b1 a2 · · · an bn ) an+1 ] = [a1 (τ (b1 · · · bn ) a2 · · · an ) an+1 ] = τ [(a1 · · · an an+1 ) b1 · · · bn ]
so that Q (a1 b1 · · · an bn an+1 ) = (a1 · · · an+1 ) b1 · · · bn , thereby proving the claim. Hence, if ∈ S(A), τ ∈ S(B) and a1 , . . . , an ∈ A and b1 , . . . , bn ∈ B, where n ≥ 2, we have Q (a1 b1 · · · an bn ) = Q (a1 b1 · · · an ) bn = (a1 · · · an ) b1 · · · bn giving ϕ,τ (a1 b1 · · · an bn ) = τ Q (a1 b1 · · · an bn ) = (a1 · · · an ) τ (b1 · · · bn ) , so that ϕ,τ is an uncoupled product state across A, B. Therefore, ∆ is contained in ∆u (A, B), as required. 3.2 Definition. D is said to have the unique C ∗ -product property across A, B if and τ have unique common extension in ∆(A, B), for all (, τ ) ∈ S(A) × S(B).
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3.3 Theorem. D has the unique C ∗ -product property across A, B if, and only if, ∆(A, B) = ∆u (A, B) and A and B faithfully independently commute. Proof. This is immediate from Theorem 3.1 on putting ∆ = ∆(A, B).
3.4 Theorem. The following statements are equivalent: (a) D has the unique C ∗ -product property across A, B and has a faithful family of GNS representations associated with product states across A, B. (b) There is a ∗-isomorphism A ⊗min B → D sending a ⊗ b → a b. Proof. Combine Proposition 2.9 with Theorem 3.3.
∗
One immediate consequence of Theorem 3.4 is that if D has the unique C product property across A, B and has a faithful family of product states across A, B then A ⊗min B identifies with D as in Theorem 3.4 (b). This answers a question raised in [3, 12]. Since any ∗-representation of a simple C ∗ -algebra is faithful, Theorem 3.4 provides the following answer to the commutation question when D is simple. 3.5 Theorem. If D is simple and has the unique C ∗ -product property across A, B then there is a ∗-isomorphism A ⊗min B → D sending a ⊗ b → a b. Identifying D with its image in D∗∗ , and A∗∗ and B ∗∗ with the respective weak -closures of A and B in D∗∗ , we have that D∗∗ is generated as a von Neumann algebra by A∗∗ and B ∗∗ . Thus, if A and B are C ∗ -independent in D, normal states on A∗∗ and τ on B ∗∗ have a common extension to a normal state on B ∗∗ (i.e., A∗∗ and B ∗∗ are W ∗ -independent in D∗∗ ). Therefore, by [3, Proposition 3], [4, Theorem2.12], and [6, Theorem 2.5] we have the following. ∗
3.6 Lemma. The following statements are equivalent: (a) A and B are C ∗ -independent in D. (b) a b = a b for all a ∈ A∗∗ and b ∈ B ∗∗ . Given a C ∗ -algebra C we use zC to denote the central projection of C ∗∗ such that C ∗∗ zC is the atomic part of C ∗∗ , and we recall that multiplication by zC is faithful on C. The atomic states of C are those states for which (zC ) = 1. 3.7 Theorem. The following statements are equivalent: (a) and τ have unique common extension in S(D) for all (, τ ) ∈ P (A)×S(B). (b) A∗∗ zA and B ∗∗ commute and A and B faithfully independently commute. (c) A∗∗ zA and B ∗∗ commute and A and B are C ∗ -independent in D. (d) zA is a central projection in D∗∗ and there is a ∗-isomorphism A ⊗β B → D zA sending a ⊗ b → a b zA , for some C ∗ -norm β on A ⊗ B.
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Proof. (a)⇒ (b) Assume (a). If ∈ P (A) and τ ∈ S(B) let ϕ,τ denote their unique common extension in S(D). Let e be a minimal projection in A∗∗ . Let ∈ P (A) such that (e) = 1. The ∗ weak -compact set (actually a face, in this case) of extensions of in S(D) is given by E() = {ϕ,τ | τ ∈ S(B)} = {ϕ ∈ S(D) | ϕ(e) = 1} and is affinely homeomorphic to S(B) via the restriction map r : E() → S(B) and may be identified with the normal state space, Sn (e D∗∗ e), of e D∗∗ e. The induced surjective positive linear isometry Ab (S(B)) → Ab (Sn (e D∗∗ e))
(ˆb → ˆb ◦ r ,
∗∗ b ∈ Bsa )
in turn induces a surjective positive linear isometry and hence, Jordan isomor∗∗ ∗∗ phism, ψ : Bsa → (e D∗∗ e)sa , where ψ(b) = e b e for all b ∈ Bsa , the latter ∗∗ equalities, for b ∈ Bsa , following from the evaluations ˆb ◦ r (ϕ,τ ) = τ (b) = ϕ,τ (e b e) = (e b e)ˆ(ϕ,τ ) , for all τ ∈ S(B). Since ψ is a Jordan homomorphism, for each projection f in B ∗∗ , e f e is also a projection, giving e f = f e. Hence, e commutes with all elements of B ∗∗ . Therefore, since A∗∗ zA is the weak∗ closed linear span of the minimal projections of A∗∗ , A∗∗ zA and B ∗∗ commute. Using Lemma 3.6 in the second equality below we have (a zA )(b zA ) = (a zA ) b = a zA · b = a zA · b zA giving that A zA and B zA are C ∗ -independent, by further use of Lemma 3.6. Moreover, the previous identity yields that b = b zA and hence zA acts faithfully on B. In other words A and B faithfully independently commute. (b)⇒(c) This is clear. (c)⇒ (d) Assume (c). Then zA is a central projection of D∗∗ , and A∗∗ zA and B ∗∗ zA are commuting C ∗ -subalgebras of D zA . As above, it follows from Lemma 3.6 that A zA and B zA are C ∗ -independent in D zA and zA acts faithfully on both A and B. Roos’s theorem now gives (d). (d)⇒(a) Let ϕ be a common extension of ∈ P (A) and τ ∈ S(B). Then ϕ(zA ) = 1 and the induced state ϕ on D zA , given by ϕ(x zA ) = ϕ(x) pulls back to a state ψ on A ⊗β B restricting to on A and τ on B so that ψ = ⊗ τ . It follows that ϕ = ∧ τ. 3.8 Corollary. Let and τ have unique common extension in S(D) for all (, τ ) ∈ P (A) × S(B). Then A and B commute if any one of the following statements is true.
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(a) zA acts faithfully by multiplication on D. (b) Each pure state of D restricts to an atomic state of A. (c) All irreducible ∗-representations of D are finite-dimensional. Proof. (a) This is immediate from Theorem 3.7 (a)⇒ (d). (b) Given (b), for all ϕ ∈ P (D) we have ϕ(zA ) = 1 so that the support projection s(ϕ) ≤ zA and hence that zD ≤ zA . Therefore A and B commute by (a). (c) This follows from (b) since, in case of (c), if ϕ ∈ P (D) with restriction ∈ S(A) then π (A) is finite-dimensional because πϕ (D) is, implying that is atomic. We close with a final observation that, in the light of Theorem 3.7 and Corollary 3.8, improves Corollary 3.3 in the case when A is abelian. 3.9 Proposition. If A is abelian and D has the unique C ∗ -product property across A, B then and τ have unique common extension in S(D) for all (, τ ) ∈ P (A) × S(B). Proof. By Lemma 2.12 (a) if A is abelian then every extension in S(D) of a pure state of A is a product state across A, B. Acknowledgements. The second author would like to express his gratitude to the Alexander von Humboldt Foundation, Bonn, for supporting his research. He also acknowledges the support of the Grant Agency of the Czech Republic, Grant no. 201/03/0455, “Noncommutative Measure Theory” and the support of the Czech Technical University, Grant. No. MSM 210000010, “Applied Mathematics in Technical Sciences”.
References [1] H. Araki, H. Moriya, Joint extension of states of subsystems for CAR system, Commun. Math. Phys. 237, 105–122 (2003). [2] J. Dixmier, C ∗ -algebras. North Holland, (1977) [3] M. Florig, S.J. Summers, On the statistical independence of algebras of observables, J. Math. Phys. 38, 1318–1328 (1997). [4] S. Goldstein, A. Luczak, I. Wilde, Statistical independence of operator algebras, Found. Physics 29, 79–89 (1999). [5] R. Haag, D. Kastler, An algebraic approach to quantum field theory, J. Math. Phys. 5, 848–861 (1964). [6] J. Hamhalter, Statistical independence of operator algebras. Ann. Inst. H. Poincar´e 67, 447–462, (1997).
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[7] J. Hamhalter, C ∗ -independence and W ∗ -independence of von Neumann algebras, Math. Nachr. 239–240, 46–56 (2002). [8] J. Hamhalter, Quantum Measure Theory, Kluwer Academic Publishers, Dordrecht, Boston, London, (2003). Book Series: Fundamental Theories of Physics, Vol. 134. [9] K. Napiorkowski, On the independence of local algebras, Rep. Math. Phys. 3, 33–35 (1972). [10] P. Kruszynski, K. Napiorkowski, On the independence of local algebras II, Rep. Math. Phys. 4, 303–306 (1973). [11] H. Roos, Independence of local algebras in quantum field theory, Commun. Math. Phys. 16, 238–246 (1970). [12] S.J. Summers, On the independence of local algebras in quantum field theory, Rev. Math. Physics 2, No. 2, 201–247 (1990). [13] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, (1977). L.J. Bunce University of Reading Department of Mathematics RG6 2AX, Reading United Kingdom email:
[email protected] J. Hamhalter Czech Technical University Faculty of Electrical Engineering Department of Mathematics Technicka 2 CS-166 27 Prague 6 Czech Republic email:
[email protected] Communicated by Klaus Fredenhagen submitted 03/12/03, accepted 26/04/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 1097 – 1115 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/061097-19 DOI 10.1007/s00023-004-0192-6
Annales Henri Poincar´ e
C ∗-Algebras of Anisotropic Schr¨ odinger Operators on Trees Sylvain Gol´enia Abstract. We study a C ∗ -algebra generated by differential operators on a tree. We give a complete description of its quotient with respect to the compact operators. This allows us to compute the essential spectrum of self-adjoint operators affiliated to this algebra. The results cover Schr¨ odinger operators with highly anisotropic, possibly unbounded potentials.
1 Introduction Given a ν-fold tree Γ of origin e with its canonical metric d, we write x ∼ y when x and y are connected by an edge and we set |x| = d(x, e). For each x ∈ Γ \ {e}, we denote by x ≡ x(1) the unique element y ∼ x such that |y| = |x| − 1 and we set x(p) = (x(p−1) ) for 1 ≤ p ≤ |x|. Let xΓ = {y ∈ Γ | |y| ≥ |x| and y (|y|−|x|) = x}, where the convention x(0) = x has been used. On 2 (Γ) we define the bounded operator ∂ given by (∂f )(x) = y =x f (y). Its adjoint is given by (∂ ∗ f )(e) = 0 and (∂ ∗ f )(x) = f (x ) for |x| ≥ 1. Let D be the C ∗ -algebra generated by ∂. In order to obtain our algebra of potentials, we consider the “hyperbolic” = Γ ∪ ∂Γ of Γ constructed as follows. An element x of the compactification Γ boundary at infinity ∂Γ is a Γ-valued sequence x = (xn )n∈N such that |xn | = n is equipped and xn+1 ∼ xn for all n ∈ N. We set |x| = ∞ for x ∈ ∂Γ. The space Γ with a natural ultrametric space structure. For x ∈ ∂Γ and (yn )n∈N a sequence in Γ we have limn→∞ yn = x if for each m ∈ N there is N ∈ N such that for the set of complex-valued each n ≥ N we have yn ∈ xm Γ. We denote by C(Γ) we can view C(Γ) as a continuous functions defined on Γ. Since Γ is dense in Γ, ∗ C -subalgebra of Cb (Γ), the algebra of bounded complex-valued functions defined we denote by V (Q) the operator of multiplication by V in on Γ. For V ∈ C(Γ), 2 (Γ). the C ∗ -algebra generated by D and C(Γ). It conLet us now denote by C (Γ) 2 tains the set K(Γ) of compact operators on (Γ). Following the strategy exposed in [6], we shall first compute its quotient with respect to the ideal of compact operators. We stress that the crossed product technique introduced in [6] in order to compute quotients cannot be used in our case. Instead, we shall use the Theorem 4.5 in order to calculate the essential spectrum of self-adjoint operators related to In this introduction we consider only the most important case, when ν > 1. C (Γ).
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→ D ⊗ C(∂Γ) such Theorem 1.1 Let ν > 1. There is a unique morphism Φ : C (Γ) that Φ(D) = D ⊗ 1 for all D ∈ D and Φ(ϕ(Q)) = 1 ⊗ (ϕ|∂Γ ). This morphism is surjective and its kernel is K(Γ). The rest of this introduction is devotedto some applications of this theorem to spectral analysis. Let ν > 1 and H = α,β aα,β (Q)∂ ∗ α ∂ β + K, where K is and aα,β = 0 for all (α, β) ∈ N2 but a finite a compact operator, aα,β ∈ C(Γ) As a consequence of the Theorem 1.1, there number of pairs. Clearly H ∈ ∗Cα(Γ). is Φ such that Φ(H) = α,β ∂ ∂ β ⊗ (aα,β )|∂Γ , and, if H self-adjoint, its essential spectrum is: σess (H) = σ aα,β (γ)∂ ∗ α ∂ β . γ∈∂Γ
α,β
This result can be made quite explicit in the particular case of a Schr¨ odinger Since ∆ is a bounded operator operator H = ∆ + V (Q) with potential V in C(Γ). We then on 2 (Γ) defined by (∆f )(x) = y∼x (f (y) − f (x)), it belongs to C (Γ). set ∆0 = ∂ + ∂ ∗ − νId (which belongs to D) and notice that ∆ − ∆0 is compact. One then gets (see [1] for instance): √ √ σess (∂ + ∂ ∗ ) = σac (∂ + ∂ ∗ ) = σ(∂ + ∂ ∗ ) = [−2 ν, 2 ν ], where σac (T ) denotes the absolute continuous part of the spectrum of a given selfadjoint operator T . On the other hand, Theorem 1.1 gives us directly σess (∂ ∗ +∂) = σ(∂ ∗ + ∂). We thus get √ √ σess (∆ + V (Q)) = σ(∆0 ) + V (∂Γ) = [−ν − 2 ν, −ν + 2 ν ] + V (∂Γ). In fact this result holds (and is trivial) in the case of ν = 1, i.e., when Γ = N. Given a continuous function on ∂Γ, the Tietze theorem allows us to extend it so one may construct a large class of Hamiltonians to a continuous function on Γ, with given essential spectra. Nevertheless, we are able to point out a concrete class with uniform behavior at infinity which form a of non-trivial potentials V ∈ C(Γ) dense family of C(Γ). Namely, for each bounded function f : Γ → R and each real α > 1 let |x| f (xk ) V (x) = , (1.1) kα k=1
because of Proposition 2.3). for x ∈ Γ (V belongs to C(Γ) where xk = x Concerning finer spectral features, based mainly on the Mourre estimate, we mention that in the case H = ∆ + V (Q), with V as in (1.1) where α ≥ 3 and such that V (∂Γ) = 0, the results of [1] can be applied (the hypotheses of the Lemmas 6 and 7 from [1] are verified since V (x) = O(|x|−α+1 ) when |x| → ∞). The aim of our work in preparation [8] is to prove that the Mourre estimate holds for more general and to develop a scattering theory for classes of Hamiltonians affiliated to C (Γ) them. Theorem 1.1 remains the key technical point for these purposes. |x|−k
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The preceding results on trees allow us to treat more general graphs. We recall that a graph is said to be connected if two of its elements can be joined by a n sequence of neighbors. Let G = i=1 Γi ∪ G0 be a finite disjoint union of Γi , each Γi being a νi -fold branching tree with νi ≥ 1 and of G0 , a compact connected graph. We endow G with a connected graph structure that respects the graph structure of each Γi and the one of G0 , such that Γi is connected to Γj (i = j) only through G0 and such that Γi is connected to G0 only through ei , the origin of Γi . The graph G is hyperbolic and its boundary at infinity ∂G is the disjoint i ) for all union ∪ni=1 ∂Γi . We now choose V ∈ C(G ∪ ∂G). One has V |Γ i ∈ C(Γ i = 1, . . . , n and we easily obtain: σess (∆ + V (Q)) =
n
√ √ [−νi − 2 νi , −νi + 2 νi ] + V (∂Γi ) .
i=1
This covers in particular the case of the Cayley graph of a free group with finite system of generators. We recall that the Cayley graph of a group G with a system of generators S is the graph defined on the set G with the relation x ∼ y if xy −1 ∈ S or yx−1 ∈ S. Let G be a free group with a system of generators S such that S = S −1 . We denote by e its neutral element and we set |S| = ν + 1. One may associate the restriction of the Cayley graph to the set of words starting with a given generator with a ν-fold branching tree having as origin the generator. Hence, the Cayley graph of G will be ∪νi=1 Γi ∪ {e} where Γi is a ν-fold branching tree with the above graph structure. R) such that V (Γ) ⊂ R (here R = We now go further by taking V ∈ C(Γ, R) R ∪ {∞} is the Alexandrov compactification of R). More precisely, V ∈ C(Γ, if and only if for each γ ∈ ∂Γ we have either limx→γ V (x) = l where l ∈ R or for each M ≥ 0 there is N ∈ N such that |V (x)| ≥ M for all n ≥ N and x ∈ γn Γ (see Proposition 2.3). We set D(V ) = {f ∈ 2 (Γ) | V (Q)f 2 < ∞}. Let T ∈ D and T0 = Φ(T ). Since T is bounded, the operator H = T + V (Q) with (i.e., its resolvent belongs domain D(V ) is self-adjoint and it is affiliated to C (Γ) −1 to C (Γ)). Indeed, we have (V (Q) + z) ∈ C(Γ) for each z ∈ C \ R, and for large such z, (H + z)−1 = (V (Q) + z)−1 (T (V (Q) + z)−1 )n , n≥0
where the series is norm convergent. Now, with the same z, we use the Theorem 1.1 and the fact that D ⊗ C(∂Γ) C(∂Γ, D) to obtain Φγ (H + z)−1 ≡ Φ (H + z)−1 (γ) = (V (γ) + z)−1 (T0 (V (γ) + z)−1 )n . n≥0
Note that (V (γ) + z)−1 = 0 if V (γ) = ∞. By analytic continuation we get Φγ ((T + V (Q) + z)−1 ) = (T0 + V (γ) + z)−1 , for all z ∈ C \ R. We used the convention (T0 + V (γ) + z)−1 = 0 if V (γ) = ∞.
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We now compute the essential spectrum of H. If V (γ) = ∞ then σ(Φγ (H)) = ∅. Otherwise, one has σ(Φγ (H)) = σ(T0 + V (γ)) = σ(T0 ) + V (γ). Hence we obtain: σess (T + V (Q)) = σ(T0 ) + V (∂Γ0 ), where ∂Γ0 is the set of γ ∈ ∂Γ such that V (γ) ∈ R. Remark. We mention an interesting question which has not been studied in this paper. In fact, one could replace the algebra D by the (much bigger) C ∗ -algebra generated by all the right translations ρa (see Subsection 3.4 for notations) and This is a natural object, since it contains consider the corresponding algebra C (Γ). all the “right-differential” operators acting on the tree (not only polynomials in ∂ and ∂ ∗ ). A combination of the techniques that we use and that of [9, 10] could allow one to compute the quotient in this case too. We also note that in [9, 10] a certain connection with the notion of crossed-product is pointed out, and this could be useful in further investigations. I would like to thank the referee for bringing to my attention the two papers of A. Nica quoted above.
2 Trees and related objects 2.1
The free mono¨ıd Γ
Let A be a finite set consisting of ν objects. Let Γ be the free mono¨ıd over A ; its elements are words and those of A letters. We refer to [3, Chapter I, §7] for a detailed discussion of these notions, but we recall that a word x is an A -valued map defined on a set of the form1 1, n with n ∈ N, x(i) being the ith letter of the word x. The integer n (the number of letters of x) is the length of the word and will be denoted |x|. There is a unique word e of length 0, its domain being the empty set. This is the neutral element of Γ. We will also identify A with the set of words of length 1. The mono¨ıd Γ will be endowed with the discrete topology. If x ∈ Γ, we denote xΓ and Γx the right and left ideals generated by x. We have on Γ a canonical order relation which is by definition: x ≤ y ⇔ y ∈ xΓ. We recall some terminology from the theory of ordered sets. If Γ is an arbitrary ordered set and x, y ∈ Γ, then one says that y covers x if x < y and if x ≤ z ≤ y ⇒ z = x or z = y. If x ∈ Γ, we denote x = {y ∈ Γ | y covers x} In our case, y covers x if x ≤ y and |y| = |x| + 1. Notice that each element x ∈ Γ\{e} covers a unique element x , its father, and each element x ∈ Γ is covered by ν elements, its sons. The set of sons of x clearly is x = {xε | ε ∈ A }. Hence: y covers x ⇔ y = x ⇔ y ∈ x . 1 We
use the notation 1, n = [1, n] ∩ N where N is the set of integers ≥ 0 and N∗ = N \ {0}.
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For |x| ≥ n, we define x(n) inductively by setting x(0) = x and x(m+1) = (x(m) ) for m ≤ n − 1. One may also notice that: |x(α) | = |x| − α, if α ≤ |x|, and for α ≤ |ab|: (ab)(α) =
ab(α) , if α ≤ |b| a(α−|b|) , if α ≥ |b|.
We remark that if ν = 1 then Γ = N and if ν > 1 then Γ is the set of monoms of ν non-commutative variables.
2.2
The tree Γ and the extended tree associated to A
Recall that a graph is a couple G = (V, E), where V is a set (of vertices) and E is a set of pairs of elements of V (the edges). If x and y are joined by an edge, one says that they are neighbours and one abbreviates x ∼ y. The graph structure allows one to endow V with a canonical metric d, where d(x, y) is the length of the shortest path in G joining x to y. The graph GΓ associated to the free mono¨ıd Γ is defined as follows: V = Γ and x ∼ y if x covers y or y covers x. It is usual to identify Γ and GΓ , the so-called ν-fold branching tree. For all x ∈ Γ, we have |x| = d(e, x). We set B(x, r) = {y ∈ Γ | d(x, y) < r} and S n = {x ∈ Γ | |x| = n}. We shall now define an extended tree by mimicking the definition of a free mono¨ıd over A . We choose o ∈ A ; this element will be fixed from now on. For associated to A is each integer r, we set Zr = {i ∈ Z | i ≤ r}. The extended tree Γ the set of A -valued maps x defined on sets of the form Zr such that {i | x(i) = o} the unique r ∈ Z such that x is a map Zr → A will be denoted is finite. For x ∈ Γ, |x| and will be called length of x. We shall identify Γ with the set {x | |x| ≥ 0 and x(i) = o if i ≤ 0} as follows: defined on Z|x| by extending if x ∈ Γ then we associate to it the element of Γ x with x(i) = o if i ≤ 0. The element e will be identified with the map e ∈ Γ such that |e| = 0 and e(i) = o, ∀i ≤ 0. Notice that the two notions of length are consistent on Γ. by concatenation, i.e., for x ∈ Γ There is a natural right action of Γ on Γ and y ∈ Γ, xy will be the function z defined on Z|x|+|y| such that z(i) = x(i), with an order for i ∈ Z|x| and z(|x| + i) = y(i) for i ∈ 1, |y| . Then we equip Γ relation by setting: x ≤ y ⇔ y ∈ xΓ. As before, y covers x if and only if x ≤ y and |y| = |x| + 1. Now, each x ∈ Γ covers a unique x ∈ Γ and each x ∈ Γ is covered by ν elements, namely those of x = {xε | ε ∈ A }. We still have: y covers x ⇔ y = x ⇔ y ∈ x . Observe that x = x|Z|x|−1 . We will set x(α) = x|Z|x|−α for all α ∈ Z. As we did it for Γ, we shall This justifies the notion of extended tree used for Γ. identify the graph GΓ with Γ.
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The boundary at infinity of Γ
We shall see in the ending remark of this subsection that the boundary at infinity of Γ can be thought as the boundary of a 0-hyperbolic space in the sense of Gromov. We prefer, however, to give a simpler presentation that is closer to the theory of p-adic numbers (see [11] for instance). In fact, if ν is prime the boundary will be the set of ν-adic integers. Definition 2.1 The boundary at infinity of Γ is the set ∂Γ = {x : N∗ → A }. For x ∈ ∂Γ, we set |x| = ∞ . be Γ ∪ ∂Γ. For x ∈ Γ, we define the sequence (xn )n∈0,|x| with values in Γ by Let Γ setting x0 = e and xn = x|1,n for n ≥ 1. Observe that the map x → (xn )n∈0,|x| For x ∈ Γ and y ∈ Γ, xy will is injective. There is a natural left action of Γ on Γ. 2 be defined on the set 1, |x| + |y| by x(i) for i ≤ |x| and by y(i − |x|) for i > |x|. with a structure of ultrametric space. We define a kind We will now equip Γ of valuation v on Γ × Γ by max{n | xn = yn } if x = y v(x, y) = (2.1) ∞ if x = y. it is easy to see that: If x, y, z ∈ Γ v(x, y) ≥ min(v(x, z), v(z, y)). Let us set on Γ:
(2.2)
y) = exp(−v(x, y)). d(x,
is an ultrametric space, i.e., a metric d) The relation (2.2) clearly implies that (Γ, We will denote, for space such that d(x, y) ≤ max(d(x, z), d(z, y)), for x, y, z ∈ Γ. y) < r}. Notice that ultrametricity implies that r) = {y ∈ Γ | d(x, r > 0, B(x, r) is closed for all x ∈ Γ and r > 0. B(x, on Γ coincides with the initial topology of Γ, the The topology induced by Γ discrete one. For x ∈ ∂Γ and n ∈ N, xn Γ
| v(x, y) ≥ n} = B(x, exp(−n + 1)) = {y ∈ Γ
n∈N is a basis of Hence for each x ∈ ∂Γ, {xn Γ} which is the closure of xn Γ in Γ. neighborhoods of x in Γ. Observe that if x ∈ Γ then x∂Γ = xΓ ∩ ∂Γ. and ∂Γ are compact spaces. Γ is a compactification of Γ. Proposition 2.2 Γ ∗
Proof. ∂Γ = A N , thus the set ∂Γ endowed with the product topology is compact. This topology coincides with the one induced by the restriction of d on ∂Γ (for 2 We
use the convention 1, ∞ = N∗ ∪ {∞}.
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x ∈ ∂Γ, the product topology gives us the same basis of neighborhoods {xn ∂Γ}n∈N ∂Γ ). as d| is compact, it suffices to remark Since ∂Γ is compact, in order to show that Γ that ∪x∈∂Γ B(x, exp(−k)) = {y Γ | |y| = k + 1} has a finite complementary in Γ, for all k ∈ N. Since Γ is dense in Γ, Γ is a compactification of Γ. Notice also that if ν > 1, the topological space ∂Γ is perfect. of continuous complex-valued functions on Γ plays an The C ∗ -algebra C(Γ) ⊂ important rˆ ole. The dense embedding Γ ⊂ Γ gives a canonical inclusion C(Γ) Cb (Γ) (Cb (Γ) is the space of bounded complex-valued functions on Γ). Moreover, we have | f |∂Γ = 0}, C0 (Γ) = {f ∈ C(Γ) (2.3) where C0 (Γ) = {f : Γ → C | ∀ε > 0, ∃M > 0 | |x| > M ⇒ |f (x)| < ε}. We shall often abbreviate C0 (Γ) by C0 . The following proposition gives us a better understanding of the functions in C(Γ). Proposition 2.3 Let E be a metrisable topological space. A function V : Γ → E → E if and only if for each x ∈ ∂Γ the extends to a continuous function V : Γ limit of V (y), when y ∈ Γ converges to x, exists. Proof. Let x ∈ ∂Γ and V (x) be the above limit. Let F be a closed neighborhood is a neighborhood of x of V (x) in E; there is k such that V (xk Γ) ⊂ F . Then xk Γ ⊂ F. and, since F is closed, we have V (xk Γ) in Γ Later on, we will need the next ultrametricity result. We will say that U =
= {xi Γ} is a covering of ∂Γ. {xi Γ} is a covering of ∂Γ if U Proposition 2.4 For each open covering {Oi }i∈I of ∂Γ, there is a disjoint and finite ⊂ Oi . covering {xj Γ}j∈J of ∂Γ such that for each j ∈ J there is i ∈ I such that xj Γ Proof. For each x ∈ ∂Γ there is i such that x belongs to the open set Oi and ⊂ Oi . Since ∂Γ is compact, there is a finite there is n = n(x, i) such that xn Γ j∈1,m such that each of its elements is a sub-covering of ∂Γ made by sets {yj Γ} subset of some Oi . But in ultrametric spaces two balls are either disjoint or one are balls, we get the result. One of them is included in the other one. Since {yj Γ} may also choose {y Γ | |y| = maxj∈1,m |yj |} as the required covering. Remark. As we said previously, this section could be presented from the perspective of hyperbolicity in the sense of Gromov, see [2, Chapter V] (a deeper investigation can be found in [4] and [7]). Let (M, d) be a metric space. For x, y ∈ M and a given O ∈ M , we define the Gromov product as: (x, y)O =
1 (d(O, x) + d(O, y) − d(x, y)). 2
(2.4)
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The space (M, d) is called δ-hyperbolic if there is δ such that for all x, y, z, O ∈ M, (x, y)O ≥ min((x, z)O , (z, y)O ) − δ. (2.5) A metric space is hyperbolic if it is δ-hyperbolic for a certain δ. In fact, if there is δ such that (2.5) holds for all x, y, z ∈ M and a given O then (M, d) is 2δhyperbolic. Classical examples of 0-hyperbolic spaces are trees (connected graphs with no cycle) and real trees (see [7] for this notion). Cartan-Hadamard manifolds, the Poincar´e half-plane and, more generally, complete simply connected manifolds with sectional curvature bounded by κ < 0 are δ-hyperbolic spaces with δ > 0. We equip the set of sequences with values in M with an equivalence relation between (un ) and (vn ) defined by the condition lim(n,m)→∞ (un , vm )O = ∞. The boundary at infinity ∂M is the set of equivalence classes. A basis of open sets of ∂M is given by O = {γ ∈ ∂M | γ is not associated to any sequence of M \ O}, where O is an open set of M . The boundary of a 0-hyperbolic space is ultrametric. In our context, if we drop the convention v(x, x) = ∞, our valuation (2.1) is exactly (2.4). Hence (2.2) implies that Γ is 0-hyperbolic. We define a geodesic ray as being γ : N → Γ such that |γ(n)| = n and γ(n + 1) ∼ γ(n). Geodesic rays are representative elements of the above equivalence classes. The two notions of boundary at infinity are identified by setting xn = γ(n).
3 Operators in 2 (Γ) 3.1
Bounded and compact operators
We are interested in operators acting on the Hilbert space 2 (Γ) = {f : Γ → C | |f (x)|2 < ∞} x∈Γ
endowed with the inner product: f, g = x∈Γ f (x)g(x). We embed Γ ⊂ 2 (Γ) by identifying x with χ{x} , where χA is the characteristic function of the set A. Observe that Γ is the canonical orthonormal basis in 2 (Γ) and each f ∈ 2 (Γ) writes as f = x∈Γ f (x)x. We denote by B(Γ), K(Γ) the sets of bounded, respectively compact operators in 2 (Γ). For T ∈ B(Γ), we will denote by T ∗ its adjoint. Given A ⊂ Γ we denote by 1A the operator of multiplication by χA in 2 (Γ). The orthogonal projection associated to {x ∈ Γ | |x| ≥ r} is denoted by 1≥r . For T ∈ Γ, we have the following compacity criterion for bounded operators T in 2 (Γ): Proposition 3.1 T ∈ K(Γ) ⇐⇒ 1≥r T −→ 0 ⇐⇒ T 1≥r −→ 0. r→∞
r→∞
Proof. If one has for example 1≥r T → 0, then T is the norm limit of the sequence of finite rank operators 1B(e,r) T , hence is compact.
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The operator ∂
We now extend x → x to a map 2 (Γ) → 2 (Γ). We set e = 0 and define the derivative of any f ∈ 2 (Γ) as: (∂f )(x) ≡ f (x) = f (y)y (x) = f (y) = f (y). y =x
y∈Γ 2
2
Thus ∂ ∈ B(Γ). Indeed, f = x∈Γ |f (x)| ≤ ν The adjoint ∂ ∗ acts on each f ∈ 2 (Γ) as follows:
y∈ x
x∈Γ
2 y∈ x |f (y)|
≤ νf 2 .
∂ ∗ f (x) = χΓ\{e} (x)f (x ). Indeed, ∂f, f = x∈Γ y∈x f (y)f (x) = x∈Γ f (x)χΓ\{e} (x)f (x ) = f, ∂ ∗ f . Moreover, ∂ ∗ f 2 = x∈Γ\{e} |f (x )|2 = ν x∈Γ |f (x)|2 = νf 2 shows that ∂∂ ∗ = νId.
(3.1) √ √ Thus ∂ ∗ / ν is isometric on 2 (Γ) and ∂ = ∂ ∗ = ν. For α ∈ N we set f (α) = ∂ α f . Thus for each x ∈ Γ, x(α) is well defined in 2 (Γ) and x(α) = 0 ⇔ α > |x|. For |x| ≥ α the notation is consistent with our old definition.
3.3
C ∗ -algebras of energy observables related to Γ
We first summarize the method used in [6] to study the essential spectrum of large families of operators. Let H be a Hilbert space and H a bounded selfadjoint operator on H . If C(H ) = B(H )/K(H ) is the Calkin C ∗ -algebra, we denote by S → S the canonical surjection of B(H ) onto C(H ) and we recall that (this is a version of Weyl’s Theorem). If C is a C ∗ -subalgebra of σess (H) = σ(H) B(H ) which contains the compact operators, then one has a canonical embedding C/K(H ) ⊂ C(H ). Thus, in order to determine the essential spectrum of an operator H ∈ C it suffices to give a good description of the quotient C/K(H ) and as element of it. As explained in [6], we can actually go further by to compute H taking H as an unbounded operator over H such that (H + i)−1 ∈ C. We shall apply this strategy in our context. Let Dalg be the ∗-algebra of operators in 2 (Γ) generated by ∂ and D the C ∗ -algebra of operators in 2 (Γ) generated by ∂. Because of (3.1), Dalg is unital. We denote by ϕ(Q) the operator of multiplication by ϕ on 2 (Γ). If C is a C ∗ subalgebra of ∞ (Γ) then we embed C in B(Γ) by ϕ → ϕ(Q). Let D, C be the C ∗ -algebra generated by D ∪ C. In this paper we shall take C = D, C. This algebra contains many Hamiltonians of physical interest, for instance Schr¨ odinger operators with potentials in C. We recall that given a graph G the Laplace operator acts on 2 (G) as follows: (∆f )(x) = (f (y) − f (x)). y∼x
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With our definitions ∆ = ∂ + ∂ ∗ − νId + χ{e} . Notice that if ν > 1 then D does not contain compact operators (see below), so ∆ ∈ / D. On the other hand, if C ⊃ C0 and V ∈ C then the Schr¨ odinger operator ∆ + V (Q) clearly belongs to D, C. We now give a new description of K(Γ). Proposition 3.2 If C0 be the C ∗ -algebra generated by D · C0 then C0 = K(Γ). Proof. For each ϕ ∈ C0 , Proposition 3.1 shows ϕ(Q) ∈ K(Γ). Hence C0 ⊂ K(Γ). For the opposite inclusion, let T ∈ K(Γ) and fix ε > 0. Proposition 3.1, shows that there is an operator T with compactly supported kernel such that T − T ≤ ε. Define δx,y ∈ K(Γ) by (δx,y f )(z) = f (y) if z = x and 0 elsewhere. We have δx,x = χ{x} (Q) ∈ C0 . As T is a linear combination of δx,y , it suffices to show that δx,y is in C0 . But this follows from δx,y = δx,x (∂ ∗ )|x| ∂ |y| δy,y . If C is a C ∗ -subalgebra of ∞ (Γ) that contains C0 , then K(Γ) ⊂ D, C. Hence, in order to apply the technique described above, we have to give a sufficiently explicit description of the quotient D, C/K(Γ). In this paper we concen which is, geometrically speaking, the most interesting trate on the case C ≡ C(Γ) will be one (see the last Remark in §2.3). The C ∗ -algebra generated by ∂ and C(Γ) and the ∗-subalgebra generated by ∂ and C(Γ) will be denoted by denoted by C (Γ) C (Γ)alg . We will need the next fundamental property. ⊂ K(Γ). Proposition 3.3 [∂, C(Γ)] one has ([∂, ϕ(Q)]f )(x) = (ϕ(y) − ϕ(x))f (y) = Proof. For each ϕ ∈ C(Γ) y =x and is defined by ψ(y) = ϕ(y)−ϕ(y ) when (∂◦ψ(Q)f )(x), where ψ belongs to C(Γ) |y| ≥ 1 and ψ(e) = 0. Observe that for γ ∈ ∂Γ we have ψ(γ) = ϕ(γ) − ϕ(γ) = 0. Hence by (2.3), ψ ∈ C0 . Proposition 3.2 implies ψ(Q) ∈ K(Γ). Remark. The algebra D is the tree analogous of the algebra generated by the momentum operator on the real line. However, these algebras are rather different: D is not commutative and the spectrum and the essential spectrum of the operators from D are not connected sets in general. For instance, one has σ(∂ ∗ ∂) = σess (∂ ∗ ∂) = {0, ν} if ν > 1. Indeed, we remind that if A, B are elements of a Banach algebra we have σ(AB) ∪ {0} = σ(BA) ∪ {0} and, as noticed below, dim Ker ∂ is infinite for ν > 1.
3.4
Translations in 2 (Γ)
Γ acts on itself to the left and to the right: for each a ∈ Γ we may define λa , ρa : Γ → Γ by λa (x) = ax and ρa (x) = xa respectively. Clearly, for a, b ∈ Γ, λa ρb = ρb λa and for any x ∈ aΓ we define a−1 x as being the y for which x = ay. For each x ∈ Γa = {y ∈ Γ | ∃z ∈ Γ s.t. y = za}, we define y = xa−1 by x = ya. We extend now these translations to 2 (Γ). The translation λa acts on each f ∈ 2 (Γ)
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as x∈Γ f (x)ax, i.e., (λa f )(x) = χaΓ (x)f (a−1 x). In the same manner, we define (ρa f )(x) = χΓa (x)f (xa−1 ). The operators λa and ρa are isometries: λ∗a λa = Id and ρ∗a ρa = Id.
(3.2)
It is easy to check that the adjoins act on any f ∈ 2 (Γ) as (λ∗a f )(x) = f (ax) and (ρ∗a f )(x) = f (xa). Moreover, λa λ∗a = 1aΓ and ρa ρ∗a = 1Γa . Note also that ∂ ∗ = |a|=1 ρa and ∂ = |a|=1 ρ∗a .
3.5
(3.3)
Localizations at infinity
In order to study C (Γ)/K(Γ) we have to define the localizations at infinity of T ∈ C (Γ) by looking at the behavior of the translated operator λ∗a T λa as a (abbreviated a → γ), for each γ ∈ ∂Γ. converges to γ in Γ If T ∈ K(Γ) then u-lima→γ λ∗a T λa = 0, where u-lim means convergence in norm. Indeed, by (3.2), (3.3) and Proposition 3.1 we get λ∗a T λa = 1aΓ T 1aΓ → 0, alg . There as a → γ. Now, we compute the uniform limit of λ∗a T λa when T ∈ C (Γ) is P , a non-commutative complex polynomial in m + 2 variables, and functions for i = 1, m, such that T = P (ϕ1 , ϕ2 , . . . , ϕm , ∂, ∂ ∗ ). We set T (γ) = ϕi ∈ C(Γ) P (ϕ1 (γ), ϕ2 (γ), . . . , ϕm (γ), ∂, ∂ ∗ ). Lemma 3.4 There is a0 ∈ Γ such that u-lima→γ λ∗a T λa = λ∗a0 T (γ)λa0 . K ∈ K(Γ) and αk , βk ∈ Γ), Proof. The Proposition give some φk ∈ C( n 3.3 and∗(3.1) n αk βk N such that T = k=1 φk (Q)∂ ∂ + K and T (γ) = k=1 φk (γ)∂ ∗ αk ∂ βk . Thus, it suffices to compute a limit of the form u-lima→γ λ∗a ϕ(Q)∂ ∗ α ∂ β λa with ϕ ∈ C(Γ). 2 We suppose |a| ≥ α and take f ∈ (Γ). We first show the result for ϕ = 1. Since (λa f )(y) = f (y), (3.4) (λ∗a ∂ ∗ α ∂ β λa f )(x) = {y|y (β) =(ax)(α) }
{y|(ay)(β) =(ax)(α) }
it suffices to show that the set {y | (ay)(β) = (ax)(α) } is independent of a if |a| ≥ α. But this is precisely what asserts the Lemma 3.5 below. The identity We now treat the general case ϕ ∈ C(Γ). (λ∗a ϕ(Q)∂ ∗ α ∂ β λa f )(x) = ϕ(ax)(λ∗a ∂ ∗ α ∂ β λa f )(x) gives us that λ∗a ϕ(Q)∂ ∗ α ∂ β λa − ϕ(γ)λ∗a ∂ ∗ α ∂ β λa ≤ ϕ(aQ) − ϕ(γ) · ∂ ∗ α ∂ β → 0 as a → γ. On the other hand, by the Lemma 3.5, ϕ(γ)λ∗a ∂ ∗ α ∂ β λa is constant for |a| ≥ α. Thus, it suffices to choose |a0 | ≥ max{αk | k = 1, . . . , n} in the statement of the lemma to end the proof.
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Lemma 3.5 For |a| ≥ α we have: ∅ S |x|+β−α {y | (ay)(β) = (ax)(α) } = (α) β x S
Ann. Henri Poincar´e
for |x| + β − α < 0, for |x| < α and |x| + β − α ≥ 0, for |x| ≥ α and |x| + β − α ≥ 0.
(3.5)
Proof. Let Jx = {y | (ay)(β) = (ax)(α) }. Then aJx
=
{ay | (ay)(β) = (ax)(α) } = {y | y (β) = (ax)(α) } ∩ aΓ
=
((ax)(α) S β (Γ)) ∩ aΓ.
We first notice that (ax)(α) S β ⊂ S |a|+|x|−α+β . If |x| − α + β < 0 then ((ax)(α) S β ) ∩ aΓ = ∅, so aJx = ∅. This implies Jx = ∅. If |x| − α + β ≥ 0 then ((ax)(α) S β ) ∩ aΓ = ∅. If we suppose that |x| < α, i.e., |(ax)(α) | < |a|, we have a ∈ (ax)(α) Γ. Let b such that a = (ax)(α) b. Thus ((ax)(α) S β ) ∩ aΓ = =
((ax)(α) S β ) ∩ (ax)(α) bΓ = (ax)(α) (S β ∩ bΓ) (ax)(α) bS β−|b| = aS β−|b| = aS β+|x|−α,
so we have aJx = aS β+|x|−α, hence Jx = S β+|x|−α . Finally, if |x| ≥ α, i.e., |(ax)(α) | ≥ |a|, one has (ax)(α) ∈ aΓ. Thus we obtain aJx = (ax)(α) S β = ax(α) S β , hence Jx = x(α) S β . Remark. As seen in the proof of Lemma 3.4, one may choose any a0 such that |a0 | ≥ deg(P ). On the other hand, we stress that the limit is not a multiplicative function of T . Indeed, u- lim λ∗a ∂ ∗ ∂λa = (u- lim λ∗a ∂ ∗ λa ) · (u- lim λ∗a ∂λa ). a→γ
a→γ
a→γ
onto its quotient Therefore, in order to describe the morphism of the algebra C (Γ) C (Γ)/K(Γ) we have to improve our definition of the localizations at infinity.
3.6
Extensions to Γ
is defined similarly to 2 (Γ). Since Γ ⊂ Γ, we have 2 (Γ) → 2 (Γ). The space 2 (Γ) 2 χ As before, we embed Γ in (Γ) by sending x on {x} and we notice that Γ is an We define ∂ : 2 (Γ) → 2 (Γ) by orthonormal basis of 2 (Γ). )(x) = f (x) = (∂f
f (y).
y =x
For α ∈ N, we set f (α) = ∂α f , notation which is consistent with our old definition of x(α) as the restriction of x to Z|x|−α . Obviously ∂ ∈ B(Γ), its adjoint ∂∗ acts as √ (∂∗ f )(x) = f (x ), ∂∗ / ν is an isometry on 2 (Γ): ∂∂∗ = νId,
(3.6)
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= ∂∗ = ν. We denote by D the C ∗ -algebra generated by ∂ and by Dalg thus ∂ Both of them are unital. the ∗-algebra generated by ∂. alg . We now make the connection between Dalg and D α Lemma 3.6 For |a| ≥ α, one has: λ∗a ∂ ∗ α ∂ β λa = 1Γ ∂∗ ∂β 1Γ .
one has Proof. For any f ∈ 2 (Γ),
α (1Γ ∂∗ ∂β 1Γ f )(x) = 1Γ (x)
1Γ (y)f (y).
{y|y (β) =x(α) }
Using the same arguments as in the proof of the Lemma 3.5, one shows that for each x ∈ Γ the set {y ∈ Γ | y (β) = x(α) } equals the r.h.s. of (3.5). Thus the above sum is the same as that of the r.h.s. of (3.4). alg . We will also need a result concerning the localization of the norm on D Lemma 3.7 If T ∈ Dalg , then T = 1Γ T1Γ . n αk Proof. Because of (3.6), we can suppose that T = k=1 ck ∂∗ ∂βk . We denote by with β the integer max{βk | k ∈ 1, n}. For each ε > 0, there is some g ∈ 2 (Γ) compact support such that g = 1 and T g ≥ T −ε. Note that if y1 , y2 , . . . , ym we are distinct points of Γ, a1 , a2 , . . . , am are complex numbers and x1 , x2 ∈ Γ, have m m m ai x1 yi 2 = |ai |2 = ai x2 yi 2 . (3.7) i=1
i=1
i=1
∗ Thus, since g has compact support, there mare x ∈ Γ, m ∈ N and ymi ∈ Γ, |yi | ≥ β, ai ∈ C, for all i ∈ 1, m such that g = k=1 ai xyi . We set f = k=1 ai eyi . Then (3.7) gives us f = g = 1. Using |yi | ≥ β, we get f ∈ 2 (Γ) and Tf ∈ 2 (Γ). Also with (3.7) we obtain for z ∈ Γ,
Tg =
m n
αk
ck ai ∂∗
∂βk xyi =
k=1 i=1
=
n
m
=
m
k=1 i=1 |z|=αk
ck ai (xyi )(βk ) z
k=1 i=1 |z|=αk
ck ai x(yi )(βk ) z =
k=1 i=1 |z|=αk n
m n
m n
ck ai e(yi )(βk ) z
k=1 i=1 |z|=αk
ck ai (eyi )(βk ) z =
n m
αk
ck ai ∂∗
∂βk eyi = Tf .
k=1 i=1
such that 1Γ T1Γ f = Tf = Tg ≥ T − ε. Hence, there is f ∈ 2 (Γ)
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4 The main results 4.1
The morphism
In the sequel, a morphism will be understood as a morphism of C ∗ -algebras. To describe the quotient C (Γ)/K(Γ), we need to find an adapted morphism. such →D Theorem 4.1 For each γ ∈ ∂Γ there is a unique morphism Φγ : C (Γ) One has K(Γ) ⊂ Ker Φγ . that Φγ (∂) = ∂ and Φγ (ϕ(Q)) = ϕ(γ), for all ϕ ∈ C(Γ). alg then by Lemma 3.4 we have Proof. We use the notations from §3.5. If T ∈ C (Γ) ∂∗ ). By u-lima→γ λ∗a T λa = λ∗a0 T (γ)λa0 . Let T(γ) be P (ϕ1 (γ), ϕ2 (γ), . . . , ϕm (γ), ∂, ∗ Lemma 3.6 and (3.6) one can choose a0 such that λa0 T (γ)λa0 = 1Γ T (γ)1Γ . Lemma 3.7 implies T(γ) = 1Γ T(γ)1Γ = λ∗a0 T (γ)λa0 = u- lim λ∗a T λa ≤ T . a→γ
Φ0 (T ) = T (γ). alg → D, Thus there is a linear multiplicative contraction Φ0γ : C (Γ) γ alg in C (Γ) allows us to extend Φ0 to a morphism Φγ : C (Γ) → The density of C (Γ) γ which clearly satisfies the conditions of the theorem. The uniqueness of Φγ is D obvious and the last assertion of the theorem follows from Proposition 3.2.
4.2
The case ν > 1
In this case, we are able to improve Theorem 4.1. We recall first that an isometry is said to be proper if it is not unitary. The operators ∂ ∗ and ∂∗ are proper isometries and the dimensions of the kernels of ∂ and ∂ are infinite: in the case of ∂, if one lets a, b be two different letters of A , and one chooses g ∈ 2 (Γa) and h ∈ 2 (Γb) such that h(xb) = g(xa) for all x ∈ Γ, then g − h is in Ker ∂. Let T be the unit circle of R2 and H 2 the closure of the subspace spanned by {einQ , n ∈ N} in 2 (T). For g ∈ L∞ (T), we define the Toeplitz operator Tg on H 2 by Tg h = PH 2 gh, where PH 2 is the projection on H 2 . We denote by T the C ∗ -algebra generated by Tu , where we u is the map u(z) = z. The next theorem is due to Coburn (see [5] for a proof). Theorem 4.2 If S is a proper isometry, then there is a unique isomorphism J of T onto S , the C ∗ -algebra generated by S, such that J (Tu ) = S. so Thus there is a unique isomorphism J of D onto D such that J (∂) = J (∂), in the case ν > 1 we can rewrite our Theorem 4.1 as follows. → D such that Theorem 4.3 Let γ ∈ ∂Γ. There is a unique morphism Φγ : C (Γ) Φγ (ϕ(Q)) = ϕ(γ) for all ϕ ∈ C(Γ) and Φγ (D) = D for all D ∈ D.
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such that J (∂) = ∂ Remark. When ν = 1, there is no isomorphism J : D → D because D is commutative. Thus, in this case, one cannot hope in a result as above. There is an other way of proving Theorem 4.3 which uses the next proposition. Proposition 4.4 If ν ≥ 1 then {∂ ∗ α ∂ β }{α,β∈N} is a basis of the vector space Dalg . α alg . One has ν > 1 if and only if {∂∗ ∂β }{α,β∈N} is a basis of space D n ∗ αi βi Proof. Let λi = 0 for all i ∈ 1, n. Assume that ∂ = 0, where i=1 λi ∂ (αi , βi ) are distinct couples. We set α = min{αi | i ∈1, n} and I = { i | αi = α}. We take x ∈ Γ such that |x| = α and we obtain i∈I λi (∂ βi f )(e) = 0. Notice that {βi }i∈I are pairwise distinct by hypothesis. Now, by taking i0 ∈ I and f the characteristic function of Sβi0 , we get that λi0 = 0 which is a contradiction. n Hence i=1 λi ∂ ∗ αi ∂ βi = 0, i.e., the family is free. Let now ν > 1 and λi = 0 n αi for all i ∈ 1, n. We suppose i=1 λi ∂∗ ∂βi = 0, with (αi , βi ) pairwise distinct. n αi and set α We fix x ∈ Γ ¯ = max{αi , i ∈ 1, n}. One has ( i=1 λi ∂∗ ∂βi f )(x) = n (α) β S ∩ x(α ) S β = ∅ if and only if y∈x(αi ) S βi f (y) = 0. Notice that x i=1 λi α − α = β − β. Taking f ∈ 2 (S |x|−α1 +β1 ), we see that one can reduce oneself o the case when there is some k such that αi − βi = k for all i ∈ 1, n. Since ¯ ¯ ¯ ¯ ¯ ¯ x(α−l) S α−k−l ⊂ x(α−1) S α−k−1 x(α) S α−k for all l ∈ 1, (¯ α − k), there is some (α) ¯ α−k ¯ (αi ) βi y0 ∈ x S \ ∪αi =α¯ x S . Then, taking f = χ{y0 } we get some i0 such that n αi βi λi0 = 0, which is a contradiction. Hence i=1 λi ∂∗ ∂∗ = 0. Finally, since when α ν = 1 one has ∂∂∗ = ∂∗ ∂ = Id, {∂∗ ∂β }α,β∈N is obviously not a basis.
4.3
Description of C (Γ)/K(Γ)
⊗C(∂Γ) →D Theorem 4.5 i) For any ν ≥ 1, there is a unique morphism Φ : C (Γ) such that Φ(∂) = ∂ ⊗ 1 and Φ(ϕ(Q)) = 1 ⊗ (ϕ|∂Γ ). This morphism is surjective and its kernel is K(Γ). → D ⊗ C(∂Γ) such ii) For ν > 1, there is a unique surjective morphism Φ : C (Γ) that Φ(∂) = ∂ ⊗ 1, Φ(ϕ(Q)) = 1 ⊗ (ϕ|∂Γ ) and Ker Φ = K(Γ). Once again, as in Remark 4.2, the statement (ii) of the theorem is false if ν = 1. As a corollary of Theorem 4.5 we obtain the following result. Proposition 4.6 If ν > 1 then D ∩ K(Γ) = {0} and if ν = 1 one has K(Γ) ⊂ D. Proof. Let ν > 1 and T ∈ D ∩ K(Γ). Theorem 4.5 gives us both Φ(T ) = T ⊗ 1 and Φ(T ) = 0 (since T is compact). For ν = 1, as in the proof of Proposition 3.2, it suffices to prove that δx,x is in D. But this is clear since δx,x = ∂ ∗ |x+1| ∂ |x+1| − ∂ ∗ |x| ∂ |x| . We devote the rest of the section to the proof of Theorem 4.5. ∂Γ such that (Φ(∂))(γ) = →D Proof. By Theorem 4.1 there is a morphism Φ : C (Γ) Since the images of ∂ and (Φ(ϕ(Q)))(γ) = ϕ(γ), for all γ ∈ ∂Γ, ϕ ∈ C(Γ).
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and since C (Γ) ∂ and ϕ(Q) through Φ belong to the C ∗ -subalgebra C(∂Γ, D), is generated by ∂ and such ϕ(Q), it follows that the range of Φ is included in We have C(∂Γ, D ) ∼ ⊗ C(∂Γ), so we get the required morphism C(∂Γ, D). = D Φ : C (Γ) → D ⊗ C(∂Γ). Now since Φ(∂) = ∂ ⊗ 1 and Φ(ϕ(Q)) = 1 ⊗ (ϕ|∂Γ ), and since any function in C(∂Γ) is the restriction of some function from C(Γ), it follows that Φ is surjective. Its uniqueness is clear. It remains to compute the kernel. As seen in Theorem 4.1, K(Γ) ⊂ Ker Φ. In the remainder of this section we shall prove the reverse inclusion. For this we need some preliminary lemmas. Lemma 4.7 Let R = ϕ(Q)∂ ∗ α ∂ β and U = {ai Γ}i∈1,n be a disjoint covering of ∂Γ. For each ε > 0 there are c1 , c2 , . . . , cm ∈ Ran(ϕ) and there is a disjoint covering U = {bj Γ}j∈1,m of ∂Γ finer than U such that 1U R − R ≤ ε, where ∗α β ∂ and U = ∪m R = m j=1 bj Γ. j=1 1bj Γ cj ∂ Proof. Let ε > 0 and denote ε/∂ ∗α ∂ β by ε . Since ϕ(∂Γ) is compact, there are γ1 , γ2 , . . . , γN ⊂ ∂Γ such that ϕ(∂Γ) ⊂ ∪N k=1 D(ϕ(γk ), ε ), where D(z, r) is the com ∩ ϕ−1 (D(ϕ(γk ), ε )) plex open disk of center z and ray r. The open sets Oi,k = ai Γ cover ∂Γ. Proposition 2.4 gives us a disjoint covering {bj Γ}j∈1,m of ∂Γ such that ⊂ Oi,k . To simplify the notafor each j ∈ 1, m there are i and k such that bj Γ tions, we will denote by γj those γk associated to bj Γ. We set U = {bj Γ}j∈1,m n and R = j=1 1bj Γ ϕ(γj )∂ ∗ α ∂ β . Recall that supx∈bj Γ |ϕ(γj ) − ϕ(x)| ≤ ε , so (R − 1U R)f 2
=
=
m | 1bj Γ (x)(ϕ(γj ) − ϕ(x))(∂ ∗ α ∂ β f )(x)|2 x∈Γ j=1 m
|(ϕ(γj ) − ϕ(x))(∂ ∗ α ∂ β f )(x)|2
j=1 x∈bj Γ
≤
m
sup |ϕ(γj ) − ϕ(x)|2
j=1 x∈bj Γ
m 2
|(∂ ∗ α ∂ β f )(x)|2
x∈bj Γ
|(∂ ∗ α ∂ β f )(x)|2
≤
ε
≤
ε ∂ ∗ α ∂ β −2 · ∂ ∗ α ∂ β 2 · f 2 = ε2 f 2 .
j=1 x∈bj Γ 2
Denoting ϕ(γj ) by cj we obtain the result. n and let ε > 0. There Lemma 4.8 Let T = k=1 ϕk (Q)∂ ∗ αk ∂ βk with ϕk ∈ C(Γ) are a compact operator K, a disjoint covering {aj Γ}j∈1,m of ∂Γ and S=
m n
1aj Γ ϕk (γj,k )∂ ∗ αk ∂ βk ,
k=1 j=1
with minj∈1,m |aj | ≥ maxk∈1,n αk and γj,k ∈ ∂Γ such that T − S − K ≤ ε.
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Proof. We denote by α = max{αk | k ∈ 1, n}. Let Tk be ϕk (Q)∂ ∗ αk ∂ βk . Setting U0 = ∪{a||a|=α} {aΓ}, we apply Lemma 4.7 inductively for k ∈ 1, n with ε/n instead of ε, U = Uk−1 and R = Tk , denoting U by Uk and R by Sk . Then, for k ∈ 1, n we get 1Uk Tk −Sk ≤ ε/k. Since Uk+1 is finer than U k for k ∈ 1, n−1, n n we obtain 1Un k=1 (Tk − Sk ) ≤ ε, hence T − 1Unc T − 1Un k=1 k ≤ ε. To S n finish the proof, we denote the compact operator 1Unc T by K, 1Un k=1 Sk by S and Un by {aj Γ}j∈1,m . We now go back to the proof of Theorem 4.5. Let T ∈ Ker Φ. For each ε > 0 alg such that T − T ≤ ε/4. By relation (3.1) and Proposition there is T ∈ C (Γ) n 3.3, we can write T = k=1 ϕk (Q)∂ ∗ αk ∂ βk + K, where K ∈ K(Γ) and ϕk ∈ C(Γ). Thus Φ(T ) ≤ ε/4. Using Lemma 4.8, we get an operator S and a compact operator K1 such that T − S − K1 ≤ ε/4. This implies that Φ(S) ≤ ε/2. Lemma 4.9 There is K2 ∈ K(Γ) such that S − K2 ≤ Φ(S). Before proving the lemma, let us remark that it implies T − K1 − K2
≤ T − T + T − S − K1 + S − K2 ≤ ε.
Hence T ∈ K(Γ). Thus Theorem 4.5 is proved.
Proof of Lemma 4.9. First, we remark that for each a ∈ Γ and α, β ≥ 0, Proposition ∗α β − 1aΓ ∂ ∗ α ∂ β 1aΓ is a compact operator. We define S = 3.3 ngives mus that 1aΓ ∂ ∂ ∗ α k βk ∂ 1aj Γ and we set K2 = S − S , which is a compact k=1 j=1 1aj Γ ϕk (γj,k )∂ operator. Since {aj Γ}j∈1,m is a disjoint covering of ∂Γ, for any f ∈ 2 (Γ): S f 2
=
=
m n | (1aj Γ ϕk (γj,k )∂ ∗ αk ∂ βk 1aj Γ f )(x)|2 x∈Γ k=1 j=1 m n
|
(1aj Γ ϕk (γj,k )∂ ∗ αk ∂ βk 1aj Γ f )(x)|2
j=1 x∈Γ k=1
≤
m n 1aj Γ ϕk (γj,k )∂ ∗ αk ∂ βk 1aj Γ 2 · 1aj Γ f 2 . j=1
k=1
Now we use (3.2) and (3.3) and get: 1aj Γ
n
n ϕk (γj,k )∂ ∗ αk ∂ βk 1aj Γ = λ∗aj ϕk (γj,k )∂ ∗ αk ∂ βk λaj .
k=1
k=1
Since |aj | ≥ max{αk | k ∈ 1, n}, Lemmas 3.6 and 3.7 give us: λ∗aj
n
ϕk (γj,k )∂ ∗ αk ∂ βk λaj
= 1Γ
n
k=1
αk
ϕk (γj,k )∂∗
∂βk 1Γ
k=1
=
n k=1
αk ϕk (γj,k )∂∗ ∂βk .
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For each j we choose γj ∈ aj ∂Γ. The family {aj Γ}j∈1,m is a disjoint covering of ∂Γ, so we have limx→γj χaj Γ (x) = 1 and limx→γj χai Γ (x) = 0 for i = j. Hence n αk Φγj (S ) = k=1 ϕk (γj,k )∂∗ ∂βk . We obtain S f 2
≤
m j=1
Φγj (S )2 · 1aj Γ f 2 ≤ sup Φγ (S )2 · f 2 . γ∈∂Γ
Finally, since K(Γ) ⊂ Ker Φ, Φ(S) = Φ(S ) = supγ∈∂Γ Φγ (S ).
Acknowledgment. I take this opportunity to express my gratitude to Vladimir Georgescu for suggesting me the subject of this work and for helpful discussions. I am also indebted to Andrei Iftimovici for comments and suggestions.
References [1]
C. Allard, R. Froese, A Mourre estimate for a Schr¨ odinger operator on a binary tree, Rev. Math. Phys 12, No.12, 1655–1667 (2000).
[2]
A. Ancona, Th´eorie du potentiel sur les graphes et les vari´et´es, Ecole d’´et´e de Probabilit´es de Saint-Flour XVIII – 1988, Lecture Notes in Mathematics 1427, Springer-Verlag, Berlin, 1990, pp. 1–112.
[3]
N. Bourbaki, El´ements de math´ematiques, Alg`ebre, chapitres 1 a ` 3, Diffusion C.C.L.S. Paris, 1970.
[4]
M. Coornaert, T. Delzant, A. Papadopoulos, G´eom´etrie et th´eorie des groupes: les groupes hyperboliques de Gromov, Lecture Notes in Mathematics 1441, Springer-Verlag, Berlin, 1990.
[5]
K.R. Davidson, C ∗ -algebra by examples, American Mathematical Society (1996).
[6]
V. Georgescu, A. Iftimovici, Crossed products of C ∗ -algebras and spectral analysis of quantum Hamiltonians, Commun. Math. Phys. 228, No. 3, 519– 560 (2002).
[7]
E. Ghys, P. De la Harpe, Sur les groupes hyperboliques d’apr`es Mikhael Gromov. Birkh¨ auser, 1990.
[8]
S. Gol´enia, Mourre estimates for anisotropic operators on trees, (in preparation).
[9]
A. Nica, C ∗ -algebras generated by isometries and Wiener-Hopf operators, J. Operator Theory 27, No. 1, 17–52 (1992).
[10] A. Nica, On a groupoid construction for actions of certain inverse semigroups, Internat. J. Math. 5, No. 3, 349–372 (1994).
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[11] A.M. Robert, A Course in p-adic Analysis. Graduate Texts in Mathematics 198, Springer-Verlag, New York, 2000. Sylvain Gol´enia D´epartement de Math´ematiques Universit´e de Cergy-Pontoise 2, avenue Adolphe Chauvin F-95302 Cergy-Pontoise Cedex France email:
[email protected] Communicated by Jean Bellissard submitted 18/12/03, accepted 29/04/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 1117 – 1135 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/061117-19 DOI 10.1007/s00023-004-0193-5
Annales Henri Poincar´ e
Soliton-Type Asymptotics for the Coupled Maxwell-Lorentz Equations Valery Imaikin∗, Alexander Komech† and Norbert Mauser
Abstract. We consider a Maxwell field translation invariantly coupled to a single charge. This Hamiltonian system admits soliton-type solutions, where the charge and the co-moving field travel with constant velocity. We prove that a solution of finite energy converges, in suitable local energy seminorms, to a certain soliton-type solution in the long time limit t → ±∞.
1 Introduction We prove soliton-type asymptotics for the Maxwell-Lorentz system of the Maxwell field coupled to a relativistic charge: each finite energy solution converges to a soliton in a long-time limit. This is a generalization of the result [9] where a similar asymptotics is proved for a scalar field. This also strengthens the result [1], where an orbital stability of solitons is proved for the Maxwell-Lorentz system with a non-relativistic charge. The generalizations have required a considerable development of methods [1, 9]. In particular, we exploit strong Huygen’s principle for the Maxwell-Lorentz equations and develop the Hamiltonian approach for canonical transformations of the equations. We consider a single relativistic charge coupled to the Maxwell field. If q(t) ∈ R3 denotes the position of the charge at a time t, then the coupled Maxwell-Lorentz equations read ˙ ∇ · E(x, t) = ρ(x − q(t)), E(x, t) = ∇ ∧ B(x, t) − ρ(x − q(t))q(t), ˙ ∇ · B(x, t) = 0, p(t) , q(t) ˙ = 1 + p2 (t)
˙ B(x, t) = −∇ ∧ E(x, t), p(t) ˙ =
(1.1)
[E(x, t) + q(t) ˙ ∧ B(x, t)]ρ(x − q(t)) d3 x.
Here and below all derivatives are understood in the sense of distributions. The last line is the Lorentz force equation and the first two lines are the inhomogeneous ∗ Supported
partly by Austrian Science Foundation (FWF) Project (P16105-N05). leave Department Mechanics and Mathematics of the Moscow State University, supported partly by Max-Planck Institute of Mathematics in the Sciences, Leipzig and by Austrian Science Foundation START Project (Y-137-TEC). † On
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Maxwell equations. E(x, t), B(x, t) is the Maxwell field, ρ is the charge distribution of the particle, on which we will comment below. We use units such that the velocity of light c = 1, ε0 = 1, and the mechanical mass of the charge m = 1. We consider all finite energy solutions to the equations (1.1). An appropriate phase space will be introduced below, but first we note that the energy integral 1 |E(x)|2 + |B(x)|2 d3 x (1.2) H(E, B, q, p) = (1 + p2 )1/2 + 2 and the total momentum integral P(E, B, q, p) = p +
E(x) ∧ B(x) d3 x
(1.3)
are conserved along sufficiently smooth solution trajectories of (1.1). It is then natural to choose as the phase space the set of all finite energy states. In fact, (1.1) can be put into Hamiltonian form. In the canonical coordinates the energy H is then the Hamiltonian of the system. The charge distribution ρ is real-valued, sufficiently smooth, radially symmetric, and of compact support, ρ, ∇ρ ∈ L2 (R3 ),
ρ(x) = ρr (|x|),
ρ(x) = 0 for |x| ≥ Rρ > 0.
(C)
As noted in [6, 8, 9] an additional important assumption is the Wiener condition (W ) ρˆ(k) = eikx ρ(x)d3 x = 0 for k ∈ R3 . It ensures that all modes of the Maxwell field couple to the charge. In particular the total charge ρ = ρˆ(0) = 0. Charge distributions satisfying both (W ) and (C) are constructed in [6, Section 10]. We will investigate the long-time behavior of all finite energy solutions to (1.1). A set of asymptotic solutions corresponds to the charge travelling with a uniform velocity, v. Up to translation they are of the form Sv (t) = (Ev (x − vt), Bv (x − vt), vt, pv )
(1.4)
with an arbitrary velocity v ∈ V := {v ∈ R3 : |v| < 1}, where ρ(y)d3 y Ev (x) = −∇φv (x) + v · ∇ Av (x), φv (x) = , 4π|v(y − x) + λ(y − x)⊥ | v Bv (x) = ∇ ∧ Av (x), Av (x) = vφv (x), pv = √ . (1.5) 1 − v2 √ Here λ = 1 − v 2 and we set x = vx + x⊥ , where x ∈ R and v⊥x⊥ ∈ R3 for x ∈ R3 . Below we call the solutions of type (1.4), (1.5) “solitons”. Let us note that in [1] solitons are studied for a non-relativistic charge. Then they exist only for
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a finite range of energies. In our case of relativistic charge, we consider all finite energy solutions, an energy of a soliton is arbitrary. Let us discuss and summarize now our main results, the precise theorems to be stated in the following section. One of the important issues of our paper is the relaxation of acceleration q¨(t) → 0 as t → ±∞ established in [8]. Our main contribution is that in this case the velocity has the limits ˙ = v± lim q(t)
t→±∞
(1.6)
together with co-moving travelling fields of type (1.4). Namely, the fields are asymptotically Coulomb travelling waves in the sense (E(q(t) + x, t), B(q(t) + x, t)) − (Ev± (x), Bv± (x)) → 0 as t → ±∞.
(1.7)
Since the energy is conserved, the convergence here is in the sense of suitable local norms. Soliton-like asymptotics of type (1.7) are proved in [12] for some translation invariant 1D completely integrable equations and in [3] for translation invariant 1D nonlinear reaction systems. Soliton-like asymptotics are also proved for small perturbations of soliton-like solutions to 1D nonlinear translation invariant Schr¨ odinger equations [2]. For a scalar field such kind of asymptotics was studied in [9]. For the Maxwell-Lorentz system (1.1) this asymptotics is proved in [7] under the condition ρ L2 1. Under the Wiener condition (W) without the smallness condition, this asymptotics is proved for the first time in the present paper. Let us note that a system of type (1.1), with a non-relativistic charge, has been considered by Bambusi and Galgani [1], where an orbital stability of the solitons is proved. We extend the orbital stability to the relativistic charge, and furthermore, prove a global attraction of all solutions to the soliton manifold, which means, in particular, its asymptotic stability. Similar global attraction is proved in [9] for a scalar field instead of the Maxwell Field. The extension to the MaxwellLorentz system (1.1) is not straightforward since it requires a detailed analysis of corresponding Hamiltonian structure and an extension of the strong Huygen’s principle. Let us give a general idea of our strategy. We transfer to Hamiltonian variables. In the case of non-relativistic charge the Hamiltonian structure is used in [1]. In the present paper we use essentially the Hamiltonian structure for the case of relativistic charge (for a scalar field this is done in [9]). Since the total momentum P, see (1.3), is conserved, we obtain the new reduced Hamiltonian depending on P as on a parameter and with the cyclic variable conjugate to P. We make this by a transfer to a moving frame as in [1, 9]. Here we prove for the first time that this transfer corresponds to a canonical transformation of the Maxwell-Lorentz equations. We will prove that the soliton with the same total momentum P is a critical point and the global minimum of the reduced Hamiltonian. Thus, initial data close to the soliton must remain close forever by conservation of energy,
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which translates into the orbital stability of soliton-type solutions. Note that for a general class of nonlinear wave equations with symmetries such orbital stability of soliton-like solutions is proved in [5]. Our argument here combine the Lyapunov function method of [1] and [9]. Orbital stability by itself is not enough. It only ensures that initial data close to a soliton remain so and does not yield the convergence of q(t), ˙ even less the convergence (1.7). Thus we need an additional, not quite obvious argument which combines the limit q¨(t) → 0 as t → ∞, from [8], with the orbital stability in order to establish the long time asymptotics. As an essential input we exploit retarded field part of the solution, apply the strong Huygen’s principle for the MaxwellLorentz equations, and estimate oscillations of Maxwell-Lorentz Hamiltonian along solutions to the perturbed Maxwell-Lorentz system, cf. Section 4.
2 Main results We first define a suitable phase space. A point in phase space is referred to as state. Let L2 denote the real Hilbert space L2 (R3 , R3 ) with the norm | · | . We introduce the Hilbert spaces F = L2 ⊕ L2 and L = F ⊕ R3 ⊕ R3 with finite norms
(E(x), B(x)) F = | E|| + | B|| and Y L = | E|| + | B|| + |q| + |p| for Y = (E(x), B(x), q, p) ∈ L.
(2.1)
L is the space of finite energy states. The energy functional H is continuous on the space L. On F and L we define the local energy seminorms by
(E(x), B(x) R = | E||R + | B||R and |||Y |||R = | E||R + | B||R + |q| + |p| for Y = (E(x), B(x), q, p) 2
(2.2)
3
for every R > 0, where | ·||R is the norm in L (BR ), BR the ball {x ∈ R : |x| < R}. Let us denote by FF , LF the spaces F , L equipped with the Fr´echet topology induced by these seminorms. Note that the spaces L and LF , FF are metrizable, but LF , FF are not complete. The system (1.1) is overdetermined. Therefore its actual phase space is a nonlinear sub-manifold of the linear space L. Definition 2.1 i) The phase space M for Maxwell-Lorentz equations (1.1) is the metric space of states (E(x), B(x), q, p) ∈ L satisfying the constraints, ∇ · E(x) = ρ(x − q) and ∇ · B(x) = 0 for x ∈ R3 .
(2.3)
The metric on M is induced through the embedding M ⊂ L. ii) Mσ for 0 ≤ σ ≤ 1 is the set of the states (E(x), B(x), q, p) ∈ M such that 0 0 ∇E(x),∇B(x) are L∞ loc outside the ball BR0 with some R = R (Y ) > 0 and |E(x)| + |B(x)| + |x| |∇E(x)| + |∇B(x)| ≤ C 0 |x|−1−σ for |x| > R0 . (2.4)
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In the sequel we consider the space M endowed with Fr´echet topology induced through the embedding M ⊂ LF . Remarks. i) M is a complete metric space, a nonlinear sub-manifold of L. The spaces M and M endowed with Fr´echet topology are metrizable. ii) M1 is dense in M, [8, Lemma 7.4]. On the other hand, since the total charge ρ = ρˆ(0) = 0, Mσ = ∅ for σ > 1 because of the Gauss theorem. By the same reason supp E(x) cannot be a compact set in contrast to supp B(x). Let us write the system (1.1) as a dynamical equation on M Y˙ (t) = F (Y (t)) for t ∈ R,
(2.5)
where Y (t) = (E(x, t), B(x, t), q(t), p(t))) ∈ M. Proposition 2.2 Let (C) hold and Y 0 = (E 0 (x), B 0 (x), q 0 , p0 ) ∈ M. Then i) The system (1.1) has a unique solution Y (t) = (E(x, t), B(x, t), q(t), p(t)) ∈ C(R, M) with Y (0) = Y 0 . ii) The energy is conserved, i.e., H(Y (t)) = H(Y 0 ) f or t ∈ R.
(2.6)
iii) The total momentum is conserved, i.e., P(Y (t)) = P(Y 0 ) f or t ∈ R.
(2.7)
|q(t)| ˙ ≤ v < 1, t ∈ R.
(2.8)
iv) The estimate holds, We refer to [8], where the statements i), ii), iv) are proved. The conservation (2.7) follows from the last equation of (3.15) of the present paper. In [8] also the following preliminary result on asymptotics is proved. Proposition 2.3 ([8]). Let (C), (W ) hold. Let Y (t) ∈ C(R, M) be the solution of the Maxwell-Lorentz equations (1.1) with initial state Y 0 ∈ Mσ with some σ > 1/2. Then q¨(t) → 0 as t → ±∞, (2.9) F
F (E(q(t) + ·, t), B(q(t) + ·, t)) − (Ev(t) (·), Bv(t) (·)) −→ 0 as t → ±∞,
(2.10)
Remark. (2.9) and (2.10) mean the convergence in the Fr´echet topology of the solution to the set of solitary waves (1.4) centered at the charge’s position. Note that (Ev(t) (x), Bv(t) (x)) is a co-moving soliton, and convergence to a certain fixed soliton was not yet proved. In the present paper we establish the convergence (except for the charge’s position) to a fixed soliton. The main results of the paper are the following two theorems. The first step is to prove an orbital stability of solitons which extends the result [1] to the relativistic charge.
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Theorem 2.4 Let (C) hold. Fix a certain v ∈ V . Let Y (t) = (E(t), B(t), q(t), p(t)) ∈ C(R, M) be a solution to the system (1.1) with initial state Y (0) = Y 0 = (E 0 , B 0 , q 0 , p0 ) ∈ M and denote δ = | E 0 (·) − Ev (· − q 0 )|| + | B 0 (·) − Bv (· − q 0 )|| + |p0 − pv | .
(2.11)
Then for every ε > 0 there exists a δ(ε) > 0 such that | E(q(t)+·, t)−Ev (·)|| +||B(q(t)+·, t)−Bv (·)|| +|p(t)−pv | ≤ ε f or all t ∈ R (2.12) provided δ ≤ δ(ε). The second step is to prove the convergence to a fixed soliton. Theorem 2.5 Let (C) and (W ) hold. Let Y 0 = (E 0 , B 0 , q 0 , p0 ) ∈ Mσ with some σ > 1/2 . Let Y (t) ∈ C(R, M) be the solution to (1.1) with Y (0) = Y 0 . Then there exist the limits (1.6) and for every R > 0 (2.13) lim | E(q(t) + · , t) − Ev± (·)||R + | B(q(t) + · , t) − Bv± (·)||R = 0 . t→±∞
Remark. Note that (2.13) follows obviously from (2.10) and (1.6). Hence, the crucial point of the proof is just the convergence (1.6).
3 Orbital stability of solitons The main idea is to derive the orbital stability from the energy conservation developing the Lyapunov function method [1].
3.1
Hamiltonian variables and dynamics
We set the system (2.5) into a Hamiltonian form. Set Es (x, t) = E(x, t)+∇ϕρ (x−q(t)), where ∇ϕρ ∈ L2 (R3 ), ∆ϕρ (x) = −ρ(x), (3.1) ϕρ is defined uniquely. Introduce a magnetic potential A(x, t) which satisfies the Coulomb gauge, B(x, t) = ∇ ∧ A(x, t), ∇ · A(x, t) = 0. (3.2) Let us introduce the charge momentum in the magnetic field as P (t) := p(t) + ρ(x − q(t))A(x, t)d3 x.
(3.3)
Assume that the fields E, B are sufficiently smooth, vanish at infinity, and the equations (1.1) hold for (E, B, q, p). Then by a straightforward computation one obtains that (A, Es , q, P ) obeys the following constraints and equations, ∇ · Es (x, t) = 0,
∇ · A(x, t) = 0,
(3.4)
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˙ t) = −Es (x, t), E˙ s (x, t) = −∆A(x, t) − Πs (ρ(x − q(t))v(t)), A(x, P (t) − ρ(x − q(t))A(x, t)d3 x q(t) ˙ = =: v(t), 2 1/2 3 1 + P (t) − ρ(x − q(t))A(x, t)d x P˙ (t) =
3
ρ(x − q(t))vk (t) · ∇Ak (x, t)d3 x,
1123
(3.5)
(3.6)
(3.7)
k=1
where Πs is the projection to the divergence-free (solenoidal) fields. Consider the functional 2 1/2 1 2 2 3 3 |Es | +|∇A| d x+ 1 + P − ρ(x − q)A(x)d x Hs (Es , A, q, P ) = . 2 (3.8) The equations (3.5), (3.6), (3.7) are Hamiltonian with the Hamiltonian functional H(Es , A, q, P ). Namely, the equations are equivalent respectively to δHs δHs ˙ E˙ s = , A=− , δA δEs ∂Hs ∂Hs ˙ , P =− . q˙ = ∂P ∂q
(3.9)
Thus we call the variables Es , A, q, P “Hamiltonian variables”. The conserved total momentum (1.3) in Hamiltonian variables reads P(Es , A, q, P ) = P + Es (x)∧(∇∧A(x))d3 x = P + Es (x)·∇A(x)d3 x, (3.10) 3 where we denote E · ∇A := k=1 Ek · ∇Ak ; in the sequel we use the second expression for P. It is easy to check that P(Es , A, q, P ) = P(E, B, q, p) and 1 Hs (Es , A, q, P ) = H(E, B, q, p) − (3.11) |∇ϕρ (x)|2 dx, 2 where variables (Es , A, q, P ) and (E, B, q, p) are connected through (3.1), (3.2), (3.3). We now introduce a phase space for the system (3.4) to (3.7) and state the existence of dynamics. Set H 0 = L2 (R3 , R3 ), H˙ 1 is the closure of C0∞ (R3 , R3 ) with respect to the norm A 1 = | ∇A|| = ∇A L2 (R3 ,R3 ) . Let Hs0 , H˙ s1 be the subspaces constituted by solenoidal vector fields, namely the closure in H 0 , H˙ 1 respectively of C0∞ vector fields with vanishing divergence. Define the phase space M0 = Hs0 ⊕ H˙ s1 ⊕ R3 ⊕ R3 ,
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where the norm of Ys = (Es , A, q, P ) is
Ys M0 = | Es | + A 1 + |q| + |P |. Proposition 2.2 implies the following result. Proposition 3.1 Let (C) hold, let Ys0 = (Es0 , A0 , q 0 , P 0 ) ∈ M0 . Then i) There exists a unique solution Ys (t) ∈ C(R, M0 ) to the system (3.4) to (3.7) with Ys (0) = Ys0 . ii) The energy and the total momentum are conserved, Hs (Ys (t)) = Hs (Ys0 ), P(Ys (t)) = P(Ys0 ), t ∈ R. iii) Consider the vector Y (t) = (E(t), B(t), q(t), p(t)) with E(t) = Es (t) − ∇ϕρ (· − q(t)), B(t) = ∇ ∧ A(t), p(t) = P (t) − ρ(x − q(t))A(x, t)d3 x, where (Es (t), A(t), q(t), P (t)) = Ys (t) is the solution to the system (3.4) to (3.7) with Ys (0) = Ys0 . Then Y (t) is the unique solution in C(R, M) of the system (1.1) with the initial data E 0 = Es0 − ∇ϕρ (· − q 0 ), B 0 = ∇ ∧ A0 , 0 0 0 q , p = P − ρ(x − q 0 )A0 (x)d3 x.
3.2
Canonical transform and reduced system
The Hamiltonian (3.8) is invariant with respect to translations in the space R3 . Hence, it is not an appropriate Lyapunov function. We exclude the translation degeneracy of the Hamiltonian reducing the system (3.4) to (3.7) by a canonical transform (cf. [1], [9]). Define the following transform of the space M0 , T (Es (x), A(x), q, P ) = (E(x), A(x), Q, P), where E(x) = Es (x + q), A(x) = A(x + q), Q = q, P = P +
Es (x) · ∇A(x)d3 x. (3.12)
The transform T : M0 → M0 is continuous and Fr´echet differentiable at points (Es (x), A(x), q, P ) with sufficiently smooth functions Es (x), A(x), but not everywhere differentiable. Since Es (x) = E(x − Q), A(x) = A(x − Q), q = Q, P = P − E(x) · ∇A(x)d3 x,
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the transform is invertible. Set H(E, A, Q, P) = Hs (T −1 (E, A, Q, P)), thus H(E, A, Q, P) 1 = 2
2
2
3
(|E| + |∇A| )d x + 1 + P −
3
E · ∇A d x −
2 1/2 ρA d x . (3.13) 3
Lemma 3.2 Let Ys (x, t) = (Es (x, t), A(x, t), q(t), P (t)) be a solution in C(R, M0 ) of the system (3.4) to (3.7). Consider (E(x, t), A(x, t), Q(t), P(t)) = T Ys = (Es (x + q(t), t), A(x + q(t), t), q(t), P (t) +
Es (x) · ∇A(x)d3 x) .
Then i) the constraints hold, ∇ · E(x, t) = 0, ∇ · A(x, t) = 0;
(3.14)
ii) (E(x, t), A(x, t), Q(t), P(t)) is the solution in C(R, M0 ) of the following Hamiltonian system, δH δH , A˙ = − , E˙ = δA δE ∂H ˙ ∂H Q˙ = , P=− . ∂P ∂Q
(3.15)
Proof. i) Follows from (3.4) by definition of E, A. ii) We check, similar to [9], that the transform T is canonical, i.e., conserves the canonical form. This can be seen from the Lagrangian viewpoint. We focus on our particular case and do not develop any general theory of infinite-dimensional Hamiltonian systems which is beyond the scope of this paper. By definition we have H(E, A, Q, P) = Hs (Es , A, q, P ) with (E, A, Q, P) = T (Es , A, q, P ). To each Hamiltonian we associate a Lagrangian through the Legendre transformation L(Es , E˙ s , q, q) ˙ = A, E˙ s + P · q˙ − Hs (Es , A, q, P ) , ˙ Q, Q) ˙ = L(E, E,
˙ + P · Q˙ − H(E, A, Q, P) , A, E
∂H δH , q˙ = , E˙ s = δA ∂P ∂H δH ˙ , Q= . E˙ = δA ∂P
These Legendre transforms are well defined because the Hamiltonian functionals ˙ Q, Q) ˙ = L(Es , E˙ s , q, q). are convex in the momenta. We claim the identity L(E, E, ˙ Clearly we have to check the invariance of the canonical 1-form, ˙ + P · Q˙ = A, E˙ s + P · q˙ . A, E
(3.16)
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For this purpose we substitute A(x) P
˙ E(x) Es · ∇A d x , Q˙
= A(q + x) , =P +
3
˙ + x) + q˙ · ∇Es (q + x) , = E(q = q˙ .
The left-hand side of (3.16) becomes then A(q + x), E˙ s (q + x) + q˙ · ∇Es (q + x) + (P + Es (x), ∇A(x)) · q˙ = A, E˙ s + P · q˙ ˙ Q, Q) ˙ = L(Es , E˙ s , q, q), after partial integration. Since L(E, E, ˙ the corresponding action functionals are identical when transformed by T . The dynamical trajectories are stationary points of the corresponding action functionals. Therefore the two Hamiltonian systems (3.9) and (3.15) are equivalent. Remark. One can also check the equations (3.15) by a straightforward computation. Since H does not depend on Q, we may think of P as of a parameter and consider the reduced Hamiltonian HP (E, A) = 1 2
2 1/2 ρA d3 x . (|E|2 + |∇A|2 ) d3 x + 1 + P − E · ∇A d3 x −
(3.17)
Then E, A satisfy the reduced Hamiltonian system δHP δHP E˙ = , A˙ = − . δA δE
3.3
(3.18)
Soliton as global minimum of reduced Hamiltonian
The solitons in the Hamiltonian variables read Es (x, t) = Es,v (x − vt), A(x, t) = As,v (x − vt), q(t) = vt, Pv = pv +
ρAv d3 x,
where v ∈ V , Es,v = Ev + ∇ϕρ , As,v = Πs Av , and Ev , Av , pv are given by (1.5). The corresponding equations are Es,v (x) = v · ∇As,v (x),
(3.19)
v · ∇Es,v (x) = ∆As,v (x) + Πs (ρ(x)v), Pv − ρ(x)Av,s (x) d3 x v= , 2 1/2 3 1 + Pv − ρ(x)As,v (x) d x
(3.20)
(3.21)
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0=
3
ρ(x)v · ∇(As,v )k (x)d3 x.
1127
(3.22)
k=1
The map v → P(v), where P(v) is the total momentum (3.10) of the soliton, is given, [10], by
1 + v2 1 |ˆ ρ(k)|2 3 1 1 1 + |v| − + d k log P(v) = √ v. (3.23) 2 k2 2|v|3 1 − |v| v 2 1 − v2 The map is differentiable and invertible, the inverse map is differentiable. Apply the canonical transform T to the soliton and obtain Ev := Es,v (x), Av := As,v (x).
(3.24)
These are stationary fields and according to the reduced Hamiltonian system (Ev , Av ) is a critical point of the reduced Hamiltonian HP(v) , where P(v) is the total momentum of the soliton. Lemma 3.3 For (E, A) ∈ Hs0 ⊕ H˙ s1 the lower bound holds, HP(v) (E, A) − HP(v) (Ev , Av ) ≥
1 − |v| (||E − Ev | 2 + A − Av 21 ), 2
(3.25)
where Ev , Av are defined by (3.24). Proof. Set E = Ev + e, A = Av + a, then ∇ · e = 0, ∇ · a = 0. We have =
HP(v) (Ev + e, Av + a) − HP(v) (Ev , Av ) 1 3 (Ev · e + ∇Av · ∇a) d x + (|e|2 + |∇a|2 ) d3 x + [1 + (pv + m)2 ]1/2 − [1 + p2v ]1/2 , 2
where m=−
(Ev · ∇a + e · ∇Av + e · ∇a)d3 x −
ρa d3 x.
Since the equations (3.19) and (3.20) hold, and ∇ · a = 0, we obtain 3 (Ev · e + ∇Av · ∇a)d x = (Ev · e − ∆Av · a)d3 x =
(Ev · e − (v · ∇Ev − Πs (ρv)) · a)d3 x = =
(v · ∇Av · e − v · ∇Ev · a + Πs (ρv) · a)d3 x
(v · ∇Av · e + v · Ev · ∇a + v · ρa − v · e · ∇a + v · e · ∇a)d3 x = −v · m − v ·
e · ∇a d3 x.
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Then 1 HP(v) (Ev + e, Av + a) − HP(v) (Ev , Av ) = 2
2
2
3
(|e| + |∇a| )d x − v ·
e · ∇a d3 x
+[1 + (pv + m)2 ]1/2 − [1 + p2v ]1/2 − v · m. The last line is non-negative, since the function f (p) := (1 + p2 )1/2 is convex and ∇f (pv ) = v. Hence, we obtain (3.25). Remark. The calculations are close to those in [9], since the algebraic structure of the Hamiltonian functional is similar.
3.4
Lyapunov function method: orbital stability
Let us finish the proof of Theorem 2.4. We denote by P 0 the total momentum of the considered solution Ys (t). There exists a soliton-like solution Ys,˜v = (Es,˜v , As,˜v , v˜t, Pv˜ ) corresponding to some v˜ ∈ V and having the same total momentum P(˜ v ) = P 0 . Then (2.11) implies |P 0 − P(v)| = |P(˜ v ) − P(v)| = O(δ) , hence also |˜ v − v| = O(δ) and | Es0 (x) − Es,˜v (x − q 0 )|| + A0 (x) − As,˜v (x − q 0 ) 1 + |P 0 − Pv˜ | = O(δ) .
(3.26)
Therefore denoting (E 0 , A0 , q 0 , P 0 ) = T Ys0 we have HP(˜v) (E 0 , A0 ) − HP(˜v) (Es,˜v , As,˜v ) = O(δ 2 ) .
(3.27)
Total momentum and energy conservation imply for (E(t), A(t), q(t), P 0 ) = T Ys (t) HP(˜v) (E(t), A(t)) = H(T Ys (t)) = HP(˜v) (E 0 , A0 ) for t ∈ R . Hence (3.27) and (3.25) with v˜ instead of v imply | E(t) − Es,˜v | + A(t) − As,˜v 1 = O(δ)
(3.28)
uniformly in t ∈ R . On the other hand, total momentum conservation implies P(˜ v ) = P (t) + E(t), ∇A(t) for t ∈ R . Therefore (3.28) leads to |P (t) − Pv˜ | = O(δ)
(3.29)
uniformly in t ∈ R . Finally (3.28), (3.29) together imply the orbital stability for solutions Ys (t) in the space of Hamiltonian variables. By Proposition 3.1, (2.12) follows.
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4 Convergence of velocity To prove Theorem 2.5 it suffices to prove the existence of the limits (1.6). We combine the orbital stability and the relaxation of the acceleration with a Hamiltonian formalism for the perturbed system (1.1). We prove (1.6) only for t → +∞ since the system is time-reversal. Introduce osc[T ;+∞) v(t) := sup |v(t1 ) − v(t2 )|. t1 ,t2 ≥T
The existence of the limits (1.6) follows from the following proposition. Proposition 4.1 Let the assumptions of Theorem 2.5 be fulfilled. Then osc[T ;+∞) v(t) → 0 as T → +∞.
(4.1)
Proof. The idea of the proof is as follows. We modify the trajectory of the charge and the field part of the solution. The new trajectory and fields satisfy a new system of equations which is a small perturbation of the system (1.1) for large t. Step 1. We introduce the above-mentioned modification of a solution. The following expansion holds, [8], E(x, t) = E(r) (x, t) + E(0) (x, t), B(x, t) = B(r) (x, t) + B(0) (x, t).
(4.2)
Here
E(r) (x, t) B(r) (x, t)
= 0
t
ds gt−s (x) ∗
E(0) (x, t) B(0) (x, t)
ρ(x − q(s)) ρ(x − q(s))q(s) ˙
= mt (x) ∗
E0 B0
;
(4.3)
,
(4.4)
where mt and gt are respectively 6 × 6- and 6 × 4-matrix-valued distributions:
K˙ t ∇ ∧ Kt −K˙ t −∇Kt mt = , gt = , 0 ∇ ∧ Kt K˙ t −∇ ∧ Kt and Kt (x) denotes the Kirchhoff kernel Kt (x) =
1 δ(|x| − |t|). 4πt
It is important that the distributions mt , gt are concentrated on the sphere {|x| = |t|}, this means the strong Huygen’s principle for the Maxwell-Lorentz system: mt (x) = 0 and gt (x) = 0 for |x| = |t|.
(4.5)
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Further, in [8] the following decay is established, 0
mt ∗ E 0 (x) ≤ C0 |t|−2 d3 y (|E 0 (y)| + |B 0 (y)|) B |y−x|=|t| d3 y (|∇E 0 (y)| + |∇B 0 (y)|). (4.6) + C1 |t|−1 |y−x|=|t|
Recall that E 0 and B 0 satisfy the constraints ∇ · E 0 (x) = ρ(x − q 0 ), ∇ · B 0 (x) = 0,
x ∈ R3
(4.7)
providing that the fields defined through (4.2) to (4.4) satisfy the system (1.1). Further, by (2.9) for every ε > 0 there exists tε such that |¨ q (t)| ≤ ε for t ≥ tε and tε → ∞ as ε → 0 .
(4.8)
Let us consider the points t1,ε = tε + 1, t2,ε = t1,ε +
Rρ Rρ , t3,ε = t2,ε + . 1−v 1−v
(4.9)
Set q3,ε = q(t3,ε ) , vε = q(t ˙ 3,ε ). Then (4.8) implies that there exists qε (t) ∈ C 2 (R) such that q(t) for t ∈ [t1,ε , +∞) , qε (t) = l(t) := q3,ε + vε (t − t3,ε ) for t ∈ (−∞, tε ] ,
(4.10)
and |¨ qε (t)| ≤ Cε for all t ∈ R
(4.11)
with C > 0 independent of ε ∈ (0, 1). Now set
Eε (x, t) Bε (x, t)
t
= −∞
ds gt−s (x) ∗
ρ(x − qε (s)) ρ(x − qε (s))q˙ε (s)
, x ∈ R3 , t > 0. (4.12)
Here the integrand, for a fixed s, is a convolution of two distributions of S , S being the space of tempered distributions. One of the distributions, gt−s (·), has a compact support by (4.5). Hence the integrand is as well a distribution of S , and this distribution depends continuously on s. Thus, the integral is understood as the Riemann integral of the continuous S -valued function on R. Step 2. We show that the modified fields satisfy the inhomogeneous Maxwell equations, coincide with soliton fields outside a certain light cone, and coincide with the retarded fields (E(r) , B(r) ) in a smaller light cone.
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Lemma 4.2 i) The fields Eε , Bε coincide with a soliton outside a light cone: Eε (x, t) = Evε (x − l(t)), Bε (x, t) = Bvε (x − l(t))
(4.13)
|x − qε (tε )| > t − tε + Rρ .
(4.14)
for ii) Eε , Bε , qε satisfy the system E˙ ε (x, t) = ∇ ∧ Bε (x, t) − ρ(x − qε (t))q˙ε (t), ∇ · Eε (x, t) = ρ(x − qε (t)), B˙ ε (x, t) = −∇ ∧ Eε (x, t),
∇ · Bε (x, t) = 0,
(4.15)
for t ∈ R, x ∈ R3 . iii) The fields Eε , Bε coincide with E(r) , B(r) in the light cone K = {|x−q(t2,ε )| < t − t2,ε }. Proof. i) Consider the soliton fields (Evε (x − l(t)), Bvε (x − l(t)) as the solution of the Cauchy problem for system (4.15) with initial data at −T , T > 0. These data equal (E −T , B −T ) = (Evε (x − l(−T )), Bvε (x − l(−T )). Let us apply the formulas of type (4.2) to (4.4) in the case, when Cauchy data are set at −T instead of 0. Then we obtain −T
t
Evε (x − l(t)) ρ(x − l(s)) E ds gt−s (x) ∗ = + mt+T ∗ , −T ρ(x − l(s))v Bvε (x − l(t)) B ε −T (4.16) since for (E −T , B −T ) the constrains of type (4.7) are satisfied with l(−T ) instead of q 0 . Here the last summand tends to zero in L2loc (R3 ) ⊕ L2loc (R3 ) (and hence in S ⊕ S ) as T → +∞. This follows by the bounds (4.6) using the formulas (1.5) and |vε | < 1. Hence, proceeding to the limit as T → +∞ we obtain the identity of distributions,
t
ρ(x − l(s)) Evε (x − l(t)) ds gt−s (x) ∗ = . (4.17) ρ(x − l(s))vε Bvε (x − l(t)) −∞ Finally, in the region (4.14) the right-hand side of (4.17) coincides with (4.12) by (4.10) and (4.5). ii) The strong Huygen’s principle (4.5) implies that for (x, t) ∈ K0 = {|x − qε (tε )| < t − tε + Rρ } and for sufficiently large tε
t
ρ(x − qε (s)) Eε (x, t) ds gt−s (x) ∗ = , (4.18) Bε (x, t) ρ(x − qε (s))q˙ε (s) −T x ∈ R3 , t ∈ R with a large T independent of (x, t) ∈ K0 . Introduce the fields −T
˜t E E Eε ∗ = + m , (4.19) t+T ˜T Bε B −T B
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˜ B) ˜ satisfy all the equations with (E −T , B −T ) like in (4.18). Then the fields (E, of (4.15) for t > −T by the same argument as in the proof of i) (the constraints of type (4.7) for initial data are satisfied). Finally, the second summand in the right-hand side of (4.19) tends to zero as T → +∞ like in (4.16). Hence, (Eε , Bε ) satisfy (4.15) for all t ∈ R. iii) follows from (4.3), (4.12), (4.10), and (4.5). Step 3. Now we show that at t = t3,ε in the ball, where the modified fields are not explicitly the soliton, they are sufficiently close to it. Indeed, we have
Eε (x, t3,ε ) − Evε (x − qε ) Bε (x, t3,ε ) − Bvε (x − qε ) t3,ε = ds gt3,ε −s (x) tε
ρ(x − qε (s)) − ρ(x − l(s)) ρ(x − qε (t))q˙ε (s) − ρ(x − l(s))vε
.
Hence, by (4.10), (4.11) we obtain that | Eε (·, t3,ε ) − Evε (· − qε )||L2 (B ε ) + | Bε (·, t3,ε ) − Bvε (· − qε )||L2 (B ε ) = O(ε), (4.20) where B ε = {x : |x − qε (tε )| ≤ 2Rρ /(1 − v) + 1 + Rρ }. Step 4. We now express the Lorentz force equation for t ≥ T := t3,ε in terms of the fields Eε , Bε . In this region qε (t) = q(t). Thus, we can change qε (t) by q(t) in the equations (4.15) for Eε , Bε : E˙ ε (x, t) = ∇ ∧ Bε (x, t) − ρ(x − q(t))q(t), ˙ ∇ · Eε (x, t) = ρ(x − q(t)), B˙ ε (x, t) = −∇ ∧ Eε (x, t), ∇ · Bε (x, t) = 0
(4.21)
for t > T . Further, one has Eε = E(r) and Bε = B(r) inside K by Lemma 4.2, iii). Thus, for t > T in supp ρ(x − q(t)) we have E = Eε + E(0) , B = Bε + B(0) and hence p(t) 3 q(t) ˙ = ˙ , p(t) ˙ = [Eε (x, t)+ q(t)∧B ε (x, t)]ρ(x−q(t)) d x+f (t), t > T, 1 + p2 (t) (4.22) where f (t) := [E(0) (x, t) + q(t) ˙ ∧ B(0) (x, t)]ρ(x − q(t)) d3 x. Let us transfer to Hamiltonian variables Es,ε , Aε , q, P , where Es,ε (x, t) = Πs Eε (x, t), Bε (x, t) = ∇ ∧ Aε (x, t), ∇ · Aε (x, t) = 0, P (t) := p(t) + ρ(x − q(t))Aε (x, t)d3 x.
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Es (x) ∧ (∇ ∧ A(x))d3 x; for a soliton (Ev , Bv ) the total momentum is P(v) = pv + Es,v (x) ∧ (∇ ∧ As,v (x))d3 x.
Recall that the total momentum is P = p +
Step 5. We establish a summable decay of f (t) and prove that Hs and P are “almost conserved” along a trajectory solution Yε (t), where Hs (Es , A, q, P ) is defined by (3.8). Lemma 4.3 For σ > 1/2 introduced in Proposition 2.3: i) The following asymptotics hold: |f (t)| = O(t−1−σ ).
(4.23)
ii) The oscillations of the Hamiltonian and the total momentum are small for large T : Hs (Yε (t)) = Hs (Yε (T )) + O(T −σ ), (4.24) P(Yε (t)) = P(Yε (T )) + O(T −σ )
(4.25)
for t > T . Proof. i) For f (t) the asymptotics follow from the explicit formulas (4.4), the bounds (4.6), the estimate (2.8), and the decay (2.4) of the initial fields. ii) By (3.11) it suffices to prove (4.24) for H(Yε (t)), where H is defined by (1.2). One has
1 d 1 + p2 + (Eε2 + Bε2 )dx = v · p˙ + Es , E˙ s + Bs , B˙ s = dt 2
v· (Eε + q˙ ∧ Bε )ρ(x − q)dx + f +Eε , ∇∧Bε −ρ(x−q)q+B ˙ ε , −∇∧Eε = v·f, since Eε , ∇ ∧ Bε − Bε , ∇ ∧ Eε = 0 similar to the proof of Proposition A.5 of [8]. Similarly, one has d P(Yε (t)) = f (t). dt Then (4.24), (4.25) follow from (4.23). Step 6. Finally, we use the orbital stability estimate (3.25). For t ≥ T := t3,ε one has P(Yε (t)) = P(˜ v (t)), where P(˜ v (t)) is the total momentum of the soliton of velocity v˜(t). From (4.25) and the differentiability of the map P(v) → v, the inverse map to (3.23), it follows that osc[T,+∞) v˜(t) → 0 as T → +∞.
(4.26)
By the statement i) of Lemma 4.2 and (4.20) one has v˜(t3,ε ) − vε = O(ε). Together with (4.26) this implies the bound |˜ v (t)| ≤ v 1 < 1 for t ≥ T . Now apply the
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estimate (3.25) and get 1 − |˜ v (t)| (||Es,ε (· + q(t), t) − Es,˜v(t) | 2 + Aε (· + q(t), t) − As,˜v (t) 2 ) 2 ≤ HP(˜v (t)) (Es,ε (· + q(t), t), Aε (· + q(t), t)) − HP(˜v(t)) (Es,˜v (t) , As,˜v(t) ).
(4.27)
Lemma 4.4 The right-hand side of (4.27) is arbitrary small uniformly in t ≥ T for sufficiently small ε and sufficiently large T . From this lemma it follows that osc[T,+∞) | Es,ε (· + q(t), t)|| → 0
and
osc[T,+∞) Aε (· + q(t), t) → 0
as T → +∞. Indeed, Es,ε (· + q(t2 ),t2 ) − Es,ε (· + q(t1 ),t1 ) = (Es,ε (· + q(t2 ),t2 ) − Es,˜v (t2 ) ) − (Es,ε (· + q(t1 ),t1 ) − Es,˜v (t1 ) ) + (Es,˜v (t2 ) − Es,˜v(t1 ) ). For t1 , t2 > T the first and the second summands are small by (4.27) and the lemma, the third is small by (4.26), since the soliton field Ev depends continuously on v in L2 . For the field A the argument is similar. Together with (4.25) this implies osc[T,+∞) p(t) → 0 as T → +∞ and hence (4.1) follows. Proposition 4.1 is proved. ˜ Proof of Lemma 4.4. Denote P(t) = P(˜ v (t)), Φ(t) = (Es,ε (· + q(t), t), Aε (· + ˜ ˜ q(t), t)), Φ(t) = (Es,˜v (t) , As,˜v(t) ). We claim that HP(t) ˜ (Φ(t)) − HP(t) ˜ (Φ(t)) is close ˜ )) and the last expression is small due to (4.13) and to H ˜ (Φ(T )) − H ˜ (Φ(T P(T )
P(T )
(4.20). Thus, it is sufficient to prove that HP(t) ˜ (Φ(t)) is close to HP(T ˜ ) (Φ(T )) and ˜ ˜ HP(t) ˜ (Φ(t)) is close to HP(T ˜ ) (Φ(T )). One has HP(t) ˜ (Φ(t)) − HP(T ˜ ) (Φ(T )) = HP(t) ˜ (Φ(t)) − HP(T ˜ ) (Φ(t)) + HP(T ˜ ) (Φ(t)) − HP(T ˜ ) (Φ(T )), ˜ ˜ this is small due to (4.24), (4.25). For HP(t) ˜ (Φ(t)) − HP(T ˜ ) (Φ(T )) the argument is similar. Acknowledgments. V. Imaikin and A. Komech are supported partly by research grants of DFG (436 RUS 113/615/0-1(R)) and RFBR (01-01-04002).
References [1] D. Bambusi, L. Galgani, Some rigorous results on the Pauli-Fierz model of classical electrodynamics, Ann. Inst. H. Poincar´e, Phys. Theor. 58, 155–171 (1993). [2] V.S. Buslaev, G.S. Perelman, On the stability of solitary waves for nonlinear Schr¨ odinger equations, Amer. Math. Soc. Trans. (2) 164, 75–98 (1995).
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[3] J. Fleckinger, A. Komech, Soliton-like asymptotics for 1D kinetic-reaction systems, Russian J. Math. Physics 5, no.3, 35–43 (1997). [4] H. Goldstein, “Classical Mechanics”, Addison-Wesley, 1951. [5] M. Grillakis, J. Shatah, W.A. Strauss, Stability theory of solitary waves in the presence of symmetry, I; II. J. Func. Anal. 74, 160–197 (1987); 94, 308–348 (1990). [6] A.I. Komech, H. Spohn, M. Kunze, Long-time asymptotics for a classical particle interacting with a scalar wave field, Comm. Partial Diff. Equs. 22, no.1/2, 307–335 (1997). [7] V.M. Imaikin, A.I. Komech, H. Spohn, Soliton-type asymptotics and scattering for a charge coupled to the Maxwell field, Russian J. Math. Phys. 9, no. 4, 428–436 (2002). [8] A.I. Komech, H. Spohn, Long-time asymptotics for the coupled MaxwellLorentz equations, Comm. Partial Diff. Equs. 25, no.3/4, 559–584 (2000). [9] A.I. Komech, H. Spohn, Soliton-like asymptotics for a classical particle interacting with a scalar wave field, Nonlinear Anal. 33, no.1, 13–24 (1998). [10] M. Kunze, H. Spohn, Adiabatic limit for the Maxwell-Lorentz equations, Ann. Inst. H. Poincar´e, Phys. Theor. 1, no.4, 625–653 (2000). ´ [11] J.L. Lions, “Probl`emes aux Limites dans les Equations aux D´eriv´ees Partielles”, Presses de l’Univ. de Montr´eal, Montr´eal, 1962. [12] S.P. Novikov, S.V. Manakov, L.P. Pitaevskii, V.E. Zakharov, “Theory of Solitons: The Inverse Scattering Method”, Consultants Bureau, 1984. Valery Imaikin, Alexander Komech and Norbert Mauser Wolfgang Pauli Institute c/o Institute of Mathematics University Vienna Nordbergstrasse 15 A-1090 Vienna Austria email:
[email protected] email:
[email protected] email:
[email protected] Communicated by Viencent Rivasseau submitted 01/12/03, accepted 26/05/04
Ann. Henri Poincar´e 5 (2004) 1137 – 1157 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/061137-21 DOI 10.1007/s00023-004-0194-4
Annales Henri Poincar´ e
On the Vacuum Polarization Density Caused by an External Field Christian Hainzl Abstract. We consider an external potential, −λϕ, due to one or more nuclei. Following the Dirac picture such a potential polarizes the vacuum. The polarization density, ρλ vac , as derived in physics literature, after a well-known renormalization procedure, depends decisively on the strength of λ. For small λ, more precisely as long as the lowest eigenvalue, e1 (λ), of the corresponding Dirac operator stays in the gap of the essential spectrum, the integral over the density ρλ vac vanishes. In other words the vacuum stays neutral. But as soon as e1 (λ) dives into the lower continuum the vacuum gets spontaneously charged with charge 2e. Global charge conservation implies that two positrons were emitted out of the vacuum, this is, a large enough external potential can produce electron-positron pairs. We give a rigorous proof of that phenomenon.
1 Introduction In 1934 Dirac and Heisenberg realized that accepting the Dirac picture of electrons filling up the negative energy states, called vacuum, consequently implies that a charged nucleus thrown into the vacuum causes a redistribution of the Dirac sea, an effect denoted as vacuum polarization. Uehling and Serber in 1935 [27, 25], long before standard renormalization procedure, demonstrated that such an indicated production of virtual electron-positron pairs give rise to a modification of the Coulomb potential and thus causes energy shifts of bound electrons. Concerning the traditional Lamb shift, known as the splitting of the 2s1/2 and 2p1/2 -state in hydrogen, this effect only accounts for about 2.5 percent. However the Uehling potential represents the dominating radiative correction in muonic atoms which emphasizes the importance of vacuum polarization (VP). Notice, whereas interaction with a photon field can be treated non-relativistically there is no non-relativistic equivalence for VP. It is a purely relativistic effect. Within the framework of QED, VP is treated by means of perturbation theory as developed by Dyson, Feynman, and Schwinger. Only recently Hainzl and Siedentop demonstrated in [11] that the effective one-particle Hamiltonian obtained from VP can be handled non-perturbatively and gives rise to a self-adjoint operator. The effective potential we gain is in fact the same as the physicists obtain after mass and charge renormalization (neglecting photon terms) and use to calculate the hyperfine structure of bound states. We refer to [20, Section 4] for a nice review concerning the influence of VP on the Lamb shift of heavy atoms.
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The main goal of the present paper is to study the vacuum polarization density caused by an external field, i.e., by one or more nuclei. As foreseen by physicists, e.g., [10, 9], the behavior of the density turns out to depend on the lowest eigenvalue of the corresponding Dirac operator. As long as this eigenvalue stays isolated the integral over the density vanishes, that means the vacuum stays neutral. But as soon as that eigenvalue touches the lower continuum the vacuum gets spontaneously charged, i.e., an electron, more precisely two electrons due to degeneracy of the “ground state”, are trapped in the vacuum and two positrons are emitted. In other words large fields can produce electron-positron pairs. Such a situation can be realized by heavy ion collision.
1.1
Model
The free Dirac operator is given by 1 D0 := α · ∇ + β i
(1)
in which α, β denote the 4 × 4 Dirac matrices. The underlying Hilbert space is given by H = L2 (Γ) with Γ = R3 × {1, 2, 3, 4}. We pick units in which the electron mass is equal to one. We regard the case of one, or more, smeared nuclei with density n ∈ L1 (R3 ) ∩ L∞ (R3 ), non-negative, and assuming R3 n = 1. We remark that it is an experimental fact that the nucleus cannot shrink to a point. In fact a point nucleus creates instability if one includes polarization effects, as shown in [11, Section 3.5]. The corresponding electric potential reads ϕ = | · |−1 ∗ n.
(2)
and the operator to be studied is given by Dλϕ := D0 − λϕ,
(3)
where λ ≥ 0 is a√ parameter and can be thought of as αZ, α the fine structure constant, e := − α the charge of an electron, and −Ze the charge of the nucleus (nuclei). In the following we want to allow any value of λ. Due the smearing out of the Coulomb singularity the case of large values of λ does not influence the behavior of the essential spectrum as well as the selfadjointness as it would be the case of the Coulomb potential. The following Lemma is well known, e.g., Weidmann [28, Theorem 10.37]. Lemma 1 Let ϕ = | · |−1 ∗ n, n ∈ L1 (R3 ) ∩ L∞ (R3 ), non-negative. Then, ∀λ ≥ 0, Dλϕ = D0 − λϕ is self-adjoint with domain H 1 (Γ) and the essential spectrum of Dλϕ is given by (4) σess (Dλϕ ) = (−∞, −1] ∪ [1, ∞).
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Throughout the paper we will denote the spectrum of Dλϕ by σ(Dλϕ ) and ei (λ) as the corresponding eigenvalues. The following is well known: For fixed λ there is an infinite number of eigenvalues which accumulate at 1 and each ei (λ) depends continuously on λ. For small values of λ all eigenvalues stay in the gap (−1, 1) of the essential spectrum. However, for each i one finds a λi such that for limλ→λi ei (λ) = −1, i.e., the eigenvalue ei (λ) dives into the lower continuum. We do not at all discuss what happens to the eigenvalues after reaching the continuum. In fact one knows from [2] that below −1 there are no embedded eigenvalues. However, the behavior of the eigenvalues after reaching (−∞, −1] won’t play any role. Our theorems only depend on the number of eigenvalues, counting multiplicity, that vanish in the lower continuum. Namely, due to our assumption ϕ ≥ 0, all eigenvalues are monotonously decreasing (this is a consequence of [23, Theorem XII.13] and the fact that each eigenvalue has non-positive derivative). That means they will not reappear after having reached −1. The fact that each eigenvalue reaches −1 for a large enough parameter can be seen by, e.g., a Theorem of Dolbeaut-Esteban-S´er´e [3]. We will see that whenever an eigenvalue dives into the “sea of occupied states”, i.e., (∞, −1], a specific number of e− e+ pairs are created depending on the degeneracy of the dived eigenvalue.
1.2
Vacuum polarization density
As already mentioned above, according to Dirac the vacuum consists of electrons occupying the negative energy states of the free Dirac operator. If one puts a nucleus into the vacuum, then the electrons rearrange and one ends up with virtual electron-positron pairs. In other words the vacuum gets polarized, see e.g., [9, page 257], for a picture describing this phenomenon, and [11] for a “mathematical” derivation of the vacuum polarization density, which follows the idea of the early papers in QED [5, 12, 29, 17, 8]. For a review about the old-fashioned way of QED we refer to [18]. The operator describing this polarization effect is given by
where
Qλϕ := P−λϕ − P−0 ,
(5)
P−λϕ := χ(−∞,−1] (Dλϕ ).
(6)
Physically speaking we project onto the occupied states of the Dirac sea. Remark 1 Notice, in the case that the lowest eigenvalue of Dλϕ , e1 (λ), is strictly positive, our definition is equivalent to [11, Equation (12)], apart from a minus sign which is chosen to adapt to the definition in the physics literature. Usually the first idea to define a density via Qλϕ would simply be taking the diagonal of the Kernel. Unfortunately, the operator Qλϕ is not trace class. The question how to extract from Qλϕ a physically meaningful density was first posed
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in the thirties by Dirac [4, 5] and Heisenberg [12] and in more recent literature this procedure is known as charge renormalization (see e.g., [8, 6]). As in [11] we use Cauchy’s formula to express the Qλϕ in terms of the respective resolvents (Kato [14], Section VI,5, Lemma 5.6) Qλϕ = P−λϕ − P−0 = :=
1 2π
dη
−∞ Pγλϕ − Pγ0 ,
where Pγλϕ :=
∞
1 1 − 2 2π
1 1 − D0 − γ + iη Dλϕ − γ + iη
∞
dη −∞
Dλϕ
1 , − γ + iη
(7)
(8)
with −1 < γ < ej (λ), ej (λ) being the lowest isolated eigenvalue of Dλϕ . Notice that the second equality in (7) is a consequence of the fact that D01−z is holomorphic with respect to z in the complex strip between (−1, 1) and Dλϕ1 −z between (−1, ej (λ)). We decompose Qλϕ into 4 terms: Qλϕ = λQ1 + λ2 Q2 + λ3 Q3 + λ4 Qλ4 ,
(9)
where 1 1 ∞ 1 Q1 := ϕ , dη 2π −∞ D0 + iη D0 + iη ∞ 1 1 1 1 Q2 := ϕ ϕ , dη 2π −∞ D0 + iη D0 + iη D0 + iη ∞ 1 1 1 1 1 ϕ ϕ ϕ , Q3 := dη 2π −∞ D0 + iη D0 + iη D0 + iη D0 + iη Qλ4 := 1 ∞ 1 1 1 1 1 ϕ 0 ϕ λϕ ϕ 0 ϕ 0 . dη 0 2π −∞ D − γ + iη D − γ + iη D − γ + iη D − γ + iη D − γ + iη (10) The first three terms we consider by means of its Fourier representation. A simple ˆ 1 to variable transform iη → iη + γ does not change the kernel of the operators Q ˆ 3 which is the reason why we suppressed the γ in the denominator. The first term Q is treated in detail in [11, Section 3.2]. There, by a well known renormalization procedure following Weisskopf [29] and Pauli and Rose [21], we extracted the corresponding physical density 4πˆ n(k)C(k) (x), (11) ρλ1 (x) := eλF −1 |k|2
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where (see [11, Equation (21)]) 1 1 C(k)/k 2 = dx(1 − x2 ) log[1 + k 2 (1 − x2 )/4] 2 0
1 + 4/k 2 + 1 1 4 5 4 2 = − + (1 − 2 ) 1 + 2 log , 3 k k 3 1 + 4/k 2 − 1 k 2
(12)
which was first explicitly written down by Uehling [27] and Serber [25] and later by Schwinger [24] and others (see also [13, 16, 10]). Observe in [11, Eq. (52)] that renormalization consists of subtracting an operator with infinite diagonal from Q1 . From what remains one defines, in (11), the diagonal corresponding to Q1 . This subtraction reflects the main difficulty concerning the proof of our main Theorem. The second and third term in (10) have a well-defined integrable diagonal when using the Fourier representation. Additionally the density corresponding to Q2 vanishes, either through integration over η or due to the fact that the Dirac matrices are traceless. Quite generally, if we expand trC4 Qλ4 into an infinite sum, each term with an even number of ϕ vanishes. The density corresponding to Q3 is given by ρλ3 (x) := eλ3 (2π)−3
dp R3
dq R3
4
ˆ 3 (p, σ; q, σ) ei(p−q)·x Q
(13)
σ=1
ˆ 3 denotes the kernel of the Fourier representation where Q ˆ 3 (p, q) = 1 Q 2π
∞
dη −∞
R3
dp1
R3
dp2 (Dp + iη)−1 ◦ ϕ(p ˆ − p1 ) ◦ (Dp1 + iη)−1
◦ ϕ(p ˆ 1 − p2 ) ◦ (Dp2 + iη)−1 ◦ ϕ(p ˆ 2 − q) ◦ (Dq + iη)−1
(14)
Dr := α · r + β. Since Q3 might not be trace class, we define, for simplicity, with ˆ 3 (p, p)dp. ρλ3 (x)dx := eλ3 trC4 Q λ The operator Q4 will be shown to be trace class in Lemma 3, so we can define ρλ4 quite general via the diagonal of λ4 trC4 Qλ4 , ρλ4 (x) := eλ4 trC4 Qλ4 (x, x)
(15)
Therefore the renormalized density reads ρλvac (x) := ρλ1 (x) + ρλ3 (x) + ρλ4 (x).
(16)
Before formulating our main theorem it is necessary to introduce the counting function d(λ) which counts the number of eigenvalues which dived in the lower continuum for parameters smaller equal λ. ¯ := {#eigenvalues, with multiplicity, that reached − 1 for parameters λ ≤ λ}. ¯ d(λ) (17)
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Theorem 1 Let n ∈ L1 (R3 ) ∩ L∞ (R3 ) and non-negative, ϕ = n ∗ R3
ρλvac (x)dx = ed(λ).
1 |·| .
Then (18)
Theorem 1 exactly reflects the picture which is presented by physicists, e.g., Greiner et al. [9, 10]: As long as the external potential, respectively λ, is so weak that the lowest eigenvalue, e1 (λ), of Dλϕ is in the gap (−1, 1) the vacuum stays neutral (and consists only of virtual electron-positron pairs). As soon as the lowest eigenvalue dives into the essential spectrum, (−∞, −1], i.e., the sea of occupied states, the vacuum immediately gets charged with charge 2e (assuming that the ground state energy of Dλϕ is twice degenerate, due to the spin). This can be interpreted in the following way: when the unoccupied bound state dives in the sea of occupied states it traps two electrons which stay in the potential well of the nucleus (nuclei). Due to Dirac’s picture two “holes” emerge which are repelled and emitted as positrons out of the vacuum. Consequently we end up with real electron-positron, e− e+ , pairs. This effect of spontaneously emitted positrons is verified in experiment by collision of heavy nuclei, which when approaching each other create an effective field strong enough to let the lowest eigenvalue dive into the continuum (see [22]). Remark 2 In more recent physics literature, compare, e.g., [9, Equation (7.23)] or [20, Equation (230)], the VP-density is “formally” denoted as the diagonal of the operator e trC4 [P−λϕ − P+λϕ ], (19) 2 with P+λϕ := 1 − P−λϕ . Since trC4 [P+0 − P−0 ] = 0 and −P+λϕ + P+0 = P−λϕ − P−0 we see that (19) coincides with our initial operator e trC4 [P−λϕ − P−0 ]. The proof of Theorem 1 will mainly be based on two ingredients: A work of Avron, Seiler, and Simon [1] concerning the index of pairs of projectors (see also [7]) and arguments of Kato [14]. The proof of Theorem 1 will be given in Section 3. In Section 2 we show that for tr[P−λϕ − P−0 ]2m+1 , m ≥ 1, a result similar to (18) holds.
2 Result on tr[P−λϕ − P−0 ]2m+1 , with m ≥ 1 Recall that the vacuum polarization is in fact described by the operator Qλϕ = P−λϕ − P−0 . Renormalization is inevitable, since that operator is not trace class. Nevertheless, due to Klaus and Scharf [15] it is at least an Hilbert-Schmidt opera
2m+1 tor. Due to [1] (in fact this follows already from Effros [7]) the traces of Qλϕ , m ≥ 1, are equal. Therefore it is self-evident to ask for their behavior.
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Theorem 2 Let n ∈ L1 (R3 ) ∩ L∞ (R3 ), non-negative, ϕ = n ∗
1143 1 |·| .
Then, ∀m ≥ 1,
tr[P−λϕ − P−0 ]2m+1 = d(λ),
(20)
where d(λ) is defined as in (17). Proof. Notice that, since ϕ(k) ˆ =n ˆ (k) 4π k2 , 2 log(2 + | · |) ˆ k 2 log(2 + |k|)|ϕ(k)| 2 . ≤ |ˆ n| p 2 dk 1 + |k| | · | (1 + | · |) q R3
(21)
Take q = 43 , p = 4, then the second term on the right-hand side is finite, as well as by Hausdorff-Young inequality 3 n 28 ≤ C8/7
n 28/7 < ∞.
|ˆ n|2 4 = ˆ
(22)
Therefore the potential ϕ is regular in the sense of Klaus and Scharf [15], cf. [19, Equation (1.7)], namely the operator Qλϕ ∈ S2 (H), i.e., Qλϕ is a Hilbert-Schmidt operator. Consequently Qλϕ ∈ Sm (H) for any m ≥ 2. To prove the Theorem we first look at the set of all λ ≥ 0 such that the lowest eigenvalue, e1 (λ), corresponding to Dλϕ fulfills e1 (λ) > −1.
(23)
This is an open set so that we can always find a γ, with −1 < γ < e1 (λ) and 1 1 1 ∞ λϕ − dη (24) Q = = Pγλϕ − Pγ0 2π −∞ D0 − γ + iη Dλϕ − γ + iη We are going to show that for m ≥ 1 tr[Pγλϕ − Pγ0 ]2m+1 = 0
(25)
on the set {λ|e1 (λ) > γ}. Since γ can be chosen arbitrarily close to −1 we infer that tr[P−λϕ − P−0 ]2m+1 = 0 (26) on {λ|e1 (λ) > −1}. To this aim we recall some results from Avron, Seiler, and Simon [1] (see also [7]) concerning the index of pairs of projections: Regard the family of orthogonal projections Pγλϕ , λ ≥ 0. Since Pγλϕ − Pγµϕ ∈ S2 (H), [1, Proposition 3.2] implies that all pairs (Pγλϕ , Pγµϕ ) are Fredholm. Combining [1, Theorem 3.1] and [1, Theorem 4.1] we obtain that for m, l ≥ 1 2m+1 2l+1 tr Pγλϕ − Pγµϕ = tr Pγλϕ − Pγµϕ = ind(Pγλϕ , Pγµϕ ) =dim(KerPγµϕ ∩ RanPγλϕ ) − dim(KerPγλϕ ∩ RanPγµϕ ) is an integer.
(27)
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Remark 3 More generally, a pair (P, Q) of orthogonal projections is called Fredholm, if the operator T = QP , as an operator from RanP → RanQ, is Fredholm. The corresponding index ind(P, Q) is defined as ind(P, Q) := ind T = dim(KerT ) − dim(RanT )⊥ .
(28)
Next we come back to the proof of (25). Observe that on {λ|e1 (λ) > γ} Pγλϕ is a continuous family with respect to the operator norm. Namely, using (8) and expanding the resolvent we get
∞
1 1
λϕ
µϕ − γ + iη D D − γ + iη −∞ ∞ 1 ≤ |λ − µ| ϕ dη , (29) 1/2 1/2 2 2 −∞ (δ1 + η ) (δ22 + η 2 )
Pγλϕ − Pγµϕ ≤ |λ − µ| ϕ
dη
with δ1 := min{γ + 1, e1(µ) − γ} and δ2 := min{γ + 1, e1 (λ) − γ}. Notice that since e1 (λ) is continuous the integral in the right-hand side of (29) can be bounded uniformly on a small enough closed neighborhood of each λ in {λ|e1 (λ) > γ}. Due to (27), Pγλϕ − Pγµϕ < 1 implies ind(Pγλϕ , Pγµϕ ) = 0. Using [1, Theorem 3.4 (c)], (30) ind(Pγλϕ , Pγ0 ) = ind(Pγλϕ , Pγµϕ ) + ind(Pγµϕ , Pγ0 ), the continuity of Pγλϕ immediately gives that ind(Pγλϕ , Pγ0 ) = 0 on the whole set {λ|e1 (λ) > γ}. Together with (27) we arrive at (25). Summarizing, the argument given above was based on the fact that on the set {λ|e1 (λ) > −1}, Pγλϕ can be continuously deformed into Pγ0 . Throughout the rest of the paper we will repeat this argument several times. In the following we consider the case that an eigenvalue has dived into the lower continuum. We know that there are no eigenvalues below −1. However, for notational simplification we treat them as if they stay embedded. ¯ ≤ −1 and e2 (λ) ¯ > −1, and γ with −1 < γ < e2 (λ). ¯ ¯ such that e1 (λ) Fix now λ ¯ Additionally we choose a λ < λ such that −1 < e1 (λ ) < γ and a γ with −1 < γ < e1 (λ ). We know 2m+1 ¯ ¯ ¯ = tr[Pγλϕ − Pγ0 ]2m+1 = ind(Pγλϕ , Pγ0 ). tr Qλϕ
(31)
Due to [1, Theorem 3.4 (c)] ¯
¯
ind(Pγλϕ , Pγ0 ) = ind(Pγλϕ , Pγλ ϕ ) + ind(Pγλ ϕ , Pγλ ϕ ) + ind(Pγλ ϕ , Pγ0 ).
(32)
The first and third term in the right-hand side in (32) vanish which can be seen by repeating the argument given above. Namely due to our choice of parameters ¯ Pγλϕ can be continuously deformed into Pγλ ϕ . As well Pγλ ϕ can be continuously deformed into Pγ0 , which equals Pγ0 .
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Electron-Positron Pair Production
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Concerning the second term in the right-hand side of (32) we note that by Cauchy’s formula we obtain
Pγλ ϕ − Pγλ ϕ = Pe1 (λ )
(33)
where Pe1 (λ ) is the projector on the eigenspace corresponding to the eigenvalue e1 (λ ). Consequently ind(Pγλ ϕ , Pγλ ϕ ) = tr[Pe1 (λ ) ]. (34) ¯ whence By means of our definition (17) of d(λ), obviously tr[Pe (λ ) ] = d(λ), 2m+1 ¯ ¯ = d(λ). tr Qλϕ
1
(35)
Repeating this argument whenever an eigenvalue dives into the lower continuum, (−∞, −1], we arrive at the statement of the theorem. Notice, due to continuity in λ the argument works no matter how many eigenvalues “meet” at −1.
3 Proof of Theorem 1 Summarizing the proof of Theorem 2, we exploited the fact that P−λϕ build a continuous family of projectors on the non-connected intervals [0, λ1 ) ∪ (λ1 , λ2 ) ∪ · · · ∪ (λi , λi+1 ) . . .
(36)
where λi denotes parameters where an eigenvalue reaches −1. As long as λ, µ belong to a connected interval the index of the corresponding projection vanishes, ind(P−λϕ , P−µϕ ) = 0,
(37)
but if λ moves to a different not connected interval the index jumps by an integer value. In order to prove Theorem 1 we first recall the definition of the density ρλvac (x) = ρλ1 (x) + ρλ3 (x) + ρλ4 (x),
(38)
the terms on the right-hand side being defined in (11), (13), (15). By means of our n C(k) explicit choice of ρλ1 via Fourier transform ρˆλ1 (k) = eλ4πˆ k2 and the fact that C(k) lim|k|→0 k2 = 0 we immediately obtain ρλ1 (x)dx = ρˆλ1 (0) = 0. (39) R3
Therefore, our goal in the following will be to show that for all λ ρλ3 (x)dx = 0, ρλ4 (x)dx = ed(λ). R3
(40)
R3
It still remains to show that Qλ4 is trace class which works analogously to [11, Lemma 3].
1146
Ch. Hainzl
Lemma 2
Ann. Henri Poincar´e
tr |Qλ4 | = Qλ4 1 ≤ C µ ϕ 44 ,
(41)
with an appropriate constant C µ depending on µ := min{γ + 1, ei (λ) − γ}, ei (λ) denoting the lowest isolated eigenvalue of Dλϕ . Proof. Let ei (λ) be the lowest isolated eigenvalue of Dλϕ , then as usually, we 1 choose a γ with −1 < γ < ei (λ). Using (10) we obtain (apart from a factor 2π )
Qλ4 1 ∞ 1 1 1 1 1 ϕ 0 ϕ λϕ ϕ 0 ϕ 0 ≤ dη 0 D − γ + iη D − γ + iη D − γ + iη D − γ + iη D − γ + iη 1 −∞ ∞ 1 1 1 1 1 ϕ 0 ϕ 0 ϕ 0 ϕ 0 ≤ dη 0 D − γ + iη D − γ + iη D − γ + iη D − γ + iη D − γ + iη 1 −∞ 0 1 , (42) (D × − γ + iη) λϕ D − γ + iη 1 with (D0 − γ + iη) Dλϕ −γ+iη
≤ 1 + λ ϕ Dλϕ1 −γ which depends on µ. Moreover, 1
≤ 1 + µ−1 , with (D0 + iη) D0 −γ+iη
1 1 1 1 1 ϕ ϕ ϕ ϕ dη D0 − γ + iη D0 − γ + iη D0 − γ + iη D0 − γ + iη D0 − γ + iη −∞ 1 ∞ 1 1 1 1 1 −1 5 ≤ dη D0 + iη ϕ D0 + iη ϕ D0 + iη ϕ D0 + iη ϕ D0 + iη (1 + µ ) −∞ 1 3 ∞ 1 1 1 −1 5 ≤ dη ϕ D0 + iη ϕ D0 + iη D0 + iη (1 + µ ) . (43) −∞ 4 4
∞
Applying an inequality of Simon [26, Theorem 4.1],
f (x)g(−i∇) 4 ≤ (2π)−3/4 f 4 g 4 ,
(44)
to the factors in (43), gives 1 1
4 ≤ 1/4 3/4 ϕ 4 1/ | · |2 + 1 + η 2 4 + iη 2 π 1 1 1
4 ≤ 1/4 3/4 ϕ 4 1/(| · |2 + 1 + η 2 ) 4 .
ϕ 0 D + iη D0 + iη 2 π
ϕ
D0
(45)
Putting all together and evaluating the integrals (cf. [11, Lemma 3]) we arrive at
Qλ4 1 ≤ C µ ϕ 44 , with an appropriate C µ .
(46)
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Electron-Positron Pair Production
1147
In the following we will proceed analogously to the proof of Theorem 2. We will circumvent the problem that P−λϕ − P−0 is not trace class by defining a family of trace class operators K ε converging strongly to ϕ. We define K ε via its Fourier representation ˆ ε (p, q) := fε (p)ϕˆε (p − q)fε (q), (47) K with fε (p) := χ(1/ε − |p|),
ϕε (x) := ϕ(x)χ(1/ε − |x|),
(48)
χ denoting the Heaviside step function. Obviously fε → 1 and ϕε → ϕ pointwise when ε → 0. The family of operators ε
DλK := D0 − λK ε
(49) ε
turn out to converge strongly to Dλϕ . For convenience we define QλK via an appropriate γ chosen corresponding to Dλϕ , ε
ε
QλK := PγλK − Pγ0 ,
(50) ε
leaving off the subscript γ, since it will not cause ambiguities. Furthermore PγλK − Pγ0 will be trace class so we can repeat the arguments given in the proof of Theorem 2. Since we already removed the “bad” part of P−λϕ −P−0 by charge renormalization (in [11]), that is the part of Q1 which prevents Pγλϕ − Pγ0 from being trace class, it suffices to show that Qε3 (respectively Qλ,ε 4 ) converge (in trace norm) to Q3 (respectively Qλ4 ). Qε3 and Qλ,ε are terms we obtain by expanding (50). 4 λ 3 ˆ Recall ρ3 (x)dx = eλ R3 trC4 Q3 (p, p)dp. First we state a few useful properties of K ε . ε
Lemma 3 (a) For all ε > 0, K ε is trace class and K ε ≥ 0. Moreover, σess (DλK ) = (−∞, −1] ∪ [1, ∞). ε (b) DλK → Dλϕ strongly as ε → 0. ε ε (c) PγλK − Pγ0 is trace class for all ε > 0 if γ ∈ σ(DλK ). Proof. (a) The fact that K ε is trace class is a direct consequence of Lemma 5. In Fourier representation we can decompose ˆ ε (p, q) = L∗ε Lε (p, q), K
(51)
√ ϕε (p). By our choice of fε with Lε (p, p ) = fε (p)hε (p − p ), hε (p) = (2π)−3/4 and ϕε R3
R3
|Lε (p, q)|2 dpdq < ∞
(52)
whence K ε is trace class. Equation (51) immediately implies K ε ≥ 0. The comε pactness of K ε yields σess (DλK ) = (−∞, −1] ∪ [1, ∞) by Weyl’s Theorem.
1148
(b)
Ch. Hainzl
Ann. Henri Poincar´e
For ψ ∈ H 1 (Γ) ε
lim [DλK − Dλϕ ]ψ = λ lim [K ε − ϕ]ψ = 0,
ε→0
ε→0
(53)
∗ ε ˆ → ϕ∗ since Kˆε K ˆ ϕˆ in the sense of distributions, and these operators are bounded. ε (c) Let eεi (λ) be the lowest isolated eigenvalue of DλK . Then, with −1 < γ < eεi (λ), ε 1 1 1 ∞ PγλK − Pγ0 = − dη 2π −∞ D0 − γ + iη DλK ε − γ + iη 1 1 1 ∞ K ε λK ε . (54) dη 0 =λ 2π −∞ D − γ + iη D − γ + iη
Consequently ε
PγλK − Pγ0 1 ≤ λ K ε 1
1 2π
∞
1
dη −∞
((1 +
γ)2
+
1/2 η2 )
1/2 , δ¯2 + η 2
(55)
1
with δ¯ := min{γ +1, eεi (λ)−γ} using DλK ε 1−γ+iη ≤ (δ¯2 +η 2 )−1/2 and D0 −γ+iη
2 2 −1/2 . ≤ (1 + γ) + η
Let us fix an arbitrary λ such that e1 (λ) > −1 and γ with −1 < γ < e1 (λ). ε Since DλK → Dλϕ strongly, a Theorem of Kato [14, VIII-5, Theorem 5.1] tells us λK ε that σ(D ) is asymptotically concentrated in any open set containing σ(Dλϕ ). Thus we can find a δ small enough such that γ < e1 (λ) − δ and a corresponding ε0 ε such that for all ε ≤ ε0 , σ(DλK ) is concentrated in a δ-neighborhood of σ(Dλϕ ), ε ε in particular ei (λ) > e1 (λ) − δ for each eigenvalue eεi (λ) of DλK . ε Thus we are able to guarantee that PγλK can be continuously deformed into 0 Pγ . Therefore we can argue analogously to the proof of Theorem 2 combined with ε the trace class property of PγλK − Pγ0 to obtain ε
ε
tr[PγλK − Pγ0 ] = ind(PγλK , Pγ0 ) = 0.
(56)
Expanding the resolvent this implies ∞ ∞ 1 1 1 1 1 Kε 0 + λ2 tr Kε 0 Kε 0 dη 0 dη 0 0 = λ tr D + iη D + iη D + iη D + iη D + iη −∞ −∞ ∞ 1 1 1 1 Kε 0 Kε 0 Kε 0 + + λ3 tr dη 0 D + iη D + iη D + iη D + iη −∞ ∞ 1 1 λ4 tr Kε 0 K ε× dη 0 D − γ + iη D − γ + iη −∞ 1 1 1 × λK ε Kε 0 Kε 0 D − γ + iη D − γ + iη D − γ + iη := λ tr Qε1 + λ2 tr Qε2 + λ3 tr Qε3 + λ4 tr Qλ,ε 4 . (57)
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Electron-Positron Pair Production
1149
Observe, this holds in particular in a small neighborhood of 0. Thus, since the fourth term on the right-hand side is of order O(λ4 ), each term in (57) vanishes separately. In particular we have: tr Qε3 = 0,
tr Qλ,ε 4 = 0,
(58)
the latter one on the set {λ|e1 (λ) > −1}. Assume for a moment we have already shown λ,ε 3 ε λ 4 eλ lim tr Q3 = ρ3 (x)dx, eλ lim tr Q4 = ε→0
ε→0
R3
R3
ρλ4 (x)dx,
(59)
then by (58) obviously R3 ρλ3 (x)dx = 0 and R3 ρλ4 (x)dx = 0 whence Theorem 1 on {λ|e1 (λ) > −1}. In order to prove (59) we formulate an auxiliary Lemma: Lemma 4 (a) There exists a non-negative function g ∈ L1 (R3 ), such that ˆ ε3 (p, p) ≤ g(p) trC4 Q
(60)
uniformly in ε. (b) Let ei (λ) be the lowest isolated eigenvalue of Dλϕ . Fix γ and δ¯ with −1 + δ¯ < ¯ Furthermore fix ε0 such that for all ε ≤ ε0 , σ(DλK ε ) is in a γ < ei (λ) − δ. ¯ δ-neighborhood of σ(Dλϕ ). Then µ 4
Qλ,ε 4 1 ≤ C ϕ 4
(61)
uniformly in ε ≤ ε0 , where C µ is an appropriate constant depending on µ := ¯ ei (λ) − δ¯ − γ}, and min{γ + 1 − δ, λ lim tr Qλ,ε 4 = tr Q4 .
ε→0
(62)
Proof. (a) We will proceed similarly to [11, Lemma 4]. For completeness we will repeat some parts of the proof. The “eigenfunctions” of the free Dirac operator in momentum space are σ · p e τ 1 τ = 1, 2, N+ (p) −(1 − E(p))e τ (63) uτ (p) := 1 σ · p eτ τ = 3, 4 N− (p) −(1 + E(p))e τ
with eτ := (1, 0)t for τ = 1, 3 and eτ := (0, 1)t for τ = 2, 4 and N+ (p) = 2E(p)(E(p) − 1), N− (p) = 2E(p)(E(p) + 1).
(64)
1150
Ch. Hainzl
Ann. Henri Poincar´e
The indices 1 and 2 refer to positive “eigenvalue” E(p) and the indices 3 and 4 to negative −E(p). Using Plancherel’s theorem we get ˆ ε3 (p, p) = trC4 Q
4
ˆ ε3 |uτ0 (p) = uτ0 (p)|Q
τ0 =1
R3
dp1
dp2
R3
4 τ0 ,τ1 ,τ2 ,=1
ˆ ε |uτ1 (p1 )uτ1 (p1 )|K ˆ ε |uτ2 (p2 )uτ2 (p2 )|K ˆ ε |uτ0 (p) × uτ0 (p)|K
×
1 (2π)4
∞
dη −∞
1 , (65) (iaτ0 E(p) − η)(iaτ1 E(p1 ) − η)(iaτ2 E(p2 ) − η)(iaτ0 E(p) − η)
with aτ = 1 for τ = 1, 2 and aτ = −1 for τ = 3, 4. The integral over η is seen to vanish by Cauchy’s theorem, if all four aτj have the same sign. In fact we only treat one case. The others then work analogously. Set aτ2 = −1, aτ0 = aτ1 = 1.
(66)
Using fε ≤ 1 the first factor in (65) can be estimated by
ˆ ε |uτ1 (p1 ) uτ0 (p)|K
τ0 =1,2
ˆ ε |uτ2 (p2 ) uτ1 (p1 )|K
τ1 =1,2
ˆ ε |uτ0 (p) uτ2 (p2 )|K
τ2 =3,4
σ · pσ · p + (1 − E(p))(1 − E(p )) 1 1 ˆ 1 − p2 )ϕ(p ˆ 2 − p)| trC2 ≤ |ϕ(p ˆ − p1 )ϕ(p N− (p2 )2 N+ (p)2 N+ (p1 )2 × σ · p1 σ · p2 + (1 − E(p1 ))(1 + E(p2 )) σ · p2 σ · p + (1 + E(p2 ))(1 − E(p)) ˆ 1 − p2 )ϕ(p ˆ 2 − p)| ≤ c|ϕ(p ˆ − p1 )ϕ(p
|p · p2 − (E(p2 ) − 1)(1 + E(p))| + |p ∧ p2 | . N− (p2 )N+ (p) (67)
(c being a generic constant.) Since 1 2π
∞
dη −∞
1 (iE(p) − η)(iE(p1 ) − η)(−iE(p2 ) − η)(iE(p) − η) =
1 2(E(p) + E(p1 ))2 E(p)
(68)
our term of interest (65) is bounded by a constant times
R3
dp1
R3
dp2 |ϕ(p ˆ − p1 )ϕ(p ˆ 1 − p2 )ϕ(p ˆ 2 − p)| ×
|p · p2 − (E(p2 ) − 1)(1 + E(p))| + |p ∧ p2 | . (69) 2N− (p2 )N+ (p)(E(p) + E(p1 ))2 E(p)
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Substituting p2 → p2 + p, p1 → p1 + p2 + p we get |(69)| ≤
R3
dp1 ×
R3
dp2 |ϕ(p ˆ 1 + p2 )ϕ(p ˆ 1 )ϕ(p ˆ 2 )|
|p · (p2 + p) − (E(p2 + p) − 1)(1 + E(p))| + |p ∧ (p2 + p)| . (70) 2N− (p2 + p)N+ (p)(E(p) + E(p1 + p2 + p))2 E(p)
Since |p · (p2 + p) − (E(p2 + p) − 1)(1 + E(p))| + |p ∧ p2 | ≤ 4|p||p2 | we obtain as an upper bound the function dp1 dp2 |ϕ(p ˆ 1 + p2 )ϕ(p ˆ 1 )ϕ(p ˆ 2 )||p2 | g¯(p) := R3
R3
1 , N− (p2 + p)E(p)3
(71)
which is obviously in L1 (R3 ), whence (a) is proven. (b) Analogously to (42) and (43) we get (apart from a constant) ∞
Qλ,ε
≤ dη 1 4 −∞ 1 1 1 1 1 ε ε ε ε K K K K × ε D0 − γ + iη 0 λK 0 0 D − γ + iη D − γ + iη D − γ + iη D − γ + iη 1 ∞ 1 1 1 1 1 ε ε ε ε K K K K ≤ dη D0 + iη D0 + iη D0 + iη D0 + iη D0 + iη 1 −∞ ε 1 (1 + µ−1 )5 . (72) × 1 + K λK ε D −γ The first term in the third line is trace class, so we can evaluate it in Fourier representation. Since fε ≤ 1 we are in the situation of Lemma 2 and end up with
ε −1 (1 + µ−1 )5 ϕ 44 (73)
Qλ,ε 4 1 ≤ c 1 + K µ which implies (61) since K ε is uniformly bounded. In order to prove (62) it suffices to show λ
Qλ,ε 5 − Q5 1 →ε→0 0,
since tr Q4 = 0, with 1 1 1 1 1 ∞ 1 ϕ ϕ ϕ ϕ . Q4 = dη 2π −∞ D0 + iη D0 + iη D0 + iη D0 + iη D0 + iη
(74)
In fact, due to Lemma 2, Q4 is trace class and using the Fourier representation one easily sees that its trace vanishes. Indeed each operator Q2n with an even number
1152
Ch. Hainzl
Ann. Henri Poincar´e
of potentials has vanishing trace, which is well known as Furry’s Theorem. For convenience we denote 1 ∞ λ,ε Q5 = dηNηε Mηε Mηε Mηε Mηε 2π −∞ (75) 1 ∞ λ Q5 = dηNη Mη Mη Mη Mη , 2π −∞ 1 1 where Nηε = DλK ε 1−γ+iη K ε D0 −γ+iη , Mηε = K ε D0 −γ+iη , Nη = 1 and Mη = ϕ D0 −γ+iη . A straightforward calculation gives λ ε
Qλ,ε 5 − Q5 1 ≤ sup Mη − Mη 4 η
1 2π
∞
−∞
1 1 ϕ Dλϕ −γ+iη D0 −γ+iη
dη Nηε Mηε Mηε Mηε 4/3
+ Nηε Mηε Mηε Mη 4/3 + Nηε Mηε Mη Mη 4/3 + Nηε Mη Mη Mη 4/3 1 ∞ ε dη Mη Mη Mη Mη 1 . (76) + sup Nη − Nη ∞ 2π −∞ η
Notice, since trace ideals fulfill Sp ⊂ S1 , for p > 1, we can easily estimate both integrals in (76) in analogy to (61) and (41). Obviously 1/4 1 ˆ ε ] [ϕˆ − K . (77)
Mηε − Mη 4 ≤ dpdqfε (p, q) := 4 p2 + 1 Recalling the definition of K ε in (47), we see that fε (p, q) →ε→0 0 pointwise, as well as 2 1 1 ∈ L1 (R3 × R3 ), |ϕ| ˆ ∗ |ϕ| ˆ (p − q) 2 fε (p, q) ≤ 2 p +1 q +1 which implies, by dominated convergence theorem, that sup Mηε − Mη 4 →ε→0 0. η
Notice that 1 1 → λϕ − γ + iη D − γ + iη
strongly
(78)
1 1 →ϕ 0 D0 − γ + iη D − γ + iη
in S4 ,
(79)
DλK ε and Kε
1 both uniformly in η. Together with the fact that ϕ D0 −γ+iη is compact (even in S4 ), i.e., it can be approximated in norm by a finite rank operator, we conclude
sup Nηε − Nη ∞ →ε→0 0 η
which yields (62).
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Electron-Positron Pair Production
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Now we are ready to prove (59). Obviously, due to our definition (47) and (48), ˆ ε (p, p) → trC4 Q ˆ 3 (p, p) trC4 Q 3
(80)
pointwise as ε → 0. By means of Lemma 4 (a) and the dominated convergence theorem ˆ ε (p, p)dp = eλ3 ˆ 4 trC4 Q tr (p, p)dp = ρλ3 (x)dx. (81) eλ3 lim Q 3 C 3 ε→0
R3
R3
R3
By means of Lemma 4 (b) we obtain 4 λ eλ4 lim tr Qλ,ε 4 = eλ tr Q4 =
ε→0
R3
ρλ4 (x)dx,
(82)
whence (59). ¯ ≤ −1 and e2 (λ) ¯ > −1. (As in the previous ¯ such that e1 (λ) Fix again λ ¯ section we use the notation e1 (λ) for convenience. The argument works whatever happens to the eigenvalue after reaching the lower continuum. Let us remark, that for our results only the features of the eigenvalues before “diving” play a role.) We can find a δ > 0 small enough such that the following holds: We can choose ¯ − δ. Additionally we choose a λ < λ ¯ such that a γ with −1 + δ < γ < e2 (λ) −1 + δ < e1 (λ ) < γ − δ and a γ with −1 + δ < γ < e1 (λ ) − δ. Moreover we find, ¯ ε due to Kato [14, VII-5, Theorem 5.1], an ε0 such that for all ε ≤ ε0 , σ(DλK ) is in ε ¯ a δ-neighborhood of σ(Dλϕ ) as well as σ(Dλ K ) in a δ-neighborhood of σ(Dλ ϕ ). We can write ¯
ε
¯
ε
ε
ε
ε
ε
tr[PγλK − Pγ0 ] = tr[PγλK − Pγλ K ] + tr[Pγλ K − Pγλ K ] + tr[Pγλ K − Pγ0 ]. ¯
ε
(83) ε
By our choice of parameters PγλK can be continuously deformed into Pγλ K , as ε well as Pγλ K into Pγ0 which equals Pγ0 . Consequently the first and the third term on the right-hand side of (83) vanish. Due to Cauchy’s formula
ε
ε
tr[Pγλ K − Pγλ K ] = tr[Pe1 (λ ) ],
(84)
due to the fact that by our choice of parameters the eigenspace of the set {eεj (λ )|γ < eεj (λ ) < γ} has the same dimension as the eigenspace of e1 (λ ). Recall Pe1 (λ ) denotes the projector on the eigenspace corresponding to e1 (λ ). By definition (17)
Whence
¯ tr[Pe1 (λ ) ] = d(λ).
(85)
¯ ε ¯ tr[PγλK − Pγ0 ] = d(λ).
(86)
Expanding the left-hand side as in (57) and using the fact that we already know that the first three terms vanish we see ¯ ¯4 tr Qλ,ε ¯ λ 4 = d(λ).
(87)
1154
Ch. Hainzl ¯
Ann. Henri Poincar´e
¯
λ By means of Lemma 4 (b) limε→0 tr Qλ,ε 4 = tr Q4 , so we infer ¯ ¯ ρλ4 (x)dx = ed(λ).
(88)
R3
Repeating this argument whenever an ei (λ) dives into (−∞, −1] we arrive at the theorem.
A
Criterion for a kernel to be trace class
It is well known that given an integral operator via a kernel K(x, y), the fact that dxK(x, x) < ∞ does not at all guarantee that K is trace class. Rn For a specific class of kernels we give a sufficient condition for the corresponding operator to be trace class. Lemma 5 Let K(x, y) = f1 (x)g(x − y)f2 (y), 2
n
1
(89)
n
with f1 , f2 ∈ L (R ) and gˆ ∈ L (R ). Then K is trace class. Proof. We can write g(k)|1/2 sgn(ˆ g(k)). gˆ(k) = |ˆ g (k)|1/2 |ˆ
(90)
Define g |1/2 ](x), h1 (x) := (2π)−3/2 F −1 [|ˆ
h2 (x) := F −1 [|ˆ g |1/2 sgn(ˆ g )](x),
(91)
such that g ] = (2π)−3/2 F −1 [|ˆ g |1/2 ] ∗ F −1 [|ˆ g |1/2 sgn(ˆ g )] = h1 ∗ h2 . g = F −1 [ˆ
Therefore
dzL1 (x, z)L2 (z, y),
K(x, y) =
(92) (93)
Rn
with
L1 (x, z) = f1 (x)h1 (x − z) L2 (z, y) = h2 (z − y)f2 (y).
Observe
j
2
dxdz|L (x, z)| = Rn
Rn
Rn
Rn
dxdz|fj (x)|2 |hj (z)|2 = fj 22 ˆ g 1 < ∞
(94)
(95)
for j = 1, 2, which implies the Lemma. Acknowledgments. The author acknowledges support through the European Union’s IHP network Analysis & Quantum HPRN-CT-2002-00277. He thanks H. Kalf for valuable explanations and H. Siedentop for continuing support, advises, and warm hospitality at the LMU-Munich. Furthermore he thanks N. Szpak, V. Shabaev, R. Frank, E. S´er´e, and R. Seiringer for important comments.
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References [1] J. Avron, R. Seiler, B. Simon, The Index of a Pair of Projectors, J. Funct. Anal. 120, 220–237 (1994). [2] A. Berthier, V. Georgescu, On the Point Spectrum of Dirac operators, J. Funct. Anal. 71, 309–338 (1987). [3] J. Dolbeault, M.J. Esteban, E. S´er´e, On the eigenvalues of operators with gaps. Application to Dirac operators, J. Funct. Anal. 174, 208–226 (2000). [4] P.-A.-M. Dirac, Th´eorie du positron, In Cockcroft, J. Chadwick, F. Joliot, J. Joliot, N. Bohr, G. Gamov, P.A.M. Dirac, and W. Heisenberg, editors, Structure et propri´et´es des noyaux atomiques. Rapports et discussions du septi`eme conseil de physique tenu a ` Bruxelles du 22 au 29 octobre 1933 sous les auspices de l’institut international de physique Solvay. Publi´es par la commission administrative de l’institut., pages 203–212. Paris: GauthierVillars. XXV, 353 S., 1934. [5] P.A.M. Dirac, Discussion of the infinite distribution of electrons in the theory of the positron, Proc. Camb. Philos. Soc. 30, 150–163 (1934). [6] F.J. Dyson, The radiation theories of Tomonaga, Schwinger, and Feynman, Physical Rev. (2) 75, 486–502 (1949). [7] E. Effros, Why the circle is connected, Math. Intelligencer 11 (1), 27–35 (1989). [8] J.D. French and V.F. Weisskopf, The Electromagnetic Shift of Energy Levels, Phys. Rev., II. Ser. 75, 1240–1248 (1949). [9] W. Greiner, J. Reinhardt, Quantum Electrodynamics, Springer-Verlag, Berlin and Heidelberg and New York and Tokyo, 2 edition, 1994. [10] W. Greiner, B. M¨ uller, and J. Rafelski, Quantum Electrodynamics of Strong Fields, Texts and Mongraphs in Physics. Springer-Verlag, Berlin and Heidelberg and New York and Tokyo, 1 edition, 1985. [11] Ch. Hainzl, and H. Siedentop, Non-perturbative Mass and Charge Renormalization in Relativistic No-photon QED, Commun. Math. Phys. 243, 241–260 (2003). [12] W. Heisenberg, Bemerkungen zur Diracschen Theorie des Positrons, Z. Phys. 90, 209–231 (1934). [13] J.M. Jauch and F. Rohrlich, The theory of photons and electrons. The relativistic quantum field theory of charged particles with spin one-half, AddisonWesley Publishing Company, Inc., Cambridge, Massachusetts, 1955.
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[14] Tosio Kato, Perturbation Theory for Linear Operators, volume 132 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1 edition, 1966. [15] M. Klaus and G. Scharf, The regular external field problem in quantum electrodynamics, Helv. Phys. Acta 50 (6), 779–802 (1977). [16] M. Klaus and G. Scharf, Vacuum polarization in Fock space, Helv. Phys. Acta 50 (6), 803–814 (1977). [17] Norman M. Kroll and Willis E. Lamb jun, On the Self-Energy of a Bound Electron, Phys. Rev., II. Ser. 75, 388–398 (1949). [18] Peter W. Milonni, The Quantum Vacuum: An Introduction to Quantum Electrodynamics, Academic Press, Inc., Boston, 1 edition, 1994. [19] G. Nenciu, G. Scharf, On regular external fields in quantum electrodynamics, Helv. Phys. Acta 51, 412–424 (1977). [20] P.J. Mohr, G. Plunien, G. Soff, QED corrections in Heavy Atoms, Phys. Rep 293, 227–369 (1998). [21] W. Pauli and M.E. Rose, Remarks on the Polarization Effects in the Positron Theory, Phys. Rev., II. Ser. 49, 462–465 (1936). [22] J. Reinhardt, B. M¨ uller, W. Greiner, Theory of positron production in heavyion collision, Phys. Rev. A 24 (1), 103–128 (1981). [23] Michael Reed and Barry Simon, Methods of Modern Mathematical Physics, volume 4: Analysis of Operators, Academic Press, New York, 1 edition, 1978. [24] Julian Schwinger, Quantum Electrodynamics II. Vacuum Polarization and Self-Energy, Phys. Rev., II. Ser. 75, 651–679 (1949). [25] Robert Serber, Linear modifications in the Maxwell field equations, Phys. Rev., II. Ser. 48, 49–54 (1935). [26] Barry Simon, Trace Ideals and their Applications, volume 35 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1979. [27] E.A. Uehling, Polarization effects in the positron theory, Phys. Rev., II. Ser. 48, 55–63 (1935). [28] J. Weidmann, Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics, 68. Springer-Verlag, New York-Berlin, 1980. ¨ [29] V. Weisskopf, Uber die Elektrodynamik des Vakuums auf Grund der Quantentheorie des Elektrons, Math.-Fys. Medd., Danske Vid. Selsk. 16(6), 1–39 (1936).
Vol. 5, 2004
Electron-Positron Pair Production
Christian Hainzl CEREMADE Universit´e Paris-Dauphine Mar´echal de Tassigny F-75775 Paris France and Laboratoire de Math´ematiques Paris-Sud-Bˆ at. 425 F-91405 Orsay Cedex France email:
[email protected] Communicated by Rafael D. Benguria submitted 18/12/03, accepted 08/04/04
To access this journal online: http://www.birkhauser.ch
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Ann. Henri Poincar´e 5 (2004) 1159 – 1180 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/061159-22 DOI 10.1007/s00023-004-0195-3
Annales Henri Poincar´ e
Perturbative Test of Single Parameter Scaling for 1D Random Media Robert Schrader, Hermann Schulz-Baldes and Ara Sedrakyan Abstract. Products of random matrices associated to one-dimensional random media satisfy a central limit theorem assuring convergence to a gaussian centered at the Lyapunov exponent. The hypothesis of single parameter scaling states that its variance is equal to the Lyapunov exponent. We settle discussions about its validity for a wide class of models by proving that, away from anomalies, single parameter scaling holds to lowest order perturbation theory in the disorder strength. However, it is generically violated at higher order. This is explicitly exhibited for the Anderson model.
1 Introduction and main result One-dimensional quantum systems with a single channel can very efficiently be described by 2 × 2 transfer matrices. In a disordered medium, the transfer matrices are chosen to be random. The one-dimensional Anderson model is the proto-type for this class of models. The most important physical phenomena in these random media is localization due to multiple coherent wave scattering. It goes along with positivity of the Lyapunov exponents associated with products of the random matrices and the Lyapunov exponent is then interpreted as the inverse localization length of the system. Moreover, the fluctuations around this asymptotic behavior are gaussian. More precisely, Anderson, Thouless, Abraham and Fisher [ATAF] stated that the Landauer conductance follows asymptotically (in the system size) a log-normal distribution centered at the Lyapunov exponent. As pointed out by Johnston and Kunz [JK], this was a rediscovery of a mathematical result by Tutubalin [Tut] (refined by Le Page [LeP]). The paradigm of single parameter scaling [AALR, ATAF] is then that there is only one parameter describing the asymptotic behavior of the random system. For a one-dimensional model this means that the Lyapunov exponent and the variance of the gaussian should be in some relation and in fact simply be equal [ATAF]. The validity of single parameter scaling in this sense has been analyzed in various particular situations [ATAF, SAJ, CRS, DLA, ST]. The main result of the present work can roughly be resumed as follows: in a wide class of one-dimensional random models single parameter scaling is valid only perturbatively in a weak disorder regime and never holds in a strict sense (exceptional parameter values excluded). For this purpose, we study a general class of one-parameter families of random transfer matrices exhibiting a so-called critical energy. This parameter is the
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effective size of the randomness. In the Anderson model, it is the coupling constant of the disordered potential while in the random dimer model [DWP] it is the distance in energy from the critical energy. We then develop a rigorous perturbation theory in this parameter. In case of the Lyapunov exponent, we do not appeal to the random phase approximation [ATAF, SAJ], but rather show that phase correlations give a contribution to lowest order perturbation theory (which, however, vanishes for the Anderson model). This generalizes arguments of [Tho, PF, JSS]. On the other hand, the perturbation theory for the variance is new to our best knowledge. One has to sum up the phase correlation decay by using adequate counter terms. This rigorous analysis is made possible by a result of Le Page (see Section 6). Furthermore, we extend the techniques of [SS] in order to calculate the scaling exponent of the expectation value of the Landauer conductance (this is sometimes also called a generalized Lyapunov exponent). This generalizes results of [Mol]. Comparing the coefficients to lowest (namely second) order perturbation theory away from Kappus-Wegner type anomalies [KW], we obtain that the Lyapunov exponent and the variance are equal while the scaling exponent of the averaged Landauer conductance is twice this value, just as predicted by [ATAF]. Calculation of higher orders is possible, but cumbersome in general. For the Anderson model it becomes feasible and is carried out in Section 9. We obtain that the next (namely forth) order contributions are not the same for the Lyapunov exponent and the variance and as a consequence they are not equal (but close) in the regime of weak disorder. For the regime of strong disorder, even large discrepancies have been observed numerically [SAJ]. Deviations from single parameter scaling even to lowest order were exhibited at the band center of the Anderson model, which is the prime example of a Kappus-Wegner anomaly [ST]. The Lloyd model analyzed in [DLA] does not fit in our framework because there the random variables do not have finite moments. The example of the Anderson model also allows to show that there does not exist a universal analytic function expressing the variance in terms of the Lyapunov exponent. After this brief introduction, let us describe our results more precisely. The transfer matrices are supposed to be elements of the following subgroup of the general linear group Gl(2, C): 0 −1 J = . (1) U(1, 1) = {T ∈ Mat2×2 (C) | T ∗ JT = J} , 1 0 Using the conjugation ∗
C JC = ı Γ ,
1 C = √ 2
ı 1
ı −1
,
Γ =
1 0
0 −1
.
one sees that U(1, 1) is also isomorphic to the subgroup of matrices T ∈ Gl(2, C) satisfying T ∗ ΓT = Γ. This representation appears in some applications, but for our
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purposes it is more convenient to work with (1) because it contains the standard rotation matrix and the real subgroup SL(2, R). We will study families (Tλ,σ )λ∈R,σ∈Σ of transfer matrices in U(1, 1) depending on a random variable σ in some probability space (Σ, p) as well as a real coupling parameter λ. The dependence on λ is supposed to be smooth. Definition 1 The value λ = 0 is a critical point of the family (Tλ,σ )λ∈R,σ∈Σ if for all σ, σ ∈ Σ: (i) [ T0,σ , T0,σ ] = 0 ,
(ii) |Tr(T0,σ )| < 2 .
(2)
Critical points appear in many applications like in the Anderson model and the random dimer model [DWP, JSS, Sed], but also continuous random Schr¨ odinger operators where the transfer matrix is calculated from a single-site S-matrix [KS]. Condition (i) assures that there is no non-commutativity at λ = 0 (even though the matrices may be random), while by condition (ii) the matrices T0,σ are conjugated to rotations so that there is no a priori hyperbolicity in the system. The example of the Anderson model is studied in more detail in Section 9. Associated to a given semi-infinite code (σn )n≥1 is a sequence of matrices (Tλ,σn )n≥1 . Codes are random and chosen independently according to the product law p⊗N . Averaging w.r.t. p⊗N will be denoted by E. We will suppose that all up to the 5th moment of p exist. In order to shorten notations, we will also write Tλ,n for Tλ,σn . Of interest is the asymptotic behavior of the random products Tλ (N ) =
N
Tλ,n .
n=1
It is first of all characterized by the Lyapunov exponent γ(λ) =
1 E log (Tλ (N )) . N →∞ N lim
(3)
The central limit theorem for products of random matrices now states that 1 −→ √ ( log(Tλ (N )e) − N γ(λ) ) N →∞ Gσ(λ) N
(4)
where Gσ(λ) is the centered Gaussian law of variance σ(λ) and the convergence is in distribution independently of the initial unit vector e. This was first proven by Tutubalin under the hypothesis that the measure p has a density [Tut]. Le Page then proved it for arbitrary measures p [LeP]. Both proofs can be found in [BL]. As already discussed above, the single parameter scaling assumption is the equality σ(λ) = γ(λ) [ATAF, CRS]. Apart from the Lyapunov exponent γ(λ) and the variance σ(λ), we are going to analyze the growth exponent of the average of the Landauer conductance defined by γˆ (λ) =
1 log (E Tr(Tλ (N )∗ Tλ (N ))) . N →∞ 2N lim
(5)
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It follows immediately from Jensen’s inequality that γˆ (λ) ≥ γ(λ). Our main results are resumed in the following: Theorem 1 Introduce the phase ησ ∈ [0, 2π) by cos(ησ ) = 12 Tr(T0,σ ). Suppose E(e2ıjησ ) = 1 for j = 1, 2, 3. Then there is a constant D ≥ 0, given in equation (22) below, such that near the critical point, γ(λ) = D λ2 + O(λ3 ) ,
γˆ (λ) = 2 D λ2 + O(λ3 ) .
(6)
If moreover D > 0, then σ(λ) = D λ2 + O(λ3 ) .
(7)
In Section 4, we also give criteria insuring that D > 0. The remainder of the paper contains the proof of this theorem as well as an analysis of higher orders for the Anderson model.
2 Normal form of transfer matrices near critical point If T ∈ U(1, 1), then T ∗ JT = J implies that det(T ) = e2ıξ for some ξ ∈ [0, π). Hence e−ıξ T ∈ SU(1, 1) = {T ∈ U(1, 1)| det(T ) = 1}. This means U(1, 1) = U(1)× SU(1, 1). Note also that Tr(T ) = det(T )Tr(T ) for T ∈ U(1, 1) so that Tr(T ) ∈ R for T ∈ SU(1, 1). One easily verifies that this implies SU(1, 1) = SL(2, R) is a real subgroup of Gl(2, C). Its Lie algebra is well known: a b a, b, c ∈ R . sl(2, R) = c −a The eigenvalues of T ∈ SL(2, R) always come in pairs κ, 1/κ where κ = Tr(T )/2 + ı 1 − Tr(T )2 /4 and are hence complex conjugate of each other if | Tr(T )| < 2 (which is the case of all transfer matrices near a critical energy). In this situation, the matrix diag(κ, 1/κ) is notin SL(2, R), however, the associcos(η) − sin(η) with κ = eıη is in SL(2, R) and ated rotation matrix Rη = sin(η) cos(η) can be attained with an adequate conjugation. Hence for any T ∈ U(1, 1) with |Tr(T )| < 2, there exists M ∈ SL(2, R) and η such that M T M −1 = eıξ Rη . Let us now consider the family (Tλ,σ )λ∈C,σ∈Σ in U(1, 1) satisfying (2). Because they commute and have a trace less than 2 for λ = 0, they can simultaneously be conjugated to a rotation at that point. Using a Taylor expansion in λ, we therefore obtain the following: M Tλ,σ M −1 = eıξσ (λ) Rησ exp λPσ + λ2 Qσ + O(λ3 ) .
(8)
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Here e2ıξσ (λ) = det(Tλ,σ ) so that λPσ +λ2 Qσ ∈ sl(2, R) for all λ ∈ R. In particular, Tr(Pσ ) = Tr(Qσ ) = 0. The constant C in (6) only depends on the rotation angles ησ and on Pσ through the constant 1 1 v = √ βσ = v|Pσ |v , . (9) −ı 2 Note that βσ = v|Pσ∗ |v, |βσ |2 = 14 Tr(|Pσ |2 + Pσ2 ) and that Pσ2 = (Pσ∗ )2 is a multiple of the identity. Moreover, v and v are the eigenvectors of all rotations Rσ . Given σn , let us now denote the associated random phases, rotations and perturbations by ξn (λ), ηn , Rn , Pn , Qn , thus suppressing the dependence on the random variable σn . In order to use the normal form (8) for the calculation of the Lyapunov and Landauer exponents, let us insert M −1 M in between each pair of transfer matrices. As it only gives boundary contributions, one may also insert an M to the left and an M −1 to the right. Hence,
N 1 −1 E log M Tλ,n M , γ(λ) = lim N →∞ N n=1
as well as a similar formula for γˆ(λ). Now it is clear that there is no use in carrying along the phases eıξσ (λ) in (8) if one is interested in calculating γ(λ) and γˆ (λ). Therefore we may set from now on ξσ (λ) = 0. This is equivalent to supposing that Tλ,σ ∈ SL(2, R). Moreover, one may factor out the subgroup {1, −1}, namely even work with the projection of M Tλ,σ M −1 into PSL(2, R) = SL(2, R)/{1, −1}.
3 Further preliminaries and notations Unit vectors in R2 will be denoted by: cos(θ) , eθ = sin(θ)
θ ∈ [0, 2π) .
(10)
Each transfer matrix Tλ,σ induces an action on unit vectors via eSλ,σ (θ) =
M Tλ,σ M −1 eθ . M Tλ,nM −1 eθ
(11)
Using the vector v of (9), this is equivalent to e2ıSλ,σ (θ) = 2
v|M Tλ,σ M −1 |eθ 2 v|M Tλ,σ M −1 |eθ . = M Tλ,σ M −1 eθ 2 v|M Tλ,σ M −1 |eθ
(12)
Now given an initial condition θ0 , a random sequence of phases θn associated to a code (σn )n∈N is iteratively defined by θn = Sλ,σn (θn−1 ) .
(13)
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A probability measure ν on S 1 is called invariant for this random dynamical system if
f ∈ C(S 1 ) , dν(θ) f (θ) = dν(θ) Eσ f (Sλ,σ (θ)) , where Eσ denotes the average w.r.t. p. Due to a theorem of Furstenberg [BL] ν exists and is unique whenever the Lyapunov exponent is positive. Positivity of the Lyapunov exponent for λ = 0 is guaranteed by condition (iii) of Definition 1 (we leave it to the reader to verify that the subgroup generated by the transfer matrices Tλ,σ is then non-compact so that Furstenberg’s criterium is satisfied [BL]). In the sequel, Eν will mean averaging w.r.t. ν as well as the whole code (σn )n∈N . Next let us turn to the Lyapunov exponent. According to [BL, A.III.3.4] it is given by
N 1 (14) γ(λ) = lim E log M Tλ,n M −1 eθ . N →∞ N n=1
As one can also insert the invariant measure, it is also given by the so-called Furstenberg formula:
γ(λ) = dν(θ) Eσ log M Tλ,σ M −1 eθ . Finally, one can use the random phase dynamics (13) in order to rewrite (14) as γ(λ) =
N 1 Eν log M Tλ,n M −1 eθn−1 . N →∞ N n=1
lim
(15)
4 Asymptotics of the Lyapunov exponent Let us introduce the random variable γn = log M Tλ,n M −1 eθn−1 .
(16)
Then the results cited in the previous section imply γ(λ) =
N 1 Eν γn = lim E (γn ) . n→∞ N →∞ N n=1
lim
(17)
Our first aim is to derive a perturbative formula for γn . Replacing (8), using that Rn is orthogonal and expanding the logarithm shows γn =
λ λ2 ˜ n + |Pn |2 + Pn2 )|eθn−1 eθn−1 |P˜n |eθn−1 + eθn−1 |(Q 2 2 λ2 − eθn−1 |P˜n |eθn−1 2 + O(λ3 ) . (18) 4
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where we have used that (Pn2 )∗ = Pn2 is a multiple of the identity and set P˜n = ˜ n = Qn + Q∗ . Now for any T ∈ Mat2×2 (R), one has Pn + Pn∗ and Q n eθ |T |eθ =
1 Tr(T ) + e v|T |v e2ıθ . 2
Hence using the definition (9) and the remark following it, we deduce λ2 |βn |2 + e γn = 2 λ2 2 4ıθn−1 λ2 2ıθn−1 2 2ıθn−1 ˜ β e v|(|Pn | + Qn )|v e − + + O(λ3 ) . (19) λβn e 2 n 2 The so-called random phase approximation consists in supposing that the angles θn−1 are distributed according the Lebesgue measure (i.e., ν is the Lebesgue measure). Then only the non-oscillatory term in (19) would contribute so that one would get γ(λ) = 12 λEσ (|βσ |2 ) + O(λ3 ). In general, however, this is erroneous. Replacing (19) into (16), one has to calculate the following oscillatory sums (as in [JSS]). Lemma 1 For j = 1, 2, set Ij (N ) = E
N −1 1 2jıθn e . N n=0
Suppose Eσ e2jıησ = 1 for j = 1, 2. Then λ Eσ βσ e2ıησ I1 (N ) = + O(λ2 , N −1 ) , 1 − Eσ (e2ıησ )
I2 (N ) = O(λ, N −1 ) .
Proof. It follows from (12) and (8) that e2ıθn = 2 e2ıηn
v|(1 + λPn )|eθn−1 2 + O(λ2 ) . (1 + λ Pn )eθn−1 |(1 + λPn )eθn−1
(20)
In particular, this implies e2ıθn = e2ı(ηn +θn−1 ) + O(λ) , so that, replacing this in each term, one obtains Ij (N ) = Eσ e2jıησ Ij (N ) + O(λ, N −1 ) . The induction hypothesis therefore implies that Ij (N ) = O(λ, N −1 ). In order to calculate the contribution of O(λ) to I1 (N ), let us expand (20). Some algebra shows that e2ıθn = e2ı(ηn +θn−1 ) − λ e2ı(ηn +θn−1 ) 2ıθn−1 βn − 2v|Pn |v − e−2ıθn−1 βn + O(λ2 ) . (21) e
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From the last three terms, those containing still an oscillatory factor e2ıθn or e4ıθn will not contribute to leading order λ due to the above. Thus we deduce I1 (N ) = Eσ e2ıησ I1 (N ) + λ Eσ e2ıησ βσ + O(λ2 , N −1 ) .
This implies the result. 2ıjθ
2ıθ
As limN →∞ Ij (N ) = Eν (e ), the lemma shows that Eν (e ) = O(λ) and calculates the lowest order contribution. If βσ is centered (as for the Anderson model), one even has Eν (e2ıθ ) = O(λ2 ). This shows in particular how far the invariant measure ν is away from the Lebesgue measure and that phase correlations are indeed present. Now we can proceed with the calculation of γ(λ). Carrying out the algebra shows γ(λ) = D λ2 + O(λ3 ) where 1 Eσ (βσ ) Eσ (βσ e2ıησ ) D = Eσ (|βσ |2 ) + e . (22) 2 1 − Eσ (e2ıησ ) The second summand is due to phase correlations. It is important, e.g., in the random polymer model [JSS]. In the Anderson model treated in Section 9, phase correlations only contribute to the forth order in λ. Because γ(λ) ≥ 0, the coefficient D defined by (22) has to be non-negative. More precisely, we prove: Proposition 1 Suppose Eσ e2ıησ = 1. Then D is always non-negative. D vanishes if and only if one of the following two mutually excluding cases occurs: (i) Both e2ıησ and βσ are p-a.s. constant. (ii) Eσ e2ıησ = 0 and βσ is a constant multiple of 1 − e2ıησ . Proof. Assume first that e2ıησ is a.s. constant. Then Eσ e2ıησ = e2ıησ = 1 and it follows from e(1 − eıϕ )−1 = 1/2 that 2 D = Eσ (|βσ |2 ) − |Eσ (βσ )|2 . By the Cauchy-Schwarz inequality, D ≥ 0 and D = 0 if and only if βσ is a.s. constant. Now let us assume that e2ıησ is not a.s. constant so that |Eσ e2ıησ | < 1. The proposition then follows from the following lemma by setting H = L2 (p), ψ1 = 1 and ψ2 = e2ıησ . Lemma 2 Let ψ1 and ψ2 be two linearly independent unit vectors in a Hilbert space H, implying |ψ1 |ψ2 | < 1. Then the quadratic form Q(ψ) = ψ|ψ +
1 1 ψ|ψ2 ψ1 |ψ + ψ|ψ1 ψ2 |ψ 1 − ψ1 |ψ2 1 − ψ2 |ψ1
on H is positive semi-definite. It is positive definite if and only if ψ1 |ψ2 = 0. If ψ1 |ψ2 = 0, then Q(ψ) = 0 if and only if ψ is a multiple of ψ1 − ψ2 . Proof. Let K be the two-dimensional subspace of H spanned by ψ1 and ψ2 and K⊥ its orthogonal complement. For an arbitrary vector ψ, we write ψ = ψ + ψ with
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ψ ∈ K and ψ ∈ K⊥ . Then Q(ψ) = Q(ψ ) + ψ 2 . Hence it suffices to assume ψ = ψ ∈ K, i.e., we may restrict Q to K. If we introduce the orthonormal basis 1 e2 = (ψ2 − ψ1 |ψ2 ψ1 ) 1 − |ψ1 |ψ2 |2
e1 = ψ1 ,
in K, then with respect to this orthonormal basis the quadratic form Q on K is represented by the 2 × 2 matrix √ 2 2 Q =
1+| ψ1 |ψ2 | (1− ψ1 |ψ2 )(1− ψ2 |ψ1 )
1−| ψ1 |ψ2 | 1− ψ2 |ψ1
√
1−| ψ1 |ψ2 |2 1− ψ1 |ψ2
.
1
Now its trace is obviously strictly positive, while its determinant satisfies 2 |ψ1 |ψ2 |2 ≥ 0. (1 − ψ1 |ψ2 )(1 − ψ2 |ψ1 )
det Q =
Hence both eigenvalues of Q are strictly positive whenever ψ1 and ψ2 are not orthogonal, while one eigenvalue is strictly positive and the other one zero if they are orthogonal. For this zero the corresponding eigenvector of Q is eigenvalue 1 proportional to the vector . This concludes the proof of the lemma. −1
5 Perturbation theory for the variance Using telescoping as in (17), the variance is given by 1 N →∞ N
σ(λ) = lim
N
Eν (γn − γ) (γk − γ)
n,k=1
N N −n 1 2 2 2 γn γn+m − γ Eν γn − γ + 2 = lim , N →∞ N n=1 m=1 where we wrote γ for γ(λ) for notational simplicity. Let us show that the sum over m is convergent even if N → ∞. To this aim, we denote by En the expectation over all σm with m ≥ n such that with the previous notation E1 = E. From the remark following Proposition 2 in Section 6 it follows that En (γn+m ) converges exponentially fast to γ for m → ∞. Moreover, the summands of the sum over n converge in expectation so that, if ν is the unique invariant measure (integration w.r.t. θ0 ), the variance is given by ∞ ∞ 2 2 2 2 2 (γm − γ) . σ(λ) = Eν γ1 − γ + 2 γ1 γm − γ = Eν γ1 − γ + 2γ1 E2 m=2
m=2
(23)
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Hence we need to evaluate the sums appearing in the following lemma. Its proof is deferred to Section 6. Lemma 3 Suppose Eσ e2ıjησ = 1 for j = 1, 2, 3 and D > 0. Then ∞
E2
(γm − γ) = λ e
m=2
Eσ (βσ ) e2ı(θ0 +η1 ) + O(λ2 ) . 1 − Eσ (e2ıησ )
As γ = O(λ2 ), the first summand γ 2 gives no contribution. Therefore we obtain 2 λ Eσ (βσ ) 2ı(θ0 +η1 ) γ e σ(λ) = Eν γ12 + e + O(λ3 ) . 1 1 − Eσ (e2ıησ ) Let us calculate the first contribution supposing that ν is the invariant measure. Then extracting the linear coefficient in λ from (19) and using Lemma 1 Eν (γ12 )
Similarly
=
lim
N →∞
N 1 Eν γn2 N n=1 N 2 1 Eν βn e2ıθn−1 + βn e−2ıθn−1 + O(λ3 ) N n=1
=
λ2 4
=
λ2 Eσ |βσ |2 + O(λ3 ) . 2
lim
N →∞
λ Eν γ1 e2ı(θ0 +η1 ) = Eσ βσ e2ıησ + O(λ2 ) . 2
This implies σ(λ) = γ(λ) + O(λ3 ) .
6 Estimates on the correlation decay The main purpose of this section is to prove Lemma 3. We choose a slightly more general formulation, however, allowing to treat also other quantities. The first aim is to show how the estimates of [BL, Theorem A.V.2.5] (due to Le Page [LeP]) can be made quantitative when combined with Lemma 1. Throughout this section, we suppose that D > 0 and, for sake of notational simplicity, λ ≥ 0. Let us introduce a distance on S 1 by δ(θ, ψ) =
1 − eθ |eψ 2 = eθ ∧ eψ ,
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where the second norm is in Λ2 R2 . For α ∈ (0, 1], the space of H¨ older continuous functions Cα (S 1 ) is given by the continuous functions f ∈ C(S 1 ) with finite H¨older norm f α = max{f ∞ , mα (f )}, where mα (f ) =
sup θ,ψ∈S 1
|f (θ) − f (ψ)| . δ(θ, ψ)α
Then we define S, P : Cα (S 1 ) → Cα (S 1 ) by
(Sf )(θ) = E(f (Sλ,σ (θ))) ,
(Pf )(θ) =
dν(ψ) f (ψ) .
Here P should be thought of as the projection on the constant function with value given by the average w.r.t. the unique invariant measure. Both S and P depend on λ. Proposition 2 There exist a constants c, d > 0 such that for α = dλ 3
(S N − P)(f )α ≤ f α e−c λ
N
.
Proof. Let us introduce ζ : SL(2, R) × S 1 × S 1 → R by δ(ST (θ), ST (ψ)) ζ(T, (θ, ψ)) = log , δ(θ, ψ) where ST : S 1 → S 1 is defined as in (11) by eST (θ) = T eθ /T eθ . Then ζ is a cocycle, namely it satisfies ζ(T T, (θ, ψ)) = ζ(T , (ST (θ), ST (ψ))) + ζ(T, (θ, ψ)) . Moreover, one has ζ(T, (θ, ψ)) = log
Λ2 T eθ ∧ eψ T eθ T eψ eθ ∧ eψ
= − log(T eθ ) − log(T eψ ) ,
because Λ2 T eθ ∧ eθ+ π2 = 1. Therefore |ζ(M Tλ,n M −1 , (θ, ψ))| ≤ c0 λ and the cocycle property implies |ζ(Tλ (n), (θ, ψ))| ≤ c1 λ n . Furthermore, invoking Lemma 1 shows that E ζ(Tλ (n), (θ, ψ)) ≤ −2D λ2 n + c2 λ3 n + λ , where D > 0 is the coefficient given in (22). Defining the sequence of angles ψn as in (13), but with initial condition ψ0 = ψ, one can now infer from ex ≤ 1+x+x2 e|x| /2
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that δ(θn , ψn )α E ≤ δ(θ, ψ)α
Ann. Henri Poincar´e
1 + α E ζ(Tλ (n), (θ, ψ)) 1 2 α E ζ(Tλ (n), (θ, ψ))2 eα |ζ(Tλ (n),(θ,ψ))| 2 1 2 1 − 2D α λ n + c2 α (λ3 n + λ) + c21 (αλn)2 ec1 αλn . 2 +
≤
Now one has on the one hand, n
mα ((S − P)(f )) =
sup θ,ψ∈S 1
E f (θn ) − f (ψn ) δ(θ, ψ)α ≤ mα (f )
sup E
θ,ψ∈S 1
δ(θn , ψn )α δ(θ, ψ)α
,
and, furthermore, using the invariance of ν and then δ(θ, ψ) ≤ 1, n (S − P)(f )∞ = sup dν(ψ) E f (θn ) − f (ψn ) θ∈S 1
≤ mα (f ) 1 − 2 α D λ2 n + c2 α (λ3 n + λ) + c1 2 (αλn)2 ec1 αλn .
Therefore, we can deduce (S n − P)(f )α ≤ mα (f ) 1 − α D λ2 + c2 (λ3 + λn−1 ) + c21 (αλn)2 ec1 αλn . Now we choose n = [c3 /λ] + 1 (as usual [x] denotes the integer part of x ∈ R) and α = c4 λ with adequate c3 , c4 > 0 and note that mα (f ) ≤ mα (f ) for α ≤ α, so that for some c5 > 0 c 3 +1. (S n − P)(f )α ≤ f α (1 − c5 λ2 ) , n= λ Finally, we note that S N − P = (S n − P)N/n because SP = PS = P so that iterating the last inequality completes the proof. When applied to the function f (θ) = Eσ (log(M Tλ,σ M −1 eθ )) (which is H¨ older continuous for any α) this estimate directly implies that E(γn ) converges exponentially fast to γ when n → ∞. Furthermore, using mα (f ) ≤ ∂θ f ∞ , one now gets: Corollary 1 There is a constant c > 0 such that E
∞ f (θm ) − Eν (f (θ)) ≤ c max{f ∞, ∂θ f ∞ } λ−3 . m=1
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We will now consider an algebra A of functions which are analytic on some neighborhood of {0} × S 1 ⊂ C2 and have a power series of the form λk Fk,l z 2l = λk Fk (z) , (24) F (λ, z) = k≥0
k≥0
|l|≤k
with complex coefficients Fk,l . The only elementary fact needed here is that F ·G ∈ A whenever F, G ∈ A. We will mainly be interested in the values on R × S 1 and there also write F (λ, θ) = F (λ, e2ıθ ). Let us give two examples for elements in A. One is the dynamics defined in (11), namely (λ, θ) → e2ı(Sλ,σ (θ)−θ) is a function in A because it is a quotient of analytic functions in (λ, θ) of the form (24). Taking its jth power, one obtains the representation j λk ck,l (σ) e2ı l θ , e2ıjSλ,σ (θ) = e2ıj(θ+ησ ) 1 + (25) k≥1
|l|≤k
j (σ). Comparing with (21), we see that for adequate complex coefficients ck,l 1 1 c1,1 (σ) = −βσ , c1,0 (σ) = 2 v|Pσ |v and c11,−1 (σ) = −βσ . Our second example is the function F (λ, θ) = Eσ (log(M Tλ,σ M −1 eθ )). With this function, one has Em (γm ) = F (λ, θm−1 ) and γ(λ) = Eν (F (λ, θ)). This function appears in Lemma 3. Generalizing Lemma 3, we are led to evaluate (perturbatively in λ) the summed up correlation decay for an arbitrary function F ∈ A: ∞
Cor(F )(λ) = E2
F (λ, θm ) − Eν (F (λ, θ)) .
m=1
From Corollary 1 follows the a priori estimate |Cor(F )(λ)| ≤ C λ−3 . For simplicity, let us now calculate Cor(F )(λ) to order λ, namely discard terms of order O(λ2 ). This is the situation covered by Lemma 3. Consider G(λ, θ) = F (λ, θ) − 4 k 5 5 k=0 λ Fk (θ) = O(λ ) so that ∂θ G(λ, . )∞ = O(λ ). Combined with Corollary 1, it follows: Cor(F )(λ) = E2
∞
G(λ, θm ) +
m=1
=
=
4 k=0 4 k=1
4 k=0
k
λ Fk (θm ) −
4
k
λ Eν (Fk (θ)) − Eν (G(λ, θ))
k=0
∞ Fk (θm ) − Eν (Fk (θ)) + O(λ2 )
k
λ E2
m=1
λk
|l|≤k
Fk,l E2
∞ 2ılθm e − Eν (e2ılθ ) + O(λ2 ) . m=1
The appearing sum over m is finite (again due to Corollary 1) and, moreover, we can calculate its value perturbatively.
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Lemma 4 Let Jj = E2
∞
e2ıjθm − Eν (e2ıjθ ) ,
j∈Z.
m=1
Suppose Eσ e2ıjησ = 1 for j = 1, . . . , 4. Then J4 = O(λ−2 ), J3 = O(λ−1 ), J2 = O(1) and 1 e2ı(θ0 +η1 ) + O(λ) . J1 = (26) 1 − Eσ (e2ıησ ) Proof. We use (25) in order to express e2ıjθm in terms of e2ıjθm−1 , but truncate the expansion at O(λK ). Using again Corollary 1, we deduce K ∞ j e2ıj(θm−1 +ηm ) 1 + Jj = E2 λk Eσ (ck,l (σ))e2ılθm−1 m=1
− Eσ e2ıjησ Eν e2ıjθ 1 +
k=1
K k=1
λk
|l|≤k
j Eσ (ck,l (σ))e2ılθ + O(λK−2 )
|l|≤k
K j = Eσ e2ıjησ Jj + e2ıj(θ0 +η1 ) + Eσ e2ıjησ λk Eσ (ck,l (σ)) Jj+l k=1
+ O(λK−2 , λ)
=
|l|≤k
K 1 j e2ıj(θ0 +η1 ) + Eσ e2ıjησ λk Eσ (ck,l (σ)) Jj+l 1 − Eσ (e2ıjησ ) k=1
|l|≤k
+ O(λK−2 , λ) , where O(λK−2 , λ) = O(λK−2 ) + O(λ) and in the second equality we used the fact Eν (e2ıjθ ) = O(λ) (due to Lemma 1). As we know that J0 = 0 and that the a priori estimate Jj = O(λ−3 ) holds, this calculation shows that Jj = O(λ−2 ) if 2ıjη σ Eσ e = 0. The estimate J3 = O(λ−1 ) now follows by choosing K = 1 and replacing J4 = O(λ−2 ) and J2 = O(λ−2 ). The same way one deduces J2 = O(λ−1 ) and J1 = O(λ−1 ). Choosing K = 2, a similarly argument shows in the next step J2 = O(1) and J1 = O(1). In order to establish (26), let us choose K = 3. The first term in the last square bracket gives the desired contribution, while the sum is O(λ) because J2 = O(1), J3 = O(λ−1 ) and J3 = O(λ−2 ). In order to calculate Cor(F )(λ), let us recall that J0 = 0 and J−j = Jj . Thus Cor(F )(λ) = λ F1,1 J1 + λ F1,−1 J1 + O(λ2 ) , with J1 given by Lemma 4. For the function F (λ, θ) = Eσ (log(M Tλ,σ M −1 eθ )), equation (19) implies F1,1 = 12 Eσ (βσ ) = F1,−1 . This gives Lemma 3.
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7 Some identities linked to the adjoint representation The adjoint representation of SL(2, R) on its Lie algebra is defined by AdT (t) = T tT −1, t ∈ sl(2, R). It leaves invariant the quadratic form q(t, s) = 12 Tr(ts) of signature (2, 1) (note that q(t, t) = − det(t)). A basis B = {b1 , b2 , b3 } of sl(2, R) is given by 1 0 0 1 0 −1 b1 = , b2 = , b3 = . 0 −1 1 0 1 0 It is orthonormal w.r.t. the scalar product s|t = 12 Tr(s∗ t). Moreover, denoting the coordinate map w.r.t. this basis by K B : sl(2, R) → R3 and the standard scalar product in R3 also by x | y for x, y ∈ R3 , we have 1 0 0 q(t, s) = K B (t)| Γ2,1 |K B (s) , Γ2,1 = 0 1 0 . 0 0 −1 B B −1 . This means (AdB Finally, let us set AdB T = K AdT (K ) T )j,k = bj |AdT (bk ). Note that J = b3 ∈ SL(2, R) and that AdB = −Γ . Thus T ∗ JT = J implies 2,1 J B ∗ B B (AdT ) Γ2,1 AdT = Γ2,1 . Hence AdT is an element of the Lorentz group SO(2, 1) B given by all A ∈ Mat3×3 (R) satisfying A∗ Γ2,1 A = Γ2,1 . As AdB 1 = Ad−1 = 1, the adjoint representation gives an isomorphism PSL(2, R) ∼ = SO(2, 1). Important for the sequel is that the eigenvalues µ1 , µ2 , µ3 of A ∈ SO(2, 1) satisfy µ1 µ2 µ3 = ±1. One of these eigenvalues, say µ3 , must be real, while the other two may either be real as well or be complex conjugates of each other. a ˆ ˆb Next it follows from a short calculation that for T = ∈ SL(2, R) cˆ dˆ ˆ ˆ a ˆd + bˆ c dˆˆb − a ˆcˆ a ˆcˆ + ˆbdˆ dˆ ˆc − a AdB ˆˆb 12 (dˆ2 − ˆb2 − cˆ2 + a ˆ2 ) 12 (dˆ2 − ˆb2 + cˆ2 − a ˆ2 ) . T = 1 ˆ2 1 ˆ2 2 2 2 2 2 ˆ ˆ ˆ ˆ dˆ c+a ˆb 2 (d + b − cˆ − a ˆ ) 2 (d + b + cˆ + a ˆ2 )
In particular, for a rotation Rη by η we get cos(2η) − sin(2η) B AdRη = sin(2η) cos(2η) 0 0
0 0 . 1
The e2ıη , e−2ıη , 1 with respective eigenvectors v1 = (e1 − ıe2 ) √ eigenvalues are √ / 2, v2 = (e1 + ıe2 )/ 2, v3 = e3 . We will also need the adjoint representation of the normal form (8): λ2 AdR exp(λP +λ2 Q) = AdR 1 + λ adP + λ2 adQ + (adP )2 + O(λ3 ) , 2 (27)
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where adt (s) = [t, s] for t, s ∈ sl(2, R). Let us write out more explicit formulas in a b the representation w.r.t. the basis B. For P = ∈ sl(2, R), c −a 0 b−c b+c c−b 0 −2a , adB (28) P = b + c −2a 0 and
4bc −2a(b + c) 2 2 (adB ) = −2a(b + c) 4a − (c − b)2 P 2a(b − c) b 2 − c2
2a(c − b) . c2 − b 2 (b + c)2 + 4a2
(29)
Finally let P : C2 ⊗C2 → C2 ⊗C2 be the permutation operator Pφ⊗ψ = ψ⊗φ. It can readily be checked that 3 1 det(bj ) bj ⊗ bj . P = 1 ⊗ 1 − 2 j=1 Multiplying this identity from the left by T ⊗ J and from the right by 1 ⊗ T −1 J, one gets for T ∈ SL(2, R), 3 1 . T ⊗ Tt P = −J ⊗ J + det(bj ) (AdB (30) T )k,j bk J ⊗ Jbj 2 j,k=1
This is useful for the calculation of the Landauer conductance because T w2 = w| Tr1 (T ⊗ T t P)|w , where Tr1 is the partial trace over the first component of C2 ⊗ C2 . Replacing (30), the first term vanishes because Tr(J) = 0. Moreover, Tr(bk J) = −2 δk,3 so that T w2 =
3
(AdB T )3,j (−1) det(bj ) w|Jbj |w .
j=1
Let us define a map g : C2 → R3 by w1 w2 + w1 w2 det(b1 ) w|Jb1 |w g(w) = − det(b2 ) w|Jb2 |w = |w2 |2 − |w1 |2 , det(b3 ) w|Jb3 |w |w1 |2 + |w2 |2 Then one has
T w2 = e3 | AdB T |g(w) .
w=
w1 w2
.
(31)
Since g(v) = e3 , this implies in particular e3 . T v2 = e3 | AdB T |
(32)
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8 Calculation of the averaged Landauer resistance The Landauer resistance ρλ (N ) of a system of length N is defined by ρλ (N ) = E Tr |Tλ (N )|2 = 2 E Tλ (N )v2 , with v as in (9). Because |Tλ (N )|2 is positive and has unit determinant, one has ρλ (N ) ≥ 2 for all N . Using the identity (32) and then the representation property B B AdB ST = AdS AdT iteratively, the expectation value appearing in the Landauer resistance can readily be calculated: N e3 , ρλ (N ) = 2 e3 | (E (AdB Tσ )) |
so that γˆ(λ) = limN →∞ log(ρλ (N ))/(2N ). Replacing the normal form (8) (recall that the phases disappear right away in the definition of the Landauer conductance): N t e3 | E AdB |AdB e3 . ρλ (N ) = 2 (AdB M −1 ) Rσ exp(λPσ +λ2 Qσ +O(λ3 )) M Next we need to do (non-degenerate, in λ) perturbation theory of the eigenvalues µ1 , µ2 , µ3 of E AdB Rσ exp(λPσ +λ2 Qσ +O(λ3 )) , which according to (27) is up to O(λ3 ) given by B B B B 2 E(AdB Rσ ) + λ E(AdRσ adPσ ) + λ E(AdRσ adQσ ) +
λ2 B 2 E(AdB Rσ (adPσ ) ) , 2
2ıησ Let us note that the eigenvalues of E(AdB ), µ2 = Rησ ) are µ1 = E(e −2ıησ ), µ3 = 1 with the eigenvectors v1 , v2 and v3 respectively. Therefore, E(e unless ησ is independent of σ, the matrix E(AdB Rησ ) is not an element of SO(2, 1) and two of its (complex conjugate) eigenvalues are strictly within the unit circle.
e3 = 0, one has µ3 = We first focus on the eigenvalue µ3 . Because e3 |adB P | 1 + O(λ2 ) and the eigenvector is v3 + O(λ2 ). Second order perturbation theory now shows µ3
=
λ2 B 2 v3 | E (AdB v3 Rσ (adPσ ) )| 2 1 B (1 − |v3 v3 |) + λ2 v3 |E(AdB Rσ adPσ ) 1 − E(AdB Rσ )
1 +
B E(AdB v3 + O(λ3 ) . Rσ adPσ )|
√ v1 = √12 (b+c+2ıa) = ı 2 βσ , Recalling (b+c)2 +4a2 = 4 |βσ |2 and using v3 |adB Pσ | we deduce E(βσ ) E(βσ e2ıησ ) µ3 = 1 + 2 λ2 E(|βσ |2 ) + 2 e (33) + O(λ3 ) . 1 − E(e2ıησ )
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Now let us analyze the eigenvalues µ1 and µ2 . If | E(e2ıησ )| < 1 (as for the dimer model), they are strictly within the unit disc and remain there also for λ sufficiently small. As ρλ (N ) ≥ 2, we conclude that µ3 ≥ 1 (this also implies D ≥ 0, like in Proposition 1, because otherwise ρλ (N ) → 0 for N → ∞). It follows that the scaling exponent of the Landauer resistance defined in (5) is given solely by µ3 , namely γˆ (λ) = 12 (µ3 − 1) + O(λ3 ) so that, when comparing with (22), γˆ (λ) = 2 γ(λ) + O(λ3 ) as claimed in Theorem 1. In the case where ησ = η independently of σ (as in the Anderson model), one has D ≥ 0. First order perturbation theory (in λ) shows that µ1,2 move along the the unit circle, while in second order they lie inside of the unit circle. Thus the same argument as above applies to deduce Theorem 1.
9 Higher orders for the Anderson model This section serves two purposes: it provides an example to which the general theory applies and we moreover outline (algebra is left to the reader) how it can be extended to calculate the next higher order in perturbation theory. The implications of the results have already been discussed in the introduction. The one-dimensional Anderson model is a random Jacobi matrix given by the finite difference equation −ψn+1 + λ vn ψn − ψn−1 = E ψn . Here |E| < 2 is a fixed energy and vn are centered real i.i.d. random variables with finite moments. This equation is rewritten as usual using transfer matrices: λvn − E −1 ψn ψn+1 . = Tλ,n , Tλ,n = 1 0 ψn ψn−1 As above, we also write vσ for one of the random variables such that vn = vσn and Tλ,σn = Tλ,n . For the basis change to the normal form of the transfer matrix Tλ,σ , let us introduce 1 sin(k) 0 E = −2 cos(k) , M = . − cos(k) 1 sin(k) It is then a matter of computation to verify M Tλ,σ M
−1
= Rk (1 + λ Pσ ) ,
Pσ
vσ = − sin(k)
0 0 1 0
.
Comparing with (8), we see that the rotation Rk by the angle k is not random in this example, that Pσ is nilpotent so that exp(λPσ ) = 1 + λPσ and that Qσ = 0. Furthermore ı vσ , v|Pσ |v = −βσ . βσ = 2 sin(k)
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One has βσ2 = −|βσ |2 , βσ4 = |βσ |4 and eθ |P˜σ |eθ = βσ (e2ıθ − e−2ıθ ) ,
eθ | |Pσ |2 |eθ = |βσ |2 (2 + e2ıθ + e−2ıθ ) . (34) As E(βσ ) = 0, one can immediately deduce from (22) the well-known formula [Tho, PF, Luc] for the Lyapunov exponent, namely γ(λ) = 12 λ2 E(|βσ |2 )+O(λ3 ) = 2
σ| ) 3 2ıjk λ2 8E(|v = 1 for j = 1, 2 (the latter condition excludes sin2 (k) + O(λ ) as long as e band edges and the Kappus-Wegner anomaly at the band center E = 0).
In order to calculate the 4th order of the Lyapunov exponent and the variance, we need higher-order expansions than (21) and (18). We will assume e2ıjk = 1 for j = 1, . . . , 4. After some algebra, e2ıSλ,σ (θ)
v|Rk (1 + λPσ )|eθ v|Rk (1 + λPσ )|eθ = e2ı(θ+k) 1 − λβσ (e2ıθ + 2 + e−2ıθ )
=
+λ2 βσ2 (e4ıθ + 3e2ıθ + 3 + e−2ıθ ) + O(λ3 ) ,
(35)
and, using (34), γλ,σ (θ)
= log((1 + λPσ )eθ ) 1 4ıθ 1 e + e2ıθ + = e λβσ e2ıθ − λ2 βσ2 2 2 1 6ıθ e + e4ıθ + e2ıθ +λ3 βσ3 3 3 1 1 +λ4 βσ4 − e8ıθ − e6ıθ − e4ıθ − e2ıθ − . + O(λ5 ) (36) 4 2 4
Using an argument similar to Lemma 1, we deduce from (35) and its square that Eν (e2ıθ ) =
λ2 E(|βσ |2 ) + O(λ3 ) , 1 − e−2ık
Eν (e4ıθ ) =
λ2 E(|βσ |2 ) + O(λ3 ) . 1 − e−4ık
Moreover, Eν (e6ıθ ) = O(λ2 ) and Eν (e8ıθ ) = O(λ2 ). Following the argument of Section 4, we therefore obtain from (36) and e(1 − eıϕ )−1 = 12 3 1 1 E(|βσ |2 )2 − E(|βσ |4 ) + O(λ5 ) . γ(λ) = λ2 E(|βσ |2 ) + λ4 2 4 4 This coincides with the fourth-order contribution obtained in [Luc] by using complex energy Dyson-Schmidt variables. In order to calculate the variance according to formula (23), one first needs to go through the arguments of Section 6. One finds E2
∞ 2ıθm e − Eν (e2ıθ ) = m=1
e2ıθ0 1 − λβ1 (e2ıθ0 + 2 + e−2ıθ0 ) + O(λ2 ) , e−2ık − 1
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a similar expression for the correlation sum of e4ıθ , and that E2 is up to O(λ4 ) equal to 2
2
λ E(|βσ | ) e
∞
m=2 (γm
e2ıθ0 (1 − λβ1 (e2ıθ0 + 2 + e−2ıθ0 )) e−2ık − 1 1 e4ıθ0 (1 − 2λβ1 (e2ıθ0 + 2 + e−2ıθ0 )) + 2 e−4ık − 1
− γ)
.
Finally, after having squared (36), Eν (γ12 ) − γ 2 =
1 2 1 1 λ E(|βσ |2 ) − λ4 E(|βσ |4 ) − λ4 E(|βσ |2 )2 + O(λ5 ) , 2 8 2
and Eν
2 γ1 E2
∞
(γm − γ)
m=2
=
7 4 λ E(|βσ |2 )2 + O(λ5 ) . 8
Combining these results according to (23), we obtain 3 1 2 1 2 4 2 2 4 E(|βσ | ) − E(|βσ | ) + O(λ5 ) . σ(λ) = λ E(|βσ | ) + λ 2 8 8 Therefore we see that Lyapunov exponent and variance are only equal to lowest order in perturbation theory. Finally let us also argue that there cannot exist a universal analytic function f such that σ = f (γ). Indeed, if f (x) = f1 x + f2 x2 + O(x3 ), then σ(λ) = f (γ(λ)) for the Anderson model implies in order λ2 that f1 = 1, but in order λ4 there is already a problem due to the prefactors of E(|βσ |4 ). Acknowledgments. This work profited from financial support of the SFB 288. H. S.B. would like to thank T. Kottos for pointing out references [DLA] and [ST].
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P. Bougerol, J. Lacroix, Products of Random Matrices with Applications to Schr¨ odinger Operators, (Birkh¨ auser, Boston, 1985).
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A. Cohen, Y. Roth, B. Shapiro, Universal distributions and scaling in disordered systems, Phys. Rev. B 38, 12125–12132 (1988).
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[DLA]
L.I. Deych, A.A. Lisyansky, B.L. Altshuler, Single parameter scaling in 1D Anderson localization. Exact analytical solution, Phys. Rev. B 64, 224202–2242013 (2001).
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R. Johnston, H. Kunz, The conductance of a disordered wire, J. Phys. C: Solid State Phys. 16, 3895–3912 (1983).
[KW]
M. Kappus, F. Wegner, Anomaly in the band centre of the onedimensional Anderson model, Z. Phys. B 45, 15–21 (1981).
[KS]
V. Kostrykin, R. Schrader, Scattering theory approach to random Schr¨ odinger operators in one dimension, Rev. Math. Phys. 11, 187–242 (1999).
[LeP]
E. Le Page, Th´eor`emes limites pour les produits de matrices al´eatoires, Lect. Notes. Math. 929, 258–303 (1982).
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L. Molinari, Exact generalized Lyapounov exponents for one-dimensional disordered tight binding models, J. Phys. A 25, 513–520 (1992).
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L. Pastur, A. Figotin, Spectra of Random and Almost-Periodic Operators, (Springer, Berlin, 1992).
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H. Schomerus, M. Titov, Band-center anomaly of the conductance distribution in one-dimensional Anderson localization, Phys. Rev. B 67, 100201–100204 (2003).
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T. Sedrakyan, Localization-delocalization transition in a presence of correlated disorder: The random dimer model, Phys. Rev. B 69, 85109–85114 (2004).
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A.D. Stone, D.C. Allan, J.D. Joannopoulos, Phase randomness in the one-dimensional Anderson model, Phys. Rev. B 27, 836–843 (1983).
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R. Schrader Institut f¨ ur Theoretische Physik Freie Universit¨ at Berlin D-14195 Berlin Germany email:
[email protected]
H. Schulz-Baldes Institut f¨ ur Mathematik Technische Universit¨ at Berlin D-10623 Berlin Germany email:
[email protected]
A. Sedrakyan Yerevan Physics Institute Yerevan 36 Armenia email:
[email protected] Communicated by Yosi Avron submitted 15/03/04, accepted 23/04/04
To access this journal online: http://www.birkhauser.ch
Ann. Henri Poincar´e 5 (2004) 1181 – 1205 c Birkh¨ auser Verlag, Basel, 2004 1424-0637/04/061181-25 DOI 10.1007/s00023-004-0196-2
Annales Henri Poincar´ e
Order by Disorder, without Order, in a Two-Dimensional Spin System with O(2) Symmetry Marek Biskup, Lincoln Chayes and Steven A. Kivelson Abstract. We present a rigorous proof of an ordering transition for a two-component two-dimensional antiferromagnet with nearest and next-nearest neighbor interactions. The low-temperature phase contains two states distinguished by local order among columns or, respectively, rows. Overall, there is no magnetic order in accord with the classic Mermin-Wagner theorem. The method of proof employs a rigorous version of “order by disorder,” whereby a high degeneracy among the ground states is lifted according to the differences in their associated spin-wave spectra.
1 Introduction 1.1
Background
For two-dimensional spin systems, the celebrated Mermin-Wagner theorem [32,34] (and its extensions [11,26]) precludes the possibility of the spontaneous breaking of a continuous internal symmetry. However, this result does not prevent such models from exhibiting phase transitions. For example, in the usual XY-model there is a low-temperature phase, known as the Kosterlitz-Thouless phase [28], characterized by power-law decay of correlations and, of course, vanishing spontaneous magnetization [22, 31]. The existence and properties of this phase have been of seminal importance for the understanding of various low-dimensional physical phenomena, e.g., 2D superconductivity and superfluidity, 2D Josephson arrays, 2D melting, etc. It it widely believed that no such phase exists for O(n)-models with n ≥ 3 although rigorous arguments for (or against) this conjecture are lacking. Of course, among such models there are other pathways to phase transitions aside from attempting to break the continuous symmetry. One idea is to inject additional discrete symmetries into the model and observe the breaking of these “small” symmetries regardless of the (global) status of the “big” one. As an example, at each r ∈ Zd (where d ≥ 2) let us place a pair (σr , πr ) of n-component unit-length spins whose interaction is described by the Hamiltonian H = −J1 (σr · σr + πr · πr ) − J2 (σr · πr )2 , (1.1) r,r
r
where r, r denotes a pair of nearest neighbors on Zd and J1 , J2 > 0. Obviously, this model has O(n) symmetry (rotating all spins) as well as a discrete Z2 symme-
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try (relative reflection between the σ’s and the π’s). It is not hard to show that at low temperatures, regardless of the global status of the σ’s and π’s, there is coexistence between a phase where the σ’s and π’s are locally aligned with one another and one where they are locally antialigned. (Note that this is based purely on energy considerations – the said alignments are the only minimizers of the second term in the Hamiltonian.) A model similar to the one defined above was analyzed in [37] where the corresponding conclusions were indeed established. We remark that these results hold even if d = 2 (and even if n > 2). Another “circumvention” is based on the adaptation of the large-entropy methods to systems which happen to have continuous symmetry. These are distinguished from the more commonly studied systems in and of the fact that there is no apparent order parameter signaling the existence of a low-temperature phase. The key idea dates back to [12, 29] where some general principles were spelt out that guarantee a point of phase coexistence. Let us consider an attractive system where there is an energetically favored alignment which confines the spin configurations to a small portion of the spin space. Suppose that there are many other less favored alignments with an approximately homogeneous energy. Under these conditions, a first-order transition at some (intermediate) value of temperature is anticipated. This kind of transition was established for specific systems (including the q-state Potts model) in [12,29], see also [38]. The general philosophy can easily be adapted to spin systems with a continuous symmetry, e.g., as in [2, 7, 8] where some related problems were discussed. To illustrate these matters let us consider an example from [2]. Here we have a two-component spin of length one at each site of Z2 which we parametrize by an angular variable θr ∈ (−π, π]. Let V (x) denote the function which equals negative one if |x| < and zero otherwise, and let V (θr − θr ), (1.2) H =J r,r
where, of course, the arguments of V are√interpreted modulo 2π. Then, at some parameter value J = Jt obeying eJt ≈ , coexistence occurs between a phase where nearly all neighboring spins are closely aligned and one where, locally, spins exhibit hardly any correlation. We reiterate that the use of n = 2 and d = 2 is not of crucial importance for proofs of statements along these lines. Indeed, in [15,16], similar results have been established in much generality. In all of the above examples a moment’s thought reveals that no violation of the Mermin-Wagner theorem occurs. Indeed, this theorem does not preclude a phase transition, it only precludes a phase transition which is characterized by breaking of a (compact) continuous internal symmetry.
1.2
Foreground
The purpose of this note is to underscore another route “around” the MerminWagner theorem. The distinction here, compared to all of the above-mentioned,
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is that it may take the reader two moments to realize that our results are also in accord with the Mermin-Wagner theorem. Not unrelated is the fact that in our example the mechanism for ordering is relatively intricate. Let us go right to the (formal) Hamiltonian which reads H =J
S r · S r+ˆex +ˆey + S r · S r+ˆex −ˆey + Jγ S r · S r+ˆex + S r · S r+ˆey . (1.3) r
r
Here r denotes a site in Z2 and the S r are unit-length two-component spins, i.e., S r ∈ R2 , with |S r | = 1, for each r ∈ Z2 . The vectors ˆex and ˆey are unit vectors in the x and y lattice directions while J (the overall interaction strength) and γ (the relative strength of nearest neighbor couplings) are positive numbers. Notice the sign of the coupling – there is antiferromagnetism all around. In order to analyze the ground states, let us focus on the cases γ 1. (Later we will only require γ < 2.) Notice, especially in this limit, that the interaction splits the lattice into even and odd sublattices. For the ground-state problem, say in an even-sided finite volume with periodic boundary conditions, it is clear that both of the sublattices will be Ne´el (i.e., antiferromagnetically) ordered. However, once this Ne´el order is in place, it is clear that the energetics are insensitive to the relative orientation of the spins on the two sublattices. Specifically, the spin at any site r couples antiferromagnetically to the sum of S r+ˆex , S r+ˆey , S r−ˆex and S r−ˆey which, in any Ne´el state, is exactly zero. Thus we conclude that the set of ground states, i.e., the “order-parameter space,” cf. [33], of this model exhibits an O(2) ⊗ O(2) symmetry. For convenience we will regard the first factor of O(2) ⊗ O(2) as acting on all spins and the second as acting on the relative orientations of (the spins on) the two sublattices. The upshot of this work (precise theorems will be stated in Section 2.1) is that, at small but positive temperatures, the order parameter space is reduced to Z2 . Although the first O(2) is restored as required by the MerminWagner theorem, the remaining Z2 is a remnant of the second O(2). Consequently, at low temperatures, there are two Gibbs states: one where there is near alignment between nearest-neighbor spins in every lattice column and the other featuring a similar alignment in every lattice row. So the continuous O(2) ⊗ O(2) symmetry is evidently broken; we have Gibbs states in which all that acts is the single O(2) factor. And all of this in two dimensions! Having arranged for the requisite two moments via procrastination, we will now reveal why this does not violate the Mermin-Wagner theorem. The answer is that the enhanced O(2)⊗O(2) symmetry was never a symmetry of the Hamiltonian – this is both the hypothesis and the driving force of the derivations of the MerminWagner theorem. Indeed, the large symmetry was only a symmetry of the ground state space and as such there is no a priori reason to expect its persistence at finite temperatures. So everything is all right. To further confuse matters, let us remark that although the “Z2 remnant” – the one that does get broken – was not an internal symmetry of the Hamiltonian, it is, somehow, more organic than the
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Figure 1: An example of the ground state of the Hamiltonian (1.3) on a finite grid. Here both sublattices exhibit Ne´el state with spins alternating between 30◦ and 210◦ on one sublattice and between 110◦ and 290◦ on the other. Any other ground state can be obtained by an independent rotation of all spins in each sublattice. O(2) group that contained it. This particular Z2 may be interpreted as the natural enactor of one of the lattice symmetries (here a 90◦ -rotation) which are typically associated with antiferromagnets. The last observation is supported by the fact that there is an order parameter associated with the above phase transition. Indeed, consider the object nr = (S r+ˆex − S r+ˆey ) · S r
(1.4)
whose expectation is zero at sufficiently high temperatures and non-zero (in appropriate states) at low temperatures. (In another context, this sort of symmetry breaking has been referred to as Ising nematic ordering [1, 27].) To summarize (in case all of this has been confusing), here we have a true long-range order but we avoid conflict with the Mermin-Wagner theorem because the O(2) ⊗ O(2)symmetry was never a true symmetry of the model.
1.3
Order by disorder
In accordance with the title, the mechanism behind this ordering is called “order by disorder” (or, in the older vernacular, “ordering due to disorder”). This concept is, as of late, extremely prevalent in the physics literature; most of the recent work concerns quantum large-S systems where finite S plays the role of thermal fluctuations, but the origin of this technique can be traced to the study of classical systems, see [39, 40] and [24]. In particular, in the latter reference, it is exactly the present model that was studied and this has since been referred
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to as the canonical model of order by disorder. The key words are “spin waves” and “stabilization by finite temperature excitations,” neither of which should be unfamiliar to the mathematical physicist but which, until recently, have not been exploited in tandem. Let us proceed with the key ideas; we will attend to the obligatory citations later. For ease of exposition, let us imagine that somehow even at finite temperatures the two sublattices remain locked in their Ne´el states. Thus there is an angle, φ , which measures the relative orientation of the states on the two sublattices. Next we perform a spin-wave calculation to account for the thermal perturbations about the ground state with fixed φ . Although said instructions may have profound implications in other contexts, for present purposes this simply means “pitch out all interactions beyond quadratic order and perform the resulting Gaussian integral.” The upshot of such a calculation is a quantity, the spin-wave free energy, which should then be minimized as a function of φ . As we will see this minimum occurs exactly when the states are either horizontally or vertically aligned, i.e., φ = 0◦ or φ = 180◦ . The reader may question the moral grounds for the working assumption of finite temperature Ne´el order which is the apparent basis of the spin-wave calculation. Of course, the cheap way out – the final arbitrator – is the fact that herein is a rigorous proof. However, the spin-wave conclusions are not so difficult to understand. Foremost, we reemphasize that the outcome is decided purely on the basis of free energetics. A cursory examination of the calculational mechanics then reveals that in fact only two ingredients are really needed. The first is that Ne´el order is present locally – which is certainly true at very low temperatures. The second boils down to the statement that the thermodynamic properties in these sorts of magnets are unaffected – to first approximation – if the system is restricted to configurations that have magnetic order. In particular, the long wavelength excitations which are ultimately responsible for the break-up of ordering in two dimensions contribute insignificantly to the free energy. Now let us discuss the historical perspective of the present paper. The first phase in understanding this sort of problems is coming to terms with the degeneracy of the ground-state space. When these situations arise, there is a selection at finite temperature according to the ability that each state has to harbor excitations. The simplest cases, namely a finite number of ground states and a small effective activity (e.g., a large “mass”) for the excitations have been understood by physicists for a long time and are now the subject of essentially complete mathematical theorems [35, 36, 41]. Many interesting situations with infinitely many ground states were introduced in late 1970s and early 1980s, see, e.g., [17, 40]. Here intricate and/or mysterious calculations are invoked to resolve the degeneracies – often resulting in phantasmagorical phase diagrams, see e.g. [18] – but the upshot in these situations is pretty much the same. In particular, with excruciating effort, some cases can now be proclaimed as theorems [6, 10]. However, the cornerstone of any systematic analysis (either mathematical or physical) is the existence of a substantial gap in the energy spectrum separating those excitations
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which resolve the ground-state degeneracy from the excitations that are readily available to all ground states. The degenerate ground-state problems look very different for the classical O(n)-spin models. Indeed, the continuous nature of the spins in combinations with their internal degrees of freedom almost inevitably lead to a gapless excitation spectrum. Although this sounds a lot harder, the necessary computations turn out to be far more palatable. To our knowledge, the first such example, studied in [39], was a frustrated FCC antiferromagnet. The system is quite similar to the one discussed here but with the ordering caused, mostly, by quantum effects. In [24], studying exactly the model in (1.3), it was demonstrated that these techniques also apply to classical systems. In the present work we will transform these classical finite-temperature derivations into a mathematical theorem. The proofs are quite tractable; all that is really required are some error estimates for the Gaussian approximations and a straightforward contour argument. To ease our way through the latter we will employ the method of chessboard estimates. In some concurrent work [3,4], a similar analysis is used to resolve some controversies concerning models of transition-metal oxides. However, in these “TMO-problems,” the ground-state spaces have additional intricacies so the beauty and simplicity of the method is obscured. To make our historical perspective complete let us also relate to the existing mathematical work on systems with continuous spins. A general approach to continuous spins with degeneracies has been developed in [13, 42]. Here the method of resolution appears to be not terribly dissimilar to ours; e.g., there are quadratic approximations, Gaussian integrals, error estimates, etc. However, only a finite number of ground states are considered and we suspect that a detailed look at the “curvature conditions” will reveal that again there is a substantial mass gap in the excitation spectrum. Finally, from an earlier era, there are the methods based on infrared bounds [14, 19–21]. However, the reflection symmetries required to get these arguments started do not seem to hold in the system defined by (1.3). And even if they did, due to the infrared divergence, this would only provide misleading evidence – a la Mermin-Wagner – that the model under consideration has no phase transition.
2 Main results 2.1
Phase coexistence
To state our results on phase coexistence in the model under consideration, we will first recall the concept of infinite-volume Gibbs measures. We begin with finite-volume counterparts thereof, also known as Gibbs specifications. Let S = (S Λ , S Λc ) be a spin configuration where S Λ and S Λc denote the corresponding restrictions to Λ and Λc , respectively. Let HΛ (S Λ , S Λc ) be the restriction of (1.3) (S c ) to pairs of sites at least one of which is in Λ. Then we let µΛ Λ be the measure
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on configurations in Λ defined by (S Λc )
µΛ
(dS Λ ) =
e−βHΛ (S Λ ,S Λc ) ΩΛ (dS Λ ). ZΛ (S Λc )
(2.1)
Here ΩΛ denotes the product Lebesgue measure on the unit circle, one for each r ∈ Λ. Following the “DLR-philosophy,” see [23], the infinite-volume Gibbs measures are those measures on full configurations on Z2 whose conditional probability in a finite volume Λ given the configuration in the complement is exactly the object in (2.1). In accord with the standard terminology, see [23], we will say that there is a phase coexistence for parameters J, γ and β if there exists more than one infinitevolume Gibbs measure for the interaction (1.3) and inverse temperature β. To adhere with mathematical-physics notation, we will refer to the Gibbs measures as Gibbs states and we will denote the expectations with respect to such states by symbol −β . Now we are in a position to state the main result of this paper. Theorem 2.1 Consider the model as defined above with fixed J ∈ (0, ∞) and γ ∈ (0, 2). Then there exists a β0 ∈ (0, ∞) and a function β → (β) satisfying (β) → 0 as β → ∞ such that the following holds: For each β ≥ β0 there exist two distinct (x) (y) Gibbs states −β and −β such that S r · S r (α) + 1 ≤ (β) β
(2.2)
whenever r, r are next-nearest neighbors in Z2 , and S r · S r (α) − 1 ≤ (β) β
(2.3)
whenever r, r ∈ Z2 are such that r = r + ˆeα . Let us informally describe the previous result. First, on both even and odd sublattice of Z2 we have a (local) antiferromagnetic order. The distinction between (x) the two states is that in −β the nearest-neighbor spins on Z2 are aligned in (y)
the x direction and antialigned in the y direction, while in −β the two alignment directions are interchanged. In particular, it is clear that the order parameter nr , (x) defined in (1.4), has positive expectation in the x-state −β and negative expec(y)
tation in the y-state −β . Since, as mentioned previously, Gibbsian uniqueness guarantees that nr β = 0 at sufficiently high temperatures, we have a bone fide phase transition of the “usual” type. Despite the existence of multiple low-temperature Gibbs states, we emphasize that no claim has been made about the actual direction that the spins will be aligned to. On the contrary, we have the following easy corollary of the aforementioned Mermin-Wagner theorem:
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Theorem 2.2 Consider the model as defined above with J, γ ∈ R fixed and let −β be any infinite-volume Gibbs state at inverse temperature β. Then −β is invariant under the simultaneous rotation of all spins and, in particular, S r β = 0 for all r ∈ Z2 . The authors do not see any significant obstruction of Theorem 2.1 (appropriately modified) in the cases n > 2 and d > 2. For the case under consideration, namely, n = 2 and d = 2, it may be presumed that there is a slow decay of correlations at sufficiently low temperatures. Here it is conceivable that, with great effort, this could be proved on the basis of technology that is currently available [9,22,30]. The anticipation is that for d ≥ 3 and n ≥ 2 there are actual sublattice Ne´el states while for d = 2 and n > 2 the decorrelations should be exponential. However, we do not expect to see a proof of any statement along these lines in the near future.
2.2
Outline of the proof
We proceed by an informal outline of the proof of our main result (Theorem 2.1). The argument hinges on the following three observations: (1) Suppose ∆ is a number that satisfies βJ∆2 1.
(2.4)
Then the (angular) difference of any typical pair of next -nearest neighbor spins will not deviate by more than ∆ from the energetically optimal configuration. (2) In situations when (1) applies and under the additional assumption that ∆ also satisfies βJ∆3 1, (2.5) then all important contributions to the free energy of the system will come from a quadratic – or spin-wave – approximation to the Hamiltonian. (3) Finally, if F (φ ) denotes the spin-wave free energy above the ground state where one sublattice is rotated by angle φ relative to the other (see Fig. 1.2), then F (φ ) is minimized only at φ = 0◦ or φ = 180◦. (The mathematical statements corresponding to (1–3) above are formulated as Theorems 3.1 and 3.2 in Section 3.1.) We observe that the necessary ∆ as stipulated by (2.4–2.5) defines a running scale – not too big and not too small – which obviously tends to zero as β → ∞. Here is how these observations will be combined together to establish longrange order: We partition the lattice in blocks of side B. On the basis of (1) above, every block will with high probability exhibit a near ground-state configuration, which by (2–3) will have the sublattices either nearly aligned or nearly antialigned. Then we need to show that each of the two possibilities are stable throughout the entire system. For that we will resort to a standard Peierls’ argument. Here the
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crucial observation (see Lemma 4.6) is that two “good” blocks with different type of alignment between sublattices are necessarily separated by a “surface” of “bad” blocks – that is those which either contain energetically charged pair of nearestneighbor spins or whose spin-wave free energy exceeds the absolute minimum by a positive amount. Appealing to chessboard estimates (see Section 4.1), the probability of a particular “surface” can be factorized – as a bound – into the product over the constituting blocks. It turns out that the energetically frustrated “bad” blocks are suppressed once βJ∆2 log B, (2.6) while the entropically frustrated blocks are suppressed once the excess spin-wave free energy times B 2 is sufficiently large. Under the conditions (2.5–2.6) and B 1 the entropy of the above “surfaces” can be controlled. The desired phase coexistence then follows by standard arguments. A couple of remarks are in order: Due to the perfect scaling properties of Gaussian distributions the suppression extracted from the spin-wave calculation is independent of β – the desired decay is achieved solely by choosing B sufficiently large. Large β is needed only to suppress large deviations away from the “perfect” ground states. Notwithstanding, for (technical) ease of exposition we will have to make B increase slowly with β; see (4.19) for the precise relation of ∆, B and β. The various steps of the proof are laid out in the following order: In Section 3 we carry out the harmonic approximation and provide the needed control of the spin-wave free energy. In Section 4 we invoke chessboard estimates and some straightforward bounds to control the contour expansion. The actual proof of Theorem 2.1 comes in Section 4.3.
3 Spin-wave calculations As mentioned above, the underpinning of our proof of the main result is (the outcome of) a spin-wave free-energy calculation. This calculation involves simply working with the harmonic approximation of the Hamiltonian (1.3) for deviations away from a fixed ground state. The calculation itself is straightforward although special attention must be paid to the “zero mode.” For reasons that will become clear in Section 4 – and also to make discrete Fourier transform readily available – all of the derivations in this section will be carried out on the lattice torus TL of L × L-sites. Here, for technical convenience, we will restrict L to multiples of four so that we can assure an equal status of the two Ne´el states.
3.1
Harmonic approximation
We will begin by an explicit definition of the torus Hamiltonian. Here and henceforth we will parametrize the spins by angular variables θ = (θr ) which are related to the S r ’s by the usual expression S r = (cos θr , sin θr ). (Of course, the θr ’s are
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always to be interpreted only modulo 2π.) Up to irrelevant constants, the corresponding torus Hamiltonian HL can then be written as HL (θ) = J 2 + cos(θr − θr+ˆex +ˆey ) + cos(θr − θr+ˆex −ˆey ) r∈TL
+ Jγ
cos(θr − θr+ˆex ) + cos(θr − θr+ˆey ) .
(3.1)
r∈TL
The spin-wave calculations are only meaningful in the situations where each of the sublattices is more or less aligned with a particular Ne´el state. To describe the overall and relative orientation of the spins on the even and odd sublattices we will need two angles θ and φ , respectively. Depending on the parities of the coordinates of r, we will write the θr for r = (x, y) in terms of the deviation variables ϑr as follows: θ , x, y-even, θ + φ , x-odd, y-even, (3.2) θr = ϑr + θ + π, x, y-odd, x-even, y-odd. θ + φ + π, Obviously, only the relative angle φ will appear in physically relevant quantities; the overall orientation θ simply factors out from all forthcoming expressions. The principal object of interest in this section is the finite-volume free energy, which will play an important role in the estimates of “entropically-disfavored” block events in Section 4. For reasons that will become clear later, we will define this quantity by the formula βJ L2 /2 1 dθr . (3.3) FL,∆ (φ ) = − 2 log e−βHL (θ) χL,∆ (θ) L 2π r∈TL
Here dθr is the Lebesgue measure on unit circle and χL,∆ (θ) = χL,∆ (θ; φ , θ ) is the indicator that the deviation quantities ϑ, defined from θ as detailed in (3.2), satisfy |ϑr | < ∆ for all r ∈ TL . The factors of βJ 2π have been added for later convenience. The goal of this section is to (approximately) evaluate the thermodynamic limit of the quantity FL,∆ (φ ) and characterize where it achieves its minima. As is standard in heuristic calculations of this sort, we will first replace the Hamiltonian (3.1) by its appropriate quadratic approximation. We will express the resulting quantity directly in variables ϑr : IL,φ (ϑ) =
βJ (ϑr − ϑr+ˆex +ˆey )2 + (ϑr − ϑr+ˆex −ˆey )2 2 r∈TL
βJ + γ cos(φ ) (ϑr − ϑr+ˆex )2 + (ϑr − ϑr+ˆey )2 . 2 r∈TL
(3.4)
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This approximation turns the integral in (3.3) into a Gaussian integral. As we will see later, here the indicator in (3.3) can be handled in terms of upper and lower bounds which allow “diagonalization” of the covariance matrix by means of Fourier variables. The result, expressed in the limit L → ∞, is the following momentum integral: dk 1 log Dk (φ ), (3.5) F (φ ) = 2 [−π,π]2 (2π)2 where Dk (φ ) = |1−ei(k1 +k2 ) |2 +|1−ei(k1 −k2 ) |2 +γ cos(φ ) |1−eik1 |2 +|1−eik2 |2 . (3.6) Here k1 and k2 are the Cartesian components of vector k. The quantity F (φ ) has the interpretation – justified via the preceding derivation – as the spin-wave free energy. As is checked by direct calculation, for γ ∈ [0, 2) we have Dk (φ ) > 0 almost surely with respect to the (normalized) Lebesgue measure on [−π, π]2 . Having sketched the main strategy and defined the relevant quantities, we can now pass to the statements of (admittedly dry) mathematical theorems. First, we express the conditions under which the above approximate calculation can be performed: Theorem 3.1 Given > 0 and γ ∈ [0, 2), there exists δ = δ(, γ) > 0 such that if βJ, ∆ and δ satisfy the bounds βJ∆3 ≤ δ then
βJ∆2 ≥ 1/δ,
(3.7)
lim supFL,∆ (φ ) − F (φ ) ≤
(3.8)
and
L→∞
holds for every φ ∈ (−π, π]. The proof is postponed to Section 3.2. Having demonstrated the physical meaning of the function φ → F (φ ), we can now characterize its absolute minimizers: Theorem 3.2 For all γ ∈ (0, 2), the absolute minima of function φ → F (φ ) occur (only) at the points φ = 0◦ and φ = 180◦ . Proof. The proof is an easy application of Jensen’s inequality. Indeed, let a ∈ [0, 1] be the number such that 2a − 1 = cos(φ ). Then we can write Dk (φ ) = aDk (0◦ ) + (1 − a)Dk (180◦ ).
(3.9)
Since Dk (0◦ ) is not equal to Dk (180◦ ) almost surely with respect to dk (this is where we need that γ > 0), the concavity of the logarithm and Jensen’s inequality imply that F (φ ) > aF (0◦ ) + (1 − a)F (180◦ ) whenever a = 0, 1. This shows that
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the only absolute minima that F can have are 0◦ and 180◦. Now F is continuous (under the assumption that γ < 2) and periodic, and so there exists at least one point in (−π, π] where it attains its absolute minimum. But F (0◦ ) = F (180◦ ) and so φ → F (φ ) is minimized by both φ = 0◦ and φ = 180◦ .
3.2
Proof of Theorem 3.1
Throughout the proof we will fix J ∈ (0, ∞) and γ ∈ [0, 2) and suppress these from our notation whenever possible. Since everything is founded on harmonic approximation of the Hamiltonian, the starting point is some control of the error that this incurs: Lemma 3.3 There exists a constant c1 ∈ (0, ∞) such that the following holds: For any ∆ ∈ (0, ∞), any θ , φ ∈ (−π, π] and any configuration θ = (θr ) of angle variables on TL , if the corresponding ϑ = (ϑr ) satisfy |ϑr | < ∆ for all r ∈ TL , then βHL (θ) − IL,φ (ϑ) < c1 (1 + γ)βJ∆3 L2 . (3.10) Proof. We begin by noting that |ϑr | < ∆ for all r ∈ TL implies that |ϑr − ϑr | < 2∆ for all pairs of nearest and next-nearest neighbors r, r ∈ TL . This and the uniform bound |x|3 , (3.11) cos(a + x) − cos(a) − sin(a)x − 12 cos(a)x2 ≤ 6 show that, at the cost of an error as displayed in (3.10), we can replace all trigonometric factors in (3.1) by their second-order Taylor expansion in differences of ϑr . Hence, we just need to show that these Taylor polynomials combine into the expression for IL,φ . It is easily checked that the zeroth order Taylor expansion in ϑr exactly vanishes. This is a consequence of the fact that for ϑ ≡ 0 we are in a ground state where, as argued before, both sublattices can be independently rotated. This means we can suppose that θ = φ = 0◦ in (3.1) at which point it is straightforward to verify that HL (θ) actually vanishes. Similarly easy it is to verify that the quadratic terms yield exactly the expression for IL,φ . It thus remains to prove that there are no linear terms in ϑr ’s. First we will note that all next-nearest neighbor terms in the Hamiltonian certainly have this property because there we have θr − θr+ˆex ±ˆey ≈ 0 or π, at which points the derivative of the cosine vanishes. Hence we only need to focus on the nearest-neighbor part of the Hamiltonian – namely, the second sum in (3.1) – which we will temporarily denote by HLnn . Here we will simply calculate the derivative of HLnn with respect to ϑr : ∂ HLnn (θ) = sin(θr+ˆex − θr ) + sin(θr+ˆey − θr ) ∂ϑr ϑ≡0 − sin(θr − θr−ˆex ) + sin(θr − θr−ˆey ) ,
(3.12)
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where the θr on the right-hand side should be set to the “ground-state” values. To make the discussion more explicit, suppose that r has both coordinates even. Then an inspection of (3.2) shows that the first sine is simply sin(φ ) while the second sine evaluates to sin(φ + π) = − sin(φ ). The net contribution of these two terms is thus zero. Similarly, the third and the fourth sine also cancel out. The other possibilities for r are handled analogously. Using the harmonic approximation of the Hamiltonian, let us now consider the corresponding Gaussian equivalent of the integral in (3.3): βJ L2 /2 L,∆ (ϑ) dϑr , (3.13) QL,∆(φ ) = e−IL,φ (ϑ) χ 2π r∈TL
where dϑr is the Lebesgue measure on R and χ L,∆ (ϑ) is the indicator that |ϑr | < ∆ for all r ∈ TL . Our next goal is to evaluate the effect of this indicator, which we will accomplish by proving an upper and lower bound on QL,∆ (φ ). We commence with the easier of the two, the upper bound: Lemma 3.4 For all β ∈ (0, ∞), all ∆ ∈ (0, ∞) and all φ ∈ (−π, π], lim sup L→∞
log QL,∆ (φ ) ≤ −F (φ ). L2
(3.14)
Proof. The argument is relatively straightforward so we will be correspondingly brief. (A more verbose argument along these lines can be found in [3].) Pick a λ > 0. We will invoke the exponential Chebyshev inequality in the form 1 2 1 (3.15) |ϑr |2 . χ L,∆ (ϑ) ≤ e 2 βJλ∆L exp − βJλ 2 r∈TL
Next we plug this bound into (3.13), diagonalize IL,φ by passing to the Fourier components ϑk = L−1 r∈TL ϑr eir·k and perform the Gaussian integrals with the result 2 1 1 QL,∆ (φ ) ≤ e 2 βJλ∆L . (3.16) [λ + D (φ )]1/2 k k∈T L
TL
−1
Here = {2πL (n1 , n2 ) : ni = 1, 2, . . . , L} is the reciprocal lattice and Dk (φ ) is as defined in (3.6). The result now follows by taking logarithm, dividing by L2 and invoking the limits L → ∞ followed by λ ↓ 0 – with the last limit justified by the Monotone Convergence Theorem. The corresponding lower bound is then stated as follows: Lemma 3.5 For all β ∈ (0, ∞), all ∆ ∈ (0, ∞), all φ ∈ (−π, π] and all λ > 0 satisfying βJ∆2 λ > 1, we have log QL,∆ (φ ) 1 lim inf , (3.17) ≥ −F (φ , λ) + log 1 − L→∞ L2 βJ∆2 λ
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where F (φ , λ) is given by the same integral as in (3.5) with Dk (φ ) replaced by λ + Dk (φ ). Proof. Again, we will be fairly succinct. Let λ > 0. We begin by considering the Gaussian measure defined by Pλ (dϑ) =
βJ L2 /2 1 1 2 exp −I βJλ (ϑ) − |ϑ | dϑr L,φ r QL (φ , λ) 2 2π r∈TL
r∈TL
(3.18) where QL (φ , λ) is an appropriate normalization constant. Let Eλ denote the corresponding expectation. A simple bound shows that we have QL,∆ (φ ) ≥ QL (φ , λ)Eλ ( χ∆,L ),
(3.19)
which reduces the desired estimates to two items: a calculation of the integral QL (φ , λ) and a lower bound on Eλ ( χ∆,L ). The first problem on the list is dispensed with similarly as in the proof of Lemma 3.4, so we just state the result: log QL (φ , λ) = −F (φ , λ). L→∞ L2 lim
(3.20)
As far as the second item on the list is concerned, here we use that by the results of [5] the magnitudes of the Gaussian field with distribution (3.18) are positively correlated. (An alternative proof of this fact uses reflection positivity.) Invoking the product structure of χ ∆,L and translation invariance of Pλ , we thus have L2 χ∆,L ) ≥ Pλ |ϑ0 | < ∆ , Eλ (
(3.21)
where ϑ0 is the variable at the origin of the torus. It remains to bound Pλ (|ϑ0 | < ∆) from below, which we will do by estimating the complementary probability from above. We will pass to the Fourier components ϑk defined as in the proof of Lemma 3.4. Under the measure (3.18), these components have zero mean, the random variables ϑk and ϑ∗k for different k and k are uncorrelated (a consequence of translation invariance), while for the autocorrelation function we get 1 1 1 ≤ . Eλ |ϑk |2 = βJ λ + Dk (φ ) βJλ
(3.22)
This allows us to use the (quadratic) Chebyshev inequality to derive Pλ (|ϑ0 | ≥ ∆) ≤
Eλ (|ϑ0 |2 ) 1 Eλ (|ϑk |2 ) 1 . = ≤ 2 2 2 ∆ L ∆ βJ∆2 λ k∈TL
(3.23)
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Inserting this into (3.21) and applying (3.19) and (3.20), the rest of the proof boils down to taking logs, dividing by L2 and letting L → ∞. Now we are ready to prove the principal approximation theorem: Proof of Theorem 3.1. We just assemble together the previously discussed ingredients. First, our constraints (3.7) imply that ∆ ≤ δ 2 and so we can assume that ∆ < π. Under this condition the integrals in (3.3) and (3.13) are over the same set of ϑr ’s and so by Lemma 3.3 we have the uniform bound log Q (φ ) L,∆ − F (φ ) (3.24) ≤ c1 (1 + γ)βJ∆3 . L,∆ L2 Second, Lemmas 3.4–3.5 ensure that log Q (φ ) 1 L,∆ F (φ ) − F (φ , λ) + log 1 − ≤ . (3.25) − F (φ ) lim sup L2 βJ∆2 λ L→∞ By the assumptions in (3.7), given an > 0 we can choose δ > 0 such that the right-hand side of (3.24) is smaller than /2. On the other hand, since F (φ , λ) increases to F (φ ) as λ ↓ 0 and since β∆2 ≥ 1/δ, we can certainly choose a λ > 0 (satisfying βJ∆2 λ > 1) and adjust δ such that also the right-hand side of (3.25) is less than /2. Combining these observations, the desired bound (3.8) is proved. Remark 1. Physically motivated readers will notice that in both Lemmas 3.4 and 3.5 we have introduced a “mass” into the spin-wave spectrum before (or while) removing the indicator χ L,∆ . The primary reason for this is the bad behavior of the zero Fourier mode for which the “spin-wave Hamiltonian” IL,φ provides no decay in the Gaussian weight.
4 Proof of phase coexistence Having discussed the spin-wave approximations (which will be essential for the arguments in this section), we are now ready to start with the proof of phase coexistence. Our basic tool in this section will be the chessboard estimates, so we will begin by introducing the notation needed for applications of this technique.
4.1
Chessboard estimates
As mentioned previously, in order use chessboard estimates, for technical reasons, we have to confine our technical considerations to toroidal geometries. Again we will use TL to denote the torus of L × L sites (as in Section 3 we restrict L to multiples of four). We will consider several events which will all take place in a box ΛB of (B + 1) × (B + 1) sites (which, for definiteness, we will assume to be placed with its lower-left corner at the torus “origin”). Since we want to be able to cover TL by translates of ΛB , we will assume that L is an even multiple of B. Thus, if A is an event in ΛB , then its translate by t1 B lattice units in the x-direction
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and t2 B units in the y-direction will be denoted by τt (A), where t = (t1 , t2 ). Here t takes values in a factor torus, namely, t ∈ TL/B . Note that events in the “neighboring” translates of ΛB may both depend on the shared side of the corresponding boxes. Let PL,β denote the Gibbs measure on TL defined from the appropriate torus version of the Hamiltonian (1.3) and inverse temperature β. Specifically, using the “spin-version” of the Hamiltonian (3.1), the Radon-Nikodym derivative of PL,β with respect to the a priori spin measure ΩTL is e−βHL (S) /ZL,β , where ZL,β is the corresponding partition function. The statement of the chessboard estimates will be considerably easier if we restrict our attention to reflection symmetric events, which are those A for which S ∈ A implies that the corresponding reflection S in any coordinate plane passing through the center of ΛB satisfies S ∈ A. For these events we will also define the constrained partition function 1τt (A) . (4.1) ZL,β (A) = ZL,β t∈TL/B
L,β
Here 1τt (A) is the indicator of τt (A) and −L,β denotes the expectation with respect to PL,β . Then we have: Theorem 4.1 (Chessboard estimates) Consider the Gibbs measure PL,β as defined above. Let A1 , . . . , Am be a collection of (not necessarily distinct) reflectionsymmetric events in ΛB and let t1 , . . . , tm be distinct vectors from TL/B . Then PL,β
m
m 2
ZL,β (Aj ) (B/L) τtj (Aj ) ≤ . ZL,β j=1 j=1
(4.2)
Proof. This is the standard chessboard estimate implied by the reflection positivity condition [19–21]. Here we consider reflection positivity in planes “through” sites, which holds in our case because we have only nearest and next-nearest neighbor interactions. Unfortunately, as often happens with chessboard estimates, we may not be able to estimate directly the quantity ZL,β (A) for the desired event under consideration. Instead, we will decompose A into a collection of more elementary events for which this estimation is easier. Here chessboard estimates can be used to establish the following standard (and often implicitly used) subadditivity property: Lemma 4.2 (Subadditivity) Let the torus TL and the block ΛB be as above and let us consider reflection-symmetric events A and (Ak )k∈K in ΛB . If A ⊆ k∈K Ak , then 2 2 ZL,β (Ak )(B/L) . (4.3) ZL,β (A)(B/L) ≤ k∈K
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Proof. See, e.g., Lemma 6.3 in [3]. Our succinct recount of the chessboard estimates is now complete. Readers wishing to obtain more details on this and related topics are referred to (still succinct) Section 6.1 of [3] or the classic references [19–21] and [38].
4.2
Good and bad events
Here we introduce the notion of good and bad blocks and events. Roughly speaking, a block is good if all spins on both sublattices are tolerably close to a Ne´el state and where the relative orientation of the two Ne´el states is near one of the two optimal values predicted by the spin-wave approximation. The bad blocks will of course be all those that are not good. Both these notions will involve two parameters: the spin-deviation scale ∆ encountered already in Section 3, and the scale κ marking the distance to a spin-wave minimum which is still considered good. We will keep κ small but fixed, while ∆ will have to be decreased (and the block scale B will have to be increased, albeit only slowly) as β goes to infinity. The precise definition is as follows: Definition 1 We say that a translate of ΛB by Bt, where t ∈ TL/B , is a good block, or that the good block event occurred in this translate if there exist two angles θ and φ such that: (1) The angle φ satisfies either |φ | ≤ κ or |φ − 180◦ | ≤ κ. (2) The collection of deviation angles ϑ = (ϑr ) defined from the angle variables θ = (θr ) and the angles θ and φ via (3.2) obeys |ϑr | < ∆
(4.4)
for all r ∈ TL . Let G0 be the notation for good-block event with φ ≈ 0◦ and let G180 be the good-block event for φ in the κ-neighborhood of 180◦. The complementary bad-block event will be denoted by B. We remark that all these events depend only on the spin configuration (angle variables) in ΛB . Remark 2. It is clear that if either G0 or G180 occurs (and if κ, ∆ 1), then the spins in ΛB are indeed well behaved in the sense of (2.2–2.3) in Theorem 2.1. Explicitly, if r, r ∈ ΛB is any pair of next-nearest neighbors, then S r · S r is very close to negative one. Moreover, on G0 we have S r · S r ≈ 1 when r = r + ˆex and S r · S r ≈ −1 for r = r + ˆey , while the opposite relations hold on G180 . (Once κ, ∆ 1, the requisite error is proportional to κ for next-nearest neighbors and to ∆2 for the nearest neighbors.) Thus, the first step in obtaining (2.2–2.3) will be to show that any particular block is of a given type of goodness with probability tending to one as β → ∞.
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Our goal is to use chessboard estimates to show that, with overwhelming probability, any given block is good and that, if one block is good with a known type of goodness, any other given block (regardless of the spatial separation) will exhibit the same type of goodness. As it turns out, on the basis of Theorem 4.1, both of these will boil down to an efficient estimate of the quantity ZL,β (B) defined in Section 4.1. Unfortunately, here we will have to introduce a further partitioning: We let BE denote the event that, for some next-nearest neighbor pair r, r ∈ ΛB , we have |θr − θr | − π ≥ ∆ . (4.5) 2B This event marks the presence of an energetic “catastrophy” somewhere in the block. As we will see, the complementary part of B, BSW = B \ BE
(4.6)
denotes the situations where the energetics – and the spin-wave approximation – are good but where the configuration is not particularly near either of the spinwave free-energy minima. The event BSW will be further split according to the relative angle between the two near-Ne´el states on even and odd sublattices. Specifically, we let φi , i = (i) 1, . . . , s, be s angles uniformly spaced on the unit circle. Then we let BSW denote the event that the block ΛB is bad but such that there exists an angle θ for which the deviation angles ϑ = (ϑr ) defined using θ and φ = φi satisfy |ϑr | < ∆ at each r ∈ ΛB . (Note that the second part is essentially the definition of the good block with the additional stipulation that φ = φi in part (1) of Definition 1.) It (i) remains to show that the BSW indeed cover BSW : Lemma 4.3 Let s be such that s∆ > 4π. Then BSW ⊆
s
(i)
BSW .
(4.7)
i=1
Proof. Consider a configuration of angle variables θ = (θr ) such that BSW occurs. Since this rules out the occurrence of BE , we have π−
∆ ∆ < |θr − θr | < π + 2B 2B
(4.8)
for any next-nearest neighbor pair r, r ∈ ΛB . But any two sites on the even sublattice in ΛB can be reached in less than B steps and so θr for any even r ∈ ΛB is within ∆/2 of θ0 or θ0 + π, depending on the parity of r in the sublattice. Hence, the overall deviations from the appropriate Ne´el state in direction θ = θ0 , where θ0 is the variable at the torus “origin,” do not exceed ∆/2 throughout the even sublattice. Similar considerations apply to the odd sublattice where we use the positive x-neighbor of the origin to define the angle θ + φ .
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It remains to show that the above implies that the spin configuration is (i) contained in one of the events BSW . Let i = 1, . . . , s be the unique index such that φi ≤ φ < φi+1 , where φs+1 is to be interpreted as φ1 . Then |φ − φi | < 2π/s which by our assumption is less than ∆/2. Consequently, all spins on the even sublattice are within ∆ of either θ or θ + π, depending on the parity, while those on the odd sublattice are within ∆ of either θ + φi or θ + φi + π, again depending (i) on the parity. In particular, the event BSW occurs, thus proving (4.7).
4.3
Proofs of Theorems 2.1 and 2.2
As alluded to in the paragraph before (4.5), the computational part of the proof boils down to estimates of the partition functions for events BE and BSW . These will be provided in next two lemmas. We begin with the event BE : Lemma 4.4 There exists δ > 0 and constants c2 , c3 ∈ (0, ∞) such that if βJ ∈ (0, ∞) and ∆ ∈ (0, 1) satisfy the bounds (3.7), then we have Z (B ) (B/L)2 2 2 2 L,β E lim sup ≤ 4B 2 (c3 βJ)B /2 e−c2 βJ∆ /B . ZL,β L→∞
(4.9)
Proof. When BE occurs, the exists a next-nearest neighbor bond in ΛB where the associated angle variables satisfy (4.5). An easy calculation shows that the energy this bond contributes to the Hamiltonian in (3.1) – note that the latter assigns zero energy to the Ne´el ground states – exceeds the J-multiple of ∆ ∆ = 2 sin2 2B . (4.10) 1 + cos π − 2B Bounding the sine from below by a linear function, which is justified because ∆/B ≤ π, the right-hand side is not less than a numerical constant times (∆/B)2 . We thus get 2 2 2 (4.11) ZL,β (BE )(B/L) ≤ 4B 2 e−c2 βJ∆ /B , where c2 ∈ (0, ∞) is a constant and where 4B 2 ≥ 2B(B + 1) bounds the number of ways to choose the “excited” bond in each translate of ΛB . Our next task is to derive a lower bound on the full partition function. A simple way to get such a bound is to insert the indicator that all angle variables θr are within ∆ of one of the spin-wave free energy minima, say, 0◦ . This gives ZL,β ≥
2π L2 /2 βJ
e−L
2
FL,∆ (0◦ )
,
(4.12)
where FL,∆ is as in (3.3). Fix > 0 and let δ > 0 be as in Theorem 3.1. Then our assumptions on β, ∆ and δ and the conclusion (3.5) tell us that 2
lim inf (ZL,β )1/L ≥ L→∞
2π 1/2 βJ
◦
e−F (0
)−
.
(4.13)
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Let us write the right-hand side as (c3 βJ)−1/2 , where c3 is a positive constant independent of β and ∆. Raising this bound to the B 2 power and combining it with (4.11) the bound (4.9) is now proved. Next we will attend to a similar estimate for the event BSW : Lemma 4.5 For each κ 1 and each γ ∈ (0, 2) there exist numbers ρ(κ) > 0 and δ > 0 such that if ∆ κ and if βJ, ∆ and δ satisfy the bounds in (3.7), then Z
lim sup
L,β (BSW )
(B/L)2
ZL,β
L→∞
2
≤ 8π∆−1 e−ρ(κ)B .
(4.14)
Proof. Let φi , i = 1, . . . , s, be s angles uniformly spaced on the unit circle. Suppose that s and ∆ satisfy 4π < s∆ < 8π. In light of the decomposition (4.7) and the subadditivity property from Lemma 4.2, it suffices to show that, under the conditions of the lemma, lim sup
Z
L→∞
(i) (B/L)2 L,β (BSW )
ZL,β
≤ e−ρ(κ)B
2
(4.15)
for every i = 1, . . . , s. First we note that for φi nearer than κ − ∆ to either 0◦ or 180◦ we au(i) (i) (i) tomatically have BSW ⊂ G0 ∩ G180 . But then BSW = ∅ because the event BSW is a subset of B. By our assumption that ∆ κ we just need to concentrate only on i = 1, . . . , s such that φi is at least, say, κ/2 from 0◦ or 180◦ . Here (i) 2π L2 /2 ) multiple of the integral in we will use that ZL,β (BSW ) is exactly the ( βJ (3.3) with φ = φi , while ZL,β can be bounded from below by a similar quantity for φ = 0◦ , i.e., Z
(i) 1/L2 L,β (BSW )
ZL,β
≤ exp −FL,∆ (φi ) + FL,∆ (0◦ ) .
(4.16)
Let now > 0 – whose size is to be determined momentarily – and choose δ > 0 so that Theorem 3.1 holds. Then the quantities FL,∆ (φ ) on the right-hand side are, asymptotically as L → ∞, to within of the actual spin-wave free energy. Hence, we will have lim sup L→∞
Z
(i) 1/L2 L,β (BSW )
ZL,β
≤ exp −F (φi ) + F (0◦ ) + 2 .
(4.17)
This proves (4.15) with ρ(κ) given as the minimum of F (φi ) − F (0◦ ) − 2 over all relevant i. To show that ρ(κ) is positive for κ 1, we first recall that Theorem 3.2 guarantees that F (φ ) is minimized only by φ = 0◦ , 180◦ . Since all of the relevant φi are bounded away from these minimizers by at least κ/2, choosing = (κ) > 0 sufficiently small implies ρ(κ) > 0 as desired.
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Apart from the above estimates, we will need the following simple observation: Lemma 4.6 Let κ 1. Then for any be two neighboring vectors t1 , t2 ∈ TL/B , (4.18) τt1 (G) ∩ τt2 (G) = τt1 (G0 ) ∩ τt2 (G0 ) ∪ τt1 (G180 ) ∩ τt2 (G180 ) . In other words, any two neighboring good blocks are necessarily of the same type of goodness. Proof. Since G = G0 ∪ G180 , the set on the right is a subset of the set on the left. The opposite inclusion is a simple consequence of the fact that neighboring blocks share a line of sites along their boundary. Indeed, suppose the shared part of the boundary is parallel with the y axis. For κ 1, Definition 1 requires that the neighboring boundary spins are nearly aligned in a G180 -block and nearly antialigned in a G0 -block. Hence, the type of goodness must be the same for both blocks. Now we are ready to prove our main result: Proof of Theorem 2.1. As is usual in the arguments based on chessboard estimates, the desired Gibbs states will be extracted from the torus measure PL,β defined in Section 4.1. Throughout the proof we will let β be sufficiently large and let ∆ scale as a (negative) power of β with exponent strictly between 1/3 and 1/2, and B grow slower than any power of β, e.g., as in 5
∆ = β − 12
and B = log β.
(4.19)
We note that these relations (eventually) ensure the validity of the bounds (3.7) for any given δ > 0 and thus make the bounds in Lemmas 4.4–4.5 readily available. First we will show that in any typical configuration from PL,β most blocks are good. Let ηL denote the sum of the ratios on the left-hand side of (4.9) and (4.14), i.e., Z (B ) (B/L)2 Z (B ) (B/L)2 L,β E L,β SW + , (4.20) ηL = ZL,β ZL,β and let η = lim supL→∞ ηL . By Theorem 4.1 and Lemma 4.2, the probability of a good block is then asymptotically in excess of 1 − η. On the basis of Lemmas 4.44.5, η is bounded by the sum of the right-hand sides of (4.9) and (4.14) which under the assumptions from (4.19) can be made as small as desired by increasing β appropriately. It remains to show that blocks with distinct types of goodness are not likely to occur in one configuration. To this end let us first observe that, once κ is small, no block can simultaneously satisfy both events G0 and G180 . Invoking also Lemma 4.6, in any given connected component of good blocks the type of goodness is homogeneous throughout the component. (Here the notion of connectivity is defined via TL/B , i.e., blocks sharing a line of sites in common, but other definitions would work as well.) We conclude that two blocks exhibiting distinct types of
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goodness must be separated by a closed surface (here ∗-connected) consisting of bad blocks. We will now employ a standard Peierls’ estimate. For any t ∈ TL/B the event G0 ∩ τt (G180 ) is contained in the union of events that the respective blocks are separated by a ∗-connected surface involving, say, m bad blocks. Using our choice of η, Lemma 4.2 and Theorem 4.1, the probability of any surface of this size is bounded by η m . Estimating the number of such surfaces by cm , for some sufficiently large c < ∞, and noting that m is at least 4, we get PL,β G0 ∩ τt (G180 ) ≤ (cη)m . (4.21) m≥4
Obviously, the right-hand side tends to zero as η ↓ 0. Thus, informally, not only are most blocks good, but most of them are of particular type of goodness. To finish the argument, we can condition on a block farthest from the origin to be, say, of G180 -type. This tells us, uniformly in L, that with overwhelming probability the block at the origin is of type G180 and similarly for the other type of goodness. The conditional state still satisfies the DLR condition for subsets not intersecting the block at the “back” of the torus. Taking the limit L → ∞ establishes the existence of two distinct infinite-volume Gibbs states which clearly satisfy (2.2–2.3) with (β) directly related to η and the various other parameters (cf Remark 2 in Section 4.2). Proof of Theorem 2.2. This is, of course, just a Mermin-Wagner theorem. Indeed, the Hamiltonian (1.3) satisfies the hypotheses of, e.g., Theorem 1 in [26], which prohibits breaking of any (compact) continuous internal symmetry of the model. Acknowledgments. The research of M.B. and L.C. was supported by the NSF under the grant NSF DMS-0306167; the research of S.K. was supported by the DOE grant DE-FG03-00ER45798.
References [1] Ar. Abanov, V. Kalatsky, V.L. Pokrovsky and W.M. Saslow, Phase diagram of ultrathin ferromagnetic films with perpendicular anisotropy, Phys. Rev. B 51, 1023–1038 (1995). [2] K.S. Alexander and L. Chayes, Non-perturbative criteria for Gibbsian uniqueness, Commun. Math. Phys. 189 no 2, 447–464 (1997). [3] M. Biskup, L. Chayes and Z. Nussinov, Orbital ordering in transition-metal compounds: I. The 120-degree model, to appear in Commun. Math. Phys. . [4] M. Biskup, L. Chayes and Z. Nussinov, Orbital ordering in transition-metal compounds: II. The orbital compass model, in preparation.
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Marek Biskup and Lincoln Chayes Department of Mathematics UCLA Los Angeles, California USA email:
[email protected] email:
[email protected] Communicated by Jennifer Chayes submitted 07/10/03, accepted 28/04/04
Steven A. Kivelson Department of Physics UCLA Los Angeles, California USA email:
[email protected]
